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An Introduction to Effective Field TheoryThinking Effectively About Hierarchies of Scalec C.P. BURGESS

iPrefaceIt 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-scalestructure. 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 ofNature: most of the details of small distance physics are irrelevant for the description oflonger-distance phenomena.Our description of Nature’s laws use quantum field theories, which share this propertythat short distances mostly decouple from larger ones. E ective Field Theories (EFTs) arethe tools developed over the years to show why it does. These tools have immense practicalvalue: knowing which scales are important and why the rest decouple allows hierarchiesof scale to be used to simplify the description of many systems. This book provides anintroduction to these tools, and to emphasize their great generality illustrates them usingapplications from all parts of physics – relativistic and nonrelativistic, and few-body tomany-body.The book is broadly appropriate for an introductory graduate course, though some topicscould be done in an upper-level course for advanced undergraduates. It should interestphysicists interested in learning these techniques for practical purposes and to those whoenjoy the beauty of the unified picture of many areas of physics.An introductory understanding of quantum and classical field theory is assumed, forwhich an appendix provides a basic summary of the main features. To reconcile the needsof readers with di ering backgrounds — from complete newbies through to experts seeking applications outside their own areas — sections are included requiring di ering amountsof sophistication.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 andlabels sections that carry key ideas that should not be missed by any student of e ectivetheories. 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 studentsto digest, but which is reasonably self-contained and worth a bit of spadework. Finally,readers wishing to beat their heads against sections containing more challenging topicsshould seek out those marked with .The lion’s share of the book is aimed at applications, since this most e ectively bringsout both the utility and the unity of the approach. It also provides a pedagogical frameworkfor introducing the technical issues that arise in new situations, such as how the treatmentof relativistic systems di er from nonrelativistic ones, or how to handle dissipation orsystems with time-dependent backgrounds, or are not in their ground state in other ways.Since many of these applications are independent of one another, a course can be built bypicking and choosing amongst those that are of most interest to the reader.

iiAcknowledgementsThis book draws heavily on the insight and goodwill of many people: in particular myteachers of quantum and classical field theory – Bryce De Witt, Willy Fischler, Joe Polchinski 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 subjectsand his comments over the years on many of the topics discussed herein.I owe a debt to Alexander Penin and Ira Rothstein for clarifying issues to do with nonrelativistic EFTs; to John Donoghue for many insights on gravitational physics; to ThomasBecher 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 forhelping unravel multiple scales in scalar cosmology; to Mike Trott for help understandingthe subtleties of power-counting and SMEFT; to Peter Adshead, Richard Holman, Greg Kaplanek, Louis Leblond, Jerome Martin, Sarah Shandera, Gianmassimo Tasinato, VincentVennin and Richard Woodard for understanding EFTs in de Sitter space and their relationto open systems, and to Ross Diener, Peter Hayman, Doug Hoover, Leo van Nierop, RyanPlestid, Markus Rummel, Matt Williams, and Laszlo Zalavari for helping clarify how EFTswork for massive first-quantized sources.Collaborators and students too numerous to name have continued to help deepen myunderstanding in the course of many conversations about physics.CERN, ICTP and the Institute Henri Poincaré, which have at various times providedme with a friendly place in which to focus undivided time on writing, will forever have aspecial place in my heart for doing so. They are joined there by McMaster University andPerimeter Institute, whose flexible work environments allowed me to take on this projectin the first place.Heaven holds a special place for Simon Capelin and his fellow editors, both for encouraging the development of this book and for their enormous patience in awaiting itsdelivery.Most importantly, I am grateful to my late parents for their gift of an early interest inscience, and to my immediate family (Caroline, Andrew, Ian, Matthew and Michael) fortheir continuing support and tolerance of time taken from them for physics.

ContentsPrefaceAcknowledgementsList of illustrationsList of tablespage iiiixxivPart I Theoretical framework1 Decoupling and hierarchies of scale1.1}An illustrative toy model1.1.1 Semiclassical spectrum1.1.2 Scattering1.1.3 The low-energy limit1.2 The simplicity of the low-energy limit }1.2.1 Low-energy e ective actions1.2.2 Why it works1.2.3 Symmetries: linear vs nonlinear realization1.3 SummaryExercises2 Effective actions2.1 Generating functionals - a review2.1.1 Connected correlations2.1.2 The 1PI (or quantum) action 2.2 The high-energy/low-energy split }2.2.1 Projecting onto low-energy states2.2.2 Generators of low-energy correlations 2.2.3 The 1LPI action2.3 The Wilson action }2.3.1 Definitions2.4 Dimensional analysis and scaling }2.4.1 Dimensional analysis2.4.2 Scaling2.5 Redundant interactions }2.6 93333394043444849

