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 practicalcalculations. 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] - $sd4x(F’[A])2 . (2.20) 1 Eq. (2.20) is the starting point of our entire discussion.We define the gen- eratingfunctional W,[J,“] of the Greenfunctions in the gaugespecified by F to be
(2.21) PUB-7F134-THY
86 B.W. Lee
Pleasenote that the aboveis a definition. Wehave not answeredyet how Green functions in different gaugesare related to eachother, or to the physicalS-ma- trix. We shall return to thesequestions in a future lecture.
2.2. Faddeev-Popov ghosts
As we havenoted in the precedingsection, AF[A] hasthe structure of a determinant.Such a determinantoccurs frequently in path integrals. Considera complex scalarfield p interactingwith a prescribedexternal po- tential V(x). The vacuum-tovacuumamplitude is written
eiw =Nj[dlp] [dvt]exp{iJd4x$(x)[-82 -p2+ V(x)]~(x))
- @et M(x,y))-’ , (2.22) where
M(x,y) = [-a2 - j.2 t V(x)]S4(x - y) - (2.23)
On the other hand,we can evaluateIV in perturbationtheory: it is a sum of vacuumloop diagramsshown in the following figure: v W=V o+v ov+v 0 v +-.
This result can be understoodin the following way. We write
(det M(x,y))-’ = (det MO(x,y))-’
X tdet [S4(x - Y) + A,& - Y) W)J)-* , (2.24) where q$GY) =t-a2 - P2P4(X-y) , (2.25)
1 Y * O-26) -a2-p2tie I> PUB-76/34-THY
Gaugetheories 87
(The ie prescriptionin eq. (2.26) follows from the Euclidicity postulateinher- ent in the definition of path integrals.See refs. [1,2],The first factor on the right-handside of eq. (2.24) may be absorbedin the normalizingfactor N. The secondfactor may be evaluatedwith the aid of the formula det( 1 + t) = exp Tr In(l t L) . Thus,
iw= -Tr In(1 +A,V), (2.27)
which showsvery clearly ?Vas a sum of loops. Next, what if p and(p; wereanticommuting fields? Nothing much changes, except that eachclosed loop acquiresa minus sign.Thus, if p andV? areanti- commuting fields, we have eiCV = N l [dvl[drltlexp(iSd4xl.lt(x)[-a2 -P ’ + W)l~(x)~
- det M(x, y) - exp {Tr In( 1 + AF V)) .
The above.isa heuristic argumentof how integralsover anticommuting c-num- bersshould be defined to be usefulin the formulation of field theory. In fact, Berezindefines the integralover an elementof Grassmann.algebra ci as
dcr = 0 p dCicj = liij . s s It then follows
dc eci”ijci - (det A)‘j2 . (2.28) .v i i
(Prohlern:prove this statement.)We shall not dwell upon the integrationover Grassmannalgebra any further, but ratherrefer you to Berezin’streatise to be cited at the end of this lecture. A nice mnemonicfor the rulesof integration over anticommutingc-numbers is, as told to me by JeanZinn-Justin, that “in- tegrationis equivalentto derivation”. For our purpose,we can write PUB-76134-THY
or symbolically
AFL41=ffj [@Ib-hlexp IiEMFvI , (2.30) wherega(x), Q(X) areelements of Grassmannalgebra. Note that the phaseof the exponent,$.MFq is purely conventional. The generatingfunctional WF[$] of eq. (2.21) can now be written as
exp{ilV&~]} =“$ [dAd[dq]
wherethe effective action Se, is givenby
The fields t, n areusually called the Faddeev-Popovghost fields. They areun- physicalscalar fields which anticommuteamong themselves. (Sometimes it is convenientto think of t ashermitian conjugateof n, but it is not necessary.) In the Lorentz gauge,where we shall write
the term in the effective action bilinearin 2:and 7 is
s d4x $5 Dib qb(x) =$d4x ap .&(x)D;~ nb(x) cc = d4x[V&(x)ar r&(X) - ga’.&(x)F,b&x)Vb(x)] . (2.33)
Even when we regard$ and v as a conjugatepair, the interaction of eq. (2.33) is not hermitian.The sole raisond ’e^treof this term is to createthe determi- nantal factor, eq. (2.29). 2.3. Feynnranrules To describethe Feynmanrules for constructingGreen functions in pertur- bation theory, it is more convenientto couple & and nQalso to their own sources0, andpi, which areanticommuting c-numbers. We define PUB-76/34-THY
Gaugetheories 89
, )I wherewe havesuppressed gauge group indices. The Feynmanrules are obtainedfrom eq. (2.34) in the usual way. We will reviewbriefly the derivationof the Feynmanrules in a simplerexample, an interactingreal scalarfield V. The action S[cp]is dividedinto two parts,
w = qJ[IPI + s, M * (2.35) whereSO [p] is the part quadraticin the field (B,and has the form
S&l =~d44f(acd2 - b2v21. (2.36)
The generatingfunctional i W[J] of the connectedGreen functions is givenby
eiwfJ] =J[dlp]exp{iS[lp) + isd4xJq}
[dpjexp IiS0 [up]+ i d4xJp} _ (2.37) N r s Therefore,we must now compute
eiwo[J1 = NJ [dq]exp {iSo [p] + ild4xJq} . (2.38)
SinceSo is quadraticin p, we can perform the integration. The functional integralin eq. (2.38) gainsa well-definedmeaning by the Euclidicity postulate, that the Greenfunctions eq. (2.37) generatesmust be the analytic continuationsof the well-behavedEuclidean Green functions. Weobtain
ei~o[JI = exp{-jji (2.39) s d4xd4~JW& -Y)JW, where PUB-76/‘34-THY
90 Lt.W. Lee
Eq. (2.37), or
exp {i W[J]) = exp exp{il.Vo[s]) , (2.37’) may be transformedinto a perhapsmore tractableform by the useof the for- mula
F(-i$-)G(x)=C(-i$-)F’Cy)e’X.Yly;O,
which can be provedby Fourier analysis(see ref. [2]),
d4x d4u *,(x - v) &q &y J
X exp OS, [p] + i~d4xJlpl~7=0 . (2.4 I)
In much the sameway, we can developthe Feynmanrules for the gauge theory from eq. (2.34). To be concrete,let us adopt the Lorentz gauge, F, = -ap-4:. We defineSO to be
taP~Qall~=ti:.ptpt-fl-JP.AP . (2.42) 3 The remainderof the action S consistsof the cubic and quartic interactions of the gaugefields and the interaction of the gaugefield with the ghost fields. The Feynmanpropagator for the gaugebosons satisfies $,a, (1 -k )IAF ”(x -u) =g;a4(x -u) , (2.43) and is given by
A;!‘(x - r) =s - d4k e (W4
Note that in this gaugethe ghostfield ,$Yalways appears as ap $?. The generat- ing functional IV, [J,,fl,@] can be written as PUB-76/‘34-THY
Gauge theories 91
exp{ij~LIJP,P,P=’fll exp (is d4xd4+%w’)6~~ &$+--r)
6 6 exp(i tS1Pp, -5 rll +DF(x-)w@@ 1)
(2.45) whereDF(x - u) is the Feynmanpropagator for a massIessscalar field,
& ,-ik: b-y) DF’x -‘~=S~2 k2 $ ic ’ n That is, in this gauge,the Faddeev-Popovghosts are massless.
2.4. Mixed transformations
A few remarkson the integrationover elementsof the Grassmannaigcbra: Sincewe wish to maintain the integrationrule
undera changeof variables,
ci=A& fi ci=,& EjdetA, i=l
we must have
7 dci = (det A)-’ y dEi -
Further,let us considera mixed multiple integral of the form
$rhci I-I de,, i tJ
whereB ’s areelements of the Grassmannalgebra. We considera changeof in- tegrationvariables of the form PUB-76134-THY
B. IV. Lee
and ask how’the Jacobianmust be defined.We considerfirst the change k 0) + 01,a
x =AY t drle - CUB+~Y
= (A - CfB-'p)y + aB-'e e
Note that a! andp areanticommuting. Thus,
Now we perform the transformation(y, 8) + 0, cp), B=Dy+Btp. Since0 and cpare anticommuting numbers,
As a result, we have the rule
s n dxi n dUP= j n dri n dqP det(A - cuB-l@(det B)-’ I
The aboveresult is in accordwith the definition of a “‘generalizeddeter- minant” or Arnowitt, Nath and Zumino. They define the determinantof the matrix
by detC=expTrlnC, with the convention that
Tr C=Cii - CP,, .
