1 Loops and Renormalization
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Effective Quantum Field Theories Thomas Mannel Theoretical Physics I (Particle Physics) University of Siegen, Siegen, Germany
Generating Functionals Functional Integration Renormalization Introduction to Effective Quantum Field Theories Thomas Mannel Theoretical Physics I (Particle Physics) University of Siegen, Siegen, Germany 2nd Autumn School on High Energy Physics and Quantum Field Theory Yerevan, Armenia, 6-10 October, 2014 T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Overview Lecture 1: Basics of Quantum Field Theory Generating Functionals Functional Integration Perturbation Theory Renormalization Lecture 2: Effective Field Thoeries Effective Actions Effective Lagrangians Identifying relevant degrees of freedom Renormalization and Renormalization Group T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Lecture 3: Examples @ work From Standard Model to Fermi Theory From QCD to Heavy Quark Effective Theory From QCD to Chiral Perturbation Theory From New Physics to the Standard Model Lecture 4: Limitations: When Effective Field Theories become ineffective Dispersion theory and effective field theory Bound Systems of Quarks and anomalous thresholds When quarks are needed in QCD É. T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Lecture 1: Basics of Quantum Field Theory Thomas Mannel Theoretische Physik I, Universität Siegen f q f et Yerevan, October 2014 T. Mannel, Siegen University Effective Quantum -
TASI 2008 Lectures: Introduction to Supersymmetry And
TASI 2008 Lectures: Introduction to Supersymmetry and Supersymmetry Breaking Yuri Shirman Department of Physics and Astronomy University of California, Irvine, CA 92697. [email protected] Abstract These lectures, presented at TASI 08 school, provide an introduction to supersymmetry and supersymmetry breaking. We present basic formalism of supersymmetry, super- symmetric non-renormalization theorems, and summarize non-perturbative dynamics of supersymmetric QCD. We then turn to discussion of tree level, non-perturbative, and metastable supersymmetry breaking. We introduce Minimal Supersymmetric Standard Model and discuss soft parameters in the Lagrangian. Finally we discuss several mech- anisms for communicating the supersymmetry breaking between the hidden and visible sectors. arXiv:0907.0039v1 [hep-ph] 1 Jul 2009 Contents 1 Introduction 2 1.1 Motivation..................................... 2 1.2 Weylfermions................................... 4 1.3 Afirstlookatsupersymmetry . .. 5 2 Constructing supersymmetric Lagrangians 6 2.1 Wess-ZuminoModel ............................... 6 2.2 Superfieldformalism .............................. 8 2.3 VectorSuperfield ................................. 12 2.4 Supersymmetric U(1)gaugetheory ....................... 13 2.5 Non-abeliangaugetheory . .. 15 3 Non-renormalization theorems 16 3.1 R-symmetry.................................... 17 3.2 Superpotentialterms . .. .. .. 17 3.3 Gaugecouplingrenormalization . ..... 19 3.4 D-termrenormalization. ... 20 4 Non-perturbative dynamics in SUSY QCD 20 4.1 Affleck-Dine-Seiberg -
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. -
Regularization and Renormalization of Non-Perturbative Quantum Electrodynamics Via the Dyson-Schwinger Equations
University of Adelaide School of Chemistry and Physics Doctor of Philosophy Regularization and Renormalization of Non-Perturbative Quantum Electrodynamics via the Dyson-Schwinger Equations by Tom Sizer Supervisors: Professor A. G. Williams and Dr A. Kızılers¨u March 2014 Contents 1 Introduction 1 1.1 Introduction................................... 1 1.2 Dyson-SchwingerEquations . .. .. 2 1.3 Renormalization................................. 4 1.4 Dynamical Chiral Symmetry Breaking . 5 1.5 ChapterOutline................................. 5 1.6 Notation..................................... 7 2 Canonical QED 9 2.1 Canonically Quantized QED . 9 2.2 FeynmanRules ................................. 12 2.3 Analysis of Divergences & Weinberg’s Theorem . 14 2.4 ElectronPropagatorandSelf-Energy . 17 2.5 PhotonPropagatorandPolarizationTensor . 18 2.6 ProperVertex.................................. 20 2.7 Ward-TakahashiIdentity . 21 2.8 Skeleton Expansion and Dyson-Schwinger Equations . 22 2.9 Renormalization................................. 25 2.10 RenormalizedPerturbationTheory . 27 2.11 Outline Proof of Renormalizability of QED . 28 3 Functional QED 31 3.1 FullGreen’sFunctions ............................. 31 3.2 GeneratingFunctionals............................. 33 3.3 AbstractDyson-SchwingerEquations . 34 3.4 Connected and One-Particle Irreducible Green’s Functions . 35 3.5 Euclidean Field Theory . 39 3.6 QEDviaFunctionalIntegrals . 40 3.7 Regularization.................................. 42 3.7.1 Cutoff Regularization . 42 3.7.2 Pauli-Villars Regularization . 42 i 3.7.3 Lattice Regularization . 43 3.7.4 Dimensional Regularization . 44 3.8 RenormalizationoftheDSEs ......................... 45 3.9 RenormalizationGroup............................. 49 3.10BrokenScaleInvariance ............................ 53 4 The Choice of Vertex 55 4.1 Unrenormalized Quenched Formalism . 55 4.2 RainbowQED.................................. 57 4.2.1 Self-Energy Derivations . 58 4.2.2 Analytic Approximations . 60 4.2.3 Numerical Solutions . 62 4.3 Rainbow QED with a 4-Fermion Interaction . -
