Theoretical Physics to Face the Challenge of LHC

Ecole´ de Physique des Houches Session XCVII, 1–26 August 2011 Theoretical Physics to Face the Challenge of LHC

Edited by Laurent Baulieu, Karim Benakli, Michael R. Douglas, Bruno Mansouli´e, Eliezer Rabinovici, and Leticia F. Cugliandolo

3 3 Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries c Oxford University Press 2015 The moral rights of the authors have been asserted Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2014953047 ISBN 978–0–19–872796–5 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work. Ecole´ de Physique des Houches Service inter-universitaire commun à l’Université Joseph Fourier de Grenoble eta ` l’Institut National Polytechnique de Grenoble Subventionné par l’Université Joseph Fourier de Grenoble, le Centre National de la Recherche Scientifique, le Commissariatal’ ` Energie´ Atomique

Directeur: Leticia F. Cugliandolo, Sorbonne Universités, Université Pierre et Marie Curie, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, Paris, France Directeurs scientifiques de la session XCVII: Laurent Baulieu, Sorbonne Universités, Université Pierre et Marie Curie, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, Paris, France Karim Benakli, Sorbonne Universités, Université Pierre et Marie Curie, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, Paris, France Michael R. Douglas, Department of Physics and Astronomy, Rutgers University, USA Bruno Mansoulié, Institut de Recherches sur les lois Fondamentales de l’Univers, CEA Saclay, France Eliezer Rabinovici, Racah Institute of Physics, Hebrew University, Jerusalem, Israel Leticia F. Cugliandolo, Sorbonne Universités, Université Pierre et Marie Curie, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, Paris, France Previous sessions

I 1951 Quantum mechanics. Quantum field theory II 1952 Quantum mechanics. Statistical mechanics. Nuclear physics III 1953 Quantum mechanics. Solid state physics. Statistical mechanics. Elementary particle physics IV 1954 Quantum mechanics. Collision theory. Nucleon-nucleon interaction. Quantum electrodynamics V 1955 Quantum mechanics. Non equilibrium phenomena. Nuclear reactions. Interaction of a nucleus with atomic and molecular fields VI 1956 Quantum perturbation theory. Low temperature physics. Quan- tum theory of solids. Ferromagnetism VII 1957 Scattering theory. Recent developments in field theory. Nuclear and strong interactions. Experiments in high energy physics VIII 1958 The many body problem IX 1959 The theory of neutral and ionized gases X 1960 Elementary particles and dispersion relations XI 1961 Low temperature physics XII 1962 Geophysics; the earths environment XIII 1963 Relativity groups and topology XIV 1964 Quantum optics and electronics XV 1965 High energy physics XVI 1966 High energy astrophysics XVII 1967 Many body physics XVIII 1968 Nuclear physics XIX 1969 Physical problems in biological systems XX 1970 Statistical mechanics and quantum field theory XXI 1971 Particle physics XXII 1972 Plasma physics XXIII 1972 Black holes XXIV 1973 Fluids dynamics XXV 1973 Molecular fluids XXVI 1974 Atomic and molecular physics and the interstellar matter XXVII 1975 Frontiers in laser spectroscopy XXVIII 1975 Methods in field theory XXIX 1976 Weak and electromagnetic interactions at high energy XXX 1977 Nuclear physics with heavy ions and mesons XXXI 1978 Ill condensed matter XXXII 1979 Membranes and intercellular communication XXXIII 1979 Physical cosmology Previous sessions vii

XXXIV 1980 Laser plasma interaction XXXV 1980 Physics of defects XXXVI 1981 Chaotic behavior of deterministic systems XXXVII 1981 Gauge theories in high energy physics XXXVIII 1982 New trends in atomic physics XXXIX 1982 Recent advances in field theory and statistical mechanics XL 1983 Relativity, groups and topology XLI 1983 Birth and infancy of stars XLII 1984 Cellular and molecular aspects of developmental biology XLIII 1984 Critical phenomena, random systems, gauge theories XLIV 1985 Architecture of fundamental interactions at short distances XLV 1985 Signal processing XLVI 1986 Chance and matter XLVII 1986 Astrophysical fluid dynamics XLVIII 1988 Liquids at interfaces XLIX 1988 Fields, strings and critical phenomena L 1988 Oceanographic and geophysical tomography LI 1989 Liquids, freezing and glass transition LII 1989 Chaos and quantum physics LIII 1990 Fundamental systems in quantum optics LIV 1990 Supernovae LV 1991 Particles in the nineties LVI 1991 Strongly interacting fermions and high Tc superconductivity LVII 1992 Gravitation and quantizations LVIII 1992 Progress in picture processing LIX 1993 Computational fluid dynamics LX 1993 Cosmology and large scale structure LXI 1994 Mesoscopic quantum physics LXII 1994 Fluctuating geometries in statistical mechanics and quantum field theory LXIII 1995 Quantum fluctuations LXIV 1995 Quantum symmetries LXV 1996 From cell to brain LXVI 1996 Trends in nuclear physics, 100 years later LXVII 1997 Modeling the earths climate and its variability LXVIII 1997 Probing the Standard Model of particle interactions LXIX 1998 Topological aspects of low dimensional systems LXX 1998 Infrared space astronomy, today and tomorrow LXXI 1999 The primordial universe LXXII 1999 Coherent atomic matter waves LXXIII 2000 Atomic clusters and nanoparticles LXXIV 2000 New trends in turbulence LXXV 2001 Physics of bio-molecules and cells LXXVI 2001 Unity from duality: Gravity, gauge theory and strings viii Previous sessions

LXXVII 2002 Slow relaxations and nonequilibrium dynamics in condensed matter LXXVIII 2002 Accretion discs, jets and high energy phenomena in astrophysics LXXIX 2003 Quantum entanglement and information processing LXXX 2003 Methods and models in neurophysics LXXXI 2004 Nanophysics: Coherence and transport LXXXII 2004 Multiple aspects of DNA and RNA LXXXIII 2005 Mathematical statistical physics LXXXIV 2005 Particle physics beyond the Standard Model LXXXV 2006 Complex systems LXXXVI 2006 Particle physics and cosmology: the fabric of spacetime LXXXVII 2007 String theory and the real world: from particle physics to astrophysics LXXXVIII 2007 Dynamos LXXXIX 2008 Exact methods in low-dimensional statistical physics and quantum computing XC 2008 Long-range interacting systems XCI 2009 Ultracold gases and quantum information XCII 2009 New trends in the physics and mechanics of biological systems XCIII 2009 Modern perspectives in lattice QCD: quantum ﬁeld theory and high performance computing XCIV 2010 Many-body physics with ultra-cold gases XCV 2010 Quantum theory from small to large scales XCVI 2011 Quantum machines: measurement control of engineered quantum systems XCVII 2011 Theoretical physics to face the challenge of LHC Special Issue 2012 Advanced data assimilation for geosciences

Publishers – Session VIII: Dunod, Wiley, Methuen – Sessions IX and X: Herman, Wiley – Session XI: Gordon and Breach, Presses Universitaires – Sessions XII–XXV: Gordon and Breach – Sessions XXVI–LXVIII: North Holland – Session LXIX–LXXVIII: EDP Sciences, Springer – Session LXXIX–LXXXVIII: Elsevier – Session LXXXIX– : Oxford University Press Preface

Every Les Houches Summer School has its own distinct character. The objective of the August 2011 session “Theoretical physics to face the challenge of LHC” was to describe, to an audience of advanced graduate students and postdoctoral fellows, the areas in high-energy physics in which profound new experimental results are hopefully on the verge of being discovered at LHC at CERN. This was to be done with the expectation that contact with new fundamental theories on the nature of fundamental forces and the structure of spacetime will be made. The students benefited from lectures by, and interacted with, many of the leaders in the field. The school was held in a summer of tense anticipation. Exciting new results from high-energy colliders were in the air, whether about the long anticipated discovery of the Higgs particle or about a “divine” surprise, evidence for the existence of supersymmetry in nature. For some years, the community of theorists had split into several components: those doing phenomenology, those dealing with highly theoretical problems, and some trying to explore if it was possible to bridge the two. In this school, we celebrated the reunification of these groups—at least for a few years. The talks given by experimentalists accurately pointed out how intensively and how precisely the newborn collider has verified all theoretical predictions that were at the frontline of the revolutionary experimental discoveries of the 1970s, 1980s, and 1990s. They detailed many of the ingenious and pioneering techniques developed at CERN for the detection and data analysis of several billions of proton–proton collisions. During the entire period of the school, the students received daily news about the progress of these searches. A trip to the CERN facilities was organized, with visits to the LHC and ATLAS detector control rooms, as well as the CMS detector coordination room and the Cosmic Antimatter Detector control room coordinated with the space laboratory. The talks given by theoreticians were about many of the attempts to go beyond the Standard Model that yield beautiful new physical insights yet to be observed experimentally. The students were very active during the talks and had interesting interactions. The organizers and speakers encouraged them to pose unrestricted questions during and after the lectures. In addition, we had a “Wisdom Tree” session during which Michael Douglas, Juan Maldacena, and Bruno Mansoulié shared their thoughts on any subject the students desired. We also held the traditional “Gong Show” in which every participant could speak about his or her work for three minutes. The cocktail of theorists and experimentalists proved to be most interesting. More precisely, the topics covered in the school were as follows. In the first morning, Jean Iliopoulos and Luis Alvarez-Gaumé gave an introduction to the school. Jean Iliopoulos recalled the historical path taking us from the Standard x Preface

Model to considering possible extensions, and Luis Alvarez-Gaumé summarized the achievements of string theory and the present open problems. Lyndon Evans reviewed the physics challenges faced in the design of the LHC in order to achieve the desired rate of highest-energy collisions. He shared with the audience the difficult road leading from envisaging how to built a Large Hadron Collider and actually doing it. Massimo Giovanozzi gave an account of how the accelerator had been commissioned, how the setback caused by a hardware failure was overcome, how the LHC functioned in the Summer of 2011, and what were the plans for its future upgrade. Dan Green took us from the accelerator to the giant detectors surrounding it. He described the requirements for the detectors and the different choices made in their design. Bruno Mansoulié guided the audience along the way from the registration of the events in the detectors to their analysis. He explained the difficulties involved in correct identification of the signals. Yves Sirois and Louis Fayard discussed the available LHC data and their implications for the Higgs boson searches at CMS and ATLAS, respectively, while Karl Jakobs summarized the constraints derived on new physics. Michelangelo Mangano explained the methods needed to compute the expected backgrounds without whose detailed knowledge one could not extract the new discoveries. Nima Arkani-Hamed and David Kosower explained new techniques recently developed to perform in a more efficient way the calculations of amplitudes, in particular for the underlying QCD processes. Gia Dvali described how unitarity is realized in effective field theories in particle physics and its implication for graviton scattering. Juan Maldacena reviewed our theoretical knowledge on quantum gravity. He described how the amazing correspondence between field theories on the boundary and gravity theories in a bulk with a negative cosmological constant arises. This comes under the umbrella of AdS/CFT. He outlined the state of the art in the field, which is a very concrete realization of the concept of holography. Jan de Boer went into the details of basic examples of AdS/CFT duality. Yaron Oz explained how to obtain hydrodynamics equations from the study of black hole solutions and described the emergence of an amazing correspondence between features of gravity and fluids. Gian Giudice discussed the most popular supersymmetric extension of the Standard Model. Gerard ’t Hooft unveiled his ideas for addressing the black hole physics information paradox. He described several consequences of the spontaneous breaking of a local conformal invariance and how this helps to obtain concrete complementarity maps between different sets of observables in the presence of a black hole. Zohar Komargodski described the possible definition of a c-function and his proof of the associated a-theorem. Alex Pomarol described the implementation of electroweak symmetry-breaking mechanisms in different extensions of the Standard Model. Preface xi

Karim Benakli described in detail the basics of supersymmetry breaking. Luis Ibáñez reviewed the implementation of the phenomenologically viable supersymmetric extensions of the Standard Model in string theory. Michael Douglas discussed the problems of classifying the possible models and estimating their frequency in the landscape of string vacua. Laurent Baulieu explained the use of a twisted supersymmetry algebra in supergravity and its consequences. Eliezer Rabinovici described his work with JoséLuis Barbón on various types of big crunches using the AdS/CFT correspondence, and the surprising result that some big crunches can be equivalently described by a nontrivial infrared theory living on a singularity-free de Sitter space. Gabriele Veneziano summarized some 25 years of work on the transplanckian- energy collisions of particles, strings, and branes. He discussed different regimes in these processes, recovering physical expectations (e.g., gravitational deflection and tidal excitation) at large distance and exposing new phenomena when the string length exceeds the gravitational radius of the collision’s energy. He also presented recent attempts to approach the short-distance regime where black hole formation is expected to occur. Daniel Zwanziger explained formal aspect of the Gribov problem in QCD. Altogether, it is the general feeling of the organizers, lecturers, and students that the School was a success, striking the right balance between the exciting physics that is currently coming out of LHC, while covering important recent developments in the theory of elementary particles. More than one of the speakers reminisced on their days as students and postdocs in Les Houches, and were grateful for this chance to return at the moment where LHC is starting to unveil a yet-unknown domain of energy. We were happy to have the chance to maintain such a longstanding tradition, and are confident that our students will make important contributions in the coming era. We hope some of them will have the chance to return as lecturers in their turn. As organizers, we express our gratitude to the local staff of the Les Houches School for their help, as well as to the funding agencies (CEA, CERN, CNRS, European Science Foundation, IN2P3, and the Les Houches School of Physics) that made possible the organization of this event.

Laurent Baulieu Karim Benakli Michael R. Douglas Bruno Mansouli´e Eliezer Rabinovici Leticia F. Cugliandolo

Contents

List of participants xviii

1 The Large Hadron Collider Lyndon EVANS 1 1.1 Introduction 3 1.2 Main machine layout and performance 4 1.3 Magnets 12 1.4 Radiofrequency systems 17 1.5 Vacuum system 20 1.6 Cryogenic system 24 1.7 Beam instrumentation 27 1.8 Commissioning and operation 30 Acknowledgments 33 References 33 2 The LHC machine: from beam commissioning to operation and future upgrades Massimo GIOVANNOZZI 35 2.1 LHC layout, parameters, and challenges 37 2.2 Digression: the chain of proton injectors 47 2.3 Proton beam commissioning and operation 48 2.4 Future upgrade options 52 Acknowledgments 55 References 55 3 The LHC detectors and the ﬁrst CMS data Dan GREEN 67 3.1 EWSB and LHC 69 3.2 LHC machine 69 3.3 Global detector properties 71 3.4 The generic detector 73 3.5 Vertex subsystem 76 3.6 Magnet subsystem 77 3.7 Tracking subsystem 78 3.8 ECAL subsystem 80 3.9 HCAL subsystem 82 3.10 Muon subsystem 85 3.11 Trigger/DAQ subsystems 88 References 89 xiv Contents

4 About the identification of signals at LHC: analysis and statistics Bruno MANSOULIE91´ 4.1 Introduction 93 4.2 Search for the Standard Model Higgs boson decaying into WW (in ATLAS) 93 4.3 Backgrounds 94 4.4 Global model of the analysis 97 4.5 Statistics 100 4.6 Global Higgs analysis 103 4.7 Conclusion 106 References 106 5 Introduction to the theory of LHC collisions Michelangelo L. MANGANO 107 5.1 Introduction 109 5.2 QCD and the proton structure at large Q 2 110 5.3 The final-state evolution of quarks and gluons 119 5.4 Applications 129 5.5 Outlook and conclusions 137 References 138 6 An introduction to the gauge/gravity duality Juan M. MALDACENA 141 6.1 Introduction to the gauge/gravity duality 143 6.2 Scalar field in AdS 148 5 6.3 The N = 4 super Yang–Mills/AdS 5× S example 152 6.4 The spectrum of states or operators 157 6.5 The radial direction 159 Acknowledgments 161 References 161 7 Introduction to the AdS/CFT correspondence Jan de BOER 163 7.1 About this chapter 165 7.2 Introduction 165 7.3 Why AdS/CFT? 166 7.4 Anti-de Sitter space 169 7.5 Correlation functions 171 7.6 Mapping between parameters 172 7.7 Derivation of the AdS/CFT correspondence 172 7.8 Tests of the AdS/CFT correspondence 174 7.9 More on finite temperature 177 7.10 Counting black hole entropy 179 7.11 Concluding remarks 180 Acknowledgments 182 References 182 Contents xv

8 Hydrodynamics and black holes Yaron OZ 185 8.1 Introduction 187 8.2 Field theory hydrodynamics 187 8.3 Relativistic hydrodynamics 188 8.4 Nonrelativistic fluid flows 192 8.5 Holographic hydrodynamics: the fluid/gravity correspondence 194 Acknowledgments 197 References 197 9 Supersymmetry Gian F. GIUDICE 199 10 Spontaneous breakdown of local conformal invariance in quantum gravity Gerard ’t HOOFT 209 10.1 Introductory remarks 211 10.2 Conformal symmetry in black holes 211 10.3 Local conformal invariance and the stress–energy–momentum tensor 217 10.4 Local conformal symmetry in canonical quantum gravity 220 10.5 Local conformal invariance and the Weyl curvature 225 10.6 The divergent effective conformal action 228 10.7 Nonconformal matter 233 10.8 Renormalization with matter present 237 10.9 The β functions 239 10.10 Adding the dilaton field to the algebra for the β functions 242 10.11 Discussion 245 10.12 Conclusions 248 Acknowledgments 250 References 250 11 Renormalization group flows and anomalies Zohar KOMARGODSKI 255 11.1 Two-dimensional models 257 11.2 Higher-dimensional models 260 References 270 12 Models of electroweak symmetry breaking Alex POMAROL 273 12.1 Introduction 275 12.2 The original technicolor model: achievements and pitfalls 276 12.3 Flavor-changing neutral currents and the top mass 277 12.4 Electroweak precision tests 277 12.5 Composite PGB Higgs 278 12.6 Little Higgs 279 xvi Contents

12.7 The AdS/CFT correspondence, Higgsless and composite Higgs models 280 12.8 LHC phenomenology 281 References 286 13 String phenomenology Luis IBA´ NEZ˜ 289 13.1 Branes and chirality 291 13.2 Type II orientifolds: intersections and magnetic fluxes 294 13.3 Local F-theory GUTs 298 13.4 The effective low-energy action 300 13.5 String model building and the LHC 307 References 310 14 The string landscape and low-energy supersymmetry Michael R. DOUGLAS 315 14.1 The goal of fundamental physics 317 14.2 Low-energy supersymmetry and current constraints 321 14.3 The gravitino and moduli problems 324 14.4 The set of string vacua 326 14.5 Eternal inflation and the master vacuum 329 14.6 From hyperchemistry to phenomenology 331 Acknowledgments 335 References 335 15 The description of N =1,d = 4 supergravity using twisted supersymmetric fields Laurent BAULIEU 339 15.1 Introduction 341 15.2 N =1,d = 4 supergravity in the new minimal scheme 342 15.3 Self-dual decomposition of the supergravity action 345 15.4 Twisted supergravity variables 346 15.5 The supergravity curvatures in the U(2) ⊂ SO(4)-invariant formalism 350 15.6 The 1.5-order formalism with SU(2)-covariant curvatures 351 15.7 Vector supersymmetry and nonvanishing torsion 356 15.8 Matter and vector multiplets coupled to supergravity 358 15.9 Conclusions and outlook 361 Appendix A: The BSRT symmetry from horizontality conditions 361 Appendix B: Tensor and chirality conventions 363 Appendix C: The action of γ matrices on twisted spinors 363 Appendix D: Algebra closure on the fields of matter and vector multiplets 364 Acknowledgments 365 References 366 Contents xvii

16 AdS crunches, CFT falls, and cosmological complexity Jos´eLuisBARBON´ and Eliezer RABINOVICI 367 16.1 Introduction 369 16.2 AdS crunches and their dS duals 370 16.3 Facing the CFT crunch time is complementary 375 16.4 Attempt at a ‘thin-thesis’ 383 16.5 Falling on your sword 389 16.6 Conclusions 392 Acknowledgments 394 References 394 17 High-energy collisions of particles, strings, and branes Gabriele VENEZIANO 397 17.1 Motivations and outline 399 17.2 Gravitational collapse criteria: a brief review 400 17.3 The expected phase diagram 402 17.4 The small-angle regime: deﬂection angle and tidal excitation 404 17.5 The string-gravity regime: a precocious black hole behaviour? 406 17.6 The strong-gravity regime: towards the large-angle/collapse phase? 407 17.7 High-energy string–brane collisions: an easier problem? 410 17.8 Summary and outlook 412 Acknowledgments 413 References 413 List of participants

Organizers

BAULIEU Laurent Sorbonne Universités, Université Pierre et Marie Curie, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, Paris, France BENAKLI Karim Sorbonne Universités, Université Pierre et Marie Curie, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, Paris, France DOUGLAS Michael R. Department of Physics and Astronomy, Rutgers University, New Jersey, USA MANSOULIEB´ runo Institut de Recherches sur les Lois Fondamentales de l’Univers, CEA Saclay, France RABINOVICI Eliezer Racah Institute of Physics, Hebrew University, Jerusalem, Israel CUGLIANDOLO Leticia F. Sorbonne Universités, Université Pierre et Marie Curie, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, Paris, France Lecturers

ALVAREZ-GAUM´ EL´ uis Theory Division, CERN, Geneva Switzerland ARKANI-HAMED Nima Institute for Advanced Studies, Princeton University, New Jersey, USA BAULIEU Laurent Sorbonne Universités, Université Pierre et Marie Curie, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, Paris, France BENAKLI Karim Sorbonne Universités, Université Pierre et Marie Curie, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, Paris, France DE BOER Jan Physics Department, University of Amsterdam, The Netherlands DOUGLAS Michael R. Department of Physics and Astronomy, Rutgers University, New Jersey, USA List of participants xix

DVALI Gia Department of Physics, New York University, USA EVANS Lyndon Imperial College, London, UK and CERN, Geneva, Switzerland FAYARD Louis Laboratoire de l’Accélérateur Linéaire, Université de Paris-Sud Orsay, France GIOVANOZZI Massimo Beams Department, CERN, Geneva, Switzerland GIUDICE Gian Theory Division, CERN, Geneva, Switzerland GREEN Dan Fermilab, USA IBANEZ˜ Luis Departamento de F´ısica Teórica, and Instituto de F´ısica Teórica, Universidad Autónoma de Madrid, Spain ILIOPOULOS Jean Laboratoire de Physique Théorique, Ecole Normale Supérieure, Paris, France JAKOBS Karl Albert-Ludwigs-Universität Freiburg, Germany KOMARGODSKI Zohar Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot, Israel KOSOWER David Service de Physique Théorique, CEA Saclay, France MALDACENA Juan Institute for Advanced Studies, Princeton University, New Jersey, USA MANGANO Michelangelo Theory Division, CERN, Geneva, Switzerland MANSOULIEB´ runo Institut de Recherches sur les Lois Fondamentales de l’Univers, CEA Saclay, France OZ Yaron Physics Department, Tel Aviv University, Israel POMAROL Alex Departament de F´ısica, Universitat Autònoma de Barcelona, Spain RABINOVICI Eliezer Racah Institute of Physics, Hebrew University, Jerusalem, Israel SIROIS Yves Ecole´ Polytechnique, Paris, France ’T HOOFT Gerard Physics Department, Utrecht University, The Netherlands xx List of participants

VENEZIANO Gabriele Coll`ege de France, Paris, France and Theory Division, CERN, Geneva, Switzerland ZWANZIGER Daniel Physics Department, New York University, USA Participants

ALBA Vasyl Institute for Theoretical and Experimental Physics, Russian Federation ALONSO Rodrigo Departamento de F´ısica Teórica, Univerisdad Autónoma de Madrid, Spain AL-SAYEGH Amara Department of Physics, American University of Beirut, Lebanon ASSEL Benjamin Laboratoire de Physique Théorique, Ecole´ Normale Supérieure, Paris, France AUZZI Roberto High Energy Physics Group, The Hebrew University, Jerusalem, Israel BARYAKHTAR Masha High Energy Theory Group, Stanford University, California, USA BENTOV Yoni Department of Physics, University of California at Santa Barbara, USA BERASALUCE-GONZALEZ´ Mikel Insituto de F´ısica Teórica, Universidad Autónoma de Madrid, Spain BESSE Adrien Laboratoire de Physique Théorique, Université de Paris-Sud Orsay, France BURDA Philipp Institute for Theoretical and Experimental Physics, Russian Federation CHAPMAN Shira Department of Physics and Astronomy, Tel Aviv University, Israel CONSTANTINOU Yiannis Department of Physics, University of Crete, Greece DE ADELHART Toorop Reinier The National Institute for Nuclear Physics and High Energy Physics, Amsterdam, The Netherlands FRELLESVIG Hjalte Axel Niels Bohr Institute, Copenhagen, Denmark GIANNUZZI Floriana Dipartimento di Fisica, Università di Bari and INFN, Italy GUDNASON Sven Bjarke High Energy Physics Group, The Hebrew University, Jerusalem, Israel List of participants xxi

HEISENBERG Lavinia Département de Physique Théorique, Université de Genève, Switzerland HOWE Kiel High Energy Theory Group, Stanford University, California, USA IATRAKIS Ioannis Department of Physics, University of Crete, Greece KHMELNITSKIY Andrey Faculty of Physics, Lüdwig Maximilians Universtät, Munich, Germany KOL Uri Department of Physics and Astronomy, Tel Aviv University, Israel KUDRNA Matej Academy of Sciences, Czech Republic LOU Hou Keong Department of Physics, Princeton University, New Jersey, USA MOELLER Jan Fakultät für Physik, Universität Bielefeld, Germany NAJJARI Saereh Institute of Theoretical Physics, Warsaw University, Poland NEIMAN Yaakov School of Physics and Astronomy, Tel Aviv University, Israel NOGUEIRA Fernando Department of Physics, University of British Columbia, Canada ORGOGOZO Axel Laboratoire de Physique Théorique et Hautes Enegies, Université Pierre et Marie Curie, Paris, France PLENCNER Daniel Faculty of Physics, Lüdwig Maximilians Universtät, Munich, Germany POZZOLI Valentina Ecole Polytechnique, Paris, France REDIGOLO Diego Faculté des Sciences, Université Libre de Bruxelles, Belgium RETINSKAYA Ekaterina Skobeltsyn Institute of Nuclear Physics, Moscow State University, Russian Federation REYS Valentin Laboratoire de Physique Théorique et Hautes Energies, Université Pierre et Marie Curie, Paris, France ROSEN Christopher Department of Physics, University of Colorado at Boulder, USA SAFDI Benjamin Physics Department, Princeton University, New Jersey, USA xxii List of participants

SALAS Hernandez´ Clara Instituto de F´ısica Teórica, Universidad Autńoma de Madrid/CSIC, Spain SALVIONI Ennio CERN–Università di Padova, Italy SCHMELL Christoph Institut für Physik, Johannes Guttenberg Universität-Mainz, Germany SJORS Stefan Department of Physics, Stockholm University, Sweden STAMOU Emmanuel Department of Physics, Technische Universität München, Germany STEFANIAK Tim Bethe Center for Theoretical Physics, Universität Bonn, Germany STORACE Stefano Physics Department, New York University, USA TAN Hai Siong Physics Department, University of California at Berkeley, USA TARONNA Massimo Scuola Normale Superiore, Italy THAMM Andrea CERN, Geneva, Switzerland TOBIOKA Kohsaku Institute for the Physics and Mathematics of the Universe, University of Tokyo, Japan UBALDI Lorenzo Physikalisches Institut, Universität Bonn, Germany WALTERS William Mathematical Sciences, University of Liverpool, UK WITASZCZYK Przemek Jagiellonian University, Cracow, Poland ZARO Marco Centre for Cosmology, Particle Physics and Phenomenology, Université Catholique de Louvain, Belgium 1 The Large Hadron Collider

Lyndon Evans

Imperial College London, UK and CERN Geneva, Switzerland

Theoretical Physics to Face the Challenge of LHC. Edited by L. Baulieu, K. Benakli, M. R. Douglas, B. Mansouli´e, E. Rabinovici, and L. F. Cugliandolo. c Oxford University Press 2015. Published in 2015 by Oxford University Press. Chapter Contents

1 The Large Hadron Collider 1 Lyndon EVANS

1.1 Introduction 3 1.2 Main machine layout and performance 4 1.3 Magnets 12 1.4 Radiofrequency systems 17 1.5 Vacuum system 20 1.6 Cryogenic system 24 1.7 Beam instrumentation 27 1.8 Commissioning and operation 30 Acknowledgments 33 References 33 Introduction 3

The Large Hadron Collider (LHC) is the most complex scientific instrument ever built for particle physics research. It will for the first time give access to the TeV energy scale. To achieve this, a number of technological innovations have been necessary. The two counter-rotating proton beams are guided and focused by superconducting magnets with a novel two-in-one structure to save cost and allow the machine to be installed in an existing tunnel. The very high field of more than 8 T in the dipoles can only be achieved by cooling them below the transition temperature of liquid helium to the superfluid state. More than 80 tons of superfluid helium is needed to cool the whole machine. In its first year of operation, it has been shown to behave in a very reliable and predictable way. Single-bunch currents 30% above the design value have already been achieved, and the luminosity has increased by five orders of magnitude in the first 200 days of operation. In this chapter, a brief description of the design principles of the major systems is given and some of the results of commissioning and first operation discussed.

1.1 Introduction The LHC is a two-ring superconducting hadron accelerator and collider installed in the existing 26.7 km tunnel that was constructed between 1984 and 1989 for the CERN Large Electron Positron (LEP) collider. The LEP tunnel has eight straight sections and eight arcs and lies between 45 and 170 m below the surface on a plane inclined at 1.4%, sloping towards Lake Léman. Approximately 90% of its length is in molasse rock, which has excellent characteristics for this application, and 10% is in limestone under the Jura mountain. There are two transfer tunnels, each approximately 2.5 km in length, linking the LHC to the CERN accelerator complex that acts as injector. Full use has been made of the existing civil engineering structures, but modifications and additions have also been needed. Broadly speaking, the underground and surface structures at Points 1 and 5 for ATLAS and CMS, respectively, are new, while those for ALICE and LHCb, at Points 2 and 8, respectively, were originally built for LEP. The approval of the LHC project was given by the CERN Council in December 1994. At that time, the plan was to build a machine in two stages, starting with a centre-of-mass energy of 10 TeV, to be upgraded later to 14 TeV. However, during 1995–96, intense negotiations secured substantial contributions to the project from nonmember states, and in December 1996, the CERN Council approved construction of the 14 TeV machine in a single stage. The LHC design depends on some basic principles linked with the latest technology. Since it is a particle–particle collider, there are two rings with counter-rotating beams, unlike particle–antiparticle colliders, which can have both beams sharing the same ring. The tunnel in the arcs has a finished internal diameter of 3.7 m, which makes it extremely difficult to install two completely separate proton rings. This hard limit on space led to the adoption of the twin-bore magnet design that was proposed by John Blewett at the Brookhaven Laboratory in 1971. At that time, it was known as the “two-in-one” superconducting magnet design [1] and was put forward as a cost-saving measure [2, 3], but in the case of the LHC, the overriding reason for adopting this solution was the lack of space in the tunnel. 4 The Large Hadron Collider

In the later part of the twentieth century, it became clear that higher energies could only be reached through better technologies, principally through superconductivity. The first use of superconducting magnets in an operational collider was in the ISR, but always at 4–4.5 K [4]. However, research was moving towards operation at 2 K and lower, to take advantage of the increased temperature margins and the enhanced heat transfer at the solid–liquid interface and in the bulk liquid [5]. The French Tokamak Tore II Supra demonstrated this new technology [6, 7], which was then proposed for the LHC [8] and brought from the preliminary study to the final concept design and validation in six years [9]. In a chapter of this length, it is impossible to describe in detail all the different systems needed to operate the LHC. Instead, we concentrate on the principal new technologies developed for the machine. A detailed description of the machine as built can be found in the LHC Design Report [10], which is in three volumes. This chapter ends with a brief description of commissioning and the first year of operation.

1.2 Main machine layout and performance 1.2.1 Performance goals The aim of the LHC is to reveal the physics beyond the Standard Model with centre- of-mass collision energies of up to 14 TeV. The number of events per second generated in the LHC collisions is given by

Nevent = Lσevent , (1.1) where σevent is the cross section for the event under study and L the machine luminosity. The machine luminosity depends only on the beam parameters and can be written for a Gaussian beam distribution as

2 Nb nb frev γr L = ∗ F, (1.2) 4πεnβ where Nb is the number of particles per bunch, nb the number of bunches per beam, frev the revolution frequency, γr the relativistic gamma factor, εn the normalized transverse beam emittance, β∗ the beta function at the collision point, and F the geometric luminosity reduction factor due to the crossing angle at the interaction point (IP). F is given by −1/2 θ σ 2 F = 1+ c z , (1.3) 2σ∗ where θc is the full crossing angle at the IP, σz the root mean square (RMS) bunch length, and σ∗ the transverse RMS beam size at the IP. The expression (1.3) assumes round beams, with σz << β, and equal beam parameters for both beams. The exploration of rare events in the LHC collisions therefore requires both high beam energies and high beam intensities. Main machine layout and performance 5

The LHC has two high-luminosity experiments, ATLAS [11] and CMS [12], both aiming at a peak luminosity of L =1034 cm−2 s−1 for proton operation. There are also two low-luminosity experiments: LHCB [13] for B physics, aiming at a peak luminosity of L =1032 cm−2 s−1, and TOTEM [14] for the detection of protons from elastic scattering at small angles, aiming at a peak luminosity of L =2×1029 cm−2 s−1 with 156 bunches. In addition to the proton beams, the LHC will also be operated with ion beams. The LHC has one dedicated ion experiment, ALICE [15], aiming at a peak luminosity of L =1027 cm−2 s−1 for nominal lead–lead ion operation. The high beam intensity required for a luminosity of L =1034 cm−2 s−1 excludes the use of antiproton beams, and hence excludes the particle–antiparticle collider configuration of a common vacuum and magnet system for both circulating beams, as used for example in the Tevatron. To collide two counter-rotating proton beams requires opposite magnetic dipole fields in both rings. The LHC is therefore designed as a proton–proton collider with separate magnet fields and vacuum chambers in the main arcs and with common sections only at the insertion regions where the experimental detectors are located. The two beams share an approximately 130-m-long common beam pipe along the interaction regions (IRs). As already mentioned, there is not enough room for two separate rings of magnets in the LEP/LHC tunnel, and so the LHC uses twin-bore magnets consisting of two sets of coils and beam channels within the same mechanical structure and cryostat. The peak beam energy depends on the integrated dipole field around the storage ring, which implies a peak dipole field of 8.33 T for the 7 TeV energy. This can only be achieved with “affordable” niobium–titanium (NbTi) superconductor by lowering the temperature to 1.9 K, below the phase transition of helium from a normal to a superfluid state.

1.2.2 Performance limitations Beam–beam limit When the beams collide, a proton in one beam is aﬀected by the electromagnetic ﬁeld of the other beam. The maximum particle density per bunch is limited by the nonlinearity of this beam–beam interaction, the strength of which is measured by the linear tune shift, given by N r ξ = b p , (1.4) 4πεn

2 2 in which rp = e /(4πε0mpc ) is the classical proton radius. Experience with existing hadron colliders indicates that the total linear tune shift summed over all IPs should not exceed 0.015. With three proton experiments requiring head-on collisions, this implies that the linear beam–beam tune shift for each IP should satisfy ξ<0.005.

Maximum dipole field and magnet quench limits The maximum beam energy that can be reached in the LHC is limited by the peak dipole field in the storage ring. The nominal field is 8.33 T, corresponding to an energy 6 The Large Hadron Collider of 7 TeV. Operating at this very high field level requires that the magnets be cooled in a bath of superfluid helium at 1.9 K.

Energy stored in the circulating beams and in the magnetic ﬁelds A total beam current of 0.584 A corresponds to a stored energy of approximately 362 MJ. In addition to the energy stored in the circulating beams, the LHC magnet system has a stored electromagnetic energy of approximately 600 MJ, yielding a total stored energy of more than 1 GJ. This stored energy must be absorbed safely at the end of each run or in the case of a malfunction or an emergency. The beam dumping system and the magnet system therefore provide additional limits for the maximum attainable beam energies and intensities.

Heat load Although synchrotron radiation in hadron storage rings is small compared with that generated in electron rings, it can still impose practical limits on the maximum attainable beam intensities if the radiation has to be absorbed by the cryogenic system. In addition to the synchrotron-radiation heat load, the LHC cryogenic system must absorb the heat deposition from luminosity-induced losses, impedance-induced losses (resistive wall eﬀect), and electron-cloud bombardment.

Field quality and dynamic aperture Field quality errors compromise the particle stability in the storage ring, and hence loss-free operation requires a high field quality. A characterizing feature of superconducting magnets is the decay of persistent currents and their “snap back” at the beginning of the ramp. Achieving small beam losses therefore requires tight control of the magnetic field errors during magnet production and during machine operation. Assuming fixed limits for the beam losses (set by the quench levels of the superconducting magnets), the accuracy of the field quality correction during operation and its limitation on machine performance can be estimated.

Collective beam instabilities The interaction of the charged particles in each beam with each other via electromagnetic ﬁelds and the conducting boundaries of the vacuum system can result in collective beam instabilities. Generally speaking, the collective eﬀects are a function of the vacuum system geometry and its surface properties. They are usually proportional to the beam currents and can therefore limit the maximum attainable beam intensities.

1.2.3 Lattice layout The basic layout of the LHC follows the LEP tunnel geometry (see Fig. 1.1). The LHC has eight arcs and eight straight sections. Each straight section is approximately 528 m long and can serve as an experimental or utility insertion. The two high-luminosity Main machine layout and performance 7

Low β (pp) High luminosity

RF CMS and future Dump experiments

Octant 5 Octant 6

Octant 4

Octant 7 Cleaning Cleaning

Octant 3

Octant 2

Octant 8 Octant 1

ALICE LHC-B Injection

Injection ATLAS Low β Low β (ions) (B physics) Low β (pp) High luminosity

Fig. 1.1 Schematic layout of the LHC (Beam 1 clockwise; Beam 2 anticlockwise). experimental insertions are located at diametrically opposite straight sections: the AT- LAS experiment is located at Point 1 and the CMS experiment at Point 5. Two more experimental insertions are located at Points 2 and 8, which also include the injection systems for Beams 1 and 2, respectively. The injection kick occurs in the vertical plane, with the two beams arriving at the LHC from below the LHC reference plane. The beams cross from one magnet bore to the other at four locations. The remaining four straight sections do not have beam crossings. Insertions at Points 3 and 7 each contain two collimation systems. The insertion at Point 4 contains two radiofrequency (RF) accelerating systems: one independent system for each LHC beam. The straight section at Point 6 contains the beam dump insertion, where the two beams can be safely extracted from the machine if needed. Each beam features an independent abort system. The arcs of the LHC lattice are made of 23 regular arc cells. The arc cells are 106.9 m long and are made out of two 53.45-m-long half cells, each of which contains one 5.355-m-long cold mass (6.63-m-long cryostat), a short straight section (SSS) 8 The Large Hadron Collider assembly, and three 14.3-m-long dipole magnets. The LHC arc cell has been optimized for a maximum integrated dipole ﬁeld along the arc with a minimum number of magnet interconnections and with the smallest possible beam envelopes. The two apertures of Ring 1 and Ring 2 are separated by 194 mm. The two coils in the dipole magnets are powered in series, and all dipole magnets of one arc form one electrical circuit. The quadrupoles of each arc form two electrical circuits: all focusing quadrupoles in Beams 1 and 2 are powered in series, and all defocusing quadrupoles in Beams 1 and 2 are powered in series. The optics of Beam 1 and Beam 2 in the arc cells are therefore strictly coupled via the powering of the main magnetic elements. A dispersion suppressor (DS) is located at the transition between an LHC arc and a straight section, yielding a total of 16 DS sections. The aim of the DS is threefold: • Adapt the LHC reference orbit to the geometry of the LEP tunnel. • Cancel the horizontal dispersion arising in the arc and generated by the separation/recombination dipole magnets and the crossing angle bumps. • Facilitate matching the insertion optics to the periodic optics of the arc.

1.2.4 High-luminosity insertions (IR1 and IR5) Interaction regions 1 and 5 house the high-luminosity experiments of the LHC and are identical in terms of hardware and optics, except that the crossing angle is in the vertical plane at Point 1 and in the horizontal plane at Point 5. The small β-function values at the IPs are generated between quadrupole triplets that leave ±23 m free space about the IP. In this region, the two rings share the same vacuum chamber, the same low-β triplet magnets, and the D1 separation dipole magnets. The remaining matching section and the DS consist of twin-bore magnets with separate beam pipes for each ring. From the IP up to the DS insertion, the layout comprises the following: • A 31-m-long superconducting low-β triplet assembly, operated at a temperature of 1.9 K and providing a nominal gradient of 205 T/m. • A pair of separation/recombination dipoles separated by approximately 88 m. • The D1 dipole located next to the triplet magnets, which has a single bore and consists of six 3.4-m-long conventional warm magnet modules yielding a nominal field of 1.38 T. • The following D2 dipole, which is a 9.45-m-long twin-bore superconducting dipole magnet, operating at a cryogenic temperature of 4.5 K with a nominal field of 3.8 T. The bore separation in the D2 magnet is 188 mm and is thus slightly smaller than the arc bore separation. • Four matching quadrupole magnets. The first quadrupole following the separation dipole magnets, Q4, is a wide-aperture magnet operating at a cryogenic temperature of 4.5 K and yielding a nominal gradient of 160 T/m. The remaining three quadrupole magnets are normal-aperture quadrupole magnets, operating at a cryogenic temperature of 1.9 K with a nominal gradient of 200 T/m. Main machine layout and performance 9

Fig. 1.2 Schematic layout of the right side of IR1 (distances in m).

1.2.5 Medium-luminosity insertion in IR2 The straight section of IR2 (see Fig 1.3) houses the injection elements for Ring 1, as well as the ion beam experiment ALICE. During injection, the optics must satisfy the special constraints imposed by the beam injection for Ring 1, and the geometrical acceptance in the interaction region must be large enough to accommodate both beams in the common part of the ring, with a beam separation of at least 10σ.

1.2.6 Beam-cleaning insertions in IR3 and IR7 The IR3 insertion houses the momentum-cleaning systems (capturing oﬀ-momentum particles) of both beams, while IR7 houses the betatron-cleaning systems (to control the beam halo) of both beams. Particles with a large momentum oﬀset are scattered by the primary collimator in IR3 and particles with large betatron amplitudes are scattered by the primary collimator in IR7. In both cases, the scattered particles are absorbed by secondary collimators. Figures 1.4 and 1.5 show the right sides of IR3 and IR7, respectively.

Fig. 1.3 Schematic layout of the matching section on the left side of IR2 (distances in m).

Fig. 1.4 Schematic layout of the matching section on the right side of IR3 (distances in m). 10 The Large Hadron Collider

Fig. 1.5 Schematic layout of the matching section on the right side of IR7 (distances in m).

In IR7, the layout of the long straight section between Q7L and Q7R is mirror- symmetric with respect to the IP. This allows a symmetrical installation for the collimators of the two beams and minimizes the space conﬂicts in the insertion. Start- ing from Q7 left, the superconducting quadrupole Q6 is followed by a dogleg structure made of two sets of MBW warm single-bore wide-aperture dipole magnets (two warm modules each). The dogleg dipole magnets are labelled D3 and D4 in the LHC sequence, with D3 being the dipole closer to the IP. The primary collimators are located between the D4 and D3 magnets, allowing neutral particles produced in the jaws to point out of the beam line and most charged particles to be swept away. The inter- beam distance between the dogleg assemblies left and right from the IP is 224 mm, i.e. 30 mm larger than in the arc. This increased beam separation allows a substantially higher gradient in the Q4 and Q5 quadrupoles, which are not superconducting because of the heavy irradiation from the collimators. The space between Q5 left and right from the IP is used to house the secondary collimators at appropriate phase advances with respect to the primary collimators.

1.2.7 RF insertion in IR4 IR4 (see Fig. 1.6) houses the RF and feedback systems, as well as some of the LHC beam instrumentation. The RF equipment is installed in the old ALEPH (LEP) cavern, which provides a large space for the power supplies and klystrons. In order to provide the transverse space for two independent RF systems for Beam 1 and Beam 2, the separation must be increased to 420 mm. This is achieved by two pairs of dogleg dipole magnets labelled D3 and D4 in the LHC sequence, with D3 being the dipole magnets closer to the IP. In contrast to IR3 and IR7, the dogleg magnets in IR4 are superconducting, since the radiation levels are low.

Fig. 1.6 Schematic layout of the matching section on the right side of IR4 (distances in m). Main machine layout and performance 11

1.2.8 Beam abort insertion in IR6 IR6 (see Fig. 1.7) houses the beam abort systems for Beams 1 and 2. Beam abort from the LHC is done by kicking the circulating beam horizontally into an iron septum magnet, which deﬂects the beam in the vertical direction away from the machine components to absorbers in a separate tunnel. Each ring has its own system, and both are installed in IR6. To minimize the length of the kicker and of the septum, large drift spaces are provided. Matching the β-functions between the ends of the left and right DS requires only four independently powered quadrupoles. In each DS, up to six quadrupoles can be used for matching. The total of 16 quadrupoles is more than suﬃcient to match the β-functions and the dispersion and to adjust the phases. There are, however, other constraints to be taken into account concerning apertures inside the insertion. Special detection devices protect the extraction septum and the LHC machine against losses during the abort process. The TCDS absorber is located in front of the extraction septum and the TCDQ in front of the Q4 quadrupole magnet downstream of the septum magnet.

1.2.9 Medium-luminosity insertion in IR8 IR8 houses the LHCb experiment and the injection elements for Beam 2. The small β- function values at the IP are generated with the help of a triplet quadrupole assembly that leaves ±23 m of free space around the IP. In this region, the two rings share the same vacuum chamber, the same low-β triplet magnets, and the D1 separation dipole magnet. The remaining matching section and the DS consist of twin-bore magnets with separate beam pipes for each ring. From the IP up to the DS insertion, the layout comprises the following:

• Three warm dipole magnets to compensate the deﬂection generated by the LHCb spectrometer magnet. • A 31-m-long superconducting low-β triplet assembly operated at 1.9 K and providing a nominal gradient of 205 T/m. • A pair of separation/recombination dipole magnets separated by approximately 54 m. The D1 dipole located next to the triplet magnets is a 9.45-m-long single-bore superconducting magnet. The following D2 dipole is a 9.45-m-long

Fig. 1.7 Schematic layout of the matching section on the right side of IR6 (distances in m). 12 The Large Hadron Collider

Fig. 1.8 Schematic layout of the matching section on the right side of IR8 (distances in m).

double-bore superconducting dipole magnet. Both magnets are operated at 4.5 K. The bore separation in the D2 magnet is 188 mm, and is thus slightly smaller than the arc bore separation. • Four matching quadrupole magnets. The ﬁrst quadrupole following the separation dipole magnets, Q4, is a wide-aperture magnet operating at 4.5 K and yielding a nominal gradient of 160 T/m. The remaining three matching-section quadrupole magnets are normal-aperture quadrupole magnets operating at 1.9 K with a nominal gradient of 200 T/m. • The injection elements for Beam 2 on the right side of IP8. In order to provide suﬃcient space for the spectrometer magnet of the LHCb experiment, the beam collision point is shifted by 15 half RF wavelengths (3.5 times the nominal bunch spacing ≈ 11.25 m) towards IP7. This shift of the collision point has to be compensated before the beam returns to the DS sections and requires a nonsymmetric magnet layout in the matching section.

1.3 Magnets 1.3.1 Overview The LHC relies on superconducting magnets that are at the edge of present technology. Other large superconducting accelerators (Tevatron-FNAL, HERA-DESY, and RHIC-BNL) all use classical NbTi superconductors, cooled by supercritical helium at temperatures slightly above 4.2 K, with fields below or around 5 T. The LHC magnet system, while still making use of the well-proven technology based on NbTi Ruther- ford cables, cools the magnets to a temperature below 2 K, using superfluid helium, and operates at fields above 8 T. Since the electromagnetic forces increase with the square of the field, the structures restraining the conductor motion must be mechanically much stronger than in earlier designs. In addition, space limitations in the tunnel and the need to keep costs down have led to the adoption of the “two-in-one” or “twin-bore” design for almost all of the LHC superconducting magnets. The two- in-one design accommodates the windings for the two beam channels in a common cold mass and cryostat, with magnetic flux circulating in the opposite sense through the two channels. Magnets 13

1.3.2 Superconducting cable The transverse cross section of the coils in the LHC 56-mm-aperture dipole magnet (Fig. 1.9) shows two layers of different cables distributed in six blocks. The cable used in the inner layer has 28 strands, each having a diameter of 1.065 mm, while the cable in the outer layer is formed from 36 strands, each of 0.825 mm diameter. The filament size chosen is 7 µm for the strand of the inner layer cable and 6 µm for the strand of the outer layer cable. They are optimized to reduce the effects of the persistent currents on the sextupole field component at injection. The residual errors are corrected by small sextupole and decapole magnets located at the end of each dipole.

1.3.3 Main-dipole cold mass The LHC ring accommodates 1232 main dipoles: 1104 in the arc and 128 in the DS regions. They all have the same basic design. The geometric and interconnection characteristics have been targeted to be suitable for the DS region, which is more demanding than the arc. The cryodipoles are a critical part of the machine, both from the machine performance point of view and in terms of cost. Figure 1.10 shows the cross section of the cryodipole. The successful operation of the LHC requires that the main dipole magnets have practically identical characteristics. The relative variations of the integrated field and the field shape imperfections must not exceed ∼ 10−4, and their reproducibility must be better than 10−4 after magnet testing and during magnet operation. The reproducibility of the integrated field strength requires close control of coil diameter and length and of the stacking factor of the laminated magnetic yokes, and possibly fine tuning of the length ratio between the magnetic and nonmagnetic parts of the yoke. The structural stability of the cold mass assembly is achieved by using very rigid collars, and by opposing the electromagnetic forces acting at the interfaces between

Fig. 1.9 Conductor distribution in the dipole coil cross section (X axis in mm on left). Picture of cables and strand on right. 14 The Large Hadron Collider

Alignment target Main quadripole bus-bars Heat-exchanger pipe Superinsulation Superconducting coils Beam pipe Vacuum vessel Beam screen Auxiliary bus-bars Shrinking cylinder / He I-vessel Thermal shield (55−75 K) Nonmagnetic collars Iron yoke (cold mass, 1.9 K) Dipole bus-bars Support post

Fig. 1.10 Cross section of cryodipole (lengths in mm). the collared coils and the magnetic yoke with the forces set up by the shrinking cylinder. A prestress between coils and retaining structure (collars, iron lamination, and shrinking cylinder) is also built in. Because of the larger thermal contraction coeﬃcient of the shrinking cylinder and austenitic steel collars with respect to the yoke steel, the force distribution inside the cold mass changes during cool down from room temperature to 1.9 K.

1.3.4 Dipole cryostat The vacuum vessel consists of a long cylindrical standard tube with an outer diameter of 914 mm (36 inches) and a wall thickness of 12 mm. It is made from alloyed low- carbon steel. The vessel has stainless steel end flanges for vacuum-tight connection via elastomer seals to adjacent units. Three support regions feature circumferential reinforcement rings. Upper reinforcing angles support alignment fixtures. An ISO- standard flanged port is located azimuthally on the wall of the vessel at one end. In normal operation, the vessel will be under vacuum. In case of a cryogenic leak, the pressure can rise to 0.14 MPa absolute, and a sudden local cooling of the vessel wall to about 230 K may occur. The steel selected for the vacuum vessel wall is tested to demonstrate adequate energy absorption during a standard Charpy test at −50 ◦C. A front view of the cryodipole is shown in Fig. 1.11. In the main dipoles, the magnetic field is up in one aperture and down in the other. In straight sections 2 and 8, the beams are separated into two apertures with special superconducting dipoles D1 with a single aperture and D2 with two apertures, where the field direction is identical for both. Such special dipoles are also used in the Magnets 15

Vacuum vessel

MLI: 4.5–10 K

Cold mass

Thermal shield assembly

MLI: 50–55 K

Bottom tray assembly

Fig. 1.11 LHC dipole cryomagnet assembly.

RF insertion at Point 4, where the beams are separated further to make way for the cavities. All these special dipoles were designed and built in the USA at Brookhaven Laboratory.

1.3.5 Short straight sections of the arcs Figure 1.12 shows a perspective view and Fig. 1.13 illustrates the cross section of an SSS. The cold masses of the arc SSSs contain the main quadrupole magnets (MQ) and

Machine Vacuum vessel interconnect Auxiliary bus-bar tube Vacuum barrier Helium vessel {inertia tube} Interconnect with ring tunnel interconnection box QRL Phase Corrector separator MSCB Beam vacuum pumping manifold

Support post Lattice quadrupole Multilayer insulation

MQ Thermal shield Machine Octupole interconnect MO BPM Housing for LHS−SSS reference drawing MQ diodes Beam tube

Current feed-through Quadrupole cryostat technical service module

Fig. 1.12 Short straight section (SSS) with jumper. 16 The Large Hadron Collider

Alignment target Main quadrupole bus-bars Heat-exchanger pipe Superinsulation Superconducting coils Beam pipe Inertia tube / He II-vessel Iron yoke Vacuum vessel Thermal shield Auxiliary bus-bars Austenitic steel collars Beam screen Instrumentation wires Filler piece Dipole bus-bars Support post

Fig. 1.13 Cross section of SSS at quadrupole cold mass inside cryostat. various corrector magnets. On the upstream end, these can be octupoles (MO), tuning quadrupoles (MQT), or skew quadrupole correctors (MQS). On the downstream end, the combined sextupole–dipole correctors (MSCB) are installed. Because of the lower electromagnetic forces than in the dipoles, the two apertures do not need to be combined, but are assembled in separate annular collaring systems.

1.3.6 Insertion magnets The insertion magnets are superconducting or normally conducting and are used in the eight insertion regions of the LHC. Four of these insertions are dedicated to experiments, while the others are used for major collider systems (one for the RF, two for beam cleaning, and one for beam dumping). The various functions of the insertions are fulﬁlled by a variety of magnets, most based on the technology of NbTi superconductors cooled by superﬂuid helium at 1.9 K. A number of standalone magnets in the matching sections and beam separation sections are cooled to 4.5 K, while in the radiation areas, specialized normally conducting magnets are installed.

1.3.7 Matching-section quadrupoles The tuning of the LHC insertions is provided by the individually powered quadrupoles in the matching and DS sections. The matching sections consist of standalone quadrupoles arranged in four half-cells, but the number and parameters of the magnets are specific for each insertion. Apart from the cleaning insertions, where specialized normally conducting quadrupoles are used in the high-radiation areas, all matching Radiofrequency systems 17 quadrupoles are superconducting magnets. Most of them are cooled to 4.5 K, except the Q7 quadrupoles, which are the first magnets in the continuous arc cryostat and are cooled to 1.9 K. CERN has developed two superconducting quadrupoles for the matching sections: the MQM quadrupole, featuring a 56-mm-aperture coil, which is also used in the DS sections, and the MQY quadrupole, with an enlarged, 70 mm, coil aperture. Both quadrupoles use narrow cables, so that the nominal current is less than 6 kA, substantially simplifying the warm and cold powering circuits. Each aperture is powered separately, but a common return is used, so that a three-wire bus-bar system is sufficient for full control of the apertures. In the cleaning insertions IR3 and IR7, each of the matching quadrupoles Q4 and Q5 consists of a group of six normally conducting MQW magnets. This choice is dictated by the high radiation levels due to scattered particles from the collimation system, and therefore the use of superconducting magnets is not possible. It features two apertures in a common yoke (2-in-1), which is atypical for normally conducting quadrupole magnets, but is needed because of transverse space constraints in the tunnel. The two apertures may be powered in series in a standard focusing/defocusing configuration (MQWA), or alternatively in a focusing/focusing configuration (MQWB) to correct asymmetries of the magnet. In a functional group of six magnets, five are configured as MQWA, corrected by one configured as MQWB.

1.3.8 Low-β triplets The low-β triplet is composed of four single-aperture quadrupoles with a coil aperture of 70 mm. These magnets are cooled with superﬂuid helium at 1.9 K using an external heat-exchanger system capable of extracting up to 10 W/m of power deposited in the coils by the secondary particles emanating from the proton collisions. Two types of quadrupoles are used in the triplet: 6.6-m-long MQXA magnets designed and developed by KEK, Japan and 5.7-m-long MQXB magnets designed and built by Fermilab, USA. The magnets are powered in series with 7 kA, with an additional inner loop of 5 kA for the MQXB magnets. Together with the orbit correctors MCBX, skew quadrupoles MQSX, and multipole spool pieces supplied by CERN, the low-β quadrupoles are completed in their cold masses and cryostated by Fermilab. The cryogenic feed-boxes (DFBX), providing a link to the cryogenic distribution line and power converters, are designed and built by Lawrence Berkeley National Laboratory, USA. Alongside the LHC main dipoles, the high-gradient wide-aperture low-β quadrupoles are the most demanding magnets in the collider. They must operate reliably at 215 T/m, sustain extremely high heat loads in the coils and high radiation dose during their lifetime, and have a very good ﬁeld quality within the 63 mm aperture of the cold bore.

1.4 Radiofrequency systems 1.4.1 Introduction The injected beam is captured, accelerated, and stored using a 400 MHz superconducting cavity system, and the longitudinal injection errors is damped using the same 18 The Large Hadron Collider system. This choice deﬁnes the maximum allowable machine impedance, in particular for higher-order modes in cavities [16]. Transverse injection errors will be damped by a separate system of electrostatic deﬂectors that will also ensure subsequent transverse stability against the resistive wall instability [17]. All RF and beam feedback systems are concentrated at Point 4 and extend from the UX45 cavern area into the tunnel on either side. The beam and machine parameters that are directly relevant to the design of the RF and beam feedback systems are given in Table 1.1. At nominal intensity in the SPS, an emittance of 0.6 eV s has now been achieved, giving a bunch length at 450 GeV of 1.6 ns. The phase variation along the batch due to beam loading is within 125 ps. This emittance is lower than originally assumed, and, as a result, an RF system in the LHC of 400.8 MHz can be used to capture the beam with minimal losses, accelerate,

Table 1.1 The main beam and RF parameters

UNITS INJECTION, COLLISION, 450 GeV 7TeV Bunch area (2σ)* eV s 1.0 2.5 Bunch length (4σ)* ns 1.71 1.06 Energy spread (2σ)* 10−3 0.88 0.22 Intensity per bunch 1011 p 1.15 1.15 Number of bunches 2808 2808 Normalized RMS transverse emittance, V/H µm 3.75 3.75 Intensity per beam A 0.582 0.582 Synchrotron radiation loss/turn keV — 7 Longitudinal damping time h — 13 Intrabeam scattering growth time: H h 38 80 L h 30 61 Frequency MHz 400.789 400.790 Harmonic number 35 640 35 640 RF voltage/beam MV 8 16 Energy gain/turn (20 min ramp) keV 485 RF power supplied during kW ∼ 275 acceleration/beam Synchrotron frequency Hz 63.7 23.0 Bucket area eV s 1.43 7.91 RF (400 MHz) component of beam current A 0.87 1.05

* The bunch parameters at 450 GeV are an upper limit for the situation after filamentation, ∼100 ms after each batch injection. The bunch parameters at injection are described in the text. Radiofrequency systems 19 and finally store it at top energy. Higher frequencies, while better for producing the short bunches required in store, cannot accommodate the injected bunch length. There will be some emittance increase at injection, but this will be within the capabilities of the RF system for acceleration and, in particular, will be below the final emittance needed in storage. This final emittance is defined by intrabeam scattering lifetime in the presence of damping due to synchrotron radiation, RF lifetime, and instability threshold considerations. Controlled emittance increase is provided during acceleration by excitation with band-limited noise, the emittance increasing with the square root of the energy to optimize both narrow and broadband instability thresholds [16]. The final emittance at 7 TeV (2.5 eV s) and a maximum bunch length given by luminosity considerations in the experiments lead to a required maximum voltage of 16 MV/beam. There are many advantages in having a separate RF system for each beam. However, the standard distance between beams in the machine, 194 mm, is insufficient. Consequently, the beam separation is increased in the RF region to 420 mm by means of special superconducting dipoles. With the increased separation and also by staggering the cavities longitudinally, the “second” beam can pass outside the cavity. However, it must still pass through the cryostat.

1.4.2 Main 400 MHz RF accelerating system (ACS) The two independent RF systems must each provide at least 16 MV in coast, while at injection about 8 MV is needed. The frequency of 400 MHz is close to that of LEP, 352 MHz, allowing the same proven technology of niobium-sputtered superconducting cavities to be applied. The present design, using single-cell cavities each with 2 MV accelerating voltage, corresponding to a conservative field strength of 5.5 MV/m, minimizes the power carried by the RF window. A large tuning range is required to compensate the average reactive beam component. Each RF system has eight cavities, with R/Q = 45Ω and of length λ/2, grouped by four with a spacing of 3λ/2inone common cryostat [18]. Each cavity is driven by an individual RF system with klystron, circulator, and load. Complex feedback loops around each cavity allow precise control of the field in each cavity, which is important for the unforgiving high-intensity LHC proton beam. The use of niobium sputtering on copper for construction of the cavities has the important advantage over solid niobium that susceptibility to quenching is very much reduced. Local heat generated by small surface defects or impurities is quickly con- ducted away by the copper. During the low-power tests, the 21 cavities produced all reached an accelerating field of twice nominal without quenching. The niobium- sputtered cavities are insensitive to the Earth’s magnetic field, and special magnetic shielding, as needed for solid-niobium cavities, is not required. Four cavities, each equipped with their helium tank, tuner, higher-order mode (HOM) couplers, and power coupler, are grouped together in a single cryomodule (see Fig. 1.14). The conception of the cryomodule is itself modular; all cavities are identical and can be installed in any position. If a problem arises with a cavity, it can be replaced. The cavity is tuned by elastic deformation, by pulling on a harness using stainless steel cables that are 20 The Large Hadron Collider

Fig. 1.14 Four-cavity cryomodule during assembly. wound around a shaft. A stepping motor, ﬁxed to the outside of the cryostat, drives the shaft. The motor therefore works in normal ambient conditions and can be easily accessed for maintenance or repair.

1.5 Vacuum system 1.5.1 Overview The LHC has three vacuum systems: the insulation vacuum for the cryomagnets, the insulation vacuum for helium distribution (QRL), and the beam vacuum. The insulation vacua before cool down do not have to be better than 10−1 mbar, but at cryogenic temperatures, in the absence of any signiﬁcant leak, the pressure will stabilize at about 10−6 mbar. The requirements for the beam vacuum are much more stringent, driven by the required beam lifetime and background at the experiments. Rather than quoting equivalent pressures at room temperature, the requirements at cryogenic temperature are expressed as gas densities normalized to hydrogen, taking into account the ionization cross sections for each gas species. Equivalent hydrogen gas densities should 15 −3 remain below 10 H2 m to ensure the required 100 hour beam lifetime. In the 13 −3 interaction regions around the experiments, the densities will be below 10 H2 m to minimize the background to the experiments. In the room-temperature parts of the beam vacuum system, the pressure should be in the range 10−10 − 10−11 mbar. Vacuum system 21

A number of dynamic phenomena have to be taken into account in the design of the beam vacuum system. Synchrotron radiation will hit the vacuum chambers, in particular in the arcs, and electron clouds (multipacting) could aﬀect almost the entire ring. Extra care has to be taken during the design and installation to minimize these eﬀects, but conditioning with beam will be required to reach nominal performance.

1.5.2 Beam vacuum requirements The design of the beam vacuum system takes into account the requirements of 1.9 K operation and the need to shield the cryogenic system from heat sources, as well as the more usual constraints set by vacuum chamber impedances. Four main heat sources have been identiﬁed and quantiﬁed at nominal intensity and energy:

• synchrotron light radiated by the circulating proton beams (0.2 W/m per beam, with a critical energy of about 44 eV); • energy loss by nuclear scattering (30 mW/m per beam); • image currents (0.2 W/m per beam); • energy dissipated during the development of electron clouds, which will form when the surfaces seen by the beams have a secondary electron yield that is too high.

Intercepting these heat sources at a temperature above 1.9 K has necessitated the introduction of a beam screen. The more classical constraints on the vacuum system design are set by the stability of the beams, which sets the acceptable longitudinal and transverse impedance [19, 20], and by the background conditions in the interaction regions. The vacuum lifetime is dominated by the nuclear scattering of protons on the residual gas. The cross sections for this interaction at 7 TeV depend on the gas species [21, 22] and are given in Table 1.2, together with the gas density and pressure (at 5 K) compatible with the requested 100 hour lifetime. This number ensures that the contribution of beam–gas collisions to the decay of the beam intensity is small

Table 1.2 Nuclear scattering cross sections at 7 TeV for diﬀerent gases and corresponding densities and equivalent pressures for a 100 h lifetime

GAS NUCLEAR SCATTERING GAS DENSITY (m−3) PRESSURE (Pa) CROSS SECTION (cm2) FOR 100 h LIFETIME AT 5 K FOR 100 h LIFETIME

−26 14 −8 H2 9.5 × 10 9.8 × 10 6.7 × 10 He 1.26 × 10−25 7.4 × 1014 5.1 × 10−8 −25 14 −8 CH4 5.66 × 10 1.6 × 10 1.1 × 10 −25 14 −8 H2O5.65 × 10 1.6 × 10 1.1 × 10 CO 8.54 × 10−25 1.1 × 1014 7.5 × 10−9 −24 13 −9 CO2 1.32 × 10 7 × 10 4.9 × 10 22 The Large Hadron Collider compared with other loss mechanisms; it also reduces the energy lost by scattered protons in the cryomagnets to below the nominal value of 30 mW/m per beam.

1.5.3 Beam vacuum in the arcs and dispersion suppressors The two beams are conﬁned in independent vacuum chambers from one end of the continuous cryostat to the other, extending from Q7 in one octant to Q7 in the next octant. Cold bores with an inner diameter of 50 mm, part of the cryomagnets, are connected together by so-called cold interconnects, which compensate for length variations and alignment errors. A beam-position monitor, with an actively cooled body, is mounted on each beam in each SSS (i.e. at each quadrupole). An actively cooled beam screen is inserted into the cold bores of all magnets (see Fig. 1.15). A racetrack shape has been chosen for the beam screen, which optimizes the available aperture while leaving space for the cooling tubes. The nominal horizontal and vertical apertures are 44.04 and 34.28 mm, respectively. Slots, covering a total of 4% of the surface area, are perforated in the ﬂat parts of the beam screen to allow condensing of the gas on surfaces protected from the direct impact of energetic particles (ions, electrons, and photons). The pattern of the slots has been chosen to minimize longitudinal and transverse impedance, and the size has been chosen to keep the RF losses through the holes below 1 mW/m. A thin copper layer (75 µm) on the inner surface of the beam screen provides a low-resistance path for the image current of the beam. A sawtooth pattern on the inner surface in the plane of bending helps the absorption of synchrotron radiation. The beam screen is cooled by two stainless steel tubes with an inner diameter of 3.7 mm and a wall thickness of 0.53 mm, allowing the extraction of up to 1.13 W/m in nominal cryogenic conditions. The helium temperature is regulated to 20 K at the output of the cooling circuit at every half-cell, resulting in a temperature of the cooling tubes between 5 and 20 K for nominal cryogenic conditions. The cooling tubes

Fig. 1.15 Conceptual design of the LHC beam screen. Vacuum system 23 are laser-welded onto the beam-screen tube and ﬁtted at each end with adaptor pieces that allow their routing out of the cold bore without any fully penetrating weld between the helium circuit and the beam vacuum. Sliding rings with a bronze layer towards the cold bore are welded onto the beam screen every 750 mm to ease the insertion of the screen into the cold bore tube, to improve the centring, and to provide good thermal insulation. Finally, since the electron clouds can deposit signiﬁcant power into the cold bore through the pumping slots, the latter are shielded with copper–beryllium shields clipped onto the cooling tubes. The net pumping speed for hydrogen is reduced by a factor of two, which remains acceptable.

Cold interconnects Beam vacuum interconnects ensure the continuity of the vacuum envelope and of the helium ﬂow, as well as a smooth geometrical transition between beam screens along the 1642 twin-aperture superconducting cryomagnets installed in the continuous cryostat. The physical beam envelope must have a low electrical resistance for image currents and must minimize coupled bunch instabilities. It must also have a low inductance for the longitudinal single-bunch instability. The maximum DC resistance allowed at room temperature for a complete interconnect is 0.1 mΩ. To meet these requirements, a complex interconnect module integrates a shielded bellows to allow thermal expansion, as well as for compensation of mechanical and alignment tolerances between two adjacent beam screens. The shielding of the bellows is achieved by means of a set of sliding contact ﬁngers made out of gold-plated copper–beryllium, which slide on a rhodium-coated copper tube.

1.5.4 Beam vacuum in the insertions Room-temperature chambers alternate with standalone cryostats in the IRs. The room-temperature part includes beam instrumentation, accelerating cavities, experiments, collimation equipment, and the injection and ejection kickers and septa, as well as some of the dipole and quadrupole magnets where superconducting elements are not used. In these regions, the vacuum systems for the two beams sometimes merge, notably in the four experimental insertions, but also in some special equipment like the injection kickers and some beam stoppers.

Beam screen The beam screen is only required in the cold bores of the standalone cryostats. It is derived from the arc type, but uses a smaller (0.6 mm) steel thickness and comes in various sizes, to match different cold-bore diameters. The orientation of the beam screen in the cold bore is adapted to the aperture requirements, which means that the flat part with the cooling tubes can be either vertical or horizontal. The saw-tooth structure has been abandoned for these beam screens, since synchrotron radiation hitting the screen in these regions is at most 1/10 as intense as in the arc, and because fitting the saw teeth at the appropriate location of the beam screen would be too expensive. 24 The Large Hadron Collider

Room-temperature beam vacuum in the field-free regions The baseline for the room-temperature beam vacuum system is to use 7-m-long oxygen-free silver-bearing (OFS) copper chambers, with an inner diameter of 80 mm and fitted with standard DN100 Conflat flanges. The thickness of the copper is 2 mm, and the chambers are coated with TiZrV non-evaporable getter (NEG) [23]. After ac- tivation at low temperature (200 ◦C), the getter provides distributed pumping and low outgassing to maintain a low residual gas pressure, as well as a low secondary electron emission yield to avoid electron multipacting. The chambers are connected by means of shielded bellows modules, some of them including pumping and diagnostic ports.

1.6 Cryogenic system 1.6.1 Overview The superconducting magnet windings in the arcs, the dispersion suppressors, and the inner triplets will be immersed in a pressurized bath of superfluid helium at about 0.13 MPa (1.3 bar) and a maximum temperature of 1.9 K [24]. This allows a sufficient temperature margin for heat transfer across the electrical insulation. As the specific heats of the superconducting alloy and its copper matrix fall rapidly with decreasing temperature, the full benefit in terms of the stability margin of operation at 1.9 K (instead of at the conventional 4.5 K) may only be gained by making effective use of the transport properties of superfluid helium, for which the temperature of 1.9 K also corresponds to a maximum in the effective thermal conductivity. The low bulk viscosity enables the coolant to permeate the heart of the magnet windings. The large specific heat (typically 105 times that of the conductor per unit mass, 2×103 times per unit volume), combined with the enormous heat conductivity at moderate flux (3000 times that of cryogenic-grade oxygen-free high-thermal conductivity (OFHC) copper, peaking at 1.9 K) can have a powerful stabilizing action on thermal disturbances. To achieve this, the electrical insulation of the conductor must preserve sufficient porosity and provide a thermal percolation path, while still fulfilling its demanding dielectric and mechanical duties. The cryogenic system must be able to cope with the load variations and a large dynamic range induced by the operation of the accelerator, as well as being able to cool down and fill the huge cold mass of the LHC, 37 × 106 kg, within a maximum delay of 15 days while avoiding thermal differences in the cryomagnet structure higher than 75 K. The cryogenic system must also be able to cope with resistive transitions of the superconducting magnets, which will occasionally occur in the machine, while minimizing loss of cryogen and system perturbations. It must handle the resulting heat release and its consequences, which include fast pressure rises and flow surges. The system must limit the propagation to neighbouring magnets and recover in a time that does not seriously detract from the operational availability of the LHC. A resistive transition extending over one lattice cell should not result in a down time of more than a few hours. It must also be possible to rapidly warm up and cool down limited lengths of the lattice for magnet exchange and repair. Finally, the cryogenic system must be Cryogenic system 25 able to handle, without impairing the safety of personnel or equipment, the largest credible incident of the resistive transition of a full sector. The system is designed with some redundancy in its subsystems.

1.6.2 General architecture The main constraints on the cryogenic system result from the need to install it in the existing LEP tunnel and to re-use LEP facilities, including four refrigerators. The limited number of access points to the underground area is reflected in the architecture of the system. The cooling power required at each temperature level will be produced by eight refrigeration plants and distributed to the adjacent sectors over distances up to 3.3 km. To simplify the magnet string design, the cryogenic headers distributing the cooling power along a machine sector, as well as all remaining active cryogenic components in the tunnel, are contained in a compound cryogenic distribution line (QRL). The QRL runs alongside the cryomagnet strings in the tunnel and feeds each 106.9-m-long lattice cell in parallel via a jumper connection (Fig. 1.16). The LHC tunnel is inclined at 1.41% with respect to the horizontal, thus giving height differences of up to 120 m across the tunnel diameter. This slope generates hydrostatic heads in the cryogenic headers and could generate flow instabilities in two- phase, liquid–vapour, flow. To avoid these instabilities, all fluids should be transported over large distances in a monophasic state, i.e. in the superheated-vapour or supercritical region of the phase diagram. Local two-phase circulation of saturated liquid can be tolerated over limited lengths, in a controlled direction of circulation. Equipment is installed as much as possible above ground to avoid the need for further excavation, but

Fig. 1.16 Transverse cross section of the LHC tunnel. 26 The Large Hadron Collider

Point 5 Sector refrigeration Point 4 Point 6 3.3 km plant

Point 3 Point 7

Point 2 Point 8

Point 1

Fig. 1.17 General layout of the cryogenic system. certain components have to be installed underground near the cryostats. For reasons of safety, the use of nitrogen in the tunnel is forbidden, and the discharge of helium is restricted to small quantities only. Figure 1.17 shows the general layout of the cryogenic system, with ﬁve “cryogenic islands” at access Points 1, 2, 4, 6, and 8, where all refrigeration equipment and ancillary equipment is concentrated. Equipment at ground level includes electrical substations, warm compressors, cryogen storage (helium and liquid nitrogen), cooling towers, and cold boxes. Underground equipment includes lower cold boxes, 1.8 K refrigeration unit boxes, interconnecting lines, and interconnection boxes. Each cryogenic island houses one or two refrigeration plants that feed one or two adjacent tunnel sectors, requiring distribution and recovery of the cooling ﬂuids over distances of 3.3 km underground.

1.6.3 Temperature levels In view of the high thermodynamic cost of refrigeration at 1.8 K, the thermal design of the LHC cryogenic components aims at intercepting the main heat influx at higher temperatures—hence the multiple-staged temperature levels in the system. The temperature levels are as follows: • 50–75 K for the thermal shield protecting the cold masses; • 4.6–20 K for lower-temperature interception and for cooling the beam screens that protect the magnet bores from beam-induced loads; • 1.9 K quasi-isothermal superfluid helium for cooling the magnet cold masses; • 4 K at very low pressure for transporting the superheated helium flow coming from the distributed 1.8 K heat-exchanger tubes across the sector length to the 1.8 K refrigeration units; • 4.5 K normal saturated helium for cooling some insertion region magnets, RF cavities, and the lower sections of the high-temperature superconductor (HTS) current leads; • 20–300 K cooling for the resistive upper sections of the HTS current leads [4]. Beam instrumentation 27

To provide these temperature levels, use is made of helium in several thermodynamic states. The cryostats and cryogenic distribution line (QRL) combine several techniques for limiting heat inﬂux, such as low-conduction support posts, insulation vacuum, multilayer reﬂective insulation wrapping, and low-impedance thermal contacts, all of which have been successfully applied on an industrial scale.

1.7 Beam instrumentation An accurate and complete set of beam instrumentation is essential for eﬃcient commissioning and operation of the LHC. This instrumentation includes beam- position measurement all around the ring, beam-loss monitors, current and proﬁle measurement, and specialized instrumentation to measure beam properties such as chromaticity and tune.

1.7.1 Beam-position measurement The majority of the LHC beam-position monitors (BPMs; 860 of the 1032) are of the arc type (see Fig. 1.18), consisting of four 24-mm-diameter button electrode feed- throughs mounted orthogonally in a 48 mm inner-diameter beam pipe. The electrodes are curved to follow the beam-pipe aperture and are retracted by 0.5 mm to protect the buttons from direct synchrotron radiation from the main bending magnets. Each electrode has a capacitance of 7.6 ± 0.6 pF, and is connected to a 50 Ω coaxial, glass-ceramic, ultrahigh-vacuum feedthrough. The inner triplet BPMs in all interaction regions are equipped with 120 mm, 50 Ω directional stripline couplers (BPMS×), capable of distinguishing between counter- rotating beams in the same beam pipe. The locations of these BPMs (in front of Q1, in the Q2 cryostat, and after Q3) were chosen to be as far as possible from parasitic crossings to optimize the directivity. The 120 mm stripline length was chosen to give a signal similar to the button electrode, thereby allowing the use of the same acquisition electronics as for the arcs. All cold directional couplers use an Ultem dielectric for use in a cryogenic environment, while the warm couplers use a Macor dielectric to allow bake-out to over 200 ◦C.

Copper–beryllium contact N connector

Glass ceramic seal 50 Ώ coaxial line

RF contact

Fig. 1.18 (a) 24 mm button electrode. (b) Mounted beam-position monitor bodies. 28 The Large Hadron Collider

The cleaning insertions at Points 3 and 7 are equipped with warm 34-mm-diameter button electrode BPMs (BPMW) ﬁtted either side of the MQWA magnets. The electrodes are an enlarged version of the arc BPM button. The same button electrodes are also used for the cold BPMs in the matching sections either side of the four interaction regions, as well as for the warm BPMs located near the D2 magnets and either side of the transverse damper (ADTV/H). The BPMCs, installed at Point 4, are combined monitors consisting of one BPM using standard 24 mm button electrodes for use by the orbit system and one BPM using 150 mm shorted stripline electrodes for use in the transverse damper system.

1.7.2 Beam-current measurement Beam-current transformers of two diﬀerent kinds provide intensity measurements for the beams circulating in the LHC rings, as well as for the transfer lines from the SPS to the LHC, and from the LHC to the dumps. The transformers are all installed in sections where the vacuum chamber is at room temperature and where the beams are separated. The fast beam-current transformers (FBCTs) are capable of integrating the charge of each LHC bunch. This provides good accuracy for both bunch-to-bunch measurements and average measurements, intended mainly for low-intensity beams, for which the accuracy of the DC current transformers (DCCTs) will be limited. For redundancy, two transformers, with totally separated acquisition chains, will be placed in each ring. These will be located at Point 4. The measurement precision for the pilot beam of 5 × 109 protons in a single bunch is around 5% (worst case 10%), and for the nominal beam below 1%. The DCCTs are based on the principle of magnetic ampliﬁers and measure the mean intensity or current of the circulating beam, and they can be used to measure the beam lifetime. Because of their operational importance, two of the devices will be installed in each ring. Currently a resolution of 2 µA can be reached, but 1 µAis targeted, corresponding to 5 × 108 circulating particles.

1.7.3 Beam-loss system The loss of a very small fraction of the circulating beam may induce a quench of the superconducting magnets or even physical damage to machine components. The detection of lost beam protons allows protection of the equipment against quenches and damage by generating a beam-dump trigger when the losses exceed thresholds. In addition to quench prevention and damage protection, the loss detection allows the observation of local aperture restrictions, orbit distortion, beam oscillations, and particle diffusion. The loss measurement is based on the detection of secondary shower particles using ionization chambers located outside the magnet cryostats. The secondary particle energy flux depends linearly on the initiating proton parameters. To observe a representative fraction of the secondary particle flux, detectors are placed at likely loss locations. The calibration of the damage and quench level thresholds with respect to the measured secondary particle energy deposition is simulation-based. Beam instrumentation 29

1.7.4 Transverse profile measurement User requirements led to the definition of four functional modes to be mapped on the different types of hardware monitors: • A single-pass monitor of high sensitivity (pilot beam) with a modest demand on accuracy and few restrictions on the beam blowup due to the traversal. • A “few-pass monitor” (typically 20 turns) dedicated to the intermediate to nominal intensity range of the injected beam for calibration or matching studies. The blowup per turn should be small compared with the effect to be measured. • A circulating beam monitor, working over the whole intensity range. No blowup is expected from such a monitor. • A circulating beam tail monitor optimized to scan low beam densities. In this mode, one may not be able to measure the core of the beam. The measurement should not disturb the tail density significantly. The monitor types include wire scanners, residual gas ionization monitors and synchrotron light monitors using light from D2-type superconducting dipoles. Synchro- tron light monitors are also under development for using light from superconducting undulators in each ring. Point 4 is the default location for all instrumentation.

1.7.5 Tune, chromaticity, and betatron coupling Reliable measurement of betatron tune and of the related quantities tune spread, chromaticity, and betatron coupling will be essential for all phases of LHC running from commissioning through to ultimate performance luminosity runs. For injection and ramping, the fractional part of the betatron tune must be controlled to ±0.003, while in collision, the required tolerance shrinks to ±0.001. With the exception of Schottky scans and the “AC-dipole” excitation outside the tune peak, all tune measurement techniques involve some disturbance to the beam. The resulting emittance increase, while acceptable for some modes of running, has to be strongly limited for full-intensity physics runs. Diﬀerent tune measurement systems are therefore installed.

General tune measurement system This uses standard excitation sources (single-kick, chirp, slow swept frequency, and noise). It operates with all ﬁlling patterns and bunch intensities and will be commissioned early after LHC start-up. Even with oscillation amplitudes down to 50 µm, a certain amount of emittance increase will result, limiting the frequency at which measurements can be made. It is therefore unsuitable for generating measurements for an online tune feedback system.

High-sensitivity tune-measurement system The beam is excited by applying a signal of low amplitude and high frequency to a stripline kicker. This frequency is close to half the bunch spacing frequency (40 MHz for the nominal 25 ns bunch spacing). The equivalent oscillation amplitude is a few micrometres or less for a β-function of about 200 m. A notch ﬁlter in the transverse 30 The Large Hadron Collider feedback loop suppresses the loop gain at this frequency, where instabilities are not expected to be a problem. If the excitation frequency divided by the revolution frequency corresponds to an integer plus the fractional part of the tune, then coherent betatron oscillations of each bunch build up turn by turn (resonant excitation). A batch structure with a bunch every 25 ns “carries” the excitation frequency as side- bands of the bunch spacing harmonics. A beam position pickup is tuned to resonate at one of these frequencies.

1.8 Commissioning and operation By 10 September 2008, seven of the eight sectors had been successfully commissioned to 5.5 TeV in preparation for a run at 5 TeV. Owing to lack of time, the eighth sector had only been taken to 4 TeV. Beam commissioning started by threading Beam 2, the counterclockwise beam, around the ring, stopping it at each long straight section sequentially to correct the trajectory. In less than an hour, the beam had completed a full turn, witnessed by a second spot on a ﬂuorescent screen intercepting both injected and circulating beams (Fig. 1.19). Very quickly, a beam circulating for a few hundred turns could be established. Figure 1.20 shows the capture process when the RF cavities are switched on. Each horizontal line on the mountain range display records the bunch intensity every 10 turns. Without the RF, the beam debunches as it should in about 250 turns, or 25 ms. A ﬁrst attempt was made to capture the beam, but, as can be seen, the injection phase was completely wrong. Adjusting the phase allowed a partial capture, but at a slightly wrong frequency. Finally, adjusting the frequency resulted in a perfect capture.

Fig. 1.19 Beam on turns 1 and 2. Commissioning and operation 31

Fig. 1.20 Clockwise from top left: no RF, with debunching in about 250 turns (i.e. roughly 25 ms); ﬁrst attempt at capture, at exactly the wrong injection phase; capture with corrected injection phase but wrong frequency; capture with optimum injection phasing, correct reference.

The closed orbit could then be corrected. Figure 1.21 shows the first orbit correction, where—remarkably at this early stage—the RMS orbit is less than 2 mm. It can be seen that in the horizontal plane, the mean orbit is displaced radially by about a millimetre, indicative of an energy mismatch of about 10−3. On 30 March 2010, the first collisions were obtained at a centre-of-mass energy of 7 TeV. Since then, operating time has been split between machine studies and physics data-taking. In view of the very large stored energy in the beams, particular attention has to be given to the machine protection and collimation systems. More than 120 collimators are arranged in a hierarchy of primary, secondary, and tertiary collimators. Tight control of the orbits in the region of the collimators is achieved with a feedback system. The collimation system also works very efficiently. Figure 1.22 shows a loss map around the ring obtained by provoking beam loss. The losses are located precisely where they should be, with a factor-of-10 000 difference between the losses on the collimators and those in the cold regions of the machine. 32 The Large Hadron Collider

Fig. 1.21 Corrected closed orbit on B2. There is an energy oﬀset of about −0.9 permil due to the capture frequency.

0.1 Cold Collimator 0.01 Factor 10,000 Warm IR2 0.001 IR1 IR8 IR5 0.0001

1e-05

1e-06 Beam loss (Gy/s) 1e-07

1e-08

1e-09 0 5000 10000 15000 20000 25000 s (m)

Fig. 1.22 Loss maps for collimation.

The machine performance at this early stage is very impressive. A single-beam lifetime of more than 1000 hours has been observed—an order of magnitude better than expected—proving that the vacuum is considerably better than expected, and also the noise level in the RF system is very low. The nominal bunch intensity of 1.1 × 1011 has been exceeded, and the β∗ at the experimental collision points has been squeezed to 2 m. The closed orbit can be kept to better than 1 mm RMS with very good reproducibility. During the 200 days of running in 2010, the luminosity has increased by ﬁve orders of magnitude, reaching 2.1 × 1032 cm−2 s−1 with a bunch separation of 150 ns. More than 6 pb−1 of integrated luminosity has been produced in each of the two big detectors. References 33

Acknowledgments The LHC is the most complex scientiﬁc instrument ever constructed. It has taken 15 years to build, and many problems have been encountered on the way. These have all been overcome thanks to the resourcefulness and resilience of the people who built it, both inside CERN and in our collaborating laboratories around the world. Now the machine is moving into its operational phase. I am conﬁdent that an equally competent team will exploit it to its full potential in the coming years.

References [1] J. P. Blewett, 200 GeV intersecting storage accelerators, 8th Int. Conf. on High- Energy Accelerators, CERN, Geneva, 1971, 501–504. [2] E. J. Bleser, Superconducting magnets for the CBA project, Nucl. Instrum. Methods Phys. Res. A235 (1985) 453–463 (see footnote on p. 435). [3] CBA Brookhaven colliding beam accelerator, Newsletter 2 (Nov. 1982) 27–31. [4] J. Billan et al., The eight superconducting quadrupoles for the ISR high-luminosity insertion, 11th Int. Conf. on High-Energy Accelerators, CERN, Geneva, 1980. Birkhäuser, Basel (1980), pp. 848–852. [5] G. Bon Mardion, G. Claudet, P. Seyfert, and J. Verdier, Helium II in low- temperature and superconductive magnet engineering, Adv. Cryo. Eng. 23 (1978) 358–362. [6] R. Aymar, G. Claudet, C. Deck, R. Duthil, P. Genevey, C. Leloup, J. C. Lot- tin, J. Parrain, P. Seyfert, A. Torossian, and B. Turck, Conceptual design of a superconducting tokamak: Torus II Supra, IEEE Trans. Magn. MAG15-1 (1979) 542–545. [7] G. Claudet and R. Aymar, Tore Supra and helium II cooling of large high-field magnets,Adv.Cryo.Eng.35A (1990) 55–67. [8] G. Claudet, F. Disdier, P. Lebrun, M. Morpurgo, and P. Weymuth, Prelimin- ary study of superfluid helium cryogenic system for the Large Hadron Collider, Proc. ICFA Workshop on Superconducting Magnets and Cryogenics, Brookhaven National Laboratory, USA, 1986, BNL 52006, pp. 270–275. [9] J. Casas Cubillos, A. Cyvoct, P. Lebrun, M. Marquet, L. Tavian, and R. van Weelderen, Design concept and first experimental validation of the superfluid helium system for the Large Hadron Collider (LHC) project at CERN, Cryogenics 32(ICEC Suppl.) (1992) 118–121. [10] The LHC Design Report, CERN-2004-003 (3 vols.) (June 2004). [11] ATLAS Technical Proposal, CERN/LHCC/94-43, LHCC/P2 (Dec. 1994). [12] CMS Technical Proposal, CERN-LHCC-94-38, LHCC/P2 (Dec. 1994). [13] LHCB Technical Proposal, CERN-LHCC-98-4 (Feb. 1998). [14] Total cross section, elastic scattering and diffractive dissociation at the LHC, CERN/LHCC 99-7 (Mar. 1999). [15] ALICE Technical Proposal, CERN-LHCC-95-71 (Dec. 1995). [16] E. Shaposhnikova, Longitudinal beam parameters during acceleration in the LHC, LHC Project Note 242. 34 The Large Hadron Collider

[17] D. Boussard, W. Höfle, and T. Linnecar, The LHC transverse damper (ADT) performance specification, SL-Note-99-055. [18] D. Boussard and T. Linnecar, The LHC superconducting RF system, Cryogenic Engineering and International Cryogenic Materials Conf., Montreal, Canada, July 1999, and LHC Project Report 316. [19] D. Angal-Kalinin and L. Vos, Coupled bunch instabilities in the LHC, 8th European Particle Accelerator Conf., Paris, 2002. [20] D. Brandt and L. Vos, Resistive wall instability for the LHC: intermediate review, LHC Project Report 257 (2001). [21] K. Eggert, K. Honkavaara, and A. Morsh, Luminosity considerations for the LHC, LHC Note 263 (1994). [22] O. Grobner, The LHC vacuum system, PAC’97, 1997, Vancouver, Canada. [23] C. Benvenuti et al., Vacuum properties of TiZrv non-evaporable getter films for the LHC vacuum system, Vacuum 60 (2001) 57–65. [24] P. Lebrun, Superconductivity and cryogenics for the Large Hadron Collider, CERN, LHC Project Report 441 (27 Oct. 2000). 2 The LHC machine: from beam commissioning to operation and future upgrades

Massimo Giovannozzi

Beams Department, CERN, Geneva, Switzerland

2 The LHC machine: from beam commissioning to operation and future upgrades 35 Massimo GIOVANNOZZI

2.1 LHC layout, parameters, and challenges 37 2.2 Digression: the chain of proton injectors 47 2.3 Proton beam commissioning and operation 48 2.4 Future upgrade options 52 Acknowledgments 55 References 55 LHC layout, parameters, and challenges 37

This chapter reports on the current status of the CERN Large Hadron Collider. General machine parameters are reviewed and the beam commissioning process is presented, showing the evolution of the performance over recent years. The highlights of the powerful complex of injectors is described, in order to provide a global picture of the impressive performance of CERN’s flagship machine, relying on both the aston- ishing quality of the LHC itself and on the incredible flexibility of the injectors. The focus is on proton physics performance, with emphasis on the different possible scenarios leading to an upgrade of LHC performance. Finally, the future of the machine will be discussed briefly. The main reference on the LHC machine is [1] together with references cited therein. Here, a selection (possibly not complete, owing to the extent of the various topics presented) of additional publications is given for details. The reader interested in lead ions will find the relevant information in [2–24].

2.1 LHC layout, parameters, and challenges The LHC in a nutshell is a two-ring, high-energy, high-luminosity, pp collider [1, 25, 26]. The choice of the particles is imposed by the requirement of providing a very high luminosity L deﬁned as

2 Nb Mfrev γr L = ∗ F, (2.1) 4 π n β where Nb is the number of charges in each bunch, M is the number of bunches in each beam, frev is the revolution frequency, γr is the relativistic γ factor, n = βrγr ∗ is the normalized rms beam emittance (βr being the relativistic β factor), and β is the value of the beta function at the collision point. The so-called geometrical factor F takes into account the eﬀect of a crossing angle at the interaction point (IP), which reduces the volume overlap between the two colliding bunches. It is expressed as 1 F = , (2.2) ∗ 2 1+(θc σz/2 σ )

∗ where θc is the value of the crossing angle at the IP, and σz and σ are the longitudinal and transverse rms beam sizes, respectively. It should be mentioned that (2.1) neglects the so-called hourglass effect, namely the change in volume overlap due to the variation of the beta function over the length of the colliding bunches. Such an effect is small for the LHC. From the definition of the luminosity, several quantities can be varied in order to maximize L. The bunch charge is certainly one of those. If this approach is followed, then the use of antiprotons is immediately ruled out. These particles need to be produced artificially and this production rate is very low. Furthermore, since they are produced as secondary beams, the natural emittances are in general very large, thus leading to the generation of beams with rather low brightness.1 To mitigate these

1 The brightness is defined as the number of particles in unit volume of phase space. 38 The LHC: from beam commissioning to operation and future upgrades drawbacks, complex beam manipulations, such as beam cooling (see [27] and references therein) and beam accumulation [28–34], would be required. It is worth noting that this complex approach was used for the CERN Super Proton Synchrotron (SPS) at the time of the pp collider [35] and for the Fermilab Teva- tron [36]. However, both colliders had luminosities much smaller than that required for the LHC. The use of multibunch beams is another key way to increase L, and is therefore assumed in order to reach the nominal LHC performance. The need to have a high-energy machine and the fact that its bending radius is imposed by the decision to re-use the tunnel of the former Lepton Positron (LEP) collider [37–39], immediately fixes the value of the magnetic field required to guide the charged particles along the nominal closed orbit. Such a field is beyond the reach of normal-conducting dipoles. Therefore, superconducting magnets had to be considered and are indeed at the heart of the whole machine. A comparison of the geometry of the LEP and LHC machines, including the required bending fields, is shown in Table 2.1. The choice of the particles, namely protons for both beams, immediately imposes a requirement for two separate magnetic channels. In fact, counter-rotating beams of the same charge require opposite-sign magnetic bending fields, and opposite-sign focusing or defocusing magnetic fields as well. In turn, this choice imposes a challenge to the designers of the superconducting magnets: in order to fit into the limited size of the LEP tunnel, the two superconducting magnets need to share the same cryostat to save space. This leads to the so-called two-in-one design as seen in Fig. 2.1, with deep implications for the crosstalks between the fields in the two apertures and therefore for the technological challenges for the magnet design [40–42]. The LHC ring features an eightfold symmetry with eight arcs and eight long straight sections (LSS), in the middle of which the insertion region (IR) can be located (see Fig. 2.2). Four such IRs house the experiments: ATLAS (IR1), Alice (IR2), CMS (IR5), and LHCb (IR8). The remaining four IRs are dedicated to special machine systems, such as radiofrequency (RF) and beam instrumentation (IR4), beam dump (IR6), and collimation (IR3 for momentum and IR7 for betatron collimation). The experimental IRs 2 and 8 also house the injection systems for Beam 1 (IR2) and Beam 2 (IR8).2 As an example, the layout of IR1 is shown in Fig. 2.3. Starting from the left- hand side, the IP is visible, then three quadrupoles (Q1, Q2, and Q3) making the

Table 2.1 An overview of geometry and bending ﬁelds required for LEP and LHC

LEP LHC Bending radius (m) 3096.175 2803.95 Momentum (GeV/c) 104 7000 B-ﬁeld (T) 0.11 8.33

2 Beam 1 is the clockwise beam, and Beam 2 is the counterclockwise beam. LHC layout, parameters, and challenges 39

Fig. 2.1 Cross section of an LHC main superconducting dipole (Credit: CERN-LHC-PHO- 2001-187). inner triplet, i.e. the structure that provides the strong focusing required at the IP. Note that Q2 is split into two magnets, while Q1 and Q3 are each a single magnet. Then, moving towards the arc, one encounters the D1 separation dipole. It consists of six normal conducting magnets that start separating in the horizontal plane the two beams travelling in the same beam pipe. The angle imparted by D1 is cancelled by the D2 superconducting separation dipole, which brings the two beams to the nominal horizontal separation of 194 mm characteristics of the arc geometry. Two absorbers are present in this region: the TAS protects the triplet quadrupoles from the debris generated at the collision point, while the TAN protects the D2 from the neutral particles that are not deflected by the magnets upstream. Finally, three quadrupoles (Q4, Q5, and Q6) provide the transition optics to the elements of the dispersion suppressor, before the continuous cryostat region at the quadrupole Q7. Correspondingly, the optical parameters for the horizontal and vertical planes as well as the dispersion functions are plotted in Fig. 2.4 for injection (a) and collision (b) energy. For the injection case, β∗ (i.e. the value of the beta function at the IP) is 11 m, while in the collision case, it is 0.55 m. Note the huge difference in the value of βmax (which is achieved in Q2) for the two configurations. The same layout is used for the other high-luminosity insertion in point 5. The arcs are made of regular structures, the so-called FODO cells (see Fig. 2.5). Each such structure comprises six main dipoles and two main quadrupoles. This periodic structure provides the necessary bending and focusing strength, based on the principle of alternating gradient or strong focusing [43]. Furthermore, a complex 40 The LHC: from beam commissioning to operation and future upgrades

CMS

Low-β pp

LSS5

DSL DSR RF ARC Dump

DSR ARC LSS4 DSL Octant 5 DSL Octant 6 LSS6 DSR ARC Octant 4 ARC

DSR Octant 7 DSL

Momentum Betatron cleaning LSS3 Beam 1 Beam 2 LSS7 cleaning

DSL Octant 3 DSR ARC ARC Octant 2 DSL LSS8 DSR Octant 8 Octant 1 DSR LSS2 ARC DSL Low-β (ions) B - physics

injection ARC injection DSR DSL ALICE LSS1 LHC-b

Low-β pp

ATLAS

T12 T18

Fig. 2.2 Layout of the LHC ring. (Figure reproduced from [1], c 2004 CERN).

LSS DS 1.9K4.5K 4.5K 4.5K 4.5K 4.5K 4.5K 4.5K 1.9K DF BL Q6,Q5 Q1 Q2 Q3 D1 D2Q4 Q5 Q6 Q4,D2 Q7 TAS MQIA MQIB MQIADFBI MBIW TAN MBRC MQY MQML MQML DFBA MQM MB IP1 BEAM 1 DQR 194 BEAM 2

Q7,Q8,Q9,Q10 1.8 1.8 6.37 2.715 5.5 1 5.5 3.215 6.37 0.766 3.44 2.413 MB 19.05 19.05 3.4 3.5 8.475 11.32 3.4 0.68 22.965 22.965 30.67 5.607 24.23 69.703 9.45 5.228 3.4 22.837 4.8 27.1 4.8 29.214 7.167 268.904

Fig. 2.3 Layout of the LHC high-luminosity IR1. (Figure reproduced from [1], c 2004 CERN). system of correctors—the so-called spool pieces, which we small corrector magnets (sextupole, octupole, and decapole) installed at the extremities of the main dipoles— have been designed to provide the necessary compensation for the unavoidable ﬁeld imperfections of the superconducting dipoles. Other sets of correctors are installed close to the main quadrupoles to correct tune (quadrupolar correctors), and linear LHC layout, parameters, and challenges 41

(a) 300 0.20 Horizontal beta function Vertical beta function Horizontal dispersion 0.15 250 0.10 200 0.05

150 0.00

−0.05 (m) Dispersion Beta function (m) 100 −0.10 50 −0.15

0 −0.20 −300 −200 −100 0 100 200 300 Distance (m)

(b) 5000 0.20 Horizontal beta function 4500 Vertical beta function Horizontal dispersion 0.15 4000 0.10 3500 0.05 3000

2500 0.00

2000

−0.05 Dispersion (m) Beta function (m) 1500 −0.10 1000 −0.15 500

0 −0.20 −300 −200 −100 0 100 200 300 Distance (m)

Fig. 2.4 Optical functions at injection (a) and at collision (b). The origin of the horizontal axis is the IP and the distance extends to the Q7 magnet (left/right of the IP). coupling (skew quadrupolar correctors), and chromaticity (sextupole magnets), and to combat instabilities (octupole magnets). A summary of the main beam parameters is presented in Table 2.2 The LHC machine is a real technological challenge [1, 44]. The heart of the machine, the system of superconducting magnets, is at the forefront of current technology, given 42 The LHC: from beam commissioning to operation and future upgrades

106.90 m

MQ MBA MBB MBA MQ MBB MBA MBB MCS BPM MCS MCS MCS MCS MSCB MCDO MCDO MSCB MCDO

MQT: trim quadrupole MO, MQT, MQS MCS: spool piece sextupole

MQS: skew trim quadrupole MCDO: spool piece octupole + decapole

MO: lattice octupole

MSCB: sextupole (skew sextupole) + orbit corrector

Fig. 2.5 Layout of the LHC FODO cell. (Figure reproduced from [1], c 2004 CERN).

