Quantum Field Theory Lecture Notes
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A New Renormalization Scheme of Fermion Field in Standard Model
A New Renormalization Scheme of Fermion Field in Standard Model Yong Zhoua a Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080, China (Jan 10, 2003) In order to obtain proper wave-function renormalization constants for unstable fermion and consist with Breit-Wigner formula in the resonant region, We have assumed an extension of the LSZ reduction formula for unstable fermion and adopted on-shell mass renormalization scheme in the framework of without field renormaization. The comparison of gauge dependence of physical amplitude between on-shell mass renormalization and complex-pole mass renormalization has been discussed. After obtaining the fermion wave-function renormalization constants, we extend them to two matrices in order to include the contributions of off-diagonal two-point diagrams at fermion external legs for the convenience of calculations of S-matrix elements. 11.10.Gh, 11.55.Ds, 12.15.Lk I. INTRODUCTION We are interested in the LSZ reduction formula [1] in the case of unstable particle, for the purpose of obtaining a proper wave function renormalization constant(wrc.) for unstable fermion. In this process we have demanded the form of the unstable particle’s propagator in the resonant region resembles the relativistic Breit-Wigner formula. Therefore the on-shell mass renormalization scheme is a suitable choice compared with the complex-pole mass renormalization scheme [2], as we will discuss in section 2. We also discard the field renormalization constants which have been proven at question in the case of having fermions mixing [3]. The layout of this article is as follows. -
Penguin Decays of B Mesons
COLO–HEP–395 SLAC-PUB-7796 HEPSY 98-1 April 23, 1998 PENGUIN DECAYS OF B MESONS Karen Lingel 1 Stanford Linear Accelerator Center Stanford University, Stanford, CA 94309 and Tomasz Skwarnicki 2 Department of Physics, Syracuse University Syracuse, NY 13244 and James G. Smith 3 Department of Physics, University of Colorado arXiv:hep-ex/9804015v1 27 Apr 1998 Boulder, CO 80309 Submitted to Annual Reviews of Nuclear and Particle Science 1Work supported by Department of Energy contract DE-AC03-76SF00515 2Work supported by National Science Foundation contract PHY 9807034 3Work supported by Department of Energy contract DE-FG03-95ER40894 2 LINGEL & SKWARNICKI & SMITH PENGUIN DECAYS OF B MESONS Karen Lingel Stanford Linear Accelerator Center Tomasz Skwarnicki Department of Physics, Syracuse University James G. Smith Department of Physics, University of Colorado KEYWORDS: loop CP-violation charmless ABSTRACT: Penguin, or loop, decays of B mesons induce effective flavor-changing neutral currents, which are forbidden at tree level in the Standard Model. These decays give special insight into the CKM matrix and are sensitive to non-standard model effects. In this review, we give a historical and theoretical introduction to penguins and a description of the various types of penguin processes: electromagnetic, electroweak, and gluonic. We review the experimental searches for penguin decays, including the measurements of the electromagnetic penguins b → sγ ∗ ′ and B → K γ and gluonic penguins B → Kπ, B+ → ωK+ and B → η K, and their implications for the Standard Model and New Physics. We conclude by exploring the future prospects for penguin physics. CONTENTS INTRODUCTION .................................... 3 History of Penguins .................................... -
PHY 2404S Lecture Notes