This definition allows the following relation to be valid: PUB-76/34-THY
Gauge theories 93
and thereforethe product property of the determinant
det(C1C2) = det C, det Cz .
TO seethis, we set Ci = exp Ji, SO that det Ci = exp Tr Jp NOW Cl Cl =exp{J1 tJztJ& whereJ12is the Baker-Hansdorffderies of commutators. But Tr[JI! Jz] = 0, erc., so that det(CI C,) = exp Tr(J, + J2). Now the ma- trix Ccan be decomposeduniquely into the form ST, where
S= T=
and
a =A -CUB-‘/~, b=arB-I,
u=P, z= B. Thus det C= det S det T= (det a)(det z)-’ . (The last follows from the definition det C= exp[(ln C)ii - (In C)PP].)
2.5. Problems
2.5.I. One shouldrepeat the foregoingarguments for quantumelcctrodynam- its, to obtain the usualFeynman rules in the Lorentz gauge.Let us note that for OL= 0, one getsthe photon propagatorin the Landaugauge; for OL= I, that in the Feynmangauge. What happens to the Faddeev-Popovghost fields in those cases?
2.5.2.Just for the sakeof exercise,quantize electrodynamics in the gauge ,F= aflAP + XA:. Derivethe Feynmanrules.
2.5.3. Show thatJ u dci n dci exp {ciMV ci)- det M, wherec andc ’ are anticommuting. * i PUB-76/‘34-THY
94 B. W.Lee
3. Surveyof renormalizationschemes
3.I. Necessity for a gauge-invariant regufarization
In this lecture, we will developtwo subjectsthat areneeded to understand later Lectures.These are reguIarization and renormalizationof Greenfunctions in quantumfield theory in general,and of Greenfunctions in a gaugetheory generatedby the expression(2.34), in particular. The Greenfunctions generatedby eq. (2.34) areplagued by the ultraviolet infinities encounteredin any realistic quantumfield theory. We aregoing to developa method of eliminating thesedivergences by redetinitions, or renor- malizationsof basicparameters and fields in the theory, in such a way that the gaugeinvariance of the original Lagrangianis unaffectedin so doing. The gaugeinvariance of the action.hasvarious implications on the structure of Greenfunctions of the theory. The precisemathematical expressions which aresatisfied by Greenfunctions due to the gaugesymmetry of the underlying action areknown as the Ward-Takahashi(WT) identities. What we will show is that theseidentities remainform invariant under renormalizationwhich elimi- nate the divergences.This point, that renormalizationcan be carriedout in a way that preservesthe WT identities, is of utmost importancefor the follow- ingreasons.First, it puts such a stringentconstraint on the theory and the re- normalizationprocedure that the renormalizedtheory becomesunique, once the underlyingrenormalizable theory is given.Second, and perhaps more to the point, the unitarity ,of the renormalizedS-matrix is shownby the WT iden- tities satisfied by the renormahzedGreen functions. The latter point requires clarification. In a perturbativeapproach, non-Abelian gauge theories suffer from such severeinfrared singularitiesthat nobody has succeededin defining a sensible S-matrix in this framework. Consensusis that a sensiblegauge theory arises only in a non-perturbativeapproach, wherein gauge fields and other matter fields carrying non-Abeliancharges do not manifest themselvesas physical par- ticles. Physically, this conjectureis at the heart of the hope that color-quark confinementmight arisenaturally from a non-Abeliangauge theory of strong interactions.There is an exception to this, and this is the casewhen the gauge symmetry is spontaneouslybroken. In fact, this latter possibility is directly responsiblefor the revivalof interest in non-Abeliangauge theories a fe.w yearsago, in conjunction with efforts to unify electromagneticand weak in- teractionsin a non-Abeliangauge theory. In this casethere is no difficulty in defining the physical S-matrix, and the unitarity of such a theory is assuredby the renormalizedversion of the WT identities. Even in unbrokengauge theory, PUB-76134-THY
Gauge theories 95 the S-matrix can be definedup to somelower orderin perturbationtheory, andhere again the unitarity of the S-matrix is a consequenceof the WT iden. ti ties. Why is the unitarity such a big issuein gaugetheory? After all, one does not worry that much about the unitarity, say, in a self-interactingscalar boson theory. The reasonis that the quantizationprocedure we adoptedmakes use of a non-positivedefinite Hilbert space,as we can readily seefrom the struc- ture of the gaugeboson propagator, eq. (2.44). Further, the Greenfunctions, of the theory contain singularitiesarising from the Faddeev-Popovghosts be- ing on the massshell. Thus, in order that the theory makessense, these un- physical “particles”, correspondingto the ghost fields and the longitudinal componentsof gaugefields, must decouplefrom the physicalS-matrix. The renormalizedWT identities arenecessary in showingthis. The WT identities areusually derivedby a formal manipulationof eq. (2.34). However,the Greenfunctions generatedby eq. (2.34) arenotoriously ill-defined objectsdue to ultraviolet divergences.It is thereforenecessary to invent a mcansof “regularizing” the Feynmanintegrals which define them without destroyingsymmetry propertiesof the Greenfunctions, so that as long aswe keepa regularizationparameter finite, the integralsare well-defined. It is only then that we can attach concretemeaning to the WT identities. After renormalization,the “regulator” may be removed,and if the renormalization is to be successful,the renormaiizedGreen functions must be finite and inde- pendent of the reguiarization parameter. A well-known regularizationscheme in quantumelectrodynamics is the Pauii-Viilarsscheme, in which oneadds unphysical fields of variablemasses to the Lagrangianin a gaugeinvariant way. After gaugeinvariant renormalization the variablemasses are let go to infinity, and renormaiizedquantities are shown to be finite in this limit. In non-Abeliangauge theory, this deviceis not avaii- able,but an ‘dternativeprocedure, wherein the dimensionaIityof space-timeis continuously varied,was invented by the geniusof ‘t Hooft and Veltman. In the next section,we will give a brief summaryof the renormalization theory a la Bogoliubov,Parasiuk, Hcpp andZimmermann. This will be follow- ed by an introduction to the dimensionalreguiarization of ‘t Hooft andVeit- man.
3.2. BPHZ renormalization
In this section we wig givea brief surveyof renormalizationtheory devel- opedand perfectedin recentyears by Bogoiiubov,Parasiuk, Hepp and Zim- mermann(BPHZ). Nothing will be proved,but we will try to give definitions ‘andtheorems in a precisemanner. PUB-76134-THY
96 8. W. Lee
First, we will givesome definitions. The interaction Lagrangianis a sum of terms1r which is a product of 6, bosonfields andf,: fermion fields with di de- rivatives.The vertex of the ith type arisingfrom JZihas the index 6, definedas
ai = b, t zfi t di - 4 = dim(Xi) - 4 . (3.1)
Let I’ be a one-particleirreducible (LPI) diagram(i.e., a diagramthat cannot be madedisconnected by cutting only one line). Let EB andE, be the num- bersof externalboson and fermion lines, IB andIP the numbersof internal bosonand fermion lines,ni the numberof verticesof the ith type. Then
Ee+21B=&bi, (3.2) i
EF t2$ =CH(.f;:. (3-3) i
The superficialdegree of divergenceof P is the degreeof divergenceone would naively guessby counting the powersof momentain the numeratorand denominatorof the Feynmanintegral. It is
D(r)=&idit21,t31,-4V+4, (3.4) i
.the last two termsarising from the fact that at eachvertex thereis a four-di- mensionaldelta function which allows one to expressone four-momentumin termsof other momenta,except that one delta function expressesthe conser- vation of externalmomenta. Making use of eqs.(3. I), (3.2) and(3.3), we can write eq. (3.4) as
D=&$-EB-,1EI:t4, (3.5) i or DtEBt;EF-4=&$.. (3.6) i
The purposeof renormalizationtheory is to give a definition of the finite part of the Feynmanintegral correspondingto T’, Fr = jl; j-dkl . . . dk, Ir 9 (3.7) + whereI, is a product of propagatorsAF andvertices P, PUB-76/34-THY
Gauge theories 97
(3.8)
The finite part of F, will be denotedby J, and written
Jr = f$ Jdkl . . . dkLR, . (3.9) *+ Weshall describeBogoiiubov ’sprescription of constructingR, from I,. Let us first considera simplecase, in which r is primitively divergent.The diagramI ’ is primitively divergentif it is proper(i.e., IPI), superficially diver- gent(i.e., D(F) 2 0) and becomesconvergent if any line is brokenup. In this case,we may usethe originalprescription of Dyson. Wewrite
Jr = jdkl .. . dkL(l -t’&. , i.e.,
R, =(I -t’)& .
