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Calculation of the non-abelian chiral anomaly on the lattice
Nuclear Physics B289 (1987) 645-672 North-Holland, Amsterdam
C A L C U L A T I O N OF T H E N O N - A B E L I A N C H I R A L A N O M A L Y ON THE LATI'ICE A. COSTE and C. KORTHALS ALTES Centre de Physique Th~orique, Section 2, CNRS Luminy, 13288 Marseille Cedex 9, France
O. NAPOLY Service de Physique Th~orique, CEN-Saclay, 91191 Gif-sur- Yoette Cedex, France
Received 18 December 1986
We check the validity of Wilson and Kogut-Susskind lattice regularizations of the Dirac fermion action by calculating perturbatively the non-abelian chiral anomaly on the lattice. We show that the result does not depend on the detailed expression of the coupling term of the lattice fermions to the chiral gauge field, taken as an external field. In two dimensions we recover the consistent expression of the anomaly, up to counterterms which we have computed. In four dimensions, we obtain the correct normalization factors of the anomaly, leaving the problem of the counterterms open.
1. Introduction
The notion of chirality has played an important role in the conception of present day particle theory, particularly for our understanding of strong interactions. The chiral invariance of the quarks is easily implemented on the classical level via the Dirac action. On the quantum level, one has to introduce regulator devices which are at the origin of chiral anomalies. The so-called abelian anomaly [1] is responsible for dynamical effects such as ~ r ° ~ ,{~, decay and 7' mass when the quarks are coupled to Q E D and Q C D gauge fields respectively. When the gauge group is enlarged to the full chiral flavour group, regularization gives rise to the non-abelian anomaly, first computed by Bardeen [2]. Its main consequence is to provide a constraint on the fermionic content of renormalizable chiral gauge theories used in model building. Among the regularization devices, lattice regularization is of special importance since it allows one to perform non-perturbative calculations for QCD, and in fact has been designed for this purpose. However, the standard weak coupling perturbation theory ought to be recovered as well. Checking this point [3], and in particular the abelian anomaly [4], has been of great benefit for the problem 0550-3213/87/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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A. Coste et al. / Chiral anomaly on the lattice
of finding correct lattice fermion formulations. It is now understood [5] that chiral anomalies originate from the fact that, in order to avoid the species doubling of the naive lattice quark action
Sc( q, ~1) = a d E ~( x )7~,V~,q(x ),
(1.1)
x,o,
where
1
V~q(x) = ~a ( q(x +/2) - q(x - 12))
(1.2)
is the lattice first derivative and a the lattice spacing, it is necessary to add to this action a term Sb(q, ~/) which breaks explicitly the U(ny) × U(nf) chiral invariance of S¢(q, el). Popular choices for S b are as follows:
S b = - ½ra a+l • ~ ( x ) A q ( x )
(Wilson),
x,p,
S b ~___~u 1 _d+l
E ~ ( x ) B , A q(x)
(Kogut-Susskind),
(1.3)
x,p,
where A is a lattice second derivative: 1
A q(x) = --~(q(x +/21 - 2 q ( x ) + q ( x - / 2 ) )
(1.4)
and B~ is a matrix in spin and flavour spaces [6] such that
which is possible only for nf a multiple of 2 d/2, say
n/= ~/. 2 a/2 .
(1.5)
S b is an irrelevant term which vanishes in the naive a ~ 0 continuum limit, it can give rise to anomalies through divergent fermion loops. It remains to be seen whether it generates the correct non-abelian anomaly. In 2 and 4 dimensions, Wess-Zumino consistency conditions [7] uniquely determine [8] the expression of the non-abelian anomaly, up to counterterms, provided it is local, Lorentz invariant and vanishes for chiral transformations which are global in space-time, thus reducing the problem to that of computing the overall coefficient. Unfortunately, we cannot use this uniqueness theorem in the framework Although
A. Coste et al. / Chiral anomaly on the lattice
647
of lattice perturbation theory. The last two properties will clearly be seen to fail while we will prove the locality of the anomaly by power counting. Therefore, checking that the non-abelian anomaly is reproduced by lattice fermion regularizations needs an explicit calculation. The purpose of this paper is to present such a calculation both for Wilson and Kogut-Susskind fermions. In sect. 2, we formulate the definition of the non-abelian anomaly in the language of the lattice functional integral and we derive three important theorems, dealing mainly with locality, which greatly simplify its calculation. In particular, the first one sets the freedom in the choice of the lattice expression for the coupling between the fermion and the chiral gauge fields. The detailed proof of these theorems is contained in appendix A. In sect. 3, we derive the Feynman rules used in the perturbative calculation. This is done in two ways. The first one allows one, in 2 dimensions, to identify diagrammatically the counterterms, rendering their computation unnecessary. It enables us to directly identify and compute the 2d anomaly. We have not yet been able to generalize this method to 4 dimensions. The second method, which is more economical in the number of graphs but does not separate the anomaly from the counterterms, is applicable in any number of dimensions. In sect. 4 are reported the results that we have obtained, using the second method, for Wilson and Kogut-Susskind fermions. In two dimensions, the calculation is complete: we found the correct anomaly plus terms which combine themselves into the chiral variation of a local counterterm whose expression is given. In four dimensions, this last step is still missing. We intend to complete the 4d calculation by either one of the methods. Finally, we discuss the problem of the presence of a regularized axial symmetry in the Kogut-Susskind formulation, and we show that it leads to no contradiction with the usual vector conserving expression [2] of the non-abelian anomaly. Appendices B and C contain technicalities relevant to sects. 3 and 4 respectively. A brief account of the results of this paper can be found in our ref. [9]. 2. The non-abelian anomaly on the lattice
In this section, we will follow closely Bardeen's notations [2] and set up a lattice version of his work. We first introduce in the fermionic action S v = Sc + Sb a smooth continuum external field, taking values in the chiral U(ne) algebra:
V
(x) = W ( x ) +
(2.1)
There are many ways to couple this external field such that the classical action fddx71Ip(V±)q is reproduced by the lattice action SF(q, Ft, V~) in the naive a ~ 0 continuum limit. In this paper, we have chosen to gauge the chiral symmetric part So(q, 71) of the action and to leave the symmetry breaking term Sb(q, Ft) unchanged:
SF(q, 7t, V~) = So(q, 71, V~) + Sb(q, 71),
(2.2)
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A. Coste et al. / Chiral anomaly on the lattice
with
Sc( q, ~1,V~) = a e E Yt(x )Y~,~7~'(U )q( x ),
(2.3)
where the covariant lattice first derivative 1
V~'(U)q(x)=-~a (U~(x)q(x + f t ) - U~-l(x-ft)q(x-ft))
(2.4)
is given in terms of the link integrated variable:
Of course there are other possibilities of coupling the external field. By the end of this section we will show when and in what sense they are equivalent. We then define the effective action by:
W(V~; a, m) = - l o g Z(V~; a, m)
(2.6)
from the lattice functional integral:
Z(V~;a,m)= f[dqdq]exp(-(Sv(V~,q,q)+
Sm(q,q))}.
(2.7)
Being regularized, the effective action can be expanded in powers of the lattice cut-off 1/a and splitted into divergent and finite pieces
W(V~; a, m) = Woo(V~; a, m) + WF(V~, m) + O(a).
(2.8)
The mass parameter m has to be introduced in order to take care of the logarithmic divergences proportional to log(am) in W~, and of possible infrared singularities. V~ being a continuum external field, the anomaly 3W can be defined as the variation of the effective action,
3W(V~, 3o~+; a, m) = W(V~ + 3V~; a, m) - W(V~; a, m)
(2.9)
under a local chiral variation of V~
3V~(x) = -O~'(3~o+(x)) - i [ V ~ ( x ) , 3 w + ( x ) ] ,
(2.10)
with 3¢0+ = 3o~V+ ~53o~A the infinitesimal parameter. Since it is the variation of a well-defined functional, the anomaly 3W as well as its finite part 3W v, fulfills
649
A. Coste et al. / Chiral anomaly on the lattice
Wess-Zumino consistency conditions [7], namely in short, 88~,,+(SW(Sto+)) - 8~,o+(8W(8~o~_)) = 8W([8~o+, &o~_]).
(2.11)
As we will see, only the finite part of the anomaly, 6WF, is interesting and, therefore, it must be compared to the consistent expression of the non-abelian anomaly, first obtained by Bardeen. Taking advantage of the invariance of the lattice fermionic measure l-lx[d q d~](x) under a local chiral redefinition of the quarks, one easily derives the following Ward identity:
~W=(SSF(q,~,V~; 8to +)),
(2.12)
with
6SF=SF(q+6q,~+6~,V~+6V~)-SF(q,
Zt, V~)
(2.13)
and
{
6q(x)=iSw+(x)q(x) ~(x) -i~(x)8w_(x).
(2.14)
Notice that in eq. (2.12), we disregard the non-anomalous contribution (SSm) coming from the mass term, which vanishes when m-~ 0. With the action SF(q, FI,V~) written in eqs. (2.2), (2.3), one has obviously 6Sc = 0
(2.15)
BW=(6Sb(q,q; Bw +)).
(2.16)
and therefore:
This equation, which is the starting point of our calculation, clearly shows that S b is the source of the chiral anomaly on the lattice. (8Sb) will be computed in perturbation theory by expanding the full quark propagator according to the following splitting of the action
SF( q, ~, V~) = St( q, ~) + ¢/'(q, Ft, V~).
(2.17)
The free propagator is thus obtained by inverting the action of a free quark with mass m,
So( q, ~; m) = SF( q, ~) + Sm(q, q)
(2.18)
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A. Coste et a L /
Chiral anomaly on the lattice
and the vertex from ~ ( q , ~, V~). It follows from eq. (2.17) that Y/"(q, ~, V~)I v,+=0 = 0
(2.19)
so that we actually perform a weak coupling expansion. Before presenting the calculation itself, we need to collect a few results which simplify our task. All these results follow from power counting and are derived in appendix A. The first one deals with the arbitrariness in the choice of the lattice action, as discussed in the beginning of this section: (i) Any action S~(q, gl, V~) having the two following properties:
(a) S~.(q, q, V~) ~
fd"xq(x)O(w.+_)q(x)in the naive continuum limit a ~ 0, (2.20)
(b)
S,~( q, Ft, V~ = O) = SF( q, q) ,
(2.21)
yields an effective action W'(V~; a, m) which differs from W(V~; a, m) obtained from SF( q, g/, V~), only by a local polynomial functional of V~. It should be clear that the anomaly is defined up to the addition of the variation 8R(V~, &o+) of a local polynomial counterterm R(V~) of the field V~. It is therefore enough to show that the correct result for the anomaly is obtained from one action verifying (a) and (b), namely from SF(q, ~, V~). Notice that, because of condition (b), we cannot prove a priori that the Wilson and Kogut-Susskind cases differ by a counterterm, hence the need of two separate calculations. The other results are as follows: (ii) Woo(V~; a, m) is a local polynomial functional of the field V~. Its contribution to the anomaly 8W(V~,Sw+; a, rn) can therefore be cancelled out by the addition of the local counterterm R = - Wo~ to the original action. So, we only need to compute the finite piece of the anomaly
m) = ( Sb(q, O, 8,0
(2.22)
(iii) The anomaly 8Wv, as defined by the above equation, is also a local functional of V~ and &0+. It is moreover free from infrared divergences when m ---, 0. The consequence of these three results is that it is enough to calculate the finite part of the anomaly at zero mass 8WF(V~, 860+; m = 0) given by eq. (2.22), for our choice of action SF( q, 7t, V~ ).
A. Coste et al. / Chiral anomaly on the lattice
651
3. The perturbative calculation of the anomaly We now turn to the actual calculation of the anomaly from eq. (2.22). This section contains many technical details which may be necessary in order to complete the 4d calculation. In order to make it readable we split it into three subsections. We first work out the formula (2.22) into a form where it becomes a sum of Feynman diagrams. In the second subsection, we use the fact that the external fields vary slowly with respect to the lattice spacing a, i.e. we expand further each diagram in powers of aqi where the q/s are the extemal momenta of the vertices. The ensuing set of graphs is then handled in two alternative ways in the last subsection, in order to separate the canonical anomaly from gauge variations of local counterterms. We will treat the Kogut-Susskind and Wilson cases together: the relevant quantities will be labelled by the superscript KS or W. 3.1. FORMULA FOR EQ. (2.22)
Starting from the action Sb~s and S w defined in (1.3) one easily finds:
8sbKS(q,~t,&o +)=aa(½ia) • (a,~l(x)B~&o+(x)q(x) X, I3,
- gl(x ) 8oa_( x ) B.A.q( x ) ) ,
(3.1)
8SW(q,q, 8w ±)= -aa(½ia) r E ( A,~(x)8~o+(x)q(x) -?t(x)&o_(x)Auq(x)). X,p,
(3.2) Introducing the matrices B and S over lattice sites and internal space defined by
(3.3)
8S b = tT" B • q,
SF(q, q, V+_)+ Sm(q, q)
--- q. • -q,
(3.4)
we get
(3.5)
{~Sb) = -Tr(S-tB). --1 can be evaluated perturbatively -1 = ( ~ 0 -[- ~ / ) - 1 =
(-- 1)n-I(SolV)
"-1So I ,
(3.6)
n=l
where the matrices S O and V are defined by
So(q,
=
So.q,
Y/'(q,~,V+_)=~.V.q, with the notations of eqs. (2.17), (2.18).