PHY 2404S Lecture Notes Michael Luke Winter, 2003 These notes are perpetually under construction. Please let me know of any typos or errors. Once again, large portions of these notes have been plagiarized from Sidney Coleman’s field theory lectures from Harvard, written up by Brian Hill. 1 Contents 1 Preliminaries 3 1.1 Counterterms and Divergences: A Simple Example . 3 1.2 CountertermsinScalarFieldTheory . 11 2 Reformulating Scattering Theory 18 2.1 Feynman Diagrams with External Lines off the Mass Shell . .. 18 2.1.1 AnswerOne: PartofaLargerDiagram . 19 2.1.2 Answer Two: The Fourier Transform of a Green Function 20 2.1.3 Answer Three: The VEV of a String of Heisenberg Fields 22 2.2 GreenFunctionsandFeynmanDiagrams. 23 2.3 TheLSZReductionFormula . 27 2.3.1 ProofoftheLSZReductionFormula . 29 3 Renormalizing Scalar Field Theory 36 3.1 The Two-Point Function: Wavefunction Renormalization .... 37 3.2 The Analytic Structure of G(2), and1PIGreenFunctions . 40 3.3 Calculation of Π(k2) to order g2 .................. 44 3.4 The definition of g .......................... 50 3.5 Unstable Particles and the Optical Theorem . 51 3.6 Renormalizability. 57 4 Renormalizability 60 4.1 DegreesofDivergence . 60 4.2 RenormalizationofQED. 65 4.2.1 TroubleswithVectorFields . 65 4.2.2 Counterterms......................... 66 2 1 Preliminaries 1.1 Counterterms and Divergences: A Simple Example In the last semester we considered a variety of field theories, and learned to calculate scattering amplitudes at leading order in perturbation theory. Unfor- tunately, although things were fine as far as we went, the whole basis of our scattering theory had a gaping hole in it. -
UNIVERSITETET I OSLO UNIVERSITY of OSLO FYSISK
A/08800Oé,1 UNIVERSITETET i OSLO UNIVERSITY OF OSLO The rols of penjruin-like interactions in weak processes') by 1 J.O. Eeg Department of Physics, University of OoloJ Box 1048 Blindern, 0316 Oslo 3, Norway Report 88-17 Received 1988-10-17 ISSN 0332-9971 FYSISK INSTITUTT DEPARTMENT OF PHYSICS OOP-- 8S-^V The role of penguin-like interactions in weak pro-esses*) by J.O. Eeg Department of Physics, University of Oslo, Box 1048 Blindern, 0316 Oslo 3, Norway Report 88-17 Received 1988-10-17 ISSN 0332-3571 ) Based on talk given at the Ringberg workshop on Hadronio Matrix Elements and Weak Decays, Ringberg Castle, April 18-22, 1988. Organized by A.J. Buras, J.M. Gerard BIK W. Huber, Max-Planck-Institut flir Physik und Astrophysik, MUnchen. The role of penguin-like interactions in weak processesprocesses . J.O. Eeg Dept. of Physics, University of Oslo, P.O.Box 1048 Blindern, N-0316 Oslo 3, Norway October 5, 1988 Abstract Basic properties of penguin diagrams are reviewed. Some model dependent calculations of low momentum penguin loop contributions are presented. CP- violating effects generated when penguin-like diagrams are inserted in higher loops are described. 1 Introduction The history of so-called penguin diagrams has. mainly been connected to the expla nation of the A/ = 1/2 rule and the CP-violating parameter e' in K —> -KIT decays. Many reviews discussing weak non-leptonic decays have been given1,2,3,4,6, and all the details will not be repeated here. As far as possible I will avoid to overlap with other speakers. -
Introduction to Flavour Physics
Introduction to flavour physics Y. Grossman Cornell University, Ithaca, NY 14853, USA Abstract In this set of lectures we cover the very basics of flavour physics. The lec- tures are aimed to be an entry point to the subject of flavour physics. A lot of problems are provided in the hope of making the manuscript a self-study guide. 1 Welcome statement My plan for these lectures is to introduce you to the very basics of flavour physics. After the lectures I hope you will have enough knowledge and, more importantly, enough curiosity, and you will go on and learn more about the subject. These are lecture notes and are not meant to be a review. In the lectures, I try to talk about the basic ideas, hoping to give a clear picture of the physics. Thus many details are omitted, implicit assumptions are made, and no references are given. Yet details are important: after you go over the current lecture notes once or twice, I hope you will feel the need for more. Then it will be the time to turn to the many reviews [1–10] and books [11, 12] on the subject. I try to include many homework problems for the reader to solve, much more than what I gave in the actual lectures. If you would like to learn the material, I think that the problems provided are the way to start. They force you to fully understand the issues and apply your knowledge to new situations. The problems are given at the end of each section. -