On Topological Vertex Formalism for 5-Brane Webs with O5-Plane
DIAS-STP-20-10 More on topological vertex formalism for 5-brane webs with O5-plane Hirotaka Hayashi,a Rui-Dong Zhub;c aDepartment of Physics, School of Science, Tokai University, 4-1-1 Kitakaname, Hiratsuka-shi, Kanagawa 259-1292, Japan bInstitute for Advanced Study & School of Physical Science and Technology, Soochow University, Suzhou 215006, China cSchool of Theoretical Physics, Dublin Institute for Advanced Studies 10 Burlington Road, Dublin, Ireland E-mail: [email protected], [email protected] Abstract: We propose a concrete form of a vertex function, which we call O-vertex, for the intersection between an O5-plane and a 5-brane in the topological vertex formalism, as an extension of the work of [1]. Using the O-vertex it is possible to compute the Nekrasov partition functions of 5d theories realized on any 5-brane web diagrams with O5-planes. We apply our proposal to 5-brane webs with an O5-plane and compute the partition functions of pure SO(N) gauge theories and the pure G2 gauge theory. The obtained results agree with the results known in the literature. We also compute the partition function of the pure SU(3) gauge theory with the Chern-Simons level 9. At the end we rewrite the O-vertex in a form of a vertex operator. arXiv:2012.13303v2 [hep-th] 12 May 2021 Contents 1 Introduction1 2 O-vertex 3 2.1 Topological vertex formalism with an O5-plane3 2.2 Proposal for O-vertex6 2.3 Higgsing and O-planee 9 3 Examples 11 3.1 SO(2N) gauge theories 11 3.1.1 Pure SO(4) gauge theory 12 3.1.2 Pure SO(6) and SO(8) Theories 15 3.2 SO(2N -
Quantum Electrodynamics in Strong Magnetic Fields. 111* Electron-Photon Interactions
Aust. J. Phys., 1983, 36, 799-824 Quantum Electrodynamics in Strong Magnetic Fields. 111* Electron-Photon Interactions D. B. Melrose and A. J. Parle School of Physics, University of Sydney. Sydney, N.S.W. 2006. Abstract A version of QED is developed which allows one to treat electron-photon interactions in the magnetized vacuum exactly and which allows one to calculate the responses of a relativistic quantum electron gas and include these responses in QED. Gyromagnetic emission and related crossed processes, and Compton scattering and related processes are discussed in some detail. Existing results are corrected or generalized for nonrelativistic (quantum) gyroemission, one-photon pair creation, Compton scattering by electrons in the ground state and two-photon excitation to the first Landau level from the ground state. We also comment on maser action in one-photon pair annihilation. 1. Introduction A full synthesis of quantum electrodynamics (QED) and the classical theory of plasmas requires that the responses of the medium (plasma + vacuum) be included in the photon properties and interactions in QED and that QED be used to calculate the responses of the medium. It was shown by Melrose (1974) how this synthesis could be achieved in the unmagnetized case. In the present paper we extend the synthesized theory to include the magnetized case. Such a synthesized theory is desirable even when the effects of a material medium are negligible. The magnetized vacuum is birefringent with a full hierarchy of nonlinear response tensors, and for many purposes it is convenient to treat the magnetized vacuum as though it were a material medium. -
14 Renormalization 1: Basics
14 Renormalization 1: Basics 14.1 Introduction Functional integrals over fields have been mentioned briefly in Part I de- voted to path integrals.