Gauge Anomalies in an Effective Field Theory*
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Quantum Field Theory*
Quantum Field Theory y Frank Wilczek Institute for Advanced Study, School of Natural Science, Olden Lane, Princeton, NJ 08540 I discuss the general principles underlying quantum eld theory, and attempt to identify its most profound consequences. The deep est of these consequences result from the in nite number of degrees of freedom invoked to implement lo cality.Imention a few of its most striking successes, b oth achieved and prosp ective. Possible limitation s of quantum eld theory are viewed in the light of its history. I. SURVEY Quantum eld theory is the framework in which the regnant theories of the electroweak and strong interactions, which together form the Standard Mo del, are formulated. Quantum electro dynamics (QED), b esides providing a com- plete foundation for atomic physics and chemistry, has supp orted calculations of physical quantities with unparalleled precision. The exp erimentally measured value of the magnetic dip ole moment of the muon, 11 (g 2) = 233 184 600 (1680) 10 ; (1) exp: for example, should b e compared with the theoretical prediction 11 (g 2) = 233 183 478 (308) 10 : (2) theor: In quantum chromo dynamics (QCD) we cannot, for the forseeable future, aspire to to comparable accuracy.Yet QCD provides di erent, and at least equally impressive, evidence for the validity of the basic principles of quantum eld theory. Indeed, b ecause in QCD the interactions are stronger, QCD manifests a wider variety of phenomena characteristic of quantum eld theory. These include esp ecially running of the e ective coupling with distance or energy scale and the phenomenon of con nement. -
Effective Field Theories, Reductionism and Scientific Explanation Stephan
To appear in: Studies in History and Philosophy of Modern Physics Effective Field Theories, Reductionism and Scientific Explanation Stephan Hartmann∗ Abstract Effective field theories have been a very popular tool in quantum physics for almost two decades. And there are good reasons for this. I will argue that effec- tive field theories share many of the advantages of both fundamental theories and phenomenological models, while avoiding their respective shortcomings. They are, for example, flexible enough to cover a wide range of phenomena, and concrete enough to provide a detailed story of the specific mechanisms at work at a given energy scale. So will all of physics eventually converge on effective field theories? This paper argues that good scientific research can be characterised by a fruitful interaction between fundamental theories, phenomenological models and effective field theories. All of them have their appropriate functions in the research process, and all of them are indispens- able. They complement each other and hang together in a coherent way which I shall characterise in some detail. To illustrate all this I will present a case study from nuclear and particle physics. The resulting view about scientific theorising is inherently pluralistic, and has implications for the debates about reductionism and scientific explanation. Keywords: Effective Field Theory; Quantum Field Theory; Renormalisation; Reductionism; Explanation; Pluralism. ∗Center for Philosophy of Science, University of Pittsburgh, 817 Cathedral of Learning, Pitts- burgh, PA 15260, USA (e-mail: [email protected]) (correspondence address); and Sektion Physik, Universit¨at M¨unchen, Theresienstr. 37, 80333 M¨unchen, Germany. 1 1 Introduction There is little doubt that effective field theories are nowadays a very popular tool in quantum physics. -
Casimir Effect on the Lattice: U (1) Gauge Theory in Two Spatial Dimensions
