Abers Lee Gauge Theories

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COURSE3

GAUGE THEORIES

BenjaminW. LEE

Fermi National Accelerator I,aboratory,

Batavia, III. 60510

R. Balinn and J. Zinn-Justin. ed r, Les Houches. Session XXVIII, 1975 - Mkthodes en

theories des champs /Methods in fietd thmry



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Contents

1. Yang-Mills ficlds

1.1. Introductory remarks

1.2. Problem: Coulomb gauge

2. Perturbation expansion for quantized gauge heories

2.1. General linear gauges

2.2. Paddecv-Popov ghosts

2.3. Feynman rules

2.4. Mixed transformations

2.5. Problems

3. Survey of renormalization schemes

3.1. Necessity for a gaugcinvari ant rcgularization

3.2. BPllZ re normalization

3.3. The regularization sch eme of ‘t.Hooft and Vcltman

3.4. Problem

4. The Ward-Takahashi identities

4.1. Notations

4.2. Bcccbi-Rouet-Stora transformation

4.3. The Ward-Takahashi identities for the generating functional of Green

functions

4.4. The Ward-Taka hashi identities: inclusion of ghos t sources

4.5. The Ward-Takahashi idenlities for the generating functional of proper

vertices

4.6. Problems

5. Renormalization of pure gauge heories

5.1. Renormalization equation

5.2. Solution to renormalization equation

6. Renormalization of theorie s with spontaneo usly broken symmetry

6.1. Inclusion of scalar ficids

. 6.2. Spontaneously broken gaugesymmetry

6.3. Gauge ndependence of the S-matrix

References

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79

1. Yang-Mills ields

I. 1. Introductory remarks

Professor addeev asdiscussedhe quantizationproblemof a system

which is described y a singularLagra ngian. or the following, we shall as-

sume hat the student s familiar with the path ntegral ormalism, and he

quantizationof the Yang-Mills heory. The following remarksare ntended o

agree n notations.

The Yang-MillsLagran gian, ithout matter fields, may be written as

For simplicity we shall assumetheunderlyinggauge ymmetry s a simple

compactLie groupG, with structure constantsfabc.

The Lagrangian1.1) is invariant under he gaugeransformations,

= U(e)[L,Ap) -f o-‘(e)all U(E)] t+(E) >

(I-2)

where he E arespace-time ependent arameters f the groupG, U-‘(E)

= M(E) and he t’ are he generators.

These augeransformations orm a group, .e., if g’g = g”, then

(Problem:prove his statement.)

The nfinitesimal versionof the gaugeransformation s

L,,SA; =

+,a/ - fabcA;ebLC ,

Of

SA” =

P

-.-iaMea +fahcebA;.

(1.3)

It is precisely his free domof red efming ields without altering he Lagran-

gian that lies n the heart of the subtlety in quantizinga gauge heory. In the

language f the operator ield theory, to quantizea dynamicalsystemonehas

to find a set of initial valuevariables, ’s andq’s, which arecomplete, n the



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80

B.

v.

Lee

sense hat their values at time zero determine he valuesof thesedynamicai

variables t all times. t is only in this case hat the imposition of canonical

commutation relationsat time zero will determi necommutatorsat all times

anddefine a quantum heory for a gauge heory. This can neverbe done be-

causewe can alwaysmakea gauge ransformationwhich vanishes t time zero.

That is, it is impossible o find a completeset of in itial-valuevariablesn a

gauge heory unlesswe remove his fr eedomof g auge ransformations.

To quantizea gaugeheory, t is necessaryo choose,a auge, hat is, im-

poseconditions which eliminate he f reed omof making gaugeransforma-

tians, and see f a completeset of initial-valuevariables xist.

Ther e s a specialgauge, alled he axial gauge,n which the quantization

isparticularly simple. t is definedby the gauge ondition that

?pd”(x) = 0

P

, (1.4)

whereTJs an arbitrary four-vector. n this gauge, he vacuum-to- vacuum m-

plitude can be written as

ejW=Nf[cU~] II S@“(x)-rj)

w

whereN is a normalizing actor.

There s in principle no reasonwhy eq. (1.5) cannot be used o generate

Gre en unctions, by the usualdeviceof addinga source erm n the action.

That is, WC define he generatingunctional of the connectedGreen unctions

WAL’;l by

X exp {iJd4x{J?(x) + $(x)A pa(x)]) .

(l-6)

However, he Feynman uleswould not be manifestly Lorentz covariant n

this gaug e nd t is desirable o developquan tum heory of the Yang-Mills

fields n a wider classof gauges.

As Professor addee v xplained,eq. (1.6) is an njunction that the

path in-

tegral s to

be

performednot over all variationsof A:(x), but overdistinct or-

bits of A:(x) under he action of the gaug e roup.To implemen t his idea,a



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81

“hypersurface”waschosenby the gauge ondition 11 A a = 0, so that the hy-

persurfacen the manifold of all field intersectseachorbit only once.The

problemwe poseourselvess how to evaluate q. ( 1.6) f we are o choose

hypersurface ther than the one or the axial gauge.

1.2. Problem: Coulomb gauge

The gauge efinedby ViAf = 0 is called he Coulombgauge.n this gauge

the two space-like ransverseomponents f Ai are he q’s, and he two space-

like transverse omponents f F& are he p’s,

Express he Lagrangian I. 1) n termsof the Coulombgauge ariables ;

0; ri”f; f"; F;: 81;. Referring o Professor addeev’secture,com-

for this gauge.

."“n

2. Perturbation

xpansion

for quantized

auge

theories

2.1. Gener al linear g(wgcs

The foregoingexample, he axial gauge ondition, is but one of the ways o

eliminate the possibility of gauge ransformations uring he period he tem-

poral development f a quantizedsystemof gauge fields s studied.Clearly

this is not the only way, and n fact, we could define a gauge y the equati on

F= [A;, cp]= 0

for all a ,

(2.1)

provided hat, givenA,“,

andother fields which we shall collcctivcly call p

there s one and only o negauge transformationwhich makeseq. 2.1) true.

For convenience, e shall dealonly with the casesn whichFa is linear n the

boson ieldsAi and p. In this lecture,however,we shall beconcerned rimari-

ly with the nstance n which Fa depends n AZ alone.

Beforeproceedingurther, let us pause erebriefly to reviewa few facts

about gloup representations.et g,g’ E G. Thengg’ E C and



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The nvariant Hurwitz measure ver he groupG is an ntegrationmeasure f

the groupmanifold which is invariant n the sensehat

dg’ = d($g) .

(2.3)

If we parametrize l(g) in the neighb orhood f the dentity as

W(g) = 1 + i@L, + O(f2) ,

(2.4)

then we may choose

dg= ? de= g=l.

0.5)

Consider ow the integral

where A,h(x))g denotes he g-transformof A:(x), as defined by eq. (1.5).

The quantity AF[Ai] is gaug envariant, n the sense hat

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-$ g 6wa (AiybW”l)

9

= A,’ [A;],

(2.7)

wherewe madeuseof eq. (2.3).

According o eq . (l.S), we can write the vacuum-tovacuumamplitude as

where5 = f d4xJ(x) is the action. Since



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83

we’may ewrite eq. (2.8) as

eiW =N

j@4AF[Al~fl

X

dg(x)~ S(F”IAg(dl)

X by 601 -AbO))expWWII.

,

(2.10)

In the ntegrandwe canmake a gaugeransformationA:(x) +

(A,b(x))g-‘.

Under he gauge ransformationof eq. (l-5), the metric [dA], the action

S[A]

andAF[A] remain nvariant,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 assumehat

Then we have

which s a constant ndependent

fA.

Therefore, his constantmay be ab

sorbed n N, and

,iW =N

&-~]A~[A] II 6(F”[A])eiS[Al.

(2.12)

This is the vacuum-to-vacuummplitudeevaluatedn the gauge pecifiedby

eq. (2. I>.

Let us evaluateAF

[A].

Since n 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



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gauge ransformationon A so that F= [A] = 0. For g in the neighborhood f

the dentity, then

(2.13)

whereDp is the covariantderivative

qb =6a,bap -gf,&; .

(2.14)

Therefore, rom eq. (2.6), we see hat

where

That is,

(2.16)

(2.17)

Mere,we can afford to be sloppy about the normalization actors,

as

long as

they do not de pend n the field variables ,,.

