Gauge Theory of Weak Decays

Cambridge University Press 978-1-107-03403-7 — Gauge Theories of Weak Decays Andrzej Buras Frontmatter More Information

This is the first advanced, systematic, and comprehensive look at weak decays in the framework of gauge theories. Included is a large spectrum of topics, both theoretical and experimental. In addition to explicit advanced calculations of Feynman diagrams and the study of renormalization group strong interaction effects in weak decays, the book is devoted to the Standard Model Effective Field Theory, dominating present phenomenology in this field, and to new physics models with the goal of searching for new particles and interactions through quantum fluctuations. This book will benefit theorists, experimental researchers, and PhD students working on flavor physics and weak decays as well as physicists interested in physics beyond the Standard Model. In its concern for the search for new phenomena at short-distance scales through the interplay between theory and experiment, this book constitutes a travel guide to physics far beyond the scales explored by the Large Hadron Collider at CERN.

Andrzej J. Buras is one of the most cited particle theorists in Europe and the most cited theoretical ﬂavor physicist worldwide. He has written extensively on weak decays through lecture notes and review articles. He has been awarded the Smoluchowski-Warburg Medal of German and Polish Physics Societies, a Senior Carl von Linde Fellowship at TUM-IAS, an Advanced ERC Grant, and the 2020 Max Planck Medal. He is an ordinary member of the Bavarian Academy of Sciences and foreign member of two Academies in Poland.

Gauge Theory of Weak Decays

The Standard Model and the Expedition to New Physics Summits

ANDRZEJ J. BURAS Technical University Munich

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www.cambridge.org Information on this title: www.cambridge.org/9781107034037 DOI: 10.1017/9781139524100 © Andrzej J. Buras 2020 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2020 Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Buras, Andrzej J. (Andrzej Jerzy), 1946– author. Title: Gauge theory of weak decays : the standard model and the expedition to new physics summits / Andrzej J. Buras. Description: [New York, New York] : [Cambridge University Press], [2020] | Includes bibliographical references and index. Identifiers: LCCN 2019040866 (print) | LCCN 2019040867 (ebook) | ISBN 9781107034037 (hardback) | ISBN 9781139524100 (epub) Subjects: LCSH: Weak interactions (Nuclear physics) | Gauge fields (Physics) | Standard model (Nuclear physics) | Particles (Nuclear physics)–Flavor. | Mathematical physics. Classification: LCC QC794.8.W4 B84 2020 (print) | LCC QC794.8.W4 (ebook) | DDC 539.7/544–dc23 LC record available at https://lccn.loc.gov/2019040866 LC ebook record available at https://lccn.loc.gov/2019040867 ISBN 978-1-107-03403-7 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

To Gurli, Robert, Karin and Allan, Franziska and Ute Freya, Falk, Elisabeth, Inga and Janosch

Contents

Preface page xiii Acknowledgments xvi List of Abbreviations xviii

Introduction 1 Grand View of the Standard Model 1 Grand View of New Physics 3 The Grand View of the Expedition and the Strategy 8 Introducing Main Players 17 How to Use This Book Optimally 20

Part I Basics of Gauge Theories 23

1 Fundamentals 25 1.1 Preliminaries 25 1.2 Lagrangians for Scalar Fields 26 1.3 First Encounter with Symmetries 27 1.4 Checking the Symmetries of Scalar Lagrangians 28 1.5 Promotion of a Global U(1) Symmetry to a Local U(1) Symmetry 29 1.6 A Closer Look at the U(1) Gauge Theory 30 1.7 Nonabelian Global Symmetries 32 1.8 Promotion of a Global Nonabelian Symmetry to a Local One 37 1.9 Lagrangians for Fermions 40 1.10 Spontaneous Symmetry Breakdown (SSB) 45 1.11 Higgs Mechanism 53

