PHY 2404S 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. -
Reference International Centre for C), Theoretical Physics
IC/88/346 REFERENCE INTERNATIONAL CENTRE FOR C), THEORETICAL PHYSICS ZETA FUNCTION REGULARIZATION OF QUANTUM FIELD THEORY A.Y. Shiekh INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION IC/88/346 §1 Introduction International Atomic Energy Agency and United Nations Educational Scientific and Cultural Organization In generalizing quantum mechanics to quantum field theory one is generally INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS confronted with a scheme plagued by infinities. If one is to make sense of such an ailment, one must find a way to deal with those infinities. In the usual interpretation this difficulty is viewed as stemming from not having yet accounted for the self- interactions which modify the masses and coupling constants to their observed values. A sensible strategy, in order to be useful, must result in only a finite ZETA FUNCTION REGULARIZATION number of physical constants being left, undetermined. Not all theories can be dealt OF QUANTUM FIELD THEORY * with in this way; which has the beneficial consequence of eliminating certain, otherwise prospective, theories of nature. This follows the line of reasoning that physics is more than descriptive, but predictive by virtue of being limited by the A.Y. Shiekh "" requirement of consistency. International Centra foe Theoretical Physics, Tri.fste, Ltaly, In order to proceed with such a theory, the infinities must be restrained (regulated) while the self-interactions are accounted for. Visiting a different space- time dimension is a popular way to regulate (dimensional regularization) as it explicitly maintains the theory's covariance, which may otherwise be difficult to hold on to during reckoning for the self-interactions (renormalization). -
Renormalization in Nonrelativistic Quantum Mechanics
Renormalization in Nonrelativistic Quantum Mechanics ∗ Sadhan K Adhikari†and Angsula Ghosh Instituto de F´ısica Te´orica, Universidade Estadual Paulista, 01405-900 S˜ao Paulo, S˜ao Paulo, Brasil July 30, 2018 Abstract The importance and usefulness of renormalization are emphasized in nonrelativistic quantum mechanics. The momentum space treatment of both two-body bound state and scattering problems involving some potentials singular at the origin exhibits ultravi- olet divergence. The use of renormalization techniques in these problems leads to finite converged results for both the exact and perturbative solutions. The renormalization procedure is carried out for the quantum two-body problem in different partial waves for a minimal potential possessing only the threshold behavior and no form factors. The renormalized perturbative and exact solutions for this problem are found to be consistent arXiv:hep-th/9706193v1 26 Jun 1997 with each other. The useful role of the renormalization group equations for this problem is also pointed out. PACS Numbers 03.65.-w, 03.65.Nk, 11.10.Gh, 11.10.Hi ∗To Appear in Journal of Physics A †John Simon Guggenheim Memorial Foundation Fellow. 1 1 Introduction The ultraviolet divergences in perturbative quantum field theory can be eliminated in many cases by renormalization to define physical observables, such as charge or mass, which are often termed the physical scale(s) of the problem [1, 2, 3, 4]. Ultraviolet divergences appear in exact as well as perturbative treatments of the nonrelativistic quantum mechanical two-body problem in momentum space interacting via two-body potentials with certain singular behavior at short distances [5, 6, 7, 8, 9, 10, 11] in two and three space dimensions. -
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. -
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. -
Toward Construction of a Consistent Field Theory with Poincaré
Toward construction of a consistent field theory with Poincare´ covariance in terms of step-function-type basis functions showing confinement/deconfinement, mass-gap and Regge trajectory for non-pure/pure non-Abelian gauge fields ∗ Kimichika Fukushima † Advanced Reactor System Engineering Department, Toshiba Nuclear Engineering Service Corporation, 8, Shinsugita-cho, Isogo-ku, Yokohama 235-8523, Japan Hikaru Sato Emeritus, Department of Physics, Hyogo University of Education, Yashiro-cho, Kato-shi, Hyogo 673-1494, Japan This article is a review by the authors concerning the construction of a Poincar´ecovariant (owing to spacetime continuum) field-theoretic formalism in terms of step-function-type basis functions without ultraviolet divergences. This formalism analytically derives confinement/deconfinement,mass-gap and Regge trajectory for non-Abelian gauge fields, and gives solutions for self-interacting scalar fields. Fields propagate in spacetime continuum and fields with finite degrees of freedom toward continuum limit have no ultraviolet divergence. Basis functions defined in a parameter spacetime are mapped to real spacetime. The authors derive a new solution comprised of classical fields as a vacuum and quantum fluctuations, leading to the linear potential between the particle and antiparticle from the Wilson loop. The Polyakov line gives finite binding energies and reveals the deconfining property at high temperatures. The quantum action yields positive mass from the classical fields and quantum fluctuations produces the Coulomb potential. Pure Yang-Mills fields show the same mass-gap owing to the particle-antiparticle pair creation. The Dirac equation under linear potential is analytically solved in this formalism, reproducing the principal properties of Regge trajectoriesata quantum level. -