Casimir effect on the lattice: U(1) gauge theory in two spatial dimensions M. N. Chernodub,1, 2, 3 V. A. Goy,4, 2 and A. V. Molochkov2 1Laboratoire de Math´ematiques et Physique Th´eoriqueUMR 7350, Universit´ede Tours, 37200 France 2Soft Matter Physics Laboratory, Far Eastern Federal University, Sukhanova 8, Vladivostok, 690950, Russia 3Department of Physics and Astronomy, University of Gent, Krijgslaan 281, S9, B-9000 Gent, Belgium 4School of Natural Sciences, Far Eastern Federal University, Sukhanova 8, Vladivostok, 690950, Russia (Dated: September 8, 2016) We propose a general numerical method to study the Casimir effect in lattice gauge theories. We illustrate the method by calculating the energy density of zero-point fluctuations around two parallel wires of finite static permittivity in Abelian gauge theory in two spatial dimensions. We discuss various subtle issues related to the lattice formulation of the problem and show how they can successfully be resolved. Finally, we calculate the Casimir potential between the wires of a fixed permittivity, extrapolate our results to the limit of ideally conducting wires and demonstrate excellent agreement with a known theoretical result. I. INTRODUCTION lattice discretization) described by spatially-anisotropic and space-dependent static permittivities "(x) and per- meabilities µ(x) at zero and finite temperature in theories The influence of the physical objects on zero-point with various gauge groups. (vacuum) fluctuations is generally known as the Casimir The structure of the paper is as follows. In Sect. II effect [1{3]. The simplest example of the Casimir effect is we review the implementation of the Casimir boundary a modification of the vacuum energy of electromagnetic conditions for ideal conductors and propose its natural field by closely-spaced and perfectly-conducting parallel counterpart in the lattice gauge theory. -
On the Trace Anomaly of a Weyl Fermion in a Gauge Background
On the trace anomaly of a Weyl fermion in a gauge background Fiorenzo Bastianelli,a;b;c Matteo Broccoli,a;c aDipartimento di Fisica ed Astronomia, Universit`adi Bologna, via Irnerio 46, I-40126 Bologna, Italy bINFN, Sezione di Bologna, via Irnerio 46, I-40126 Bologna, Italy cMax-Planck-Institut f¨ur Gravitationsphysik (Albert-Einstein-Institut) Am M¨uhlenberg 1, D-14476 Golm, Germany E-mail: [email protected], [email protected] Abstract: We study the trace anomaly of a Weyl fermion in an abelian gauge background. Although the presence of the chiral anomaly implies a breakdown of gauge invariance, we find that the trace anomaly can be cast in a gauge invariant form. In particular, we find that it does not contain any odd-parity contribution proportional to the Chern-Pontryagin density, which would be allowed by the con- sistency conditions. We perform our calculations using Pauli-Villars regularization and heat kernel methods. The issue is analogous to the one recently discussed in the literature about the trace anomaly of a Weyl fermion in curved backgrounds. Keywords: Anomalies in Field and String Theories, Conformal Field Theory arXiv:1808.03489v2 [hep-th] 7 May 2019 Contents 1 Introduction1 2 Actions and symmetries3 2.1 The Weyl fermion3 2.1.1 Mass terms5 2.2 The Dirac fermion8 2.2.1 Mass terms9 3 Regulators and consistent anomalies 10 4 Anomalies 13 4.1 Chiral and trace anomalies of a Weyl fermion 13 4.1.1 PV regularization with Majorana mass 14 4.1.2 PV regularization with Dirac mass 15 4.2 Chiral and trace anomalies of a Dirac fermion 16 4.2.1 PV regularization with Dirac mass 16 4.2.2 PV regularization with Majorana mass 17 5 Conclusions 18 A Conventions 19 B The heat kernel 21 C Sample calculations 22 1 Introduction In this paper we study the trace anomaly of a chiral fermion coupled to an abelian gauge field in four dimensions. -
On the Limits of Effective Quantum Field Theory