Part II The Standard Model 57

2 The Standard Model of Electroweak and Strong Interactions 59 2.1 Particle Content and Gauge Group of the SM 59 2.2 Short Overview: Lagrangian of the SM 60 2.3 Spontaneous Symmetry Breakdown in the SM 62 2.4 Gauge Boson Self-Interactions 65 2.5 Flavor Structure of the SM 66 2.6 Lepton Flavor Violation 79

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2.7 Quantumchromodynamics (QCD) 80 2.8 Final Remarks 84

Part III Weak Decays in the Standard Model 85

3 Weak Decays at Tree Level 87 3.1 Muon Decay 87 3.2 Leptonic Decays of Charged Mesons 90 3.3 Semileptonic Decays of Charged Mesons 94 3.4 The Determination of |Vcb| and |Vub | 96 3.5 Leptonic and Semileptonic Decays of Neutral Mesons 98 3.6 Nonleptonic Decays of Mesons 98 3.7 Summary and Motivation 101

4 Technology beyond Trees 103 4.1 Loop Calculations 103 4.2 Renormalization 116 4.3 Renormalization Group Equations 123

5 Short-Distance Structure of Weak Decays 130 5.1 Operator Product Expansion in Weak Decays 130 5.2 Current-Current Operators beyond Leading Order 149

6 Efective Hamiltonians for FCNC Processes 179 6.1 Overture: General View of FCNC Processes 179 6.2 Calculations of Basic One-Loop Functions 189 6.3 ∆F = 2 Transitions 206 6.4 The World of Penguins 212 6.5 B → Xsγ Decay 229 6.6 b → sℓ+ℓ− and d → sℓ+ℓ− Transitions 244 6.7 d → sνν¯ and b → sνν¯ Transitions 249

7 Nonperturbative Methods in Weak Decays 253 7.1 General View 253 7.2 Dual QCD Approach 254 7.3 Lattice QCD Results 274 7.4 QCD Factorization for Exclusive B Decays 277 7.5 Heavy Quark Effective Theory (HQET) and Heavy Quark Expansions (HQE) 281 7.6 Other Nonperturbative Methods 282

8 Particle-Antiparticle Mixing and CP Violation in the Standard Model 283 8.1 Particle-Antiparticle Mixing 283

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8.2 Bq Decays into CP Eigenstates 299 8.3 Classiﬁcation of CP Violation 304 8.4 Standard Analysis of the Unitarity Triangle (UT) 309 8.5 The Angles α, β,andγ from Bd,s Decays 317 8.6 B → πK Decays 325

9RareB and K Decays in the Standard Model 329 ∗ 9.1 B¯ → K¯ ℓ+ℓ− and B¯ → K¯ ℓ+ℓ− 329 + − + − + − 9.2 Bs,d → µ µ and Bs,d → τ τ , e e 347 + + 9.3 B → τ ντ 355 ∗ 9.4 B¯ → Dℓν¯l, B¯ → D ℓν¯l, Bc → J/ψℓν¯l,andΛb → Λcℓν¯l 357 + + 0 9.5 K → π νν¯ and KL → π νν¯ 367 ∗ 9.6 B → K νν¯, B → Kνν¯,andB → Xsνν¯ 375 + − 0 + − 9.7 KL,S → µ µ and KL → π ℓ ℓ 384

10 ε′/ε in the Standard Model 388 10.1 Preliminaries 388 10.2 Basic Formulas 390 10.3 Hadronic Matrix Elements 392 10.4 A Convenient Formula for ε′/ε in the SM 395 10.5 Numerical Analysis of ε′/ε 399

11 Charm Flavor Physics 402 11.1 Preliminaries 402 0 11.2 D0 − D¯ Mixing 402 11.3 CP Asymmetries in D Decays 404 11.4 Connection between D and K Physics 408

12 Status of Flavor Physics within the Standard Model 410 12.1 Successes 410 12.2 Summary of the Anomalies 410 12.3 Implications for the Wilson Coefﬁcients 412

Part IV Weak Decays beyond the Standard Model 415

13 First Steps beyond the Standard Model 417 13.1 Preliminaries 417 13.2 ∆F = 2 Transitions 418 13.3 ∆F = 1 Nonleptonic Operators 427

