Invariance of fermion determinant under large gauge transformations

Volume 189, number 4

PHYSICS LETTERS B

14 May 87

INVARIANCE OF FERMION DETERMINANT UNDER LARGE GAUGE TRANSFORMATIONS H. BANERJEE, G. B H A T T A C H A R Y A a n d J.S. B H A T T A C H A R Y Y A Saha Institute o f Nuclear Physics, 92, Acharya Prafulla Chandra Road, Calcutta 700 009, India

Received 29 October 1986; revised manuscript received 9 January 1987

We argue that in the perturbative framework the natural symmetry of the fermionic determinant is the perturbative gauge transformation (p.g.t.) which differs from the usual gauge transformation of the effective action through the absence of terms independent of the coupling constant. Calculated in a non-perturbative framework appropriate for large gauge function, the sum of these latter terms vanish. In three dimensions the invariance of the full fermion determinant under large gauge transformations is thus ensured due to the invariance under p.g.t, of the Chern-Simons term arising in some perturbative regularisations.

In three s p a c e - t i m e dimensional gauge theories the fermion effective action acquires in some perturbative regulansation schemes, e.g., Pauli-Villars, a local t e r m [1,2], the C h e r n - S i m o n s (CS) term, which changes b y i n n u n d e r large gauge transformations with h o m o t o p y index n. The gauge theory, therefore, becomes inconsistent a n d the euclidean p a t h integral vanishes [2]. It is true that in o d d s p a c e - t i m e d i m e n s i o n s the i n t r o d u c t i o n o f a f e r m i o n mass t e r m through the P a u l i - V i l l a r s regulator spoils invariance u n d e r parity [2,3], as, indeed, is evident from the CS t e r m ,1. W h a t , however, concerns us here is the gauge n o n - i n v a n a n c e o f the CS t e r m which appears in a manifestly gauge-invariant regularisation scheme a n d whether this is to be necessarily interpreted as the b r e a k d o w n in the full fermionic d e t e r m i n a n t o f invariance u n d e r large gauge transformations. It is our a i m here to give a consistent i n t e r p r e t a t i o n o f invariance u n d e r gauge transformations, b o t h large a n d small, within the p e r t u r b a t i v e f r a m e w o r k a n d ~ The answer to the interesting question [ 1] whether there exist regularisation schemes which respect both parity and gauge invariance seems to be in the affirmative. Thus in a non-perturbative approach deWitt [4] has shown with (-function regularisation that the fermionic determinant in odd space-time dimensions does not acquire any local term in the phase nor does it change sign under large gauge transformations. In perturbative approach dimensional regularisation does the job [ 5]. We shall, in what follows, consider the fundamental fermion to be massive and use the notation AuA~T a where T~ are the generators of the symmetry group. 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V, ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )

argue that the appearance o f the CS t e r m does not spoil the large gauge invariance o f the full fermionic determinant. The genesis o f gauge symmetry, we recall, is the equivalence o f the t r a n s f o r m a t i o n A u-~A~v = U - 1Au U - (i/e) U -10u U

(1)

to a similarity t r a n s f o r m a t i o n o f the D i r a c o p e r a t o r D(A)-(P-m+eA)

(2)

,

a n d the formal invariance o f the f e r m i o n determin a n t u n d e r a similarity t r a n s f o r m a t i o n detD(AU)=det[U-iD(A)U]=detD(A)

.

(3)

W h a t could possibly have spoiled this formal invariance o f the infinite d i m e n s i o n a l d e t e r m i n a n t is the ultraviolet divergence. It is, therefore quite puzzling that a gauge i n v a r a n t regularisation scheme like Pauli-Villars which respects the invariance o f the fermion d e t e r m i n a n t when the gauge t r a n s f o r m a t i o n is " s m a l l " should fail to do so for large gauge transformations. M o r e precisely, in the P a u l i - V i l l a r s scheme the effective action W = Tr I n D ( A ) = Wcs + gauge i n v a r i a n t terms

(4)

contains the C h e r n - S i m o n s t e r m Wcs [ 3] e2 [" 3

Wcs=8--~Jd xe

uI.'2

2

tr{AuF~x-~eAuA~Aa} ,

(5) 431

Volume 189, number 4

PHYSICS LETTERS B

which is invariant under small gauge transformations but changes by in if the homotopy index of the "large" function U is unity. In this paper we argue that the full fermion determinant is, in fact, invariant under the transformation (1) even when the gauge function Uis large. The observation which plays a key role in our arguments is that though for the full determinant the gauge transformation (1) is a symmetry, however, for the piece det Dp, which is amenable to perturbative treatment and is defined through the decomposition det D = det Dnp det/9p ,

