The absence of strong interaction corrections to the axial anomaly in the model is proved in a cutoff independent way using Zimmermann's normal product algorithm.
1. Introduction
In 1969, Adler [1] suggested that there are no higher order corrections to the axial anomaly [13]. This suggestion was supported later by Adler and Bardeen [4] with convincing cutoff dependent arguments in the framework of spinor electrodynamics and in a simple version of the GellMann and L6vy o model [5] coupled to the electromagnetic field.
In the case of the a model the arguments proposed by Adler and Bardeen are, however, much weaker than in the case of spinor electrodynamics. In fact, Adler and Bardeen do not prove the renormalizability of the model and use Ward identities without being sure that they are not affected by the renormalization procedure. Unfortunately the more relevant case is actually the former because, using the AdlerBardeen result in the framework of the model, it is possible to compute the low energy value of the ~ o ~ 27 amplitude.
Recently, Zee [6] and, independently, Lowenstein and Schroer [7] have proved the absence of radiative corrections to the axial anomaly using the CallanSymanzik equation [8]. In particular the proof given by Lowenstein and Schroer using the Zimmermann's normal product algorithm (NPA) [9] does not involve any cutoff procedure. Using the method of Lowenstein and Schroer we prove in this paper the AdlerBardeen theorem in the simplified version of the a model in which the rc is an isoscalar meson and only one fermion field (say, the proton field) exists.
The paper is organized as follows. First we state the renormalization rules for the o model using the NPA (Section 2). Then we derive an equation analogous to the CallanSymanzik equation for our model using the method developed by Lowenstein [10] (Section 3). In Section 4 we discuss the consequences of the coupling to an external electromagnetic field and we prove the theorem.
The renormalization rules for the a model are widely discussed in the existing literature [1113]. However, the a model with spinors has never been treated before in the framework of the NPA, therefore we have to study it in some detail.
We consider a truncated version of the a model which contains only a proton field go) a neutral pseudoscalar (~) and a scalar meson (a). By definition of the a model, an axial current j~ (x) exists that satisfies the Ward identity: % , q ' ~ ( x ) X > + =  c < ~ c ( x ) x >
m + + i F.,~ ,~(x  x 3 < ~ ( x ) X ~ > +
1  i ~,j 6 ( x  yj) <(a(x) + F ) X ~ > +  ½ 1 k6(x  z;) ((vp(x)7s)'k X ~ > + 1 where < >+ means the vacuum expectation value of the covariant time ordered product, X is any product of fields:
n :,n p p
X = ~i a(xi)~j ~(yj)~l ~v(z,)~,Ilk tP(zJk
1 1 1 1
(
We shall first show that the existence of the identity (
Let us begin to show how Eq. (
Let us consider the vacuum functional of the model So[J~, J~, ~7,F/]. Here J~(x), J~(x), rl(x), ~(x) are the external sources of the re, a, ~p, ~ fields respectively [14]. The functional generating the connected parts of the timeordered Green's functions is:
Z[J~,J~,11,~] =  i log So [J~, J~, 11,~].
Integrating (
So + So t'5 ~(x) = 0 and the equation for Z: , dx {J,~(x)(b j ~
Z + F) (c + J ~ ( x ) ) ~ Z
(
6 r1(x) = aL(xf z ; a
6 Z ( x )  a j a x ) z ;
~ e(x) = ag(x~z ; ~(x) = z all(x) and to perform the Legendre transformation: WEn, S, T, C~]
= z  ~ d x ( n ( x ) £ ( x ) + Z(x) (L(x) + c) + ~(x) ~(x) + ~(~) ~'(x)). W+ ~dxcS(x) is the functional generating the proper vertices [15]. Since w =  £ ( x ) ;
w =  J A x ) ; ,5  ,arI (x)
,5 6CU(x) w =  ~ ( x ) ; w ,5~'(xY =  ~ ( x )
,5

aS(x)
~
we get, from Eq. (
W(S(x)+F)
,5 + 2 ( ~ ( x ) , ,  3 ~,5i  w + w ~ 3 ~,~~'(x))} = o.