14 Standard Model Efective Field Theory 433 14.1 Basic Framework 433 14.2 Full Set of Dimension-6 Operators 437

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14.3 Rotations in the Flavor Space 439 14.4 Renormalization Group Equations 443 ℓ+ℓ− 14.5 SU(2)L Correlations between b → sνν¯ and b → s 450 14.6 General Procedure and Useful Results 453 14.7 ε′/ε beyond the SM 459

15 Simplest Extensions of the SM 466 15.1 Minimal Flavor Violation 466

15.2 2HDMMFV 478 15.3 Beyond MFV: Models with U(2)3 Symmetry 480 15.4 Beyond MFV: Z′ Boson 483 15.5 Beyond MFV: Z Boson with FCNC 505 15.6 Beyond MFV: Right-Handed W′ 513 15.7 Beyond MFV: Neutral Scalars and Pseudoscalars 515 15.8 Beyond MFV: Charged Scalar Exchanges 528 15.9 Beyond MFV: Colored Gauge Bosons and Scalars 529

16 Speciic Models 533 16.1 Preliminaries 533 16.2 331 Models 534 16.3 Vector-Like Quarks and Leptons 543 16.4 Leptoquark Models 556

17 Beyond Quark Flavor Physics 580 17.1 General View 580 17.2 Lepton Flavor Violation 583 17.3 Electric Dipole Moments (EDMs) 599

17.4 Anomalous Magnetic Moments (g − 2)µ,e 615 17.5 Neutrino Oscillations 621

18 Grand Summary of New Physics Models 622 18.1 Preliminaries 622 18.2 General Observations on B Physics Anomalies 624 18.3 Tree-Level Mediators 626 18.4 Kaon Physics 626 18.5 Generalities 627

19 Flavor Expedition to the Zeptouniverse 628 19.1 Preliminaries 628 19.2 Basic Requirements for a Successful Zeptouniverse Expedition 629 19.3 Classifying Correlations between Various Observables 630 19.4 DNA Charts 634 19.5 Can We Reach the Zeptouniverse with Rare K and Bs,d Decays? 639

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20 Summary and Shopping List 641

Appendix A Dirac Algebra, Spinors, Pauli and Gell-Mann Matrices 647 A.1 Dirac Algebra and Spinors 647 A.2 Pauli and Gell-Mann Matrices 648 A.3 Fierz Identities 649

Appendix B Feynman Rules of the Standard Model 651 B.1 Preliminaries 651 B.2 Gauge Boson Propagators 651 B.3 Fermion and Scalar Propagators 652 B.4 Fermion–Gauge Boson Couplings 652 B.5 Fermion–Goldstone (Higgs) Boson Couplings 652 B.6 Gauge Boson Self-Interactions 653 B.7 Gauge Boson–Goldstone (Higgs) Interactions 653

Appendix C Massive Loop Integrals 654 C.1 Integrals with Two Propagators 654 C.2 Integrals with Three Propagators 655 C.3 Integrals with Four Propagators 656 C.4 More Complicated Integrals 657