RUNHETC-2019-15 On the Limits of Effective Quantum Field Theory: Eternal Inflation, Landscapes, and Other Mythical Beasts Tom Banks Department of Physics and NHETC Rutgers University, Piscataway, NJ 08854 E-mail: [email protected] Abstract We recapitulate multiple arguments that Eternal Inflation and the String Landscape are actually part of the Swampland: ideas in Effective Quantum Field Theory that do not have a counterpart in genuine models of Quantum Gravity. 1 Introduction Most of the arguments and results in this paper are old, dating back a decade, and very little of what is written here has not been published previously, or presented in talks. I was motivated to write this note after spending two weeks at the Vacuum Energy and Electroweak Scale workshop at KITP in Santa Barbara. There I found a whole new generation of effective field theorists recycling tired ideas from the 1980s about the use of effective field theory in gravitational contexts. These were ideas that I once believed in, but since the beginning of the 21st century my work in string theory and the dynamics of black holes, convinced me that they arXiv:1910.12817v2 [hep-th] 6 Nov 2019 were wrong. I wrote and lectured about this extensively in the first decade of the century, but apparently those arguments have not been accepted, and effective field theorists have concluded that the main lesson from string theory is that there is a vast landscape of meta-stable states in the theory of quantum gravity, connected by tunneling transitions in the manner envisioned by effective field theorists in the 1980s. -
Chiral Gauge Theories Revisited
CERN-TH/2001-031 Chiral gauge theories revisited Lectures given at the International School of Subnuclear Physics Erice, 27 August { 5 September 2000 Martin L¨uscher ∗ CERN, Theory Division CH-1211 Geneva 23, Switzerland Contents 1. Introduction 2. Chiral gauge theories & the gauge anomaly 3. The regularization problem 4. Weyl fermions from 4+1 dimensions 5. The Ginsparg–Wilson relation 6. Gauge-invariant lattice regularization of anomaly-free theories 1. Introduction A characteristic feature of the electroweak interactions is that the left- and right- handed components of the fermion fields do not couple to the gauge fields in the same way. The term chiral gauge theory is reserved for field theories of this type, while all other gauge theories (such as QCD) are referred to as vector-like, since the gauge fields only couple to vector currents in this case. At first sight the difference appears to be mathematically insignificant, but it turns out that in many respects chiral ∗ On leave from Deutsches Elektronen-Synchrotron DESY, D-22603 Hamburg, Germany 1 νµ ν e µ W W γ e Fig. 1. Feynman diagram contributing to the muon decay at two-loop order of the electroweak interactions. The triangular subdiagram in this example is potentially anomalous and must be treated with care to ensure that gauge invariance is preserved. gauge theories are much more complicated. Their definition beyond the classical level, for example, is already highly non-trivial and it is in general extremely difficult to obtain any solid information about their non-perturbative properties. 1.1 Anomalies Most of the peculiarities in chiral gauge theories are related to the fact that the gauge symmetry tends to be violated by quantum effects. -
Renormalization and Effective Field Theory
Mathematical Surveys and Monographs Volume 170 Renormalization and Effective Field Theory Kevin Costello American Mathematical Society surv-170-costello-cov.indd 1 1/28/11 8:15 AM http://dx.doi.org/10.1090/surv/170 Renormalization and Effective Field Theory Mathematical Surveys and Monographs Volume 170 Renormalization and Effective Field Theory Kevin Costello American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair MichaelA.Singer Eric M. Friedlander Benjamin Sudakov MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 81T13, 81T15, 81T17, 81T18, 81T20, 81T70. The author was partially supported by NSF grant 0706954 and an Alfred P. Sloan Fellowship. For additional information and updates on this book, visit www.ams.org/bookpages/surv-170 Library of Congress Cataloging-in-Publication Data Costello, Kevin. Renormalization and effective fieldtheory/KevinCostello. p. cm. — (Mathematical surveys and monographs ; v. 170) Includes bibliographical references. ISBN 978-0-8218-5288-0 (alk. paper) 1. Renormalization (Physics) 2. Quantum field theory. I. Title. QC174.17.R46C67 2011 530.143—dc22 2010047463 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. -