The factor A,[A] can be evaluated rom eq. (2.17) for variouschoicesof

I? The examplewe will consider s the so-called orentzgauge,



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Gauge heories 85

FU

= afin; + P(x),

whereC”(X) s an arbitrary unction of space-time. nd er ire nfinitesimal

gaugeransformation 1.3)

ofA:, F,

changes y

ar

SFQ=--(6

g

a -gf

ab P

AC)eb

ubcp ’

so that

bz,XlM~lb,J4 = -aiD”%“(x - y) -

Y

The appearancef the delta functionaJJJ6 Fa

A])

makeseq. (2.12) not

very amenableo practicalcalculations.We could havechosen sgauge ondi-

tion:

F”[A]-c”(x)=O,

(2.18)

with an arbitrary space-timeunction co,

nsteadof eq. (2.1). The determinant

AF[A]

is still the same s before, hat is, is givenby eq. (2.17)‘ andclearly the

left-handside of eq. (2.12) is independent f co. Thus, we may ntegrate he

right-hand ide of the equation

eiw

=N[[dA]AF[A] n 6(F”[A] - cO)eis~AI

overc,(x) with a suitableweight, specifically with

exp(zjd”x c:(x)) )

(2.19)

where01s a r eal parameter, nd obtain

eiw =NJ[d4]AF[A]exp

iS[A] - $sd4x(F’[A])2 .

1

(2.20)

Eq. (2.20) is th e starting point of our entir e discussion.We define he gen-

erating unctional W,[J,“] of the Gre en unctions in the gau ge pecifiedby

F

to be

(2.21)



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86 B.W.Lee

Please ote that t he above s a definition. Wehavenot answered et how Gr een

functions n different gauges re elated o eachother, or to the physicalS-ma-

trix. We shall return to thesequestions n a future lecture.

2.2. Faddeev-Popov ghosts

As we havenoted n the preceding ection,AF[A] has he structure of a

determinant.Such a determinantoccurs requently n path ntegrals.

Consider complex scalar ield p interactingwith a prescribed xternal po-

tential V(x). The vacuum-to vacuum mplitude s 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 evaluate V n perturbat ion heory: it is a sum of

vacuum oop diagr ams hown n the following figure:

v

W=V o+v ov+v

0

v +-.

This result can be understoodn 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) = -a2 - P2P4(Xy) ,

(2.25)

1

-a2-p2tie

I>

*

O-26)



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Gauge heories

87

(Th e e pr escription n eq. (2.26 ) follows from the Euclidicity postulate nher-

ent in the definition of path ntegrals.See efs. [1,2],The first factor on the

right-hand ide of eq. (2.24) may be absorbedn the normalizing actor N. T he

second actor 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 ?V sa sum of loops.

Next, what if p an d p; wereanticommuting ields?Nothing much changes,

except that e achclosed oop acquires minus sign.Thus, f p andV? areanti-

commuting ields, we have

eiCV = N

l [dvl drltlexp(iSd4xl.lt(x)[-a2P’ + W)l~(x)~

- det M(x, y) - exp Tr In( 1 + AF V)) .

The above.is heuristic argument f how ntegralsoveranticommuting -num-

bersshould be defined o be useful n the formulation of field theory. In fact,

Berezindefines he integralover an elementof Grassmannalgebrai as

s

dcr = 0 p

s

dCicj = liij .

It then follows

.v

dc

i

eci”ijci - (det

A)‘j2 . (2.28)

i

(Prohlern:prove his statement.)We shall not dwell upon he integrationover

Grassman n lgebra ny further, but rather efer you to Berezin’s reatise o be

cited at the end of this lecture. A nice mnemonic or the rulesof integration

over anticommutingc-numberss, as old to me by JeanZinn-Justin, hat “in-

tegratio n s equivalent o derivation”.

For our purpose,we can write



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or symbolically

AFL41 fj [@Ib-hlexpiEMFvI

(2.30)

where a(x), Q(X) areelements f Gr assmann lgebra.

Note that the phase f the exponent

$.MFq

s purely conventional.

The generatingunctional WF[$] of eq. (2.21) can now be written as

exp{ilV&~]} =“$ [dAd[dq]

where he effective action Se, is givenby

The fields t, n a reusually called the Fad deev-Popovhost ields. They areun-

physicalscalar ields which anticommuteamong hemselves.Sometimest is

convenient o think of t ashermitian conjugateof n, but it is not necessary.)

In the Lorentz gauge,wherewe shall write

the term n the effective action bilinear n 2:and7 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)

Evenwhen we regard andv as a conjugatepair, the interaction of eq. (2.33)

is not hermitian.The sole aisond’e^tre f this term s to create he determi-

nantal actor, eq. (2.29).

2.3. Feynnran ules

To describe he Feyn man ules or constructingGreen unctions n pertur-

bation theory, t is more convenient o coupl e & and nQalso o their own

sources , andpi, which areanticommutingc-numbers.We define



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Gaugeheories

89

)I

,

wherewe havesuppressedauge roup ndices.

The Feynman ulesareobtained rom eq. (2.34) n the

usual

way. We will

reviewbriefly th e derivationof the Feynman ules n a simplerexample,an

interacting eal scalar ield V. The action S[cp] s divided nto two parts,

w = qJ IPI s, M *

(2.35)

whereSO p] is the part quadratic n the field (B,andhas he form

S&l =~d44f(acd2 - b2v21.

(2.36)

The generatingunctional

i

W[J] of the connecte dGree n unctions s givenby

eiwfJ] =J[dlp]exp{iS[lp) + isd4xJq}

r

[dpjexp

IiS0 [up]+ i

d4xJp} _

(2.37)

N

s

Therefore,we must now compute

eiwo[J1 = NJ

[dq]exp

{iSo [p] + ild4xJq} .

(2.38)

SinceSo is quadratic n p, we can perform he integration.

The functional integral n eq. (2.38) gainsa well-definedmeaning y the

Euclidicity postulate, hat the Gre en unctions eq. 2.37) generates ust be

the analytic continuationsof the well-behaved uclideanGree n unctions.

We obtain

ei~o[JI = exp{-jji

s

d4xd4~JW& -Y)JW,

(2.39)

where



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90

Lt.W. Lee

Eq. (2.37), or

exp

i W[J]) =

exp

exp{il.Vo s]) ,

(2.37’)

may be transformednto a perhapsmore tractable orm 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 ef. [2]),

J

d4x d4u *,(x - v) &q

&y

X expOS, [p] + i~d4xJlpl~7=0 .

(2.4 I)

In much the sameway, we can develop he F eynman ules or the gauge

theory from eq. (2.34). To be concrete, et us adopt the Lorentz gauge,

F, = -ap-4:. We defineSO o be

taP~Qall~=ti:.ptpt-fl-JP.AP .

3

(2.42)

The remainder f the action S consistsof the cubic and quartic nteractions

of the gauge ields and the nteraction of the gaugeield with the ghost ields.

The Feynmanpropagatoror the gauge osons atisfies

$,a, 1 -k AF”(x -u) =g;a4(x -u) ,

( )I

(2.43)

and s given by

d4k

A; ‘(x - r) =s - e

(W4

Note that in this gauge he ghost ield ,$Y lwaysappears sap $?.The generat-

ing functional IV, [J,,fl,@] can be written as



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Gauge heories

91

exp{ij~LIJP,P,P’fllexp s

d4xd4+%w’)6~~ &$+--r)

6 6

+DF(x )w@@

1)

expi S1 p, 5 ll

(2.45)

where

DF(x

- u) is the Feynmanpropagator or a massIesscalar ield,

& ,-ik: b-y)

DF’x -‘~=S~2 k2 $ ic ’

n

That is, in this gauge,he Faddeev-Popovhostsaremassless.

2.4. Mixed transformations

A few remarks n the ntegrationover elementsof the Grassmann igcbra:

Sincewe wish to maintain he integration ule

undera change f variables,

ci=A&

fi ci=,& EjdetA,

i=l

we must have

7 dci = (de A)-’

y dEi -

Further, et us considera mixed multiple integral of the form

$rhci I-I de,,

i

tJ

whereB’s areelements f the Grassmann lgebra.We consider change f in-

tegration ar iables f the form



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B. V. Lee

and ask how’the Jacobianmust be defined.We consider irst 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 cp reanticommutingnumbers,

As a result, we have he rule

s n dxi n dUP= j n dri n dqP det(A - cuB-l@(detB)-’ I

The above esult s in accordwith the definition of a “‘generalizedeter-

minant” or Arnowitt, Nath and Zumino. They define the determinantof the

matrix

by

detC=expTrlnC,

with the convention hat

Tr C=Cii - CP,,

This definition allows the following relation to be valid:



Gauge heories

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93

and herefore he product property of the determinant

det(C1C2) = det C, det Cz .

TO see his, we set Ci = exp

Ji,

SO that det Ci = e xp Tr

Jp

NOW

Cl Cl

=exp{J1 tJztJ& whereJ12

s the Baker-Hansdorff eriesof commutators.

But Tr[JI

Jz]

= 0, erc., so that det(CI C,) = exp

Tr(J, + J2). Now

the ma-

trix Ccan be decomp osedniquely nto 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 ast follows from the definition det C= exp[(ln C)ii - (In C)PP].)

2.5.

Problems

2.5. . One should epeat he foregoingargumentsor quantumelcctrodynam-

its, to o btain the usualFeyn man ules n the Lorentz gauge. et us note that

for OL 0, one gets he photon propagatorn the Landau auge;or OL I, that

in the Feynman auge.Whathappens o the Faddeev-Popovhost ields n

those cases?

