PRI TCHETT, WALECKA, AND ZUCKER

The cross section can therefore be written helicities gives

d'~g a' cos'(-', 8) mq—((k4/k*') I3,I'

d~2dQ~dQ, */4' eP sin4(28)- W

+I km/2@2+(W2/m2) tan2(-', 8)j(l3+'I'y I3 'I')

+(k'/2k*')2 Re(3+')~(3 ')+(k'/k*')

X L(W'/m') tan'(-'8)+k'/k*']'~'v2

Xlm3,*(3+'+3—') j . (C26)Now from I,

kZZdQq*

c XgXg4z.

x IO~I»(w, k) l~,o)12

=s Z 2 —(4k*a) '2(»+1)&x &2 ko J

(C30)

3~s~i"'= (4k*v) '"E(2~+1)&~, x, ,~.'(—y, —8~,)*JX(4I &~(W k')

I xgXg), (C27)

(3,)g,g, ——(k*/k, ) (4k*q)—'~' P (2J+1)

kZZdQq*

LIXax,+'I'+ I3),~, 'I'j4x

X nx, g,'(—yn —8.yn)*(~~I2'(W, k')

I ~~0) (C»)

The relation between the angles is obtained by com-paring Fig. 4 of I with Fig. 14:

+I(& I2'(W, k')lz, —1)l'j=2 (~+2)L I

2'3i~'+ I'+I

2'~i~'+ I 'j (C31)

ey —e/;q ) fy —2X QIrq ~ (C29)

If only the electron is detected, one must integrate overthe final pion direction, and using the orthogonality ofthe 2) functions the interference terms in (C26) goout. Finally, summing and averaging over nucleon

where the eigenstates of parity have been introduced.The resulting cross section for detection of only theelectron is that given in Eq. (3.31) of I. The resonantamplitudes for producing the I~X) channel in thecoupled-channel calculations are obtained by multiply-ing Eq. (5.31) by cose.

PH YS ICAL R EVI E%' VOLUM E 184, NUM BER 5 25 AUGUST 1969

Anomalous Ward Identities in Spinor Field TheoriesWILLIAM A BARDEEN f

Institute for Advanced Study, Princeton, %no Jersey 0$540('Received 24 February 1969)

W'e consider the model of a spinor Geld with arbitrary internal degrees of freedom having arbitrarynonderjvative coupling to external scalar, pseudoscalar, vector, and axial-vector fields. By carefully de6ningthe S matrix in the interaction picture, the vector and axial-vector currents associated with the externalvector and axial-vector Acids are found to satisfy anomalous Ward identities. H we require that the vectorcurrents satisfy the usual Ward identities, the divergence of the axial-vector current contains well-deredanomalous terms. These terms are explicitly calculated.

I. INTRODUCTION

HE presence of anomalous terms in the %ardidentities for currents dered in a number of

spinor field theories has been noted by severalauthors. ' ' The existence of these terms may be traced

~Research sponsored by the Air Force Once of ScientificResearch, U. S. Air Force, under AFOSR Grant No. 68-1365.

f Present address: Department of Physics, Stanford University,Stanford, California 94305.' J. Steinberger, Phys. Rev. 76, 1180 (1949); J. Schwinger,ibid. 82, 664 (1951).

S. L. Adler, Phys. Rev. 177, 2426 {1969).

to the local products of field operators which are sosingular as to prohibit the naive use of the field equa-tions. In a version of the 0. model, the anomalous termsin the %'ard identity for the neutral isospin currenthave led to a low-energy theorem for the decay x' —+ pp. '

In this paper, we consider a theory of a spinor fieldwith an arbitrary number of internal degrees of freedomcoupled to external scalar, pseudoscalar, vector, and

3 C. R. Hagen, Phys. Rev. 176, 2622 (1969); R. Jackie andK. Johnson, ibid. 182, 1457 (1969); R. Brandt, ibid. 180, 1490(1969); K. Wilson, ibid. 181, 1909 {1969);J.S.Bell and R. Jackiw,Nuovo Cimento 60, 47 (1969).

184 ANOMALOUS WAR D I DE NTI TI ES I N SP I NOR F I EL D THEORIES 1849

axial-vector fields. In the interaction picture, the Smatrix is carefully defined using a symmetric ~ separa-tion on the spinor loops. The terms which would besingular in the limit as e goes to zero are isolated. Thecurrents are defined by variation of the 5 matrix with

respect to the external vector and axial-vector fields.

By examining the Ward identities for these currents,anomalous terms are found to arise from the smaller

loops which are, at least, linearly divergent.A renormalized 5 matrix for the loops is defined by

the subtraction of contact terms which remove thedivergent terms and many of the anomalous termsin the %ard identities. However, not all of theanomalous terms can be removed in this way. In par-ticular, the choice of counter term which preserves theusual Ward identities for the vector currents provideswell-defined anomalous terms in the axial-vector currentWard identities. The result of Adler' is obtained for theneutral isospin current coupled to external photons.When charged currents are present, additional termsare found.

Ke indicate the following plan for this paper. InSec. II, we define the theory and discuss some of itsgeneral properties. The terms which are singular in thelimit as e goes to zero are found in Sec. III. In Sec. IV,the Ward identities are examined and the anomalousterms isolated. The renormalized S matrix is definedin Sec. V, where the resultant anomalous terms in theWard identities are also discussed.

II. DEFINITIONS

We propose to consider a theory consisting of aquantized spinor field with arbitrary internal degreesof freedom having arbitrary nonderivative couplingsto external scalar, pseudoscalar, vector, and axial-vector fields. This theory is described by the Lagrangiandensity

~()=~()L'v ~+r()j~(),where

v'=v"~p& 7~ =vs.

The field g(z) is a Dirac spinor field and is a columnvector in the internal space. The function I'(z) is givenby

r(z) = —G~,(z)+~„V, (z),where

P+(z) =F+Z(z)+iysil(z),V (z)=V (z)+& a"(z).