14 May 87

1 Wp[AV]=Trln 1-~p_ m Xe(UT1AU-~U-lOU)I =Trln

( lq

+Trln(1-

1

p_m_iU_ldueU-aA i p_mU-ld

U)

(6)

=-WUp[A]+Wnp[U] . Dnp = / ~ - m ,

Dp---(1

+l---~eA)

(7)

/~-m

the natural symmetry is instead the perturbative gauge transformation (p.g.t.)

Au.-~U-IA~zU,

(8)

1)~lk-iU-' OU.

(9)

Invariance under p.g.t, is realised term by term in the perturbative expansion of the piece Wp contributed by det Bp to the effective action

( Wp[A] = Tr lnkl

+)-_--_--~

=Tr~"(-1)'-'(1--LeA) \l)--m =Tr~('l

2

n--1

1 " (U-I-~-~eAU)

(11)

Thus the perturbative gauge transformation of the effective action Wp[A] differs from ordinary gauge transformation only through terms independent of the coupling e which appears in the latter. The two transformations are clearly equivalent if and only if

Wn,[ U] =0. In order to give flesh and substance to our formal arguments we need to regularise the ultraviolet divergences. It should, however, be clear that in a scheme like Pauli-Villars, where the regularization is implemented term by term in the perturbation series for Wp [A ], invariance under p.g.t, will, in general, be realised. Invariance under the orginary gauge transformation, however, depends on the additional condition that Wnp[ U] vanishes. The vanishing of Wnp [ U] may be achieved perturbatively, i.e. every term in a perturbative expansion of Wnp[ U] vanishes. This possibility is realised when U is abelian U = e ~, with A an ordinary function ([A, [p-m, A]] =0). WnP[ e~Z] = T r Z ( - 1n) n-' (~---~mdA)"

n

1

=Tr~ !- l )"-l (lk-m-iU-l su =-W~[A] .

l

AU) (10)

Here "Tr" stands for the trace in the configuration space and the symmetry and spinor indices. The difference between p.g.t, and ordinary gauge transformation is readily recognised from the identity 432

+iAda(~im,A)n-21~1-m] =0

(12)

where we have used the commutation relation dA = i [ p - m , A]. Perturbative arguments can be used to

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PHYSICSLETTERSB

demonstrate the vanishing of Wnp[ U] alSO when U is nonabelian, provided that it is "small", i.e. U can be generated by successive application of infinitesimal transformations

U~=l+ie(x),

(13)

where e(x) is an infinitesimal Lie algebra valued function. In this case e (x) provides the small parameter and keeping, as usual, terms of only first order in e(x) in the perturbative expansion of Wnp[ U,] we obtain Wnp[ U,] = --i Tr p - ~ m [ p - m, e] = 0 .

(14)

A problem may arise with the series expansion of W~p[ U] if the gauge function U is large. In this case, alas, we do not have any infinitesimal function at our disposal. If now, the series expansion does not vanish in a particular regularisation scheme, the nonvanishing result, which is necessarily a local expression, may be wrongly diagnosed as the sign of the nonvanishing of the full fermion determinant under ordinary gauge transformation. This is precisely what happens with Pauli-Villars regularisation in three dimensions. Under an ordinary gauge transformation (1) the change in the local Chern-Simons term (5) arising in Pauli-Villars regularisation of Wp [A ] is given by

14 May 87

formal operations require the sum rather than the individual terms of the perturbative series for Wnp [ U] to vanish. W~p[ U] = Tr In ( 1 -

=Trln

i U-10U)

U - l ( p - m ) U - T r l n (p-m)

=0.

(16)

In order to answer unambiguously the question of gauge invariance it is, therefore, necessary to evaluate W,p[ U] non-perturbatively. We recognise Wnp[ U] =ln J[ U ] ,

(17)

where J[ U] is the jacobian for the transformation

T~U~,

~--,~U -t

(18)

of the fermionic measure in the path integral

I:fdT/d(/exp(ff/(iO-m)qJd3x).

