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COURSE 3 GAUGE THEORIES Benjamin W COURSE3 GAUGE THEORIES BenjaminW. LEE Fermi National Accelerator I,aboratory, Batavia, III. 60510 R. Balinn and J. Zinn-Justin. edr, Les Houches. Session XXVIII, 1975 - Mkthodes en theories des champs /Methods in fietd thmry PUB-76/34-THY Contents 1. Yang-Mills ficlds 19 1.1. Introductory remarks 79 1.2. Problem: Coulomb gauge 81 2. Perturbation expansion for quantized gauge theories 81 2.1. General linear gauges 81 2.2. Paddecv-Popov ghosts 86 2.3. Feynman rules 88 2.4. Mixed transformations 91 2.5. Problems 93 3. Survey of renormalization schemes 94 3.1. Necessity for a gaugcinvariant rcgularization 94 3.2. BPllZ renormalization 95 3.3. The regularization scheme of ‘t.Hooft and Vcltman 99 3.4. Problem 107 4. The Ward-Takahashi identities I07 4.1. Notations 107 4.2. Bcccbi-Rouet-Stora transformation 109 4.3. The Ward-Takahashi identities for the generating functional of Green functions 111 4.4. The Ward-Takahashi identities: inclusion of ghost sources 112 4.5. The Ward-Takahashi idenlities for the generating functional of proper vertices 114 4.6. Problems 117 5. Renormalization of pure gauge theories 117 5.1. Renormalization equation 117 5.2. Solution to renormalization equation 120 6. Renormalization of theories with spontaneously broken symmetry 124 6.1. Inclusion of scalar ficids 124 . 6.2. Spontaneously broken gaugesymmetry 127 6.3. Gauge independence of the S-matrix 130 References 135 PUB-76/34-THY Gauge theories 79 1. Yang-Millsfields I. 1. Introductory remarks ProfessorFaddeev has discussed the quantizationproblem of a system which is describedby a singularLagrangian. For the following, we shall as- sumethat the studentis familiar with the path integralformalism, and the quantizationof the Yang-Millstheory. The following remarksare intended to agreeon notations. The Yang-MillsLagrangian, without matter fields, may be written as For simplicity we shall assume,the underlying gauge symmetry is a simple compactLie groupG, with structure constantsfabc. The Lagrangian(1.1) is invariant under the gaugetransformations, = U(e)[L,Ap) -f o-‘(e)all U(E)] t+(E) > (I-2) wherethe E arespace-time dependent parameters of the groupG, U-‘(E) = M(E) and the t’ are the generators. Thesegauge transformations form a group,i.e., if g’g = g”, then (Problem: prove this statement.) The infinitesimal versionof the gaugetransformation is L,,SA; = +,a/ - fabcA;ebLC , Of SA” = -.-iaMea +fahcebA;. P (1.3) It is preciselythis freedomof redefmingfields without altering the Lagran- gian that lies in the heart of the subtlety in quantizinga gaugetheory. In the languageof the operatorfield theory, to quantizea dynamicalsystem one has to find a set of initial valuevariables, p ’s andq ’s, which arecomplete, in the PUB-76134-THY 80 B. iv. Lee sensethat their values at time zero determinethe valuesof thesedynamicai variablesat all times. It is only in this casethat the imposition of canonical commutation relationsat time zero will determinecommutators at all times and define a quantumtheory for a gaugetheory. This can neverbe done be- causewe can alwaysmake a gaugetransformation which vanishesat time zero. That is, it is impossibleto find a completeset of initial-valuevariables in a gaugetheory unlesswe removethis freedomof gaugetransformations. To quantizea gaugetheory, it is necessaryto choose,agauge, that is, im- poseconditions which eliminate the freedomof making gaugetransforma- tians, and seeif a completeset of initial-valuevariables exist. Thereis a specialgauge, called the axial gauge,in which the quantization isparticularly simple.It is definedby the gaugecondition that ?pd”P(x) = 0 , (1.4) whereTJ is an arbitrary four-vector.In this gauge,the vacuum-to-vacuumam- plitude can be written as ejW=Nf[cU~] II S@“(x)-rj) w whereN is a normalizingfactor. Thereis in principle no reasonwhy eq. (1.5) cannot be usedto generate Greenfunctions, by the usualdevice of addinga sourceterm in the action. That is, WC define the generatingfunctional of the connectedGreen functions WAL ’;l by X exp {iJd4x{J?(x) + $(x)A pa(x)]) . (l-6) However,the Feynmanrules would not be manifestly Lorentz covariantin this gaugeand it is desirableto developquantum theory of the Yang-Mills fields in a wider classof gauges. As ProfessorFaddeev explained, eq. (1.6) is an injunction that the path in- tegralis to be performednot over all variationsof A:(x), but over distinct or- bits of A:(x) under the action of the gaugegroup. To implement this idea,a PUB-76/34-THY Gaugetheories 81 “hypersurface”was chosen by the gaugecondition 11-A a = 0, so that the hy- persurfacein the manifold of all field intersectseach orbit only once.The problemwe poseourselves is how to evaluateeq. ( 1.6)if we are to choosea hypersurfaceother than the one for the axial gauge. 