F is a constant matrix in the internal space and servesto give the spinor fields arbitrary masses. The fieldsZ(z), II(z), VI'(z), and A&(z) are all matrices in the in-ternal space and may be written

Z (z) = Xz Z (z), II(z) = Xn II,(z),V (z)=),v V.(z), A (z)=X A.(z).

(4)

The fields Z (z), II (z), V (z), and A, (z) are the ex-

Pro. 1.Feynman diagram forthe external fields coupling toan external spinor line.

&n &n-i Pl PP

ternal scalar, pseudoscalar, vector, and axial-vectorfields, respectively, and Xz, X&, P z, and P& are theirrespective coupling matrices.

We will discuss this theory in the interaction picturewhere the free Lagrangian density is given by

&o()=A)(. ~ —~ox()and the interaction Lagrangian density is given by

z,(.) =y(,)r (,)y(.) . (6)

S=T exp i dz Zr(z) (7)

where T is the time-ordering operator. There are twotypes of Feynman diagrams for this 8 matrix. Thediagram where the external fields couple to a free spinorline is shown in Fig. 1. These diagrams are perfectlywell defined by the S matrix in Eq. (7). The externalfields may also couple to a spinor loop, as shown inFig. 2. These diagrams are not all well defined, since theloops with four or fewer internal spinor lines have formaldivergences.

To define the loops, we use a symmetric e separationat the vertices, which is effected by the replacement of«(z) by

f&'(z) =&I z+—«z)& z—

I

2n 2n)

where e is a spacelike four-vector, and n is the numberof external lines on the loop. In addition, a symmetricaverage of e over the spacelike directions is to be per-formed after the loop is calculated. The loop is thendefined by the limit as e' —& 0, if it exists. For the loopswhich are singular in the limit as e' goes to zero, wemust first isolate and remove the singular terms before

Fro. 2. Feynman diagram forthe external fields coupling toa spinor loop.

2

k;

4 S. G. Gasiorowicz, Elementary Particle Physics Qohn %iley RSons, Inc., New York, 1966).

The mass term Mo in the free Lagrangian is chosen tobe proportional to the unit matrix in the internal space,and r(z) is defined by

r(.) = r(.)+m, .In addition to the external fields, r(z) contains the mass-splitting counter term M0 —GP'.

Using the conventional transformation to the inter-action picture, 4 the S matrix is formally given by

l84'BARDEE&A.

( );g/hvar"(*sed tP isolate th J„OX =roced. ure u;tion

tp the }imit. The p. III. This pres«p '

passing to' 'ven in Sec '

sjpn for theerms is g»

he expressiosingu}ar t

f the }oops. Ttp deine]pop in g'p 2 js given by

c

ds Zr(s)~( ) ),~o)k(x) exP

y850

(10),p/bA "(x

) „p k ds|'r(s)r &(„),„,.~.V(* e"p 'd, „C—i(2~) '&"S,.(r) =(-')-„(')"

—r ..".(M, —a,)- r(.,)».Xtr((Mp —Ir) 'Fs . . .p—

k' sk . 9

tion i

A. Four-Vertex Loop

lo arit m''thmically divergent.ou -vert t m.ex loop is g tFrom Erl. (9), we write t e p-s

}pop thehen the current co place

wlmn

) t be rep ace

~ .(y&+" Issk)1

t verte»n Eq: .8).

(z1—z 'e"Curre

s given in q} interaction

„) . . dke'"separa

'we have de n

&

interacting

ed vertex ased in t e in

X

this section,'

d s inor 6e in

In "of a quantize sp

. III we isolatePiet ~

externa fje ds''

the }jmit as

re a theory o. In Sec.ith a,rbitrary e

'-dependent in tw} ich remain

f the & separathe terms

jn the de6nitio oumb ers of ver-

pes to zero.

NCESDgfERGE

necessaryre}ated, as

sp that looPsd needed for t e

ere de6ne d b a symmetric

~

p}ytic

'ot the only pne p . In Sec.

~

d b then taking thet} is }imit

identities.f the loops is no

'the explicit

erage on ~ an ythan four vertices,

This

is deinition oe of reserving

s with mPregra] is

Thisthe advantage o p

}}be useful lnzero Fpr loops

k inside the }pop .ta

oops This pr pHowever, th's

exists and mayf the smal}er jpop

'we

but itha, st e .rp erty wi . ze

a beta eninaln

symrri yin Secs. III

an anoma}pus}imjt dpes no

-de endent. Intwp-

of t e o 'and IV. ~ '

.pt exjst for t e

thjs sectio

iscus»pns~ ive rise to ma y ' c. Iy

which re~ain —

the four-, thre -,

of the loops w gQ be seen in e ' terms;

ergent terms '"

de6nition od identities, as w

h }pops in orderiso}ate the div g

j}}be necessary'

entities as ppssi 'and pne-

ence, it wipf the Nard j e

'erformed

th ough the ad it o} ctor current~ a

h ex-The vec«r

t i.x with respand axia-ve

ect to t eof the Smar'} }ternal vector and axia -ve

sk . 11p— 'r(sk)(Mp —kk) 'F(sk)) . 1

S '(F) =(—I)-,'(i)'

—0) 'I'(s)(Mo —It) 'F s o— —' s . 1sk (Mp —

k p

Z4 4 4 d$ je ik I (zg—zI )d dskc i(27r) —'j' ks' 'dZg dZ2 dZ3 Z4 -ik3 (z4-za)-»s ( 3—a) dP e-' ~dk2e-' '

symmetric av g

sqI (zg—z1)

}pop becomes

dZ4 dy'e P.dqle

—sCI z

dory

dZ2 1ZS Z4s, (r) =(-1)-;(2.)- —ql. (z3—zR)qge

)'