Table 2.2 Main parameters of the LHC machine

BEAM DATA INJECTION COLLISION Proton energy (GeV) 450 7000

γr 479.6 7461 11 Nb 1.15 × 10 M 2808 Longitudinal emittance (eV s) 1.0 2.5

n = γr (μm rad) 3.5 3.75 Circulating beam current (A) 0.582 Stored energy per beam (MJ) 23.3 362

Peak-luminosity-related data

RMS bunch length (cm) 11.24 7.55 RMS beam size at IP1 and IP5 (μm) 375.2 16.7 Geometric reduction factor F 0.836 Peak luminosity in IP1 and IP5 1.0 × 1034 (cm−2 s−1) Peak luminosity per bunch crossing 3.56 × 1030 in IP1 and IP5 (cm−2 s−1) LHC layout, parameters, and challenges 43 the quality required for use in an accelerator application. Another challenge was the required transfer of the production of the entire ensemble of magnets to industry, as it was not possible to build them at the laboratory [45–47]. The quality assurance process was perfectly transferred to industry, so that the required standards were achieved without any drift in performance during production and with no systematic differences among the various manufacturers. The superconducting magnets require a powerful cryogenic system, which was the first to be developed on the large scale required for the 27 km ring [48, 49]. The vacuum system, comprising three complete subsystems (for the beam vacuum, for the insulation vacuum of the magnets, and for the vacuum system in the cryogenic line), all reaching an unprecedented scale of complexity, should also be mentioned [50–52]. Last but not least, another key challenge is the machine protection system that is needed to deal with the stored energy in the magnets and in the beams [53–55]. In each main dipole, about 7 MJ are stored in the magnetic field, while the nominal beams store 362 MJ. In case of problems, such as the quench of a magnet or abnormal beam behaviour, the machine protection system is supposed to provide a safe means to dispose of the beam and to manage the resistive transition of the magnets without damaging the machine. For the first time in an accelerator project, fail-safe criteria had to be implemented in the design of several components in order to achieve the required level of protection and performance. The quench protection system (QPS) is a distributed system that provides the necessary protection against magnet quench [56–58], while the LHC beam dumping system (LBDS) [59] is located in IR6 and provides a safe way to dispose of the beams in case of problems. In addition, the LHC machine is also a challenge in terms of beam physics [1]. A selection of the physical phenomena and processes that affect the proton beams and need to be kept under control to achieve the nominal performance are now briefly reviewed. To achieve the required performance, tight control of the machine optics is mandatory. Failing this, the machine aperture would not be enough to ensure safe operation ∗ with the large values of βmax that are generated when β is squeezed down. This implies a detailed knowledge of the magnetic properties of the several families of normal and superconducting quadrupoles that control the machine’s optics. This information was obtained thanks to a massive programme of magnetic measurements during the production and installation stage of the LHC ring [60, 61]. Further, the knowledge acquired had to be fully used in the control system. To this end, a magnetic model, the so-called field description of the LHC (FiDeL) was developed and successfully implemented during the beam commissioning stage and the ensuing operation [62–68]. In addition, powerful techniques have been developed to measure the machine’s optics and perform corrections whenever necessary [69–72]. These techniques were extremely successful in correcting the β beating, i.e., the difference between the nominal and the measured optical parameters, to less than 10% [73], the target value being about 17%, i.e. a factor two better than estimated. Unlike normally conducting magnets, whose field quality is determined by the pole shape profile, superconducting devices generate a magnetic field by shaping the 44 The LHC: from beam commissioning to operation and future upgrades current distribution. For technical reasons, it is not possible to achieve perfect field quality, and unavoidable perturbations are to be expected. If the usual expansion in multipoles is used, ∞ − x + iy n 1 B + iB = B (b + ia ) , (2.3) y x ref n n R n=1 r where n = 1 stands for a dipole field, an,bn are the skew and normal components, respectively, Rr is a reference radius, and x, y stand for the horizontal and vertical coordinates, respectively, then superconducting dipoles have systematic b3,b5,b7 components and quadrupoles b6,b10. These components require careful optimization as they introduce nonlinear effects in the beam dynamics, possibly inducing beam losses due to particles that are pushed to high amplitude by these effects. All this was studied in detail in the design phase by analytical estimates and massive numerical simulations. Specifications on the tolerable multipolar components were calculated [74] and imposed at the production level [75–80]. Corrector magnets (the spool pieces) were envisaged to minimize even the residual imperfections further. Finally, at the installation level, when the field quality of each magnet was known, the sorting algorithm devised earlier [81–84] was applied to provide the last optimization of the machine performance [85]. Thanks to these measures, both linear and nonlinear effects are not an issue in the current operational phase of the LHC [86]. The LHC superconducting magnets do not tolerate beam losses, as the lost protons will deposit energy in the magnets, leading to quenches. Therefore, tight control of beam loss is mandatory. Furthermore, the high-amplitude particles, forming the so-called primary halo, must be prevented from generating background problems in the experiments. In addition, passive protection of the machine aperture using the information from the beam loss monitors is necessary. These are the key functions of the powerful collimation system that is installed in the LHC [87–94]. To achieve the required performance, a multistage system was designed, in which a series of collimators of different materials, were used to intercept and absorb protons in the beam halo. In Fig. 2.6, the hierarchy of apertures of the various collimator stages is shown for injection (a) and collision (b) energies.3 It is worth pointing out that the aperture hierarchy is essential to ensuring the correct operation of the whole system, which in turn imposes very tight constraints on the machine reproducibility and stability. Furthermore, the choice of material for the collimators faces a dilemma: low-Z materials would be preferred in terms of robustness, but these materials are poor conductors and thus have a negative impact on the machine impedance [95–97]. The opposite occurs for high-Z materials. A tradeoff has been found using different materials for the various type of collimators (primary, secondary, etc.). A staged approach has been selected to profit from incremental improvement of the materials to increase the overall system performance. Two LHC insertions are devoted to the collimation system, with two separate functions: one to intercept particles with large momentum deviation (IR3) and a second to intercept

3 The collimator aperture at top energy will be only a few millimetres. LHC layout, parameters, and challenges 45

5.7σ 6.7σ 10.0σ (a) ARC ARC ARC IP and Triplets and IP

Primary Secondary Absorber Tertiary Physics absorbers (robust) (robust) (W metal) (W metal) (Cu metal) 6.0+σ 7.0+σ 10.0+σ 8.5+σ 10.0+σ (b) ARC ARC ARC IP & Triplets IP

Primary Secondary Absorber Tertiary Physics absorbers (robust) (robust) (W metal) (W metal) (Cu metal)

Fig. 2.6 Principle of the collimation system: hierarchy of the collimators aperture at injection (a) and collision (b) energies. (courtesy R. Assmann-CERN). particles with large betatronic amplitude (IR7). These insertions thus provide the special optics that are required to ensure optimal system performance. Another subtle effect is that of the so-called electron cloud [98–118]. The circulating proton beams are extracting electrons from the surface of the beam screen. Two different processes are involved: protons hitting the surface can extract electrons (this effect is dominant at lower energies); alternatively, synchrotron radiation can also extract electrons from the beam screen surface (this effect is dominant at higher energies). Furthermore, the subsequent proton bunches can accelerate the extracted electrons, which can then hit the surface and extract even more electrons in an avalanche phenomenon (see Fig. 2.7 (a)), provided that conditions on the bunch spacing are satisfied. The net result is to generate a standing cloud of electrons that reaches a saturation level (Fig. 2.7 (b)). The interaction of the cloud with the proton bunches can lead to strong deformation of the bunch structure (Fig. 2.7 (c)), eventually driving the beam to instability and generating emittance growth and/or beam losses. In both cases, this is detrimental to the luminosity of the LHC. Mitigation measures had to be studied and implemented. They included the following: special coating of the warm regions with a special non-evaporable getter (NEG) [119] to reduce the secondary emission yield; reduction of the reflectivity of the beam screen surface by means of a sawtooth surface [1, 120]; installation of solenoids around the vacuum chambers to trap the low-energy electrons, preventing 46 The LHC: from beam commissioning to operation and future upgrades

(a) γγγLOST or REFLECTED Secondary electron 10 ev 10 ev

5 ev Secondary electron

5 ev 5 ev

Photoelectron 200 ev 200 ev

2 kev 2 kev 200 ev

Time 20 ns 5 ns 20 ns 5 ns

(b) Bunch

2.5 Direction in which the bunch moves 2

1.5 Electrons Bunch 1

ρ (arb. units) 0.5 After a few passages through the 0 electron clould 10.50 21.5 2.5 3 3.5 4 4.5 t (μs)

Fig. 2.7 (a) Principle of the electron cloud buildup. (courtesy F. Ruggiero-CERN) (b) Evo- lution of the electron cloud density due to the passage of two batches of 72 bunches each. (courtesy D. Schulte-CERN) (c) Interaction of the electron cloud with the proton bunch. (courtesy G. Rumolo-CERN). the development of the avalanche phenomenon [121–126]; surface conditioning of the beam screen in the arcs by means of the cloud itself (beam scrubbing) [100, 118]. In the end, this is the only way of reducing the effect in the LHC arcs, even if it has the drawback of requiring beam time, which will be at the expense of time for the physics programme. In general, collective effects, arising from the electromagnetic interaction of the beam particles among themselves, with their environment, and with the other beam, will ultimately limit the performance of the LHC [127–150]. Depending on the beam intensity and on the bunch filling pattern, they give rise to parasitic losses, can cause beam instabilities or degrade the beam quality by emittance growth, and can lead to poor lifetimes of all or some specific bunches. Intense efforts have been put into theoretical understanding, implementation of mitigation measures, and stabilization processes with the aim of preventing such harmful effects on beam quality. Currently, the LHC parameters are such that these effects are already appearing, and efforts are being devoted to the observation and the analysis of the detected behaviours with a view to improving machine performance. Digression: the chain of proton injectors 47

2.2 Digression: the chain of proton injectors The high-brightness proton beam required for the LHC [151] is generated by a chain of accelerators comprising a linac and three synchrotrons. Their performance has been pushed to serve the LHC, and upgrades have been implemented over the years [152, 153]. The travel through the chain of accelerators starts from a hydrogen bottle providing the protons by ionizing the gas. Next, the particles are accelerated by Linac 2 up to a kinetic energy of 50 MeV. Protons are then transferred to the Proton Synchrotron Booster (PSB) [154], which brings them to 1.4 GeV kinetic energy. The extraction energy has been upgraded twice since the construction of this machine, namely from the original value of 800 MeV to 1 GeV and then to 1.4 GeV in order to mitigate space-charge issues for the special beams required by the LHC. The PSB is made of four superimposed rings stacked vertically, as can be seen in Fig. 2.8. The linac beam has to be deflected in order to fill the four rings. Similarly, the PSB bunches need to be merged back in the vertical plane when extracted towards the Proton Synchrotron (PS) (for a recent account of this machine, see [155] and references therein). This ring accelerates the protons up to 26 GeV total energy and plays a key role in the generation of the longitudinal properties of the bunches for the LHC. In fact, the bunch length and bunch spacing at extraction from the PSB and injection in the PS cannot fulfil the LHC requirements, since these would exceed the space- charge limits tolerable. Therefore, only six bunches are injected, leaving one empty bucket. These bunches, which are much more intense than a single LHC bunch and are also much longer, are then split in the longitudinal plane by means of a complex RF gymnastic [156].4 At injection, each single bunch is split into three, while at top energy, two double splittings are applied, such that 12 bunches are produced out of one injected bunch (see Fig. 2.9(a)). The spacing is also the nominal, namely 25 ns. The evolution of the longitudinal bunch profile during a triple splitting is shown in Fig. 2.9(b).

Fig. 2.8 The PS-Booster, showing the four superimposed rings. (Credit: CERN-PHOTO- 201405-098).

4 Note that in parallel with the development of these new techniques, another essential tool was developed, namely the tomographic reconstruction of the longitudinal phase space [157, 158], which is currently used in daily operation to monitor the quality of the LHC beams produced by the PS machine. 48 The LHC: from beam commissioning to operation and future upgrades

(a) (b) 320 ns beam gap PS ejection: 72 bunches 72 bunches on h =84 in 1 turn Quadruple splitting at 25 GeV Acceleration 18 bunches to 25 GeV on h =21 Triple splitting at 1.4 GeV

PS injection: 6 bunches

3+3 bunches on h =7 1 trace / 356 revolutions ( ~ 800 μ s) in 2 batches 50 ns/div Empty bucket

Fig. 2.9 Principle of the longitudinal bunch splitting performed in the PS ring. The sequence of triple and double splittings is shown (a), together with the evolution of the longitudinal proﬁle of one bunch during the triple splitting (b). The various traces represent the time evolution (from bottom to top) of the bunch proﬁle during splitting. (courtesy R. Garoby-CERN).

The traces represent the bunch profile at different times (from bottom to top) and the generation of three bunches out of a single Gaussian is clearly visible. The net result is the production of a batch of 72 bunches. It is clear that by cleverly combining double and triple splitting, it is possible to generate different bunch spacings at extraction from the PS. For instance, the 50 ns bunch spacing variant of the LHC beam, which is currently being used for physics, is derived from the nominal scheme by simply inhibiting the last double splitting at the flat top. The groups of 72 PS bunches are then transferred to the Super Proton Synchrotron (SPS) [159] before the final acceleration towards the LHC injection. The PS batches are injected in the SPS in sequences of two, three, or four, and these will be properly placed in the LHC buckets in order to provide the desired collision schemes in the four LHC experiments.

2.3 Proton beam commissioning and operation The beam commissioning of the LHC machine started as early as August 2008,5 even before the rings were completely ready for beam. The first step was a series of injection tests, in which single-bunch beams were transferred from the SPS to the injection lines into the LHC and through the sectors already connected to the injection points and ready to receive beams. These tests were essential to the full preparation for the beam, and allowed a number of crucial tests to be made, such as magnet alignment, mechanical aperture verification, and (what should never be underestimated) polarity tests for the different magnet families [162–175]. The official day for establishing circulating beam was 10 September 2008, and in a few hours both beams were circulating with a good lifetime of a few hours.

5 The hardware commissioning of the rings had already started before that date [160, 161]. Proton beam commissioning and operation 49

Unfortunately, such a bright start was abruptly stopped by the event that occurred on 19 September, which caused severe damage to several magnets in Sector 3–4. A hectic period of analysis to identify the cause and to define the appropriate repair and mitigation measures started, and the machine was back in operation just over a year later [170]. In November 2009, two beams were again circulating in the machine and accelerated to 1.18 TeV, which was at that time a record energy for a particle accelerator. After the usual winter stop, during which additional tests were performed to permit safe operation at even higher energies, the LHC was restarted, and on 30 March 2009, the first collisions at 3.5 TeV took place. This event marks the beginning of a period of operation intertwined with beam commissioning. The strategy was based on a gradual increase of the beam power, with periods of performance levelling to provide time to study the machine behaviour in the new conditions, before moving on to the next milestone. Also, in order to simplify as much as possible the control of the machine and avoid problems, only one beam parameter was changed at a time. The official goal for the year 2010 was set so as to reach a luminosity of 1 × 1032 cm−2 s−1 [176–178]. Such a value corresponds to only 1% of the nominal value. Nevertheless, it implies a stored beam energy of 30 MJ, while beforehand the machine was operated with only 0.17 MJ and the Tevatron was operating at 2 MJ level. Initially, the bunch intensity was increased, with only a limited number of bunches, in order to avoid complications due to the needs of the crossing schemes. Also, the value of β∗ was reduced, thus leading to an exploration of the uncharted territory of dynamic change of the optics in the experimental insertions. In the second part of the year, during August/September 2010, it was decided to start the commissioning of bunch trains. A bunch train is a sequence of bunches with a constant bunch spacing and undergoing the same beam–beam effects. This strategy implied that once the potential physical issues were solved for a single bunch train, the addition of more trains would be rather straightforward, thus allowing a faster pace in the improvement of the machine performance. This stage in the beam commissioning is visible in the plateau in Fig. 2.10(a). The performance ramp-up following the successful completion of this phase is impressive and clearly seen in Fig. 2.10(b) in which the evolution of the peak luminosity is plotted for the entire year. For the year 2011 [179–182], the performance goal was set in terms of the integrated luminosity, namely 1 fb−1. To achieve this result, several improvements had to be implemented. First, the bunch spacing had to be reduced to 50 ns from 150 ns used at the end of 2010. Then the bunch intensity had to be pushed, and, finally, the β∗ decreased to 1 m. The machine performance, in terms of turnaround and availability, was such that it was possible to reach the goal for the year by June (see Fig. 2.11(a)). Overall, the total integrated luminosity for 2010 exceeded 5.5 fb−1. The situation for the year 2012 foresaw a goal of around 15 fb−1, requiring an even smaller β∗, larger intensity per bunch, and, also, a higher collision energy. The evolution of the peak luminosity in the first half of 2012 is shown in Fig. 2.11(b). All these changes allowed peak luminosities in excess of 0.65 × 1034 cm−2 s−1 to be reached, as can be seen in Fig. 2.12, where the evolution of the instantaneous luminosity of a typical fill is shown. 50 The LHC: from beam commissioning to operation and future upgrades

(a) (b) 250 ) ATLAS online luminosity s = 7 Tev ) ATLAS online luminosity s = 7 Tev –1 5 LHC delivered all –1 LHC delivered s

LHC delivered stable –2 200 Peak luminosity: 2.1 × 1032 cm–2 s–1 Bunch Trains Set Up ATLAS ready recorded 4 cm 30 150 3 High-intensity period 100 2 Low-intensity period 50 1 Total integrated luminosity (pb Peak luminosity (10 0 0 29/03 26/04 24/05 21/06 19/07 16/08 13/09 26/03 26/04 27/05 27/06 28/07 28/08 28/09 29/10 Day in 2010 Day in 2010

Fig. 2.10 (a) Evolution of the integrated luminosity in the ﬁrst period of 2010. The ﬂat part of the distribution in August/September corresponds to the setting up of the bunch trains. (b) Overall evolution of the peak intensity in 2010. The goal for 2010 is achieved at the end of the year. The dramatic increase in luminosity due to the introduction of the bunch trains is clearly visible.

(a) (b)

–1 ATLAS: 5.575 fb–1 ATLAS: 2.993 fb –1 3000 ALICE: 0.001 fb–1 ALICE: 0.005 fb –1 CMS: 5.725 fb–1 CMS: 3.207 fb –1 5000 LHCb: 1.212 fb–1 LHCb: 0.391 fb Latest fill included: 2663 ) Latest fill included: 2267 ) 2500 –1 –1 s Proton–proton: s = 8 Tev 4000 Proton–proton: = 7 Tev L –1 2000 All experiments: L = 6.591 fb–1 All experiments: Del = 12.517 fb Del

3000 1500

2000 1000 Integrated luminosity (pb Integrated luminosity (pb Integrated 1000 500

0 0 26/03 25/04 25/05 24/06 24/07 23/08 22/09 22/10 05/04 12/04 19/04 26/04 03/05 10/05 17/05 24/05

Fig. 2.11 (a) Evolution of the peak luminosity in 2011. The goal for 2011 of 1 fb−1 was achieved in June, and the total delivered luminosity reached more than 5.5 fb−1 for ATLAS and CMS. (b) Evolution of the peak luminosity for the ﬁrst part of the year 2012.

For reference, the main beam and optical parameters for the years 2011 and 2012 are listed in Table 2.3, together with the corresponding nominal values. The table indicates that a number of parameters are below the nominal value (e.g. energy, β∗, number of bunches, and factor F ), which explains why the nominal value of the luminosity has not yet been achieved. Nevertheless, some other parameters are well beyond their nominal value, as is the case for the beam emittance and the bunch intensity, which explains why the luminosity is already at two-thirds of the nominal value. It should be stressed that the higher bunch charge and smaller emittance are a result of Proton beam commissioning and operation 51 )

–1 ATLAS s ALICE –1 6000 CMS 5000 LHCb

4000

3000

2000 Instantaneous luminosityInstantaneous (µb

1000

0 04h 06h 08h 10h 12h 14h 16h Date: 2012-05-26

Fig. 2.12 Evolution of the peak luminosity during Fill 2669. During this ﬁll, about 130 pb−1 were collected for ATLAS and CMS. The instantaneous luminosity for LHCb does not feature the characteristic decay, because of a levelling procedure applied (see later).

Table 2.3 Main beam and optical parameters of the LHC machine for the nominal case and for the 2011 and 2012 physics runs

LHC NOMINAL 2011 2012 Proton energy (TeV) 7.0 3.5 4.0 Number of bunches 2808 1380 1380 Bunch spacing (ns) 25 50 50 11 Nb (10 ) 1.15 1.4 1.5–1.6

n = γr (μm) 3.75 2.5 2.5 β∗ (cm) 55 100 60 Factor F 0.84 0.91 0.81 Peak luminosity (1034) 1.0 0.32 0.6–0.7 Number of events/crossing 25 16 32 the gymnastics used to generate the beam in the injectors and, in particular, in the PSB and PS. The ﬂexibility to achieve better-than-nominal beam parameters is available only for the 50 ns bunch spacing, but is almost completely lost for the nominal 25 ns case. The price to pay for the increased bunch charge and very large current peak luminosity is the higher-than-nominal number of events per crossing, which are challenging the experimental apparatus, even if, for the time being, these values are still acceptable. 52 The LHC: from beam commissioning to operation and future upgrades

2.4 Future upgrade options The complexity of the LHC machine and the long time between the first studies and the construction and operation phases triggered very early questions about possible upgrade options (for an overview of the topic, see [183] and references therein, as well as [184–188]). The first study of these options dates back to 2002 [189]. The rationale behind the strategy presented then is that the triplet magnets have a finite lifetime due to the radiation damage induced by the collision debris. Therefore, rather than simply replacing the magnets, one should aim at improving the overall machine performance. Later on, another argument was taken into account, namely that the nominal LHC parameters might be difficult to achieve. Hence, one of the goals of the upgrade might have been the consolidation of the machine in view of achieving or even slightly surpassing the nominal performance. These considerations led to the definition of a staged approach to the LHC upgrade, namely a so-called Phase 1 upgrade [190], aimed at a modest increase in machine performance (a factor of 2–3 with respect to the nominal in terms of peak luminosity, i.e. (2−3)×1034 cm−2 s−1); then a Phase 2 [191] aiming at a vigorous upgrade (a factor of 10 with respect to the nominal performance, i.e. 1035 cm−2 s−1). This second upgrade should be carried out in parallel with an upgrade of the experiment detectors. Additional boundary conditions set for the Phase 1 upgrade consisted of limiting the changes to the machine layout to the strict minimum, i.e. the triplets. The studies performed [192–194] showed that the minimum β∗ achievable would be about 30 cm. However, a number of serious limitations, essentially in terms of mechanical aperture leading to beam aperture bottlenecks, arose from the parts of the machine that were not to be upgraded. Furthermore, the smaller-than-nominal β∗ called for improved corrections of the chromatic aberrations in order to avoid too strong a dependence of the optical parameters on the momentum offset of the particles. This resulted in an over-constrained optical configuration with hardly any flexibility for realistic operational conditions. In addition, the strategy based on two upgrades was critically reviewed in light of the experience with previous machines, such as HERA [195, 196] and the Tevatron [197, 198]. In the evaluation of the performance reach, the stop due to the implementation of the upgrade and the following period of beam re-commissioning was compared with the alternative of continuous operation of the existing machine. The outcome of the analysis was that the optimal scenario should envisage only one upgrade of the LHC machine. Furthermore, in 2010, a critical review of the performance of the LHC injectors was carried out, to identify improvements of the existing machines in terms of continued consolidation and, possibly, some additional upgrade [199], but without considering building new machines as in the standard scenario considered until then (for additional details, see e.g. [200–203]). Therefore, by mid 2010, the CERN strategy was redefined. Three projects were defined: a high-luminosity LHC upgrade (HL-LHC); an LHC injectors upgrade project (LIU); are a consolidation project to ensure the reliable functioning of the CERN accelerators over a timescale of 25 years. Future upgrade options 53

13–14 Tev collision energy

injector Cryolimit 10 × Nominal Splice upgrade HL-LHC 8 Tev interaction luminosity installation consolidation cryogenics regions Point 4

2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023

dispersion Button collimators, suppression R2E project collimation, R2E project Experiment beam 2 × Nominal luminosity Experiment experiment Nominal pipe Nominal luminosity upgrade luminosity upgrade 70% phase 1 Radiation damage

Fig. 2.13 Timeline of the HL-LHC and LIU projects over 10 years as deﬁned in 2010 and reviewed in 2011. (courtesy L. Rossi-CERN).

The timeline of these projects was defined in the framework of a 10-year planning of CERN activities, including periods of long stops for the implementation of the various upgrades approved. The overall scheme is shown in Fig. 2.13, and even if some details might change, and shifts in time are possible, the overall picture is not likely to change. In this schedule, the installation date for HL-LHC matches very well what is obtained by considerations based on the time required to half the statistical error of the physical results (the so-called halving-time) [183]. The goals of the HL-LHC project are twofold [183]: • an instantaneous luminosity of 5 × 1034 cm−2 s−1 with levelling; • an integrated luminosity of about 250 fb−1 per year, reaching 3000 fb−1 in 12 years. Even if the first goal is rather conservative, one should consider that for the first time the performance reach is not specified in terms of peak luminosity, but rather in terms of levelled luminosity. This means that the upgrade is based on techniques that allow the luminosity to be kept constant over a period of time, while the intensity is decreasing because of the proton burn-off. This in turn implies that the peak luminosity that is virtually required to reach such a levelled luminosity is much higher. To level the luminosity, there are essentially three possibilities: • Vary the transverse separation of the beams. This requires precise control of the magnets generating the separation scheme. It is routinely done for the LHCb experiment (see Fig. 2.12). • Vary the crossing angle. This can be achieved either by controlling the magnets generating the crossing angle or by using the so-called crab cavities [204–211], which are RF devices that provide a transverse kick that varies according to the longitudinal position in the bunch. These devices are part of the baseline of the HL-LHC project because they also restore the head-on collision and hence cancel the effect of the crossing angle affecting the geometrical factor F without altering the beam–beam separation in the common pipe region. Therefore, no additional aperture is needed to accommodate the initial large crossing angle. 54 The LHC: from beam commissioning to operation and future upgrades

Such a large aperture would be required if orbit correctors were to be used instead. A mitigation of this problem can be achieved with transversely elliptical beams, also called ﬂat beams. • Vary β∗. This requires precise control of the optics and orbit, but no additional beam aperture is required with respect to the case of β∗ squeezed down to the minimum possible value. It has never been used routinely in any of the colliders built so far.

It is clear that to achieve higher luminosity, the injectors also need to be upgraded. The usual approach consists of reducing β∗, but, unlike Phase 1, the constraint of replacing only the triplets has been removed and the layout of the high-luminosity insertions can now be reviewed. The constraint imposed by the chromatic aberrations remains. To overcome it, a novel optics concept has been developed [212–214] and has notably been already successfully tested in the current machine [215]. In spite of this very encouraging step, many challenges remain, including a few in the domain of beam dynamics, beam–beam, and collective effects in these high-power beams and the collimation system performance. The technological challenges should not be underestimated. It is planned to build the new triplet quadrupoles using Nb3Sn superconductors. These devices have never achieved, as yet, the quality required for accelerator applications, which means that a vigorous R&D programme is needed to meet the standards for HL-LHC [216]. Taking advantage of the LHC, as currently operating, an intensive programme of experiments have been planned in order to probe, wherever possible, some of the conditions foreseen for the HL-LHC. For reference, a list of possible beam parameters for the upgrade are reported in Table 2.4 (see also Refs. [217, 218]). The baseline consists of the 25 ns option, with 50 ns as a backup scenario since, for a given luminosity, the pile-up is doubled for this option. It is worth emphasizing the enormous effect of the geometrical factor F generated by the crossing angle required to control the beam–beam effects. This explains why crab cavities are an essential ingredient for HL-LHC, since they allow efficient use of the protons stored in the machine. On a longer timescale, two other upgrade scenarios are being considered. One is focused on an energy upgrade of the LHC (HE-LHC) [219–221]. A target energy of 16.5 TeV, corresponding to 20 T main dipole magnets, is the key parameter. At such an energy, the beam dynamics will be dominated by synchrotron radiation, even for protons. This would have a strong impact not only on the dynamics, but also on the cooling system, and synchrotron radiation damage must be seriously considered. It would also require a new LHC injector, delivering protons to the HE-LHC at 1 TeV. We conclude by mentioning that a second scenario exists, focusing on lepton– proton collisions (LHeC) [222–232], as was done in the HERA machine. Two options have been considered: a ring–ring and a linac–ring. The first envisages a second ring being installed in the LHC tunnel for circulating the electron beam. This option is technically relatively straightforward, even if it requires additional bypasses around the existing experiments for the HL-LHC, together with challenging installation work in the environment of an operating accelerator infrastructure. The second configuration References 55

Table 2.4 Main beam and optical parameters of the nominal LHC and HL-LHC. The peak luminosity for the HL-LHC options has to be considered as the levelled value of the luminosity

LHC NOMINAL HL-LHC HL-LHC 25 ns 25 ns 50 ns Number of bunches 2808 2808 1404 11 Nb (10 ) 1.15 2.0 3.3 Beam current (A) 0.58 1.01 0.83

σz (cm) 7.5 7.5 7.5

n = γr (μm) 3.75 2.5 3.0 β∗ (cm) 55 15 15 Factor F 0.84 0.30 0.33 Peak luminosity (1034) 1.0 6.0 7.4 Virtual luminosity (1034) 1.2 20.0 22.7 Number of events/crossing 25 123 247

foresees the construction of a new linear accelerator for the electron beam acceleration that intersects with the LHC machine in IP2. Several options have been considered for this linear accelerator: pulsed linac, recirculating linac, and energy-recovery linac (ERL) conﬁgurations. The energy of the electrons would be 60 GeV for both options. Recently, a panel of experts reviewed both options described in the conceptual design report and stressed that the ﬁrst is less favourable than the second and will therefore be abandoned.

Acknowledgments It goes without saying that the achievements reported in this chapter are the result of the joint eﬀorts of several people at CERN and elsewhere. In particular, I would like to express my warm thank yous to all colleagues of the Accelerator Physics Group and of the Operations Group for several discussions and for material used in the preparation of this chapter. My warmest thanks go to E. McIntosh for invaluable comments on the original version of the manuscript.

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Dan Green

Fermilab, USA

3 The LHC detectors and the ﬁrst CMS data 67 Dan GREEN

3.1 EWSB and LHC 69 3.2 LHC machine 69 3.3 Global detector properties 71 3.4 The generic detector 73 3.5 Vertex subsystem 76 3.6 Magnet subsystem 77 3.7 Tracking subsystem 78 3.8 ECAL subsystem 80 3.9 HCAL subsystem 82 3.10 Muon subsystem 85 3.11 Trigger/DAQ subsystems 88 References 89 LHC machine 69

The design of the LHC detectors follows from the requirement of confronting electroweak symmetry breaking (EWSB) in a decisive fashion. The LHC accelerator also meets those requirements. The simple discussion of the subsystems of a “generic” LHC detector presented here explains why the detectors have the parameters they do.

3.1 EWSB and LHC There are many theoretical speculations, but one thing is certain: WW scattering at a center-of-mass (CM) energy of ∼1.8 TeV violates perturbative unitarity. Therefore, one must design the LHC and the LHC detectors to comprehensively study WW scattering at ∼1 TeV.

3.2 LHC machine Given the existing LEP tunnel, 26 km around, one wants to use the most aggressive magnets to achieve the highest-energy pp machine. With only dipoles, a 7 TeV beam requires 5.6 T magnets because of the need for lenses, straight sections, injection, abort and radiofrequency (RF). The LHC magnets are 8.3 T, and such a field requires superfluid helium at 1.8 K (critical current density). Each magnet has 12 kA, L =0.1h, or 7 MJ/magnet. With 1200 dipoles, this is ∼8 GJ which implies the need for quench protection For RF purposes, bunches are short, ∼10 cm, separated by >7.5 m (25 ns), with a total of <3466 bunches (2808 design, and an abort gap is needed). Given 14 TeV pp CM energy (>1 TeV parton–parton CM energy), a sufficiently large interaction rate is needed. The number of protons in a bunch is limited by Coulomb repulsion, depending quadratically on collision rate. The number of bunches is limited by the RF and the bunch length. The revolution frequency is approximately constant. The beam size at the interaction point is limited by the strength of the “low-beta” quadrupoles, their aperture, and the collimation. Putting the factors together, the LHC design luminosity is

L ∼ 1034 cm−2 s−1. (3.1)

The vector bosons decay to either quark or lepton pairs. However, the enormous backgrounds that exist at the LHC owing to strongly produced QCD processes make the detection of leptonic decays experimentally favored. This explains why LHC detectors tend to focus on lepton detection. The branching ratio for a W to decay to a muon plus neutrino is ∼1/9. A crude estimate of the cross section for electroweak W + W production at the LHC with subsequent W decay to muons is α2 σ(p + p → W + + W − → μ+ + ν + μ− +¯ν ) ∼ W B2. (3.2) μ μ sˆ μ This estimate gives a 5 fb cross section times branching ratio squared for a W -pair mass of 1 TeV. To have sufficient statistical power in studying this process, the LHC should provide 100 fb−1/yr. Taking a running time T of 107 s/yr (∼30% of the calendar year), 70 The LHC detectors and the first CMS data there will be ∼790 W + W events produced per year with a mass above 1 TeV that decay into the experimentally favorable final state containing two muons. A similar event sample will be available in the two-electron final state and twice that in the muon-plus-electron final state. The high luminosity that is required of the LHC because of the need to explore terascale physics means that the detectors will be exposed to high particle rates. Therefore, LHC experiments will require fast, radiation-hard, and finely segmented detectors. It is assumed in what follows that all the detectors can be operated at a speed that can resolve the time between two successive RF bunches: 25 ns at the LHC. Note that the total inelastic cross section is ∼100 mb, as shown in Fig. 3.1. With a luminosity as defined above, the total inelastic reaction rate R = σL =109 s−1 = 1 GHz. Each crossing contains ∼25 inelastic events at full luminosity. This leads to

Fermilab CERN LHC

σ E710 9 tot 10 UA4/5

σ 1 mb b b 107

UA1 –1 105 s –2 cm 1 μb 34 10

= 103 ℓ

σ(W ℓν) CDF, D0 (p p–) (proton–proton) σ 1 nb 10 UA1/2

σ – Events/s for t t p p– m ( ) top = 175 GeV CDF, D0 – 10–1 (p p) 1 pb

σ –3 Higgs 10 m = 500 GeV MAX H σ × BRγγ : mH = 100 GeV SM σ × BR4l : mH = 180 GeV 0.001 0.01 0.1 1.0 10 100 √s (TeV)

Fig. 3.1 The cross sections for several processes as functions of CM energy. Global detector properties 71 experimental issues, but the luminosity that is required at the LHC and the size of the inelastic cross section make a “pileup” of NI = 25 inelastic events in each bunch crossing unavoidable.

3.3 Global detector properties In designing a generic LHC detector, it is crucial to understand what part of phase space the produced particles occupy. Note that single-particle relativistically invariant phase space for a particle of mass M, momentum P, and energy E is

dP d4Pδ(P 2 − M 2)= = πdydP2 , E T E = M cosh y, M2 = M 2 + P 2 , (3.3) T T T θ y → η = − log tan ,P>>M. 2 Therefore, if the produced particles in a typical inelastic reaction are described by single-particle phase space, they can be expected to be found uniformly distributed in rapidity y. The momentum transverse to the proton beam directions is denoted by PT . For light particles, with M/P << 1, the rapidity y can be approximated by the pseudorapidity η. Most of the produced particles are pions, which have a mass of 0.14 GeV, and their mean transverse momentum at the LHC is expected to be ∼0.8 GeV. Therefore, the angular variable, the pseudorapidity η, is a good approximation to y in most cases. Indeed, production data at η ∼ 0 (i.e., near 90◦ in the p + p CM system) display a roughly uniform distribution. Note also that the density D =(1/σ)(dσ/dη) of particles in the region and the width of that region rise slowly with increasing CM energy. A Monte Carlo example is shown in Fig. 3.2.

4*10–9

y y max max 3*10–9 pp (pb) 2*10–9 /dy /dy σ d

10–9

0.0 –10 –5 0 5 10 yg

Fig. 3.2 Rapidity distribution of low-transverse-momentum gluon jets at the LHC in the g + g → g + g reaction. 72 The LHC detectors and the ﬁrst CMS data

The kinematic limit for the 7 TeV incident protons is |y| < 9.6. Clearly, requiring that the LHC detectors detect most of the produced particles means that a rapidity coverage of ∼|y| < 5 is needed. That coverage is suﬃcient to detect the majority of the particles produced in normal inelastic events. A generic detector should cover a full rapidity range of 10. The particle density per unit rapidity will be about nine: six charged pions and three neutral pions under the assumption that all produced particles are pions and all charge states of the pions are equally probable. The mean transverse momentum of the pions will be ∼0.8 GeV. Therefore, in each time-resolved RF bunch crossing, there are 2250 particles in the complete detector coverage. Thus, we have

D ∼ 9 π/unit of y :6π± +3π0, P ∼0.8 GeV, (3.4) T NI (2ymax)D = 2250 particles, 1.8 TeV total PT .

The high-mass-scale processes of greatest interest populate phase space somewhat diﬀerently. For example, a hypothetical 2 TeV mass recurrence of the Z boson decaying into electrons preferentially appears at small rapidities as shown in Fig. 3.3. Therefore, the generic detector will put more stress on the low-|y| regions of phase space and deploy precision detectors to cover rapidities |y| < 2.5. In fact, the more precise generic

× 10−5 2

1.8

1.6

1.4 u

+ 1.2 u

1 (arb. units) dy /

σ 0.8 ' (2 Tev) d Z 0.6

0.4

0.2

0 −5 −4 −3 −2 −1 0 1 2 3 4 5 yμ

Fig. 3.3 Rapidity distribution for a sequential Z boson with mass 2 TeV. The generic detector 73

0.09

q W, Z 0.06 H0

0.03 q W, Z

–3 0.0 3

Fig. 3.4 Rapidity distribution of the forward/backward-going “tag” jets in the VBF process. detectors are limited to the region |y| < 2.5 because of their rate limitations and the risk of radiation damage both to the detectors and to the front-end electronics. The larger-|y| region is covered only by radiation-hardened calorimetry. A study of vector boson fusion (VBF) will allow the isolation of WW, ZZ,and WZ scattering, as illustrated in Fig. 3.4. These measurements may be crucial to the understanding of EWSB. The virtual emission of a W by an initial-state quark in both of the protons will lead to the scattered W + W plus the two quarks recoiling after W emission. This process is a critical component of the LHC physics program because it has a signature that allows experimenters to tag and isolate vector boson scattering.

3.4 The generic detector The required detector coverage in angle follows from the properties of inelastic events and from the VBF process. The generic detector consists of several subsystems with specific roles in particle detection and identification and is described below. The idea here is simply to observe from the physics requirements what a typical general purpose detector might look like. It is generic in the sense that specific subsystem detector design choices are not made, but rather the general needs of the several detector subsystems are examined. The overall layout of the detector is shown in Fig. 3.5. The overall coverage in rapidity of 2 × 5 units has been motivated earlier. This coverage should be “hermetic,” which means that all produced particles should be detected and their positions and momenta should be well measured. If this is achieved, then the production of neutrinos may be inferred by vectorially adding all the observed particle transverse momenta in the final state, because the initial state contains almost no transverse momentum. If an imbalance remains, the existence of “missing transverse momentum” (MET) implies that an undetected, noninteracting, neutral stable particle or particles have been produced that carried off the MET. 74 The LHC detectors and the first CMS data

4.0

η = 1. 5

3.0

HCAL r(m) 2.0

η = 2.5 ECAL 1. 0

η = 5.0

1.0 2.0 3.0 4.0 5.0 6.0 Vertex Tracker z(m) pixel strips

Fig. 3.5 The layout of the generic detector, showing the vertex, tracking, Electromagnetic Calorimeter (ECAL), and Hadronic Calorimeter (HCAL) subsystems.

There are several specific subsystems that are indicated. A general purpose detector, such as those installed at the LHC, aims to measure all the particles of the Standard Model (SM) from each interaction as well as possible. This follows because, whatever form the physics beyond the SM takes, the final-state particles will ultimately decay to the bosons, quarks, gluons, charged leptons, and neutrinos of the SM. The role of the subsystems is shown in Table 3.1. The detector focuses on the “barrel” or wide-angle region, because heavy new particles are kinematically expected to be produced there (Fig. 3.3). Coverage by the vertex, tracking, and electromagnetic calorimetry is limited to |y| < 2.5 because of the fierce radiation field that exists at smaller angles. That field must, however, be confronted because of the high luminosity needed to explore EWSB at the terascale and the need for hermetic coverage (Figs. 3.2 and 3.4). The size and location of the detector subsystems follow from the physics requirements. The vertex subsystem exists to measure the secondary decay vertices of the heavy quarks and leptons. Therefore, the vertex subsystem is as near to the LHC vacuum beam pipe as is possible. The tracking subsystem is immersed in the magnetic field of a large solenoid (with a uniform field in the z direction). By measuring the trajectory of the charged particles produced in the collisions, the charges, positions, and momenta of all produced charged particles are determined. A radius of 1 m is allocated to the vertex and tracking systems to achieve this objective with sufficient accuracy. The generic detector 75

Table 3.1 Particles of the SM detected and identiﬁed in detector subsystems

PARTICLES SIGNATURE GENERIC SUBSYSTEM u, c, t → W + b quarks d, s, b Jet of hadrons (λ0) Calorimeter g ECAL + HCAL e,γ Electromagnetic shower(X0) Calorimeter ECAL Tracker ν ,ν ,ν e μ τ Missing transverse energy (MET) Calorimeter W →μ + ν μ ECAL+HCAL μ, τ → μ + ν +¯ν τ μ Only ionization (dE/dx) Muon absorber and Z → μ + μ detectors Tracker c, b,τ Secondary decay vertices Vertex + Tracker

The size of the calorimetry is dictated by the characteristic distance that is needed to initiate an interaction, either electromagnetic (ECAL) or hadronic (HCAL). These sizes are

(X0)Pb =0.56 cm, (3.5) (λ0)Fe =16.8cm.

The showers in ECAL are fully developed and contained in about 20 radiation lengths X0. Allowing space for shower sampling and readout, 20 cm in depth is used for ECAL. The HCAL follows in depth with 1.5 m of steel, or 8.9 nuclear absorption lengths λ0. The focus in angular coverage of all the subsystems is in the “barrel” region, |η| < 1.5. Coverage by the vertex, tracking, and ECAL subsystems goes only to |η| < 2.5. In the region 2.5 < |η| < 5.0, the radiation field at the LHC precludes all but very radiation-resistant calorimetry. This forward calorimeter region is not shown explicitly, but is thought to reside at z = 10 m, a large distance that reduces the radiation dose. If it were stationed at z ∼ 3.2 m, the dose would be ∼9.8 times larger, and a substantial added radiation resistance would then be required of the specific technology chosen for the calorimetry. This location is indicated in Fig. 3.5. The high rates of background processes, as shown in Fig. 3.1, imply that the leptons must be measured in robust and redundant systems. For example, muons are measured in the tracking systems and then again in redundant specialized muon systems that have lower rates consisting of fairly pure muons because almost all other SM particles have been absorbed in the thick calorimeters. Electrons are also measured redundantly, in their case using the tracking systems and the electromagnetic calorimeters. 76 The LHC detectors and the first CMS data

In this way, the generic detector allows for clean lepton identiﬁcation and measurement even though the cross sections of greatest interest, such as those of Higgs decays, are of order femtobarns or smaller, while the inelastic backgrounds are of order millibarns, a factor of at least 1012 larger.

3.5 Vertex subsystem The subsystem at smallest transverse radius r is the vertex subsystem. It consists of pixels of silicon deployed from radius 10 to 20 cm in three layers in the generic model. The vertex detector is used to eﬃciently ﬁnd and identify the secondary decay vertices of the heavy quarks and leptons as shown in Table 3.1. The lifetimes of these particles set the scale for the pixel size:

(cτ)τ =87µm,

(cτ)b ∼ 475 µm, (3.6) 0 ± (cτ)c ∼ (123, 312) µm(D ,D ).

Assuming a pixel size of δz = δs= 200 µm, where s is the distance in the azimuthal direction (on a cylindrical vertex pixel layer) and z is along the beam direction, the vertex detector spatial resolution is suﬃcient to identify and measure the decay vertices. A CMS result for tagging is shown in Fig. 3.6.

CMS Preliminary 2010

10–1

10–2 SSVHE SSVHP –3 TCHE 10 TCHP JP Light-flavour efficiency Light-flavour JBP 10–4

10–5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 b efficiency

Fig. 3.6 Plot of the CMS b-tag efficiency versus the light-tag efficiency. A typical operating point has 60% b-tag efficiency and 99% light-quark rejection, as indicated by the shaded circle. Magnet subsystem 77

Note that analogue readout of the energy deposited in neighboring pixels yields better spatial resolution by interpolation than simple yes/no information. The probability for a pixel to be struck by a charged pion in a single bunch crossing is δz δs P ∼ D N . (3.7) pix c I r 2πr

With a density Dc of 6 charged pions per unit of y and NI = 25 inelastic events per bunch crossing at full luminosity, the pixel occupation probability at the innermost layer at a radius r =10cmis∼0.000095. The low occupation probability is achieved at a cost of the independent readout of many silicon pixels. A rough estimate is that, for three layers extending over |z| < 80 cm, there are 8000 pixels per layer in z and 4700 pixels in azimuth, for a total of 110 million pixels covering |η| < 2.5. The sparse occupation of the pixel system makes it an ideal location to start finding track patterns. Indeed, the required high rates at the LHC were what caused ATLAS and CMS to adopt pixel detectors, which was the first time they had been deployed at hadron colliders. Clearly, there is a tradeoff in radius between the number of pixels and the radiation field. If the vertex system could start at a radius of 5 cm, then the number of pixels would be reduced to 28 million and the radiation would be increased fourfold because of the r2 behavior of the number of pixels and the 1/r2 behavior of the dose. A smaller radius also reduces the position error made in extrapolating from the inner pixel layer to the actual decay vertex, because of multiple scattering in the beam vacuum pipe and the detectors, which limits the accuracy of the angular track measurements. The radiation field can be approximately evaluated using the number of charged tracks leaving ionization in the silicon detectors, ignoring increases due to secondary interactions of those tracks, to photon conversion, or to their multiple passage through the detectors as a result of particles being curled up in the magnetic field:

∼ 1 dEion (Dose)pix σI LT Dc 2 . (3.8) 2πr d(ρSiδr)

The dose is the energy deposited per weight and 1 Mrad = 6.2 × 1010 GeV/g. The 9 number of inelastic events per second is σI L =10 Hz and the exposure time is taken 7 to be a “year,” with T = 1 yr deﬁned to be 10 s. With a densityDc of 6 charged particles per interaction and vertex detectors at r = 10 cm, the charged-particle ﬂuence is 9.5×1014 π± cm−2 yr−1. Each minimum ionizing particle at normal incidence deposits −1 −2 dEion/d(ρSiδr)=1.66 MeV g cm , or 0.12 MeV in a δr = 300 µm thick detector. The estimated dose is then 2.5 Mrad/yr for the inner vertex detectors. Clearly, smaller radii will be very challenging.

3.6 Magnet subsystem The vertex and tracking subsystems are assumed to be immersed in a strong solenoid field. Roughly speaking, the current I flowing through n turns/length of conductor leads to a field B = μ0nI. Suppose there were conductors of size 2 cm in z, all stacked 78 The LHC detectors and the first CMS data by 4 in r for a total of n = 200 turns/m. Then, to achieve a field of 5 T, 20.8 kA of current is required. LHC magnets are a large step in stored energy above what has been previously achieved in the Tevatron (CDF and D0) as regards the solenoids used in the detectors. In the generic detector, a field of 5 T is assumed.

3.7 Tracking subsystem The requirement that the LHC experiments decisively confront EWSB has been seen to imply a complete study of WW scattering at a mass of ∼1.0 TeV. In that case, the transverse momentum of the W is ∼0.5 TeV and that of the lepton for the two-body W decay is ∼0.25 TeV. The radial size of the tracker is then set by the need to measure the lepton momentum well up to this mass scale. The approximate transverse momentum impulse imparted to a particle of charge e in traversing a transverse distance r in a magnetic ﬁeld B is

(ΔPT )B = erB =0.3rB, (3.9) where the units are GeV, T, and m, and an electronic charge is assumed. The bend is in the azimuthal direction, and the angle through which the particle is bent,ΔφB, is approximately the impulse divided by the transverse momentum, (ΔPT )B/PT . The error in the inverse transverse momentum is therefore 1 dPT d(ΔφB) ∼ ds d = 2 = 2 , (3.10) PT PT (ΔPT )B er B where ds is the spatial resolution in the azimuthal direction. Note that the quality factor for the momentum resolution scales as 1/r2B, which argues for a strong magnetic field and a large radius for the cylindrical tracking subsystem. Assuming a 5 T magnetic field extending over r = 1 m, the magnetic impulse is 1.5 GeV. The bend angle is then 6 milliradians for the 0.25 TeV lepton from the W decay. The tracker spatial resolution is estimated by assuming the digital readout of strips√ of 400 µm width. The root mean square (RMS) resolution is then ds ∼ 400/ 12 µm = 115 µm. The momentum will then be measured to 1.9% at 0.25 TeV. Later, it is shown that a 1.2% momentum accuracy is needed in order not to degrade the Z-peak natural width, which implies analogue readout of the tracker detector. The momentum is measured to 10% accuracy at 1.30 TeV, which makes a sufficiently good measurement at the terascale. A smaller radius or lower-field tracking system would require a more accurate measurement than the ds value quoted above. At a lower momentum scale, there is another limitation on the measurement accuracy. The multiple-scattering transverse impulse (ΔPT )MS due to scattering in the tracker material compared with the magnetic field impulse sets a limit of dP (ΔP ) T ∼ T MS , P (ΔP ) T T B (3.11) Li (ΔPT )MS = Es , 2X0 Tracking subsystem 79 where the scattering energy Es = 21 MeV, X0 is the radiation length, and the sum is over all material in the tracker. For a tracker-plus-vertex system having silicon detectors of 400 µm thickness in 11 layers (8 layers in four stations with small-angle stereo in the tracker and 3 vertex layers), the scattering impulse is 3.2 MeV. There are two terms in the momentum resolution, dPT /PT = cPT ⊕ d, which are folded in quadrature. There is a term due to measurement error, c =0.000078 GeV−1, (3.10), and a term due to multiple scattering, d =0.0021, (3.11), which limits the low-momentum measurements. The crossover transverse momentum at which the errors are equal is 27 GeV in this example, and the total resolution at this momentum is ∼0.3%. Clearly, keeping the contribution due to multiple scattering low allows one to improve the momentum measurement accuracy at low momentum, where the fractional momentum resolution is a constant. Incidentally, it should be pointed out that only counting the detectors themselves and ignoring power leads, cooling leads, and other material is very optimistic. For a complete understanding of the momentum scale and resolution, a detailed understanding of the tracker material throughout the system is needed. One can use photon conversions for probing high-Z materials and nuclear interactions for low-Z material. Results from CMS are shown in Fig. 3.7. Most of the information provided by the tracker is contained in the measurement of the azimuthal coordinates (the “bend plane”), while the helical paths are less demanding in the z direction. Therefore, to reduce costs, the generic tracking detector is fashioned of silicon strips, 10 cm long in z and 400 µm wide in s. Each of the four stations has a strip layer oriented along z as the long axis and a second at small angle to the first. Using (3.7), the probability for a strip to be occupied ranges from 0.006 to 0.00096 as the strip radius goes from 40 to 100 cm. This level of occupation is sufficiently sparse to allow for robust track pattern recognition and trajectory fitting.

30 0.03 CMS Preliminary 2010 4 z 10 | |<26 cm (cm) y Data 20 0.025 Simulation: conversions

103 Simulation: fakes 10 0.02 Conversions/0.2 cm Conversions/0.2 0 0.015 102 –10 0.01

10 –20 0.005

CMS Preliminary 2010 CMS Preliminary –30 0 0 10 20 30 40 50 60 –30 –20 –10 0 10 20 30 Conversion radius (cm) for |z|< 26 cm nucl. int., Data √s = 7 TeV x (cm)

Fig. 3.7 Material in the CMS tracker as determined from photon conversions and hadronic interactions. 80 The LHC detectors and the ﬁrst CMS data

The total number of strips in one layer at an average radius of 70 cm is ∼52 in z, |z| < 2.6 m, times ∼11 000 in r. Therefore, in eight layers, there are roughly 4.6 million independent strips to record (and purchase).

3.8 ECAL subsystem The generic electromagnetic calorimeter has the role of measuring the energy and position of the photons and electrons created in the LHC collisions. It is the first of the calorimeters to be struck by outgoing particles in the generic detector. Calorimeters measure energy by initiating interactions of the incident particles and completely absorbing the resulting energy, which appears as a geometrically growing “shower” of particles. Compared with the tracking and vertex systems, which absorb only the tiny ionization energy deposited in them, the calorimeter readout is “destructive” in that the showering particle is totally absorbed. The ECAL supplies a redundant measurement of the electrons. The tracking system first measures the electron momentum, charge, position, and direction. The ECAL measures the electron position and energy a second time. The electrons are important in the study of vector boson interactions because they are decay products of the W and Z. Redundancy is needed in order to be able to cleanly study the rare interactions of vector bosons using the final-state electrons in a background of strongly produced neutral pions and the photons resulting from their decay. A figure of merit is set by the sharpness of the Z resonant peak:

Γ =2.5GeV,M=91.2 GeV, Z Z dE Γ (3.12) < Z =1.2%. E ECAL 2.36MZ The development of the ECAL shower is parametrized using the depth L in radiation-length units, t = L/X0, and the energy in critical-energy units, y = E/Ec. The incident energy is E0 and a ∼ 1 + (ln y)/2. The shower development in depth is then

dE E b(bt)a−1e−bt = 0 , dt Γ(a) (3.13) a − 1 t = ,b∼ 0.5. max b A typical shower proﬁle has a rapid rise owing to the geometric shower growth. Because the energy of the electrons and photons in the shower is shared over ever more particles as the shower grows with depth, the average particle energy falls with depth until a point comes where radiative shower growth stops, at the “shower maximum” at a depth tmax where the average particle energy is near the critical energy. The shower then dies out at greater depths through loss of energy due to ionization and photoelectric absorption. This behavior is due to the geometric growth behavior of the number of particles in the shower: ECAL subsystem 81

tmax E/Ec ∼ Ns ∼ 2 , ln(E/E ) (3.14) t ∼ c . max ln 2 With fine sampling and a very uniform medium, the stochastic fluctuation of the number√Ns of shower particles can largely determine the energy resolution dE/E ∼ 1/ Ns ∼ Ec/E . For E = 0.25 TeV, the number for Pb is 34 245, with a 0.54% fluctuation, which meets the energy resolution requirement.√ The stochastic coefficient for the energy resolution is estimated to be a = Ec =8.5%, where GeV energy units are used. The transverse shower development is also important, since it defines the requirement for the ECAL tower size and also provides additional particle identification capability. The Molière radius rM is the radius of a cylinder within which 90% of the shower energy is deposited. In Pb, the radius is rM =1.6 cm, and that sets the tower size. Finer tower segmentation is not called for, since no new information in the dense core of the shower can reasonably be extracted. Particle identification follows from the fact that hadron showers are much wider transversely, with a radius ∼λ0. In the generic detector, the barrel ECAL is at rE =1.2mand|zE | < 2.8 m, covering the range |η| < 1.5. The ECAL towers are defined by the Molière radius in lead, δη = δφ ∼ 2rM /rE =0.027. The barrel has 175 towers in z and 236 in azimuth. If there are three depth segments read out independently to measure the shower development in depth, then there are a total of 124 000 ECAL readout channels in the barrel alone. An event display of a photon + jet final state in CMS is shown in Fig. 3.8.

CMS Experiment at LHC, CERN Data recorded: Tue Jul 6 02:19:02 2010 CEST Photon Run/Event: 139458 / 397605499 p c T = 76.1 GeV/ Lumi section: 365 η = 0.0 φ = 1.9 rad

Anti-kT 0.5 PFjet p c T = 72.0 GeV/ η = 0.0 φ = 1.2 rad

Fig. 3.8 Event display for a CMS candidate event with a photon plus a jet in the ﬁnal state. 82 The LHC detectors and the ﬁrst CMS data

Note that the ECAL towers are displayed summed over 25 contiguous towers, which shows how ﬁne the tower segmentation really is. The radiation ﬁeld is severe, since, in contrast to the tracking, the entire particle energy is now absorbed by the medium. The radiation dose in the barrel due solely to neutral-pion absorption by the ECAL medium is ∼ 1 PT (Dose)ECAL σI LT D0 2 . (3.15) 2πrE ρEΔtX0

With three neutral pions per unit of rapidity, D0, depositing 0.8 GeV into ECAL with a density ρE of Pb over a region ΔtX0 in depth, Δt ∼ 7, there is a dose of ∼0.097 Mrad/yr in the ECAL barrel. The dose in the end cap is, however, much higher. Fundamentally, the increase is due to the higher-energy pions in the end cap (at ﬁxed transverse momentum, Eγ = PT / sin θ) and the smaller radii covered by the ECAL end cap, which is located at zE. The dose ratio between the barrel and end cap for ECAL is approximately (Dose) 2 endcap ∼ zE 1 3 . (3.16) (Dose)barrel rE θ

For the end cap, the minimum angle is at |ηE| =2.5, where the dose ratio is about 42, which means that the end-cap ECAL has a yearly dose of about 4 Mrad. This rapid increase of dose with rapidity sets the limit in the generic ECAL on the angular coverage outﬁtted with precision electromagnetic calorimetry.

3.9 HCAL subsystem The Hadronic Calorimeter (HCAL) measures the energy of the strongly interacting quarks and gluons by absorbing the jets of particles into which these fundamental particles fragment (Table 3.1). In addition, the “hermetic” calorimetry measures the energy of all the secondary particles within a range of |y| <, 5, which is sufficiently complete coverage (Fig. 3.2) that a large missing transverse energy (MET) indicates neutrinos in the final state and thus provides both particle identification and a measurement of the neutrino transverse energy. In principle, the requirements on the HCAL are very stringent, again being set by the natural width of the W boson, where a dijet resonance is now reconstructed from the quark rather than the leptonic decay of the W . In practice, such precise energy performance, with an implied stochastic coefficient of ∼17.4%, at an energy of 0.25 TeV, is not attainable. One reason is the small number of particles in a hadronic shower, with the resulting large stochastic fluctuations in that number. The analogue of the critical energy for the ECAL is the threshold energy to make additional secondary pions, Eth ∼ 2mπ =0.28 GeV. That energy scale leads to an estimate for the number of particles in the hadronic shower of ∼ E/Eth, which implies a stochastic coefficient of 53%, or 3.35% fractional energy resolution dE/E for E =0.25 TeV, far from the requirement set by the natural resonance width of 1.1%: HCAL subsystem 83

200 12 11 99% 10 ) I

150 9 λ 95% 8 7 100 6 Depth in iron ( Depth in iron Depth in iron (cm) Depth in iron / Bock parameter 5 / CDHS data / CCFR data 4 50 3 510 50100 500 1000 Single hadron energy (GeV)

Fig. 3.9 Depth of calorimetry needed to contain a hadronic shower as a function of the energy of the hadron.

W + → u + d,¯ c +¯s,

ΓW /MW =2.6% , (3.17)

(dE/E)HCAL ∼ 1.1%. The total depth of the HCAL need not be as deep as that needed to fully contain the ECAL shower, because the hadron calorimetry does not achieve the desired 1.1% precision energy measurement and so leakage of shower energy at the back of the HCAL is not as strictly limited as in the ECAL case. A plot of the depth needed as a function of hadron energy to limit the shower leakage is shown in Fig. 3.9. The hadronic shower is contained transversely in a cylinder of radius ∼λ0. Assum- ing that overlapping hadronic showers cannot be resolved, a size δz = δs =15cmis chosen in the generic HCAL. The generic HCAL begins at radius rH =1.6m. The HCAL towers then subtend a rapidity and azimuthal interval

δη = δφ =0.094 ∼ λ0/rH . (3.18) The larger transverse tower size leads to a greater probability for an HCAL tower to be 2 occupied by a charged pion, (Dc = 6)(NI = 25)(δη) /2π =0.21. The mean transverse energy in an HCAL tower is then 0.16 GeV = 0.21 × 0.8 GeV. The number of barrel HCAL towers in z is 10 m/0.15 m = 67 and in azimuth also 67, or 4490 HCAL readout towers in the barrel. With three depth segments read out independently, there are 13 470 towers in total. Note that the showers in both the ECAL and the HCAL develop longitudinally on the scale of the radiation length, so that a complete knowledge of the hadronic shower should logically have that depth segmentation. However, that level of detail has not been attempted in the LHC calorimeters, although it is planned for in ILC/CLIC calorimetry. With regard to radiation, the HCAL is at a larger barrel radius than the ECAL, but there are twice as many charged as neutral pions. The hadronic shower energy 84 The LHC detectors and the ﬁrst CMS data is more spatially spread out than the electromagnetic shower, where for hadrons the initial interaction point has a mean of one and an RMS of one λ0. Therefore, the full width of the deposited energy is roughly Δλ0 ∼ 2λ0. Taking the factors together, the dose in the HCAL is about one-third that of ECAL at the same rapidity. However, the need to cover angles as small as |y| = 5 arose from the requirement that the generic detector measure most of the inclusive inelastic pions and also measure the forward jets from the VBF process. This requirement places a large radiation

CMS data (836 nb−1) 3 10 Fit Excited quark String √s = 7 TeV 2 η η η 10 | 1, 2| < 2.5 and | | < 1.3 M 11>220 GeV k R 10 Anti- T = 0.7 CaloJets S (1 TeV) (pb/GeV) 1 dm / S (1.6 TeV) σ q* (0.7 TeV) d 10−1 q* (1 TeV) 10−2 CMS preliminary 10−3 500 1000 1500 2000 Dijet mass (GeV)

Fig. 3.10 CMS search for dijet resonances from early 2010 data.

30 type 1 caloE/T(Data) type 1 caloE/T (MC) E/ 25 type 2 calo T (Data) type 2 caloE/T (MC) tcE/ T (Data) E/ 20 tc T (MC) E/ ) (GeV) pf T (Data)

X,Y E/ / pf T (MC) E 15

10 (Calibrated σ

5 √S = 7 TeV CMS preliminary 2010 0 0 50 100 150 200 250 300 350 400

Calibrated pfσET (GeV)

Fig. 3.11 Resolution in missing transverse momentum in CMS for dijet events using pure calorimetry and using “particle flow” combining calorimetry and tracking information. Muon subsystem 85 burden on the forward calorimetry. It covers the range in |y| from 2.5 to 5.0. At |y| = 4, the average pion energy is 22 GeV. Secondary particles deposit approximately 77 W of power, which heats the forward calorimetry. The luminosity can be measured in the forward calorimeter with a thermometer! At |η| = 5, the polar angle is 0.75◦. At a location z = 10 m for the forward calorimetry, the radius in the (x, y) plane is only 13 cm, and a “tower” with the same δη as the barrel HCAL would have an extent of only 1.2 cm in r and 7.7 cm in azimuth s. Clearly, such fine segmentation, given the physical size of a hadronic shower, is not very useful. Indeed, the pileup probability for a tower of transverse size ∼ λ0 is quite a bit higher than that for the barrel HCAL. Going to a smaller z, as indicated in Fig. 3.5, only makes the situation worse. The radiation dose in the forward calorimetry may be roughly estimated to be ∼ 1 PT (Dose)forward σI LT D0 2 3 . (3.19) 2πzF θ ρforwardΔtX0 Taking only the most densely deposited neutral energy to be the full radiation dose and the z for the calorimetry, zF , to be 10 m, and assuming the calorimeter to be made of steel, the dose is approximately 280 Mrad/yr at |y| = 5. The size of the dose sets the limit on the calorimetric coverage to |y| < 5 because of the need for long-term survivability of the forward calorimetry. Nevertheless, the HCAL is an indispensable tool for measuring jets in the detector, as indicated in Table 3.1. A spectrum of jet–jet masses and the accompanying limits on new dijet resonances is shown in Fig. 3.10. Even in the first running of the LHC, the HCAL allows an exploration of such resonances at the several TeV mass scales. In Fig. 3.11, the resolution in missing transverse momentum is shown for standalone calorimetry and for the combined use of calorimetry and tracking. Tracking offers better resolution at low momentum than calorimetry alone. This is “particle flow” in the current jargon.

3.10 Muon subsystem A major physics goal at the LHC is to explore vector boson interactions. Leptonic decays of the bosons are the favored mode, since the lepton backgrounds are low and the leptons can be cleanly and redundantly measured. The muon vector momentum and position are measured first in the tracking system. The muons then deposit only ionization energy in the ECAL and HCAL, while the other final-state particles are almost completely absorbed. In fact, the calorimetry can perform a useful muon identification role if the muons are isolated from other final-state particles by measuring the “minimum ionization particle” ionization energy in the HCAL. Ideally, in the muon system that follows the calorimetry in depth, only muons exist along with noninteracting neutrinos. Muons at low transverse momentum arise from the decays of inclusively produced pions and kaons. These muons can largely be removed as a background by requiring a good tracker fit to the hypothesis of no decay “kink” in the found track. 86 The LHC detectors and the first CMS data

For example, in the decay π → μ + ν, the muon transverse momentum relative to the initial pion direction is only ∼ 0.03 GeV. Therefore, a 10 GeV pion that decays in the tracker will have a 3 milliradian “kink” in the full track. In discussing the generic tracker, an angular resolution of 0.12 milliradians was quoted. Therefore, decays can be removed using the tracker up to a few hundred GeV. Since the probability to decay in the tracker falls with increasing momentum, pion and kaon decays are not a major problem at the high momentum scales that are of major interest to the LHC experimenters. A second source of muons is heavy flavor decay, such as b → c + μ + ν, which occurs much more promptly and within or before passage through the vertex detectors. The calculated LHC cross section in leading order for production of a b-quark pair where one b decays into a muon with transverse momentum greater than 10 GeV is σμ ∼ 60µb. At the LHC design luminosity, that cross section corresponds to a rate Rμ = σμL ∼ 0.6 MHz, which is small with respect to the total inelastic LHC rate of 1 GHz but too large with respect to an acceptable trigger rate. The requirements for the muon system are similar to those for the tracking system. There should be a good momentum measurement up to ∼0.25 TeV for the muons. In addition, the system must produce a trigger to reduce the rate of background muons from the heavy flavor decays. The ionization energy loss in the calorimetry provides a lower momentum cutoff for the muons. The generic HCAL contains 1.5 m of steel, which stops all muons with a momentum less than ∼1.7 GeV. However, that cutoff is not sufficient to reduce the muon rate in the muon system to an acceptable level. Therefore, the muon system must measure the muon transverse momentum accurately, set a threshold value, say 20 GeV, and report it to the trigger system. In the muon system, nearly all the particles are muons, but they are of low transverse momentum and arise from heavy flavor decays. Since the ionization-range cutoff is in energy, while the intrinsic muon rate is controlled by the transverse momentum, the rates in the forward muon systems are much larger than the rates at wide angles. Beam halo from upstream interactions in the LHC accelerator also makes the forward muon region more difficult. For these reasons, the generic muon system is thought to cover the same limited rapidity range as the tracking system, |y| < 2.5. The steeply falling muon spectrum from heavy flavor decays means that the muon trigger system momentum resolution is very important. Poor resolution lets in a large number of triggers from lower-momentum muons measured to have a higher momentum because of finite resolution, thus increasing the trigger rate. There is a complication for very-high-momentum muons. At the LHC, their momenta are sufficiently large that radiation is an issue. The muon is a heavy electron, so that it experiences the same forces as the electron but has a much reduced acceleration. Radiation scales as the square of the acceleration, so the critical energy for a muon is much larger, but still finite, compared with the critical energy for an electron; typical values in steel are ∼300 GeV: 2 (Ec)µ ∼ (Ec)e(mµ/me) . (3.20) The particle rates outside the calorimetry are sufficiently low that drift chambers with large drift distances can be used, as opposed to the other subsystems in the Muon subsystem 87 generic detector. If the chambers are operated in “air,” then the accuracy required is similar to that for the tracker and position resolution, and alignment specifications are stringent. If the chambers are operated in the flux return yoke of the solenoid, then the redundant muon momentum measurement is multiple-scattering-limited. In that case, the fractional momentum error is constant for momenta below which the alignment errors make a negligible contribution:

dP E 1 = √s √ . (3.21) P 2 eB LX0 For example with B = 2 T in steel with an L = 2 m return yoke, the muon momentum is measured to 13%. The tracking provides a redundant momentum measurement with <10% momentum error for muons with momenta <1.3 TeV in comparison. At still higher momenta, the tracker resolution increases as P , so the muon system will improve the tracker muon momentum resolution, provided that good alignment of

Fig. 3.12 Event display of a cosmic-ray muon in CMS, showing the redundant measurements in the tracking and muon subsystems. 88 The LHC detectors and the ﬁrst CMS data

J ψ ρ,ω φ / 105

ψ’ Y (1,2,3S) 104 ) 2 c 103

Z 102

Events/(GeV/ CMS preliminary 10 √s L −1 = 7 TeV, int = 1.25 pb

110102 μ+μ– mass (GeV/c2)

Fig. 3.13 Resonances observed in CMS in the dimuon ﬁnal state. the muon chambers can be maintained so that the measurement remains multiple- scattering-dominated. An event display of a cosmic ray in CMS is shown in Fig. 3.12, where the muon trajectory in both the tracker and the muon subsystem is evident. Data on dimuon resonances from CMS are shown in Fig. 3.13. The resonances vary from low-mass ones discovered 50 years ago to the Z, found in the 1980s.

3.11 Trigger/DAQ subsystems The inclusive rate at the LHC design luminosity is 1 GHz. The number of interactions capable of being stored to permanent media is ∼100 Hz. Therefore, a reduction in rate by a factor of 10 million is needed. This is accomplished in steps. First, the rate is reduced by “triggering” on leptons and jets above some transverse momentum thresholds. Imposing these thresholds on the events reduces the rate to ∼100 kHz. In order to not incur dead time, a front-end “pipeline,” which stores all the data until the first-stage trigger decision is made, is provided for each channel of data that is read out. There are ∼100 million independent channels of information, mostly analogue, in the generic detector. Typically, after this initial decision, the full event is sent from the detector front ends off the detector by means of digital optical fiber data transmission to a set of digital electronics accessible to the experimenters. More incisive trigger decisions that reduce the event rate to ∼100 Hz are then made. Even after suppression of detector elements with no hits or with signals below some low threshold, the large number of particles per event and the large number of References 89 events piling up lead to a typical event size of 1 Mb. Assuming a data-taking run of ∼4 months per year, 1000 Petabytes/yr—or about one million DVDs—are stored for offline analysis. The trigger and data acquisition techniques for both ATLAS and CMS are more or less specific to the experiment and thus are not susceptible to a “generic” analysis. In fact, after the first level of triggering, commercial off-the-shelf modules are mostly deployed. Examples are telecommunications switching networks used to assemble the full event using input from the different subsystems and “farms” of commodity PCs that are used for online high-level triggering and for offline analysis.

References The LHC detectors and accelerator are well documented. ATLAS and CMS are described in [1–4], while the LHC is documented in [5, 6]. A full technical description of the detectors and accelerator is given in [7], and the speciﬁc detector choices and capabilities are described in [8].

[1] ATLAS experiment, and links therein. [2] Compact Muon Solenoid, . [3] CMS, . [4] ATLAS Experiment, . [5] The Large Hadron Collider, . [6] LHC Design Report, . [7] The CERN Large Hadron Collider: accelerator and experiments, J. Instrum. 3 (2008) SO8001–SO8007. [8] D. Froidevaux and P. Sphicas, General-purpose detectors for the Large Hadron Collider, Annu. Rev. Nucl. Part. Sci. 56 (2006) 375–440.

4 About the identiﬁcation of signals at LHC: analysis and statistics

Bruno Mansoulie´

CEA IRFU Saclay, France

4 About the identiﬁcation of signals at LHC: analysis and statistics 91 Bruno MANSOULIE´

4.1 Introduction 93 4.2 Search for the Standard Model Higgs boson decaying into WW (in ATLAS) 93 4.3 Backgrounds 94 4.4 Global model of the analysis 97 4.5 Statistics 100 4.6 Global Higgs analysis 103 4.7 Conclusion 106 References 106 Search for the Standard Model Higgs decaying into WW (in ATLAS) 93

As the vast majority of the students at the 2011 Les Houches Summer School were theoreticians, it was felt appropriate to give them a short guide through a “modern” analysis, typical of the analysis performed on the LHC data. The main ingredients are reviewed: irreducible and reducible backgrounds, background estimates from Monte Carlo or data-driven, signal and control regions, global ﬁt, extraction of limits and signal, and the look-elsewhere eﬀect.

4.1 Introduction This chapter presents very briefly the main ingredients and steps of the analysis used in the search for the Standard Model Higgs boson decaying into WW (l, l, νν)inthe Atlas experiment, presented during the summer of 2011. This analysis has since been published in [1]. The presentation does not go into the details of the analysis, but rather tries to use it as an example of all analyses at LHC, emphasizing some delicate and important points. Indeed, any given analysis is often a complex construction, involving the classification of the data into several subcategories and the heavy use of Monte Carlo simulations. Most important is the global modelling of the whole analysis, used to derive background estimates, include systematic errors of all sorts, and finally extract results from statistical analysis. We then present the statistical aspect of the global search for the Higgs, and in particular we discuss the so-called “look-elsewhere effect”.

4.2 Search for the Standard Model Higgs boson decaying into WW (in ATLAS) The search for the Standard Model Higgs boson (SM Higgs) is of course of prime importance at the LHC. Given the Higgs production cross section and decay branching ratios (BRs), the highest sensitivity will be attained for a Higgs with a mass between 130 and 200 GeV, searched for in the WW decay channel where both W bosons decay into a lepton (e, μ) and a neutrino. Figure 4.1 shows the cross section multiplied by BR for the channels that can be considered for a search. WW in l, l, νν benefits from a large BR of the Higgs to WW, a sizeable BR of the W into leptons (10% each), and a reasonable experimental signature given by the two leptons (ee, eμ,orμμ) and the missing transverse energy (ET miss) from the escaping neutrinos. However, with two neutrinos in the final state, the invariant mass of the final state cannot be reconstructed. Hence, the signal will not show as a narrow mass peak on top of a smooth background continuum, unlike decay modes like γγ or ZZ → 4l, but rather as an excess in a broad signal region, i.e. a counting experiment. The selection requests two well-identified leptons (e or μ) with transverse momenta pT larger than 25 and 15 GeV, and ET miss in excess of 25 or 40 GeV, depending on the supposed Higgs mass. The total cross section is about 200 fb, and after final cuts, it is reduced to 25 fb; i.e. there is a global efficiency for the signal of about 12%. − With the available integrated luminosity (1 fb 1), the expected number of events is very small (25), and it is very easy to provide much larger numbers of fully 94 Identification of signals at LHC: analysis and statistics

10 s = 7 TeV SM BR (pb) × σ 1 WW l±νqq

WW l+νl−ν− 10−1

ZZ l+l−qq

10−2 ZZ l+l−νν−

H τ+τ− ZZ l+l−l+l− 10−3 VBF H τ+τ− l = e, μ WH l±νbb ν = ν , ν , ν γ γ e μ τ ZH l+l−bb q = udscb 10−4 100 200 300 400 500 M H (GeV)

Fig. 4.1 Cross section multiplied by branching ratio for the main Higgs decay modes. (ATLAS Experiment c 2011 CERN). simulated events. The only question is which theoretical physics to put in. Monte Carlo generators are available based on next-to-leading order (NLO) calculations, and the cross sections are available at next-to-next-to leading order (NNLO) and used for normalization.

4.3 Backgrounds

Despite the good signature offered by two leptons and ET miss, several backgrounds can generate events in the signal region. The diversity of origins of these backgrounds offers a good example of a search for a very small signal among a large number of processes with an enormous range of cross sections. Trivially enough, the total background is the sum of all contributions: Σσi × i. with σi the cross section and i the efficiency for each background component i. I would like to emphasize that in this sum, terms of the same order of magnitude can be formed from very different σi and i.

4.3.1 WW continuum The first background is of course the non-Higgs SM production of W pairs (continuum), followed by the same leptonic decays. Event by event, it is indistinguishable from the signal, and is thus an irreducible background. This continuum belongs to the class “small cross-section, large efficiency”. Since no mass peak can be expected, the handles to increase the signal-to-background ratio are mostly related to event kinematics, and not to detector performance for particle identification or measurement. Backgrounds 95

It was noted early that the angular correlation of the charged leptons is different in the Higgs decay and in the continuum process. Indeed, for the scalar Higgs, the W spins are opposite, and from the V–A decay of the W ’s, the charged leptons tend to go in the same direction. At leading order (LO) and NLO, the continuum comes from quark–antiquark initial states, with no W polarization. Hence the leptons tend to go back to back because of the boost given by the W momenta. At the LHC, this is best seen in the transverse plane, and the Δφ distribution can be used to enrich the signal at small values of Δφ. However, it was discovered that at NNLO there is a qualitatively different contribution from the gluon–gluon initial state, and that it induces some W polarization in the same way as that of the Higgs. It is of prime importance to estimate this contribution correctly, as well as the resulting angular distribution. The Higgs is mostly produced by a gluon–gluon initial state, through a top quark loop. Because of this different initial state, it is more often accompanied by jets than the WW pairs of the continuum. This fact can be best exploited by separating the analysis into subchannels: WW +0jet,WW +1jet,WW + n(> 1) jets. In this last category, WW + 2 jets helps also to select a different process of Higgs production (vector boson fusion), but, with a small event number, this mode was not used at this stage of the search. This is a general feature of the recent analyses: dividing the analysis into subcategories. Of course, if a category contains a small part of the events but with a much larger signal-to-background ratio (S/B), one would tend to impose cuts that just keep this category. But if the S/B is significantly higher—but not much—one would lose in global significance because of the loss of the information contained in the other events. Dividing into subcategories that have a different S/B allows one to optimize globally the statistical power of the analysis. In this class of “small cross-section, large efficiency”, it is very easy to provide large numbers of simulated events for this background, as for the signal.

4.3.2 Top–antitop Top quark pair production is a background to many searches at the LHC. The total cross section is large (165 pb), and the final states very diverse. Each top quark decays into a W boson and a b quark, and then the W and b each have a choice of decay modes, hadronic or (semi)leptonic. In this search, the most dangerous contribution to the WW + jets channel comes from the decay chain tt¯ → WWb¯b → lνlνb¯b. For other analyses that need to identify b-quark jets, the detector has the capability of measuring very precisely the origin of charged tracks in the Inner Detector, and of looking for a secondary vertex, since the b lifetime allows it to fly away from the primary vertex by a few millimetres. In our case, this can be used to veto b quarks. However, the veto efficiency cannot be 100%, giving rise to backgrounds when one or two b-jets are outside the detector acceptance or have not been identified as b’s. − For 1 fb 1, about 165 000 tt¯ pairs have been produced in ATLAS data. If we want to safely simulate the inclusive process, we need at least 10 times more events, to be 96 Identification of signals at LHC: analysis and statistics sure that every decay mode and configuration will be represented in the final Monte Carlo sample after all the analysis cuts. With the present computing systems (i.e. the LHC Grid etc.), it is possible to provide a few millions of simulated events for important processes like tt¯. But if we want to trust the simulation to estimate the contribution of this process to the signal region, we need to be sure that the accuracy of the simulation is good to a precision that is typically the ratio of the inclusive cross section of this background to that of the signal: 165 000/25 = order of 10−4. Several factors enter—branching ratios, kinematics, b-jet identification—and it would be hazardous to take blindly the result of the simulation. The method is then to measure as many of these factors as possible in the data themselves: this is called a data-driven method. This can be done for example by checking that the cut flow is well reproduced in the Monte Carlo events starting from the initial event samples, down to levels of selection where the background still dominates any possible signal. One can also use control regions, selecting events with a configuration close to that of the signal but different—for example, inverting one or several of the selection cuts. In the case of the tt¯ background, control regions are easily made by inverting the b veto into a b selection. However, one should beware that this method assumes that the ratio of “control-like” to “signal-like” events is again well described in the Monte Carlo simulation. In fact, there is no rigorous proof that data-driven methods give the right background estimates. It always requires the analyser (and reader!) physicist’s commonsense that there is no strange unnoticed correlation that would make the extrapolation from control region to signal region invalid.

4.3.3 Z bosons Z bosons are copiously produced at LHC: the cross section is about 30 nb at 7 TeV, and the branching ratio into each lepton species is 3%. The decays into ee or μμ can be efficiently rejected in the HWW analysis by cuts removing pairs with an invariant mass close to the Z mass, and also by requesting a large ET miss. This looks simple, and such well-identified events do not look dangerous. But there are tails in the ET miss distribution coming from detector effects. ET miss is obtained by making the vector sum (in the transverse plane) of all calorimetric deposits in the event. But how are these calibrated? Is the calibration the same for a quark jet or a gluon jet or a photon? What is the role of pileup events that add calorimeter energies everywhere? Muons (in particular the muons from Z decay) deposit some ionization energy in the calorimeter. Usually this energy is small and its most probable value can be corrected for. But what about the so-called “catastrophic muon energy losses”? Can we always identify these and take them into account? In some cases, a pileup event deposits energy close to a muon: if we identify it wrongly as muon-induced, this will bias the energy balance and generate spurious ET miss. Etc., etc. The most dangerous contribution of the Z bosons comes from their decay into ττ, which can subsequently decay leptonically. The cross section for these events is still − 200 pb. For 1 fb 1, it is possible to simulate these 200 000 events. But are we sure of the correct description of these events by the Monte Carlo simulation to a few parts in 100 000? Indeed, it has been found that the selection of signal-like events is sensitive to Global model of the analysis 97 the spin correlations between the τ’s and to the role of polarization in their subsequent decay. Care should be taken that the Monte Carlo simulation correctly describes these very fine effects.

4.3.4 W +jets W production corresponds to a very large cross section (about 10 nb). In a significant part of the events, the W is accompanied by one or more jets. The probability that a jet is identified as a lepton is very low: the detectors have been designed for a good lepton identification efficiency with a high jet rejection. Hence this is the typical, dangerous case of a large background submitted to a high rejection. Generally, these events are rejected by requesting that the leptons be isolated, which is not the case for particles inside jets. One should be aware that this event sample contains different subcategories with very different properties. For example W + light jets and W + c jets or W + b jets. The probabilities for these jets to fake a lepton are very different: b or c mesons can decay semileptonically into a real lepton. These can be partially rejected by requesting a small track impact parameter. Light jets can fake electrons because γ rays from π0 decays can convert in the layers of the inner detector, or because a charged pion is very close to one or several γ’s, etc. The probability for this to happen depends on the jet energy and location in the detector. Despite all efforts to model jet production and heavy flavour production and decay, and to model the detector very finely, this is a case where the Monte Carlo simulation cannot be used because of the very large numbers involved. It is hardly possible to simulate 108 events, and, even if it were, one would not trust the simulation to describe reality to a precision of a few parts in 108. Hence the W + jets background is estimated with data-driven methods. Typically, a control region is defined containing W + jets events with some selection of jets to make them closer to the signal configuration. The probability that these jets fake a lepton is evaluated on a different sample (QCD jets), and applied to the control region to estimate the contribution to the signal region. Several cross checks are used to control correlations, contribution of special cases, etc.

4.4 Global model of the analysis 4.4.1 The data model Given the above, the analysis plan has to deal with many separate data samples: • Nine independent channels: • three modes: ee, eμ, μμ; • foreachmode:0jet,1jet,> 1 jet. • For each channel: • one signal region; • two background control regions (tt¯, W + jets). The extraction of a possible Higgs signal consists in a global fit over these 27 data regions. This calculation will try to fit the event content of these regions by using either the expected contributions from the backgrounds only, or the same plus the 98 Identification of signals at LHC: analysis and statistics contributions from a SM Higgs production. The statistical analysis will evaluate if, and by how much, the fit with a Higgs is better than the fit with background only. This can be compared with another famous analysis: the search for Higgs decaying into two photons. The simplest that can be done is simply to select events with two photons and plot the distribution of their invariant mass. A signal would be seen as a peak on top of a smooth background. This could be considered as a fully data- driven background estimation, with the control regions taken as side bands or a fit to the (data) spectrum. In reality, the two photon analysis is also refined by splitting the sample according to the number of jets, etc., but the principle of searching for a peak above a smooth background remains: it is not necessary to describe in detail this background by simulations and/or data-driven methods. In the WW analysis, the absence of a strong discriminator makes the complicated modelling and fitting necessary. Note the importance of the global model: some backgrounds are estimated directly from the Monte Carlo simulation; others are estimated from control regions in the data themselves. √ Table 4.1 Cross sections at the centre-of-mass energy of s =7TeV for background processes. W → ν and the Z/γ∗ → cross sections are single-flavour cross sections

PROCESS GENERATOR CROSS SECTION σ (pb) [× BR] Inclusive W → ν ALPGEN 10.5 × 103 Inclusive W → τν PYTHIA 10.5 × 103 ∗ 2 Inclusive Z/γ → (M > 40 GeV) ALPGEN 10.7 × 10 ∗ 2 Inclusive Z/γ → ττ (Mττ > 60 GeV) PYTHIA 9.9 × 10 ∗ 3 Inclusive Z/γ → (10

From ATLAS Experiment c 2011 CERN. Global model of the analysis 99

The signal efficiency and the amount and efficiency of many backgrounds are evaluated by Monte Carlo simulations. Many event samples are generated, fully simulated by a large program based on GEANT IV, and reconstructed by the same software chain as the data. To give a flavour, Table 4.1 shows the main Monte Carlo data sets that are used [2].

4.4.2 Nuisance parameters The analysis must of course take into account the possibility of systematic errors in all the quantities measured or estimated. In Table 4.2 shows the main systematic

Table 4.2 Experimental sources of systematic uncertainty per object or event

SOURCE OF UNCERTAINTY TREATMENT IN ANALYSIS Jet energy resolution (JER) ∼ 14% Jet energy scale (JES) Takes into account close-by jets effect, jet flavour composition uncertainty and event pile-up uncertainty in addition to global JES uncertainty Global JES < 10% for pT > 15 GeV and |η| < 4.5 Pile-up uncertainty 2-5% for |η| < 2.1 and 3–7% for 2.1 <η| < 4.5 Electron selection efficiency Separate systematics for electron identification, reconstruction, and isolation, added in quadrature Total uncertainty of 2–5%, depending on η and ET Electron energy scale Uncertainty better than 1%, depending on η and ET Electron energy resolution (Small) constant term varied within its uncertainty, 0.6% of the energy at

Muon selection eﬃciency 0.3–1 % as a function of |η| and pT

Muon momentum scale η-dependent scale oﬀset in pT ,upto∼ 0.13%

Muon momentum resolution pT -andη-dependent resolution smearing functions, ≤ 5% b-tagging eﬃciency pT -dependent scale factor uncertainties, 6–16% b-tagging mis-tag rate up to 21 % as a function of pT Missing transverse energy 13.2% uncertainty on topological cluster energy Electron and muon pT changes owing to smearing propagated to MET Eﬀect of out-of-time pileup: ET Miss smeared by 5 GeV in 1/3 of Monte Carlo events Luminosity 3.7%

From ATLAS Experiment c 2011 CERN. 100 Identification of signals at LHC: analysis and statistics uncertainties [2]. The list of parameters that carry a systematic error is very long, from the integrated luminosity to the jet energy scale and lepton reconstruction efficiency, to cite only a few. Each of these has been determined by separate studies. For example, lepton reconstruction and identification efficiencies are determined from data samples of J/ψ and Z. These parameters with systematic errors are treated as nuisance parameters:inthe global fit, they are allowed to vary inside their systematic error. The model describes in detail the impact of each nuisance parameter on the various variables and in turn on the event content of all regions. In this way, with the proper model, correlations are automatically taken into account. For example, a variation of the jet energy scale (JES) is propagated to the measured jet energies, which changes the number of events above a certain energy cut. But it is also propagated to the measurement of ET miss in which jets enter.

4.5 Statistics 4.5.1 Inferences At this stage the selection cuts are fixed, the data model is defined (the event content of all data regions is known), and the nuisance parameters are defined and their error margin specified. The objective part of the analysis is complete. The last step is to apply an algorithm to synthesize the information to infer the presence or absence of a Higgs signal. This is usually presented in terms of statistical significance, but the reader (and especially young theorists!) should be aware that there is no unique definition of this, and that despite a nice mathematical presentation, all inference methods contain some part of arbitrariness and/or subjectivity [3]. In this chapter, 1 will not describe in detail the mathematics used to infer confidence levels: these can be found very clearly explained in [4]. However, in the same spirit as above, I would like to help theorists understand the philosophy behind the maths. It is well known that there are two main types of methods for inference: Bayesian and deterministic. In Bayesian methods, one makes a hypothesis concerning the distribution of the background, an a priori probability (“prior”). The data then bring in additional information, and the result, through Bayes’ law, can be interpreted in turn as a probability. The question is that the final result depends on the prior. After days of glory during the LEP era, the tendency is now rather in favour of the following, deterministic, method, although Bayesian studies are still in use. In practice, Bayesian results obtained with reasonable priors are very close to the deterministic ones. In deterministic methods, we build a quantity (a test statistic) that indicates whether the data is more compatible with the background-only hypothesis or with the background-plus-signal hypothesis. This is the method used as a baseline here. The reference model is the global fit of data to the expected background, fitting also the values of all nuisance parameters within their errors (marginalization of the nuisance parameters). The same fit is then performed adding a signal with strength μ compared with the SM Higgs signal. The likelihood with signal is compared with that Statistics 101 without signal, all after marginalization of the nuisance parameters. The test statistic is taken as the ratio of the two likelihoods. From these fits, we infer answers to two main kinds of questions: • Concerning background only: What is the probability that the background alone has fluctuated up to the level seen in the data? • Concerning background and possible signal, for each mass hypothesis, a statement on a potential signal: What is the signal strength that can be excluded at a given confidence level? (A signal strength μ means that the tested signal is μ times that of the SM Higgs.)

4.5.2 p0 plot

The first statement is reflected in the p0 plot of Fig. 4.2. Let us first concentrate on the “observed” line. From the definition (probability that the background fluctuates up to the level seen in data), the dips correspond to mass hypotheses where there is some sign of a signal or a signal-like fluctuation. The probability p0 indicated is the local probability, i.e. the probability of a background fluctuation at this precise mass hypothesis. In the plot of Fig. 4.2, we see a broad depression from 110 to 160 GeV, with p0 around 5%. Let us now turn to the “expected” line. Given the statistics in the data (number of events) and the possible systematic errors, we would like to know what would the “observed” line be in the presence of a signal. At each mass hypothesis, we can generate a large number of pseudo-experiments, each of which represents one occurrence of the real experiment, with the backgrounds and signal of the SM Higgs. In each pseudo-experiment, events are drawn randomly in all regions, with the same 0 p 1 H→WW(*) →lνlν

10−1 2σ 10−2 3σ 10−3 Observed

10−4 Expected 4σ −5 10 ∫L dt = 2.05 fb−1 s 10−6 = 7 TeV 5σ 10−7 100 150 200 250 300

mH (GeV)

Fig. 4.2 p0 plot: probability that the background fluctuates more than the data for each mass hypothesis. (ATLAS Experiment c 2011 CERN). 102 Identification of signals at LHC: analysis and statistics average numbers as that of the data, and a set of values of the nuisance parameters is drawn randomly from their distribution. Then each pseudo-experiment i is analysed as the real experiment, and p0,i(m) is derived for each mass hypothesis. At each mass, we define the “expected” p0(m) as the median of the p0,i(m) among the pseudo-experiments. In practice, generating a large number of pseudo-experiments for all mass points, varying all nuisance parameters etc., would be extremely time-consuming computer- wise. One can instead use asymptotic formulas that model analytically the fluctuations of the parameters and event numbers. This is what is shown in the plot in Fig. 4.2. We see that the “expected” curve reaches very low values at around 160 GeV. This means that the experiment is very sensitive to a signal in this mass region: in this hypothesis, we would have seen so many events that it would be very improbable to interpret them as a fluctuation of the background. As this is not what is observed, we expect this mass range to be excluded by the actual data. In contrast, the wide dip at low mass is about what we would expect from the presence of a Higgs at 130 GeV, where the “observed” curve meets the “expected” one.

4.5.3 CLs plot

The second statement is reﬂected in the famous CLs plot in Fig. 4.3, also called the “green and yellow bands” plot. For each mass hypothesis, the “observed” line gives the value of μ which can be excluded by the observed data at 95% conﬁdence level (CL). A value of 1 means that the SM Higgs can be excluded (in this channel) at 95% CL. Hence the exclusion region is simply the region of the x-axis where the “observed” line is below 1.

102

SM (*) H→WW →lνlν σ /

σ Observed Expected ∫ L dt = 2.05 fb−1 ± 1σ s = 7 TeV 10 ± 2σ 95% CL limit on on limit CL 95%

10−1 120 140 160 180 200 220 240 260 280 300

mH (GeV)

Fig. 4.3 CLs plot: 95% exclusion level, observed, and expected (median, ±1σ band, ±2σ band). The step at 220 GeV is due to the change in analysis cuts at this mass value. (ATLAS Experiment c 2011 CERN). Global Higgs analysis 103

Now we are interested in knowing if the “observed” line belongs to the class of lines that could normally be expected in the presence of background only, or if it has a distinctive feature (like the presence of a signal). Here again we use the pseudo-experiments, and analyse each of them, i, in the same way as the real one, deriving for each mass hypothesis a 95% CL value μ95,i. From the set of pseudo- experiments, at each mass hypothesis we can derive the median of these μ95,i,which we call the “expected” value, and the percentile values equivalent to −1σ, +1σ (the dark (“green”) band in Fig. 4.3) and −2σ, +2σ (the light (“yellow”) band). With these definitions, one can see directly if the “observed” line behaves as a “normal” experiment or not. If the “observed” line is below the “expected”, it means that the actual data enabled one to exclude a lower value of μ than expected: presumably there were fewer events observed than expected. This can be interpreted as a downward fluctuation of the background if the background estimate is correct, or as an overestimate of the background. If the “observed” line is above the “expected”, it means that the data did not enable one to exclude a value as low as the expected one. Thus can be interpreted as an upward fluctuation of the background, or the presence of a signal, or an underestimate of the background. Again here, we see that the “observed” line is above the “expected” line in a rather wide region between 110 and 160 GeV, at the 1σ or 2σ level. Indeed, the mass resolution of this channel is poor, and the “correlation length” along the x-axis is large.

4.6 Global Higgs analysis As the Higgs branching ratios vary with mass, some channels are more sensitive than others in diﬀerent mass ranges. In total, about 12 decay modes can be used, with very diﬀerent statistical power, depending on the mass hypothesis. At the time of the lecture on which this chapter is based, only one mode (WW) had sensitivity for the SM Higgs in some range, with all the other modes giving a μ95 value larger than 1 for all masses. At the time of writing of this chapter, combined analyses have been published by ATLAS and CMS. The discussion here is based on the ATLAS combination published in [5], based on all the 2011 data (almost 5 fb−1).

4.6.1 Global combination The method for combining different channels is quite similar to the method for combining the information from different data regions in the previous analysis. All signal and control regions are fed into a global model, as well as all nuisance parameters. The fit then tries to adapt the data to background or signal+ background expectations, in each case with marginalization of the nuisance parameters. The resulting p0 and CLs plots are shown in Fig. 4.4(a) and (b). A third plot, theμ ˆ plot (Fig. 4.4(c)), shows for each mass the best-fit value of the signal strength μ, always relative to the signal of the SM Higgs. The shaded band in this plot is an estimate of the ±1σ variation that could be expected for the experiment (strictly speaking, this calculation is not exact for the lowest, negative values ofμ ˆ; hence the exact definition of the band is written on the plot). 104 Identification of signals at LHC: analysis and statistics

μ ∫ −1 10 L dt ~ 1.04−4.9 fb Observed s= 7 TeV Background. expected ± 1σ ± 2σ

95% CL Limit on on Limit CL 95% 1

(a) CL limits 10−1 s 0 200 300 400 500 p 1

10−1 Local 10−2 2σ −3 10 3σ 10−4 −5 4σ 10 Signal expected 10−6 Observed σ 10−7 (b) 5 2 μ^ 1.5 1 0.5 0 −0.5 Observed −1 (c) −2 ln λ(μ) < 1 −1.5 110 150 200 300 400 500 600

mH (GeV)

Fig. 4.4 Combined Higgs search: (a) CLs;(b)p0;(c)μˆ. (ATLAS Experiment c 2011 CERN).

The plots show asymptotic results, but several points in mass have been cross- checked with a large number of pseudo-experiments, and the diﬀerences are reported in [5] (especially so in the vicinity of the apparent excess). From the CLs plot, we see that the SM Higgs is excluded at 95% CL in a large mass range 131–468 GeV (the “observed” line is below 1) except for a tiny region at 245 GeV. Note that the line shows a very local dip at 114 GeV, which can be interpreted as a downward ﬂuctuation of the background, but which enables one to exclude the 114.5 GeV value hinted at by LEP data. There is also a large dip around 360 GeV where observed is more than 2σ below expected. Global Higgs analysis 105

On the p0 plot, we see some dips corresponding to excesses in the data, that with −4 the lowest p0 being for a mass hypothesis of 125 GeV with p0 = 210 , and some other local excesses with higher probability of a background fluctuation. From theμ ˆ plot, we see that the signal strength is indeed compatible with zero everywhere except at 125 GeV. The best-fit value is 1.5 at 125 GeV: if it is really due to a 125 GeV Higgs, we have been a bit “lucky”, but the 1σ band includes 1, meaning that the number of events is well compatible with a SM Higgs. Indeed, the three plots display the same features seen from different points of view: exclusion limit, probability of a background fluctuation, best-fit signal strength. In particular, the structure at 125 GeV is the most “signal-like” of the excesses, with a local probability of a background fluctuation of 210−4, i.e. a 3.6 σ effect.

4.6.2 Look-elsewhere effect If one has no a priori preferred mass hypothesis, the interesting question is rather: “What is the probability that the background fluctuates up to the level of the data anywhere in the explored mass range?” This probability is obviously larger than any local probability. This is referred to as the look-elsewhere effect (LEE).TheLEEhas always been considered intuitively, but it is only recently that experiments have tried to quantify it when presenting their results. Consider a histogram of a variable (e.g. an invariant mass) with 10 independent bins, and suppose that the background has a flat distribution. The data consist of 1000 events. For background only, in each bin, the distribution of the content is given by the Poisson law with average 100 (very close to a Gaussian with average 100 and σ=10). To test the presence of a “signal” in a single bin with content c, we calculate the probability pc of a background fluctuation beyond c, i.e. the sum of the Poisson probabilities from c to infinity. To translate into “standard deviations”, we just recall that in the Gaussian approximation the integral beyond c is the integral of the Gaussian beyond (c−100)/10 standard deviations. Now the probability that any of the 10 bins had such a fluctuation is obviously 10 × pc, since the bins are independent. Of course, in a real invariant mass distribution, the bins are not independent. Neighbouring bins are correlated if they are closer than the resolution of the measurement. Roughly speaking, if the resolution is constant over the mass range considered, the number of independent bins is the abscissa range of the histogram Δm divided by the resolution δm. Hence the old rule of thumb to estimate the global probability of a background fluctuation was to take pc × Δm/δm. Obviously this naive estimate could not apply in more complicated cases, when the resolution depends on the mass, or in cases like that here where the analysis is based on a fit more complicated than a single- variable histogram. Procedures have been found recently that allow one to calculate a global probability in a consistent way [6]. In the example above of the combined Higgs search, the global probability that a background fluctuation appears anywhere in the mass range with a local p0 as low as the observed one is 1.4%, corresponding to a 2.2σ effect. However, it should be noted that just around the corner lies the question of arbitrariness in inferring the presence or absence of a new signal: we know how to quantify 106 Identification of signals at LHC: analysis and statistics the global probability of a background fluctuation in one plot, and even in one analysis. But large experiments make hundreds of plots and dozens of analyses. What is the probability that any of these display an excess? First there is the question of naturalness of the analysis: was it defined before data-taking? Of course, the best is a blind analysis, or one where we defined the analysis on a first set of data, and just look at a second (independent) set of data without touching anything. Here, for the SM Higgs search, the modes and the analyses are naturally dictated by the SM itself and the analysis is more or less “unique”. But for searches beyond the SM, we should keep in mind the number of analyses, as an additional LEE.

4.7 Conclusion Analyses in large modern particle physics experiments are often quite complex and difficult to comprehend globally. However, a lot of effort has been devoted to standard- izing the methods, rationalizing the background and error calculations, systematizing the statistical treatment, and synthesizing the results. The Higgs search is of course of such a importance that all the tools have been refined and utilized to extract as much information as possible from the data, while keeping a safe estimate of the uncertainties. This chapter has attempted to help young theorists in understanding what is going on in an analysis and what are the delicate points and possible pitfalls, and finally to help them interpret correctly the most usual statistical plots.

References (∗) [1] The ATLAS Collaboration. Search for the√ Higgs boson in the H → WW → +ν−ν¯ decay channel in pp collisions at s = 7 TeV with the ATLAS detector. Phys.Rev.Lett.108 (2012) 111802 [arXiv:1112.2577 [hep-ex]]. [2] The ATLAS Collaboration. Search for the Standard Model Higgs boson in the H → WW(∗) → νν decay mode with the ATLAS detector. ATLAS-CONF-2011-111 (2011). [3] K. Popper, The Logic of Scientiﬁc Discovery, 1934 (as Logik der Forschung, English translation 1959). [4] G. Cowan, K. Crammer, E. Gross, and O. Vitells. Asymptotic formulae for likelihood-based tests of new physics. Eur. Phys. J. C71 (2011) 1554 [arXiv 1007.1727v2 [physics. data-anal]], and references within. [5] The ATLAS Collaboration. Combined search√ for the Standard Model Higgs boson using up to 4.9 fb−1 of pp collision data at s = 7 TeV with the ATLAS detector at the LHC. Phys. Lett. B710 (2012) 49–66 [arXiv:1202.1408 [hep-ex]]. [6] E. Gross and O. Vitells. Trial factors for the look elsewhere eﬀect in high energy physics. Eur. Phys. J. C70 (2010) 525–530 [arXiv:1005.1891 [physics. data-anal]]. 5 Introduction to the theory of LHC collisions

Michelangelo L. Mangano

Theory Division CERN, Geneva, Switzerland

5 Introduction to the theory of LHC collisions 107 Michelangelo L. MANGANO

5.1 Introduction 109 5.2 QCD and the proton structure at large Q 2 110 5.3 The ﬁnal-state evolution of quarks and gluons 119 5.4 Applications 129 5.5 Outlook and conclusions 137 References 138 Introduction 109

5.1 Introduction The frontier of high-energy physics is being redefined by the experiments at CERN’s Large Hadron Collider (LHC), the proton–proton collider operating at a centre-of-mass energy of 8 TeV (to become ∼14 TeV by 2014). The main goal of these experiments is to unveil the nature of electroweak symmetry breaking, via the detection of the Higgs boson and of other new particles associated with possible extensions of the Stand- ard Model (SM). While strong interactions between quarks and gluons—described by quantum chromodynamics (QCD)—have no direct relation to electroweak phenomena, their role in making the LHC physics programme possible cannot be underestimated. The processes leading to the creation of Higgs bosons, and other interesting particles, are all driven by the interactions among quarks and gluons, and it is these interactions that will define the production rate of the new phenomena, as well as of the numerous SM processes that will act as irreducible backgrounds to their detection. A solid and quantitative understanding of the role of strong interactions in LHC physics is therefore essential for full exploitation of the LHC’s discovery and measurement potential. Most of the collisions between protons are the result of the complex long- distance QCD dynamics responsible for holding quarks together, and their quantitative description still lacks a robust first-principles understanding. On the other hand, the processes leading to the production of heavy elementary particles arise from the short- distance interactions among the pointlike proton constituents. These hard interactions can be described by the perturbative regime of QCD, making it possible to perform quantitative, first-principles predictions. This chapter will illustrate the basic principles underlying the use of perturbative QCD in predicting the structure of hard processes in high-energy hadronic collisions. The starting point, in Section 5.2, will be a discussion of the factorization formula, which is the basis for the description of all hard processes in terms of universal functions parametrizing the density of quarks and gluons inside the proton. In Section 5.3, I discuss the evolution of the perturbative final states, made of quarks and gluons, toward physical systems made of hadrons. Finally, Section 5.4 collects several applications and examples of comparisons between the theoretical predictions and present data. These will provide a picture of the success of this theoretical framework, giving good confidence on the reliability of its future applications to the study of LHC collisions. The treatment will be introductory, and the emphasis will be on basic and intuitive physics concepts. Given the large number of papers that have contributed to the development of the field, it is impossible to provide a complete and fair bibliography. I therefore limit my bibliography to some review books and to references to some of the key results discussed here. For an excellent description of the early ideas about quarks and their dynamics, the classic reference is Feynman’s book [1]. The emergence of QCD as a theory for strong interactions is nicely reviewed in [2]. For a general, but rather formal, introduction to QCD, see [3]. For a more modern and pedagogical introduction, in the context of an introductory course on field theory, I refer to the excellent book by Peskin and Schroeder [4]. For a general introduction to collider physics, see [5]. For QCD applications to the LEP, Tevatron, and LHC, see [6] and, 110 Introduction to the theory of LHC collisions specifically for the LHC, [7]. Explicit calculations, including the nitty-gritty details of next-to-leading-order (NLO) calculations and renormalization, are given in great detail for several concrete cases of interest in [8]. Many of the ideas used in this review are inspired by the very physical perspective presented in [9]. While I shall be making occasional reference to perturbative calculations and to Monte Carlo generator tools, I shall not provide a systematic review of the state of the art in these very active areas. For recent status reports, see [10, 11].