The operationt r must be defined to cancelthe infinity in J,. Jr. is a function of EF t EB - 1 = E - 1 externalmomenta pl, .. . , pEeI,
The operation(1 - tr) onfis definedby subtractingfromf the first D(T’) + 1 termsin a Taylor expansionabout pi = 0,
O(P1, .*- , P&q) =f(O, -0-> 0) + *-* (3.10)
E-1
id ) v ’ where(I = D(r). The operation(I - I r, amountsto makingsubtractions in the integrandI,, the numberof subtractionsbeing determined by the super- ficial degreeof divergenceof the integral. Somemore definitions:,A renormalizationpart is a properdiagram which is superficially divergent(D > 0). Two diagrams(subdiagrams) are disjoint, y1 n y2 # 4 if they haveno lines or verticesin common.Let (7, 9. . . , yC} bea set of mutually disjoint connectedsubdiagrams of r. Then PUB-76/‘34-THY
98 B. W. Lee is definedby contractingeach 7 to a point and assigningthe value 1 to the correspondingvertex. Weare now in a position to describeBogoliubov ’sR operation: (i) if F is not a renormalizationpart (i.e., D(r) Q -l)?
R,=R,, (3.11)
(ii) if r is a renormalizationpart (D(7) > 0),
R, ~(1 -tr)&, (3.12) whereRF is definedas
(3.13) andOr = -PET, Ehere the sum is over all possibledifferent setsof (7i). This definition of R,. in termsOCR? appears to be recursive;in perturbation theory thereis no problem;the R, appearingin the definition of Rr. is neces- sarily of lower order. It is possibleto “solve” eq. (3.13). We refer the interestedreader to Zim- mermann’slectures and merelypresent the result. Again we needsome more definitions beforewe can do this:Two diagrams71 and 72 are said to over- lap, 71 0 72, if none of the following holds:
A P-forestII is a hierarchyof subdiagramssatisfying (a)-(c) below: (a) elementsof U arerenormalization parts;(b) any two elementsof U, 7’ and 7n are non-overlapping;(c) U may be empty. A r-forest U is futl or normal re- spectively dependingon whetherU contains r itself or not. The theoremdue to Zimmermannis
(3.14) whereI; extendsovcc all possible(full, normal and empty) r forests,and in the product [I(-IX) the factors areordered such that th standsto the left of t” if h > u. If h f? (I = 9, the orderis irrelevant.A simple exampleis in order. Considerthe diagramin fig. 3.1. The forests areI$ (empty); 71 (full); 72 (nor- mal); 71,~~ (full). Eq. (3.14) can be written in this caseas PUB-76/34-THY
Gauge theories 99
------1 .( r-7 i I ‘I -; Ql I i Y, Q: - --A Fig. 3.1. Example of the BPHZ definition of subdiagramsin a particular contribution to the four-point function in a Aa4 coupling theory.
R, = (1 - t’: - t’fz t PI t’YZ)lr = (1 - tYl)(l - tr2)Zr .
Note that in the BPHprogram, the R-operationis performedwith respect to subdiagramswhich consistsof vertices,andall propagatorsin I?which con- nect thesevertices. I3y the BPH defmition, the subdiagramy2 abovedoes not containrenormalization parts other than itself and in this sensethe present treatment differs from Salam’sdiscussion. In formulating the BPHtheorem it is necessaryfirst to reguIarizethe prop- agatorsin cq. (3.9) hy somedevice such as
A&J) re; A&J;~, 4 = -i l da exp[ia(p* - m2 tie)] , I and defineI,$-, E) asin eq. (3.9) in termsof A&, E), and then construct R,(r, E) by the R-operation.The BPH theoremstates that R, existsas Y-+ 0 andE -+ O+,as a boundaryvalue of an analytic function in the externalmo ments.Another theorem,the proof of which can be found in the book by Bogoliubovand Shirkov, sect. 26, andwhich is combinatoricin nature,states that the subtractionsimplied by the (1 - tr) prescriptionin the R-operation can be formally implementedby addingcounterterms in the Lagrangian. A theory which hasa finite numberof renormalizationparts is calledrenor- malizable.A theory in which all 6i areless than, or equalto zerois renormaliz- able.In this casethe index of a subtractionterm in the R-operationis bounded byD+En +&!+- 4 which is at most equalto zero by eq. (3.5). In sucha the- ory, only a finite numberof renormalizationcounterterms to the Lagrangian suffice to implementthe R-operation.
3.3. The regularization scheme of ‘t Hooft and Veltman
Kecently, ‘t Hooft and Veltmanproposed a schemefor reguIarizingFeyn- PUB-76/34-THY
100 B. IV.Lee
manintegrals which preservesvarious symmetries of the underlyingLagran- gian.This methodis applicableto electrodynamics,and non-Abeliangauge theories,and dependson the idea of analytic continuation of Feynmaninte- gralsin the numberof space-timedimensions. The critical observationshere are that the global or local symmetriesof thesetheories are independent of space-timedimensions, and that Feynmanintegrals are convergent for suff-- ciently small, or complex IV, whereN is the “complex dimension” of space- time. Let us first reviewthe nature of ultraviolet divergenceof a Feynmandia- gram.For this purpose,it is convenientto parametrizethe propagatorsas
A,(p2) = $ i datexp[iol(p2 - nz2+ ie)] . (3.15). 0 Makinguse of this representation,we can write a typical Feynmanintegral as
X exp {i C ai((7i2- .rf + ie)} , (3.16) i
whereI is the numberof in ternal propagatorsin r, L the numberof loops, and 1,) .. . , 1, may take any valuesfrom I to L. The momentumqi carriedby the jth propagatoris a linear function of loop momentaki and externalmomenta p,nBThe exponenton the right-handside of eq. (3.16) can thereforebe writ- ten as I I C a&qf- mf+ ie) = I’ CkiAij(a)kj t ic m ki Bi,(a)p, - 7 “i(mf - ie) i=l i,i ,
+kT+ktk-B*p- C c+(mf - ie) ,
wherek is a column matrix with entrieswhich arefour-vectors. The matrices A andB arehomogeneous functions of fi?st degreein CY’S,and A is symmetric. Ubon translatingthe integrationvariables
k+k’=k tA+Bp PUB-76/34-THY
Gauge theories 101
anddiagonalizing the matrix A by an orthonormaltransformation on k’, we can perform the loop integrationsover ki in eq. (3.16). The result is a sum over termseach of which hasthe form
X %+-~[~p~C(or)~p t C c+(rnf - if)]}, (3.17) i
where TX/I . . . v is a tensor,typically a product of gP09s,Ai is the ith eigen- valueof the matrix A, andSi is a positive numberwhich is determinedby the tensorialstructure of F,. Note that Ai is homogeneousof first degreein ois. The matrix C is
C= BTA-‘B
andis also a homogeneousfunction of first degreein [Y’SIn this parametriza- tion, the ultraviolet divergencesof the integralappear as the singularitiesof the integrandon the right-handside of eq. (3.17) arisingfrom the vanishingof somefactors IIi[Ai(~)JSi as someor all (Y’Sapproach to zero in certain orders, for example,