~ (k+k+k+k )) t ((M n) r()—an ma n e of variablesand make the change o var

(12)

ce the loop integra' '

e I'=4ee t e ation variable P=k

terms which would

=P+q), kk=P+qk, k-

q'

~ ~ poes to zebe singular in

—sqS (z4—z3)

era e on

dq e Iqg'

3+q4X dq4e '"

—sqI (zu—z1). . .dzl z4 dI' e'P' dqre 'q '=(—1)kk(2s.) " ds) s4

qi+q2+q

—P—q,)- r(s,)(M,—P—q,)- r., M,——Xtr((Mp-

dq4e 'q4' 'q4

— (zX—z4) g

4 )M — ' s )PF(s )Pr(x )PI'(s ))M ' —P')-' tr(PF s4 s,X Mo — -4

4 pp)p

r(,) .r(z)p"r(z) }

dp e p''(Mo' p )

)+(1/24) try r(z)&"

=(— 4

X{ try„r(s) 7" "

(,),r(z)y" r(z) }

i)-'(2 ) ' d',, »r(.)v«')' ('

)+(1/96) try«, „r(,),„r(z)v"«s= —pc 1(p) d.{(1/48) '"""

(14)

t.ion s:ere we de6ne ependent fun

2 2 1

the fpllowing

dp e4P'(Mo'C,(,) = —i(2z) 'dp e*&'(Mo''2~ '4( )

Thre~ e Vertex Lo p

ta eof thecyclc utatipnx loo d t~ing advan gr to those prx loo may e

f r fpur-vertex p~ be written as

formations»m~ .ent three-verte

sing trans .the linearly diverg

.p . (zg—za)

ices t e'Iz. (z3 zs) dk 3edk2e ' '

in the ind rmetry in

;I,i. (z2—zi))—4]p dkye+ p(r) (—1)s(4)

(M k,) 'r( i)}g,—~r(z, )(Mo —k')

. . „+4,+p &I4 tr{(Mo—"'se (Xe

;q3. (zi—z3)~

q2, (z3—zg) d$3e. (z —zi) dg2edp ez~' dq, e "'Z2 dZ3I)— pr-"2) ~p dz, d

, „,Pr(„)pr(zp)M, r(zi)}

»1, .r(z)ip ar(z)})Mor( )+"d,{.t~„r(.)v"r('=iCi(o)

A(a g) (a„A)&~

THEORIES 185&IN SPIN OR FIELD~ARD I DEN TI TIESANOM A LOU

where

C Two-Vertex LooP

determined to betwo-vertex loop ls etically divergent two-The singular part of the quuadra

&& (&I+&a) l2e—zan (zi——~&1 ( a—zi)—'24r 4]' dk4e ' 'dzp ds2[ )s (r) =(—1)-, (4) 1

q4+ qp)—zqa (zi—za) g !4pl' («I «4) dqpe 4dj' e»'&

dydee—'qi'2~ -S

dzy dz2=(—

zy. )e

r — —'r(zp) (Mo —1'rr) r(zr) }Xtr{(Mp—4) 'r zp o—

1).( )

M,r(.,)M.r(s,)}p — p -p tr{Pr(s,)Pr(s,)+

I' zp +PqpPr(zp)pqpPr(zp) }]+2(Mo'—

+(Mo' —P') ' tr

dz tr&„r z q —'p dzpMp trr(z)r(z)C, . dz trq„r(z)q r(z)}—iC ()

s «a r(z)]. (16)+(1/24) try„a„r(s)q»z 4 p trp ar )r+-,'M, trr(s)i~. ar(s) —„tr~

loo is given byThe one-vertex oop'

D. One-Vertex Loop

Cp 4 ds trI'(z).—k) 'I'(z) = —iMpcp(p)dk e""tr(Mp — ——— Cp 4s'( )=(—= —1)j ds[ —i(2n-)-4] — = — Cp

BAR OFF NILLIAMi852

(18)

oes to zerent in the limit as e g

r —3d. =g S,"(r).S,(r) =S.(r-3d. =

—17). The sum of13) and (15)—1e ex ressions

ent art of nvenience, repehave, or coE. (1), h

r.)}these terms is given in q.

ds 1/48 tr y„r(s)v"r(s)v. r(s)v"r s

have

M C pr s —C(p)z3f 'CS,(f') =sz[Cz(p)+Mo C ds tr(y„r(s)y )}

r s)~"r(s)V"r(s)}—;.«V.I' ~ ~+(1/96) tr(v„r(sh„s p

"~"r(s)}—z'p «(v &r s v.—(1/24) tr(y„B„F(s)yp —' gr s &. (19)

'h remain e-dePen-t,d th, term~ wh"" 'have»»amailer loop 1

dia rams by;ngular part o

f ~ single loctly evaluating t

~e degne the sum od

IV. WARDARD IDENTITIES substitutionV+~(x) ~ V p(x)+a~~ x,

(23)

[see Fzg.. 2 and

1' — . —slezz (z1—zzz)—ik1 (zg—z1) ~ . .s" v ~~,r)=(-1)(')" d"7 ds„[ i(2zr) —45" dkze

) 'r(sz)} (24)—k. ,)-' yr, a, ———Iz ) ' 8~(s.)(Mp-" ' »" tr Mp —„—ye~s (4+ + ((Xe"'

h d'ces we obtainrmutation oft ein icb arts an a c clicpe

—'n —z ) i~ ~ (k1+" +k„)/n

n integration y p

—ik» (z1—z„)et&

Using an in

ik1' (zR z1) 4, d e— zs' —z ) s44 ~dkieS"y B~,r)=(—1)(i)" dszf Y

—z(2zr)