Lecture 12 from Correlation Functions to Amplitudes
Lecture 12 From Correlation Functions to Amplitudes Up to this point we have been able to compute correlation functions up to a given order in perturbation theory. Our next step is to relate these to amplitudes for physical processes that we can use to compute observables such as scattering cross sections and decay rates. For instance, assuming given initial and final states, we want to compute the amplitude to go from one to the other in the presence of interactions. In order to be able to define initial and final states we consider a situation where the interaction takes place at a localized region of spacetime and the initial state is defined at t ! −∞, whereas the final state is defined at t ! +1. These asymptotic states are well defined particle states. That is, if we have a two-particle state of momentum jp1p2i, this means that the wave- packets associated with each of the particles are strongly peaked at p1 and p2, and are well separated in momentum space. This assumption simplifies our treatment of asymptotic states. The same will be assumed of the final state jp3 : : : pni. We will initially assume these states far into the past and into the future are eigenstates of the free theory. The fact is they are not, even if they are not interacting say through scattering. The reason is that the presence of interactions modifies the parameters of the theory, so even the unperturbed propagation from the far past or to the far future is not described by free eigenstates. However, the results we will derive for free states will still hold with some modifications associated with the shift of the paramenters of the theory (including the fields) in the presence of interactions. -
Direct CP Violation in B Meson Decays
Faculty of Sciences School of Physical Sciences Direct CP Violation in B Meson Decays Liam Christopher Hockley Supervisors: Professor Anthony Thomas Associate Professor Ross Young February 2020 Contents Frontmatter xi 1 Introduction 1 1.1 The Standard Model . 2 1.1.1 The Electroweak Force . 3 1.1.2 Quantum Chromodynamics . 4 1.2 Chirality . 6 2 CP Violation 9 2.1 Symmetries . 9 2.1.1 Parity . 9 2.1.2 Charge Conjugation . 12 2.2 C and P Violation . 13 2.2.1 Parity Violation in the Decays of 60Co Nuclei . 14 2.2.2 Charge Conjugation Violation in the Decays of Pions . 15 2.2.3 Time Reversal and CPT . 16 2.3 CP Violation . 17 2.3.1 CP Violation in Kaons . 17 2.3.2 CP Violation in B Mesons . 18 2.4 Matter-antimatter Asymmetry . 20 2.5 Belle Results . 21 3 Effective Hamiltonian 23 3.1 CKM Matrix . 23 3.2 Four Fermion Interactions . 29 3.3 Operator Product Expansion . 33 3.4 Tree and Penguin Operators . 34 3.4.1 Tree Operators . 34 3.4.2 Penguin Operators . 35 3.5 Effective Hamiltonian . 36 3.6 Wilson Coefficients . 37 4 CP Asymmetry 41 4.1 Formalism . 41 4.2 Choice of Diagrams . 44 4.3 Tree and Penguin Amplitudes . 48 5 Hadronic Matrix Elements 49 5.1 Naive Factorisation . 49 5.1.1 Fierz Identities . 50 5.2 Factorisation Approximation . 52 iii iv Contents 5.3 Form Factors . 54 5.3.1 Motivation . 54 5.3.2 Matrix Element Parameterisation . 55 5.4 Matrix Element Contraction . -
Penguin-Like Diagrams from the Standard Model
Penguin-like Diagrams from the Standard Model Chia Swee Ping High Impact Research, University of Malaya, 50603 Kuala Lumpur, Malaysia Abstract. The Standard Model is highly successful in describing the interactions of leptons and quarks. There are, however, rare processes that involve higher order effects in electroweak interactions. One specific class of processes is the penguin-like diagram. Such class of diagrams involves the neutral change of quark flavours accompanied by the emission of a gluon (gluon penguin), a photon (photon penguin), a gluon and a photon (gluon-photon penguin), a Z- boson (Z penguin), or a Higgs-boson (Higgs penguin). Such diagrams do not arise at the tree level in the Standard Model. They are, however, induced by one-loop effects. In this