5.2 QCD and the proton structure at large Q2 An understanding of the structure of the proton at short distances is one of the key ingredients in predicting cross sections for processes involving hadrons in the initial state. All processes in hadronic collisions, even those intrinsically of electroweak nature such as the production of W/Z bosons or photons, are in fact induced by the quarks and gluons contained inside the hadrons. In this section, I shall introduce some important concepts, such as the notions of partonic densities of the proton and of parton evolution. These are the essential tools used by theorists to predict production rates for hadronic reactions. We shall limit ourselves to√ processes where a proton–(anti)proton pair collides at large centre-of-mass energy ( S, typically larger than several hundred GeV) and undergoes a very inelastic interaction, with momentum transfers between the participants in excess of several GeV. The outcome of this hard interaction could be the simple scattering at large angle of some of the hadron’s elementary constituents, their annihilation into new massive resonances, or a combination of the two. In all cases, the final state consists of a large multiplicity of particles, associated with the evolution of the fragments of the initial hadrons, as well as of the new states produced. As discussed below, the fundamental physical concept that makes the theoretical description of these phenomena possible is ‘factorization’, namely the ability to isolate separate independent phases of the overall collision. These phases are dominated by different dynamics, and the most appropriate techniques can be applied to describe each of them separately. In particular, factorization allows one to decouple the complexity of the proton structure and of the final-state hadron formation from the elementary nature of the perturbative hard interaction among the partonic constituents. Figure 5.1 illustrates how this works. As the left proton travels freely before coming into contact with the hadron coming in from the right, its constituent quarks are held together by the constant exchange of virtual gluons (e.g. gluons a and b in the picture). These gluons are mostly soft, because any hard exchange would cause the constituent quarks to fly apart, and a second hard exchange would be necessary to reestablish the balance of momentum and keep the proton together. Gluons of high virtuality (gluon c in the picture) therefore prefer to be reabsorbed by the same quark, within a time inversely proportional to their virtuality, as prescribed by the uncertainty principle. The state of the quark is, however, left unchanged by this process. Altogether, this suggests that the global state of the proton, although defined by a complex set of gluon exchanges between quarks, is nevertheless determined by interactions that have QCD and the proton structure at large Q2 111

c H HP e

d f g b

a UE

Fig. 5.1 General structure of a hard proton–proton collision.

a timescale of the order of√ 1/mp. When seen in the laboratory frame, where the proton is moving√ with energy S/2, this time is furthermore Lorentz-dilated by a factor γ = S/2mp. If we disturb a quark with a probe of virtuality Qχmp, the time frame for this interaction is so short (1/Q) that the interactions of the quark with the rest of the proton can be neglected. The struck quark cannot negotiate with its partners a coherent response to the external perturbation: it simply does not have the time to communicate to them that it is being kicked away. On this timescale, only gluons with energy of the order of Q can be emitted, something that, to happen coherently over the whole proton, is suppressed by powers of mp/Q (this suppression characterizes the ‘elastic form factor’ of the proton). In Fig. 5.1, the hard process is represented by the rectangle labelled HP. In this example a head-on collision with a gluon from the opposite hadron leads to a qg → qg scattering with a momentum exchange of the order of Q. This and other possible processes can be calculated from first principles in perturbative QCD, using elementary quarks and gluons as external states. When the constituent is suddenly deflected, the partons that it had recently radiated cannot be reabsorbed (as happened to gluon c earlier), because the constituent is no longer there waiting for the partons to come back. This is the case, for example, of the gluon d emitted by the quark, and of the quark e from the opposite hadron; the emitted gluon became engaged in the hard interaction. The number of ‘liberated’ partons will depend on the hard scale Q: the larger the value of Q, the more sudden the deflection of the struck parton, and the fewer the partons that can reconnect before its departure (typically only partons with virtuality larger than Q). After the hard process, the partons liberated during the evolution before the collision and the partons created by the hard collision will themselves emit radiation. The radiation process, governed by perturbative QCD, continues until a low-virtuality scale is reached (the boundary region labelled with a dotted line H in Fig. 5.1). To describe this perturbative evolution phase, proper care has to be taken to incorporate quantum coherence effects, which in principle connect the probabilities of radiation off different 112 Introduction to the theory of LHC collisions partons in the event. Once the low-virtuality scale is reached, the memory of the hard-process phase has been lost, once again as a result of different timescales in the problem, and the final phase of hadronization takes over. Because of the decoupling from the hard-process phase, the hadronization is assumed to be independent of the initial hard process, and its parametrization, tuned to the observables of some reference process, can then be used in other hard interactions (universality of hadronization). Nearby partons merge into colour-singlet clusters (the grey blobs in Fig. 5.1), which then decay phenomenologically into physical hadrons. To complete the picture, we need to understand the evolution of the fragments of the initial hadrons. As shown in Fig. 5.1, this evolution cannot be entirely independent of what happens in the hard event, because at least colour quantum numbers must be exchanged to guarantee the overall neutrality and conservation of baryon number. In our example, the gluons f and g, emitted early on in the perturbative evolution of the initial state, split into qq¯ pairs, which are shared between the hadron fragments (whose overall interaction is represented by the oval labelled UE, for ‘underlying event’) and the clusters resulting from the evolution of the initial state. The above ideas are embodied in the following factorization formula, which represents the starting point of any theoretical analysis of cross sections and observables in hadronic collisions: dσ d σˆjk(Q) = fj(x1,Q)fk(x2,Q) F (Xˆ → X; Q), (5.1) dX ˆ ˆ j,k X d X where • X is some hadronic observable (e.g. the transverse momentum of a pion or the invariant mass of a combination of particles,); • the sum over j and k extends over the parton types inside the colliding hadrons; • the function fj(x, Q) (known as the parton distribution function, PDF) parametrizes the number density of parton type j with momentum fraction x in a proton probed at a scale Q (more later on the meaning of this scale); • Xˆ is a parton-level kinematical variable (e.g. the transverse momentum of a parton from the hard scattering); • σˆjk is the parton-level cross section, differential in the variable Xˆ; • F (Xˆ → X; Q) is a transition function, weighting the probability that the partonic state defining Xˆ gives rise, after hadronization, to the hadronic observable X. In the rest of this section, I shall cover the above ideas in some more detail. While I shall not provide you with a rigorous proof of the legitimacy of this approach, I shall try to justify it qualitatively to make it sound at least plausible.

5.2.1 The parton densities and their evolution As mentioned above, the binding forces responsible for the quark conﬁnement are due to the exchange of rather soft gluons. If a quark were to exchange just a single hard virtual gluon with another quark, the recoil would tend to break the proton apart. QCD and the proton structure at large Q2 113

Fig. 5.2 Gluon exchange inside the proton.

It is easy to verify that the exchange of gluons with virtuality larger than Q is then proportional to some large power of mp/Q, where mp is the proton mass. Since the gluon coupling constant gets smaller at large Q, exchange of hard gluons is significantly suppressed.1 Consider in fact the picture in Fig. 5.2. The exchange of two gluons is required to ensure that the momentum exchanged after the first gluon emission is returned to the quark, and the proton maintains its structure. The contributions of hard gluons to this process can be approximated by integrating the loop over large momenta: 4 d q ∼ 1 6 2 . (5.2) Q q Q 2 At large Q, this contribution is suppressed by powers of (mp/Q) , where the proton mass mp is included as being the only dimensionful quantity available (one could use here the fundamental scale of QCD, ΛQCD, but numerically this is anyway of the order of a GeV). The interactions keeping the proton together are therefore dominated by soft exchanges, with virtuality Q of the order of mp. Owing to Heisenberg’s uncertainty principle, the typical timescale of these exchanges is of the order of 1/mp: this is the time during which fluctuations with virtuality of the order of mp can survive. In the laboratory system, where the proton travels with energy E, this time is Lorentz-dilated ∼ 2 to τ γ/mp = E/mp. If we probe the proton with an off-shell photon, the interaction takes place during the limited lifetime of the virtual photon, which, once more from the uncertainty principle, is given by the inverse of its virtuality. Assuming that the virtuality Q mp, once the photon gets ‘inside’ the proton and meets a quark, the struck quark has no time to negotiate a coherent response with the other quarks, because the time scale for it to ‘talk’ to its partners is too long compared with the duration of the interaction with the photon itself. As a result, the struck quark has no option but to interact with the photon as if it were a free particle. Let us look in more detail at what happens during such a process. In Fig. 5.3, we see a proton as it approaches a hard collision with a photon of virtuality Q. Gluons emitted at a scale q>Qhave the time to be reabsorbed, since their lifetime is very short. Their contribution to the process can be calculated in perturbative QCD, since the scale is large and in the domain where perturbative calculations are meaningful. Since after being reabsorbed the state of the quark remains the same, their only effect is an overall renormalization of the wavefunction, and they do not affect the quark density. A gluon emitted at a scale q

1 The fact that the coupling decreases at large Q plays a fundamental role in this argument. Were this not true, the parton picture could not be used! 114 Introduction to the theory of LHC collisions

Fig. 5.3 Gluon emission at different scales during the approach to a hard collision. quark, the quark has been kicked away by the photon, and is no longer there. Since the gluon has taken away some of the quark momentum, the momentum fraction x of the quark as it enters the interaction with the photon is different from the momentum it had before, and therefore its density f(x) is affected. Furthermore, when the scale q is of the order of 1 GeV, the state of the quark is not calculable in perturbative QCD. This state depends on the internal wavefunction of the proton, which perturbative QCD cannot predict. We can, however, say that the wavefunction of the proton, and therefore the state of the ‘free’ quark, are determined by the dynamics of the soft- gluon exchanges inside the proton itself. Since the timescale of this dynamics is long compared with the timescale of the photon–quark interaction, we can safely argue that the photon sees to good approximation a static snapshot of the proton’s inner configuration. In other words, the state of the quark had been prepared long before the photon arrived. This also suggests that the state of the quark will not depend on the precise nature of the external probe, provided the timescale of the hard interaction is very short compared with the time it would take for the quark to readjust itself. As a result, if we could perform some measurement of the quark state using, say, a virtual-photon probe, we could then use this knowledge on the state of the quark to perform predictions for the interaction of the proton with any other probe (e.g. a virtual W or even a gluon from an opposite beam of hadrons). This is the essence of the universality of the parton distributions. The above picture leads to an important observation. It appears in fact that the distinction between which gluons are reabsorbed and which are not depends on the scale Q of the hard probe. As a result, the parton density f(x) appears to depend on Q. This is illustrated in Fig. 5.4. The gluon emitted at a scale μ has a lifetime short enough to be reabsorbed before a collision with a photon of virtuality Q<μ, but too long for a photon of virtuality Q>μ. When going from μ to Q, therefore, the partonic density f(x) changes. We can easily describe this variation as follows:

Q > µ q < µ

xin x = yxin x = xin

Fig. 5.4 Scale dependence of the gluon emission during a hard collision. QCD and the proton structure at large Q2 115 1 Q 1 2 2 f(x, Q)=f(x, μ)+ dxin f(xin,μ) dq dy P (y,q ) δ(x − yxin), (5.3) x μ 0

Here we obtain the density at the scale Q by adding to f(x)atthescaleμ (which we label as f(x, μ)) all the quarks with momentum xin >xthat retain a proton- 2 momentum fraction x = y/xin by emitting a gluon. The function P (y,Q ) describes the ‘probability’ that the quark emits a gluon at a scale Q, keeping a fraction y of its momentum. This function does not depend on the details of the hard process—it simply describes the radiation of a free quark subject to an interaction with virtuality Q. Since f(x, Q) does not depend upon μ (μ is just used as a reference scale to construct our argument), the total derivative of the right-hand side with respect to μ should vanish, leading to the following equation: 1 df (x, Q) ⇒ df (x, μ) dy 2 2 =0 2 = f(y,μ) P (x/y, μ ). (5.4) dμ dμ x y

Dimensional analysis and the fact that the gluon emission rate is proportional to the QCD coupling squared allow us to further write

α 1 P (x, Q2)= s P (x), (5.5) 2π Q2 from which the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) equation follows [12–14]: 1 df (x, μ) αs dy 2 = f(y,μ) Pqq(x/y). (5.6) d log μ 2π x y

The so-called splitting function Pqq(x) can be calculated in perturbative QCD. The subscript qq is a labelling convention indicating that x refers to the momentum fraction retained by a quark after emission of a gluon. More generally, one should consider additional processes. For example, one should include cases in which the quark interacting with the photon comes from the splitting of a gluon. This is shown in Fig. 5.5: the diagram (a) is the one we considered above; the diagram (b) corresponds to processes where an emitted gluon has the time to split

(a) (b)

Fig. 5.5 The processes leading to the evolution of the quark density. 116 Introduction to the theory of LHC collisions

(a) (b)

Fig. 5.6 The processes leading to the evolution of the gluon density. into a qq¯ pair, and it is one of these quarks that interacts with the photon. The overall evolution equation, including the eﬀect of gluon splitting, is given by 1 dq(x, Q) αs dy x x = q(y,Q) Pqq + g(y,Q) Pqg , (5.7) dt 2π x y y y where t = log Q2. For external probes that couple to gluons (such as an external gluon, coming, for example, from an incoming proton), we have a similar evolution of the gluon density (see Fig. 5.6): dg(x, Q) α 1 dy x x = s g(y,Q)P + q(y,Q) P . (5.8) dt 2π y gg y gq y x q,q¯

At the leading perturbative order (LO), the explicit calculation of the splitting 2 functions Pij(x) (see, e.g. [8]) then gives the following expressions:

1+x2 P (x)=P (1 − x)=C , (5.9) qq gq F 1 − x 1 2 2 Pqg(x)= x +(1− x) , (5.10) 2 1 − x x P (x)=2C + + x(1 − x) , (5.11) gg A x 1 − x

2 − where CF =(NC 1)/2NC and CA =2NC are the Casimir invariants of the fundamental and adjoint representation of SU(NC )(NC = 3 for QCD). In the following, we shall derive some general properties of the PDF evolution and give a few concrete examples. The factorization given by (5.1) cannot of course hold for all phenomena in hadronic collisions. Processes like elastic scattering or generic soft collisions cannot be described in terms of quarks and gluons, since they are driven by the long-distance structure of the proton. Furthermore, modifications to this simple factorization relation can be required to describe processes like hard diffraction. The DGLAP evolution of parton densities, furthermore, requires modifications with respect to what shown here when x

2 The expressions given here are strictly valid only for x = 1. Slight modiﬁcations are required to extend them to x =1. QCD and the proton structure at large Q2 117 is so small that αs log(1/x) is of order 1, or when the gluon density becomes so large that gluon recombination becomes possible. Since the processes we shall consider in our applications do not cover these cases, we shall not further discuss these aspects in this review.

5.2.2 Example: quantitative evolution of parton densities A complete solution for the evolved parton densities in x space can only be obtained from a numerical analysis. This work has been done by several groups (for a review see [15, 16]), and is continually being updated [17–21] by including the most up-to- date experimental results used for the determination of the input densities at a fixed scale. Figure 5.7(a) shows the up-quark valence momentum density at various scales Q, as obtained from one of these studies. Note the softening at larger scales and the clear log Q2 evolution. As Q2 grows, the valence quarks emit more and more radiation, since they change direction over a shorter amount of time (larger acceleration). They therefore lose more momentum to the emitted gluons, and their spectrum becomes softer. The most likely momentum fraction carried by a valence up quark in the proton goes from x ∼ 20% at Q =3GeVtox 10% at Q = 1000 GeV. Notice finally that the density vanishes at small x. Figure 5.7(b) shows the gluon momentum density. This grows at small x, with an approximate g(x) ∼ 1/x1+δ behaviour, and δ>0 slowly increasing at large Q2. This low-x growth is due to the 1/x emission probability for the radiation of gluons, which was discussed Section 5.2.1 and which is represented by the 1/x factors in the Pgq(x) 2 and Pgg(x) splitting functions. As Q grows, we find an increasing number of gluons at small x, as a result of the increased radiation off quarks, as well as off the harder gluons.

(a) (b) 2 2 xUval (x, Q ) xG (x, Q ) 100 103 102 101 10–1 100 10–1 dot-dash: Q = 1000 Gev dots: Q = 100 Gev 10–2 10–2 dot-dash: Q = 1000 Gev dashes: Q = 10 Gev dots: Q = 100 Gev 10–3 solid: Q = 3 Gev dashes: Q = 10 Gev solid: Q = 3 Gev 10–4 10–3 10–5 10–4 10–3 10–2 10–1 100 10–4 10–3 10–2 10–1 100 x x

Fig. 5.7 (a) Valence up-quark momentum-density distribution for diﬀerent scales Q. (b) Gluon momentum-density distribution. 118 Introduction to the theory of LHC collisions

(a)2 (b) xUsea (x, Q ) 101 103 102 10–0 101 10–1 100 10–2 dot-dash: Q = 1000 Gev 10–1 dots: Q = 100 Gev 10–2 Q = 1000 Gev 10–3 dashes: Q = 10 Gev solid: xG(x) solid: Q = 3 Gev –3 10 dashes: xUsea(x) 10–4 dots: xCharm(x) 10–4 dot-dash: xUval(x) 10–5 10–5 10–4 10–3 10–2 10–1 100 10–4 10–3 10–2 10–1 100 x x

Fig. 5.8 (a) Sea up-quark momentum-density distribution for diﬀerent scales Q.(b) Momentum-density distribution for several parton species at Q = 1000 GeV.

Figure 5.8(a) shows the evolution of the up-quark sea momentum density. Shape and evolution match those of the gluon density, a consequence of the fact that sea quarks come from the splitting of gluons. Since the gluon-splitting probability is proportional to αs, the approximate ratio sea/gluon ∼ 0.1, which can be obtained by comparing Figs. 5.7 and 5.8, is perfectly justified. Finally, the momentum densities for gluons, up-sea, charm, and up-valence distributions are shown, for Q = 1000 GeV, in Fig. 5.8(b). Note here that usea and charm are approximately the same at very large Q and small x, as will be discussed in more detail in Section 5.3. The proton momentum is carried mostly by valence quarks and by gluons. The contribution of sea quarks is negligible. Parton densities are extracted from experimental data. Their determination is therefore subject to the statistical and systematic uncertainties of the experiments and of the theoretical analysis (e.g. the treatment of nonperturbative effects, and the impact of missing higher-order perturbative corrections). Techniques have been introduced recently to take these uncertainties, into account and to evaluate their impact on concrete observables. A summary of such an analysis [22] is given in Fig. 5.9. What is plotted is the uncertainty bands for partonic luminosities3 corresponding to the gg and qq¯ initial-state channels. The partonic flux is given as a function ofs ˆ, the partonic CM invariant mass. Obvious features include the growth in uncertainty of the gg density at large mass, corresponding to the lack of data covering the large-x region of the gluon density. Notice, in particular, that the uncertainty bands reported by different groups do not necessarily overlap: this reflects different choices in the input sets of experimental data. In the future, the LHC data will be used in these global fits, leading to an important reduction in the uncertainties.

3 For the deﬁnition of parton luminosity see Section 5.4.1. The ﬁnal-state evolution of quarks and gluons 119

(a) (b) ∑ qq √s q( ) luminosity at LHC ( = 7 TeV) ∑q(qq) luminosity at LHC (√s = 7 TeV) 1. 2 1. 2

1. 15 MSTW08 NLO 1. 15 MSTW08 NLO CTEQ6.6 HERAPDF1.0 1. 1 CT10 1. 1 ABKM09 NNPDF2.1 GJR08 1.05 1.05

1 1

0.95 0.95

0.9 0.9

0.85 0.85 WZ WZ 0.8 (68% CL) 2008 NLO MSTW to Ratio 0.8 10−3 10−2 10−1 10−3 10−2 10−1 √ŝ / s √ ŝ / s

1. 15 MSTW08 NLO 1. 15 MSTW08 NLO CTEQ6.6 HERAPDF1.0 1. 1 CT10 1. 1 ABKM09 NNPDF2.1 GJR08 1.05 1.05

1 1

0.95 0.95

0.9 0.9

0.85 0.85 Ratio to MSTW 2008 NLO (68% CL) 2008 NLO MSTW to Ratio (68% CL) 2008 NLO MSTW to Ratio (68% CL) 2008 NLO MSTW to Ratio 0.8 120 180240 0.8 −3 −2 120 180 240 −1 −3 −2 M (GeV) tt −1 M (GeV) tt 10 10 H 10 10 10 H 10 √ŝ / s √ŝ / s

Fig. 5.9 Uncertainty in the parton luminosity functions at the LHC [22].

5.3 The final-state evolution of quarks and gluons We discussed in Section 5.2 the initial-state evolution of quarks and gluons as the proton approaches the hard collision. We study here how quarks and gluons evolve after emerging from the hard process, and finally transform into hadrons, neutralizing their colours. We start by considering the simplest case, namely e+e− collisions, which provide the cleanest environment in which to study applications of QCD at high energy. This is the place where theoretical calculations have today reached their greatest accuracy, and where experimental data are the most precise, especially thanks to the huge statistics accumulated by LEP, LEP2, and SLC. The key process is the annihilation of the e+e− pair into a virtual photon or Z0 boson, which will subsequently decay to a qq¯ pair. Therefore, e+e− collisions have the big advantage of providing an almost point-like source of quark pairs, so that, in contrast to the case of interactions involving hadrons in the initial state, we at least know very precisely the state of the quarks at the beginning of the interaction process. Nevertheless, it is by no means obvious that this information is sufficient to predict the properties of the hadronic final state. We know that this final state is clearly not simply a qq¯ pair, but some high-multiplicity set of hadrons. For example, as shown in Fig. 5.10, the average multiplicity of charged hadrons in the decay of a Z0 is 120 Introduction to the theory of LHC collisions )

ch (a) (b)

n −1 OPAL data OPAL data ( 10

P JETSET 7.2 JETSET 7.2

10−2

10−3

−4 Whole event Single 10 hemisphere

10−5 0 20 40 60 20 40 60 n n ch ch

Fig. 5.10 Charged-particle multiplicity distribution in Z0 decays approximately 20. It is therefore not obvious that a calculation done using the simple picture e+e− → qq¯ has anything to do with reality. For example, one may wonder why we do not need to calculate σ(e+e− → qqg...g...¯ ) for all possible gluon multiplicities to get an accurate estimate of σ(e+e− → hadrons). And since in any case the final state is not made of q’s and g’s, but of π’s, K’s, ρ’s, etc., why would σ(e+e− → qqg...g¯ ) be enough? The solution to this puzzle lies both in a question of time and energy scales and in the dynamics of QCD.√ When the qq¯ pair is produced, the force binding q andq ¯ is + − proportional to αs(s)( s being the e e centre-of-mass energy). Therefore it is weak, and q andq ¯ behave to good approximation like free particles. The radiation emitted in the first instants after the pair creation is also perturbative, and it will stay so until a time after creation of the order of (1 GeV)−1, when radiation with wavelengths (1 GeV)−1 starts being emitted. At this scale, the coupling constant is large, and nonperturbative phenomena and hadronization start to play a role. However, as we shall show, colour emission during the perturbative evolution organizes itself in such a way as to form colour-neutral, low-mass, parton clusters highly localized in phase space. As a result, the complete colour neutralization (i.e. the hadronization) does not involve long-range interactions between partons far away in phase space. This is very important, because the forces acting among coloured objects at this timescale would be huge. If the perturbative evolution were to separate far apart colour-singlet qq¯ pairs, the final-state interactions taking place during the hadronization phase would totally upset the structure of the final state. In this picture, the identification of the perturbative cross section σ(e+e− → qq¯) with observable, high-multiplicity hadronic final states is realized by jets, namely The final-state evolution of quarks and gluons 121 collimated streams of hadrons that are the final result of the perturbative and nonperturbative evolution of each quark. The large multiplicity of the final states, shown in Fig. 5.10, corresponds to the many particles that emerge from the collinear emissions of many gluons from each quark. The dynamics of these emissions leads these particles to grossly follow the direction of the primary quark, and the emergent bundle, the jet, inherits the kinematics of the initial quark. This is shown in Fig. 5.11(a). Three-jet events, shown in Fig. 5.11(b), arise from the O(αs) corrections to the tree-level process, namely to diagrams such as those shown in Fig. 5.12. An important additional result of this ‘pre-confining’ evolution, is that the memory of where the local colour-neutral clusters came from is totally lost. So we expect the properties of hadronization to be universal: a model that describes hadronization at a given energy will work equally well at some other energy. Furthermore, so much time has passed since the original qq¯ creation that the hadronization phase cannot significantly affect the total hadron production rate. Perturbative corrections due to the emission of the first hard partons should be calculable in perturbation theory, providing a finite, meaningful cross section. The nature of nonperturbative corrections to this picture can be explored. One can 2 prove, for example, that the leading correction to the total rate Re+e− is of order F/s , ∝ | a μνa| 4 where F 0 αsFμν F 0 is the so-called gluon condensate. Since F = O(1 GeV ), these nonperturbative corrections are usually very small. For example, they are

Fig. 5.11 Experimental pictures of 2- and 3-jet ﬁnal states from e+e− collisions.

+ − Fig. 5.12 O(αs) corrections to the tree-level e e → qq¯ process. 122 Introduction to the theory of LHC collisions √ O(10−8)attheZ0 peak. Corrections scaling like Λ2/s or Λ/ s can nevertheless appear in other less inclusive quantities, such as event shapes or fragmentation functions. We now come back to the perturbative evolution, and shall devote the ﬁrst part of this section to justifying the picture given above.

5.3.1 Soft-gluon emission Emission of soft gluons plays a fundamental role in the evolution of the final state [6, 9]. Soft gluons are emitted with large probability, since the emission spectrum behaves like dE/E, typical of bremsstrahlung as familiar in QED. They provide the seed for the bulk of the final-state multiplicity of hadrons. The study of soft-gluon emission is simplified by the simplicity of their couplings. Being soft (i.e. long-wavelength) they are insensitive to the details of the very-short-distance dynamics: they cannot distinguish features of the interactions that take place on timescales shorter than their wavelength. They are also insensitive to the spin of the partons: the only feature to which they are sensitive is the colour charge. To prove this let us consider soft-gluon emission in the qq¯ decay of an off-shell photon:

pj pj

ka (5.12) k a pi pi

− i μ a μ i a Asoft =¯u(p) (k)(ig) Γ v(¯p) λij +¯u(p)Γ (ig) (k) v(¯p) λij p/ + k/ p/¯ + k/ g g = u¯(p) (k)(p/ + k/)Γμ v(¯p) − u¯(p)Γμ (¯p/ + k/) (k) v(¯p) λa . 2p · k 2¯p · k ij

I have used the generic symbol Γμ to describe the interaction vertex with the photon to stress the fact that the following manipulations are independent of the speciﬁc form of Γμ. In particular, Γμ can represent an arbitrarily complicated vertex form factor. Neglecting the factors of k/ in the numerators (since k p, p¯, by deﬁnition of soft) and using the Dirac equations, we get p · p ¯ A = gλa − A . (5.13) soft ij p · k p¯ · k Born

We then conclude that soft-gluon emission factorizes into the product of an emission factor and the Born-level amplitude. From this exercise, one can extract general Feynman rules for soft-gluon emission: The ﬁnal-state evolution of quarks and gluons 123 a

p j p i (5.14)

An analogous exercise leads to the g → gg soft-emission rules: a c b (5.15)

Consider now the ‘decay’ of a virtual gluon into a quark pair. One more diagram should be added to those considered in the case of the electroweak decay. The fact that the quark pair is no longer in a colour-singlet state makes things a bit more interesting: pj pj pj k a

k a ka pi pi pi → Q p p ¯ k=0 igf abc λc + g(λb λa) − g (λaλb) A ij Qk ij pk ij pk Born Q p ¯ p Q = g(λa λb) − + g(λb λa) − . (5.16) ij Qk pk ij pk Qk The two factors correspond to the two possible ways colour can ﬂow in this process: i b i a a

b j j (5.17) The basis for this representation of the colour ﬂow is the following diagram, which makes explicit the relation between the colours of the quark, antiquark, and gluon entering a QCD vertex: j a i

(5.18) 124 Introduction to the theory of LHC collisions

We can therefore represent the gluon as a double line, with one line carrying the colour inherited from the quark and the other carrying the anticolour inherited from the antiquark. In the ﬁrst diagram in (5.17), the antiquark (colour label j) is colour- connected to the soft gluon (colour label b), and the quark (colour label i) is connected to the decaying gluon (colour label a). In the second case, the order is reversed. The two emission factors correspond to the emission of the soft gluon from the antiquark and from the quark line, respectively. When squaring the total amplitude and summing over initial and ﬁnal-state colours, the interference between the two pieces is suppressed by 1/N 2 relative to the individual squares:

N 2 − 1 |(λaλb) |2 = tr λaλbλbλa = C = O(N 3), (5.19) ij 2 F a,b,i,j a,b N 2 − 1 C (λaλb) [(λbλa) ]∗ = tr(λaλbλaλb)= C − A = O(N). ij ij 2 F 2 a,b,i,j a,b 1 − 2N (5.20)

As a result, the emission of a soft gluon can be described, to leading order in 1/N 2, as the incoherent sum of the emission from the two colour currents. The ability to separate these emissions as incoherent sums is the basic fact that allows a sequential, Markovian description of the parton-shower evolution as implemented in numerical simulations of the hadronic ﬁnal state of hard collisions. The neglect of subleading 1/N 2 contributions is therefore an intrinsic approximation of all such approaches.

5.3.2 Angular ordering for soft-gluon emission The results presented above have important consequences for the perturbative evolution of the quarks. A key property of soft-gluon emission is the so-called angular ordering (for an overview of colour coherence and its relation to angular ordering, see [9, 23]). This phenomenon consists in the continuous reduction of the opening angle at which successive soft gluons are emitted by the evolving quark. As a result, this radiation is confined within smaller and smaller cones around the quark direction, and the final state will look like a collimated jet of partons. In addition, the structure of the colour flow during the jet evolution forces the qq¯ pairs that are in a colour-singlet state to be close in phase space, thereby achieving the pre-confinement of colour-singlet clusters alluded to at the beginning of this section.

(a) (b)

θ k μ2 α I

Fig. 5.13 Radiation off qq¯ pair produced by an off-shell photon. The final-state evolution of quarks and gluons 125

A simple derivation of angular ordering, which more directly exhibits its physical origin, can be obtained as follows. Consider Fig. 5.13(a), which shows a Feynman diagram for the emission of a gluon from a quark line. The quark momentum is denoted by l and the gluon momentum by k, the opening angle between the quark and antiquark is denoted by θ, and the angle between the nearest quark and the emitted gluon by α. We shall work in the double-log enhanced soft k0 << l0 and collinear α<<1 region. The internal quark propagator p =(l + k) is oﬀ-shell, setting the timescale for the gluon emission:

1 l0 1 Δt = → Δt . (5.21) ΔE (k + l)2 k0α2

In order to resolve the quarks, the transverse wavelength of the gluon λ⊥ =1/E⊥ must be smaller than the separation between the quarks b(t) θ Δt, giving the constraint 1/(αk0) <θΔt. Using the results of (5.21) for Δt, we arrive at the angular ordering constraint α<θ. Gluon emissions at an angle smaller than θ can resolve the two individual colour quarks and are allowed; emissions at greater angles do not see the colour charge and are therefore suppressed. In processes involving more partons, the angle θ is defined not by the nearest parton, but by the colour-connected parton (e.g. the parton that forms a colour singlet with the emitting parton). Figure 5.13(b) shows the colour connections for the qq¯ event after the gluon has been emitted. Colour lines begin on quarks and end on antiquarks. Because gluons are colour octets, they contain the beginning of one line and the end of another, as we showed in (5.17). If one repeats now the exercise for emission of one additional gluon, one will find the same angular constraint, but this time applied to the colour lines defined by the previously established antenna. As shown in Section 5.3.1, the qqg¯ state can be decomposed at the leading order in 1/N into two independent emitters, one given by the colour line flowing from the gluon to the quark, the other by the colour line flowing from the antiquark to the gluon. So the emission of the additional gluon will be constrained to take place either within the cone formed by the quark and the gluon or within the cone formed by the gluon and the antiquark. Either way, the emission angle will be smaller than the angle of the first gluon emission. This leads to the concept of angular ordering, with successive emission of soft gluons taking place within cones which get smaller and smaller, as in Fig. 5.14 The fact that colour always flows directly from the emitting parton to the emitted one, the collimation of the jet, and the softening of the radiation emitted at later stages ensure that partons forming a colour-singlet cluster are close in phase space. As a result, hadronization (the nonperturbative process that will bind together colour-singlet parton pairs) takes place locally inside the jet and is not a long-distance phenomenon connecting partons far away in the evolution tree: only pairs of nearby partons are involved. In particular, there is no direct link between the precise nature of the hard process and the hadronization. These two phases are totally decoupled and, as in the case of the partonic densities, one can infer that hadronization factorizes from the hard process and can be described in a universal (i.e. hard-process-independent) fashion. 126 Introduction to the theory of LHC collisions

Fig. 5.14 Collimation of soft gluon emission during the jet evolution.

The inclusive properties of jets (e.g. the particle multiplicity, jet mass, and jet broad-√ ening) are independent of the hadronization model, up to corrections of order (Λ/ s)n (for some integer power n, which depends on the observable), with Λ 1 GeV. The final picture, in the case of a deep irelastic scattering (DIS) event, therefore appears as in Fig. 5.15. After being deflected by the photon, the struck quark emits the first gluon, which takes away the quark colour and passes on its own anticolour to the escaping quark. This gluon is therefore colour-connected with the last gluon emitted before the hard interaction. As the final-state quark continues its evolution, more and more gluons are emitted, each time leaving their colour behind and transmitting their anticolour to the emerging quark. Angular ordering forces all these gluons to be close in phase space, until the evolution is stopped once the virtuality of the quark becomes of the order of the strong-interaction scale. The colour of the quark is left behind, and when hadronization takes over it is only the nearby colour-connected gluons that are transformed, with a phenomenological model, in hadrons. This mechanism for the transfer of colour across subsequent gluon emissions is similar to what happens when one places a charge near the surface of a dielectric medium. This will become polarized, and a charge will appear on the medium opposite end. The appearance of the charge is the result of a sequence of local charge shifts, whereby neighbouring atoms become polarized, as in Fig. 5.16. Notice that the transfer of colour between the final state jet and the jet associated to the initial state leads to the existence of one colour-singlet

Fig. 5.15 The colour ﬂow diagram for a DIS event. The ﬁnal-state evolution of quarks and gluons 127

e– A+ e– A+ e– A+ e– A+ e– A+

Fig. 5.16 Charge transfer in a dieletric medium, via a sequence of local polarizations. cluster containing partons from both jets. Therefore, the particles arising from this cluster cannot be associated with either jet: they will be hadrons whose source is a combination of two of the hard partons in the event. This feature will be present in any hard process: it is the unavoidable consequence of the neutralization of the colour of the partons involved in the hard scattering.

5.3.3 Hadronization The application of perturbation theory to the evolution of a jet, with the sequential emission of partons, governed by QCD splitting probabilities and angular ordering to enforce quantum mechanical quantum coherence, will stop once the scale of the emissions reaches values in the range of 1 GeV. This is called the infrared cutoff.The are two reasons why we need to stop the emission of gluons at this scale. To start with, we cannot control with perturbation theory the domain below this scale, where the strong coupling constant αs becomes very large. Furthermore, we know that the number of physical particles that can be produced inside a jet must be finite, since the lightest object we can produce is a pion, and energy conservation sets a limit to how many pions can be created. This is different from what happens in a QED cascade, where the evolution of an accelerated charge can lead to the emission of an arbitrary number of photons. This is possible because the photon is massless, and can have arbitrarily small energy. The gluons of a QCD cascade, on the contrary, must have enough energy to create pions. When the perturbative evolution of the jet terminates, we are left with some number of gluons. As shown in Section 5.3.2 and displayed in Fig. 5.15, these gluons are pairwise colour-connected. As two colour-connected gluons travel away from each other, a constant force pulls them together. Phenomenological models (see [6] for a more complete review) are then used to describe how this force determines the evolution of the system from this point on. What I shall describe here is the so-called cluster model [24], implemented in the HERWIG Monte Carlo generator [25, 26], but the main qualitative features are shared by other alternatives, such as the Lund string approach [27], implemented in the PYTHIA generator [28, 29]. Most of the hadrons emerging from the evolution of a jet are known to be made of quarks; glueballs, i.e. hadrons made of bound gluons, are expected to exist, but their production is greatly suppressed compared with that of quark-made particles. For this reason, the first step in the description of hadronization is to assume that the force among gluons will rip them apart into a qq¯ pair, and that these quarks will act as seeds for the hadron production. The breakup into quarks is not parametrized using the 128 Introduction to the theory of LHC collisions

DGLAP g → qq¯ splitting function, since we are dealing here with a nonperturbative transition. One therefore typically employs a pure phase-space ‘decay’ of the gluon into the qq¯ pair, introducing as phenomenological parameters the relative probabilities of selecting the various active flavours (up, down, strange, etc.). The quark qi from one gluon (i representing the flavour) then forms a colour-singlet pair with the antiquark q¯j emerging from the breakup of the neighbouring gluon. This colour-singlet qiq¯j pair cannot, however, directly form a hadron, since in general the quarks will still be moving apart, and the invariant mass of the pair will not coincide with the mass of an existing physical state. As they separate subject to a constant force, however, their kinetic energy turns into a linearly rising potential energy. The potential energy accumulated in the system will be able to convert into a new quark–antiquark pair qkq¯k once its value exceeds the relevant mass threshold. We are now left with two colour-singlet pairs: qiq¯k and qkq¯j. When converting the potential energy into mass and kinetic energy of the newly produced pair, one can use the freedom in selecting the spatial kinematics of qk andq ¯k such that both qiq¯k and qkq¯j invariant masses coincide with some resonance with the proper flavour. The residual energy of the system is then assumed to be entirely kinetic, and the two resonances fly away free. Once again, one can associate phenomenological parameters with the probabilities of selecting flavours k of a given type. Since the pair of flavour indices ik does not specify uniquely a hadron (e.g. a ud¯ system could by a π+ or a ρ+, as well as many other objects), the model has a further set of rules and/or parameters to select the precise flavour type. For example, a phenomenologically successful description of the π/ρ ratio is obtained by simply assuming a production rate proportional to the number of spin

0.9 Q = 35 GeV 0.8 Q = 91.2 GeV 0.7 Q = 189 GeV Q = 500 GeV 0.6 Q = 1000 GeV 0.5

0.4

0.3

0.2

0.1

1 10 M (GeV)

Fig. 5.17 Invariant mass distribution of clusters of colour-singlet quarks after nonperturbative gluon splitting, as obtained with the HERWIG generator [30]. The spectra for final states corresponding to different centre-of-mass energies are normalized to the same area, displaying the energy independence of the shapes. Applications 129 states (one for the scalar pion, three for the vector rho) and to a Boltzmann factor exp(−M/T), where M is the resonance mass and T is a universal parameter, to be fitted from data. Furthermore, one can introduce the possibility of converting the potential energy into a diquark–antidiquark pair, namely (qkq)(¯qkq¯). The resulting hadrons qiqkq andq ¯jq¯kq¯ will be a baryon–antibaryon pair. The measurement of hadron multiplicities from Z0 decays is used to tune the few phenomenological parameters of the model, and these parameters can be used to describe hadronization at different energies and in different high-energy hadron- production processes. The internal consistency of this assumption is supported by Fig. 5.17 [30], which shows the invariant mass distribution of clusters of colour-singlet quarks, after the nonperturbative gluon splitting, for e+e− collisions at different centre- of-mass energies. All curves are normalized to 1, and they all overlap very accurately. This confirms the validity of the implementation of factorization in the Monte Carlo simulation: higher initial energies provide more room for the perturbative evolution, leading to more splitting and more emitted radiation; but the structure and distribution of colour-singlet clusters at the end of evolution are independent of the initial energy, and the same model of hadronization can be applied.

5.4 Applications In hadronic collisions, all phenomena are QCD-related. The dynamics is more complex than in e+e− or DIS, since both beam and target have a nontrivial partonic structure. As a result, calculations (and experimental analyses) are more complicated. Perturbative corrections to the Born-level amplitudes (leading order, LO) require the calculation of a large number of loop and high-order tree-level diagrams. For example, the first correction to the four LO gg → gg diagrams for the jet cross section in the purely gluonic channel requires the evaluation of the one-loop insertions in the four LO diagrams, plus the calculation of the 25 LO diagrams for the gg → ggg process. Next-to-leading-order (NLO) calculations like this are nevertheless available today for most processes with up to two or three final-state partons, and new techniques have emerged recently that are pushing the frontier of NLO results even further (for a recent review, see [10]). Next-to-next-to-leading order (NNLO) results, which require the evaluation of two-loop corrections as well, have so far been completed only for processes with only one final-state particle at the Born level, such as production of massive gauge bosons (W, Z) or of a Higgs boson. For these reasons, the accuracy of QCD calculations in hadronic collisions is typically less than in the case of e+e− or DIS processes. Nevertheless, pp¯ or pp collider physics is primarily discovery physics, rather than precision physics (there are exceptions, such as measurement of the W mass and of the properties of b-hadrons, but these are not QCD-related measurements). As such, knowledge of QCD is essential both for the estimate of the expected signals and for the evaluation of the backgrounds. Tests of QCD in pp¯ collisions con- firm our understanding of perturbation theory, or, when they fail, point to areas where our approximations need to be improved (see, for example, the theoretical advances prompted by the measurements of ψ production at CDF). 130 Introduction to the theory of LHC collisions

In this section, I shall brieﬂy review some applications to the most commonly studied QCD processes in hadronic collisions: the production of gauge bosons and of jets. More details can be found in [6, 7].

5.4.1 Drell–Yan processes While the Z boson has recently been studied with great precision by LEP experiments, it was actually discovered, together with the W boson, by the CERN experiments UA1 and UA2 in pp¯ collisions. W physics was studied in great detail at LEP2, but the best direct measurements of its mass by a single group still belong to pp¯ experiments (CDF and D0 at the Tevatron). Until a new, future, e+e− linear collider is built, the monopoly of W studies will remain in the hands of hadron colliders, with the Tevatron, and soon with the start of the LHC experiments. Precision measurements of W production in hadronic collisions are important for several reasons: • This is the only process in hadronic collisions that is known to NNLO accuracy. • The rapidity distribution of the charged leptons from W decays is sensitive to the ratio of the up- and down-quark densities, and can contribute to our understanding of the structure of the proton. • Deviations from the expected production rates of highly virtual W ’s (pp¯ → W ∗ → eν) are a possible signal of the existence of new W bosons, and therefore of new gauge interactions. The tail of the invariant mass distribution of the W , furthermore, provides today’s most sensitive determination of the W width. The production rate for the W boson is given by the factorization formula dσ(pp¯ → W + X)= dx1 dx2 fi(x1,Q) fj(x2.Q) dσˆ(ij → W ). (5.22) i,j

The partonic cross sectionσ ˆ(ij → W ) can be easily calculated, giving the following result [5, 6]: √ 2 σˆ(q q¯ → W )=π |V |2 G M 2 δ(ˆs − M 2 )=A M 2 δ(ˆs − M 2 ), (5.23) i j 3 ij F W W ij W W wheres ˆ is the partonic centre-of-mass energy squared and Vij is an element of the Cabibbo–Kobayashi–Maskawa matrix. The delta function comes from the 2 → 1 phase space, which forces the centre-of-mass energy of the initial state to coincide with the W mass. It is useful to introduce the two variables sˆ τ = ≡ x1x2, (5.24) Shad 1 E + pz 1 x y = log W W ≡ log 1 , (5.25) − z 2 EW pW 2 x2 where Shad is the hadronic centre-of-mass energy squared. The variable y is called the rapidity. For slowly moving objects it reduces to the standard velocity, but, in Applications 131 contrast to the velocity, it transforms additively even at high energies under Lorentz boosts along the direction of motion. Written in terms of τ and y, the integration measure over the initial-state parton momenta becomes: dx1 dx2 = dτ dy. Using this expression and (5.23) in (5.22), we obtain the following result for the LO total W production cross section: πA 1 dx τ πA σ = ij τ f (x) f ≡ ij τL (τ), (5.26) DY M 2 x i j x M 2 ij i,j W τ i,j W ¯ where the function Lij (τ) is usually called the partonic luminosity. In the case of ud collisions, the overall factor in front of this expression has a value of approximately 6.5 nb. It is interesting to study the partonic luminosity as a function of the hadronic centre-of-mass energy. This can be done by taking a simple approximation for the parton densities. Following the indications of the figures presented in Section 5.3, we 1+δ shall assume that fi(x) ∼ 1/x , with δ<1. Then 1 dx 1 x 1+δ 1 1 dx 1 1 L(τ)= 1+δ = 1+δ = 1+δ log (5.27) τ x x τ τ τ x τ τ and δ ∼ −δ 1 Shad Shad σW τ log = 2 log 2 . (5.28) τ MW MW The Drell–Yan cross section therefore grows at least logarithmically with the hadronic centre-of-mass energy. This is to be compared with the behaviour of the Z production cross section in e+e− collisions, which is steeply diminishing for values of s well above the production threshold. The reason for the different behaviour in hadronic collisions is that while the energy of the hadronic initial state grows, it will always be possible to find partons inside the hadrons with the appropriate energy to produce the W directly on shell. The number of partons available for the production of a W increases with the increase in hadronic energy, since the larger the hadron energy, the smaller will be the value of hadron momentum fraction x necessary to produce the W . The increasing number of partons available at smaller and smaller values of x then causes the growth in the total W production cross section. A comparison between the best available predictions for the production rates of W and Z bosons in hadronic collisions and the LHC experimental data[31] is shown in Fig. 5.18. The experimental uncertainty is already smaller than the spread of the predictions based on different PDF sets, and will therefore add important constraints to improving the knowledge of PDFs. Since the LHC is a pp collider, the initial state contains more up quarks than down quarks, and therefore the production rates and the distributions of W + bosons will be different from those of W − bosons. For the total rates, this is clearly visible in Fig. 5.18. The rapidity asymmetry between positive and negative leptons produced in W ± decays is shown in Fig. 5.19, where the ATLAS data are compared against the predictions of several PDF sets. Once again, the precision of the data strongly constrains the PDFs. 132 Introduction to the theory of LHC collisions

ATLAS ATLAS

6.5 11 ) (nb) ) (nb) v ν l + l 6 10

+ W

W –1 –1 ∫L dt = 33–36 pb ∫L dt = 33–36 pb 5.5 x BR( x BR( ± + s 9 s Data 2010 (√ = 7 TeV) total uncertainty W Data 2010 (√ = 7 TeV) total uncertainty W tot tot σ σ MSTW08 sta sys acc MSTW08 sta sys acc uncertainty uncertainty 5 HERAPDF1.5 HERAPDF1.5 ABKM09 68.3% CL ellipse area ABKM09 68.3% CL ellipse area JR09 8 JR09

3.5 4 4.5 0.8 0.9 1 tot – – tot + – σ – x BR(W l ν) (nb) σ x BR(Z/γ* l l ) (nb) W Z

Fig. 5.18 Comparison of W and Z cross sections measured by the ATLAS experiment at the LHC [31] with the predictions of several PDF sets.

v ℓ v W± l±ν √s p ℓ E M W± l ±ν √s p E M at LHC ( = 7 TeV) with T > 20 GeV, /T > 25 GeV, T > 40 GeV at LHC ( = 7 TeV) with T > 20 GeV, / > 25 GeV, T > 40 GeV 0.3 0.3

–1 –1 0.28 ATLAS(ℓ= µ), L = 31 pb 0.28 ATLAS(ℓ= µ), L = 31 pb 0.26 0.26 0.24 0.24 0.22 0.22 PDFs (with NLO K-factors) PDFs (with NLO K-factors) 0.2 0.2 MSTW08 (90% CL) MSTW08LO 0.18 MSTW08 (90% CL) 0.18 MRSTLO* 0.16 MRSTLO* 0.16 MRST04NLO

Lepton charge asymmetry MRST01LO Lepton charge asymmetry 0.14 CTEQ6.6 0.14 CT09MC2 CTEQ6.6 HERAPDF1.0 0.12 0.12 CTEQ6L1 0.1 0.1 0 0.5 11.5 2 2.5 3 0 0.5 11.5 2 2.5 3 η η Lepton pseudorapidity l ℓl Lepton pseudorapidity l ℓlz

Fig. 5.19 The lepton charge asymmetry at the LHC compared with the predictions of various PDF sets [22].

5.4.2 Jet production Jet production is the hard process with the largest rate in hadronic collisions. The measurement of highest-energy jets allows one to probe the shortest distances ever reached. The leading mechanisms for jet production are shown in Fig. 5.20. The two-jet inclusive cross section can be obtained from the formula (H1) (H2) dσˆij→k+l dσ = dx1 dx2 fi (x1,μ) fj (x2,μ) dΦ2, (5.29) dΦ2 ijkl which has to be expressed in terms of the rapidity and transverse momentum of the quarks (or jets), in order to make contact with physical reality. The two-particle phase space is given by Applications 133

Fig. 5.20 Representative diagrams for the production of jet pairs in hadronic collisions.

d3k dΦ = 2πδ((p + p − k)2) , (5.30) 2 2k0(2π)3 1 2 and, in the centre-of-mass frame of the colliding partons, we get

1 dΦ = d2k dy 2 δ(ˆs − 4(k0)2 ), (5.31) 2 2(2π)2 T where kT is the transverse momentum of the ﬁnal-state partons. Here y is the rapidity of the produced parton in the parton centre-of-mass frame. It is given by

y − y y = 1 2 , (5.32) 2 where y1 and y2 are the rapidities of the produced partons in the laboratory frame (in fact, in any frame). We also introduce

y1 + y2 1 x1 sˆ y0 = = log ,τ= = x1 x2 . (5.33) 2 2 x2 Shad We have

dx1 dx2 = dy0 dτ . (5.34)

We obtain 1 (H1) (H2) dσˆij→k+l 1 2 dσ = dy0 fi (x1,μ) fj (x2,μ) 2 2 dy d kT , (5.35) Shad dΦ2 2(2π) ijkl which can also be written as dσ 1 (H1) (H2) dσˆij→k+l 2 = 2 fi (x1,μ) fj (x2,μ) . (5.36) dy1 dy2 d kT Shad 2(2π) dΦ2 ijkl 134 Introduction to the theory of LHC collisions

The variables x1 and x2 can be obtained from y1, y2,andkT by the equations y + y y = 1 2 , (5.37) 0 2 y − y y = 1 2 , (5.38) 2 2kT xT = √ , (5.39) Shad y0 x1 = xT e cosh y, (5.40) −y0 x2 = xT e cosh y. (5.41) For the partonic variables, we needs ˆ and the scattering angle θ in the parton centre- of-mass frame, since sˆ sˆ t = − (1 − cos θ) ,u= − (1 + cos θ) . (5.42) 2 2 Neglecting the parton masses, we can show that the rapidity can also be written as θ y = − log tan ≡ η, (5.43) 2 with η usually being referred to as pseudorapidity. The leading-order Born cross sections for parton–parton scattering are reported in Table 5.1.

2 Table 5.1 Cross sections for light-parton scattering. The notation is p1 p2 → kl, sˆ =(p1 +p2) , 2 2 tˆ=(p1 − k) , uˆ =(p1 − l)

dσˆ Process dΦ2 2 2 qq → qq 1 4 sˆ +û 2ˆs 9 tˆ2 1 1 4 sˆ2+û2 sˆ2+tˆ2 8 sˆ2 qq → qq + 2 − 2 2ˆs 9 tˆ2 uˆ 27 uˆtˆ → 1 4 tˆ2+û2 qq¯ q q¯ 2ˆs 9 sˆ2 1 4 sˆ2+û2 tˆ2+û2 8 uˆ2 qq¯ → qq¯ + 2 − 2ˆs 9 tˆ2 sˆ 27 sˆtˆ 1 1 32 tˆ2+û2 8 tˆ2+û2 qq¯ → gg − 2 2 2ˆs 27 tûˆ 3 sˆ 1 1 tˆ2+û2 3 tˆ2+û2 gg → qq¯ − 2 2ˆs 6 tûˆ 8 sˆ 2 2 2 2 gq → gq 1 − 4 sˆ +û + uˆ +ˆs 2ˆs 9 sûˆ tˆ2 1 1 9 tûˆ sûˆ sˆtˆ gg → gg 3 − 2 − − 2 2 2ˆs 2 sˆ tˆ2 uˆ Applications 135

5.4.3 Comparison of theory and experimental data Predictions for jet production at colliders are available today at NLO in QCD (see the review in [7]). One of the preferred observables is the inclusive ET spectrum. The agreement between data and theory at the LHC is excellent, as shown in Fig. 5.21. An accurate comparison of data and theory, should it exhibit discrepancies at the largest values of ET , could provide evidence for new phenomena, such as the existence of a quark substructure [33]. But how do we know that these discrepancies are not due to poorly known quark or gluon densities at large x? In principle, one could incorporate the jet data into a global ﬁt to the partonic PDFs, and verify whether it is possible to modify them so as to maintain agreement with the other data, and at the same time to also ﬁt satisfactorily the jet data themselves. On the other hand, doing this would prevent us from using the jet spectrum as a probe of new physics. In other words, we might be hiding away a possible signal of new physics by ascribing it to the PDFs. Is it possible to have a complementary determination of the PDF at high x that could constrain the possible PDF systematics of the jet cross section and at the same time leave the high-ET tail as an independent and usable observable?

CMS L =34 pb–1 s =7 TeV

1011 IyI< 0.5 (×3125) 1010 0.5 ≤ IyI < 1.5 (×625) 1 ≤ IyI < 1.5 (×125) 109 1.5 ≤ IyI < 2 (×25) 108 2 ≤ IyI < 2.5 (×5) ) 7 –1 10 2.5 ≤ IyI < 3 106

(pb GeV 105 dy T 104 dp / σ

2 3

d 10

102 NLO NP 10 (PDF4LHC)

1 Exp. uncertainty 10–1 Anti-kT R = 0.5

20 30 100 200 1000

PT (Gev)

Fig. 5.21 Data versus theory comparison for the inclusive jet spectrum measured by the CMS experiment at the LHC [32]. 136 Introduction to the theory of LHC collisions

This is indeed possible, by fully exploiting the kinematics of dijet production and the wide rapidity coverage of the collider detectors. One could in fact consider final states where the dijet system is highly boosted in the forward or backward region. For example, one could consider cases where x1 → 1andx2 1. In this case, the 2 invariant mass of the dijet system would be small (since Mjj = x1x2S S), and we know from lower-energy measurements that at this scale jets must behave like pointlike particles, following exactly the QCD-predicted rate. These final states are characterized by having jets at large positive rapidity. One can therefore perform a measurement with forward jets, and use these data to fit the x1 → 1 behaviour of the quark and gluon PDFs without the risk of washing away possible new-physics effects. At that point, the large-x PDFs thus constrained can be safely applied to the kinematical configurations where both x1 and x2 are large, namely the highest-ET final states, and, if any residual discrepancy between data and theory is observed,

104 W→ℓν + jets Data 2010, √s = TeV ALPGEN SHERPA 103 PYTHIA jets) (pb) BLACKHAT-SHERPA jet N

≥ ATL AS + W ( σ 102

∫L dt = 36 pb–1 10 k R anti- T jets, = 0.4 pjet y jet T , > 30 GeV, I I<4.4

1 Theory/data

0 ≥0 ≥1 ≥2 ≥3 ≥4 N Inclusive jet multiplicity jet

Fig. 5.22 Production rates of W plus multijets, measured by ATLAS at the LHC [35], compared against various theoretical predictions. Outlook and conclusions 137 infer the possible presence of new physics. The agreement seen in Fig. 5.21 in the data at large rapidity suggests that the PDF parametrizations at large x are reliable. A closer study of correlations among the two leading jets in the events can then lead to strong constraints on the possible composite structure of quarks, as seen for example in the ATLAS study of [34]. Of great phenomenological interest, in particular in the context of the study of backgrounds to the production of top quarks, of the Higgs boson, and of several signals of new physics, is the associated production of W or Z bosons and jets. LHC data and theoretical predictions are so far in very good agreement, as shown in Fig. 5.22. Similar excellent agreement between data and QCD has been observed for a large number of observables, ranging from the inner structure of jets, to the spectra of high-pT photons, to the production rates of bottom and top quarks.

5.5 Outlook and conclusions In spite of the intrinsic complexity of the proton, the factorization framework allows a complete, accurate and successful description of the rates and properties of hard processes in hadronic collisions. One of the key ingredients of this formulation is the universality of the phenomenological parametrizations of the various nonperturbative components of the calculations (the partonic densities and the hadronization phase). This universality makes it possible to extract the nonperturbative information from the comparison of data and theory in a set of benchmark measurements, and to apply this knowledge to different observables or to different experimental environments. All the differences will be then accounted for by the purely perturbative part of the evolution. Many years of experience at the Tevatron collider, at HERA, and at LEP have led to an immense improvement of our understanding, and to the validation, of this framework, and put us today in a solid position to reliably anticipate in quantitative terms the features of LHC final states. LEP, in addition to testing with great accuracy the electroweak interaction sector, has verified at the percent level the predictions of perturbative QCD, from the running of the strong coupling constant, to the description of the perturbative evolution of single quarks and gluons, down to the nonperturbative boundary where strong interactions take over and cause the confinement of partons into hadrons. The description of this transition, relying on the factorization theorem that allows one to consistently separate the perturbative and nonperturbative phases, has been validated by the study of LEP data at various energies, and their comparison with data from lower-energy e+e− colliders, allowing the phenomenological parameters introduced to model hadronization to be determined. The factorization theorem supports the use of these parameters for the description of the hadronization transition in other experimental environments. HERA has made it possible to probe with great accuracy the short-distance properties of the proton, with the measurement of its partonic content over a broad range of momentum frac- tions x. These inputs, from LEP and from HERA, beautifully merge into the tools that have been developed to describe proton–antiproton collisions at the Tevatron, where the agreement between theoretical predictions and data confirms that the key assumptions of the overall approach are robust. 138 Introduction to the theory of LHC collisions

The accuracy of the perturbative input relies on complete higher-order calculations, as well as possible resummations of leading contributions to all orders of perturbation theory. Today’s theoretical precision varies from the few-percent level of W and Z inclusive cross sections, which are known to NNLO, to the level of 10% for several processes known to NLO (inclusive jet cross sections, top quark production, and production of pairs of electroweak gauge bosons), up to very crude estimates for the most complex, multijet ﬁnal states, where uncertainties at the available leading order can be as large as factors of 2 or more. At the LHC, the statistics and systematics for measurements such as the total W , Z,ortt¯ cross sections are expected to be reduced to the few-percent level, comparable with the theoretical accuracy. These measurements, and others, will allow us to stress-test our modelling of basic phenomena at the LHC and to validate and improve their reliability, laying the foundations for a successful exploration of the new landscapes that will be uncovered at the energy frontier.

References [1] R. P. Feynman, Photon–Hadron Interactions (Benjamin, New York, 1972). [2] D.J. Gross, arXiv:hep-th/9809060; G. ’t Hooft, arXiv:hep-th/9808154. [3] T. Muta, Foundations of QCD (World Scientiﬁc, Singapore, 1998). [4] M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley, Reading, MA, 1995). [5] V. Barger and R. J. N. Phillips, Collider Physics (Addison-Wesley, Redwood City, CA, 1997). [6] R. K. Ellis, W. J. Stirling and B. R. Webber, QCD and Collider Physics (Cambridge University Press, Cambridge, 1996). [7] J. M. Campbell, J. W. Huston, and W. J. Stirling, Rep. Prog. Phys. 70 (2007) 89 [arXiv:hep-ph/0611148]. [8] R. Field, Applications of Perturbative QCD (Addison-Wesley, Redwood City, CA, 1989). [9] Yu. L. Dokshitzer, V. A. Khoze, A. H. Mueller, and S. I. Troyan, Basics of Perturbative QCD (Editions Fronti`eres, Gif-sur-Yvette, 1991). [10] L. J. Dixon, arXiv:0712.3064 [hep-ph]. [11] M. A. Dobbs et al., arXiv:hep-ph/0403045. [12] V. N. Gribov and L. N. Lipatov, Sov. J. Nucl. Phys. 15 (1972) 438 [Yad. Fiz. 15 (1972) 781]. [13] G. Altarelli and G. Parisi, Nucl. Phys. B126 (1977) 298. [14] Y. L. Dokshitzer, Sov. Phys. JETP 46 (1977) 641 [Zh. Eksp. Teor. Fiz. 73 (1977) 1216]. [15] M. Dittmar et al., arXiv:0901.2504 [hep-ph]. [16] H. Jung et al., arXiv:0903.3861 [hep-ph]. [17] P. M. Nadolsky et al., Phys. Rev. D78 (2008) 013004 [arXiv:0802.0007 [hep-ph]]. [18] A. D. Martin, W. J. Stirling, R. S. Thorne, and G. Watt, arXiv:0901.0002 [hep-ph]. [19] S. Alekhin, K. Melnikov, and F. Petriello, Phys. Rev. D74 (2006) 054033 [arXiv:hep-ph/0606237]. References 139

[20] R. D. Ball et al. [NNPDF Collaboration], Nucl. Phys. B809 (2009) 1 [Erratum B816 (2009) 293] [arXiv:0808.1231 [hep-ph]]. [21] B. C. Reisert [ZEUS Collaboration], arXiv:0809.4946 [hep-ex]. [22] G. Watt, JHEP 1109 (2011) 069 [arXiv:1106.5788 [hep-ph]]. [23] Y. L. Dokshitzer, V. A. Khoze, S. I. Troian, and A. H. Mueller, Rev. Mod. Phys. 60, 373 (1988). [24] B. R. Webber, Nucl. Phys. B238 (1984) 492. [25] G. Marchesini and B. R. Webber, Nucl. Phys. B310 (1988) 461. [26] G. Corcella et al., JHEP 0101 (2001) 010 [arXiv:hep-ph/0011363]. [27] B. Andersson, G. Gustafson, G. Ingelman, and T. Sjostrand, Phys. Rep. 97 (1983) 31. [28] T. Sjostrand, P. Eden, C. Friberg, L. Lonnblad, G. Miu, S. Mrenna, and E. Norrbin, Comput. Phys. Commun. 135 (2001) 238 [arXiv:hep-ph/0010017]. [29] T. Sjostrand, S. Mrenna, and P. Skands, JHEP 0605 (2006) 026 [arXiv:hep- ph/0603175]. [30] S. Gieseke, A. Ribon, M. H. Seymour, P. Stephens, and B. Webber, JHEP 0402 (2004) 005 [arXiv:hep-ph/0311208]. [31] G. Aad et al. [ATLAS Collaboration], Phys. Rev. D85 (2012) 072004 [arXiv:1109.5141 [hep-ex]]. [32] S. Chatrchyan et al. [CMS Collaboration], Phys. Rev. Lett. 107 (2011) 132001 [arXiv:1106.0208 [hep-ex]]. [33] E. Eichten, K. D. Lane, and M. E. Peskin, Phys. Rev. Lett. 50 (1983) 811. [34] G. Aad et al. [ATLAS Collaboration], New J. Phys. 13 (2011) 053044 [arXiv:1103.3864 [hep-ex]]. [35] G. Aad et al. [ATLAS Collaboration], Phys. Rev. D85 (2012) 092002 [arXiv:1201.1276 [hep-ex]].

6 An introduction to the gauge/gravity duality

Juan M. Maldacena

Institute for Advanced Study, Princeton, New Jersey, USA

6 An introduction to the gauge/gravity duality 141 Juan M. MALDACENA

6.1 Introduction to the gauge/gravity duality 143 6.2 Scalar ﬁeld in AdS 148 5 6.3 The N = 4 super Yang–Mills/AdS 5× S example 152 6.4 The spectrum of states or operators 157 6.5 The radial direction 159 Acknowledgments 161 References 161 Introduction to the gauge/gravity duality 143

6.1 Introduction to the gauge/gravity duality In these lectures, we explain the gauge/gravity duality [1–3]. The gauge/gravity duality is an equality between two theories: on one side, we have a quantum field theory in d spacetime dimensions. On the other side, we have a gravity theory on a (d + 1)- dimensional spacetime that has an asymptotic boundary that is d-dimensional. It is also sometimes called AdS/CFT, because the simplest examples involve anti-de Sitter spaces and conformal field theories. It is often called gauge/string duality. This is because the gravity theories are string theories and the quantum field theories are gauge theories. It is also referred to as “holography” because one is describing a (d+1)- dimensional gravity theory in terms of a lower-dimensional system, in a way that is reminiscent of an optical hologram, which stores a three-dimensional image on a two- dimensional photographic plate. It is called a “conjecture,” but by now there is a lot of evidence that it is correct. In addition, there are some derivations based on physical arguments.∗ The simplest example involves an anti-de Sitter spacetime. So, let us start by describing this spacetime in some detail. Anti-de Sitter spacetime AdS is the simplest solution of Einstein’s equations with a negative cosmological constant. It is the Lorentzian analog of hyperbolic space, which was historically the first example of a non-Euclidean geometry. In a similar way, AdS/CFT gives the simplest example of a quantum mechanical spacetime. The metric in AdS can be written as 2 2 2 2 2 dr 2 2 ds = L −(r +1)dt + + r dΩ − , (6.1) AdSd+1 r2 +1 d 1 where the last term is the metric of a unit sphere, Sd−1. L is the radius of curvature. Note that near r =0,AdS looks like flat space. As we go to larger values of r,wesee that g00 and the metric on the sphere grow. The growth of g00 canbeviewedasarising gravitational potential.√ In fact, a slowly moving massive particle feels a gravitational potential V ∼ −g00. If a particle is set at rest at a large value of r, it will execute an oscillatory motion in the r direction, very much like a particle in a harmonic oscillator potential. This gravitational potential confines particles around the origin. A massive particle with finite energy cannot escape to infinity, r = ∞. A massless geodesic can go to infinity and back in finite time. One way to see this is to look at the Penrose diagram of AdS. We can factor out a factor of 1 + r2 in the metric (6.1), and define a new radial coordinate, x,viadx = dr/(1 + r2), which now has a finite range. Thus, the Penrose diagram of AdS is a solid cylinder; see Fig. 6.1(a). The vertical direction is time, and the boundary is at r = ∞, which is a finite value of x.TheSd−1 is the spatial section of the surface of the cylinder. The metric in (6.1) has an obvious R × SO(d) symmetry. AdS has further symmetries. The full symmetry group is SO(2,d). These symmetries can be made more manifest by viewing AdS as the hyperboloid

∗Full acknowledgement to Juan Maldacena, The Guage/Gravity Duality, Gary T Horowitz, 2012, c Cambridge University Press 2012, reproduced with permission. 144 An introduction to the gauge/gravity duality

S d−1 Time AdS d+1

Boundary

. r ∞ r = 0 =

(a) (b)

Fig. 6.1 (a) Penrose diagram for anti-de Sitter space. It is a solid cylinder. The vertical direction is time. The boundary contains the time direction and a sphere Sd−1, represented here as a circle. (b) A massive geodesic (solid line) and a massless geodesic (dashed line).

− 2 − 2 2 ··· 2 − 2 Y−1 Y0 + Y1 + + Yd = L (6.2) in R2,d. This description is useful for realizing the symmetries explicitly. However, in this hyperboloid, the time direction, t in (6.1), is compact (it is just the angle in the [−1, 0] plane). However, in all physical applications, we want to take this time direction to be noncompact. These isometries of AdS are very powerful. Let us recall the situation in flat space. If we have a massive geodesic in flat space, we can always boost to a frame where it is at rest. In AdS, it is the same: if we consider the oscillating trajectory of a massive particle, then we can “boost” to a frame where that particle is at rest. Thus, the moving particle does not know that it is moving and, despite appearances, there is no “center” in AdS. The Hamiltonian is part of the symmetry group (as in the Poincaré group) and there are several choices for a Hamiltonian. Once we choose one Hamiltonian, for example the one that shifts t in (6.1), then we choose a “center” and a notion of lowest-energy state, which is a particle sitting at this “center.” In some applications, it is useful to focus on a small patch of the boundary and treat it as R1,d. In fact, there is a choice of coordinates where the AdS metric takes the form

2 2 2 −dt + d x − + dz ds2 = L2 d 1 . (6.3) z2

Here the boundary is at z = 0 and we have slices that display the Poincaré symmetry group in d dimensions (one time and d − 1 spatial dimensions). In fact, if we take t → ix0, we get hyperbolic space, now sometimes called Euclidean AdS! In these Introduction to the gauge/gravity duality 145 coordinates, we can also see clearly another isometry that rescales the coordinates (t, x, z) → λ(t, x, z). These coordinates have a horizon at z = ∞ and cover only a portion of (6.1). These coordinates are convenient when we want to consider a CFT living in Minkowski space R1,d−1. The AdS/CFT relation postulates that all the physics in an asymptotically anti-de Sitter spacetime can be described by a local quantum field theory that lives on the boundary. The boundary is given by R × Sd−1. The isometries of AdS act on the boundary. They send points on the boundary to points on the boundary. This action is simply the action of the conformal group in d dimensions, SO(2,d). Thus, the quantum field theory is a CFT. In fact, the rescaling symmetry of (6.3) translates into a dilatation on the boundary. The boundary theory is thus scale-invariant. It has no dimensionful parameter. Usually theories that are scale-invariant are also conformally invariant. These are theories where the stress–energy–momentum tensor is traceless. The conformal group includes the Poincaré group and the dilatation, as well as “special conformal transformations” that will not be too important for us here. Notice that the conformal symmetry makes sure that we can choose an arbitrary radius for the boundary Sd−1, so we can set it to one. In fact, if we have a conformal field theory, the tracelessness of the stress tensor implies that a field theory on a space with a metric b 2 b gμν or ω (x)gμν is basically the same (up to a well-understood conformal anomaly). Here we are talking about the metric on the boundary, where the field theory lives. This metric is not dynamical, it is fixed. How can it be that a (d+1)-dimensional bulk theory is equivalent to a d-dimensional one? In fact, let us attempt to disprove this. A skeptic would argue as follows. Let us do a simple count of the number of degrees of freedom. Since the bulk has one extra dimension, we seem to have a contradiction. In fact, we could consider the number of degrees of freedom at large energies, in the microcanonical ensemble. To compute it, we can introduce an effective temperature. In a theory with massless fields (or a theory d−1 with no scale), we expect the entropy to go like S ∼ Vd−1T . So if the boundary theory is a CFT on R × S3, then, for temperatures large compared with the radius of S3 (T 1), we expect the entropy to grow like

S ∝ cTd−1, (6.4) where c is a dimensionless constant that measures the effective number of fields in the theory. For free fields, it can be explicitly computed, as we will do later in an example. On the other hand, from the bulk point of view, it seems that we also have a theory with massless particles, which are the gravitons. We could also have extra fields, but, for the time being, let us include only the gravitons, which give a lower bound to the entropy. The entropy of these gravitons is certainly bigger than the entropy from the region where r ∼ 1. In that region, which has a volume of order one, we get

d Sgas of gravitons >T , (6.5) because it has d spatial dimensions. For large enough T , we see that (6.5) is bigger than (6.4). So we appear to have a contradiction with the basic claim of AdS/CFT. 146 An introduction to the gauge/gravity duality

However, we are forgetting something essential: it is crucial that the bulk theory contains gravity. And gravity gives rise to black holes. And black holes give rise to bounds on entropy. Black holes in AdS have the metric ⎡ ⎤ ⎢ 2gm dr2 ⎥ ds2 = L2 ⎣− r2 +1− dt2 + + r2 dΩ2 ⎦, (6.6) AdSd+1 rd−2 2gm d−1 r2 +1− rd−2 where m is proportional to the mass and g is proportional to the Newton constant in units of the AdS radius, Gd+1 g ∝ N . (6.7) Ld−1 d+1 The gas of gravitons extends up to rz ∼ T and has a mass of the order of m ∼ T . For large T , we can neglect the 1 in (6.6) when computing the Schwarzschild radius: d ∼ ∼ d+1 rs gm gT . We see that the Schwarzschild radius is greater than the size of the system for temperatures that are high enough, T>1/g. Thus, the computation in (6.5) breaks down for such high temperatures. At large enough energies, we compute the entropy in terms of the black hole entropy. This entropy grows like the area of ∼ d−1 the horizon S rs /g. One can see that the Hawking temperature for large black d−1 holes is T ∝ rs. The entropy of the black hole is SBH ∼ T /g.Thisisnowofthe expected form, (6.4), with − 1 Ld 1 c ∝ ∝ AdS . (6.8) g GN,d+1 Thus, AdS/CFT connects the entropy of a black hole with the ordinary thermal entropy of a field theory. This has two very important applications. First, with regard to conceptual issues about the entropy of black holes, it gives a statistical interpretation of black hole entropy. In addition, since it displays the black hole as an ordinary thermal state in a unitary quantum field theory, we see that these black holes are consistent with quantum mechanics and unitary evolution. Second, it allows us to compute the thermal free energy, and other thermal properties, in quantum field theories that have gravity duals. The number of fields scales like the inverse Newton constant. Notice that g, (6.8), measures the effective gravitational coupling at the AdS scale. It is the dimensionless constant measuring the effective nonlinear interactions among gravitons. Thus, if we want a weakly coupled bulk theory, we need the field theory to have a large number of fields. This is a necessary, but not sufficient, condition. One important feature of a weakly coupled theory is the existence of a Fock-space structure in the Hilbert space. Namely, we can talk about a single particle, two particles, etc. Their energies are, up to small corrections, proportional to the sum of the energies of each of the particles. The dual quantum field theory has to have a similar structure. In fact, this structure emerges quite naturally in large-N gauge theories. A large-N gauge theory is a gauge theory based on the gauge group SU(N) (or U(N)) with fields in the adjoint representation. In this case, we can form gauge-invariant operators by taking μν μν traces of the fundamental fields, such as Tr[Fμν F ], Tr[Fμν DρDσF ], etc. These are Introduction to the gauge/gravity duality 147 all local operators where the fields are all evaluated at the same point in spacetime. In addition, one could have double-trace operators, such as the product of the two operators we mentioned above. When we act with one such operator on the field theory vacuum, we create a state in the field theory. In a general CFT (even if it does not have a known gravity dual), we have a map between states on the cylinder R×Sd−1 and operators on the plane Rd. The dimension of the operator is equal to the energy of the corresponding state. (The scaling dimension tells us how the operator scales under the scaling transformation that we mentioned after (6.3).) This follows from the fact that we can go to a Euclidean cylinder. The Euclidean cylinder and the plane differ by an overall Weyl transformation of the metric d(log r)2 + dΩ2 = r−2(dr2 + r2 dΩ2). Thus, they are equivalent in a CFT. An operator at the origin of the plane creates a state at fixed r that can be viewed as a state of the field theory on the cylinder; see Fig. 6.2. This state–operator mapping is valid for any CFT. AdS/CFT relates a state of the field theory on the cylinder to a state of the bulk theory in global coordinates (6.1). Since the symmetries on both sides are the same, we can divide the states, or operators, according to their transformation laws under the conformal group. Such representations are characterized by the spin of the operator and its scaling dimension Δ. A simple example is the stress tensor operator Tμν . This operator creates a graviton in AdS. The dimension of the stress tensor is d. Single-trace operators are associated with single-particle states in the bulk. Multitrace operators correspond to multiparticle states in the bulk. There exists a general argument, based on a simple analysis of Feynman diagrams, that shows that the dimensions of multitrace operators are the sums of the dimensions of each single-trace component, up to corrections of order 1/N 2. In fact, the same analysis of Feynman diagrams shows that the large-N limit of general gauge theories gives a string theory [4]. The argument does not specify precisely the kind of string theory we are supposed to get—it only says that we can organize the diagrams in terms of diagrams we can draw on a sphere plus those on the torus, etc. Each time we increase the genus of the surface, we get an additional power 2 of 1/N . This looks like a string theory with a string coupling gs ∼ 1/N . The strings we have in the bulk are precisely the strings that are suggested by this argument.

Sd−1 r log r

S d−1 (a) (b)

Fig. 6.2 (a) A conformal ﬁeld theory on the Euclidean plane Rd. We can act with various operators at r =0. These create certain states on Sd−1, which are given by performing the path integral of the ﬁeld theory in the interior of the Sd−1 with the operators inserted. (b) Because of the Weyl symmetry of the theory, we can rescale the metric and view it as the metric on a cylinder R × Sd−1. States of the theory on this cylinder are in one-to-one correspondence to operators on the plane. This is a general property of CFTs and completely independent of AdS/CFT. 148 An introduction to the gauge/gravity duality

The preceding argument says that the large-N limit is necessary to have a weakly coupled gravity theory. This does not mean that we are restricted to the linearized solutions. In a weakly coupled gravity theory, we can consider full classical nonlinear solutions of the equations, such as the black hole solutions we discussed above. Weak coupling means that we can neglect quantum gravity corrections or loop diagrams. In string theory, the graviton is the lowest mode of oscillation of a string. The gravitational coupling we discussed above is related to the strength of interaction between strings. However, we have another condition for the validity of the gravity approximation. In gravity, we treat the graviton as a pointlike particle and we ignore all the massive string states. The typical size of the graviton is given by the string length ls, an additional parameter beyond the Planck scale. In order to ignore the rest of the string states, we need L AdS 1 (6.9) ls for gravity to be a good approximation. This condition is simply saying that the typical size of the space should be much bigger than the intrinsic size of the graviton in string theory. This condition is important because in many concrete examples, we have to make sure that this condition is met, otherwise gravity will give the wrong answers, even if we have a large number of fields! If (6.9) is not valid, then we should consider the full string theory in AdS. A salient feature of string theory is that there are massive string states of higher spin S>2. In fact, in a large-N gauge theory, we S μν can easily write down single-trace operators with higher spin, such as Tr[Fμν D+F ], where D+ is a derivative along a null direction. Such operators have relatively small scaling dimensions at weak coupling, in this case Δ = 4+S. These give rise to particles of spin S whose bulk mass is comparable to the inverse AdS radius. Such light string states render the Einstein gravity approximation invalid. Thus, in order to trust the gravity approximation, the field theory should necessarily be strongly interacting. This is a necessary, but not sufficient, condition. The coupling should be strong enough to give a large dimension to all the higher-spin single-trace operators of the theory. Such masses are set by the parameter (6.9). In concrete examples, we find that this quantity, (6.9), is proportional to a positive power of the effective ’t Hooft coupling of 2 the theory, gYMN. Here gYM is the coupling constant of the gauge theory. The extra factor of N comes in because color-correlated particles can exchange N gluons, which 2 enhances their interactions at large N. By taking a large value of gYMN, it is possible to give a large dimension to the higher-spin states. We are left with a light graviton, and other lower-spin states. In this case, we expect their interactions to be those of Einstein gravity.

6.2 Scalar field in AdS In order to be more precise about the correspondence between states in AdS and states in the boundary, it is necessary to do the quantum mechanics of a particle in AdS. Equivalently, we quantize the corresponding field in AdS. In this subsection, we consider a massive scalar field in AdS with an action Scalar field in AdS 149 √ S = dd+1x g (∇φ)2 + m2φ2 . (6.10)

Let us compute the energy spectrum in AdS in global coordinates, (6.1). Let us only focus on the ground state. We expect it to have zero angular momentum. So we make an ansatz for the wavefunction φ = e−iωtF (r), with the boundary condition that F (r) → 0 at infinity. By setting ∇2φ − m2φ =0,wegetanω-dependent equation for F (r) with two boundary conditions, one at infinity and one at the origin. This is an eigenvalue problem that gives quantized frequencies. It is possible to check that 1 φ = e−iΔt (6.11) (1 + r2)Δ/2 with d d2 Δ= + +(mL)2 (6.12) 2 4 is a solution of the equation of motion with the right boundary conditions. We identify this solution as the ground state, since it has no oscillations in the radial direction. The energy is ω = Δ. The energies of all other states differ from this by an integer, ωn =Δ+n. This is due to the fact that we can get all the other states from the action of the conformal generators, and thus their energies are determined by the conformal algebra. We get a wavefunction localized near the center of AdS. In the large-mass limit mL 1, we see that it becomes sharply localized at r = 0, as we expect for a classical particle in AdS.FormL of order one, the wavefunction is extended over a region of order one in r, which corresponds to proper distances of order of the AdS radius. A particle of zero mass, m = 0, has an integer energy Δ = d. The case of the graviton gives an equation that is similar to that of a massless field, and also leads to Δ = d, as expected from the dimension of the stress tensor. It seems from (6.12) that the dimensions of operators are bounded below by d, which is the dimension of a marginal operator in the field theory. However, there are two effects that allow us to go to lower dimensions. First, there are some allowed “tachyons” in AdS. Namely, it is possible for a field to have −d2/4 ≤ (mL)2 < 0. In other words, if a field is only slightly tachyonic, it is allowed [5]. The reason that it does not lead to an instability comes from the boundary conditions. These force the field to have some kinetic energy in the radial direction that overwhelms the negative energy of the mass term. In fact, we can check from (6.12) that such states have positive energies. A second fact is that in the range −d2/4 ≤ (mL)2 < 1 − d2/4, we can have a second quantization prescription [6]. To understand this, note that if we choose the other sign for the square root in (6.12), then (6.11) is another solution of the equation of motion. For tachyons, both solutions decay as r →∞.Sowecan set boundary conditions that remove any of these. The quantization leading to (6.12) corresponds to removing the solution that decays more slowly as r →∞. Let us now do a different computation that will further elucidate the relation between bulk fields and boundary operators. It is convenient to go to Euclidean space 150 An introduction to the gauge/gravity duality and choose the Poincaré coordinates (6.3). We can consider the problem of computing the path integral of this scalar field theory with fixed boundary conditions at the boundary. The quantum gravity problem in AdS contains such a problem: we have to do this for all the fields of the theory, including the graviton. Let us consider the classical, or semiclassical, contribution to this problem. This is given by finding the classical solution that obeys the boundary condition and evaluating the action for this solution. If we set zero boundary conditions, then the field is zero and the action is zero. If we set nonzero boundary conditions, the classical action gives us something interesting. Since we have translational symmetry along the boundary directions, we go to Fourier space and write φ = eikxf(k, z). The wave equation becomes d2f 1 df (mL)2 +(1− d) − k2 + f =0. (6.13) dz2 z dz z2

Near the boundary, for small z, there are two independent solutions behaving as f = zΔ,orf ∼ zd−Δ. We will put a boundary condition on the largest component of the solution. Since that component of the solution depends on z, we put a boundary condition at z = and set the boundary condition to be

d−Δ φ(x, z)|z= = φ0(x) . (6.14)

The solution of (6.13) that decays at z →∞is d2 f(k, z)=eikzzd/2K (kz) ,ν= +(mL)2, (6.15) ν 4 where K is a Bessel function. In order to satisfy the boundary conditions, we set

f(k, z) φ(k, z)=φ (k) d−Δ . (6.16) 0 f(k, )

We now insert this into the action (6.10). We can integrate by parts and use the equations of motion. The computation then reduces to a boundary term. For each Fourier mode, we get [7]

1 zd f(k, z) S = φ (− k) d−Δ zd φ(k, z)| = φ (−k)φ ( k) d−2Δ z 0 d z z= 0 0 f(k, ) Γ(−ν) = φ (−k)φ ( k) −2ν Polynomial[k2 2] −|k|2ν 2−2ν 2ν . (6.17) 0 0 Γ(ν)

Note that the first term contains divergent terms when → 0. These terms are analytic in momentum and, on Fourier transformation, give terms that are local in position space. These terms were to be expected since the boundary conditions we are considering are such that the field grows toward the boundary. From the field theory point of view, these divergences can be viewed as ultraviolet divergences. On the other hand, Scalar field in AdS 151 the last term in (6.17) gives a nonlocal contribution in position space and represents the interesting part of the correlator. Transformed back to position space, this gives 2νΓ(Δ) φ (x)φ (y) S = − Ld−1 ddxddy 0 0 . (6.18) πd/2Γ(ν) |x − y|2Δ

The AdS/CFT dictionary states that this computation with fixed boundary conditions is related to the generating function of correlation functions for the corresponding operator in the field theory [2, 3]. In other words, for a field φ related to the single-trace operator O, we have the equality

d d xφ0(x)O(x) ZGravity[φ0(x)] = ZField Theory[φ0(x)] = e . (6.19)

The leading approximation to the gravity answer is given by evaluating the classical action and is given by e−S, with S as in (6.18). Correlation functions of operators are then given by δ δ O(x1) ···O(xn) = ··· ZGravity[φ0(x)]. (6.20) δφ0(x1) δφ0(x1) In the quadratic approximation, the gravity answer is given by (6.18), and the correlation functions factorize into products of two-point functions. We can include interactions in the bulk. For example, we can have a φ3 bulk interaction. Then the leading approximation is given by considering the classical, but nonlinear, solution with these boundary conditions and evaluating the corresponding action. This can be computed perturbatively by evaluating Feynman–Witten diagrams in the bulk [3]. For each single-trace operator, we have a corresponding field in the bulk with a certain boundary condition. Among these fields is the graviton, associated with the stress tensor. The generating function of correlation functions of the stress tensor b is obtained by considering the field theory on a general boundary geometry gμν (x). At the classical level, we find a solution of Einstein’s equations Rμν ∝ gμν , with b gμν as a boundary condition. We insert this in the action and obtain the quantity b ∼ −SE [gcl] ZGravity[gμν (x)] e . We can also view this quantity as the Hartle–Hawking wavefunction of the universe in the Euclidean region. As a first step, one can expand Einstein’s equations to quadratic order and compute the two-point function. The action for each polarization component is similar to that of a massless scalar field. In this case, the -dependent factor in (6.14) drops out. So it makes sense to compute the absolute normalization of the two-point function. This two-point function of the stress tensor is another measure of the degrees of freedom of the theory. It is proportional to the overall coefficient in the Einstein action, which is the quantity c introduced earlier, 2d (6.8). In other words, we schematically have Tμν (x)Tσδ(0) = ctμνσδ/|x| , where tμνσδ is an x-dependent tensor taking into account the fact that the stress tensor is traceless and conserved. In fact, since the classical gravity action contains c as an overall factor, we conclude that, to leading order in 1/c, and in the gravity approximation, all stress tensor correlators are proportional to c. They are universal for any field theory that has a gravity dual, for each spacetime dimension. Such correlators are not universal in 152 An introduction to the gauge/gravity duality quantum field theory (except in two dimensions, d = 2). The universality arises only in the gravity approximation, and it is removed by higher-derivative corrections to the action. Stringy corrections give rise to these higher-derivative corrections. Similarly, the coefficient that appears in this computation is equal to the coefficient appearing in the computation of the thermal free energy (up to universal constants). Again, this does not hold for general field theories in d>2. It does hold in d =2. Another interesting case is a gauge field in AdS. This corresponds to a conserved current in the boundary theory. Thus a gauge symmetry in the bulk corresponds to a global symmetry on the boundary. We have an exactly conserved charge in the boundary theory. Because of black holes, the only way to ensure that we have a conserved charge is to have a gauge symmetry in the bulk.

5 6.3 The N = 4 super Yang–Mills/AdS 5× S example The preceding discussion was completely general. To be specific, let us discuss one particular explicit example of a dual pair. We will first discuss the field theory, then the gravity theory. We will show how various objects match on the two sides. We consider a four-dimensional field theory that is similar to quantum chromodynamics. In quantum chromodynamics, we have a gauge field Aμ that is a (traceless) 3 × 3 matrix in the adjoint representation of SU(3). The action is − 1 4 μν S = 2 d x Tr[Fμν F ],Fμν =[∂μ + Aμ,∂ν + Aν ]. (6.21) 4gYM We can consider the generalization of this theory to a gauge group SU(N), or U(N), where Aμ is now an N ×N matrix. In QCD, we also have fermions that transform in the fundamental representation. Here we will do something different. We add fermions that transform in the adjoint representation. The reason is that we would like to construct a theory that is supersymmetric. Supersymmetry is a powerful tool to check many of the predictions of the duality. The existence of the duality does not rely on supersymmetry, but it is easier to find a dual pair when we have supersymmetry. Supersymmetry is a symmetry that relates bosons and fermions. In a supersymmetric theory, the bosons and their fermionic partners are in the same representation of the gauge group. If we simply add a Majorana fermion in the adjoint representation, we get an N =1 supersymmetric theory. This theory is not conformal quantum mechanically—it has a beta function, as in the theory with no fermions. If instead we add four fermions, χα, and six scalars, φI , all in the adjoint, and with special couplings, we get a theory which has maximal supersymmetry, an N = 4 supersymmetric theory [8]. The Lagrangian of this theory is completely determined by supersymmetry and the choice of the gauge group. It has the schematic form 1 S = − d4x Tr F 2 +2(D ΦI )2 + χ/Dχ + χ[Φ,χ] − [ΦI , ΦJ ]2 4g2 μ YM IJ θ + Tr[F ∧ F ]. (6.22) 8π2 5 The N = 4 super Yang–Mills/AdS5× S example 153

2 We have two constants: the coupling constant gYM and the θ angle. All relative coefficients in the Lagrangian are determined by supersymmetry. The fields are all in a single supermultiplet under supersymmetry. This theory is classically and quantum mechanically conformally invariant. In other words, its beta function is zero. Thus, unlike QCD, it does not become more weakly coupled as we go to high energies. The coupling is set once and for all. If it is weak, it is weak at all energies; if it is strong, it is strong at all energies. The effective coupling constant is

2 λ = gYMN. (6.23)

The extra factor of N arises as follows. If we have two fields whose color and anticolor are entangled, or summed over, then there are N gluons that can be exchanged between them that preserve this entanglement. The theory has an SO(6), or SU(4), R-symmetry that rotates the six scalars into each other, and also rotates the fermions. An R-symmetry is a symmetry that does not commute with supersymmetry. This is the case here because bosons and fermions are in different representations of SU(4). Now let us discuss the gravity theory. It is a string theory, which gives rise to a quantum mechanically consistent gravity theory. Since we started from a supersymmetric gauge theory, we also expect to have a supersymmetric string theory. There are well-known supersymmetric string theories in 10 dimensions. In particular, there is one theory that contains only closed oriented strings, called the type IIB theory. This string theory reduces at long distances to a gravity theory. It is a supergravity theory called, not surprisingly, type IIB supergravity [9]. This is a theory that contains the metric, plus other massless fields required by supersymmetry. In particular, this theory contains a 5-form field strength Fμ1...μ5 , completely antisymmetric in the indices. It is also constrained to be self-dual: F5 = ∗F5. It is analogous to the 2-form field strength Fμν of electromagnetism. In four dimensions, we can have charged black hole solutions that involve the metric and the electric (or magnetic) 2-form field strength. In particular, the near-horizon solution of an extremal black hole has the geometry 2 2 AdS2 × S , with a 2-form flux on the AdS2 (or the S ) for an electrically (or a mag- netically) charged black hole. Something similar arises in 10 dimensions. There is a 5 solution of the equations of the form AdS5 × S with a 5-form along both the AdS5 and S5 directions. We have both electric and magnetic fields owing to the self-duality constraint on F5. The Dirac quantization condition says that magnetic fluxes on an S2 are quantized. In the string theory case, the flux on the S5 is also quantized:

F5 ∝ N. (6.24) S5

This number is the same as the number of colors of the gauge theory. The equations of motion of 10-dimensional supergravity that are relevant for us follow from the action √ 1 10 − 2 S = 7 8 d x g(R F5 ) (6.25) (2π) lp 154 An introduction to the gauge/gravity duality plus the self-duality constraint F5 = ∗F5. The equations of motion relate the radii 5 4 4 of AdS5 and S to N. In fact, we find that both radii are given by L /p =4πN. −1/4 In string theory, we also have the string length, given by ls = gs lp. This sets the −1 string tension T =(2πlp) . Here gs determines the interaction strength between strings. It is given by the vacuum expectation value of one of the massless fields of φ the 10-dimensional theory, gs = e . The gravity theory has another massless scalar field χ. This second field is an axion, with a periodicity χ → χ +2π. These two fields 2 are associated with the two parameters gYM and θ that we had in the Lagrangian. It is natural to identify θ with the expectation value, or boundary condition, for χ and 2 2 gYM to the string coupling gs: gYM =4πgs. The precise numerical coefficient can be set by the physics of D-branes [10], or by using the S-duality of both theories. After 5 doing this, one can write the AdS5 and S radii in terms of the Yang–Mills quantities:

4 4 L 2 L 4 =4πgsN = gYMN = λ, 4 =4πN. (6.26) ls lp

As we discussed in general, in order to have a weakly coupled bulk theory, we need N 1. In addition, in order to trust the Einstein gravity approximation, we need a large eﬀective coupling. Thus, we have the following situation:

2 gYMN 1 : gravity is good, gauge theory is strongly coupled; 2 gYMN 1 : gravity is not good, gauge theory is weakly coupled.

In these two extreme regimes, it is easy to do computations using one of the two descriptions. The ’t Hooft limit [4], which gives planar diagrams, corresponds to N →∞with 2 gYMN fixed. It is sometimes useful to take the ’t Hooft limit first and obtain a free string theory in the bulk and then vary the ’t Hooft coupling λ from weak to strong, so that we change the AdS radius in string units. The string is governed by a two- dimensional field theory whose target space is AdS (plus the S5 and some fermionic dimensions). This two-dimensional field theory is weakly coupled if the AdS radius is large and strongly coupled when the radius is small or the gauge theory is weakly 2 ∼ coupled. For values of order one, gYMN 1, it is necessary to use the full string theory description or solve the full planar gauge theory. The N = 4 super Yang–Mills theory has an S-duality symmetry that exchanges weak and strong coupling. One is tempted to go to strong coupling and then use S- duality in order to get a weakly coupled theory again. This does not work. The bulk theory also has an S-duality symmetry. These two S-duality symmetries are in one-to- one correspondence. So, in order to test whether we can trust the gravity description, first we apply S-duality on both sides to send gs < 1 and then we apply the criterion stated above. It is interesting to return to the problem of comparing the thermal free energy of the gauge theory and the gravity theory. This time, we keep track of the numerical R3 × 1 coefficients. We consider the field theory in Sβ. The free energy at weak coupling is given by the usual formula 5 The N = 4 super Yang–Mills/AdS5× S example 155 3 d k 1 −β|k| − βF = V nbosons log + nfermions log(1 + e ) (2π)3 (1 − e−β|k|) π2 = VN2T 3, (6.27) 6 2 where β =1/T andwehaveusednbosons = nfermions =8N . At strong coupling, we consider the Euclidean black brane solution, with τ ∼ τ + β, ⎡ ⎤ ⎢ 4 2 2 2 ⎥ 2 2 ⎢ z dτ dz dx ⎥ ds = L ⎣ 1 − + + ⎦, (6.28) z4 z2 z4 z2 0 2 − z 1 4 z0 which is simply related to the large-mass limit of (6.6). We can relate β = πz0 by demanding no singularity at z = z0, as usual. The entropy is given by the usual Bekenstein–Hawking formula [11]

8 2 Area L VS5 π 2 3 S = = 3 = VN T . (6.29) 4GN 4GN,10z0 2 From the entropy, we can simply compute the free energy. We get π2 − βF = S/4= VN2T 3. (6.30) 8 We see that there is a factor of 3/4 difference between (6.30) and (6.27). This does not represent a disagreement with AdS/CFT. On the contrary, it is a prediction of how the free energy changes between weak and strong coupling. From general large-N arguments, we expect the free energy to have the form F (λ, N) 1 = f (λ)+ f (λ)+··· . (6.31) F (λ =0,N) 0 N 2 1 3 We expect f0(λ) to go smoothly between f0 =1atλ =0andf0 = 4 at λ 1. In fact, the leading corrections from both values have been computed and they go in the naively expected direction [12, 13]. In this example, the function f0 becomes√ constant at large λ. There are examples where this function goes as f0 ∼ 1/ λ for large λ [14, 15]. If we are interested in computing the free energy of super Yang–Mills at strong coupling, we can do it by using the gravity result (6.30). The existence of the S5 is related to the SO(6) symmetry of the theory. The Killing 5 vectors generating the S isometries give rise to gauge fields in AdS5. These are the gauge fields associated with global symmetries that we expected in general. Of course, in other gauge/gravity duality examples, one can also have global symmetries that are not associated with a Kaluza–Klein gauge field. There are many observables that have a simple geometric description at strong coupling. In fact, the strings (and the branes) of string theory can end on the boundary, 156 An introduction to the gauge/gravity duality and they correspond to various! types of operators in the boundary theory. For example, a Wilson loop operator Tr[Pe C A] can be computed in terms of a string in the bulk that ends on the boundary along the contour C. At strong coupling, the leading approximation is given just by the area of the surface that ends on this contour. At finite coupling, we need to carry out the worldsheet quantization of this theory. In other words, we need to sum over all surfaces that end on this contour. Certain Wilson loops can be computed exactly using techniques that rely on supersymmetry, confirming the predictions of the duality [16]. The gauge theory contains scalar fields. The potentials for these scalar fields have flat directions. That is, it is possible to give expectation values to the fields in such a way that the vacuum energy continues to be zero. This spontaneously breaks the conformal symmetry. At high energies, the conformal symmetry is restored, but it is broken at low energies. These flat directions correspond to expectation values for the scalar fields that are diagonal matrices. As a simple example, we can set Φ1 = diag(a, 0,...,0) and all the rest to zero. This breaks the gauge group from U(N) → U(1)×U(N −1). In the gravity dual, it correspond to setting a D3-brane at a position z ∼ 1/a in the Poincaré coordinates (6.3). One would expect the gravitational potential to push the brane toward the horizon. This force is precisely balanced by an electric repulsion that is provided by the presence of the electric 5-form field strength. The masssless fields living on this D3-brane correspond to the fields in the U(1) factor. The massive W bosons arising from the Higgs mechanism correspond to strings that go from the brane to the horizon. It is interesting that one can find solutions that correspond to general vacuum expectation values. We write

ds2 = f −1/2(−dt2 + d x 2)+f 1/2(d y 2), l4 (6.32) f =4π p . | y − y |4 i i

Here x is a three-dimensional vector and y is a six-dimensional vector. The yi are related to the vacuum expectation values of the scalar fields, Φ = diag( y1, y2,..., yN ). This solution looks like a multicentered black brane. In principle, we cannot trust the solution near a single center, since the curvature is very large. However, in situations where we have many coincident centers, we can trust the solution. For example, if we break U(2N) → U(N) × U(N) by giving the expectation value Φ1 = diag(a,...,a,0,...,0), with Na’s, we can trust the solution everywhere. In the ultraviolet, for large | y|, we have a single AdS geometry, which splits into two AdS throats with smaller radii as we go to lower values of | y|. This describes the corresponding flow in the gauge theory from the ultraviolet to the infrared, where we have two decoupled CFTs. This is an example of a geometry that is only asymptotically AdS near the boundary but is different in the interior. It is instructive to consider the solution (6.32) with [17]

4πNl4 f =1+ p . (6.33) |y|4 The spectrum of states or operators 157

ND-branes AdS S5 5 x (a) (b)

Fig. 6.3 (a) The geometry of the black 3-brane solution (6.32) with (6.33). Far away, we have 5 10-dimensional ﬂat space. Near the horizon, we have AdS5 × S . (b) The D-brane description. D-brane excitations are described by open strings living on them. They can start and end on any of N D-branes, so we have N 2 of them. At low energies, they give rise to a U(N) gauge theory: N =4super Yang–Mills.

This enables us to give a physical derivation of the gauge/gravity duality for this 1/4 example [1]. This solution goes to 10-dimensional flat space for | y| N lp.It represents an extremal black 3-brane; see Fig. 6.3. It is extended along 1 + 3 of the spacetime dimensions, labeled by t and x, and it is localized in six of them, labeled by y. The near-horizon geometry of this black D3-brane is obtained by going to small values of y and dropping the 1 in (6.33). When the string coupling is very small, gsN 1, this system can be described as a set of N D3-branes. D3-branes are solitonic defects that exist in string theory [18]. They are described in terms of an extremely simple string theory construction. This construction tells us that we get N = 4 super Yang– Mills at low energies. In fact, it is easy to understand the scalar fields: they come from the motion of the branes in the six transverse dimensions. We can view the gauge fields as arising from supersymmetry. A system of N identical branes is expected to have an ordinary SN permutation symmetry. However, for these branes, this symmetry is enlarged into a full U(N) gauge symmetry. Thus, we have two descriptions for the brane: first as a black brane and second as a set of D-branes. We can now take the low- energy limit of each of these descriptions. The low-energy limit of the D-branes gives us the N =4U(N) super Yang–Mills theory. The low-energy limit on the gravity side corresponds to going very close to the horizon of the black 3-brane. There, the large redshift factor (the fact that f −1/2 → 0) gives a very low energy to all the particles 5 living in that near-horizon region. This region is simply AdS5 × S .Assumingthat these two descriptions are equivalent, we get the gauge/gravity duality.

