Volume 152B, number 3,4 PHYSICS LETTERS 7 March 1985

INTEGRABILITY OF CHIRAL NONLINEAR SIGMA MODELS WITH A WESS-ZUMINO TERM

M.C.B. ABDALLA 1 The Niels Bohr Institute, University of Copenhagen, Blegdamsve] 1 7, DK-21 O0 Copenhagen O, Denmark

Received 10 October 1984 Revised manuscript received 17 December 1984

It is shown that the inclusion of a Wess-Zumino term to the chiral model does not spoil the classical (and quantum) integrability of the system. It is possible to construct its conserved classical (and quantum) nonlocal charge.

Nonlinear sigma models are an important theo- retical laboratory of quantum field theory [1 ], espe- cially those defined on a riemannian symmetric space, which is the integrability condition of the classical theory [2]. In general the quantum integrability is spoiled by anomalies [3]. A general criterion for quantum integrability is obtained by looking at the gauge group H: there is no anomaly if the Lie algebra ~f has no nontrivial ideal [3]. This means that chiral models, where the fundamental field g(x) takes value on a group G, and transforms under a direct product G × G, is quantum mechanically integrable. The S- matrix can be exactly computed. In the case G = SU(N), the action reads

S = Tr f d2x 8ug+(x)~Ug(x), (1)

together with the constraint g+(x)g(x) = 1, where g(x) is an element of SU(N). In this case the part icle- particle scattering is given by [4] * x

_ sh[~-(0 + 2rri/N)] u I (0) - sh[~-(0 - 21ri/N)]

[ V(1/N + 0/27ri)P(1 - 0/27ri) ]2 × \P(0/27ri)P(1 + 1 / N - 0/2~i)] " (2)

If now we sum a Wess-Zumino term to the principal chiral model [eq. (1)], the physical interpretation of

1 Work supported by CAPES, Brazil. ,1 In ref. [4], there is a misprint in ul(O) and $2(0 ).

0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

the model changes drastically. One writes the action as

S = f d 2 x Tr(4~2 a ~ g - l ~ g

neU v t ' . 1 ~ ) + ~n J o t g - -~ gg- 1 Ougg- 1 B,g . (3)

The one-loop 3-function is given by [5]

~C/k, n) = -(X2N/4rr) [ 1 - (~2n/47r) 2] , (4)

and the equations of motion are summarized by the conservation equations

~) ls [ J U - - (nX 2 /47r)eUVJv] = 0 , (5a)

JU = g - 10 Ug, (5b)

and

0 u [iu + (nX2/47r)eUViv] = 0 , (6a)

iu = ~#gg- 1 (6b)

The 3-function has a nonperturbative zero at X 2 = 147r/n I. At this point one can show that the model is equivalent to an n-plet of free fermions, in the fun- damental representation of G, for G = SU(N) or O(N). This means that the zero of the 3-function re- mains at higher orders. For general X, we can show that the model displays higher conservation laws. Classically, due to the equations of motion, and the following identities:

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Volume 152B, number 3,4 PHYSICS LETTERS 7 March 1985

buJ v - -avJ u + [Ju,Jv] = 0 , (7a)

bui v -- bvi u - [iu, iv] = 0 , (7b)

we can prove that the following nonlocal charges

o = fdyl dy2 e(Y 1 -- Y2) [S0 + (n?t2/41r)J1 ] ( t ,Y l )

X [J0 + (n?t2/4rr)J1 ] (t ,Y2)

+ 2 [1 - n2?t4/(47r)2] f J l ( t , y ) d y , (8)

and

a' = f dyl dy2 eCvl - y2)[ io - (n?t2/47r)il ] ( t ,y l )

X [i0 - (n?t2/47r)ill (t, Y2)

[ 1 - - n2X4/(47r) 2 ] f i 1 (t, y ) dy (9) + 2

are conserved [2]. For the quantization of the model, we use methods which are well known now, namely we study the Wilson expansion of the commutator of two currents [3],

[Ju (x + e), Jr(x)] = Co u v(e)J o (x) + D ° u p (e) 0 oJp (x ) , (10a)

[iu(x + e), iv(x)] 'o • ' o o • = Cuv(e)tp(x ) + Duo (e)3oto(x) . (10b)

Note that on the RHS there is no anomaly due to a theorem which can be applied to chiral fields [3]. The theorem states that there is no other operator of dimension 2 which can enter the RHS of eq. (10) above. This means that quantum charges can be de- fined by renormalizing the second term in the quan- tum non local charge, and it should read:

( f dYl dY2 e(Yl - Y 2 ) Q = lim 6~o \ lyl-y21<6

X [J0 + ( n}t2 [47r)J1 ] (t, Yl )[J0 + (n?~2/47r)J1] (t ,Y2)

+z, f [J1 +(nX2/47r)Jo]( t ,y )dy) , (11)

where the relative coefficients o f J 1 and J0 in the second term can be adjusted by summing the charge

of SU(N) relations, namely

f dy [Jo + (n~2/47r)J1 ] (t, y ) . (12)

The existence of the classical nonlocal charge implies the integrability of the model. This result can also be obtained by the integrability condition of the Lax pair ,2 . The quantization was possible due to the fact that there is no possible anomaly since this is a model where the fields take value on the group G and trans- forms under the direct product of G X G.

Note that in the case of nonabelian bosonization, n a m e l y }k 2 = 47r/n, where the model is equivalent to free massless fermions, the nonlocal charge is trivial- ly conserved, due to the equations of motion. In fact, this should happen, since the very existence of the conserved nonlocal charge implies nontrivial scatter- ing: due to the difference of in and out expressions of the charge [4], it cannot commute with a unity S-matrix.

In order to define the exact S-matrix we must know the asymptotic states of the theory. However, this is a nontrivial task in the present model, due to the existence of infra-particles [6] in this model [7]. The nonexistence of a mass gap is known from the Bethe ansatz for the fermion theory equivalent to the present model. An S-matrix can be obtained, but it is not clear which particle scattering is described [7 ]. From the equivalence to fermions some hints for the asymptotic states (and consequently for the S-matrix) can be obtained [5]. Work in this direc- tion is presently under way.

Note added. The classical integrability of the sys- tem (7a) with auJ u + aeuvauJ v = 0 was discussed in ref. [8]. I thank H.J. de Vega for a letter concerning this point.

,2 I acknowledge a discussion with P. Wiegmann in this and the following.

References

[1 ] A. D'Adda, P. Di Vecchia and M. Lfischer, Nucl. Phys. B146 (1978) 63;B152 (1979) 125; E. Witten, Nucl. Phys. B149 (1979) 285 ; E. Abdalla, Lecture notes in physics (Springer, Berlin), to be published.

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[2] H. Eichenherr and M. Forger, Nucl. Phys. B155 (1979) 381;B164 (1980) 528.

[3] E. AbdaUa, M. Forger and M. Gomes, Nucl. Phys. B210 [FS6] (1982) 181.

[4] E. Abdalla, M.C.B. Abdalla and A. Lima-Santos, Phys. Lett. 140B (1984) 71.

[5] E. Witten, Commun. Math. Phys. 92 (1984) 455; E. Abdalla and M.C.B. AbdaUa, Non abelian bosonization in the operator language, preprint NBI-HE-84-11.

[6] B. Schroer, Fortschr. Phys. 11 (1963) 1. [7 ] A.M. Polyakov and P.B. Wiegmann, Phys. Lett. 141B

(1984) 223. [8] H.J. de Vega, Phys. Lett. 87B (1979) 233.

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