paper, we present an exact calculation of the penguin diagram vertices in the ‘tHooft-Feynman gauge. Renormalization of the vertex is effected by a prescription by Chia and Chong which gives an expression for the counter term identical to that obtained by employing Ward-Takahashi identity. The on- shell vertex functions for the penguin diagram vertices are obtained. The various penguin diagram vertex functions are related to one another via Ward-Takahashi identity. From these, a set of relations is obtained connecting the vertex form factors of various penguin diagrams. Explicit expressions for the gluon-photon penguin vertex form factors are obtained, and their contributions to the flavor changing processes estimated. Keywords: Standard model, penguin diagrams, neutral change of quark flavours, rare processes. PACS: 11.30.Hv, 12.15.-y, 14.70.-e INTRODUCTION The Standard Model (SM) of elementary particles is based on the SU(3)×SU(2)×U(1) gauge group. -
Quantum Field Theory Koichi Hamaguchi Last Updated: July 31, 2017 Contents
2017 S-semester Quantum Field Theory Koichi Hamaguchi Last updated: July 31, 2017 Contents x 0 Introduction 1 x 0.1 Course objectives . 2 x 0.2 Quantum mechanics and quantum field theory . 2 x 0.3 Notation and convention . 3 x 0.4 Various fields . 4 x 0.5 Outline: what we will learn . 4 x 0.6 S-matrix, amplitude M =) observables (σ and Γ) . 5 x 1 Scalar (spin 0) Field 16 x 1.1 Lorentz transformation . 16 x 1.1.1 Lorentz transformation of coordinates . 16 x 1.1.2 Lorentz transformation of quantum fields . 17 x 1.2 Lagrangian and Canonical Quantization of Real Scalar Field . 18 x 1.3 Equation of Motion (EOM) . 22 x 1.4 Free scalar field . 23 x 1.4.1 Solution of the EOM . 23 x 1.4.2 Commutation relations . 25 x 1.4.3 ay and a are the creation and annihilation operators. 25 x 1.4.4 Consistency check . 29 x 1.4.5 vacuum state . 30 x 1.4.6 One-particle state . 30 x 1.4.7 Lorentz transformation of a and ay ................... 31 x 1.4.8 [ϕ(x); ϕ(y)] for x0 =6 y0 .......................... 32 x 1.5 Interacting Scalar Field . 34 x 1.5.1 What is ϕ(x)?............................... 34 x 1.5.2 In/out states and the LSZ reduction formula . 36 x 1.5.3 Heisenberg field and Interaction picture field . 42 x 1.5.4 a and ay (again) . 45 x 1.5.5 h0jT(ϕ(x1) ··· ϕ(xn)) j0i =? ...................... -
Quantum Field Theory: Spin Zero
Quantum Field Theory Part I: Spin Zero Mark Srednicki Department of Physics University of California Santa Barbara, CA 93106 [email protected] This is a draft version of Part I of a three-part introductory textbook on quantum field theory. arXiv:hep-th/0409035v1 2 Sep 2004 1 Part I: Spin Zero 0) Preface 1) Attempts at Relativistic Quantum Mechanics (prerequisite: none) 2) Lorentz Invariance (prerequisite: 1) 3) Relativistic Quantum Fields and Canonical Quantization (2) 4) The Spin-Statistics Theorem (3) 5) The LSZ Reduction Formula (3) 6) Path Integrals in Quantum Mechanics (none) 7) The Path Integral for the Harmonic Oscillator (6) 8) The Path Integral for Free Field Theory (3, 7) 9) The Path Integral for Interacting Field Theory (8) 10) Scattering Amplitudes and the Feynman Rules (5, 9) 11) Cross Sections and Decay Rates (10) 12) The Lehmann-K¨all´en Form of the Exact Propagator (9) 13) Dimensional Analysis withh ¯ = c = 1 (3) 14) Loop Corrections to the Propagator (10, 12, 13) 15) The One-Loop Correction in Lehmann-K¨all´en Form (14) 16) Loop Corrections to the Vertex (14) 17) Other 1PI Vertices (16) 18) Higher-Order Corrections and Renormalizability (17) 19) Perturbation Theory to All Orders: the Skeleton Expansion (18) 20) Two-Particle Elastic Scattering at One Loop (10, 19) 21) The Quantum Action (19) 22) Continuous Symmetries and Conserved Currents (8) 23) Discrete Symmetries: P , C, T , and Z (22) 24) Unstable Particles and Resonances (10, 14) 25) Infrared Divergences (20) 26) Other Renormalization Schemes (25) 27) Formal Development of the Renormalization Group (26) 28) Nonrenormalizable Theories and Effective Field Theory (26) 29) Spontaneous Symmetry Breaking (3) 30) Spontaneous Symmetry Breaking and Loop Corrections (19, 29) 31) Spontaneous Breakdown of Continuous Symmetries (29) 32) Nonabelian Symmetries (31) 2 Quantum Field Theory Mark Srednicki 0: Preface This is a draft version of Part I of a three-part introductory textbook on quantum field theory. -