Contentsiv3 Power counting and matchingLoops, cuto s and the exact RG 3.1.1 Low-energy amplitudes3.1.2 Power counting using cuto s3.1.3 The exact renormalization group3.1.4 Rationale behind renormalization }3.2 Power counting and dimensional regularization }3.2.1 EFTs in dimensional regularization3.2.2 Matching vs integrating out3.2.3 Power counting using dimensional regularization3.2.4 Power-counting with fermions3.3 The e ective-action logic }3.4 SummaryExercises3.14 SymmetriesSymmetries in field theory 4.1.1 Unbroken continuous symmetries4.1.2 Spontaneous symmetry breaking4.2 Linear vs nonlinear realizations }4.2.1 Linearly realized symmetries4.2.2 Nonlinearly realized symmetries4.2.3 Gauge symmetries4.3 Anomaly matching 4.3.1 Anomalies4.3.2 Anomalies and EFTs4.4 SummaryExercises4.15 Boundaries5.1 ‘Induced’ boundary conditions5.2 The low-energy perspective5.3 Dynamical boundary degrees of freedom5.4 SummaryExercises6 Time dependent systems6.16.26.3Sample time-dependent backgrounds }6.1.1 View from the EFTEFTs and background solutions }6.2.1 Adiabatic equivalence of EFT and full evolution6.2.2 Initial data and higher-derivative instabilities Fluctuations about evolving backgrounds 6.3.1 Symmetries in an evolving 27129134135

Contentsv6.3.2 Counting Goldstone states and currents 6.4 SummaryExercisesPart II Relativistic applications7 Conceptual issues7.1}The Fermi theory of weak interactions7.1.1 Properties of the W boson7.1.2 Weak decays7.2 Quantum Electrodynamics7.2.1 Integrating out the Electron7.2.2 Muons and the Decoupling Subtraction scheme 7.2.3 Gauge/Goldstone equivalence theorems7.3 Photons, gravitons and neutrinos7.3.1 Renormalizable interactions }7.3.2 Strength of nonrenormalizable interactions }7.3.3 Neutrino-photon interactions 7.4 Boundary e ects7.4.1 Surface polarization7.4.2 Vacuum polarization and Casimir energy 7.4.3 Boundary currents and quantum Hall systems 7.5 SummaryExercises8 QCD and chiral perturbation theory 8.1Quantum Chromodynamics8.1.1 Quarks and hadrons8.1.2 Asymptotic freedom8.1.3 Symmetries and their realizations8.2 Chiral perturbation theory8.2.1 Nonlinear realization8.2.2 Soft-pion theorems8.2.3 Including baryons8.2.4 Loops and logs8.3 191195198201203203

Contentsvi9 The Standard Model as an effective theory9.1Particle content and symmetries9.1.1 The Lagrangian9.1.2 Anomaly cancellation9.2 Nonrenormalizable interactions9.2.1 Dimension-five interactions9.2.2 Dimension-six interactions9.3 Naturalness issues9.3.1 Technical and ’t Hooft naturalness9.3.2 The electroweak hierarchy problem 9.3.3 The cosmological constant problem9.4 SummaryExercises10 General Relativity as an effective theory 10.1 Domain of semi-classical gravity10.2 Time-dependence and cosmology10.2.1 Semiclassical perturbation theory10.2.2 Slow-roll suppression10.3 Turtles all the way down?10.3.1 String theory10.3.2 Extra dimensions10.4 SummaryExercisesPart III Nonrelativistic Applications11 Conceptual issues11.1 Integrating out antiparticles11.2 Nonrelativistic scaling11.2.1 Spinless fields11.2.2 Spin-half fields11.3 Coupling to electromagnetic fields11.3.1 Scaling11.3.2 Power counting11.4 SummaryExercises12 Electrodynamics of non-relativistic particles12.1 Schrödinger from Wilson12.1.1 Leading powers of 1/m12.1.2 Matching12.1.3 Thomson 268271275276277277277279286