Appendix D Numerical Input 661

Appendix E Analytic Solutions to SMEFT RG Equations 663

References 666 Index 715

Preface

The present theory of elementary particles and of their interactions known under the name of the Standard Model (SM) [1–3] is a relativistic quantum field theory with a specific local gauge symmetry dictated by nature. It incorporates the theory of strong interactions (quantum chromodynamics, QCD) and the unified theory of electroweak interactions (quantum flavor dynamics, QFD). It is a theoretically adequate description of leptons, quarks and their fundamental interactions in accordance with the principles of unitarity, causality, Lorentz invariance, quantum mechanics, and gauge invariance. As such it combines the main achievements of the physics of the twentieth century and includes the Maxwell theory of electromagnetism formulated already in 1865. The dream of physicists for many centuries was to find a “world formula” by help of which all the phenomena surrounding us could be described and explained. In particle physics the first step toward such a formula is the Lagrangian of the SM, which summarizes the particle content of the present theory and describes the basic structure of particle interactions as we know them at present. As the SM is unable to explain all phenomena around us, other world formulas in the form of Lagrangians appear in the literature. They generally contain all particles and interactions of the SM, but in addition new particles and new interactions are present in these models. As of 2020, we do not know which of these models will turn out to be the next step toward our dream, but it is exciting that in the coming years we might know it. To a person, not familiar with the subject, these world formulas look like hieroglyphics that appear to us when we visit Egyptian pyramids. Yet, to trained particle physicists, similar to trained archaeologists, these formulas reveal a vast amount of information about the physics of elementary particles. In these formulas the basic dynamics of particle interactions in a given model are compactly encoded, and it is in principle only a matter of will and time to translate this information into the more common language and to derive testable predictions. In practice the derivation of physical predictions from the Lagrangian of the SM and of its extensions is often not an easy task. It requires generally certain mathematical skills and the knowledge of sophisticated field theoretical methods. Moreover on many occasions the tools that we presently have at our disposal are not yet powerful enough to allow for accurate predictions. Fortunately there exist also many quantities for which accurate calculations have been already made and can be already compared with experimental results. Such a comparison is in fact most exciting as it allows testing a given theory. But this is only possible if experimental results for a given quantity are also available, and this is not always the case. In fact, some phenomena take place very rarely, and this requires large and often expensive experiments to measure the probability for seeing them. xiii

xiv Preface

The first main goal of this book is to describe weak decays of mesons, the bound states of quarks and antiquarks and of leptons within the SM. Starting from the Lagrangian of this model we will derive, as far as it is possible, predictions for decay widths, branching ratios, and various observables that can be used to confront this theory with experimental data. Therefore, in addition to more technical, field theoretical aspects of the SM, we will also devote a large part of this book to phenomenological applications of the derived formulas and to the comparison of the SM predictions with experimental data. The physics of weak decays is a fascinating subject and constitutes an important part of the SM and particle physics in general. There are several reasons for this: • This sector probes in addition to weak interactions also electromagnetic and strong interactions at short- and long-distance scales. As such it involves the dominant part of the dynamics of the SM. • It contains most of the free parameters of the SM and consequently plays an important role in their determination. • The occurrence of a large class of processes that take place only as “loop effects” automatically tests the quantum structure of the theory. • The renormalization group effects known also from statistical physics play here an important role in view of the vast difference between the weak interaction O(100 GeV) and strong interaction O(1GeV) scales. • The nature of the violation of various symmetries such as charge conjugation (C), parity (P), time reversal (T), and CP can be studied very efficiently. • It is an ideal laboratory for nonperturbative techniques. • Very importantly, it is a “window” through virtual effects to very short distances, which may shed light on some outstanding questions in particle physics such as the origin of masses and the number of fermion generations as well as hierarchies in the strength of their interactions. In particular weak decays could help us to identify the origin of the matter-antimatter asymmetry observed in the universe, which is necessary for our existence and which is not understood within the SM. They could also shed some light on dark matter. The last point brings us to the second main goal of this book: the presentation of weak decays within the most popular extensions of the SM. In particular we will develop efficient methods that will allow us to distinguish the predictions of these extensions for weak decays from the SM ones. This may help us to identify new physics (NP) at very short-distance scales well beyond the reach of the Large Hadron Collider (LHC). But even if the LHC would discover NP in the coming years, a detailed study of the properties of discovered new particles and interactions cannot be made only through high-energy collisions but requires very strong involvement of low-energy processes like weak decays of mesons where these new phenomena manifest themselves through quantum effects. Similar to the SM, its extensions have also been the subject of intensive research in the last forty years. This implies that it is a challenge to describe adequately only the most important advances in this field in one book. Therefore as in any serious expedition, a strategy for reaching our goals is unavoidable. Before presenting this strategy and the related map of our expedition, it is necessary to present two Grand Views: the first one

