DOCSLIB.ORG

Renormalization, Non-Abelian Gauge Theories and Anomalies Lecture

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Renormalization, Non-Abelian Gauge Theories and Anomalies Lecture Advanced Quantum Field Theory : Renormalization, Non-Abelian Gauge Theories and Anomalies Lecture notes Amsterdam-Brussels-Geneva-Paris Doctoral School \Quantum Field Theory, Strings and Gravity" Brussels, October 2017 Adel Bilal Laboratoire de Physique Th´eoriquede l'Ecole´ Normale Sup´erieure PSL Research University, CNRS, Sorbonne Universit´es,UPMC Paris 6 24 rue Lhomond, F-75231 Paris Cedex 05, France Abstract These are lecture notes of an advanced quantum field theory course intended for graduate students in theoretical high energy physics who are already familiar with the basics of QFT. The first part quickly reviews what should be more or less known: functional inte- gral methods and one-loop computations in QED and φ4. The second part deals in some detail with the renormalization program and the renormalization group. The third part treats the quantization of non-abelian gauge theories and their renormalization with spe- cial emphasis on the BRST symmetry. Every chapter includes some exercices. The fourth part of the lectures, not contained in the present notes but based on arXiv:0802.0634, discusses gauge and gravitational anomalies, how to characterise them in various dimen- sions, as well as anomaly cancellations. Contents 1 Functional integral methods 1 1.1 Path integral in quantum mechanics . .1 1.2 Functional integral in quantum field theory . .2 1.2.1 Derivation of the Hamiltonian functional integral . .2 1.2.2 Derivation of the Lagrangian version of the functional integral . .3 1.2.3 Propagators . .5 1.3 Green functions, S-matrix and Feynman rules . .7 1.3.1 Vacuum bubbles and normalization of the Green functions . .7 1.3.2 Generating functional of Green functions and Feynman rules . 10 1.3.3 Generating functional of connected Green functions . 12 1.3.4 Relation between Green functions and S-matrix . 14 1.4 Quantum effective action . 16 1.4.1 Legendre transform and definition of Γ[']......................... 16 1.4.2 Γ['] as quantum effective action and generating functional of 1PI-diagrams . 17 1.4.3 Symmetries and Slavnov-Taylor identities . 20 1.5 Functional integral formulation of QED . 21 1.5.1 Coulomb gauge . 21 1.5.2 Lorentz invariant functional integral formulation and α-gauges . 23 1.5.3 Feynman rules of spinor QED . 26 1.6 Exercises . 28 1.6.1 Linear operators on L2(R4)................................ 28 1.6.2 Green's function and generating functional in 0 + 1 dimensions . 29 1.6.3 Z[J], W [J] and Γ['] in φ4-theory . 30 1.6.4 Effective action from integrating out fermions in QED . 30 2 A few results independent of perturbation theory 31 2.1 Structure and poles of Green functions . 31 2.2 Complete propagators, the need for field and mass renormalization . 34 2.2.1 Example of a scalar field φ ................................. 34 2.2.2 Example of a Dirac field . 37 2.3 Charge renormalization and Ward identities . 38 2.4 Photon propagator and gauge invariance . 40 2.5 Exercices . 43 2.5.1 Two-particle intermediate states and branch cut of the two-point function . 43 2.5.2 A theory of two scalars, one charged and one neutral . 44 3 One-loop radiative corrections in φ4 and QED 45 3.1 Setup . 45 3.2 Evaluation of one-loop integrals and dimensional regularization . 