(3.7) (3.8)
652
A. Coste et al. / Chiral anomaly on the lattice a)
b) c)
$-1
=
,
W
=
I.
-1
,
IB
+
4.
Tr ($-~le)
:
...
__©
: __©
. <3-.
<3(.
Fig. 1. Diagrammatic representation of (a) the full quark propagator, the free quark propagator, the vertex and the anomalous vertex; (b) the perturbative expansion of the full propagator; (c) the perturbative expansion of the anomaly.
Hence (3Sb) = ~ D(")(B,V),
(3.9)
n=l
with
D(")(8, V) = ( - 1)"Tr(~olB(~oiV)"-i).
(3.10)
This expansion is represented diagrammatically in fig. 1. After a Fourier transformation one gets: D(")(B,V) - _ ( - 1 ) " / . d / d q i - dq,,3L(ql+''" (2~r) ,,a JBZ ""
+q,)
× tr[A ( / ) ~ ( l , ql)A (l + Ol)g/'(l + 01, q2) ... xW'(l+
a,,_2,q,,_l)A(l+
O,_i)zgU(l+ O , - i , q , ) ] ,
(3.11)
where the loop momentum ! and the external momenta q~ are integrated over a Brillouin zone (BZ) ( ] - rr/a, ~/a]) '~. We set Q,= - ( q i + "'"
+qi)
and ad E eikx
~L(~) (2~)~
(3.12)
A. Coste et al. / Chiral anomaly on the lattice
qn
ql
q n - l ~ - l ~ D (n) ( [13,\V )
q2 ' t÷ O2
i' "
X
653
Qi : - (ql + "" +qi )
,
,
q~
Fig. 2. Momentum flow in the diagram D ("~ given by eq. (3.11).
for the lattice Dirac distribution. Our conventions for the momentum flow are given in figs. 2 and 3. The quark propagator is defined by
( o)x,y
=
(2~r
z
dk eik(y-x)A(k),
leading to AKS(k) = a
AW(k) = a
F.,[y~isin(ak~) + B~,(1- cos(ak~))] + ma (F~,4 sinZ(½ak~) + m2a 2 } ~[y~isin(ak~) + r(1 - cos(ak~))] + ma {F~.sin2(ak~) + (rE~(1 - cos(ak~)) + ma)2} "
(3.13)
(3.14)
The anomalous vertex (see fig. 3a) is defined by
~(l, q) = E eilyBy,xe-i(l-q)x,
(3.15)
x,y
leading to
i
~KS(l, q) = - a E { B,3"~ + (q)(1 - cos(air) ) ,u
-3"d~_(q)B~(1-cos(a(l~-G)))
},
i ~W(l, q ) = a ~.(3"d~+(q)(1-cos(al.))-3"~_(q)(1-cos(a(l~-q~)))), t~
(3.16) (3.17)
A. Coste et al. / Chiral anomaly on the lattice
654
I
'll~ (I, q )
--,
q
Fig. 3. Conventions for the momentum flow in (a) the anomalous vertex (cf. eq. (3.15)); (b) the vertex (cf. eq. (3.19)).
where 8to±(q) is the Fourier transform of 8to _+(x)
8-'~+_(q ) = ad~_, eiqxsto ± ( x ) .
(3.18)
x
In the same way, the vertex W(k, q) is defined by (see fig. 3b):
YV'(k, q) = E
(3.19)
eikyVy,x e-i(k-q)x,
x,y
which gives ~¢//"KS(k, q) = ~ : W ( k , q) = ½]~_,y~(ei"k~lYV~+(q) - e-iak~lYV~_(q)),
(3.20)
jz
with
lfv~+(q)=aaEeiqx!(Bx
U-l(x)),
a
l,V~_(q) = aa~_,eiqxl (n - U ~ ( x - / 2 ) ) . x
(3.21)
a
In the limit a ~ 0, the loop integral in D (") is possibly divergent. T o extract the degree of divergence one keeps track of the leading powers in a and l:
A(t)-t -1, ~ ( l , q) - al 2, ~ r ( l + Q, q) - 1°.
(3.22)
Therefore -1-n
dall 2
D("~(R, V) - aJ
-
for n < d + 2
/a oga
(3.23) for n > d + 2.
655
A. Coste et al. / Chiral anomaly on the lattice
a)
L-'-Q >
=
L >
÷
L t >×; Q
l .
t I ×;× Q Q
•
. . .
Fig. 4. Diagrammatic representation of (a) the Taylor expansion of the propagator A(l+ Q) (cf. eq. (3.25)); (b) the expansion of the vertex ¢¢'(/+ Q, q) (cf. eq. (3.28)); (c) the expansion of the anomalous vertex 8 ( l , q) (cf. eqs. (3.30) and (3.31)).
As expected, only the first ( d + 1) diagrams contribute to the anomaly when a ~ 0, and we can write d=l
8W~ + 8 W v = • D~")CB,v).
(3.24)
n=l
3.2. EXPANSION OF FEYNMAN DIAGRAMS IN POWERS OF aqi
D ~1>can be readily computed and in fact contributes only to 8W~. The diagrams D ~'), for 2 ~
0a
A(I, Q) =z~(l) + E Q ~ - ~ , ~t
(I) + . . . ,
(3.25)
t~
which is nothing but the Taylor expansion in powers of Q. For the vertex ~P(I + Q, q) given by eq. (3.20) it is convenient to realize that:
1 ~ (q) = __+½(1~°) (q) + a l ~ l ) ( q ) + a 21,V~(2)(q) _ a 31~9>(q)) + O ( a 4 ) ,
(3.26)
where the l ~ i ) ( q ) ' s are related through a Fourier transform, like in eq. (3.18), to the
656
A. Coste et al. / Chiral anomaly on the lattice
following expressions: l.V<°)(x) = 2iV~+(x),
w~
i o~v~+(x) + (v~+(x)) 2,
W~(2)(x) = ~(i 02V~+(x) + 20~V~+(x)V~+(x) + V~+(x) O.V~+(x) - i(V~+(x))3), ~ 2 + v~+(x) o~v~+(x) o~v+(x))
VV(3)(x)=~(iO2V~+(x)+3a~V~+(x)V~+(x)+3(
-3i o.v~+(x)(V~+(x)) ~- 2/v+~(x) o.v~+(x)V~+(x) (3.27)
-i(V~+(x)) 2 0.V~+(x)-(V~+(x)4)).
The vertex ~/'(l + Q, q) can then be expanded as follows (see fig. 4b) 3
Y//'(l+ Q, q) = E ai~/'(i)( l, q, Q) + O(a4),
(3.28)
i=O
with gp (o)( 1, q, Q) = ½~./~ I~7~(°)( q)cos(al~), ~¢~ (1)( l, q, Q ) = 1 E "/'~(il~r(1) ( q )
- Q~ff.(o)( q))sin(al~),
zeus(2)(l, q, Q) = ½E "/~(1~(2)(q) - iQ~ffTf)(q) - ½Q217v~(°)(q))cos(al.), ~(3)( 1, q, Q ) = ½• "/"( i l.v.(3)( q ) - Q~I~(2)( q ) - ½iQ~lTV~(1)(q) #z
+ ~Q31~(°)(q))sin(al~). (3.29) Finally the expansion of the anomalous vertex is: i ~Ks(/, q) = a Y'.{(B~8~+(q)-So3_(q)B~)(1-cos(al~)) #
+So3 ( q ) B . ( a q . s i n ( a l . ) - 5 ( 1 aq.) 2cos(al.) ~(aq.)3sin(al.))},
(3.30)
ir ~W(l, q ) = a E{(So3+(q)-So3_(q))(1-cos(al.)) t~
+8o3_(q)( aq~sin(al.) - ½( aq.)2cos(al~) 1 3 • -g(aq~) sm(al~,))}.
(3.31)
A. Coste et aL / Chiral anomaly on the lattice
o: D':'=
=
{0) --
©
,o,--©
(1)
O(2 2)
=
,o,
o"';
657
©
11) . . . . .
(0)
,o, --
~' ( 0 1
Fig. 5. The diagrams contributing to the finite part of the 2d anomaly with the method of subsect. 3.3(a).
It is easy to see that each order in one of these expansions decreases the degree of divergence of D (") by one power. Using the diagrammatic representation of fig. 4, we are thus able to isolate the diagrams which contribute to the finite part 8W F of the anomaly (see for instance the 2d diagrams in fig. 5). 3.3. S E P A R A T I O N OF C A N O N I C A L A N O M A L Y
In order to cope with their calculation one has to use the symmetries of the theory at best. For this we set up two different diagrammatics which we present now: (a) "'Combinatorial diagrammatics". The idea is to use algebraic properties of the breaking term S b to derive vertex identities leading to cancellations between diagrams. This is reminiscent of diagrammatic form of Ward identities in continuum field theory. Such identities arise from similarities between some terms in the expansions of vertices and propagators. Explicitly we can write the massless propagator: A(l,m=0)=a
D
'
(3.32)
with .~KS= A~w= i Z y~sin(alr),
(3.33)
/z
/~ KS = y: B. (1 - cos(alr,)), /t
/~w = r Y', (1 - cos( al, ) ),
D =
_:i2 +
ew = 1,
( 3.34)
h2,
e Ks = -- 1 .
(3.35)
Then the lowest order term in the anomalous vertex reads ,~(1)(l, q) =
i
^
ae(B(l)SYo+(q)
-
8o5+(q)/~(t))
(3.36)
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A. Coste et al. / Chiral anomaly on the lattice
and we have an "anomalous vertex identity"
(8~+(q)B(1)-B(1)8~_(q)) A ( l ) ~ 1 ) ( l , q)A (l) = ia
D(I)
(3.37)
Its derivation, which makes use of the following properties
(i) (a)
/~ 2 and .~2 are scalars, ^ ^
(iii)
^
^
AB = eBA ,
(3.38)
proceeds in two steps; firstly:
= (h~,0 h - 8 ~ + ~ 2 ) ~
+ (h2~o~+-hs~ ~ ) ~ = 0.
Secondly:
~( ~ ( ~ -
~-~)~ ÷ ~I ~ -
~ ~)~ }
Since it involves only one factor 1 / D ( l ) , the r.h.s, of eq (3.37) looks very much like a propagator. Together with identities relating Q-expansions of propagators and vertices, it allows us to identify anomaly and variations of counterterms as linear combinations of diagrams. The advantage of this method is that it leads to the same combinations both for Wilson and Kogut-Susskind cases, an account of which we have in the fact that (3.37) does not depend on e. Moreover these combinations can, at least in two dimensions, be identified purely graphically. We give details on this graphical identification for d = 2 in appendix B. (b) "'Hermitian diagrammatics". The idea is to make use a formal antihermiticity property of the action (2.2). Let us define the conjugation t by:
(i) (ii)
(V~)t= V",
(A~)*= -A ~
r* = - r ,
(V~+)*= V~_,
m* = - m ,
(3.39)
(iii) t acts as usual hermitian conjugation on i factors and Dirac matrices, in particular
A. Coste et al. / Chiral anomaly on the lattice
659
Under this conjugation, one has =
so that the action is antihermitian: St = -S.
(3.40)
This hermitian conjugation is formal since (i) is not compatible with the axial-vector source A~'(x) being in the algebra of U(n/). Of course, it is well known [10] that no hermitian conjugation, with (A~) t = A ~', can make the chiral Dirac operator (anti-) hermitian (in the continuum as well as on the lattice) so that the euclidean effective action W = logdet(0 ) has an imaginary part which precisely contains the chiral anomaly. Here, the operation t is used consistently as a trick in order to simplify the form of the anomalous vertex. Indeed, with the additional rule (iv)
8¢*v= 8o~v,
8o:a = -8o~A = (8~+) t = 8o~_,
the anomalous vertex matrix B is antihermitic and can be written B = x(A - At),
(3.41)
with ~AKSq =
ad(ia) ~_, ASI(x)B~8~+(x)q(x),
(3.42)
x,/t
qAWq =
--ad(ia) r E A~q(x)8o~+(x)q(x).
(3.43)
X,/L
Plugging (3.40) and (3.41) into (3.5) we get
(SSb)
= -ReTr($-IA).