xv Preface

dealing with the SM and the second one dealing with some aspects of NP. These grand views will be brief and superficial. They are only meant to help nonexperts to understand at least roughly the map of this book, which will be presented subsequently and most importantly to motivate any reader to join the author in this expedition. In the latter context let me remark that the failure of the LHC to discover any new particles until now resulted in some frustration in the particle physics community. Often at conferences the speakers show pictures of the Sahara, meaning that until a new very high energy collider is built, no new particles will be discovered. My view, as a flavor physicist, is much more optimistic. It is represented well by the photo on the book’s front cover. We are standing on the mainland representing the SM. There is an energy gap represented by the water, which we have to cross in order to reach eventually the NP summits in the far distance. They represent different possibilities for the grander theory. But in order to find out which summits will answer all our questions, we have to cross the glacier and all of its crevasses, which represent various sophisticated technologies in flavor physics necessary to conquer these summits. But the main reason for showing this photo is its stunning beauty similar to the beauty of flavor physics.

Acknowledgments

This book was written between the summer 2013 and the summer 2019. Initially, it was planned to be complete in 2015, but fortunately it did not happen as in the last four years many exciting things have happened in flavor physics, which allowed me to include them in this book. First of all there are four people whom I want to thank most for helping me reaching my goals. Jennifer Girrbach-Noe: We started writing this book together. Jennifer being much stronger in LATEX than me set up all the files, which I used until I completed this book. She also wrote the grand view of the Standard Model and also the grand view of new physics. Part II has been written fully by her, although at view places I added new material and made a few modifications in her writing. But this part of the book should be credited to her. She also produced several figures and read roughly 25 percent of the book, which was the status at the end of 2014. Unfortunately, in January 2015, Jennifer decided not to continue reseach in particle physics, so the rest of the book was written by me alone. Yet, these first steps made with her were very important, and foremost I want to thank her. As Jennifer was one of the most efficient collaborators among about 120 collaborators I had in my research and we wrote twenty papers within three and a half years together, I am sure the book would be better than it is now if she had continued this expedition with me. Robert Buras-Schnell: After Jennifer left, Robert was responsible for almost all figures in the book. Moreover, he was of great help in any LATEX issues. Without him the progress in writing this book would be very slow. Many thanks to Robert. Jason Aebischer: Jason was the only one who read almost the full book before it was sent to Cambridge University Press for printing. In particular he demonstrated Swiss precision in reading Chapter 14, where one can get easily lost in the indices present in the renormalization group equations of the SMEFT. He checked many equations in the full book, found misprints, and made suggestions that in my view improved the clarity of the text. Many thanks to Jason. Christoph Bobeth: The sections on leptoquarks and vectorlike quarks resulted from our intensive study of these models in the context of the SMEFT. Despite two joint publications on these models a significant fraction of these two sections was not published and is presented here for the first time. Moreover, Christoph and Jason helped in doing some numerics relevant for the book. But there are still several of my colleagues who helped me with advices on the literature, updates of our previous joined papers, and also updates of their own papers, which I xvi

xvii Acknowledgments

asked them to do in connection with my book. These are in alphabetic order: Monika Blanke, Gerhard Buchalla, Marcin Chrzaszcz, Vincenzo Cirigliano, Andreas Crivellin, Sebastien´ Descotes-Genon, Svjetlana Fajfer, Fulvia de Fazio, Jean-Marc Gerard,´ Martin Jung, Alexander Lenz, Emilie Passemar, Janusz Rosiek, Luca Silvestrini, David Straub, and Robert Szafron. In particular, I beneﬁted enormously from Robert’s expertise in