If we assume that all the terms of the effective Lagrangian are Zimmermann's normal products N4, then their coefficients are proportional to functional derivatives of W.
Indeed let us consider the most general effective Lagrangian for the re,
0. and ~p fields:
l + b l
£°af = N4 i(1 + a ) ~ (Y~  A~p~p  igl~PVs~P~  g 2 ~ 2 c r +    2   (~urc) z
+ l ~ b 2 ( a u a ) 2
~_~L1r2  B~2 a 2   ~ l ~ 2 a   2 z a 3   2 3 ~4
(
The values at zero external momenta of the superficially divergent proper vertices are given by the coefficients of the Lagrangian. If W is the generator of the proper vertices, we have, for example 1: ¼Tr 7, 0P. ~ ( _ p ) ,5~(p) ~o=p=O (t0) ¼Tr~[ = ~ tt a ~ ' (  p )
W
6 1 a ~'(p) ,,=p=o } =  A .
We can solve gtobally Eq. (
g2 2 ]  2D ~gmo.3  3 0 ~  2 0"2( 7~2 t 0"2)  E gm2 0_4 /~2+B 2 62 + C
(
n t lI, dxe ' 1 ' ~ i i w=(2~)~a *q'
~6O(ql)
W =  3 6 D
m2 ,
Wo:o . . . .
3
W
=  (36D + 24E) qo=O # aO(o) an(p) aZ(p)wp=~=o poles of the p r o t o n and zc propagators and the position of the pole of the o" propagator, and we have: _ IgL( 5_t+ ] g LaZ(0) \ a n ( 0 ) / w]~,=o The parameters D, E, d, f are proportional to the value at zero external m o m e n t a of superficially convergent vertices, therefore they are known functions of the physical parameters m,/~, 5, and g.
We will now show that in the theory defined by the effective
Lagrangian given in Eq. (
We shall start by showing that the current satisfies the corresponding integrated Ward identity. Indeed, using the method developed by Lowenstein [15], we have:
n o = .f d x ~ ° ~ ( x ) X 5 + = ~ Y~, (~(x,) x ~ 5 +
1
1 { ~1
2(,
f l k X ~
In the righthand side of Eq. (
+ caaarc + c4~tprc + ics~yslpa] (x)X)+ , cl =  ~ g  @#~P~' fiFl(O) MT(p) 6 Z (  p ) ,=+=o, ¢2 ~ c3 = 6g m [ _ ~ ( 6 i 4 60 t~z(o) \ a n ( o ) ]
] W+=o'
W
~=0
,
c4 = 
4gLet us assume for a moment that c4 = c5 then, choosing for d, f , D, E
the values given in Eq. (
Suppressing the integration and bringing to the lefthand side all
the new terms appearing at the righthand side which have the form
8~,(.j(sl)U(x)X>+ we obtain Eq. (
If the integrated Ward identity is verified this happens order by order in h (i.e. in the loop number). The Feynman diagrams corresponding to c¢ and cs are superficially convergent, thus to a given order in h they contain renormalization corrections of lower orders. Now it is easy to convince oneself that if the integrated Ward identity is true to nth order in h then c,, = c s to (n + 1)th order, this implies that the Ward identity is true to (n + 1)th order. Since the integrated Ward identity is valid to 0th order (a = A = d = b = f = B = C = D = E = 0) we can conclude that c4 = c5 to any order in h.
We now study the CallanSymanzik equations for the model defined in Section 2. Lowenstein [10] has shown that in the framework of the N P A a class of generalized CallanSymanzik equations for the ~p4 model and for a massive vector meson model [7] can be obtained in a straightforward manner. Indeed the differentiation of a Green's function with respect to the parameters of the theory is equivalent to the insertion of new vertices in the corresponding Feynman diagrams. The independent vertex insertions (DVO's) correspond to the different terms in the Lagrangian. If there are more differential operations than DVO's, one gets directly linear relations among the differential operations (these are the CallanSymanzik equations). Since our Lagrangian contains 13 terms, we have to exploit the content of the Ward identities in order to use this method.