LSZ Reduction Formula and the S-Matrix
PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Dr. Anosh Joseph, IISER Mohali LECTURE 34 Wednesday, March 18, 2020 (Note: This is an online lecture due to COVID-19 interruption.) Topic: LSZ Reduction Formula and the S-Matrix. LSZ Reduction Formula and the S-Matrix The Lehmann-Symanzik-Zimmermann (LSZ) reduction formula is a general relation between cor- relation functions and the S-matrix elements. In the previous lecture we showed that the poles of the two-point correlation functions are related to the physical (renormalized) masses of particles and strengths of bare fields (wave function renormalization constants). The LSZ reduction formula for n > 2 correlation functions Gn(x1; x2; ··· ; xn) ≡ hΩjT φ(x1)φ(x2) ··· φ(xn)jΩi; relates the poles of their Fourier transforms Z Z 4 ip1x1 4 ipnxn Gn(p1; p2; ··· ; pn) = d x1e ··· d xne Gn(x1; x2; ··· ; xn); (1) to the S-matrix elements. 2 2 We encounter a pole when any of the n momenta pi approaches the mass shell, pi = Mphys. The most interesting pole, the simultaneous pole, occurs when all n momenta go on-shell. For simplicity, let us consider the case of scalar fields. Let us consider the following kinematics 0 0 pi ! +E(pi); i = 1; 2; ··· n and kj ! +E(kj); j = 1; 2; ··· m: (2) PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Then the LSZ reduction formula is n Z m Z Y 4 ipi·xi Y 4 −ikj ·yj d xie d yje hΩjT fφ(x1) ··· φ(xn)φ(y1) ··· φ(ym)gjΩi i=1 j=1 0 1 n p ! m p Y i Zφ Y i Zφ ∼ @ A hp1 ··· pnjSjk1 ··· kmi: (3) p2 − m2 + i k2 − m2 + i 0 i=1 i j=1 j each pi ! +Epi 0 each kj ! +Ekj The LSZ formula tells us that we can compute the S-matrix elements as follows. -
Redalyc.Gupta-Bleuler Quantization for Massive and Massless Free
Brazilian Journal of Physics ISSN: 0103-9733 [email protected] Sociedade Brasileira de Física Brasil Dehghani, M. Gupta-Bleuler quantization for massive and massless free vector fields Brazilian Journal of Physics, vol. 39, núm. 3, septiembre, 2009, pp. 559-563 Sociedade Brasileira de Física Sâo Paulo, Brasil Available in: http://www.redalyc.org/articulo.oa?id=46413562009 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative Brazilian Journal of Physics, vol. 39, no. 3, September, 2009 559 Gupta-Bleuler quantization for massive and massless free vector fields M. Dehghani∗ Department of Physics, Ilam University, Ilam, Iran (Received on 28 April, 2009) It is shown that the usual quantum field theory leads to an ultraviolet divergence in the vacuum energies and an infrared divergence in the two-point functions of the massive and massless vector fields. Using a new method of quantization it is shown that the vacuum energies converge, and the normal ordering procedure is not necessary. Also the propagators are calculated, which are automatically renormalied. Keywords: vector fields, quantization, Gupta-Bleuler 1. INTRODUCTION as well. The results for the Casimir energy and Casimir force are the same given in [13, 14]. Particles of spin-1 are described as the quanta of vector In section 2 a brief review of the massive vector field quan- fields. Such vector bosons play a central role as the medi- tization, using usual quantum field theory is presented. -
Non-Renormalization Theorems Without Supersymmetry
CALT-2015-024 Non-renormalization Theorems without Supersymmetry Clifford Cheung and Chia-Hsien Shen Walter Burke Institute for Theoretical Physics California Institute of Technology, Pasadena, CA 91125∗ (Dated: May 11, 2015) We derive a new class of one-loop non-renormalization theorems that strongly constrain the running of higher dimension operators in a general four-dimensional quantum field theory. Our logic follows from unitarity: cuts of one-loop amplitudes are products of tree amplitudes, so if the latter vanish then so too will the associated divergences. Finiteness is then ensured by simple selection rules that zero out tree amplitudes for certain helicity configurations. For each operator we define holomorphic and anti-holomorphic weights, (w, w) = (n − h, n + h), where n and h are the number and sum over helicities of the particles created by that operator. We argue that an operator Oi can only be renormalized by an operator Oj if wi ≥ wj and wi ≥ wj , absent non-holomorphic Yukawa couplings. These results explain and generalize the surprising cancellations discovered in the renormalization of dimension six operators in the standard model. Since our claims rely on unitarity and helicity rather than an explicit symmetry, they apply quite generally. INTRODUCTION where n(A) and h(A) are the number and sum over helic- ities of the external states. Since A is physical, its weight Technical naturalness dictates that all operators not is field reparameterization and gauge independent. The forbidden by symmetry are compulsory—and thus gen- weights of an operator O are then invariantly defined by erated by renormalization. Softened ultraviolet diver- minimizing over all amplitudes involving that operator: gences are in turn a telltale sign of underlying symmetry.