46 3.3 Wick rotation . 50 3.4 Vacuum polarization . 52 3.5 Electron self energy . 55 3.6 Vertex function . 57 3.6.1 Cancellation of the divergent piece . 58 3.6.2 The magnetic moment of the electron: g − 2 ....................... 59 3.7 One-loop radiative corrections in scalar φ4 ............................. 61 3.8 Exercices . 64 3.8.1 The simplest (non-trivial) one-loop integral . 64 3.8.2 Vacuum polarization in QED . 65 Adel Bilal : Advanced Quantum Field Theory -2 Lecture notes - October 10, 2017 4 General renomalization theory 66 4.1 Degree of divergence . 66 4.2 Structure of the divergences . 69 4.3 Bogoliubov-Parasiuk-Hepp-Zimmermann prescription and theorem . 72 4.4 Summary of the renormalization program and proof . 73 4.5 The criterion of renormalizability . 74 4.6 Exercices . 75 4.6.1 Yukawa theory . 75 4.6.2 Quantum gravity . 75 4.6.3 Non-renormalizable interactions from integrating out a massive field . 76 5 Renormalization group and Callan-Szymanzik equations 77 5.1 Running coupling constant and β-function: examples . 77 5.1.1 Scalar φ4-theory . 77 5.1.2 QED . 80 5.2 Running coupling constant and β-functions: general discussion . 82 5.2.1 Several mass scales . 82 5.2.2 Relation between the one-loop β-function and the counterterms . 83 5.2.3 Scheme independence of the first two coefficients of the β-function . 85 5.3 β-functions and asymptotic behaviors of the coupling . 86 5.3.1 case a : the coupling diverges at a finite scale M ..................... 86 5.3.2 case b : the coupling continues to grow with the scale . 86 5.3.3 case c : existence of a UV fixed point . 87 5.3.4 case d : asymptotic freedom . 88 5.3.5 case e : IR fixed point . 89 5.4 Callan-Symanzik equation for a massless theory . 90 5.4.1 Renormalization conditions at scale µ ........................... 90 5.4.2 Callan-Symanzik equations . 92 5.4.3 Solving the Callan-Symanzik equations . 94 5.4.4 Infrared fixed point and critical exponents / large momentum behavior in asymptotic free theories . 95 5.5 Callan-Symanzik equations for a massive theory . 96 5.5.1 Operator insertions and renormalization of local operators . 96 5.5.2 Callan-Symanzik equations in the presence of operator insertions . 97 5.5.3 Massive Callan-Symanzik equations . 98 5.6 Exercices . 99 5.6.1 β-function in φ3-theory in 6 dimensions . 99 5.6.2 3-point vertex function in an asymptotically free theory . 99 6 Non-abelian gauge theories: formulation and quantization 100 6.1 Non-abelian gauge transformations and gauge invariant actions . 100 6.2 Quantization . 104 6.2.1 Faddeev-Popov method . 105 6.2.2 Gauge-fixed action, ghosts and Feynman rules . 107 6.2.3 BRST symmetry . 108 6.3 BRST cohomology . 111 6.4 Exercices . 115 6.4.1 Some one-loop vertex functions . 115 6.4.2 More general gauge-fixing functions . 115 6.4.3 Two ghost - two antighost counterterms . 115 Adel Bilal : Advanced Quantum Field Theory -1 Lecture notes - October 10, 2017 7 Renormalization of non-abelian gauge theories 116 7.1 Slavnov-Taylor identities and Zinn-Justin equation . 116 7.1.1 Slavnov-Taylor identities . 116 7.1.2 Zinn-Justin equation . 117 7.1.3 Antibracket . 118 7.1.4 Invariance of the measure under the BRST transformation . 118 7.2 Renormalization of gauge theories theories . 120 7.2.1 The general structure and strategy . 120 7.2.2 Constraining the divergent part of Γ . 122 7.2.3 Conclusion and remarks . 127 7.3 Background field gauge . ..