ar, car2 <... where(rl, r2 , .. . , rJ) is a permutationof (1, 2, *.. , I). See,for instance,a more detailedand careful discussionof Hepp. The ‘t Hooft-Veltman regularizationconsists in defining the integralF, in n dimensions,n > 4 (one-timeand (II - I)-spacedimensions) while keeping externalmomenta and polarizationvectors in the first four dimensions(i.e., in the physical space),performing the ?I - 4 dimensionalintegrals in the space orthogonalto the physical space,and then continuing the result in ?z.(For sin- gle-loopgraphs one may perform all n integrationstogether.) For sufficiently smalln, or complex n, the subsequentfour-dimensional integrations are con- vergent. To seehow it works, considerthe integral PUB-76/34-THY 102 B. W.Lee X n (kd * el)exp[i cc-+($ - rnf + if)], (3.18) i where,now, the ki aren-dimensional vectors. As beforewe can expressthe qi as linear functions of the kj and the external momentapi, wherethe pi have only first four-componentnon-vanishing. From now, we shall denotean n-di- mensionalvector by (k,K)A wherek^ is the projection of k onto the physical space-timeand K = k - k. Thus,p = ($, 0). Eq. (3.18) may be written as a sum of termsof the form (3.19) The integralsover kj can tie performedimmediately, usingthe formulas d”-4KK h’ . . . K 42r exp(-jA K2) s % Q2 )( (jA)-n/2+2-r , wherethe summationis over the elementsu of the symmetricgroup on 2~ob- jects (aI, ‘~2,. .. , a>), and =n-4. %% ThusF, of eq. (3.19) will have the form PUB-76/34-THY Gaugetheories 103 wheref(n) is a polynomial in n andri is a non-negativeinteger depending on the structure of II K, - K, in eq. (3.19). For sufficiently small n < 4, the sin- gularitiesof the integrandas some or all 0;‘sgo to zero disappear. The reasonsthis regularizationpreserves the Ward-Takahashiidentities of the kind which will be discussedare, firstly, that the vector manipulations suchas k'"(2p +q = [(p +k)S- m2] -(p2- m2), or partial fractioning of a product of two propagators,which arenecessary to verify theseidentities “by hand”, arevalid in any dimensions,and, secondly, that the shifts of integrationvariables, dangerouswhen integrals are divergent, arejustified for small enough,or complexn, sincethe integralin questionis convergent. The divergencein the original integralis manifestedin the polesof Fr(fz) at n = 4. Thesepoles are removed by the R-operation,so that Jr(n) as defined by the R-operationis finite and well-definedas IZ + 4. Actually, to our know- ledgethe proof of this hasnot appearedin the literature,except for the origi- nal discussionof ‘t Hooft and Veltman. Hepp’sproof, for example,does not really apply here,since the analytical discussionof Heppis not tailored for this kind of regularization.However, the argumentof ‘t Hooft and Veltman is sufficiently convincingand we haveno reasonto believewhy a suitablemodi- ftcation of Hepp’sproof, for example,of the BPHZ theoremshould not go through with the dimensionalregularization. The abovediscussion is fine for theorieswith bosonsonly. Whenthere are fermionsin the theory, a complication may arise.This hasto do with the oc- currenceof the so-calledAdler-Bell-Jackiw anomalies. The subjectof anoma- lies in Ward-Takahashiidentities hasbeen discussed thoroughly in two excel- lent lecturesby Adler, and by Jackiw, and we shall not go into any further de- tails here.In short, the Adler-Bell-Jackiwanomalies may occur whenthe veri- fication of certain Ward-Takahashiidentities dependson the algebraof Dirac gammamatrices with y5, suchas -y,,y5 + y5 7, =.O.Typically, this happens whena propervertex involving an odd numberof axial vector currentscannot PUB-76134-THY 104 B.W. Lee be regularizedin a way that preservesall the Ward-Takahashiidentities on such a vertex, and asa consequencesome of the Ward-Takahashiidentities haveto be broken.The occurrenceof theseanomalies is not a metter of not being cleverenough to ilevisea properregularization scheme: for certainmodels sucha schemeis impossibleto devise.The dimensionalregularization does not help in such a case,due to the fact that 75 and the completely antisymmetric tensordensity ehPypare unique to four dimensionsand do not allow a logical- ly consistentgeneralization to H dimensions.When there are anomalies in a spontaneouslybroken gaugetheory, the unitarity of the S-matrix is in jeopardy since,as we shall see,the unitarity of the S-matrix, i.e., cancellationof spuri- ous singularitiesintroduced by a particular choice of gaugeis inferredfrom the Ward-Takahashiidentities. Grossand Jackiw haveshown that, in an Abe- lian gaugetheory, thc.occurrenceof anomaliesruns afoul of the dual require- mentsof ’unitarity and renormalizabilityof the theory. Thus, a satisfactory theory should be free of anomalies.Fortunately, it is possibleto construct modelswhich are anomaly-free,by a judicious choiceof fermion fields to be included in the model. Thereare two ‘lemmas” which make the aboveassertion possible. One is that the anomaliesare not “renor- mafized”, which in particular meansthat the absenceof anomaliesin lowest orderinsures their absenceto all orders.This wasshown by Adler and Bardeen in the context of an SU(3) versionof the u-model,and by Bardeenin a more generalcontext which encompassesnon-Abelian gauge theories. The secondis the observationthat all anomaliesare related;in particular,if the simplest anomalyinvolving the vertex of threecurrents is absentin a model,so areall other anomalies.This can be inferred from an explicit construction of all anom- aliesby Bardeen,or from a moregeneral and elegantargument of Wessand Zumino. Let us concludewith a simpleexample of dimensionalregularization: the vacuumpdarization in scalarelectrodynamics. The Lagrangianis and the relevantvertices are shown in fig. 3.2. Thereare two diagramswhich contribute to the vacuumpolarization, shown in fig. 3.3. The sum of these contributions is d”k 1t2k+?4,(2k +d, - 2((k +d2-p2kw,] Z=e2j- (3.20) pv Weuse the exponentialparametrization of the propagatorsto obtain PUB-76/34-THY Gaugetheories 105 Fig. 3.2. Photon-scalar meson vertices in chargedscalar electrodynamics. k Fig. 3.3. Second order vacuum polarization diagrams in charged scalar electrodynamics. X [Ox: +dp +P)” - 2((k +pj2 -p2jgpv]. (3.21) The exponentis proportional to (CY+ P)k2 + 2k . pa + ap2 - (a + /.I)(/.? - ie) so we may write =-e2bgp, - p2g,,) 0r da 0i $3(s)2j Enn Xexpi (a+/3)k2+ SP2 -(cY+p)(p2-ie) i 1 PUB-76/34-THY (3.22) The first term is explicitly gaugeinvariant and only logarithmicaffydivergent, so that a subtraction will makeit convergent.it is the secondterm that re- quiresa carefulhandling. We needthe formulas d”k - exp(iAk2) = Gw (2&Q” ’ 7 / k2 exp(iXk2) = $ (-in) i expb 4’““), (2&X)” s (3.23) so that tfie secondterm, 12, is X exp i -$p2~-(atp)(&ie) A.- I 11 (a+P) x (i(l -in)-[-.&+(otp)p~)’ e iml4 = -2ie 2g -----ldadfl6(1 -u-~jj~ei~l”ap’-P2f”l pv (2&r)” 0 if x [il-l(1 - $2) + i(orpp2 -/.l’)] . (3.24) PUB-76/‘34-THY Gauge theories 107 For sufficiently small n, n < 2, the A-integration is convergent, and - dh ,ik(A+iel(I -2’ +i~) oJ hrtl2-I = p dh -& {h’-“/2 exp[iX(A t ic)]) = 0. (3.25) 0 So the dimensionalregularization gives the gaugeinvariant result, 12=0. 3.4. Problem Repeatthe vacuumpolarization calculation in spinorelectrodynamics using the dimensionalregularization. 