—&, -' "(3d,—a,)- r(.,)}Mo —0 ) 'i(k„z-o—.-'A

z—k.)~(s„)(M,—Iz„,)- ".Xtr((

—i n — i~ k1+ "+km)fn—ikzz (Z1—Zzz)ei~dkye' '(z~ ")~ ~ ~de [ Z(2zr) 4)—" dip,—e* 4. '= (—1)z"

d dentities for where=X A (x)&Kg yzA. p x .

the War i e

A.y x = y

we examinethe Intro-

becomes

II A tdfor the n-vertex loopfth gdfor the loops because o

se cfepe zfe upo

J„.(x)= -z S

p . g r)i 8/b—8 it (x)]s,"(y 8

x S,"(y Bit+,r).i[8/b 8pA-;(x

anom

h % d d

h currents are giv

a

I' sznce thesensider t e

can containrenceso t ec

attached to a free p'e

ee s inor ine,

8pJ„(x)= z[b/ 8 x) S,"y.x =i[b/W. ,-(x)]S,.(& a

the expression

I

K (9

(4M'p —kz)-'r (s,)}

.[—z(2zr) ]

(..) —z~(s,)r(s„)]'r s~z) . (Mp —kz) '[F s, iXtr((Mp —A„z) 'F(s, . — —4 r s, z s„

( — ' s„—zM~(s„)]Xtr( p—M —Iz ) '[iiVoA s„—

—ikn (z1—zn)eie. 1 ~ ~ ~(k1+" +kzz —1) /(n —1)—ik1 (zg—z1). . .dkge—' 1'+(—1)i" dsz dg

6 3/0 (1957).'

Nuovo Cimento~ V. Takahas, gp

184 THEOOR I ESS IN SP INOR FNARD II DEN TI T I ESA NO M ALOUS

+ — i s.[ —i(2~) 4-j" dk, e-'"~ (*~*»+(—1)i" ds) ds. —i

4 (4+" +&a—s)/(~ —&)1

~ )- A-( )F(-)3&. 25

ie (kI+" +kn) f& g«'(kI+''' n— (n—1)

—I() 'LF(s i s„—i~ ~ ~

ments.

b —q /. '

(25) becomesb —q8&. Equationp to replacei~ q by —q &.an integration y

'n b parts in oan use an

'

~ ZI—Z„) i~ (kI+".+k.»skn (4 n dP ~—ikI (Z~ZI). . .d „( i(2—)r)'$" dk)e * &~."(v.8A = —1)i" dsr . „— ' " dk)eF) (

'PiMpA (z ) iMo—A+(z )5 ()— —' s,' iMpA s. —' s. (M() —I())-'F(s ))Xtr{(Mp—0 ) ' iMpA s„—'

Xte

r — 'F s z) (MpXtr{(Mp —k„)

. In the third term, we make t e oand s„ integrations. In the t lre erform the k„an s„'d term above, we per

n —1

In the secon

k =Psn 5Z 1p ~ ~ ~

p

n—I''k —~ (ZI-Zn-I)—skI. (Zm—ZI). . .) —i(27r) 4$" ' dk, e—' &~ ~ ~

'F(s.-)) (Mo —/I(&) 'LF s) iA+ s, i-+" +P„- ) /( —) ) tr{ (MXe" ('

~P—sqn-I (zn zn-I) dp &4 ~dS d e 'qI' dq &t,—sqn-I. z„—+(—1) 2 ) " ds) ds~ q)e

«&+ "+p. »»/~ e (p-)+" +p i) —pr/(p ))j-XLe—(pt+ "+p -x) Bp/ —(p(

X r{tr (Mo Pq„—F F)+D,"(A+,F).S,"-'(FiA, —iA F,=8,"(iM,(A A,), I)+, ——

(-)- A-( )F(..)j&'F(s--)) .o— — -' s ( s„D n .(M—p—P—q&)

—'LF(s, i

(26)

divergence. r qrom Eq (

I' (zn Zn —I)&qn-I' ndg„dqqe —'qI'dS qg' ( I—zI) ~ . . dq g—eqn-I' np "(&+,&)=(-~)(2 )-'"f&z ". ss. qe

'teraction picture)encein t eln er

hil h h' d

erst telm on gl

x Nard identity is g'rm corresponds to

the e-vertexterm co

0

(m+" +p~-)) 8//(n —))j+"'+qn-I)'~Q/a ~ qI(pl

s )i (s ) iA (s,)—F(s„)]j. (2&). Mp —P—q~)-'LF sr i s

(a) /p=4:

Xtr{(Mp—P—q„, -''F(s=r) (

in square brac eq kets is of order &.

toe b ot.igthtth t

ore divergent will survive ln

"P 'o o ~ ed

gh

sint e r

t 'I h

which wo

g yFor n&, e4 the loops are su ciens aller loops and obtain

s "F(s)+8"A (s)y.F(s)~F(s)y„F s' s +8&A+(s)y.F(z)y„F(s)y"F s +s tr{8"A+(s)y„F(s)y.F(s)y'(sD, '(A+, F)=— i ds tr

KVI LLIAM A. BAR BEEN l84

(b) m=3:

D.'(A+, F)=—

(c) n=2:

ds tr{-,'M, a A, (s)~„F(s)F(s)+z'M,B~A,(s)F(z)~„F(s)+,'M, B~A (z)F(s)q„F(z)(2s.)4

+(1/24)M LA (s) —A (s)]I (s)r 8 F(z)+r ia A (z)7 „F(z)r~ 8(z) ——ia "A (z)r"F(z)r B„F(z)

3—X6 'iB A+( z)y„F( )sy"B„F( )s+$&&6 'ia A+(s)y„F(s)y BF(s)+s)&6 'ia A+(s)y "F(s)y„a,F(s)}; (28)

D,m(A, F) = ~(sC,( )s+M p Cz(e)]i dz tr(a"A+(s)y„F(s)}

(d) m=1:

+ i ds tr(sMoiLA+(s) —A (z)]a'F(s)+(1/24)a~A+(z)~„82F(z) };(2s)'

D, '(A+, F)=0.%e now consider the Ward identity satisfied by the sum of all single loop diagrams. Using the S matrix defined

in Eq. (18) and noting thats, (& BA„F)=s, (iM, (A —A,), F)=D, (A„F)=0,

we Gnd

S,(~ BA„f)=P S,.(~ aA, F)= P fS,-(iM, (A —A,), F)+S,.-'(FiA, —iA F, F)+D,.(A„F)}s 1 fL~2

where

= Q fs,"((F M0)iA+—iA (I' —Ms), F—)+D,"(A+,F)}a~1

=s,(FiA —iA 1', F)+D,(A,F),

D,(A„r)= P D..(A+, F)=Z D,.(A„F).

