Z. Phys. C 67, 707-709 (1995) ZEITSCHRIFT FOR PHYSIK C �9 Springer-Verlag 1995

Path integral solubility of a general two-dimensional model

Ashok Das, Marcelo Ho t t*

Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA

Received: 23 January 1995

Abstract. The solubility of a general two dimensional model, which reduces to various models in different limits, is studied within the path integral formalism. Various subtleties and interesting features are pointed out.

There are a number of 1 + 1 dimensional field theoretic models that can be solved exactly. The solubility of these models have been studied from various points of view [1-11]. Normally, these models are formulated in terms of a fermion field with a vector or axial-vector or a chiral coupling. More recently, however, there has been an inter- est in a model [12, 13] where the fermion has both vector and axial vector couplings of arbitrary strength. This model reduces to all other known models in various limits. In this article, we show how this model can be solved in its generality within the path integral formalism. We com- pare our results with those obtained through a point- splitting regularization [ 13] and point out various charac- teristics of the model.

To begin with, let us consider a fermion in 1 + 1 dimension interacting with an external spin 1 field, de- scribed by

S : i~Tu(Ou - - i(1 + rTs)Au) ~

= i~Tu(Ou -- i(tlu~ + r % ~ ) A ~ ) ~ (1)

Here 'r' is an arbitrary real parameter and we have used the familiar identities of (1 + 1)dimensions in the last line of(l). (See [10] for notations, identities and details.). Let us define

3~ = (r/.., + re.~ }A ~ (2)

*On leave of absence from UNSEP, Campus de Guaratinguet&, P.O. Box 205, CEP: t2.500, Guaratinguetfi, S.P., Brazil

and note that in 1 + 1 dimensions, we can write

J'u = ~?~ + %~c~P (3)

so that the Lagrangian in (1) takes the form

= toy (o , - i f t , ) O = i ~ y " ( O , , - iO,~ - q s o , p ) O (4)

It is clear now that if we define

~t = ei(~+TsP)~t '

= ~'e-i{~- ,,o) (5)

then the Lagrangian in (4) reduces to a free theory, namely,

i~7" (c3u i0ua iys0up)0 .-, u , . . . . '~ 7 a ,0 (6)

In the path integral formalism, the Jacobian under the field redefinition in (5) is nontrivial [14] and we obtain the effective action by evaluating this Jacobian [9-11].

The evaluation of the Jacobian is straightforward and can be read off from [10]. However, we would like to emphasize that for the present case, we can define

/~D = %~]~ (7)

which leads to the identity

~D 7"/~ u = yu(r//l, + ~ 7 s A u )

with

and one can use the Euclidean Dirac operator

D~ = Yu(Ou - irly~Au - z~Yu'/sAu ) (9)

to evaluate the Jacobian for the change of variables. We note here that it is this operator which provides the most general regularization which is consistent.

For an infinitesimal field redefinition

= ~'e-i(~(x)- ~#(~)) (i0)

708

the Jacobian with the regularization in (9) can be read off --+ A# ) and from [10] (simply replace A# A# and As# -v

has the form

In the generating functional, the ferminoic fields can be integrated out to give the effective action in (16) with the substitution

[ -] d xe(rle(x)eu~F#~ + {g(x)eu~Fu~ ) (11) J = e x p _ ~ 5 2 ~D

which when rotated to Minkowski space has the form

J = exp ~ dZx(tle(x)a#ft" + ~g(x)c~#ft #'~) (12)

The anomaly equations for the vector and the axial-vector currents, then, follow to be (j~ = ~7#0,j~ = ~7s7#0)

A. ~ A. + eB~ (18) As a result, we can write

Zvor(Au, J~,) = N [. @B#e/s~ Au' J~')

where

with

1 ) + B,Q u + ~ A.RI~A~

(19)

(20)

= __ -- #v ~- GJ'b ~ G 71# ~- ~ (G A# + re o.A,) 7C 7~

a,j~a = _ #-_ a#fl , .v = __~ ( rGA, + e ' "GA d 7"C IZ

(13)

These, of course, reduce to the well known results [10] when r = 0 and we note that for j u = j ) - - rj"A = ~TU(1 + r?5)~,

r a#j" -- _1 ('I + &Z)GA# + _ e~GA~ (14) 7"g

( e2 ) p#v = y/#v [ ] _]_ # I -}- 7 (C At- /~ r2)

- 1 + - - (1 + r 2) E] - t OuOv

e2r

7~ - - - (~"a" + e,~? ") G E] i

Qu = yu__~ce ( _ ( d + w2)A u

Herej # is the current of our theory in (1) and we note that it is anomalous for r r 0 for any choice of regularization.