4. The Ward-Takahashi identities 4.1. Notations Oneof the problemsin discussinggauge theories is that notationswill get cumbersomeif we are to put explicitly space-timevariables, Lorenb indices andgroup representation indices. We will thereforeuse a highly compactno- tation. For simplicity in notation, we will assumethat the gaugegroup in questionis a simpleLie group.Extension to a product of simpleLie groups, suchas SU(2) X U(I), is not too difficult. Wewill agreeto denoteall fields by r&. Again for simplicity in notation we will assume#j to be bosons.Inclusion of fermionsdoes not presentany diff- culty in our discussions,but we will haveto be mindful of their anticommut- ing nature.Thus, for the gaugefield A;(x) i standsfor the groupindex a, the Lorentzindex 1-1,and the space-timevariable x. Summationand integration overrepeated indices will be understood.Thus (piz=sd4x q A;(x)A~~(x) + 0.., (4-I) wherethe dotted portion includescontribution from other speciesof fields. PUB-76/34-THY 108 B. W. Let The infinitesimal local gaugetransformation may be written as l#Jp$;=$it(A; tt~~j)ea, (4.2) whereO. = 8,(x,) is the space-timedependent parameter of the groupG. We choose@i to be real, SO that t; = ~“‘(xa - xp4(xn - Xi) (4.3) is real antisymmetric, where7; is the representationof the generatorLa of G in the basis&. The’inhomogeneousterm q is non-vanishingonly for the gaugefields hi” =$ars4(xi - xa)ija, b for & = A:@$ = 0 otherwise. (4.4) Weshall also define Notice that (4.5) where f le =pw(xo - Xb)64(Xa - xc) , $g - SinceI$! ’ is non-vanishingonly whenj refersto a gaugefield, let us write i = (CJA, Xi), i = (d, V,xi>. Then t; = gpy.pd64(Xi - xa)S4(xa- Xi) , PUB-76/‘34-THY Gauge theories 109 g(t; ij! - t; A.) =fabcj-d4x E”(xa - x)S4(x - xi) -& s4(x - xb) t S4(Xb- x)S4(x - Xi)-& S4(x - xa)1 =pbcld4x S4(x - xi) a [“‘(x~ - x)S4(x - xb)] i3Xfl =pbbe&sd4x S4(x - xj)b4(x -mx,)S4(x - xb) , which is equaltog times the right-handside of eq. (4.5). The gaugeinvariance of the action can be statedin a compactform, (4.7) The linear gaugecondition we discussedin lecture 2 may be written as Fa[@I = fai @ 3i (4-g) where fai = a; for Qi = A$xi) , (4.9) a; = tPbap64(xi - xa) . (4.10) Enthis notation, the effective action is written as f&-jt’h f>d = s[@l -&F; [@I+ ‘$Mab[@hb 3 (43) with 4.2. Becchi-Rouet-Stora transformation The WT identities for gaugetheories have been derived in a numberof dif- ferent ways. The most convenientway that I know of is to considerthe re- PUB-76/34-THY 110 B. IV.Lee sponseof the effective action, eq. (4.1 l), to the so-calledBecchi, Rouet, Stora(BRS) transformation.It is a global transformationof anticommuting type which leavesthe effective action invariant. Here1 shall follow a very ele- gant discussionof Zinn-Justin givenat the Bonn SummerInstitute in 1974. The BRS transformationfor non-Abeliangauge theory is definedas “$=Di”QX, (4.13a) when?6 X is an anticommutingconstant. Note that if we identify O. = T)~Sh, we seeimmediately that the action S[$] is invariantunder (4.13a). There are two important propertiesof the BRS transformationwhich we shall describe in turn. (i) The transfonnatiorrs on 9t and pa are nilpotent, i.e., s2ei=0, (4.14) s2qa= 0. (4.15) Proof: Eq. (4.14) follows from “(D;Q=O. (4.16) Indeed Since?jO and vb anticommute, the coefficient of vaqn in the first term on the right-handside may be antisymmetrizedwith respectto a and b. It vanishesas a consequenceof eq. (4.5). To show eq. (4.15), we note that (4. I 7) by the Jacobi identity. QED. PUB-76/34-THY Gauge theories Ill (iij The BRS transformations leave the effective action SEn of eq. 14.II) invariant Proof: As noted aboveS[$] is invariantunder eq. (4.13a).We further note that s =~FaMabqbsA-~Fa~D;qb6hi0 (4.19) P by the definition of Mab, eq. (4.12). For,lateruse, we remarkfinally that the metric [d&dta dn,] is invariant under the BRS transformationof eqs.(4.13). I want you to verify it. 4.3. The Ward-Takahashi identities for the generating functio?zal of Green functions We will first derivethe Ward-Takahashiidentity satisfiedby lV,[JI of eq. (2.3 l), Z,[~~eiWr;I’l=NS[d~d5:dq]exp{id,ff[~,~,q]+iS-~i). (4.20) We first note that, accordingto the rule of integrationover anticommuting numbers (4.21) we have (4.22) s [WdEdq] 4 ‘,exp CiSeff Ml + iJi 3> = 0 , becauseSe-r contains t andq only bilinearly. In eq. (4.22) we makea change of variablesaccording to the BRS transformations(4.13). Since a changeof integrationvariables does not changethe valueof an integral,we have O=j[d@d[dq] -~a[$j+iJiE,D,!‘[@]qb exp(iS,ff+iJi$i) I (4.23) I PUB-76/34-THY 112 B. W. Lee Eq. (4.23) is the WT identity as first derivedby Slavnovand Taylor. We can rewrite it in a differential form involvingZF. We define (ZFlba z Ni s Id&Wtl Carib exp We, + iJ&$ e (4.24) It is the ghost propagatorin the presenceof externalsources Ji- It satisfies M (4.25) ab [LsiSJ 1(zj =&ti ne. i+ I wIl1leave the derivation of eq. (4.25) asarrexercise. Eq. (4.23) can be writ- ten as aFa c+& 1ZF[J] - J,D,!’ [+& 1tzj7)ba =o* (4.26) Eq. (4.26) is in the form written down by &n-Justin and Lee. It is the WT identity for the generatingfunctional of Greenfunctions, and as suchit is rathercumbersome for the discussionof renormaiizability,since, as we have seenin lecture 3, the renormalizationprocedure is phrasedin terms of (single- particle irreducible) propervertices. Nevertheless, eq. (4.26) wasused to de- duce “by hand” consequencesof gaugesymmetry or renormalizationparts by Zinn-Justin and myself. We do not haveto do this, sincewe know better now. Eq. (4.26) will be useful in discussingthe unitarity of the S-matrix later, how- ever. ‘4.4. The Wurd-Takuhushi identities: inclusion of ghost sources To discussrenormalization of gaugetheories, we haveto considerproper verticessome of whoseexternal lines areghosts. For this reason,the ghost fields 5 and q shouldhave their own sources.We thereforereturn to eq. (2.34), X exp[iCQ#, t, 4 + U +@a + J#,.i,l . (4.27) For the ensuingdiscussion, it is more convenientto consideran object (4.28) and define PUB-76134-THY Gaugetheories 113 (4.29) In eq.(4.28), Ki and-La are sources for the compositeoperators Dffr$]qa and $?f,a Q, Q, respectively.It follows from eqs.(4.16) and(4.17) that Z is in- variant under the BRS transformations(4.13). Note further that Ki is of anti- commutingtype* and (4.30) (4.3 1) The invarianceof Z is expressedas or (4.32) Weneed one more equation, g=ftig. (4.33) i Let us examinethe consequencesof eqs.(4.32) and(4.33). We perform a changeof variables (4.34a) (4.34b) &$=-iFa& (4.34c) Eq. (4.32) tells us that Z is invariant undersuch a transformation,and the in- tegrationmeasure [d&d$dp] is also,thanks to PUB-76/34-THY 114 B.W. Lee A?&(-) (4.3%) “qiKi~ ’ (4.35b) Thus, the changeof variables(4.34) in eq. (4.29) leadsto X exp[i{X + t - /I + /3t - 9 +Jr.$>] = 0. (4.36) Next, the equationof motion for n is (4.37) Combiningeqs. (4.33) and(4.37), we obtain (4.38) Eqs.(4.36) and (4.38) are the basisfor derivingthe WT identity for the generatingfunctional of propervertices. 4.5. The Ward-Takahashi identities for the generating functional of proper vertices The generatingfunctional of propervertices is obtainedfrom IV, W=-itnZ, by a Legendretransformation. We define (4.39a) (4.39b) PUB-76/34-THY Gauge theories 115 (4.39c) wherewe haveused the samesymbols for the expectationvalues of fields as for the integrationvariables. The generatingfunctional for propervertices is P-C?. (4.40) As usual,we havethe relationsdual to eqs.(4.39), (4.41a) (4.41b) (4.41c) It is easyto verify that if Wand F dependon parametersQ, suchas K or L in our case,which arc not involvedin the Legendretransformation, they satis- f-Y (4.42) From eqs.(4.36) and (4.38) we can derivetwo equationssatisfied by F, (4.43) (4.44) It is important to observethe correspondencebetween eqs. (4.32) and (4.33), andeqs. (4.43) and(4.44). If we now define (4.45) PUB-76/34-THY 116 B. W. Lee f&a%,=%. (4.47) a The functional r carriesa net ghostnumber zero, wherewe definethe ghostnumber Np as N&II = 1 3 NJK]=-1 s N,LEl = -1, NJL] =-2, NJ4 =o. ClearlyI ’ may be expandedin termsof the ghostnumber carrying fields: Substituting the expression(4.48) in eqs.(4.46) and(4.47), differentiating with respectto Q, andsetting all ghostnumber carrying fields equalto zero, we obtain (4.49) (4=50) Theseare the equationsfirst derivedby me from (4.26) by a complicated functional manipulation.These are the fully dressedversions of eqs.