(29)

(30)

The first term on the right-hand side of Eq. (29) corresponds to the usual operator divergence, while the secondterm, gives the anomalous divergence for the loops as defined in Sec. II. The anomalous divergence D, is given bythe sum of the contributions in Eq. (28): We obtain the result

D.(A+,f) = p D."(A+,F) = i dz tr((1/36)a A+(s)y„f'(s)y„f(s)y"F(z)+(I/36)a A (z)y, F(z)y„f(z)y"f (z)a-z (2s)4

+(1/36) B~A,(z)q„F(s)~ I (z)~„r(s)+-,'iA (z)~ BF(z)~ aF(s) —&iA (s)B.f (z)~„a f (z)„„—(I/72)B„~.F(s)a ~ LF(s)iA (s) —iA (z) F(s) —~ BA,(z)]

+(I/72)V ar(s)V BEF(z)iA,(z) —iA (z)F(z) —~ aA, (s)]}

+ M,i dz tr( ——„F(z)i~ BLF(s)iA,(s) —iA (z)F(s) —q BA,(z)](2s.)'

—,', LF(s)'A+(s) iA (s)f'(s) —yaA (s)]—y af'(z)}

7r2

+ C (E')+2Mp Cg(r)+ Mp 'L ds tr(| „'r BA+(z)r&F(s)},4 (2s.)' (31)

where we have replaced F by I'+MD and used integrations by parts in z to combine some of the terms.This completes the discussion of the %'ard identities for the loops defined with symmetric e separation. As was

noted jn the beginning of this section, the Nard identities for the currents attached to external spinor lines containnp anomalous terms. Since these two types of diagrams are the only ones present in this theory, we may use theresult in Eq. (29) to evaluate the divergences of the operator currents defined in Eq. (10). The anomalous terms

184 ANOMALOUS XVARD I DENTITIES IN SPI NOR FIELD THEORIES 1855

will be given in terms of D„using relations of the form given in Eq. (23). In the interaction picture, we find

8~1„(x)=8~ y(x)y, ) v~(x) exp i ds ~r(z)

P(x)P,r,iF(xgP(x)+i D, (lt+, I') exp i ds Zz(z)bii. (x)

8"Jg„(x)=8" P(x)y„yqXgQ(x) exp i ds Zr(z)

(32)

f(x)(Xg y5, if—(x)}P(x)+i D, (A+,F) exp i ds Zr(z)bAg (x)

The equivalent form for the divergence equations,written in the "Heisenberg picture, "would be

8 J (x)=8 (g(x)y„Xv Q(x)},= (q(x)P.,;ir(x)]g(x)},

+i/8/bA (x)]D,(~,F),8~J,„.(x)= 8~(q(x)q„qg, .g(x) },

= ( q(x) (7,.v„- 'F(x) }q(x)},+i [b/ibA5 (x)]D,(~,F),

where Q(x) are the Heisenberg spinor field operators,and the local product of the spinor fields is defined tocorrespond to symmetric e separation in the interactionpicture.

In this section, we have shown how anomalous termsarise in the Ward identities and divergence equationsfor the currents in a theory defined by symmetric e

separation. These terms arose from the singular natureof the smaller loop diagrams, Because of the way wehave defined the loops, the anomalous divergencecontains many terms and may be seen in Eq. (31).In Sec. V, ere will redefine the loops in order to de-termine a "minimal" anomalous divergence.

U. RENORMALIZATION

The definition for the spinor loops given in Sec. IIis not the only one we could have used. The absorptiveparts of the loops are unaffected by the addition oflocal polynomials in the external fields and their deriva-tives. In this section we define renormalized loops bythe addition of counterterms to the single-loop S

=Sa(I'i&—ih I', I')+Da(&, f'), (35)

where

D„(b,f') =R(f'ih ih f y8—~, )F—+D,(h, f'). (36)

Da(h+, F) gives the anomalous divergence for the loopsdefined by the subtraction of the counterterm R(1').

In order that the loops be finite in the limit as e2 goesto zero, we must include counterterms which removethe singular terms given in Eq. (19).We define

matrix given in Kq. (18). The counterterms will servetwo purposes. They should remove the terms, isolatedin Sec. III, which were singular in the limit as e goesto zero. In addition, the counterterms will be chosento remove many of the anomalous terms in Wardidentities found in Sec. IV.