2.5.2.Just for the sakeof exercise, uantizeelectrodynamicsn the gauge

,F= aflAP

+

XA:.

Derive he Feynma n ules.

2.5.3. Show hatJ u dci n dci exp {ciMV ci)- det M, wherec andc’ are

anticommuting. *

i



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94

B. W.Lee

3. Surveyof renormalization chemes

3.

. Necessity for a gauge-invariant regufarization

In this lecture, we will develop wo subjects hat areneededo understand

later Lectures. heseare eguIarization nd renormalization f Gree n unctions

in quantum ield theory n general, nd of Gr een unctions n a gauge heory

generated y the expression 2.34), in particular.

The Green unctions generated y eq. (2.34) areplagu ed y the ultraviolet

infinities encounteredn any realistic quan tum ield theory. We aregoing o

develop method of eliminating hesedivergences y redetini ions, or renor-

malizationsof basicpara meters nd ields in the theory, n such a way that the

gaugenvariance f the original Lagrangians unaffected n so doing.

The gaugenvarianceof the action.has ar ious mplications on the structure

of G reen unctions of the theory. The precisemathematical xpressions hich

aresatisfiedby G reen unctions due to the gauge ymmetry of the underlying

action areknown as he Ward-TakahashiWT) dentities. What we will show s

that these dentities remain orm invariant under enormalizationwhich elimi-

nate he divergences. his point, that renormalization an be carriedout in a

way that preserveshe WT dentities, is of utmost importa nce or the follow-

ingreason s. irst, it puts such a stringentconstraint on the theory and the re-

normalizationprocedure hat the renormalized heory becomes nique, once

the underlying enormalizableheory is given.Second, ndperhapsmore to

the point, the unitarity ,of the renormalized -matrix s shownby the WT iden-

tities satisfiedby the renorma hzed reen unctions. The atter point requires

clarification.

In a perturbativeapproach, on-Abeliangauge heories uffer from such

severenfrared singularities hat nobod y has succeede dn defining a sensible

S-matrix n this framework. Consensu ss that a sensible aug e heory arises

only in a non-perturbative pproach,whereingaugeields and other matter

fields carrying non-Ab elian harge s o n ot manifest hemselves s physical par-

ticles. Physically, this conjecture s at t he heart of the hope hat color-q uark

confinementmight arisenaturally from a non-Abelian augeheory of strong

interactions.Ther e s an exception to this, and his is the casewhen the gauge

symmetry s spontaneously r oken. n fact, this latter possibility is directly

responsibleor the revivalof interest n non- Abelian auge heoriesa fe.w

yearsago, n conjunction with efforts to unify e lectromagnet ic ndweak n-

teractions n a non-Abelia n aug e heory. In this case here s no difficulty in

defining the physicalS-matrix, and he unitarity of such a theory is assured y

the renormalized ersionof the WT identities. Even n unbrokengauge heory,



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Gauge heories

95

the S-matrix can be definedup to some ower order n perturbation heory,

andhereagain he unitarity of the S-matrix s a consequencef the WT iden.

ti ties.

Why is the unitarity such a big ssue n g auge heory?After all, one does

not wor ry that much about the unitarity, say, n a self-interacting calarboson

theory. The reasons that the quantizationprocedurewe adoptedmakesuse

of a non-positive efinite Hilbert space, swe can readily see rom the struc-

ture of the gauge osonpropagator, q. (2.44). Further, the Green unctions,

of t he theory contain singularities rising rom the Fad deev-Popovhostsbe-

ing on the mass hell.Thus, n order hat the theory makes ense, heseun-

physical “particles”, cor respond ingo the ghost ields and he ongitudinal

components f gaugeields, must decouple rom the physicalS-matrix. The

renormalizedWT identities arenecessaryn showin g his.

The WT identities areusually derivedby a formal manipulationof eq.

(2.34). However, he Gre en unctions generated y eq. (2.34) arenotoriously

ill-defined objectsdue to ultraviolet divergences.t is therefore ecessaryo

invent a mcansof “regularizing” he Feynman ntegralswhich define hem

without destroying ymmetry propertie sof the Gre en unctions, so that as

long aswe keepa regularization arameterinite, the ntegralsarewell-defined.

It is only th en that we can attach concretemeani ng o the WT identities. After

renormalization, he “regulator” may be removed, nd f the renormalization

is to b e successful, he ren ormaiizedGreen unctions must be inite and nde-

penden of the reguiarizaion parameter.

A well- known regularization ch emen quantumelectrodynamicss the

Pauii-Viilars cheme,n which oneaddsunphysical ields of variablemasseso

the Lagrangiann a gaugenvariant way. After ga ugenvariant renormalizatio n

the variablemasses re et go to infinity, and renormaiized uantitiesareshown

to be inite in this limit. In no n-Abeli an auge heory, this device s not avaii-

able,but an dternativeprocedure,wherein he dimensionaIityof space-times

continuously varied,was nventedby th e geniusof ‘t Hooft and Vel man.

In the next section,we will give a brief summar yof th e renormalizatio n

theory a a Bogoliubov,Parasiuk,Hcpp andZimme rman n. his will be follow-

ed by an nt roduction to the dimensional eguiarization f ‘t Hooft andVeit-

man.

3.2. BPHZ renormalization

In this section we wig givea brief surveyof renormalizati on heory devel-

opedand perfected n recentyearsby Bogoiiubov,Parasiuk,HeppandZim-

merman n BPHZ). Nothing will be proved,but we will try to give definitions

‘and heoremsn a precisemanner.



PUB-76134-THY

96 8. W. Lee

First, we will givesomedefinitions. The interaction Lagrangians a su m of

terms1r which is a product of 6, boson ields andf,: fermion fields

with di

de-

rivatives.The vertex of the th type arising rom JZi as he ndex 6, definedas

ai =

b,

t zfi t

di

- 4 = dim(Xi) - 4 .

(3.1)

Let I’ be a one-particlerreducible LPI) diagram i.e., a diagram hat cannot

be madedisconnected y cutting only one ine). Let EB andE, be the num-

bersof externalbosonand ermion ines, B an d P the numbers f internal

bosonand ermion ines,ni the numberof verticesof the th type. Then

Ee+21B=&bi,

i

(3.2)

EF t2$ =CH(.f;:.

(3-3)

i

The superficialdegree f divergence f P is the degree f divergence ne

would naivelyguess y counting the powersof momenta n the numeratorand

denominator f the Feynman ntegral. t is

D(r)=&idit21,t31,-4V+4,

i

(3.4)

.the ast two termsarising rom the fact that at eachvertex there s a four-di-

mensional elta unction which allows one to express n e our-momen tumn

termsof other momenta,except that one delta function expresseshe conser-

vation of externalmomenta.Makinguseof 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 purpose f renormaliza tion heory is to give a definition of the finite

part of the Feynman ntegral correspondingo T’,

Fr = jl; j-dkl . dk, Ir 9

(3.7)

+

where , is a product of propagators F andvertices

P,



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Gauge heories

97

(3.8)

The finite part of

F,

will be denotedby J, and written

Jr

= f$

Jdkl

. dkLR, .

*+

(3.9)

Weshall describeBogoiiubov’s rescriptionof constructing

R,

from I,.

Let us first consid era simplecase, n which r is primitively divergent. he

diagr am ’ is primitively divergent f it is proper i.e., IPI), superficially diver-

gent i.e.,

D(F)

2 0) andbecomes onvergentf any ine s brokenup. In this

case,we may use he originalprescriptionof Dyson. Wewrite

Jr =

jdkl . dkL(l -t’&. ,

i.e.,

R, =(I -t’)& .

The operation r must be defined o cancel he nfinity in

J,.

Jr. is a function

of

EF

t

EB

- 1 =

E

- 1 externalmomentapl, . , pEeI,

The operation 1 - tr) onfis definedby subtracting romf the first

D(T’) +

1

terms n a Taylor expansion bout 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, amounts o makingsubtractions n

the ntegrand ,, the numberof subtractionsbeingdetermined y the super-

ficial deg ree f divergen ce f the ntegral.

Somemor e definitions:,A renormalizationpart is a properdiagramwhich

is superficially divergent

D

> 0). Two diagramssubdiagrams)redisjoint,

y1 n y2 # 4 if they haveno lines or vertices n co mmon .Let (7, 9 , yC} bea

set of mutually disjoint connecte d ub diagrams f r. The n



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98

B. W. Lee

is definedby contractingeach7 to a point a nd assigninghe value 1 to the

correspondingertex.

We arenow in a position to de scribeBogoliubov’sR operation:

(i) if F is not a renormalization art (i.e., D(r) Q -l)?

R,=R,,

(ii) if r is a renormalization ar t (D(7) > 0),

R, ~(1 -tr)&,

whereRF is definedas

(3.11)

(3.12)

(3.13)

andOr = -PET, Ehere he sum s over all possibledifferent setsof (7i).

This definition of R,. in termsOCR?appearso be recursive;n perturbation

theory there s no problem; heR, appearingn the definition of Rr. is neces-

sarily of lower order.

