Evaluating fermion determinants through the chiral anomaly

ANNALS

OF PHYSICS

158, 67-77

(1984)

Evaluating through

Ferrnion Determinants the Chiral Anomaly*

RAFAEL

I. NEPOMECHIE

Department of Physics, Brandeis Universit),, Waltham, Massachusetrs 02254 Received

January

20. 1984

A class of fermion operators whose determinants can be calculated exactly has recently been noted. We observe that typically such operators can be chirally rotated into the free Dirac operator; hence. their determinants are given by the chiral anomaly. Four-dimensional fermion determinants of this type are computed: the appearance of the Wess-Zumino anomaly term 1s noted. T : 1964 Academic Press, Inc.

1. INTRODUCTION In a recent paper, Alvarez 111demonstrates that there is a large class of fermion operators whose determinants can be computed exactly using only knowledge of their short-distance behavior. In particular, in two dimensions, a modified Dirac operator in an arbitrary non-abelian gauge field background can be seen to belong to this class; and hence, its determinant is calculable. This fact was independently realized by Polyakov and Wiegman [2]. Here we provide a simple argument explaining why such a fermion operator has a calculable determinant: Roughly speaking, on performing a local chiral rotation, this operator becomesthe free Dirac operator: since the path integral fermion measure is not invariant under the chiral rotation 13, 41, the operator’s determinant is determined by the chiral anomaly. The above argument, which is valid in arbitrary (even) dimensions, is made precise in Section 2. As an explicit four-dimensional example, we evaluate in Section 3 the determinant of the Dirac operator with a Y+ A coupling to a background nonabelian chiral field. Our expression for the value of this determinant includes the fourdimensional Wess-Zumino anomaly term 15, 61, as to be expected from the earlier work in two dimensions [l, 21; the remaining terms are of the type generated as oneloop radiative corrections of the (non-renormalizable) quantized chiral model (see, e.g., 171).’ Various authors 191have also recently studied fermions in a background chiral field, emphasizing the case of a chiral model soliton 181as the background. * Research supported in part by DOE Contract DE-AC03-76ER03232-A01 I Such terms also serve to stabilize classical soliton solutions of the chiral

I. model

18 I.

67 0003-4916/84 All

$7.50

Copyright ‘T 1984 by Academic Press. Inc. rights of reproduction in any form reserved.

68

RAFAELI.NEPOMECHIE

Finally, in Section 4 we briefly discuss the fermion determinant for a gauged chiral model. Appendix A contains a concise review of non-linear realizations and the chiral model, with emphasis on the coupling to fermions. Appendix B lists our convections.

2. FERMION DETERMINANTS

AND THE CHIRAL

ANOMALY

In this section, we describe how the determinants of certain fermion operators are determined through the chiral anomaly. Following [ 11, we consider the model with action Z[ Yt, Y, (, t] = Tr Ytiy,[a,

+ U; “‘(a8 Ui”)]

Y

E Tr Y’D, Y, where Y = Y:(x)

(i = l,..., N) is a Dirac spinor field; also,

with <. J z C,n,, A, are the hermitian generators of SU(N), introduced for later convenience. Moreover, D, E iU:12y, 3, CJ:”

is a hermitian

and t is a parameter UC)

differential operator. Writing tp2(dtr

u:“) = vi + A;

(2)

with VL E iV:,l,, AL E Aha& ys, one sees that this model describes fermions with a V + A coupling to a chiral model. This is explained in more detail in Appendix A. Our approach is motivated by the following observation: by performing the finite chiral transformation Y -+ Y” = Ui”Y, Y’ + Ytlt = Y’Ui’*, the action (1) becomes the free-field action: I[ Y+, Y, <, t] = Z[ y/t,+, Y”, 01.

In order to calculate det D,, we proceed by infinitesimals:

Let

Y’=u~~2Y,(1+6tr.~ys)Y, (3)

yy’+= y+ug2. Then z[Y+, Y, C,t] =z[Y/+,

Y’,[, t-c%].

(4)

EVALUATING

FERMION

69

DETERMINANTS

Thus, *

Here we have used the fact 13, 41 that under the infinitesimal fermion measure transforms as g

y/It&

yy'

=g

ytg

chiral rotation

ye-*TrStS..~Y5,

(3). the

(6)

where the trace, when suitably evaluated, yields the chiral anomaly. Thus,

This agrees with the result found in [I]; our derivation has the advantage of directly exhibiting the relationship to the chiral anomaly. In fact, (8) is nothing but a convenient form of the anomalous chiral Ward identity. To evaluate the r.h.s. of (8), we proceed in standard fashion: the regulated trace is Tr{+,e-““j.