We will show that six "differential operators" exist that leave unchanged the integrated Ward identities. They will thus be expressible in terms of symmetric linear combinations of the 13 DVO's.
Then we will show that the symmetric DVO's are only five and furthermore that only one symmetric vertex insertion of degree smaller than four exists (the actual degree being two). From these results the existence of two independent generalized CaUanSymanzik equations immediately follows.
We now study how the differential operators change the Ward
identity (
R(F) W(g,m,#,6)=O (where F = ~ ) (17) (18) (19) (20) (21.a) (21.b) (21.c) (21.d) if we multiply g by 1 + ~/we have to first order in i/: Comparing Eq. (17) and Eq. (18) we get:
R(F(1  t/)) W(g(1 + t/), m, ~, 6) = 0 .
V(aeR) W+ R(g OgW) = 0.
F ( O v R ) W = F S d x ~ W =  R F[dx 14" =R(FA,,W)
R((g (?g+ F A,,)W) = 0 R((m ~,,  FAo)W) = 0,
R(/~ gu W) = 0,
R(5 c~a W) = 0.
N ~ W = , d x ( n(x) 6 w + x(x) 6 w ) multiplies each vertex by the number of the corresponding boson legs. Since: 5 [RNnN, RJW=RNBW=T.[ d x ~ x }  W =  R F A ~ W . In much the same way we can show that the operator Np which multiplies each vertex by the number of proton legs satisfies the equation:
R(Np W) = 0.
For s = m, #, 6, g Lowenstein has shown that:
Axial A n o m a l y in a M o d e l R((NB+ F A~) W) = 0.
13 s 0~W = ~ j (s ~ cj) Aj W
13 N~W= ~ j bjAj W , 1 13 N p W = ~ j pjAj W 1
I3 FA~ W = ZJ f~Aj W.
1
R A(s) W = 0. where the c~ ( j = 1 ... 13) are the coefficients of the 13 terms of the Lagrangian and the DVO's Aj represent the insertions of the corresponding vertices.
It is also easy to see that [10] (23) (24) (25) (26.a) (26.b) (26.c) (27) (28) and by Zimmermann's identity: By Eqs. (21.ad), (23), (24), (25), (26), we know that the operators m~?mFA . #c~, 6c?~, g~g+FA~, N B + F A ~ and Nv correspond to linear combinations A(s) of the A~'s such that: We now determine the number of independent A(s). Suppose we change the coefficients of the effective Lagrangian in such a way that the generators of the proper vertices W(fil, ..., fl0 corresponding to the new Lagrangian v zeoff(l~l, ..., ~ )  ~ o~.o~ ~_ Y~i B~N~[OI~] 1 satisfies the equation R(F)W(fl~, ..., fl~)= 0 to first order in the fliS'. Then to each O~s) there corresponds a A~s) since [O~jR W(fi~ . . . . , fl~)]~=0 = [R(O~j W(fll,..., fl,))]¢=0 = R(A}s) W) = 0. (29)
We know that, at fixed F, all the coefficients of the Lagrangian are
known functions of five parameters (those which are fixed by the
normalization conditions) hence we can infer that the number of
independent AlS)'s is five. An alternative procedure to find the AlS)'s is to use the
equation:
0 = ~ dx OuQ;°)U(x)A~s)x>+ =  c 5 dx(~z(x)AlS)X>+  c', ~ dx<rc(x)X> +
+ i 2i' (rt(xv) A~s) X ~ . ) > +  i ZJ ((a(y/) + F) A!s3X @ > +
1 1
(30)
(31)
(32)
(33)
1
where AIsl = i5 dxN4[Ol s)] (x), which is equivalent to Eq. (29). Indeed if
Z(fil, ...,fly) is the generator of the connected Green's functions for
the Lagrangian ¢.~eff(~l, ..., ~v} it turns out from Eqs. (
w(fl~, ..., fl~) + ~ a x c ( f l , , ..., flO ~(x) generates the proper vertices.]