It is possible o “solve” eq. (3.13). We refer the nterested ea der o Zim-

mermann’secturesand merelypresent he result. Again we needsomemore

definitions beforewe can do this:Two diagrams 1 and72 are said o over-

lap, 71 0 72, if none of the following holds:

A P-forest I is a hierarchyof subdiagramsatisfying a)-(c) below:

(a) elementsof U are eno rmalization arts;(b) any two elements f U, 7’ and

7n arenon-overlapping;c) U may be empty. A r-forest U is futl or nor mal re-

spectively depen ding n whetherU contains r itself or not. The theoremdue

to Zimmermanns

(3.14)

where ; extendsovcc all possible full, normal and empty) r forests,and n

the product [I(-IX) the factors areordered uch hat th stands o the left of

t” if h > u. If h f? (I = 9, the order s irrelevant.A simple example s in order.

Consider he diagram n fig. 3.1. The forests are $ empty); 71 (full); 72 (n or-

mal); 71,~~ (full). Eq. (3.14) can be written in this caseas



Gauge heories

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99

- -----1

.( r-7 i

Q

I

I

‘ -; Ql

i Y,

: - --A

Fig. 3.1. Example of the BPHZ definition of subdiagra ms n 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, he R-operation s perfor medwith respect

to subdi agrams hich consists f vertices,and ll propagatorsn I?which con-

nect these ertices. 3y he BPH defmition, the subdiagram2 above oesnot

contain enormalization artsother than tself and n this sensehe present

treatment differs from S alam’s iscussion.

In formulating he BPH heo rem t is necessaryirst to reguIarizehe prop-

agatorsn cq. (3.9) hy some evicesuchas

A&J) re; A&J;~, 4 = -i l da exp[ia(p* - m2 tie)] ,

I

and define ,$-, E) as n eq. (3.9) in t ermsof A&, E), and hen construct

R,(r, E) by the R-operation.The BPH heorem tates hatR, existsasY-+0

andE + O+,asa boundary alueof an analytic unction in the externalmo

ments.Another theorem, he proof of which can be found n the book by

Bogoliubov ndShirkov, sect. 26, a ndwhich s combinatoric n nature,states

that the subtractions mplied by the (1 - tr) prescription n the R-operation

can be ormally implemented y adding ountertermsn the Lagrangian.

A theory which hasa finite numbe rof renormalization arts s called enor-

malizable.A theory n which all 6i are ess han, or equal o zero s renormaliz-

able. n this case he index of a subtraction erm n the R-operation s bounded

byD+En +& +-

4 which s at most equal o zero by eq. (3.5). In sucha the-

ory, only a finite numbe rof renormalization ounterterms o the Lagrangian

suffice to implement he R-operation.

3.3. The regularization scheme of ‘t Hooft and Ve ltman

Kecently, t Hooft and Veltmanproposed schemeor reguIarizing eyn-



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100

B. V.Lee

man ntegralswhich preservesarioussymmetries f the underlyingLagran-

gian.This method s applicable o electrodynamics, ndnon-Abelian auge

theories,and depends n the dea of analytic continuation of Feynman nte-

grals n the numberof sp ace-time imensions. h e critical observations ere

are hat

the

global or local symmetri esof these heoriesare ndependent f

space-time imensions, n d hat Feynman ntegralsareconvergent or suff--

cien ly small, or complex V, wher eN is the “complex dimension”of space-

time.

Let us first review he natureof ultraviolet divergence f a Fey nmandia-

gram.For this purpose,t is convenien t o parametr izehe propagators s

A,(p2) = $ i datexp[iol(p2 - nz2+ ie)] .

0

(3.15).

Makinguseof this representation, e can write a typical Feynman ntegral as

X exp {i C ai((7i2 .rf + ie)} ,

i

(3.16)

where is the numberof in ternal propagatorsn r, L the numberof loops, and

1,) . , 1, may take any values rom I to

L.

The momentumqi carriedby the

jth propagators a inear function of loop mome nta

i

and externalmomenta

p,nBThe exponenton the right-hand ide of eq. (3.16) can hereforebe writ-

ten as

I

I

C a&qf- mf+ ie) = I

i=l

’ CkiAij(a)kj t i

i,i

c ki Bi,(a)p, - 7 “i(mf - ie)

,

m

+kT+ktk-B*p-

C c+(mf - ie) ,

where is a column matrix with entrieswhich are our-vectors.The matrices

A

and

B

arehomogeneousunctions of fi?st deg reen CY’S,ndA is symmetric.

Ubon translating he ntegrationvariables

k+k’=k tA+Bp



PUB-76/34-THY

Gauge heories

101

anddiagonalizing he matrix A by an orthonormal ransformation n k’, we

can perform the oop integrationsover

ki

in eq. (3.16). The result s a sum

over ermseachof which has he form

X %+-~[~p~C(or)~p t C c+(rnf - if)]},

i

(3.17)

where TX/I . . v

is a tensor, ypically a product of gP09s, i is the th eigen-

valueof the matrix A, andSi s a positive numbe rwhich is determined y the

tensorialstructureof

F,.

Note that Ai is homoge neousf first degreen

ois. The matrix C is

C= BTA-‘B

and s also a homogene ousunction of first degreen [Y’S n this parame triza-

tion, the ultraviolet divergencesf the ntegralappear s he singularities f

the ntegrandon the right-hand ide of eq. (3.17) arising rom the vanishing f

some actors IIi[Ai(~)JSi as someor all (Y’S pproach o zero n certain orders,

for example,

ar, car2 <...<iY

‘J

,

where rl, r2

. , rJ) is a permutationof (1, 2, *.. , I). See, or instance,a more

detailedandcareful discussion f Hepp.

The t Hooft-Vel tman regulari zation onsists n defining he ntegral

F,

in

n dimensions, > 4 (one-timeand II - I)-spacedimensions)while keeping

externalmomentaand polarizationvectors n the first four dimensions i.e.,

in the physical space), erforming he ?I 4 dimensionalntegrals n the space

orthogonal o the physical space,and hen continuing he result n ?z. For sin-

gle-loopgraphs

ne

may perform all n integrations ogether.)For sufficiently

smalln, or complexn, the subsequentour-dimensionalntegrationsarecon-

vergent.

To seehow it works, consider he ntegral



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102

B. W.Lee

X n (kd * el)exp[i cc-+($ -

rnf + if)],

i

(3.18)

where,now, the

ki

aren-dimen sional ectors.As beforewe can express he qi

as inear functions of thekj and the external momenta i, where he pi have

only first four-component on-vanishing. rom now, we shall denotean n-di-

mensional ector by (k,K) wherek^ s the pr ojection of

k

onto the physical

A

space-time ndK = k - k. Thus,p = ($, 0). Eq. (3.18) may be written as a sum

of termsof the form

(3.19)

The ntegralsoverkj can tie perf ormed mmediately, using he formulas

s

d”-4KK h’

%

Q2

K

42r exp(-jA K2)

)( (jA)-n/2+2-r

,

where he summation s over he elementsu of the symmetricgroupon 2~ob-

jects (aI, ‘~2, , a>), an d

%%

=n-4.

ThusF, of eq. (3.19) will have he form



Gaugeheories

PUB-76/34-THY

103

where (n) is a polynomial n n and i is a non-negativentegerdepending n

the structure of II K, - K, in eq. (3.19). For sufficiently small n < 4, the sin-

gularitiesof the ntegrand ssomeor all 0;‘sgo to zero disappear.

The reasonshis regularization reserveshe Ward-Takaha shidentities of

the kind which will be discussed re, irstly, that the vector manipulations

suchas

k'"(2p +q = [(p +k)S- m2] -(p2- m2),

or partial ractioning of a product of two propagators, hich arenecessaryo

verify these dentities “by hand ”, arevalid n a ny dimensions, nd,secondly,

that the shifts of integrat ionvariables, angerouswhenntegrals redivergent,

are ustified for small enough , r complexn, since he ntegral n question s

convergent.

The divergencen the original ntegral s manifestedn the polesof

Fr(fz)

at n = 4. Thesepolesare emovedby the R-operation, o that Jr(n) as defined

by the R-operation s finite a nd well-defined s Z+ 4. Actually, to our know-

ledge he proof of this hasnot appearedn the literature,except or the origi-

nal discussion f ‘t Hooft a nd Veltman. Hepp’sproof, for exampl e,doesnot

really apply here,since he analytical discussion f Hepp s not tailored or

this kind of regularization.However, he argume nt f ‘t Hooft and Veltman s

sufficiently convincingandwe haveno reason o believewhy a suitablemodi-

ftcation of Hepp’sproof, for example,of the BPHZ he orem houldnot go

throughwith the dimensional egularization.