(9)

For small E, the diagonal part of the heat kernel for Df has the asymptotic (xl e-f

1 lx> = c4nEjd,2 [a;(x)

+ az:(x, + &*a:(x) + . ..j.

where d is the dimension of spacetime. Hence, the renormalized 1 (47p Combining minant

this with

i

expansion

(10)

trace is

ddx tr i . lys a&,(x).

(8), we obtain a simple expression

ddx tr c + 1y, a&(x)

for the renormalized

deter-

+ constant.

Although we have treated the special case (l), it is clear that there is a class of models that can be approached in this way. Indeed, it is not essential that the fermion ’ For

simplicity

we ignore

possible

zero modes.

70

RAFAELLNEPOMECHIE

operator in question be trivial up to a chiral transformation; we in fact make use only of the less restrictive condition (4), which is equivalent to Alvarez’ requirement $t=

MU

(13)

where f is some t-independent function. An example of this more general type is the model of fermions with a gauged V + A coupling to a chiral field, which we study in Section 4. 3. AN EXAMPLE

IN FOUR DIMENSIONS:

THE CHIRAL

We now explicitly compute the fermion determinant four dimensions, which by (12) amounts to evaluating WC, t] = - &

MODEL

for the chiral model (1) in

ji dr j d4x tr <. 3L~~a;(x).

Clearly, first we must determine a;, which arose in the asymptotic expansion of the heat kernel for 03. Recall (see, e.g., [ 101) that the general operator A--G=+E,

Qpa,,+G,,

(15)

where aP (but not E) is a differential operator, has the corresponding a2 coefficient a2 = &[a,,,

g,,]’ + fE2 - bk3’E.

(16)

Our operator 05 can be cast in the form (15), with G, = VP + ia,,A:,

(17)

E=-D;A:-2A;A;.

Substituting into (16) yields the desired, although lengthy, expression for ai. Remembering that A; is proportional to ys, and noting the identities (A. 12) and (B.2)-(BS), we find that only a small number of terms survive the trace (14), so that3 WC tl = - &

j’dz Tr r. A+

E,,,,A,AJ,A,

0

-&jIdrTr [i-h5(f D*(D.A)+4A,(D*A)A,-;{A=,(D*A)l + + {D,A,,A,A,l

(18)

)I .

3 Our evaluation of tr ySc . kz, is in fact a special case of a more Balachandran et al. 141; our result is consistent with theirs.

general

calculation

performed

by

EVALUATING

FERMION

71

DETERMINANTS

(For clarity, we have ceased to explicitly indicate the r dependence of A,, D,, etc.) The first term in (18) is the Wess-Zumino term [S, 61. To see this, we use the identity (A.l) to rewrite it as

(19) On the other hand, recall Witten’s

expression Rr

5B

161 for the Wess-Zumino

term

w,

(20)

where w is a five-form which can locally (but not globally) be expressed as w = dll. and B is the five ball, so that aB = S4. Performing a small variation, kO=j

iko=j B

6(dA)=j B

6A.

d(&)=l B

(21)