(19)

Formally, the jacobian is trivial, because J ( U ) = det U det U =exp[Tr ~-Tr

1

~],

Wcs[A v] - Wcs[A] U = e i~ , - 8leaPt Tr[e0~(0p

U U'IA:,)

2~-½(U-I Oo~UU -1 013U

U-10y U)] .

(15)

For large U it is the second term on the right-hand side of (15) that has a non-vanishing contribution. But this term does not depend on the coupling e and hence in the decomposition (11 ) it can arise only from the series expansion of W.p[U]. The nonvanishing contribution to Who[ U] from (15 ) still leaves open the question whether the entire series for Wnp[ U] vanishes. It may only imply the non-commutativity of the limiting procedure of removing the Pauli-Villars regularisation by taking the regulator mass M to infinity and summing the infinite series for W.p[ U]. We emphasise that gauge symmetry and

(20)

with ~b lie-algebra valued. Note the difference from the jacobian for Chiral gauge transformation in even dimensional gauge theories. There the contributions from d~/and d~ transformations add up in the exponent instead of cancelling as in (20). In order to make the formal operations in (20) meaningful we need to regularise Tr[~]. We do this with (-function regularisation and show that not only J[ U] is trivial but [ Tr ~ ] reg = 0.

In order to regularise the formally divergent expression T r ( ~ ) , we use complete basis functions ~m(X) that are eigenfunctions [3] of the operator (p+eA)2-m 2, i.e., [ (/~+ cA) z - m:]~m =22m~m.

(21)

Consequently 433

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PHYSICSLETTERSB

Trfl)=fd3xTr(fl)(X)~m'm(X)'*m(X)) , The regularised Tr (qb) reg is defined [ 4 ] by the following expression: [ Tr q~ ] r e g = I d 3 x tr [ q) (X) ~(0; X, X) ],

(22)

,/

where (23) m

Am

"

It is well known [ 4] that in odd dimensions ~(0; x, x) = 0 .

(24)

Thus [ Tr • ] reg 0 in three dimensions. To summarise, we suggest that invariance of the full fermion determinant under gauge transformation (1) is achieved in a perturbative framework as follows =

14 May 87

satisfy. We conclude with a few remarks. The local Chern-Simons term (5) gives a mass, known as the topological mass [ 3 ] to the gauge boson. Our arguments suggest that this mass term is consistent with gauge transformations both large and small and is, in this respect, very similar to the gauge invariant mass term in the Schwinger model in two dimensions. There is, however, a crucial difference. Whereas the mass term in the Schwinger model is dictated by gauge invariance and is obtained in any gauge invariant regularisation scheme, the Chern-Simons term (5) is peculiar to the Pauli-Villars scheme and is absent in dimensional or (-function regularisation [4], of the full fermion determinant in three dimensions. In the perturbative framework, the nonappearance of the Chern-Simons term has been demonstrated [5] with dimensional regularisation of the effective action and also with regularisation of the induced fermion current in the point-split method with phase-factor preserving gauge invariance.

Det(Ik-m-iU-l dU+eU-1AU) = D e t ( / ~ - m) + Wnp[ U] + =Det(p-m) +

WVp(A)

Discussions with A.Chatterjee, S. Mallik, P. Majumdar and P. Mitra are gratefully acknowledged.

WVp[A] (25)

with W,p [ U] = 0, as obtained in a non-perturbative scheme like C-function regularisation. The Pauli-Villars result is consistent with invariance under large gauge transformation simply because it is invariant under the perturbative gauge transformation W~ [A ] reg = Wp [A ] reg. This is to be contrasted with the conventional criterion Wp[AU]reg=Wp[A]reg, for invariance under ordinary gauge transformations, which the Chern-Simon term obtained in the Pauli-Villars scheme fails to

434

References [1] A.N. Redlich, Phys. Rev. Lett. 52 (1984) 18; Phys. Rev. D 29 (1984) 2366. [2] L. Alvarez-Gaum6and E. Witten, Nucl. Phys. B 234 (1984) 269. [3] S. Deser, R. Jackiw and S. Templeton, Phys. Rev. Lett. 48 (1982) 975; Ann. Phys. (NY) 140 (1982) 372. [4] B.S.deWitt, in: Les HouchesLectures 1983 (North-Holland, Amsterdam, 1984) p. 600. [ 5] H. Banerjee, G.Bhattacharyaand J.S. Bhattaeharyya, S1NP report SINP-TNP-85-27 (1985); S. Rao andR. Yahalom, Phys. Lett. B 172 (1986) 227.