(3.44)
This leads to the diagrammatic expansion (SSb) = Re( n=i~D(")(A' V)) '
(3.45)
with (-1)" D ( , ) ( A , V ) - (-~--r-~-7t f, zdldql...dq, XTr[A (l)~¢(l,
6L(ql + ... +q,)
ql)A(l + Q1)Y¢/'(I+ QI, q2).., g:(l + Q,-1, q,)], (3.46)
A. Costeet aL/ Chiralanomaly on the lattice
660
and ,~¢KS(l, q) =
dW(l, q) =
-2i a
~,B,~"~+(q)(1-cos(al,)),
(3.47)
/x
2i ar~,,8o~+(q)(1-cos(al,)).
(3.48)
The difference with eqs. (3.16) and (3.17) is that d ( l , q) needs not be expanded in q, leading to only one anomalous vertex. This saves a number of diagrams and is the main advantage of this method. The anomaly is now given by [d+l ) AW~ + 8WF = Re/,~--1D(')(A'V)
(3.49)
and the diagrams contributing to 8WF are respectively 3 and 21 in 2 and 4 dimensions (see fig. 6).
+
~WF(d=2) = --'~(1)+
. _ . ~ (0)
---¢~10) (0)
~WF(d=Z,) = ---~(3) + ---¢~(2) (0) (2) (1}
"
-1~(11
• I1 ~
(01
11)
.~(0)
~(1)
~(0)
111 (0)
(1) :~/Iol
(0) ,._.,/(oi
{0) ,..y/(, 1
{0)
(21
.
101
+
" 111
:.,/(oi ~'101 i(o)
+ ._.~
(0)
(1} .~
101 1o1 (01
•
~' 101
,,..~ (o)
"101
" 101
,...,/(oi x(O)
101
T o;-,°, Fig. 6. The diagrams contributing to the finite part of the anomaly with the method of subsect, 3.3(b), in d = 2 and d = 4 .
A. Coste et al. / Chiral anomaly on the lattice
661
The results reported in the next section have been derived in the framework of this second method. 4. The results In the two-dimensional case, the calculation yields, for m = 0:
8WF( V~, 8o~+) = X . s~¢+ 8R ,
(4.1)
where ~¢ is the consistent 2d non-abelian anomaly [11]
i f d2xie~tr(753~+ O~V+) ~¢= 4~r
(4.2)
written in the L - R or V - A symmetric way. The coefficient X is given by the trigonometric integrals:
~kw
2r 4
~r
33C 2
2"
= ---~-LdOldO2--~-(Sc1-2sl),
(4.3)
with the notations
c i = cos 0i ,
s i = sin Oi,
for i = 1, d
and d
3=
]~(1-c,),
D=
~ s2 + i=1 /
(4.4)
r 2 3 2.
R is the following local counterterm RKs=
f d2x{c2tr(V~+ V~+)+ #
2 ca .... tr(V"+B, IV~-B~2) ~VlV2
+
E
ie..c""~tr(ysV~+IB, V"2B,]l btl~t*2 \ 1 -2/]
~
~1 ]22 I)lV2
Rw= f d2x ( c l ( r ) E \
tr(V~+ V~+) + c2(r)Etr(V~+ V~_) /x
/z
+¢3(r)Ei ,,tr(v,v + w_)}, ~v
(4.5)
A. Costeet al. / Chiralanomaly on the lattice
662
where the coefficients c, c ~1"2, c v1"2 and cl(r ), c2(r), c3(r ) are also given by 2d trigonometric finite integrals. It is not essential to give their complete expression here; it is nevertheless interesting to note that they are such that, on one hand R Ks is not Lorentz invariant and on the other hand R w is r-dependent. These counterterms of course depend on the choice of the lattice action SF(q, ?I, V~), as stated in sect. 2, result (i). On the contrary, it is very important to verify that the correct normalization of the anomaly is recovered, i.e. to show that xKs = Xw = 1.
(4.6)
This can be done by calculating analytically the integrals of eqs. (4.3) by a method very similar to one used previously in various calculations of the abelian anomaly on the lattice (see e.g. refs. [4]). Not only this method allows us to calculate also the normalization of the 4-dimensional anomaly, but it is already necessary in order to recast the various contributions to 8W F, different from ~¢, under the form of the variation of the counterterm R. We believe that it may be worthwhile to illustrate it with an example: for this purpose, we present the calculation of XKs in appendix C. In 4 dimensions, the calculation becomes much more lengthy and we only isolated the two members of the standard non-abelian anomaly 1
~¢1 =
48i~r2
f d4x e.,ootr(758~° + O~'V~- O°V~, )
(4.7)
fdgxe,,ootr(ys~o+½iO"(V+V°+V+))
(4.8)
1
~¢2 =
48i~r 2
and computed the coefficients multiplying them in 8W F. The diagrams contributing to each of these two terms are drawn in fig. 7. With the notations of eqs. (4.4), these coefficients are given by the following integrals:
XlS = 3 f 4~"2 -~r
( 1 - c i ) 2! clc2("3(2s2-¢~¢4)
d40 [ i i=1
84
'
Xw = _ --' ~ry6 f,~_ d40 rZSaClCEC3(c4D_4c4s24_4rZ~S2)D 4
~
,
(4.9)
(o) (o)
(oi
, . . , j (oi - - -
"~ 101
"~ {01
"
"
101
Fig. 7. The diagramscontributing to -~i and ,~2 (cf. eqs.(4.7) and (4.8)) in the 4d anomaly.
A. Coste et al. / Chiral anomaly on the lattice
663
for ~¢1, and
_ _
~kW ~"
_
_
2~r2 -'~ d48 i _
f
( 1 - c i ) 2 CLC2C3S42 2 3 c 4 -
d,Or232c,c2c3s 2
1 22 f qT
(
Dc4
s2
4)/
(c4+r23) E s 2
--or
5,
D'
(4.10)
i=1
for ~¢2- These integrals can again be calculated analytically by the method mentioned above. The coefficients are all found equal to 1, so that we recover the correct consistent anomaly •~¢= -~¢1 q ,J~2 •
(4.11)
To complete this result, it must be shown that the remaining terms in 3 W F combine themselves into the variation of a local counterterm, as they do in 2 dimensions. We hope to achieve this in the future.
5. Discussion and outlook
Lattice regularization has been set up in order to probe low-energy features of QCD. One of its successes is the evidence [12,13] that chiral symmetry breaking occurs spontaneously, accompanied by massless pions. For this, the use of KogutSusskind fermions has been crucial, allowing at least one axial symmetry to be unbroken in the action. To this axial generator corresponds one conserved current and this raises the question of knowing whether the associated pion undergoes electromagnetic decay through the triangle anomalous diagram. We will now answer this question by showing that the difference between (a) the picture described by Kogut-Susskind fermions where one axial symmetry is exactly conserved and vector symmetries, including the electric charge, are explicitly broken, and (b) the usual picture where the electric charge is conserved and the axial symmetry is anomalous, is taken care of by the addition of a suitable local counterterm. For this, we first take for granted the result, only partially derived in d = 4 in the last section, that Kogut-Susskind fermions, with the action SF(q, Ft, V~) of eq. (2.2), generate the correct non-abelian anomaly, namely
3 W v =.~¢+ 3R .
(5.1)
Then, we consider the possibility of using, instead of SF( q, ~, V~), the action
S~(q, Ft, V~) = So(q, FT,V~) + Sb( q, q, V~),
(5.2)
664
A. Coste et aL / Chiral anomaly on the lattice
defined by eq. (2.3) and by replacing the second derivative in a covariant second derivative: 1
A(U)q(x) = ~5(U,(x)q(x +/2) -
2q(x)
Sb(q, ~) in eq. (1.3) by
+ u~-l(x -/2)q(x
In this action, the residual flavour invariance of SF( q, ~t), namely [6] X U(~f) a corresponding to the generators 3(o+ obeying 3,0+B,= B,&0_
for
-/2)).
(5.3)
H F = u(,~:)
~ = 1 . . . . . d,
v
(5.4)
is gauged, ie. becomes a local invariance. This invariance goes over to the corresponding effective action W'(V~). As a consequence, the anomaly 3W'(V~, &o+) simply vanishes for &0+ in the algebra of H v. Since we know from the result (i) in sect. 2, that = w(v
) +
R'(v.+),
it follows from eq. (5.1) that there exists a counterterm
Q(V~) such
that
( ~ + 3Q)G+~nF = 0.
(5.5)
It is easy to see that a solution of this equation is given by 1
Q = ~e,,f
d2x Y',tr(ysB~V~+B~V~-)
for d = 2
ot
1
Q 384i~Te..o.fd"x~tr{vsB.V~-B.(V+O~V~++iV:V°+V~+) - ~tysB,,V_B,V+B,,V_B,,V+}
for d = 4, (5.6)
with A 'OB=A(OB)- (OA)B. These counterterms are Lorentz invariant. Notice that, in the Wilson case, H v = U(nf) v so that a solution of eq. (5.5) is just the counterterm introduced by Bardeen in order to cancel the anomaly of the vector currents, namely:
{~:~----~ie~fd2xtr(TsV~+V~-)
9~-~ie.,.of d4xtr(vs(V~_(V+"J°V"+)+ ¼V~_V+V°+V°+)}
for d = 2 for d = 4. (5.7)
A. Coste et al. / Chiral anomaly on the lattice
665
For the sake of Monte Carlo calculation of weak matrix elements, one would like to know, in the future, how the QCD gauging of Kogut-Susskind fermions will affect our results, and also, like for Wilson fermions [14], what are the proper weak hadronic currents to be used and how they are renormalized. We are grateful to L. Alvarez-Gaumr, P. Chiappetta, R. Dykgraef, K. Fabricius, T. Gonzalez-Arroyo, G. 't Hooft, T. Jolicoeur, L. Maiani, G. Martinelli, A. Namazie, J. Smit, R. Stora and D. Verstegen for useful discussions and T. Jolicoeur for reading of the manuscript. A.C. thanks the Fondation de France for financial support and O.N. thanks the CERN Theory Division for financial support and hospitality while this work was being done.
Appendix A This appendix is devoted to the derivation of the statements (i) to (iii) of sect. 2, which deal respectively with the locality of A W - I V ' - IV, Wo~ and 8WF. All these functionals, defined in the main text, can be written as a sum of diagrams, analogous to D(~)(A, V) (see eq. (3.46)), but with different vertices: their relevant properties can be demonstrated by power counting. The general features of the perturbation expansion and power counting are exposed in sect. 3 in view of computing the anomaly 8W. We therefore begin with the proof of the locality and infrared finiteness of the anomaly, and in order to simplify the discussion, we treat together 8WF and 8Woo. Notice that the locality of ~W~ follows also from the one of Woo which will be shown afterwards. In eq. (3.49), (SWoo + 8WF) has been written as the sum of a finite number of diagrams D("~(A, V). We want to show now that each diagram is a local polynomial in 8o~+ and V~. For this we use the expansions of the propagator A(I+ Q) in eq. (3.25) and of the vertex ~¢/r(l + Q, q) in eq. (3.28) which allow one to factorize the loop integral fBzdl in eq. (3.46) from the integral over the external momenta
fdql ...dq~. D~"~(A, V) is thus expressed as the sum of integrals of the form:
I (~) = constfBzdql...dqn 8(ql + " " + qn)P(ql ..... qn)R(~'d~+(ql), l'V~(i)(q2)... ) ×F(am, a ) ,
(A.1)
where P and R are polynomial functions and F(am, a) is the result of the loop integration multiplied by an explicit power of a. F(am, a) contains all the information on the ultraviolet as well as infrared behaviours of 1 (n). We can therefore take the limit a---, 0 in the integral over the external momenta q. By inverse Fourier transform, the polynomial P(ql,-.-, qn) is traded for a polynomial of derivatives
666
A. Coste et al. / Chiral anomaly on the lattice
P(iOx, ..... i Ox. ) acting on the fields contained in R. The presence of 8(q 1 + "" " + qn) then ensures that we get a local expression of the form:
I(n)=const×F(a, am) X
f dy~a(8oa+(y),V~(y)),
(A.2)
where ~ is a polynomial in 8~+(y), V~(y) and their derivatives. The locality of (6Wo~ + ~WF) will follow if we can show that the expansion of D(n)(A, V) stops at a finite order. For this purpose, it is convenient to reformulate the power counting rules by rescaling the loop momentum by a, thus introducing the variable
l= al
(A.3)
to be integrated over [-~r, ~r]. In the loop integral, the ultraviolet singularity is traded for a singularity for/*~ 0, according to the leading behaviours a - 7' f2
s~C(l,q) - --, a ~q/'(l + Q, q) - l°a°. The degree of divergence of
D(n)(A,V), already obtained in eq. (3.22),
an " D(")(A,V)
(A.4)
d^[2
Jm~d l F
ir(1/a)d+l-n -"~alog(am) ~a
is recovered:
for n < d + 2 for n = d + 2 for n > d + 2.