1.2. Problem: Coulomb gauge The gaugedefined by ViAf = 0 is called the Coulombgauge. In this gauge the two space-liketransverse components of Ai are the q’s, and the two space- like transversecomponents of F& arethe p’s, Expressthe Lagrangian( I. 1) in termsof the Coulombgauge variables A!; 0; ri”f; f"; F;: 81;. Referringto ProfessorFaddeev ’lecture,s com- for this gauge. ."“n 2. Perturbationexpansion for quantizedgauge theories 2.1. General linear g(wgcs The foregoingexample, the axial gaugecondition, is but one of the ways to eliminate the possibility of gaugetransformations during the periodthe tem- poral developmentof a quantizedsystem of gauge fields is studied.Clearly this is not the only way, andin fact, we could define a gaugeby the equation F= [A;, cp]= 0 for all a , (2.1) providedthat, givenA, “, and other fields which we shall collcctivcly call p thereis one and only one gauge transformationwhich makeseq. (2.1) true. For convenience,we shall dealonly with the casesin which Fa is linear in the bosonfields Ai and p. In this lecture,however, we shall be concernedprimari- ly with the instancein which Fa dependson AZ alone. Beforeproceeding further, let us pausehere briefly to reviewa few facts about gloup representations.Let g,g’ E G. Thengg ’ E C and PUB-76/34-THV The invariant Hurwitz measureover the groupG is an integrationmeasure of the groupmanifold which is invariantin the sensethat dg’ = d($g) . (2.3) If we parametrizerl(g) in the neighborhoodof the identity as W(g) = 1 + i@L, + O(f2) , (2.4) then we may choose dg= ? de= g=l. 0.5) Considernow the integral where(A,h(x))g denotesthe g-transformof A:(x), as defined by eq. (1.5). The quantity AF[Ai] is gaugeinvariant, in the sensethat A,’ [(-4j)gJ =j c dg’(x) n S(P[(A;(x))g ’gJ) 0,x =s n d@‘&(x) n 6 (P [@,h(x)@‘]) X ax = $ l-J &‘I-$ g9 6wa [(AiybW”l) = A,’ [A;], (2.7) wherewe madeuse of eq. (2.3). According to eq. (l.S), we can write the vacuum-tovacuumamplitude as where5 = f d4xJ(x) is the action. Since PUB-76134-THY Gaugetheories 83 we’mayrewrite eq. (2.8) as eiW =N j@4AF[Al~fl dg(x)~ S(F”IAg(dl) X X by 601 -AbO))expWWII. (2.10) , In the integrandwe canmake a gaugetransformation A:(x) + (A,b(x))g-‘. Under the gaugetransformation of eq. (l-5), the metric [dA], the action S[A] andAF[A] remaininvariant, so eq. (2.10) may be written as ei’V=NJ[dA]A,[A]tinx s(Fg[A(x)])eiS[Aj , (2.1 J) Let us assumethat Then we have which is a constantindependent ofA. Therefore,this constantmay be ab sorbedin N, and ,iW =N &-~]A~[A] II 6(F”[A])eiS[Al. (2.12) This is the vacuum-to-vacuumamplitude evaluated in the gaugespecified by eq. (2. I>. Let us evaluateAF [A]. Sincein eq. (2.12) this is multiplied by II6 (F, [A]), we needonly to know AF[A] for A which satisfieseq. (2.1). Let us makea PUB-76134-THY gaugetransformation on A so that F= [A] = 0. For g in the neighborhoodof the identity, then (2.13) whereDp is the covariantderivative qb =6a,bap -gf,&; . (2.14) Therefore,from eq. (2.6), we seethat where (2.16) That is, (2.17) Mere,we can afford to be sloppy about the normalizationfactors, as long as they do not dependon the field variablesA,,. The factor A,[A] can be evaluatedfrom eq. (2.17) for variouschoices of I? The examplewe will consideris the so-calledLorentz gauge, PUB-76134-THY Gauge theories 85 FU = afin; + P(x), whereC ”(X) is an arbitrary function of space-time.Under tire infinitesimal gaugetransformation (1.3) ofA:, F, changesby ar SFQ=--(6 a -gf AC)eb g ab P ubcp ’ so that bz,XlM~lb,J4 = -aiD”%Y“(x - y) - The appearanceof the delta functionaJJJ6 (Fa [A]) makeseq. (2.12) not very amenableto practical calculations.We could havechosen as gauge condi- tion: F”[A]-c”(x)=O, (2.18) with an arbitrary space-timefunction co, insteadof eq. (2.1). The determinant AF[A] is still the sameas before,that is, is givenby eq. (2.17)‘ andclearly the left-handside of eq. (2.12) is independentof co. Thus, we may integratethe right-handside of the equation eiw =N[[dA]AF[A] n 6(F”[A] - cO)eis~AI (2.19) overc,(x) with a suitableweight, specifically with exp(zjd”x c:(x)) ) where01 is a real parameter,and obtain eiw =NJ[d4]AF[A]exp iS[A] - $sd4x(F’[A])2 . (2.20) 1 Eq. (2.20) is the starting point of our entire discussion.We define the gen- eratingfunctional W,[J,“] of the Greenfunctions in the gaugespecified by F to be (2.21) PUB-7F134-THY 86 B.W. Lee Pleasenote that the aboveis a definition. Wehave not answeredyet how Green functions in different gaugesare related to eachother, or to the physicalS-ma- trix. We shall return to thesequestions in a future lecture.
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