Contentsvii12.2 Multiple particle species12.2.1 Atoms and the Coulomb potential12.2.2 Dipole approximation12.2.3 HQET12.2.4 Particle-antiparticle systems12.3 Neutral systems12.3.1 Polarizability and Rayleigh scattering12.3.2 Multipole moments12.4 SummaryExercises13 First-quantized methods13.1 E ective theories for lumps13.1.1 Collective coordinates }13.1.2 Nonlinearly realized Poincaré symmetry 13.1.3 Including spin13.2 Point-particle EFTs: Electromagnetism13.2.1 Electromagnetic and gravitational couplings13.2.2 Boundary conditions I13.2.3 Thomson scattering revisited13.3 PPEFT and central forces13.3.1 Boundary conditions II13.3.2 Contact interaction13.3.3 Inverse-square potentials: Fall to the centre13.3.4 Nuclear e ects in atoms13.4 SummaryExercisesPart IV Many-Body Applications14 Goldstone bosons in nonrelativistic systems14.1 Magnons14.1.1 Antiferromagnetism14.1.2 Ferromagnetism14.2 Low-energy Superconductors14.2.1 Implications of the Goldstone mode14.2.2 Landau-Ginzburg theory14.3 Phonons14.3.1 Goldstone counting revisited14.3.2 E ective action14.3.3 Perfect fluids14.4 366367372376377382383383385388390390

Contentsviii15 Degenerate systems15.1 Fermi liquids15.1.1 EFT near a Fermi surface15.1.2 Irrelevance of fermion self-interactions15.1.3 Marginal interactions15.2 Superconductivity and fermion pairing15.2.1 Phonon scaling15.2.2 Phonon-Coulomb competition15.3 SummaryExercises16 EFTs and open systems16.1 Thermal fluids16.1.1 Statistical framework 16.1.2 Evolution through conservation16.2 Open systems16.2.1 Density matrices 16.2.2 Reduced time evolution}16.3 Mean fields and fluctuations16.3.1 The mean/fluctuation split}16.3.2 Neutrinos in matter16.3.3 Photons: mean-field evolution 16.3.4 Photons: scattering and fluctuations 16.4 Late times and perturbation theory16.4.1 Late-time resummation16.4.2 Master equations16.5 endix AConventions462Appendix BMomentum eigenstates and scattering476Appendix CQuantum Field Theory: a Cartoon486Appendix DFurther reading522531537ReferencesIndex

e shape of the toy model’s potential V( R , I ), showing its sombrero shapeand the circular line of minima at v.The tree graphs that dominate R I scattering. Solid (dotted) lines represent R ( I ), and ‘crossed’ graphs are those with external lines interchanged relative to those displayed.The tree graphs that dominate the I I scattering amplitude. Solid (dotted)lines represent R and I particles.A sampling of some leading perturbative contributions to the generating functional Z[J] expressed using eq. (2.11) as Feynman graphs. Solid lines arepropagators ( 1 ) and solid circles represent interactions that appear in S int .1-particle reducible and 1PI graphs are both shown as examples at two loopsand a disconnected graph is shown at four loops. The graphs shown use onlyquartic and cubic interactions in S int .The Feynman rule for the vertex coming from the linear term, S lin , in theexpansion of the action. The cross represents the sum S / 'a Ja .The tree graphs that dominate the (@µ @µ )2 (panel a) and the (@µ @µ )3 (panels b and c) e ective interactions. Solid lines represent propagators whiledotted lines denote external fields.One-loop graphs that contribute to the (@µ @µ )2 interaction in the Wilson and1LPI actions using the interactions of eqs. (1.24) and (1.25). Solid (dotted)lines represent (and ) fields. Graphs involving wave-function renormalizations of are not included in this list.The tree and one-loop graphs that contribute to the (@µ @µ )2 interaction inthe 1LPI action, using Feynman rules built from the Wilson action. All dottedlines represent particles, and the ‘crossed’ versions of (b) are not drawnexplicitly.The graph describing the insertion of a single e ective vertex with E externallines and no internal lines.Graphs illustrating the two e ects that occur when an internal line is contracted to a point, depending on whether or not the propagator connects distinct vertices (left two figures) or ties o a loop on a single vertex (right twofigures). In both cases a double line represents the di erentiated propagator.The two options respectively correspond to the terms [ S W , int / (p)][ S W , int / (and 2 S W , int / (p) ( p) appearing in the Wilson-Polchinski relation, eq. (3.25)of the text.One-loop graphs that contribute to the @µ @µ kinetic term in the Wilson and678202432353856p)]61

