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An Introduction to Effective Field Theory An Introduction to Effective Field Theory Thinking Effectively About Hierarchies of Scale c C.P. BURGESS i Preface It is an everyday fact of life that Nature comes to us with a variety of scales: from quarks, nuclei and atoms through planets, stars and galaxies up to the overall Universal large-scale structure. Science progresses because we can understand each of these on its own terms, and need not understand all scales at once. This is possible because of a basic fact of Nature: most of the details of small distance physics are irrelevant for the description of longer-distance phenomena. Our description of Nature’s laws use quantum field theories, which share this property that short distances mostly decouple from larger ones. E↵ective Field Theories (EFTs) are the tools developed over the years to show why it does. These tools have immense practical value: knowing which scales are important and why the rest decouple allows hierarchies of scale to be used to simplify the description of many systems. This book provides an introduction to these tools, and to emphasize their great generality illustrates them using applications from many parts of physics: relativistic and nonrelativistic; few- body and many-body. The book is broadly appropriate for an introductory graduate course, though some topics could be done in an upper-level course for advanced undergraduates. It should interest physicists interested in learning these techniques for practical purposes as well as those who enjoy the beauty of the unified picture of physics that emerges. It is to emphasize this unity that a broad selection of applications is examined, although this also means no one topic is explored in as much depth as it deserves. The book’s goal is to engage the reader’s interest, but then to redirect to the appropriate literature for more details. To this end references in the main text are provided mostly just for the earliest papers (that I could find) on a given topic, with a broader – probably more useful – list of textbooks, reviews and other sources provided in the bibliography. There will be inevitable gems about which I am unaware or have forgotten to mention, and I apologize in advance to both their authors and to you the reader for their omission. An introductory understanding of quantum and classical field theory is assumed, for which an appendix provides a basic summary of the main features. To reconcile the needs of readers with di↵ering backgrounds — from complete newbies through to experts seek- ing applications outside their own areas — sections are included requiring di↵ering amounts of sophistication. The background material in the appendices is also meant to help smooth out the transitions between these di↵erent levels of difficulty. The various gradations of sophistication are flagged using the suits of playing cards: , , and in the titles of the chapter sections. The flag indicates good value and } ~ | } labels sections that carry key ideas that should not be missed by any student of e↵ective theories. flags sections containing material common to most quantum field theory classes, ~ whose familiarity may warm a reader’s heart but can be skipped by aficianados in a hurry. The symbol indicates a section which may require a bit more digging for new students to digest, but which is reasonably self-contained and worth a bit of spadework. Finally, ii readers wishing to beat their heads against sections containing more challenging topics should seek out those marked with . | The lion’s share of the book is aimed at applications, since this most e↵ectively brings out both the utility and the unity of the approach. The examples also provide a pedagogical framework for introducing some specific techniques. Since many of these applications are independent of one another, a course can be built by starting with Part I’s introductory material and picking and choosing amongst the later sections that are of most interest. Acknowledgements This book draws heavily on the insight and goodwill of many people: in particular my teachers of quantum and classical field theory – Bryce De Witt, Willy Fischler, Joe Polchin- ski and especially Steven Weinberg – who shaped the way I think about this subject. Special thanks go to Fernando Quevedo for a life-long collaboration on these subjects and his comments over the years on many of the topics discussed herein. I owe a debt to Patrick Labelle, Sung-Sik Lee, Alexander Penin and Ira Rothstein for clarifying issues to do with nonrelativistic EFTs; to John Donoghue for many insights on gravitational physics; to Thomas Becher for catching errors in early versions of the text; to Jim Cline for a better understanding of the practical implications of Goldstone boson infrared e↵ects; to