The abovediscussions fine for theorieswith boson s nly. When hereare

fermions n the theory, a complication may arise.This has o d o with the oc-

currence f the so-called dler-Bell-Jackiw nomalies. he subjectof anoma-

lies n Ward-Takaha shidentities hasbeendiscussedhoroughly n two excel-

lent lecturesby Adler, a nd by Jackiw, and we shall not go nto any further de-

tails here. n short, the Adler-Bell-Jackiw nomali esmay occur when he veri-

fication of certain Ward-Takahashidentities depen ds n the algebra f Dirac

gammamatriceswith y5, suchas -y,, 5 + y5 7, =.O.Typically, this happen s

whena propervertex nvolving an odd numberof axial vector currentscannot



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104

B.W. Lee

be regularizedn a way that preserves ll the Ward-Takahashidentities on such

a vertex, and asa consequenceomeof the Ward-Takahashidentitieshave o

be broken.The occurrence f theseanomaliess not a metter of not being

clevereno ugh o ilevisea proper egularization cheme: or certainmodels

sucha schemes impossible o devise. he dimensional egularization oesnot

help n such a case,due to the fact that 75 and the completely antisymmetric

tensordensity eh Pyp

reunique o four dimensions nd do not allow a logical-

ly consistentgeneralizationo H dimensions.When hereareanomaliesn a

spontaneously roken gauge heory, the unitarity of the S-matrix s in jeopardy

since,as we shall see, he unitarity of the S-matrix, .e., cancellation f spuri-

ous singularities ntroduced by a particular choice of gauges inferred ro m

the Ward-Takahashidentities. GrossandJackiw haveshown hat, in an Abe-

lian gauge heory, thc.occurrence f anomalies uns afoul of the dual require-

mentsof’unitarity and renormalizabilityof the theory.

Thus, a satisfactory he ory should be ree of an omalies. ortunately, t is

possible o construct model swhich are anomaly-free, y a udicious choiceof

fermion fields to be ncluded n the model. Thereare wo ‘lemmas” which

make he aboveassertion ossible.One s that the anomalies renot “renor-

mafized”, which n particular means hat the absence f anomaliesn lowest

order nsures heir absenceo all orders.This wasshownby Adler and Bardeen

in the context of an SU(3) versionof the u-model,and by Bardeenn a more

general ontext which encompasseson-Abeliangauge heories.The seconds

the observatio n hat all anomalies re elated; n particular, f the simplest

anomaly nvolving the vertex of threecurrents s absent n a model,so areall

other anomalies. his can be nferred rom an explicit constructionof all anom-

aliesby Bardeen, r from a moregeneral n d elegant r gument f Wess nd

Zumino.

Let us concludewith a simpleexampleof di mensional egularization: he

vacuumpdarization in scalarelectrodynamics. h e Lagrangians

and the relevant erticesareshown n fig. 3.2. Thereare wo diagrams hich

contribute to the vacuumpolarization,shown n fig. 3.3. The sum of these

contributions s

d”k

Z=e2j-

1t2k+?4,(2k +d, - 2((k +d2 p2kw,]

pv

(3.20)

We use he exponentialparametrization f the propagatorso obtain



Gaugeheories

PUB-76/34-THY

105

Fig. 3.2. Photon-scalar meson vertices in chargedscalar electrodynamics.

k

Fig. 3.3. Second order vacuum polarization diagrams n charged scalar electrodynamics.

X [Ox: +dp +P)” - 2((k +pj2 -p2jgpv]. (3.21)

The exponen t s proportional

o

(CY P)k2 + 2k . pa + ap2 - (a + /.I)(/.? - ie)

so we may write

= e2bgp,p2g,,)da $3s)2j E

0 0

nn

Xexpi (a+/3)k2+

i

SP2 -(cY+p)(p2-ie)

1



PUB-76/34-THY

(3.22)

The first term s explicitly gaugenvariant andonly logarithmicaffydivergent,

so that a subtractionwill make t convergent.t is the second erm that re-

quiresa carefulhandling.We need he formulas

d”k

- exp(iAk2) =

Gw

(2&Q” ’

b

2 exp(iXk2) = $ (-in) i exp

7 /

4’““),

(2&X)”

s

(3.23)

so that tfie second erm, 12, s

X exp i

I

-$p2~-(atp)(&ie) A.-

11

(a+P)

x (i(l -in)-[-.&+(otp)p~)’

= -2ie

2

e ml4

g

-----ldadfl6(1 -u-~jj~ei~l”ap’-P2f”l

pv (2&r)”

0 if

x [il-l(1 - $2) + i(orpp2 -/.l’)] .

(3.24)



Gauge heories

For sufficiently small n, n < 2, the A-integration is convergent, and

- dh

J

,ik(A+iel(I -2’ +i~)

o hrtl2-I

=

p dh -& {h ’-“/2 exp[iX(A t ic)]) = 0.

0

PUB-76/‘34-THY

107

(3.25)

So the dimensional egularization ives he gaugenvariant esult,

12=0.

3.4.Problem

Repeat he vacuumpolarizationcalculation n spino relectrodynamics sing

the dimensional egularization.

4. T he Ward- Takahashi identities

4.1. Notations

Oneof the problems n discussing augeheories s that notationswill get

cumbersomef we are o put explicitly space-timeariables, oren b indices

andgroup epresentationndices.We will thereforeusea highly compactno-

tation. For simplicity in notation, we will assu mehat the gauge roup n

question s a simpleLie group.Extension o a product of simpleLie groups,

suchasSU(2) X U(I), is not too difficult.

We will agree o denoteall fields by r&.Again or simplicity in notation we

will assume j to be bosons. nclusion of fermionsdoesnot presentany diff-

culty in our discussions, ut we will have o be mindful of their anticommut-

ing nature.Thus, or the gauge ield A;(x)

i

stands or the group ndex a, the

Lorentz ndex 1-1,nd the space-time ariable . Summatio n nd ntegration

over epeatedndiceswill b e understood.Thus

(piz sd4x q A;(x)A~~(x) + 0..

(4-I)

where he dotted portion includescontribution from other species f fields.



PUB-76/34-THY

108 B. W. Let

The nfinitesimal local gaug eransformationmay be written as

l#Jp$;=$it(A; tt~~j)ea,

(4.2)

whereO. = 8,(x,) is the space-time ependent ara meter f the groupG. We

choose@i o be real,SO that

t; = ~“‘(xa - xp4(xn - Xi)

(4.3)

is realantisymmetric, where7; is the representation f the generator a of G

in the basis&. The’inhomogeneouserm q is non-vanishing nly for the

gaugeields

hi”

=$ars4(xi - xa)ija

,

b for & = A:@$

= 0 otherwise.

Weshall also define

Notice that

(4.4)

(4.5)

where

le =pw(xo - Xb)64(Xa - xc) ,

$g - <gk =f”“‘t; s

(4.6)

pk being he structure constantof G. The proof of eq. (4.5) is simple: we

will just show

Since

$ ’ is non-vanishing nly when refers o a gaugeield, let us write

i

= (CJA, Xi), i = (d, V, i>. Then

t; = gpy.pd64(Xi xa)S4(xa Xi)



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Gauge heories

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 equal og times he right-hand ide of eq. (4.5).

The gaugenvariance f the action can be stated n a compact orm,

(4.7)

The inear gauge ondition we discussedn lecture 2 ma y be written as

Fa @I ai@i

(4-g)

where

fai = a;

for Qi = A$xi) ,

a; = tPbap64(xi - xa) .

En his no tation, the effective action is written as

f&-jt’h f>d = s[@l -&F; [@I+ ‘$Mab @hb 3

with

(4.9)

(4.10)

(43)

4.2. Becchi-Rouet-Stora transformation

The WT dentities or gaug e heorieshavebeenderived n a numberof dif-

ferent ways.The most convenientway that I know of is to consider he re-



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110

B. V.Lee

sponse f the effective action, eq. (4.1 ), to the so-calledBecchi,Rouet,

Stora BRS) transformation. t is a global ransformationof anticommuting

type which leaves he effective action nvariant. Here1 shall ollow a very ele-

gant discussion f Zinn-Justingivenat the Bonn Summer nstitute in 1974.

The BRS transformation or non-Abelian augeheory s definedas

“$=Di”QX,

(4.13a)

when? X s an anticommutingconstant.Note that if we dentify O. = T)~S,

we see mmediate ly that the action S[$] is invariantunder 4.13a).Thereare

two important propertiesof the BRS transformationwhich we shall describe

in turn.

(i) The transfonnatiorrs on 9t and pa are nilpotent, i.e.,

s2ei=0,

s2qa= 0.

Proof:

Eq. (4.14) follows from

“(D;Q=O.

Indeed

(4.14)

(4.15)

(4.16)

Since?jO n d vb anticommute, he coefficient of vaqn in the first term on the

right-hand ide may be antisymmetrizedwith r espect o

a

and b. It vanishes s

a consequence f eq. (4.5).

To show eq. (4.15), we note that

(4. I 7)

by the Jacobi dentity. QED.



PUB-76/34-THY

Gauge heories

Ill

(iij The BRS transformations leave the effective action SEn of eq.

14. I)

invariant

Proof: As note d aboveS[$] is invariantundereq. (4.13a).We urther note

that

s<Mab@h&lb)0.