(g − 2)µ,e. This book was written entirely at the TUM Institute for Advanced Study. The fantastic atmosphere at this Institute, created by the directors Gerhard Abstreiter and Ernst Rank and by the wonderful, in many respects, IAS-Team, was very helpful in writing this book. Particular thanks go also to the Clusters of Excellence: Universe and Origins,for ﬁnancial support and Stephan Paul for being such a wonderful boss after my retirement in 2012. Many thanks to my secretary Elke Hutsteiner for help in preparing PowerPoint presentations in the last 20 years that were indirectly important for writing this book. Particular thanks go to Simon Capelin, Roisin Munnelly, Henry Cockburn, Sarah Lambert, and Dinesh S. Negi from Cambridge University Press for many advices and help during the writing of my book. Very special thanks go to Karen Slaght, an impressive copyeditor, who improved my English. Many thanks to Sapphire Duveau and in particular to Neena S. Maheen for the great help during the ﬁnal preparations of the book for printing and to the LATEX team for an impressive job while introducing corrections. Finally, I would like to thank my family, in particular my wife Gurli and my daughter Karin, for the great encouragement for completing this book during all these years and to Robert as already mentioned earlier. Special thanks go to my second son, Allan Buras, for providing the stunning photo on the front cover, which he took during one of his expeditions to the far north.

Abbreviations

Our conventions, notations, and useful general formulas are collected at the end of the book. Here we just list those abbreviations that are used frequently in our book.

2HDM = Two Higgs-Doublet Model ADM = Anomalous Dimension Matrix BSM = Beyond the Standard Model ChPT = Chiral Perturbation Theory CKM = Cabibbo-Kobayashi-Maskawa CLFV = Charged Lepton Flavor Violation CMFV = Constrained Minimal Flavor Violation CP = CP-invariance CPV = CP-invariance Violation DHP = Double Higgs-penguins DR = Dimensional Regularization DRED = Dimensional Reduction DQCD =DualQCD EDM = Electric Dipole Moment EFT = Effective Field Theory EWP = Electroweak Penguin EWSB = Electroweak Spontaneous Symmetry Breakdown FBP = Flavor Blind Phases FC = Flavor Changing FCNC = Flavor Changing Neutral Currents FLAG = Flavor Lattice Averaging Group FR = Feynman rules FSI = Final State Interactions GIM = Glashow-Iliopoulos-Maiani GUT = Grand Uniﬁed Theory HFLAV = Heavy Flavor Averages HP = Higgs-penguins HQE = Heavy Quark Expansion HV = ’t Hooft-Veltman I.B. = Isospin Breaking KG = Klein-Gordon LCSR = Light-Cone Sum Rules

xviii

xix List of Abbreviations

LD = Long Distance LEFT = Low Energy Effective Field Theory LFU = Lepton Flavor Universality LFUV = Lepton Flavor Universality Violation LFV = Lepton Flavor Violation LH = Left-handed LHC = Large Hadron Collider LHS = Left-handed Scenario LHT = Littlest Higgs Model with T-Parity LO = Leading Order LQCD = Lattice QCD LR = Left-Right LQ = Leptoquark MFV = Minimal Flavor Violation MS = Minimal Scheme MS = Modiﬁed Minimal Scheme MSSM = Minimal Supersymmetric Standard Model NDR = Naive Dimensional Regularization NL = Non-Leptonic NLO = Next to Leading Order NNLO = Next to Next Leading Order NP = New physics OPE = Operator Product Expansion PBE = Penguin-Box Expansion PDG= Particle Data Group PMNS = Pontecorvo-Maki-Nakagawa-Sakata QCD = Quantumchromodynamics QCDF = QCD Factorization QCDP = QCD Penguins QED = Quantumelectrodynamics QFT = Quantum Field Theory RG = Renormalization Group RH = Right-handed RHS = Right-handed Scenario RS = Renormalization Scheme SCET = Soft-collinear effective theory SD = Short Distance SL = Semi-leptonic SM = Standard Model SM4 = Standard Model with 4 Generations SMEFT = Standard Model Effective Field Theory SSB = Spontaneous Symmetry Breakdown SUSY = Supersymmetry

xx List of Abbreviations

UV = Ultraviolet UT = Unitarity Triangle UUT = Universal Unitarity Triangle VLL = Vectorlike Lepton VLQ = Vectorlike Quark WC = Wilson Coefﬁcient