† In brief, time ordering properties and Gaussian properties generalize immediately from paths to field integrals. The sin- gular nature of Green’s functions in field theory introduces new problems which have no counterpart in path integrals. This problem is addressed by regularization and is briefly presented in the appendix. The fundamental difference between quantum mechanics (systems with a finite number of degrees of freedom) and quantum field theory (systems with an infinite number of degrees of freedom) can be labelled “radiative corrections”: In quantum mechanics “a particle is a particle” character- ized, for instance, by mass, charge, spin. In quantum field theory the concept “particle” is intrinsically associated to the concept “field.” The particle is affected by its field. Its mass and charge are modified by the surrounding fields, its own, and other fields interacting with it. An early example of this situation is Green’s calculation of the motion of a pendulum in fluid media. The mass of the pendulum in its equation of motion is modified by the fluid. Nowadays we say the mass is “renor- malized.” The remarkable fact is that the equation of motion remains valid provided one replaces the “bare” mass by the renormalized mass. Green’s example is presented in section 14.2. Although regularization and renormalization are different concepts, † Namely in section 1.1 “The beginning”, in section 1.3 “The operator formalism”, in section 2.3 “Gaussian in Banach spaces”, in section 2.5 “Scaling and coarse- graining”, and in section 4.3 “The anharmonic oscillator”. -
Sum Rules and Vertex Corrections for Electron-Phonon Interactions
Sum rules and vertex corrections for electron-phonon interactions O. R¨osch,∗ G. Sangiovanni, and O. Gunnarsson Max-Planck-Institut f¨ur Festk¨orperforschung, Heisenbergstr. 1, D-70506 Stuttgart, Germany We derive sum rules for the phonon self-energy and the electron-phonon contribution to the electron self-energy of the Holstein-Hubbard model in the limit of large Coulomb interaction U. Their relevance for finite U is investigated using exact diagonalization and dynamical mean-field theory. Based on these sum rules, we study the importance of vertex corrections to the electron- phonon interaction in a diagrammatic approach. We show that they are crucial for a sum rule for the electron self-energy in the undoped system while a sum rule related to the phonon self-energy of doped systems is satisfied even if vertex corrections are neglected. We provide explicit results for the vertex function of a two-site model. PACS numbers: 63.20.Kr, 71.10.Fd, 74.72.-h I. INTRODUCTION the other hand, a sum rule for the phonon self-energy is fulfilled also without vertex corrections. This suggests that it can be important to include vertex corrections for Recently, there has been much interest in the pos- studying properties of cuprates and other strongly corre- sibility that electron-phonon interactions may play an lated materials. important role for properties of cuprates, e.g., for The Hubbard model with electron-phonon interaction 1–3 superconductivity. In particular, the interest has fo- is introduced in Sec. II. In Sec. III, sum rules for the cused on the idea that the Coulomb interaction U might electron and phonon self-energies are derived focusing on enhance effects of electron-phonon interactions, e.g., due the limit U and in Sec. -
Relativistic Wave Equations Les Rencontres Physiciens-Mathématiciens De Strasbourg - RCP25, 1973, Tome 18 « Conférences De : J
Recherche Coopérative sur Programme no 25 R. SEILER Relativistic Wave Equations Les rencontres physiciens-mathématiciens de Strasbourg - RCP25, 1973, tome 18 « Conférences de : J. Leray, J.P. Ramis, R. Seiler, J.M. Souriau et A. Voros », , exp. no 4, p. 1-24 <http://www.numdam.org/item?id=RCP25_1973__18__A4_0> © Université Louis Pasteur (Strasbourg), 1973, tous droits réservés. L’accès aux archives de la série « Recherche Coopérative sur Programme no 25 » implique l’ac- cord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utili- sation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Relativistic Wave Equations R. Seiler Institut für Theoretische Physik Freie Universität Berlin Talk présentée at the Strassbourg meeting^November 1972 (Quinzième rencontre entre physiciens théoriciens et mathé maticiens dans le cadre de recherche coopérative sur pro gramme n° 25 du CNRS). III - 1 - Introduction In general quantum field theory, formulation of dynamics is one of the most important yet most difficult problems. Experience with classical field theories - e.g. Maxwell theory - indicatesthat relativistic wave equations, in one form or another are going to play an important role. This optimism is furthermore supported by the analysis of models, e. g. Thirring model ^ 4 1 or <j theory both in 2 space time dimensions, and of renormalized perturbation theory. Asa step towards full quantum theory one might consider the so called external field problems. -
Extracting Supersymmetry-Breaking Effects from Wave-Function
