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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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Gauge heories
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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Gaugeheories
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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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:
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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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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.
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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
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- -----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
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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
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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
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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
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(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 - $2) + i(orpp2 -/.l’)] .
(3.24)
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For sufficiently small n, n < 2, the A-integration is convergent, and
- dh
J
,ik(A+iel(I -2’ +i~)
o hrtl2-I
=
p dh -& {h ’-“/2 exp[iX(A t ic)]) = 0.
0
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107
(3.25)
So the dimensional egularization ives he gaugenvariant esult,
12=0.
3.4.Problem
Repeat he vacuumpolarizationcalculation n spino relectrodynamics sing
the dimensional egularization.
4. T he Ward- Takahashi identities
4.1. Notations
Oneof the problems n discussing augeheories s that notationswill get
cumbersomef we are o put explicitly space-timeariables, oren b indices
andgroup epresentationndices.We will thereforeusea highly compactno-
tation. For simplicity in notation, we will assu mehat the gauge roup n
question s a simpleLie group.Extension o a product of simpleLie groups,
suchasSU(2) X U(I), is not too difficult.
We will agree o denoteall fields by r&.Again or simplicity in notation we
will assume j to be bosons. nclusion of fermionsdoesnot presentany diff-
culty in our discussions, ut we will have o be mindful of their anticommut-
ing nature.Thus, or the gauge ield A;(x)
i
stands or the group ndex a, the
Lorentz ndex 1-1,nd the space-time ariable . Summatio n nd ntegration
over epeatedndiceswill b e understood.Thus
(piz sd4x q A;(x)A~~(x) + 0..
(4-I)
where he dotted portion includescontribution from other species f fields.
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108 B. W. Let
The nfinitesimal local gaug eransformationmay be written as
l#Jp$;=$it(A; tt~~j)ea,
(4.2)
whereO. = 8,(x,) is the space-time ependent ara meter f the groupG. We
choose@i o be real,SO that
t; = ~“‘(xa - xp4(xn - Xi)
(4.3)
is realantisymmetric, where7; is the representation f the generator a of G
in the basis&. The’inhomogeneouserm q is non-vanishing nly for the
gaugeields
hi”
=$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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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.
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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$i) I (4.23)
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112
B. W. Lee
Eq. (4.23) is the WT identity as irst der ivedby SlavnovandTaylor. We can
rewrite t in a differential form involvingZF. We define
(ZFlba z Ni
s
Id&Wtl Caib exp We, + iJ&$ 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)
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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
$g=o,
90
=D;qa+f q q 6.
ahc b c&qa
(5.19)
i
Eq. (5.19) is a direct consequ encef eqs. 5.16), (5.17) that the BRS transfor-
mation on 4 andqa s nilpotent. Next we note that
(5.20)
wherewe haveused he fact that
(5.21)
Direct computationsyield
(5.22)
6 30 sqo) r(o)
SD; (())
“3
50, “3 - =
K@.
1
qa63 “3’
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B.IK Lee
“Qo
s50)
-
s2r(0)
s r(o)
s 2 r(o)
s%
qo) = -
(
Qf.
-++----
sKjs’tl, sV* 6L*6Qa
1
Thus
which,proveseq. (5.18).
The act that 9 is nifpotent means hat, in general, 9 for arbitrary
9 = Y(i$, & 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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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
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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 $* transformsunder he gaugeransformation
as
6. Renormalization f
theories
with spontaneously
roken symmetry
6.1. hclusion of scalar fields
In chapter5, we detailed he renorrhalization f pure gaug eheories.Let us
considernow a theory of gau ge osonsand scalarmesons. et
4j = (.qxj, s&)) 9
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Gauge heories
where , arescalarmesons, nd et
be the vector which defies the gauge. he action for the scalar ields s of the
form
wher e Ds), is the covariantderivativeacting on S, V(s) s a G-invarian tquar-
tic polynomial in s andof dimensionat most zero.
Weshall write
(6.2)
[W, ta = [6M, ”] =0,
(6.3)
whereM& is the renor malize$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
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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 = -$- 1’2(c3r ,
a= u
s
( 1
which leaves
invariant, and shift the si fields by
s; = Sk
- u,w P
andchoose
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An important lesson to be learned here is that in a generalineargauge, calar
fields can developgauge-dependentacuumexpectation alues,which are n-
nocuous rom the renormalization oint of view.
6.2. Spontaneously broken ga uge symmetry
Let us consider he casewhereM’ is not positive semi-definite.t is by now
well known that undersuchcircumstances pontaneous reakdown f the
gaug e ymmetry takesplace,and someof the scaIar ields and someof the
(transverse).gaugeosons ombine o form massive ector bosons.We will give
herea very brief discussion f the Higgsphenomenon.
We define V0 by
If M’ is not positive semi-definite,So= 0 is no longera minimum of the po-
tential V,. Let sa = 11,be the absoluteminimum of Vo,
(6.7)
“2Vo
q$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
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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
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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 $J @? Wecan write
the generatingunctional as
Z[iJ =~~[WJexpW[~l+ it@ d~~)l-
Wecan express in termsof Z,,
(6.15)
where
F(4) = c#J)- b3
Let us considera four-poin t function genera ted y Z[i],
GJ1,2,3,4] =(-Q4
S4-mJ
Sit lPSWA3)6jW
-
Whateq. (6.16) tells us may be pictured as oliows:
(6.16)
wherewe haveshown but a classof diagramshat emergen the expansion f
the right-hand ide of eq. (6.16). The part of the diagram nclosed y a dotted
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132 B. W. Lee
squares a Green unction generated y Z O q. Let us now consider he two
point functions
Aj
and AJ generated y Z[j] andZu [J],
+ . ..a
So, f we examine he propagators earp2.= pf, we find
zi
lim Ai = -
P2+
P”-P,2 7
where he ratio
ZJ
lim AJ=-------
P2-$
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)
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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
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B.M Lee
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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
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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,
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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.
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[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,
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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).
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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)