6.4 The spectrum of states or operators In this case, we can construct a complete dictionary between the massless fields in the bulk and operators in the field theory. The massless 10-dimensional fields can be expanded in spherical harmonics on the S5. In addition, they fill supermulti- plets. It is interesting to note that we have 32 supercharges in the bulk theory. 158 An introduction to the gauge/gravity duality

With this large number of supercharges, a generic supermultiplet would contain states with spins greater than two. However, we can have special BPS multiplets with spins only up to two. Thus, all the massless particles of the 10-dimensional theory should be in special BPS multiplets. In 10 flat dimensions, this is only possible if the particles 5 are massless. In AdS5 × S , this is only possible if the AdS energy is fixed in terms of the SO(6) charge. In the field theory, these are in multiplets that contain the operators Tr[Φ(I1 ΦI2 ···ΦIJ )], where the SO(6) indices are symmetrized and the traces extracted. These operators are in the same representation as the spherical harmonics on S5 with angular momentum J. Their dimension is Δ = J, at all values of the coupling because it is a BPS state. Here we see the power of supersymmetry allowing us to compute these dimensions for all values of the coupling. These operators correspond to a special field in the bulk theory, which is a deformation of the S5 and the 5-form field strength. The other supergravity fields are related to this one by supersymmetry. It is interesting to consider the fate of other operators. As we mentioned above, we can consider higher-spin operators. It is simpler to understand the mechanism that gives them a large dimension by considering operators with large charges. Let us consider Z =Φ1 + iΦ2 and the operator Tr[ZJ ]. If we now add some derivatives, such as Tr[D+ZZZD+ZZZ ···], then at weak coupling the dimension of the operator is the same, independently of the order. As we turn on the coupling, the Hamiltonian, or the dilatation operator, starts moving these derivatives. In some sense, we can view the chain of Z’s as defining a lattice. The fact that only planar diagrams contribute implies that the interactions are short-range on this lattice. The range increases as we increase the order in perturbation theory. So the derivatives start moving around, and they gain a kinetic energy that depends on their “momentum” along the chain of Z’s. More explicitly, the operators that diagonalize the Hamiltonian (of the dilatation operator) have the schematic form ipl l J−l−2 O ∼ e Tr[D+ZZ D+ZZ ]+··· . (6.34) l

We have used the cyclicity of the trace to set the first derivative on the first spot. The dots represent extra terms that can appear when l or J − l − 2 are small, and are important to precisely quantize the momentum. The momentum p is quantized 2 with an expression of the form pn ∼ 2πn/J + o(1/J ), where the subleading term depends on the extra terms that appear when the two derivatives “cross” each other. A derivative with zero momentum can be pulled out of the trace and act as an ordinary derivative. This is just an element of the conformal group, and it does not give rise to spin, but to ordinary orbital angular momentum in AdS. Thus, in order to get spin, we need derivatives that have some momentum, and thus some kinetic energy. This kinetic energy increases as λ increases. Thus, for large λ, all the states that have nonzero momentum acquire a large energy. This is especially true for a short string (small J), where the momentum has to be relatively large, due to the momentum quantization condition. This was just a qualitative argument. The exact computation of these energies requires considerable technology and it employs a deep “integrability” symmetry of the planar gauge theory, or the corresponding string theory [19]. For the The radial direction 159 lightest spin-4 state (Konishi multiplet), these energies have been computed for any λ in [20]. They behave as expected and are in complete agreement with AdS/CFT. That is, they go from a value of order one at weak coupling to the strong-coupling answer, which is Δ = 2λ1/4. This strong-coupling answer is computed as follows. When the AdS radius is large in string units, the massive string states feel almost as if they were 2 2 in flat space. The lightest massive string state in flat space has mass m =4/ls [21]. Using (6.12) and (6.26), this gives Δ ∼ Lm ∼ 2λ1/4.

6.5 The radial direction One of the crucial elements of the gauge/gravity duality is the emergence of an extra “radial” dimension, the z coordinate in (6.3). Let us discuss this in more detail and in some generality. In ordinary physics, we are used to particles that typically are massive. These particles have a quantum state described by the three spatial positions. Even if they have internal constituents, as protons or atoms do, then (ignoring spin) we can describe the state by simply giving its spatial momentum or position. Of course, its energy is determined by its mass. In a scale-invariant theory, we cannot have massive particles. Naively, we would say that all the particles are massless. On the other hand, these massless particles interact in a nontrivial way and they cannot be viewed as good asymptotic states. This is true even in large-N gauge theories. However, in large-N gauge theories, we have some weakly interacting excitations. These are the objects that are created by acting with single-trace operators on the vacuum. They are characterized by the 4-momentum of the operator. Notice that we have one more component of the momentum, as compared with the case of an ordinary massive μν particle. For a simple operator, such as Tr[Fμν F ], the state created at zero coupling with a given 4-momentum is a pair of gluons that sum up to the total 4-momentum. As we increase the coupling, we start to produce more and more gluons via a showering 2 μ process. For these CFT states, we can specify the value of k = kμk arbitrarily. In the CFT, we can view these as objects that have a position and, in addition, a size. The size is one more continuous variable that we need to specify to characterize the state. This is the reason why we need to give four continuous quantum numbers to specify a state in the four-dimensional CFT. These states are not particles in the CFT—they are particles in AdS. We can say that the size of the state in the CFT is related to the position along the radial direction in AdS; see Fig. 6.4. In the coordinates (6.3), the size is proportional to z. In fact, particles in AdS are a simple way to parametrize the representations of the conformal group. In other words, unitary representations of the conformal group correspond, through a one-to-one mapping, to particles, or fields, in AdS together with a boundary condition. This is a completely general mathematical result. We saw this explicitly above for the case of a scalar field. A representation is characterized by the value of the scaling dimension, which in turn determines the mass of the field. Now, if this is so general, why don’t all theories have gravity duals? Well, to some extent, we can say that they all do. However, the gravity dual could be a strongly coupled theory in the bulk. Large-N theories give weakly coupled string duals. 160 An introduction to the gauge/gravity duality

z Boundary

Fig. 6.4 The size/radius correspondence. In the CFT, we have excitations that have a size. We see the same object with two diﬀerent sizes, related by a dilatation. They correspond to two particles with the same proper size in AdS but located at diﬀerent values of the radial position of AdS.

However, they can be highly stringy. We need some additional conditions that ensure an approximate locality of the interactions in the bulk. In particular, we need locality within an AdS radius. A necessary condition is that all the higher-spin ﬁelds have large anomalous dimensions. It might be that this is a suﬃcient condition, but it has not been clearly demonstrated from the axioms of CFTs—although, of course, it is expected to hold if we assume bulk locality. Even though we have focused here on CFTs, the gauge/gravity duality is also valid for nonconformal theories [22–24]. In these cases, the metric has the form ds2 = w(z)2(dx2 + dz2). As in the conformal case, w increases rapidly when we

Warp Warp factor factor

Gravitational Gravitational potential potential

z z Radial direction z 0 Radial direction (a) (b)

Fig. 6.5 (a) The behavior of the warp factor, or gravitational potential in the AdS case. It rises to infinity at the boundary and goes to zero toward the interior. A particle is pushed toward ever larger values of z. In the field theory, this corresponds to an excitation expanding in size. (b) Warp factor in a theory with a mass gap. The warp factor has a minimum, and excitations minimize their energy by sitting at z0. Typically, the space ends, in smoothly, at z = z0. In the dual field theory, excitations have a preferred size, like the size of a proton in QCD. References 161 approach the boundary. In these cases, the size is also related to the z direction. However, since we do not have a precise scaling symmetry, the physical behavior of boundary objects of different sizes is different. The same happens in the bulk: particles at different positions in the z direction see a geometry with different properties. In some cases, the geometric description fails for small or large z. In particular, this happens when the gauge theory coupling becomes weak in the ultraviolet or infrared [22]. In such cases, the gravity description is a good approximation only for distance (or energy) scales such that the gauge theory coupling is large. One can consider quantum field theories with a mass gap. The corresponding gravity configurations are such that the warp factor has a minimum value at some position z0. In several cases, one finds 5 that the space ends at z0 because some other dimensions (similar to the S above) are shrinking smoothly at z = z0. A massive particle minimizes its energy by sitting at z0. Of course, its wavefunction is concentrated around z0; see Fig. 6.5. The precise shape of the warp factor has to be computed by solving the bulk equations.

Acknowledgments I thank G. Horowitz for comments on the draft. This work was supported in part by U.S. Department of Energy Grant DE-FG02-90ER40542.

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[15] N. Drukker, M. Marino, P. Putrov, Commun. Math. Phys. 306, 511–563 (2011) [arXiv:1007.3837 [hep-th]]. [16] V. Pestun, Commun. Math. Phys. 313, 71–129 (2012) [arXiv:0712.2824 [hep-th]]. [17] G. T. Horowitz, A. Strominger, Nucl. Phys. B360, 197–209 (1991). [18] J. Polchinski, Phys. Rev. Lett. 75, 4724–4727 (1995) [arXiv:hep-th/9510017]. [19] N. Beisert et al., Lett. Math. Phys. 99, 3 (2012) [arXiv:1012.3982 [hep-th]]. [20] N. Gromov, V. Kazakov, P. Vieira, Phys. Rev. Lett. 104, 211601 (2010) [arXiv:0906.4240 [hep-th]]. [21] J. Polchinski, “String Theory, Vol.I,” Cambridge University Press, Cambridge (1998). [22] N. Itzhaki, J. M. Maldacena, J. Sonnenschein, S. Yankielowicz, Phys. Rev. D58, 046004 (1998) [arXiv:hep-th/9802042]. [23] E. Witten, Adv. Theor. Math. Phys. 2, 505–532 (1998) [arXiv:hep-th/9803131]. [24] I. R. Klebanov, M. J. Strassler, JHEP 0008, 052 (2000) [arXiv:hep-th/0007191 [hep-th]]. 7 Introduction to the AdS/CFT correspondence

Jan de Boer

Institute for Theoretical Physics, University of Amsterdam, Amsterdam, The Netherlands

7 Introduction to the AdS/CFT correspondence 163 Jan de BOER

7.1 About this chapter 165 7.2 Introduction 165 7.3 Why AdS/CFT? 166 7.4 Anti-de Sitter space 169 7.5 Correlation functions 171 7.6 Mapping between parameters 172 7.7 Derivation of the AdS/CFT correspondence 172 7.8 Tests of the AdS/CFT correspondence 174 7.9 More on ﬁnite temperature 177 7.10 Counting black hole entropy 179 7.11 Concluding remarks 180 Acknowledgments 182 References 182 Introduction 165

7.1 About this chapter The AdS/CFT correspondence, one of the most exciting discoveries in theoretical physics of the past 15 years, provides a framework to study strongly coupled field theories using weakly coupled gravity, and at the same time gives us a nonperturbative definition of quantum gravity in anti-de Sitter space. A huge number of papers have been written, on the subject, and one can only scratch the surface of the field in an introductory chapter like this. Moreover, this chapter will not give a comprehensive self-contained overview of the material that was covered in the lectures at the Summer School. Rather, it is intended as a summary of the main points, together with some pointers to the literature. A long list of reviews has been compiled on the String Theory Wiki at http://www.stringwiki.org/w/index.php?title=The AdS/CFT correspondence.

7.2 Introduction The AdS/CFT correspondence is one of the most significant results that string theory has produced. It refers to the existence of amazing dualities between theories with gravity and theories without gravity, and is also sometimes referred to as the gauge theory/gravity correspondence. The prototype example of such a correspondence, as originally conjectured by Maldacena [1], is the exact equivalence between type IIB 5 string theory compactified on AdS5 × S and four-dimensional N = 4 supersymmetric Yang-Mills theory. The notation AdS5 refers to an anti-de Sitter space in five dimensions, and S5 refers to a five-dimensional sphere. Anti-de Sitter spaces are max- imally symmetric solutions of the Einstein equations with a negative cosmological constant. The large symmetry group of five-dimensional anti-de Sitter space matches precisely with the group of conformal symmetries of the N = 4 super Yang–Mills theory, which has long been known to be conformally invariant. The term “AdS/CFT correspondence” has its origin in this particular example, CFT being an abbreviation for conformal field theory. Since then, many other examples of gauge theory/gravity dualities have been found, including ones where string theory is not compactified on an anti-de Sitter space and where the dual field theory is not conformal. Nevertheless, all these dualities are often still referred to as examples of the AdS/CFT correspondence. At first sight, it is quite startling that a duality between a theory with gravity and a theory without gravity could exist. But what is a consistent theory of gravity anyway? String theory provides a consistent framework to compute finite quantum corrections to classical general relativity, but the full nonperturbative structure of string theory is not very well understood. The AdS/CFT correspondence relates a theory with gravity in d dimensions to a local field theory without gravity in d − 1 dimensions. If true, it implies that actual quantum degrees of freedom of gravity cannot resemble the degrees of freedom of a local field theory defined on the same space. That this has to be true follows, among other things, from the existence of black holes, as we briefly review in Section 7.3. If the degrees of freedom in gravity were local, then one could imagine the possibility of arbitrarily large volumes with fixed energy density. However, we know 166 Introduction to the AdS/CFT correspondence that already in classical gravity this is not true: a fixed energy density in a sufficiently large volume will collapse into a black hole. In fact, hints of the existence of gauge theory/gravity duality have been around for quite some time, most notably in the case of three dimensions.1 Gravity in three dimensions can, modulo various subtleties, be described by a Chern–Simons theory [2, 3]. Chern–Simons theory is a topological field theory and when put on a manifold with a boundary reduces to a (1 + 1)-dimensional field theory on the boundary [4, 5]. For compact gauge groups, a precise relation between Hilbert spaces and correlation functions can be established. For noncompact gauge groups such as those relevant for (2 + 1)-dimensional gravity, there is still an equivalence at the level of the actions, but the precise map between Hilbert spaces and correlation functions is not quite as well understood. Nevertheless, the duality between Chern–Simons theory and two- dimensional conformal field theory has many features in common with the AdS/CFT duality. It is very hard to directly prove the equivalence between type IIB string theory 5 on AdS5 × S and four-dimensional N = 4 super Yang–Mills theory. For one thing, as alluded to above, we do not have a good definition of nonperturbative type IIB string theory. Even at string tree level, we do not (yet) know how to solve the theory completely. From this perspective, perhaps a more appropriate perspective is to view N = 4 super Yang–Mills theory as the definition of nonperturbative type IIB string 5 theory on the AdS5 × S background. This implies in particular that the nonperturbative quantum degrees of freedom of string theory do not resemble those it seems to contain at low energies. A weaker form of the AdS/CFT correspondence is obtained by restricting to low 5 energies on the string theory side. At low energies, type IIB string theory on AdS5 ×S 5 reduces to type IIB supergravity on AdS5 × S . The corresponding limit on the gauge 2 theory side is one where both N and gYMN become large, where N is the rank of the U(N) gauge group of the N = 4 supersymmetric gauge theory (note that N and N are 2 different and should not be confused!) and gYM is the gauge coupling constant. The 5 equivalence between type IIB supergravity on AdS5 × S and N = 4 gauge theory in 2 the large-N, large-gYMN limit has now been tested in incredibly many ways.

7.3 Why AdS/CFT? The AdS/CFT correspondence is related to several ideas in physics. The ﬁrst idea comes from lattice gauge theory, which in the continuum limit is supposed to yield Yang–Mills theory. For more details, we refer to the book by Polyakov [6]. The basic variables are group-valued variables associated with all the links of a lattice, and the partition function looks roughly like

1 In three or fewer dimensions, gravity has no local propagating degrees of freedom, so perhaps it is not a good example; still, it will turn out that this case has many similarities with the story in dimensions above three. Why AdS/CFT? 167 ⎡ ⎤ ⎣ ⎦ Z ∼ exp −β Tr(U1U2U3U4) ,

{Ui} plaq where the sum is over all plaquettes (oriented squares in the lattice) and U1,U2,U3,U4 are the four group variables associated with the side of the square, in order. One can now examine this theory at strong coupling, i.e., small β. The dominant contributions, as follows from group theory, come from surfaces at whose edges one needs to put propagating quarks and antiquarks. A surface contributes ∼βA to the partition function, where A is the surface area. The dominant contributions are therefore minimal-area surfaces, exactly as in string theory. Not only do we obtain a connection to string theory, but the strings are seen to correspond to qq¯ fluxtubes and the area law corresponds to a linear confining potential for the fluxtube, i.e., confinement. The second idea, namely, that large-N gauge theory might be equivalent to a string theory, is due to ’t Hooft[7]. Suppose that one has a theory where the fundamental degrees of freedom M transform in the adjoint representation of a U(N) gauge group, so that under a gauge transformation,

M → UMU−1, with U ∈ U(N). A simple example of such a theory would be a matrix model with action 1 S ∼ Tr (∂M)2 + c M k . g2 k YM k

The idea is now to study this matrix model in perturbation theory and to keep track of the powers of N that appear. As easy diagrammatic way to do this is to use so-called double-line notation, where a propagator is simply represented by two parallel lines. Each parallel line corresponds to one of the delta functions in

i k∼ i k Mj Ml δl δj and therefore represents a single index that takes values 1,...,N. Similarly, one can represent vertices as places where double lines meet each other and are glued together in a cyclic way. With this double-line notation, the number of factors of N is easily determined, since it is the number of closed single lines in a diagram, because a closed single line represents a single index that is being summed over. Moreover, it is a classic result of graph theory, basically dating back to Euler, that each double-line graph can be put on a Riemann surface in such a way that no lines cross each other. The genus h (i.e., the number of holes) of this Riemann surface is

2 − 2h = #loops − #propagators + #vertices. 168 Introduction to the AdS/CFT correspondence

2 By reorganizing perturbation theory, keeping track of powers of N and of gYM,one then discovers that the perturbative expansion of a large-N gauge theory in 1/N and 2 gYMN can be organized as 2−2h Z = N fh(λ), (7.1) h≥0

≡ 2 where λ gYMN is the so-called ’t Hooft coupling. All of this is explained in detail in the original paper by ’t Hooft [7]. This is similar to the loop expansion in string theory, 2g−2 Z = gs Zg, (7.2) g≥0 with the string coupling gs equal to 1/N . Through some peculiar and not completely understood mechanism, Feynman diagrams of the gauge theory are turned into surfaces that represent interacting strings. Feynman diagrams represent skeletons of the Riemann surfaces, and the large-N limit seems to somehow close up the holes in the Riemann surface. A toy example of this phenomenon is discussed, for example, in [8]. Apparently, this is also somehow happening in the AdS/CFT correspondence. A third important idea that something like AdS/CFT could exist is the invention of the concept of holography [9, 10]. This idea has its origin in the study of the thermodynamics of black holes. It was shown by Bekenstein and Hawking [11] that black holes can be viewed as thermodynamic systems with a temperature and an entropy. The temperature is directly related to the black body radiation emitted by the black hole, whereas the entropy is given by S = A/4G, with G the Newton constant and A the area of the horizon of the black hole. With these definitions, Einstein’s equations of general relativity are consistent with the laws of thermodynamics. Since in statistical physics entropy is a measure of the number of degrees of freedom of a theory, it is rather surprising to see that the entropy of a black hole is proportional to the area of the horizon. If gravity behaved like a local field theory, one would expect an entropy proportional to the volume. A consistent picture is reached if gravity in d dimensions is somehow equivalent to a local field theory in d−1 dimensions. Both have an entropy proportional to the area in d dimensions, which is the same as the volume in d − 1 dimensions. The analogy between this situation and a hologram, which stores all the information about a three-dimensional object in a two-dimensional picture, led to the use of the name holography in this context. The AdS/CFT correspondence is holographic because it states that quantum gravity in five dimensions (forgetting the compact 5-sphere) is equivalent to a local field theory in four dimensions. A fourth, incredibly sketchy, idea is the observation that the renormalization group equations in quantum field theory are local equations. Suppose we consider a family of quantum field theories parametrized by an ultraviolet cutoff z. Theories whose cutoffs do not differ too much are quite similar to each other, so one could imagine interpreting z as an extra coordinate and the renormalization group equations as the equations that provide the dynamics in this extra direction. However, this theory with one extra dimension cannot be an ordinary field theory, since that would have Anti-de Sitter space 169 too many degrees of freedom. However, as our discussion of the holographic principle shows, the theory with the one extra dimension could be gravitational. Suppose that this extra theory is indeed gravitational and that the metric takes the form

ds2 = f 2(z) dz2 + g2(z)(−dt2 + dxi dxi). (7.3)

This choice respects Poincaré invariance, but has some nontrivial dependence in the z direction. Assume that the coordinates xi describe a finite volume V . To simplify the discussion, we will also assume the quantum field theory is a conformal field theory, i.e., that it is scale-invariant. Then the entropy of this field theory, with cutoff z0, ∼ d−1 scales like S Vz0 . Let us compare this with the entropy of a black hole in the fiducial metric (7.3), which extends to z0. That entropy would be proportional to the d−1 horizon area at radius z0, which scales as g(z0) V . This suggests that we should take g(z) ∼ z.Ifg(z)=z, we see another feature of the metric (7.3), namely, that the second part of the metric is scale-invariant under t → λt, xi → λxi,z → λ−1z,which is precisely how these quantities should scale in a conformal field theory. To make the full metric invariant under this rescaling, we need to choose f(z) ∼ 1/z. This then finally leads to the metric

dz2 ds2 = + z2(−dt2 + dxi dxi). (7.4) z2 which happens to be precisely the metric for AdSd+1. The above is of course not meant as a serious derivation of AdS/CFT, but rather should be viewed as a heuristic motivation. Many of the arguments above do not rely on conformal invariance and apply to more general quantum field theories, and one may therefore wonder how crucial conformal invariance is. The AdS/CFT correspondence has meanwhile been extended to various other theories, but it appears that generic quantum field theories cannot have a weakly coupled gravitational dual, and even generic scale-invariant quantum field theories will not have weakly coupled gravitational duals. A nice discussion of the very special features that quantum field theories need to have in order to have a weakly coupled gravitational dual can be found in [12].

7.4 Anti-de Sitter space To describe the correspondence in some more detail, we ﬁrst need to describe the geometry of anti-de Sitter space in some more detail. Five-dimensional anti-de Sitter space AdS5 can be described as the ﬁve-dimensional manifold

− 2 − 2 2 2 2 2 2 X0 X1 + X2 + X3 + X4 + X5 = L (7.5) embedded in a six-dimensional space with metric

2 − 2 − 2 2 2 2 2 ds = dX0 dX1 + dX2 + dX3 + dX4 + dX5 . 170 Introduction to the AdS/CFT correspondence

From this way of writing the metric, it is clear that the isometry group of AdS5 is SO(2, 4), which is exactly the conformal group in 3 + 1 dimensions. By picking a particular parametrization of the solutions of (7.5) in terms of ﬁve coordinates, we obtain the metric 2 2 2 2r μ ν ds = L dr + e (ημν dx dx ) . (7.6)

This is exactly the metric we saw before in (7.4), but with z replaced by er.The parameter L is just a scale factor. The limit where the radial coordinate r goes to infinity and the exponential factor blows up is called the boundary of anti-de Sitter space. This boundary is the place where the dual field theory lives. One can indeed verify that string theory excitations in anti-de Sitter space extend all the way to the boundary [13]. In this way, one obtains a map from string theory states to states in the field theory living on the boundary. We see that er sets the scale of the Minkowski part of the metric, and it turns out that er can quite literally also be viewed as a scale of the dual field theory (see, e.g., [14]). The metric (7.6) is often called planar AdS5 or Poincaré AdS5. One can verify that it is geodesically incomplete and does not cover all of (7.5). One can find a different set of coordinates that yields the metric

dr2 ds2 = − (1 + r2) dt2 + r2 dΩ2. (7.7) 1+r2 3

This metric is called global AdS5, and it is geodesically complete. There is an important difference between planar and global AdS5. The first is dual to a conformal field theory on a plane and the second to a conformal field theory that lives on a “cylinder” S3 × R. The most striking difference between the two is that the Hamiltonian for the cylinder has a discrete spectrum, whereas the Hamiltonian for the plane has a continuous spectrum. To see this in more detail, we notice that the Euclidean cylinder and the Euclidean plane are conformally equivalent to each other:

1 dτ 2 + dΩ2 = (du2 + u2 dΩ2), (7.8) 3 u2 3 with τ = log u. Thus, time translations τ → τ + const act on the plane as rescalings: the Hamiltonian on the cylinder is the same as the dilatation operator on the plane. Because of this, we very often use the eigenvalues of the dilatation operator (called anomalous dimensions) to characterize states and operators on the plane rather than the Hamiltonian on the plane itself. Another way to see that the spectrum of the Hamiltonian on the plane is continuous is that, by scale invariance, any eigenstate with energy E canbemappedtoadiﬀerent state with energy λ−1E by simply rescaling xμ → λxμ. Correlation functions 171

7.5 Correlation functions The AdS/CFT correspondence in the form in which it was proposed in [1] did not yet provide a detailed map between AdS and CFT quantities. Such a map was given in [15, 16] and makes the correspondence much more explicit. To describe it, we first consider the AdS side, and, in particular, we consider a free field with mass m propagating in anti-de Sitter space. The field equation

(2 + m2)φ = 0 (7.9) has two linearly independent solutions that behave respectively as

e−Δr and e(Δ−4)r (7.10) as r →∞, where

Δ(Δ − 4) = m2. (7.11)

Consider now a solution of the supergravity equations of motion with the boundary condition that the ﬁelds behave near r = ∞ as

μ ∼ 0 μ (Δ−4)r φi(r, x ) φi (r, x )e . (7.12) Then the map between AdS and CFT quantities is given by " # − 4 0 exp[ Γsugra(φi)] = exp d xφi Oi , (7.13) where the left-hand side is the supergravity action evaluated on the classical solution given by φi, and the right-hand side is a generating function for correlation functions in super Yang–Mills theory. Actually, we should really use the full string theory partition function subject to the relevant boundary conditions on the left-hand side, to which the supergravity approximation is only the saddle-point approximation. However, for many applications, the above formula suffices. We also see that there should be an operator Oi in Yang–Mills theory for every field φi in anti-de Sitter spaces. With some further work, one can show that this operator needs to have conformal dimension Δi. Finally, note that we restricted attention to scalar fields in (7.13). The full AdS/CFT correspondence should of course involve all the AdS degrees of freedom, not just the scalar ones. As one can see, to compute a two-point function, one needs to know the Green function of a field in anti-de Sitter space. For a scalar field, the Green function G(x, y) is a function of one variable d(x, y) only, since there is only one SO(2, 4)- invariant function that can be made from two coordinates in AdS5. The resulting Green function is of hypergeometric type and can be solved explicitly; see, e.g., [21] for a detailed discussion. It simplifies dramatically if one of the two points is taken to the boundary of AdS5: if we choose coordinates on planar AdS5 of the type 1 ds2 = (dz2 + η dxμ dxν), (7.14) z2 μν 172 Introduction to the AdS/CFT correspondence where the boundary is at z = 0, then the Green function looks like 2 2 −Δ (z2 − z1) +(x2 − x1) G(z1,x1; z2,x2) ∼ , (7.15) z1z2 which is the so-called bulk-boundary propagator [15]. If we take both points to the boundary and strip off some powers of z1 and z2, we recognize in (7.15) the two-point function of two operators of conformal weight Δ:

∼ 1 O(x1)O(x2) 2Δ . (7.16) |x1 − x2| We have obviously been quite sketchy; for a more detailed derivation of this two-point function, we refer to the literature.

7.6 Mapping between parameters To illustrate the fact that the AdS/CFT duality is an example of a strong/weak- coupling duality, we give the relations between the parameters of both theories. 5 String theory on AdS5 × S has a dimensionless string coupling constant gs,which measures the string interaction strength relevant for string splitting and joining, a dimensionful string length ls, which sets the size of ﬂuctuations of the string worldsheet, 5 and another dimensionful parameter L, the radius of curvature of AdS5 and S that appears in (7.6). Four-dimensional N = 4 super Yang–Mills theory with gauge group U(N)has, 2 beside the rank N of the gauge group, a dimensionless coupling constant gYM. The identiﬁcation of the parameters reads

2 4 2 gs = gYM, (L/ls) =4πgYMN =4πλ. (7.17) By comparing the expansions (7.1) and (7.2), we now see that the AdS/CFT correspondence is indeed an example of a weak/strong-coupling duality. Depending on the choice of parameters, either the AdS theory or the CFT provides a weakly coupled description of the system, but never both at the same time. Gauge theory is a good 2 2 description for small gYMN and small gYM, whereas string theory is good for large 2 2 gYMN and small gYM. Therefore, the AdS/CFT correspondence can be applied in two directions. We can use string theory to learn about gauge theory, and we can use gauge theory to learn about string theory.

7.7 Derivation of the AdS/CFT correspondence The derivation of the AdS/CFT correspondence given in [1] crucially involves the notion of D-branes. D-branes are certain extended objects in string theory that were introduced by Polchinski [17]. They are labeled by the number of dimensions of the object, so that a D0-brane is like a particle, a D1-brane is like a string, a D2-brane is like a membrane, etc. There are two ways to think about D-branes. On the one hand, Derivation of the AdS/CFT correspondence 173 they are solitonic solutions of the equations of motion of low-energy closed string theory. On the other hand, they are objects in open string theory with the property that open strings can end on them. Open strings have a finite tension, and their center of mass cannot be taken arbitrarily far away from the D-brane. As a consequence, the degrees of freedom of the open string can effectively only propagate in a direction parallel to the brane: one says that they are confined to the brane, or that they live on the brane. The open string spectrum can be reproduced directly from the soliton in the closed string description via a collective coordinate quantization. As a very crude analogy, one can think about two ways to describe a monopole. On the one hand, one can think of it as the ’t Hooft–Polyakov monopole, in which case it is an extended soliton solution of the Yang–Mills–Higgs equations of motion. On the other hand, one can view a monopole as a point particle on which magnetic field lines can end. Both descriptions have their advantages, as do the open and closed string descriptions of D-branes. To derive the AdS/CFT correspondence, one starts with a stack of N D3-branes. This has a description in terms both of open and of closed strings. Next, one takes a suitable low-energy limit of the system, which involves taking ls → 0. The open string description reduces to N =4U(N) super Yang–Mills theory, whereas the closed string 5 description reduces to string theory on AdS5 × S . Thus, the AdS/CFT duality arises as a consequence of the duality between open and closed strings. This duality is most easily visualized by thinking of a cylinder. It can be viewed either as the worldsheet of a closed string moving along an interval or, equivalently, as an open string moving along a circle. Although the above description may sound fairly simple, there are various subtleties in taking the low-energy limit. The closed string description of N D3-branes that extend along the 0123 direction and that are located at r = 0 in the remaining 456789 directions with r the radial variable of six-dimensional spherical coordinates reads

−1/2 − 2 i i 1/2 2 2 2 H ( dt + dx dx )+H (dr + r dΩ5), (7.18) where 4πg N4 H =1+ s s . (7.19) r4 The low-energy limit, at the operational level, amounts to simply removing the 5 constant term in H, yielding AdS5 × S . What is perhaps rather peculiar is that the branes with which we started out were located at r = 0. After taking the decoupling limit, however, we often think of the gauge theory as living at r = ∞ instead—at least, this is where we insert delta-function sources in order to compute correlation functions of local operators. In the original D-brane description, we computed correlation functions of local operators by inserting sources at r = 0 instead—so how does the field theory move from r =0tor = ∞? It is difficult to explain this in detail here, but if we study scattering of waves in the background (7.18) by solving free field equations, we find that those waves experience an effective potential that has a sharp peak whose distance from the brane 174 Introduction to the AdS/CFT correspondence is proportional to the frequency ω, while the height is proportional to ω−4 [18]. It is the region between the potential barrier and the D-branes that becomes AdS5.Froma closed string, S-matrix point of view, it is more natural to impose boundary conditions somewhere on the barrier as opposed to r = 0, and this is why one eventually ends up with delta-function boundary conditions at r = ∞. I am not aware of a place in the literature where this is spelled out in great detail, and it could be an interesting exercise to do so.

7.8 Tests of the AdS/CFT correspondence 7.8.1 Spectrum of operators In order for (7.13) to hold, there should for every gauge-invariant operator in the gauge 5 theory exist a corresponding closed string field on AdS5 × S , whose mass is related to the scaling dimension of the operator according to (7.11). It is difficult to test this statement in full generality, because we don’t know the spectrum of super Yang–Mills theory at strong coupling, nor do we know the string spectrum at strong coupling. However, we do know the spectrum of a subset of the operators of super Yang–Mills theory at strong coupling, namely, the so-called BPS operators. BPS operators O have the property that the corresponding state |ψ = O|0 is annihilated by some Hermitian supersymmetry generator Q, which satisfies Q2 = Δ+J, with J some U(1) generator.2 If a state |ψ is annihilated by Q,itobeys

0=|Q|ψ|2 =(Δ+J)||ψ|2, (7.20) and therefore Δ = −J. The eigenvalues of J are quantized, and therefore, as long as there is no phase transition at finite coupling constant, the weak- and strong-coupling values of Δ have to be identical. In 10-dimensional string theory, there are massless and massive fields. The dimen- ∼ 2 1/4 sions of the operators corresponding to massive fields scale as (L/ls) (gYMN) , and vary continuously with the Yang–Mills coupling constant. Such operators therefore cannot be BPS, and it is difficult to compare them with operators in the gauge theory. Massless fields, on the other hand, have scaling dimensions that are independent of 2 gYMN, and these should therefore correspond to BPS operators. It has been verified [15] that there is indeed a precise match between massless fields in string theory and BPS operators in the gauge theory. A first necessary condition for such a matching to 5 be possible is that the global symmetries match. The space AdS5 × S has isometry group SO(2, 4) × SO(6), whereas N = 4 super Yang–Mills theory is invariant under the conformal group SO(2, 4) and the R-symmetry group SO(6). The bosonic global symmetries therefore do indeed agree. In addition, there are fermionic symmetries

2 The reason why Δ appears is that the Hamiltonian of a conformal field theory has a continuous spectrum when quantized on R4, but on a cylinder S3 ×R it has a discrete spectrum, with eigenvalues given by Δ. Tests of the AdS/CFT correspondence 175 that also agree, and, together with the bosonic symmetries, these form the supergroup PSU(2, 2|4). One can verify directly that the massless fields in string theory fall in the same multiplets of PSU(2, 2|4) as the BPS operators of super Yang–Mills theory. A lot of progress has been made in studying the spectrum of operators in N =4 super Yang–Mills theory for large N but finite values of the ’t Hooft coupling. The anomalous dimension operator appears to be one of an infinite series of conserved charges that the theory possesses. In other words, the theory at large N appears to be integrable. An impressive amount of work has been done in order to explore the integrability in detail (see, e.g., [19] for an extensive review). So far, all results that have been obtained strongly support the validity of the AdS/CFT correspondence for large N.

7.8.2 Correlation functions Correlation functions have been considered in great detail (see e.g. [21] for a review of early work). The results are quite encouraging, although one has to face the same problem as when comparing the spectra of operators; namely, one can only compare quantities that can be computed in the field theory at strong coupling. This originally restricted one to the set of correlation functions that can be compared with those that satisfy nonrenormalization theorems in the field theory, so that the strong-coupling answer is a straightforward extrapolation of the weak-coupling answer. Such correlation functions have been compared with the string theory answer, so far always with success. In addition, the AdS/CFT correspondence has suggested new nonrenormalization theorems, some of which remain to be proven in field theory (but see, e.g., [20]), for extremal correlation functions of the form OΔ(x)OΔ1 (x1) ...OΔn (xn) , (7.21) $ where Δ = i Δi. More recently, the integrability that has been found, and that we mentioned above, has led to the computation of many more correlation functions for on-shell operators, in other words amplitudes. Once more, we refer the reader to the overview [19] for further details.

7.8.3 Wilson loops Another interesting quantity to compare is the vacuum expectation value of a Wilson loop, " % # μ Tr Pexp Aμdx . (7.22) C A Wilson loop in ﬁeld theory can be expanded in terms of local operators, and, together with (7.13), one could in principle use this to determine which computation in supergravity (or string theory) one would have to do to compute this same expectation value. However, it turns out that the string theory calculation has a direct geometric interpretation [22, 23]. A Wilson loop is described by a string worldsheet in AdS, which extends all the way to the boundary of AdS and approaches there the curve C. 176 Introduction to the AdS/CFT correspondence

A classically stable string worldsheet is a minimal-area surface, so that the computation of the Wilson loop vacuum expectation value reduces to a minimal-area problem. For a rectangular Wilson loop, one can solve the minimal-area problem and in particular ﬁnd the quark-quark energy as a function of their separation r:

4π2(2g2 N)1/2 E = − YM . (7.23) 1 4 Γ( 4 ) r

2 Interestingly, the ’t Hooft coupling gYMN appears with a fractional power, and therefore this result cannot remain valid at weak coupling, where the answer will be qualitatively diﬀerent. The result in (7.23) is an example of a nontrivial prediction of the AdS/CFT correspondence. Wilson loops play a prominent role in computing amplitudes in N = 4 super Yang– Mills theory as well, and are important in many aspects of the underlying integrable structure of the large-N theory. Once more, further details can be found in [19]. It is amusing to notice that the fundamental string in AdS is the same as the QCD string of Yang–Mills theory. In some sense, we have come a full circle toward the original motivation for string theory, namely, as an eﬀective theory for the strong interactions.

7.8.4 Finite temperature So far, the ﬁeld theory, N = 4 super Yang–Mills theory, has been a theory at zero temperature, and it has been obtained in the derivation of the AdS/CFT correspondence as the low-energy limit of the degrees of freedom associated with a stack of D3-branes. It turns out that this setup can be generalized to nonzero temperature by employing so-called near-extremal D3-branes. In particular, One can compute the free energy as a function of temperature, using either the ﬁeld theory or the dual gravitational description. The results are

π2 π2 F = − N 2VT4,F= − N 2VT4. (7.24) sugra 8 SYM 6

There is no disagreement, since the first result is valid at strong coupling in the gauge theory and the second at weak coupling. As the coupling varies, one expects a smooth transition between these two results (for a discussion, see, e.g., [24]), but the possibility of a phase transition at a finite value of the coupling has not been ruled out completely. More generally, we can directly consider a finite-temperature version of the AdS/CFT correspondence, by replacing the boundary geometry R4 in (7.6) by S3 ×S1, where the S1 represents periodic imaginary time, so that the system is indeed at finite temperature [25]. The five-dimensional geometry that solves the string equations of motion and has this boundary is no longer anti-de Sitter space, but a space with a different metric. Actually, there are two different 5-manifolds with conformal boundary S3 × S1. One is a finite-temperature version of Euclidean AdS and the other describes a Schwarzschild black hole in AdS. More on finite temperature 177

The finite-temperature version of AdS dominates the supergravity path integral at low temperatures. In this geometry, the expectation value of the Wilson loop vanishes, and the center of the gauge group is unbroken, as in a confining phase. The second solution has a nonzero expectation value for the Wilson loop, and the center is broken, as in a deconfining phase. Altogether, this provides a tantalizing geometric picture of the confinement/deconfinement transition: it is mapped to a topology-changing process in gravity!

7.8.5 Glueballs and the string tension So far, successful tests of AdS/CFT have relied heavily on supersymmetry in order to compare answers at weak and strong coupling. Of course, for more realistic applications, it is desirable to consider cases with less or no supersymmetry. One way to break supersymmetry is to consider supersymmetric Yang–Mills at finite temperature and take the T →∞limit. Naively, the field theory reduces to some nonsupersymmetric extension of three-dimensional QCD. To test this, one can try to compute quantities in this version of AdS/CFT and compare them with lattice results in three-dimensional QCD. In particular, one can compute glueball masses using field equations in AdS, and string tensions using the Wilson loop. There is considerable disagreement with lattice results. The string theory in question has many additional states with a mass of the order of T ∼ ΛQCD, which is also the mass scale of the glueballs. At present, no controlled way to remove the cutoff and reach a weak-coupling limit has been found. The string that appears in the AdS/CFT correspondence yields results that differ from the strong-coupling lattice results, and these in turn differ from the continuum lattice results. For a detailed overview, see [26]. The latter two are separated by a phase transition, but whether there are any such phase transitions in AdS/CFT remains an open problem.

7.9 More on finite temperature Let us return to D3-branes at finite temperature and derive (7.24) in some more detail. The relevant geometry to describe D3-branes at finite temperature is very similar to that of a black hole, which also has a finite (Hawking) temperature. The usual Schwarzschild solution contains a factor 1 − 2MG/r, with GM/r being simply the linearized Newton potential. Forces mediated by massless particles in dimension d give rise to a potential that scales like GM/rd−3; it is a great exercise to obtain this qualitative behavior directly from the Feynman diagram for tree-level exchange of a massless particle. For a black hole in d dimensions, we therefore expect to find a factor 1 − 2MG/rd−3. Since D3-branes have 6 transverse dimensions they should behave like black holes in 6 + 1 dimensions, and one expects a factor 1 − c/r4 for some c. Indeed, the metric for D3-branes at finite temperature, after taking the decoupling limit, reads 178 Introduction to the AdS/CFT correspondence 1 dz2 ds2 = −f(z)dt2 + dxi dxi + , (7.25) z2 f(z) with

4 − z f(z)=1 4 , (7.26) z0 where z0 is a free parameter related to the temperature of the D3-branes. To ﬁnd the relation between z0 and the temperature, we rotate√ the metric (7.25) to Euclidean signature and introduce a new coordinate ρ = z0 − z. This Euclidean metric near ρ = 0 looks exactly like the two-dimensional plane in polar coordinates, and it is exactly the two-dimensional plane if we make Euclidean time into a periodic variable. We leave it as an exercise to show that its period needs to be πz0. Therefore, the temperature reads T =1/πz0. We can now compute the entropy of ﬁnite-temperature D3-branes. According to the Bekenstein–Hawking entropy formula, this entropy (per unit area) is equal to

R3 S = 3 , (7.27) 4G5z0

4 1/4 where we have reinstated the radius of curvature R =(4πgsNs) .IntypeIIB 6 8 2 string theory, the 10-dimensional Newton constant G10 =8π sgs , but it is outside the scope of this chapter to review this. The five-dimensional Newton constant is then fixed by reducing the 10-dimensional theory over a 5-sphere of radius R, giving −1 3 5 −1 G5 = π R G10 , where we have used the fact that the volume of a unit 5-sphere 3 1 2 2 3 is π . Putting in all the numbers, we find that the entropy density S = 2 π N T , and, using dE = TdS, we finally reproduce the energy density given above:

3π2 E = N 2T 4. (7.28) 8 The weakly coupled computation can be done as follows. Consider a free boson in aboxofsizeL3. According to standard statistical mechanics, the relevant partition function is & 1 Z = . (7.29) B 1 − e−β2π|n|/L n

The product is over vectors n with integer entries that parametrize the wavenumbers of the bosonic ﬁeld in the box. We next take the log of this expression, write the log of the denominators as a Taylor series, and replace the sum over n by an integral since we are interested in the large-volume limit, to obtain L3 ∞ e−βln βF = − 4πn2 dn . (7.30) B (2π)3 l 0 l>0 Counting black hole entropy 179

This is easily evaluated using ζ(4) = π4/90, with the result

L3 π2 βF = − , (7.31) B β3 90 so that the energy E = ∂β(βF) becomes

π2 E = L3T 4. (7.32) B 30 For fermions, the result is the almost the same, except that an extra factor (−1)l+1 appears on the right-hand side of (7.30), which reduces the result by a factor of 7/8. Now N = 4 super Yang–Mills has 8N 2 bosonic degrees of freedom: 6N 2 from the six scalar ﬁelds in the adjoint of U(N) and 2N 2 from the physical polarizations of the U(N) gauge ﬁeld. Similarly, and by supersymmetry, there are 8N 2 fermionic degrees of freedom. Including the factor of 7/8, the total energy is therefore equal to 15N 2 times that of a single boson, and we get

π2 E = L3T 4. (7.33) free 2 The diﬀerence between (7.28) and (7.33) is a factor of 3/4, and, when converted into free energies, we recover (7.24).

7.10 Counting black hole entropy There is a long history of trying to provide a microscopic interpretation of black hole entropy in string theory, starting with [27]. A very important example of such an interpretation is provided by the duality between AdS3 and two-dimensional CFTs, which underlies the majority of microscopic entropy counting calculations that have been performed to date. Black holes in AdS3 are very simple; they look like

dr2 ds2 = −(r2 − M) dt2 + + r2 dφ2. (7.34) r2 − M The Bekenstein–Hawking entropy of such a black hole is √ 2π M S = . (7.35) 4G3

By AdS3/CFT2 duality, this black hole should be dual to a CFT at some finite temperature. As in the case of finite-temperature D3-branes, the temperature is readily obtained by examining the periodicity of imaginary time. In the case of D3-branes discussed in Section 7.9, there was a mismatch of a factor of 3/4 when comparing the entropy of the weakly coupled field theory and that of the D3-brane system. What makes two-dimensional CFTs very special is that the behavior 180 Introduction to the AdS/CFT correspondence of the partition function at large temperature is universal and independent of whether the theory is at weak or strong coupling. One can understand this by considering the two-dimensional CFT on a 2-torus with circumferences β and L, which describes a two- dimensional CFT on a circle of size L at finite temperature β. The high-temperature limit is one where β L. However, there is a priori nothing that tells us that one of the circles of the torus is Euclidean time and the other is space. We might as well interpret the circle of size L as time and the circle of size β as space. Moreover, by scale invariance, the theory will depend only on the ratio of these two quantities. If we change the interpretation of the two circles, the same torus now describes the CFT—but at a very low temperature. At very low temperature, the theory is dominated by its ground state, and the partition function is easy to evaluate. This high–low temperature duality is also known as modular invariance, and is a very powerful feature of two-dimensional CFTs. The main subtlety in this computation is related to the fact that two-dimensional CFTs on the cylinder have a Casimir energy, which one needs to take into account. In any case, in this way one finds the famous Cardy formula, which states that as a function of the two scaling dimensions3 Δand Δ,¯ the number of states grows as S =2π cΔ/6+2π cΔ¯ /6, (7.36) where the only property of the entropy that appears is the central charge c, which is a measure of the number of degrees of freedom. Strikingly, this answer does not depend on any coupling constants. If one translates the M and G3 in (7.35) into c and Δ, Δ,¯ one finds perfect agreement with (7.36)! Although this computation may not appear to be very profound, the above is an amazing result and a very strong test of AdS/CFT on the one hand and the validity of the microscopic interpretation of the Bekenstein–Hawking entropy formula on the other.

7.11 Concluding remarks In the past few years, we have seen many applications of AdS/CFT to strongly coupled problems such as QCD and various condensed matter systems. In such applications, we view AdS/CFT as a phenomenological tool that is supposed to address qualitative features only, since it is extremely unlikely that the CFTs that appear in AdS/CFT also appear in nature. For example, AdS/CFT works when the CFT has a very particular strong-coupling and large-N limit where only ﬁnitely many degrees of freedom survive at low energies—and systems in nature tend not to have a large N but rather N =1orN = 3 as in QCD. Still, it is conceivable that some interesting features of strongly coupled systems are captured well by AdS/CFT. Consider, for example, QCD. There are of course many diﬀerences between QCD and N = 4 Super Yang–Mills theory (SYM)

3 In fact, Δ + Δ¯ is the scaling dimension and also the energy as measured by the Hamiltonian on the cylinder, whereas Δ − Δ¯ is a (half-)integer that measures the spin of the state. Concluding remarks 181

1. QCD is not conformal, although it becomes approximately conformal at high T , whereas N = 4 SYM is exactly conformal. 2. QCD conﬁnes below a critical temperature Tc, whereas N = 4 SYM does not. 3. QCD breaks chiral symmetry below Tc, whereas N = 4 SYM does not. 4. QCD does not have supersymmetry, whereas N = 4 SYM does, although this is broken at ﬁnite T . 5. QCD is asymptically free, whereas N = 4 SYM must be strongly coupled to be able to use gravity. 6. In QCD, N = 3, whereas in AdS/CFT N must be very large. However, various quantities may not depend strongly on N, so this need not be a major problem. 7. In QCD, Nf ∼ Nc, whereas in N = 4 SYM, we often can only study Nf Nc.

There have been various attempts to construct gravity duals that are more realistic representations of QCD, including some that also exhibit confinement. These will not be discussed here, but we would like to mention one situation where AdS/CFT seems to capture some qualitative features of actual QCD dynamics, and where several of the differences we listed above are not that important, namely, a quark–gluon plasma of QCD. In a quark–gluon plasma, QCD is believed to be fairly strongly coupled, and lattice computations suggest that it is also approximately scale-invariant in this regime. This is therefore the regime where QCD looks as similar to a strongly coupled CFT as possible. Are there interesting quantities that one can compute and compare? One such quantity is the ratio η/s of the shear viscosity divided by the entropy density. This is a quantity that describes a particular feature of a fluid. The term shear viscosity refers to the fact that if one has a fluid between two parallel plates, and keeps one plate at rest while moving the other plate, keeping the distance between the plates fixed, one experiences some resistance. This resistance is the shear viscosity and, as the name suggests it is related to dissipation. Shear viscosity is one of the parameters that appears in the equations of fluid dynamics. It can be measured experimentally roughly as follows: at the relativistic heavy ion collider in Brookhaven, heavy ions such as gold are collided with each other, leading to production, for a short time, of a quark–gluon plasma. This plasma expands and cools, and eventually hadronizes. By measuring how many particles are emitted in each direction, one can reconstruct part of the time evolution of the plasma, and this turns out to be described well by the equations of fluid mechanics. In particular, one can measure the shear viscosity, which turns out to be a remarkably small number, η/s ∼ 0.1. There is no known fluid for which this quantity is smaller, and the quark– gluon plasma is perhaps the most perfect fluid seen in nature. One can also compute this quantity using AdS/CFT. According to standard results from linear response theory, the shear viscosity can be extracted from the two-point function of the stress tensor. This two-point function can be computed using standard AdS/CFT techniques, and surprisingly the result for η/s is determined only by the near-horizon physics of the black hole that is dual to the finite-temperature plasma. Moreover, the answer is independent of the details of the theory that lives in AdS and it is given by 182 Introduction to the AdS/CFT correspondence

η 1 = . (7.37) s 4π

This is very close to the number that is measured experimentally, strongly suggesting that ﬁnite-temperature N = 4 SYM theory is in the same universality class as the quark–gluon plasma. We refer the interested reader to [28] for a much more detailed discussion of applications of holography to QCD, and also to the relevant reviews listed on the String Theory Wiki mentioned in Section 7.1 for a discussion of applications to other areas of physics such as condensed matter theory.

Acknowledgments I would like to thank the organizers of the Les Houches Summer School 2011 for giving me the opportunity to present these lectures and all participants for their enthusiastic participation.

References [1] J. M. Maldacena, “The large N limit of superconformal ﬁeld theories and supergravity,” Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)] [arXiv:hep-th/9711200]. [2] E. Witten, “(2+1)-dimensional gravity as an exactly soluble system,” Nucl. Phys. B 311, 46 (1988); “Topology changing amplitudes in (2+1)-dimensional gravity,” Nucl. Phys. B323, 113 (1989). [3] A. Achucarro and P. K. Townsend, “A Chern–Simons action for three-dimensional anti-de Sitter supergravity theories,” Phys. Lett. B180, 89 (1986). [4] E. Witten, “Quantum ﬁeld theory and the Jones polynomial,” Commun. Math. Phys. 121, 351 (1989). [5] S. Elitzur, G. W. Moore, A. Schwimmer, and N. Seiberg, “Remarks on the canonical quantization of the Chern–Simons–Witten theory,” Nucl. Phys. B326, 108 (1989). [6] A. M. Polyakov, “Gauge Fields And Strings,” Harwood, Chur (1987). [7] G. ’t Hooft, “A planar diagram theory for strong interactions,” Nucl. Phys. B72, 461 (1974). [8] H. Ooguri and C. Vafa, “Worldsheet derivation of a large N duality,” Nucl. Phys. B641, 3 (2002) [arXiv:hep-th/0205297]. [9] G. ’t Hooft, “Dimensional reduction in quantum gravity,” arXiv:gr-qc/9310026. [10] L. Susskind, “The world as a hologram,” J. Math. Phys. 36, 6377 (1995) [arXiv:hep-th/9409089]. [11] J. Bekenstein, “Black Holes And Entropy,” Phys. Rev. D7, 2333 (1973); “Gen- eralized Second Law of Thermodynamics in black hole physics,” Phys. Rev. D9, 3293 (1974); S. W. Hawking, “Black Holes And Thermodynamics,” Phys. Rev. D13, 191 (1976). References 183

[12] S. El-Showk and K. Papadodimas, “Emergent spacetime and holographic CFTs,” JHEP 1210, 106 (2012) [arXiv:1101.4163 [hep-th]]. [13] J. de Boer, H. Ooguri, H. Robins, and J. Tannenhauser, “String theory on AdS3,” JHEP 9812, 026 (1998) [arXiv:hep-th/9812046]. [14] L. Susskind and E. Witten, “The holographic bound in anti-de Sitter space,” [arXiv:hep-th/9805114]. [15] E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. 2, 253 (1998) [arXiv:hep-th/9802150]. [16] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge theory correlators from non-critical string theory,” Phys. Lett. B428, 105 (1998) [arXiv:hep-th/9802109]. [17] J. Polchinski, “Dirichlet-branes and Ramond–Ramond charges,” Phys. Rev. Lett. 75, 4724 (1995) [arXiv:hep-th/9510017]. [18] I. R. Klebanov, “World volume approach to absorption by nondilatonic branes,” Nucl. Phys. B496, 231 (1997) [arXiv:hep-th/9702076]. [19] N. Beisert et al., “Review of AdS/CFT integrability: an overview,” Lett. Math. Phys. 99, 3 (2012) [arXiv:1012.3982 [hep-th]]. [20] M. Baggio, J. de Boer, and K. Papadodimas, “A non-renormalization theorem for chiral primary 3-point functions,” JHEP 1207, 137 (2012) [arXiv:1203.1036 [hep-th]]. [21] E. D’Hoker and D. Z. Freedman, “Supersymmetric gauge theories and the AdS/CFT correspondence,” arXiv:hep-th/0201253. [22] S. J. Rey and J. Yee, “Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity,” Eur. Phys. J. C22, 379 (2001) [arXiv:hep- th/9803001]. [23] J. M. Maldacena, “Wilson loops in large N ﬁeld theories,” Phys. Rev. Lett. 80, 4859 (1998) [arXiv:hep-th/9803002]. [24] I. R. Klebanov, “TASI Lectures: Introduction to the AdS/CFT Correspondence,” arXiv:hep-th/0009139. [25] E. Witten, “Anti-de Sitter space, thermal phase transition, and conﬁnement in gauge theories,” Adv. Theor. Math. Phys. 2, 505 (1998) [arXiv:hep-th/9803131]. [26] M. Caselle, “Lattice gauge theories and the AdS/CFT correspondence,” Int. J. Mod. Phys. A15, 3901 (2000) [arXiv:hep-th/0003119]. [27] A. Strominger and C. Vafa, “Microscopic origin of the Bekenstein–Hawking entropy,” Phys. Lett. B379, 99 (1996) [arXiv:hep-th/9601029]. [28] J. Casalderrey-Solana, H. Liu, D. Mateos, K. Rajagopal, and U. A. Wiedemann, “Gauge/String Duality, Hot QCD and Heavy Ion Collisions,” Cambridge Univer- sity Press, Cambridge (2014); an earlier, shorter version of this book is available at arXiv:1101.0618 [hep-th].

8 Hydrodynamics and black holes

Yaron Oz

Raymond and Beverly Sackler Faculty of Exact Sciences School of Physics and Astronomy, Tel-Aviv University, Ramat-Aviv, Israel

8 Hydrodynamics and black holes 185 Yaron OZ

8.1 Introduction 187 8.2 Field theory hydrodynamics 187 8.3 Relativistic hydrodynamics 188 8.4 Nonrelativistic fluid flows 192 8.5 Holographic hydrodynamics: the fluid/gravity correspondence 194 Acknowledgments 197 References 197 Field theory hydrodynamics 187

8.1 Introduction According to the Holographic Principle [1], the degrees of freedom of a quantum theory of gravity in a volume of space V are encoded on its boundary A.Thus, the Holographic Principle relates quantum gravity in d + 1 spacetime dimensions to quantum field theory without gravity in one lower space dimension. The quantum field theory at the boundary is not necessarily local. In the framework of the AdS/CFT correspondence [2, 3], the quantum theory of gravity is superstring (M-) theory on asymptotically anti-de Sitter (AdS) space, and the theory on the boundary is a local conformal field theory (CFT). The AdS/CFT correspondence has been generalized in various directions, and there are now a large number of examples of non-AdS/non-CFT relations. Some of the non- CFT theories are QCD-like gauge field theories. Generically, when the field theory is strongly coupled, the gravitational description is weakly coupled; that is, the curved geometry has a small curvature. On the other hand, when the field theory is weakly coupled, the gravitational description is strongly coupled. A particularly interesting regime of the quantum field theory is the hydrodynamic one, where the theory is in local thermal equilibrium. The AdS/CFT correspondence relates the field theory hydrodynamics to perturbations of black hole (brane) gravitational backgrounds. Indeed, an important experimental framework to which the correspondence has been applied is the description of the QCD plasma produced at RHIC and LHC. This plasma exhibits strong-coupling dynamics, αs(TRHIC) ∼ O(1). The AdS/CFT correspondence provides a real-time nonperturbative framework, and one typically uses the strong-coupling properties of a CFT plasma as a reference point for describing the strongly coupled QCD plasma. A well-known quantity is the ratio of the shear viscosity η to the entropy density s, which for a generic strongly coupled gauge field theory is low. Indeed, such a low ratio is a generic property of the gravitational description [4]. Hydrodynamic simulations at low shear viscosity to entropy ratio are consistent with RHIC data [5]. In this chapter, we discuss the fluid/gravity correspondence for relativistic and nonrelativistic fluid flows. In Section 8.2, we present the hydrodynamic framework from the field theory point of view. In Section 8.3, we outline the dual gravitational description for relativistic fluids, before turning to consider the nonrelativistic case in Section 8.4. In Section 8.5, we give further details of the fluid/gravity correspondence, discussing the bulk geometry and the dynamics of the black hole horizon.

8.2 Field theory hydrodynamics In the hydrodynamic regime, the system is in local thermal equilibrium. This is en- forced by frequent collisions between the particles. Thus, the hydrodynamic regime is characterized by a short mean free path (correlation length), much smaller than the characteristic scale of variations of the macroscopic fields. Since it is dominated by collisions, an appropriate description is that of a collective fluid-type flow rather than a particle (kinetic theory) one. Hydrodynamics is the theory of this fluid flow. 188 Hydrodynamics and black holes

The postulated eﬀective degrees of freedom in the hydrodynamic regime are charge densities ρ( x, t), which may be understood as nonequilibrium thermal averages. The hydrodynamic equations are the conservation laws, which govern the evolution of the charge densities:

i ∂tρ + ∂ij =0. (8.1)

Since the charge densities in the hydrodynamic regime are slowly varying functions, constitutive relations express ji in terms of ρ and its derivatives. These relations are material-dependent. If we take ji = −D∂iρ (Fick’s law), plug in (8.1), and Fourier transform with respect to the space coordinates x,weget

2 ρ( k, t)=e−Dk tρ( k, t =0). (8.2)

D is called the transport coeﬃcient and depends on the material (microscopic theory). The relation (8.2) is the characteristic behavior of hydrodynamic modes: it has a lifetime τ(k)=1/Dk2, which is inﬁnite in the long-wavelength limit k → 0. Indeed, since ρ is a conserved quantity, it cannot disappear locally but can only relax slowly over the entire system. The dispersion relation for the hydrodynamic mode is ω = −iDk2. It shows up as a pole in the retarded thermal correlation function of the microscopic theory

S( x, t)=ρ( x, t),ρ(0, 0)eq, (8.3) where ...eq is the thermal equilibrium average, and we have assumed ρ( x, t)eq =0. S( x, t) describes the ﬂuctuations of the charge density ρ and can be measured experimentally. In the gravitational description of hydrodynamics, the poles of the retarded thermal correlation function correspond to the quasinormal modes of the black hole, which characterize its late time relaxation to equilibrium after a perturbation.

8.3 Relativistic hydrodynamics

We will work in four flat spacetime dimensions with a Lorentzian metric ημν = diag(−1, 1, 1, 1). We define the hydrodynamic expansion parameter (Knudsen number) Kn as Kn ≡ lcor/L 1, where lcor is the correlation length of the fluid and L is the characteristic scale of variations of the macroscopic fields. The macroscopic fields are the energy density (x), the pressure p(x), the charge densities ρa(x)and chemical potentials μa(x), the temperature T (x), the entropy density s(x), and finally μ i μ the local four-velocity field u (x)=(γ,γβ ) satisfying uμu = −1. These local fields are not independent. We can choose an independent set, for instance (x), ρa(x), and uμ(x), and determine the other local fields by using the equation of state and the thermodynamic relations

a d = Tds+ μ dρa, (8.4a) a dp = sdT + ρa dμ . (8.4b) Relativistic hydrodynamics 189

Hydrodynamics is a generalization of thermodynamics, in which the charge densities are upgraded to local currents. The hydrodynamic evolution equations are the conservation laws of these currents:

μν μ ∂μT =0,∂μJa =0. (8.5)

μν μ T is the stress–energy tensor, and Ja ,a =1, ..., n, are symmetry currents, which can be abelian such as the electromagnetic charge current or nonabelian such as ﬂavor symmetry currents in QCD. The constitutive relations have the form of series for the stress–energy tensor and the symmetry currents: ∞ ∞ μν μν μ μ T (x)= T(l) (x),Ja = Ja(l)(x), (8.6) l=0 l=0

μν μ ∼ l where T(l) ,Ja(l) (Kn) . The expansion in the Knudsen number is a derivative expansion, where l counts the number of derivatives. One introduces a local entropy current sμ, whose divergence is required to be non-negative with hyphen order by order in the derivative expansion:

μ ∂μs ≥ 0. (8.7)

This local form of the Second Law of Thermodynamics constrains the transport co- eﬃcients of the theory. Note, however, that this requirement still lacks a microscopic ﬁeld theory interpretation.

8.3.1 Ideal hydrodynamics Keeping only the ﬁrst term l = 0 in the series gives ideal hydrodynamics, which is essentially determined by thermodynamics. We have

μν μ ν μν T(0) = u u + pP , (8.8a) μ μ Ja(0) = ρau , (8.8b) μ μ s(0) = su , (8.8c) where P μν = ημν + uμuν. There is one less conservation equation than the number μ of fields , p, ρa,u , and so one more equation is needed in order to have a complete description of the system. This is called the equation of state, and it allows us to choose μ a set of independent variables, for instance, , ρa, and u . In CFT hydrodynamics, μ Tμ = 0, and the equation of state reads =3p. In this case, there is only one dimensionful quantity, the temperature T ,and , p ∼ T 4. The ideal conformal fluid stress–energy tensor can be recast, up to an overall constant coefficient, in the form

μν 4 μν μ ν T(0) = T (η +4u u ) . (8.9) At the ideal hydrodynamics order, there is no dissipation, and the hydrodynamic μ conservation laws imply that ∂μs(0) =0. 190 Hydrodynamics and black holes

8.3.2 Viscous hydrodynamics First-order dissipative hydrodynamics is obtained by keeping also the l = 1 term in the series. When getting out of equilibrium, there is an ambiguity in the definition of the charge densities due to the possibility of adding derivative terms. Fixing this ambiguity requires a choice of frame. In one such frame, the Landau frame, the fluid velocity is an eigenvector of the stress–energy tensor, and the energy density is its μν μ eigenvalue: T uν = − u . In this frame, the local rest frame of the flow is where the − μ energy density is at rest. The charge density ρa = uμJa . The first-order stress–energy tensor and the symmetry currents satisfy

μν μ uμT(1) =0,uμJa(1) =0. (8.10) They take the forms μν − μν − α μν T(1) = 2ησ ζ(∂αu )P , (8.11a) μb J μ = −Tσ P μν ∂ , (8.11b) a(1) ab ν T where 1 σμν = P μαP νβ∂(αuβ) − ∂ uαP μν (8.12) 3 α is the shear tensor. There are three types of transport coefficients at this order: the shear viscosity η, the bulk viscosity ζ, and the conductivity matrix σab. Note that in the conformal case, ζ = 0. The transport coefficients can be calculated from the retarded Green functions of the microscopic thermal field theory using linear response theory and the Kubo formulas. The entropy current at this order reads μa sμ = − J μ . (8.13) (1) T a(1) It is no longer conserved at the first viscous order, and we have

η ζ μa μb ∂ Sμ = σ σμν + (∂ uμ)2 + Tσ P μν ∂ ∂ ≥ 0. (8.14) μ 2T μν T μ ab μ T ν T The positivity requirements imposed on the transport coefficients in (8.14) can be verified in the microscopic quantum field theory.

8.3.3 Quantum anomalies and chiral eﬀects The hydrodynamic description exhibits interesting chiral eﬀects when a global μ symmetry current Ja of the microscopic theory is anomalous: 1 ∂ J μ = C μνρσF a F b . (8.15) μ a 8 abc μν ρσ Relativistic hydrodynamics 191

μ μ μ Cabc is the coefficient of the triangle anomaly of the currents Ja , Jb ,andJc .Theform of an anomalous symmetry hydrodynamic current is modified by a term proportional to the vorticity of the fluid, 1 ωμ ≡ μνλρu ∂ u . (8.16) 2 ν λ ρ This was first discovered in the context of the fluid/gravity correspondence [6]. The global anomalous symmetry hydrodynamic current takes the form μ jμ = ρ uμ + σ b Eμ − TPμν ∂ b + ξ ωμ + ξ(B)Bbμ. (8.17) a a a b ν T a ab

μ μν aμ 1 μνλρ a b Here, Ea = Fa uν and B = 2 uν Fλρ, while ρa, T , μa,andσa are the charge densities, temperature, chemical potentials, and conductivities of the medium. The anomaly coeﬃcients can be calculated from the requirement that the entropy current μ must have a positive divergence, ∂μs ≥ 0 [7, 8]. They read 2ρ 1 ξ = C μbμc +2β T 2 − a C μbμcμd +2β μbT 2 , (8.18a) a abc a + p 3 bcd b ρ 1 ξ(B) = C μc − a C μcμd + β T 2 . (8.18b) ab abc + p 2 bcd b

The coefficients βa are related to the gravitational anomaly [9], that is, to the triangle μ anomaly diagram of the current Ja and two stress–energy tensors. There are proposals for detecting experimental signatures of the axial current triangle diagram anomaly in a hydrodynamic description of high-density QCD. At RHIC and LHC, the chemical potentials are small compared with the energy density; hence, b c (B) c ξa = Cabcμ μ and ξab = Cabcμ . The chiral magnetic effect corresponds to the generation of an electric current in the direction of the magnetic field, j ∼ μAB ,using Cabc of two electric currents and one axial current in (8.17), with μA being the axial chemical potential. The chiral vortical effect corresponds to the generation of a baryon number current in the direction of the vorticity vector j ∼ μAμB ω,usingCabc of two baryon number currents and one axial current in (8.17), with μB being the baryon chemical potential. The chiral magnetic effect and the chiral vortical effect yield charge separation and baryon number separation in the measured spectrum of hadrons [10]. Another possible experimental signature is based on the idea that the axial charge density, in a locally uniform flow of massless fermions, is a measure of the alignment between the fermion spins. When the QCD fluid freezes out and the quarks bind to form hadrons, aligned spins result in spin-excited hadrons. The ratio between spin- excited and low-spin hadron production and its angular distribution may therefore be used as a measurement of the axial charge distribution. An enhancement of spin- excited hadron production along the rotation axis of the collision is predicted [11]. New chiral effects arise in superfluid hydrodynamics [12–14], that is, when there are spontaneously broken symmetries. This is a relevant framework for QCD at high 192 Hydrodynamics and black holes densities and low temperatures. One possible observable effect is the chiral electric effect [13], which is the generation of an electric current perpendicular to the electric field: aμ a μνρσ b c JCEE = c bc uν ξρEσ, (8.19) aμ ab μ compared with the standard electric conductivity term JConduct = σ Eb . Here, cabc are the transport coefficients proportional to the triangle anomaly diagram coefficient a Cabc,andξμ is the phase gradient of the broken symmetry, which is proportional to the velocity of the superfluid part.

8.4 Nonrelativistic fluid flows 8.4.1 The incompressible Navie–Stokes equations Consider the equations of relativistic CFT hydrodynamics without charges. At the first viscous order, they take the form 1 u ∂ T μν = ∂ uμ +3D ln T = σ σμν , (8.20a) ν ν μ 2πT μν 1 P ∂ T μν = a + P μ∂ ln T = P μ(∂ σα − 3σαa ), (8.20b) σν μ σ σ μ 2πT σ α μ μ α μ where D = u ∂μ, aσ = Duσ, and we have used the Einstein gravity relation η/s = 1/4π. The incompressible Navier–Stokes equations can be obtained in the nonrelativistic limit of relativistic hydrodynamics [15, 16]. We expand uμ =(1+ 2 i i 2 v /2+ ···,v ), and T = T0(1 + p + ···), with the scaling v ∼ ε, ∂i ∼ ε, ∂t ∼ ε , and p ∼ ε2, where ε ∼ 1/c. Equation (8.20a) gives the incompressibility condition i ∂iv = 0. Equation (8.20b) gives i j i i i ∂tv + v ∂jv = −∂ p + νΔv , (8.21) where ν =1/4πT0 is the kinematic viscosity.

8.4.2 Turbulent flows Turbulence refers to a state in which a large number of interacting degrees of freedom deviate far from equilibrium. Most nonrelativistic fluid flows in nature are turbulent. We define the dimensionless Reynolds number LV R = , (8.22) e ν where L and V are, respectively a characteristic scale and velocity of the flow. Tur- 3 bulence develops when Re ∼ 10 or larger. In nature, Re is generically large since the kinematic viscosity is generically low. For instance, the kinematic viscosity of water at room temperature is ν 10−6 m2 s−1. Consider a random external force that excites a three-dimensional fluid at a scale L. In general, the statistics of the flow velocity generated by the excitation is not calculable, because of the nonlinear nature of the fluid equations. An important Nonrelativistic fluid flows 193 question arises, however: What are the universal characteristics of the flow statistics? In 1941, Kolmogorov suggested that the velocity statistics in the inertial range of turbulence, at a scales r, with

viscous scale r force scale, is scale-invariant. In order to quantify this, we define the velocity difference at distance r as r δv(r) ≡ [ v( r, t) − v(0,t)] · . (8.23) | r| Scale invariance means that the probability density function P (δv(r)) satisfies δv(r) P (δv(r))δv(r)=F , (8.24) rh where the function F is independent of the force and h is a real number. Thus, the turbulent flow should exhibit a self-similar (fractal) structure. We define the structure functions

n Sn ≡[δv(r)] . (8.25)

S2 is the energy contained in the fluid modes with wavenumbers larger than 1/r. 1 The Kolmogorov dimensional scaling argument fixes h = 3 by the requirement for a constant energy flux [δv(r)]2/[rδv(r)] in the turbulent (direct) cascade. Note that in the direct cascade, the excitation (force) of the fluid is at a large lengthscales, while the dissipation is at small lengthscales. The Fourier transform of the energy E(k) has the Kolomogorov scaling

E(k) ∼ k−5/3. (8.26)

However, it is now well established numerically and experimentally that scale invariance is broken in direct cascades, and one expects a multifractal structure in turbulence and a spectrum of anomalous exponents in the structure functions. In particular, the energy spectrum deviates from the Kolmogorov scaling (8.26). There is experimental and numerical evidence that in the inertial range of distance scales, the flows exhibit ξn the behavior Sn(r) ∼ r . Only in the case n = 3 does the Kolmogorov scaling ξ3 =1 agree with the data. The major open problem of turbulence is to calculate the anomalous exponents ξn. These anomalous exponents are encoded in the geometry of the black hole horizon [17]. One can distinguish between direct and inverse cascades of the fluid flow. In the inverse cascade (such as in two-dimensional incompressible fluids), turbulence occurs at lengthscales exceeding the force scale, and energy is transferred throughout the scales. Here there is numerical and experimental evidence that turbulence exhibits scale-invariant statistics [18] and perhaps also conformal invariance [19]. In order to describe this structure, the black hole horizon should exhibit statistically a fractal structure. 194 Hydrodynamics and black holes

Note also that the study of the universal structure of turbulence in relativistic flows is only at its beginning and very little is known. It is possible to derive a relativistic scaling relation that reduces to Kolmogorov’s relation for S3 [20]; however, we are still missing even a proper definition of the relativistic higher correlation functions analogous to Sn. An important issue in the hydrodynamic description is whether, starting with appropriate initial conditions, where the velocity vector field and its derivatives are bounded, the system can evolve such that it will exhibit within a finite time a blowup of the derivatives of the vector field. Physically, such singularities, if present, indicate a breakdown of the effective hydrodynamic description at long distances and imply that some new degrees of freedom are required. The issue of hydrodynamic singularities has an analogue in gravity: Given appropriate Cauchy data, will the evolving spacetime geometry exhibit a naked singularity, i.e., a blowup of curvature invariants and the energy density of matter fields at a point not covered by a horizon? See [21] for a first step in relating the hydrodynamics and gravity singularities using the Penrose inequality [22].

8.5 Holographic hydrodynamics: the ﬂuid/gravity correspondence 8.5.1 Bulk geometry Consider the ﬁve-dimensional Einstein equations with a negative cosmological constant:

Emn ≡ Rmn +4gmn =0. (8.27)

These equations have a thermal equilibrium asymptotically AdS solution, the boosted black brane. In Eddington–Finkelstein coordinates, the metric reads

(0) m n − μ − 2 μ ν 2 μ ν gmn dx dx = 2uμ dx dr r f(br)uμuν dx dx + r Pμν dx dx , (8.28)

m μ where x =(r, x ),μ =0, ..., 3, uμ is the 4-velocity boost vector, T =1/πb is the temperature, and f(r)=1− 1/r4. The metric (8.28) is a solution of (8.27) when b and uμ are constants. It has an event horizon (null hypersurface) with planar topology at r = b−1, whose normal is m =(r,μ)=(0,uμ). The Bekenstein–Hawking entropy density s =(4b3)−1 and the normal combine to give the entropy current μ 3 −1 μ S =(4b ) u , where we set GN =1. The metric (8.28) is no longer a solution when b(xα)anduμ(xα) are not constants, and needs to be corrected. We seek a solution of the Einstein equations (8.27) by the method of variation of constants:

(0) (1) ··· gmn = gmn + gmn + , (8.29)

(1) α μ α where gmn includes first derivatives of b(x )andu (x ). Imposing AdS asymptotics and regularity at the horizon, we find that the momentum-constraint Einstein equa- r μν tions Eμ = 0 give the boundary CFT hydrodynamic equations ∂μT = 0 as a series Holographic hydrodynamics: the fluid/gravity correspondence 195 expansion in derivatives [23]. The ratio of the shear viscosity to the entropy density that is calculated from the gravitational background takes the universal value η/s =1/4π as a consequence of the requirement that there be no naked singularities. Consider the metric (8.28) and the timelike boundary r →∞, where nm denotes its normal. The boundary stress–energy tensor takes the form [24] ν ∼ 4 ν − ν ν Tμ lim r Kμ Kδμ +3δμ , (8.30) r→∞ ν ∇ ν where Kμ = μn is the extrinsic curvature and K is its trace. It is straightforward to check that this is the stress–energy tensor of ideal CFT hydrodynamics. The boundary stress–energy tensor with the derivative corrections (8.29) is the viscous stress–energy tensor of the CFT hydrodynamics.

8.5.2 Horizon dynamics The CFT hydrodynamic equations can be viewed also as the Gauss–Codazzi equations governing the dynamics of the event horizon [25]. In particular, the null horizon focusing equation is equivalent to the entropy balance law of the hydrodynamic ﬂuid [25, 26]. Here we will discuss the ideal-order CFT hydrodynamics. The viscous case is analyzed in [25].

Geometry of null surfaces As above, the coordinates of the bulk spacetime are xm =(r, xμ). We now consider xμ as coordinates on the horizon H and r as a coordinate transverse to it. We choose r = 0 as the location of the horizon. We denote the null normal to the horizon by m. In fact, it is both normal and tangent to the horizon and tangent to its null m μ generators. In components, l =(0, ). The pullback of the bulk metric gmn into H is the degenerate horizon metric γμν . Its null directions are the generating light rays of ν μ H, i.e., γμν = 0. The Lie derivative of γμν along is the second fundamental form 1 θ = L γ . (8.31) μν 2 μν

(H) We can decompose θμν into a shear tensor σμν and the expansion θ: 1 θ = σ(H) + θγ . (8.32) μν μν 3 μν

In fact, θμν = 0 at the ideal ﬂuid order that we will be working with. Since γμν is degenerate, we cannot use it to deﬁne an intrinsic connection on the null horizon, as could be done for spacelike or timelike hypersurfaces. The bulk spacetime’s connection does induce a notion of parallel transport in H, but only along its null generators. This structure is not fully captured by γμν ; instead, it is encoded ν n by the extrinsic curvature Θμ , which is the horizon restriction of ∇m :

ν ν Θμ = ∇μ . (8.33) 196 Hydrodynamics and black holes

For a non-null hypersurface, the extrinsic curvature at a point is independent of the induced metric. For null hypersurfaces, this is not so. Indeed, lowering an index ν of Θμ with γμν , we get the shear/expansion tensor θμν : ρ Θμ γρν = θμν . (8.34) ν This expresses the compatibility of the parallel transport defined by Θμ with the hori- ν μ zon metric γμν . Contracting Θμ with yields the surface gravity κ, which measures the non-affinity of μ: ν μ ν Θμ = κ . (8.35) ν At the ideal fluid order, Θμ can be written as ν ν μ Θμ = cμ ,cμ = κ. (8.36)

As we will see, the covector cμ encodes in its four components all the ﬂuid data, that is, the temperature and the velocity 4-vector. The null Gauss–Codazzi equations that will give the ideal-order Navier–Stokes equations can be written as ν ν cμ∂ν S +2S ∂[νcμ] =0, (8.37) where we have deﬁned the area entropy current Sμ = vμ,andv is a scalar density equal to the horizon area density. Sμ is the gravitational analog of the hydrodynamic entropy current, which we denoted by sμ.

Hydrodynamic equations

Considering the boosted black brane metric, we can easily check that cμ ∼ Tuμ and ν 3 μ S ∼ T u . Indeed, the covector cμ contains all the hydrodynamic information. Plug- ging these values into the Gauss–Codazzi equations (8.37) and projecting along uμ, we ﬁnd the conservation of the entropy current: 3 μ ∂μ(T u )=θ =0. (8.38) This means that the horizon is nonexpanding at the ideal order. Projecting transverse to uμ, we get the ideal relativistic Euler equation μ aσ + Pσ ∂μ ln T =0. (8.39)

Bulk viscosity We can generalize the fluid/gravity correspondence to charged and nonconformal a hydrodynamics by including bulk gauge Aμ and scalar φi fields, respectively. Now the bulk viscosity is nonzero. Remarkably, it is possible to derive a simple formula for the ratio of bulk to shear viscosities [27] 2 ζ dφH dφH = s i + ρa i , (8.40) η ds dρa i a a where s is the entropy density, ρ are the charges associated with the gauge fields Aμ, H φi are the values of the scalar fields on the horizon, and the derivatives are taken with couplings and mass parameters held fixed. References 197

8.5.3 Outlook The application of the ﬂuid/gravity correspondence to the study of relativistic and nonrelativistic turbulence is the main open challenge. The universal statistical structure of turbulence is encoded in the geometry of the black hole horizon. Clearly, to describe this, the horizon has to exhibit a fractal structure (in the inverse cascade) and a multifractal structure (in the direct cascade). To see this, one is likely to need to use a random force that will make the horizon a random surface.

Acknowledgments It is a great pleasure to thank the organizers and participants of the wonderful and stimulating Les Houches Summer School. I would also like to thank my students and collaborators in this ongoing research. This work is supported in part by the ISF Center of Excellence.

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Gian F. Giudice

Theory Division CERN, Geneva, Switzerland

9 Supersymmetry 199 Gian F. GIUDICE Supersymmetry 201

In the lectures at the Ecole´ de physique des Houches, I presented the motivations for low-energy supersymmetry, the construction of realistic models, the various schemes for generating soft terms (gravity mediation, gauge mediation, anomaly mediation, and gaugino mediation), their collider phenomenology, and their implications for dark matter. The subject is well established and there are excellent reviews and textbooks that fully cover this material. Therefore, rather than documenting the full content of my lectures, I direct readers to appropriate references, where they can find well- organized and exhaustive introductions to supersymmetry. For some textbooks on supersymmetry, see • M. Dine, “Supersymmetry and String Theory: Beyond the Standard Model,” Cambridge University Press, Cambridge, 2007. • J. Terning, “Modern Supersymmetry: Dynamics and Duality,” Oxford University Press, Oxford, 2006. • P. Binétruy, “Supersymmetry: Theory, Experiment, and Cosmology,” Oxford University Press, Oxford, 2006. • H. Baer and X. Tata, “Weak Scale Supersymmetry: From Superfields to Scattering Events,” Cambridge University Press, Cambridge, 2006. • M. Drees, R. M. Godbole, and P. Roy, “Theory and Phenomenology of Sparti- cles: An Account of Four-Dimensional N = 1 Supersymmetry in High-Energy Physics,” World Scientific, Singapore, 2004. • N. Polonsky, “Supersymmetry: Structure and Phenomena. Extensions of the Standard Model,” Springer-Verlag, Berlin, 2001. • S. Weinberg, “The Quantum Theory of Fields. Volume III: Supersymmetry,” Cambridge University Press, Cambridge, 2000. • J. Wess and J. Bagger, “Supersymmetry and Supergravity, 2nd edn,” Princeton University Press, Princeton, 1992. For collections of review articles on various topics related to supersymmetry, see • “Perspectives on Supersymmetry,” ed. G. L. Kane, World Scientific, Singapore, 1998. • “Perspectives on Supersymmetry II,” ed. G. L. Kane, World Scientific, Singapore, 2010. For some reviews on supersymmetry breaking and the various schemes for generating soft terms, see • Y. Shirman, “TASI 2008 Lectures: Introduction to Supersymmetry and Super- symmetry Breaking,” [arXiv:0907.0039 [hep-ph]]. • M. A. Luty, “2004 TASI Lectures on Supersymmetry Breaking,” [hep-th/ 0509029]. • J. Terning, “TASI 2002 Lectures: Nonperturbative Supersymmetry,” [hep-th/ 0306119]. • G. F. Giudice and R. Rattazzi, “Theories with gauge mediated supersymmetry breaking,” Phys. Rep. 322 (1999) 419 [hep-ph/9801271]. • J. A. Bagger, “Weak scale supersymmetry: theory and practice” [hep-ph/ 9604232]. 202 Supersymmetry

Here, instead, I will present the part of my lectures on more recent results, which cannot be found in standard textbooks. I will focus on the lessons that we have learned from preliminary LHC results on Higgs searches. I will concentrate mostly on the Higgs boson, rather than on new physics, not only because the mechanism of electroweak (EW) symmetry breaking is one of the priorities in the LHC program, but also because we have new data on the Higgs and it is exciting to think about where they lead us. The phenomenon of EW symmetry breaking had already been established before the LHC. After LEP, we had ample evidence for gauge structure in interactions (including the triple gauge boson couplings γWW and ZWW) and for the existence of longitudinal components of W and Z. Combining this information with knowledge of gauge boson masses, we conclude that propagating particles do not share the full symmetry of interactions, and thus that the EW symmetry is spontaneously broken. This means that every known phenomenon in particle physics (at least before December 13, 2011) can be described by the Lagrangian

2 1 μν μ v † μ v L = − Tr Fμν F + ifγ¯ Dμf + Tr DμΣ D Σ − √ f¯LΣλf fR +h.c., (9.1) 4 4 2 iT aπa Σ ≡ exp , (9.2) v where πa are the longitudinal polarizations of W and Z,andv = 246 GeV. The first two terms in (9.1) describe the kinetic terms and gauge interactions of the Standard Model (SM) particles. The last two terms contain the effect of the longitudinal polarizations and the mass terms that arise when gauge symmetry is realized nonlinearly. Although the Lagrangian (9.1) was fully satisfying from an experimental point of view, even before December 13 every theorist knew that it could not be the full story. Scattering amplitudes of longitudinal gauge bosons grow like (E/4πv)2, signaling loss of perturbative unitarity, and thus the onset of new phenomena, at E ≈ 4πv =3TeV. One of the goals of the LHC was to discover what the new phenomenon is. It is well known that the simplest option is given by a single real scalar field h,whichformsa complete SU(2) doublet together with πa. So three-quarters of the Higgs had already been found, and the LHC discovered the missing quarter. However, there is no strong motivation, other than simplicity, for choosing a single h, and nature may have good reasons to make different choices. In this respect, hunting for the Higgs is not just looking for the last missing piece of the SM, but it means exploring unknown territory and identifying the nature of the new force responsible for EW breaking, which I will call the fifth force. When we examine the SM, we note that almost all of its open problems originate from Higgs interactions. The flavor problem comes from Yukawa couplings, the hierarchy problem from the Higgs bilinear, the stability problem from the Higgs quartic coupling—and one can add the cosmological constant problem from a constant term in the scalar potential. The crux of these puzzles is that the fifth force is not a gauge Supersymmetry 203 force. Therefore, it lacks the properties of uniqueness, robustness against deformations, and predictivity that are characteristic of gauge theories. In order to discuss what we have learned from Higgs data, I will identify two fundamental questions. The first is: What is the fifth force? We want to know if it is weak or strong, if it is a gauge force or associated with a fundamental scalar. The answer to this question will come from precise measurements of Higgs couplings. These measurements will play the role that precision EW data played at the time of LEP. An important difference is that in the case of Higgs couplings, deviations from the SM expectation could be large (not necessarily of one-loop size) and thus show up even at an early stage. Actually, the more natural the Higgs boson is, the more its properties must deviate from the SM. This is because a natural theory must give large corrections to the Higgs two-point function (to cure the hierarchy problem). These large corrections must also modify the Higgs production rate at the LHC and some of its decay channels, as can easily be seen by inserting two gluons (or two photons) in the Feynman diagram of the Higgs two-point function. This expectation is fully confirmed in all the examples of natural theories known to us. So, measuring the Higgs couplings is the way to probe the fifth force and may be the first way for new physics to show up. The second question is: Is the Higgs natural? This is not an idle question. Its importance goes beyond EW symmetry breaking, and its answer will influence the strategy for future directions in particle physics. Naturalness is a concept fully linked to the use of effective field theories (EFTs). EFT is the tool that we use to implement an intuitive notion: separation of scales. In simple words, separation of scales means that we don’t need to know the motion of every atom inside the Moon to compute its orbit. Or we don’t need to know about quarks to describe physics at the atomic scale. We build a stack of EFTs, one on top of the other, just like a matryoshka doll with one layer inside the other. Each EFT is appropriate to describe a certain energy regime, but it is connected to the next in the sense that free parameters in one layer can be computed in the next layer. One of the most remarkable results of modern physics has been the discovery that at each layer we find simpler physical laws, larger symmetry, and unification of concepts that seemed unrelated in the previous layer. It is amazing that nature works this way, but it is just an empirical fact. We can use the criterion of naturalness in EFT to infer the energy at which the validity of one layer ends and a new layer must set in. Whenever a next layer exists, this procedure gives a reasonable answer, as shown by various examples (electron self-energy, pion mass difference, and neutral kaon mass difference). When applied to the Higgs boson mass, this criterion gives a maximum scale for new physics at about 500 GeV. Since we have not yet found any new physics at the LHC, one may wonder what is the fate of naturalness. The issue is not yet settled. It is quite possible that new physics is just around the corner. After all, the LHC has entered the territory of naturalness, but the exploration is far from complete. The alternative is that the idea of naturalness does not apply to the Higgs because there is a failure of the EFT approach. After all, dark energy could already be taken as evidence for failure of EFT, since the naturalness of the cosmological constant suggests new physics at around 10−3 eV. Holography, gauge/gravity duality, and the AdS/CFT correspondence show 204 Supersymmetry that some theories are much richer than can be captured by a single Lagrangian. The best we can do to describe the physical content is to resort to two different Lagrangians, two dual versions. Maybe this is an indication that our theoretical tools are failing, that an EFT Lagrangian is not able to catch all the underlying physics. There could be connections between small and large scales. A numerological curiosity is that if we combine the largest possible scale (the Hubble length H−1 =1026 m) with the smallest −1 −35 (the Planck length√MP =10 m), we can reproduce the scale of the√ cosmological −3 constant (ΛCC = HMP =5× 10 eV) and the weak scale (ΛEW = ΛCCMP = 5 TeV). If behind this numerical curiosity there is some theoretical infrared/ultraviolet connection, we will never be able to catch it with an EFT. Another approach that would invalidate naturalness is the idea of the multiverse. Out of the process of eternal inflation, a multitude of universes are created, each with its own values of the fundamental constants and its own physical laws. Anthropic arguments then select the kind of universe in which we live in, or, in other words, the physical laws that govern nature. It may sound like a crazy idea to some (bordering on science fiction), but at present the multiverse yields the most convincing explanation of the cosmological constant. A lesson from the multiverse is that some of the questions that we thought to be fundamental may actually be just the result of envir- onmental conditions, and carry no more significance than the shapes of continents or the emergence of a particular animal species in Darwinian evolution. So physics has reached a fork in the road, and the LHC will tell us which path we have to follow. One path follows the road that guided us toward the extraordinary successes of particle physics in the last 100 years or so: a new layer, new symmetry, more unification that bring us closer to a single governing principle of nature. The other path is marked by the failure of naturalness, the collapse of the picture of a multilayered matryoshka doll hiding a single final truth. We have only vague ideas of where this path leads to, and maybe the multiverse is the most concrete construction along this path. It is clear that establishing the fate of Higgs naturalness has far- reaching consequences for particle physics, well beyond the problem of EW breaking. The LHC will teach us which path we have to follow. The first path is very familiar to particle physicists, and promises new discoveries on the road toward final unification. The other path may mean finding the Higgs and nothing else at the LHC. It will force us to abandon naturalness and EFT, and look for new paradigms in a holistic vision, where nature should be seen as a whole and cannot always be reduced to its smaller components. But physics is a natural science, and we want to find answers with the experimental method. While along the first path there are new phenomena to be studied and the goals are clear, how will we be able to make progress if the second path turns out to be true? This is a difficult question, to which I don’t have an answer. However, I want to show how the measured value of the Higgs mass gives us some indirect hints. For the sake of argument, let me assume that the SM with a single Higgs with mass around 125–126 GeV fully describes physics up to a very large energy scale. In this case, as shown in Fig. 9.1, we learn that our universe is in a very critical condition, at the verge of a cosmic catastrophe. It is a remarkable coincidence that we happen to live just at the boundary between two phases: the ordinary Higgs phase and a region Supersymmetry 205

(a) (b) 180 7 8 10 10 9 200 Instability Instability 10 178 1010 11 Nonperturbativity 10 150 176 1012

(GeV) 13 t 10 (GeV)

t Metastability M 1016 M 174 100 Stability Metastability 1,2,3σ 172 1019

Top mass 50 Top pole mass 170 1018 1014 Stability 0 168 0 50 100 150 200 120 122 124 126 128 130 132

Higgs mass Mh (GeV) Higgs pole mass Mh (GeV)

Fig. 9.1 Regions of absolute stability, metastability, and instability of the SM vacuum in the Mt–Mh plane. The frame in (b) zooms in the region of the preferred experimental range of Mh and Mt shown by the rectangular box in (a) (the gray areas in the center denote the allowed regions at 1, 2, and 3σ). The three boundary lines correspond to αs(MZ )=0.1184 ± 0.0007, and the density of the shading indicates the size of the theoretical error. The dotted contour lines show the instability scale Λ in GeV, assuming αs(MZ )=0.1184. where the Higgs ﬁeld slides to very large values. The condition for absolute stability is M [GeV] − 173.1 α (M ) − 0.1184 M [GeV] > 129.4+1.4 t − 0.5 s Z ± 1.0 . h 0.7 0.0007 th

Precise experimental information on the Higgs and top quark masses is needed to ascertain the ultimate fate of the universe and whether we live a stable vacuum or not. The Higgs mass can be measured very precisely at the LHC. The top mass will then become the largest source of uncertainty, and every GeV in Mt counts as a shift of 2 GeV in the Higgs mass. A reduction in the error on the top mass may become the best telescope to peek into the future of our universe. Also, the hierarchy problem can be interpreted as a sign of near-criticality between two phases. The coefficient m2 of the Higgs bilinear in the scalar potential is the order parameter that describes the transition between the symmetric phase (m2 > 0) and the 2 2 − 2 2 broken phase (m < 0). In principle, m could take any value between MP and +MP , but quantum corrections push m2 away from zero toward one of the two endpoints of the allowed range. The hierarchy problem is the observation that in our universe the value of m2 is approximately zero—in other words, its sits near the boundary between the symmetric and broken phases. Therefore, if the LHC result is confirmed, we must conclude that both m2 and λ, the two parameters of the Higgs potential, happen to be near critical lines that separate the EW phase from a different (and inhospitable) 206 Supersymmetry phase of the SM. Is criticality just a capricious numerical coincidence or is it telling us something deep? The occurrence of criticality could be the consequence of symmetry. For instance, supersymmetry implies m2 = 0. If supersymmetry is marginally broken, m2 will remain near zero, solving the hierarchy problem. But if no new physics is discovered at the LHC, we should turn away from symmetry and look elsewhere for an explanation of the near-criticality. It is known that statistical systems often approach critical behaviors as a consequence of some internal dynamics or are attracted to the critical point by the phenomenon of self-organized criticality. As long as no new physics is discovered, the lack of evidence for a symmetry explanation of the hierarchy problem will stimulate the search for alternative solutions. The observation that both parameters in the Higgs potential are quasicritical may be viewed as evidence for an underlying statistical system that approaches criticality. The multiverse is the most natural candidate to play the role of the underlying statistical system for SM parameters. If this vision is correct, it will lead to a new interpretation of our status in the multiverse: our universe is not a special element of the multiverse where the parameters have the peculiarity of allowing for life, but rather it is one of the most common products of the multiverse because it lies near an attractor critical point. In other words, the parameter distribution in the multiverse, instead of being flat or described by simple power laws (as is usually assumed) could be highly peaked around critical lines because of some internal dynamics. Rather than being selected by anthropic reasons,

160 tan β = 50 Split SUSY tan β = 4 tan β = 2 150 tan β = 1

(GeV) 140 h m

High–scale SUSY 130

Higgs mass ExperimentallyE favored 120

110 104 106 108 1010 1012 1014 1016 1018 Supersymmetry breaking scale (GeV)

Fig. 9.2 NNLO (next-to-next-to-leading order) prediction for the Higgs mass Mh in high-scale supersymmetry (lower region) and split supersymmetry (upper region) for tan β = {1, 2, 4, 50}. The thickness of the lower boundary at tan β =1and of the upper boundary at tan β =50 shows the uncertainty due to the present 1σ error on αs (dark-coloured band) and on the top quark mass (light-coloured band). Supersymmetry 207 our universe is simply a very generic specimen in the multitude of the multiverse. If you have complained that string theory has made no predictions about our universe, do not rejoice in this. Now the situation may become even worse: the multiverse is making predictions, but about other universes. What does a Higgs mass of 125–126 GeV tell us about natural theories? A Higgs mass smaller than 120 GeV would have been perfect for natural supersymmetry, while a mass larger than 130 GeV would have excluded the simplest scenarios. If the Higgs mass really is 125 GeV, right in the middle, then it looks like nature wants to tease theoretical physicists. In supersymmetry, a Higgs mass of 125 GeV can be reached— but only for extreme values of the parameters, especially those of the stop squark. Therefore, certain natural setups, where parameters are correlated, are in bad shape (for instance gauge mediation), but the idea of low-energy supersymmetry is not killed. As shown in Fig. 9.2, a Higgs mass of 125 GeV rules out the idea of split supersymmetry with a high scale, say larger than 108 GeV. However, it ﬁts very well with split supersymmetry with a low scale. Actually, the simplest model of split supersymmetry, based on anomaly mediation, predicts a hierarchy between scalars and fermions of a one-loop factor and thus looks like a very satisfactory solution. The indication for a Higgs mass in the range 125–126 GeV is the most exciting result from the LHC so far. It has important consequences for supersymmetry and other theories beyond the SM, but the most puzzling (and surprising) message that we obtained from preliminary LHC data on the Higgs is the apparent near-criticality of the parameters entering the SM Higgs potential.

10 Spontaneous breakdown of local conformal invariance in quantum gravity

Gerard ’t Hooft

Institute for Theoretical Physics and Spinoza Institute, Utrecht University, Utrecht, The Netherlands

10 Spontaneous breakdown of local conformal invariance in quantum gravity 209 Gerard ’t HOOFT

10.1 Introductory remarks 211 10.2 Conformal symmetry in black holes 211 10.3 Local conformal invariance and the stress– energy–momentum tensor 217 10.4 Local conformal symmetry in canonical quantum gravity 220 10.5 Local conformal invariance and the Weyl curvature 225 10.6 The divergent eﬀective conformal action 228 10.7 Nonconformal matter 233 10.8 Renormalization with matter present 237 10.9 The β functions 239 10.10 Adding the dilaton ﬁeld to the algebra for the β functions 242 10.11 Discussion 245 10.12 Conclusions 248 Acknowledgments 250 References 250 Conformal symmetry in black holes 211

10.1 Introductory remarks This chapter comprises the material discussed in the Les Houches Lectures given by the author in 2011. Part of the text is new, while most of it is based on notes that were sent to the Internet around that time [1, 2]. Some background information is added. We think that the insights obtained here are important and will shed new light on the issues of black hole information, quantum divergences, and symmetry properties of quantum gravity. The key word is local conformal symmetry. Basically, this symmetry has always been there, in a hidden form in canonical quantum gravity, but exploiting it adds interesting twists to what is usually told about the quantum theory of gravity and its divergences. It is shown that the black hole complementarity principle can be naturally implemented by treating local conformal invariance as an exact, but spontaneously broken, symmetry of quantum gravity. This allows us to describe the black hole either in terms of the imploding particles or entirely in terms of the Hawking particles coming out. These two complementary representations can then be obtained from one another by means of a local conformal transformation—or, in other words, the black hole scattering matrix is equivalent to a local conformal gauge transformation. Next, one may observe that perturbative canonical quantum gravity, when coupled to a renormalizable model for matter fields, indeed has this conformal symmetry built in, and this symmetry would indeed be exact if the local conformal anomalies were to cancel out. The Einstein–Hilbert action can indeed be regarded as breaking local conformal invariance only dynamically, not explicitly. We show how to disentangle the functional integral over the dilaton component of the metric field from the other integrations over the metric and the matter fields. This turns the remainder of the theory into a trivially conformally invariant system. When the residual metric is treated as a background, and if this background is taken to be flat, this leads to a novel constraint: in combination with the dilaton contributions, the matter Lagrangian should have a vanishing β function. The zeros of this β function are isolated points in the landscape of quantum field theories, and so we arrive at a denumerable, or perhaps even finite, set of quantum theories for matter, where not only the coupling constants, but also the masses, as well as the cosmological constant, are all fixed, and computable, in terms of the Planck units.

10.2 Conformal symmetry in black holes It was Hawking’s great discovery [3, 4] that, according to quantum ﬁeld theory placed in a background metric describing a black hole, black holes emit particles with a thermal distribution, where the temperature is given by

1 kT = , (10.1) 8πGN M where k is Boltzmann’s constant and GN is Newton’s gravitational constant, and furthermore, has been normalized to one. This resulted in a an intuitive picture of 212 Spontaneous breakdown of local conformal invariance in quantum gravity

Hawking Imploding radiation matter in out

Fig. 10.1 Production and decay of a black hole. Particles enter at the left, while implosion into a black hole takes place, and Hawking particles leave at the right, causing the black hole to end its life in an explosion. black holes as active, dynamical systems: a black hole on the one hand can be formed by an implosion of material bodies or by heavy collisions of two or more extremely energetic particles, while on the other hand it can decay. In principle, the decay can take place in the same channels as the formation, but in practice the decay is controlled by the laws of statistical physics: every channel is associated with a Boltzmann factor e−E/kT , and if the energy E of a state is large while its entropy is small, then decay in that state is strongly suppressed. Figure 10.1 illustrates the situation. Most descriptions of black holes treat the imploding matter in a way that is very diﬀerent from the description of the Hawking particles, but what Fig. 10.1 suggests is that one should be able to phrase a theory for black holes that is symmetric under time inversion [5]. Our ﬁrst step in that direction is to postulate a fundamental principle: the black hole complementarity principle [6]. In its most basic form, the principle states the following:

• An observer moving into a black hole can use states inside and outside the black hole to formulate exact rules of the evolution of the entire black hole and its contents. • An observer outside the black hole can use only the states of the particles outside, but also the particles in the Hawking radiation, to formulate how the evolution of the hole and its surrounding particles proceeds. • A mapping should exist to relate these two complementary descriptions.