(A.5)
With these variables, the expansion of A(l + Q) in eq. (3.25), provided one keeps at each order the leading term only, appears as an expansion in powers of aQ/k
A ( I + Q ) - ~ a A(0)+
~ l+ . . .
+AU)
+-...
(A.6)
On the other hand, the expansion of W(l+ Q, q) in eq. (3,28) is obviously an expansion in powers of a. Therefore, supposing that the propagators of D~n)(A, V) are expanded to an overall order i and the vertices to an overall order j, the corresponding contribution to D(")(A, V) behaves as follows:
[a n+i+j-d-1 [D(n)(A,V)] ~i'j) an+i+Jad+lf d a l ~ - ] a J + l l o g ( a m ) ^
~a j+l
forn+i d + 2. (A.7)
A. Costeet al. / Chiralanomaly on the lattice
D(n) (\V)
=
667
- - - ~
Fig. 8. An exampleof a diagram D(")(V) in the expansionof IV.
Since j >~ 0, it is clear that, in the limit correspond to
a ~
0,
the only non-zero contributions
i + j ~ < d + 1 - n.
(A.8)
Therefore, only a finite number of terms have to be retained in the expansion of D(')(A, V), which completes the proof of the locality of 3Woo and 8W F. Another consequence of eqs. (A.7) is the infrared finiteness of the anomaly. Indeed, divergences for m--* 0 can only come from divergent loop integrals I(am) which only occur when n + i >~ d + 2, i.e. for diagrams vanishing when a --* 0. The derivation of the locality of Woo and AW goes along the same lines and we will work it out in less details. Wo~ is the sum of diagrams D(")(V) with n propagators and n vertices W(I + Q, q) (see fig. 8). Their leading divergence is as follows:
a"
/(l/a) d-"
D(")(V) - -~'2fa daf l - l ~ g ( am )
for n < d for n = d for n > d,
(A.9)
so that only the first d diagrams contribute to W~. By expanding the propagators and the vertices like for the anomaly, each of these diagrams gives rise to series of local polynomials in V~, denoted by [D(')(V)] (i'j), among which only a finite number are retained in Wo~. Indeed, one has
[D(n)(v)] (i'j)
f a n+i+j-d a'+i+J f~ a 1 aa md l frYg7 - ~aqog(arn) ka j
for n + i < d
forn+i=d
(A.IO)
for n + i > d,
so that [D(n)(v)] (i'j) diverges only when i + j < d - n or (i = d - n, j = 0). We now turn to the proof of statement (i). Given an action S~(q, q, V~) satisfying conditions (a) and (b) in (i), we introduce the matrices $ ' and Ag defined
668
A. Coste et a L / Chiral anomaly on the lattice AS
a,
b)
AW= &S0 • &sO/xs" &S&&S _(n,n') A(~ U(~S.W)
~,
AS
""'
AS
Fig. 9. (a) Diagrammaticrepresentationof the expansionof AWgivenby eq. (A.13). (b) An exampleof a diagram D~"'"'I(AS,V)in the expansionof aW.
by:
S~( q, Ft, V~ ) = ?1" S' . q,
(A.11)
AS F = S ~ - S F = ~/. A ~ . q .
(A.12)
Conditions (a) and (b) imply respectively (a)' (b)'
ASF(q, ~, V~) -+ 0
in the naive continuum limit a -+ 0,
ASF( q, 7t, V~= O) = O.
With the notations of sect. 3, the perturbative expansion of AW follows from (see fig. 9)
aW-
14/'- W = trlog(n + AS. S - l ) =
m
(1) n-1
£
--tr((zl~-~-l)n}.
n=l
(A.13)
n
By further expanding S - 1 according to eq. (3.6), zlW can be expressed as a double sum of diagrams D("' "')(AS, V), like in fig. 9b, with (n+n') propagators, n' vertices ~/r(l + Q, q) and n vertices AS(l + Q, q) defined by
AS(k, q) = E eikYASy, xe-i(k-q)x"
(A.14)
x,y
This vertex can be treated in exactly the same way as Y¢/'(l+ Q, q) in eq. (3.28), namely it can be expanded in powers of a, at fixed l = al. Just like for g / ( l + Q, q) in eqs. (3.29), each term in this expansion is itself a finite sum of subvertices, denoted AS (e' r)(/, Q, q), with the following behaviour for a --+ 0 and l--+ 0: zIS(P")(I, Q, q) - apPQs(3-~v_~)(q),
(A.15)
A. Coste et al. / Chiralanomaly on the lattice
669
where r, s, t and u are non-negative integers, and (OtV~_)(q) is the Fourier transform of OtV~_(x). These exponents are constrained as follows d- 1 =p-s+
( d - (t + u ) ) ,
p+r>~l, u>~l,
(A.16)
where the first constraint comes from the dimensionality of AS, the second one from (a)' which states that AS(I + Q, q) ~ 0 when a ~ 0 at fixed momenta l + Q and q, and the third one from (b)'. One easily derives from these constraints the inequalities p + r >/1,
p >/0.
(A.17)
In a diagram D(n'n')(AN,V), there are n subvertices AS (p,r) and we call P and R the overall sum of the exponents p and r. They obviously satisfy: P+R>~n,
P>~0,
R>~0.
(A.18)
If we now expand the (n + n') propagators and the n' vertices g/'(l + Q, q) to an overall order i and j, the divergence of the corresponding localpolynomial contribution to D(", ")(AS, V) is given by
an+n'+P+i+j ~R ad famddl fn+n'+i
[ D(n'n')( A S , V)] (P'R)'(i'j)
f ( l / a ) d-(,+,'+P+i+j) -
~ a P+R
+Jlog(am)
kaP+R+J
forn+n'+id+R. (A.19)
Recalling that n >/1 and n' >t 0, the inequalities (A.18) clearly imply that the only non-vanishing contributions to AW, when a ~ 0, correspond to n + n ' + P + i+j<~d. They are therefore in finite number, hence the locality of AW.
(A.20)
A. Coste et al. / Chiral anomaly on the lattice
670
Appendix B Here we wish to give an idea of rearrangements occurring between diagrams within the framework of our "combinatorial diagrammatics". In d = 2, the four diagrams contributing to 8 W v are represented in fig. 5:
8 W r ( d = 2) = D}2) + D2(2) + D3(2) + D (3) .
(B.1)
The identity (3.37) can be represented graphically as ~l~ - -
-
=
~
-
g
.
stands for iS"--d+(q) and/7 stands for a B ( l ) / D ( l ) where Eq. (B.2) leads immediately to
as defined in (3.34).
D~2) = 0.
(B.3)
In order to deal with the other diagrams let us split the quantity 3LI/OQ, appearing in the expanded propagator into its V~ and B~ parts (see eq. (3.25)). Graphically we write: ~(
=
)A(
+
X
(B.4)
B
This induces a splitting
W ) - - z)
+ DJ ).
(B.5)
Then let us consider the transformation law of the lowest order vertex term. From eqs. (2.7) and (3.29) we obtain:
8( ycp O( q , Q ) ) = i ~ ( _ )( q ) yCr ° ( q , Q) _ ycr O( q , Q ) i ~ + ( q ) + q~Afl~'~(+)( q)cos(al~),
(B.6)
where A~ = i'/~. Or in a graphical way
+
7- -
(B.7)
This last graphical identity, combined with (B.2) leads to
I+]
"-'3n(2)B+ D~3) = 18 (0)
which is the last counterterm of formula (4.5).
(0) ,
(B.8)
A. Coste et aL / Chiral anomaly on the lattice
671
Then D2(2) and .-'3an(z)add up to Of ='+- -,An(=)=x f d2x tr[ O~&O(+ly.yy+ ] ,
(B.9)
and if we plug into this
Vl,'y.= 3~. - ie..y5
(B.IO)
we recognize the anomaly (4.2) and (4.3) and the first counterterm of (4.5):
Etr(V~+V~+).
aR = cf
(B.11)
/L
Appendix C
In this appendix we wish to give an idea of the method used in order to calculate or rearrange the trigonometric integrals involved in 3W F. We take the example oi 1 XKs= ~ f~,dOldO 2 ( ( 1 - Cl) 2 "{- ( 1 - c2) 2) ~c2[ t s ~ 2 _ C13)
We first notice that
[ SlC2 ] 01~,-~-1
qc2 32
2s~1c2 33
so that
f.
1
){
.
XKs = - - lim d02 {9(1021 - e ) £ , d01 ((1 - c1) 2 q-- (1 - ¢2) 2 2~r ~ o -~
1 ~r C1 C 2 1 -dT~f],deldO2[(1-cl)2+(1-c2)2]~-+~--~f"_,dO1dO2 ~,
-
~
(1 -- Cl)2C2
= ~¥f"_,dOldO 2
1
32
1 clc2 o1('1c2~ 2(1 - q) s21c2
We can then remark that
I slc2~
q c 2 3 - s2c2
=
f=-r
=
f --qr d 0 1 d 0 2
dol do2 O(101- e) 01( --~)$1e2
@(101 - e )
2 q ( 1 - q ) c 2 - s2c2 32 (1 -- Cl)2C2
= _
dOld020(lO I - ~ )
32
32 lJ
2 32
32
672
A. Coste et aL / Chiralanomaly on the lattice
with 10l = ~02 + 022 • Therefore,
e--*0\ 2~" -~rd01 d020(]0[ --e) 0 I/~s Tl c)2)~.
?,Ks= lim ( _ I f
This integral is now easy to calculate and one finds ~KS = 1. In the Wilson case, one ends up on the same expression with the denominatol D Ks = 28 replaced by D w --- (E~s~) + r 2& The term of order r 2 is irrelevant in the vicinity of 0 = 0, so that one also has Aw = 1. Similarly in dimension four, one finds the following integrals:
~kl= elilm
- ~ f d OO(1OI-~)~ D2
A 2 = lim
-
e---,0
2
=1,
4 __e)~l(0 Sl(X~S2)C2C3CaD 3 )) =1, fdoo(101
with D = D Ks or D w as the case may be. References
[1] J. Steinberger, Phys. Rev. 76 (1949) 1180;
[2] [3] [4]
[5]
[6]
[7] [8] [9] [10] [11] [12]
[13] [14]
J. Schwinger, Phys. Rev. 82 (1951) 664; L. Rosenberg, Phys. Rev. 129 (1963) 2786; S. Adler, Phys. Rev. 177 (1969) 2426; J. Bell and R. Jackiw, Nuovo Cim. 60A (1969) 47 W.A. Bardeen, Phys. Rev. 184 (1969) 1848 L. Karsten and J. Smit, Nucl. Phys. B144 (1978) 536; Phys. Lett. 85B (1979); P. Mitra and P. Weisz, Phys. Lett. 126B (1983) 355 L. Karsten and J. Smit, Nucl. Phys. B183 (1981) 103; H.S. Sharatchandra, H.J. Thun and P. Weisz, Nucl. Phys. B192 (1981) 205; M. Gockeler, Nucl. Phys. B224 (1984) 508; D. Verstegen, Nucl. Phys. B243 (1984) 65 and references therein L. Karsten, Phys. Lett. 104B (1981) 315 and the first of ref. [4]; H. Nielsen and H.M. Ninomiya, Nucl. Phys. B185 (1981) 20; Phys. Lett. 105B (1981) 219; Nucl Phys. B193 (1981) 173 P. Becher, Phys. Lett. B104 (1981) 221; F. Gliozzi, Nucl. Phys. B206 (1982) 419; H. Kluber$-Stern, A. Morel, O. Napoly and B. Petersson, Nucl. Phys. B220 [FS8] (1983) 447 J. Wess and B. Zumino, Phys. Lett. 37B (1971) 95 C. Becchi, A. Rouet and R. Stora, Renormalizable theories with symmetry breaking, in Field theory quantization and statistical physics, ed. E. Tirapegui (Riedel, 1981) A. Coste, C. Korthas Altes and O. Napoly, Phys. Lett. 179B (1986) 125 L. Alvarez-Gaum~ and E. Witten, Nucl. Phys. B234 (1983) 269 B. Zumino, in Proc. Les Houches 1983 Summer School, ed. R. Stora (North-Holland) J.M. Blairon, R. Brout, F. Englert and J. Greensite, Nucl. Phys. B180 [FS2] (1981) 439; H. Kluberg-Stern, A. Morel, O. Napoly and B. Petersson, Nucl. Phys. B190 [FS3] (1982) 504; N. Kawamoto and J. Smit, Nucl. Phys. B192 (1981) 100 E. Marinari, G. Parisi and C. Rebbi, Phys. Rev. Lett. 47 (1981) 1795; H. Hamber and G. Parisi, Phys. Rev. Lett. 47 (1981) 1792 M. Bochicchio, L. Maiani, G. Martinelli, G. Rossi and M. Testa, Nucl. Phys. B262 (1985) 331
C A L C U L A T I O N OF T H E N O N - A B E L I A N C H I R A L A N O M A L Y ON THE LATI'ICE A. COSTE and C. KORTHALS ALTES Centre de Physique Th~orique, Section 2, CNRS Luminy, 13288 Marseille Cedex 9, France
O. NAPOLY Service de Physique Th~orique, CEN-Saclay, 91191 Gif-sur- Yoette Cedex, France
Received 18 December 1986
We check the validity of Wilson and Kogut-Susskind lattice regularizations of the Dirac fermion action by calculating perturbatively the non-abelian chiral anomaly on the lattice. We show that the result does not depend on the detailed expression of the coupling term of the lattice fermions to the chiral gauge field, taken as an external field. In two dimensions we recover the consistent expression of the anomaly, up to counterterms which we have computed. In four dimensions, we obtain the correct normalization factors of the anomaly, leaving the problem of the counterterms open.