Applications of Effective Field Theory Techniques to Jet Physics by Simon
Applications of Effective Field Theory Techniques to Jet Physics by Simon M. Freedman A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Physics University of Toronto c Copyright 2015 by Simon M. Freedman Abstract Applications of Effective Field Theory Techniques to Jet Physics Simon M. Freedman Doctor of Philosophy Graduate Department of Physics University of Toronto 2015 In this thesis we study jet production at large energies from leptonic collisions. We use the framework of effective theories of Quantum Chromodynamics (QCD) to examine the properties of jets and systematically improve calculations. We first develop a new formulation of soft-collinear effective theory (SCET), the appropriate effective theory for jets. In this formulation, soft and collinear degrees of freedom are described using QCD fields that interact with each other through light-like Wilson lines in external cur- rents. This formulation gives a more intuitive picture of jet processes than the traditional formulation of SCET. In particular, we show how the decoupling of soft and collinear degrees of freedom that occurs at leading order in power counting is explicit to next-to-leading order and likely beyond. We then use this formulation to write the thrust rate in a factorized form at next-to-leading order in the thrust parameter. The rate involves an incomplete sum over final states due to phase space cuts that is enforced by a measurement operator. Subleading corrections require matching onto not only the next-to-next-to leading order SCET operators, but also matching onto subleading measurement operators. -
Lectures on Effective Field Theories
Adam Falkowski Lectures on Effective Field Theories Version of September 21, 2020 Abstract Effective field theories (EFTs) are widely used in particle physics and well beyond. The basic idea is to approximate a physical system by integrating out the degrees of freedom that are not relevant in a given experimental setting. These are traded for a set of effective interactions between the remaining degrees of freedom. In these lectures I review the concept, techniques, and applications of the EFT framework. The 1st lecture gives an overview of the EFT landscape. I will review the philosophy, basic techniques, and applications of EFTs. A few prominent examples, such as the Fermi theory, the Euler-Heisenberg Lagrangian, and the SMEFT, are presented to illustrate some salient features of the EFT. In the 2nd lecture I discuss quantitatively the procedure of integrating out heavy particles in a quantum field theory. An explicit example of the tree- and one-loop level matching between an EFT and its UV completion will be discussed in all gory details. The 3rd lecture is devoted to path integral methods of calculating effective Lagrangians. Finally, in the 4th lecture I discuss the EFT approach to constraining new physics beyond the Standard Model (SM). To this end, the so- called SM EFT is introduced, where the SM Lagrangian is extended by a set of non- renormalizable interactions representing the effects of new heavy particles. 1 Contents 1 Illustrated philosophy of EFT3 1.1 Introduction..................................3 1.2 Scaling and power counting.........................6 1.3 Illustration #1: Fermi theory........................8 1.4 Illustration #2: Euler-Heisenberg Lagrangian.............. -
Gauge Theories in Particle Physics