Illustrationsx4.14.36.17.17.27.37.47.57.67.71LPI actions using the interactions of eqs. (3.36) and (3.37). Solid (dotted)lines represent (and ) fields.A sketch of energy levels in the low-energy theory relative to the high-energyscale, M, and the relative splitting, v, within a global ‘symmetry’ multiplet.Three cases are pictured: panel (a) unbroken symmetry (with unsplit multiplets); panel (b) low-energy breaking (v M) and panel (c) high-energy M). Symmetries are linearly realized in cases (a) and (b)breaking (with v but not (c). If spontaneously broken, symmetries in case (c) are nonlinearlyrealized in the EFT below M. (If explicitly broken in case (c) there is littlesense in which the e ective theory has approximate symmetry at all.)The triangle graph that is responsible for anomalous symmetries (in fourspacetime dimensions). The dot represents the operator J µ and the externallines represent gauge bosons in the matrix element hgg J µ i, where i isthe ground state.A sketch of the adiabatic time-evolution for the energy, E(t) (solid line), ofa nominally low-energy state and the energy, M(t) (double line), for a representative UV state. The left panel shows level crossing where (modulo levelrepulsion) high- and low-energy states meet so the EFT description fails. Inthe right panel high-energy states evolve past a cuto , (dotted line), withoutlevel crossing (so EFT methods need not fail).The Feynman graph responsible for the decay ! e 3 1 at leading order inunitary gauge.The tree graph that generates the Fermi Lagrangian.The Feynman graph contributing the vacuum polarization. The circular linedenotes a virtual electron loop while the wavy lines represent external photonlines.The Feynman graph contributing the leading contribution to photon-photonscattering in the e ective theory for low-energy QED. The vertex representseither of the two dimension-eight interactions discussed in the text.The leading Feynman graphs in QED which generate the e ective four-photonoperators in the low energy theory. Straight (wavy) lines represent electrons(photons).Schematic of the energy scales and couplings responsible for the hierarchyof interactions among gravitons, photons and neutrinos. Here the blue ovalsrepresent the collection of particles at a given energy that experience renormalizable interactions with one another. Three such circles are drawn, forenergies at the electron mass, me , the W-boson mass, MW , and a hypothetical scale, Mg , for whatever theory (perhaps string theory) describes gravity atvery high energies.Feynman graphs giving neutrino-photon interactions in the Standard Model.Graph (a) (left panel): contributions that can be regarded as low-energy renormalizations of the tree-level weak interaction. Graph (b) (middle panel): contributions generating higher-dimension interactions when integrating out theW. Graph (c) (right panel): contributions obtained when integrating out the6888104142151152157161162170

Illustrationsxi7.87.97.107.118.19.19.29.3Z. Although not labelled explicitly, quarks can also contribute to the loopin panel (c). Similar graphs with more photon legs contribute to neutrino/nphoton interactions.Feynman graphs giving neutrino/single-photon interactions within the EFTbelow MW . Graph (a) (left panel): loop corrections to the tree-level Fermiinteraction. Graph (b) (middle panel): loop corrections to tree-level higherdimension e ective four-fermion/one-photon interactions. Graph (c) (rightpanel): loop-generated higher-dimension e ective two-fermion/one-photoninteractions. Similar graphs with more photon legs describe multiple-photoninteractions.Feynman graph showing how the light-by-light scattering box diagram appears in the 2 ! 3 neutrino-photon scattering problem. The dot representsthe tree-level Fermi coupling, though C and P invariance imply only the vector part need be used.Traces of longitudinal (or Ohmic) and Hall resistivity versus magnetic field,with plateaux appearing in the Hall plot. The Ohmic resistivity vanishes forthe same fields where the Hall resistivity shows the plateau behaviour. (Figuretaken from S. Girvin, Séminaire Poincaré 2 (2004) 53 – 74.)Cartoon of semiclassical Landau motion in a magnetic field, showing howorbits in the interior do not carry charge across a sample’s length while surface orbits can if they bounce repeatedly o the sample’s edge. Notice thatthe motion is chiral inasmuch as the circulation goes around the sample ina specific direction. This is a specific mechanism for the origin of surfacecurrents in quantum Hall systems, as are required on general grounds for thelow-energy EFT by anomaly matching.The Feynman graphs giving the dominant contributions to pion-pion scattering in the low-energy pion EFT. The first graph uses a vertex involving twoderivatives while the second involves the pion mass, but no derivatives.A example of a type of UV physics that can generate the dimension-fivelepton-violating operator in SMEFT.Graphs contributing to the Higgs mass in the extended UV theory. Solid (dotted) lin

An Introduction to Effective Field Theory Thinking Effectively About Hierarchies of Scale C.P. BURGESSc. 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 .

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