(4.7) and (4.12), PUB-76/34-THV Gaugetheories I17 4.6. Prob Iems 4.6.1. Convinceyourself that the measure[dddldq] is invariantunder the BRS transformation. 4.6.2. Show that Proveeq. (4.25). 4.6.3. Show that Af(c)=O, i not summed, dci ac i whereci is an anticommutingnumber. Prove eq. (4.37). 4.6.4. Derive the WT identity for Z[J, fl, $1, 5. Renormalization of pure gauge theories 5. I. Renormalization equation We areready to discussrenormalization of non-Abeliangauge theories basedon the WT identity for propervertices derived in the last lecture. Let us recall that our Feynmanintegrals are regularizeddimensionally so that for a suitably chosenn not equalto 4, all integralsare convergent. Thus, we can perform the BogotiubovR-operation after the integralhas been done, insteadof making the subtractionof eq. (3.10) in the integrandas Zimmer- manndictates. In fact this is the procedureused by Bogoliybov,Parasiuk and Hepp.Further, insteadof makingsubtractions at pi = 0, we will choosea point whereal1 momenta flowing into a renormalizationpart areEuclidian. PUB-76/‘34-THY 118 B. W.Lee For a vertex with II externallines, this point may be chosento be --pi” = a2, pi - pi = c&z - I). This is to avoidinfrared divergences.At this point the squareof a sum of any subsetof momentais alwaysnegative, so that the am- plitude is rea1and free of singularities. For simplicity we first considera puregauge ’theory.Inclusion of matter fields, suchas scalarand spinorwith renormalizableinteractions present no difficulty. In particular, couplingof gaugefields with scalarmesons will be treatedin chapter6. Wemay write down the propervertex as a sum of terms,each being a pro- duct of a scalarfunction of externalmomenta and a tensorcovariant, which is a poiynomial in the componentsof externalmomenta carrying available Lore& indices.All renormalizationparts in this theory haveeitherD = 0 or 1. The self-massof a gaugeboson is purely transverseas we shall see,so that it alsohas effectively D = 0. Thus, only the scalarfunctions associatedwith ten- sor covatiantsof lowest order aredivergent as n + 4. (Note also that vertices involving external ghostIines have lower superficial degreesof divergencethan simplepower counting indicates.This is because.C, always appears as 8 E,&,.) The basicproposition on renormalizationof a gaugetheory is the following. If we scalefields and the couplingconstant according to Q1i = z’/2(e)@i’’ K.I = .i?/2(~)K’ i’ r;a = S2(e)g’ a’ L a = zqE)L' a’ a = Z(E)c? , X(e). g’ g= (5.1) +z(e)z1’2(e) ’ wheree = n - 4 is the’regularizationparameter, and chooseZ(e), X(E) and Z(e) appropriately,then is a finite (that is, as e = D - 4 4 0) functional of its arguments$ ‘, 5’. $, K’, L’ and or’. Under the renormalizationtransformation of (5.1). eqs.(4.45) and (4.46) become PUB-76/34-THY Gauge theories 119 We will expandloopwise, Wehave Supposethat our basicproposition is true up to the (n - 1) loop appioxima- tion. That is, up to this order, all divergences are removed by rescahg OF fields and parametersas in eq. (5.1). We supposethat we havedetermined the renormalizationconstants up to this order, (x)n-, = 1 +X(r) + .** + X(n-1) " P-7) Wehave to show that the divergencesin the n-loop approximationare also removedby suitably chosenz,r , 2n andx,r. Following Zinn-Justin,we introduce the symbol wherethe superscriptr denoteshere the quantities rcnormalizedup to the (rr - 1) loop approximation.We can write eq. (5.8) as with (5.10) PUB-76/34-THY 120 B. W.Lee The right-handside of eq. (5.9) involvesonly quantitieswith lessthan II loops, it is finite by the induction hypothesis.Further, divergencesin subdiagramsof r(,, areremoved by renormalizationsup to (n - 1) loops.Thus, the only re- mainingdivergences in r(,l) are the overall ones.Let us denoteby Odiv the divergentpart. If we adjust finite parts of I& appropriately,we have (5.Ii) (5.12) 5.2. Solution to renormalization equation The divergent part off&) is a solution of the functional differential equa- tions (5.1 I), (X1.2).We recall that (5.13) (5.14) (5.15) wherethk functional operator9 is givenby 9= clot61 3 (5.16) (5.17a) (5.17b) PUB-76/34-THY Gaugetheories 121 From now on, for the interest of notational economy,we will drop the super- script r, until further notice. An important aid in solvingeq. (5.14) is the observationthat p=0. (5.18) We will prove this in steps.First, we verify by direct computationthat $g=o, =D;qa+f ahcq b q c&qa6. (5.19) 90 i Eq. (5.19) is a direct consequenceof eqs.(5.16), (5.17) that the BRS transfor- mation on 4 andqa is nilpotent. Next we note that (5.20) wherewe haveused the fact that (5.21) (5.22) Direct computationsyield 6‘ 30 sqo) s *r(o) SD6;r(()) 50,= “3 [email protected] =qa63 “3’ “3 1 PUB-76/34-THY 122 B.IK Lee s2r(0) s r(o) s 2 r(o) “Qoqo) = - -s 50) -++---- ‘Qf. sKjs’tl, sV* 6L*6Qa s% ( 1 Thus which,proves eq. (5.18). The fact that 9 is nifpotent meansthat, in general,99 for arbitrary 9 = Y(i$, & 1, K, L) is a solutibn of eq. (5.14) 8(sw).= 0 * (5.23) The questionis whether thereare other solutionsnot of the form ~7. This questionalso arisesin renormalizationof gaugeinvariant operators, and has beenstudied, in particular, by Kluberg-Sternand Zuber.They alsoadvanced a conjecture: they suggestedthat the generalsolution to eq. (5.14) is of the form W(‘,)P” =GM+99[$,t,tl,KLl, (5.24) whereG[$] is a gaugeinvariant functional, Recently, Jogtekarand I wereable to prove this, mostly by the effort of the first author. The proof is tedious,and I believethat it can be improvedas to rigor, eleganceand length. For this reason,I will not presentthe proof. It is easyto seethat the form (5.24) satisfieseq. (5.14) and that G[$] of eq. (5.25) is not expressibleas gS, in this caseat least. It is the completenessof eq. (5.24) which requiresproof. Eq. (5.15) [or (5.4)] is immediately solved.It meansthat r&J fh Es~5 K, Ll= r&l [A O,V,Kj f a;$, L] + ‘jQ’[ti* -5V, L.1,(5.26) whereQ ’ is transverse:3% Qj = 0 - PUB-76/34-THY Gaugetheories 123 The quantity (l&} div is a local functional of its arguments.If we asignto K and L the dimensionsD(K) = 2 andD(L) = 2, then Z. hasthe uniform di- nmsion 0, and SOdoes Irfn)ldiv. It hasNg [r’fil,] = O. SinceNg [ 91 = t.1, it follows that iVg[ 9]= -1 in eq. (5.24). In order that the right-handside of eq. (5.24) is local, both 9 and Qmust be separatelylocal. The most general form of {r{,r)}div satisfying the aboverequirements is wherecr, p, y arein generaldivergent, i.e., e-dependent,‘constants.Using the explicit form of 9, eqs.(5.17) we can write (5.28) In eq. (5.27) G[$] is equalto S[I$).This is so becausethe action is the only local functional of dimensionfour which satisfieseq. (5.25). Because (5.29) Thus, combiningeqs. (5.28), (5.29), we obtain (5.30) Recall that in cq. (5.30) d, f, 77,K, L andg are renormalizedquantities up to the (n - 1) loop approximagon.We shall denotethem by (#),-t, etc. If we now define(Z),, (Z),, (X), by PUB-76/34-THY 124 93.IV. Lee etc. , (5.32) (5.33) and choose+), :(,+ xtn) to be %) = -4 a * lw) , ?(a = 37 + IN4 9 (5.34) %) - q,z) - :2(n) = 2+j, then {~fnj)drv is eliminated: I’&, is a finite functional in terms of (@j,,, . .. I and ($),,. Furthermore,since (@), = (Zji’2$, .. . , eqs.