(4.18)

by eq. (4.16) and he definition of Mab, eq. (4.12). Thus

=~FaMabqbsA-~Fa~D;qb6hi0

P

(4.19)

by the definition of Mab , eq. (4.12).

For,lateruse,we rema rk inally that the metric [d&dta dn,] is invariant

under he BRS transformationof eqs. 4.13). I wan t you to verify it.

4.3. The Ward- Takahashi dentities for the generating functio?zal of Green

functions

We will first derive h e Ward-Takahashidentity satisfiedby lV,[JI of

eq. (2.3 ),

Z,[~~eiWr;I’l=NS[d~d5:dq]exp{id,ff[~,~,q]+iS-~i).

(4.20)

We irst note that, according o the rule of integrationover anticommuting

numbers

(4.21)

we have

s

[WdEdq]4‘,expCiSeff l + Ji 3> = 0 ,

(4.22)

because e-r ontains and

q

only bilinearly. In eq. (4.22) we makea change

of variables ccording o the BRS transformations 4.13). Sincea change f

integrationvariables oesnot change he valueof an ntegral,we have

O=j[d@d[dq]

-~a[$j+iJiE,D, ‘[@]qb

I

exp(iS,ff+iJi$i) I (4.23)



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Eq. (4.23) is the WT identity as irst der ivedby SlavnovandTaylor. We can

rewrite t in a differential form involvingZF. We define

(ZFlba z Ni

s

Id&Wtl Caib exp We, + iJ&$ e

(4.24)

It is the ghost propagatorn the presence f externalsources i- It satisfies

M

[ 1

s zj&inei+

b iSJ

(4.25)

I wIl1 eave he derivation of eq. (4.25)

asarrexercise. q. (4.23) can be writ-

ten as

aFa +&

c 1

F[J] - J,D, ’ +&

[ 1

zj7)ba

(4.26)

Eq. (4.26) is in the form written down by &n-Justin and Lee. t is the WT

identity for the generatingunctional of Gree n unctions, and assuch t is

rathercumb erso me or th e discussion f renormaiizabil ity,since,as we have

seen n lecture 3, the renormalization rocedures phrasedn terms of (single-

particle rreducible) propervertices.Nevertheless, q. (4.26) wasused o de-

duce“byhand” consequencesf gauge ymmetry or renormalization arts by

Zinn-Justin and myself. We do not have o do this, sincewe know better now.

Eq. (4.26 ) will be useful n discussinghe unitarity of th e S-matrix ater, how-

ever.

‘4.4. The Wurd-Takuhushi identities: inclusion of ghost sources

To discuss enormalizationof gaugeheories,we have o considerproper

verticessomeof whoseexternal ines areghosts.For this reason, he ghost

fields 5 and q shouldhave heir own sources.We herefore eturn to eq. 2.34),

X exp[iCQ#, , 4 + U +@a+ #,.i,l .

(4.27)

For the ensuingdiscussion,t is more convenient o consideran object

(4.28)

and define



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113

(4.29)

In

eq.

4.28), Ki and Laaresourcesor the compositeoperators

ffr$]qa

and

$?f,a Q, Q, respectively. t follows from eqs. 4.16) and 4.17) that Z is in-

variant under he BRS transformations4.13). Note further that Ki is of anti-

commuting ype* and

The nvarianceof Z is expressed s

or

Weneedone more equation,

g=ftig.

i

(4.30)

(4.3 1)

(4.32)

(4.33)

Let us examine he consequencesf eqs. 4.32) and 4.33). We perform a

change f variables

(4.34a)

(4.34b)

&$=-iFa& (4.34c)

Eq. (4.32) tells us that Z is invariant undersucha transformation,and the n-

tegrationmeasu re d&d$dp ] is also, hanks o



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114

A?&(-)

“qiKi~ ’

B.W. Lee

Thus, he change f variables 4.34) in eq. (4.29) eads o

X exp[i{X + t - /I + /3t - 9 +Jr.$>] = 0 .

Next, the equationof mot ion for n is

Combiningeqs. 4.33) and 4.37), we obtain

(4.3%)

(4.35b)

(4.36)

(4.37)

(4.38)

Eqs. 4.36) and 4.38) are he basis or deriving he WT dentity for the

generatingunctional of prope rvertices.

4.5.

The Ward-Takahashi identities for the generating functional of proper

vertices

The generatingunctional of propervertices s obtained rom IV,

W=-itnZ,

by a Legendreransformation.We define

(4.39a)

(4.39b)



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115

(4.39c)

wherewe haveused he same ymbols or the expectation a luesof fields as

for the ntegrationvariables. he generatingunctional for propervertices s

As usual,we have he relationsdual to eqs. 4.39),

P-C?.

(4.40)

(4.41a)

(4.41b)

(4.41c)

It is easy o verify that if Wand F de pend n parame ters , suchasK or L

in ou r case,which arc not involved n the Legendreransformation, hey satis-

f-Y

From eqs. 4.36) and 4.38) we can derive wo equations atisfiedby F,

(4.42)

(4.43)

(4.44)

It is important to observe he corresponde nceetween qs.

4.32)

and 4.33),

andeqs. 4.43) and 4.44).

If we now define

(4.45)



116

B. W. Lee

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f&a%,=%.

a

(4.47)

The functional r carries net ghostnumberzero, wherewe define he

ghostnumberNp as

N&II = 1 3

NJK]=-1 s

N,LEl = -1,

NJL] =-2,

NJ4 =o.

Clearly ’ may be expandedn termsof the ghostnumber arrying ields:

Substituting he expression4.48) in eqs. 4.46) and 4.47), differentiating

with respect o Q, a ndsetting all ghostnumbercarrying ields equal o zero,

we obtain

(4.49)

(4=50)

These re he equations irst d erivedby me rom (4.26) by a complicated

functional manipulation.These re he fully dressed ersions f eqs. 4.7) and

(4.12),



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I17

4.6. Prob Iems

4.6.1. Convince ourself that the measuredddldq ] is invariantunder he

BRS ransformation.

4.6.2. Show hat

Proveeq. 4.25).

4.6.3. Show hat

dci ac

f(c)=O, i

not summed,

i

where i is an anticommutingnumber.Proveeq. (4.37).

4.6.4. Derive he WT identity for Z[J, fl, $1,

5. Renormalization of pu re gauge theories

5. I. Renormalization equation

We are eady o discuss enormalization f non-Abelian auge heories

based n the WT identity for properverticesderived n the ast ecture.

Let us recall that our Feynman ntegralsare egularized imensionally o

that for a suitably chosen not equal o 4, all integralsareconvergent. hus,

we can perform the BogotiubovR-operation fter the integralhasbeendone,

insteadof making the subtractionof eq. (3.10) in the ntegrand sZimmer-

manndictates. n fact this is the proc edur e sedby Bogoliybov,Parasiuk nd

Hepp.Further, nsteadof makingsubtractions t pi = 0, we will choose

point whereal1momen ta lowing into a renormalization art areEuclidian.



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B. W.Lee

For a vertex with II external ines, this point may be chosen o be --pi” = a2,

pi - pi = c&z - I).

Th

s is to avoid nfr ared divergences. t this point the

square f a sum of any subsetof momenta s alwaysnegative, o that the am-

plitude s rea1 nd ree of singularities.

For simplicity we first consider pur egauge’theory.nclusion of ma tter

fields, suchasscalarand spinorwith renormalizablenteractionspresentno

difficulty. In particular, couplingof gaugeields with scalarmesonswill be

treated n chapter6.

Wemay write do wn the propervertex asa sum of terms,eachbeinga pro-

duct of a scalar unction o f externalmomentaand a tensorcovariant,which

is a poiynomial n the components f externalmomentacarryingavailable

Lore& indices.All renormal ization arts n this theory haveeitherD = 0 or 1.

The self-mass f a ga uge oson s purely transverse s we shall see, o that it

alsohaseffectively

D

= 0. Thus,only the scalar unctions associated ith ten-

sor covatiantsof lowest orderaredivergentasn +

4.

(Note also hat vertices

involving external ghost ineshave ower superficialdegrees f divergencehan

simplepower counting ndicates.This is becauseC,alwaysappears s8 E, ,.)

The basicproposition on renormalization f a gauge heory s the following.

If we scale ields and the couplingconstantaccording o

Q1

i

= z’/2(e)@’

i’

r;

a

= S2(e)g’

a’

K. = .i?/2(~)K’

I i’

L

a

= zqE)L'

a’

a

= Z(E)c? ,

g=

X(e). g’

+z(e)z1’2(e) ’

(5.1)

wheree = n - 4 is the’regularization arameter, nd choose (e), X(E) and

Z(e) appropriately, hen

is a finite (that is, as e = D - 4 4 0) functional of its arguments ‘, 5’. $, K’,

L’ and or’. Under he renormalization ransformationof (5.1). eqs. 4.45) and

(4.46) become



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119

We will expand oopwise,

Wehave

Suppose hat our basicproposition s true up to the (n - 1) oop appioxima-

tion. That is, up to this order, all divergences are removed by rescahg OF

fields and parameters s n eq. (5.1). We suppose hat we ha vedetermined he

renormalization onstantsup to this order,

(x)n-, = 1 +X(r) + .** + X(n-1) "

P-7)

Wehave o show that the divergencesn the n-loop approximationarealso

removed y suitably chosen ,r 2n andx,r.