Claudia de Rham, Luis Lehner, Adam Solomon, Andrew Tolley and Mark Trodden for helping better understand applications to time-dependent systems; to Subodh Patil and Michael Horbatsch for helping unravel multiple scales in scalar cosmol- ogy; to Mike Trott for help understanding the subtleties of power-counting and SMEFT; to Peter Adshead, Richard Holman, Greg Kaplanek, Louis Leblond, Jerome Martin, Sarah Shandera, Gianmassimo Tasinato, Vincent Vennin and Richard Woodard for understand- ing EFTs in de Sitter space and their relation to open systems, and to Ross Diener, Peter Hayman, Doug Hoover, Leo van Nierop, Duncan Odell, Ryan Plestid, Markus Rummel, Matt Williams, and Laszlo Zalavari for helping clarify how EFTs work for massive first- quantized sources. Collaborators and students too numerous to name have continued to help deepen my understanding in the course of many conversations about physics. CERN, ICTP, KITP Santa Barbara and the Institute Henri Poincare´ have at various times provided me with pleasant environs in which to focus undivided time on writing, and with stimulating discussions when taking a break from it. The book would not have been fin- ished without them. The same is true of McMaster University and Perimeter Institute, whose flexible work environments allowed me to take on this project in the first place. Heaven holds a special place for Simon Capelin and his fellow editors, both for encour- aging the development of this book and for their enormous patience in awaiting it. Most importantly, I am grateful to my late parents for their gift of an early interest in science, and to my immediate family (Caroline, Andrew, Ian, Matthew and Michael) for their continuing support and tolerance of time taken from them for physics. Contents Preface page i Acknowledgements ii List of illustrations x List of tables xvi Part I Theoretical framework 1 1 Decoupling and hierarchies of scale 4 1.1 An illustrative toy model } 5 1.1.1 Semiclassical spectrum 5 1.1.2 Scattering 6 1.1.3 The low-energy limit 8 1.2 The simplicity of the low-energy limit } 8 1.2.1 Low-energy e↵ective actions 9 1.2.2 Why it works 10 1.2.3 Symmetries: linear vs nonlinear realization 12 1.3 Summary 15 Exercises 15 2 Effective actions 17 2.1 Generating functionals - a review ~ 17 2.1.1 Connected correlations 20 2.1.2 The 1PI (or quantum) action 21 2.2 The high-energy/low-energy split } 25 2.2.1 Projecting onto low-energy states 25 2.2.2 Generators of low-energy correlations 27 2.2.3 The 1LPI action 28 2.3 The Wilson action } 32 2.3.1 Definitions 32 2.4 Dimensional analysis and scaling } 38 2.4.1 Dimensional analysis 39 2.4.2 Scaling 42 2.5 Redundant interactions } 43 2.6 Summary 47 Exercises 48 iii iv Contents 3 Power counting and matching 50 3.1 Loops, cuto↵s and the exact RG 51 3.1.1 Low-energy amplitudes 52 3.1.2 Power counting using cuto↵s 54 3.1.3 The exact renormalization group 58 3.1.4 Rationale behind renormalization } 63 3.2 Power counting and dimensional regularization } 64 3.2.1 EFTs in dimensional regularization 64 3.2.2 Matching vs integrating out 67 3.2.3 Power counting using dimensional regularization 70 3.2.4 Power-counting with fermions 73 3.3 The Big Picture } 75 3.3.1 Low-energy theorems 75 3.3.2 The e↵ective-action logic } 76 3.4 Summary 78 Exercises 79 4 Symmetries 81 4.1 Symmetries in field theory ~ 81 4.1.1 Unbroken continuous symmetries 83 4.1.2 Spontaneous symmetry breaking 86 4.2 Linear vs nonlinear realizations } 89 4.2.1 Linearly realized symmetries 90 4.2.2 Nonlinearly realized symmetries 93 4.2.3 Gauge symmetries 98 4.3 Anomaly matching 104 4.3.1 Anomalies~ 105 4.3.2 Anomalies and EFTs 108 4.4 Summary 112 Exercises 113 5 Boundaries 115 5.1 ‘Induced’ boundary conditions 116 5.2 The low-energy perspective 118 5.3 Dynamical boundary degrees of freedom 121 5.4 Summary 123 Exercises 124 6 Time dependent systems 126 6.1 Sample time-dependent backgrounds } 126 6.1.1 View from the EFT 128 6.2 EFTs and background solutions } 129 6.2.1 Adiabatic equivalence of EFT and full evolution 130 6.2.2 Initial data and higher-derivative instabilities | 132 v Contents 6.3 Fluctuations about evolving backgrounds 137 6.3.1 Symmetries in an evolving background 138 6.3.2 Counting Goldstone states and currents | 141 6.4 Summary 144 Exercises 145 Part II Relativistic applications 147 7 Conceptual issues (relativistic systems) 150 7.1 The Fermi theory of weak interactions } 150 7.1.1 Properties of the W boson 150 7.1.2 Weak decays 152 7.2 Quantum Electrodynamics 154 7.2.1 Integrating out the Electron 155 7.2.2 E m and Large Logs 161 e | 7.2.3 Muons and the Decoupling Subtraction scheme 164 7.2.4 Gauge/Goldstone equivalence theorems 166 7.3 Photons, gravitons and neutrinos 168 7.3.1 Renormalizable interactions } 169 7.3.2 Strength of nonrenormalizable interactions
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