There remains to be studied how many independent vertex insertions A(S)=i~dxN[O(S)](x) with 5 < 4 exist in the model which satisfy Eq. (30). For 5 = 2 we consider:
AtoS)= Ti ~ dx N2 [~(n 2 + a 2) + #a2] (x).
By Zimmermann's formula, we obtain:
[ dx<(N3[S,J (x)  N 4 [ S J (x))A(oS)X>+ = ~ dx{<N4.[ClgFl~ t C2~30"
+ C3a3rc + c4~Ip~ + icsgp75tp~] (x) A~)X>+
+ (d~ ~ + 4 0 <N2 D ~ ] (x) X> +}
[compare with Eq. (
1  ~ l j / ( a ( Y j ) + ~m ~j A (os ) x ",,(~y'~\)/+  ½ { ~ k ((~(Z'k)75)~kAtoS)X~(2:k)~>+ } : + Z,, <(~v~(z,))~,A~o~)X~>+ ( / +
B) j"dx<,~(x)~(2X>+ m ~ .[ ax<~(x)X> + + g(1  d g  a, 0 5 ax<N~ D = ] (x)X> +. g
Axial A n o m a l y in cr M o d e l
In order to obtain Eq. (
We can conclude that only one symmetric insertion (A~)) exists of degree c5< 4.
Now it is easy to obtain the generalized CallanSymanzik equations. Indeed, using Zimmermann's identity, we can write: (34.a) (34.b) (35) (36) (37) (38) and by Eqs. (21.ad), (23) and (24)
5 A(os) : ZJ rjA}s) 1
5 m 0,,  F A, = Ej Sj ~,4j(S) , 1 5 0. = Y,j tjA~s~ , 1 5 0~ = Y~j ujA~ s) , There are consequently two independent linear relations among the quantities (34.ab) which we may take to be: where and
( D + t N n + u N p ) W = ( 1  1  h  t ) F A ~ W + v A ( o S ) W , ( D ' + t ' N B + u ' N e ) W = ( 1
 t '  h '  t ' ) F A ~ W + v ' A ( o s ) w D = 2 Oz + hg O~+ I(6 0a  m Ore), D' = A Oz + h' g Oo + l' ~ ~~, m c'~,),
2 0k = m 0,, +/~ 0, + 6 0a.
The values of the coefficients h, l, t, u up to second order in 9 are computed in the Appendix.
Before concluding this section, it is convenient to see how Eq. (30) transforms if we suppress the integration. The new terms which appear in the lefthand side have the form
 ~ { (Al~(x)AI ~ X> + + < J,ts/l,(x)X) + } where i = O, ..., 5; bringing them to the righthand side we obtain: a.{O~(x)a x>+ + q;~(x)x> +} = (c(=(x)a x)+ + c'(=(x)X)+) n /71 + i ~ , 6(x  x,) (rc(x)A X~~ao) +  i ~ . 6(x  yj) ((a(x) + F)A X @ ) + 1 1
P
 ½ Y~, {a(~  z',) <(~ (x) ~,)~, A X~=i>~>+ + a (x  z,) <(~5 W(x))~, A X ~
1
where
and
5
0
We now write Eq. (
z[c~,, fl] z [ ~ , fl] + z[~,fl]a@*(x) ,,,7(x)).
We shall now discuss the coupling of the a model to an external electromagnetic field. We will first study the changes of the Ward identities due to the electromagnetic field. Then using the modified Ward identities we shall show that the proper vertices containing photon legs (39) >+ } (41) satisfy the CallanSymanzik equations (35) and (36). Finally, using Eq. (35), we shall prove the theorem.