Following Zinn-Justin,we ntroduce he symbol

where he superscript denotes er e he quantities cnormalized p to the

(rr - 1) oop approximation.We can write eq. (5.8) as

with

(5.10)



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B. W.Lee

The right-hand ideof e q. (5.9) involvesonly quantitieswith less han II loops,

it is finite by the nduction hypothesis.Further,divergencesn subdiagrams f

r(,, are emoved y renormalizations p to (n - 1) oops.Thus, the only re -

mainingdivergencesn r(,l) are he overall ones.Let us denoteby Odiv the

divergent art. If we adjust inite parts of I& appropriately,we have

(5. i)

(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 hat

where hk functional operator9

9= clot61 3

is givenby

(5.13)

(5.14)

(5.15)

(5.16)

(5.17a)

(5.17b)



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121

Fro m now on, for the nterest of notation al economy,we will d rop the super-

script r, until furth er notice.

An important aid n solvingeq. (5.14) s the observati on hat

p=0.

(5.18)

We will prove his in steps.First, we verify by direct computation hat

$g=o,

90

=D;qa+f q q 6.

ahc b c&qa

(5.19)

i

Eq. (5.19) is a direct consequ encef eqs. 5.16), (5.17) that the BRS transfor-

mation on 4 andqa s nilpotent. Next we note that

(5.20)

wherewe haveused he fact that

(5.21)

Direct computationsyield

(5.22)

6 30 sqo) r(o)

SD; (())

“3

50, “3 - =

K@.

1

qa63 “3’



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122

B.IK Lee

“Qo

s50)

-

s2r(0)

s r(o)

s 2 r(o)

s%

qo) = -

(

Qf.

-++----

sKjs’tl, sV* 6L*6Qa

1

Thus

which,proveseq. (5.18).

The act that 9 is nifpotent means hat, in general, 9 for arbitrary

9 = Y(i$, & 1, K, L) is a solutibn of eq. (5.14)

8(sw).= 0 *

(5.23)

The question s whether hereareother solutionsnot of the form ~7. This

questionalso arisesn renormalization f gaug envariantoperators, nd has

been tudied, n particular, by Kluberg-Stern ndZuber.They alsoadvanced

conjecture: hey suggestedhat the general olution to eq. (5.14) s of the

form

W(‘,)P”

=GM+99[$,t,tl,KLl,

whereG[$] is a gaugenvariant unctional,

(5.24)

Recently, Jogtekarand wereable o prove his, mostly by the effort of the

first autho r. The proof is tedious,and believe hat it can be mprovedas o

rigor, elegance nd ength. For this reason, will not present he proof. It is

easy o see hat th e form (5.24) satisfieseq. (5.14) and hat G[$] of eq. (5.25)

is not expressible s gS, in this caseat least. t is the completen essf eq.

(5.24) which requires roof.

Eq. (5.15) [or (5.4)] is immediately solved. t mean s hat

r&J fh Es 5K, Ll= r&l [A O,V,Kj f a;$, L] +

‘jQ’[ti*

-5V, L.1,

5.26)

whereQ’ is transverse: %Qj = 0 -



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Gaugeheories

123

The quantity (l&}

div is

a ocal

functional

of its arguments.f we asign

o

K and L the dimensions (K) = 2 andD(L) = 2, then Z. has he uniform di-

nmsion 0, andSO oes rfn)ldiv. It hasNg r’fil,] = O.SinceNg 91 = t.1, it

follows that Vg 9]= -1 in eq. (5.24). In ord er hat the right- hand ide of

eq. (5.24) is local, both 9 and Qmust be separatelyocal. The most general

form of {r{,r)}div satisfying he above equirementss

wherecr,p, y are n gen eral ivergent,.e., e-dependent,‘constants.sing he

explicit form of 9, eqs. 5.17) we can write

(5.28)

In eq . (5.27) G[$] is equal o S[I$).This is so becausehe action s the only

local functional of dimension our which satisfies q. (5.25).

Because

Thus, combiningeqs. 5.28), (5.29), we obtain

(5.29)

(5.30)

Recall that in cq. (5.30) d, f, 77 , , L andg are renormalized uantitiesup to

the (n - 1) loop approximagon.Weshall deno te hem by (#),-t, etc.

If we now define Z),, (Z),, (X), by



PUB-76/34-THY

124

93. V. Lee

<a,= +Q-l+qn)

etc. ,

and enormalize he fields and coupling constantaccording o

(5.3 I)

etc. ,

(5.32)

(5.33)

and choose+), :(,+ xtn) to be

%)

= -4 a * lw) ,

?(a = 37 + IN4 9

%)

- q,z) - :2(n) = 2+j,

(5.34)

then {~fnj)dr v is elimina ted: I’&,

is a finite functional n termsof (@j,,, . I

and $),,. Furthermore, ince @), = (Zji’2$, . , eqs. 5.3), (5.4) areako

true for the newly renormnlized uantities.This completes he induction.

Note further that

(5.35)

is finite. The renormalized ield $* transformsunder he gaugeransformation

as

6. Renormalization f

theories

with spontaneously

roken symmetry

6.1. hclusion of scalar fields

In chapter5, we detailed he renorrhalization f pure gaug eheories.Let us

considernow a theory of gau ge osonsand scalarmesons. et

4j = (.qxj, s&)) 9



PUB-76/34-THY

Gauge heories

where , arescalarmesons, nd et

be the vector which defies the gauge. he action for the scalar ields s of the

form

wher e Ds), is the covariantderivativeacting on S, V(s) s a G-invarian tquar-

tic polynomial in s andof dimensionat most zero.

Weshall write

(6.2)

[W, ta = [6M, ”] =0,

(6.3)

whereM& is the renor malize$mass atrix for the scalarmesons.We shahas-

.sumeor the mome nt that Mr is a positive semi-definitematrix.

Let us discuss enormalizati on.Almost everythingwe discussedn the last

sectionholds true. In particular we have

WC

n

ldiv = G [$I + 9 W> E, l, K t3 ,

wherewe have written @i= {A,, sa) andKi = {K,, &},A, being he gauge

fields, A, = A;(x), t = (a,p, x). Now we have

(6.4)

where Y’ is a G-invariantquartic polynomia l n s. This term s eliminatedby

renormalizations f couplingconstantsappearingn V(s). 9 takes he form



PUB-76/34-THY

126

B. IV. Lee

where

$

is a G-covar iant oefficient. (It could be $, or somethin g lse,such

as he

d-type

coupling n SU(3), for example.)This gives

{r&p= +pl) & +ia$] ‘a

1 [

a

t a

2

Gag

where

Thesedivergences reeliminated f we renormalize , &q, K,,

La

andg as be-

fore and

%

= 3/z 9

s a’

K - &. l12p

Q- z

>

p = -$- 1’2(c3r ,

a= u

s

( 1

which leaves

invariant, and shift the si fields by

s; = Sk

- u,w P

andchoose



PUB-76/34-THY

Gaugeheories

An important lesson to be learned here is that in a generalineargauge, calar

fields can developgauge-dependentacuumexpectation alues,which are n-

nocuous rom the renormalization oint of view.

6.2. Spontaneously broken ga uge symmetry

Let us consider he casewhereM’ is not positive semi-definite.t is by now

well known that undersuchcircumstances pontaneous reakdown f the

gaug e ymmetry takesplace,and someof the scaIar ields and someof the

(transverse).gaugeosons ombine o form massive ector bosons.We will give

herea very brief discussion f 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

q$sp s=u

= ‘Nl$ ,

CR,, positive definite.

G-invariance f the potential Vo is expressed s

s ta

6 Vo

-=o.

Q d 6s

P

Differentiating this with respect o s7 and settings = U, andmakinguseof

eqs. 6.7), (63, we obtain

(6.9)

Therefore, hereareasmany eigenvectors orrespondingo the eigenvalue

zero as hereare inearly independent ectorsof the form C&U,. If the dimen -

sion of G is N and he little groupg which leaves invariant hasdimensionm,

thereare V - ~11igenvaluesf 7K2 which vanish.

For fltture use, t is useful to definea vector 1: by



128

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B. W. Lee

where~4; s to be defined. Sinceall representations, xcept the identity repre-

sentation, of a Lie group are aithful, there areN - m independent ectors of

this form. Now we define

(6.10)

This matrix is of a bloc; diagonal orm; mor eover f 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 space panned y vec-

tors of the form

t~pt$.

This space s N - m dimensional,

trP=N- tn.