The renormalized single-loop S matrix is given by

S,(1')= S,(F)—R(F), (34)

where S,(I') is defined in Eq. (18) to be the sum of allsingle-loop diagrams with symmetric e separation, andR(I') is a local polynomial in the external fields and theirderivatives. The renormalized S matrix satisfies thefollowing Ward identity, corresponding to the onesatisfied by the e-separated S matrix in Eq. (29):

S (y 8~,F)=S,(y 8~,P) R(y 8~,F)—=S,(f'i~ —iaaf', 1')+D,(~,f') —R(q 8~,I')=S,(I' ~—~I', I')+R(I' b —~I' —v 8~, I')

+ D,(~„r)

Ã2

Ri(P) =- Cz(e)+bfo Ci(e)+ M,' i dz tr(y„F(z)y&F(z)} Ci(e)i —ds tr((1/48)y„f(z)yi'F(z)y„F(z)y"F(z)8 (2s)'

+(I/96)y„F(z)y„f'(z)r F(z)y"F(z) ——„7„F(z)y I (z)iy. 8f'(z)

—(I/24)y„8, F(z)y"8"F(z)—~ try 8F(z)y 8f(z)}. (37)

By comparing Eq. (37) with Kq. (19),we see that Sa,(F)=S,(I') —Ri(I') is finite in the limit as e goes to zero. Thecontribution of Ri to the anomalous divergence may be calculated using Eq. (36). The second term in Eq. (37)

WILLIAM A, BARDEEN 184

does not contribute, while the first term gives

~2R,(f'i«+ —i« I' —y 8«+, I')= —— C2(e)+MD Ci('E)+ Mo' i dztr{&„p 8«+(s)p"f'(s)}.

4 (2')'(38)

This contribution exactly cancels the third term in the expression for the anomalous divergence D, in Eq. (31).We now consider additional counterterms which will be used to remove some of the terms remaining in the

anomalous divergence Da, . We define the counterterm R2(I') by

1 x2 1R,(f') =—— Moi ds tr{I'(s)iy af'(z)}+ i dz tr{y„a,r(s)y"8"I'(s) y—a r(s)y af'(s)

24 (2m)4 144 (2n.)4

1 Ã2

+— Goi dz tr{y V (s)P+(s)y V (z)P+(z)+2y U. (s)y V+(s)P (z)P+(s)}12 (27r) 4

1 x'ds tr{2 V „(z)V+ (z) V „(s)V+"(z)+ V+„(s)V+„(s)V+ (z) V+"(s)}, (39)

72 (2x)4

where P+(s) and V+&(s) are given in terms of I'(s) in Eq. (2). The contribution of these counterterms to the anom-alous divergence may be calculated using Eq. (36). For the S matrix defined by

s„+,(r) =s,(r) -R,(f)-R,(r),we find the associated anomalous divergence for the loops has the form

D„+„,(«„r)=R,(fv« i«r —~ 8—«„r)+R,(r i« —i«r —~ 8«„r)+D,(«„f)1

i dsie„...try5{2i«+(s)8&V+"(s)a'V~'(z) —8&«+(z)V+"(s)U+ (s)V+'(s)},6 (27r)4

(40)

(41)

where e„„,is the totally antisyi~~etric tensor, with&oz23= 1.

The loops, as defined in Kq. (40), have a number ofinteresting properties. They are all finite in the limit ase' goes to zero; therefore, we may pass to this limit. Theadditional, finite counterterm R2 allows the anomalousdivergence to take on the particularly simple formgiven in Eq. (41).We note that all of the terms contain-ing scalar and pseudoscalar fields have been removedfrom the anomalous divergence. In a theory with scalarand pseudoscalar external fields, only "matrix elements"or three or more currents are affected by the anomalousdivergence. In addition, we note that the anomalousdivergence is proportional to the pseudotensor e„„., so

that only matrix elements having an abnormal parityrelation are affected.

We briefly mention the construction of the full Smatrix and the operator divergence equations with theloops defined by Eq. (40). In the interaction picture,the full S matrix is given by

Sz,+ii,= T exp i ds Zr(s) exp) —Ri(r) —R~(r)j.(42)

The currents are obtained from the S matrix in Eq.(42) by variation with respect to the external vectorand axial-vector fields, as done in Kq. (10). We obtain

fII

J„(x)= P(x)y„Xv+(x)+i t R (y 8«+, r)+Rs(y 8«+, I")j exp i ds Zr(z)88~«(x)

&&exp) —Ri(I') —Rq(r) j={g(x)y„X& i'(x) }&,+»,

J,„(x)= tt (x)y„X~&(x)+i- — (Ri(y 8«+,r)+R2(y 8~,I')j exp i ds Zr(s)88~«p(x)

Xexp/ —Ri(F)—R2(I')j

(43)

184 ANOMALOUS WARD I DENTI TIES IN SP I NOR F I EL 0 THEORIES 1857

The divergence equations, corresponding to those in Eq. (33), are given by

a&J„(x)= a&{itt(x)y„kv Q(x)}s,+s,= {g(x)Llw. ,ir(x)$$(x) }„, ,jiLa/ah. (x)jD, „,(~,r)= {g(x) Ll~v, ir(x) jQ(x) }s,+R,——,'Lz'/(2s)'$ie„„. , tr {2ihv y, a" V+( x) a' V+'(x)

(44)+Xv y5(a&V~"(x) V~ (x) V+'(x') —V+"(x)a"V+'(x) V+'(x)+ V+&(x)V~"(x)a V~'(*)j},

a"J5„(x)= a"{g(x)y„y5Xg g(x) }s,+s,= {itt(x) {X„7„.-, i I'(—x) }Q(x)}a,+a,+i)8/aA; (x)jDs,+s,(+, I')= {hatt(x){Xi y.-, i r—(x) }Q(x) }a,+s, ', (—s'-/(2s)'joe„„„ tr{2ih~ a&V~"(x)a V+'(x)

+~~ t a"V+"(x) V+ (x) V+'(x) —V+"(x)a"V+'(*)V+'(x)+ V+"(x) V+"(*)a'V+'(x) j}.In the derivation of the renormalized loops in Eq. (40), we have treated the vector and axial-vector fields sym-

metrically. Therefore, both the vector and axial-vector currents contain anomalous divergences. In some theories,such as electrodynamics or the gluon model, it would be useful to define the loops so that the vector currents obeythe usual divergence equations, while the axial-vector currents have the only anomalous divergences. We are ableto do this by again redefining the loops by the addition of the following counter term'.