Load more

Recommended publications
  • An Introduction to Quantum Field Theory
    AN INTRODUCTION TO QUANTUM FIELD THEORY By Dr M Dasgupta University of Manchester Lecture presented at the School for Experimental High Energy Physics Students Somerville College, Oxford, September 2009 - 1 - - 2 - Contents 0 Prologue....................................................................................................... 5 1 Introduction ................................................................................................ 6 1.1 Lagrangian formalism in classical mechanics......................................... 6 1.2 Quantum mechanics................................................................................... 8 1.3 The Schrödinger picture........................................................................... 10 1.4 The Heisenberg picture............................................................................ 11 1.5 The quantum mechanical harmonic oscillator ..................................... 12 Problems .............................................................................................................. 13 2 Classical Field Theory............................................................................. 14 2.1 From N-point mechanics to field theory ............................................... 14 2.2 Relativistic field theory ............................................................................ 15 2.3 Action for a scalar field ............................................................................ 15 2.4 Plane wave solution to the Klein-Gordon equation ...........................
    [Show full text]
  • Arxiv:Cond-Mat/0203258V1 [Cond-Mat.Str-El] 12 Mar 2002 AS 71.10.-W,71.27.+A PACS: Pnpolmi Oi Tt Hsc.Iscoeconnection Close Its High- of Physics
    Large-N expansion based on the Hubbard-operator path integral representation and its application to the t J model − Adriana Foussats and Andr´es Greco Facultad de Ciencias Exactas Ingenier´ıa y Agrimensura and Instituto de F´ısica Rosario (UNR-CONICET). Av.Pellegrini 250-2000 Rosario-Argentina. (October 29, 2018) In the present work we have developed a large-N expansion for the t − J model based on the path integral formulation for Hubbard-operators. Our large-N expansion formulation contains diagram- matic rules, in which the propagators and vertex are written in term of Hubbard operators. Using our large-N formulation we have calculated, for J = 0, the renormalized O(1/N) boson propagator. We also have calculated the spin-spin and charge-charge correlation functions to leading order 1/N. We have compared our diagram technique and results with the existing ones in the literature. PACS: 71.10.-w,71.27.+a I. INTRODUCTION this constrained theory leads to the commutation rules of the Hubbard-operators. Next, by using path-integral The role of electronic correlations is an important and techniques, the correlation functional and effective La- open problem in solid state physics. Its close connection grangian were constructed. 1 with the phenomena of high-Tc superconductivity makes In Ref.[ 11], we found a particular family of constrained this problem relevant in present days. Lagrangians and showed that the corresponding path- One of the most popular models in the context of high- integral can be mapped to that of the slave-boson rep- 13,5 Tc superconductivity is the t J model.
    [Show full text]
  • 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
    [Show full text]
  • Quantum Field Theory*
    Quantum Field Theory y Frank Wilczek Institute for Advanced Study, School of Natural Science, Olden Lane, Princeton, NJ 08540 I discuss the general principles underlying quantum eld theory, and attempt to identify its most profound consequences. The deep est of these consequences result from the in nite number of degrees of freedom invoked to implement lo cality.Imention a few of its most striking successes, b oth achieved and prosp ective. Possible limitation s of quantum eld theory are viewed in the light of its history. I. SURVEY Quantum eld theory is the framework in which the regnant theories of the electroweak and strong interactions, which together form the Standard Mo del, are formulated. Quantum electro dynamics (QED), b esides providing a com- plete foundation for atomic physics and chemistry, has supp orted calculations of physical quantities with unparalleled precision. The exp erimentally measured value of the magnetic dip ole moment of the muon, 11 (g 2) = 233 184 600 (1680) 10 ; (1) exp: for example, should b e compared with the theoretical prediction 11 (g 2) = 233 183 478 (308) 10 : (2) theor: In quantum chromo dynamics (QCD) we cannot, for the forseeable future, aspire to to comparable accuracy.Yet QCD provides di erent, and at least equally impressive, evidence for the validity of the basic principles of quantum eld theory. Indeed, b ecause in QCD the interactions are stronger, QCD manifests a wider variety of phenomena characteristic of quantum eld theory. These include esp ecially running of the e ective coupling with distance or energy scale and the phenomenon of con nement.