The Jacobian for the finite field redefinition in (5) is again straightforward following [10] and we obtain

R #~ = _ _

I i ~ d2x(~lu~ + rgu~)(r/vz + rsvT ) J = exp -

A ~( a"av a aaG\ l

which leads to the effective action

Z[Au] = N ~ ~-@6~e i[.d~xe

= N ' e x p [ - ~ S d 2 x ( t l # . +rau,)(rl~ + rG~)

A ~ (ti UO~ a%p~ @?o ~ -5- +

(16) X

It is straightforward to check that this generating func- tional yields the anomaly equation (14). This can be checked to coincide with the result obtained through the point-splitting regularization [13].

Next, let us consider the general model described by [12, 13]

a =

~TOT = - - ~ (auB~ -- GB.)(O#B ~ - a~B #) + T BuB#

+ i~7#(~?# -- i(1 + r75)(A # + eS#))~p + J.B # (17)

+ (1 + t "2) 0#0 �9 A )

[] - + r(d## ~ + J ~ U ) G D - * A ~

0#0~ 1 _(~ + rlrZ)rl#V+ (1 + r z ) - ~ - 7r

+ r(e~M ~ + c~0#)G[] -*) (21)

The action in (20) is quadratic in B# and hence the generating functional is easily obtained to be

[ i ] ZToT(Au, Jt~) = N' exp --~ y d2x(Q#P-;~IQ v + AuR#*A~)

(22)

Note that if we define

p~-i = ar/#~ + b~.~?~ + c(e~#0~ + e ~ . ) 0 " (23)

then, from

Pu~P~;) = 6"a (24}

we can determine

(e2r/=) 2 [] +/~o z + (e2/~)(~ +/~r 2) -~- t~oZ+(ea/~)(<+r/r2)_(e2/rO{l+r2)

1 2

[~] -~- /~phys

b = ( [ ] + (e2/~)(1 + re)) 1

2 u~ - (e~/n)(~ + ~r 2) ( N + m ~ . ) []

eZr 1 1 (25) = 2 c n(po 2 -- (e2 /n) ( r /+ ~r2)) ( N + tryhys) []

We can also rewrite

2 2 (1 - (lle2/Tz/~02)(1 - - r2))(1 "~- (@Z/:Z#02)(1 -- r2)) Yl~phys = # 0 (1 -- (ez / =l~2 ) (rl -t- ~r2) )

(26)

which coincides with the result obta ined through the point-spl i t t ing regularizat ion [13]. The p r o p a g a t o r for the Bu-field is now seen to be

1 [- ( [ ] + (e2/n)(1 + rZ)) a.a ~ Du~ -- ( D + m2hys) L r/"v + (b t2 -- (e2/n)(r/ + ~r2)) [ ]

e2r/7~ (a~0~ + a~0~)0~[53 - 1 ] (27)

+ { ~ - (e2/~I{~ + ~r~)t We end our discussion by not ing that the t e rm quad-

ratic in Qu in (22) gives rise to a te rm which is quadra t ic in Au. (See definition in (21).) Consequent ly, the te rm quad- ratic in Au will have a s tructure of the form

1 - - (1 X / i # I x z~ v 2 ~ -2 ~ ~ , , ~ (28)

in the exponent of the generat ing functional. Conse- quent ly the anoma ly equa t ion derived f rom this generat- ing functional will differ f rom that in (14). The reason for this is not ba rd to understand. Since both A~ and B u

709

couple to the same current, the one- loop d iagrams contri- buting to the anoma ly will have two parts.

While (14) contains the cont r ibut ion f rom the first d i ag ram alone, it is the second d iagram which is respon- sible for the modif icat ion in the anoma ly (also in (28)).

This work is suppor ted in par t by the U.S. D e p a r t m e n t of Energy Gran t No. DE-FG-02-91ER40685. M.H. would like to thank the F u n d a 9 i o de A m p a r o a Pesquisa do Estado de Silo Paulo for the financial support .

R e f e r e n c e s

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Ann. Phys. (N.Y.) 26, 1 (1964) 5. L.S. Brown: Nuovo Cimento 29, 617 (1963) 6. C.R. Hagen: Nuovo Cimento 51B, 169 (1967); Nuovo Cimento

51A, 1033 (1967) 7. C.R. Hagen: Ann. Phys. (N.Y.) 81, 67 (1973) 8. R. Jackiw, R. Rajaraman: Phys. Rev. Lett. 54, 1219 (1985) 9. R. Roskies, F.A. Schaposnik: Phys. Rev. D23, 558 (1981)

10. A. Das, V.S. Mathur: Phys. Rev. D33, 489 (1986) 11. A. Das: Phys. Rev. Lett. 55, 2126 (1985) 12. A. Bassetto, L. Griguolo, P. Zanca: Phys. Rev. 50D, 1077 (1994) 13. C.R. Hagen: University of Rochester preprint UR-1382 14. K. Fujikawa: Phys. Rev. Lett. 42, 1195 (1979); K. Fujikawa,

Phys. Rev. D21, 2848 (1980); ibid D22, 1499(E) (1980)

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