Eq. (6.9) may be written as

We renormalizc the gauge ields, ghost fields and gauge oupling constant

as before, and renormslizes, according o

sa

=Z’/J(s” +u’

a (Y a

+su )

a 5

and determine6 U, = (6

.ta)l

+ (6 & t .._by the condition that the diver-

gences f the form -(6 S[$]/S s,)A(e)u, in the n-loop approxkation be can-

celled by the displacementof the renormalized ields s:, (6 u&. (See he dis-

cussion n subsect.6.1.) The renormalized acuumexpectation value z& is to

be determined by the condition

s2rr

6 f#J;s ;

I

=u’

positive semi-definite



Gaugeheories

PUB-76/34-THY

128

We equate he gaugeixing term C, J o (numeri cally)

Then the terms n the action quadratic n renormalizedields andcoupling

constants excluding enormaiizati on ounter terms)are

wherewe havesuppressedhe superscript altogether.The propagatorsor

the gauge osom,scalars nd ghost ields are, espectively,

[Aj$“(k,c~)]~~=~ ( ’ ) ,

k2-p2+ie cb

which can be written, in a representat ionn which p2 is block diagonal, s

01

[&12a)

the atter holding or a, b beingoneof the

m

indicescorrespondingo genera-

tors of th e little grou pg;

[AF(k2s)], = 1 J’lq k2

( -

A2 + ,),

tz~(~)ab(kl_:2+iQ)hlpe9

6.12b)



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B. W. Lee

tAdk2p a&

= ( k2

_

b,z

+ , ).,

(6.12~)

If the theory is to b e sensible, n d gau genvariant, then the poleswhose

locationsdependon the gauge arameter Y annot be physical,and he parti-

cles correspondingo such polesmust decouple rom theSmatr ix. If this is

the case,aswe shall show, hen thereareN -

m

massive ector bosons,

m

massless auge osons,and N -

m)

lessscalarbosons han we started out

with. This is the Higgsphenomenon.

is a theory of this kind renormalizable? he answers yes, becausehe

Feynman ulesof the theory, ncluding the propagators boveare hoseof a

renormalizableheory, and the WT identity, eq. (6.3), and he ensuingdiscus-

sion n chapter 5 and subsect.6.1 hold true whetheror not#is positive def-

inite. That is, by the methodsdiscussed, e can construct a finite I” in this

casealso.The expansion oefficients of T” about $:= uf, where

62r’

646 b;Q=uZ

positive definite,

then are he reducibleverticesof the renormali zed heory. I shall not describe

the detailsof the renormalization rogramsince hey havebeendescribedn

many papers,most recently n my pa per ref. [SO]), but the principle nvolved

should be clear.

But an additional remark s in order: the divergent arts of variouswave-

function an d coupling-consta nt enormalizat ion onstantsare ndependent f

MF.

This has o do with the fact that theseconstantsareat most ogarithmi-

cally divergent,and nsertion of the scalarmassoperators whosedimension s

2) renders hem finite. For detailedargume nts, ee ef. [50].

6.3. Gau ge indepeizdemze

of

the S-matrix

What remains o be done s to demonstrate hat the unphysicalpoles n the

propagatorsineq. (6.12). which dependon the parameter \ nd someof which

correspond o negativemetric particles,do not cause nwantedsingularities n

the renormalized matrix. Weshall do this by proving hat the renormalized

S-matrix s independent f the gaugeixing parameter X. o ensure hat the S-



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131

matrix is well-defined, shall assumehat after the spontaneous reakdown

of gau ge ymmetry here s at most onemassless auge oson n the theory.

Beforeproceedingo the proof, the following ihustration s useful. For

simplicity let us considera X@4 heory. The generatingunctional of Gree n

functions s

where

Whathappensf we nsteadcouple he externalsource o $J @? Wecan write

the generatingunctional as

Z[iJ =~~[WJexpW[~l+ it@ d~~)l-

Wecan express in termsof Z,,

(6.15)

where

F(4) = c#J)- b3

Let us considera four-poin t function genera ted y Z[i],

GJ1,2,3,4] =(-Q4

S4-mJ

Sit lPSWA3)6jW

-

Whateq. (6.16) tells us may be pictured as oliows:

(6.16)

wherewe haveshown but a classof diagramshat emergen the expansion f

the right-hand ide of eq. (6.16). The part of the diagram nclosed y a dotted



PUB-76134-TH’r’

132 B. W. Lee

squares a Green unction generated y Z O q. Let us now consider he two

point functions

Aj

and AJ generated y Z[j] andZu [J],

+ . ..a

So, f we examine he propagators earp2.= pf, we find

zi

lim Ai = -

P2+

P”-P,2 7

where he ratio

ZJ

lim AJ=-------

P2-$

P2 -$

u = (zj/zJ)1’2

is givendiagrammaically by

0=1*-i- e+...

The renormalized -matrix s defined by

S’(k,, .

>=n

k;-$ _

lim ----gyG(kl....),

i=l k&,2

(6.17)

(6.18)

(6.19)

whereG”s the momentumspaceGreen unction. Let us consider he unrenor-

malizedS-matrix defined rom (?j,

Si”(kl, ..e

) = n lim(kf - pz)Gj(kl, . ) . (6.20)

Clearly only thesediagramsOf ej in which ther e arepoles n all momentum

variables t $ will survive he amputatio n process. In fig. 6.1, therearepoles

in k: andk: at cl,“.) Thus

S;(kl, . ) = crN’2SJ(k, 9 a.. ) ) (6.2 1)



PUB-76/34-THY

Gaugeheories

133

whereN is the numberof the externalparticle;. It follows from eqs. 6.1Q-

(6.21) that

SpSfES’,

(6.22)

and we reachan mp or an conclusion: f two Z’s differ only in the external

source erm, both of them yield the same enormalized -matrix.

We now co me back to the originalproblem,and ask what ha ppenso

ZF [.Il if an nfinitesimal chang es made n

F,

We aredealingwith unrenormalized ut dimensionally egularized uantities

in eq. (6.23). To first order n

AF, we

have

Now, makinguse of the WT dentity ( 5.23),

s

[&%dVl IF, - iJi~~D~[~]~~) exp{i~~~ +iJ&} = 0,

we can write eq. (6.24) as

Since

:AF,

[ 1

& iJi=-.z,

i



134

B.M Lee

PUB-76134-THY

we obtain

z

F+AF - ‘F

=iJiN

s

[dtdrld@Jexp{iSeK iJ&.i,

X G-iAFabPJ~a~~I~J~&~.

But, eq. (6.25) mea ns hat to lowest order n AF,

‘F+AF

=fVj [d$dEdqJexpis,, + iJiai} ,

where

(6.25)

(6.26)

Thus,an ir@itesimal changen the gauge ondition correspondso changing

the source erm by an nfinitesimalamount. But we havealready hown hat

the renormahze d -matrix s invariantunder sucha change Thus

WlF+AF WIF :

(6.27)

A few final remarks: i) In’ he previous ectureswhenwe discussedenor-

malization,we defined he renormatization onstants n respect o their diver-

gentparts. The wave-function eno rmalization onstantsused n this lecture

aredefined by the on-shell ondition (6.17). These wo are elated o ea ch

other by a finite mdtiplicative factor. To see his, observe hat we can make

the propagato rsinite by the renormalization ounter termsdefined n the

ptevious ectures.The propagators o renormal ized o not in ge neral atisfy

the on-shell ondition

hm Ak(p’) = L --

P2-$ P”-P,2 ’

but a finite, final renormalizatio n uffices o m ake hem do so. ii) We can de-

fine the couplingconstants o b e the valueof a relevant ertex when all physi-

cal external ines areon mass hell.Then



Gauge heories

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13s

References

Lecture

For the geneialdiscussion f path ntegral ormalismapplied o gauge he-

ories,seeProf. Fadde ev’sectures,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).

Original iterature on the quantizationof gaugeields ncludes

(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 g auge nvariant 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 irst studiedby

111 R. Arnowitt and S.I. bkkler, Phys. Rev. 127 (1962) 1821.

In conjunction with L. Fadde ev’sectures ncluded n 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 he ones discussed in

this chapter. n particular he following papers iscuss uantization nd/or re-

normalizationof gauge heories n gauges uadratic n fields:

[13] G. ‘t Hooft and M. Veltman, Nucl. Phys. B50 (1972) 318.

[14J S. Joglekar, Phys. Rev. DlO (1974) 4095.

Differentiation and ntegrationwith respect o anticommuting -numbers

arestudiedand axiomatized n

[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 renormalization heory 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. Bogoliu bov an d 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 a nd H. Pendleton (MJT Press,Cambridge, 1970) p. 395.

The dimensional eguhuization,n the form discussed ere, s due o:

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 reiated egularizationmethod analytic regularization)s discussed

in:

[25] E.R. Speer, Generalized Feynman amplitudes (Princeton Univ. Press,Princeton,

1969).

Excellent reviewson the Adler-BelEJackiw nomalies re:

[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 complete ist of anoma ly ertices, nvolving only currents not pio ns)

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