    [Show full text]
  • Feynman Diagrams, Momentum Space Feynman Rules, Disconnected Diagrams, Higher Correlation Functions
    PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Dr. Anosh Joseph, IISER Mohali LECTURE 05 Tuesday, January 14, 2020 Topics: Feynman Diagrams, Momentum Space Feynman Rules, Disconnected Diagrams, Higher Correlation Functions. Feynman Diagrams We can use the Wick’s theorem to express the n-point function h0jT fφ(x1)φ(x2) ··· φ(xn)gj0i as a sum of products of Feynman propagators. We will be interested in developing a diagrammatic interpretation of such expressions. Let us look at the expression containing four fields. We have h0jT fφ1φ2φ3φ4g j0i = DF (x1 − x2)DF (x3 − x4) +DF (x1 − x3)DF (x2 − x4) +DF (x1 − x4)DF (x2 − x3): (1) We can interpret Eq. (1) as the sum of three diagrams, if we represent each of the points, x1 through x4 by a dot, and each factor DF (x − y) by a line joining x to y. These diagrams are called Feynman diagrams. In Fig. 1 we provide the diagrammatic interpretation of Eq. (1). The interpretation of the diagrams is the following: particles are created at two spacetime points, each propagates to one of the other points, and then they are annihilated. This process can happen in three different ways, and they correspond to the three ways to connect the points in pairs, as shown in the three diagrams in Fig. 1. Thus the total amplitude for the process is the sum of these three diagrams. PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 ⟨0|T {ϕ1ϕ2ϕ3ϕ4}|0⟩ = DF(x1 − x2)DF(x3 − x4) + DF(x1 − x3)DF(x2 − x4) +DF(x1 − x4)DF(x2 − x3) 1 2 1 2 1 2 + + 3 4 3 4 3 4 Figure 1: Diagrammatic interpretation of Eq.
    [Show full text]
  • An Introduction to Supersymmetry
    An Introduction to Supersymmetry Ulrich Theis Institute for Theoretical Physics, Friedrich-Schiller-University Jena, Max-Wien-Platz 1, D–07743 Jena, Germany [email protected] This is a write-up of a series of five introductory lectures on global supersymmetry in four dimensions given at the 13th “Saalburg” Summer School 2007 in Wolfersdorf, Germany. Contents 1 Why supersymmetry? 1 2 Weyl spinors in D=4 4 3 The supersymmetry algebra 6 4 Supersymmetry multiplets 6 5 Superspace and superfields 9 6 Superspace integration 11 7 Chiral superfields 13 8 Supersymmetric gauge theories 17 9 Supersymmetry breaking 22 10 Perturbative non-renormalization theorems 26 A Sigma matrices 29 1 Why supersymmetry? When the Large Hadron Collider at CERN takes up operations soon, its main objective, besides confirming the existence of the Higgs boson, will be to discover new physics beyond the standard model of the strong and electroweak interactions. It is widely believed that what will be found is a (at energies accessible to the LHC softly broken) supersymmetric extension of the standard model. What makes supersymmetry such an attractive feature that the majority of the theoretical physics community is convinced of its existence? 1 First of all, under plausible assumptions on the properties of relativistic quantum field theories, supersymmetry is the unique extension of the algebra of Poincar´eand internal symmtries of the S-matrix. If new physics is based on such an extension, it must be supersymmetric. Furthermore, the quantum properties of supersymmetric theories are much better under control than in non-supersymmetric ones, thanks to powerful non- renormalization theorems.