1. Introduction
The notion of chirality has played an important role in the conception of present day particle theory, particularly for our understanding of strong interactions. The chiral invariance of the quarks is easily implemented on the classical level via the Dirac action. On the quantum level, one has to introduce regulator devices which are at the origin of chiral anomalies. The so-called abelian anomaly [1] is responsible for dynamical effects such as ~ r ° ~ ,{~, decay and 7' mass when the quarks are coupled to Q E D and Q C D gauge fields respectively. When the gauge group is enlarged to the full chiral flavour group, regularization gives rise to the non-abelian anomaly, first computed by Bardeen [2]. Its main consequence is to provide a constraint on the fermionic content of renormalizable chiral gauge theories used in model building. Among the regularization devices, lattice regularization is of special importance since it allows one to perform non-perturbative calculations for QCD, and in fact has been designed for this purpose. However, the standard weak coupling perturbation theory ought to be recovered as well. Checking this point [3], and in particular the abelian anomaly [4], has been of great benefit for the problem 0550-3213/87/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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A. Coste et al. / Chiral anomaly on the lattice
of finding correct lattice fermion formulations. It is now understood [5] that chiral anomalies originate from the fact that, in order to avoid the species doubling of the naive lattice quark action
Sc( q, ~1) = a d E ~( x )7~,V~,q(x ),
(1.1)
x,o,
where
1
V~q(x) = ~a ( q(x +/2) - q(x - 12))
(1.2)
is the lattice first derivative and a the lattice spacing, it is necessary to add to this action a term Sb(q, ~/) which breaks explicitly the U(ny) × U(nf) chiral invariance of S¢(q, el). Popular choices for S b are as follows:
S b = - ½ra a+l • ~ ( x ) A q ( x )
(Wilson),
x,p,
S b ~___~u 1 _d+l
E ~ ( x ) B , A q(x)
(Kogut-Susskind),
(1.3)
x,p,
where A is a lattice second derivative: 1
A q(x) = --~(q(x +/21 - 2 q ( x ) + q ( x - / 2 ) )
(1.4)
and B~ is a matrix in spin and flavour spaces [6] such that
which is possible only for nf a multiple of 2 d/2, say
n/= ~/. 2 a/2 .
(1.5)
S b is an irrelevant term which vanishes in the naive a ~ 0 continuum limit, it can give rise to anomalies through divergent fermion loops. It remains to be seen whether it generates the correct non-abelian anomaly. In 2 and 4 dimensions, Wess-Zumino consistency conditions [7] uniquely determine [8] the expression of the non-abelian anomaly, up to counterterms, provided it is local, Lorentz invariant and vanishes for chiral transformations which are global in space-time, thus reducing the problem to that of computing the overall coefficient. Unfortunately, we cannot use this uniqueness theorem in the framework Although
A. Coste et al. / Chiral anomaly on the lattice
647
of lattice perturbation theory. The last two properties will clearly be seen to fail while we will prove the locality of the anomaly by power counting. Therefore, checking that the non-abelian anomaly is reproduced by lattice fermion regularizations needs an explicit calculation. The purpose of this paper is to present such a calculation both for Wilson and Kogut-Susskind fermions. In sect. 2, we formulate the definition of the non-abelian anomaly in the language of the lattice functional integral and we derive three important theorems, dealing mainly with locality, which greatly simplify its calculation. In particular, the first one sets the freedom in the choice of the lattice expression for the coupling between the fermion and the chiral gauge fields. The detailed proof of these theorems is contained in appendix A. In sect. 3, we derive the Feynman rules used in the perturbative calculation. This is done in two ways. The first one allows one, in 2 dimensions, to identify diagrammatically the counterterms, rendering their computation unnecessary. It enables us to directly identify and compute the 2d anomaly. We have not yet been able to generalize this method to 4 dimensions. The second method, which is more economical in the number of graphs but does not separate the anomaly from the counterterms, is applicable in any number of dimensions. In sect. 4 are reported the results that we have obtained, using the second method, for Wilson and Kogut-Susskind fermions. In two dimensions, the calculation is complete: we found the correct anomaly plus terms which combine themselves into the chiral variation of a local counterterm whose expression is given. In four dimensions, this last step is still missing. We intend to complete the 4d calculation by either one of the methods. Finally, we discuss the problem of the presence of a regularized axial symmetry in the Kogut-Susskind formulation, and we show that it leads to no contradiction with the usual vector conserving expression [2] of the non-abelian anomaly. Appendices B and C contain technicalities relevant to sects. 3 and 4 respectively. A brief account of the results of this paper can be found in our ref. [9]. 2. The non-abelian anomaly on the lattice
In this section, we will follow closely Bardeen's notations [2] and set up a lattice version of his work. We first introduce in the fermionic action S v = Sc + Sb a smooth continuum external field, taking values in the chiral U(ne) algebra:
V
(x) = W ( x ) +
(2.1)
There are many ways to couple this external field such that the classical action fddx71Ip(V±)q is reproduced by the lattice action SF(q, Ft, V~) in the naive a ~ 0 continuum limit. In this paper, we have chosen to gauge the chiral symmetric part So(q, 71) of the action and to leave the symmetry breaking term Sb(q, Ft) unchanged:
SF(q, 7t, V~) = So(q, 71, V~) + Sb(q, 71),
(2.2)
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A. Coste et al. / Chiral anomaly on the lattice
with
Sc( q, ~1,V~) = a e E Yt(x )Y~,~7~'(U )q( x ),
(2.3)
where the covariant lattice first derivative 1
V~'(U)q(x)=-~a (U~(x)q(x + f t ) - U~-l(x-ft)q(x-ft))
(2.4)
is given in terms of the link integrated variable:
Of course there are other possibilities of coupling the external field. By the end of this section we will show when and in what sense they are equivalent. We then define the effective action by:
W(V~; a, m) = - l o g Z(V~; a, m)
(2.6)
from the lattice functional integral:
Z(V~;a,m)= f[dqdq]exp(-(Sv(V~,q,q)+
Sm(q,q))}.
(2.7)
Being regularized, the effective action can be expanded in powers of the lattice cut-off 1/a and splitted into divergent and finite pieces
W(V~; a, m) = Woo(V~; a, m) + WF(V~, m) + O(a).
(2.8)
The mass parameter m has to be introduced in order to take care of the logarithmic divergences proportional to log(am) in W~, and of possible infrared singularities. V~ being a continuum external field, the anomaly 3W can be defined as the variation of the effective action,
3W(V~, 3o~+; a, m) = W(V~ + 3V~; a, m) - W(V~; a, m)
(2.9)
under a local chiral variation of V~
3V~(x) = -O~'(3~o+(x)) - i [ V ~ ( x ) , 3 w + ( x ) ] ,
(2.10)
with 3¢0+ = 3o~V+ ~53o~A the infinitesimal parameter. Since it is the variation of a well-defined functional, the anomaly 3W as well as its finite part 3W v, fulfills
649
A. Coste et al. / Chiral anomaly on the lattice
Wess-Zumino consistency conditions [7], namely in short, 88~,,+(SW(Sto+)) - 8~,o+(8W(8~o~_)) = 8W([8~o+, &o~_]).
(2.11)
As we will see, only the finite part of the anomaly, 6WF, is interesting and, therefore, it must be compared to the consistent expression of the non-abelian anomaly, first obtained by Bardeen. Taking advantage of the invariance of the lattice fermionic measure l-lx[d q d~](x) under a local chiral redefinition of the quarks, one easily derives the following Ward identity:
~W=(SSF(q,~,V~; 8to +)),
(2.12)
with
6SF=SF(q+6q,~+6~,V~+6V~)-SF(q,
Zt, V~)
(2.13)
and
{
6q(x)=iSw+(x)q(x) ~(x) -i~(x)8w_(x).
(2.14)
Notice that in eq. (2.12), we disregard the non-anomalous contribution (SSm) coming from the mass term, which vanishes when m-~ 0. With the action SF(q, FI,V~) written in eqs. (2.2), (2.3), one has obviously 6Sc = 0
(2.15)
BW=(6Sb(q,q; Bw +)).
(2.16)
and therefore:
This equation, which is the starting point of our calculation, clearly shows that S b is the source of the chiral anomaly on the lattice. (8Sb) will be computed in perturbation theory by expanding the full quark propagator according to the following splitting of the action
SF( q, ~, V~) = St( q, ~) + ¢/'(q, Ft, V~).
(2.17)
The free propagator is thus obtained by inverting the action of a free quark with mass m,
So( q, ~; m) = SF( q, ~) + Sm(q, q)
(2.18)
650
A. Coste et a L /
Chiral anomaly on the lattice
and the vertex from ~ ( q , ~, V~). It follows from eq. (2.17) that Y/"(q, ~, V~)I v,+=0 = 0
(2.19)
so that we actually perform a weak coupling expansion. Before presenting the calculation itself, we need to collect a few results which simplify our task. All these results follow from power counting and are derived in appendix A. The first one deals with the arbitrariness in the choice of the lattice action, as discussed in the beginning of this section: (i) Any action S~(q, gl, V~) having the two following properties:
(a) S~.(q, q, V~) ~
fd"xq(x)O(w.+_)q(x)in the naive continuum limit a ~ 0, (2.20)
(b)
S,~( q, Ft, V~ = O) = SF( q, q) ,
(2.21)
yields an effective action W'(V~; a, m) which differs from W(V~; a, m) obtained from SF( q, g/, V~), only by a local polynomial functional of V~. It should be clear that the anomaly is defined up to the addition of the variation 8R(V~, &o+) of a local polynomial counterterm R(V~) of the field V~. It is therefore enough to show that the correct result for the anomaly is obtained from one action verifying (a) and (b), namely from SF(q, ~, V~). Notice that, because of condition (b), we cannot prove a priori that the Wilson and Kogut-Susskind cases differ by a counterterm, hence the need of two separate calculations. The other results are as follows: (ii) Woo(V~; a, m) is a local polynomial functional of the field V~. Its contribution to the anomaly 8W(V~,Sw+; a, rn) can therefore be cancelled out by the addition of the local counterterm R = - Wo~ to the original action. So, we only need to compute the finite piece of the anomaly
m) = ( Sb(q, O, 8,0
(2.22)
(iii) The anomaly 8Wv, as defined by the above equation, is also a local functional of V~ and &0+. It is moreover free from infrared divergences when m ---, 0. The consequence of these three results is that it is enough to calculate the finite part of the anomaly at zero mass 8WF(V~, 860+; m = 0) given by eq. (2.22), for our choice of action SF( q, 7t, V~ ).
A. Coste et al. / Chiral anomaly on the lattice
651
3. The perturbative calculation of the anomaly We now turn to the actual calculation of the anomaly from eq. (2.22). This section contains many technical details which may be necessary in order to complete the 4d calculation. In order to make it readable we split it into three subsections. We first work out the formula (2.22) into a form where it becomes a sum of Feynman diagrams. In the second subsection, we use the fact that the external fields vary slowly with respect to the lattice spacing a, i.e. we expand further each diagram in powers of aqi where the q/s are the extemal momenta of the vertices. The ensuing set of graphs is then handled in two alternative ways in the last subsection, in order to separate the canonical anomaly from gauge variations of local counterterms. We will treat the Kogut-Susskind and Wilson cases together: the relevant quantities will be labelled by the superscript KS or W. 3.1. FORMULA FOR EQ. (2.22)
Starting from the action Sb~s and S w defined in (1.3) one easily finds:
8sbKS(q,~t,&o +)=aa(½ia) • (a,~l(x)B~&o+(x)q(x) X, I3,
- gl(x ) 8oa_( x ) B.A.q( x ) ) ,
(3.1)
8SW(q,q, 8w ±)= -aa(½ia) r E ( A,~(x)8~o+(x)q(x) -?t(x)&o_(x)Auq(x)). X,p,
(3.2) Introducing the matrices B and S over lattice sites and internal space defined by
(3.3)
8S b = tT" B • q,
SF(q, q, V+_)+ Sm(q, q)
--- q. • -q,
(3.4)
we get
(3.5)
{~Sb) = -Tr(S-tB). --1 can be evaluated perturbatively -1 = ( ~ 0 -[- ~ / ) - 1 =
(-- 1)n-I(SolV)
"-1So I ,
(3.6)
n=l
where the matrices S O and V are defined by
So(q,
=
So.q,
Y/'(q,~,V+_)=~.V.q, with the notations of eqs. (2.17), (2.18).