A more daring assumption is now added to this:

• Both observers may use rules that obey the principles of causality and locality.[1]

Often, this is believed to be impossible, but it is here that we can make progress. Assuming causality and locality means that we can identify light cones, and if any mapping exists between the data of these two diﬀerent observers, then these light cones must be related. Sets of data that are spacelike-separated as seen by one observer must also be spacelike-separated for the other observer, and the same holds for timelike separations. And so we conclude that our two complementary observers agree about the location of the light cones. Conformal symmetry in black holes 213

Does this mean that they can both use the same metric tensor gμν ( x, t)? No, because if we write 2 gμν = ω ( x, t) gμν , (10.2) then both metrics generate the same light cones. Lightlike vectors μ μ ν 2 x obey gμν x x = 0, so factors ω in the metric leave these equations invariant (the square in (10.2) has been put there only for later convenience). Note that we are not performing further coordinate transformations, so modifications of the metric of the form (10.2) are physically different and nontrivial. Only if ω happened to have the form C/(x − a)2 in flat space would this transformation transform a vacuum into a vacuum; in other cases, spaces with Gμν = 0 will transform into spaces with Gμν = 0. Therefore, particles, and their energy–momentum tensors, are not invariant under such transformations. What we now propose is that these transformations are nevertheless physically significant. From now on, local conformal transformations are defined to be transformations that modify the metric tensor as in (10.2); these do not involve coordinate transformations at all. In contrast, global conformal transformations will be defined as an inversion in the spacetime coordinates, with, in addition, possible Poincaré transformations, such that there is an associated local conformal transformation of the form ω = C/(x − a)2. In a global conformal transformation, the vacuum is invariant, but for a local one, in general, it is not. We claim that the symmetry (10.2) is spontaneously broken,ina way very reminiscent of the Brout–Englert–Higgs mechanism [7]. We also claim that ω2( x, t) is not as easily measurable as one might think. An observer who inspects a region of spacetime might be tempted to use light rays, but 2 then, of course, (s)he measures gμν ( x, t) modulo factors w ( x, t). Then, one might say, well, let us measure ω2( x, t) by using clocks and measuring rods; these do depend on ω2( x, t). The problem with that, however, is that if we have a vacuum, then this may be well defined, but if we assume that we may be sitting inside a dense medium, then clocks and measuring rods may be strongly affected by the medium in ways that are difficult to control. Once we know that we are in a vacuum—but how do we know this?—we can use light signals and leave our clocks and measuring rods at infinity. Well, indeed, if they are at infinity, and we have measured a (more or less) flat gμν , assuming that we are close to a vacuum state, then singular factors C/x2 can be excluded, and then our measurement becomes unambiguous. As soon as we drop the information that we are close to a vacuum, we lose control over ω2( x, t). Our two complementary observers may indeed use different values of ω2( x, t)for their description of spacetime. They are not both surrounded by a state that they would call a vacuum, and this may be the cause of a possible disagreement. A slowly decaying black hole may be described by a metric dr2 ds2 = M 2( t˜) − (1 − 2/r)dt2 + + r2 dΩ2 , (10.3) 1 − 2/r where dΩ stands for the angular component, and r and t are scaled by factors M( t˜); the time variable t˜ may be set equal to t or either the advanced or the retarded 214 Spontaneous breakdown of local conformal invariance in quantum gravity time variable. Since the mass M varies only very slowly with time, this makes little difference, as the time differences grow only logarithmically with the separation from the horizon. Now, the observer traveling into a black hole at time t = t0 will only see M( t0 ), while the outside observer will see him squeezed against the horizon, slowly shrinking to a vanishing size. This is a big disagreement! Clearly, the disagreement can be restricted to a conformal factor ω = M( t˜)/M ( t0 ). The conventional view of what happens at a black hole [8] is illustrated in Fig. 10.2(a). For simplicity, the imploding matter is assumed to be in the form of a thin imploding spherical shell, for which case Einstein’s equations are easily solved exactly. The region from which no escape is possible is shaded. Hawking particles are generated near the event horizon. All Hawking particles seem to originate from a spot that can be recovered by solving the field equations backward in time. This gives the geometric shape of the “cradle of Hawking’s radiation” in the form of a caustic. Only in the case of ideal spherical symmetry is this caustic just a single point. If we want a time-reversal-symmetric picture, we should include the backreaction of this Hawking radiation. Often, investigators try to argue that this backreaction is absent[9–11], but this is only according to the view of an observer falling into the

(a) (b) Final caustic

Matter out Horizon

Common Singularity region

Hawking trajectory

Singularity

Imploding

matter Matter in

Initial caustic Horizon caustic

Fig. 10.2 (a) Conventional spacetime picture of a black hole formed by imploding matter. (b) Conformal picture, obtained by stretching the previous picture. Information entering the “common region” may seem to emerge conformally transformed to near the Planck scale, close to the “final caustic.” In reality, it is believed, this information leaves the hole much earlier with the Hawking particles. Conformal symmetry in black holes 215 hole. External observers, of course, do notice the backreaction, simply by noting that the black hole shrinks. This difference, due to the fact that the two observers do not agree about the vacuum state, allows us to draw a time-reversal-symmetric picture as perceived by an outside observer. Thus, the time-reversal-invariant picture that emerges is as illustrated in Fig. 10.2(b). Here, the lightlike geodesics that all complementary observers agree about are sketched for a black hole that decays in a short amount of time (for macroscopic black holes, the picture must be stretched very far in the time direction). Two things have changed in this second picture. One, the outside observer is given the option either to consider the outgoing matter as visible or invisible Hawking radiation or to consider ingoing matter as visible or invisible Hawking radiation. We postulate that the light cones in all these options remain the same. Since in the standard picture in Fig. 10.2(a), Hawking radiation, optically, appears to originate in a regular region of spacetime, the same must be true for the ingoing matter in the time-reversed picture. Therefore, in that picture, what used to be the onset of a singularity must somehow have been transformed into a regular region of spacetime as well. This regular region has also been transformed into a very tiny region; ω( x, t) there is very small. Note that M( t˜) reached the value zero. Therefore, the second thing that must have happened here is a local conformal transformation. The singularity has been transformed to be reduced to infinitesimal size. The local conformal transformation is particularly dramatic in an idealized situation sketched in Fig. 10.3. Here, we have taken a special case where the Hawking radiation is emitted in a single shell of matter. Although the probability for this is suppressed by a huge Boltzmann factor, it is not zero, and therefore it is legitimate to consider the transition amplitude from a collapsing shell to an expanding shell of Hawking matter. The light cones then have an internal region in common that is

(a) (b) (c) Singularity Future horizon Past horizon Singularity

Fig. 10.3 (a) The in-conﬁguration of a black hole. (b) The out-conﬁguration. Dashed lines are the horizons in both pictures. In this case, the ingoing matter and the outgoing matter form a single shell; there is a region at the center that both pictures seem to have in common (c). 216 Spontaneous breakdown of local conformal invariance in quantum gravity

Time OUT-gauge

D A′ D B′ A B

A B C C A′ B′ IN-gauge

Fig. 10.4 Conformal transformation of the inside region of an imploding black hole (“IN- gauge”) to the inside region of an exploding black hole (“OUT-gauge”). The letters show how the various regions are mapped. See text. vacuum both for the conventional ingoing observer and for an observer who only sees the radiation coming out. The only conformal transformation that can do the job here is the inversion; see Fig. 10.4. The circle at the left is mapped to the point t →∞at the right; the cross at the right is mapped to the point t →−∞at the left. In this figure, the complementary pictures are referred to as the IN-gauge and the OUT-gauge. What this means is that the conformal factor ω2( x, t) is chosen such that in the IN-gauge, the component T ++ of the matter energy–momentum tensor is chosen to vanish (ω2 is then such that G++ = 0), and in the OUT-gauge, T −− =0. Indeed, these are gauge conditions on ω( x, t); the black hole S-matrix is thus seen to coincide with a matrix representing a gauge transformation for the conformal factor. See further Section 10.3. The letters in the figure show how the various regions are mapped. Regions A and B are not transformed; here, ω = 1. Region C + D is conformally inverted; Eq. (10.4). In the IN-gauge, the backreaction of the ingoing particles, the thick solid lines, is seen to work on the metric. Hawking particles are the dashed lines. In the time-reversed picture, only the outgoing particles are seen to have an effect on the metric; the imploding matter that gave rise to the black hole has now been plutoed to the status of Hawking radiation—reversed in time. The conformal inversion in region C + D is

μo 2 μ μ − δ xout xin =( 1) 2 , (10.4) |(xout) | where the square is of course the Lorentz-invariant square of the coordinate xout.We have chosen the sign here such that the timelike causal order is preserved. All vectors in the relevant region are timelike with respect to the chosen origin. Figure 10.4 shows the in- and outgoing matter not exactly crossing the singular points of the mapping (the circle and the cross). This is to indicate more clearly the effect of the conformal transformation. In reality, the shaded regions in the picture, regions A and B, are to be squeezed to become infinitesimally thin, so that the bulk of the matter indeed reaches these singularities; but then the conformal transformation is seen to become highly singular. This is indeed what we expect to be the case; the shells of matter are very close to the shells x2 = 0, which implies that the coefficient is very small, close to the Planck scale. Local conformal invariance and the stress–energy–momentum tensor 217

(a)Time = ∞ (b)Time = ∞ (c) Time = ∞ Hawking r = 0 radiation Apparent Singularity singularity r = 0 Imploding matter r = 0 r = ∞ r = 0 r = ∞ Exploding r = ∞ Local matter light cone Singularity r = 0 Hawking matter in Time = −∞ Time = −∞ Time = −∞

Fig. 10.5 (a) Conventional Penrose diagram for a decaying black hole. (b) Its timereverse. (c) The new Penrose diagram for a decaying black hole is the same as that for Minkowski spacetime. The singularity at the center is only there as seen by a distant observer. Local observers can remove it by choosing appropriate local conformal transformations.

We see cosmic censorship [11, 12] at work. In terms of the conventional metric gμν , any naked singularity is hidden behind a horizon. Then, whenever we encounter such a singularity, we can adjust ω in Eq. (10.2) in such a way that a clock shows infinite time when approaching this singularity. Therefore, it now occurs at t = ∞. In practical examples, when we apply local conformal transformations, the singularity will also be smeared out so that it effectively disappears, and furthermore also the horizon disappears, as we see in Fig. 10.4. The Penrose diagram [13] for a black hole is usually sketched [8] as shown in Fig. 10.5(a). Its time-reversed form would be as in Fig. 10.5(b). Note, that now, by using conformal transformations, we can replace both of them by the trivial Penrose diagram, that of flat Minkowski spacetime, Fig. 10.5(c). To obtain this Penrose diagram, a nearly singular function ω( x, t) had to be invoked. As we shall see later, such transformations generate highly peaked expressions for the Ricci scalar (which is not conformally invariant). This means that the outside observer decides that some violent μ explosion took place at the horizon, generating a very large T μ at that point. The local observer, using a different ω( x, t), sees no such singular behavior. We return to this topic in the next section.

10.3 Local conformal invariance and the stress–energy–momentum tensor The above considerations suggest that the scale factor ω(x) is ambiguous, yet it is needed to describe clocks and rulers in the macroscopic world, and it is also needed if we wish to compute the Riemann and Ricci curvatures, because they depend on the entire metric gμν ,notjustˆgμν . Clearly, the world that is familiar to us is not scale-invariant. Without ω, we cannot define distances and we cannot define matter, 218 Spontaneous breakdown of local conformal invariance in quantum gravity but we can define the geometry of the light cones, and, ifg ˆμν is nontrivial, it should describe some of the physical phenomena that are taking place. It should be possible to write down equations for it without directly referring to ω. To what extent will ω(x) be determined byg ˆμν (x), if we impose some physical conditions? For instance, we can impose the condition that the spacetime singularities are all moved to t →∞, so that the interiors of black holes at the moment of collapse, are re-interpreted as the exteriors of decaying black holes, as in Figs. 10.3 and 10.4. After that is done, global conformal transformations are no longer possible, because they would move infinities from the boundary back to finite points in spacetime, which we do not allow. But we do wish to impose some extra “gauge” condition on ω(x), to fix its value, allowing us to define what clocks, rulers, and matter are. A natural thing to impose is that all particles appear to be as light as possible. If μ we had only noninteracting, massless particles, then Tμ = 0. Therefore, let us start with what seems to be a natural gauge to choose: the vanishing of the induced Ricci scalar:

2 R( ω gˆμν )=0. (10.5)

Wheng ˆμν (x) is given, we can compute (in four spacetime dimensions)

−2 αβ Rμν = Rˆμν + ω (4 ∂μω∂ν ω − gˆμν gˆ ∂αω∂βω)

−1 αβ + ω (−2Dμ ∂νω − gˆμν gˆ Dα ∂βω), (10.6)

2 −1 μν ω R = Rˆ − 6 ω gˆ Dμ∂νω. (10.7)

2 Here, we write Rμν (ω gˆμν )=Rμν and Rμν (ˆgμν )=Rˆμν , and similarly for the Ricci scalars. The covariant derivative Dμ treats the ﬁeld ω as a scalar with respect to the metricg ˆμν . The condition that (10.7) vanishes gives us the equation

1 gˆμνD ∂ ω(x)= Rωˆ (x). (10.8) μ ν 6 This happens to be a linear equation for ω(x), which is the reason for having included the square in its definition, in (10.2), since ω is not restricted to be infinitesimal. Now let ω(x) obey (10.8) in a region whereg ˆμν = ημν ,sothatRˆ = 0, and consider the Fourier transform ω(k)ofω(x), which, at k = 0, we now take to be infinitesimal. The wavevector k obeys k2 = 0. Let this wavevector be in the + direction. Then, according to (10.6), the only nonvanishing component of the Ricci tensor, and hence also the only nonvanishing component of the stress–energy–momentum tensor, is the ++ component. Thus, we have shells of massless particles traveling in the x− direction. Adding all possible Fourier components, we see that in the infinitesimal case, our transition to nontrivial ω(x) values leads us from the vacuum configuration to the case that massless particles are flying around. As long as ω(k)atk = 0 stays infinitesimal, these massless particles are noninteracting. Local conformal invariance and the stress–energy–momentum tensor 219

In the case of an evaporating black hole, these massless particles are the Hawking particles. Our complementarity transformation, the transformation that modifies the values of ω(x) while keeping (10.8) valid, switches on and off the effects these Hawking particles have on the metric.1 Often, it will seem to be more convenient to impose that we have flat spacetime, 2 i.e., ω gˆμν = ημν at infinity, rather than imposing R = 0 everywhere. In the case of an implosion followed by an evaporating black hole (for simplicity emitting a single shell of matter as in Section 10.2), this would allow us to glue the different conformal regions together, to obtain Fig. 10.6. This then gives us a region of large Ricci scalar curvature near the horizon. Ricci scalar curvature is associated with a large burst of pressure in the matter stress–energy–momentum tensor. This means that an observer who uses this frame sees a large explosion near the horizon that sends the ingoing material back out; that is, the black hole explodes classically. The local observer would be wondering what causes this “unnatural” explosion, which, at the very last moment, avoided the formation of a permanent black hole; in our theory, however, the local observer would see no reason to use this function ω, so she would not notice any causality violation. Only the distant, outside observer would use this ω, concluding that, indeed, the black hole is not an eternal one, since it evaporates.

Flat

Matter out

R =/ 0 R =/ 0 Schwarzschild Schwarzschild

Time Matter in Flat

Space

Fig. 10.6 Combining the diﬀerent conformal frames into one metric for the imploding and subsequently evaporating black hole gives us a region of large Ricci scalar values R near the horizon (here, the two circles). Note that the metric outside the collapsing and evaporating shells is Schwarzschild, while inside the shells it is ﬂat. Demanding ω to be continuous across the shells implies that the r coordinate should be matched there (see [16]).

1 It is often claimed that Hawking particles have little or no eﬀect on the stress–energy–momentum operator on the horizon [9, 10]. This, however, is true only for the expectation values in the states used by the ingoing observer (the Hartle–Hawking states). The outgoing observer uses Boulware states, for which the Tμν diverges [14, 15]. 220 Spontaneous breakdown of local conformal invariance in quantum gravity

The metric described in Fig. 10.6 is related to the metrics described in [16]. In that paper, care was taken that no conical singularity should arise when ingoing and outgoing shells meet. Here, we do allow this singularity, interpreting it as a region of large R values, which we now demand not to be locally observable.

10.4 Local conformal symmetry in canonical quantum gravity The Einstein–Hilbert action of the generally covariant theory of gravity reads √ 1 S total = d4x −g R + L mat , (10.9) 16πGN where the matter lagrangian L mat is written in a generally covariant manner using mat the spacetime metric gμν (x). In this section, we begin studying the case where L is globally conformally symmetric, which means that under a spacetime transformation μ μ x − a xμ = C + bμ, (10.10) (x − a)2 we have a transformation law for the matter fields such that 2 mat mat gμν (x )=λ(x) gμν (x), −g(x ) L (x )= −g(x) L (x), (10.11) S mat = S mat, so that in n dimensions, L mat (x)=λn L mat(x). (10.12) For the conformal transformation (10.10) we have λ(x)=C/(xμ − aμ)2, which leaves flat spacetime flat, but for curved background spacetimes, where we drop the condition of flatness, λ(x) may be any function of xμ. There are several examples of such conformally invariant matter systems such as N = 4 super Yang–Mills theory in n =4 spacetime dimensions. We will concentrate on n =4. We begin by temporarily assuming conformal invariance of the matter fields, only for convenience; later we will see that allowing matter fields to be more general will only slightly modify the picture. In canonical gravity, the quantum amplitudes are obtained by functionally integrating the exponent of the entire action over all components of the metric tensor at all spacetime points xμ: mat iS total Γ= Dgμν (x) Dϕ (x) e . (10.13)

Although one usually imposes a gauge constraint to reduce the size of function space, this is not formally necessary. In particular, one has to integrate over the common factor ω(x) of the metric tensor gμν (x), when we write

def 2 gμν (x) = ω (x)ˆgμν (x), (10.14) whereg ˆμν (x) may be subject to some arbitrary constraint concerning its overall factor. For instance, in any coordinate frame, one may impose Local conformal symmetry in canonical quantum gravity 221

det(ˆg)=−1, (10.15) besides imposing a gauge condition for each of the n = 4 coordinates. The quantity gˆμν (x) in (10.15) does not transform as an ordinary tensor but as what could be called a “pseudo” tensor, meaning that it scales unconventionally under coordinate transformations with a nontrivial Jacobian. ω(x) is then a “pseudo” scalar. The preﬁx “pseudo” was put in quotation marks here because, usually, pseudo means that the object receives an extra minus sign under a parity transformation; it is therefore preferred to use another phrase. For this reason, we replace “pseudo” by meta, using the words metatensor and metascalar to indicate ﬁelds that transform as tensors or scalars, but with prefactors containing unconventional powers of the Jacobian of the coordinate transformation. Rewriting

Dgμν (x)= Dω(x) Dgˆμν (x), (10.16)

2 while imposing a gauge constraint that only depends ong ˆμν , not on ω, one may mat consider first to integrate over ω(x) and then overg ˆμν (x)andϕ (x). This has peculiar consequences, as we will see. In the standard perturbation expansion, the integration order does not matter. Also, if dimensional regularization is employed, the choice of the functional metric in the space of all fields ω(x) andg ˆμν (x) is unambiguous, since its effects are canceled against all other quartic divergences in the amplitudes (any ambiguity is represented by integrals of the form dnk Pol(k) that vanish when dimensionally renormalized). ω(x) acts as a Lagrange multiplier. Again, perturbation expansion tells us how to handle this integral: in general, ω(x) has to be chosen to lie on a complex contour. The momentum integrations may be carried out in Euclidean (Wick-rotated) spacetime, but, even then, ω(x) must be integrated along a complex contour, which will later (see Section 10.10) be determined to be

ω(x)=1+iα(x),αreal. (10.17)

If ω(x) itself had been chosen real, then the Wick-rotated functional integral would diverge exponentially, so that ω would no longer function properly as a Lagrange multiplier. If there had been no further divergences, one would have expected the following scenario: • The functional integrand ω(x) only occurs in the gravitational part of the action, since the matter ﬁeld is conformally invariant (nonconformal matter does contribute to this integral, but these would be subdominating corrections; see later).

2 μν A ﬁne choice would be, for instance, ∂μgˆ = 0. Of course, the usual Faddeev–Popov quantization procedure is assumed. 222 Spontaneous breakdown of local conformal invariance in quantum gravity

• After integrating over all scale functions ω(x), but not yet overg ˆμν , the resulting eﬀective action in terms ofg ˆμν should be expected to become scale-invariant; i.e. if we were to splitg ˆμν again as in (10.14),

? 2 gˆμν (x) =ˆω (x)gˆμν , (10.18) no further dependence onω ˆ(x) should be expected. • Therefore, the effective action should now describe a conformally invariant theory, both for gravity and for matter. Because of this, the effective theory might be expected to be renormalizable, or even finite! If any infinities do remain, one might again employ dimensional renormalization to remove them. However, this expectation is jeopardized by an apparent difficulty: the ω integration is indeed ultraviolet-divergent [17]. In contrast to the usual procedures in perturbation theories, it is not associated with an infinitesimal multiplicative constant (such as the coupling constant in ordinary perturbation theories), and so a renormalization counterterm would actually represent an infinite distortion of the canonical theory. Clearly, renormalization must be carried out with much more care. Later, in Section 10.6, we suggest various scenarios. First, the main calculation will be carried out, in Section 10.5. Then, the contributions from conformal matter fields are considered, and subsequently the effect of nonconformal matter, by adding mass terms. Finally, we will be in a position to ask questions about renormalization (dimensional or otherwise), and the constraints they may impose on the matter field interactions. We end with some concluding remarks. In conventional canonical quantum gravity, one may split up the quantum gravity functional integral into an integral over the conformally invariant metricg ˆμν and an integral over the conformal factor ω( x, t). Calculations related to the conformal term in gravity, and their associated anomalies, date back from the early 1970s and have been reviewed, for example, in a nice paper by Duff [18]. In particular, we here focus on footnote 4 in that paper. First, we go to n spacetime dimensions, in order later to be able to perform dimensional renormalization. For future convenience (see (10.20)), we choose to then replace the parameter ω by ω2/(n−2), so that (10.14) becomes

4/(n−2) gμν (x)=ω gˆμν (x). (10.19)

In terms ofg ˆμν and ω, the Einstein–Hilbert action (10.9) now reads − n 1 2 4(n 1) μν mat S = d x −gˆ ω Rˆ + gˆ ∂μω∂νω + L (ˆgμν ) . (10.20) 16πGN n − 2 This shows that the functional integral over the ﬁeld ω(x) is a Gaussian one, which can be performed rigorously: it is a determinant. The conformally invariant matter Lagrangian is independent of ω, although we shall later consider mass terms for matter (this would still allow us to do the functional integral over ω, but for now we wish to avoid the associated complications, assuming that, perhaps, at scales close to the Planck scale the ω dependence of matter might dwindle). Local conformal symmetry in canonical quantum gravity 223

We use the caret ( ˆ ) to indicate all expressions deﬁned by the metatensorg ˆμν , such as covariant derivatives, as if it were a true tensor. Note that, in “Euclidean gravity,” the ω integrand has the wrong sign. This is why ω must be chosen to be on the contour (10.17). In practice, it is easiest to do the functional ω integration perturbatively, by writing gˆμν (x)=ημν + κhμν (x),ημν = diag(−1, 1, 1, 1),κ= 8πGN , (10.21) and expanding in powers of κ√(although later we will see that that expansion can sometimes be summed). A factor −gˆ 8(n − 1)/16πGN (n − 2) in (10.20) can be absorbed in the deﬁnition of ω.3 This turns the action (10.20) into n − 2 S = dnx −gˆ 1 gˆμν ∂ ω∂ ω + 1 Rωˆ 2 + L mat(ˆg ) . (10.22) 2 μ ν 2 4(n − 1) μν

Regardless of the ω contour, the ω propagator can be read oﬀ from the action (10.22):

1 P (ω)(k)=− , (10.23) k2 − iε where kμ is the momentum. The iε prescription is the one that follows from the conventional perturbative theory. We see that there is a kinetic term (perturbed by a possible nontrivial spacetime dependence ofg ˆμν ), and a direct interaction, “mass” term proportional to the background scalar curvature Rˆ:

n − 2 → 1 Rˆ n−→4 R,ˆ (10.24) 4(n − 1) 6

The most important diagrams contributing to the eﬀective action for the remaining 2 ﬁeldg ˆμν are those indicated in Fig. 10.7, which include the terms up to O(κ ). The “tadpole” (Fig. 10.7(a)), does not contribute if we apply dimensional regularization, since there is no mass term in the single propagator that we have, (10.23). So, in this approximation, we have to deal with the 2-point diagram only.

(a) (b)

Fig. 10.7 Feynman diagrams for the ω determinant.

3 Note that, therefore, Newton’s constant disappears completely (its use in (10.21) is inessential). This a characteristic feature of this approach. 224 Spontaneous breakdown of local conformal invariance in quantum gravity

We can compute the integral

1 3 − 1 − dnk π 2 n+ 2 23 n(q2) 2 n 2 F (q) def= = . (10.25) 2 − 2 1 − 1 − 1 Eucl k (k q) Γ( 2 n 2 )sinπ(2 2 n)

Now we will also need integrals containing extra factors kμ in the numerator. Therefore, we define 1 dnkk···k k ···k def= . (10.26) 2 − 2 F (q) Eucl k (k q) Then, 1 k = q , (10.27) μ 2 μ 1 k k = (nq q − q2δ ), (10.28) μ ν 4(n − 1) μ ν μν 1 k k k = (n +2)q q q − q2(δ q + δ q + δ q ) , (10.29) μ ν λ 8(n − 1) μ ν λ μν λ νλ μ λμ ν 1 k k k k = [(n + 2)(n +4)q q q q μ ν α β 16(n − 1)(n +1) μ ν α β 2 − q (n + 2)( δμν qαqβ +fiveterms) 4 + q (δμν δαβ + δμαδνβ + δμβδνα)], (10.30) where the five terms are simply the remaining five permutations of the previous term. These expressions can now be used to compute all diagrams that contribute to the ω determinant, but the calculations are lengthy and not very illuminating. More important are those parts that diverge as n → 4. The expression (10.25) for F(q) diverges at n → 4, so that all integrals in (10.26) diverge similarly. By using general covariance, one can deduce right away that the divergent terms must all combine in such a way that they depend only on the Riemann curvature. Dimensional arguments then suffice to conclude that the coefficients must be local expressions in the squares of the curvature. The key calculations for the divergent parts were performed in 1973 [19]. There, it was found that a Lagrangian of the form √ L − − 1 μν 1 2 = g 2 g (x) ∂μϕ∂νϕ + 2 M(x) ϕ (10.31) will generate an effective action, whose divergent part is of the form √ div n div div −g 1 μν − 1 2 1 1 2 S = d x Γ (x), Γ = 8π2(4−n) 120 (Rμν R 3 R )+ 4 (M + 6 R) (10.32) (we use here a slightly modified notation, implying, among others, a sign switch in the definition of the Ricci curvature, and a minus sign as [19] calculated the Lagrangian L +ΔL, ΔL = −Γ div needed to obtain a finite theory). Local conformal invariance and the Weyl curvature 225

In our case, we see that, in the Lagrangian (10.22) (with the dynamical part of ω imaginary), √ 1 −gˆ 1 M = − R,ˆ Γ div = (Rˆ Rˆμν − Rˆ2), (10.33) 6 960π2(4 − n) μν 3 since the second term in (10.32) cancels out exactly. Indeed, it had to cancel out, as we will see shortly. To see what the divergence here means, we use the fact that the mass dependence of a divergent integral typically takes the form C m 1 C(n)mn−4Γ(2 − 1 n) → 1+(n − 4) log → C log Λ + + finite, 2 4 − n Λ 4 − n where m stands for a mass or an external momentum k, and Λ is some reference mass, such as an ultraviolet cutoff. Thus, the divergent expression 1/(4 − n) generally plays the same role as the logarithm of an ultraviolet cutoff Λ.

10.5 Local conformal invariance and the Weyl curvature We consider an eﬀective theory that not only has general covariance,

gˆμν → gˆμν + Dˆ μuν + Dˆ νuμ, (10.34) where uμ(x) are the generators of inﬁnitesimal coordinate transformations and Dˆ μ is the covariant derivative with respect tog ˆμν , but now also has a new kind of gauge invariance, namely, local conformal invariance, which we write in inﬁnitesimal notation, for convenience:

gˆμν → gˆμν + λ(x)ˆgμν . (10.35) Note that this transformation is quite distinct from scale transformations in the coordinate frame, which of course belong to (10.34) and as such are always an invariance of the usual theory. In short, we now have a theory with a ﬁve-dimensional local gauge group. Theories of this sort have been studied in detail [20, 21]. ˆα The Riemann tensor R βμν transforms as a decent tensor under the coordinate transformations (10.34), but it is not invariant (or even covariant) under the local scale transformation (10.35). Now, in four spacetime dimensions, we can split up the 20 independent components of the Riemann tensor into the 10-component Ricci tensor ˆ ˆα Rμν = R μαν (10.36) and the components orthogonal to that, called the Weyl tensor [13, 22],

Wˆ μναβ = Rˆμναβ 1 + (−g Rˆ + g Rˆ + g Rˆ − g Rˆ ) 2 μα νβ μβ να να μβ νβ μα 1 + (g g − g g )R,ˆ (10.37) 6 μα νβ να μβ 226 Spontaneous breakdown of local conformal invariance in quantum gravity which has the remaining 10 independent components. The Weyl tensor has been defined in such a way that it is traceless in every respect; all contractions of two indices yield zero. The transformation rules under coordinate transformations, (10.34), are as usual; all these curvature fields transform as tensors. To see how they transform under (10.35), first note how the connection fields transform:

ˆ → ˆ 1 − O 2 Γαμν (1 + λ)Γαμν + 2 (ˆgαν ∂μλ +ˆgαμ∂νλ gˆμν ∂αλ)+ (λ ), (10.38) from which we derive

ˆ → ˆ 1 ˆ − ˆ − ˆ ˆ Rαβμν (1+λ)Rαβμν + 2 (ˆgαν Dβ∂μλ gˆαμDβ∂νλ gˆβνDα∂μλ+ˆgβμDα∂νλ). (10.39) From this, we ﬁnd how the Ricci tensor transforms:

ˆ → ˆ − ˆ − 1 ˆ 2 ˆ → ˆ − − ˆ 2 Rμν Rμν Dμ∂ν λ 2 gˆμν D λ, R R(1 λ) 3D λ. (10.40) The Weyl tensor (10.37), being the traceless part, is easily found to be invariant (apart from the canonical term):

Wˆ αβμν → (1 + λ)Wˆ αβμν . (10.41)

Since the inverseg ˆμν and the determinantg ˆ of the metric transform as

gˆμν → (1 − λ)ˆgμν , gˆ → (1 + 4λ)ˆg, (10.42) we establish that exactly the Weyl tensor squared yields an action that is totally invariant under local scale transformations in four spacetime dimensions (remember thatg ˆμν is used to contract the indices): 1 L = C −gˆ Wˆ Wˆ αβμν = C −gˆ(Rˆ Rˆαβμν − 2Rˆ Rˆμν + Rˆ2). (10.43) αβμν αβμν μν 3 Now the integral of 1 εμναβεκλγδRˆ Rˆ = Rˆ Rˆμνκλ − 4Rˆ Rˆμν + Rˆ2 (10.44) 4 μνκλ αβγδ μνκλ μν

1 αβγδ is a topological invariant (related to the topological invariant 4 ε Fαβ Fγδ in Yang– Mills theory). Therefore, (10.43) can be further reduced to 1 L =2C −gˆ(Rˆ2 − Rˆ2), (10.45) μν 3 to serve as our locally conformally invariant Lagrangian. The constant C may be any dimensionless parameter. Note that according to (10.40), neither the Ricci tensor nor the Ricci scalar are invariant; therefore, they are locally unobservable at this stage of the theory. Clearly, in view of Einstein’s equation, matter, and in particular its stress–energy–momentum tensor, are locally unobservable Local conformal invariance and the Weyl curvature 227 in the same sense. This will have to be remedied at a later stage, where we must work on redefining what matter is at scales much larger than the Planck scale. Thus, we have verified that, indeed, the action (10.33) is the only expression that we could have expected there (apart from its overall constant), since we integrated out the scale component of the original metric gμν . Demanding locality immediately leads to this expression. The anomaly is the one called ‘type B’ in [26]. Type A has the form of the Gauss–Bonnet term (10.44), which we ignore as long as we work in a topologically flat spacetime. The finite part of the one-loop amplitudes must contain logarithmic expressions, as described in [23]. In fact, gravity theories with this action as a starting point have been studied extensively [20], and the suspicion has been expressed that such theories might be unitary, in spite of the higher time derivatives in the action. Model calculations show [21] that unitarity can be regained if one modifies the Hermiticity condition, which is equivalent to modifying the boundary conditions of functional amplitudes in the complex plane. Effectively then, the fields become complex, and their operators non- Hermitian. The question whether such procedures are acceptable is strongly debated. Note that we have sound physical reasons for the contour rotation (10.17), which is why similar contour rotations as carried out in refs [20, 21] are difficult to dismiss first hand, and we give the cited work the benefit of the doubt. Before following such a route further, we would have to understand the underlying physics. In (10.33), we arrived at the conformal action with an essentially infinite coefficient in front. Before deciding what to do with this infinity, and to obtain more insight in the underlying physics, let us study the classical equations that correspond to this action. To this end, consider an infinitesimal variation hμν on the metric:g ˆμν → gˆμν +δgˆμν , δgˆμν = hμν . The infinitesimal changes in the Ricci tensor and scalar are ˆ 1 ˆ ˆ α ˆ ˆ α − 2 − ˆ α δRμν = 2 (DαDμhν + DαDνhμ D hμν Dμ∂ν hα), (10.46) ˆ − αβ ˆ ˆ ˆ αβ − ˆ 2 α δR = h Rαβ + DαDβh D hα. (10.47) Using the Bianchi identity μ 1 DμR ν = 2 ∂νR, (10.48) the variation of the Weyl action (10.43), (10.45) is then found to be − n − αβ 2R δS = 2C d x ghˆ αβ, with 1 1 2 2R = Dˆ 2Rˆ − Dˆ Dˆ Rˆ − g Dˆ 2Rˆ − 2RˆμRˆ +2Rˆμν Rˆ − RˆRˆ . αβ αβ 3 α β 6 αβ α μβ αμβν 3 αβ (10.49) The classical equations of motion for the Ricci tensor as they follow from the Weyl action are therefore 2R αβ =0. (10.50)

To see their most salient features, let us linearize in Rˆμν and ignore connection terms. 228 Spontaneous breakdown of local conformal invariance in quantum gravity

We get 1 Rˆ − Rδˆ def= S ,∂S = ∂ S ,∂2S − ∂ ∂ S =0. (10.51) μν 6 μν μν μ μν ν αα μν μ ν αα Deﬁning λ(x) by the equation

2 def ∂ λ = − Sαα, (10.52) we ﬁnd that the solution Sμν of (10.51) can be written as

2 Sμν = −∂μ∂νλ + Aμν , with ∂ Aμν =0,Aαα =0,∂μAμν =0. (10.53)

From (10.40), we notice that the free function λ(x) corresponds to the local scale degree of freedom (10.35), while the equation for the remainder Aμν tells us that the Einstein tensor, after the scale transformation λ(x), can always be made to obey the 2 d’Alembert equation ∂ Gμν = 0, which is basically the ﬁeld equation for the stress– energy–momentum tensor that corresponds to massless particles.4 Thus, it is not true that the Weyl action gives equations that are equivalent to Einstein’s equations, but rather that they lead to Einstein equations with only massless matter as their source.

10.6 The divergent effective conformal action Let us finally address our real problem, the divergence of the effective action (10.33) as n → 4. This really spoils the beautiful program we outlined at the beginning of Section 10.5. One can imagine five possible resolutions of this problem.

10.6.1 Option A1: Cancelation against divergences due to matter Besides scalar matter fields, one may have Dirac spinors and/or gauge fields that also propagate in the conformal metricg ˆμν ( x, t). These also lead to divergences. Ignoring interactions between these matter fields, one indeed finds that these fields contribute to the divergence in the effective action (10.33) as well. In fact, all these divergences take the same form of the Weyl action (10.45), and they each just add to the overall coefficient. So, with a bit of luck, one might hope that all these coefficients added up might give zero. That would certainly solve our problem. It would be unlikely that the finite parts of the effective action would also completely cancel out, so we would end up with a perfectly conformally invariant effective theory. A curious problem would have to be addressed, which is that the effective action scales as the fourth power of the momenta of the conformalg ˆμν fields, so there should be considerable concern that unitarity is lost. One might hope that unitarity can be saved by observing that the theory is still based on a perfectly canonical theory where we started off with the action (10.9).

4 Not quite, of course. The statement only holds when these particles form classical superpositions of plane waves such as an arbitrary function of x − t. The divergent eﬀective conformal action 229

Unfortunately, this approach is ruled out for a very simple reason: the matter fields can never cancel out the divergence because they all contribute with the same sign! This is a rather elaborate calculation, of a kind already carried out in the early 1970s [24, 25]. As we are only interested in the part due to the action of scalar, spinor, and vector fields on a conformal background metricg ˆμν , we recapitulate the result of this calculation: it is found that, if the matter fields consist of N0 elementary scalar fields, 1 N1/2 elementary Majorana spinor fields (or 2 N1/2 complex Dirac fields), and N1 real Maxwell or Yang–Mills fields (their mutual interactions are ignored), then the total coefficient C in front of the divergent effective action √ −gˆ 1 Seff = C dnx (Rˆμν Rˆ − Rˆ2) (10.54) 8π2(4 − n) μν 3 is 1 1 1 C = (1 + N )+ N + N . (10.55) 120 0 40 1/2 10 1 Here, the first 1 is the effect of the metascalar component ω of gravity itself. All contributions clearly add up with the same sign. This, in fact, could have been expected from simple unitarity arguments, but as such arguments famously failed when the one- loop β functions for different particle types were considered, it is preferred to do the calculation explicitly. In any case, option A1 is excluded.

10.6.2 Option A2: Add the effects of gravitinos and the gravitational field itself 3 Gravitinos have spin 2 and the gravitational field has spin 2. Why shouldn’t we add these [26, 27] to the anomalies in (10.55)? Indeed, the contribution of gravitinos to − 233 this particular anomaly is known to be 720 N3/2 , so wouldn’t this solve the problem? The problem with this is that supergravity is not power-counting renormalizable, so it must be handled differently. Second, it comes with the spin-2 contributions, claimed 53 to be + 45 N2.

10.6.3 Option B: Make the integral finite with a local counterterm, of the same form as (10.54), but with opposite sign This is the option most physicists who are experienced in renormalization would certainly consider as the most reasonable one. However, a combination of two observations casts serious doubts on the viability of this option. First, in conventional theories where renormalization is carried out, this is happening in the context of a perturbation expansion. The expression that has to be subtracted has a coefficient in front that behaves as gα , (10.56) (4 − n)β where g is a coupling strength, and the power α is usually greater than the power β. If we agree to stick to the limit where first g is sent to zero and then n is sent to 4, the total coefficient will still be infinitesimal, and as such not cause any violation 230 Spontaneous breakdown of local conformal invariance in quantum gravity of unitarity, even if it does not have the canonical form. This is exactly the reason why the consideration of noncanonical renormalization terms is considered acceptable when perturbative gravity is considered, as long as the external momenta of in- and outgoing particles are kept much smaller than the Planck value. Here, however, Newton’s constant has been eliminated, so there is no coupling constant that makes our counterterm small, and of course we consider all values of α the momenta. The Weyl action is quadratic in the Riemann curvature R βμν and therefore quartic in the momenta. As stated when we were considering option A, one might hope that the original expression we found can be made compatible with unitarity because it itself follows from the canonical action (10.9). The counterterm itself cannot be reconciled with unitarity (assuming that the curvature operators have to be Hermitian; see Section 10.5). In fact, not only does it generate a propagator of the form 1/(k2 − iε)2, which at large values of k2 is very similar to the difference of two propagators, 1/(k2 − iε) − 1/(k2 + m2 − iε), where the second would describe a particle with indefinite metric, but also the combination with the total action would leave a remainder of the form 1 [(k2)n/2 − μn−4(k2)2] → (k2)2 log(k2/μ2), (10.57) 4 − n where μ is a quantity with the dimension of a mass that defines the subtraction point. The effective propagator would take a form such as

1 , (10.58) (k2 + m2 − iε)2 log(k2/μ2) which develops yet another pole, at k2 ≈ μ2. This is a Landau ghost, describing something like a tachyonic particle, violating most of the principles that one would like to obey in quantizing gravity. All these objections against accepting a noncanonical renormalization counterterm are not totally exclusive [28], but they are suﬃcient reason to search for better resolutions. For sure, one would have to address the problems, and as yet this seems to be beyond our capacities.

10.6.4 Option C If one follows what actually happens in conventional, perturbative gravity, one might be very much tempted to conclude that it is incomplete: one has to include the contribution of theg ˆμν field itself to the infinity! Only in this way would one obtain the complete renormalization group equations for the coefficient C in the action (10.54). What is more, in some supergravity theories, the conformal anomaly then indeed cancels out to zero. [27].5 However, arguing this way would not at all be in line with the entire approach advocated here: first integrate over the ω field and only then over the fieldsg ˆμν . We here discuss only the integral over ω, with perhaps in addition the matter fields, and this should provide us with the effective action forg ˆμν . If that is no

5 I thank M. Duff for this observation. The divergent effective conformal action 231 longer conformally invariant, we have a problem. Treating C as a freely adjustable, running parameter, even if it turns out not to run anymore, would be a serious threat to unitarity, and would bring us back to perturbative gravity as a whole, with its well-known difficulties. In addition, an important point then comes up: how does the measure of theg ˆμν integral scale? Not only might this also be impossible to reconcile with conformal invariance, but also it might be difficult (if not impossible) to calculate: the measure is only well defined if one fixes the gaugea ` la Faddeev–Popov, and this we wish to avoid, at this stage. We neither wish to integrate overg ˆμν , nor fix the gauge there. In conclusion therefore, we dismiss option C as well. Therefore, yet another option may have to be considered.

10.6.5 Option D: No counterterm is added at all. We accept an infinite coefficient in front of the Weyl action To see the consequences of such an assumption, just consider the case that the coefficient K = C/(4 − n) is simply very large. In the standard formulation of the functional integral, this means√ that the quantum fluctuations of the fields are to be given coefficients going as 1/ K.Theclassical field values can take larger values, but they would act as a background for the quantized fields, and not take part in the interactions themselves. In the limit K →∞, the quantum fluctuations would vanish, and only the classical parts would remain. In short, this proposal would turn theg ˆμν components of the metric into classical fields! There are important problems with this proposal as well: classical fields will not react upon the presence of the other, quantized, fields such as the matter fields. There- fore, there is no backreaction of the metric. This proposal, then, should be ruled out because it violates the action = reaction principle in physics. Furthermore, the reader may already have been wondering about gravitons. They are mainly described by the parts ofg ˆμν that are spacelike, traceless, and orthogonal to the momentum. If we were to insist thatg ˆμν is classical, would this mean that gravitons are classical? It is possible to construct a gedanken experiment with a device that rotates gravitons into photons; this device would contain a large stretch of very strong, transverse magnetic fields. Turning photons into gravitons and back, it would enable us to do quantum interference experiments with gravitons. This then would be a direct falsification of our theory. However, the suspected classical behavior of gravitons comes about because of their interactions with the logarithmically divergent background fluctuations. If the usual renormalization counterterm of the form (10.54) is denied to them, this interaction will be infinite. The magnetic fields in our graviton–photon transformer may exhibit fluctuations that are fundamentally impossible to control; gravitons might still undergo interference, but their typical quantum features, such as entanglement, might disappear. Yet, there may be a different way to look at option D. In previous publications [29], the author has speculated about the necessity to view quantum mechanics as an emergent feature of Nature’s dynamical laws. “Primordial quantization” is the procedure where we start with classical mechanical equations for evolving physical variables, after which we attach basis elements of Hilbert space to each of the possible 232 Spontaneous breakdown of local conformal invariance in quantum gravity configurations of the classical variables. Subsequently, the evolution is re-expressed in terms of an effective Hamiltonian, and further transformations in this Hilbert space might lead to a description of the world as we know it. This idea is reason enough to investigate this last option further. There is one big advantage from a technical point of view. Sinceg ˆμν is now considered to be classical, there is no unitarity problem. All other fields, both the metascalar field ω (the “dilaton”) and the matter fields, are described by renormalizable Lagrangians, so that no obvious contradictions arise at this point. How bad is it that the action = reaction principle appears to be violated? The metric metatensor does allow for a source in the form of an energy–momentum tensor, as described in (10.50)–(10.53). This, however, would be an unquantized source. We get a contradiction if sources are described that evolve quantum mechanically: the background metric cannot react. In practice, this would mean that we could just as well mandate that

gˆμν = ημν ; (10.59) in other words, we would live in a flat background where only the metascalar component of the metric evolves quantum mechanically.6 Could a nontrivial metric tensorg ˆμν be emergent? This means that it is taken either to be classical or totally flat beyond the Planck scale, but it gets renormalized by dilaton and matter fields at much lower scales. This may be the best compromise between the various options considered. Spacetime is demanded to be conformally flat at scales beyond the Planck scale, but virtual matter and dilaton fluctuations generate theg ˆμν as we experience it today. A problem with this argument, unfortunately, is that it is difficult to imagine how dilaton fluctuations could generate a nontrivial effective metric. This is because, regardless of the values chosen for ω(x), the light cones will be those determined byg ˆμν alone, so that there is no “renormalization” of the speed of light at all. We therefore prefer the following view. Consider a tunable choice for a renormalization counterterm in the form of the Weyl action (10.54), described by a subtraction point μ.Ifμ were chosen to be at low frequencies, so at large distance scales, then the Landau ghost (10.58) would be at low values of k2 and therefore almost certainly ruin unitarity of the amplitudes. Only if μ were chosen as far as possible in the ultraviolet would this ghost stay invisible at most physical lengthscales, so the further away we push the subtraction point, the better, but perhaps the limit μ →∞must be taken with more caution. 20 Suppose μ is very large (of order e10 for instance), but not infinite? In this case, the graviton propagator (10.58) would stay very small, so that violation of unitarity would be suppressed by factors such as 10−20. The graviton propagator would become −20 quartic at energies beyond MPlanck × 10 , implying that the gravitational force is effectively screened at distances smaller than the Standard Model mass scale, which has not yet been tested experimentally. A more mainstream standpoint would be the following.

6 This idea goes back to, among others, Nordstr¨om [30]. Nonconformal matter 233

10.6.6 Option E: The action (10.9) no longer properly describes the situation at scales close to the Planck scale

At |k|≈MPlanck, we no longer integrate over ω(k), which has two consequences: a natural cutoff at the Planck scale and a breakdown of conformal invariance. Indeed, this would have given the badly needed scale dependence to obtain a standard interpretation of the amplitudes computed this way. Note that, in our effective action (10.67) below, all dependence on Newton’s constant has been hidden in an effective quartic interaction term for the φ field. That could have been augmented with a “natural” quartic interaction already present in the matter Lagrangian, so we would have lost all explicit references to Newton’s constant. Now, with the explicit breakdown of conformal invariance, we get Newton’s constant back. Adopting this standpoint, it is also easy to see how a subtraction point wandering to infinity, as described in option D, could lead to a classical theory forg ˆμν . It simply corresponds to the classical limit. Letting the subtraction point go to infinity is tan- tamount to forcing MPlanck to infinity, in which limit, of course, gravity is classical. Only if we were to embrace option D fully, would we insist that the physical scale is not determined by MPlanck this way but by adopting some gauge convention at a boundary at infinity. This is the procedure demanded by black hole complementarity. The price paid for option E is that we lose the fundamental advantages of exact conformal invariance, which are a calculable and practically renormalizable effective interaction and a perfect starting point for a conformally invariant treatment of the black hole correspondence principle as was advocated in Section 10.2. The idea advocated in this chapter is not to follow option E representing what would presumably be one of the mainstream lines of thought. With option E, we would have ended up with just another parametrization of nonrenormalizable, perturbative, quantum gravity. Instead, we are searching for an extension of the canonical action (10.9) that is such that the equivalent of the ω integration can be carried out completely.

10.7 Nonconformal matter To generalize to nonconformal matter fields, the easiest case to consider is a scalar field φ(x). Conformally invariant scalar fields are described by the action

√ 1 Lφ = − 1 −g(gμν ∂ φ∂ φ + Rφ2), (10.60) conf 2 μ ν 6 where the second term is a well-known necessity for complete conformal invariance. Indeed, substituting the splitting (10.14), we ﬁnd that the ﬁeld φ(x) must be written as ω−1φˆ(x), and then √ 1 ∂ ω ∂ ω −gR= ω2 −gˆRˆ − 6∂ −gˆ gˆμν ∂ ω − 6 −gˆ gˆμν μ ν , (10.61) μ ω ν ω ω √ ∂ ω ∂ ω −ggμν ∂ φ∂ φ = −gˆ gˆμν ∂ φˆ − μ φˆ ∂ φˆ − ν φˆ , (10.62) μ ν μ ω ν ω 234 Spontaneous breakdown of local conformal invariance in quantum gravity

1 1 Lφ = − −gˆ(ˆgμν ∂ φ∂ˆ φˆ + Rˆφˆ2) + derivative. (10.63) conf 2 μ ν 6 The extra term with the Ricci scalar is in fact the same as the insertion (10.24) in (10.22). Inserting this as our matter Lagrangian leaves everything in Sections 10.4 and 10.5 unaltered. Now, however, we introduce a mass term: √ Lφ,mass Lφ − 1 − 2 2 = conf 2 gm φ . (10.64) After the split (10.14), this turns into Lφ,mass Lφ − 1 − 2 2 ˆ2 = conf 2 gmˆ ω φ . (10.65) Thus, an extra term proportional to ω2 arises in (10.22). But, as it is merely quadratic in ω, we can still integrate this functional integral exactly.7 At n → 4, and remembering that we had scaled out a factor 6/κ2 in going from (10.20) to (10.22), the quantity M in Eq. (10.31) is now replaced by 1 1 M = − Rˆ + κ2m2φˆ2, (10.66) 6 6 and plugging this into the divergence equation (10.32) replaces (10.33) by √ −gˆ 1 1 1 Γ div = (Rˆ Rˆμν − Rˆ2)+ (κ2m2φˆ2)2 , (10.67) 8π2(4 − n) 120 μν 3 144

2 where κ =8πGN . Indeed, the extra term is a quartic interaction term and as such again conformally invariant. κ2m2 is a dimensionless parameter and, usually, it is quite small. The two terms in Eq. (10.67) have to be treated in quite a different way. As was explained in Section 10.5, the first term would require a noncanonical counterterm, which we hesitate to add just like that, so it presents real problems that will have to be addressed. This difficulty does not play any role for the second term. Its divergent part can be renormalized in the usual way by adding a counterterm representing a quartic self-interaction of the scalar field. There will be more subtle complications due to the fact that renormalization of these nongravitational interaction terms in turn often (but not always) destroys scale invariance. As for the matter fields, we return to their contributions in Sections 10.8–10.10. It is important to conclude from the discussion this section that nonconformal matter does not affect the formal local conformal invariance of the effective action after integration over the metascalar ω field. Also, the nonconformal parts, such as the mass terms, do not have any effect on the dangerously divergent term in this effective action.

7 Note that a cosmological constant would add a term CΛω4 to the action, so that the ω integration can then no longer be done exactly. Therefore, for the time being, we omit the cosmological constant. Nonconformal matter 235

In Sections, 10.4–10.6, it was argued that the functional integral in canonical quantum gravity, EH mat mat mat i[S (gμν )+S (gμν ,φ )] Dgμν Dφ e , (10.68) where S EH is the Einstein–Hilbert action and S mat is the action of the matter fields, here abbreviated as φ mat, should be considered to be taken in two steps, 2 gμν ≡ ω gˆμν , det(ˆgμν )=−1, Dgμν = Dgˆμν Dω, (10.69) and the integral over the dilaton field ω should be taken together with the integrations over the matter fields φ mat. Rewriting the Einstein–Hilbert action (including a possible cosmological constant) in terms of ω andg ˆμν , one finds, in four dimensions, 1 S EH = d4x (Rωˆ 2 +6ˆgμν ∂ ω∂ ω − 2Λω4), (10.70) 2κ2 μ ν

2 where κ =8πGN ,andRˆ is the Ricci scalar associated withg ˆμν . Now, let us include those parts of the matter Lagrangian that do not seem to be conformally invariant (though we intend to show that, in our formalism, all interactions are locally conformally invariant). It is convenient to split the Lagrangian of the matter ﬁelds into conformally invariant kinetic parts, mass terms, and interaction terms. In a simpliﬁed notation (later we will be more precise), we have

mat φ = {Aμ(x),ψ(x), ψ¯(x),ϕ(x)}, (10.71) 1 1 1 Lkin = − gˆμαgˆνβF F − gˆμν ∂ ϕ∂ ϕ − Rϕˆ 2 − ψγ¯ μDˆ ψ, (10.72) 4 μν αβ 2 μ ν 12 μ 1 Lmass = − m2ω2ϕ2 − ψωm¯ ψ. (10.73) 2 s d

Here, Fμν = ∂μAν − ∂νAμ; the kinetic term for the Dirac fields is shorthand for the a corresponding expression using a vierbein fielde ˆμ for the metricg ˆμν with its associated connection field. ms stands for the scalar masses and md for the Dirac masses. The 1 ˆ 2 term 12 Rϕ could be removed by a field redefinition, but is kept here for convenience, making the scalar Lagrangian conformally invariant. Interaction between the matter fields must now be written in the form 1 1 Lint = − λϕ4 − ψy¯ ϕ ψ − g ϕ3ω, (10.74) 4! i i 3! 3 where the Yukawa couplings yi could be matrices in the indices labeling the fermion species, and λ and g3 could be 4- and 3-index tensors in the scalar field indices. If we now rescale the ω field, κ ω(x)=√ ω˜(x), (10.75) 6 236 Spontaneous breakdown of local conformal invariance in quantum gravity we notice two things. First, the action for theω ˜ field is now nearly identical to the kinetic term for the ϕ field, both having the same conformal dimension, 1 1 1 L EH = gˆμν ∂ ω∂˜ ω˜ + Rˆω˜2 − κ2Λω4, (10.76) 2 μ ν 12 36 and, second, the mass terms turn into conformally invariant quartic coupling terms between matter fields and theω ˜ field,

Lmass − 1 2 2 2 2 − ¯ = 2 κ˜ ms ω˜ ϕ κm˜ d ψ ωψ,˜ (10.77) √ 4 whereκ ˜ = κ/ 6= 3 πGN has the dimension of an inverse mass. Also, the scalar 3-ﬁeld coupling, which originally was not conformally invariant, now turns into a conformally invariant 4-ﬁeld coupling (see (10.74)),

1 − κg˜ ϕ3ω,˜ (10.78) 3! 3 and a new quartic interaction term for theω ˜ field is generated by the cosmological constant, 1 − κ˜2Λ˜ω4. (10.79) 6 It is important to observe that the Lagrangian (10.76) for the dilaton fieldω ˜ has an overall sign opposite to that of ordinary scalar fields ϕ. For any other field theory, this would be disastrous because it would violate causality. Here, however, the unconventional sign is a necessary consequence of the canonical structure of the theory. Since it is an overall sign, it has no net effect on the Feynman rules; this we will exploit by rotating the field in the complex plane,

ω˜(x) ≡ iη(x), (10.80) so that the new field η(x) will be indistinguishable from other scalar fields, with one important exception: the conformal interaction terms from the original scalar mass terms ms and the Dirac mass terms md in (10.77), as well as the interaction from the original 3-field interaction, (10.78), acquire unconventional factors −1ori. Another important thing to observe is that there is no term at all in the Lagrangian 8 that could serve as a kinetic term for theg ˆμν field. Any such kinetic terms should arise solely from higher-order effects due to the interactions with the matter fields. Naturally, since the entire Lagrangian is now conformally invariant, we should expect the effective Lagrangian forg ˆμν to be conformally invariant as well. It is emphasized in Section 10.4 however that, as is well known [18], this conformal invariance is mutilated by conformal anomalies.

8 Of course, this term reappears if one substitutes η = −i + O(κη˜), whereη ˜ is a small oscillating ﬁeld, by expanding Rˆ. Renormalization with matter present 237

In our present approach, we now propose to postpone any attempts to describe the functional integrals overg ˆμν . Not only do we have anomalies there, but also there are difficulties with unitarity and Landau ghosts.9 As was explained in Section 10.4, the effective interactions with matter and dilaton fields generate an action forg ˆμν that largely coincides with the familiar conformally invariant action obtained from the square of the Weyl curvature, but with an infinite numerical coefficient, which would have to be renormalized. The difficulties associated with that are sufficient reason for us now to postpone this sector of the theory entirely. At first sight, one may well find logical objections to such a procedure: why not also first integrate overg ˆμν before examining whether the amplitudes obtained obey conformal constraints? We argue, however, that theg ˆμν integration is very different from the rest;g ˆμν determines the location of the local light cones, so that it determines the causal relationships between points in spacetime. It may well be that quantum interference of states with light cones at different places will require treatments that differ in essential ways from the standard functional integral. In any case, it is worth investigating what happens if we follow this procedure. The implications are quite remarkable, as we will show.

10.8 Renormalization with matter present mat a Let us assume that the matter fields φ consist of Yang–Mills fields Aμ, Dirac fields ψ,¯ ψ, and scalar fields ϕ, the latter three sets being in some (reducible or irreducible, chiral or nonchiral) representation of the local Yang–Mills gauge group. For brevity, we will write complex scalar fields as pairs of real fields, and if Weyl or Majorana fermions occur, the Dirac fields can be replaced by pairs of these.10 Let us rewrite the Lagrangian for matter interacting with gravity more precisely than in the previous section: 1 1 L(ˆg ,η,φmat)=− Ga Ga − ψ¯ γˆμDˆ ψ − gˆμν (D ϕD ϕ + ∂ η∂ η) μν 4 μν μν μ 2 μ ν μ ν 1 1 − Rˆ(ϕ2 + η2) − V (ϕ) − iV (ϕ)η + κ˜2m2η2ϕ2 − Λ˜η4 12 4 3 2 i i − ¯ 5 5 ψ(yiϕi + iyi γ ϕi + iκm˜ dη)ψ, (10.81) where Gμν is the (non-Abelian) Yang–Mills curvature, and Dμ and Dˆ μ are covariant derivatives containing the Yang–Mills fields;γ ˆμ and Dˆ μ also contain the vierbein fields 5 and connection fields associated withg ˆμν ; the Yukawa couplings yi, yi , and fermion mass terms md are matrices in terms of the fermion indices. The scalar self-interactions V3(ϕ)andV4(ϕ) must be third- and fourth-degree polynomials in the fields ϕi:

9 In [20, 21] claims are made that unitarity can be restored. This, however, requires the integration contours to be rotated in the complex plane such thatg ˆμν becomes complex. Such approaches are interesting and may well serve as good starting points for generating promising theories, but they will not be pursued here. 10 A single Weyl or Majorana fermion then counts as half a Dirac ﬁeld. 238 Spontaneous breakdown of local conformal invariance in quantum gravity

1 1 V (ϕ)= λϕ4 = λijkϕ ϕ ϕ ϕ , (10.82) 4 4! 4! i j k 1 V (ϕ)= g˜ijkϕ ϕ ϕ , g˜ =˜κg . (10.83) 3 3! 3 i j k 3 3

2 ˜ In (10.81), like md,themi δij are also mass matrices, in general. Furthermore, Λ stands 1 2 for 6 κ˜ Λ. Of course, all terms in (10.81) must be fully invariant under the Yang–Mills gauge rotations. They must also be free of Adler–Bell–Jackiw anomalies [31]. Now that the dilaton field η has been included, the entire Lagrangian has been made conformally invariant. It is so by construction, and no violations of conformal invariance should be expected. This invites us to consider the β functions of the theory. Can we conclude at this point that the β functions should all vanish? Let us not be too hasty. In the standard canonical theory, matter fields and their interactions are renormalized. Let us consider dimensional renormalization, and the associated anomalous behavior under scaling. In 4 − ε dimensions, where ε is infinitesimal, the scalar field dimensions are those of a mass raised to the power 1 − ε/2, so that the couplings λ have dimension ε. This means that in most of the terms in the Lagrangian (10.81), the integral powers of η will receive extra factors of the form η±ε or η±ε/2, which will then restore exact conformal invariance at all values for ε.If we follow standard procedures, we accept that η is close to −i, so that singularities at η → 0orη →∞are not considered to be of any significance. Indeed, the limit η → 0 may be seen to be the small-distance limit. This is the limit where gravity goes wrong anyway, so why bother? However, now one could consider an extra condition on the theory. Let us assume that the causal structure, that is, the location of the light cones, is determined by gˆμν and that there exist dynamical laws forg ˆμν . This was seen to be a very useful starting point for a better understanding of black hole complementarity, as was shown in the preceding sections. The laws determining the scale ω(x) should be considered to be dynamical laws, and the canonical theory of gravity itself would support this: formally, the functional integral over the η fields is exactly the same as that for other scalar fields. In view of the above, we do think it is worthwhile to pursue the idea that the η field must be handled just as any other scalar component of the matter fields; but then all fractional powers, in the ε → 0 limit, which would lead to log η terms, must clearly be excluded. Renormalization must be done in such a way that no traces of logarithms are left behind. Certainly then, a scale transformation, which should be identical to a transformation where the fields η are scaled, should not be associated with anomalies. Implicitly, this also means that the region η → 0 is now assumed to be regular. This is the small-distance region, so that our theory does indeed say something nontrivial about small distances. This is why our theory leads to new predictions that eventually should be testable. Predictions follow from the demand that all β functions of the conformal “theory” (10.81) must vanish. Note that one set of terms is absent from (10.81): the terms linear in ϕ and hence cubic in η. This, of course, follows from the fact that, usually, no terms linear in the scalar fields are needed in the standard matter Lagrangians; such terms can easily be The β functions 239 removed by translations of the fields: ϕi → ϕi + ai for some constants ai.Thus,the classical Lagrangian is stationary when the fields vanish: ϕ = 0 is a classical solution. In our present notation, this observation is equivalent to the observation that fields may be freely transformed into one another without modifying the physics. One such transformation is a rotation of one of the scalar fields, say ϕ1, with the η fields: ' ϕ1 → ϕ1 cosh α1 + iη sinh α1, (10.84) η → η cosh α1 − iϕ1 sinh α1, where α1 stands for the original shift of the field ϕ1. The transformation is taken to − 1 2 2 1 2 − 2 be a hyperbolic rotation because the “kinetic term” 2 (∂η + ∂ϕ )= 2 (∂ω˜ ∂ϕ ) in (10.81) has to be invariant. In most cases, these transformations need not be considered, since terms linear in ϕ will in general not be gauge-invariant. Notice also that the Yang–Mills fields are not directly coupled to the η field. In the “classical limit,” η →−i, we can see why this is so. Since not η butω ˜ is real, the invariant quantity is ϕ2 − ω2. Rotating ϕ fields with η fields would therefore be a noncompact transformation, and Yang–Mills theories with noncompact Lie groups usually do not work. There is food for thought here, but we will not yet pursue this line. When constructing a complete theory, the next step should be to consider just any configuration ofg ˆμν ( x, t), formulate the renormalized theory in this background, and finally consider functional integrals overg ˆμν . Unfortunately, this is still too difficult. In a nontrivialg ˆμν background, there will be anomalies depending on the derivatives ofg ˆμν ; there are divergences [2] proportional to the Weyl curvature squared, which ˆ 2 − 1 ˆ2 can be seen to correspond to the field combination Rμν 3 R , and subtracting those leads to new conformal anomalies [18]. It was suggested in Section 10.6, option D,to keep the conformal infinity, which would turn gravitons into “classical” particles, or, more precisely, particles that cannot interfere quantum mechanically. To avoid further complications, we now look at the case wheng ˆμν ( x, t)=ημν , or spacetime is basically flat, although we do keep the field η( x, t).

10.9 The β functions Thus, we return to a theory to which all known quantum field theory procedures can be applied, the only new thing being the presence of an extra, gauge-neutral, spinless field η, and the perfect local scale invariance of the theory. We have arrived at the Lagrangian (10.81), and we wish to impose on it the condition that all its β functions vanish, since conformal invariance has to be kept. As the theory is renormalizable, the number of β functions is always exactly equal to the number of freely adjustable parameters. In other words, we have exactly as many equations as there are freely adjustable unknown variables, so that all coupling constants, all mass terms, and also the cosmological constant should be completely fixed by the equations βi = 0. They are at the stationary points. Masses come in 2 the combinationκm ˜ i and the cosmological constant in the combinationκ ˜ Λ, so all dimensionful parameters of the theory will be fixed in terms of Planck units. 240 Spontaneous breakdown of local conformal invariance in quantum gravity

In principle, there is no reason to expect that any of these fixed points should be extremely close to any of the axes, are not exactly on top of them. Neither masses nor the cosmological constant can be expected to be unnaturally small, at this stage of the theory. In other words, as yet, no resolution of the hierarchy problem is in sight: why are many of the physical mass terms 40 orders of magnitude smaller than the Planck mass, and the cosmological constant more than 120 orders of magnitude? We have no answer to that in this paper, but we will show that the equations are quite complex, and exotic solutions cannot be excluded. The existence of infinitely many solutions cannot be excluded. This is because one can still adjust the composition and the rank(s) of the Yang–Mills gauge group, as well as one’s choice of the scalar and (chiral) spinor representations.11 These form infinite, discrete sets. However, many choices turn out not to have any nontrivial, physically acceptable fixed point at all: the interaction potential terms V (ϕ) must be real and properly bounded, for instance. Searches for fixed points then automatically lead to vanishing values of some or more of the coupling parameters, which means that the symmetries and representations have not been chosen correctly. Every advantage has its disadvantage. Since all parameters of the theory will be fixed, we cannot apply perturbation theory. However, we can make judicious choices of the scalar and spinor representations in such a way that the existence of a fixed point for the gauge coupling to these fields can be made virtually certain. The β function for SU(N) gauge theories with Nf fermions and Ns complex scalars in the elementary representation is

16π2 β(g)=−ag3 − bg5 + O(g7), (10.85) 11 2 1 a = N − N − N , (10.86) 3 3 f 6 s 2 b = O(N ,NNf ,NNs). (10.87)

± 1 Choosing one scalar extra, or one missing, we can have a as small as a = 6 , while a quick inspection of the literature [33, 34] reveals that, in that case, b may still have either sign,

b = ±O(N 2), (10.88) depending on further details, such as the ratio of fermions and scalars, the presence of other representations, and the values of the Yukawa couplings. Choosing the sign of a opposite to that of b, one then expects that a ﬁxed point can be found at

g2 = −b/a = O(1/N 2). (10.89)

This, we presume, is close enough to zero that the following procedure may be assumed to be reliable. Let there be ν physical constants, the νth being the gauge coupling g, which is determined by (10.89). If we take all other coupling parameters

11 which of course must be free of Adler–Bell–Jackiw anomalies [31, 32]. The β functions 241 to be of order g or g2, then the β-function equations are reliably given by the one-loop expressions only, which we will give below. Now these are ν − 1 equations for the ν − 1 remaining coupling parameters, and they are now inhomogeneous equations, since the one coupling, g2, is already fixed. All we have to do now is find physically acceptable solutions. We have already seen that nonabelian Yang–Mills fields are mandatory; we will quickly discover that, beside the η fields, both fermions and other scalar matter fields are indispensable to find any nontrivial solutions. One trivial, yet interesting, solution must be mentioned: N = 4 super Yang–Mills. We take its Lagrangian, and add to that the η field while postulating that this η field does not couple to the N = 4 matter fields at all. Then indeed, all β functions vanish [35]. However, since the η field is not allowed to couple, the physical masses are all strictly zero, which disqualifies the theory physically. Note, however, that the cosmological constant is also rigorously zero. Perhaps the procedure described above can be applied by slightly modifying the representations in this theory, so that a solution with masses close to zero, and in particular a cosmological constant extremely close to zero, emerges. The one-loop β functions are generated by an algebra [36], in which one simply has to plug the Casimir operators of the Yang–Mills Lie group, the types of the representations, the quartic scalar couplings and the fermionic couplings. If we take the scalar fields ϕi and η together as σi, the generic Lagrangian can be written as 1 1 L = − Ga Ga − (D σ )2 − V (σ) − ψ¯ γD − (S + iγ5P )σ ψ, (10.90) 4 μν μν 2 μ i i i i ¯ where σi and ψ, ψ are in general in reducible representations of the gauge group, Dμ is the gauge-covariant derivative, V (σ) is a gauge-invariant quartic scalar potential, and Si and Pi are matrices in terms of the fermion flavor indices. Everything must be gauge-invariant and the theory must be anomaly-free [31, 32]. a La The covariant derivatives contain the Hermitean representation matrices Tij, Uαβ , Ra and Uαβ : ≡ a a Dμσi ∂μσi + iTijAμσj, (10.91) 1 D ψ ≡ ∂ ψ + i(U LaP L + U RaP R)Aa ψ ,PL,R ≡ (1 ± γ5), (10.92) μ α μ α αβ αβ μ β 2 The gauge coupling constant(s) g are assumed to be included in these matrices T and U. The operators P L and P R are projection operators for the left- and right-handed chiral fermions. The group structure constants f abc are also assumed to include a factor g,and they are defined by

[T a,Tb]=if abcT c, [U La,ULb]=if abcU Lc, [U Ra,URb]=if abcU Rc. (10.93) Casimir operators Cg,Cs,andCf will be deﬁned as

apq bpq ab a b ab La Lb Ra Rb ab f f = Cg , Tr (T T )=Cs , Tr (U U + U U )=Cf . (10.94) 242 Spontaneous breakdown of local conformal invariance in quantum gravity

All β functions are given by writing down how the entire Lagrangian (10.90) runs as a function of the scale μ [36]: μ∂ L L 1 L = β( )= 2 Δ , (10.95) ∂μ 8π 1 11 1 2 ΔL = − Ga Gb Cab − Cab − Cab − ΔV − ψ¯(ΔS + iγ5ΔP )δ ψ. 4 μν μν 3 g 6 s 3 f i i i (10.96) Here, 1 3 3 ΔV = V 2 − V (T 2σ) + (δT aT bσ)2 4 ij 2 i i 4 2 2 2 2 + σiVj Tr (SiSj + PiPj) − Tr (S + P ) +Tr[S, P ] , (10.97) where i 2 Vi = ∂V (σ)/∂σ ,Vij = ∂ V (σ)/∂σi∂σj .

It is convenient to deﬁne the complex matrices Wi as

Wi = Si + iPi. (10.98) Then, 1 1 ΔW = W W ∗W + W W ∗K + W W ∗W i 4 k k i 4 i k k k i k 3 3 − (U R)2W − W (U L)2 + W Tr (S S + P P ). (10.99) 2 i 2 i k k i k i If we now write the collection of scalars as {σi = ϕi,σ0 = η}, taking due notice of the factors i in all terms odd in η, we can apply this algebra to compute all β functions of the Lagrangian (10.81). The values of the various β functions depend strongly on the choice of the gauge group, the representations, the scalar potential function, and the algebra for the Yukawa terms, and there are very many possible choices to make. However, the signs of most terms are ﬁxed by the algebra (10.96)–(10.99). By observing these signs in the equation (ΔL = 0), we can determine which are the most essential algebraic constraints they impose on possible solutions. As we will see, they are severely restrictive.

10.10 Adding the dilaton field to the algebra for the β functions Consider the dilaton field η added to the Lagrangian, as in (10.81). This requires extending the indices i,j,... in the Lagrangian (10.90) to include a value 0 referring to the η field. The unusual thing is now that the terms odd in η are purely imaginary, while all terms in (10.81) are of dimension 4. Let us split off this special component. In (10.90), we then write ˜ 4 − 1 2 2 2 V (σ)=Lη 2 mi η ϕi + iV3(ϕ)η + V4(ϕ), (10.100) 5 S0 = imd,P0 =0,Si = yi,Pi = yi . (10.101)

Here, md is the fermionic mass matrix, yi are matrices representing the scalar Yukawa 5 2 couplings, and yi are the pseudoscalar Yukawa coupling matrices. mi δij is the scalar Adding the dilaton field to the algebra for the β functions 243 mass matrix, which is allowed to have negative eigenvalues, so we allow the Higgs mechanism to take place. We henceforth choose modified Planck units by setting 2 4 κ˜ = 3 πGN =1. Note that in the Standard Model, there is no gauge-invariant Dirac mass matrix and no gauge-invariant cubic scalar interaction, so there md and V3 are zero, but we will need to be more general. We write W = S + iP , and now W˜ = S − iP . The algebra (10.96)–(10.99) is then found to become ΔV (σ)=ΔV 0(ϕ)(a) −1 2 − 1 2 2 2 V3i V3ϕi Tr (mdyi)+ ( mi ϕi ) (b) 2 4 3 1 +iη − V (T 2ϕ) + V V + ϕ V Tr (y y + y5y5)+V Tr (m y ) 2 3i i 2 3ij 4ij i 3j i j i j 4i d i −4Tr(m (y ϕ )3)+2Tr([m ,y5ϕ ][y ϕ ,y5ϕ ]) − 2Tr[(y5ϕ )2(m y + y m )ϕ ] d i i d i i j j k k i i d j j d j − 2 − 2 − 2 2 2V3imi ϕi V3 Tr md mj ϕj ϕi Tr (mdyi) (c) 1 1 +η2 − V 2 − V Tr (m y ) − m2V 2 +2Tr((m y ϕ )2)+4Tr(m2(y ϕ )2) 4 3ij 3i d i 2 i 4ii d i i d i i 2 5 2 − 5 2 − 2 5 5 3 2 2 +2 Tr (md(yi ϕi) ) Tr ([md,yi ϕi] ) mj ϕiϕj Tr (yiyj + yi yj )+ mi ϕi(T ϕ)i 2 4 2 2 2 2 − ˜ 2 2 +2mi ϕi +Tr(md)mi ϕi 6Λmi ϕi (d) 3 − 1 2 − 2 3 ˜ +iη mi V3ii mi ϕi Tr (mdyi)+4Tr(md yiϕi)+4Λϕi Tr (mdyi) (e) 2 1 +η4 36Λ˜ 2 − 4ΛTr(˜ m2)+ m4 − Tr (m4) , (f) d 4 i d i (10.102) where V3i = ∂V3/∂ϕi, etc. and we apply the following summation convention: double indices are summed over starting with 1, except the index in the scalar mass matrix 2 mi , which is only summed over if it occurs twice elsewhere as well, or if this is explicitly indicated. ΔV 0 is the expression that we already had, in (10.97). The β coefficients for the Yukawa couplings follow from adding the index 0 to (10.99): 1 ΔW =ΔW 0 − (m2 W + W m2)+m W˜ m − m Tr (m S ) i i 4 d i i d d i d d d i (10.103) 3 1 2 1 2 −iΔW =Δm = − m3 − m Tr (m2)+ (y2 + y5 )m + m (y2 + y5 ) 0 d 2 d d d 4 k k d 4 d k k − 5 5 5 5 + yk md yk yk md yk + yk Tr (yk md)+iγ yk Tr (yk md), (10.104) 244 Spontaneous breakdown of local conformal invariance in quantum gravity

0 where ΔWi stands for the standard β function for the corresponding dimension-4 interaction terms. Before demanding that the β functions all vanish, we observe that we can still allow for an infinitesimal field transformation of the form (10.84) in the original Lagrangian. This adds to the counterterms: ∂V (σ) ∂V (σ) δV (σ)=iαi η − ϕi ∂ϕi ∂η 2 2 = αiV3ϕi + iη αi V4i + mj ϕj ϕi 2 − 3 − 2 − ˜ + η αi ( V3i)+iη αi mi ϕi 4Λϕi , (10.105) and a similar rotation in the Yukawa couplings. This can be used to eliminate the term (e) in (10.102) by adjusting αi; it corresponds to a field shift in the nongravitational case. In most cases, however, such as in the Standard Model, the terms in (e) are forced to vanish anyhow owing to gauge invariance. In a similar way, an infinitesimal chiral rotation among the fermions can be used to eliminate the last term in (10.104). Thus, after the term (e) has been made to vanish by hand, the demand that all β functions vanish, for all values of η, applies in particular to the terms (a)–(d) and (f) in (10.102) and to (10.103) and (10.104). We have already assumed that the nonabelian Yang–Mills field coupling(s) g have a small but nonvanishing fixed point. Through the effects of the group matrices T a,ULa, and U Ra, the coupling(s) g determine the values of the other parameters, by as many coupled nonlinear equations as there are unknowns. It follows that in this theory, we must have nonabelian Yang–Mills fields. In contrast, abelian U(1) components are not allowed, since these do not have fixed points close to the origin. Next, let us consider the requirement that the term (f) in (10.102) vanishes: 1 36Λ˜ 2 =Tr(m4)+4ΛTr˜ (m2) − m4. (10.106) d d 4 i i The right-hand side of this equation resembles a supertrace. Since its sign must be positive, we read off right away that there must be fermions. If, furthermore, we like to have a very small or vanishing cosmological constant Λ, we clearly need that the sum of the fourth power of the Dirac fields (approximately) equals the sum of the fourth powers of the masses of the real scalar particles divided by 4. Can we do without the scalar fields ϕi? Equation (10.106) would have a solution, although the cosmological constant would come out fairly large. However, now there 5 is only one more equation to consider: (10.104), with all Yukawa couplings yi and yi replaced by 0. That gives 3 ? m3 + m Tr (m2) =0. (10.107) 2 d d d Whenever md has a real, nonvanishing eigenvalue, this would imply that the trace of 2 md is negative, an impossible demand. Therefore, our theory also must have scalars ϕi, beside the dilaton field η. Discussion 245

It appears that in today’s particle models, not only the cosmological constant Λ˜ but also the mass terms are quite small, in the units chosen, which are our modified Planck units. Also, if there is a triple scalar coupling V3(ϕ), it appears to be small as well. This is the hierarchy problem, for which we cannot offer any solution other than suggesting that we may have to choose a very complex group structure—as in the landscape scenarios often proposed in superstring theories. Perhaps the small numbers in our present theory are all related. If the masses are indeed all small, then the only large terms in our equations are those that say how the coupling constants and masses run with scale. Our theory suggests that they might stop running at some scale; in any case, a light Higgs particle indeed follows from the demand that the Higgs self-coupling is near an ultraviolet fixed point. The author has not (yet) succeeded in finding a physically interesting prototype model with a nontrivial fixed point; this is a very complex, but interesting, technical problem. Let us briefly set out a strategy. We search for a solution where all mass terms, and of course also the cosmological constant, are small. Start with a theory that has nearly, but not quite, a set of β functions that vanish at one loop. Assume that it has a fixed point near the origin. It is known that (10.96)–(10.99) allow this. For simplicity, let us assume that there are no triple scalar couplings, V3 = 0, and no gauge-invariant scalars, so that the terms (c) and (e) in (10.102) are forbidden by gauge invariance. If we deviate slightly from the fixed point, then the term (b) dictates that we must be at a point where the β 4 function for the scalar self-couplings is negative. This gives us a first guess for mi ,the 2 absolute values of the scalar masses, but not the signs of mi . 5 Knowing the approximate values of the Yukawa couplings yk and yk allows us to fix the Dirac mass matrix md using (10.104). Since we want these masses to be small, we must assume that the pseudoscalar Yukawa couplings largely cancel against the scalar ones in this equation. Then, (10.103) can be obeyed by moving slightly away from the original fixed point in that direction. The only remaining equation is then the vanishing of the term (d), the running 2 2 4 of the scalar mass-squared terms. Knowing V4ii and Tr (md), and assuming that mi 2 is very small, then gives us an equation for mi . Notice that its sign is still free, so that there is some freedom here. Various further attempts to refine this procedure may well lead to interesting models with fixed points. We do note that, apparently, 5 relatively large pseudoscalar Yukawa couplings yi are wanted. Also, we are talking about primary, gauge-invariant Dirac masses, which have to be there because of the term (f) is (10.102), while they do not occur in the presently known Standard Model.

10.11 Discussion Consider a large region of spacetime, filled with light particles scattered here and there. Suppose we ask what this state looks like after a very large Lorentz boost. This question is significant for instance if we consider the neighborhood of a black hole 246 Spontaneous breakdown of local conformal invariance in quantum gravity horizon: a (large) time boost for an external observer corresponds to a (large) Lorentz boost for a local observer. A problem with that is that light particles in one frame transform to extremely energetic ones in the boosted frame. Their energies may become so large that the ensuing backreaction upon the metric may no longer be ignored. As we saw in Section 10.3, the effects on the spacetime metric due to light particles moving nearly with the speed of light can be efficiently represented by a conformal factor in the metric.12 This means that large Lorentz transformations may work as usual only on spaces with ag ˆμν metric, but generate delicate gravitational corrections in the scale function ω(x). This may be seen as a different way to phrase our motivation for claiming that ω(x) should be considered as locally unobservable, in particular when black hole horizons are considered. This way, the Lorentz group can be kept as a perfect symmetry near the Planck scale, but only for theg ˆμν part of the metric. Our theory avoids the need to treat spacetime as “emergent” [37, 38]. Rather, we concentrate upon the demands of causality and locality. The light cones therefore come first, and the scale function ω is of secondary importance. Describing matter in ag ˆμν metric will still be possible as long as we restrict ourselves to conformally invariant field theories, which may perhaps be not such a bad constraint when describing physics at the Planck scale. Of course, that leaves us the question of where Nature’s mass terms come from, but an even more urgent problem is to find the equations for the gravitational field itself, considering the fact that Newton’s constant GN is not scale-invariant at all. The Einstein–Hilbert action is not scale-invariant. Here, we cannot use the Riemann curvature or its Ricci components, but the Weyl component is independent of ω, so that may somehow have to be used. We suspect that, eventually, scales enter into our world in the following way. In- formation is now strictly limited to move along the light cones, since only lightlike geodesics are well defined, not the timelike or spacelike ones. It is generally believed that the amount of information moving around in Nature is limited to exactly one bit in each surface element of size 4 ln 2 squared Planck lengths. Turning this observation around, one might assume that, whatever the equations are, they define information to flow around. The density of this information flow may well define the Planck length locally, and with that all scales in Nature. Obviously, this leaves us with the problem of defining what exactly information is, and how it links with the equations of motion. The notion of information might not be observer-independent, since the scale factor ω isn’t. Quantum mechanics will probably require that all these bits of information form distinct elements of a basis for Hilbert space, as described in [39, 40]. Our theory derives constraints from the fact that matter fields interact with gravity. The basic assumption could be called a new version of relativity: the scalar matter fields should not be fundamentally different from the dilaton field η( x, t). Since there are no singular interactions when a scalar field tends to zero, there is no reason to

12 The configurations described in Section 10.3 are actually limited to superpositions of flat shells of massless matter; this would leave a problem for massless particles that are pointlike. In a quantum theory, this happens to be not such a bad restriction, if we assume that the particles form plane waves, but we will not expand further on this issue here. Discussion 247 expect any singularity when η( x, t) tends to zero at some point in spacetime. Standard gravity theory does have singularities there: this domain refers to the short-distance behavior of gravity, which is usually considered to be “not understood.” What if the short-distance behavior of gravity and matter fields is determined by simply demanding the absence of a singularity? Matter and dilaton then join smoothly together in a perfectly conformally invariant theory. This, however, only works if all β functions of this theory vanish: its coupling parameters must be at a fixed point. There are only discrete sets of such fixed points. Many theories have no fixed point at all in the domain where physical constants are real and positive—that is, stable. Searching for nontrivial fixed points will be an interesting and important exercise. Indeed, all physical parameters, including the cosmological constant, will be fixed and calculable in terms of the Planck units. This may be a blessing and a curse at the same time. It is a blessing because it removes all dimensionless freely adjustable real numbers from our theory; everything is calculable, using techniques known today; there is a strictly discrete set of models, where the only freedom we have is the choice of gauge groups and representations. It is difficult to tell how many solutions there are—the number is probably infinite. This result is also a curse, because the values these numbers have in the real world is a strange mix indeed: the range of absolute values covers some 122 orders of magnitude: ˜ O −122 2 ≈ × −36 Λ= (10 ),μHiggs 3 10 . (10.108) The question where these various hierarchies of very large, or small, numbers come from is a great mystery called the “hierarchy problem.” In our theory, these hierarchies will be difficult to explain, but we do emphasize that the equations are highly complex, and possibly theories with large gauge groups and representations have the potential to generate such numbers. Our theory is a “top-down” theory, meaning that it explains masses and couplings at or near the Planck domain. It will be difficult to formulate any firm predictions about physics at energies as low as the TeV domain. Perhaps we should expect large regions on a logarithmic scale with an apparently unnatural scaling behavior. There is in principle no supersymmetry, although the mathematics of supersymmetry will be very helpful for constructing the first nontrivial models. What is missing furthermore is an acceptable description of the dynamics of the remaining partsg ˆμν of the metric field. In option D in Section 10.6, it was suggested that this dynamics may be non-quantum mechanical, although this does raise the question of howg ˆμν can backreact on the presence of quantum matter. Standard quantum mechanics possibly does not apply tog ˆμν because the notion of energy is absent in a conformal theory, and consequently the use of a Hamiltonian may become problematic. A Hamiltonian can only be defined after coordinates and conformal factor have been chosen, while this is something one might prefer not to do. The author believes that quantum mechanics itself will have to be carefully reformulated before we can really address this problem. Our theory indeed is complex. We have found that the presence of non-Abelian Yang–Mills fields, scalar fields, and spinor fields is required, while U(1) gauge fields 248 Spontaneous breakdown of local conformal invariance in quantum gravity are forbidden (at least at weak coupling, since the β function for the charges here is known to be positive). Because of this, one “prediction” stands out: there will be magnetic monopoles, although presumably their masses will be of the order of the Planck mass. There is one other firm prediction: the constants of nature will indeed be truly constant. Attempts to experimentally observe variations in constants such as the fine structure constant or the proton–electron mass ratio, with time, or position in distant galaxies, are predicted to yield negative results.

10.12 Conclusions Our research was inspired by recent ideas about black holes (Section 10.2). There, it was concluded that an effective theory of gravity should exist where the metascalar component either does not exist at all or is integrated out. This would enable us to understand the black hole complementarity principle, and, indeed, turn black holes effectively indistinguishable from ordinary matter at tiny scales. A big advantage of such constructions would be that, because of the formal absence of black holes, we would be allowed to limit ourselves to topologically trivial, continuous spacetimes for a meaningful and accurate, nonperturbative description of all interactions. This is why we searched for a formalism where the metascalar ω is integrated out first. Let us briefly summarize here how the present formulation can be used to resolve the issue of an apparent clash between unitarity and locality in an evaporating black hole. An observer going into the hole does not explicitly observe the Hawking particles going out. (S)he passes the event horizon at Schwarzschild time t →∞,andfrom his/her point of view, the black hole at that time is still there. For the external observer, however, the black hole has disappeared at t →∞. Owing to the backreaction of the Hawking particles, energy (and possibly charge and angular momentum) has been drained out of the hole. Thus, the two observers appear to disagree about the total stress–energy–momentum tensor carried by the Hawking radiation. Now, this stress–energy–momentum tensor was constructed in such a way that it had to be covariant under coordinate transformations, but this covariance only applies to changes made in the stress–energy–momentum when creation and/or annihilation operators act on it. About these covariant changes, the two observers do not disagree. It is the background subtraction that is different, because the two observers do not agree about the vacuum state. This shift in the background’s source of gravity can be neatly accommodated by a change in the conformal factor ω(x) in the metric seen by the two observers. This we see particularly clearly in Rindler space. Here, we can generate a modification of the background stress–energy–momentum by postulating an infinitesimal shift of the parameter λ(x) in (10.40) and (10.53). It implies a shift in the Einstein tensor Gμν (and thus in the tensor Tμν )oftheform

2 Gμν → Gμν − Dμ∂νλ + gμν D λ. (10.109) Conclusions 249

If now the transformation λ is chosen to depend only on the lightcone coordinate x−, then

2 G−− → G−− − ∂−λ, (10.110) while the other components do not shift. Thus we see how a modification only in the energy and momentum of the vacuum in the x+ direction (obtained by integrating − − G−− over x ) is realized by a scale modification λ(x ). In a black hole, we choose to modify the pure Schwarzschild metric, as experienced by an ingoing observer, by multiplying the entire metric with a function ω2(t) that decreases very slowly from 1 to 0 as Schwarzschild time t runs to infinity. This then gives the metric of a gradually shrinking black hole as seen by the distant observer. Where ω has a nonvanishing time derivative, this metric generates a nonvanishing Einstein tensor, and hence a nonvanishing background stress–energy–momentum. This is the stress–energy–momentum of the Hawking particles. Calculating this stress–energy–momentum yields an apparently disturbing surprise: it does not vanish at spacelike infinity. The reason for this has not yet been completely worked out, but presumably lies in the fact that the two observers disagree not only about the particles emerging from the black hole, but also about the particles going in, and indeed an infinite cloud of thermal radiation filling the entire universe around the black hole. All of this is a sufficient reason to suspect that the conformal (metascalar) factor ω(x) must be declared to be locally unobservable. It is fixed only if we know the global spacetime and after choosing our coordinate frame, with its associated vacuum state. If we did not specify that state, we would not have a specified ω. In “ordinary” physics, quantum fields are usually described in a flat background. Then the choice for ω is unique. Curiously, it immediately fixes for us the sizes, masses, and lifetimes of all elementary particles. This may sound mysterious, until we realize that sizes and lifetimes are measured by using light rays, and then it is always assumed that these light rays move in a flat background. When this background is not flat, becauseg ˆμν is nontrivial, then sizes and time stretches become ambiguous. We now believe that this ambiguity is a very deep and fundamental one in physics. Although this could in principle lead to a beautiful theory, we do hit a real obstacle, which is, of course, that gravity is not renormalizable. This “disease” still plagues our present approach, unless we turn to rather drastic assumptions. The usual idea that one should just add renormalization counterterms wherever needed is found to be objectionable. So, we turn to ideas related to the “primitive quantization” proposal of [29]. Indeed, this quantization procedure assumes a basically classical set of equations of motion as a starting point, so the idea would fit beautifully. Of course, many other questions are left unanswered. Quite conceivably, further research might turn up more alternative options for a cure to our difficulties. One of these, of course, is superstring theory. Superstring theory often leads one to avoid asking certain questions at all, but eventually the black hole complementarity principle will have to be considered, just as the question of the structure of Nature’s degrees of freedom at distance and energy scales beyond the Planck scale. 250 Spontaneous breakdown of local conformal invariance in quantum gravity

Acknowledgments The author thanks C. Kounnas and C. Bachas at the ENS in Paris, S. Giddings [41, 42], R. Bousso, C. Taubes, M. Duﬀ, and P. Mannheim for discussions, and P. van Nieuwenhuizen for his clariﬁcations concerning the one-loop pole terms.

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11 Renormalization group ﬂows and anomalies

Zohar Komargodski

Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot, Israel

11 Renormalization group ﬂows and anomalies 255 Zohar KOMARGODSKI

11.1 Two-dimensional models 257 11.2 Higher-dimensional models 260 References 270 Two-dimensional models 257

This chapter is a brief summary of lectures given at the 2011 Les Houches School “Theoretical Physics to face the Challenge of LHC.” Similar lectures were delivered at the 29th Jerusalem Winter School in Theoretical Physics. The aim of the lectures was to review various aspects of renormalization group ﬂows and anomalies.

11.1 Two-dimensional models We consider Euclidean two-dimensional theories that enjoy the isometry group of R2. Namely, the theories are invariant under translations and rotations in the two space directions. If the theory is local, it enjoys an energy–momentum tensor operator Tμν μ that is symmetric and conserved: Tμν = Tνμ and ∂ Tμν = 0. These equations should be interpreted as operator equations; namely, they must hold in all correlation functions except, perhaps, at coincident points. Here we will study theories where, if possible, these equations hold in fact also at coincident points. In other words, we study theories where there is no local gravitational anomaly. See [1] for some basic facts about theories violating this assumption. We can study two-point correlation functions of this operator. This correlation function is highly constrained by symmetry and conservation. It takes the following most general form in momentum space: 1 2 2 2 Tμν (q)Tρσ(−q) = qμqρ − q ημρ)(qν qσ − q ηνσ + ρ ↔ σ f(q ) 2 (11.1) 2 2 2 +(qμqν − q ημν )(qρqσ − q ηρσ)g(q ).

The most general two-point function is therefore ﬁxed by two unknown functions of the momentum squared. (Note that above we have stripped out the trivial delta function enforcing momentum conservation.) Now let us make a further assumption, that the theory is scale-invariant (but not necessarily conformally invariant). This allows us to ﬁx the two functions f and g up to a constant:1 b d f(q2)= ,g(q2)= . (11.2) q2 q2

μ μ − We can now calculate the two-point function Tμ (q)Tμ ( q) :

μ μ − 2 Tμ (q)Tμ ( q) =(b + d)q . (11.3)

μ μ ∼ (2) Transforming back to position space, this means that Tμ (x)Tμ (0) (b+d) δ (x). In particular, at separated points, the correlation function vanishes. This means that μ the trace itself is a vanishing operator: Tμ = 0 (since it creates nothing from the vacuum, it must be a trivial operator).

1 A logarithm is disallowed since it violates scale invariance. (The rescaling of the momentum would not produce a local term.) 258 Renormalization group ﬂows and anomalies

The operator equation

μ Tμ = 0 (11.4) is precisely the condition for having the full conformal symmetry of R2, SO(3, 1), present. The equation (11.4) is satisfied by all correlation functions at separated points, but it may fail at coincident points. We have already seen this phenomenon in (11.3). Such contact terms signal an anomaly of SO(3, 1). Other than this potential anomaly, (11.4) means that SO(3, 1) is a perfectly good symmetry of the theory. Therefore, we see that scale-invariant theories are necessarily conformally invariant in two dimensions [2].2 More generally, we find ρ − − − 2 Tρ (q)Tμν ( q) = (b + d)(qμqν q ημν ), (11.5) which is again a polynomial in momentum and hence a contact term in position space, consistently with (11.4). One may wonder at this point whether there exist quantum field theories for which these contact terms are absent because b + d = 0. Consider the correlation function T11T11. This has support at separated points. It is proportional to b + d. Therefore, in unitary theories, it follows from reflection positivity that

b + d>0. (11.6)

So far, we have seen that scale-invariant theories in fact enjoy SO(3, 1) symmetry, but the symmetry is afflicted with various contact terms such as (11.5). To see clearly the physical meaning of this anomaly, we can couple the theory to some ambient curved space (there is no dynamics associated with the curved space, it is just a background 2 μν field). This is done to linear order via ∼ d xT hμν , where hμν is the linearized metric gμν = ημν + hμν . Hence, in the presence of a background metric that deviates only slightly from flat space, μ ∼ 2 μ ρσ ∼ 2 ρ σ 2 − ρσ 2 Tμ (0) gμν d x Tμ (0)T (x) hρσ(x) (b + d) d x ∂ ∂ δ (x) η δ (x) hρσ

∼ ρ σ − ρσ (b + d)(∂ ∂ η )hρσ. (11.7)

ρ σ ρσ The ﬁnal object (∂ ∂ − η )hρσ is identiﬁed with the linearized Ricci scalar. In principle, if we had analyzed three-point functions of the energy–momentum tensor and so forth, we would have eventually constructed the entire series expansion of the Ricci scalar. Therefore, the expectation value of the trace of the energy–momentum tensor is proportional to the Ricci scalar of the ambient space. This is the famous two-dimensional trace anomaly. It is conventional to denote the anomaly by c (and not by b + d as we have done so far). The usual normalization is c T = − R. (11.8) 24π

2 Some technical assumptions implicit in the argument above are spelled out in [2]. Two-dimensional models 259 c is also referred to as the “central charge,” but we will not emphasize this algebraic interpretation here. Our argument (11.6) translates to c>0. Using (11.8), we can present another useful interpretation of c. Consider a two- dimensional conformal ﬁeld theory compactiﬁed on a two-sphere S2 of radius a:

4a2 2 2 ds2 = (dx )2, |x|2 = (x )2. (11.9) (1 + |x|2)2 i i i=1 i=1

The Ricci scalar R =2/a2. Because of the anomaly (11.8), the partition function

− 2 L(Φ) ZS2 = [dΦ]e S (11.10) depends on a. (If the theory had been conformal without any anomalies, we would have expected the partition function to be independent of the radius of the sphere.) We ﬁnd that √ √ d c c 2 2 c S2 − S log Z = g T = gR = 2 Vol ( )= . (11.11) d log a S2 24π S2 24π a 3

Thus, the logarithmic derivative of the partition function yields the c anomaly. This particular interpretation of c will turn out to be very useful later. We will now consider non-scale-invariant theories, i.e., theories where there is some conformal ﬁeld theory at short distances, CFTUV, and some other conformal ﬁeld theory (that could be trivial) at long distances, CFTIR. Let us study the correlation functions of the stress tensor in such a case, following [3]. To avoid having to discuss contact terms (which were very important above), we switch to position space. We begin by rewriting (11.1) in position space. In terms of the complex coordinate z = x1 + ix2, the conservation equations are ∂z¯Tzz = −∂zT and ∂zTz¯z¯ = −∂z¯T , where T stands for the trace of the energy– momentum tensor. We can parametrize the most general two-point functions consistent with the isometries of R2:

F (zz,M¯ ) T (z)T (0) = , zz zz z4 G(zz,M¯ ) T (z)T (0) = , (11.12) zz z3z¯ H(zz,M¯ ) T (z)T (0) = . z2z¯2 Here, M stands for some generic mass scale of the theory. As we have seen in our analysis above, (11.1), we know that the conservation equation should bring down the number of independent functions to two. Indeed, we ﬁnd the relations F˙ = −G˙ +3G and H˙ − 2H = −G˙ + G, where X˙ ≡|z2|dX/d|z|2, leaving two real undermined functions (remember that G and F are complex). 260 Renormalization group ﬂows and anomalies

Using these relations, we find that the combination C ≡ F − 2G − 3H satisfies the following differential equation: C˙ = −6H. (11.13) However, since H is positive-definite, this equation means that C decreases monotonically as we increase the distance. Let us now identify C at very short and very long distances. At very short and very long distances, it is described by the appropriate quantities in the corresponding conformal field theories. As we have explained above, in conformal field theory, G and H are contact terms and hence can be neglected as long we do not let the operators collide. On the other hand, F ∼ c. (It is easy to verify that F is sensitive only to the combination b + d as defined in (11.2). Hence, it is only sensitive to c.) This shows that C is a monotonically decreasing function that starts from cUV and flows to cIR. Since the anomalies cUV and cIR are defined inherently in the corresponding conformal field theories, this means that the space of two-dimensional conformed field theories admits a natural foliation, and the renormalization group flow can proceed in only one direction in this foliation. No cycles of the renormalization group are allowed. One can think of c as a measure of degrees of freedom of the theory. In simple renormalization group flows, it is easy to understand that c should decrease, since we merely integrate out some massive particles. However, there are many highly nontrivial renormalization group flows where there are emergent degrees of freedom, and the result that

cUV >cIR (11.14) is a strong constraint on the allowed emergent degrees of freedom. We can integrate (11.13) to obtain a certain sum rule: 2 2 2 cUV − cIR ∼ d log |z |H ∼ d z |z |T (z)T (0) > 0. (11.15)

Since c can also be understood as the path integral over the 2-sphere, the inequality (11.14) can also be interpreted as a statement about the partition function of the massive theory on S2.

11.2 Higher-dimensional models Having understood the two-dimensional case, the main question that comes to mind is whether there exists a function in three and higher dimensions satisfying something similar to (11.14). The problem consists of identifying a candidate quantity that could satisfy such an inequality and then proving that it indeed does so. There are various ways to define quantities in higher-dimensional field theories that share some common features with c. For example, in conformal theories in two dimensions, c is equivalent to the free energy density of the system, divided by the appropriate power of the temperature. One could define a similar Higher-dimensional models 261 object in higher-dimensional field theories. However, one quickly finds that it is not monotonic [4]. This already shows that such inequalities are quite delicate, and they fail if one chooses to measure the number of effective degrees of freedom in the wrong way (albeit a very intuitive and seemingly natural way).

11.2.1 Three-dimensional models Progress with the problem of identifying a candidate quantity generalizing (11.14) has happened quite recently [5, 6]. The conjecture arose independently from studies in AdS/CFT and from studies of N = 2 supersymmetric three-dimensional theories. Any conformal ﬁeld theory on R3 can be canonically mapped to a theory on the curved space S3. This is because S3 is stereographically equivalent to ﬂat space (thus, themetriconS3 is conformal to R3). In three dimensions, there are no trace anomalies, and hence the partition function over S3 has no logarithms of the radius. (This should be contrasted with the situation in two dimensions, (11.11).) Consider

− 3 L(Φ) ZS3 = [dΦ]e S . (11.16)

This is generally divergent and takes the form (for a 3-sphere of radius a)

3 log ZS3 = c1(Λa) + c2(Λa)+F. (11.17)

Terms with inverse powers of Λ are dropped since they are not part of the continuum theory. Since this is a conformal field theory, Λ is the only scale (of course, a ficti- tious scale!). The √ constants √ c1 and c2 are nonuniversal and can be removed with the counterterms g and gR. However, no counterterm can remove F .3 Imagine a three-dimensional flow from some CFTUV to some CFTIR. We can (in principle) then compute FUV and FIR via the procedure above. The conjecture is

FUV >FIR. (11.18)

Let us outline the computation of F with simple examples. Take a free massless L 1 2 scalar = 2 (∂Φ) . To put it in a curved background while preserving conformal invariance (more precisely, Weyl invariance), we write in d dimensions 1 √ d − 2 S = d3x g (∇Φ)2 + R[g]φ2 . (11.19) 2 4(d − 1)

This coupling to the Ricci scalar is necessary to preserve Weyl invariance. Weyl invariance means that the action is invariant under rescaling the metric by any function.

3 More precisely, one can have the gravitational Chern–Simons term, but this cannot aﬀect the real part of F . We disregard the imaginary part of F in our discussion. 262 Renormalization group ﬂows and anomalies

We achieve this by accompanying the action on the metric with some action on the ﬁelds. For the action above, Weyl invariance means that the action is invariant under − 1 − g → e2σg, φ → e 2 (d 2)σφ. (11.20)

We can now compute the partition function on the 3-sphere by diagonalizing the − 3 1 −∇2 1 corresponding diﬀerential operator log ZS = 2 log det + 8 R . The Ricci scalar is related to the radius in three dimensions via R =6/α2. The eigenfunctions are of course well known. The eigenvalues are 1 3 1 λ = n + n + , n a2 2 2 and their respective multiplicities are

2 mn =(n +1) . The free energy on the 3-sphere due to a single conformally coupled scalar is therefore ∞ 1 3 1 − log ZS3 = m −2 log(μ a) + log n + + log n − . (11.21) 2 n 0 2 2 n=0

We have inserted an arbitrary scale μ0 to soak up the dependence on the radius of the sphere. Since there are no anomalies in three dimensions, we expect that there will be no dependence on μ0 eventually. This sum clearly diverges and needs to be regulated.$ We choose to regulate it − using the zeta function. We find that with this regulator, n=0 mn = ζ( 2) = 0, and therefore a logarithmic dependence on the radius is absent, as anticipated. We are left with 1 d 1 1 1 1 3ζ(3) F = − 2ζ s − 2, + ζ s, = 2 log 2 − ≈ 0.0638. scalar 2 ds 2 2 2 16 π2 We can perform a similar computation for a free massless Dirac fermion field, and we find log 2 3ζ(3) F = + ≈ 0.219. fermion 4 8π2 The absolute value of the partition function of a massless Majorana fermion is just one-half of the result above. We see that the counting of degrees of freedom is quite nontrivial. An interesting fact is that a nonzero contribution to F arises also from topological degrees of freedom. This has to be contrasted with the situation in two dimensions, where c was defined through a local correlation function and hence was oblivious to topological matter. For example, let us take Chern–Simons theory associated with some gauge group G: k 2 S = Tr A ∧ dA + A ∧ A ∧ A , (11.22) 4π M 3 Higher-dimensional models 263 where k is called the level. This theory has no propagating degrees of freedom. Indeed, the equation of motion is

0=F = dA + A ∧ A, which means that the curvature of the gauge field vanishes everywhere. Such gauge fields are called flat connections. The space of flat connections on the manifold M is fixed completely by topological properties of the manifold. The partition function of Chen–Simons theory on the 3-sphere has been discussed 1 in [7]. In particular, for U(1) Chen–Simons theory the answer is 2 log k, while for U(N)itis

− N N1 πj F (k, N)= log(k + N) − (N − j) log 2sin . (11.23) CS 2 k + N j=1

We see that the contribution from a topological sector can in fact be arbitrarily large as we take the level k to be large. Let us now check the inequality (11.18) in a simple flow. We can start from the conformal field theory described by U(1)k Chen–Simons theory coupled to Nf Dirac fermions of charge 1. This is a conformal field theory because the Lagrangian has no coupling constant that can run. (The Chen–Simons coefficient is discrete because it is topological in nature.) This conformal field theory is weakly coupled when k>>1. Hence, the F coefficient is 1 log 2 3ζ(3) F ≈ log k + N + . UV 2 f 4 8π2

Let us now deform this by a mass term. The fermions disappear, but there is a pure Chen–Simons term in the infrared with a shifted level k±Nf /2, where the sign depends on the sign of the mass term. Hence,

1 F ≈ log (k ± N /2) , IR 2 f and we can convince ourselves that in the regime where our analysis is valid,

FUV >FIR holds true. There are many more complicated examples that have been checked, all of which are consistent with the conjecture. In particular, a rather general argument for renormalization group flows in N = 2 theories can be devised [8, 9]. There is not yet a conventional field-theoretic proof of this inequality (11.18), but an ingenious construction relating (11.18) with the entanglement entropy has been given [10]. There, the inequality follows from some inequalities satisfied by the density matrix. Various issues with this construction are discussed in [11]. 264 Renormalization group flows and anomalies

11.2.2 Four-dimensional models We saw that in two dimensions, the natural monotonic property of the renormalization group evolution was tightly related to the trace anomaly in two dimensions. In three dimensions, the main role was played by the 3-sphere partition function (there are no trace anomalies in three dimensions). In four dimensions, there are two trace anomalies, and the monotonic property of ﬂows is concerned again with these anomalies. The anomalous correlation function is now

Tμν (q)Tρσ(p)Tγδ(−q − p). And again, as in our analysis in two dimensions, there are contact terms that are μ necessarily inconsistent with Tμ = 0. In four dimensions, it turns out that there are two independent trace anomalies. Introducing a background metric field, we have μ − 2 Tμ = aE4 cW , (11.24) 2 − 2 2 2 2 − 2 1 2 where E4 = Rμνρσ 4Rμν +R is the Euler density and Wμνρσ = Rμνρσ 2Rμν + 3 R is the Weyl tensor squared. These are called the a-andc-anomalies, respectively. It was conjectured in [12] (and shortly after studied extensively in perturbation theory in [13, 14]) that if the conformal field theory in the ultraviolet, CFTUV,is deformed and flows to some CFTIR, then

aUV >aIR. (11.25) The four-dimensional c-anomaly does not satisfy such an inequality (this can be seen by investigating simple examples), and also the free energy density divided by the appropriate power of the temperature does not satisfy such an inequality. In two and three dimensions, we have seen that the quantities satisfying inequalities like (11.14) and (11.18) are computable from the partition functions on S2 and S3, respectively. Similarly, in four dimensions, since the 4-sphere is conformally flat, the partition function on S4 selects only the a-anomaly. Indeed, √ √ μ 2 4 − − − ∂log r log ZS = g Tμ = a gE4 = 64π a. S4 S4 In this formula, r stands for the radius of S4. Real scalars contribute to the anomalies (a, c) = [90(8π)2]−1(1, 3), a Weyl fermion (a, c) = [90(8π)2]−1(11/2, 9), and a U(1) gauge field (a, c) = [90(8π)2]−1(62, 36). We will now present an argument for (11.25), but first it is useful to repeat the two-dimensional story from a new point of view. The main idea is to promote various coupling constants to background fields [15, 16].