FB

Taking 161 1 co=4

1 . -tr 240~’

~~~ijkl,,,LiLjL~LfL~

d5J1,

LiS UP& w

(22)

then one possible choice of A is

Ad-

4

l

.tr y,(log U) LiLjL,L, 240~’

dy’A d#A dykA dy’,

(23)

so that

&?= i

tr ys 17’ 6U E,,.,~L, L,.L, L, d”y. PB

(24)

Hence, (25) in agreement with (19). 4 The quantization of the coefficient of J2 (N > 3) has been discussedat length by a number of authors. Let us now turn to the remaining terms in (18). The r integration can be readily * In fact, that (19) is equivalent to (22) dinate. I thank D. Gepner for this remark.

can be seen by inspection,

by identifying

r as the fifth

coor-

RAFAELLNEPOMECHIE

72

performed upon making lJi/‘a, U; “‘), then dA

the following

observation:

since A, = f (U; 1’28p Ui” -

where f E

A=D,,f, dr

(26)

Similarly, (27)

so that -$D.A)=Dif+

Performing

the r integration,

[[&f],A,].

(28)

and discarding a total divergence, we obtain + (D . A)’ - (A,A,)’

. I

Using again the identities (A. 1 l), this becomes -&Tr

+ [a,(U-‘l?,U)]’

-~(U~1B,UU~1B,U)2 !

I

+(c?,L,)’

-a&,~,)

-$

]L,,L,]’

I

.

(30)

These (regulator-dependent) terms are of the type generated as one-loop radiative corrections of the quantized chiral model [7]. The terms (30), plus the Wess-Zumino term (25), comprise the complete expression for the four-dimensional fermion determinant.

4. GAUGED

CHIRAL

MODEL

In this section we briefly consider another four-dimensional determinant for the (V + A) gauged chiral model I[ Yt, Y, a,, [, t] = Tr !@@,,[a,, + U1”(G& U-“‘)I -Tr

example: the fermion Y

YYtD,Y,

(314

where now D, E iU;1’2yw gw U,e”2,

9q=a,+a,,

a,=p,+a,.

@lb)

EVALUATINGFERMION

73

DETERMINANTS

(See Appendix A for more details.) Naturally, one expects that det D, involves the “gauged” Wess-Zumino term [5, 61. That this is indeed the case can be seendirectly: Proceeding as in Section 2, we observe that

=e

-W[a,,5.f-Sfl+2SITr3.~y,

(32)

Let G,[P,] = -2 Tr Lays denote the anomaly in the presenceof the background gauge field /I,. Then (32) implies that (up to a constant) (33) since the background gauge field in question is (34) The r.h.s. of (33) coincides with an expression 15) for the gauged Wess-Zumino term.

APPENDIX

A:

NON-LINEAR

REALIZATIONS

AND THE CHIRAL

MODEL

The general theory of non-linear realizations has been described in the classic papers of Callan et al. [I 11. In our work we make use of a specific non-linear realization; as the adaptation of the general formalism to this special case is not entirely straightforward, we now present the salient points. Our approach is similar to that of 1121. 1. Transformation Laws Since we work in Euclidean space and choose to have hermitian fermion operators, we require the non-linear realization of a non-compact [ 1, 41 version of SU(N),. x 5 To this end, introduce the matrix ~wn4Iwm. qx> E e23w’a~Y,, where { . A = Q,,

(A.11

and i, are the hermitian matrix generators of SU(N) IL, 41 = ifabcfLr

f&

real.

’ For the purpose of computing an operator’s determinant, it suffices that the operator be elliptic (rather than hermitian); hence, one may alternatively give up hermiticity, and work in Euclidean space with compact chiral symmetries. I thank L. Alvarez-GaumC for bringing this to my attention.

74

RAFAEL

Under a global (x-independent)

I. NEPOMECHIE

isospin transformation, vE &U’a.,

U’(x) = VU(x) v- l, so that for u infinitesimal,

(A.2a)

c has the linear transformation Xc(x) = -0, L(xMlbc.

A global chiral (= axial) transformation

is given by (A.2b)

which on c is non-linear 6&(x) = --a,. Establishing a non-linear realization for matter fields (which we take to be fermions) is more subtle. First, introduce a Dirac spinor field v/(x) which transforms linearly under (A.2a,b) w’(x) = Vv(x)

(isospin)

(A.3a)

v’(x) = A w(x)

(chiral).

(A.3b)

Then, a field Y(x) which has a non-linear transformation Y(x) = u(x)“* Indeed, under the isospin transformation transforms linearly

law is

v/(x).

64.4)

(A.2a), U’(x)“*

= VU(x)“’

Y’(x) = W(x); however, under the chiral transformation manner, and Y’(x)

V-l, so that Y (ASa)

(A.2b), U”*

transforms in a complicated

= A(x) Y(x),

(A.5b)

where /i is an element of SU(N),, but in general depends on r(x), and hence, on x. Clearly, the factor U1’* is the essential ingredient in the construction of the non-linear realization for the matter fields. One can show [ 121 that /i satisfies U I I/* = /i U’/*A - 1 = A - 1U’/*A - 1 2. Covariant

Derivatives

and Invariant

(A.61

Actions

To construct covariant derivatives, first introduce the quantities X,s