(3.7) (3.8)
652
A. Coste et al. / Chiral anomaly on the lattice a)
b) c)
$-1
=
,
W
=
I.
-1
,
IB
+
4.
Tr ($-~le)
:
...
__©
: __©
. <3-.
<3(.
Fig. 1. Diagrammatic representation of (a) the full quark propagator, the free quark propagator, the vertex and the anomalous vertex; (b) the perturbative expansion of the full propagator; (c) the perturbative expansion of the anomaly.
Hence (3Sb) = ~ D(")(B,V),
(3.9)
n=l
with
D(")(8, V) = ( - 1)"Tr(~olB(~oiV)"-i).
(3.10)
This expansion is represented diagrammatically in fig. 1. After a Fourier transformation one gets: D(")(B,V) - _ ( - 1 ) " / . d / d q i - dq,,3L(ql+''" (2~r) ,,a JBZ ""
+q,)
× tr[A ( / ) ~ ( l , ql)A (l + Ol)g/'(l + 01, q2) ... xW'(l+
a,,_2,q,,_l)A(l+
O,_i)zgU(l+ O , - i , q , ) ] ,
(3.11)
where the loop momentum ! and the external momenta q~ are integrated over a Brillouin zone (BZ) ( ] - rr/a, ~/a]) '~. We set Q,= - ( q i + "'"
+qi)
and ad E eikx
~L(~) (2~)~
(3.12)
A. Coste et al. / Chiral anomaly on the lattice
qn
ql
q n - l ~ - l ~ D (n) ( [13,\V )
q2 ' t÷ O2
i' "
X
653
Qi : - (ql + "" +qi )
,
,
q~
Fig. 2. Momentum flow in the diagram D ("~ given by eq. (3.11).
for the lattice Dirac distribution. Our conventions for the momentum flow are given in figs. 2 and 3. The quark propagator is defined by
( o)x,y
=
(2~r
z
dk eik(y-x)A(k),
leading to AKS(k) = a
AW(k) = a
F.,[y~isin(ak~) + B~,(1- cos(ak~))] + ma (F~,4 sinZ(½ak~) + m2a 2 } ~[y~isin(ak~) + r(1 - cos(ak~))] + ma {F~.sin2(ak~) + (rE~(1 - cos(ak~)) + ma)2} "
(3.13)
(3.14)
The anomalous vertex (see fig. 3a) is defined by
~(l, q) = E eilyBy,xe-i(l-q)x,
(3.15)
x,y
leading to
i
~KS(l, q) = - a E { B,3"~ + (q)(1 - cos(air) ) ,u
-3"d~_(q)B~(1-cos(a(l~-G)))
},
i ~W(l, q ) = a ~.(3"d~+(q)(1-cos(al.))-3"~_(q)(1-cos(a(l~-q~)))), t~
(3.16) (3.17)
A. Coste et al. / Chiral anomaly on the lattice
654
I
'll~ (I, q )
--,
q
Fig. 3. Conventions for the momentum flow in (a) the anomalous vertex (cf. eq. (3.15)); (b) the vertex (cf. eq. (3.19)).
where 8to±(q) is the Fourier transform of 8to _+(x)
8-'~+_(q ) = ad~_, eiqxsto ± ( x ) .
(3.18)
x
In the same way, the vertex W(k, q) is defined by (see fig. 3b):
YV'(k, q) = E
(3.19)
eikyVy,x e-i(k-q)x,
x,y
which gives ~¢//"KS(k, q) = ~ : W ( k , q) = ½]~_,y~(ei"k~lYV~+(q) - e-iak~lYV~_(q)),
(3.20)
jz
with
lfv~+(q)=aaEeiqx!(Bx
U-l(x)),
a
l,V~_(q) = aa~_,eiqxl (n - U ~ ( x - / 2 ) ) . x
(3.21)
a
In the limit a ~ 0, the loop integral in D (") is possibly divergent. T o extract the degree of divergence one keeps track of the leading powers in a and l:
A(t)-t -1, ~ ( l , q) - al 2, ~ r ( l + Q, q) - 1°.
(3.22)
Therefore -1-n
dall 2
D("~(R, V) - aJ
-
for n < d + 2
/a oga
(3.23) for n > d + 2.
655
A. Coste et al. / Chiral anomaly on the lattice
a)
L-'-Q >
=
L >
÷
L t >×; Q
l .
t I ×;× Q Q
•
. . .
Fig. 4. Diagrammatic representation of (a) the Taylor expansion of the propagator A(l+ Q) (cf. eq. (3.25)); (b) the expansion of the vertex ¢¢'(/+ Q, q) (cf. eq. (3.28)); (c) the expansion of the anomalous vertex 8 ( l , q) (cf. eqs. (3.30) and (3.31)).
As expected, only the first ( d + 1) diagrams contribute to the anomaly when a ~ 0, and we can write d=l
8W~ + 8 W v = • D~")CB,v).
(3.24)
n=l
3.2. EXPANSION OF FEYNMAN DIAGRAMS IN POWERS OF aqi
D ~1>can be readily computed and in fact contributes only to 8W~. The diagrams D ~'), for 2 ~
0a
A(I, Q) =z~(l) + E Q ~ - ~ , ~t
(I) + . . . ,
(3.25)
t~
which is nothing but the Taylor expansion in powers of Q. For the vertex ~P(I + Q, q) given by eq. (3.20) it is convenient to realize that:
1 ~ (q) = __+½(1~°) (q) + a l ~ l ) ( q ) + a 21,V~(2)(q) _ a 31~9>(q)) + O ( a 4 ) ,
(3.26)
where the l ~ i ) ( q ) ' s are related through a Fourier transform, like in eq. (3.18), to the
656
A. Coste et al. / Chiral anomaly on the lattice
following expressions: l.V<°)(x) = 2iV~+(x),
w~
i o~v~+(x) + (v~+(x)) 2,
W~(2)(x) = ~(i 02V~+(x) + 20~V~+(x)V~+(x) + V~+(x) O.V~+(x) - i(V~+(x))3), ~ 2 + v~+(x) o~v~+(x) o~v+(x))
VV(3)(x)=~(iO2V~+(x)+3a~V~+(x)V~+(x)+3(
-3i o.v~+(x)(V~+(x)) ~- 2/v+~(x) o.v~+(x)V~+(x) (3.27)
-i(V~+(x)) 2 0.V~+(x)-(V~+(x)4)).
The vertex ~/'(l + Q, q) can then be expanded as follows (see fig. 4b) 3
Y//'(l+ Q, q) = E ai~/'(i)( l, q, Q) + O(a4),
(3.28)
i=O
with gp (o)( 1, q, Q) = ½~./~ I~7~(°)( q)cos(al~), ~¢~ (1)( l, q, Q ) = 1 E "/'~(il~r(1) ( q )
- Q~ff.(o)( q))sin(al~),
zeus(2)(l, q, Q) = ½E "/~(1~(2)(q) - iQ~ffTf)(q) - ½Q217v~(°)(q))cos(al.), ~(3)( 1, q, Q ) = ½• "/"( i l.v.(3)( q ) - Q~I~(2)( q ) - ½iQ~lTV~(1)(q) #z
+ ~Q31~(°)(q))sin(al~). (3.29) Finally the expansion of the anomalous vertex is: i ~Ks(/, q) = a Y'.{(B~8~+(q)-So3_(q)B~)(1-cos(al~)) #
+So3 ( q ) B . ( a q . s i n ( a l . ) - 5 ( 1 aq.) 2cos(al.) ~(aq.)3sin(al.))},
(3.30)
ir ~W(l, q ) = a E{(So3+(q)-So3_(q))(1-cos(al.)) t~
+8o3_(q)( aq~sin(al.) - ½( aq.)2cos(al~) 1 3 • -g(aq~) sm(al~,))}.
(3.31)
A. Coste et aL / Chiral anomaly on the lattice
o: D':'=
=
{0) --
©
,o,--©
(1)
O(2 2)
=
,o,
o"';
657
©
11) . . . . .
(0)
,o, --
~' ( 0 1
Fig. 5. The diagrams contributing to the finite part of the 2d anomaly with the method of subsect. 3.3(a).
It is easy to see that each order in one of these expansions decreases the degree of divergence of D (") by one power. Using the diagrammatic representation of fig. 4, we are thus able to isolate the diagrams which contribute to the finite part 8W F of the anomaly (see for instance the 2d diagrams in fig. 5). 3.3. S E P A R A T I O N OF C A N O N I C A L A N O M A L Y
In order to cope with their calculation one has to use the symmetries of the theory at best. For this we set up two different diagrammatics which we present now: (a) "'Combinatorial diagrammatics". The idea is to use algebraic properties of the breaking term S b to derive vertex identities leading to cancellations between diagrams. This is reminiscent of diagrammatic form of Ward identities in continuum field theory. Such identities arise from similarities between some terms in the expansions of vertices and propagators. Explicitly we can write the massless propagator: A(l,m=0)=a
D
'
(3.32)
with .~KS= A~w= i Z y~sin(alr),
(3.33)
/z
/~ KS = y: B. (1 - cos(alr,)), /t
/~w = r Y', (1 - cos( al, ) ),
D =
_:i2 +
ew = 1,
( 3.34)
h2,
e Ks = -- 1 .
(3.35)
Then the lowest order term in the anomalous vertex reads ,~(1)(l, q) =
i
^
ae(B(l)SYo+(q)
-
8o5+(q)/~(t))
(3.36)
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A. Coste et al. / Chiral anomaly on the lattice
and we have an "anomalous vertex identity"
(8~+(q)B(1)-B(1)8~_(q)) A ( l ) ~ 1 ) ( l , q)A (l) = ia
D(I)
(3.37)
Its derivation, which makes use of the following properties
(i) (a)
/~ 2 and .~2 are scalars, ^ ^
(iii)
^
^
AB = eBA ,
(3.38)
proceeds in two steps; firstly:
= (h~,0 h - 8 ~ + ~ 2 ) ~
+ (h2~o~+-hs~ ~ ) ~ = 0.
Secondly:
~( ~ ( ~ -
~-~)~ ÷ ~I ~ -
~ ~)~ }
Since it involves only one factor 1 / D ( l ) , the r.h.s, of eq (3.37) looks very much like a propagator. Together with identities relating Q-expansions of propagators and vertices, it allows us to identify anomaly and variations of counterterms as linear combinations of diagrams. The advantage of this method is that it leads to the same combinations both for Wilson and Kogut-Susskind cases, an account of which we have in the fact that (3.37) does not depend on e. Moreover these combinations can, at least in two dimensions, be identified purely graphically. We give details on this graphical identification for d = 2 in appendix B. (b) "'Hermitian diagrammatics". The idea is to make use a formal antihermiticity property of the action (2.2). Let us define the conjugation t by:
(i) (ii)
(V~)t= V",
(A~)*= -A ~
r* = - r ,
(V~+)*= V~_,
m* = - m ,
(3.39)
(iii) t acts as usual hermitian conjugation on i factors and Dirac matrices, in particular
A. Coste et al. / Chiral anomaly on the lattice
659
Under this conjugation, one has =
so that the action is antihermitian: St = -S.
(3.40)
This hermitian conjugation is formal since (i) is not compatible with the axial-vector source A~'(x) being in the algebra of U(n/). Of course, it is well known [10] that no hermitian conjugation, with (A~) t = A ~', can make the chiral Dirac operator (anti-) hermitian (in the continuum as well as on the lattice) so that the euclidean effective action W = logdet(0 ) has an imaginary part which precisely contains the chiral anomaly. Here, the operation t is used consistently as a trick in order to simplify the form of the anomalous vertex. Indeed, with the additional rule (iv)
8¢*v= 8o~v,
8o:a = -8o~A = (8~+) t = 8o~_,
the anomalous vertex matrix B is antihermitic and can be written B = x(A - At),
(3.41)
with ~AKSq =
ad(ia) ~_, ASI(x)B~8~+(x)q(x),
(3.42)
x,/t
qAWq =
--ad(ia) r E A~q(x)8o~+(x)q(x).
(3.43)
X,/L
Plugging (3.40) and (3.41) into (3.5) we get
(SSb)
= -ReTr($-IA).