(5.3), (5.4) are ako true for the newly renormnlizedquantities. This completesthe induction. Note further that (5.35) is finite. The renormalizedfield $* transformsunder the gaugetransformation as 6. Renormalizationof theories with spontaneouslybroken symmetry 6.1. hclusion of scalar fields In chapter5, we detailedthe renorrhalizationof pure gaugetheories. Let us considernow a theory of gaugebosons and scalarmesons. Let !4j = (.qxj, s&)) 9 PUB-76/34-THY Gauge theories wheres, arescalar mesons, and let be the vector which defies the gauge.The action for the scalarfields is of the form where(Ds), is the covariantderivative acting on S, V(s) is a G-invariantquar- tic polynomial in s andof dimensionat most zero. Weshall write (6.2) [W, ta] = [6M,t ”] =0, (6.3) whereM& is the renormalize$massmatrix for the scalarmesons. We shahas- .sumefor the moment that Mr is a positive semi-definitematrix. Let us discussrenormalization. Almost everythingwe discussedin the last sectionholds true. In particular we have WCn ,ldiv = G [$I + 9 W> E,rl, K t3 , wherewe have written @i= {A,, sa) andKi = {K,, &},A, beingthe gauge fields, A, = A;(x), t = (a, p, x). Now we have (6.4) where Y’ is a G-invariantquartic polynomial in s. This term is eliminatedby renormalizationsof coupling constantsappearing in V(s). 9 takes the form PUB-76/34-THY 126 B. IV. Lee whered$ is a G-covariantcoefficient. (It could be i$, or somethingelse, such as the d-type couplingin SU(3), for example.)This gives {r&p=& +pA&l) +ia$]- ‘2a Gag a 1 [ t a where Thesedivergences are eliminated if we renormalizeA, &q, K,, La andg as be- fore and = 3/z 9 K - &. l12p p = -$- 1’2(c3r , %! s a’ a= u Q- ( z s > ( s 1 which leaves invariant, and shift the si fields by s; = Sk - u,w P and choose PUB-76/34-THY Gaugetheories An important lesson to be learned here is that in a generallinear gauge, scalar fields can developgauge-dependent vacuum expectation values, which arein- nocuousfrom the renormalizationpoint of view. 6.2. Spontaneously broken gauge symmetry Let us considerthe casewhere M ’ is not positive semi-definite.It is by now well known that undersuch circumstances spontaneous breakdown of the gaugesymmetry takesplace, and someof the scaIarfields and someof the (transverse).gaugebosons combine to form massivevector bosons.We will give herea very brief discussionof the Higgsphenomenon. We define V0 by If M’ is not positive semi-definite,So = 0 is no longera minimum of the po- tential V,. Let sa = 11,be the absoluteminimum of Vo, (6.7) “2Vo = ‘Nl$ , CR,, positive definite. q$sp s=u G-invarianceof the potential Vo is expressedas 6 Vo s ta -=o. Q d 6sP Differentiating this with respectto s7 and settings = U, andmaking use of eqs.(6.7), (63, we obtain (6.9) Therefore,there are as many eigenvectorscorresponding to the eigenvalue zero as thereare linearly independentvectors of the form C&U,. If the dimen- sion of G is N and the little groupg which leavesu invariant hasdimension m, thereare iV - ~11eigenvalues of 7K2 which vanish. For fltture use,it is useful to definea vector 1: by PUB-76/34-THY 128 B. W. Lee where ~4;is to be defined. Sinceall representations,except the identity repre- sentation, of a Lie group are faithful, there areN - m independentvectors of this form. Now we define (6.10) This matrix is of a bloc; diagonalform; moreoverif a or b refer to a generator of the little groupg, (+)ab vanishes.Now form (6.11) This is a projection operator,P2 = P, onto the vector spacespanned by vec- tors of the form t~pt$. This spaceis N - m dimensional, trP=N- tn. Eq. (6.9) may be written as We renormalizc the gaugefields, ghost fields and gaugecoupling constant as before, and renormslizes, accordingto sa =Z’/J(sa ” (Y+u ’a +su a) 5 and determine6 U, = (6 t.ta)l + (6 I& t .._by the condition that the diver- gencesof the form -(6 S[$]/S s,)A(e)u, in the n-loop approxkation be can- celled by the displacementof the renormalizedfields s:, (6 u&. (See the dis- cussionin subsect.6.1.) The renormalizedvacuum expectation value z& is to be determined by the condition s2rr positive semi-definite. 6 f#J;s4; I#=u ’ PUB-76/34-THY Gaugetheories 128 Weequate the gaugefixing term C,!Jto (numerically) Then the termsin the action quadraticin renormalizedfields and coupling constants(excluding renormaiization counter terms)are wherewe havesuppressed the superscriptr altogether.The propagatorsfor the gaugebosom, scalars and ghostfields are, respectively, [Aj$“(k,c~)]~~=~ ( ’ ) , k2-p2+ie cb which can be written, in a representationin which p2 is block diagonal,as 01 [&12a) the latter holding for a, b beingone of the m indicescorresponding to genera- tors of the little groupg; [AF(k2sa)], =(1 -J ’lq (k2 - A2 + ,), tz~(~)ab(kl_:2+iQ)hlpe9(6.12b) PUB-76/34-THY 130 B. W. Lee tAdk2p a& = ( k2 _ b,z + , )., (6.12~) If the theory is to be sensible,and gaugeinvariant, then the poleswhose locations dependon the gaugeparameter LY cannot be physical, and the parti- cles correspondingto such polesmust decouplefrom the Smatrix. If this is the case,as we shall show, then thereare N - m massivevector bosons,m masslessgauge bosons, and (N - m) lessscalar bosons than we started out with. This is the Higgsphenomenon. is a theory of this kind renormalizable?The answeris yes, becausethe Feynmanrules of the theory, including the propagatorsabove are thoseof a renormalizabletheory, and the WT identity, eq. (6.3), and the ensuingdiscus- sion in chapter 5 and subsect.6.1 hold true whetheror not#is positive def- inite. That is, by the methodsdiscussed, we can construct a finite I” in this casealso. The expansioncoefficients of T” about r$:= uf, where 62r’ positive definite, 646 (b; IQ=uZ then are the reduciblevertices of the renormalizedtheory. I shall not describe the detailsof the renormalizationprogram since they havebeen described in many papers,most recently in my paper(ref. [SO]), but the principle involved should be clear. But an additional remarkis in order: the divergentparts of variouswave- function and coupling-constantrenormalization constants are independentof MF. This has to do with the fact that theseconstants are at most logarithmi- cally divergent,and insertion of the scalarmass operators (whose dimension is 2) rendersthem finite. For detailedarguments, see ref. [50]. 6.3. Gauge indepeizdemzeof the S-matrix What remainsto be done is to demonstratethat the unphysicalpoles in the propagatorsineq. (6.12). which dependon the parameterL\! and someof which correspondto negativemetric particles,do not causeunwanted singularities in the renormalizedSmatrix. Weshall do this by provingthat the renormalized S-matrix is independentof the gaugefixing parameterCX. To ensurethat the S- PM-76/34-THY Gauge theories 131 matrix is well-defined,I shall assumethat after the spontaneousbreakdown of gaugesymmetry thereis at most onemassless gauge boson in the theory. Beforeproceeding to the proof, the following ihustration is useful. For simplicity let us considera X@4theory. The generatingfunctional of Green functions is where Whathappens if we insteadcouple the externalsource to $J+ @? Wecan write the generatingfunctional as Z[iJ =~~[WJexpW[~l+iit @+ d~~)l- (6.15) Wecan expressZ in termsof Z,, (6.16) where F(4) = c#J-)- (b3 . Let us considera four-point function generatedby Z[i], GJ1,2,3,4] =(-Q4 S4-mJ Sit lPSWA3)6jW - Whateq. (6.16) tells us may be pictured asfoliows: wherewe haveshown but a classof diagramsthat emergein the expansionof the right-handside of eq. (6.16). The part of the diagramenclosed by a dotted PUB-76134-TH’r’ 132 B. W. Lee squareis a Greenfunction generatedby ZO[q. Let us now considerthe two point functions Aj and AJ generatedby Z[j] andZu [J], + . ..a So, if we examinethe propagatorsnear p2.