    [Show full text]
  • Canonical Quantization of the Self-Dual Model Coupled to Fermions∗
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Canonical Quantization of the Self-Dual Model coupled to Fermions∗ H. O. Girotti Instituto de F´ısica, Universidade Federal do Rio Grande do Sul Caixa Postal 15051, 91501-970 - Porto Alegre, RS, Brazil. (March 1998) Abstract This paper is dedicated to formulate the interaction picture dynamics of the self-dual field minimally coupled to fermions. To make this possible, we start by quantizing the free self-dual model by means of the Dirac bracket quantization procedure. We obtain, as result, that the free self-dual model is a relativistically invariant quantum field theory whose excitations are identical to the physical (gauge invariant) excitations of the free Maxwell-Chern-Simons theory. The model describing the interaction of the self-dual field minimally cou- pled to fermions is also quantized through the Dirac-bracket quantization procedure. One of the self-dual field components is found not to commute, at equal times, with the fermionic fields. Hence, the formulation of the in- teraction picture dynamics is only possible after the elimination of the just mentioned component. This procedure brings, in turns, two new interac- tions terms, which are local in space and time while non-renormalizable by power counting. Relativistic invariance is tested in connection with the elas- tic fermion-fermion scattering amplitude. We prove that all the non-covariant pieces in the interaction Hamiltonian are equivalent to the covariant minimal interaction of the self-dual field with the fermions. The high energy behavior of the self-dual field propagator corroborates that the coupled theory is non- renormalizable.
    [Show full text]
  • 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
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Solutions 4: Free Quantum Field Theory
    QFT PS4 Solutions: Free Quantum Field Theory (22/10/18) 1 Solutions 4: Free Quantum Field Theory 1. Heisenberg picture free real scalar field We have Z 3 d p 1 −i!pt+ip·x y i!pt−ip·x φ(t; x) = 3 p ape + ape (1) (2π) 2!p (i) By taking an explicit hermitian conjugation, we find our result that φy = φ. You need to note that all the parameters are real: t; x; p are obviously real by definition and if m2 is real and positive y y semi-definite then !p is real for all values of p. Also (^a ) =a ^ is needed. (ii) Since π = φ_ = @tφ classically, we try this on the operator (1) to find Z d3p r! π(t; x) = −i p a e−i!pt+ip·x − ay ei!pt−ip·x (2) (2π)3 2 p p (iii) In this question as in many others it is best to leave the expression in terms of commutators. That is exploit [A + B; C + D] = [A; C] + [A; D] + [B; C] + [B; D] (3) as much as possible. Note that here we do not have a sum over two terms A and B but a sum over an infinite number, an integral, but the principle is the same. −i!pt+ip·x −ipx The second way to simplify notation is to write e = e so that we choose p0 = +!p. −i!pt+ip·x ^y y −i!pt+ip·x The simplest way however is to write Abp =a ^pe and Ap =a ^pe .
    [Show full text]
  • Complete Normal Ordering 1: Foundations
    KCL-PH-TH/2015-55, LCTS/2015-43, CERN-PH-TH/2015-295 Complete Normal Ordering 1: Foundations a,b a,b a,c John Ellisp q, Nick E. Mavromatosp q and Dimitri P. Sklirosp q (a) Theoretical Particle Physics and Cosmology Group Department of Physics, King’s College London London WC2R 2LS, United Kingdom (b) Theory Division, CERN, CH-1211 Geneva 23, Switzerland (c) School of Physics and Astronomy, University of Nottingham Nottingham, NG7 2RD, UK [email protected]; [email protected]; [email protected] Abstract We introduce a new prescription for quantising scalar field theories (in generic spacetime dimension and background) perturbatively around a true minimum of the full quantum ef- fective action, which is to ‘complete normal order’ the bare action of interest. When the true vacuum of the theory is located at zero field value, the key property of this prescription is the automatic cancellation, to any finite order in perturbation theory, of all tadpole and, more generally, all ‘cephalopod’ Feynman diagrams. The latter are connected diagrams that can be disconnected into two pieces by cutting one internal vertex, with either one or both pieces free from external lines. In addition, this procedure of ‘complete normal ordering’ (which is an extension of the standard field theory definition of normal ordering) reduces by a substantial factor the number of Feynman diagrams to be calculated at any given loop order. We illustrate explicitly the complete normal ordering procedure and the cancellation of cephalopod diagrams in scalar field theories with non-derivative interactions, and by using a point splitting ‘trick’ we extend this result to theories with derivative interactions, such as those appearing as non-linear σ-models in the world-sheet formulation of string theory.
    [Show full text]