Path Integral in Quantum Field Theory Alexander Belyaev (Course Based on Lectures by Steven King) Contents
Path Integral in Quantum Field Theory Alexander Belyaev (course based on Lectures by Steven King) Contents 1 Preliminaries 5 1.1 Review of Classical Mechanics of Finite System . 5 1.2 Review of Non-Relativistic Quantum Mechanics . 7 1.3 Relativistic Quantum Mechanics . 14 1.3.1 Relativistic Conventions and Notation . 14 1.3.2 TheKlein-GordonEquation . 15 1.4 ProblemsSet1 ........................... 18 2 The Klein-Gordon Field 19 2.1 Introduction............................. 19 2.2 ClassicalScalarFieldTheory . 20 2.3 QuantumScalarFieldTheory . 28 2.4 ProblemsSet2 ........................... 35 3 Interacting Klein-Gordon Fields 37 3.1 Introduction............................. 37 3.2 PerturbationandScatteringTheory. 37 3.3 TheInteractionHamiltonian. 43 3.4 Example: K π+π− ....................... 45 S → 3.5 Wick’s Theorem, Feynman Propagator, Feynman Diagrams . .. 47 3.6 TheLSZReductionFormula. 52 3.7 ProblemsSet3 ........................... 58 4 Transition Rates and Cross-Sections 61 4.1 TransitionRates .......................... 61 4.2 TheNumberofFinalStates . 63 4.3 Lorentz Invariant Phase Space (LIPS) . 63 4.4 CrossSections............................ 64 4.5 Two-bodyScattering . 65 4.6 DecayRates............................. 66 4.7 OpticalTheorem .......................... 66 4.8 ProblemsSet4 ........................... 68 1 2 CONTENTS 5 Path Integrals in Quantum Mechanics 69 5.1 Introduction............................. 69 5.2 The Point to Point Transition Amplitude . 70 5.3 ImaginaryTime........................... 74 5.4 Transition Amplitudes With an External Driving Force . ... 77 5.5 Expectation Values of Heisenberg Position Operators . .... 81 5.6 Appendix .............................. 83 5.6.1 GaussianIntegration . 83 5.6.2 Functionals ......................... 85 5.7 ProblemsSet5 ........................... 87 6 Path Integral Quantisation of the Klein-Gordon Field 89 6.1 Introduction............................. 89 6.2 TheFeynmanPropagator(again) . 91 6.3 Green’s Functions in Free Field Theory . -
Lehmann-Symanzik-Zimmermann S-Matrix Elements on the Moyal Plane
Preprint typeset in JHEP style - HYPER VERSION SU-4252-916 Lehmann-Symanzik-Zimmermann S-Matrix elements on the Moyal Plane A. P. Balachandran Department of Physics, Syracuse University, Syracuse, NY 13244-1130, USA E-mail: [email protected] Pramod Padmanabhan Department of Physics, Syracuse University, Syracuse, NY 13244-1130, USA E-mail: [email protected] Amilcar R. de Queiroz Instituto de Fisica, Universidade de Brasilia, Caixa Postal 04455, 70919-970, Brasilia, DF, Brazil E-mail: [email protected] Abstract: Field theories on the Groenewold-Moyal(GM) plane are studied using the Lehmann-Symanzik-Zimmermann(LSZ) formalism. The example of real scalar fields is treated in detail. The S-matrix elements in this non-perturbative approach are shown to be equal to the interaction representation S-matrix elements. This is a new non-trivial result: in both cases, the S-operator is independent of the noncommutative deformation parameter θ and the change in scattering amplitudes due to noncommutativity is just a arXiv:1104.1629v3 [hep-th] 6 Sep 2011 µν time delay. This result is verified in two different ways. But the off-shell Green’s functions do depend on θµν. In the course of this analysis, unitarity of the non-perturbative S-matrix is proved as well. Keywords: Non-Commutative Geometry, QFT. Contents 1. Introduction 1 2. Twisted Relativistic Quantum Fields on the Moyal plane θ 3 A 3. The Untwisted and Twisted LSZ Reduction Formula 5 4. Non-perturbative Computations of the Scattering Amplitudes 7 5. Remarks 13 6. Acknowledgements 14 1. Introduction Spacetime at the Planck scale is possibly noncommutative.