Two-dimensional models revisited Imagine any renormalizable quantum field theory (in any number of dimensions) and set all the mass parameters to zero. The extended symmetry includes the full conformal group. If the number of spacetime dimensions is even, then the conformal group has Higher-dimensional models 265 trace anomalies. If the number of spacetime dimensions is of the form 4k + 2, then there may also be gravitational anomalies. We will continue to ignore gravitational anomalies here. Upon introducing the mass terms, one violates conformal symmetry explicitly. Thus, in general, the conformal symmetry is violated both by trace anomalies and by μ an operatorial violation of the equation Tμ = 0 in flat spacetime. The latter violation can always be removed by letting the coupling constants transform. Indeed, let us replace every mass scale M (either in the Lagrangian or associated with some cutoff) by Me−τ(x), where τ(x) is some background field (i.e., a function of spacetime). Then the conformal symmetry of the Lagrangian is restored if we accompany the ordinary conformal transformation of the fields by a transformation of τ. To linear order, τ(x) ∼ d μ always appears in the Lagrangian as d xτTμ . Setting τ =0,wearebackto the original theory, but we can also let τ be some general function of spacetime. The variation of the path integral under such a conformal transformation that also acts on τ(x) is thus fixed by the anomaly of the conformal theory in the ultraviolet. This idea allows us to study some questions about general renormalization group flows using the constraints of conformal symmetry. We will sometimes refer to τ as the dilaton. Consider integrating out all the high-energy modes and flow to the deep infrared. Since we do not integrate out the massless particles, the dependence on τ is regular and local. As we have explained, the dependence on τ is tightly constrained by the conformal symmetry. Since in even dimensions the conformal group has trace anomalies, these must be reproduced by the low-energy theory. The conformal field theory at long distances, CFTIR, contributes to the trace anomalies, but to match to the defining ultraviolet theory, the τ functional has to compensate precisely for the difference between the anomalies of the conformal field theory at short distances, CFTUV,and the conformal field theory at long distances, CFTIR. Let us see how these ideas are borne out in two-dimensional renormalization group flows. Let us study the constraints imposed by conformal symmetry on action functionals of τ (which is a background field). An easy way to analyze these constraints is to introduce a fiducial metric gμν into the system. Weyl transformations act on 2σ the dilaton and metric according to τ → τ + σ and gμν → e gμν . If the Lagrangian for the dilaton and metric is Weyl-invariant, then on setting the metric to be flat, one finds a conformally invariant theory for the dilaton. Hence, the task is to classify local diffeomorphism (diff) × Weyl-invariant Lagrangians for the dilaton and metric background fields. −2τ It is convenient to defineg ˆμν = e gμν , which is Weyl-invariant. √ At the level of two derivatives, there is only one diff × Weyl-invariant term: gˆRˆ. However, this is a topological term, and so it is insensitive to local changes of τ(x). Therefore, if one 2 2 starts from a diff×Weyl-invariant theory, on setting gμν = ημν ,theterm d x (∂τ) is absent because there is no appropriate local term that could generate it. The key is to recall that unitary two-dimensional theories have a trace anomaly c T μ = − R. (11.26) μ 24π 266 Renormalization group flows and anomalies

We must therefore allow the Lagrangian to break Weyl invariance, such that the Weyl variation of the action is consistent with (11.26). The action functional that reproduces the two-dimensional trace anomaly is c √ S [τ,g ]= g τR+(∂τ)2 . (11.27) WZ μν 24π We see that even though the anomaly itself disappears in flat space, (11.26), there is a two-derivative term for τ that survives even after the metric is taken to be flat. This is of course the familiar Wess–Zumino term for the two-dimensional conformal group. Let us consider now some general two-dimensional renormalization group flow from a conformal field theory in the ultraviolet (with central charge cUV) and one in the infrared (with central change cIR). We replace every mass scale according to M → Me−τ(x). We also couple the theory to some background metric. Under a simultaneous Weyl transformation of the dynamical fields and the background √ field τ(x), the theory 1 2 is noninvariant only because of the anomaly δσS = 24 cUV d x gσR. Since this is a property of the full quantum theory, it must be reproduced at all scales. An immediate consequence of this idea is that also in the deep√ infrared the effective action should 1 2 reproduce the transformation δσS = 24 cUV d x gσR. At long distances, we obtain a contribution cIR to the anomaly from CFTIR; hence, the rest of the anomaly must come from an explicit Wess–Zumino functional (11.27) with coefficient cUV − cIR.In particular, setting the background metric to be flat, we conclude that the low-energy theory must contain a term c − c UV IR d2x (∂τ)2. (11.28) 24π Note that the coefficient of this term is universally proportional to the difference between the anomalies and does not depend on the details of the flow. Higher-derivative terms for the dilaton can be generated from local diff×Weyl-invariant terms, and there is no a priori reason for them to be universal (i.e., they may depend on the details of the flow, and not just on the conformal field theories at short and long distances). Zamolodchikov’s theorem that we reviewed in Section 11.1 follows directly from (11.28). Indeed, from reflection positivity, we must have that the coefficient of the term (11.28) is positive, and thus the inequality is established. μ ··· We can be more explicit. The coupling of τ to matter must take the form τTμ + , where the corrections have more τ’s. To extract the two-point function of τ with two μ 2 derivatives, we must use the insertion τTμ twice. (Terms containing τ can be lowered once, but they do not contribute to the two-derivative term in the effective action of τ.) As a consequence, we find that

( μ 2 ) 1 e τTμ d x = ···+ τ(x)τ(y)T μ(x)T μ(y) d2xd2y + ··· 2 μ μ 1 = ···+ τ(x)∂ ∂ τ(x) (y − x)ρ(y − x)σT μ(x)T μ(y) d2y d2x + ···. 4 ρ σ μ μ (11.29) Higher-dimensional models 267

In the final line of this equation, we have concentrated entirely on the two-derivative term. It follows from translation invariance that the y integral is independent of x: 1 (y − x)ρ(y − x)σT μ(x)T μ(y) d2y = ηρσ y2T μ(0)T μ(y) d2y. (11.30) μ μ 2 μ μ To summarize, we find the following contribution to the dilaton effective action at two derivatives: 1 d2xττ d2yy2T (y)T (0). (11.31) 8

According to (11.28), the expected coeﬃcient of ττ is (cUV − cIR)/24π, and so, by comparing, we obtain Δc =3π d2yy2T (y)T (0). (11.32)

As we have already mentioned, Δc>0 follows from reﬂection positivity (which is a property of unitary theories). Equation (11.32) precisely agrees with the classic results about two-dimensional ﬂows.

Back to four dimensions We start by classifying local diﬀ×Weyl-invariant functionals of τ and a background 2σ metric gμν . Again, we demand invariance under gμν −→ e gμν and τ −→ τ + σ. −2τ We will often denoteg ˆ = e gμν . The combinationg ˆ transforms as a metric under diﬀeomorphisms and is Weyl-invariant. The most general theory up to (and including) two derivatives is 1 f 2 d4x − detg ˆ Λ+ Rˆ , (11.33) 6

μν where we have deﬁned Rˆ =ˆg Rμν [ˆg]. Since we are ultimately interested in the ﬂat- space theory, let us evaluate the kinetic term with gμν = ημν . Using integration by parts, we get S = f 2 d4xe−2τ (∂τ)2. (11.34)

We can use the ﬁeld redeﬁnition Ψ = 1 − e−τ to rewrite this as S = f 2 d4x ΨΨ. (11.35)

We can also study terms in the effective action with more derivatives. With four derivatives, there are three independent (dimensionless) coefficients: 4 − ˆ2 ˆ2 ˆ2 d x gˆ κ1R + κ2Rμν + κ3Rμνρσ . (11.36) 268 Renormalization group flows and anomalies

It is implicit that√ indices are raised and lowered withg ˆ. Recall the expressions for − 2 − 2 2 the Euler density gE4 and the Weyl tensor squared: E4 = Rμνρσ 4Rμν + R 2 2 − 2 1 2 and Wμνρσ = Rμνρσ 2Rμν + 3 R We can thus choose instead of the basis of local terms (11.36) a diﬀerent parametrization 4 − ˆ2 ˆ ˆ 2 d x gˆ κ1R + κ2E4 + κ3Wμνρσ . (11.37)

We immediately see that the κ2 term is a total derivative. If we set gμν = ημν , then −2τ gˆμν = e ημν is conformal to the flat metric, and hence also the κ3 term does not play any role as far as the dilaton interactions in flat space are concerned. Consequently, terms in the flat-space limit arise solely from Rˆ2. A straightforward calculation yields * * 2 1 d4x −gˆRˆ2* =36 d4x τ − (∂τ)2 ∼ d4x (Ψ)2 . * − 2 gμν =ημν (1 Ψ) (11.38) So far, we have only discussed diff×Weyl-invariant terms in four dimensions, but from the two-dimensional re-derivation of the c-theorem we have shown above, we anticipate that the anomalous functional will play a key role. The most general√ anomalous variation one needs to consider takes the form 4 − 2 − δσSanomaly = d x gσ cWμνρσ aE4 . The question is then how to write a functional Sanomaly that reproduces this anomaly. (Note that Sanomaly is only defined modulo diff×Weyl-invariant terms.) Without the field τ, one must resort to nonlocal expressions, but in the presence of the dilaton, one has a local action. It is a little tedious to compute this local action, but the procedure is straightforward in principle. We first replace σ on the right-hand side of the anomalous variation with τ: √ 4 − 2 − ··· Sanomaly = d x gτ cWμνρσ aE4 + . (11.39)

While the variation of this includes the sought-after terms, as the ··· indicate, this cannot be the whole answer, because the object in parentheses is not Weyl-invariant. Hence, we need to keep fixing√ this expression with more factors of τ until the procedure − 2 terminates. Note that gWμνρσ, being the square of the Weyl tensor, is Weyl- invariant, and hence we do not need to add any fixes proportional to the c-anomaly This makes the c-anomaly “abelian” in some sense. The “nonabelian” structure coming from the Weyl variation of E4 is the key to our construction. The a-anomaly is therefore quite distinct algebraically from the c-anomaly. The final expression for Sanomaly is (see [17], where the anomaly functional was presented in a form identical to what we use in this note) √ 4 μν 1 μν 2 4 Sanomaly = −a d x −g τE4 +4 R − g R ∂μτ∂ντ − 4(∂τ) τ +2(∂τ) 2 √ 4 − 2 + c d x gτWμνρσ . (11.40) Higher-dimensional models 269

Note that even when the metric is flat, self-interactions of the dilaton survive. This is analogous to what happens with the Wess–Zumino term in pion physics when the background gauge fields are set to zero, and this is also what we saw in two dimensions. Setting the background metric to be flat, we thus find that the non-anomalous terms in the dilaton-generating functional are + , 4 −4τ −τ 2 2 2 d x α1e + α2(∂e ) + α3 τ − (∂τ) , (11.41) where αi are some real coefficients. The a-anomaly has a Wess–Zumino term, leading to the additional contribution 4 2 4 SWZ =2(aUV − aIR) d x 2(∂τ) τ − (∂τ) . (11.42)

The coeﬃcient is universal because the total anomaly has to match (as we have explained in detail in the case of two dimensions). We see that if we knew the four-derivative terms for the dilaton, we could extract aUV − aIR. A clean way of separating this anomaly term from the rest is achieved by rewriting it with the variable Ψ = 1 − e−τ . Then the terms in (11.41) become α d4x α Ψ4 + α (∂Ψ)2 + 3 (Ψ)2 , (11.43) 1 2 (1 − Ψ)2 while the Wess–Zumino term (11.42) is 2(∂Ψ)2Ψ (∂Ψ)4 S =2(a − a ) d4x + . (11.44) WZ UV IR (1 − Ψ)3 (1 − Ψ)4

We see that if we consider background fields Ψ that are null (Ψ = 0), then α3 disappears and only the last term in (11.44) remains. Therefore, by computing the partition function of the quantum field theory in the presence of four null insertions of Ψ, we can extract aUV − aIR directly. Indeed, consider all$ the diagrams with four insertions of a background Ψ with 2 A momenta ki, such that i ki =0andki = 0. Expanding this amplitude to fourth order in the momenta ki, we find that the momentum dependence takes the form 2 2 2 s + t + u with s =2k1 · k2, t =2k1 · k3, u =2k1 · k4. Our effective action analysis 2 2 2 shows that the coefficient of s + t + u is directly proportional to aUV − aIR. In fact, we can even specialize to the so-called forward kinematics, choosing k1 = −k3 and k2 = −k4. Then the amplitude is only a function of s =2k1 · k2.Itis 2 possible to extract aUV − aIR from the s term in the expansion of the amplitude around s = 0. Continuing s to the complex plane, there is a branch cut for positive s (corresponding to physical states in the s-channel) and negative s (corresponding to physical states in the u-channel). There is a crossing symmetry s ↔−s, so these branch cuts are identical. To calculate the imaginary part associated with the branch cut, we utilize the optical theorem. The imaginary part is manifestly positive-definite. Using Cauchy’s 270 Renormalization group flows and anomalies

2 theorem, we can relate the low-energy coefficient of s , aUV − aIR,toanintegralover the branch cut. Fixing all the coefficients, we find A − 1 Im (s) aUV aIR = 3 . (11.45) 4π s>0 s As explained, the imaginary part Im A(s) can be evaluated by means of the optical theorem, and it is manifestly positive. Since the integral converges by power counting (and thus no subtractions are needed), we conclude

aUV >aIR. (11.46) Note the difference between the ways in which positivity is established in two and four dimensions. In two dimensions, one invokes reflection positivity of a two-point function (reflection positivity is best understood in Euclidean space). In four dimensions, the Wess–Zumino term involves four dilatons, so the natural positivity constraint comes from the forward kinematics (and hence it is inherently Minkowskian). Let us say a few words about the physical relevance of aUV >aIR. Such an inequality severely constrains the dynamics of quantum field theory, and in favorable cases can be used to establish that some symmetries must be broken or that some symmetries must be unbroken. In a similar fashion, if a system naively admits several possible dynamical scenarios, one can use aUV >aIR as an additional handle.

References [1] L. Alvarez-Gaume and E. Witten, “Gravitational anomalies,” Nucl. Phys. B234 (1984) 269. [2] J. Polchinski, “Scale and conformal invariance in quantum field theory,” Nucl. Phys. B303 (1988) 226. [3] A. B. Zamolodchikov, “Irreversibility of the flux of the renormalization group in a 2D field theory,” JETP Lett. 43 (1986) 730 [Pisma Zh. Eksp. Teor. Fiz. 43 (1986) 565]. [4] T. Appelquist, A. G. Cohen, and M. Schmaltz, “A new constraint on strongly coupled gauge theories,” Phys. Rev. D60 (1999) 045003 [arXiv:hep-th/9901109]. [5] R. C. Myers and A. Sinha, “Holographic c-theorems in arbitrary dimensions,” JHEP 1101 (2011) 125 [arXiv:1011.5819 [hep-th]]. [6] D. L. Jafferis, I. R. Klebanov, S. S. Pufu, and B. R. Safdi, “Towards the F - theorem: N = 2 field theories on the three-sphere,” JHEP 1106 (2011) 102 [arXiv:1103.1181 [hep-th]]. [7] E. Witten, “Quantum field theory and the Jones polynomial,” Commun. Math. Phys. 121 (1989) 351. [8] D. L. Jafferis, “The exact superconformal R-symmetry extremizes Z,” JHEP 1205 (2012) 159 [arXiv:1012.3210 [hep-th]]. [9] C. Closset, T. T. Dumitrescu, G. Festuccia, Z. Komargodski, and N. Seiberg, “Contact terms, unitarity, and F -maximization in three-dimensional superconformal theories,” JHEP 1210 (2012) 053[arXiv:1205.4142 [hep-th]]. References 271

[10] H. Casini and M. Huerta, “On the RG running of the entanglement entropy of a circle,” Phys. Rev. D85 (2012) 125016 [arXiv:1202.5650 [hep-th]]. [11] I. R. Klebanov, T. Nishioka, S. S. Pufu, and B. R. Safdi, “Is renormalized entanglement entropy stationary at RG fixed points?” JHEP 1210 (2012) 058 [arXiv:1207.3360 [hep-th]]. [12] J. L. Cardy, “Is there a c theorem in four-dimensions?” Phys. Lett. B215 etc (1988) 749. [13] H. Osborn, “Derivation of a four-dimensional c theorem,” Phys. Lett. B222 (1989) 97. [14] I. Jack and H. Osborn, “Analogs for the c theorem for four-dimensional renormalizable field theories,” Nucl. Phys. B343 (1990) 647. [15] Z. Komargodski and A. Schwimmer, “On renormalization group flows in four dimensions,” JHEP 1112 (2011) 099 [arXiv:1107.3987 [hep-th]]. [16] Z. Komargodski, “The constraints of conformal symmetry on RG flows,” JHEP 1207 (2012) 069 [arXiv:1112.4538 [hep-th]]. [17] A. Schwimmer and S. Theisen, “Spontaneous breaking of conformal invariance and trace anomaly matching,” Nucl. Phys. B847 (2011) 590 [arXiv:1011.0696 [hep-th]].

12 Models of electroweak symmetry breaking

Alex Pomarol

Departament de F´ısica, Universitat Aut`onoma de Barcelona, Bellaterra, Barcelona, Espanya

12 Models of electroweak symmetry breaking 273 Alex POMAROL

12.1 Introduction 275 12.2 The original technicolor model: achievements and pitfalls 276 12.3 Flavor-changing neutral currents and the top mass 277 12.4 Electroweak precision tests 277 12.5 Composite PGB Higgs 278 12.6 Little Higgs 279 12.7 The AdS/CFT correspondence, Higgsless and composite Higgs models 280 12.8 LHC phenomenology 281 References 286 Introduction 275

12.1 Introduction Although the Higgs mechanism is a simple and economical way to break the electroweak gauge symmetry of the Standard Model (SM) and at the same cure the bad high-energy behavior of the WLWL scattering amplitudes, there is an expensive prize to pay, namely, the hierarchy problem. For this reason, it is interesting to look for other ways to break the electroweak symmetry and unitarize the WLWL scattering amplitudes. An example can be found in QCD, where pion–pion scattering is unitarized by the additional resonances that arise from the SU(3)c strong dynamics. A replica of QCD at energies ∼ TeV that breaks the electroweak symmetry can then be an alternative to the Higgs mechanism. This is the so-called technicolor (TC) model [1]. In TC, there is no Higgs particle and the SM scattering amplitudes are unitarized, as in QCD, by infinite heavy resonances. One of the main obstacles to implementing this approach has arisen from electroweak precision tests (EWPT), which have disfavored this type of models. The reason is as follows. Without a Higgs, we expect the new particles responsible for unitarizing the SM amplitudes to have a mass at about 1 TeV. These same resonances give large tree-level contributions to the electroweak observables. There have been two different approaches to overcoming this problem. One assumes that either (1) there are extra contributions to the electroweak observables that make the model consistent with the experimental data or (2) the strong sector does not break the electroweak symmetry but just delivers a composite pseudo-Goldstone boson (PGB) to be identified with the Higgs. This Higgs gets a potential at the one-loop level and triggers electroweak symmetry breaking (EWSB) at lower energies. In the first case, the Higgsless approach, the EWPT are satisfied thanks to additional contributions to the electroweak observables that can come from extra scalars or fermions of the TC model, or from vertex corrections. As we will see, the cancellations needed to pass the EWPT are not large, making this possibility not so inconceivable. In the second case, the Higgs plays the role of partly unitarizing the SM scattering amplitudes. Compared with theories without a Higgs, the scale at which new dynamics is needed can be delayed, and therefore the extra resonances that ultimately unitarize the SM amplitudes can be heavier. In this case, the EWPT will be under control. This is the approach of the composite Higgs models, first considered by Georgi and Kaplan [2]. In these theories, a light Higgs arises as a PGB of a strongly interacting theory, in a very similar way as pions do in QCD. Although these scenarios offer an interesting completion of the SM, the difficulty is performing calculations in strongly coupled theories has been a deterrent against their fully exploration. Nevertheless, the situation has changed in the last few years. Inspired by the AdS/CFT correspondence [3], a new approach to building realistic and predictive Higgsless and composite Higgs models has been developed. The AdS/CFT correspondence states that weakly coupled five-dimensional (5D) theories in anti-de Sitter space (AdS) have a 4D holographic description in terms of strongly coupled conformal field theories (CFT). This correspondence gives a definite prescription for how to construct 5D theories that have the same physical behavior and symmetries as the desired strongly coupled 4D theory. This has allowed the construction of concrete 276 Models of electroweak symmetry breaking

Higgsless [4] and composite Higgs [5, 6] models that not only are consistent with the experimental constraints, but also make clear predictions for the physics at the LHC. We will brieﬂy discuss them in Section 12.7.

12.2 The original technicolor model: achievements and pitfalls Technicolor models [1] of EWSB consist of a new strong gauge sector, SU(N)or SO(N), that is assumed to confine at a low scale μIR ∼ TeV. In addition, the model u,d u,d contains (at least) two flavors of techniquarks TL and TR transforming in the fundamental representation of the strong group and as ordinary quarks under the electroweak group. As in QCD, this implies that the strong sector has a global G = × × u,d SU(2)L SU(2)R U(1)X symmetry under which TL transforms as a (2,1)1/6 and u,d R TR transforms as a (1,2)1/6 (the hypercharge is given by Y =2(T3 +X)). Assuming ¯ ∼ 3 that the TC-quarks form a condensate, TLTR μIR, the global symmetry of the strong sector G is broken down to H = SU(2)V × U(1)X . The electroweak symmetry is then broken, giving masses to the corresponding SM gauge bosons. Fermion masses 2 are assumed to arise from higher-dimensional operators such asq ¯LuRT¯RTL/M that can be induced from an extended heavy gauge sector (ETC). After the TC-quark ∼ 3 2 condensation, SM fermions acquire masses mu μIR/M . If the number of colors N of the TC group is large enough, the strong sector can be described by an infinite number of resonances [7]. The masses and couplings of the resonances depend on the model. Nevertheless, as in QCD, we can expect vector resonances transforming as a triplet of SU(2)V ; the TC-rho of mass mρ ∼ μIR. In order to see the implications of these resonances on the SM observables, it is useful to write the low-energy Lagrangian of the SM fields obtained after integrating out the strong sector (the equivalent of the QCD chiral Lagrangian). It is convenient to express this Lagrangian in an SU(2)L ×SU(2)R ×U(1)X -symmetric way. To do so, we promote the elementary SM fields to fill complete representations of SU(2)L ×SU(2)R ×U(1)X .For the bosonic sector, this means introducing extra nondynamical vectors, i.e., spurions, L R to complete the corresponding adjoint representations Wμ , Wμ ,andBμ. With the Goldstone multiplet U parametrizing the coset SU(2)L ×SU(2)R/SU(2)V , the bosonic low-energy Lagrangian is given by L 2 1| |2 cS L Rμν † ··· eff = f DμU + 2 Tr[Wμν UW U ]+ , (12.1) 4 mρ L − R where DμU = ∂μU√+ iWμ U iUWμ and f is the analog of the pion decay constant 4 that scales as f ∼ N/(4π)×mρ [7]. In (12.1), we have omitted terms of order (DU) that do not contribute to the SM gauge boson self-energies, and terms of order 2 2 4 f D /mρ that are subleading for physics at energies below mρ. The coefficient cS 2 2 − is of order one and in QCD takes the value cS = L10mρ/f 0.4. The mass of the 2 2 2 SM W arises from the kinetic term of U, which gives MW = g f /4, from which we can deduce 3 f = v 246 GeV and m 2 TeV. (12.2) ρ N Electroweak precision tests 277

2 2 2 We also obtain MW = MZ cos θW owing to the SU(2)V symmetry, which corresponds to a custodial symmetry.

12.3 Flavor-changing neutral currents and the top mass If the SM fermion masses arise from an ETC sector that generates the operators i j ¯ 2 q¯LuRTRTL/M , then this sector will also generate ﬂavor-changing neutral currents i j k l 2 (FCNC) of orderq ¯LuRq¯LuR/M that are larger than experimentally allowed. Also, the top mass is too large to be generated from a higher-dimensional operator. Solutions to these problems have been proposed (see, e.g., [8] and references therein). Nevertheless, most of the solutions cannot successfully pass EWPT.

12.4 Electroweak precision tests The most important corrections to the electroweak observables coming from TC-like models are universal corrections to the SM gauge boson self-energies, Πij(p), and nonuniversal corrections to Zb¯b, δgb/gb. The universal corrections to the SM gauge bosons can be parametrized by four quantities: S-, T-, W ,andY [9]. The first two, the most relevant ones for TC models [10], are defined as 2 2 g - - − + − S = g ΠW3B(0), T = 2 [ΠW3W3 (0) ΠW W (0)] . (12.3) MW Since T- is protected by the custodial symmetry, the Lagrangian (12.1) only generates - S.Wehave 2 - − 2 f × −3 N S = g cS 2 2.3 10 , (12.4) mρ 3 where we have extracted the result from QCD. Extra contributions to S-, beyond those of the SM, are constrained by the experimental data as shown in Fig. 12.1. They must be smaller than1 S- 2 · 10−3 at 99% CL. We see that the contribution (12.4) is at the edge of the allowed value. Models with N>3 or with an extra generation of TC- quarks, needed for realistic constructions (ETC models), are therefore ruled out. The bound S- 2 × 10−3 can be saturated only if T- receive extra positive contributions −3 ∼ 5 × 10 beyond those of the SM. Although the custodial SU(2)V symmetry of the TC models guarantees the vanishing of the TC contributions to the T- parameter, one- loop contributions involving both the top and the TC sector are nonzero. Nevertheless, in strongly interacting theories, we cannot reliably calculate these contributions and know whether they give the right amount to T-. As we said before, the generation of a top mass around the experimental value is difficult to achieve in TC models and requires new strong dynamics beyond the original sector [8]. Even if a large enough top mass is generated, an extra difficulty arises from Zb¯b. On dimensional grounds, assuming that tL,R couples with equal strength to the

1 Since TC models do not have a Higgs, we are taking the result of [9] for Mh 1TeV. 278 Models of electroweak symmetry breaking

0.01

0.009

0.008

0.007

T 0.006

0.005 Mh ~ 100 GeV Higgsless 0.004 (à la QCD) X 0.003 Mh ~ 1 Tev

0.003 0.004 0.005 0.006 0.007 0.008 S

Fig. 12.1 Experimental constraints at 68%,95%,and99% CL on the S- and T- parameters following [9], and the SM prediction as a function of the Higgs mass.

TC sector responsible for EWSB, we have the estimate δgb/gb ∼ mt/mρ 0.07, −3 which overwhelms the experimental bound |δgb/gb| 5 × 10 . Similar conclusions are reached even if tL,R couples with diﬀerent strength to the TC sector [5], unless the custodial symmetry is preserved by the bL coupling [11]. Realistic extradimensional Higgsless models can be constructed in which the above problems can be overcome, although this requires extra new assumptions and some adjustments of the parameters of the model [12].

12.5 Composite PGB Higgs By enlarging the group G, while keeping qualitatively the same properties as the Higgsless models described above, we are driven to a different scenario in which the strong sector, instead of breaking the electroweak symmetry, contains a light Higgs in its spectrum that will be responsible for EWSB. The minimal model consists of a strong sector with the symmetry-breaking pattern [5] SO(5) → SO(4). (12.5) It contains four Goldstone bosons parametrized by the SO(5)/SO(4) coset: Π/f 04 ha Σ= Σ e , Σ =(0, 0, 0, 0, 1), Π= − T , (12.6) ha 0 where ha (a =1, ..., 4) is a real 4-component vector, which transforms as a doublet under SU(2)L ∈ SO(4). This is identified with the Higgs. Instead of following the TC Little Higgs 279 idea for fermion masses described before, we can assume, inspired by extradimensional models [5], that the SM fermion couples linearly to fermionic resonances of the strong sector. This can lead to correct fermion masses without severe FCNC problems. The low-energy theory for the PGB Higgs, written in an SO(5)-invariant way, is given by L 2 1 μ T cS μν T eff = f (DμΣ) (D Σ) + 2 ΣFμν F Σ + V (Σ) + ... , (12.7) 2 mρ where Fμν is the field strength of the SO(5) gauge bosons (only the SM bosons must 2 2 2 be considered dynamical). From the kinetic term of Σ, we obtain MW = g (sh f) /4 2 2 2 ≡ 2 together with MW = MZ cos θW , where we have defined sh sin h/f, with h = ha. This implies

v = shf 246 GeV. (12.8) In this model, the contribution to S- has an extra suppression factor v2/f 2 compared with (12.4), and then for v f one can satisfy the experimental constraint. Also, δgb/gb can be under control owing to the custodial symmetry [11]. The exact value of v/f comes from minimizing the Higgs potential V (h) that arises at the loop level from SM couplings to the strong sector that break the global SO(5) symmetry. The dominant contribution comes at one-loop level from the elementary SU(2)L gauge bosons and the top quark. In the model of [6], the potential is given approximately by 2 − 2 2 V (h) αsh βshch , (12.9) where α and β are constants induced at the one-loop level. For α<βand β 0, the electroweak symmetry is broken, and, if β>|α|, the minimum of the potential is at . β − α s = . (12.10) h 2β

To have sh < 1 as required, we need α ∼ β, which can be accomplished in certain regions of the parameter space of the models. The physical Higgs mass is given by 8βs2 c2 M 2 h h . (12.11) h f 2 Since β arises from one-loop eﬀects, the Higgs is light. In the extradimensional composite Higgs models [6], one obtains f 500 GeV, mρ 2.5TeV,andMh ∼ 100–200 GeV.

12.6 Little Higgs In the last few years, similar ideas based on the Higgs as a PGB have also been put forward under the name of little Higgs (LH) models [13]. In these models, however, the gauge and fermion sector is extended in order to guarantee that Higgs mass corrections 280 Models of electroweak symmetry breaking arise at the two-loop level instead of one-loop, allowing for a better insensitivity of the electroweak scale to the strong sector scale mρ.

12.7 The AdS/CFT correspondence, Higgsless and composite Higgs models The AdS/CFT correspondence relates 5D theories of gravity in AdS to 4D strongly coupled conformal field theories [3]. In the case of a slice of AdS, a similar correspondence can also be formulated. The boundary at y = πR corresponds to an ultraviolet cutoff in the 4D CFT and to the gauging of certain global symmetries. For example, in the case we are considering where gravity and the SM gauge bosons live in the bulk, the corresponding 4D CFT will have the Poincaré group gauged (giving rise to gravity) and also the SM group SU(3)×SU(2)L ×U(1)Y (giving rise to the SM gauge bosons). Matter localized on the boundary at y = πR corresponds to elementary fields external to the CFT that interact only via gravity and gauge interactions. On the other hand, the boundary at y = 0 corresponds in the dual theory to an infrared cutoff of the CFT. In other words, it corresponds to breaking the conformal symmetry at the TeV scale. The Kaluza–Klein (KK) states of the 5D theory correspond to the bound states of the strongly coupled CFT. Although the CFT picture is useful for understanding some qualitative aspects of the theory, it is practically useless for obtaining quantitative predictions, since the theory is strongly coupled. In this sense, the 5D gravitational theory in a slice of AdS represents a very useful tool, since it allows one to calculate the particle spectrum, which would otherwise be unknown from the CFT side. Following the AdS/CFT correspondence, we can design 5D models with the properties of the strongly coupled models discussed before. For example, Higgsless models [4] consist of gauge theories in Randall–Sundrum spaces with the following symmetry pattern:

boundary at y =0: SU(2)V × U(1)X × SU(3)c,

5D bulk: SU(2)L × SU(2)R × U(1)X × SU(3)c, (12.12)

boundary at y = πR : SU(2)L × U(1)Y × SU(3)c. For composite PGB Higgs models, we have [5, 6] the following:

boundary at y =0: O(4) × U(1)X × SU(3)c,

5D bulk: SO(5) × U(1) × SU(3) , X c (12.13)

boundary at y = πR : SU(2)L × U(1)Y × SU(3)c.

In these models, the lightest KK states are the partners of the top with SM quantum numbers (3,2)7/3,1/3 and (3,1)4/3.The spectrum is shown in Fig. 12.2. Gauge boson and graviton KK states are heavier, around 2.5 and 4 TeV, respectively. LHC phenomenology 281

2.5

1 2.0 2/3 2 1/6 2 7/6

(Tev) 1.5 KK M 1.0

0.5

115 125 135 145 155 165 175 185 M h (Gev)

Fig. 12.2 KK fermion masses versus Higgs mass in the model of [6]. All fermion KK states are color triplets under the strong group. The quantum numbers under SU(2)L × U(1)Y is also given. We can see that the normalization of hypercharge in [6] is diﬀerent from ours; it’s necessary to multiply by 2 to get the hypercharges as deﬁned here.

12.8 LHC phenomenology 12.8.1 Heavy resonances at the LHC The universal feature of strongly coupled theories of EWSB or their extradimensional analogs is the presence of vector resonances, triplets under SU(2)V , with masses in the range 0.5–2.5 TeV; they are the TC-rho or KK states of the Wμ. They can be produced either by qq¯ Drell–Yan scattering or via weak boson fusion. These vector resonances will mostly decay into pairs of longitudinally polarized weak bosons (or, if possible, to a weak boson plus a Higgs), and to pairs of tops and bottoms. Studies at the LHC have been devoted to a very light TC-rho, mρ 600 GeV, that will be able to be seen for an integrated luminosity of 4 fb−1 (see, e.g., [14]). In extradimensional Higgsless and composite Higgs models, one also expects heavy gluon resonances. Their dominant production mechanism at the LHC is through uu¯ or dd¯ annihilation, decaying mostly into top pairs. The signal will be then a bump in the invariant tt¯ mass distribution. For an integrated luminosity of 100 fb−1, the reach of the gluon resonances can be up to masses of 4 TeV [15]. The most promising way to unravel some composite Higgs model is by detecting ∗ heavy fermions with electric charge 5/3(q5/3) [6]. These states are expected to be lighter than vector resonances (see Fig. 12.2). For not-too-large values of its mass ∗ mq5/3 , roughly below 1 TeV, these new particles will be mostly produced in pairs, via QCD interactions, → ∗ ∗ qq,gg¯ q5/3 q¯5/3 , (12.14) ∗ ∗ with a cross section completely determined in terms of mq5/3 . Once produced, q5/3 will mostly decay to a (longitudinally polarized) W + plus a top quark. The ﬁnal state of the process (12.14) consists then mostly of four W’s and two b-jets: 282 Models of electroweak symmetry breaking

∗ ∗ → + −¯→ + + − −¯ q5/3 q¯5/3 W tW t W W bW W b. (12.15)

Using same-sign dilepton final states, we could discover these particles for masses of 500 GeV or 1 TeV for an integrated luminosity of 100 pb−1 on 20 fb−1, respectively ∗ [16]. For increasing values of mq5/3 the cross section for pair production quickly drops, and single production might become more important; masses up to 1.5 TeV could be reached at the LHC [17]. ∗ Beside q5/3, certain composite models and LH models also predict states of electric charge 2/3or−1/3 that could also be produced in pairs via QCD interactions or singly via bW or tW fusion [18, 19]. They will decay to an SM top or bottom quark plus a longitudinally polarized W or Z, or a Higgs. When kinematically allowed, a heavier resonance will also decay to a lighter one accompanied by a Wlong, Zlong,orh. Decay chains could lead to extremely characteristic final states. For example, in one of the ∗ models of [6], the KK with charge 2/3 is predicted to be generally heavier than q5/3. ∗ If pair-produced, they can decay to q5/3, leading to a spectacular six W ’s plus two b-jets final state:

∗ ∗ → − ∗ + ∗ → − + + + − −¯ q2/3 q¯2/3 W q5/3 W q¯5/3 W W W bW W W b. (12.16)

In conclusion, our brief discussion shows that there are characteristic signatures predicted by these models that will distinguish them from other extensions of the SM. While certainly challenging, these signals will be extremely spectacular, and will provide an indication of a new strong dynamics responsible for EWSB.

12.8.2 Experimental tests of a composite Higgs As an alternative to the detection of heavy resonances, the composite Higgs scenario can also be tested by measuring the couplings of the Higgs and seeing diﬀerences from those of a SM pointlike Higgs. For small values of ξ ≡ v2/f 2, as needed to satisfy the constraint on S-, we can expand the low-energy Lagrangian in powers of h/f and obtain in this way the following dimension-6 eﬀective Lagrangian involving the Higgs doublet H: ←→ ←→ L cH μ † † cT † μ † SILH = 2 ∂ H H ∂μ H H + 2 H D H H D μH 2f 2f c λ 3 c y − 6 H†H + y f H†Hψ¯ Hψ +h.c. . (12.17) f 2 f 2 L R

This equation will be referred as the strongly interacting light Higgs (SILH) Lagran- 2 gian [20]. We have neglected operators suppressed by 1/mρ that are subleading versus 2 2 ∼ 2 those of (12.17) by a factor f /mρ N/(16π ), or operators that do not respect the global symmetry G and therefore are only induced at the one-loop level with extra suppression factors—see [20]. The coeﬃcients cH ,cT ,c6,andcy are constants of order one that depend on the particular models. In 5D composite Higgs models, they take, at tree level, the values [20] cH =1,cT =0,cy = 1 (respectively 0), and c6 = 0 (respectively 1) for the model of [6] (respectively [5]). Only the coeﬃcient cT is LHC phenomenology 283 highly constrained by the experimental data, since it contributes to the T- parameter. Nevertheless, all models with an approximate custodial symmetry give a small contribution to cT . The other operators can only be tested in Higgs physics. They modify the Higgs decay widths according to → ¯ → ¯ − Γ h ff SILH =Γ h ff SM [1 ξ (2cy + cH )] , → → (∗) − Γ(h WW)SILH =Γ(h WW )SM [1 ξcH ] , → → (∗) − Γ(h ZZ)SILH =Γ(h ZZ )SM [1 ξcH ] , → → − Γ(h gg)SILH =Γ(h gg)SM [1 ξ Re (2cy + cH )] , (12.18) → → − 2cy + cH cH Γ(h γγ)SILH =Γ(h γγ)SM 1 ξ Re + , 1+Jγ /Iγ 1+Iγ /Jγ → → − 2cy + cH cH Γ(h γZ)SILH =Γ(h γZ)SM 1 ξ Re + . 1+JZ /IZ 1+IZ /JZ

The loop functions I and J are given in [20]. Notice that the contribution from cH is universal for all Higgs couplings and therefore it does not affect the Higgs branching ratios, but only the total decay width and the production cross section. Measurement of the Higgs decay width at the LHC is very difficult and can only be reasonably done for a rather heavy Higgs, well above the two-gauge-boson threshold, which is not the case for a composite Higgs. However, for a light Higgs, LHC experiments can measure the product σh × BRh in many different channels: production through gluon, gauge boson fusion, and top-strahlung; on decay into b, τ, γ, and (virtual) weak gauge bosons. In Fig. 12.3, we show the prediction in the case of a 5D composite Higgs for the relative deviation from the SM expectation in the main channels for Higgs

1.0 σ ( h→WW ZZ c ξ c /c VBF) BR( , ) H =1/4 y H=1 σ (h) BR(h→γγ) σ (tth) BR(h→bb) 0.5 σ (VBF) BR(h→ττ), σ (h) BR(h→WW,ZZ) BR) σx 0.0 BR)/( σx (

∆ –0.5

–1.0 120 140 160 180 200 Mh (Gev)

Fig. 12.3 Deviations from the SM predictions of Higgs production cross sections σ and decay branching ratios BR deﬁned as Δ(σBR)/(σBR) = (σBR)SILH/(σBR)SM − 1. The predictions are shown for some of the main Higgs discovery channels at the LHC with production via vector boson fusion (VBF), gluon fusion (h), and top-strahlung (tth). 284 Models of electroweak symmetry breaking discovery at the LHC. At the LHC with about 300 fb−1, it will be possible to measure the product of Higgs production rate with branching ratio in the various channels with 20–40% precision [21]. This will translate into a sensitivity on |cH ξ| and |cyξ| up to 0.2–0.4, at the edge of the theoretical predictions. Since the Higgs coupling determinations at the LHC will be limited by statistics, they can beneﬁt from a luminosity upgrade, such as the SLHC. At a linear collider, such as the ILC, precisions on σh × BRh can reach the percent level [22], providing a very sensitive probe on the scale f. Deviations from the SM predictions of Higgs production and decay rates could be a hint of models with strong dynamics. Nevertheless, they do not unambiguously imply the existence of a new strong interaction. The most characteristic signals of the SILH Lagrangian have to be found in the very high-energy regime. Indeed, a peculiarity of the SILH Lagrangian is that, in spite of a light Higgs, longitudinal gauge boson scattering amplitudes grow with energy, and the corresponding interaction can become sizable. Indeed, the extra Higgs kinetic term proportional to cH ξ in (12.17) prevents Higgs exchange diagrams from accomplishing the exact cancellation, present in the SM, of the terms growing with energy in the amplitudes. Therefore, although the Higgs is light, we obtain strong WW scattering at high energies. Using the equivalence theorem [23], it is easy to derive the following high-energy limit of the scattering amplitudes for longitudinal gauge bosons: 0 0 → + − + − → 0 0 − ± ± → ± ± A ZLZL WL WL = A WL WL ZLZL = A WL WL WL WL , c s A W ±W ± → W ±W ± = − H , (12.19) L L L L f 2 c t c (s + t) A W ±Z0 → W ±Z0 = H ,AW +W − → W +W − = H , (12.20) L L f 2 L L L L f 2 0 0 → 0 0 A ZLZL ZLZL =0. (12.21)

2 This result is correct to leading order in s/f , and to all orders in ξ in the limit gSM =0, when the σ-model is exact. The absence of corrections in ξ follows from the nonlinear symmetry of the σ-model, corresponding to the action of the generator Th, associated with the neutral Higgs, under which v shifts. Therefore, we expect that corrections can O 2 arise only at (s/mρ). The growth with energy of the amplitudes in (12.19)–(12.21) is strictly valid only up to the maximum energy of our effective theory, namely mρ.The behavior above mρ depends on the specific model realization. In the case of the little Higgs, we expect the amplitudes to continue to grow with s up to the cutoff scale Λ. In 5D models, such as the holographic Goldstone, the growth of the elastic amplitude is softened by KK exchange, but the inelastic channel dominates, and strong coupling is reached at a scale ∼4πmρ/gρ. Note that the results in (12.19)–(12.21) are exactly proportional to the scattering amplitudes obtained in a Higgsless SM [23]. Therefore, in theories with an SILH, the cross section at the LHC for producing longitudinal gauge bosons with large invariant masses can be written as

→ 2 → σ (pp VLVLX)cH =(cH ξ) σ (pp VLVLX)H , (12.22) LHC phenomenology 285

→ where σ(pp VLVLX)H is the cross section in the SM without Higgs, at the leading order in s/(4πv)2. With about 200 fb−1 of integrated luminosity, it should be possible to identify the signal of a Higgsless SM with about 30–50% accuracy. This corresponds to a sensitivity up to cH ξ 0.5–0.7. In the SILH framework, the Higgs is viewed as a pseudo-Goldstone boson, and therefore its properties are directly related to those of the exact (eaten) Goldstones, corresponding to the longitudinal gauge bosons. Thus, a generic prediction of SILH is that the strong gauge boson scattering is accompanied by strong production of Higgs pairs. Indeed, we ﬁnd that as a consequence of the O(4) symmetry of the H multiplet, the amplitudes for Higgs pair production grow with the center-of-mass energy as (12.19): c s A Z0 Z0 → hh = A W +W − → hh = H . (12.23) L L L L f 2 Note that scattering amplitudes involving longitudinal gauge bosons and a single Higgs 4 vanish. This is a consequence of the Z2 parity embedded in the O(4) symmetry of the 2 2 operator OH =(∂μ|H| ) , under which each Goldstone changes sign. Nonvanishing amplitudes necessarily involve an even number of each species of Goldstones. Using (12.19), (12.20) and (12.23), we can relate the Higgs pair production rate at the LHC to the longitudinal gauge boson cross sections: + − 2σδ,M (pp → hhX) = σδ,M pp → W W X cH L L cH 1 2 δ 0 0 + 9 − tanh σδ,M pp → Z Z X . (12.24) 6 2 L L cH

Here, all cross sections σδ,M are computed with a cut on the pseudorapidity separation between the two final-state particles (a boost-invariant quantity) of |Δη| <δ,and with a cut on the two-particle invariant masss>M ˆ 2. The sum rule in (12.24) is a characteristic of SILH. However, the signal from Higgs-pair production at the LHC is not so prominent. It was suggested that, for a light Higgs, this process is best studied in the channel b¯bγγ, but the small branching ratio of h → γγ makes the SILH rate unobservable. However, in SILH, one can take advantage of the growth of the cross section with energy. Although we do not perform a detailed study here, it may be possible that with sufficient luminosity, the signal of b¯bb¯b with high invariant masses could be distinguished from the SM background. Note however that because of the high boost of the Higgs boson, the b jets are often not well separated. The case in which the Higgs decays to two real W ’s appears more promising for detection. The cleanest channel is the one with two like-sign leptons, where hh → ±±νν jets. For order-unity coefficients ci, we have described the SILH in terms of the two parameters mρ and gρ. An alternative description can be given in terms of two mass scales. These can be chosen as 4πf, the scale at which the σ-model would become fully strongly interacting in the absence of new resonances, and mρ, the scale at which new states appear. An upper bound on mρ is obtained from the theoretical naive dimensional analysis (NDA) requirement mρ < 4πf, while a lower bound on mρ comes from the experimental constraint on the S- parameter. 286 Models of electroweak symmetry breaking

Searches at the LHC, and possibly at the ILC, will probe unexplored regions of the 4πf–mρ space. Precise measurements of Higgs production and decay rates at the LHC will be able to explore values of 4πf up to 5–7 TeV, mostly testing the existence of cH and cy. These measurements can be improved with a luminosity upgrade of the LHC. Higgs-physics studies at a linear collider could reach a sensitivity on 4πf up to about 30 TeV. Analyses of strong gauge boson scattering and double-Higgs production at the LHC can be sensitive to values of 4πf up to about 4 TeV. These studies are complementary to Higgs precision measurements, since they test only the coefficient cH and probe processes highly characteristic of a strong electroweak-breaking sector with a light Higgs boson. On the other side, the parameter mρ can be probed at colliders by studying pair production of longitudinal gauge bosons and Higgs, by testing triple gauge vertices, or, more directly, by producing the new resonances. For fixed mρ, resonance production at the LHC will overwhelm the indirect signal of longitudinal gauge boson and Higgs production at large 4πf (small gρ). However, at low 4πf (large gρ), resonance searches become less effective in constraining the parameter mρ, and the indirect signal gains importance. While the search for new resonances is most favorable at the LHC, precise measurements of triple gauge vertices at the ILC can test mρ up to 6–8 TeV. With complementary information from collider data, we will explore a large portion of the interesting region of the 4πf–mρ plane, testing the composite nature of the Higgs.

Note added On July 4, 2012, a scalar with the properties of the SM Higgs was discovered at the LHC with a mass around 125 GeV. This rules out technicolor theories that are Higgsless models. However, it is consistent with composite Higgs models where a light Higgs is present in the spectrum.

References [1] S. Weinberg, Phys. Rev. D13 (1976) 974; Phys. Rev. D19 (1979) 1277; L. Susskind, Phys. Rev. D20 (1979) 2619. [2] D. B. Kaplan and H. Georgi, Phys. Lett. 136B (1984) 183; H. Georgi and D. B. Kaplan, Phys. Lett. B145 (1984) 216. [3] J. M. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231; S. S. Gubser, I. R. Kle- banov and A. M. Polyakov, Phys. Lett. B428 (1998) 105; E. Witten, Adv. Theor. Math. Phys. 2 (1998) 253. [4] C. Csaki, C. Grojean, L. Pilo, and J. Terning, Phys. Rev. Lett. 92 (2004) 101802; G. Burdman and Y. Nomura, Phys. Rev. D69, 115013 (2004); R. Barbieri, A. Pomarol, and R. Rattazzi, Phys. Lett. B591 (2004) 141. [5] K. Agashe, R. Contino, and A. Pomarol, Nucl. Phys. B719 (2005) 165. [6] R. Contino, L. Da Rold, and A. Pomarol, Phys. Rev. D75, 055014 (2007). [7] G. ’t Hooft, Nucl. Phys. B72 (1974) 461; E. Witten, Nucl. Phys. B160 (1979) 57. [8] K. Lane, arXiv:hep-ph/0202255. References 287

[9] R. Barbieri, A. Pomarol, R. Rattazzi, and A. Strumia, Nucl. Phys. B703 (2004) 127. [10] M. E. Peskin and T. Takeuchi, Phys. Rev. Lett. 65, 964 (1990); Phys. Rev. D46, 381 (1992). [11] K. Agashe, R. Contino, L. Da Rold, and A. Pomarol, Phys. Lett. B641 (2006) 62. [12] G. Cacciapaglia, C. Csaki, G. Marandella, and J. Terniry, Phys. Rev. D75 (2007) 015003. [13] N. Arkani-Hamed, A. G. Cohen, and H. Georgi, Phys. Lett. B513 (2001) 232– 240; N. Arkani-Hamed, A. G. Cohen, T. Gregoire, and J. G. Wacker, JHEP 0208 (2002) 020. [14] G. L. Bayatian et al. [ CMS Collaboration ], J. Phys. G34 (2007) 995–1579. [15] K. Agashe, A. Belyaev, T. Krupovnickas, G. Perez, and J. Virzi, Phys. Rev. D77 (2008) 015003. [16] R. Contino and G. Servant, JHEP 0806 (2008) 026; J. A. Aguilar-Saavedra, JHEP 0911 (2009) 030. [17] J. Mrazek and A. Wulzer, Phys. Rev. D81 (2010) 075006. [18] T. Han, H. E. Logan, B. McElrath, and L.-T. Wany, Phys. Rev. D67 (2003) 095004; M. Perelstein, M. E. Peskin, and A. Pierce, Phys. Rev. D69 (2004) 075002. [19] G. Azuelos et al., Eur. Phys. J. C39S2 (2005) 13. [20] G. F. Giudice, C. Grojean, A. Pomarol, and R. Rattazzi, JHEP 0706 (2007) 045. [21] M. D¨uhrssen, ATL-PHYS-2003-030. [22] J. A. Aguilar-Saavedra et al. [ECFA/DESY LC Physics Working Group], arXiv:hep-ph/0106315. [23] M. S. Chanowitz and M. K. Gaillard, Nucl. Phys. B261 (1985) 379.

13 String phenomenology

Luis Iba´nez˜

Departamento de F´ısica Teórica and Instituto de F´ısica Teórica UAM-CSIC, Universidad Autónoma de Madrid, Cantoblanco, Madrid, Spain

13 String phenomenology 289 Luis IBA´ NEZ˜

13.1 Branes and chirality 291 13.2 Type II orientifolds: intersections and magnetic ﬂuxes 294 13.3 Local F-theory GUTs 298 13.4 The eﬀective low-energy action 300 13.5 String model building and the LHC 307 References 310 Branes and chirality 291

This chapter is based on lectures given at the Les Houches Summer School in 2011, in which I reviewed a number of topics in the field of string phenomenology, focusing on orientifold/F-theory models yielding semirealistic low-energy physics. The emphasis was on the extraction of the low-energy effective action and the possible test of specific models at the LHC. The chapter is a brief summary of some of the main topics covered in the lectures, updated where appropriate.

13.1 Branes and chirality String theory (ST) is the most serious candidate for a consistent theory of quantum gravity coupled to matter. In fact, ST actually predicts the very existence of gravity, since a massless spin-2 particle, the graviton, appears automatically in the spectrum of closed string theories. Furthermore, ST has also allowed us to improve our understanding of the origin of black hole degrees of freedom and provides for explicit realizations of holography through the AdS/CFT correspondence. Remarkably, not only is ST a theory of quantum gravity, but it also incorporates all the essential ingredients of the Standard Model (SM) of particle physics: gauge interactions, chiral fermions, Yukawa couplings,. . . It is thus a strong candidate to provide us with a unified theory of all interactions, including the SM and gravitation. In the last 25 years, enormous progress has been obtained in the understanding of the space of four-dimensional (4d) string vacua [1]. From the point of view of unification, the main objective is to understand how the SM may be obtained as a low-energy limit of string theory. We would like to understand how the SM gauge group, the three quark/lepton generations, chirality, Yukawa couplings, CP violation, neutrino masses, Higgs sector, etc. may appear from an underlying string theory. The first step in that direction is learning which compactifications lead to a chiral spectrum of massless fermions at low energies. There are essentially five large classes of such chiral 4d string vacua, symbolized by the five vertices of the pentagon in Fig. 13.1. These include three large classes of type II orientifolds (IIA with O6 orientifold planes, and IIB with O3/O7 or O9/O5 orientifold planes). In addition, there are the

G M ( 2)

IIA (O6) Heterotic Chiral D =4 string theory

IIB (O3,O7) IIB (O9,O5)

Fig. 13.1 The five large classes of 4d chiral string compactifications. 292 String phenomenology well-studied heterotic vacua in Calabi–Yau (CY) manifolds. Finally, there are less studied (and difficult to handle) vacua obtained from the 11d M-theory compactified on manifolds of G2 holonomy. Different dualities connect these different corners, so the different classes of vacua should be considered as five different corners of a single underlying class of theories. It is impossible to overview all these different classes of theories, so we will concentrate on the case of the type II orientifolds whose potential for the construction of realistic SM-like compactifications has been explored in the last15years. The essential objects in chiral type II orientifolds are Dp-branes, nonperturbative solitonic states of string theory that extend over p + 1 space + time dimensions. For our purposes, Dp-branes may be considered as subspaces of the 10d space of type II string theory in which open strings are allowed to start and end. They are charged under antisymmetric tensors of the Ramond–Ramond (RR) sector of type II theory with p + 1 indices. Since in type IIA (IIB), the massless RR tensors have an odd (even) number of indices, there are Dp-branes with p even (odd) for type IIA (IIB) string theory. We will be interested in Dp-branes large enough to contain the standard Minkowski space inside, so that the relevant Dp-branes will be D4, D6, D8 in type IIA and D3, D5, D7, D9 in type IIB. In compactified theories, Gauss’s theorem will force the vanishing of the overall RR charges with respect to these antisymmetric fields. This leads to the so-called tadpole cancellation conditions, which turn out to also ensure cancellation of gauge and gravitational anomalies in the theory. In the worldvolume of Dp-branes, there live (are localized) gauge and charged matter degrees of freedom. In a single Dp-brane lives a U(1) gauge boson, and M such branes located in the same place in transverse dimensions contain an enhanced U(M) gauge symmetry with N = 4 SUSY in flat space (Fig. 13.2). The corresponding spectrum is obviously nonchiral and insufficient to yield realistic physics. In order to obtain chirality, additional ingredients must be present. In the case of type IIA models with the six extra dimensions compactified in a CY manifold, chiral fermions appear at the intersection of pairs of D6-branes, as we will describe later. In the case of type

U(M)

Fig. 13.2 Open strings ending on a stack of M parallel Dp-branes give rise to a U(M), N =4 gauge theory. Branes and chirality 293

IIB models, chiral fermions may appear at the worldvolume of D7 or D9 branes in the presence of magnetic fluxes in the compact directions. Alternatively, chirality may appear if the geometry is singular, like, for example, the case of D3-branes on ZN orbifold singularities. The other crucial ingredients in perturbative type II models are Op-orientifolds. These are geometrically analogous to Dp-branes, with the crucial difference that they are not dynamical and do not contain any field degrees of freedom in their worldvolume. They are, however, charged under the RR antisymmetric fields, and also they have negative tension compared with their Dp-brane counterparts. It is precisely these two properties that make the presence of orientifold planes useful, since their negative tension and RR charges may be used to cancel the positive contribution of Dp-branes, allowing the construction of type II vacua with zero vacuum energy (Minkowski) and overall vanishing RR charges in a compact space. Another important property of type IIA and IIB vacua is mirror symmetry. This is a symmetry that exchanges IIA and IIB compactifications by exchanging the respective underlying CY space with its mirror. For each CY manifold, one can find a mirror manifold in which the Kähler and complex structure moduli are exchanged. In simple examples (such as tori and orbifolds thereof), one can show that mirror symmetry is a particular example of T-duality. The action of T-duality in these toroidal/orbifold set- tings (to be discussed below) is nontrivial and exchanges Neumann and Dirichlet open string boundary conditions. An odd number of T-dualities along 1-cycles exchanges type IIA and IIB theories, and the dimensionalities of Dp-branes change accordingly. Thus, for example, three T-dualities on type IIB D9-branes on T6 change them into D6-branes wrapping a 3-cycle in T6. The basic rules for D-brane model building are as follows (for reviews, see, e.g., [2]). One starts with type II theory compactified on a CY (in some simple examples, one may consider T6 tori or orbifolds). One then considers possible distributions of Dp-branes containing Minkowski space and preserving N = 1 SUSY in 4d. The branes wrap subspaces (cycles) or are located at specific regions inside the CY. The configuration so far has positive energy and RR charges and is untenable if one wants to obtain Minkowski vacua. To achieve that, appropriate Op-orientifold planes will be required to cancel both the positive vacuum energy and the overall RR charges. This will require the construction of a CY orientifold. Finally, the brane distribution is so chosen that the massless sector resembles as much as possible the SM or the Minimal Supersymmetric Standard Model (MSSM). If the brane distribution respects the same N = 1 SUSY in 4d, the theory will be perturbatively stable. In the above enterprise, two approaches are possible:

• Global models. One insists on having a complete globally consistent CY compactification, with all RR tadpoles canceling. • Local models. One considers local sets of lower-dimensional Dp-branes (p ≤ 7) that are localized on some region of the CY and reproduce the SM or MSSM physics there. One does not care at this stage about global aspects of the compactification and assumes that eventually the configuration may be embedded inside a fully consistent global compact model. 294 String phenomenology

The latter is often called the bottom-up approach [3], since one ﬁrst constructs the local (bottom) model with the idea that eventually one may embed it in some global model. Note that this philosophy is not applicable to heterotic or type I vacua, since in those strings the SM ﬁelds live in the bulk six dimensions of the CY.

13.2 Type II orientifolds: intersections and magnetic fluxes In type IIA compactifications, in principle, we have D4-, D6-, and D8-branes, big enough to contain Minkowski space M4. They can span M4 and wrap respectively 1-, 3-, and 5-cycles in the CY. However, since CY manifolds do not have nontrivial 1- or 5-cycles, in IIA orientifolds only D6-branes are relevant for our purposes. It is easy to see that a pair of intersecting branes, D6a,D6b give rise to chiral fermions at their intersection from open strings starting in one and ending on the other brane (see Fig. 13.3). The mass formula for the fields at an intersection in flat space is given (in bosonized formulation) by

(r + r )2 1 3 1 M 2 = N + θ − + |θ |(1 −|θ |), (13.1) ab osc 2 2 2 i i i=1

1 where rθ =(θ1,θ2,θ3, 0) and r $belongs to the SO(8) lattice (ri = Z, Z + 2 for NS, RR sectors, respectively, with i ri = odd). The reader can check that the state − 1 − 1 − 1 1 r + rθ =( 2 + θ1, 2 + θ2, 2 + θ3, + 2 ) is massless for any value of the angles, so there is always a massless fermion at the intersection. If there are N D6a and M D6b intersecting stacks of branes, the fermion transforms in the bifundamental (N, M). There are also three scalars (e.g., r + rθ =(−1+θ1,θ2,θ3, 0)) with mass-squared that may be positive, zero, or negative, depending on the values of the angles. Tachyons are avoided for large ranges of the intersecting angles. On the other hand, for particular choices of the angles, there is a massless scalar, the partner of the chiral fermion, signaling the presence of an N = 1 SUSY, at least at the local level. To construct 4d models, one compactiﬁes type IIA string theory down to four dimensions on a CY manifold. The resulting theory has N = 2 SUSY and is not yet

X X X 5 7 9 a θ a 3 θ 1 θ b 2 b b a

X X X 4 6 8

Fig. 13.3 Open strings between D6a and D6b branes intersecting at angles yield massless chiral fermions. Here X5,...,X9 are local coordinates for the compact space. Type II orientifolds: intersections and magnetic fluxes 295 suitable for realistic model building. One then constructs an orientifold by modding the theory by ΩR, where Ω is the worldsheet parity operation and R is a Z2 antiholomorphic involution on the CY with RJ = −J and RΩ3 = Ω3 (J and Ω3 are the Kähler 2-form and the holomorphic 3-form characteristic of CY manifolds). The resulting theory now has N = 1 SUSY in 4d, and the submanifolds left fixed under the R operation are orientifold O6-planes carrying C7 RR antisymmetric field charge. To flesh out these proceses, let us consider the simplified (yet phenomenologically interesting) case of a T6 orientifold compactification [4]. Consider type IIA string theory compactified in a factorized torus T6 = T2 × 2 2 T × T . D6-branes are assumed to wrap M4 and a 3-cycle that is the direct product of three 1-cycles, one per T2 (see Fig. 13.4). These cycles are described by integers i i i i (na,ma),i =1, 2, 3 indicating the number of times na(ma)theD6a brane wraps around the horizontal(vertical) directions. For each stack of Na D6a-branes there is a U(Na) gauge group. Furthermore at the intersection of two stacks of branes D6a, D6b the exchange of open strings gives rise to massless chiral fermions in bifundamental (Na, Nb) representations. Their multiplicity is given by their intersection number 1 × 2 × 3 1 1 − 1 1 2 2 − 2 2 3 3 − 3 3 Iab = Iab Iab Iab =(namb manb )(namb manb )(namb manb ), (13.2) which is, 2 × 2 × 1 = 4 in the example of Fig. 13.4. We now construct an orientifold by modding out the theory by the worldsheet operator Ω(τ,σ)=(τ,−σ) acting on the worldsheet coordinates. Simultaneously, we act with a reflection on the three coordinates R(Xi)=−Xi, i =5, 7, 9. This geometrical reflection leaves invariant the space defined by X5 = X7 = X9 = 0 in which the O6-orientifold lives. In addition, the orientifold projection on invariant states may modify the gauge group of the branes if the latter wrap a 3-cycle that is left invariant by the orientifold. Depending on the details of the projection, one may get Sp(N)orO(N) groups. On the other hand, if the 3-cycle wrapped by the D6-brane stack is not invariant, one must add in the background extra

X1 X3

X0 X2

X5 X7 X9

X4 X6 X8

Fig. 13.4 D6-branes wrap 1-cycles in each of the three T2 and intersect at angles. Chiral bifundamental fermions are localized at the intersections. 296 String phenomenology

Table 13.1 Wrapping numbers of D6-branes in an MSSM-like conﬁguration

1 1 2 2 3 3 Nα (ni ,mi )(ni ,mi )(ni ,mi )

Na =3+1 (1, 0) (3, 1) (3, −1)

Nb =1 (0, 1) (1, 0) (0, −1)

Nc =1 (0, 1) (0, −1) (1, 0) mirror D6∗-branes siting on the reflected 3-cycle with wrapping numbers (ni, −mi). Then the configuration is also invariant, but the gauge group U(N) remains. It is easy to find choices of D6-branes with appropriate wrapping numbers (ni,mi) yielding a semirealistic chiral massless spectrum. Let as consider three stacks D6a, 2 D6b,D6c of branes on rectangular T tori with multiplicities and wrapping numbers as in Table 13.1 [5]. The D6b and D6c branes are assumed to be located at X7 =0 and X9 = 0, respectively, so that the orientifold projection yields an Sp(1) SU(2) gauge group for both of them. On the other hand, the 3 + 1 D6a branes in Table 13.1 i i → should be suplemented by their mirrors with wrapping numbers flipped as (na,ma) − i × (na, ma), and the gauge group is U(3) U(1). The complete gauge group is then U(3)×SU(2)×SU(2)×U(1), but one linear combination of the two U(1)’s is anomalous and becomes massive through a generalized Green–Schwarz mechanism. All in all, one obtains the gauge group of the minimal left–right-symmetric extension of the MSSM. The reader may check using (13.2) that these are three generations of quarks and leptons, with three right-handed neutrinos. Furthermore, if the branes D6b and D6c sit on top of each other in the first T2, there is one minimal set of Higgs fields. Choosing 2 2 3 3 Rx/Ry = Rx/Ry for the radii in the second and third tori, one can see that θ2 +θ3 =0 and there is one unbroken N = 1 SUSY. The above example is a good local model, but it is globally inconsistent. The reason is that, as it stands, it gives rise to RR tadpoles, and the overall charge with respect to the C7 RR forms does not vanish as it should in a compact space. It is easy to show that those conditions in this toroidal setting are 1 2 3 1 2 3 Nananana =16, Nanamama = 0 (+ permutations), (13.3) a a and plugging in the wrapping numbers from Table 13.1 shows that they are not obeyed. It is, however, easy to constract a Z2 × Z2 orbifold variation of this model with some additional D6-branes and orientifold planes that is supersymmetric and obeys the corresponding tadpole conditions [6]. This model is remarkably simple, and its chiral sector gets quite close to a phenomenologically interesting model, the L–R extension of the MSSM. It still has the shortcoming that, like most toroidal/orbifold models, the massless spectrum includes additional adjoint chiral multiplets of the SM gauge group. The vacuum expectation values of these adjoints parametrize the freedom to translate in parallel the positions of the branes in any of these models. The latter is a characteristic of toroidal compactifications and is in general absent in more general CY orientifolds. Type II orientifolds: intersections and magnetic fluxes 297

A second class of interesting type II compactifications comprises type IIB orientifolds. Now the internal orientifold geometric involution acts like RJ = J, RΩ3 = −Ω3. In the toroidal setting, these may be obtained as T-duals of type IIA intersecting brane models. Indeed, upon an odd number of T-dualities along the six circles in the T6, a D6-brane may transform into a D9-, D7-, D5-, or D3-brane, depending on the particular T-duality transformation. If the original D6-, brane is rotated with respect to the orientifold plane, the resulting IIB Dp-branes will in general contain a magnetic flux turned on in their worldvolume. Indeed, higher-dimensional type IIB branes in SUSY configurations (unlike the D6-branes in type IIA) may contain magnetic flux backgrounds. They in turn induce lower-dimensional Dp-brane charge i and also chirality. Let us consider [7] the case of Na D9-branes wrapped ma times on 2 i the ith T and with na units of U(1)a quantized magnetic flux: i 1 i i ma Fa = na. (13.4) 2 2π Ti

i i Interestingly enough, the (na,ma) D6 wrapping numbers are mapped under T-duality i into the magnetic integers deﬁned above. In addition, the relative angle θab of D6a–D6b branes in the ith torus is mapped into the diﬀerence

i i i − i i na θab = arctan Fb arctan Fa,Fa = i . (13.5) maRxiRyi

In the presence of a magnetic ﬂux F in a IIB brane wrapping T2, the open string boundary conditions are modiﬁed as

∂σX − F∂τ Y =0,∂σY + F∂τ X =0. (13.6)

In particular, by varying F , one interpolates between Neumann and Dirichlet boundary conditions and, for example, at formally infinite flux, they are purely Dirichlet. Thus, adding fluxes on a higher-dimensional brane induces RR charge corresponding to lower-dimensional branes. For example, D9-branes with flux num- 2 2 3 3 1 bers (1, 0)(na,ma)(na,ma) are equivalent to D7 -branes that are localized on the first T2 and wrap the remaining T2 × T2. On the other hand, D9-branes with flux numbers (1, 0)(1, 0)(1, 0) (formally infinite flux in the three T2’s) are equivalent to D3- branes. Note in particular that the semirealistic model with intersecting D6-branes as 1 2 3 in Table 13.1 are mapped into a set of three stacks of D7a,D7b ,D7c that overlap pairwise on a T2. Chirality arises in this type IIB mirror from the mismatch between L- and R-handed fermions induced by the finite flux in the second and third tori. This view of the orientifolds in terms of type IIB D7-branes overlapping on 2d spaces (T2 in the toroidal example) is particularly interesting because it admits a straightforward generalization to type IIB CY orientifolds, at least in the large- compact-volume approximation in which Kaluza–Klein field theory techniques are available. In contrast, the mirror class of models of type IIA orientifolds with intersecting D6-branes is more difficult to generalize to curved CY spaces, since the 298 String phenomenology mathematical definition of BPS D6-branes in curved space (wrapping so-called special Lagrangian 3-cycles) is more difficult to analyze. A further argument for concentrating on type IIB orientifolds with D7/D3-branes is that in the last 10 years we have learnt a great deal about how the addition of type IIB closed string antisymmetric field fluxes can fix most or all the moduli. The equivalent analysis for type IIA or heterotic vacua is at present far less developed. One generic problem in both IIA and IIB cases is the top quark problem in models with a unified gauge symmetry such as SU(5). The point is that in perturbative orientifolds, the GUT symmetry is actually U(5) and the quantum numbers of a GUT generation are 5¯−1 + 102, with Higgs multiplets 51 + 5¯−1. It is then clear that D- quark/lepton Yukawas are allowed by the U(1) symmetry, but the U-quark couplings from 10210251 are perturbatively forbidden. This U(1) symmetry is in fact anomalous and massive, but still remains as a perturbative global symmetry in the effective action. Instanton effects may violate it, but one expects the corresponding nonperturbative contributions to be small and to be relevant at most only for the lightest generations, not the top quark. Thus insisting on unification of SM groups in perturbative orientifolds gives rise to a top quark problem.

13.3 Local F-theory GUTs F-theory [8] may be considered as a geometric nonperturbative formulation of type IIB orientifolds. From the model-building point of view, its interest is twofold: (1) it provides a solution to the top quark problem of perturbative type II orientifolds with a GUT symmetry and (2) moduli fixing induced by closed string antisymmetric fluxes is relatively well understood. In loose terms, one could say that it combines advantages from the heterotic and type IIB vacua. An important massless field of 10d type IIB string theory is the complexified dila- −φ ton field τ = e + iC0. The dilaton φ controls the perturbative loop expansion and C0 is a RR scalar. The 10d theory has an SL(2, Z) symmetry under which τ is the modular parameter. The symmetry is generated by the transformations τ → 1/τ and τ → τ + i and is clearly nonperturbative (e.g., it exchanges strong and weak coupling by inverting the dilaton). F-theory provides a geometrization of this symmetry by adding two (auxiliary) extra dimensions with T2 geometry and identifying the complex structure of this T2 with the type IIB τ field. The resulting geometric construction is 12-dimensional, and one obtains N = 1 4d vacua by compactifying the theory on a complex 4-fold CY X4 that is an elliptic fibration over a six-dimensional base B3; i.e., 2 locally one has X4 T × B3. The theory contains 7-branes that appear at points in the base B3 at which the fibration becomes singular, corresponding to 4-cycles wrapped by the 7-brane. As in the case of perturbative D7-branes, there is a gauge group associated with these branes. However, unlike the perturbative case, the possible gauge groups include the exceptional ones E6,E7,andE8. This is an important property, since, as we will see momentarily, it allows for the existence of an SU(5) GUT symmetry with a large top Yukawa. A particularly interesting type of F-theory constructions comprises those involving a GUT symmetry such as SU(5) and termed F-theory GUTs (for reviews, see [9]). These are motivated by the apparent unification Local F-theory GUTs 299 of coupling constants in the MSSM. Such constructions are a nonperturbative generalization of the type IIB models with intersecting (and magnetized) D7 branes that we discussed in Section 13.2. There is a 7-brane wrapping a 4-cycle S inside B3, yielding an SU(5) gauge symmetry. As in the bottom-up approach mentioned above, one can decouple the local dynamics associated with the SU(5) brane from the global aspects of the B3 compact space. Chiral matter again appears at the intersection of pairs of 7-branes, matter curves in the F-theory language, corresponding to an enhanced degree of the singularity. These 7-branes are, however, nonperturbative and cannot simply be described in terms of perturbative open strings. A visual intuition of the appearance of matter fields in an SU(5) F-theory GUT is shown in Fig. 13.5. At the matter curves, the symmetry is locally enhanced to SU(6) or SO(10). Recalling the adjoint branchings

SU(6) −→ SU(5) × U(1), (13.7) 35 −→ 240 + 10 +[51 +c.c.],

SO(10) −→ SU(5) × U(1), (13.8) 45 −→ 240 + 10 +[104 +c.c.], one sees that in the matter curve associated with the 5-plet, the symmetry is enhanced to SU(6), whereas in that related to the 10-plets, the symmetry is enhanced to SO(10). As in the perturbative magnetized IIB orientifolds, in order to get chiral fermions there must be in general nonvanishing ﬂuxes along the U(1) and U(1) symmetries. A third matter curve with an enhanced SU(6) symmetry is also required to obtain Higgs 5-plets. Yukawa couplings appear at the intersection of the Higgs matter curve with the fermion matter curves, as illustrated in Fig. 13.6. At the intersection point, the symmetry is further enhanced to SO(12) in the case of the 10 × 5¯ × 5¯H couplings and to E6 in the case of the U-quark couplings. One may now understand why there are

7′ 7′′

U ′ (1) U(1)′′

SU SO(10) (6) SU (5) SU(5)

7GUT 7GUT ∑ ∑ 5q′′ 10q′

Fig. 13.5 The SU(5) matter ﬁelds live at matter curves corresponding to the intersection of the bulk SU(5) brane with U(1) branes. At the matter curves, the symmetry is enhanced to SU(6) and SO(10), respectively, for the multiplets 10 and 5¯. 300 String phenomenology

5 H 10 5 5 H S

Fig. 13.6 Matter curves generically intersect at points of further enhanced symmetry at which the 10 × 5¯ × 5¯H and 10 × 10 × 5¯H Yukawa couplings localize.

U-quark Yuyawa couplings in F-theory by looking at the branching of E6 adjoint into SU(5) × U(1) × U(1): E6 −→ SU(5) × U(1) × U(1) , (13.9) 78 −→ adjoints + [(10, −1, −3) + (10, 4, 0) + (5, −3, 3) + (1, 5, 3) + h.c.]. We now see that one can form a 10 × 10 × 5 coupling, which is indeed allowed by the U(1) symmetries. We will come back to the issue of Yukawa couplings in F-theory local GUTs in the Section 13.4. To make the final contact with SM physics, the SU(5) symmetry must be broken down to SU(3) × SU(2) × U(1). In these constructions, there are no massless adjoints to make that breaking, and discrete Wilson lines are also not available. Still, one can make such a breaking by the addition of an additional flux FY along the hypercharge direction of the SU(5), which has the same symmetry-breaking effect as an adjoint Higgs. Interestingly enough, this hypercharge flux may also be used to obtain doublet– triplet splitting of the Higgs multiplets 5H + 5¯H .

13.4 The effective low-energy action To make contact with the low-energy physics, we need to have information about the effective low-energy action remaining at scales well below the string scale. Here, we will concentrate on the case of field theories with N = 1 supersymmetry, which is assumed to be broken later at scales of order the electroweak (EW) scale. In this case, the action is determined by the Kähler potential K, gauge kinetic functions fa, and the superpotential W , which we will discuss in turn. For definiteness, we will concentrate on the case of the effective action for type IIB orientifolds, whose general features are also expected to apply to the F-theory case. In the massless sector of an N = 1 compactification, there are charged fields from the open string sector (to be identified with the SM fields) and closed string The effective low-energy action 301

fields giving rise to singlet chiral multiplets, the moduli. Among the latter, there is −φ the complex dilaton S = e + iC0, which is just the dimensional reduction of the i complex dilaton τ mentioned above. In addition, there are h11 Kähler moduli T and − j h21 complex structure moduli U (the minus means the number of (2, 1)-forms that are odd under the orientifold projection). The Kähler moduli parametrize the volume of (i) the manifold and also of all the 4-cycles Σ4 of the specific CY. The complex structure fields U j, on the other hand, parametrize the deformations of the CY manifolds and are (j) associated with the 3-cycles Σ3 in the CY. Specifically, one has [10] (in the simplest − h11 = 0 case) i −φ (i) (i) j T = e Vol(Σ )+iC ,U= Ω3, (13.10) 4 4 (j) Σ3 $ $ (i) (i) (i) where Vol( 4 ) is the volume of the 3-cycle 4 and the C4 are 4d zero modes of the RR 4-form C4 on the 4-cycles. The N = 1 supergravity Kähler potential associated with the moduli in type IIB orientifold compactifications may be written as [10] ∗ −3φ/2 KIIB = − log(S + S ) − 2 log[e Vol(CY )] − log −i Ω3 ∧ Ω3 , (13.11) where Vol(CY ) is the volume of the CY manifold. In the toroidal case with rectangular T6 = T2 × T2 × T2, the Kähler potential takes the simple form

3 3 − ∗ − ∗ − ∗ KIIB = log(S + S ) log(Ui + Ui ) log(Ti + Ti ), (13.12) i=1 i=1

−φ j j k k − i i where Ti = e RxRyRxRy iC4, with i = j = k = i and Ui = Ry/Rx. This is the familiar no-scale structure that also appears in heterotic N = 1 vacua. Concerning the action for the charged matter fields Φa on the 7-branes, the corresponding Kähler metrics, gauge kinetic functions, and superpotential are themselves functions of the moduli. One can write for the general form of the supergravity Kähler potential an expression (to leading order in a matter field expansion) ∗ ∗ ∗ ∗ ∗ K(M,M , Φa, Φa)=KIIB(M,M )+ Kab(M,M )ΦaΦb ab (13.13) 2 + log |W (M)+WY (M,Φa)| , where M collectively denotes the moduli S, T i,Uj, W (M) is the superpotential of the moduli, and WY (M,Φa) is he Yukawa coupling superpotential of the SM fields. We have already discussed the first term in (13.13), and we will discuss the remaining terms in what follows. 302 String phenomenology

13.4.1 The Kähler metrics Equation (13.13) includes the kinetic term for the matter fields, which is controlled by the Kähler metric Kab, which is a function of the moduli. This dependence on the moduli is dictated by the geometric origin of the field. It has been computed at the classical level for some simple cases (mostly toroidal/orbifold orientifolds), either by dimensional redaction from the underlying 10d theory or by using explicit string correlators. We are particularly interested in the Kähler metrics of fields living at intersecting 7-branes, since those are the ones that are associated with the MSSM fields in semirealistic IIB or F-theory compactifications. In the case of type IIB toroidal/orbifold orientifolds, the matter fields associated with a pair of intersecting D7i–D7j branes has a metric (neglecting magnetic fluxes for the moment) [11]

ij 1 Kab = δab ,i= j = k = i, (13.14) 1/2 1/2 1/2 1/2 ui uj tk s ∗ ∗ ∗ where ti = Ti + Ti ,ui = Ui + Ui ,ands = S + S . We thus see that the metrics of −1/2 matter fields at intersections scale like Kab t with the Kähler moduli. Toroidal orientifolds/orbifolds, however, are very special in some ways. We would rather like to see to what extent this type of Kähler metric generalizes to more general IIB CY orientifolds. In particular D7-branes wrap 4-tori whose volumes are directly related to the overall volume of the compact manifold. One would rather like to obtain information about the Kähler metric when the 7-branes wrap a local 4-cycle whose volume is not directly connected to the overall volume of the CY. An example of this is provided by the Swiss cheese type of compactifications discussed in [12]. In this more general setting, one assumes that the SM fields are localized at D7-branes wrapping small cycles in a CY whose overall volume is controlled by a large modulus 3/2 − tb (see fig.(13.7) so Vol(CY )=tb h(ti), where h is a homogeneous function of the 3 small Kähler moduli ti of degree 2 . The simplest example of this is provided by the 4 CY manifold P[1,1,1,6,9], which has only two Kähler moduli tb and t, with a Kähler potential of the form

t 2 t 1

Fig. 13.7 CY manifold with a Swiss cheese structure. The eﬀective low-energy action 303

− 3/2 − 3/2 KIIB = 2 log(tb t ). (13.15)

Here we will assume tb t and take both large so that the supergravity approximation is still valid. In the F-theory context, the analogue of these moduli t and tb would correspond to the size of the 4-fold S and the 6-fold B3, respectively. Focusing only on the Kähler moduli dependence of the metrics, we can write a large-volume ansatz for the Kähler metrics of charged matter fields at the intersections [13]: t1−ξα Kα = , (13.16) tb with ξα to be fixed. We can compute ξα by studying the behavior with respect to a scaling of t in the effective action. In particular, in N = 1 supergravity, the physical (i.e., with normalized kinetic terms) Yukawa coupling Yˆαβγ among three chiral fields (0) is related to the holomorphic Yukawa coupling Yαβγ by Y (0) ˆ K/2 αβγ Yαβγ = e 1/2 . (13.17) (KαKβKγ) On the other hand, it is well known that the perturbative holomorphic Yukawa couplings in type IIB string theory are independent of Kähler moduli. Then, using (13.15) and (13.16), we find a scaling of the physical Yukawa coupling:

(ξα+ξβ +ξγ −3)/2 Yˆαβγ t . (13.18)

The dependence on tb drops at leading order in t/tb, as expected for a model whose physics is essentially localized on the 4-cycle parametrized by t. On the other hand, we can alternatively compute the scaling behavior of the physical Yukawa coupling in terms of its computation as an overlap integral of the respective wavefunctions in S (see below) so that 2 Yˆαβγ ΨαΨβΨγ , |Ψα| =1. (13.19)

For fields localized at intersecting branes, the above normalization integrals are essentially 2d, so the wavefunctions should scale like t−1/4. On the other hand, the overlap integral for the Yukawa coupling is essentially pointlike, so that it scales like ˆ −3/4 1 Y t . Comparing this with (13.18), we conclude that all ξα = 2 , and hence the matter metrics of fields at intersecting branes have a metric with a local Kähler modulus dependence of the form t1/2 Kα = . (13.20) tb −1/2 Note that setting tb t reproduces the Kähler modulus dependence t of toroidal models, (13.14).

13.4.2 The gauge kinetic function The gauge kinetic function for the gauge group living on the D7-worldvolume may be extracted by expanding the Dirac–Born–Infeld (DBI) action of the D7-brane to second order in the gauge field strength Fa. We obtain the general expression [1] 304 String phenomenology − 2 D7 (α ) −φ J+i2πα Fa 2πα Fa fa = e Re e + i C2k e , (13.21) (2π)5 a a Σ4 Σ4 k where J is the Kähler 2-form and the second piece performs a formal sum over all RR C2k forms contributing to the integral. On expanding the exponential, the first a term produces the volume of the 4-fold Σ4 wrapped by the D7 and the second term is proportional to C4. Taking (13.10) into account, we see that the gauge kinetic function is proportional to the Kähler modulus Ta, i.e.,

fa = Ta. (13.22)

The subsequent terms in the expansion describe contributions from the worldvolume gauge magnetic flux, which will be subleading for large volume ta =ReTa. In particular, since F = n, with n an integer for quantized gauge fluxes, we can estimate Σ2 a 1/2 the flux density as Fa n(Re S/Re Ta) . The leading flux correction to the gauge kinetic function then has the form 2 2 n Re S Re fa =ReTa(1 + |Fa| ) Re Ta 1+ , (13.23) Re Ta which indeed will be subleading in the large ta limit.

13.4.3 The superpotential As indicated in (13.13), there will be a superpotential for the moduli W (M) and a second superpotential W (M,Φα) involving (moduli-dependent) Yukawa couplings. In the absence of closed string antisymmetric fluxes, the perturbative superpotential of the moduli vanishes: Wpert(M) = 0. However, in the presence of type IIB NS (RR) 3- form fluxes H3 (F3), there is an induced effective superpotential involving the complex dilaton S and the complex structure moduli U j given by [14] j Wflux(S, U )= (F3 − iSH3) ∧ Ω3, (13.24) CY where Ω3 is the CY holomorphic 3-form, which may be expanded in terms of the complex structure moduli U j in the CY. This superpotential depends only on S and the U j, and its minimization can give rise easily to the fixing of all these moduli. Furthermore, since generic CY manifolds have of the order of a hundred U j fields or more, and the fluxes F3 and H3 also have a range of possible quantized values, at a minimum there may be accidental cancelations such that there is a very tiny value for Wflux = W0. Such tiny values would be needed to understand the smallness of the SUSY breaking scale compared with a large string scale Ms not much below the Planck scale. The above fluxes are unable to fix the values of the Kähler moduli. However, in specific compactifications, there are nonperturbative effects that induce superpotential terms involving the Kähler moduli. Examples of such nonperturbative effects are instanton effects induced by Euclidean D3-instantons and gaugino condensation The effective low-energy action 305 on D7-branes wrapping appropriate 4-cycles in the CY. Such$ effects have typically − an exponentially suppressed behavior of the form W np a exp( BaTa) for some constants Ba. These effects combined with those induced by fluxes Wflux have the potential to fix all the moduli of specific CY orientifold compactifications [12, 15]. Al- though a detailed example with all the required properties, including a realistic model and nonvanishing (but very small) cosmological constant, is still lacking, it seems very likely that those ingredients have the potential to fix all moduli. Of more direct phenomenological interest are the Yukawa couplings involving SM quarks and leptons to Higgs scalars. As we have already mentioned, Yukawa couplings among Dp-brane matter fields in type IIB compactification arise from the overlap integral of the wavefunction in the extra dimensions of the three participant fields. Consider the case of D9-branes to simplify the discussion (recall that the case of D7- branes may be described in terms of D9-branes with appropriate fluxes). Suppose we have initially a D = 10 type IIB orientifold with D9 branes and a gauge group U(n). In the field theory limit, our action will be 10d super Yang–Mills, 1 i L = − Tr F MNF + Tr ΨΓ¯ M D Ψ . (13.25) 4 MN 2 M We then compactify the theory on some CY manifold and turn on magnetic fluxes that may break the gauge group to an SM-like gauge group. The 10d fields can then be expandeda ` la Kaluza–Klein (KK): μ m μ m μ m μ m Ψ(x ,y )= χ(k)(x ) ⊗ ψ(k)(y ),An(x ,y )= ϕ(k)(x ) ⊗ φ(k),n(y ), k k where xμ and ym are 4d and internal coordinates, respectively. The 4d massless spectrum may be chiral and N = 1 supersymmetric with judicious choice of magnetic fluxes. The 4d Yukawa coupling between these matter fields arise from KK reduction of the cubic coupling A × Ψ × Ψ from the 10d Lagrangian in (13.25). As illustrated in Fig. 13.8, the Yukawa coupling coefficients are obtained from the overlap integrals

QL H QR

Yukawa =

Fig. 13.8 Pictorial representation of the computation of Yukawa coupling constants as overlap integrals of zero modes. 306 String phenomenology g α† m β γ Yijk = ψi Γ ψj φkmfαβγ , (13.26) 2 CY where g is the 10d gauge coupling, α, β, γ are U(n) gauge indices, and fαβγ are U(n) structure constants; also, ψ and φ are fermionic and bosonic zero modes, respectively, and i, j, k label the different zero modes in a given charge sector, i.e., the families in semirealistic models. The Yukawa couplings are thus obtained as overlap integrals of the three zero-mode wavefunctions in the CY. In order to compute the Yukawa coupling constants, we thus need to know the explicit form of the wavefunctions on compact dimensions of the involved matter fields, quarks, leptons, and Higgs multiplets in a realistic model. However, such wavefunctions are only accessible to explicit computation for simple models such as toroidal compactifications or orbifolds thereof. Indeed, this computation has been worked out for general toroidal/orbifold models [16]. The wavefunctions turn out to be proportional to Jacobi θ-functions with a Gaussian profile, and the holomorphic Yukawa couplings also turn out to be proportional to products of Jacobi θ-functions (one per T2 factor), depending only on the complex structure moduli U j and the open string moduli (Wilson line degrees of freedom). As an example, the semirealistic model in Table 13.1 has holomorphic Yukawa couplings with the structure 1 i 1 j Y U ∼ ϑ 3 3J (2) × ϑ 3 3J(3) , ij 0 0 (13.27) 1 1 ∗ D i (2) j (3) Y ∗ ∼ ϑ 3 3J × ϑ 3 3J , ij 0 0 where ϑ is a Jacobi θ-function, J(i) are the Kähler forms of the ith torus, and i, j,and j∗ are family indices for the Q, U,andD SM chiral multiplets, respectively. These expressions yield proportional expressions for U-andD-quark Yukawa couplings, but they differ if one takes into account the generic possibility (in tori) of Wilson line backgrounds along the T6 circles. However, because of the factorized structure of the family dependence, only one quark/lepton generation acquires a mass. The corresponding Yukawa coupling is of the order of the gauge coupling constant. This may be considered as a good first approximation to the observed quark/lepton mass spectrum, and one expects further corrections to give rise to the Yukawa couplings of the lighter generations. The computation of Yukawa couplings in general curved CY manifolds is more difficult, although it becomes more tractable within the context of the bottom-up approach mentioned above. The idea is that in models in which the SM fields are localized in brane intersections, Yukawa couplings appear at points in the CY in which three such intersections (corresponding to SM and Higgs fields) meet. We already saw that in the F-theory context in Fig. 13.5. Thus, for example, the Yukawa coupling 10 × 5¯ × 5¯H in an SU(5) F-theory GUT is localized at a point of triple intersection of the three matter curves. The Yukawa coupling now has the schematic form S ψiψjφH , in which the integral, extended over the 4-fold S, is dominated by the intersection region. In such a situation, to compute the Yukawa coupling, we only need to know the wavefunctions String model building and the LHC 307 in the neighborhood of the intersection point [17]. These local wavefunctions may be obtained by solving the Dirac and Klein–Gordon equations at the local level. Inter- estingly, it is again found that only one (the third) generation gets a nonvanishing Yukawa coupling, which is also of the order of the gauge coupling constant. It has been found, however, that instanton corrections induced by distant 7-branes wrapping other 4-cycles in compact spaces in general induce the required Yukawa couplings for the lighter generations [18, 19]. Instanton effects not only give rise to superpotentials for the Kähler moduli and induce the Yukawa couplings of the lighter generations. They may also give rise to interesting terms in the SM superpotential that are forbidden in perturbation theory. In particular, in brane models of SM physics there are typically extra U(1) gauged symmetries beyond those of the SM. A classical example is U(1)B−L which often appears gauged in many string constructions, including right-handed neutrinos. This symmetry is anomaly-free, but there are often in addition U(1)’s with triangle anomalies that are canceled by the 4d version of the Green–Schwarz mechanism. All anomalous U(1)’s become massive by combining with the imaginary part of the Kähler (complex structure) moduli in type IIB (IIA) orientifolds. But, in addition, anomaly-free gauge symmetries such as U(1)B−L may also become massive in this way. This happens because, for example, in type IIB, some Im Ti transform under the corresponding gauge → a U(1)a symmetries as Im Ti Im Ti + qi Λa, with Λa the gauge parameter. This has interesting consequences for instanton physics [20]. In a type IIB orientifold, some instanton configurations correspond to Euclidean D3-branes wrapping the compact dimensions (so that they are localized in Minkowski space, as instantons should be). If they intersect the D7-branes where the SM fields live, there appear charged zero modes (from open string exchange) contributing to instanton-induced transitions. This is why this class of stringy instantons are often called charged instantons. In particular, if the D3-brane wraps a 4-cycle with Kähler modulus M (some linear combination of the Ti’s), nonperturbative operators of the general form −M e Φq1 ...Φqn , qi = 0 (13.28) i may appear [20]. These operators are gauge-invariant because the sum of the charges of the Φ chiral fields is compensated by the shift in Im M induced by the gauge transformation. An example of this is the generation of right-handed neutrino masses in MSSM-like orientifolds with a massive U(1)B−L induced by a Green–Schwarz mech- −M anism. In this case, the operator has the form e νRνR, and the noninvariance of the bilinear under U(1)B−L is compensated by a shift of the Im M. The mass is of −Re M 12 14 order e Ms, which may be in the right phenomenological ballpark 10 –10 GeV 16 for Ms 10 GeV and Re M 100. This type of charged instantons could also be important for the generation of other phenomenologically relevant terms, such as the MSSM μ-term.

13.5 String model building and the LHC With the LHC in operation, an important issue is trying to make contact between an underlying string theory and experimental data. Of course, it would be really exciting 308 String phenomenology if the string scale Ms were within reach of the LHC. We could perhaps observe some string or KK excitation as resonances in LHC data. On the other hand, a large string 16 scale Ms 10 GeV seems to be favored if one sticks to a SUSY version of the SM such as the MSSM, in which gauge couplings nicely unify at a scale of order 1016 GeV. So it is important to see whether specific classes of string compactifications may lead to low-energy predictions for SUSY breaking parameters. We have seen that in certain large classes of type II models, there is information about the structure of the low-energy effective action. In particular, in type IIB orientifolds (or their F-theory extension) with SM fields localized at intersecting D7-branes (or matter curves in F-theory GUTs), one can compute the dependence on the local Kähler modulus of the gauge kinetic function (13.22) and also of the Kähler metric (13.20). If an MSSM-like model is constructed in such a setting, one can obtain specific expressions for SUSY-breaking soft terms, assuming Kähler moduli dominance in SUSY breaking, i.e., nonvanishing auxiliary fields Ft = 0. This is a reasonable assumption within type IIB/F-theory, since in type IIB orientifolds, such nonvanishing auxiliary fields correspond to the presence of nonvanishing antisymmetric RR and NS imaginary self-dual (0, 3) fluxes [21], which are known to solve the classical equations of motion [22]. As we mentioned above, such closed string fluxes are generically present in compactifications with fixed moduli. Using standard N = 1 supergravity formulae and the above information on the effective action, one obtains soft terms with the constrained MSSM (CMSSM) structure but with the additional relationships [23] √ 2 M = 2m = − A = −B, (13.29) 3 where M is the universal gaugino mass, m the universal scalar mass, A the trilinear scalar parameter, and B the Higgs bilinear parameter. Here one assumes the presence of an explicit μ-term in the low-energy Lagrangian, so that altogether there are only two free parameters: M and μ. The universality of soft terms may be understood if an underlying GUT structure exists as in F-theory GUTs. As we have mentioned, magnetic flux backgrounds are generically present on the worldvolume of the underlying 7-branes in order to get a chiral spectrum. In the presence of magnetic fluxes, the gauge kinetic functions (see (13.23)) and the Kähler metrics may acquire small corrections to (13.22) and (13.20), i.e., S t1/2 c f = T 1+a ,K= 1+ α , (13.30) α 1/2 T tb t where a and cα are constants and S is the complex dilaton field. These corrections are suppressed in the large-t limit, corresponding to the physical weak coupling. In this limit, one may also neglect the correction to f compared with that coming from Kα. One then finds corrected soft terms of the form 1 3 m2 = |M|2 1 − ρ , (13.31) f˜ 2 2 f 1 3 m2 = |M|2 1 − ρ , (13.32) H 2 2 H String model building and the LHC 309

1 A = − M(3 − ρ − 2ρ ), (13.33) 2 H f

B = −M(1 − ρH ), (13.34)

1/2 where ρα = cα/t and the subscripts f and H refer to fluxes through the fermion matter curves and the Higgs curve, respectively. Note that in order of magnitude, 1/2 1/2 one numerically expects ρH,f 1/t αGUT 0.2. The above soft terms apply 16 at the string/unification scale Ms 10 GeV. To get the low-energy physics around the EW scale, one has to run down the soft parameters according to the renormalization group equations (RGE). Then one has to check that the boundary conditions are consistent with radiative EW symmetry breaking (REWSB) and with present low- energy phenomenological constraints. One may in addition impose the condition that the lightest neutralino is stable and provides the dark matter in the universe. The resulting scheme is extremely constrained [24]. In particular, setting the fermion flux correction to zero for simplicity, one has a theory with three free parameters (M,μ, and ρH ) and two constraints (REWSB and dark matter), or, equivalently, lines in the planes of any pair of parameters or SUSY masses. As an example Fig. 13.9 shows

(m~τ –m~~χ 0)/mτ 1 1 1 10–3 10–2 10–1 114

116

+ – –8 118 BR(Bs → μ μ ) > 1.1 x 10 –4 BR(b → sγ) < 2.85 x 10

120

mh (Gev) 122

+ – –9 BR(Bs → μ μ ) > 4.5 x 10 124

126

128

τ˜ − 0 τ˜ Fig. 13.9 Normalized mass diﬀerence (m 1 mχ1 )/m 1 as a function of the lightest Higgs mass mh in the modulus dominance scheme. Appropriate REWSB, neutralino dark matter, and + − BR(Bs → μ μ ) limits are only consistent for a Higgs mass in the 125 GeV region. (From [24]). 310 String phenomenology

− 0 the normalized mass difference (mτ˜1 mχ1 )/mτ˜1 as a function of the lightest Higgs mass mh [24]. Dots correspond to points fulfilling the central value in the result from WMAP for the neutralino relic density and dotted lines denote the upper and lower limits after including the 2σ uncertainty. The dot–dashed line represents points with a critical matter density Ωmatter = 1. The vertical line corresponds to the 2σ limit + − on the branching ratio BR(b → sγ) and the upper bound on BR(Bs → μ μ )from [25] and the recent LHCb result [26]. The gray area indicates the points compatible with the latter constraint when the 2σ error associated with the SM prediction is included. As is obvious from the figure, the dark matter condition is fulfilled thanks to a stau–neutralino coannihilation mechanism. Interestingly enough, the recent con- + − straint on BR(Bs → μ μ ) from LHCb forces the Higgs mass to a region around 125 GeV, consistent with the hints of a Higgs particle in that range as measured at CMS and ATLAS. Fixing the mass of any SUSY particle fixes the rest of the spectrum. In particular, with a lightest Higgs mass around 125 GeV, gluinos have a mass around 3 TeV, the first- and second-generation squarks around 2.7–2.8 TeV, and the lightest stop around 2 TeV. The lightest slepton is a stau with mass around 600 GeV, almost degenerate with the lightest neutralinos. The existence of gluinos and squarks of these masses can be tested at LHC running at 14 TeV and 30 fb−1 integrated luminosity. It is remarkable that a lightest MSSM Higgs mass as heavy as 125 GeV is possible in this scheme. In most SUSY schemes (including minimal gauge and anomaly mediation models and the CMSSM with squarks that are not superheavy), the lightest Higgs mass is typically around 115 GeV or so (see, e.g., [27]). In this scheme, a relatively heavy Higgs appears because the soft terms in (13.34) predict a large A-parameter with A −2m, giving rise to a large stop mixing parameter and hence a big one-loop correction to the Higgs mass. In addition, the dark matter and REWSB conditions require a large tan β 40, pushing the tree-level Higgs mass to its maximum value. This large tan β and stop mixing parameters imply that, as it stands, this simple scheme may + − soon be ruled out if LHCb finds no deviation from the SM value for BR(Bs → μ μ ). On the other hand, a Next-to-Minimal supersymmetric SM (NMSSM) version of the same model, also viable in type IIB/F-theory schemes, would remain consistent, as would R-parity violation, since it would avoid the dark matter overabundance problem. This shows how the LHC results may provide important constraints on the possible compactifications and SUSY-breaking schemes within string theory; see, e.g., [28] for other-string-derived approaches.

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“Higgs mass prediction for realistic string/M theory vacua,” arXiv:1112.1059 [hep- ph]; S. P. de Alwis, “Classical and quantum SUSY breaking eﬀects in IIB local models,” JHEP 1003 (2010) 078 [arXiv:0912.2950 [hep-th]]; T. Li, J. A. Maxin, D. V. Nanopoulos, and J. W. Walker, “A Higgs mass shift√ to 125 GeV and a multi-jet supersymmetry signal: miracle of the ﬂippons at the s = 7 TeV LHC,” arXiv:1112.3024 [hep-ph]. 14 The string landscape and low-energy supersymmetry

Michael R. Douglas

Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, New York, USA

14 The string landscape and low-energy supersymmetry 315 Michael R. DOUGLAS

14.1 The goal of fundamental physics 317 14.2 Low-energy supersymmetry and current constraints 321 14.3 The gravitino and moduli problems 324 14.4 The set of string vacua 326 14.5 Eternal inﬂation and the master vacuum 329 14.6 From hyperchemistry to phenomenology 331 Acknowledgments 335 References 335 The goal of fundamental physics 317

We brieﬂy survey our present understanding of the string landscape, and use it to discuss the chances that we will see low-energy supersymmetry at the LHC.

14.1 The goal of fundamental physics Particle physics is entering a new era. The LHC has finished its first scientific phase at 8 TeV, and the Higgs boson has finally been discovered, with a mass of 125 GeV. This landmark discovery, to which many of the lecturers and participants at this school have contributed, should inspire all particle physicists, theorists, and experimentalists to think about our field in new ways.∗ Although the Higgs boson is a signature prediction of the Standard Model, it will be some time before we know whether the real Higgs boson has exactly the properties that the Standard Model predicts. It is not easy to measure the details of interactions in a hadron collider, and many interesting decay modes are rare. There might be additional particles in the Higgs sector, even with masses below 125 GeV. Of course, many physicists have argued over the years that the Standard Model must be incomplete on theoretical grounds, and that there are many reasons to expect non-Standard Model physics at or just above the electroweak scale. This was a primary motivation to build the LHC, but so far it has given us no clear evidence for any non- Standard Model physics. In particular, colored gauginos, a signature of low-energy supersymmetry, have been excluded below about 900 GeV. Of course, they may be just around the corner, waiting to be discovered in the first 14 TeV runs. But we have no guarantees; we it may simply require many years of data taking and subtle analysis to uncover superpartners that have only electroweak interactions, or conversely to exclude them at these energies. By now it is almost a truism that string theory makes no definite predictions for LHC physics, only suggestions for rather implausible scenarios such as black hole creation, whose nondiscovery would not falsify the theory. This is not literally true, since there are potential discoveries that would give strong evidence against string theory,1 but at present there is no reason to expect them. Even if we find no “smoking gun” that speaks directly for or against the theory, there is a program that could someday lead to falsifiable predictions. It is to understand the landscape of string vacua, and derive a probability measure on the set of vacua based on quantum cosmology. From this, we can infer the probabilities that each of the various possibilities for beyond the Standard Model, cosmological, and other fundamental physics would come out of string theory. If future discoveries and (to some extent) present data come out as highly unlikely by this measure, we have evidence against string theory under the assumed scenario for quantum cosmology. This evidence might or might not be conclusive, but it would be the best we could do with the information to hand.