Taking into account the vector current conservation we add to the effective Lagrangian the coupling term: e(1 + a) N 3[~ y, ~p]A u =Jn An (42)
~(~(x),~ ~ + r(flt ..... fls)~u*°'F.~(x)F~(x) . ~ , (43) where Z [ a . , An, fl] is the generator of the connected Green's functions corresponding to the Lagrangian 5f(~iA'~) = ~ ( ~ ) + AuJU"
Fu~ = ~u A~  ~ A~,, and r
1
8 . 4 ! a"~°'[Oq~ Ok'(N3 [S~z](O)fu(q)f~(k)>P+R°e]P=q=°'
(44)
1
r ,  8 . 4 ! eu~e~[Oq° Ok~(N3[Src'] (0) AIS)L(q)L(k))~R°P]p=q= o . (45)
The last term in Eq. (43) is the axial anomaly. Performing again the
Legendre transformation (
W[au' a u, fi] 3
W[%, A,, fl] (46) f f
\ with  c~u 3~u(x) + up to first order in the fl~'s.
We now come to the CallanSymanzik equations for the proper vertices with v photon legs which are generated by:
I1i 6AU,(xi) W[0, Au, 0] ]
A~ = 0 = W~1.,.u.
(47) In this discussion we never consider the vacuum polarization vertex. Following the procedure used in order to obtain Eq. (25) and Eqs. (26.ab), we get for s = m, p, 3, 9: Zimmermann's identities (26.c) and (34.a) are modified in the following way: 13
F A , W~I...,  v T , ( I +a)*W,~...~ = ~_.4fjA~W,~...,~,
1
5
A(~)Wu~ . u.~  V. T s. ( l +. a ). l W. ~ , , , = 2 i rjAj(s) W,,...,v
1
(48.b)
(48.c)
(49)
,
?s=¼ A(os) Tr g~p 673(_p ) W 6~(p)
v=~,=o"
(54)
By applying Eq. (35) to the vertex
Since the vector current is conserved, we also have:
(51)
(52)
(55)
(56)
(
A, Wu,...uv+ v A<os)wu~...u~  v((1  l  h  t)y, + V?s) (1 + a) 1 Wu~..4,", which, by Eq. (55), becomes: ( D + t N ~ + u N e ) W u l . . . u = ( 1  I  h  O F A ~ W u l . . . u v +vA~oS)Wu~...,~. (57) Recalling that r is proportional to
[ 5 FeU~°~Oq~k~3Au(q)
6 6 6A~(k) 6 f f l (  q  k ) and applying Eq. (57) to r, we obtain: Dr = (D logF)r + F D  ;  = (1  I  h ) r  tr + (1  l  h  t) F A i r + vA(oS)r.
.l To recast Eq. (58) in a simpler form, we remark that, from Eq. (46), we have:
I 6 6  " [ d w 3  ~ O e 6 a Q ( x )
3 3 W[au, Au, O]l 5At,(y) 6a~(z) a=A=cg=O = F(f dw ?
W[0,A~,0]] Applying to the lefthand side of Eqs. (59) and (60) the wellknown low energy theorem for the vacuum expectation value of the time ordered product of the divergence of the axial current and of two electromagnetic currents, we obtain: limeu~°~Oq~k IF 3 5 k~o ~L 6£(0) 6 f I (  k  q ) q~o
6 + 6 1 ~ (  k  q )
6 3 bA~,(q) 6Av(k) time.V~3qjkJ3po ~Zo [
_ 6 6 6 5 I I (  k  q ) 5Au(q) 6Av(k)
W[0, A~, 0]]
= 0,
A=~=0
W[O,A~,fl]] 6 3 6A.(q) 6A~(k) W[0,A.,0]
= 0 .
A=~=~=o (62) From Eqs. (61), (62), we obtain FAir=  r and A(oS)r= 0. Then Eq. (58) becomes:
Dr = 0.