U1/2a,,U-1/2z

Y, E U-l/*

V,,-A,,

3, Ul/* E V, + A,,

64.7)

EVALUATING

FERMION

75

DETERMINANTS

vu= iv,,& = t(Y, +X,>= fi(a,C,)CbfabcAc + ... A,-A,,~,y,=4(Y,-X,)=(a,r.n)y,+... These transform

under (A.2a,b)

.

according to

v; =nv,/t-’

+m,/i-

A;=AA,L’

(isospin)

(A.8a)

(chiral)

(A.8b)

where (A.6) has been used in obtaining the chiral transformation the derivative expressions

laws. It follows

that

(A.9)

with A,,, V,, as defined in (A.7), all transform covariantly under both isospin and chiral (global) transformations. A simple example of an invariant action is

I d4x tr{iY+y,,

D, Y + A,A,

1.

(A.10)

Two useful identities are u-18 u

a,#-‘a,

U-2U-'12ApU'i2, -

U) = 2U-“‘(D

(A.1 la) ’ A) U”‘.

From the first of these, one immediately sees that the pion term Tr A,A, can be written in the more familiar form -i Tr a, U 3, Up ‘. Finally, we state the integrability condition of (A.7), D,A,

- D,A,

VOL.+ IA,J,.l=O.

= 0,

(A.1 lb) in (A. 10)

(A. 12a) (A.12b)

3. Gauged Chiral Model Thus far we have discussed only global isospin and chiral symmetries. Consider again the transformations (A.2a,b), but where now V= V(x), A = A(x) are arbitrary functions of x.

595jl58/1-6

76

RAFAEL

I. NEPOMECHIE

TO form covariant derivatives, first introduce the independent gauge fields U,(X), a,(x) which transform according to

v:=vv,v-~+~Ya,V-~ a;=

(local isospin)

(A.13a)

(local chiral)

(A.13b)

vu,v-1

v; =/iv&

+$,,,P

a’P =A2 BA-’ where g is an arbitrary coupling constant, and A satisfies (A.6) as before. Derivative expressions which transform covariantly under both isospin and chiral local transformations are

D,C= a,,, etc.

D, Y = (a, + gv,) Y,

(A. 14)

Observe that although the chiral transformation is specified by A(x), the gauge fields transform with A(x). One can relate rip(x), u,(x) to new fields p,(x), a,(x) which do transform directly with .4(x); namely, define (A.15)

v, + a, = iF2

where p, , a, transform as

p;= vp,v-‘+Layv-l g a’=Va Ir

(local isospin)

(A. 16a)

(local chiral).

(A.16b)

a v-l

p; =k+--1 a:=Ap,L4-‘+$4a,A-l One can then verify that these transformation passing, we note that

laws are consistent with (A-13). In

~~,+g(v,+u~)}~=(~,+U”‘[g@,+a,)+~,]~~~”)~

= (lJ”2 g; u-1’2) y where g,, = 3, + g(p, + a,) is an operator.

(A.17)

EVALUATING

APPENDIX

FERMION

B:

17

DETERMINANTS

CONVENTIONS

AND

IDENTITIES

Here we list our conventions as well as some useful identities:

(Y,- Y”l = qLL,.

t Ys= Yc3

Y:=Y,,

u p,s = 4 lu, 1v,l,

yf=

1

urrnuuo = 2ia,, + 36,,

[UP,.’u,& = 2i(-d,,u,,, - L~,~ f 4$% + LQ3) tr u,,. = 0 = tr y5uuur,

tr Y~u~~,(T,~ = -4~~~~~~~

tr urruud? = 4(6,, a,, - au&n

).

(B.1) 03.2) (B.3) (B.4) (B.5)

We work in Euclidean space. Following 141, Tr denotes trace over Dirac and internal indices and also over x; tr denotes trace over Dirac and/or internal indices only.

ACKNOWLEDGMENTS I thank

H. Schnitzer

for discussions

on chiral

model

solitons.

as well as for his interest

in this work.

Nofe added in prooJ A number of additional relevant references have come to my attention: J. Schwinger, Phys. Rev. 128 (1962), 2425; D. I. D’yakonov and M. I. l?ides. JETP Left. 38 (1983). 433; L. Bonora and P. Cotta-Ramusino, Comm. Mafh. Phys. 87 (1983), 589; A. Andrianov and L. Bonora, Nuclear Phys. B 233 (1984), 232, 247; R. Jackiw, Les Houches lectures (1983); B. Zumino. Y.-S. Wu, and A. Zee, Nuclear Phys. B 239 (1984), 477; Alvarez, Hwang. and Ingermanson (unpublished).

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

0. ALVAREZ, Nuclear Phys. B 238 (1984), 61. A. M. POLYAKOV AND P. B. WIEGMAN, Phys. Letf. B 131 (1983), 121. K. FUJIKAWA, Phvs. Rev. Left. 42 (1979), 1195. A. P. BALACHANDRAN. G. MARMO. V. P. NAIR, AND C. G. TRAHERN. Phys. Rev. D 25 (1982). 2713. J. WESS AND B. ZUMINO, Phys. Letf. B 37 (1971), 95: B. ZUMINO. Les Houches lectures, 1983. E. WITTEN. Nuclear Phys. B 223 (1983), 422, 433; Comm. Math. Phys. 92 (1984), 455. A. A. SLAVNOV. N&ear Phys. B 31 (1971), 301; I. GERSTEIN, R. JACKIW, B. W. LEE. AND S. WEINBERG, Phys. Rev. D 3 (1971), 2486. T. H. R. SKYRME, Proc. Roy. Sot. London Ser. A 260 (1961), 127. J. GOLDSTONE AND F. WILCZEK, Phys. Rev. Lett. 47 (1981), 986; A. P. BALACHANDRAN. V. P. NAIR AND C. G. TRAHERN, Phys. Rev. D 27 (1983), 1153; J. M. GIPSON, Nuclear Phys. B 231 (1984), 365.

10. P. GILKEY. J. Differential Geom. 10 (1975). 601. 11. C. G. CALLAN, S. COLEMAN, J. WESS, AND B. ZUMINO, 12. J. WESS AND B. ZUMINO, Phys. Rev. 163 (1967), 1727; (1967). 1752.

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164