= pf, we find zi ZJ lim Ai = - lim AJ=------(6.17) P2+ P”-P,2 7 P2-$ P2 -$ wherethe ratio u = (zj/zJ)1’2 (6.18) is givendiagramma tically by 0=1*-i- e+... The renormalizedS-matrix is defined by k;-$ _ S’(k,, . . . >=n lim ----gyG(kl....), (6.19) i=l k&,2 whereG ”is the momentumspace Green function. Let us considerthe unrenor- malizedS-matrix definedfrom (?j, Si”(kl, ..e ) = n lim(kf - pz)Gj(kl, . . . ) . (6.20) Clearly only thesediagrams Of ej in which there arepoles in all momentum variablesat $ will survivethe amputation process.(In fig. 6.1, thereare poles in k: andk: at cl,“.) Thus S;(kl, . . . ) = crN’2SJ(k, 9 a.. ) ) (6.2 1) PUB-76/34-THY Gaugetheories 133 whereN is the numberof the externalparticle;. It follows from eqs.(6.1 Q- (6.21) that SpSfES’, (6.22) and we reachan impor tant conclusion:if two Z’s differ only in the external sourceterm, both of them yield the samerenormalized S-matrix. We now come back to the originalproblem, and ask what happensto ZF [.Il if an infinitesimal changeis madein F, Weare dealing with unrenormalizedbut dimensionallyregularized quantities in eq. (6.23). To first orderin AF, we have Now, makinguse of the WT identity (5.23), s [&%dVl IF, - iJi~~D~[~]~~)exp{i~~~ +iJ&} = 0, we can write eq. (6.24) as Since :AF, [i& 1iJi=-.z, i PUB-76134-THY 134 B.M Lee we obtain zF+AF - ‘F =iJiN s [dtdrld@Jexp{iSeK+ iJ&.i, X G-iAFabPJ~a~~I~J~&~. (6.25) But, eq. (6.25) meansthat to lowest order in AF, ‘F+AF =fVj [d$dEdqJexp{is,, + iJiai} , (6.26) where Thus,an ir@itesimal changein the gaugecondition correspondsto changing the sourceterm by an infinitesimalamount. But we havealready shown that the renormahzedS-matrix is invariantunder sucha change!Thus WlF+AF= WIF : (6.27) A few final remarks:(i) In’ the previouslectures when we discussedrenor- malization,we definedthe renormatizationconstants in respectto their diver- gentparts. The wave-functionrenormalization constants used in this lecture aredefined by the on-shellcondition (6.17). Thesetwo arerelated to each other by a finite mdtiplicative factor. To seethis, observethat we can make the propagatorsfinite by the renormalizationcounter termsdefined in the pteviouslectures. The propagatorsso renormalizeddo not in generalsatisfy the on-shellcondition hm Ak(p’) = L!-- P2-$ P”-P,2 ’ but a finite, final renormalizationsuffices to make them do so. (ii) We can de- fine the couplingconstants to be the value of a relevantvertex when all physi- cal external lines areon massshell. Then PUB-76/34-THY Gaugetheories 13s References Lecture I For the geneialdiscussion of path integralformalism applied to gaugethe- ories,see Prof. Faddeev’lectures,s and [l J E. Abers and B-W. Lee, Phys. Reports 9C (1973) 1. [2] S. Coleman, Secret symmetry, Lectures at 1973 Int. School of Physics “&tore Majorana”, to be published. [SJ N.P. Konopleva and V.N. Popov, Kalibrovochnye polya (Atomizdat, Moscow, 1972). Originalliterature on the quantizationof gaugefields includes (4 J R.P. Feynman, Acta Phys. Polon. 26 (1963) 697. [S] B. De Witt, Phys. Rev. 162 (1967) 1195,1239. [6J L-D. Faddeev and V.N. Popov, Phys. Letters 25B (1967) 29. [7J V.N. Popov and L.D. Faddeev, Perturbation theory for gaugeinvariant fields. Kiev ITP report, unpublished. [S] S. Mandelstam, Phys. Rev. 175 (1968) 1580. (91 M. Veltman, Nucl. Phys. B21 (1970) 288. IlOJ G. ‘t Hooft. Nucl. Phys. B33 (1971) 173. The axial gaugewas first studiedby 111J R. Arnowitt and S.I. bkkler, Phys. Rev. 127 (1962) 1821. In conjunction with L. Faddeev’lecturess included in this volume,see [12] L.D. Faddeev, Theor, Math. Phys. 1 (1969) 3 [English trabslation I (1969) 11. Lecture 2 Gauge theories cnn be quantized in other gauges than the ones discussed in this chapter.In particular the following papersdiscuss quantization and/or re- normalizationof gaugetheories in gaugesquadratic in fields: [13] G. ‘t Hooft and M. Veltman, Nucl. Phys. B50 (1972) 318. [14J S. Joglekar, Phys. Rev. DlO (1974) 4095. Differentiation andintegration with respectto anticommutingc-numbers arestudied and axiomatizedin [15] F.A. Berezin, The method of second quantization (Academic Press,New York, .1966) p. 49; R. Amowitt, P. Nath and B. Zumino, Phys. Letters 56 (1975) 81. Lecture 3 For renormalizationtheory see: [ 161 F.J. Dyson, Phys. Rev. 75 (1949) 486,1736. 1171 A. Salam, Phys. Rev. 82 (1951) 217; 84 (1951) 426. (181 S. Weinberg, Phys. Rev. 118 (1960) 838. [ 191 N.N. Bogoliubov and D-V. Shirkov, Introduction to the theory of quantized fields (Interscience, NY, 1959) ch. N, and references therein. [ZOJ K. Hepp, Comm. Math. Phys. 1 (1965) 95; Th&orie de fa renormatisation (Springer, Berlin, 1969). PUB-76/34-THY 136 B. W. Lee (211 W. Zimmermann, Lectures on elementary particles and quantum field theory, ed. S. Deser, M. Grisaru and H. Pendleton (MJT Press,Cambridge, 1970) p. 395. The dimensionalreguhuization, in the form discussedhere, is due to: 1221 G. ‘t Hooft and M. Veltman, Nucl. Phys. E44 (1972) 189; 1231 C.G. Bollini, J.-J. Giambiagi and A. Gonzales Dominguez, Nuovo Cimento 31 (1964) 550; [24] G. Cicuta and E. Montaldi, Nuovo Cimento Letters 4 (1972) 329. A closely reiatedregularization method (analytic regularization)is discussed in: [25] E.R. Speer, Generalized Feynman amplitudes (Princeton Univ. Press, Princeton, 1969). Excellent reviewson the Adler-BelEJackiwanomalies are: [26] S.L. Adler, Lectures on elementary particles and quantum field theory, ed. S. Deser, hi. Grisaru and 11.Pendleton (MIT Press,Cambridge, 1970). [27] R. Jackiw, Lectures on current algebra and its applications (Princeton Univ. Press, Princeton, 1970). For a completelist of anomalyvertices, involving only currents(not pions) see: [28] W.A. bnrdeen, Phys. Rev. 184 (1969) 1848. [.29] J. Wess and B. Zumino, Phys Letters 37B (1971) 95. The following papersdiscuss the problemof anomaliesin gaugetheories: (301 D.J. Gross and R. Jackiw, Phys. Rev. D6 (1972) 477. [31] H. Ceorgi and S.L. Glashow,.Phys. Rev. D6 (1972) 429. 1321 C. Bouchiat, J. Iliopoulos and P. Meyer, Phys. Letters 38B (1972) 519. (331 W.A. Bxdeen, Proc. 16th Conf. on high-energy physics, 1972, vol. 2, p. 295. The last referencegives a concisealgorithm for dimensionalregularization valid for scalarloops. Our prescriptionagrees with it for this case. For an excellent discussionof dimensionalregularization and a diagrammatic discussionof the Ward-Takahashiidentities, see: i34] G. ‘t Hooft and M.J.G. Veltman, Diagrammar, CERN yellow preprint 1973. Lecture 4 Th’eWard-Takahashi identities werefirst discussedin the context of quan- tum electrodynamicsin 135) J.C. Ward, Phys. Rev. 78 (1950) 1824. [36] Y. Takahashi, Nuovo Cimento 6 (1957) 370. The WT identities for the non-Abeliangauge theories were first discussedin: (371 A.A. Slavnov, Theor. Math. Phys. 10 (1972) 152 [English translation 10 (1972) 99. (381 J.C. Taylor. Nucl. Phys. B10 (1971) 99. and usedextensively to study renormalizationin [39] B.W. Lee and J. Zinn-Justin, Phys. Rev. D5 (1972) 3121,3137; D7 (1973) 1049. [40) G. ‘t Hooft and M. Veltman, Nuci. Phys. B50 (1972) 318. The WT identity (5.49) (4.50) wasobtained in: 1411 B.W. Lee, Phys. Letters 46B (1974) 214; Phys. Rev. 9 (1974) 933.