(3.44)
This leads to the diagrammatic expansion (SSb) = Re( n=i~D(")(A' V)) '
(3.45)
with (-1)" D ( , ) ( A , V ) - (-~--r-~-7t f, zdldql...dq, XTr[A (l)~¢(l,
6L(ql + ... +q,)
ql)A(l + Q1)Y¢/'(I+ QI, q2).., g:(l + Q,-1, q,)], (3.46)
A. Costeet aL/ Chiralanomaly on the lattice
660
and ,~¢KS(l, q) =
dW(l, q) =
-2i a
~,B,~"~+(q)(1-cos(al,)),
(3.47)
/x
2i ar~,,8o~+(q)(1-cos(al,)).
(3.48)
The difference with eqs. (3.16) and (3.17) is that d ( l , q) needs not be expanded in q, leading to only one anomalous vertex. This saves a number of diagrams and is the main advantage of this method. The anomaly is now given by [d+l ) AW~ + 8WF = Re/,~--1D(')(A'V)
(3.49)
and the diagrams contributing to 8WF are respectively 3 and 21 in 2 and 4 dimensions (see fig. 6).
+
~WF(d=2) = --'~(1)+
. _ . ~ (0)
---¢~10) (0)
~WF(d=Z,) = ---~(3) + ---¢~(2) (0) (2) (1}
"
-1~(11
• I1 ~
(01
11)
.~(0)
~(1)
~(0)
111 (0)
(1) :~/Iol
(0) ,._.,/(oi
{0) ,..y/(, 1
{0)
(21
.
101
+
" 111
:.,/(oi ~'101 i(o)
+ ._.~
(0)
(1} .~
101 1o1 (01
•
~' 101
,,..~ (o)
"101
" 101
,...,/(oi x(O)
101
T o;-,°, Fig. 6. The diagrams contributing to the finite part of the anomaly with the method of subsect, 3.3(b), in d = 2 and d = 4 .
A. Coste et al. / Chiral anomaly on the lattice
661
The results reported in the next section have been derived in the framework of this second method. 4. The results In the two-dimensional case, the calculation yields, for m = 0:
8WF( V~, 8o~+) = X . s~¢+ 8R ,
(4.1)
where ~¢ is the consistent 2d non-abelian anomaly [11]
i f d2xie~tr(753~+ O~V+) ~¢= 4~r
(4.2)
written in the L - R or V - A symmetric way. The coefficient X is given by the trigonometric integrals:
~kw
2r 4
~r
33C 2
2"
= ---~-LdOldO2--~-(Sc1-2sl),
(4.3)
with the notations
c i = cos 0i ,
s i = sin Oi,
for i = 1, d
and d
3=
]~(1-c,),
D=
~ s2 + i=1 /
(4.4)
r 2 3 2.
R is the following local counterterm RKs=
f d2x{c2tr(V~+ V~+)+ #
2 ca .... tr(V"+B, IV~-B~2) ~VlV2
+
E
ie..c""~tr(ysV~+IB, V"2B,]l btl~t*2 \ 1 -2/]
~
~1 ]22 I)lV2
Rw= f d2x ( c l ( r ) E \
tr(V~+ V~+) + c2(r)Etr(V~+ V~_) /x
/z
+¢3(r)Ei ,,tr(v,v + w_)}, ~v
(4.5)
A. Costeet al. / Chiralanomaly on the lattice
662
where the coefficients c, c ~1"2, c v1"2 and cl(r ), c2(r), c3(r ) are also given by 2d trigonometric finite integrals. It is not essential to give their complete expression here; it is nevertheless interesting to note that they are such that, on one hand R Ks is not Lorentz invariant and on the other hand R w is r-dependent. These counterterms of course depend on the choice of the lattice action SF(q, ?I, V~), as stated in sect. 2, result (i). On the contrary, it is very important to verify that the correct normalization of the anomaly is recovered, i.e. to show that xKs = Xw = 1.
(4.6)
This can be done by calculating analytically the integrals of eqs. (4.3) by a method very similar to one used previously in various calculations of the abelian anomaly on the lattice (see e.g. refs. [4]). Not only this method allows us to calculate also the normalization of the 4-dimensional anomaly, but it is already necessary in order to recast the various contributions to 8W F, different from ~¢, under the form of the variation of the counterterm R. We believe that it may be worthwhile to illustrate it with an example: for this purpose, we present the calculation of XKs in appendix C. In 4 dimensions, the calculation becomes much more lengthy and we only isolated the two members of the standard non-abelian anomaly 1
~¢1 =
48i~r2
f d4x e.,ootr(758~° + O~'V~- O°V~, )
(4.7)
fdgxe,,ootr(ys~o+½iO"(V+V°+V+))
(4.8)
1
~¢2 =
48i~r 2
and computed the coefficients multiplying them in 8W F. The diagrams contributing to each of these two terms are drawn in fig. 7. With the notations of eqs. (4.4), these coefficients are given by the following integrals:
XlS = 3 f 4~"2 -~r
( 1 - c i ) 2! clc2("3(2s2-¢~¢4)
d40 [ i i=1
84
'
Xw = _ --' ~ry6 f,~_ d40 rZSaClCEC3(c4D_4c4s24_4rZ~S2)D 4
~
,
(4.9)
(o) (o)
(oi
, . . , j (oi - - -
"~ 101
"~ {01
"
"
101
Fig. 7. The diagramscontributing to -~i and ,~2 (cf. eqs.(4.7) and (4.8)) in the 4d anomaly.
A. Coste et al. / Chiral anomaly on the lattice
663
for ~¢1, and
_ _
~kW ~"
_
_
2~r2 -'~ d48 i _
f
( 1 - c i ) 2 CLC2C3S42 2 3 c 4 -
d,Or232c,c2c3s 2
1 22 f qT
(
Dc4
s2
4)/
(c4+r23) E s 2
--or
5,
D'
(4.10)
i=1
for ~¢2- These integrals can again be calculated analytically by the method mentioned above. The coefficients are all found equal to 1, so that we recover the correct consistent anomaly •~¢= -~¢1 q ,J~2 •
(4.11)
To complete this result, it must be shown that the remaining terms in 3 W F combine themselves into the variation of a local counterterm, as they do in 2 dimensions. We hope to achieve this in the future.
5. Discussion and outlook
Lattice regularization has been set up in order to probe low-energy features of QCD. One of its successes is the evidence [12,13] that chiral symmetry breaking occurs spontaneously, accompanied by massless pions. For this, the use of KogutSusskind fermions has been crucial, allowing at least one axial symmetry to be unbroken in the action. To this axial generator corresponds one conserved current and this raises the question of knowing whether the associated pion undergoes electromagnetic decay through the triangle anomalous diagram. We will now answer this question by showing that the difference between (a) the picture described by Kogut-Susskind fermions where one axial symmetry is exactly conserved and vector symmetries, including the electric charge, are explicitly broken, and (b) the usual picture where the electric charge is conserved and the axial symmetry is anomalous, is taken care of by the addition of a suitable local counterterm. For this, we first take for granted the result, only partially derived in d = 4 in the last section, that Kogut-Susskind fermions, with the action SF(q, Ft, V~) of eq. (2.2), generate the correct non-abelian anomaly, namely
3 W v =.~¢+ 3R .
(5.1)
Then, we consider the possibility of using, instead of SF( q, ~, V~), the action
S~(q, Ft, V~) = So(q, FT,V~) + Sb( q, q, V~),
(5.2)
664
A. Coste et aL / Chiral anomaly on the lattice
defined by eq. (2.3) and by replacing the second derivative in a covariant second derivative: 1
A(U)q(x) = ~5(U,(x)q(x +/2) -
2q(x)
Sb(q, ~) in eq. (1.3) by
+ u~-l(x -/2)q(x
In this action, the residual flavour invariance of SF( q, ~t), namely [6] X U(~f) a corresponding to the generators 3(o+ obeying 3,0+B,= B,&0_
for
-/2)).
(5.3)
H F = u(,~:)
~ = 1 . . . . . d,
v
(5.4)
is gauged, ie. becomes a local invariance. This invariance goes over to the corresponding effective action W'(V~). As a consequence, the anomaly 3W'(V~, &o+) simply vanishes for &0+ in the algebra of H v. Since we know from the result (i) in sect. 2, that = w(v
) +
R'(v.+),
it follows from eq. (5.1) that there exists a counterterm
Q(V~) such
that
( ~ + 3Q)G+~nF = 0.
(5.5)
It is easy to see that a solution of this equation is given by 1
Q = ~e,,f
d2x Y',tr(ysB~V~+B~V~-)
for d = 2
ot
1
Q 384i~Te..o.fd"x~tr{vsB.V~-B.(V+O~V~++iV:V°+V~+) - ~tysB,,V_B,V+B,,V_B,,V+}
for d = 4, (5.6)
with A 'OB=A(OB)- (OA)B. These counterterms are Lorentz invariant. Notice that, in the Wilson case, H v = U(nf) v so that a solution of eq. (5.5) is just the counterterm introduced by Bardeen in order to cancel the anomaly of the vector currents, namely:
{~:~----~ie~fd2xtr(TsV~+V~-)
9~-~ie.,.of d4xtr(vs(V~_(V+"J°V"+)+ ¼V~_V+V°+V°+)}
for d = 2 for d = 4. (5.7)
A. Coste et al. / Chiral anomaly on the lattice
665
For the sake of Monte Carlo calculation of weak matrix elements, one would like to know, in the future, how the QCD gauging of Kogut-Susskind fermions will affect our results, and also, like for Wilson fermions [14], what are the proper weak hadronic currents to be used and how they are renormalized. We are grateful to L. Alvarez-Gaumr, P. Chiappetta, R. Dykgraef, K. Fabricius, T. Gonzalez-Arroyo, G. 't Hooft, T. Jolicoeur, L. Maiani, G. Martinelli, A. Namazie, J. Smit, R. Stora and D. Verstegen for useful discussions and T. Jolicoeur for reading of the manuscript. A.C. thanks the Fondation de France for financial support and O.N. thanks the CERN Theory Division for financial support and hospitality while this work was being done.
Appendix A This appendix is devoted to the derivation of the statements (i) to (iii) of sect. 2, which deal respectively with the locality of A W - I V ' - IV, Wo~ and 8WF. All these functionals, defined in the main text, can be written as a sum of diagrams, analogous to D(~)(A, V) (see eq. (3.46)), but with different vertices: their relevant properties can be demonstrated by power counting. The general features of the perturbation expansion and power counting are exposed in sect. 3 in view of computing the anomaly 8W. We therefore begin with the proof of the locality and infrared finiteness of the anomaly, and in order to simplify the discussion, we treat together 8WF and 8Woo. Notice that the locality of ~W~ follows also from the one of Woo which will be shown afterwards. In eq. (3.49), (SWoo + 8WF) has been written as the sum of a finite number of diagrams D("~(A, V). We want to show now that each diagram is a local polynomial in 8o~+ and V~. For this we use the expansions of the propagator A(I+ Q) in eq. (3.25) and of the vertex ~¢/r(l + Q, q) in eq. (3.28) which allow one to factorize the loop integral fBzdl in eq. (3.46) from the integral over the external momenta
fdql ...dq~. D~"~(A, V) is thus expressed as the sum of integrals of the form:
I (~) = constfBzdql...dqn 8(ql + " " + qn)P(ql ..... qn)R(~'d~+(ql), l'V~(i)(q2)... ) ×F(am, a ) ,
(A.1)
where P and R are polynomial functions and F(am, a) is the result of the loop integration multiplied by an explicit power of a. F(am, a) contains all the information on the ultraviolet as well as infrared behaviours of 1 (n). We can therefore take the limit a---, 0 in the integral over the external momenta q. By inverse Fourier transform, the polynomial P(ql,-.-, qn) is traded for a polynomial of derivatives
666
A. Coste et al. / Chiral anomaly on the lattice
P(iOx, ..... i Ox. ) acting on the fields contained in R. The presence of 8(q 1 + "" " + qn) then ensures that we get a local expression of the form:
I(n)=const×F(a, am) X
f dy~a(8oa+(y),V~(y)),
(A.2)
where ~ is a polynomial in 8~+(y), V~(y) and their derivatives. The locality of (6Wo~ + ~WF) will follow if we can show that the expansion of D(n)(A, V) stops at a finite order. For this purpose, it is convenient to reformulate the power counting rules by rescaling the loop momentum by a, thus introducing the variable
l= al
(A.3)
to be integrated over [-~r, ~r]. In the loop integral, the ultraviolet singularity is traded for a singularity for/*~ 0, according to the leading behaviours a - 7' f2
s~C(l,q) - --, a ~q/'(l + Q, q) - l°a°. The degree of divergence of
D(n)(A,V), already obtained in eq. (3.22),
an " D(")(A,V)
(A.4)
d^[2
Jm~d l F
ir(1/a)d+l-n -"~alog(am) ~a
is recovered:
for n < d + 2 for n = d + 2 for n > d + 2.