∗The string landscape and low energy supersymmetry’, by Michael R. Douglas in Strings, Gauge Fields, and the Geometry Behind: The Legacy of Maximilian Kreuzer edited by Anton Rebhan, Ludmil Katzarkov, Johanna Knapp, c 2013 World Scientiﬁc. 1 This includes time-varying α [1] and probably faster-than-light neutrinos. 318 The string landscape and low-energy supersymmetry

This program is only being pursued by a few groups today and would require major advances in our understanding of both string theory and quantum cosmology before convincing predictions could be made. My guess at present is that twenty years or more will be needed, taking us beyond the LHC era. Even then, it is likely that such predictions would depend on hypotheses about quantum cosmology that could not be directly tested and might admit alternatives. It is entirely reasonable that skeptics of the landscape should reject this entire direction and look for other ways to understand string theory, or for other theories of quantum gravity. At present, we do not know enough to be confident that they are wrong. Nevertheless, the evidence at hand leads me to think that they are wrong and that this difficult path must be explored. In this chapter, I will briefly outline this program and how I see it proceeding. Although it is clearly a long-term project, I am going to go out on a limb and argue that

• String/M theory will predict that our universe has supersymmetry, broken at the 30 −100 TeV scale. If it is broken at the lower values, we may see gluinos at LHC, while if it is broken at the higher values, it will be very hard to see any evidence for supersymmetry.

This is a somewhat pessimistic claim that far outruns our ability to actually make predictions from string theory. Nevertheless, I am going to set out the argument, fully realizing that many of the assumptions as well as the supporting evidence might not stand the test of time. Indeed, we should all hope that this is wrong! To begin, we have to make the case that a fundamental theory should allow us to make any predictions of this scope. This is not at all obvious. Certainly most major scientific discoveries were not anticipated in any detail. However, the record in particle physics is far better, with examples including the positron, neutrinos, the charm quark, the third generation of quarks and leptons, the W and Z bosons, and as it now appears, the Higgs boson. The framework of quantum field theory is highly constraining, and this record of success is the evidence. Of course, quantum field theory is only constraining within certain limits. For example, there is no good argument that favors three generations of quarks and leptons over four. There are many other consistent extensions of the Standard Model that we might imagine discovering. Even the basic structure of the Standard Model, its gauge group and matter representation content, admits consistent variations. As things stand, it is entirely reasonable to claim that this structure was a choice that could not have been predicted a priori, and equally that the existence of as yet undiscovered matter cannot be excluded a priori. Quantum gravity is hoped to be more constraining, although there is no consensus yet on whether or why this is true. In the case of string/M-theory, we can benefit from over 25 years of work on string compactification. It is clear that there are several different constructions based on the different perturbative limits and compactification manifolds: heterotic on a Calabi–Yau manifold, F-theory, type II with branes, M- theory on G2 manifolds. Each involves many choices that lead to different outcomes for low-energy physics, and for this reason one cannot make definite predictions. The goal of fundamental physics 319

There is even an extreme point of view that (in some still vague sense) “all” consistent low-energy theories can be realized as string compactifications. This idea can lead in various directions: one can hope that consistency will turn out to be a more powerful constraint in quantum gravity than it was in quantum field theory [2]. On the other hand, this does not seem to be the case in six spacetime dimensions[3]. And so far, in four dimensions, no-go results are surprisingly rare. While there are too many compactifications to study individually, one can hope that a particular construction or class of compactification would lead to some generic predictions. If the broad structure of the Standard Model or some beyond the Standard Model scenario came out this way, one might hypothesize that that construction was preferred, and look for top-down explanations for this. However, no construction seems especially preferred at this point. For example, it is simple to get grand unification out of the E8 × E8 heterotic string. On the other hand, this construction does not naturally lead to three generations of matter; even if we grant that the number of generations must be consistent with asymptotic freedom, most choices of Calabi–Yau manifold and bundle have other numbers of generations. Can one do better? There are brane constructions that relate this “three” to the number of extra (complex) dimensions, but these do not realize grand unification. Which is better? It seems that any attempt to narrow down the possibilities will involve this type of weighing of different factors, and this is a strong motivation to systematize this weighing and make it more objective. While this would seem a very open-ended problem, in the context of string compactification there is a natural way to do it—namely, to count the vacua of different types, and regard high-multiplicity vacua as favored or “more natural.” This includes the considerations of tuning made in traditional naturalness arguments, and extends them to discrete and even qualitative features such as numbers of generations or comparison of different supersymmetry-breaking mechanisms. The basic outlines of such an approach are set out in [4]. And have led so far to a few general results that we will survey below. One can imagine continuing this study along formal, top-down lines to develop a quantitative picture of the landscape. While this sort of information seems necessary to proceed further, by itself it is not going to lead to convincing predictions. I like the analogy to the study of solutions of the Schrödinger equation governing electrons and nuclei, better known as chemistry [5]. The landscape of chemical molecules is very complicated, but one could imagine deducing it ab initio and working out a list of long-lived metastable compounds and their properties. However, in any real world situation (both on Earth and in astronomy), the number density of the various molecules is very far from uniform, or from being a Boltzmann distribution. One needs some information about the processes that created the local environment, be it the surface of the Earth, the interior of a star, or whatever, to make any ab initio estimate of this number density. Conversely, we see in astrophysics that fairly simple models can sometimes lead to useful estimates. One can then make statements about “typical molecules,” meaning typical for that local environment, on purely theoretical grounds. While one cannot push this analogy very far, I think it confirms the point that we need some information about the processes that created our vacuum as one of the many possibilities within the landscape, to estimate a measure and make believable 320 The string landscape and low-energy supersymmetry predictions. Doing this is a primary goal of quantum cosmology and has been discussed for over 30 years. The first question is whether one needs the microscopic details of quantum gravity to do this, or whether general features of quantum gravity suffice. There is a strong argument, based on the phenomenon of eternal inflation, that the latter is true, so that one can ask the relevant questions and set up a framework to answer them without having a microscopic formulation. Of course, their answers might depend on microscopic details; for example, one needs to know which pairs of vacua are connected by tunneling processes, and this depends on the structure of configuration space. Anyways, these general arguments are well reviewed in [6–8], while some of the most recent developments are discussed in [9, 10]. Quantum cosmology is a contentious subject in which I am not an expert. However, within the eternal inflation paradigm, starting from a variety of precise definitions for the measure factor and using the presence of many exponentially small numbers in the problem, one obtains a fairly simple working definition, the “master” or “dominant” vacuum ansatz [11–13]. This states that the a priori measure is overwhelmingly dominated by the longest-lived metastable de Sitter vacuum. The measure for other vacua is given by the tunneling rate from this “master” vacuum, which to a good approximation is that of the single fastest chain of tunneling events. Although the measure is dominated by the master vacuum, it is a priori likely (and we will argue) that observers cannot exist in this vacuum—it is not “anthropically allowed.” While there are many objections to anthropic postselection, they have been well addressed in the literature, and we will not discuss them here. Nevertheless, we must take a position on what anthropic postselection should mean in practice. The philosophically correct definition that a vacuum admit observers is impossible to work with, while simpler proxies such as entropy production [14] have not yet been developed in the detail we need for particle physics. In practice, the anthropically allowed vacua will be those that realize the Standard Model gauge group, and the first family of quarks and leptons, with parameters roughly the ones we observe. It is not obvious that even these are all anthropically selected; for example, it is argued is [15] that one does not even need the weak interactions!2 Conversely, while the precise values of quark masses are not usually considered to be selected, given the plethora of fine tunings in chemistry, it might well be that life and the existence of observers is much more dependent on the specific values of these parameters than as first appears. Besides these questions of detail, any definition of anthropic selection suffers from the objection that it is time-dependent and would be different in 1912 or 2112 than in 2012. While this is so, we would reply that all we can do in the end is to test competing theories with the evidence to hand, and one can try out all the variations on this theme in order to do this. It is quite reasonable to expect our evidence to improve with time, and perhaps our understanding of the anthropic constraints will improve as well.

2 Even granting this point, the need to get several quarks and leptons with similar small nonzero masses is far more easily met by a chiral theory such as the Standard Model than a vector-like weakless theory. Low-energy supersymmetry and current constraints 321

Granting that the master vacuum is not anthropically allowed, the measure we are interested in is thus the “distance” in this precise sense (defined using tunneling rates) from the master vacuum, restricted to the anthropically allowed vacua. Clearly it is important to find the master vacuum, and one might jump to the conclusion that string theory predicts that we live in a vacuum similar to the master vacuum. However, because of the anthropic constraint, whether this is so depends on details of the tunneling rates. The main constraint is that one needs to reach a large enough set of vacua to solve the cosmological constant problem. The more tunneling events required to do this, and the more distinct the vacua they connect, the more disparate a set of vacua will fall into this category, and the weaker the predictions such an analysis will lead to. Somewhat counterintuitively, if the master vacuum admits many discrete variations, then there are more nearby vacua, and one does not need to go so far to solve the cosmological constant problem. If this set of vacua includes anthropically allowed vacua, then these will be favored and one can imagine getting fairly definite predictions. One can already make some guesses about where to look for the master vacuum in the string landscape, as we will describe. Continuing in this speculative vein, we will argue that this favors “local models” of the Standard Model degrees of freedom, and supersymmetry breaking driven by dynamics elsewhere in the extra dimensions, gravitationally mediated to the Standard Model. This is a much-discussed class of models and, as we discuss in Section 14.3, there is a key difficulty in that their ability to solve the hierarchy problem is limited by the cosmological moduli problem, which seems to require supersymmetry breaking at or above about 30 TeV. Still, compared with the GUT or Planck scales, this is a huge advantage, and thus we predict low- energy supersymmetry but with superpartners around this scale. The argument that we have just given is not purely top-down and is closely related to the familiar arguments that if low-energy supersymmetry were the solution to the hierarchy problem, then we should see superpartners in the current LHC runs. My phenomenology is rather sketchy, and there are many other scenarios that would need to be considered to make a convincing argument. But the point here is to illustrate the claim that with some additional input from the string theory landscape, allowing us to compare the relative likelihood of different tuned features, we could make such arguments precise.

14.2 Low-energy supersymmetry and current constraints Most arguments for “beyond the Standard Model” physics are based on its potential for solving the hierarchy problem, the large ratio between the electroweak scale 19 16 MEW ∼ 100 GeV and higher scales such as MPlanck ∼ 10 GeV or MGUT ∼ 10 GeV. Low-energy supersymmetry is a much-studied scenario with various circumstantial arguments in its favor. Theoretically, it is highly constraining and leads to many generic predictions, most importantly the gauge couplings of superpartners. This is why LHC already gives us strong lower bounds on the masses of colored superpartners, especially the gluino. If there is a 125 GeV Higgs, this turns out to put interesting constraints on supersymmetric models. Recent discussions of this include [16–18], the talk [19] and [20], which reviews a line of work that has inﬂuenced my thoughts on these questions. 322 The string landscape and low-energy supersymmetry

A broad-brush analysis of the hierarchy problem can be found in [21]. Its solution by low-energy supersymmetry can be understood by restricting attention to a few ﬁelds, most importantly the top quark and its scalar partner the “stop.” In general terms, top quark loops give a quadratically divergent contribution to the Higgs mass, which is cut oﬀ by stop loops. This leads to the rough estimate

2 ∼ 2 ΛSUSY δMH1 0.15MST log , (14.1) MST where MH1 is the mass of the Higgs that couples to the top, ΛSUSY ≡ M3/2 is the 2 supersymmetry-breaking scale, and MST is the average stop mass squared. The strongest sense in which supersymmetry could solve the hierarchy problem would be to ask not just that MH comes out small in our vacuum, but that it comes out small in a wide variety of vacua similar to ours; in other words, all contributions to MH are of the same order so that no ﬁne tuning is required. This is called a natural solution, and from (14.1) it requires

MST ∼ MH in a fairly strong sense (for example [22] estimates MST 400 GeV), Although this might sound as if it is already ruled out, the stop cross section is quite a bit smaller than that of the gluino, and there are even scenarios in which the lightest stop is hard to ﬁnd because it is nearly degenerate with the top. While ΛSUSY ∼ MEW as well, there are many diﬀerent types of supersymmetry breaking, and this does not in itself require the gluino to be light. But one can get a much stronger constraint by assuming that MST is naturally low as well, as it gets mass renormalization from gluon and gluino loops. Assuming that there are no other colored particles involved, this leads to an upper bound on the gluino mass [22]:

Mg˜ 2MST.

Thus, LHC appears to be on the verge of ruling out a wide variety of natural models similar to the Minimal Supersymmetric Standard Model (MSSM). These low bounds on the masses of superpartners in natural models were already problematic before LHC for a variety of reasons, but most importantly because of the difficulty of matching precision measurements in the Standard Model. The longest- standing problem here is the absence of flavor-changing processes other than those mediated directly by the weak interactions, which translates into lower bounds for the scale of much new physics of Λ 10–100 TeV! Assuming the superpartners are not found below 1 TeV, a reasonable response is to give up on naturalness and accept some tuning of the Higgs mass. The cleanest such scenario is to grant the standard structure of low-energy supersymmetry, but push it all up to the 10–100 TeV scale. Thus, all of the superpartners and the Higgs bosons would a priori lie in the range 0.1–1 times ΛSUSY, but we then postulate an additional 10−4–10−6 fine tuning of one of the Higgs boson masses. At first sight, this has the problem that we lose the WIMP as a candidate dark matter particle. However, Low-energy supersymmetry and current constraints 323 because the gauginos have R-charge and the scalars do not, it is natural for them to acquire a lower mass after supersymmetry breaking. In a bit more detail, one very generally expects irrelevant interactions between the supersymmetry-breaking sector and the Standard Model sector to produce soft masses for all the scalars of order ΛSUSY = M3/2. In the original supergravity models, this scale was set to MEW, but this leads to many lighter particles and by now has been ruled out. One way to try to fix this is gauge mediation, in which other interactions provide larger soft masses. As far as the Standard Model is concerned, this may be good, but it suffers from the cosmological moduli and gravitino problems we will discuss in Section 14.3. One can instead try to work with ΛSUSY MEW, and argue that the naive expectations for the soft masses are incorrect. There are many ideas for this, such as sequestering [23] (see [24] for a recent string theory discussion), focus point models [25], intersection point models [26], and others. Clearly it would be important if a generic mechanism could be found, but as yet none of these have found general acceptance. Thus, we accept the generic result M0 ∼ ΛSUSY for scalar masses. However, the gaugino soft masses have other sources and, as we will discuss below, can be smaller. The extreme version of this scenario is split supersymmetry [27, 28], in which the scalars can be arbitrarily heavy while all fermionic superpartners are light. In any case, although MH is tuned, it is essentially determined by the quartic Higgs coupling and the Higgs vacuum expectation value, which we know from MZ . If the underlying model is the MSSM, then, since the quartic Higgs coupling comes from a D-term, it is determined by the gauge couplings and we get a fairly direct prediction for the Higgs 2 ≤ 2 mass. As is well known, at tree level, there is a bound MH MZ , and one must call on (14.1) just to satisfy the LEP bound MH > 113 GeV. There has been much recent discussion of the difficulty of getting MH ∼ 125 GeV to come out of the MSSM in a natural way [16–19]. If MH is fine-tuned, the loop contribution (14.1) is no longer directly measurable. However, there is a similar-looking constraint coming from the running of the quartic Higgs coupling, of the general form3

3g2M 4 Λ M 2 ∼ M 2 cos2 2β + top log SUSY . (14.2) H Z 2 2 2 16π sin βMW Mtop

In fact, for MH = 125 GeV, this turns out to predict ΛSUSY ∼ 100 TeV. This prediction is not robust; for example, by postulating another scalar that couples to the Higgses through the superpotential (the next-to-minimal supersymmetric Standard Model, NMSSM), or an extra U(1), one can change the relation between the Higgs quartic coupling and the gauge coupling. Such modiﬁcations will aﬀect the Higgs branching ratios, so this alternative will be tested at LHC in the coming years.

3 This is a bit simpliﬁed, and the precise expression depends on ones’ assumptions, see, e.g., [11, 17]. The main diﬀerence with (14.1) is that one has put in the observed electroweak parameters and thus accepted the possibility of tuning. 324 The string landscape and low-energy supersymmetry

Within this scenario, the key question for LHC physics is whether we will see the gauginos. There are many models, such as anomaly mediation [23, 29, 30], in which the expected gaugino mass is β(g) M ∼ Λ . (14.3) χ 2g2 SUSY where β(g) is the exact beta function. This prefactor is of order 10−2 for the neutralinos, so we again have a dark matter candidate if ΛSUSY ∼ 100 TeV. For the gluino, it is ∼ 1/40, and thus we are right at the edge of detection at LHC-14. With ΛSUSY ∼ 30 TeV, we would have gluino mass M3 ∼ 750 GeV, which should be seen very soon. Note that there are other string compactifications with large gaugino masses.4 a a The relevant contribution is F ∂af, where F is an F-term and f is a gauge kinetic term. Thus, gaugino masses will be large if the supersymmetry breaking F-terms are in fields, such as the heterotic string dilaton, whose expectation values strongly affect the observed gauge couplings. This is a question about the supersymmetry-breaking sector that must be addressed top-down; in Section 14.6, we will suggest that small masses are preferred. It would be very valuable from this point of view to know the lower limit on ΛSUSY in these scenarios, as, all other things being equal, it seems reasonable by naturalness to expect the lower limit (we will discuss stringy naturalness later).

14.3 The gravitino and moduli problems It has been known for a long time that light, weakly coupled scalars can be problematic for inflationary cosmology [31]. Because of quantum fluctuations, on the exit from inflation they start out displaced from any minimum of the potential, leading to possible “overshoot” of the desired minimum, and to excess entropy and/or energy. In string/M-theory compactification, moduli of the extra-dimensional metric and other fields very generally lead after supersymmetry breaking to scalar fields with gravitational strength coupling and mass M ∼ ΛSUSY, making this problem very ∼ ∼ 2 3 ∼ relevant [32, 33]. For M 1 TeV, such particles will decay around T MPlanck/M 103 s. This is very bad, since it spoils the predictions for abundances of light nuclei based on big bang nucleosynthesis, which takes place during the period 0.1 s T 100 s. One needs such particles to be either much lighter or much heavier, so that they decay at T<<0.1 s. One can also increase their couplings so that their decay reheats the universe well above the scale of nucleosynthesis—see [34] for a recent proposal of this type. There is a closely related constraint from gravitino decay [35], which can also be solved this way. Although not proven, it is quite generally stated that this constraint forces the moduli masses to satisfy

Mmoduli 30 TeV. (14.4)

4 I thank Bobby Acharya, Gordy Kane, and Gary Shiu for discussions on this point. The gravitino and moduli problems 325

This may not a priori require ΛSUSY 30 TeV, since there are other ways to lift moduli masses. For example, ﬂuxes can give masses to moduli while preserving supersymmetry [4, 36]. On the other hand, the structure of the N = 1 supergravity potential makes it generic to have at least one scalar with M M3/2, as shown in [37, 38]. This is a scalar partner of the goldstino, and it need not have gravitational strength interactions, but if supersymmetry breaking takes place in a hidden sector, this will often lead to a constraint. In addition, the gravitino has M = M3/2 by deﬁnition. If one is going to call on reheating or other physics to solve these problems, it is simplest and probably most generic for the problematic particles to be associated with a single energy scale. These various considerations all suggest that

ΛSUSY = M3/2 30 TeV. (14.5)

This is an extremely strong constraint that very much disfavors the natural solutions to the hierarchy problem and is independent of the arguments we gave in Section 14.2. This is widely recognized, and therefore there has been a major eﬀort to search for generic ways out, or at least loopholes to this bound. On the other hand, even if there are loopholes, the bound still might be generic. In the context of the string landscape, the correct attitude would then be to accept it as preferred, also allowing the vacua (and cosmological histories) that realize the loophole, but weighting them by appropriate tuning factors. We will begin this discussion below. Perhaps the main reason that (14.5) has not been more widely accepted is that there is no direct evidence at this point for the signiﬁcance of the energy scale 30–100 TeV. This is of course related to the lack of direct evidence for supersymmetry, so perhaps we should not be too bothered by it, but we should ask for some more fundamental reason why ΛSUSY ∼ 30–100 TeV should be preferred. Later, we are going to argue this from stringy naturalness, essentially that ΛSUSY should take the lowest possible value consistent with anthropic constraints. A weaker but more general claim, less dependent on anthropic constraints, is that the cosmological moduli problem will always favor a “little hierarchy” between the mass scale of “normal” matter and the supersymmetry-breaking scale. Let us start from a more general statement of the problem—namely, that important cosmological physics (in our universe, big bang nucleosynthesis) takes place at a temperature just below the mass of normal matter, and thus moduli (and the gravitino) must decay well before this happens and/or reheat the universe above this temperature. Thus, we have

Treheat >cMmatter, (14.6)

∼ −3 −1 ∼ 1/2 with a constant c 10 –10 .GivenTreheat ΛSUSY/MPlanck, this implies

3 2 ΛSUSY >cMPlanckMmatter (14.7) and the large hierarchy MPlanck Mmatter forces ΛSUSY Mmatter, but only as the one-third power of MPlanck/Mmatter and suppressed by the constant c. An even more broadbrush way of arguing would be to say that inﬂationary cosmology is already diﬃcult enough to make work at each of the relevant scales 326 The string landscape and low-energy supersymmetry

(the matter/big brag nucleosynthesis scale, the electroweak scale, and now the supersymmetry-breaking scale) that one should expect at least little hierarchies between these various scales just to simplify the problem. Whether this simplicity is of the type that Occam would have favored, or whether it has any relevance for stringy naturalness, remains unclear.

14.4 The set of string vacua The broad features of string compactification are described in [4] and many other reviews. We start with a choice of string theory or M-theory, of compactification manifold, and of the topological class of additional features such as branes, orientifolds, and fluxes. We then argue that the corresponding supergravity or string/M-theory equations have solutions, by combining mathematical existence theorems (e.g., for a Ricci-flat metric on a Calabi–Yau manifold), perturbative and semiclassical computations of corrections to supergravity, and general arguments about the structure of four-dimensional effective field theory. Perhaps the most fundamental distinction is whether we grant that our vacuum breaks N = 1 supersymmetry at the compactification (or “high”) scale or whether we can think of it as described by a four-dimensional N = 1 supersymmetric effective field theory, with supersymmetry breaking at a lower scale. Almost all work makes the second assumption, largely because there are no effective techniques to control the more general problem, nor is there independent evidence (say from duality arguments) that many high-scale vacua exist. Early work suggesting that such vacua were simply the large-ΛSUSY limit of the usual supersymmetric vacua [39, 40] was quickly refuted by a more careful analysis of supersymmetry breaking [38]. A heuristic and probably correct argument that this type of nonsupersymmetric vacuum is very rare is that stability is very difficult to achieve without supersymmetry– recently this has been shown in a precise sense for random supergravity potentials [41]. There are many versions of this question, some analogous and some dual, such as the existence of Ricci-flat metrics without special holonomy and the existence of interacting conformal field theories without supersymmetry. We will assume that metastable nonsupersymmetric vacua are not common enough to outweigh their disadvantages; of course, if a large set of them were to be discovered, this would further weaken the case for low-energy supersymmetry. Another context in which nonsupersymmetric vacua might be very important is for the theory of inflation. A natural guess for the scale of observed inflation is the GUT scale, in other words the compactification scale. The requirement of near-stability is still very constraining, however, and almost all work on this problem assumes broken N = 1 supersymmetry as well. Since the inflationary trajectory must end up in a metastable vacuum, it is hard to see how it could be very different from this vacuum anyways. Granting the need for four-dimensional N = 1 “low-scale” supersymmetry (here meaning compared with the string-theoretic scales), each of the five 10-dimensional string theories as well as 11-dimensional supergravity have a preferred The set of string vacua 327 extra-dimensional geometry that leads there. While some theories (such as type I and SO(32) heterotic) were originally thought not to contain the Standard Model, during the “duality revolution” of the 1990s it was realized that there are many more possible sources of gauge symmetry, matter, and the various interactions, such as fields localized on branes and their intersections. While the individual theories still make generic predictions, in general there is a lot of freedom to realize the Standard Model and a wide variety of additional matter sectors. An important distinction can be made between “global” models such as heterotic string compactification and “local” models such as F-theory. In a global model, realizing chiral matter requires postulating structure on the entire extra-dimensional manifold. By contrast, in a local model, chiral matter can be realized at the intersection of branes that are contained in some arbitrarily small subregion of the manifold. This is nontrivial, because chiral matter can only be realized by brane intersections that (in a certain topological sense) span all of the extra dimensions [42]. While naively this makes local models impossible—and on simple topologies such as an n-torus they would be impossible—they are possible in more complicated geometries such as resolved orbifolds and elliptic fibrations. Local models tend not to realize gauge unification, and in the simplest examples cannot realize the matter representations required for a GUT, such as the spinor of SO(10). These two problems were more or less overcome by the development of F- theory local models [43, 44]. F-theory is also attractive in that one can more easily understand the other constructions by starting from F-theory and applying dualities than the other way around. It was suggested in [43] that local models should be preferred because they admit a consistent decoupling limit. Essentially, this is a limit in which one takes the small subregion containing the local model to become arbitrarily small. Because observable scales (the Planck scale and the scale of matter) tend to be related to scales in the extra dimensions, it is more natural to get hierarchies in this limit. At present, the status of this argument is extremely unclear, since it is generally agreed that global models can realize hierarchies through dynamical supersymmetry breaking and otherwise. Later, we will discuss a different, cosmological argument that might favor local models. Because of the dualities, and the existence of topology-changing transitions in string/M-theory, the usual picture is of a single “configuration space” containing all the vacua and allowing transitions (perhaps via chains of elementary transitions) between any pair of vacua. Only special cases of this picture have been worked out; for example, it has long been known that all of the simply connected Calabi–Yau 3-folds are connected by conifold transitions. More recently, “hyperconifold transitions” have been introduced that can change the fundamental group [45, 46], but it is not known whether these connect all the non-simply connected threefolds. It is very important to complete this picture and develop concrete ways to represent and work with the totality of this configuration space. Even its most basic properties, such as any sense in which it is finite, are not really understood. Various ideas from mathematics can be helpful here; in particular, there is a theory of spaces of Riemann- ian manifolds in which finiteness properties can be proven, such as Gromov–Cheeger compactness. Very roughly, this says that if we place a few natural restrictions on the 328 The string landscape and low-energy supersymmetry manifolds, such as an upper bound on the diameter (the maximum distance between any pair of points), then the space of possibilities can be covered by a finite number of finite-size balls. These restrictions can be motivated physically and lead to a very general argument that there can only be a finite number of quasirealistic string vacua [47]; however, this does not yet lead to any useful estimate of their number. Given the topological choices of manifolds, bundles, branes, and the like, one can often use algebraic geometry to form a fairly detailed picture of a moduli space of compactifications with unbroken N = 1 supersymmetry. Various physical constraints, of which the simplest is the absence of long-range “fifth force” corrections to general relativity, imply that the scalar fields corresponding to these moduli must gain masses. To a large extent, this so-called “moduli stabilization” problem can be solved by giving the scalars supersymmetric masses. For example, background flux in the extra dimensions can lead to a nontrivial superpotential depending on the moduli, with many supersymmetric vacua [36]. The many choices of flux also make the anthropic solution of the cosmological constant problem easy to realize [48]. Moduli stabilization also determines the distribution of vacua in the moduli space, and thus the distribution of couplings and masses in the low-energy effective theory. One can make detailed statistical analyses of this distribution, which incorporate and improve the traditional discussion of naturalness of couplings [4]. While supersymmetric effects lift many neutral scalars, it is not at all clear that it generically lifts all of them, satisfying bounds like (14.4) before taking supersymmetry breaking into account. Explicit constructions such as that of [49] are usually left with one or more light scalars, and, as we discussed earlier, one can argue that this is generic [34, 38]. Another important point that is manifest in the flux sector is what I call the “broken-symmetry paradox.” Simply stated, it is that in a landscape, symmetry is heavily disfavored. One can already see this in chemistry—while the Schrödinger equation admits SO(3) rotational symmetry, and this is very important for the structure of atomic and molecular orbitals, once one shifts the emphasis to studying molecules, this symmetry does not play much of a role. While a few molecules do preserve an SO(2) or discrete subgroup, the resulting symmetry relations rarely have qualitatively important consequences beyond a few level degeneracies, and it is not at all true that molecules with symmetry are more abundant or favored in any way in chemical reactions. It was shown in [50] that discrete R-symmetries are heavily disfavored in flux compactification, and the character of the argument is fairly general. Suppose we want vacua with a ZN symmetry; then it is plausible that of the various parameters of some class of vacua including a symmetric point, order 1/N of them will transform trivially, and order 1/N will each transform in one of the N −1 nontrivial representations. But, since the number of vacua is exponential in the number of parameters, symmetry is extremely disfavored. While one can imagine dynamical arguments that would favor symmetry, since these tend to operate only near the symmetric point, it is hard to see them changing the conclusion. One virtue of this observation is that it helps explain away the gap between the many hundreds or thousands of fields of a typical string compactification (especially those with enough vacua to solve the cosmological constant problem) and the smaller Eternal inflation and the master vacuum 329 number in the Standard Model, since symmetry breaking will get rid of nonabelian gauge groups and generally lift fields. But this is very different from the usual particle physics intuition.

14.5 Eternal inflation and the master vacuum The wealth of disparate possibilities coming out of string compactification combined with the relative poverty of the data seem to force us to bring in extra structure and constraints to help solve the vacuum selection problem and test the theory. This would probably remain true even were we to discover many new particles at LHC. A good source of extra structure is cosmology, both because there is data there and because some of the key particle physics questions (such as low-energy supersymmetry) can have cosmological consequences (such as WIMP dark matter). In addition to these more specific hints, as we discussed in Section 14.1, we have real world examples of landscapes and we know there that the dynamics that forms metastable configurations plays an absolutely essential role in preferring some configurations over others. It is entirely reasonable to expect the same here. A very worrying point is that the dynamics of chemistry, and even big bang nucleosynthesis, is highly nonlinear and depends crucially on small energy differences. The problem of deducing abundances ab initio, without experimental data, is completely intractable. While this might be true of the string landscape as well, in fact the most popular scenario appears to be much simpler to analyze, since the central equations are linear. This is the idea of eternal inflation, reviewed in [6–8] and elsewhere. There is a good deal of current work on bringing this into string theory. While I am not an expert, this seems to have two main thrusts. One is to find microscopic models of inflation or, even better, a gauge dual to inflation analogous to AdS/CFT. The other is to try to make the framework sufficiently well defined to be able to make predictions, by deriving a measure factor on the set of vacua. We will simply cite [11] for a review of the status of this field and move to discussing the concrete prescription we already quoted in Section 14.1 [13] which we call the “master vacuum” prescription:

• The measure factor is overwhelmingly dominated by the longest-lived metastable de Sitter vacuum. For other vacua, it is given by the tunneling rate from this “master” vacuum, which to a good approximation is that of the single fastest chain of tunneling events.

Once we have convinced ourselves of this, evidently the next order of business is to find the master vacuum. For some measure prescriptions, this would be an absolutely hopeless task. For example, suppose we needed to find the metastable de Sitter vacuum with the smallest positive cosmological constant. By arguments from computational complexity theory [51], this problem is intractable, even for a computer the size of the universe! The problem of finding the longest-lived vacuum in this prescription could be much easier. A large and probably dominant factor controlling the tunneling rate out of a 330 The string landscape and low-energy supersymmetry metastable vacuum is the scale of supersymmetry breaking [52, 53]. The intuitive reason is simply that supersymmetric vacua are generally stable, by BPS arguments. Thus, a reasonable guess is that the master vacuum is some flux sector in a vacuum with the smallest ΛSUSY. The actual positive cosmological constant is less important, both because this factor cancels out of tunneling rates in the analysis of the measure factor and because there are so many choices in the flux sector available to adjust it. The relation to ΛSUSY also makes it very plausible that the master vacuum is not anthropically allowed. The question of how to get small ΛSUSY deserves detailed study, but it is a very reasonable guess that this will be achieved by taking the topology of the extra dimensions to be as complicated as possible, and, even more specifically, by an extra-dimensional manifold with the largest possible Euler number χ. Of course it is intuitively reasonable that complexity allows for more possibilities and thus more extreme parameter values, but there is a more specific argument, which we will now explain. The first observation is that ΛSUSY is a sum of positive terms (the sum in quadrature of D-andF -breaking terms) and thus cannot receive cancellations, so one is simply trying to make the individual D-andF -terms small. If we imagine doing this by dynamical supersymmetry breaking driven by an exponentially small nonperturbative effect, then the problem is to realize a supersymmetry-breaking gauge theory with the smallest possible coupling g2N at the fundamental scale. This coupling is determined by moduli stabilization, and is typically related to ratios of coefficients in the effective potential. These coefficients can be geometric (intersection numbers, numbers of curves, etc.) or set by quantized fluxes. To obtain a small gauge coupling, we want these coefficients to be large. In both cases, the typical size of the coefficients is controlled by the topology of the extra dimensions. For example, the maximum value of a flux is determined by a tadpole or topological constraint, which for F-theory on a Calabi–Yau 4-fold is 1 η N iN j + N = χ. (14.8) ij D3 24 i Here the N are integrally quantized values of the 4-form flux, ηij is a symmetric unimodular intersection form, and ND3 is the number of D3-branes sitting at points i in the extra dimensions. The fluxes N are maximized by taking χ large and ND3 small, allowing large ratios of fluxes. Although the other geometric quantities are much more complicated to discuss, it is reasonable to expect similar relations. Thus, we might look for the master vacuum as an F-theory compactification on the fourfold with maximal χ, which (as far as I know) is the hypersurface in weighted projective space given in [54] with χ =24· 75852. This compactification also allows a very large enhanced gauge group with rank 60 740, including 1276 E8 factors [55]. With this large number of cycles, the number of similar vacua obtained by varying fluxes and other choices should be so large (1010000 or even more), that the nearby vacua that solve the cosmological constant problem will be similar, answering the question of predictivity raised in Section 14.1. But the complexity of this compactification suggests that it might not be easy to find the precise moduli and fluxes leading to the From hyperchemistry to phenomenology 331 master vacuum. Before doing this, we need to refine the measure factor prescription, for the following reason. As stated, it assumes there is a unique longest-lived vacuum. Now, it is true that supersymmetry breaking will generate a potential on the moduli space so that de Sitter vacua will be isolated, but with this very small ΛSUSY these potential barriers will be incredibly small. At the very least, one expects the tunneling rates to other vacua on (what was) the moduli space to be large. It might be a better approximation to regard the “master vacuum” as a distribution on this moduli space given by a simple probability measure, perhaps uniform or perhaps a vacuum counting measure as in [4]. The interesting tunneling events, toward anthropically allowed vacua, would be those that increase the scale of supersymmetry breaking. One might imagine that supersymmetry breaking will be associated with a single matter sector5 (i.e., a minimal set S of gauge groups such that no matter is charged under both a group in S and a group not in S) and that these tunneling events will affect only this sector. But since the masses of charged matter depend on moduli, in parts of the moduli space where additional matter becomes light, one could get tunneling events that affect other sectors as well. We will suggest a more intuitive picture of this dynamics in Section 14.6. Much is unclear about this picture. One very basic assumption is that we can think of the cosmological dynamics using a (4 + k)-dimensional split, although of course spacetime can be much more complicated. Better justification of this point would require a better understanding of inflation in string compactification. If this can only be realized as granting such a split (as appears at present to be the case), then this would be a justification; if not, not. Another question is that since there are supersymmetric transitions between compactifications with different topology, one should not even take for granted that the master vacuum is concentrated on a single topology, although this seems plausible because such transitions change the fluxes and tadpole conditions [57].

14.6 From hyperchemistry to phenomenology Granting that the dynamics of eternal inﬂation and the master vacuum are an important part of the vacuum selection problem, it would be very useful to develop an intuitive picture of this dynamics. Let us suggest such a picture based on the assumptions stated above. The starting point is to think of the various structures that lead to the gauge– matter sectors relevant for low-energy physics—groups of cycles and/or intersecting branes—as objects that can move in the extra dimensions. The idea is that we are trying to describe a distribution on a pseudo-moduli space of nearly supersymmetric vacua, and the moduli correspond to sizes of cycles, positions of branes, and the like. Of course, the background space in which they move will not be Euclidean or indeed any ﬁxed geometry, and a really good picture must also take into account this geometrical

5 A recent paper on such sectors is [56]. 332 The string landscape and low-energy supersymmetry freedom. But, with this in mind, a picture of objects moving in a fixed six-dimensional space can be our initial picture.6 Next, the most important dynamical effect that could influence the tunneling rates is the possibility that as the moduli vary, new light fields come down in mass, perhaps coupling what were previously disjoint matter sectors. In the brane picture, this will happen when groups of branes come close together. Again, in the most general case, this can happen in other ways, such as by varying Wilson lines, but let us start with the simplest case to picture. The dynamics is thus one of structured objects (groups of cycles and intersecting branes) moving about in the extra dimensions, and perhaps interacting when they come near each other—a sort of chemistry of the extra dimensions. By analogy with the familiar word “hyperspace,” we might call this “hyperchemistry.” As in chemistry, while the structures and their possible interactions are largely governed by symmetry (here the representation theory of supersymmetry), questions of stability and rates are more complicated to determine—although hopefully not intractable. The basic objects or molecules of hyperchemistry are “clusters” of branes and cycles that intersect topologically. These translate into chiral gauge theories in the low-energy effective theory. Two groups of branes and cycles that do not intersect topologically are in different clusters; these can interact gravitationally, at long range, or by having vector-like matter become light, at short range. Although the nature and distribution of the clusters is not known in four dimensions, it has recently been determined for F-theory compactifications to six dimensions with eight supercharges [58]. It turns out that the minimal clusters give rise to certain preferred gauge theories with matter that cannot be Higgsed, for example SU(2)×SO(7)×SU(2) with half-hypermultiplets in the (2, 8, 1)⊕(1, 8, 2), or E8 with no matter. Thus, a Calabi–Yau with many cycles will give rise to a low-energy theory with many clusters. A similar picture (though with different clusters) is expected to be true for compactifications to four dimensions as well.7 As a simple picture of the dynamics, we can imagine the clusters moving around in the extra dimensions, occasionally undergoing transitions (tunneling events) that change their inner structure. Thus, we have a fixed set of chiral gauge theories, loosely coupled to each other through bulk gravitational interactions. Occasionally, two clusters will collide, leading to vector-like matter becoming light. This enables further transitions such as Higgs–Coulomb or the more complicated extremal transitions in the literature. To some extent, the details of the extra-dimensional bulk geometry would not be central to this picture—one could get away with knowing the relative distances and orientations between each pair of clusters. Our previous simplifying assumption that the clusters are moving in a fixed extradimensional geometry would imply many constraints on these parameters, which to some extent would be relaxed by

6 Although F-theory postulates a 4-fold, i.e., an eight-real-dimensional space, two of these dimensions are a mathematical device used to represent a varying dilaton–axion field. The actual extra dimensions are six-dimensional. 7 D. Morrison, private communication. From hyperchemistry to phenomenology 333 allowing the extra dimensional geometry to vary as well. In this way, our picture could accommodate all of the relevant configurations. Granting this picture, how might the master vacuum tunnel to an anthropically allowed vacuum? Now, the Standard Model is a chiral gauge theory, and we know various ways to make it up out of branes and cycles—in other words, as a cluster. It is a cluster of moderate complexity, which within F-theory can be obtained by resolving singularities of a sort that appear naturally in 4-folds. Thus, it is natural to imagine that such clusters are already present in the master vacuum. On the other hand, the master vacuum has an extremely small supersymmetry-breaking scale, probably due to dynamics in a single cluster, with no reason to have large couplings to the Standard Model cluster. Thus, the simplest dynamics that could create an anthropically allowed vacuum involves two steps: the supersymmetry-breaking cluster is modified to produce a larger scale of supersymmetry breaking, and its interactions with the Standard Model cluster are enhanced to produce the observed supersymmetry breaking. The first step is the one that should answer questions about the underlying scale ΛSUSY of supersymmetry breaking, while the second will determine its mediation to the observable sector. Regarding the first, it is reasonable to expect some high-scale vacua stabilized by 12 tuned structure in the potential as in [38], with number growing as ΛSUSY for reasons explained there. The number of these compared with low-cale vacua with ΛSUSY exponentially small is not yet clear. However, granting that the master vacuum must be one with extremely small ΛSUSY, it is already a low-scale vacuum, and thus the transition of the first step can easily be one that produces a low-scale vacuum, perhaps by varying a single flux, and thus the gauge coupling appearing in the exponential. Even if high-scale vacua can also be produced in comparable numbers, their disadvantage in solving the hierarchy problem will remain. A possible loophole would be if the mediation to the Standard Model were somehow suppressed—which seems unlikely as we argue shortly. As for the origin of the Standard Model, these pictures suggest that it would be realized by a single matter sector in a localized region of the extra dimensions—in other words a local model. This is not because it must make sense in the decoupling limit, but rather because this is the most likely way for it to be produced by cosmological dynamics. Furthermore, there is no reason that the supersymmetry-breaking sector must be near the Standard Model sector or share matter with it. This suggests that supersymmetry breaking is generically mediated by supergravity interactions. The generic estimate for scalar masses in supergravity mediation is M0 ∼ F/MPlancks. This might be smaller if the two sectors were “far apart” in the extra dimensions, but there is no known dynamics that would favor this. As we discussed in Section 14.2, other proposals for how this could be smaller such as sequestering are not presently believed to be generic in string theory. On the other hand, it is possible for the supersymmetry-breaking cluster and the Standard Model cluster to approach very closely so that the mediation is larger. In fact, they must be closer than the string scale, and thus (from brane model intuition) they will be coupled by vector- like matter, leading to a gauge-mediation scenario. While this is possible, since it is 334 The string landscape and low-energy supersymmetry continuously connected to the gravitational-mediation scenario, distinguished only by varying moduli, it requires additional tuning compared with gravitational mediation. The upshot is that gravitational mediation with M0 ∼ F/MPlancks seems favored, unless there is some reason that more of the alternative models satisfy the anthropic constraints. This question deserves close examination by those more expert in the field than myself, but I know of no major advantage in this regard. Indeed, one might expect gauge mediation to lead to small M3/2 and a cosmological moduli problem. The picture also suggests that the F-terms are of the type giving rise to small gaugino masses, since they arise in a hidden matter sector. We now recall the beyond the Standard Model part of our argument. This was to compare what seem to be the two likeliest candidate solutions of the hierarchy problem, namely, the natural supersymmetric scenario and the scenario with ΛSUSY ∼ 30–100 TeV and then an additional fine tuning. The claim is then that the additional 10−5 or so of fine tuning gained by naturalness is more than lost by the difficulty of solving the cosmological moduli problem, as well as meeting the other anthropic constraints, which are much stronger in the far more complicated natural supersymmetric theories. While this claim is hard to argue in the absence of any knowledge about higher-energy physics, if we believe we know the right class of theories to look at on top-down grounds, we can argue it. The class of extensions of the Standard Model that can be realized as local models in string/M-theory, interacting with a supersymmetry-breaking sector, is probably narrow enough to allow evaluating the bottom-up argument and making it quantitative. As it happens, for the question of whether we will see gauginos at LHC, it makes a great difference whether we expect ΛSUSY ∼ 30 TeV or 100 TeV and whether F-terms couple to observable gauge couplings, and it would be great if the arguments could reach that level of detail. To summarize the overall picture at this point, it is that we have three sources of information about how string/M-theory could describe our universe. Traditional particle phenomenology and astroparticle physics are of course bottom-up and motivate model building within broad frameworks such as quantum field theory and effective Lagrangians. Another source is top-down, the study of compactifications and their predictions for “physics” broadly construed. The results can be summarized in effective Lagrangians, tunneling rates between vacua and the like, and statistical summaries of this information for large sets of vacua. This is a “mathematical” definition of the landscape, which could in principle be developed ab initio, accepting only the most minimal real world input. Finally, there is the dynamics of early cosmology, by which the various vacua constructed in the top-down approach are created. This subject is still in its infancy—although we have pictures such as eternal inflation that might work, the details are not yet well understood, and there are variations and competing pictures yet to be explored. Simplified pictures such as hyperchemistry could help us to think physically about this dynamics. Unless the data improve dramatically, it seems to us that all three sources must be combined to make real predictions from string/M-theory. One must understand the set of vacua, or at least those near the master vacuum. One must understand the dynamics of early cosmology and presumably tunneling rates between vacua. In general, these problems will have little or nothing to do with either Standard Model or beyond the Standard Model physics, because the relevant dynamics is at completely References 335 different energy and time scales. The other part is anthropic, but, given the vagueness and difficulty of working with the anthropic principle, it is probably better to simply call it “bottom-up” and require that we match some or all of the data to hand. The main difference with the existing paradigm in phenomenology is that we can use the top-down and early cosmology information to make a well-motivated definition of naturalness, so that if reproducing the data requires postulating an unnatural vacuum, then we have evidence against the theory. All this is a long-range project, but I think we are at the point where we can begin to work on it.

Acknowledgments I thank Bobby Acharya, Raphael Bousso, Frederik Denef, Michael Dine, Dave Morri- son, Gordy Kane, Patrick Meade, Gary Shiu, Steve Shenker, and Lenny Susskind for discussions and comments on the manuscript. This research was supported in part by DOE grant DE-FG02-92ER40697.

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Laurent Baulieu

Theory Division CERN Geneva, Switzerland, and Laboratoire de Physique Théoriqueet Hautes Energies, Sorbonne Universités – Université Pierre et Marie Curie, Paris, France

15 The description of N =1, d = 4 supergravity using twisted supersymmetric ﬁelds 339 Laurent BAULIEU

15.1 Introduction 341 15.2 N =1,d = 4 supergravity in the new minimal scheme 342 15.3 Self-dual decomposition of the supergravity action 345 15.4 Twisted supergravity variables 346 15.5 The supergravity curvatures in the U(2) ⊂ SO(4)-invariant formalism 350 15.6 The 1.5-order formalism with SU(2)-covariant curvatures 351 15.7 Vector supersymmetry and nonvanishing torsion 356 15.8 Matter and vector multiplets coupled to supergravity 358 15.9 Conclusions and outlook 361 Appendix A: The BSRT symmetry from horizontality conditions 361 Appendix B: Tensor and chirality conventions 363 Appendix C: The action of γ matrices on twisted spinors 363 Appendix D: Algebra closure on the ﬁelds of matter and vector multiplets 364 Acknowledgments 365 References 366 Introduction 341

This chapter shows how one can extend the method of twisted supersymmetric ﬁelds for describing global supersymmetry, as used in the context of topological ﬁeld theories, to the case of local supersymmetry. As an example, the case of N = 1 Euclidean supergravity on a 4-manifold with an almost complex structure is considered, with its couplings to scale and vector multiplets.

15.1 Introduction Twisting is an important tool in the study of supersymmetric theories and has provided important new insights. It fundamentally means that one supercharge is singled out and used as the primary symmetry of the theory. The twist often allows for a splitting in the set of supersymmetric generators, which can be very useful. In some cases, one can find a subset of the generators that is sufficient to constrain the Lagrangian to be invariant under the full supersymmetry, while it admits off-shell closed field representations. The first examples used nontrivial R-symmetries associated with extended supersymmetries to retain full Lorentz invariance. However, it has proved useful to consider the twist of N = 1 theories, even if this means that only part of the Lorentz symmetry is explicitly realized: a Spin(7) or U(4) symmetry in dimension eight, a G2 symmetry in dimension seven. Here we consider the case of the simplest four-dimensional supergravity, to illustrate the formalism of twisted symmetry in curved space. We work with a Euclidean signature, which allows us to retain a U(2) subgroup of the rotational symmetry. This same twist has been previously considered in the theory with only global supersymmetry [1]. In the case of the N =1,d = 4 Euclidean supergravity, only a subset of the rotational symmetry is explicitly realized after the twist, and spinors are no longer present in the theory. All fields transform as tensor products of the fundamental representation of U(2) ⊂ SO(4). The fermionic part of the symmetry algebra consists of four fermionic twisted generators, one scalar, one vector, and one pseudoscalar. The translations are part of the supersymmetry algebra and appear, in the twisted formalism, in the anticommutator of the vector supersymmetry generators and the scalar or pseudoscalar generators. The twisted generators can be untwisted to recover the spinorial anticommuting generators of Poincaré supergravity. The twisted superalgebra and the superalgebra of Poincaré supergravity are related in the fact that they define the same invariant action, modulo a twist. Twisted and untwisted supergravity transformation laws can be related by a linear mapping, in a way that generalizes the case of super Yang–Mills theories [1]. The construction of the twisted superalgebra is done on a 4-manifold with a Eu- clidean signature and an almost complex structure. In this case, the Majorana spinors can be decomposed into holomorphic and antiholomorphic forms. Among the twisted fermionic generators, the scalar nilpotent one is of the main interest to us. It is formally similar to a BRST operator, and has an analogous interpretation as the twisted supersymmetry generator of topological Yang–Mills symmetry, two-dimensional quantum gravity or a topological string [2]. 342 Description of N = 1,d= 4 supergravity using twisted supersymmetric fields

The supergravity action in a twisted form is in fact determined by the invariance under this scalar supersymmetry, with an interesting decomposition occurring for both the Einstein and Rarita–Schwinger actions. Subtle phenomena arise when one requires additional invariance under the full SO(4) symmetry group. Building the twisted superalgebra produces an interesting new framework. First, we mention that supersymmetric invariants exist as nontrivial local cocycles, a property that might be of significant importance if the twisted construction can be extended to supergravities of rank N≥2. Second, the fact that invariance under the twisted scalar supersymmetry generator alone is enough to write down an action for the twisted fields might be of interest to bypass the issues raised by the lack of a system of auxiliary fields in theories such as higher-dimensional supergravities. It could be that requiring the off-shell closure of the complete Poincaré superalgebra is just too demanding. Within this approach, the super-Poincaré symmetry is not postulated, but is an emergent property once the invariance under the twisted scalar supersymmetry is imposed. In view of these hypothetical higher-dimensional generalizations, we have computed the twisted formulation for the couplings of supergravity to scalar and vector multiplets. The results are less aesthetically pleasing than those obtained for the genuine supergravity multiplet, but their existence is a plausible four-dimensional signal that twisted formulations could also be obtained in 2n ≥ 4 dimensions, with a corresponding U(2) → U(n) generalization. The scheme of the chapter is as follows. In Section 15.2, we recall some known facts about N =1,d = 4 supergravity in the new minimal scheme, focusing on the BRST formulation of its symmetries. In Section 15.3, we display a possible (anti-)self-dual decomposition of the supergravity action by exploring some properties of the Einstein and Rarita–Schwinger Lagrangians. In Sections 15.4 and 15.5, the twisted formalism is introduced through definitions of the twisted fields and the twisted operators corresponding to the symmetries of the supergravity action. The various curvatures needed to build the supergravity action are also displayed in twisted form. In Section 15.6, we use the so-called 1.5-order formalism to build the twisted scalar symmetry generator for all fields except the spin connection and give a primitive twisted form of the supergravity action. In Section 15.7, we explore the consequences of requiring the invariance of the action under the twisted vector symmetry, which eventually yields the complete twisted supergravity action. In Section 15.8, we compute the coupling to twisted supergravity of the twisted Wess–Zumino and vector multiplets. Finally, appendices give useful formulas.

15.2 N = 1, d = 4 supergravity in the new minimal scheme The N =1,d = 4 supergravity multiplet in the new minimal system of auxiliary ﬁelds [3] is

a ab e ,λ,ω ,A,B2. (15.1)

a μ Here e is the 1-form vielbein, the Majorana spinor λ = λμ dx is the 1-form gravitino, ab and ω is the spin-connection 1-form. A and B2 are auxiliary fields, with gauge N = 1,d= 4 supergravity in the new minimal scheme 343 invariances, such that the multiplet has as many bosonic and fermionic degrees of freedom both on shell and off shell, modulo the gauge invariances. The abelian 1-form gauge field A ∼ A + dc gauges chirality and B2 ∼ B2 + dΛ1, Λ1 ∼ Λ1 + dΛ0 is a gauge real 2-form. The associated curvatures are

ab ab 1 ab R = dω + 2 [ω,ω] , a a ab i a T = de + ω eb + λγ¯ λ, 2 1 ab 5 ρ = dλ + 2 ω γab + Aγ λ, (15.2) i ¯ a G3 = dB2 + 2 λγ λea, F = dA. We will often use the covariant-derivative notation D ≡ d+ω+A. We use the following expression for the N = 1 supergravity action, as in [4]: 1 a b cd ¯ 5 a ∗ I = abcde ∧ e ∧ R (ω)+iλ ∧γ γ ρ(λ, ω, A)∧ ea − 2B2∧ dA + G3∧ G3 . M4 4 (15.3) The multiplet (15.1) is an oﬀ-shell balanced multiplet with 6 bosonic degrees of freedom deﬁned modulo all gauge invariances, 12 fermionic ones, and 6 auxiliary ones, according to the following count: ea :6=16− 6 Lorentz − 4 reparametrizations, λ :12=16− 4 supersymmetries, A :3=4− 1 chiral,

B2 :3=6− 4 vector + 1 scalar. The spin connection is not an independent field, but is fixed by the (super)covariant constraint a − 1 a b c T (e, λ)= 2 Gbce e , (15.4) ab ab ≡ ab 1 ab c ab so that ω = ω (e, λ, B2) ω (e, λ)+ 2 Gc e , where ω (e, λ) is the usual spin connection seen as a function of the vielbein and gravitino. This necessary constraint expresses the fact that no first-order formalism exists for getting an off-shell closed Poincaré supersymmetry and an invariant action. The transformation laws of the various fields under supersymmetry can be expressed using a BRST symmetry operator s, where one replaces all parameters of supergravity infinitesimal transformations by local ghost fields with opposite statistics. All ghosts transform under the BRST symmetry, in such a way that s is nilpotent. The nilpotence of s is equivalent to the off-shell closure of the system of supergravity infinitesimal transformations, as shown in [4]. This BRST symmetry can be built directly (both in the minimal and new minimal set of auxiliary fields), as outlined below. 344 Description of N = 1,d= 4 supergravity using twisted supersymmetric fields

Call ξμ the vector ghost for reparametrization. The other ghosts are those of local SUSY (χ), Lorentz symmetry (Ω), the chiral U(1) symmetry (c), and the 2-form gauge 1 μ symmetry (B1 ). The ξ -dependent part of the supergravity BRST algebra decouples by redeﬁning

sˆ = s − Lξ, dˆ= d +ˆs + iφ, (15.5) where the vector ﬁeld φ is a bilinear in the supersymmetry ghost χ, i φμ = − χγ¯ μχ = sξμ − ξν∂ ξμ, (15.6) 2 ν iV is the interior derivative on the manifold for a given vector V ,andL is the Lie derivative, LV = iV d + diV . One has the important property

dˆ=exp(−iξ)(d + s)exp(+iξ), (15.7)

2 2 2 2 which ensures that (d+s) =0anddˆ = 0 are equivalent, and s =0⇔ sˆ = Lφ.The supergravity BRST transformations can be obtained by imposing constraints on the curvatures (15.2), in a way that merely generalizes the Yang–Mills case. Using ghost unification allows for a direct check of the off-shell closure by means of the Bianchi identities. In the end, one finds the following action of the BRST operators ˆ on the fields:

a ab a seˆ = −Ω eb − iχγ¯ λ, ab 5 sλˆ = −Dχ − Ω γabλ − cγ λ, − 1 − a sBˆ 2 = dB1 iχγ¯ λea, (15.8) 1 sAˆ = −dc − iχγ¯ 5γaX , 2 a sωˆ ab = −(DΩ)ab − iχγ¯ [aXb], where the spinor Xa is 1 1 X = ρ eb − G γbc + Gbcdγ5 λ. (15.9) a ab 2 abc 12 abcd The ghost transformation laws can be found in Appendix 15.A. They are such that the 2 2 closure relation s =0⇔ sˆ = Lφ is satisﬁed. The way in which the BRST symmetry transforms the supersymmetry ghost will have nontrivial consequences in the twisted formulation. By using the twist formulas of Majorana spinors as in [1, 5–9], one could analytically continue and twist by brute force these transformations in Euclidean space. We will rather try to obtain the twisted formulation in a more straightforward way, so as to unveil and better understand the mechanisms taking place in the twisted formalism. Therefore, we now proceed to our direct construction of the twisted superalgebra, keeping in mind that both untwisted and twisted formulations can be compared at any given stage. Self-dual decomposition of the supergravity action 345

As we will see, all of the information about supergravity is actually contained in the twisted scalar nilpotent generator that is hidden in the Poincar´e supersymmetry algebra. To reach this result, we need to separate both the Einstein and Rarita– Schwinger Lagrangians into parts depending only on the self-dual or the anti-self-dual parts of the spin connection.

15.3 Self-dual decomposition of the supergravity action Each of the Einstein and Rarita–Schwinger Lagrangians can be naturally split into two parts: one that depends only on the self-dual components of the spin-connection and another that depends only on the anti-self-dual ones. These two parts are equal modulo suitable boundary terms. In the case of the Einstein Lagrangian, this property has already been used for other types of twisting [9]. The Einstein Lagrangian can be written as1 1 1 L = eaebRcd = eaeb R+ − R− . (15.10) E 4 abcd 2 ab ab Since the SO(4) Lie algebra splits into two parts, the self-dual components of the ±ab ±ab ±a ±cb curvature R = dω + ω c ω depend only on the components of the spin connection ω±ab with the same self-duality. i ¯ In supergravity, the torsion is often taken to be Ta = Dea + 2 λγaλ, but to establish the equality between the two parts of the Einstein Lagrangian, it is simpler to also use b the purely bosonic torsion ta ≡ Dea, which satisﬁes the Bianchi identity Dta = Rabe . Indeed, contracting this identity with ea,wehave

a a b + − e Dta = e e (Rab + Rab), (15.11) while

a a a D(e ta)=t ta − e Dta. (15.12)

We then get 1 1 L = −eaebR− + tat − d(eat ) E ab 2 a 2 a i 1 1 i = −eaebR− − λγ¯ aλT + T aT − d eaT − eaλγ¯ λ (15.13) ab 2 a 2 a 2 a 2 a i 1 1 i =+eaebR+ + λγ¯ aλT − T aT + d eaT − eaλγ¯ λ . (15.14) ab 2 a 2 a 2 a 2 a

The second line is obtained by expressing ta in terms of Ta, remembering that a λγ¯ λλγ¯ aλ = 0 when λ is a Majorana spinor.

1 Our conventions for (anti-)self-dual tensors are collected in Appendix 15.B. 346 Description of N = 1,d= 4 supergravity using twisted supersymmetric ﬁelds

Since T a is constrained to be zero or a quantity independent of the spin connection, the expressions obtained for the Einstein action depend only on the anti-self-dual part ω−ab (in the case of (15.13)) or the self-dual part ω+ab (for (15.14)) of the spin connection. An analogous property holds for the Rarita–Schwinger Lagrangian. We can derive it using the decomposition of the gravitino on its chiral components (which are not + − ± 1 ± 5 independent for a Majorana spinor). Deﬁning λ = λ + λ with λ = 2 (1 iγ )λ, we write2

5 a + a − − a + LRS = i λγ¯ γ ρea = λ¯ γ ρ ea − λ¯ γ ρ ea, (15.15) using λ¯±γaλ± =0.3 By adding a suitable total divergence, we get

+ a − − a + − a + LRS =2λ¯ γ ρ ea − λ¯ γ λ Ta + d(λ¯ γ λ ea) (15.16)

− + + − With anticommuting Majorana fermions, we have the identity X¯ γaY = −Y¯ γaX . Since the chiral projections commute with the generators of Lorentz transformations on spinors, we simply have ρ− = D(λ−). Chiral fermions give the minimal representations of the subalgebras associated with the self-dual and anti-self-dual parts of the rotation generators, so that ρ− depends only on the anti-self-dual part of the spin connection ω−ab: − 1 −ab − ρ = d + 2 ω γab + iA λ . (15.17) The Rarita–Schwinger action can therefore be written as 5 a (ω) + a (ω−) − − a + IRS = i λγ¯ γ D λea = 2λ¯ γ D λ ea−λ¯ γ λ Ta. (15.18)

We have succeeded in expressing IE + IRS in a way that depends only on either the self-dual or the anti-self-dual part of the spin connection, whenever the constraint on the torsion is independent of the spin connection. This condition is necessary for the closure of the supersymmetry algebra acting on the vielbein.

15.4 Twisted supergravity variables In order to be able to twist the theory, we must work in a Euclidean space with an almost complex structure, i.e., a map on each tangent space J(x) with J 2 = −1, or, μ ρ − μ more explicitly, Jρ (x)Jν (x)= δν . Introducing complex coordinates zm, z¯m¯ , where m =1, 2, we can locally reduce n n n¯ − n¯ the complex structure to a diagonal one, Jm = iδm , Jm¯ = iδm¯ . Making use of a compatible metric to lower one of the indices in J, J becomes an antisymmetric tensor with Jmn¯ as the only nonvanishing components.

2 See Appendix 15.B for the details of our chirality conventions. 3 Care must be taken in Minkowski space, where the conjugation changes chirality, so that, for example, λ+ = λ¯−. Twisted supergravity variables 347

The tensor Jmn¯ can be used instead of the metric to lower and raise indices in the m mn¯ m¯ mn¯ tangent space, according to X = −iJ Xn¯ and X = iJ Xn. In order to keep our formulas as uncluttered as possible, we will use a notation similar to Einstein’s notation for contracting antiholomorphic and holomorphic SU(2) indices by means of the complex structure constant tensor, as follows:

a ab mn¯ X Ya = g XaYb = −iJ (XmYn¯ + Xn¯ Ym) ≡ XmYm¯ − Xm¯ Ym. (15.19)

The antisymmetry of the tensor Jmn¯ implies that we must be careful about the ordering of indices. It explains the minus sign appearing in the last term of (15.19). Twisting must be done in Euclidean space, where it is known that there are no Majorana spinors. We therefore forget the Majorana condition, the eﬀect of which can be recovered afterwards from a careful consideration of the Wick rotation [10]. We α associate with the spinor (λ ,λα˙ ) the following four quantities with only holomorphic or antiholomorphic indices:

(Ψm, Ψm¯ n¯ , Ψ0). (15.20)

The indices m andm ¯ take two diﬀerent values and the object Ψm¯ n¯ is antisymmetric in its indices, so that it only has one nonzero component. The twisted components of a spinor (15.20) are deﬁned from the following linear mapping, which uses Pauli matrix elements [1, 5–9]:

α Ψm = λ (σm)α1˙ , ¯ α˙ Ψm¯ n¯ = λα˙ (¯σm¯ n¯ ) 2˙ , (15.21) ¯ α˙ Ψ0 = λα˙ δ2˙ .

In Appendix 15.C, we give the expression of the twists of Γλ as functions of the twisted components of λ for some elements Γ of the Cliﬀord algebra. This construction reduces the tangent space SO(4) symmetry to an SU(2) × U(1) ⊂ SO(4) symmetry. With this change of variables, SO(4)-invariant expressions can be related to their twisted counterparts, which generally split into a sum of independently U(2)-invariant terms. For instance, the Rarita–Schwinger Lagrangian can be decomposed as follows:

¯ 5 a λγ γ ρea =(Ψ0ρm +Ψmρ0)em¯ − (2Ψm¯ n¯ ρn − Ψnρm¯ n¯ )em. (15.22)

The commuting Majorana ghost of local supersymmetry χ is twisted as follows:

χ ∼ (χm,χm¯ n¯ ,χ0), (15.23) 348 Description of N = 1,d= 4 supergravity using twisted supersymmetric ﬁelds

μ − 1 μ μ − ν μ and the vector ﬁeld in (15.6), φ = 2 iχγ¯ χ = sξ ξ ∂ν ξ , is now given by

φm = −χmχ0,φm¯ = −χm¯ n¯ χn. (15.24)

When the parameter of vector supersymmetry vanishes, χm = 0, the vector ﬁeld φ vanishes.4 A consistent interpretation of the twisted supersymmetry involves only fermionic global charges. Thus, in what follows, χm,χm¯ n¯ ,χ0 will be treated as constant ghosts. We will build a set of corresponding generators δm¯ ,δmn,δ that satisfy anticommutation relations that close independently of the equations of motion (oﬀ-shell closure), but possibly modulo bosonic gauge transformations. We will consider the operation

Q = χmδm¯ + χm¯ n¯ δmn + χ0δ. (15.25) For a vanishing gravitino field, Q is nilpotent, off-shell, and modulo bosonic gauge transformations. It turns out that the global Q invariance is a sufficiently strong condition to determine the supergravity action. In fact, it gives a Ward identity that is sufficient to control the quantum perturbative behavior of the theory generated by the Q-invariant action, once all its gauge invariances are gauge-fixed in a BRST invariant way. When the gravitino field is not zero, the closure algebra is more involved. We will see that it involves supersymmetry transformations with gravitino-field-dependent structure coefficients. In this construction, the supergravity action is, however, fully determined by the global supersymmetry operation Q. Local supersymmetry is warranted because of the systematic construction of the charges δm¯ , δmn, δ in a way that is compatible with the Bianchi identities of all field curvatures. The four generators (δ, δm¯ ,δmn) must act on all the twisted fields of the multiplet (15.1), with the following g-grading assignments:

ﬁeld grading ﬁeld grading em 0 A 0 em¯ 0 B2 0 Ψm 1 ωmn 0 Ψ0 −1 ωm¯ n¯ 0 Ψm¯ n¯ −1 ωmn¯ 0

generators grading δ 1 δm¯ −1 δmn 1

4 This condition means that χ is a pure spinor, and it is not surprising that it entails great simpliﬁcations in the formalism, as in [11]. Twisted supergravity variables 349

The commutation properties of the various ﬁelds are always obtained by computing the sum of the form degree and the grading g of ﬁelds (for instance, em is an anticommuting object since the form degree is 1 and g =0;Ψm is a commuting object since the form degree is 1 and g = 1; etc.). After having obtained a classical action that is invariant under the twisted nilpotent global supersymmetry Q, one must in principle check that it remains invariant under local supersymmetry by giving a coordinate dependence to (χ0,χm,χm¯ n¯ ). This is in fact automatically realized, since all derivatives will appear as supercovariantized ones. If we now generalize (χ0,χm,χm¯ n¯ ) into local commuting (twisted) Faddeev–Popov ghosts, we get the operator

sˆ = χ0(x)δ + χm(x)δm¯ + χm¯ n¯ (x)δmn. (15.26) Its action on the classical fields is the same as that of the standard BRST transformations in twisted form. In the flat-space N = 1 super Yang–Mills theory [1], the three nilpotent symmetry 2 generators δ and δp¯ satisfy the off-shell closure anticommutation relations δ =0, {δp¯,δq¯} =0,and{δ, δp¯} = ∂p¯. The situation is more complicated in supergravity. In this case, one indeed has the 2 propertys ˆ = Lφ, where the vector field φ has been defined in (15.24). One also has the transformation lawsχ ˆ ∼ iφΨ (see Appendix 15.A), which remains true even when the supersymmetry ghosts are assumed to be constant. This implies the following supergravity generalization: 2 δ =0, {δp¯,δq¯} =0, (15.27) {δ, δp¯} = Lp¯ − Ψp,a¯ δa¯. a=0,m,m¯ n¯ These anticommutation relations hold modulo bosonic gauge transformations. The derivative Lp¯ is the Lie derivative along the vector field dual to the vielbein component ep¯. In fact, the supersymmetry generators that occur in the expression for {δ, δp¯} occur proportionally to the components Ψp,a¯ of the gravitino field Ψa ≡ Ψm,aem¯ − Ψm,a¯ em. One thus recognizes the expected feature of supergravity: the anticommutator {δ, δp¯} closes on supersymmetry generators with field-dependent coefficients, proportionally to gravitino-field components. Therefore, the anticommutator {δ, δq¯} is expected to involve the fourth symmetry generator δpq, whose existence can be checked afterward in the twisted method. Once δ and δp¯ have been determined, the δ-andδp¯-invariant action turns out to be automatically invariant under a δpq symmetry. In the four-dimensional supergravity, the relations between δpq and the other generators δ and δp¯ are satisfied off shell. The fermionic scalar operator δ can be extended as a globally well-defined object (provided there is a complex structure). We will focus mainly on the question of its direct construction. In fact, δp¯ and δpq can only be given a geometrical interpretation on a coordinate patch. 350 Description of N = 1,d= 4 supergravity using twisted supersymmetric fields

15.5 The supergravity curvatures in the U(2) ⊂ SO(4)-invariant formalism In Section 15.3, we have shown that both the Einstein and Rarita–Schwinger actions depend only on the self-dual or anti-self-dual components of the spin connection. In SU(2) ⊂ SO(4) notation, the self-duality condition on an antisymmetric Lorentz 1 cd tensor, Fab = 2 abcdF , reads mn¯ Fm¯ n¯ = Fmn =0,Fmm¯ ≡ iJ Fmn¯ =0, (15.28) − 1 cd while the anti-self-duality condition Fab = 2 abcdF reads

Fmn¯ − iJmn¯ Fpp¯ =0. (15.29)

ab +ab −ab Thus, the spin connection ω = ω + ω ≡ (ωmn,ωm¯ n¯ ,ωmn¯ ) splits into self-dual and anti-self-dual parts,

+ab ω ∼ (0, 0,ωmn¯ − iJmn¯ ω), (15.30) −ab ω ∼ (ωmn,ωm¯ n¯ ,iJmn¯ ω), respectively, where ω ≡ iJmn¯ ωmn¯ . The SO(4) Lie algebra is the product of two SU(2) corresponding to the self-dual and anti-self-dual generators. Therefore, the anti-self-dual part of the curvature 2-form − R ∼ (Rmn,Rm¯ n¯ ,R) and its Bianchi identities depend only on the anti-self-dual part of the connection ω−ab:

R = dω +2ωmnωm¯ n¯ ,

Rmn = dωmn − ωωmn,

Rm¯ n¯ = dωm¯ n¯ + ωωm¯ n¯ , (15.31)

dR =2Rmnωm¯ n¯ − 2ωmnRm¯ n¯ ,

dRmn = Rmnω − Rωmn,

dRm¯ n¯ = Rωm¯ n¯ − Rm¯ n¯ ω.

The SO(4) symmetry acts only as this SU(2) on Ψ0 and Ψm¯ n¯ , owing to chirality properties. We can thus deﬁne the SU(2)-covariant curvatures for Ψ0 and Ψm¯ n¯ : 1 ρ = dΨ − ω − A Ψ + ω Ψ , 0 0 2 0 mn m¯ n¯ (15.32) 1 ρ = dΨ + ω + A Ψ − ω Ψ . m¯ n¯ m¯ n¯ 2 m¯ n¯ m¯ n¯ 0 Their Bianchi identities are 1 Dρ = − R + F Ψ + R Ψ , 0 2 0 mn m¯ n¯ (15.33) 1 Dρ = R + F Ψ − R Ψ . m¯ n¯ 2 m¯ n¯ m¯ n¯ 0 The 1.5-order formalism with SU(2)-covariant curvatures 351

The curvature ρm of Ψm involves only the self-dual part of the spin connection. We can skip its deﬁnition, since it is not needed in the supergravity action. The torsion involves both self-dual and anti-self-dual components of the spin connection:

Tm = dem + ωmnen¯ − ωmn¯ en +ΨmΨ0, (15.34) Tm¯ = dem¯ + ωmn¯ en¯ − ωm¯ n¯ en +Ψm¯ n¯ Ψn,

DTm = Rmnen¯ − Rmn¯ en + ρmΨ0 − Ψmρ0, (15.35) DTm¯ = Rmn¯ en¯ − Rm¯ n¯ en + ρm¯ n¯ Ψn − Ψm¯ n¯ ρn.

We now use the SU(2) notation to decompose the Einstein and Rarita–Schwinger Lagrangians as sums of terms that are separately SU(2)-invariant, using the expressions (15.13) and (15.18):

IE = − Remem¯ + Rmnem¯ en¯ + Rm¯ n¯ emen − ΨmΨ0Tm¯ − Ψm¯ n¯ ΨnTm + TmTm¯ , (15.36)

IRS = − 2ρm¯ n¯ Ψnem − 2ρ0Ψmem¯ + ΨmΨ0Tm¯ − Ψm¯ n¯ ΨnTm . (15.37)

Equations (15.36) and (15.37) are interesting. However, at ﬁrst sight, they are not yet very suggestive about the existence of a twisted scalar supersymmetry. In fact, to build the scalar supersymmetry, we depart from the method used in [4]. The so-called 1.5-order formalism, once it has been adapted to the twisted ﬁelds of supergravity, will neatly separate the various terms of the invariant actions (15.36) and (15.37).

15.6 The 1.5-order formalism with SU(2)-covariant curvatures The justification for the 1.5-order formalism for supergravity is detailed in [12]. One first builds a supersymmetry that acts on all fields except the spin connection ω.The later is taken not to transform under supersymmetry in a first step. The second-order formalism transformation law of ω is the one compatible with all Bianchi identities of the theory, including that of the Riemann curvature. In the 1.5-order formalism, it is particularly simple to obtain the twisted scalar supersymmetry on all fields but the spin connection, by imposing consistent constraints on the ghost-dependent curvatures. The ghost-dependent curvatures are obtained by the substitutions

d → dˆ= d + χ0δ1.5 + iφ, Ψ → Ψ=Ψ+ˆ χ. (15.38)

We are concerned only with the scalar supersymmetry for the moment. Thus, we only retain a constant χ0 as the only nonvanishing component in χ. Since χm =0,we 352 Description of N = 1,d= 4 supergravity using twisted supersymmetric ﬁelds

ˆ ˆ2 2 have φm = φm¯ =0andd = d + χ0δ1.5. The property d = 0 implies δ1.5 =0on all ﬁelds. The 1.5-order formalism constraints that are compatible with the Bianchi identities are

Rˆ = R, Rˆmn = Rmn, Rˆm¯ n¯ = Rm¯ n¯ , Fˆ = F,

ρˆ0 = ρ0, ρˆm¯ n¯ = ρm¯ n¯ , ρˆm = ρm, (15.39)

Gˆ3 = G3, Tˆm = Tm, Tˆm¯ = Tm¯ , where G3 is the ﬁeld strength of the 2-form B2, deﬁned in twisted form as

Gˆ3 = dBˆ 2 + Ψˆ mΨˆ 0em¯ − Ψˆ m¯ n¯ Ψˆ nem. (15.40) We now use (15.38) and pick up the term with ghost number 1 in (15.39). This gives the δ1.5 transformation laws for all ﬁelds:

δ1.5

em −Ψm em¯ 0 Ψm 0 1 − Ψ0 2 ω A Ψm¯ n¯ ωm¯ n¯ (15.41) ωmn 0 ωm¯ n¯ 0 ω 0 A 0 B2 −Ψmem¯

The curvatures transform as

δ1.5R =0,δ1.5Rmn =0,δ1.5Rm¯ n¯ =0,δ1.5F =0, 1 (15.42) δ ρ = − R + F, δ ρ = −R ,δρ =0. 1.5 0 2 1.5 m¯ n¯ m¯ n¯ 1.5 m

We can therefore build three δ1.5-invariant Lagrangians that respectively contain the three independent SU(2)-invariant pieces Remem¯ , Rmnem¯ en¯ ,andRm¯ n¯ emen of the Einstein Lagrangian:

Rmnem¯ en¯ ,

Rm¯ n¯ emen +2ρm¯ n¯ Ψnem, (15.43)

Remem¯ − 2ρ0Ψmem¯ − 2FB2. The action

I = αRmnem¯ en¯ + β(Rm¯ n¯ emen +2ρm¯ n¯ Ψnem)+γ(Remem¯ − 2ρ0Ψmem¯ − 2FB2) (15.44) is thus invariant under the transformations (15.41), for all possible values of the coeﬃcients α, β,andγ. Lorentz symmetry is obtained when α = β = γ. The 1.5-order formalism with SU(2)-covariant curvatures 353

Alternatively, in a method that is closer to the we used in [4], we can directly check the invariance of the action (15.44) by computing the following quantities, using the Bianchi identities for the curvatures:

Dˆ(Rˆmneˆm¯ eˆn¯ )=2Rˆmn(Tˆm¯ − Ψˆ m¯ p¯Ψˆ p)ˆen¯ ,

Dˆ(Rˆm¯ n¯ eˆmeˆn)=2Rˆm¯ n¯ (Tˆm − Ψˆ mΨˆ 0)ˆen,

Dˆ(Rˆeˆmeˆm¯ )=Rˆ(Tˆm − Ψˆ mΨˆ 0)em¯ − Rˆeˆm(Tˆm¯ − Ψˆ m¯ p¯Ψˆ p), (15.45) 1 Dˆ(ˆρ Ψˆ eˆ )= Rˆ + Fˆ Ψˆ − Rˆ Ψˆ Ψˆ eˆ +ˆρ ρˆ eˆ m¯ n¯ n m 2 m¯ n¯ m¯ n¯ 0 n m m¯ n¯ n m

−ρˆm¯ n¯ Ψˆ n(Tˆm − Ψˆ mΨˆ 0), 1 Dˆ(ˆρ Ψˆ eˆ )= − Rˆ + Fˆ Ψˆ + Rˆ Ψˆ Ψˆ eˆ +ˆρ ρˆ eˆ − ρˆ Ψˆ (Tˆ − Ψˆ Ψˆ ), 0 m m¯ 2 0 pq p¯q¯ m m¯ 0 m m¯ 0 m m¯ m¯ p¯ p

Dˆ(FˆBˆ2)=Fˆ(Gˆ3 − Ψˆ mΨˆ 0eˆm¯ + Ψˆ m¯ n¯ Ψˆ neˆm).

Taking the part with ghost number 1 of these equations and retaining only χ0 =0, we obtain the δ1.5 transformations of the various terms in the action:

δ1.5(Rmnem¯ en¯ )=0,

δ1.5(Rm¯ n¯ emen)=−2Rm¯ n¯ Ψmen,

δ1.5(Remem¯ )=−RΨmem¯ , (15.46)

δ1.5(ρm¯ n¯ Ψnem)=−Rm¯ n¯ Ψnem, − 1 δ1.5(ρ0Ψmem¯ )= 2 R + F Ψmem¯ ,

δ1.5(FB2)=−F Ψmem¯ , which ensure that δ1.5(I)=0. The formulas (15.45) are actually quite useful to directly compute the action of the 1.5 vector supersymmetry δp¯ , by generalizing to the case where χp =0.Usingaghost expansion as for the scalar symmetry, we get

1.5 δp¯ (Rmnem¯ en¯ )=2RmnΨm¯ p¯en¯ , 1.5 δp¯ (Rm¯ n¯ emen)=2Rp¯n¯ Ψ0en, δ1.5(Re e )=−RΨ e + Re Ψ , (15.47) p¯ m m¯ 0 p¯ m m¯ p¯ 1 δ1.5((ρ Ψ e )= − R + F Ψ + R Ψ e , p¯ m¯ n¯ n m 2 m¯ p¯ m¯ p¯ 0 m 1 δ1.5(ρ Ψ e )= − R + F Ψ + R Ψ e , p¯ 0 m m¯ 2 0 mn m¯ n¯ p¯ 1.5 − δp¯ (FB2)= F (Ψ0ep¯ +Ψm¯ p¯em). 354 Description of N = 1,d= 4 supergravity using twisted supersymmetric ﬁelds

1.5 We find that δp¯ is another symmetry of the complete action, provided that α = β = γ, in which case the SU(2) symmetry is enlarged to SO(4). However, we must be careful in the interpretation of this vector symmetry, since it α cannot be obtained by twisting the supersymmetry generators (Q ,Qα˙ ). Indeed, δ1.5 1.5 { 1.5} and δp¯ do not have the right anticommutation relations, since δ1.5,σp¯ Ψ=0,in contradiction with the twisted supersymmetry algebra (15.27). In fact the 1.5-order formalism, which is useful to determine the invariant action, does not properly define the supersymmetry generators. We must determine the ω transformations consistent with the constraints, which appear as equations of motion in the 1.5-order formalism. With the invariant action (15.44), the equations of motion of the anti-self-dual spin connection give 12 = 3 × 4 equations that can be solved algebraically to determine the 12 components of the three 1-forms ωmn, ωm¯ n¯ ,andω, as functions of e and Ψ. The precise values then depend on the parameters α, β,andγ. We can then compute the δ1.5 transformations of these functions through the chain rule to obtain the transformations of ωmn, ωm¯ n¯ ,andω. Since δ1.5 is nilpotent on e and Ψ, this procedure gives a nilpotent transformation in the second-order formalism, where ωmn,ωm¯ n¯ ,andω are not independent fields. The case of interest is for the rotationally invariant action (15.44), which has α = β = γ. In this case, the spin-connection equations of motion give

δ − I(e, Ψ,B ,ω)=e T (ω ) =0, δω 2 m m¯ δ (ω−) I(e, Ψ,B2,ω)=e[¯mTn¯] =0, (15.48) δωmn δ (ω−) I(e, Ψ,B2,ω)=e[mTn] =0. δωm¯ n¯

− Here T (ω ) is a function only of ω−:

(ω−) Tm = dem + ωmnen¯ +ΨmΨ0, − (ω ) − Tm¯ = dem¯ ωm¯ n¯ en +Ψm¯ n¯ Ψn.

These 12 equations ﬁx the 12 components of the anti-self-dual part of the spin connection, ω = ω(e, Ψ), ωmn = ωmn(e, Ψ), and ωm¯ n¯ = ωm¯ n¯ (e, Ψ), as functions of the vielbein and the twisted gravitino. These components are the anti-self-dual parts of the complete spin connection that satisfy the constraint Tm = Tm¯ =0, As a consequence of the chain rule, ω(e, Ψ), ωmn(e, Ψ), and ωm¯ n¯ (e, Ψ) transform under supersymmetry, and the 1.5-order formalism guarantees that

I = − Rmnem¯ en¯ +(Rm¯ n¯ emen +2ρm¯ n¯ Ψnem)+(Remem¯ −2ρ0Ψmem¯ −2FB2) (15.49) is still supersymmetric. To avoid the heavy calculations from the chain rule, we can use the formalism used in [4] and determine modified horizontality conditions for the field strengths Rˆ The 1.5-order formalism with SU(2)-covariant curvatures 355 and Fˆ at ghost numbers 1 and 2, such that the Bianchi identities are satisfied and the constraints are invariant. The invariance of the constraints is equivalent to the satisfaction of the chain rule. We define Rˆ = R + R(1) + R(2), (15.50) Fˆ = F + F (1) + F (2), (15.51) while keeping Tˆ = T, ρˆ = ρ, (15.52)

Gˆ3 = G3. The ghost-number-2 part of the Bianchi identity on the torsion Tˆ ensures that when χ =0,R(2) = F (2) = 0. The condition Gˆ = G implies δB = −Ψ e ,and m 3 3 2 m m¯ 1 ρˆ =(d + s)Ψˆ − ωˆ − Aˆ Ψˆ +ˆω Ψˆ = ρ , 0 0 2 0 mn m¯ n¯ 0 (15.53) 1 ρˆ =(d + s)Ψˆ + ωˆ + Aˆ Ψˆ − ωˆ Ψˆ = ρ , m¯ n¯ m¯ n¯ 2 m¯ n¯ m¯ n¯ 0 m¯ n¯ together with their respective Bianchi identities, imply R(1) =2F (1), (15.54) (1) Rm¯ n¯ =0. Finally, the part with ghost number 1 of the Bianchi identity on Tˆ, (15.34), implies (1) 1 R = − ρp[n,m]ep¯ + ρp¯[n,m]ep , mn 2 (15.55) (1) R =(ρp¯m,m¯ ep + ρpm,m¯ ep¯). These values of R1 and F 1 determine the transformation laws of ω and A,so that the second-order scalar supersymmetry transformations that leave invariant the action (15.49) are 1δ (with δ2 =0)

em −Ψm em¯ 0 Ψm 0 1 − Ψ0 2 ω A Ψ ω m¯ n¯ m¯ n¯ (15.56) − 1 ωmn 2 ρp[n,m]ep¯ + ρp¯[n,m]ep ωm¯ n¯ 0 ω ρp¯m,m¯ ep + ρpm,m¯ ep¯ 1 A 2 (ρp¯m,m¯ ep + ρpm,m¯ ep¯) B2 −Ψmem¯ 356 Description of N = 1,d= 4 supergravity using twisted supersymmetric ﬁelds

We have used a notation where ρmn, ρm¯ n¯ ,andρmn¯ are the components of the 2-form 1 ρ on the vielbein basis, i.e., ρ = 2 (ρmnem¯ en¯ + ρm¯ n¯ emen + ρmn¯ emen¯ ). The indices to the right of the comma refer to the twisted spinor indices 0, m orm ¯ n¯.

15.7 Vector supersymmetry and nonvanishing torsion

There is no vector supersymmetry δp¯ for the action (15.49) that can satisfy the oﬀ-shell closure relation {δ, δp¯} = Lp¯ − Ψp,a¯ δa¯. Indeed, suppose that such a symmetry existed. 2 2 The oﬀ-shell closure means dˆ =(d + χ0δ + χpδp¯ + iφ) = 0, with φm = −χmχ0 =0. Thus, the Bianchi identity,

dGˆ 3 = −Ψˆ mρˆ0eˆm¯ +ˆρmΨˆ 0eˆm¯ +Ψˆ mΨˆ 0Tˆm¯ −ρˆm¯ n¯ Ψˆ neˆm+Ψˆ m¯ n¯ ρˆneˆm−Ψˆ m¯ n¯ Ψˆ nTˆm, (15.57) has a nontrivial ghost-number-2 part, which is

iφG3 = χmχ0Tm¯ . (15.58) Therefore, the torsion cannot be taken identically equal to zero, which implies that the Lagrangian found in Section 15.6 must be modiﬁed by terms that have an oﬀ-shell relevance. To remain in the context of a Lorentz-invariant action, we use the following constraints on the torsion, which generalize (15.58): 1 Tm = dem + ωmnen¯ − ωmn¯ en +ΨmΨ0 = (Gmp¯q¯epeq − Gmpq¯ep¯eq), 2 (15.59) 1 T = de + ω e − ω e +Ψ Ψ = (G e e − G e e ). m¯ m¯ mn¯ n¯ m¯ n¯ n m¯ n¯ n 2 mpq¯ p¯ q¯ m¯ pq¯ p q¯ The value of the spin connection is therefore changed, and the distortion on the horizontality condition (15.50) becomes (1) 1 R = − ρp[n,m]ep¯ + ρp¯[n,m]ep + Gmnp¯Ψp , mn 2 (15.60) (1) R = ρp¯m,m¯ ep + ρpm,m¯ ep¯ + Gmp¯m¯ Ψp. The scalar supersymmetry transformations are now δ (with δ2 =0)

em −Ψm em¯ 0 Ψm 0 1 − Ψ0 2 ω A Ψ ω m¯ n¯ m¯ n¯ (15.61) − 1 ωmn 2 ρp[n,m]ep¯ + ρp¯[n,m]ep + Gmnp¯Ψp ωm¯ n¯ 0 ω ρp¯m,m¯ ep + ρpm,m¯ ep¯ + Gmp¯m¯ Ψp 1 A 2 (ρp¯m,m¯ ep + ρpm,m¯ ep¯ + Gmp¯m¯ Ψp) B2 −Ψmem¯ Vector supersymmetry and nonvanishing torsion 357

With T = 0, the variation of the action found in Section 15.6 involves new terms proportional to Tδω, with must be compensated by the variation of new terms quadratic in G. We have

δGmpq¯ = ρqp,m¯ − Gmpr¯Ψq,r¯ − 2Gmr¯q¯Ψp,r, (15.62) δGmp¯q¯ = ρp¯q,m¯ − 2Gmr¯q¯Ψp,r¯ , 1 δe = − Ψ e e e , 2 p¯nrs¯ p n r¯ s¯ (15.63) ∗ δ( G3G3)=−eGmpq¯ Grp¯q¯Ψr,m¯ +2Gmr¯q¯Ψp,r¯

∗ Here G3 denotes the Hodge dual of G3 and e is the volume form built from (em,em¯ ). From the relation (15.58) between the torsion and the 3-form G3,wehave 1 T T + ∗G G = − (G G e e e e + G G e e e e ). (15.64) m m¯ 3 3 4 mp¯q¯ m¯ rs¯ p q r s¯ mpq¯ mrs¯ p¯ q r¯ s¯ ∗ We thus add the term TmTm¯ + G3G3 to the action (15.49), which cancels the eﬀect of the variations of the spin connection given in (15.61) under the δ symmetry. The resulting invariant action is

Itot = − Rmnem¯ en¯ +(Rm¯ n¯ emen +2ρm¯ n¯ Ψnem)+(Remem¯ − 2ρ0Ψmem¯ − 2FB2) ∗ −TmTm¯ − G3G3. (15.65) Using (15.36) and (15.37), this action can be written as ∗ Itot = LE + LRS +2FB2 + G3G3 (15.66)

This is nothing more that the complete supergravity action of (15.3). This action is also invariant under δp¯ and δpq, since it is equivalent to the one determined to be invariant under the complete untwisted BRST symmetry operator in [4]. The transformations under all twisted supersymmetry generators of the ﬁelds are

δ δp¯ δpq

em −Ψm iJmp¯Ψ0 0 − em¯ 0 Ψp¯m¯ 2iJm¯ [pΨq] 1 − Ψm 0 iJpm¯ 2 ω A + ωpm¯ 0 1 − − Ψ0 2 ω A 0 ωpq 1 Ψm¯ n¯ ωm¯ n¯ 0 2Jm¯ [p|Jn¯|q] 2 ω + A − 1 − 1 ωmn X[m,n] 2 iJmp¯ (ρqn,¯ 0eq + ρqn,0eq¯) 2 Gmnp¯Ψ0 0 − 1 − ωm¯ n¯ 0 2 (ρq¯n,¯ m¯ p¯eq + ρqn,¯ m¯ p¯eq¯ + Gm¯ nq¯ Ψp¯q¯) iJm¯ [pXn,¯ |q] 1 1 − ωmn¯ 2Xm,n¯ 2 iJmp¯ (ρq¯n,¯ 0eq + ρqn,¯ 0eq¯)+ 2 (Gmp¯n¯ Ψ0 Gmqn¯ Ψp¯q¯) 0 1 − A Xm,m¯ 2 (ρq¯p,¯ 0eq + ρqp,¯ 0eq¯ Gmp¯m¯ Ψ0 + Gmqm¯ Ψp¯q¯) Xp,q

B2 −Ψmem¯ −Ψ0ep¯ − Ψp¯m¯ em −2Ψ[peq] 358 Description of N = 1,d= 4 supergravity using twisted supersymmetric fields with the twisted X spinor in (15.9) defined as 1 X = − (ρ e + ρ e + G Ψ ). (15.67) m,n 2 pn,m p¯ pn,m¯ p mnp¯ p Since these transformations are obtained directly from the Bianchi identities and the modified horizontality conditions for field strengths, the three anticommutation relations (15.27) hold true.5

15.8 Matter and vector multiplets coupled to supergravity In this section, we will compute both the scalar and vector symmetries acting on the matter ﬁelds, so we will retain (χ0,χp) = 0 when we expand the curvature equations in ghost number. The invariant actions for both multiplets can be expressed as δ-exact terms, in a way that generalizes the ﬂat-space case [1].

15.8.1 The Wess–Zumino multiplet The Wess–Zumino matter multiplet is (P, σ, H), where P is a complex scalar field, σ a Majorana spinor (higgsino), and H a complex auxiliary field, twisted into (φ, φ,¯ σ0,σm¯ ,σmn,Bm¯ n¯ ,Bmn). The various field strengths are

Pˆ = Dφˆ + Ψˆ mσm¯ , ˆ P¯ = Dˆφ¯ − Ψˆ 0σ0 − Ψˆ m¯ n¯ σmn,

Σˆ 0 = Dσˆ 0 + BmnΨˆ m¯ n¯ ,

Σˆ m¯ = Dσˆ m¯ − Bm¯ n¯ Ψˆ n, (15.68)

Σˆ mn = Dσˆ mn + BmnΨˆ 0,

Hˆmn = DBˆ mn,

Hˆm¯ n¯ = DBˆ m¯ n¯ , with the covariant derivative D explicitly deﬁned as Dφˆ = dφˆ + wAφ,ˆ ˆ ¯ ˆ¯ − ˆ¯ Dφ = dφ wAφ, 1 Dσˆ 0 = dσˆ 0 + ωˆ − w Aˆ σ0 +ˆωm¯ n¯ σmn, 2 1 Dσˆ m¯ = dσˆ m¯ + ωˆ + w Aˆ σm¯ − ωˆmn¯ σn¯ , (15.69) 2 1 Dσˆ = dσˆ − ωˆ + wAˆ σ − ωˆ σ , mn mn 2 mn mn 0 DBˆ mn = dBˆ mn − w ABˆ mn, ˆ DBˆ m¯ n¯ = dBm¯ n¯ + w ABˆ m¯ n¯ .

5 The explicit veriﬁcation is nontrivial, since it relies on the expression of the spin connection, expressed as a solution of (15.59). Matter and vector multiplets coupled to supergravity 359

To have Bianchi identities, we must have w = w +1andw = w + 2. We obtain

DˆPˆ = wFφˆ +ˆρmσm¯ − Ψˆ mΣˆ m¯ , DˆP¯ˆ = −wFˆφ¯ − ρˆ σ − ρˆ σ + Ψˆ Σˆ + Ψˆ Σˆ , 0 0 m¯ n¯ mn 0 0 m¯ n¯ mn 1 DˆΣˆ = Rˆ − (w +1)Fˆ σ + Rˆ σ + Hˆ Ψˆ + B ρˆ , 0 2 0 m¯ n¯ mn mn m¯ n¯ mn m¯ n¯ 1 DˆΣˆ = Rˆ +(w +1)Fˆ σ − Rˆ σ − Hˆ Ψˆ − B ρˆ , (15.70) m¯ 2 m¯ mn¯ n¯ m¯ n¯ n m¯ n¯ n 1 DˆΣˆ = − Rˆ − (w +1)Fˆ σ − Rˆ σ + Hˆ Ψˆ + B ρˆ , mn 2 mn mn 0 mn 0 mn 0

DˆHˆmn = −(w +2)FBˆ mn,

DˆHˆm¯ n¯ =(w +2)FBˆ m¯ n¯ . The distorted horizontality conditions that are compatible with the Bianchi identities and warrant oﬀ-shell closure are as follows:

Pˆ = P, P¯ˆ = P,¯ 1 Σˆ =Σ + Ψˆ P¯ − wG φ¯ , 0 0 p p¯ 2 mp¯m¯ 1 Σˆ =Σ − Ψˆ P + wG φ , (15.71) m¯ m¯ 0 m¯ 2 pm¯ p¯ 1 Σˆ =Σ + Ψˆ P¯ − wG φ¯ , mn mn [m n] 2 n]qq¯ ˆ ˆ ˆ Hmn = Hmn + Ψp(Σp,mn¯ + Sp,mn¯ ) − iΨpJp¯[m(Σn],0 + Sn],0), ˆ ˆ Hm¯ n¯ = Hm¯ n¯ − 2Ψ0(Σ[¯m,n¯] + S[¯m,n¯]), where 1 1 S − iJ S =2iwG J Ψ φ¯ − i wJ G Ψ φ¯ + i wJ G σ p,mn¯ p¯[m n],0 qr¯q¯ p¯[m n],r 2 p¯[m n]qp¯ q,p¯ 2 p¯[m n]qq¯ 0 1 − (w +2)Gqq¯ p¯σmn + Gmpn¯ σ0, 2 1 1 S = P − wG Ψ + P − wG Ψ − G σ [¯m,n¯] [¯m 2 qq¯[¯m n¯],0 q 2 rqr¯ [¯m,n¯]¯q qq¯[¯m n¯] 1 1 −G σ + wρ φ − wρ φ. m¯ np¯ p¯ 2 m¯ n,¯ 0 2 qq,¯ m¯ n¯ The ghost-number-1 parts of these equations give the scalar and vector transformations of the ﬁelds: 360 Description of N = 1,d= 4 supergravity using twisted supersymmetric ﬁelds

δφ =0,δp¯φ = −δp¯,

δφ¯ = δ0,δp¯φ¯ =0, 1 δσ =0,δσ = P¯ − wG φ,¯ 0 p¯ 0 p¯ 2 mp¯m¯ 1 δσ = −P − wG φ, δ σ = B , (15.72) m¯ m¯ 2 pm¯ p¯ p¯ m¯ p¯m¯ 1 δσ = −B ,δσ = i P¯ | − wG |φ¯ J , mn mn p¯ mn [m 2 qq¯[m n]¯p

δBmn =0,δp¯Bmn =(Σp,mn¯ + Sp,mn¯ ) − iJp¯[m(Σn],0 + Sn],0),

δBm¯ n¯ = −2(Σ[¯m,n¯] + S[¯m,n¯]),δp¯Bm¯ n¯ =0.

The anticommutation relations (15.27) can be explicitly veriﬁed for all ﬁelds, in a much easier way than for the supergravity multiplet (see Appendix 15.D).

15.8.2 The vector multiplet

The twisted vector multiplet is (B,ξm,ξm¯ n¯ ,ξ0,h), with B a U(1) gauge field, (ξm,ξm¯ n¯ ,ξ0) its twisted Majorana supersymmetric partner, and h a real auxiliary field. The field strengths are ˆ Fˆ = dB − (Ψˆ 0ξm + Ψˆ mξ0)em¯ − (Ψˆ pξm¯ p¯ + Ψˆ m¯ p¯ξp)em,

Ξˆ0 = Dξˆ 0 − hΨˆ 0,

Ξˆm = Dξˆ m + hΨˆ m, (15.73)

Ξˆm¯ n¯ = Dξˆ m¯ n¯ − hΨˆ m¯ n¯ , Hˆ = dh,ˆ

ˆ ˆ ˆ − 1 ˆ where D is given by Dξ0 = dξ0 2 ωξˆ 0 + Aξ0 +ˆωmnξm¯ n¯ , etc. The Bianchi identities for these ﬁeld strengths are

dˆFˆ =(Ψˆ 0Ξˆ m + Ψˆ mΞˆ 0 − ρˆ0ξm − ρˆmξ0)ˆem¯ +(Ψˆ pΞˆ m¯ p¯ + Ψˆ m¯ p¯Ξˆ p − ρˆpξm¯ p¯ − ρˆm¯ p¯ξp)ˆem

+(Ψˆ mξ0 + Ψˆ 0ξm)Tˆm¯ +(Ψˆ m¯ p¯ξp + Ψˆ pξm¯ p¯)Tˆm, 1 DˆΞˆ 0 = − Rˆ + Fˆ ξ0 + Rˆmnξm¯ n¯ − HˆΨˆ 0 − hρˆ0, (15.74) 2 1 DˆΞˆm = − Rˆ + Fˆ ξm − Rˆpm¯ ξp + HˆΨˆ m + hρˆm, 2 1 DˆΞˆm¯ n¯ = Rˆ + Fˆ ξm¯ n¯ − Rˆm¯ n¯ ξ0 − HˆΨˆ m¯ n¯ − hρˆm¯ n¯ , 2

dˆHˆ =0. Appendix A 361

The supersymmetry is deﬁned by the constraints Fˆ = F,

Ξˆ 0 =Ξ0 + FmnΨˆ m¯ n¯ ,

Ξˆ m =Ξm −Fpm¯ Ψˆ p, (15.75)

Ξˆm¯ n¯ =Ξm¯ n¯ + Fm¯ n¯ Ψˆ 0,

Hˆ = H + Ψˆ p(Ξp,¯ 0 + Gmpn¯ ξm¯ n¯ ), which give

δB = ξmem¯ ,δp¯B = ξ0ep¯ + ξm¯ p¯em,

δξ0 = h, δp¯ξ0 =0,

δξm =0,δp¯ξm = Fpm¯ − iJpm¯ h, (15.76)

δξm¯ n¯ = Fm¯ n¯ ,δp¯ξm¯ n¯ =0,

δh =0,δp¯h =Ξp,¯ 0 + Gmpn¯ ξm¯ n¯ . The algebra closure relations (15.27) are satisﬁed by all ﬁelds (see Appendix 15.D).

15.9 Conclusions and outlook The supergravity action, within the new minimal auxiliary field structure, is basically completely determined by a single (twisted) scalar supersymmetry generator, which is nilpotent and quite analogous to the one encountered in the twisted super Yang– Mills theory. By requiring the existence of a vector supersymmetry generator that anticommutes consistently with the scalar one, we find a set of generators that can be identified as the twisted version of the ordinary super-Poincaré generators. The fourth symmetry δmn occurs for free, and allows us to untwist the system into the ordinary formulation. There is an underlying localization around gravitational instantons that seems of interest in this construction. The construction can be extended to the twisted formulation of the Wess–Zumino and vector multiplets coupled to the supergravity multiplet. Generalizations to higher-dimensional supergravities could be of interest, and an analogous twist could be used to split the Poincaré symmetry of, for example, d = 10 supergravity into smaller and (hopefully) simpler sectors.