GRADUATE STUDENT SERIES IN PHYSICS Series Editor: Professor Douglas F Brewer, MA, DPhil Emeritus Professor of Experimental Physics, University of Sussex GAUGE THEORIES IN PARTICLE PHYSICS A PRACTICAL INTRODUCTION THIRD EDITION Volume 1 From Relativistic Quantum Mechanics to QED IAN J R AITCHISON Department of Physics University of Oxford ANTHONY J G HEY Department of Electronics and Computer Science University of Southampton CONTENTS Preface to the Third Edition xiii PART 1 INTRODUCTORY SURVEY, ELECTROMAGNETISM AS A GAUGE THEORY, AND RELATIVISTIC QUANTUM MECHANICS 1 Quarks and Leptons 3 1.1 The Standard Model 3 1.2 Levels of structure: from atoms to quarks 5 1.2.1 Atoms ---> nucleus 5 1.2.2 Nuclei -+ nucleons 8 1.2.3 . Nucleons ---> quarks 12 1.3 The generations and flavours of quarks and leptons 18 1.3.1 Lepton flavour 18 1.3.2 Quark flavour 21 2 Partiele Interactions in the Standard Model 28 2.1 Introduction 28 2.2 The Yukawa theory of force as virtual quantum exchange 30 2.3 The one-quantum exchange amplitude 34 2.4 Electromagnetic interactions 35 2.5 Weak interactions 37 2.6 Strong interactions 40 2.7 Gravitational interactions 44 2.8 Summary 48 Problems 49 3 Eleetrornagnetism as a Gauge Theory 53 3.1 Introduction 53 3.2 The Maxwell equations: current conservation 54 3.3 The Maxwell equations: Lorentz covariance and gauge invariance 56 3.4 Gauge invariance (and covariance) in quantum mechanics 60 3.5 The argument reversed: the gauge principle 63 3.6 Comments an the gauge principle in electromagnetism 67 Problems 73 viii CONTENTS 4 -
Nonperturbative Definition of the Standard Models
PHYSICAL REVIEW RESEARCH 2, 023356 (2020) Nonperturbative definition of the standard models Juven Wang 1,2,* and Xiao-Gang Wen3 1School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey 08540, USA 2Center of Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts 02138, USA 3Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 14 June 2019; accepted 21 May 2020; published 17 June 2020) The standard models contain chiral fermions coupled to gauge theories. It has been a longstanding problem to give such gauged chiral fermion theories a quantum nonperturbative definition. By classification of quantum anomalies (including perturbative local anomalies and nonperturbative global anomalies) and symmetric interacting invertible topological orders via a mathematical cobordism theorem for differentiable and triangulable manifolds, and by the existence of a symmetric gapped boundary (designed for the mirror sector) on the trivial symmetric invertible topological orders, we propose that Spin(10) chiral fermion theories with Weyl fermions in 16-dimensional spinor representations can be defined on a 3 + 1D lattice without fermion doubling, and subsequently dynamically gauged to be a Spin(10) chiral gauge theory. As a result, the standard models from the 16n-chiral fermion SO(10) grand unification can be defined nonperturbatively via a 3 + 1D local lattice model of bosons or qubits. Furthermore, we propose that standard models from the 15n-chiral fermion -
U(1) Gauge Extensions of the Standard Model
U(1) Gauge Extensions of the Standard Model Ernest Ma Physics and Astronomy Department University of California Riverside, CA 92521, USA U(1) Gauge Extensions of the Standard Model (int08) back to start 1 Contents • Anomaly Freedom of the Standard Model • B − L • Le − Lµ and B − 3Lτ • U(1)Σ • Supersymmetric U(1)X • Some Remarks U(1) Gauge Extensions of the Standard Model (int08) back to start 2 Anomaly Freedom of the Standard Model Gauge Group: SU(3)C × SU((2)L × U(1)Y . Consider the fermion multiplets: (u, d)L ∼ (3, 2, n1), uR ∼ (3, 1, n2), dR ∼ (3, 1, n3), (ν, e)L ∼ (1, 2, n4), eR ∼ (1, 1, n5). Bouchiat/Iliopolous/Meyer(1972): The SM with n1 = 1/6, n2 = 2/3, n3 = −1/3, n4 = −1/2, n5 = −1, is free of axial-vector anomalies, i.e. 2 [SU(3)] U(1)Y : 2n1 − n2 − n3 = 0. 2 [SU(2)] U(1)Y : 3n1 + n4 = 0. 3 3 3 3 3 3 [U(1)Y ] : 6n1 − 3n2 − 3n3 + 2n4 − n5 = 0. U(1) Gauge Extensions of the Standard Model (int08) back to start 3 It is also free of the mixed gravitational-gauge anomaly, U(1)Y : 6n1 − 3n2 − 3n3 + 2n4 − n5 = 0. Geng/Marshak(1989), Minahan/Ramond/Warner(1990) : Above 4 equations ⇒ n1(4n1 − n2)(2n1 + n2) = 0. n2 = 4n1 ⇒ SM; n2 = −2n1 ⇒ SM (uR ↔ dR); n1 = 0 ⇒ n4 = n5 = n2 + n3 = 0. Here eR ∼ (1, 1, 0) may be dropped. (u, d)L, (ν, e)L have charges (1/2, −1/2) and (uR, dR) have charges (n2, −n2).