CERN-TH/97-145 hep-ph/9706540 Extracting Supersymmetry-Breaking Effects from Wave-Function Renormalization G.F. Giudice1 and R. Rattazzi Theory Division, CERN, CH-1211, Gen`eve 23, Switzerland Abstract We show that in theories in which supersymmetry breaking is communicated by renor- malizable perturbative interactions, it is possible to extract the soft terms for the observ- able fields from wave-function renormalization. Therefore all the information about soft terms can be obtained from anomalous dimensions and β functions, with no need to fur- ther compute any Feynman diagram. This method greatly simplifies calculations which are rather involved if performed in terms of component fields. For illustrative purposes we reproduce known results of theories with gauge-mediated supersymmetry breaking. We then use our method to obtain new results of phenomenological importance. We calcu- late the next-to-leading correction to the Higgs mass parameters, the two-loop soft terms induced by messenger-matter superpotential couplings, and the soft terms generated by messengers belonging to vector supermultiplets. CERN-TH/97-145 June 1997 1On leave of absence from INFN, Sez. di Padova, Italy. 1 Introduction Theories in which supersymmetry breaking is mediated by renormalizable perturbative inter- actions have an interesting advantage over the gravity-mediated scenario. This is because in these theories the soft terms are in priciple calculable quantities, very much like g − 2inQED. Gauge-mediated models [1, 2], in particular, have also a strong motivation as they elegantly solve the supersymmetric flavour problem. In the simplest version of these theories, the gaugino and sfermion masses arise respectively from one- and two-loop finite diagrams. -
The Electron Self-Energy in the Green's-Function Approach
ISSP Workshop/Symposium 2007 on FADFT The Electron Self-Energy in the Green’s-Function Approach: Beyond the GW Approximation Yasutami Takada Institute for Solid State Physics, University of Tokyo 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan Seminar Room A615, Kashiwa, ISSP, University of Tokyo 14:00-15:00, 7 August 2007 ◎ Thanks to Professor Hiroshi Yasuhara for enlightening discussions for years Self-Energy beyond the GW Approximation (Takada) 1 Outline Preliminaries: Theoretical Background ○ One-particle Green’s function G and the self-energy Σ ○ Hedin’s theory: Self-consistent set of equations for G, Σ, W, Π, and Γ Part I. GWΓ Scheme ○ Introducing “the ratio function” ○ Averaging the irreducible electron-hole effective interaction ○ Exact functional form for Γ and an approximation scheme Part II. Illustrations ○ Localized limit: Single-site system with both electron-electron and electron-phonon interactions ○ Extended limit: Homogeneous electron gas Part III. Comparison with Experiment ○ ARPES and the problem of occupied bandwidth of the Na 3s band ○ High-energy electron escape depth Conclusion and Outlook for Future Self-Energy beyond the GW Approximation (Takada) 2 Preliminaries ● One-particle Green’s function G and the self-energy Σ ● Hedin’s theory: Self-consistent set of equations for G, Σ, W, Π, and Γ Self-Energy beyond the GW Approximation (Takada) 3 One-Electron Green’s Function Gσσ’(r,r’;t) Inject a bare electron with spin σ’ at site r’ at t=0; let it propage in the system until we observe the propability amplitude of a bare electron with spin σ at site r at t (>0) Æ Electron injection process Reverse process in time: Pull a bare electron with spin σ out at site r at t=0 first and then put a bare electron with spin σ’ back at site r’ at t. -
Self-Consistent Green's Functions with Three-Body Forces
Self-consistent Green's functions with three-body forces Arianna Carbone Departament d'Estructura i Constituents de la Mat`eria Facultat de F´ısica,Universitat de Barcelona Mem`oriapresentada per optar al grau de Doctor per la Universitat de Barcelona arXiv:1407.6622v1 [nucl-th] 24 Jul 2014 11 d'abril de 2014 Directors: Dr. Artur Polls Mart´ıi Dr. Arnau Rios Huguet Tutor: Dr. Artur Polls Mart´ı Programa de Doctorat de F´ısica Table of contents Table of contentsi 1 Introduction1 1.1 The nuclear many-body problem . .3 1.2 The nuclear interaction . .9 1.3 Program of the thesis . 13 2 Many-body Green's functions 15 2.1 Green's functions formalism with three-body forces . 17 2.2 The Galistkii-Migdal-Koltun sum rule . 21 2.3 Dyson's equation and self-consistency . 24 3 Three-body forces in the SCGF approach 31 3.1 Interaction-irreducible diagrams . 32 3.2 Hierarchy of equations of motion . 40 3.3 Ladder and other approximations . 54 4 Effective two-body chiral interaction 61 4.1 Density-dependent potential at N2LO . 63 4.2 Partial-wave matrix elements . 72 5 Nuclear and Neutron matter with three-body forces 89 5.1 The self-energy . 93 5.2 Spectral function and momentum distribution . 99 5.3 Nuclear matter . 105 5.4 Neutron matter and the symmetry energy . 111 6 Summary and Conclusions 119 A Feynman rules 129 i Table of contents B Interaction-irreducible diagrams 137 C Complete expressions for the density-dependent force 141 C.1 Numerical implementation .