1Ra(r) =— i dz ir.„„„tr{ i V "(z)V—~"(z)ia'V+'(z)y.-

6 (2s-)4 —V-"()V "()V+'()V+'() .-+lV+"()V-"()V+()V+'()»} (43)The loops are then defined by

s&„&„„(r)=s,(r) —z,(r) —z, (r) —z,(r),and the associated anomalous divergence is given by

(46)

Ds,+R~s, (it+, r) =Ez(riA+ —iA I' —y aA+, r)+Da,+s,(h+, I')

1dzz„„., trrit;(z){ ,'Fvt'"(z)Fv"-(z)+ ,',Fg~"(z)Fg—"(z)+-,'iA"(z)A "(z)Fv"(z)+-,'iFv""()A (z)A'(z)

4m'

+', iA "(z)Fv-"'(z)A'(z) (8/3)A "—(z)A" (z)A~(z)A'(z)} (47)where trr means the trace only over the internal degrees of freedom. The field-strength tensors Fv""(z) and Fz&"(z)are of the Yang-Mills type and are invariant under vector and axial-vector gauge transformations. They aregiven by

Fv~" (z) = a~V"(z) a"V~(—z) i[V~(—z), V"(z)j i[A~(—z), A "(z)j,F&~"(z)= a~A "(z)—a"A~(z) —i LV( )z, A(z) j—iLA~(z), V"(z)]. (48)

a"J„(x)= a"{&(x)y„Xv&(x)}ii,+z~s,= {&(x)Ll~, 'r(x)]&(x)}, ,+ t b/bA (x))D„„,,(il, r)= {4(x)o~v ir(x) jC(x)}a,+a,+a. ,

a"J~. (*)= a"{4(x)v.v~v~ C(x)}s,+a~a,= {g(*){X-V„—r( )}g(x)}„„,„,+iLa/u. ,-(x)jD„„„,,(~,r)= {Q(x){X~ys, ir(x)}Q(x—)}s,+z,+s,+(I/4n')e„„„ trrLl~~ {'Fv""(x)Fv '(x)+ ,'~Fa~"-(x)Fg '(x)—+zzi A "(x)A"(x)Fv"(x)+ z2iF vt'"(x) A (x)A '(x)+-', iAI'(x) Fv" (x)A '(x) —(8/3) A "(x)A "(x)A'(x) A '(x) }].

(49)

The full S matrix and the operator currents are defined by the addition of Ei(r) to the expressions in Eqs. (42)and (43), respectively. The operator divergence equations for the vector and axial-vector currents correspondingto those in Eq. (44) may be determined using the expression for the anomalous divergence in Eq. (47). We obtainthe result

The divergence equations given in Eq. (49) representthe end product of our entire calculation. The anomalousterm in the divergence of axial-vector current is,indeed, the minimal anomalous divergence. The ad-dition of further counterterms would either destroythe normal divergence of the vector current or give

' C. N. Yang and R. L. Mills, Phys. Rev. 96, 19i {1959).

additional terms in the axial-vector-current divergenceequations. To see this fact, we examine the possiblecounterterms that could be added to the S matrix inEq. (46). In order to preserve the vector-current diver-gence equation, the vector 6elds can only enter thecounterterm through the field-strength ten sors inEq. (48). To have any eifect on the a,nomalous diver-

9; I Ll I A»I A. BAR BEEN

gence in Eq. (49), the counterterm must be proportionalto the pseudotensor ~„„„'therefore, the product of thefields must be odd under parity and even under chargeconjugation. Under these operations, the fields trans-form according to

P: V~ V, 3 —+—A, Fv —+Fv, Fg —+—Fg,C V —+ —V~ A —+ A~ Fv —& —Fv~ F~ ~ F~~, (50)

where V~ means the transpose of the matrix indices in

the internal space.The possible counterterms are given by

R,(V,A) ~f d„E...[R t (F."'(z)F; (s)(

+R42 trr{Fr""(z)}trr{F~"(s)}+iR4, trr{Fg""(z)A (z)A'(z) }

+244 trr{FQ""(z)A'(s)} trrA'(z)]

+terms that are higher order in thefields or their derivatives. (51)

The first two terms in Eq. (51) may be present, butthey do not give any contribution to the anomalousdivergence. The third term can be eliminated, since itis odd under charge conjugation. Note that we have notincluded terms in Eq. (51) which would violate parity.The fourth term may be present and can contributeterms to the anomalous divergence. However, this termdoes not have the coupling structure of a spinor loopand cannot be used to cancel completely any of theanomalous terms in Eq. (49). For completeness, wegive the form of the contribution that this countertermwould give to the anomalous axial-vector-currentdivergence:

Dr(,(it,-, V,A)

=&,(fA,ix,),(V,i2.j a~., V,A—)-=844' dg e„„,„2trl iAg s 3 z,Fv"" s

'Xtrr{A '(z)}+2 trr{iitz(z)} trr{Fv""(z)A (s)A '(z) }—-', trr{A5(z)} trr{F~ "(z)Fg"(z)}J. (52)

The terms which are higher order in the fields or theirderivatives in Eq. (51) will not give contributions to theanomalous divergence of the form given in Eq. (49)and would simply add higher-order terms to the anoma-lous divergence.

We should note that all of the counterterms that wehave used to define the loops are consistent withWeinberg's theorem' for the asymptotic behavior of the

~ S. steinberg, Phys. Rev. 118, 838 {1960).The asymptotic be-haviors of the n-point functions as determined by steinberg arethat the two-point function may be quadratically divergent, thethree-point function may be linearly divergent, and the four-pointfunction may be logarithmically divergent. The counter termsR&, R&, and Rs in Eqs. (37), (39),and (45) do obey these asymptoticbounds.

vertex functions. Therefore, if these definitions of theloops were used in an otherwise renormalizable, inter-acting field theory, the addition of these countertermswould not destroy the renormalizability.