(A.5)
With these variables, the expansion of A(l + Q) in eq. (3.25), provided one keeps at each order the leading term only, appears as an expansion in powers of aQ/k
A ( I + Q ) - ~ a A(0)+
~ l+ . . .
+AU)
+-...
(A.6)
On the other hand, the expansion of W(l+ Q, q) in eq. (3,28) is obviously an expansion in powers of a. Therefore, supposing that the propagators of D~n)(A, V) are expanded to an overall order i and the vertices to an overall order j, the corresponding contribution to D(")(A, V) behaves as follows:
[a n+i+j-d-1 [D(n)(A,V)] ~i'j) an+i+Jad+lf d a l ~ - ] a J + l l o g ( a m ) ^
~a j+l
forn+i
A. Costeet al. / Chiralanomaly on the lattice
D(n) (\V)
=
667
- - - ~
Fig. 8. An exampleof a diagram D(")(V) in the expansionof IV.
Since j >~ 0, it is clear that, in the limit correspond to
a ~
0,
the only non-zero contributions
i + j ~ < d + 1 - n.
(A.8)
Therefore, only a finite number of terms have to be retained in the expansion of D(')(A, V), which completes the proof of the locality of 3Woo and 8W F. Another consequence of eqs. (A.7) is the infrared finiteness of the anomaly. Indeed, divergences for m--* 0 can only come from divergent loop integrals I(am) which only occur when n + i >~ d + 2, i.e. for diagrams vanishing when a --* 0. The derivation of the locality of Woo and AW goes along the same lines and we will work it out in less details. Wo~ is the sum of diagrams D(")(V) with n propagators and n vertices W(I + Q, q) (see fig. 8). Their leading divergence is as follows:
a"
/(l/a) d-"
D(")(V) - -~'2fa daf l - l ~ g ( am )
for n < d for n = d for n > d,
(A.9)
so that only the first d diagrams contribute to W~. By expanding the propagators and the vertices like for the anomaly, each of these diagrams gives rise to series of local polynomials in V~, denoted by [D(')(V)] (i'j), among which only a finite number are retained in Wo~. Indeed, one has
[D(n)(v)] (i'j)
f a n+i+j-d a'+i+J f~ a 1 aa md l frYg7 - ~aqog(arn) ka j
for n + i < d
forn+i=d
(A.IO)
for n + i > d,
so that [D(n)(v)] (i'j) diverges only when i + j < d - n or (i = d - n, j = 0). We now turn to the proof of statement (i). Given an action S~(q, q, V~) satisfying conditions (a) and (b) in (i), we introduce the matrices $ ' and Ag defined
668
A. Coste et a L / Chiral anomaly on the lattice AS
a,
b)
AW= &S0 • &sO/xs" &S&&S _(n,n') A(~ U(~S.W)
~,
AS
""'
AS
Fig. 9. (a) Diagrammaticrepresentationof the expansionof AWgivenby eq. (A.13). (b) An exampleof a diagram D~"'"'I(AS,V)in the expansionof aW.
by:
S~( q, Ft, V~ ) = ?1" S' . q,
(A.11)
AS F = S ~ - S F = ~/. A ~ . q .
(A.12)
Conditions (a) and (b) imply respectively (a)' (b)'
ASF(q, ~, V~) -+ 0
in the naive continuum limit a -+ 0,
ASF( q, 7t, V~= O) = O.
With the notations of sect. 3, the perturbative expansion of AW follows from (see fig. 9)
aW-
14/'- W = trlog(n + AS. S - l ) =
m
(1) n-1
£
--tr((zl~-~-l)n}.
n=l
(A.13)
n
By further expanding S - 1 according to eq. (3.6), zlW can be expressed as a double sum of diagrams D("' "')(AS, V), like in fig. 9b, with (n+n') propagators, n' vertices ~/r(l + Q, q) and n vertices AS(l + Q, q) defined by
AS(k, q) = E eikYASy, xe-i(k-q)x"
(A.14)
x,y
This vertex can be treated in exactly the same way as Y¢/'(l+ Q, q) in eq. (3.28), namely it can be expanded in powers of a, at fixed l = al. Just like for g / ( l + Q, q) in eqs. (3.29), each term in this expansion is itself a finite sum of subvertices, denoted AS (e' r)(/, Q, q), with the following behaviour for a --+ 0 and l--+ 0: zIS(P")(I, Q, q) - apPQs(3-~v_~)(q),
(A.15)
A. Coste et al. / Chiralanomaly on the lattice
669
where r, s, t and u are non-negative integers, and (OtV~_)(q) is the Fourier transform of OtV~_(x). These exponents are constrained as follows d- 1 =p-s+
( d - (t + u ) ) ,
p+r>~l, u>~l,
(A.16)
where the first constraint comes from the dimensionality of AS, the second one from (a)' which states that AS(I + Q, q) ~ 0 when a ~ 0 at fixed momenta l + Q and q, and the third one from (b)'. One easily derives from these constraints the inequalities p + r >/1,
p >/0.
(A.17)
In a diagram D(n'n')(AN,V), there are n subvertices AS (p,r) and we call P and R the overall sum of the exponents p and r. They obviously satisfy: P+R>~n,
P>~0,
R>~0.
(A.18)
If we now expand the (n + n') propagators and the n' vertices g/'(l + Q, q) to an overall order i and j, the divergence of the corresponding localpolynomial contribution to D(", ")(AS, V) is given by
an+n'+P+i+j ~R ad famddl fn+n'+i
[ D(n'n')( A S , V)] (P'R)'(i'j)
f ( l / a ) d-(,+,'+P+i+j) -
~ a P+R
+Jlog(am)
kaP+R+J
forn+n'+i
Recalling that n >/1 and n' >t 0, the inequalities (A.18) clearly imply that the only non-vanishing contributions to AW, when a ~ 0, correspond to n + n ' + P + i+j<~d. They are therefore in finite number, hence the locality of AW.
(A.20)
A. Coste et al. / Chiral anomaly on the lattice
670
Appendix B Here we wish to give an idea of rearrangements occurring between diagrams within the framework of our "combinatorial diagrammatics". In d = 2, the four diagrams contributing to 8 W v are represented in fig. 5:
8 W r ( d = 2) = D}2) + D2(2) + D3(2) + D (3) .
(B.1)
The identity (3.37) can be represented graphically as ~l~ - -
-
=
~
-
g
.
stands for iS"--d+(q) and/7 stands for a B ( l ) / D ( l ) where Eq. (B.2) leads immediately to
as defined in (3.34).
D~2) = 0.
(B.3)
In order to deal with the other diagrams let us split the quantity 3LI/OQ, appearing in the expanded propagator into its V~ and B~ parts (see eq. (3.25)). Graphically we write: ~(
=
)A(
+
X
(B.4)
B
This induces a splitting
W ) - - z)
+ DJ ).
(B.5)
Then let us consider the transformation law of the lowest order vertex term. From eqs. (2.7) and (3.29) we obtain:
8( ycp O( q , Q ) ) = i ~ ( _ )( q ) yCr ° ( q , Q) _ ycr O( q , Q ) i ~ + ( q ) + q~Afl~'~(+)( q)cos(al~),
(B.6)
where A~ = i'/~. Or in a graphical way
+
7- -
(B.7)
This last graphical identity, combined with (B.2) leads to
I+]
"-'3n(2)B+ D~3) = 18 (0)
which is the last counterterm of formula (4.5).
(0) ,
(B.8)
A. Coste et aL / Chiral anomaly on the lattice
671
Then D2(2) and .-'3an(z)add up to Of ='+- -,An(=)=x f d2x tr[ O~&O(+ly.yy+ ] ,
(B.9)
and if we plug into this
Vl,'y.= 3~. - ie..y5
(B.IO)
we recognize the anomaly (4.2) and (4.3) and the first counterterm of (4.5):
Etr(V~+V~+).
aR = cf
(B.11)
/L
Appendix C
In this appendix we wish to give an idea of the method used in order to calculate or rearrange the trigonometric integrals involved in 3W F. We take the example oi 1 XKs= ~ f~,dOldO 2 ( ( 1 - Cl) 2 "{- ( 1 - c2) 2) ~c2[ t s ~ 2 _ C13)
We first notice that
[ SlC2 ] 01~,-~-1
qc2 32
2s~1c2 33
so that
f.
1
){
.
XKs = - - lim d02 {9(1021 - e ) £ , d01 ((1 - c1) 2 q-- (1 - ¢2) 2 2~r ~ o -~
1 ~r C1 C 2 1 -dT~f],deldO2[(1-cl)2+(1-c2)2]~-+~--~f"_,dO1dO2 ~,
-
~
(1 -- Cl)2C2
= ~¥f"_,dOldO 2
1
32
1 clc2 o1('1c2~ 2(1 - q) s21c2
We can then remark that
I slc2~
q c 2 3 - s2c2
=
f=-r
=
f --qr d 0 1 d 0 2
dol do2 O(101- e) 01( --~)$1e2
@(101 - e )
2 q ( 1 - q ) c 2 - s2c2 32 (1 -- Cl)2C2
= _
dOld020(lO I - ~ )
32
32 lJ
2 32
32
672
A. Coste et aL / Chiralanomaly on the lattice
with 10l = ~02 + 022 • Therefore,
e--*0\ 2~" -~rd01 d020(]0[ --e) 0 I/~s Tl c)2)~.
?,Ks= lim ( _ I f
This integral is now easy to calculate and one finds ~KS = 1. In the Wilson case, one ends up on the same expression with the denominatol D Ks = 28 replaced by D w --- (E~s~) + r 2& The term of order r 2 is irrelevant in the vicinity of 0 = 0, so that one also has Aw = 1. Similarly in dimension four, one finds the following integrals:
~kl= elilm
- ~ f d OO(1OI-~)~ D2
A 2 = lim
-
e---,0
2
=1,
4 __e)~l(0 Sl(X~S2)C2C3CaD 3 )) =1, fdoo(101
with D = D Ks or D w as the case may be. References
[1] J. Steinberger, Phys. Rev. 76 (1949) 1180;
[2] [3] [4]
[5]
[6]
[7] [8] [9] [10] [11] [12]
[13] [14]
J. Schwinger, Phys. Rev. 82 (1951) 664; L. Rosenberg, Phys. Rev. 129 (1963) 2786; S. Adler, Phys. Rev. 177 (1969) 2426; J. Bell and R. Jackiw, Nuovo Cim. 60A (1969) 47 W.A. Bardeen, Phys. Rev. 184 (1969) 1848 L. Karsten and J. Smit, Nucl. Phys. B144 (1978) 536; Phys. Lett. 85B (1979); P. Mitra and P. Weisz, Phys. Lett. 126B (1983) 355 L. Karsten and J. Smit, Nucl. Phys. B183 (1981) 103; H.S. Sharatchandra, H.J. Thun and P. Weisz, Nucl. Phys. B192 (1981) 205; M. Gockeler, Nucl. Phys. B224 (1984) 508; D. Verstegen, Nucl. Phys. B243 (1984) 65 and references therein L. Karsten, Phys. Lett. 104B (1981) 315 and the first of ref. [4]; H. Nielsen and H.M. Ninomiya, Nucl. Phys. B185 (1981) 20; Phys. Lett. 105B (1981) 219; Nucl Phys. B193 (1981) 173 P. Becher, Phys. Lett. B104 (1981) 221; F. Gliozzi, Nucl. Phys. B206 (1982) 419; H. Kluber$-Stern, A. Morel, O. Napoly and B. Petersson, Nucl. Phys. B220 [FS8] (1983) 447 J. Wess and B. Zumino, Phys. Lett. 37B (1971) 95 C. Becchi, A. Rouet and R. Stora, Renormalizable theories with symmetry breaking, in Field theory quantization and statistical physics, ed. E. Tirapegui (Riedel, 1981) A. Coste, C. Korthas Altes and O. Napoly, Phys. Lett. 179B (1986) 125 L. Alvarez-Gaum~ and E. Witten, Nucl. Phys. B234 (1983) 269 B. Zumino, in Proc. Les Houches 1983 Summer School, ed. R. Stora (North-Holland) J.M. Blairon, R. Brout, F. Englert and J. Greensite, Nucl. Phys. B180 [FS2] (1981) 439; H. Kluberg-Stern, A. Morel, O. Napoly and B. Petersson, Nucl. Phys. B190 [FS3] (1982) 504; N. Kawamoto and J. Smit, Nucl. Phys. B192 (1981) 100 E. Marinari, G. Parisi and C. Rebbi, Phys. Rev. Lett. 47 (1981) 1795; H. Hamber and G. Parisi, Phys. Rev. Lett. 47 (1981) 1792 M. Bochicchio, L. Maiani, G. Martinelli, G. Rossi and M. Testa, Nucl. Phys. B262 (1985) 331