Appendix A: The BSRT symmetry from horizontality conditions The supergravity transformations can be expressed as BRST transformations, in a way that merely generalizes the Yang–Mills case (ghost unification, horizontality equations for the curvatures, etc.) [4]. Denote the BRST operator of the supergravity transformation by s and its ghost by ξ. The other ghosts are those of local SUSY (χ), Lorentz symmetry (Ω), the chiral U(1) symmetry (c), and the 2-form gauge symmetry 1 (B1 ). We get the usual transformation laws of classical fields by changing the ghosts into local parameters, with the opposite statistics. Their off-shell closure property is equivalent to the nilpotency of the graded differential operator s. The difficult part of 362 Description of N = 1,d= 4 supergravity using twisted supersymmetric fields the supergravity BRST symmetry is its dependence on the supersymmetry ghost χ. The reparametrization invariance can be absorbed by redefinings ˆ ass ˆ = s − Lξ, with μ ν μ 1 μ 2 sξ = ξ ∂νξ + 2 χγ¯ χ. With this property, the off-shell closure relation s =0is 2 equivalent tos ˆ = Lχγ¯ μχ. Reparametrization invariance is decoupled by the operation exp(−iξ), when classical and ghost fields are unified into graded sums, a property that was found for the study of gravitational anomalies but turns out to be very useful for the construction of supergravity BRST symmetries. For the N =1,d= 4 supergravity in the new minimal scheme, the action of the operators ˆ is as follows:

a ab a seˆ = −Ω eb − iχγ¯ λ, ab 5 sλˆ = −Dχ − Ω γabλ − cγ λ, − 1 − a sBˆ 2 = dB1 iχγˆ λea, (15.A.1) 1 sAˆ = −dc − iχγ¯ 5γaX , 2 a sωˆ ab = −(DΩ)ab − iχγ¯ [aXb], b− 1 bc 1 bcd 5 where the spinor Xa is Xa = ρabe ( 2 Gabcγ + 12 abcdG γ )λ. Xa vanishes when we use the equations of motion of the gravitino and of the (propagating) auxiliary fields. 2 2 The property s = 0, equivalent tos ˆ = Lχγ¯ μχ, is warranted by the ghost transformation laws [4]. At the root of these equations, is a unification between classical fields and ghosts [4], which is analogous to the one that occurs when analyzing anomalies by descent equations. In fact, everything boils down to computing constraints on the curvatures, which satisfy the following Bianchi identities: 1 1 Tâ ≡ deˆ a +(ω +Ω)abe + i(λ¯ +¯χ)γa(λ + χ)=− Ga ebec, b 2 2 bc 1 ρˆ ≡ dˆ(λ + χ)+(ω +Ω+A + c)(λ + χ)= ρ eaeb, 2 ab 1 1 Gˆ ≡ dˆ(B + B1 + B2)+ i(λ¯ +¯χ)γa(λ + χ)ea = G eaebec, (15.A.2) 3 2 1 0 2 6 abc 1 Râb ≡ dˆ(ω +Ω)+(ω +Ω)2 = Rab − iχγ¯ [aXb] − iχγ¯ cχGab, 4 c 1 1 Fˆ ≡ dˆ(A + c)=F − iχγ¯ 5γaX − iχγ¯ aχ Gbcd. 2 a 24 abcd By expansion at ghost number 1, we find the transformation laws in (15.8) and at ghost number 2 those of the ghosts:

sχˆ = −iφλ − Ωχ − cχ, 1 scˆ = −i A − iχγ¯ aχ Gbcd, φ 24 abcd 1 sBˆ 1 = −i B − dB2 − iχγ¯ aχe , (15.A.3) 1 φ 0 2 a 2 − 1 sBˆ = iφB1 , 1 1 sˆΩab = −i ωab − i [Ω, Ω]ab − iχγ¯ cχGab. φ 2 2 c Appendix C 363

Appendix B: Tensor and chirality conventions The normalization of the completely antisymmetric four-index symbol with tangent- space indices is

0123 =1. (15.B.1)

Once twisted, this is taken to be

112¯ 2¯ =1. (15.B.2)

The dual of an antisymmetric Lorentz tensor is

1 F˜ = F cd. (15.B.3) ab 2 abcd

The self-dual and anti-self-dual parts of Fab are

1 F ± = (F ± F˜ ). (15.B.4) ab 2 ab ab

2 We take γ5 such that (γ5) = −1 and deﬁne the chiral projections

1 ± iγ λ± = 5 λ, 2 (15.B.5) 1 ± iγ λ¯± = λ¯ 5 , 2 in order to have λ = λ+ + λ− and λ¯ = λ¯+ + λ¯−. Then, we have the useful identity

+ a + + a + + a + + a + λ¯ γ λ = λ¯ γ5γ γ5λ = −iλ¯ γ (−i)λ = −λ¯ γ λ =0, (15.B.6) and similarly λ¯−γaλ− = 0. Finally, once in twisted form, the chiral projections of spinor separate its various components according to

+ λ ∼ (0, Ψp, 0), − λ ∼ (Ψ0, 0, Ψm¯ n¯ ).

Appendix C: The action of γ matrices on twisted spinors

The action of a γ matrix on a twisted spinor with components (Ψ0, Ψm, Ψm¯ n¯ ) is deﬁned as follows:

0 p¯ pq γmΨ iΨm −Jmp¯Ψ0 0 (15.C.1) γm¯ Ψ 0 iΨm¯ p¯ 2Jm¯ [pΨq] 364 Description of N = 1,d= 4 supergravity using twisted supersymmetric ﬁelds

Similarly, the action of a γ matrix on a twisted spinor with components (σ0,σm¯ ,σmn), such as that appearing in the Wess–Zumino muliplet, is

0 p p¯q¯ γmσ 0 iσmp −Jm[¯pσq¯] (15.C.2) γm¯ σ iσm¯ −Jpm¯ σ0 0

These conventions allow us to retrieve the Cliﬀord algebra for the twisted γ matrices:

{γm,γn} =0,

{γm¯ ,γn¯ } =0,

{γm,γn¯ } = −iJmn¯ ≡ gmn¯ .

We also deﬁne the γab matrices in twisted form as

γmn¯ = γmγn¯ − γn¯ γm,

γmn = γmγn − γnγm,

γm¯ n¯ = γm¯ γn¯ − γn¯ γm¯ , which act on the two kinds of twisted spinors according to the following tables

0 p p¯q¯ γ Ψ 0 0 −2J J Ψ mn m[¯p q¯]n 0 (15.C.3) γm¯ n¯ Ψ 2Ψm¯ n¯ 0 0 γmn¯ Ψ iJmn¯ Ψ0 2iJpn¯ Ψm − iJmn¯ Ψp −iJmn¯ Ψp¯q¯

0 p¯ pq γ σ 2σ 0 0 mn mn (15.C.4) γm¯ n¯ σ 0 0 −2Jm¯ [pJq]¯nσ0 − − 3i i γmn¯ σ iJmn¯ σ0 2 Jmp¯σn¯ + 2 Jmn¯ σp¯ iJmn¯ σpq

Appendix D: Algebra closure on the fields of matter and vector multiplets In this appendix, we give some examples of the anticommutation relations (15.27) on some matter fields of the Wess–Zumino and vector multiplets. Starting with the φ and φ¯ fields of the Wess–Zumino multiplet, we need their transformation laws under the pseudoscalar symmetry in order to check (15.27). These are obtained in the same way as the scalar and vector symmetry transformation laws, namely, by isolating the part of ghost number 1 in the horizontality conditions on ˆ Pˆ = P and P¯ = P¯ and keeping χmn = 0. This yields

δmnφ =0,δmnφ¯ = σmn. (15.D.1) Appendix D 365

The transformation laws in (15.72) allow us to compute straightforwardly δ2φ =0, {δ ,δ }φ = − (B + B )=0, (15.D.2) p¯ q¯ p¯q¯ q¯p¯ 1 {δ, δp¯}φ = Pp¯ + wGmp¯m¯ φ 2 1 = ∂ φ + wA + wG φ +Ψ σ p¯ p¯ 2 mp¯m¯ p,m¯ m¯ gauge = ∂p¯φ + δ (A, G)φ − Ψp,a¯ δa¯φ. a=0,m,m¯ n¯ where the last equality is a consequence of (15.72) and (15.D.1). Similarly, for φ¯, 2 δ φ¯ = δσ0 =0, {δ ,δ }φ¯ =0, (15.D.3) p¯ q¯ 1 {δ, δ }φ¯ = P¯ − wG φ¯ p¯ p¯ 2 mp¯m¯ 1 = ∂ φ¯ − wA + wG φ¯ − (Ψ σ +Ψ σ ) p¯ p¯ 2 mp¯m¯ p,¯ 0 0 p,¯ m¯ n¯ mn gauge = ∂p¯φ¯ + δ (A, G)φ¯ − Ψp,a¯ δa¯φ,¯ a=0,m,m¯ n¯ again using (15.72) and (15.D.1) for the last equality. TurningtotheB ﬁeld of the vector multiplet, the horizontality condition on its ﬁeld strength Fˆ = F allows us to compute

δmnB = ξnem, (15.D.4) and the transformation laws (15.76) of the vector multiplet ﬁelds yield 2 δ B = δ(ξmem¯ )=0,

{δp¯,δq¯}B = ξ0Ψp¯q¯ + ξp¯q¯Ψ0 + ξ0Ψq¯p¯ + ξq¯p¯Ψ0 =0, (15.D.5)

{δ, δp¯}B = hep¯ + Fm¯ p¯em − ξm¯ p¯Ψm + Fpm¯ em¯ − iJpm¯ hem¯ + ξmΨp¯m¯

= Fpm¯ em¯ −Fp¯m¯ em − ξm¯ p¯Ψm + ξmΨp¯m¯

= ∂p¯B − (Ψp,¯ 0ξm +Ψp,m¯ ξ0) em¯ − (Ψp,q¯ ξm¯ q¯ +Ψp,¯ m¯ q¯ξq) em = ∂p¯B − Ψp,a¯ δa¯B, a=0,m,m¯ n¯ where, for the last equality, we have used the B transformations given by (15.76) and (15.D.4).

Acknowledgments I thank my collaborators M. Bellon and S. Reys, with whom I worked out the construction. 366 Description of N = 1,d= 4 supergravity using twisted supersymmetric ﬁelds

References [1] L. Baulieu and G. Bossard, Reconstruction of N =1supersymmetry from topological symmetry, Phys. Lett. B632 (2006) 138 [arXiv:hep-th/0507004]; L. Baulieu, SU(5)-invariant decomposition of ten-dimensional Yang–Mills supersymmetry, Phys. Lett. B69 (2011) 63 [arXiv:1009.3893[hep-th]]. [2] L. Baulieu and I. M. Singer, The topological sigma model, Commun. Math. Phys. 125 (1989) 227; L. Baulieu and I. M. Singer, Conformally invariant gauge fixed actions for 2-D topological gravity, Commun. Math. Phys. 135 (1991) 253; L. Baulieu and I. M. Singer, Topological Yang–Mills symmetry,Nucl.Phys.Proc. Suppl. 5B (1988) 12; L. Baulieu, H. Kanno and I. M. Singer, Special quantum field theories in eight-dimensions and other dimensions, Commun. Math. Phys. 194 (1998) 149 [arXiv:hep-th/9704167]. [3] M. S. Sohnius and P. C. West, An alternative minimal off-shell version of N =1 Supergravity, Phys. Lett. B105 (1981) 353; The tensor calculus and matter coupling of the alternative minimal auxiliary field formulation of N =1supergravity, Nucl. Phys. B198 (1982) 493; A. Galperin, V. Ogievetsky and E. Sokatchev, On matter couplings in N =1supergravities,Nucl.Phys.B252 (1985) 435. [4] L. Baulieu and M. P. Bellon, p-forms and supergravity: gauge symmetries in curved space,Nucl.Phys.B266 (1986) 75. [5] A. Johansen, Twisting of N =1SUSY gauge theories and heterotic topological theories,Int.J.Mod.Phys.A10 (1995) 4325 [arXiv:hep-th/9403017]. [6] E. Witten, Supersymmetric Yang–Mills theory on a four manifold,J.Math.Phys. 35 (1994) 5101 [arXiv:hep-th/9403195]. [7] A. D. Popov, Holomorphic analogs of topological gauge theories, Phys. Lett. B473 (2000) 65 [arXiv:hep-th/9909135]; T. A. Ivanova and A. D. Popov, Dressing symmetries of holomorphic BF theories, J. Math. Phys. 41 (2000) 2604 [arXiv:hep-th/0002120]. [8] C. Hofman and J. -S. Park, Cohomological Yang–Mills theories on Kähler 3 folds, Nucl. Phys. B600 (2001) 133 [arXiv:hep-th/0010103]; J.-S. Park, N =2topological Yang–Mills theory on compact Kähler surfaces, Commun. Math. Phys. 163 (1994) 113 [arXiv:hep-th/9304060]; J. -S. Park, Holomorphic Yang–Mills theory on compact Kähler manifolds, Nucl. Phys. B423 (1994) 559 [arXiv:hep-th/9305095]. [9] L. Baulieu and A. Tanzini, Topological symmetry of forms, N =1supersymmetry and S-duality on special manifolds, J. Geom. Phys. 56 (2006) 2379 [arXiv:hep- th/0412014]. [10] H. Nicolai, A possible constructive approach to (super Φ3) in four-dimensions. 1. Euclidean formulation of the model,Nucl.Phys.B140 (1978) 294. 5 [11] N. Berkovits, Perturbative Super-Yang–Mills from the Topological AdS(5)×S sigma model,JHEP0809 (2008) 088 [arXiv:0806.1960 [hep-th]]. [12] P. Van Nieuwenhuizen, Supergravity, Phys. Rep. 68 (1981) 189. 16 AdS crunches, CFT falls, and cosmological complexity

Jos´e Luis Barbon´ 1 and Eliezer Rabinovici2

1Institute de F´ısica Te´orica IFT UAM-CSIC, Universidad Ant´onoma de Madrid, Cantoblanco, Madrid, Spain 2Racah Institute of Physics, The Hebrew University, Jerusalem, Israel

16 AdS crunches, CFT falls, and cosmological complexity 367 Jos´eLuisBARBON´ and Eliezer RABINOVICI

16.1 Introduction 369 16.2 AdS crunches and their dS duals 370 16.3 Facing the CFT crunch time is complementary 375 16.4 Attempt at a ‘thin-thesis’ 383 16.5 Falling on your sword 389 16.6 Conclusions 392 Acknowledgments 394 References 394 Introduction 369

We discuss aspects of the holographic description of crunching AdS cosmologies. We argue that crunching FRW models with hyperbolic spatial sections are dual to semiclassical condensates in deformed de Sitter CFTs. De Sitter-invariant condensates with a sharply defined energy scale are induced by effective negative-definite relevant or marginal operators, which may or may not destabilize the CFT. We find this result by explicitly constructing a ‘complementarity map’ for this model, given by a conformal transformation of the de Sitter CFT into a static time-frame, which reveals the crunch as an infinite potential-energy fall in finite time. We show that, quite generically, the crunch is associated with a finite-mass black hole if the de Sitter O(d, 1) invariance is an accidental IR symmetry, broken down to U(1) × O(d) in the UV. Any such regularization cuts off the eternity of de Sitter spacetime. Equivalently, the dimension of the Hilbert space propagating into the crunch is finite only when de Sitter is not eternal.

16.1 Introduction The resolution of cosmological singularities has been a permanent fixture in the ‘to do’ list of string theory since its quantum gravitational interpretation was launched. Suc- cessful singularity resolutions by various instances of ‘stringy geometry’ always apply to timelike singularities that may be regarded as ‘impurities’ in space. These singularities are resolved by a refinement in the quantum description of the impurity, often by identifying additional light degrees of freedom supported at the singularity locus.∗ Spacelike singularities looking like bangs and crunches in General Relativity have so far resisted close scrutiny. At the most naive level, they represent a challenge to the very notion of Hamiltonian time evolution. In the case of spacelike singularities censored by black hole horizons, they posed a famous historic challenge to S-matrix unitarity. The advent of notions such as black hole complementarity, holography, and the anti-de Sitter/conformal field theory (AdS/CFT) correspondence have resulted in a conceptual framework in which black hole singularities should be resolved by a refinement of the hole’s quantum description, albeit nonperturbative and highly nonlocal in this case. On general grounds, any problem that can be successfully embedded into an AdS/CFT environment should admit an honest Hamiltonian answer. In this sense, the occurrence of crunches in the interior of AdS black holes poses no serious threat to the CFT Hamiltonian description. The black hole and everything inside is described by a finite-dimensional subspace of the full Hilbert space, and thus the crunch must be ‘deconstructed’ within this class of states, even if the details of such a deconstruction remain largely unknown. Potentially more serious is the situation where a crunch engulfs a globally defined, asymptotically AdS spacetime, for then the whole CFT Hilbert space seems destined for the crunch. Such a radical situation occurs in the future development

∗Journal of High Energy Physics, ‘AdS crunches, SFT falls and cosmological complemetarity’, April 2011, 2011:44, by JoséL.F.Barbón & Eliezer Rabinovici c SISSA 2011. With kind permission of Springer Science+Business Media. 370 AdS crunches, CFT falls, and cosmological complexity of Coleman–de Luccia (CdL) bubbles nucleated inside a false AdS vacuum [1]. The interior of the bubble contains a crunching cosmology that eventually engulfs the whole AdS spacetime, right up to the boundary, not to mention the complicated pattern of multibubble nucleation and collision. This poses a ‘clear and present danger’ for the CFT Hamiltonian picture. In this chapter, we discuss various aspects of this conundrum. We first review the construction of highly symmetric bubble-like backgrounds with a crunchy destiny and AdS asymptotic behavior. These backgrounds include the CdL single-bubble configurations, but can be much more general, and are characterized by an exact O(d, 1) symmetry. On the dual holographic side, this symmetry can be realized by specifying the CFT states as living on an eternal d-dimensional de Sitter (dS) spacetime. A central observation in this chapter is the identification of a simple ‘complementarity map’ that relates the formulation of the CFT in de Sitter spacetime to another description appropriate for an ‘observer’ who falls into the crunch in finite time. The large amount of symmetry determines this map to be a conformal transformation to the same CFT defined on a static Einstein spacetime. Using results from [2], we argue that the crunching states are seen in this frame as infinite negative-energy falls, their precise nature depending on whether the condensate was stable or unstable in the de Sitter-frame description. We discuss the implications of this statement for the interpretation in terms of a ‘cosmological complementarity’ using in particular a regularization that strictly reverts the model into a standard case of black hole complementarity [3].

16.2 AdS crunches and their dS duals In this section, we review the basic issues arising in the construction of CFT duals of AdS crunching cosmologies. We begin with a description of the relevant geometries and subsequently introduce the corresponding dual CFT structures. Many details of relevance to these constructions can be found in [4–9].

16.2.1 The crunches We shall refer to ‘AdS crunches’ as a particular class of Friedmann–Robertson–Walker (FRW) cosmologies with O(d, 1)-invariant spatial sections (i.e., d-hyperboloids Hd), 2 − 2 2 dsFRW = dt + G(t) ds Hd . (16.2.1) The profile function G(t) solves Einstein’s equations with negative cosmological constant and a generic O(d, 1)-invariant matter distribution modeled by a set of fields ϕ(t) depending only on the FRW time coordinate t. In general, for smooth initial conditions at some fixed time t = t0, this spacetime has curvature singularities both in the future (crunch) and the past (bang). If the matter contribution is small, the metric is close to 2 an exact AdSd+1 in FRW parametrization, corresponding to G(t)=sin t,atleastfor a long time. In the pure AdS case, the points t = 0,π are only coordinate singularities signaling the Killing horizons associated with the hyperbolic sections becoming null at this locus. AdS crunches and their dS duals 371

The behavior of the FRW patch in pure AdS suggests that a crunching cosmology of type (16.2.1) could be given initial smooth data by matching across a zero of the function G(t), tuned with a locally static matter distribution. This is precisely the case if the FRW cosmology is regarded as the interior of an expanding bubble, in a generalization of the classic work by Coleman and de Luccia [1]. It is convenient to parametrize such backgrounds in terms of the Euclidean versions with O(d +1) isometries. Let us consider the metric

2 2 2 dsball = dρ + F (ρ) dΩd, (16.2.2) satisfying the field equations with an O(d+1)-symmetric matter distribution ϕ(ρ). We term it the ‘ball’ on account of its O(d+1) symmetry, even if it may be noncompact in ≈ 2 ≈ 1 2 general. Smoothness at the center of the ball requires F (ρ) ρ and ϕ(ρ) ϕ0+ 2 ϕ0ρ as ρ → 0. 2 2 2 2 Writing dΩd = dθ +cos θdΩd−1, we generate a Lorentz-signature metric with O(d, 1) symmetry by the analytic continuation θ = iτ. We call this metric the ‘bubble’: 2 2 − 2 2 2 2 2 dsbubble = dρ + F (ρ) dτ +cosh τdΩd−1 = dρ + F (ρ) dsdSd , (16.2.3) where the group O(d, 1) acts on global de Sitter sections dSd. By construction, the matter fields are de Sitter-invariant functions ϕ(ρ), so all features of the metric and matter fields expand like a de Sitter spacetime, i.e., we have a generalized notion of an ‘expanding bubble’. This bubble background is time-symmetric around τ = 0, where it can be formally matched to the Euclidean O(d + 1)-invariant ‘ball’. Therefore, we may interpret this construction as a time-symmetric cosmology with bang and crunch, or as a crunching cosmology that evolves from a particular initial condition obtained from some quantum-cosmological tunneling event, àlaHartle–Hawking [10]. At ρ =0 thedSd sections become null and they may be further extended as the nearly null Hd sections of the FRW patch. By mimicking the pure AdS case, we can achieve this matching by the coordinate redefinition ρ = itandy = τ + iπ/2: 2 − 2 2 2 2 − 2 2 dsFRW = dt + G(t) dy +sinh ydΩd−1 = dt + G(t) dsHd , (16.2.4) where the smooth matching requires G(t) ≈ t2 ≈−F (it) near t = 0. For the rest of the fields, ϕ(ρ) continues to an O(1,d)-invariant function ϕ(t) with small-t behavior ≈ − 1 2 ϕ(t) ϕ0 2 ϕ0 t . Hence, the result is an FRW model with negative spatial curvature (16.2.1), which eventually crunches, barring fine-tuning.

16.2.2 The duals We now restrict further the form of the ball metric to asymptote AdS, F (ρ) → sinh2 ρ as ρ →∞. Using the Euclidean AdS/CFT rules [11], this background describes a certain large-N master ﬁeld for a perturbed dual CFT on the Euclidean d-sphere Sd: L L gO O QFT = CFT + − , (16.2.5) d d d O ΔO d S S O S (Λ ) 372 AdS crunches, CFT falls, and cosmological complexity

FRW

patch

Hd τ = const

Bubble patch dSd Lorentzian

τ = 0

Euclidean

Sd Ball patch

Fig. 16.1 Schematic diagram showing the analytic construction of the crunching FRW space- d time, with H spatial sections, matching to the ‘bubble’ patch with dSd timelike sections, and finally the Euclidean ‘ball’ background, with Sd sections. The Euclidean piece is O(d +1)- invariant, whereas the Lorentz-signature piece is O(d, 1)-invariant. The gray line signals the radius ρ = ρO ∼ log ΛO, where the metric begins with differ significantly from the asymptotic AdS form. Equivalently, it is associated with the energy scale ΛO in the dual QFT. where ΔO is the conformal dimension of the perturbing operator, and the dimensionless source terms gO are determined by the boundary values of the different fields ϕO according to the rule −ρ ΔO −d gO = lim e ΛO ϕO(ρ), (16.2.6) ρ→∞ with ΔO the conformal dimension of the operator O and ΛO the energy scale set by the operator perturbation. All bulk dimensions are measured in units of the asymptotic AdS radius of curvature, and all CFT length dimensions are measured in units of the Sd radius. For the case considered here, with O(d+1) symmetry, the gO are actually constant on Sd and therefore represent couplings rather than sources. Consistency at the level of the field equations requires that only the couplings associated with relevant or marginal operators may be nonvanishing, since otherwise the backreaction would destroy the assumed AdS asymptotic behavior of the metric. This condition can actually be extended to include also marginal operators, i.e., we have gO =0forΔO ≥ d with no loss of generality, since a marginal perturbation can be conventionally folded into the definition of LCFT. AdS crunches and their dS duals 373

Using the UV/IR correspondence, scales ΛO 1 are associated with characteristic geometric features at radii ρO ∼ log ΛO, and thus we may speak of the radial development of the gravity solution as depicting a certain renormalization group ﬂow. If the theory has no other large dimensionless parameters, we may encounter a number of diﬀerent qualitative scenarios according to the overall geometrical features of the solution:

1. The flow may become singular inside the ball at a radius of order ρO. In some cases, the singularity admits a geometric resolution, for example by a vanishing cycle in an additional compact factor of the spacetime, producing a ‘confining hole’ at the center of the ball. The Lorentzian versions of such models would define confining gauge theories on de Sitter, examples of which can be found in [8] and references therein. Irrespectively of an eventual resolution of the singularity, any such background that does not reach smoothly ρ = 0 cannot be Wick-rotated into an FRW patch and thus does not provide crunch models. 2. A flow with ΛO 1 may evolve into an approximate IR fixed point at the origin of the ball, thus depicting a flow between a UV fixed point CFT+ and an IR fixed point CFT− . This corresponds to a ‘thin-walled bubble’ of AdS− inside AdS+ with radiusρ ¯ ∼ log ΛO, so that the bulk fields at the ρ = 0 origin of the ball 2 come close to a local minimum of the supergravity potential with mϕ(AdS−) > 0. 3. The flow below the threshold ΛO may continue without approaching an IR fixed point or a gap before it reaches ρ = 0. In this case, we have roughly a bubble with ‘thick walls’, whose interior differs significantly from AdS, but may still define O(d, 1)-invariant crunches by the procedure outlined above, provided the background is smooth at the origin.1 An interesting case of a ‘thick-walled’ bubble is the situation with ΛO 1, where the deformation is so small compared with the scale of the Sd sphere that the ‘bubble’ is but a small ‘hump’ around ρ = 0 (cf. the appendix of [9]). A well-studied class of solutions of this form can be found in [4], using a truncated four-dimensional supergravity model with one scalar field close − 2 2 ≤ to a local BF-stable [12] maximum of the potential, d /4

1 The so-called ‘thin-wall’ approximation refers to the idealization in which the bubble’s shell is regarded as having zero thickness. In this chapter, we will refer to a ‘thin-walled bubble’ whenever the interior contains an approximate AdS− metric, and a ‘thick-walled bubble’ corresponds to a bubble with no recognizable AdS metric in its interior. This terminology is meant to give a qualitative description of the bubble, rather than a strict implementation of the ‘thin-wall approximation’. 374 AdS crunches, CFT falls, and cosmological complexity special situations, we may have ‘flows’ that are normalizable at the boundary, so that no operators, relevant or marginal, are turned on. For example, any smooth flow that starts at the boundary from a local minimum of the supergravity potential, 2 mϕ(AdS+) > 0, must necessarily be a normalizable deformation of pure AdS, because any amount of non-normalizable deformation backreacts strongly at ρ = ∞ and destroys the AdS asymptotic behavior. These normalizable flows are the CdL bounces, which break the conformal symmetry spontaneously, and thus come in continuous families (moduli spaces) associated with bulk translations of the O(d + 1)-invariant solution. In addition, there are dilute limits with (at least) approximate multibubble solutions. In our conventions, we regard a marginal coupling as part of the unperturbed CFT. Therefore, any non-normalizable master field that induces a marginal operator in the CFT Lagrangian may be regarded as a normalizable master field of the deformed CFT. This means that flows associated with marginal operators can be conventionally treated as CdL-type backgrounds. The defining feature of the CdL instantons is their normalizable nature at the outer boundary, irrespective of their detailed structure in the interior; i.e., we may have CdL instantons realized as ‘bubbles of nothing’ [13], as nongeometric impurities, or as smooth bubble-like backgrounds. It is the last class that allows us to study crunches in their real-time development, although they only have a putative ‘true vacuum’ inside for the case of thin walls (see [14] for a tour around the various pitfalls of the theory of CdL tunneling). The Wick rotation of any of these Euclidean ‘ball metrics’ (i.e., master fields) into the bubble spacetime defines a large-N state in the same QFT formulated on the Wick d rotation of the S slices, i.e., the de Sitter slices. We have the real-time QFT on dSd with action gO L = L − O. (16.2.7) QFT CFT ΔO −d d d d (ΛO) dS dS Orelevant dS

Geometrical features at radii ρO ∼ log ΛO 1 correspond to fixed energy scales for the QFT on de Sitter spacetime, measured in units of the Hubble constant of dSd. For instance, we may have a confining theory in de Sitter space, which corresponds in the bulk to an expanding a ‘bubble of nothing’ (in this case associated with a non-normalizable deformation of asymptotic AdS). A case of more interest for the purpose of discussing crunches is the state obtained by Wick rotation of the ‘domain wall flow’, which defines a state looking like the vacuum of a CFT+ in the UV and as the vacuum of a CFT− in the IR. In the bulk, we simply see a thin-walled bubble of AdS− expanding exponentially into the external AdS+. In this last situation, there is an alternative CFT representation of the bubble interior, in terms of the IR CFT fixed point: we just write the same expression as in (16.2.7), replacing the UV CFT+ by the IR CFT−, gO L = L − O, (16.2.8) QFT− CFT− ΔO −d d d d (ΛO) dS dS Oirrelevant dS Facing the CFT crunch time is complementary 375 and restricting the sum over perturbing operators to the infinite tower of irrelevant operators, according to the operator content of the CFT− fixed point (cf. [9]). This gives an approximate Wilsonian description of the bubble’s interior with UV cutoff ΛO, after the bubble’s wall and whatever lies outside have been ‘integrated out’. In particular, the bubble could be sitting inside an asymptotic (d + 1)-dimensional de Sitter or Minkowski spacetime, and the IR description would be very similar, provided the AdS− fixed point exists. Thus, universality of the Wilsonian flow around an IR fixed point explains the fact that all nearly AdS crunching FRW cosmologies look roughly the same, irrespective of the initial conditions. The downside of this Wilsonian description is that details of the UV completion are hidden in the properties of the full tower of irrelevant operators in (16.2.8). The crunch singularities of the FRW patch start at the boundary and ‘propagate inwards’ into the bulk. Therefore, the crunch makes its first appearance in the deep- UV regime of the dual QFT, and its properties are potentially very sensitive to the details of the UV completion.

16.3 Facing the CFT crunch time is complementary The Lagrangian (16.2.7) gives a seemingly well-defined holographic representation of the bubble patch, i.e., a de Sitter-invariant state of a deformed CFT in de Sitter spacetime. The Lagrangian (16.2.8) gives also a well-defined, albeit approximate, description of the bubble quantum mechanics in the case that an approximate AdS− interior exists. Both of these ‘CFT-on-dS’ pictures have a causal patch in the bulk that leaves out the crunch, since it happens ‘after’ the end of de Sitter time τ = ∞. On the other hand, the τ = 0 surface of the bubble patch is a valid Cauchy surface for the complete crunching manifold in the bulk. This means that the τ = 0 state of the theory (16.2.7) does contain all the relevant information to probe the crunch. In order to expose this information, we need a ‘complementarity transformation’ which in this case is essentially determined by the symmetries of the problem. To find the complementarity map, notice that an ‘infalling’ observer is characterized by meeting the crunch in finite time. In standard black hole states, only sufficiently IR probes have the chance to fall into the black hole (i.e., thermalize) and become eligible ‘to meet the crunch’. This is actually a complication for the problem of finding the complementarity map in the QFT, since it entangles it with the renormalization group. In the case of the AdS cosmological crunches, however, the singularity is visible from the boundary, and thus it can be probed by arbitrarily UV states of the theory. So one strategy is to simply find a time variable in the boundary that, unlike de Sitter time, sees the crunch coming in finite time. Any dSd slice at constant ρ corresponds to an accelerating, asymptotically null trajectory that hits the AdS+ boundary in finite static time. This suggests that we should use static AdS time in order to describe states in such a way that they ‘fall’ into the crunch in finite time (i.e., infalling observers). Static time is defined as the time coordinate t adapted to the asymptotic timelike Killing vector ∂/∂t in AdS+. This Killing vector exists with a good approximation in the ‘exterior’ of the 376 AdS crunches, CFT falls, and cosmological complexity bubble wall trajectory. Hence, we may parametrize the near-boundary metric of the deformed AdS in a neighborhood of t = τ =0as

2 2 2 2 dr 2 2 ds ≈−dt (1 + r )+ + r dΩ − . (16.3.1) bubble 1+r2 d 1 This metric defines states looking in the UV like the vacuum of a CFT on the Einstein × d−1 2 space Ed = R S . Let the metric on the Einstein space be given by dsE = − 2 2 dt + dΩd−1, which is conformally related to the dSd metric as 1 ds2 =Ω2(t) ds2 , Ω(t)=coshτ = , (16.3.2) dS E cos t where t = Ω−1(τ) dτ = 2 tan−1 [tanh(τ/2)]. The conformal transformation thus defined maps the ‘eternity’ of de Sitter time into a finite interval of Einstein time, and therefore the associated Hamiltonian ‘meets’ the crunch in finite t time. The extension beyond the interval −π/2

Fig. 16.2 The AdS+ region of the bubble patch, coordinated along the O(d, 1)-invariant dS- frame (a) and the U(1) × O(d)-invariant E-frame (b). The crunch and its corresponding bang by time-reversal symmetry are separated by an inﬁnite amount of dS-frame time τ. In E-frame time, their time separation is ﬁnite, Δt = π.

2 Conformal maps interpreted as ‘complementarity transformations’ appear in [15], under a slightly diﬀerent guise. Facing the CFT crunch time is complementary 377

16.3.1 Crashing the CFT by irresponsible driving We can now rewrite the model (16.2.7) in the Einstein (E)-frame by acting with the conformal transformation (16.3.2). Since a given operator transforms as O→ −ΔO Ω(t) O we ﬁnd that the model (16.2.7) at ﬁxed values of the couplings gO can be equivalently rewritten on the Einstein manifold as3 LQFT = LCFT − JO(t) O, (16.3.3) d d d E E Orelevant E

where

gO JO(t)= , ΛO(t)=Ω(t)ΛO. (16.3.4) ΔO −d (ΛO(t)) If only marginal operators are turned on, this transformation is a symmetry of the deformed CFT, and the E-frame Lagrangian has no explicit time dependence. On the other hand, each relevant operator that is turned on at the UV fixed point CFT+ breaks the conformal symmetry, and that is reflected in the E-frame theory as an explicit time-dependent ‘driving’ term proportional to JO(t)O. Notice that Ω(t) diverges as t → t, and, since d − ΔO > 0 for a relevant operator, the E-frame driving term JO(t) ‘blows up’ in finite time. This conclusion holds for any O(d, 1)-invariant and relevant perturbation of the de Sitter field theory (such time-dependent deformations were invoked in [16, 17] to avert the occurrence of multibubble solutions). So far, this analysis leaves out the case of CdL bubbles, which include the case of an exactly marginal deformation, and will be studied in Section 16.3.2. We now argue that the value of the relevant deformation, for example whether gO may be ‘sufficiently positive’ or ‘sufficiently negative’ does have a bearing on the physical interpretation in terms of a crunching cosmology. We shall give two heuristic arguments in this section, and another one in Section 16.3.2, showing that crunching backgrounds are associated with CFT deformations contributing a sufficiently negative potential energy. In order to gain some intuition about such relevant driving terms, we can look at a simple model of a classical conformal field theory with similar physical phenomena to those described here. Consider the O(N) sigma model defined on a four-dimensional de Sitter space of unit Hubble constant, and perturbed by a mass operator 2 L − 1 · μ 2 2 2 2 φ = ∂μ φ ∂ φ + φ + g4 φ + g2 Λ φ , (16.3.5) dS4 dS4 2 where we set Λ 1 in order to be able to neglect the Hubble contribution to the mass 2 and let g4 > 0 ensure the global stability of the model. The mass operator φ is relevant but nonleading for large values of φ. For a positive-definite mass perturbation, g2 > 0,

3 In this discussion, we neglect the effect of conformal anomalies, a subject of great interest for future work. 378 AdS crunches, CFT falls, and cosmological complexity the theory acquires a mass gap of order Λ and has a trivial IR limit. For a negative- definite mass perturbation, g2 < 0, the minimum of the potential is found at the vacuum condensate |φ |vac ∝ Λ, and its low-lying excitations are the familiar Goldstone bosons with N − 1 degrees of freedom, the slow de Sitter expansion introducing just small thermal corrections to these vacuum states.4 Thus, this simple model with Λ 1 allows us to emulate two scenarios of the classification in Section 16.2.2. Namely, it gives an example of scenario (1) for g2 > 0 (a gapped phase), and a dS-invariant domain-wall flow, or scenario (2), for g2 < 0. When written in the E-frame variables, this theory has Lagrangian 2 L − 1 · μ 1 2 2 2 2 φ = ∂μ φ ∂ φ + φ + g4 φ + g2 Λ (t)φ , (16.3.6) E4 E4 2 2 where now the field theory lives on a static 3-sphere but the mass operator is time- dependent, with a scale Λ(t)=ΛΩ(t) diverging at t = t. For g2 > 0, i.e., with no condensate in the dS description, the theory (16.3.6) exhibits an increasing gap that eventually decouples all states from the static-sphere vacuum. On the other hand, for g2 < 0, the growing tachyonic perturbation sends the typical field values of the symmetry-breaking ground state to infinity, with ever increasing kinetic energy and (negative) potential energy. Given the stationary ‘condensate’ field configuration in the dS theory, φ vac , with ∂τ φ vac = 0, the corresponding solution in the E-frame theory is Ω(t)φvac , which diverges as t →±π/2. This suggests that any coherent quantum state that is peaked around some nonzero dS-invariant configuration in the dS theory will be mapped in the E-frame theory to a time-dependent coherent state whose support in field space is transferred to large field values. We shall refer to this transfer of ‘power’ in coherent states from the IR to the UV as the ‘CFT fall’. Note that this happens in the E-frame formalism (16.3.6) as a result of the driving term being negative-definite, even if the system is stable at any finite value of the time variable. On the other hand, in the de Sitter variables (16.3.5), we are simply describing a certain stationary state looking like a thermal excitation of the stable symmetry-breaking vacuum. Essentially the same physics is obtained in any model in which the formation of a condensate is controlled by the sign of a relevant operator, with a less-relevant one (not necessarily marginal) protecting the UV stability. This example suggests that a positive-energy driving term, which is large in Hubble units, should be associated with an ever-increasing mass gap, dual to the expanding ‘confinement bubble of nothing’ in the bulk. Conversely, a state with nontrivial IR content, such as one having soft degrees of freedom of an infrared CFT−, should be associated with driving operators JO(t) O with a negative contribution to the potential energy. In our qualitative discussion of the O(N) model, we have chosen the case Λ 1to emphasize a phase structure with either a clear mass gap or a clear bosonic condensate

4 The negative energy density at the g2 < 0 vacuum does not mean that we lose de Sitter, since gravity is not dynamical, i.e., Newton’s constant is zero in the CFT. Facing the CFT crunch time is complementary 379

2 as a ground state, in a way that is visible in the classical approximation. For g2Λ of the order of the Hubble scale squared, or smaller, the question of whether there is a condensate or a gap depends on the nature of the quantum corrections in the de Sitter background. For weakly coupled field theories, we do not expect a vacuum condensate to survive the thermal de Sitter bombardment for Λ < 1, so that the critical value of ∗ the mass coupling g2 separating gapped phases from condensates (perhaps metastable or even unstable ones) should be strictly negative and O(1). On the other hand, for the purposes of studying holographic duals of crunches, we are interested in semiclassical states arising as large-N master fields, and here large-N effects might allow for small- field condensates surviving the dS thermal bath, corresponding to a positive critical ∗ value of the mass deformation, g2 > 0, in our toy example. ∗ For the particular case of the O(N) model on de Sitter, the value of g2 at large N is an interesting question that deserves further study (cf. [18]). For a CFT admitting a large-N gravity dual, we can get this information from the gravity solution. Consider, for example, a model like that described in the appendix of [9]. Here we have a flow that is ‘stopped’ by the finite-size effects of Sd before it goes nonlinear; i.e., we have ΛO 1 or scenario (3) of Section 16.2.2. If the gravity solution turns on a relevant boundary operator, the field ϕ(ρ) starts at the boundary of the ball at a local (BF- stable) AdS+ maximum of the bulk potential, which we denote conventionally ϕ+ =0. If the smooth ϕ(ρ) solution stays small throughout the ‘ball’, ϕ(ρ = 0) remains close to the value of ϕ at the maximum, and it makes no difference in which direction we perturb away from ϕ+ = 0 (see Fig. 16.3). d If we now gradually increase ΛO past the inverse size of the sphere S , the gravity solution becomes nonlinear before reaching ρ = 0 and, in a potential with two extrema, such as that depicted in Fig. 16.3, it does make a difference in which direction we flow away from ϕ+ = 0. In particular, flowing to negative values of ϕ, we enter the basin of attraction of an AdS− minimum, so that, if the initial conditions and the detailed form of the potential are just right, we may have a domain-wall solution with ΛO 1. Conversely, flowing to positive ϕ, the scalar field will run away and the solution will eventually develop a singularity. Since resolved ‘confining holes’ are seen in the (d + 1)-dimensional gravity theory as singularities, this is a typical case of a gapped phase. In summary, for ΛO 1, we recover the phenomenology of the O(N) model with large Λ, suggesting that this is a generic situation; namely, large, positive-definite, relevant deformations lead to gapped phases, whereas large, negative-definite, relevant deformations lead to condensates. It is only the latter that can be used to investigate crunches. The supergravity picture also indicates that ‘small’, ΛO 1, flows of relevant operators on Sd do generate small, stable condensates on the dS-CFT, which can be used as holographic duals of crunches [9]. The gravity solution ensures that this condensate exists independently of the sign of the microscopic coupling, provided the flow is sufficiently weak throughout the ball. While the large ΛO condensates are visible in a weakly coupled description of the dS-CFT, the small ΛO condensates arise as a peculiar property of strongly coupled dS-CFTs admitting gravity duals. 380 AdS crunches, CFT falls, and cosmological complexity

Fig. 16.3 A generic bulk potential with two extrema has inequivalent flows for the two directions of ϕ, provided these flows enter the nonlinear regime. The arrows indicate the inward evolution of the bulk scalar field ϕ(ρ), starting from an initial value ϕ+ =0at the boundary of the ball. In the figure, the direction of negative fields may lead to smooth domain wall flow, whereas the direction of positive fields leads to scalar runaway, which generally produces singularities in the (d +1)-dimensional gravity description.

We shall return once more to these issues in Section 16.4, aided by a phenomenological eﬀective action, allowing us to map out the diﬀerent scenarios of Section 16.2.2.

16.3.2 The CdL falls Our interpretation of crunches as CFT falls can be made considerably sharper in the case that we describe the bubble in the strict thin-wall approximation. In this setup, the Euclidean background consists of an AdS− ball of ﬁnite radiusρ ¯, surrounded by an exact AdS+ metric. The physical parameters of the background are summarized by a charge q, controlling the jump in the AdS cosmological constant when entering the bubble, and a tension parameter σ for the thin shell. The condition for the Lorentzian bubble to expand is that 0 <σ

It was shown in [2] that this operator can be captured given the thin-wall dynamical data, i.e., the tension σ and charge q of the effective brane building the bubble’s shell. The dynamics of thin-walled vacuum bubbles, defined in terms of junction conditions [19, 20], can be summarized in the effective brane action (cf. [2])

I = −σ Vol [shell] + d · q · Vol [bubble], (16.3.7) where the first term is of Nambu–Goto type and the second term, proportional to the volume of the bubble, is of Wess–Zumino type. A general brane configuration can be parametrized by a collective radial field ρ(x), where x coordinates the conformal boundary spacetime, where the CFT is defined. Using then the results of [21] and [2], we can find the derivative expansion of (16.3.7) after a convenient field redefinition from ρ(x) to a canonically normalized field φ(x). The leading terms for smooth and large bubbles are 1 d − 2 L [φ]=− (∂φ)2 − R φ2 − λφ2d/(d−2) + O φ2(d−4)/(d−2),∂4 , (16.3.8) eff 2 8(d − 1) d where we recognize the conformal mass coupling to the background curvature and the classical marginal interaction of a conformal scalar field in d dimensions, with coupling

− d − 2 2d/(d 2) q λ = σ2d/(d−2) 1 − . (16.3.9) 2 σ

Note that, precisely for the BPS-violating case corresponding to expanding bubbles, 0 <σ

(d−2)/2 φ˜bubble(t)=Ω(t) φ,¯ so that now φ¯ is the time-symmetric t = 0 turning point of the ‘CFT-fall’, or the nucleation point if we think of the real-time bubble as emerging from a pure AdS+ background by a spontaneous tunneling process [2]. It is clear that any such CdL bubble can be nucleated anywhere inside AdS+, so that any classical solution of (16.3.10) is metastable to further nucleation of Fubini bubbles, including the O(d, 1)-invariant conﬁguration φ = φ¯ that we have hitherto considered (cf. [2, 16]). In general, unstable operators can be expected to be marginal only in an approximate sense. In the absence of extra degrees of freedom, the λφ2d/(d−2) theory is not exactly conformally invariant, since λ renormalizes logarithmically.5 Take for example the well-known case of d = 4, where negative λ is asymptotically free, leading to the corrected potential φ4 λφ4 → λ(φ) φ4 = − , (16.3.12) log (φ/ΛIR)

5 This is true both in perturbation theory and in the bulk picture when λ is associated with a double-trace operator [26]. Attempt at a ‘thin-thesis’ 383 where ΛIR is the RG-invariant strong-coupling scale in the IR. Hence, under quantum corrections, the fall is not exactly conformal, the instantons are not exactly de- ﬁned as normalizable solutions, and they exist only in an approximate sense. The corrected background is actually marginally relevant, and we are back to the discussion in Section 16.2, where the driving source is given by the term (16.3.12) with a 6 time-dependent IR Landau pole Λφ4 (t)=ΛIRΩ(t). Incidentally, this argument gives additional evidence supporting our prior interpretation of driven crunches as associated with negative energy falls, at least for the case of marginally relevant operators, whose eﬀects can be approximated by those of a marginal operator.

16.4 Attempt at a ‘thin-thesis’ Given that a marginally relevant operator can produce effects qualitatively similar to those of an exactly marginal operator, it is of interest to pursue the dynamical description of more general bubbles in terms of effective Lagrangians of the type (16.3.10) and (16.3.11). In particular, let us consider bubbles in the generic situation where there are relevant operators turned on in the dual dS-CFT. If the bubbles are thin-walled, i.e., an approximate AdS− patch is visible in the interior, we can expect the global dynamics of the bubble to be well described by a single collective field φ(x), of the type introduced in Section 16.3.2. Even for thick bubbles, we may expect the qualitative energetics to be accounted for by a collective field controlling the overall size and shape of a large and smooth bubble. After integrating out the remaining degrees of freedom of the full dS-CFT, we can write down a phenomenological Landau–Ginzburg model for φ whose leading operators, in a large-φ and long-distance expansion, are completely determined by conformal symmetry: ⎛ ⎞ 1 d(d − 2) L [ φ ]=− ⎝ (∂φ)2 + φ2 + λ (Λ )d−Δ O(Δ)(φ)+...⎠, eff 2 8 Δ Δ eff dSd dSd Δ≤d (16.4.1) O(Δ) 2Δ/(d−2) where eff (φ)=φ is the effective operator of classical conformal dimension Δ ≤ d, accounting for the leading effects of any relevant or marginal operators OΔ that may be turned on at the UV fixed point CFT+, as in (16.2.7). In principle, the Lagrangian (16.4.1) can be rigorously derived in the large-N limit, starting from a given supergravity background in ‘the ball’. The analysis in [2], using the extreme thin-wall approximation, captures just the marginal coupling and the curvature- induced mass term. Computing systematic corrections to the extreme thin-wall approximation should yield the couplings λΔ of the conformal symmetry-breaking operators with Δ

6 The time-dependent IR Landau pole Λφ4 (t) eventually becomes larger than unity at times |ts − −1 t|∼(ΛIR) ; i.e., slightly before the crunch, the E-frame description becomes nonperturbative on the scale of the Sd−1 sphere. 384 AdS crunches, CFT falls, and cosmological complexity be absolutely stable. The example of Section 16.3, corresponding to CdL bubbles in the thin-wall approximation, had λd = λ<0, leaving a globally unstable direction. By allowing logarithmic effective operators as in (16.3.12), we can also use the model to discuss marginally relevant falls (cf. [24]). Nongeneric CFTs may have λd = 0, like, for instance, N = 4 Super Yang–Mills projected along the Coulomb branch. In that situation, the global stability depends on the leading less-relevant operator. Despite the similarity to (16.3.5) and its obvious generalizations, it is important to notice that the model (16.4.1) has a very different status. While (16.3.5) is a microscopic (UV) definition of a toy deformed CFT, we have to think of (16.4.1) as a large-N effective Lagrangian for the single collective mode φ(x). Therefore, the classical approximation to (16.4.1) describes all large-N quantum dynamics of the underlying CFT, for the particular case of states looking like bulk bubbles. Notice also that the effective couplings λΔ appearing in (16.4.1) are functions of the microscopic couplings gO featuring in (16.2.7) or the toy model (16.3.5). However, they are related by a priori complicated dynamics, and no simple relation exists between, say, the sign of a given effective coupling λΔ and the analogous microscopic coupling gΔ, except perhaps when these couplings are sufficiently large, and we have a configuration that is well-approximated by a thin-walled bubble. The essential qualitative behavior can be illustrated by a simplified case with a single relevant operator inducing a scale Λ, and a single marginal operator of coupling λ: − 1 2 d(d 2) 2 2d/(d−2) Leff[ φ ]=− (∂φ) + φ + λφ dSd dSd 2 8 d−Δ 2Δ/(d−2) + λΔ Λ φ + ... . (16.4.2)

We choose λ>0 to ensure global stability of the CFT+ fixed point, and we assume that neither λ nor |λΔ| is parametrically large, so that the only energy hierarchy in the system is controlled by the scale Λ. dS-invariant bubbles or, equivalently, large-N dS-invariant states in the dS theory are described in this approximation as extrema of the effective potential in (16.4.2). For λΔ ≥ 0, the only dS-invariant solution is the dS vacuum, with no scalar condensate φ¯ = 0. Nontrivial condensates require λΔ < 0, corresponding to a negative-definite, relevant perturbation in the effective theory. In this situation, there are still different scenarios depending on the value of Δ and the strength of the relevant perturbation, compared with the Hubble scale. For very relevant perturbations, Δ

(d − 2)(d − Δ) α ≡ . 2(d − 2 − Δ) Attempt at a ‘thin-thesis’ 385

This type of stable dS condensate is reminiscent of scenario (3) in the classification of Section 16.2.2; i.e., it is qualitatively similar to the case studied in the appendix of [9], where the condensate is so small that it is stabilized by the positive curvature-induced mass of dS. We remind the reader that while this condensate requires a negative-definite effective coupling λΔ < 0, this could be compatible with a small range of positive values of the microscopic relevant coupling gΔ (cf. the discussion in Section 16.3.1). On the other hand, for less relevant operators, d>Δ >d− 2, it is the strong- perturbation limit, Λ 1, that yields interesting solutions. We have a stable minimum ¯ (d−2)/2 ¯ α at φ− ∼ Λ and a local maximum near the origin, φ+ ∼ Λ 1. Finally, for Δ = d − 2, both the small-Λ minimum and the large-Λ maximum degenerate to φ¯ = 0 in this crude approximation, whereas the large-Λ stable minimum (d−2)/2 is present at φ¯− ∼ Λ . Any dS-invariant solution at a minimum of (16.4.2) is a stable background modeling a crunch with a well-defined status as a stationary state in the dS-CFT. For Λ 1, we have the domain-wall scenario (2), in the classification of Section 16.2.2. For Λ 1, we have seen that we can parametrize a model of type (3) in the same list. The situation is different for the dS-invariant states at a local maximum of the ¯ α (16.4.2) potential, such as φ+ ∼ Λ for less relevant operator perturbations. At the purely classical level, this state is interpreted as a bulk CdL bubble. Thus, this is an example of a normalizable perturbation of a bulk background with relevant operators turned on at the boundary. The dynamical implications are entirely similar to the previous discussion of CdL bubbles associated with marginal operators; i.e., any such small-field condensate will degrade by further uncontrolled bubble nucleations and collisions. Unlike the case of the absolutely unstable dS theories of Section, 16.3, there is now a finite amount of potential energy available in the dS-frame potential, since the dS theory is absolutely stable for λ>0.7 This means that the condensate-free dS vacuum decays in this theory toward a ‘superheated’ dSd state—at least when probed on distance scales smaller than the Hubble length (above the Hubble length, thermalization of the scalar field fluctuations will not be as effective). The reheating temperature is of order Λ 1, much larger than the starting dS temperature. Remarkably, there are bulk descriptions of superheated dSd spaces precisely for the static patch, in terms of bulk hyperbolic black holes (see [8] and references therein for a recent discussion of these holographic duals in their de Sitter incarnation).

16.4.1 Going down your own way

To any classical solution φcl(x) of (16.4.2), we can associate a classical solution (d−2)/2 φ˜cl(x)=Ω(t) φcl(x) of the ‘driven’ E-frame theory

7 A similar model was proposed in [2] using a negative-definite marginally relevant operator to introduce the local instability, and an exactly marginal operator of the type studied in [27] to stabilize the system. 386 AdS crunches, CFT falls, and cosmological complexity − 2 ˜ 1 ˜ 2 (d 2) ˜ 2 ˜ 2d/(d−2) Leff[ φ ]=− (∂φ ) + φ + λ φ Ed Ed 2 8 (16.4.3) d−Δ 2Δ/(d−2) +λΔ Λ(t) φ˜ + ... , where Λ(t)=Ω(t)Λ is the by now familiar time-dependent scale that effects the ‘driving’. This theory has a time-dependent potential with a growing gap at the origin for λΔ > 0 and a negative well becoming infinitely deep in finite time for any λΔ < 0. ¯ The E-frame description of a nontrivial dS-invariant state φbubble = φ =0isa (d−2)/2 time-dependent bubble field φ˜bubble(x)=Ω(t) φ¯ exercising what we call the ‘CFT fall’, i.e. it goes to infinity in time π/2 from its t = 0 turning point at φ¯,having started at infinity at t = −π/2. While the kinetic energy of this field in the E-frame diverges as t → t, its E- frame potential energy density also diverges to negative values.Eventhevalueatthe turning point t = 0 has a nonpositive energy density, because the positive mass term 1 − ¯2 in (16.4.3) is smaller than the dS-frame mass term by an amount 4 (d 2)φ . The E-frame fall of a stable dS-invariant state looks quite different from that of a CdL-like state. In the case of a configuration sitting at a minimum of the dS potential, its E-frame or ‘infalling’ representation involves a rolling field that is ‘gently held’ by the simultaneous fall of the E-frame potential well (cf. Fig. 16.4). It is interesting to ask if this E-frame field configuration sits at the instantaneous minimum of the time- dependent E-frame potential, or if rather it ‘rolls’ to some extent, relative to the overall fall of the potential itself. The answer is that the φ˜bubble(t) configuration is slightly shifted from the instantaneous minimum of the driven potential, the displacement becoming smaller as time goes by. This is again a consequence of the slight mismatch

Fig. 16.4 Picture a stable fall. In (a), the dS-invariant state, φbubble = φ¯,ispicturedasthe locally stable black dot in the dS-frame. In (b), the E-frame state φ˜bubble(t) executes a combined motion: there is a slow roll down the E-frame potential, starting at φ¯ at t =0and approaching the minimum, while simultaneously the whole potential well falls down to infinity as t → t. Attempt at a ‘thin-thesis’ 387 of curvature-induced masses, i.e., the factor of d(d−2) versus (d−2)2 between (16.4.2) and (16.4.3), an entirely analogous effect to the more dramatic case of the marginal fall discussed in Section 16.3, where the potential did not acquire time dependence in the E-frame, but the field configuration did. We can explicitly illustrate the effect in the case of a large (Λ 1) four-dimensional mass deformation, Δ = 2 and d =4. We find Λ2 − 1 1/2 φ¯ = 2λ for the dS-frame solution. The E-frame field is

φ˜bubble(t)=Ω(t) φ,¯ while the instantaneous minimum sits at Ω2(t)Λ2 − 1/2 1/2 φ˜ (t)= . min 2λ

Hence, φ˜bubble(t) is slightly larger than φ˜min(t), the mismatch approaching zero as t → t, and being maximal at t =0.

Fig. 16.5 Picture of a CdL fall. The stationary state φ bubble = φ¯ of the dS-frame CFT sits at a local maximum (black dot in (a)). In the E-frame CFT (b), we have a time-dependent falling state φ˜bubble(t). The potential changes slightly, so that the ‘sphaleron’ point φ¯ of the dS-frame is mapped to the turning point of the E-frame conﬁguration. If the negative slope is controlled by a marginal operator, the E-frame potential is time-independent. If the slope is controlled by a relevant operator, there is typically a receding minimum further down the potential, which may be reached by the moving dot only at the crunch time t = t.

The stated qualitative diﬀerences between the CdL-type falls and the ‘stable’ falls refer to the behavior of classical dS-invariant conﬁgurations at the extrema of the 388 AdS crunches, CFT falls, and cosmological complexity dS-CFT potential. We may also consider perturbations around these extrema, of d−1 the form φ(τ,Ωα)=φ¯ + δφ(τ,Ωα), where Ωα parametrizes S . At the linearized level, δφ solves a conformally invariant Klein–Gordon equation on dS, with long-time asymptotic behavior

. 1 − d 4m2 δφ(τ,Ω ) ∼ exp(γ±τ) ,γ± = 1 ∓ 1 − , α 2 (d − 1)2 where m is the effective mass of the linearized field δφ. Here we see crucial differences between the stable and unstable dS-invariant vacua. Stable vacua have m2 > 0and Re(γ±) < 0, leading to decaying perturbations in the dS-frame. After transformation to the E-frame, the harmonic fluctuations are still suppressed relative to the zero mode in the t → t limit. In particular, the gradient energy never dominates over the kinetic energy of the field zero mode. This means that the crunch is quite homogeneous, except for possible nonlinearities generated at an intermediate time scale. 2 On the other hand, for the case of CdL solutions, m < 0 and Re(γ+) > 0, so that there is an unstable solution leading to the growth of harmonic perturbations on the Sd−1 sphere. This leads to a rapid dominance of the energy by the spatial gradient terms. Over at the E-frame, the gradient modes get excited in the fall and dominate the energy density, a case of ‘tachyonic preheating’ (see [28] for a review). A recent detailed analysis of such processes for a CdL fall can be found in [29]. These simple considerations regarding nonhomogeneous configurations show that while crunches associated with stable CFT falls ‘have no hair’, in the sense that the homogenous solution is linearly stable as t → t, crunches associated wtih CdL falls are quite ‘hairy’, even at the level of single-bubble dynamics.

16.4.2 Rules of engagement? We finish this section with some general considerations on the overall physics picture implied by the effective Landau–Ginzburg approach. One would like to identify definite field-theoretical hallmarks for the crunch, and we have highlighted several candidates, none of which is perfect, but each of which manifested an important aspect of the crunching phenomenon. An indication in the E-frame was an infinite energy fall, which came in essentially two flavors. In one class of ‘CFT fall’ the E-frame potential is itself time-dependent but stable for any time before the crunch. The value of the potential at its minimum is negative and becomes more and more negative as the time approaches the crunch time. The field has an expectation value that shifts to the UV—that is, to larger and larger values. A potential in the dS-frame that is negative at its minimum and that has a next-to-leading operator with a large negative coupling would be mapped to the desired form in the E-frame. The other type of potential is one that is unbounded for any time in both Einstein and de Sitter frames. It may be more palatable to accept that the crunches associated Falling on your sword 389 with the stabilized potentials can exist in a theory of gravity, even if those associated with the second type of potentials are classically well defined for a finite time interval.8 These scenarios are particularly clear-cut when the energy scale associated with the condensate is much larger than the Hubble scale. This corresponds in the bulk with a thin-walled bubble of large size compared with the AdS radius of curvature. In this case, it is natural to expect that the sign of the effective coupling λO is correlated with the sign of the microscopic coupling gO, so that a large and negative value of d−ΔO gO(ΛO) is expected to be a hallmark for the crunch in the microscopic specifica- d−ΔO tion of the CFT. Conversely, a large and positive gO(ΛO) would be a hallmark of a gapped theory with no crunch in its bulk version. When the energy scale associated with the condensate is not large in Hubble units, the geometrical picture becomes murkier. If the slightly negative operator is sufficiently relevant, we have found small condensates with thin walls in the Landau–Ginzburg model, mimicking the expected behavior of supergravity solutions with bubbles that are very small in units of the AdS curvature radius. Such supergravity solutions are, strictly speaking, outside the realm of the thin-wall approximation, and therefore reducing the number of effective fields to one single collective mode becomes questionable. Yet it is in such circumstances that a crunch can be reliably identified in the bulk within a valid linear approximation of the supergravity equations. This linearity implies that there cannot be a strict correlation between the signs of λO and gO in these cases. All examples of crunches considered in this chapter seem to conform to a general rule, namely one needs a large-N worth of massless degrees of freedom at the Hubble scale, coexisting with a semiclassical condensate, which itself could be large or small in Hubble units. Expectation values of collective fields measuring the condensate may then serve as ‘order parameters’ for the existence of the crunch. The gapless character of the condensates seems to be essential. Semiclassical condensates exist in gapped theories, such as gluon condensates in a confining AdS/CFT model, and yet such theories provide unambiguous crunching models only when the confining scale is small in Hubble units, so that the glueballs still look essentially massless at the Hubble scale.

16.5 Falling on your sword In this section, we introduce a regularized version of the single-bubble crunching dynamics that simply makes the fall finite, further elaborating on the discussion in [2, 4]. We will see that the result is a conventional black hole final state, thus adding support to our complementarity interpretation of the map between Einstein and de Sitter frames. From the point of view of the CFT in E-time frame, the crunch is associated with a finite-time fall, down an infinite cliff of negative potential energy. The geometry codifies

8 Beyond the N = ∞ limit, which corresponds to the classical approximation to (16.3.11), the fate of the CdL-fall model depends on poorly understood multibubble dynamics [16, 20]. Even at the level of the 1/N expansion, the quantum evolution of a single bubble suffers from ambiguities on time scales arbitrarily close to t = 0 (D. Marolf, unpublished). 390 AdS crunches, CFT falls, and cosmological complexity a state in the QFT that is transferring support to UV modes, with all the characteristic energy scales diverging proportionally to Ω(t). The infinite character of the fall is a consequence of the O(d, 1) symmetry of the state, since any dS-invariant configuration φbubble = φ¯, with constant φ¯, is mapped to a configuration whose time dependence is just dictated by the blowing-up Weyl factor Ω(t). Hence, any regularization of the fall must break dS symmetry close to the boundary. In the dS-time frame, this means breaking the eternal character of de Sitter spacetime. When the crunch is a result of an explicit driving term, the breaking of the O(d, 1) symmetry must be prescribed by hand, simply declaring that the driving source JO(t) stops growing before the pole at t = t. If we stop the bubble as some fixed radius in the E-frame metric, rmax =ΛE, the state breaks the O(d, 1) symmetry to an U(1) × O(d) symmetry, which is the global symmetry of the CFT on the Einstein manifold. In dS- frame variables, the stopping of the bubble is equivalent to terminating the ‘eternity’ of de Sitter, with a limiting time

τmax ∼ log(ΛE/ΛO).

The long-time evolution in the E-frame will take the system to a generic state with the unbroken U(1) × O(d) symmetry. If the potential energy released is large enough, those states look like a locally thermal state in the QFT, or a black hole in the bulk description. We may estimate the energy and entropy of this final black hole in the following way. Since the driving operator O has dimension ΔO, the expectation value at long ΔO times in the E-frame is of order O ∼ Neff (Λ E) , where Neff is the central charge of the CFT, or number of effective field species in the UV limit. Therefore, the amount of potential energy released is of order

gO ΔO d H ∼ N (Λ ) ∼ gO N (Λ ) . driving ΔO −d eff E eff E (ΛE) If all this energy is eventually thermalized in the E-frame QFT, it will take the form d Neff (Teff) , which gives an effective ‘reheating’ temperature

1/d Teﬀ ∼ (gO) ΛE .

If Teﬀ 1, the bulk representation of this state will be a large AdS black hole of entropy

d−1 (d−1)/d d−1 SBH ∼ Neff(Teff) ∼ Neff (gO) (ΛE) , (16.5.1) where all quantities are normalized dimensionally to the unit size of the E-frame sphere. Since gO < O(1) as part of the Wilsonian convention in defining the effective energy scales, we find that the entropy of the resulting black hole is always bounded by the maximal information capacity of the E-frame QFT, defined with a Wilsonian cutoff at scale ΛE:

SBH

In the case of CdL bubbles, or spontaneous decay, the same arguments apply, except that now the O(d, 1) symmetry is broken to U(1) × O(d) by postulating a stabilization potential in the E-frame Hamltonian HCFT at field values of order φmax ∼ ΛE. For the situation studied in Section 16.4, in the thin-wall approximation, the estimates about the properties of the final state apply with gO ∼|λ|1 and Neff the fraction of degrees of freedom carried by the bubble’s shell, so that the information-theoretic bound (16.5.2) is far from saturated. In general, the details of the approach to the typical thermalized state are much more involved here, since due attention must be paid to multibubble collisions that occur within the O(d, 1)-invariant region, r<ΛE. The fundamental aspect of the regularization is to regard the O(d + 1) group (or its Lorentzian counterpart) as an accidental symmetry emerging below a scale ΛE. Above this scale, the symmetry of the state is only the U(1) × O(d) group of the

Fig. 16.6 Causal diagram of the regularized crunch model. The shaded region is the part of the bulk with approximate O(d, 1) symmetry. The bubble patch is shown by light shading and the FRW patch by dark shading. The unshaded part is the bulk region realizing the UV symmetry group U(1) × O(d). The ingoing ﬂux of energy from the bubble collision at the UV wall bounds the FRW region and forms the ﬁnal black hole. As ΛE →∞, this bolt of energy diverges, as does the black hole size, producing the FRW crunch. 392 AdS crunches, CFT falls, and cosmological complexity

Einstein space. Going back to the original problem posed by (16.2.1), the resolution of the crunch by a UV breaking of the O(d, 1) symmetry puts the finger of blame on the large isometry of the FRW model on noncompact spatial sections. If this symmetry is broken to U(1) × O(d) outside a compact set, the crunch is tamed in the sense that it resides inside a black hole. However, it would be wrong to claim that the crunch inside the black hole is the precise regularization of the cosmological crunch of (16.2.1). A glance at the causal diagram in Fig. 16.6 shows that the cosmological crunch is associated with the boundary of the region with O(d, 1) symmetry, i.e., the null surface produced by the ‘backflow’ from the collision of the bubble with the UV wall. Inside the final black hole event horizon, this surface coincides with the apparent horizon of the black hole.

16.6 Conclusions Singularities have been encountered time and again in physics. In most cases, they marked the limits of validity of the approximation involved. The drive to discover what lies beyond the approximation originated from the knowledge that these singularities are not present in the phenomena described. In the study of gravity, a new way emerged to understand such singularities: one considered the possibility that they are cloaked by horizons. This was actually realized in systems that have a holographic description. The finite entropy that could be absorbed by a crunch inside a black hole in AdS is captured in principle by the holographic boundary observer. In this paper, we have shown explicitly how such a complementarity may work for a class of cosmological crunches. We see that even such singularities in which an infinite amount of entropy flows into the crunch can be decoded by holographic observers, provided they are prepared to wait for a de Sitter eternity to complete the measurement. We have also shown that if in some way it turns out that dS spaces are not eternal [30], then these singularities will involve only finite entropy and will be regulated, as crunches in black holes are, by the system itself. We have distilled these lessons from an analysis of certain negative-curvature FRW backgrounds with a crunching future endpoint and O(d, 1) symmetry. By embedding these FRW cosmologies inside expanding AdS bubbles, one can view these backgrounds as the evolution of certain O(d, 1)-invariant states looking like Bose condensates of a perturbed CFT on de Sitter spacetime. The same quantum state can be codified in the Hilbert space of another perturbed CFT defined on a static Einstein spacetime. This alternative representation of the initial state uses the same basic CFT, but now perturbed by time-dependent couplings. These two descriptions of the initial state are related by a conformal transformation that we interpret as a ‘complementarity map’ in the sense in which this term is used in discussions of black hole information theory. For those dS-invariant states whose bulk development has a crunch, and the condensate has a sharply defined energy scale, we argue that the E-frame (infalling) description consists of an infinite negative-energy fall, where the quantum state coherently shifts its support to the UV under the action of an unbounded Hamiltonian, either through an explicit time-dependent driving term or through an Conclusions 393 unbounded potential. We view this behavior, which we term the ‘CFT fall’, as the CFT landmark of a crunch. When transformed back into the dS-frame picture, the O(d, 1)-invariant state is seen as a large-N master field for a dS-CFT perturbed by sufficiently negative-definite relevant or marginal operators. As a peculiarity of the large-N limit, we can have semiclassical condensates that are very small compared with the Hubble scale. In this case, the form of the gravity solution shows similar qualitative behavior, although it is not possible to sharply trace the ‘crunchy’ character to the sign of the microscopic coupling. Still, if an effective Landau–Ginzburg description should exist in terms of a single collective field, it is expected to follow the general picture above. The O(d, 1)-invariant condensates that harbor crunches in their bulk description fall into two broad classes, depending on whether they are stable or unstable states in the deformed dS-CFT. In the first case, they give self-consistent holographic models for the crunch. In the second case, corresponding to bulk backgrounds with the interpretation of CdL bubbles, the dynamics is heavily corrected by finite-N effects involving multibubble nucleation and collisions, and even the quantum 1/N corrections pose a consistency challenge. We have also investigated an explicit regularization of the CFT fall, introducing a hard UV wall with the manifest U(1) × O(d) symmetry of the E-frame theory. The O(d, 1) symmetry is then regarded as an accidental IR symmetry. Since the CFT fall transfers the support of the state to the UV, the reduced symmetry of the wall determines the long-time dynamics of the system, so that it approaches the generic state with U(1) × O(d) symmetry, i.e., a locally thermalized state, or a black hole in bulk parlance. Within this interpretation, the FRW crunch is regularized as the null ‘backflow’ of debris from the collision of the CFT state with the UV wall. Upon removal of the UV wall, it is the diverging energy of this back flow that produces the crunch. As a byproduct, this regularization establishes a direct relation between finiteness of the black hole entropy and the non-eternal character of the de Sitter state. At the same time, it shows that our ‘cosmological complementarity map’ can be regarded as a limit of the more standard, and yet more mysterious, black hole complementarity map. Looking at our results in the large, it is tempting to promote the scenario depicted here to the rank of a general rule, namely, that the quantum description of spacelike singularities is always going to involve Hamiltonian ‘falls’, most likely of the ‘stable’ type, which seem well defined enough. This would apply strictly to the case in which an infinite-dimensional Hilbert space can ‘fall into’ a crunch or ‘come from’ a bang. For finite-entropy crunches/bangs, one expects any continuous-time Hamiltonian description to be at best approximate (cf. [31]). At this point, we may ask: Who is afraid of singular Hamiltonians? The ‘infalling’ Hamiltonians described here act for a finite time and therefore must be singular. From the holographic point of view, it makes no sense to go beyond the crunch, and thus there is no need for the Hamiltonian description to be regular at t = t.In any case, the examples discussed in this chapter show vividly how an infinite fall propagated by an unbounded Hamiltonian can be a valid ‘history’ of a completely regular state. 394 AdS crunches, CFT falls, and cosmological complexity

We have one state and two noncommuting operator algebras that measure it. In one operator algebra, described as a de Sitter-CFT, the ‘history’ is that of a stationary state, whereas in the other operator algebra, characterized as a driven Einstein-CFT, we have a singular Hamiltonian for a ﬁnite time. They are completely equivalent for the interval −t

Acknowledgments We are indebted to T. Banks, R. Brustein, J. Mart´ınez–Magán, and S. Shenker for useful discussions. We thank the Galileo Galilei Institute for hospitality. E. Rabinovici wishes to thank Stanford University for hospitality. The work of J. L. F. Barbón was partially supported by MEC and FEDER under a grant FPA2009-07908, the Span- ish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042), and Comunidad Autónoma de Madrid under grant HEPHACOS S2009/ESP-1473. The work of E. Ra- binovici was partially supported by the Humbodlt Foundation, a DIP grant H-52, the American–Israeli Bi-National Science Foundation, and the Israel Science Foundation Center of Excellence.

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Gabriele Veneziano

Coll`ege de France, Paris, France, and Theory Division CERN, Geneva, Switzerland

17 High-energy collisions of particles, strings, and branes 397 Gabriele VENEZIANO

17.1 Motivations and outline 399 17.2 Gravitational collapse criteria: a brief review 400 17.3 The expected phase diagram 402 17.4 The small-angle regime: deﬂection angle and tidal excitation 404 17.5 The string-gravity regime: a precocious black hole behaviour? 406 17.6 The strong-gravity regime: towards the large-angle/collapse phase? 407 17.7 High-energy string–brane collisions: an easier problem? 410 17.8 Summary and outlook 412 Acknowledgments 413 References 413 Motivations and outline 399

We summarize some 25 years of work on the transplanckian-energy collisions of particles, strings, and branes, seen as a theoretical laboratory for understanding how gravity and quantum mechanics can be consistently combined in string theory. The ultimate aim of the exercise is to understand whether and how a consistent quantization of gravity can solve some long-standing paradoxes, such as the apparent loss of information in the production and decay of black holes at a semiclassical level. Considerable progress has been made in understanding the emergence of General Relativity expectations and in evaluating several kinds of quantum string corrections to them in the weak-gravity regime while keeping unitarity manifest. While some progress has also been made in the strong-gravity/gravitational collapse domain, full control of how unitarity works in that regime is still lacking.

17.1 Motivations and outline Progress in fundamental physics has often been based on stepping up in energy, either experimentally or through theoretical (gedanken) experiments. The realization, for instance, that a theory of the weak interactions based on fundamental massive vector particles would violate unitarity at very (and at the time unrealistically) high energies was one of the ingredients that led eventually to the idea of spontaneous symmetry breaking and to the present electroweak standard model. When dealing with candidate theories of quantum gravity, the need for very (unrealistically?) high energies is even more obvious. Indeed, while classical gravity—or classical string theory—has no intrinsic energy scale, quantum and string gravity do: the Planck mass MP and the string mass scale Ms, respectively. Both are presumably too high for real experiments, except for those that presumably occurred naturally in the early universe. In this chapter, we will use very high (i.e. transplanckian) energy to expose in a clear way some of the deep conceptual problems that seem to emerge when one tries to combine the basic principles of General Relativity with those of Quantum Mechanics, the two great revolutions that took place in physics about a century ago. The issue of a possible loss of information/quantum coherence in processes where a black hole is produced and then evaporates has been the subject of much debate since Hawking’s observation [1] that black holes should emit an exactly thermal spectrum of light particles, known as Hawking radiation. Progress coming from string theory, in particular on the microscopic understanding of black hole entropy [2] and on the AdS/CFT correspondence [3], have lent strong support [4] to the “no-information- loss” camp. In spite of these developments, several issues still remain unclear. One would like to understand, for instance, how exactly information is retrieved and what this implies on the properties of the pure ﬁnal state that a given pure initial state generates through its unitary evolution. There is a second, less fundamental but phenomenologically interesting, reason for studying high-energy collisions. Models have been proposed [5] in which gravity, being sensitive to some “large” extra dimensions of space, becomes strong at an eﬀective energy scale MD that is much smaller than the “phenomenological” four-dimensional 400 High-energy collisions of particles, strings, and branes

19 Planck energy MP ∼ 10 GeV. Assuming MD to be not too far from the few TeV scale, a variety of interesting strong-gravity signals could be expected [6] when the LHC turns on. We will start by recalling briefly some well-known results on gravitational collapse criteria in classical General Relativity (CGR). When naively applied to our collision problem, they lead naturally to a two-dimensional “phase diagram” with a “phase transition” curve separating a “dispersion” region from a “collapse” one. Furthermore, the dispersion region can be divided into two subregions depending on the relative sizes of the impact parameter b and the string length ls. These two subregions will then be discussed, in succession, and an explicitly unitary ansatz for the S-matrix will be presented. We will then present the first serious attempt to move into the collapse region of parameter space. Here success is much more limited, in particular in what concerns control of inelastic unitarity. Finally, we will turn our attention to a supposedly simpler problem, that of the high-energy scattering of a light closed string off a stack of D- branes, describe analogies and differences with respect to the previous problem, and present the progress made so far. We shall conclude with a brief summary and outlook.

17.2 Gravitational collapse criteria: a brief review There are many analytic as well as numerical CGR results on whether some given initial data should lead to gravitational collapse or to a completely dispersed final state. The two phases would be typically separated by a critical hypersurface in the parameter space of the initial states. The problem has some amusing analogies with the physics of phase transitions in statistical systems. In fact, the approach to criticality resembles that of phase transitions (the order of the transition and critical exponents can be defined by analogy). For pure gravity, in a classic work, Christodoulou and Klainerman [7] have found a finite-measure parameter-space region bordering Minkowski spacetime and lying on the dispersion side of the critical surface. In the case of spherical symmetry, regions on the collapse side have been found, analytically by Christodoulou [8] and numerically by Choptuik and collaborators [9]. In the absence of any special symmetry, Christodoulou [10] has identified another collapse-bound region. It is characterized by a lower bound on incoming energy per unit advanced time holding uniformly over the full solid angle. More precisely, denote by μ(v,θ,φ) the incoming (null) energy per unit advanced time v and solid angle dΩ entering the system during a “short” interval δ ≡ δv. Then, if δ k M(θ, φ, δ) ≡ dv μ(v,θ,φ) > , (17.2.1) 0 8π a closed trapped surface (CTS) forms with a Schwarzschild radius RS >k−O(δ) > 0. Unfortunately, such a beautiful criterion is not useful for two-body collisions, the energy being concentrated in two narrow back-to-back cones, but the general idea of the method still applies. This consists in the identification of a CTS at a certain Gravitational collapse criteria: a brief review 401 point in the system’s evolution. It should be stressed that such criteria can only be of the sufficiency type. Identification of a CTS guarantees collapse, whereas the opposite situation does not lead to any firm conclusion, since it can be the result of one’s inability to follow the evolution of the system for sufficiently long times. Examples of criteria that can be established in the case of two-body collisions are as follows: • Point-particle collisions: • b = 0: For a head-on collision of pointlike massless particles, Penrose [11] has argued that there is a lower bound on the fraction of the incoming energy that goes into forming a black hole: √ √ MBH >E/ 2 ∼ 0.71 s. (17.2.2) • b = 0: Eardley and Giddings [12] have generalized the above result to a collision at a generic impact parameter b. One example in D =4is √ RS ≤ 1.25 (RS ≡ 2G s). (17.2.3) b cr • The Eardley and Giddings approach was generalized further to extended sources in [13]. One example is the central collision of two homogeneous null discs of radius L, where one finds R S ≤ 1 . (17.2.4) L cr Finally, we should mention that Choptuik and Pretorius [14] have obtained new numerical results for a highly relativistic axisymmetric situation (their results will be compared with ours in Section 17.6). So far, our considerations have been purely classical. What happens when we go from the classical to the quantum problem? We can certainly prepare pure initial states that√ correspond, roughly, to the classical data by specifying the centre-of-mass energy s of the collision and, instead of the impact parameter, the total angular momentum J ∼ bE. We can then ask several interesting questions: • Does a unitary S-matrix (evolution operator) always describe the evolution of the system? • If so, does such an S-matrix develop singularities as one approaches a critical (parameter-space) surface? • If so, what happens in its vicinity? Does the nature of the final state change as one goes through it? • Is there a relation between the classical and quantum critical surfaces? • What happens to the final state deep inside the collapse region? Does it resemble at all Hawking’s thermal spectrum for each initial pure state? All these questions are obviously related to the information paradox/puzzle. As already mentioned, there can be more phenomenological motivations for studying transplanckian-energy (TPE) collisions, i.e. finding signatures of string/quantum 402 High-energy collisions of particles, strings, and branes gravity at the LHC. This is in principle possible in Kaluza–Klein models with large extra dimensions, in brane-world scenarios, and in general if the true quantum gravity scale can be lowered to the TeV scale. However, even in the most optimistic situation, the LHC will be quite marginal for producing black holes, let alone semiclassical ones. The question then is this: • Can there be some precursors of BH behaviour even below the expected black hole-production threshold? Perhaps surprisingly, the answer will turn out to be positive. TPE string collisions represent a perfect theoretical laboratory for studying these questions within a framework that claims to be a fully consistent quantum theory of gravity. We can hardly imagine a simpler pure initial state that could lead to black hole formation and whose unitary evolution we would like to understand/follow. TPE is obviously need in order to have a chance of forming (and studying) a semiclassical black hole, i.e. one with RS lP . As it turns out, TPE also simplifies the theoretical analysis by allowing the use of some semiclassical approximation.

17.3 The expected phase diagram √ Collisions of light particles at superplanckian energies (E = s MD) have received considerable attention since the late 1980s. While in [15, 16] the focus was on D =4 collisions in the ﬁeld theory limit, two groups have carried out the analysis within superstring theory, possibly allowing for a number of “large” extra dimensions. In the approach due to Gross, Mende, and Ooguri (GMO) [17], one starts from a genus-by- genus analysis of ﬁxed-angle scattering, and then attempts an all-genus resummation. In the work by Amati, Ciafaloni, and myself (ACV) [18–20], one starts from an all- order eikonal description of small-angle scattering and then attempts to push the results towards larger and larger angles. In the approach by Fabbrichesi et al. [21], one relates, at arbitrary scattering angle, the semiclassical S-matrix to a boundary term and tries to estimate it in terms of classical solutions. The picture that has emerged (see e.g. [22] for some reviews) is best explained by working in impact parameter (b =2J/E) space, rather than in scattering angle (θ) or momentum transfer. We can thus represent the various regimes of superplanckian collisions by appealing to an (E,b) plane or, equivalently but more conveniently, to 1 an (RS,b) plane, where

1/(D−3) RS(E) ∼ (GDE) (17.3.1) is the Schwarzschild radius associated with the centre-of-mass energy E. Since both coordinates in this plane are lengths, we can also mark on its axes two (process- independent) lengths: the Planck length lD and the string length ls. We shall use the following deﬁnitions:

1 We shall denote by GD the D-dimensional Newton constant and simply by G the four-dimensional one. The expected phase diagram 403 √ −1 D−2 2−D ls = 2α = Ms ,GD = lD = MD , (17.3.2) where α is the open-string Regge-slope parameter (equal to twice that of the closed string). We will assume string theory to be (very) weakly coupled:

−2/(D−2) −2/(D−2) ls =(gs) lD lD, i.e. MD = Ms(gs) Ms, (17.3.3) where gs is the string coupling constant. We shall keep gs (and hence lD/ls = Ms/MD) ﬁxed and very small. By deﬁnition of transplanckian energy, RS >lD. Since we also restrict ourselves to b>lD, a small square near the origin is not considered. The rest of the diagram is divided essentially into three regions (see Fig. 17.1):

• The first region, characterized by b>max(ls,RS), is the easiest to analyse and corresponds to small-angle quasi-elastic scattering. • The second region, RS > max(b, ls), is the most difficult: this is where we expect black hole formation to show up. Unfortunately, in spite of much effort (see e.g. [20, 21, 23]), not much progress has been achieved in the way of going through the b = RS >ls boundary of Fig. 17.1. • Finally, the third region (b, RS

b 1

1s 3

2 BH

1P 1s

Fig. 17.1 Phase diagram of transplanckian scattering showing three diﬀerent kinematic regions and two diﬀerent paths towards the regime of large black hole production. Here lP = lD. 404 High-energy collisions of particles, strings, and branes

hole whose radius would be smaller than ls. Instead, it is found [19] that a maximal D−3 deflection angle θmax ∼ (RS/ls) is reached at an impact parameter of order ls (for which the colliding strings graze each other). If one considers scattering at angles larger than such θmax, one finds (see, e.g., [22]) an exponential suppression of the cross section, in very good agreement with the results obtained by GMO [17] through their very different approach. This region merges into the previous one at RS ∼ ls, corresponding to the expected energy threshold for black hole ∼ −2 production at E = Eth gs Ms, in agreement with the so-called correspondence [24, 25] between black holes and fundamental strings.

17.4 The small-angle regime: deﬂection angle and tidal excitation General arguments, as well as explicit calculations, suggest the following form for the elastic S-matrix at TPE: ∼ Acl ≡ 2iδ(E,M,b) S(E,b) exp i e , − G s b4−D R 2(D 3) λ2 λD−2 D S s P ··· δ(E,M,b)= 1+O + O 2 + O D−2 + , ΩD−4 b b b √ d/2 D−3 16πGD s ≡ 2π RS = , Ωd . (17.4.1) (D − 2)ΩD−1 Γ(d/2)

Here we have written explicitly the leading term at large b, followed by “corrections” that are controlled either by the ratio ls/b (so-called string corrections) or by the ratio RS/b (so-called classical corrections), while corrections proportional to the Planck length are systematically ignored for the reasons already mentioned. The ﬁrst kind of corrections will be discussed here; the second kind will be discussed in Section 17.6. Note, however, that both kinds of corrections may become relevant even when the corresponding ratio is still very small. The reason is that, unlike the leading term, corrections are not necessarily real. Whenever they contain an imaginary part, the resulting damping of the elastic S-matrix becomes important as soon as the overall quantity in the exponent (including the large pre-factor) becomes of order one.

17.4.1 The point-particle limit at large b This regime can be described by resumming the leading-eikonal ﬁeld theoretical diagrams (crossed ladders included). This restores elastic unitarity since the large partial-wave amplitude, by exponentiation, becomes a harmless rapidly varying phase. This fact can actually be used to justify a saddle-point approximation when we transform back the S-matrix from impact parameter to (centre-of-mass) scattering angle (or momentum transfer q) space. The small-angle regime: deﬂection angle and tidal excitation 405

The position of the saddle point is given by √ RD−3 D−3 8πGD s ∼ S bs = , (17.4.2) θ ΩD−2 θ and can be seen as the generalization of Einstein’s deflection formula for ultrarela- tivistic collisions and extended to arbitrary D. It corresponds precisely to the relation between impact parameter and deflection angle in the (Aichelburg–Sexl) metric [26] generated by a relativistic point-particle of energy E. Note that this effective metric is not put in by hand: it is an “emergent” semiclassical metric generating the expected deflection. We conclude the discussion of this regime with the following important remark: high-qT is not necessarily short-distance in high-energy gravitational scattering! In- deed, as can be seen from (17.4.2), at fixed θ, larger E probes larger b. The reason 4−D is simple: because of the eikonal exponentiation, GDsb / also gives the average/dominant loop number. The total momentum transfer q ∼ θE is thus shared among O(s ∼ E2) exchanged gravitons to give

D−4 qb θ 3−D qind ∼ ∼ (R/b) ∼ , (17.4.3) GDs b bs thus recovering the uncertainly-principle relation and the conclusion that the process is soft, even at large q, provided that b is suﬃciently large.

17.4.2 String corrections at large b Even at large b, graviton exchange can excite one or both strings. The physical reason [27] is that a string moving in a nontrivial metric feels tidal forces as a result of its ﬁnite size. A simple argument gives the critical impact parameter bt below which this tidal excitation phenomenon kicks in (in agreement with what was found by direct calculation by ACV!). It is indeed parametrically larger than ls:

− 2 1/(D 2) ∼ GDsls bt . (17.4.4)

These eﬀects are neatly captured, at the leading eikonal level, by replacing the impact parameter b by a shifted impact parameter, displayed by each string’s position operator (stripped of its zero modes) evaluated at τ = 0 (equal to the collision time) and averaged over σ. This leads to a unitary operator eikonal formula for the S-matrix:

S(E,b) → Sˆ(E,b) = exp[2iδ(E,b − Xu + Xd)], (17.4.5) where 2π 2π − ≡ 1 ˆ − ˆ δ(E,b Xu + Xd) 2 dσu dσd : δ b + Xd(σd, 0) Xu(σu, 0) : 4π 0 0 (17.4.6) 406 High-energy collisions of particles, strings, and branes

Several quantities can be computed from this general formula—for instance the mass distribution of the excited strings. One ﬁnds [18] that such a distribution grows roughly like the density of string states (i.e. exponentially) up to a certain maximal 2−D excitation mass (Mmax ∼ GDslsb ) and then sharply cuts oﬀ. Also, in order to conserve probability, the cross section for each exclusive process (including the elastic one) is exponentially damped. For instance,

∼ − 2 2−D σel exp( GDsλsb ). (17.4.7)

In other words, the whole process of tidal excitations produces a sort of statistical (though nonthermal) ensemble of ﬁnal states, while strictly maintaining quantum coherence.

17.5 The string-gravity regime: a precocious black hole behaviour? Because of (good old Dolen–Horn–Schmit) duality, even single-graviton exchange does not give a real scattering amplitude. The imaginary part is due to formation of closed strings in the s-channel. Since the imaginary part of the tree amplitude lacks the Coulomb singularity, it is exponentially damped at large impact parameter. This is why this contribution is irrelevant in region I but important in region III. The average number of closed strings produced is given by the average number of cut gravi-reggeons according to the AGK cutting rules [28] (see [29] for details). This is still given by the eikonal phase, but this time by its imaginary part: 2 GDsls s NCGR =4Imδ = D−2 = O 2 , (17.5.1) (Yλs) M∗ √ where M∗ ∼ Ms/gs ∼ MsMth will turn out to play an important role in the following. Note that M∗ coincides, up to numerical factors, with the mass of D0-branes (see e.g. [30]), possibly just as a coincidence. If we forbid particle production, we are again penalized by an exponential factor. For instance, the elastic cross section is damped as 2 ∼ − − GDsls σel exp( 8Imδ) = exp D−2 . (17.5.2) (Yls)

Amusingly, such a suppression (which coincides with (17.4.7) at b ∼ ls) resembles a − statistical factor√ exp( Sbh), where Sbh is the Bekenstein–Hawking entropy of a black hole of mass s. Such a suppression is known to appear in the elastic amplitude for the scattering of a particle oﬀ a classical black hole when the impact parameter goes below that of classical capture (see e.g. [31]). For D = 4 the agreement holds up to a numerical factor, while for D>4 the functional dependence is diﬀerent. For any value of D, however, agreement with a black-hole-type suppression holds (up to factors O(1)) as one approaches Eth. The strong-gravity regime: towards the large-angle/collapse phase? 407

Let us now look instead at the typical final state that roughly saturates the cross section. The total energy will be shared among NCGR CGRs, giving an average energy per CGR: √ 2 s M∗ ECGR = ∼ √ . (17.5.3) NCGR s The average energy of the final states thus decreases as the energy is increased within our window. This is quite unlike what one is accustomed to in particle physics, but is similar to what we√ expect in black hole physics! Once more, for D =4the functional dependence on s of ECGR follows that of a thermal spectrum with a Hawking temperature TH ∼ /RS, while for D>4 the functional dependence is different. For all values of D, however, agreement (up to factors O(1)) with Hawking temperature expectations occurs at Eth. An “antiscaling” behaviour, corresponding to √ 2 2 −2 E CGR s = M∗ = Ms gs , (17.5.4)

−2 holds [32] for any D in the energy window M∗

17.6 The strong-gravity regime: towards the large-angle/collapse phase? We shall now address the more difficult question of how to deal with what we called classical corrections, those characterized by the expansion parameter (R/b)2(D−3) and which become important when the scattering angle becomes O(1). It was understood quite early on [20] that these corrections are associated with (the resummation of) a particular class of diagrams, those in which the gravitons exchanged between the two fast-moving particles only interact (if at all) at tree level. It is of course not unexpected that classical corrections are related to tree diagrams, but in our context we have to be precise about the meaning of a “tree diagram”. In the following discussion, we shall limit ourselves to the point-particle limit and to D = 4. While the second limitation looks easy to dispose of (actually, some of the infrared problems we shall encounter disappear for D>4), the first limitation is hard to circumvent. Ideally, one would like to find, for the classical corrections, a simple recipe, like that of (17.4.5), for taking the string-size corrections into account. No such a generalization has been found so far. Standard reasoning strongly suggests that the sum of all (connected and discon- nected once the external lines are removed) diagrams simply gives the exponential of the connected diagrams, the latter therefore being directly related to the phase of the semiclassical S-matrix. Power counting confirms this assumption. If n gravitons are exchanged and interact with a connected tree, we find

2n−1 n 2(n−1) 2(n−1) Acl(E,b) ∼ G s ∼ GsR → Gs(R/b) , (17.6.1) which is the expected scaling for D =4. 408 High-energy collisions of particles, strings, and branes

Summing tree diagrams should amount to solving a classical field theory, and the question therefore is: Which is the effective field theory describing transplanckian scattering? There is actually a good candidate for it: the effective action proposed long ago by Lipatov [33]. Unfortunately, this is still too complicated to attempt any kind of (analytic or numerical) solution. In [23], ACV proposed a simplified version of Lipatov’s action in which the longitudinal dynamics is frozen and factored out. One is then left with a much more manageable D = 2 effective action (corresponding to the coordinates transverse to the collision axis) containing four fields: the ++ and −− components of the metric, sourced by the energy–momentum tensor of the two fast particles, together with a complex field φ representing the two physical polarizations of physical gravitons. In D = 4, one polarization is affected by infrared divergences, which in principle can be cured by considering infrared-safe observables, but will be actually neglected for the sake of simplicity. Thus, the complex scalar field gets replaced by a real one, for a total of three real fields. The action, for whose explicit form we refer to [23], contains up to four derivatives but is otherwise nice looking. Its equations of motion are rather simple and admit a perturbative expansion in our “small” parameter RS/b. The semiclassical approximation amounts to solving the equations of motion and to computing the classical action on the solution after imposing suitable boundary, reality, and regularity conditions on it [23]. The resulting problem is still too hard for analytic study, but can be dealt with numerically. In an impressive paper, Marchesini and Onofri [34] managed to solve directly the resulting partial differential equations (PDEs) by fast Fourier transform methods and concluded that real, regular solutions only exist if

b>bc ∼ 2.28RS , (17.6.2) to be compared with the CTS lower bound of [12] bc > 0.80RS. Thus the “ACV value” for bc is a factor 2.85 or so above the CTS lower bound by Eardley and Giddings. For an analytic approach, we have to turn to an even simpler problem: axisymmetric beam–beam collisions [23, 35]. The data in this case lie in two functions, R1(r),R2(r), that correspond to the gravitational radii associated with the centre- of-mass energy in each beam below a distance r from the symmetry axis. This is a simpler, yet rich, problem, for several reasons: • The sources contain several parameters, and we can look for critical surfaces in their multidimensional space. • The CTS criterion is simple (see below). • Numerical results are coming in (see e.g. [14]). • The “bad” infrared-singular polarization is not produced. • Last but not least: PDEs become ODEs. Here we give a short account of the main results obtained in this problem [35]: • ACV versus CTS criterion—some general results: The strong-gravity regime: towards the large-angle/collapse phase? 409

1. The criterion for the existence of a CTS in the case of axisymmetric beam–beam collisions reads as follows [13]: if there exists an rc such that

2 R1(rc)R2(rc)=rc , (17.6.3) then we can construct a CTS, and therefore a black hole must form. In [35], the following theorem was proved: • Whenever the Kohlprath–Veneziano criterion (17.6.3) holds, the ACV field equations do not admit regular real solutions. Thus the CTS criterion implies the one based on the ACV equations, but not necessarily the other way around! 2. At the opposite end, a sufficient criterion for the existence ACV can be given:2 if the inequality R (r)R (r) 4r4 R2 2r2 2 1 2 ≤ 1 − log 1+ (17.6.4) R2 (3R2 +2r2)2 2r2 3R2 holds at all r, then the ACV equations admit regular, real solutions. • Three particular examples: 1. Particle scattering off a ring-shaped beam with all the energy concentrated at r = b (thus R2(r)=Rθ(r − b)). This case can be dealt with analytically, since it leads to a simple cubic equation that has real solutions if and only if √ 3 3 b2 > R2 ≡ b2 . (17.6.5) 2 S c

Thus we ﬁnd (b/RS)c ∼ 1.61, to be compared with the CTS prediction [13] (b/RS)c ≥ 1. The following two examples can be solved numerically with Mathematica: 2. The collision of two homogeneous beams of radius L. One ﬁnds R R CTS S ∼ 0.47, while S < 1.0 . (17.6.6) L c L c

3. Two beams having a Gaussian proﬁle of width L. Here our Lc turns out to be a factor of about 2.70 above the CTS lower bound. In conclusion, while the ACV-based results are never in contradiction—and even in qualitative agreement with—those based on the CTS criterion, there is typically a factor of 2–3 discrepancy between them. This could be due either to our rough approximations or to the CTS criteria being too loose (or to a combination of both). An amusing coincidence appears to point in this latter direction. In an interesting paper, Choptuik and Pretorius [14] analysed a “similar” situation numerically (the relativistic central collision of two solitons of ﬁxed mass and transverse size). They found black hole formation to occur at a critical Lorentz boost parameter γc (i.e.

2 P.-L. Lions, private communication. 410 High-energy collisions of particles, strings, and branes basically Rc) that is a factor 2–3 below the naive CTS value (but still in the relativistic regime where a connection to our process is possibly justified). The conclusions that can be drawn on string–string collisions in the strong-gravity regime can be summarized by saying that the above results are encouraging (even better than one could have expected), but also that real control over the different approximations is lacking, in particular on the freezing of longitudinal dynamics. This is probably at the origin of some puzzles we find3 in connection with gravitational radiation at b R. It seems that the fraction of incoming energy getting lost in gravitational radiation becomes O(1) already in the small-angle regime. Unfortu- nately, this is a regime in which there appears to be no reliable general relativistic calculation. An even bigger problem is an apparent violation of unitarity in this regime. Since the solution is no longer real below bc, a new elastic-unitarity deficit appears, which, unlike the previous ones (related to the opening of identified inelastic channels), has no simple physical interpretation.4 Recent work [36] suggests that, perhaps, one should not use regular complex solutions below bc, but rather stick to the reality of the solution, abandoning the constraint of regularity at r = 0. As a consequence, the action computed on the solution blows up (because of the above singularity) for b

17.7 High-energy string–brane collisions: an easier problem? In order to make some progress on the small-b regime, we have recently turned our attention to a hopefully easier problem: the high-energy collision of a light closed string off a stack of Dp-branes [37]. The brane configuration was chosen to be the simplest possible: N infinitely extended parallel branes spanning p + 1 directions inside the ambient (9 + 1)-dimensional spacetime. By playing with the residual (p+1)-dimensional Lorentz invariance, we can always go to a frame in which the closed string moves in a hyperplane perpendicular to the brane system and impinges on it, carrying an energy E, at an impact parameter b. The brane system conserves energy, but can absorb a transverse momentum q. Let us start with some general comments: As in the case of string–string collisions, we are not assuming any metric: calculations are done in flat spacetime (the N Dp-branes being introduced via the boundary state formalism [38]). There are, also in this case, three relevant scales in the problem:

• the impact parameter b related to the (orbital) angular momentum J = bE of the incoming string, with J>> (justifying a semiclassical treatment); • the scale Rp of the (expected) emerging geometry (see below); • the string length ls.

3 M. Ciafaloni and G. Veneziano, unpublished. 4 Taking it as an indication of black hole formation would seem too good to be true! High-energy string–brane collisions: an easier problem? 411

These three lengthscales lead to a phase diagram resembling that of Fig. 17.1, but with the collapse region replaced by one of capture of the closed string by the brane system. Since the stack of D-branes is inﬁnitely heavy, we expect the emergent metric to be well described by the classical metric produced by the branes themselves. This is known to be given by 2 1 α β i j ds = ηαβ dx dx + H(r)(δij dx dx ), (17.7.1) H(r) where the indices α,β,... run along the Dp-brane worldvolume, the indices i,j,... 2 i j indicate the transverse directions, r = δijx x ,and √ 7−p 7−p Rp 7−p gsN( 2πls) H(r)=1+ ,Rp = . (17.7.2) r (7 − p)Ω9−p

Note that the ratio Rp/ls can be tuned by varying gsN (with gs 1,N 1). At very high energy, gravity, in the form of graviton exchange, dominates. Yet we can neglect closed string loops below an Emax that goes to infinity with N. This problem is expected to be easier than the two-particle/string collisions problem. This is because the closed string, though very energetic, still acts as a probe of the geometry induced by the infinitely extended (hence infinitely heavy) brane system. We therefore expect a negligible backreaction on the geometry. At the disc (tree) and annulus (one-loop) level, an effective classical brane geometry emerges from the calculation under the reasonable assumption that the exponentiation persists at higher loop level. This is seen, again, through the classical deflection formulae that are satisfied at the saddle point of the b-integral. Furthermore, unlike in the case of string–string scattering, the agreement with general relativistic expectations can be checked explicitly at next-to-leading order in the deflection angle, and an extension to all orders appears to be within reach. Indeed, a nontrivial calculation of a subleading term in the annulus diagram gives the following next-to-leading expression for the deflection angle: − − √ Γ((8− p)/2) R 7 p 1 Γ ((15 − 2p)/2) R 2(7 p) Θ = π p + p + ... , p Γ((7− p)/2) b 2 Γ(6− p) b (17.7.3) in agreement with the expansion of the exact (though somewhat implicit) classical expression. Tidal effects can also be computed, and they turn out to be in complete agree- 2 ment with what one would obtain (to leading order in Rp/b and (ls/b) ) by quantizing the string in the D-brane metric. Indeed, one can justify, at high energy and leading 2 order in (ls/b) , a “Penrose pp-wave limit” for the D-brane metric and then study the nonlinear σ-model describing string fluctuations in that background. Amusingly, at this order the result is the same as the one we described for a string moving in the Aichelburg–Sexl metric—something that can be understood in terms of a Lor- entz contraction of the metric along the direction of the incoming energetic string. 412 High-energy collisions of particles, strings, and branes

Once more, these eﬀects become relevant below a critical impact parameter bt that is parametrically larger than the string scale:

− π √ Γ((8− p)/2) b8 p = α πs(7 − p) R7−p . (17.7.4) t 2 Γ((7− p)/2) p

The tidal excitation spectrum has been double-checked by considering the tree-level exclusive process [39] and the full microscopic understanding of the allowed transition is being clariﬁed [40]. It is also interesting to look at the imaginary part of the tree-level diagram in analogy with what we have already discussed for string–string collisions. In both cases, closed strings (corresponding to a reggeized graviton) are exchanged in the t-channel. However, in this latter case, the imaginary part is due to formation of heavy open strings in the s-channel. As one goes to impact parameters smaller than ls, this imaginary part becomes very relevant and should damp the elastic process in favour of copious production of many open strings living on the branes. Going to higher and higher energies, the average number of these open strings produced goes up so fast that the average mass of each open string produced will eventually go to zero. It is tempting to assume that the dynamics of these massless open strings will be described by a conformal ﬁeld theory living on the branes. For p = 3, where the metric near the boundary is AdS5, we may hope to make contact with the famous ADS/CFT correspondence [3] within a truly S-matrix approach.

17.8 Summary and outlook I hope to have passed the message that transplanckian energy collisions in flat spacetime are an ideal theoretical laboratory for studying several conceptual issues (cf. the information puzzle) arising from interplay of quantum mechanics and gravity within a fully consistent framework. Highlighting the main achievements so far: • At sufficiently large distances, we have been able to reproduce classical expectations (gravitational deflection, tidal effects) and extend them to the case of extended objects within a unitarity-preserving semiclassical description; • When string-size effects dominate, we found no evidence for black hole formation (again in agreement with classical expectations), but, instead, a rapid increase in the multiplicity and a corresponding softening of the final state resembling a smooth transition to a Hawking-evaporation-like regime. • In the regime of strong gravitational fields, our successes are still limited. Amusingly, a drastic approximation of the dynamics appears to reproduce at a semiquantitative level expectations based on CTS collapse criteria. • No firm conclusion can be drawn on this regime without more work. Some features of the present approach may not survive a more complete treatment (e.g. on longitudinal dynamics, which was frozen in the present approach). Many issues remain to be settled (in particular the saturation of unitarity), possibly because of our drastic approximations. References 413

• A general pattern seems to emerge whereby, at the quantum level, the transition between the dispersive and collapse phases is smoothed out by quantum mechanics. As some critical value of the impact parameter is approached, the nature of the final state changes smoothly from that characteristic of a dispersive state to one reminiscent of Hawking radiation (very high multiplicity and very low energies). • Transplanckian-energy string collisions off a stack of D-branes seem to offer a new tool to study all these issues within an easier setup. We have already seen how classical expectations from an effective metric are reproduced both through deflection formulae and from tidal excitations at leading and next-to-leading order. Generalization to higher (all) orders looks within reach. • Extension to the classical-capture regime should be possible, and will allow an understanding of how quantum coherence is preserved through the production of a coherent multi-open-string state living on the branes. • In the case of D3-branes, we hope that this gedanken experiment will shed some new light on the AdS/CFT correspondence within an S-matrix framework.

Acknowledgments I wish to thank the organizers of the beautiful Les Houches Summer School for the invitation and for the gentle pressure they exercised to have my lectures written up.

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