The result that we have obtained for the anomalousaxial-vector current in Kq. (49) may be directly relatedto the recent result of Adler' concerning the anomalousdivergence of the neutral axial-vector isospin current.In these theories, the external axial-vector field is notpresent, and the external vector field is the photonfield. The anomalous divergence is given in Eq. (49)by making the following substitutions:

F&&"(x) = eg 'F,""(x), A "(x)= 0, (53)

where X& is the charge matrix for the spinor field. Theanomalous divergence becomes

B&J5„'(x)=JP(x)+ (1/4z. )no

)&tr Xr„9.9.~~„„.,F„~"(x)F,"(x), (54)

where X~' is the coupling matrix for the neutral axial-vector isospin current to the spinor field, and ao ——e02/47r.

This is the result quoted by Adler.

VI«CONCLUSION

In this paper, we have studied the theory of a quan-tized spinor field with arbitrary internal degrees of free-dom having arbitrary coupling to external scalar,pseudoscalar, vector, and axial-vector fields. By go-ing into the interaction picture, we were able to care-fully define and make finite all S-matrix elements.The vector and axial-vector currents were defined bya variation of the S matrix with respect to the externalvector and axial-vector fields. Because of the singularnature of the smaller spinor loops, the Ward identitiessatisfied by these currents were found to contain anoma-lous terms. By considering all possible S-matrix ele-ments, we were able to write the divergence equationsfor the operator currents defined in this theory, asgiven in Eq. (33).

The anomalous terms in the divergence equationsfor the currents could be cast in a particularly simplefrom by a redefinition of the spinor loops. By treatingthe vector and axial-vector currents symmetrically, thecurrents were found to satisfy the divergence equationsin Eq. (44). If we required that the vector currentshave the normal divergences, the divergences of theaxial-vector currents contained the minimal anomalousterms given in Kq. (49). These anomalous terms wereminimal in the sense that any further redefinition of theS matrix would either destroy the normal vector-current divergences or simply give additional terms inthe anomalous axial-vector-current divergences.

The result that we have obtained for the minimalanomalous divergence of axial-vector current is notactually dependent upon our original definition of theS matrix. Any other definition which makes the Smatrix well defined and finite could be used as a starting

184 ANOMALOUS KARD I DENTI TIES I N S P I NOR F I EL D THEORIES 1859

point. The Ward identities could then be calculated,and we would, in general, expect to find many anomalousterms, as we did in the case of the 5 matrix defined bysymmetric e separation. However, by again adding theappropriate counterterms, we would find the sameminimal anomalous divergence given in Eq. (49).Therefore, while the form of counterterms Ei, E2,and E3 does depend upon the original definition of the5 matrix, the form of the minimal anomalous diver-gence does not.

We wish to comment briefly on the form of the mini-mal anomalous divergence obtained in Eq. (49). Wenote that all of the anomalous terms dependent uponthe scalar and pseudoscalar fields have been removed.Therefore, in a theory having only scalar and pseudo-scalar external fields, the Ward identities for the matrixelements of three or more currents are the only onesaffected by the presence of the anomalous terms. Inaddition, the anomalous terms are all proportional tothe pseudotensor e„„„,and hence will affect only thosematrix elements of currents having an abnormalparity relation.

Anomalous terms in the axial-vector-current Wardidentities can arise in two ways. In Fig. 3, we illustratethe types of diagrams which were found to have anom-alous Ward identities. The Ward identities for thetriangle and box diagrams involve terms which arelinearly, or more highly, divergent leading to the exis-tence of anomalous terms when the loop integrationvariable is translated. Anomalous terms may alsoarise when the smaller loops are redefined to satisfythe correct vector-current Ward identities. Therefore,while the Ward identities for the pentagons contain noanomalous terms arising directly from the lineardivergences, they do contain anomalous terms when thebox diagrams are defined to have the correct vector-current Ward identities. The diagrams which do haveanomalous Ward identities may be seen by referringto the expression for the anomalous divergence in Eq.(49). For neutral currents, only the A VV and AAAtriangle diagrams will have anomalous Ward identities.For charged currents, anomalous Ward identities alsoarise for the VVVA and VAAA box diagrams and for

FK. 3. Loop diagrams whichmay have anomalous Wardidentities.

the VVVVA, VVAAA, and AAAAA pentagondiagrams.

The anomalous terms in the divergence of the axial-vector current can contribute to low-energy theoremsfor off-shell matrix elements of the naive divergenceoperator. If these low-energy theorems are combinedwith a smoothness assumption for matrix elements ofthe naive divergence operator, the existence of theanomalous terms may lead to physical consequences.For these cases, we note that the full divergence oper-ator will not be, in general, a smooth operator. "Forexample, the anomalous terms in the Ward identityfor the A A A A A pentagon may add anomalous termsto the low-energy theorems used in the (partiallyconserved axial-vector current) calculation of five-pion scattering. ' Adler' has shown that the anomalousWard identity for the A VV triangle yields a low-energytheorem for the decay m' —+ pp in a truncated versionof the cr model. We note that this low-energy theoremhas been shown to be exact to any finite order in per-turbation theory for the truncated o. model by Adlerand Bardeen. ' Using the results of this paper, the calcu-lation may be extended to the full o. model which in-cludes charged mesons.

ACKNOWLEDGMENTS

The author would like to thank Dr. S. L. Adler forintroducing him to this problem and for many stimu-lating discussions. The author would also like to thankDr. I. C. Kaysen for the hospitality of the Institute forAdvanced Study, where the work was performed.After the completion of this manuscript, the authorlearned that I. Gerstein and R. Jackiw have madean independent study of anomalous Ward identitiesfor triangle diagrams with mesons having SU(3) XSU(3)couplings. However, v e disagree with their conclusionsconcerning the box and pentagon diagrams.

' The author would like to thank R. F. Dashen for raising thispoint.

9 S. L. Adler and W. A. Bardeen, Phys. Rev. 182, 151S (1969).