For reasons of dimensionality r is a function of q and of the mass ratios. In terms of the variables g, x = ( ~ 2 / m 2 )  l / s , y=(6m/#2), Eq. (63) becomes:
(H(g, x)g ~o+ L(g, x) ~x) r(g, x) = 0 where the fixed parameter y is omitted. F r o m the Appendix, we obtain: oo H(g, x) = h(g,x) _ ~,. ~ . it.,,.gmx., with Ho.o = 1, (65) 4 o o L(g, x) = 2 oo oo
. ° . . . . ng ~ , with (6o) (61) (63) (64) since H, L, and r are formal power series in 9 whose coefficients are analytic functions of x a r o u n d x = 0 2, we put: and we obtain, from Eq. (64): cO oO CO oo Z., ZmZ°, 0 l 0 0 which implies that for any M and N: rm,.=0 (67) (68) (69) M N Ern En(mrm, nHMm,Nn+( tzJc"1)rm,n+lLMm,Nn)=O.
t o F o r M = 1, N = 0 , Eq. (68) gives q,oHo,o = 0, for M = t, N = I , we have rl, l(Lo, 1 + Ho,o) = 0 and taking into account the relations obtained from M = 1 up to N = N  1 we get for M = 1, N = N : r 1,g(Ho, o + N L o , 1) = O. If we now increase M, we obtain for arbitrary values of M and N: (MHo, o + N Lo, 1)rM, N = 0 (since of course M H o , o + N L o , 1 never vanishes). Equation (69) implies that r = ro which does not depend on y. Thus the AdlerBardeen theorem is proved.
The proof of the AdlerBardeen theorem for the a model is analogous to the one given by Zee and by Lowenstein and Schroer in the case of spinor electrodynamics with some differences which are due to the structures of the models.
Indeed a "true" CallanSymanzik equation does not exist in our case. By '"true", we mean an equation which does not contain derivatives with respect to mass ratios. It is interesting to point out that in the case of the omodel without fermions a "true" CallanSymanzik equation does exist. The basic difference between the two models is that in the symmetric limit the p r o t o n is massless.
Acknowledgements.I am deeply indebted to Professor R. Stora for constant encouragement and help. I wish also to thank Professors J. S. Bell and V. Glaser for interest and discussions.
2 For x = 0 and y ~ 1, all the particles of the theory are stable.
To compute the coefficients of Eq. (35) we apply it to some simple vertices. To zeroth order in g we immediately obtain: To second order we start considering the vertex: lice 3H3(p) 3II(3p) W] p=q~=O = l + b where b ~ 0(02). Since 2~(1 + b ) = 0 we have, from Eq. (35) hg 8gb + l(~5~  m D,,)b + 2t(t + b) = ( 1  h  l  t ) ~  [ poop. 6F/(p) ' ~ H (  p ) W] p=q~=O + ~ ~p Op. 3II(p) 3II(p) Jp=~o=o selecting the terms which are O<g2)we obtain:
1 2 ' = 8[~ p ~ " 3MI(p) M I (  p ) 3 ( F A ~ + ( ~   y 2 ) A ( ° s ) ) w t , = ~ , = o (1"4) which can be written in the form: 2t =  i [.[ (24~d)q
Op8p,(FA~+(~_l~a)A(oS))l(q,p)]p=o8 (1.5) where I(q,p) is the integrand corresponding to the sum of Feynman diagrams in Fig. 1: (1.1) (A.2) (A.3)
Appendix l=h=t=u=O,
I "'i" I,z~ I
D 3
Op,I(q,p)]p=o t is completely determined by the nonintegrable part of Ov3v~I(q,p) (for the integrable part we can extract 0k from the integral and obtain zero). Thus the only contribution to t comes from DI: t =  1 i ~  9~2 (2~}4 m~,.OpOpT.r {75 _p+lq m 75
da =   i 9 2 ~(m2)~ m In much the same way, applying Eq. (35) to
q2' p2 75t}]p=0 (1.6) (1.7) (A.8) (A.10)
a 3 a~0(o) a,P(o) w 5¢,(o)}]~,=o since the divergent parts of the two diagrams in Fig. 3 cancel. h + t + 2 u = 0 and h = 4 (  ~  ) 2 (A.11)
D ,///
o i ?. . u%r// x,\ ~r
D10 It then follows: (A.12) For (62/m2) = ~ / = 0 cancel.