A New Approach for Bosonization of Massive Thirring Model in Three Dimensions

We develop a new approach for bosonization based on the direct comparison of current correlation functions and apply it to the case of the Massive Thirring Model in three dimensions in the weak coupling regime, but with an arbitrary mass. Explicit bosonized forms for the lagrangian and the current are obtained in terms of a vector gauge field. Exact results for the corresponding expressions are also obtained in the case of a free massive fermion. Finally, a comment on the derivation of the current algebra directly from the bosonized expressions is included. ∗ On leave of absence from S.N.Bose National Centre for Basic Sciences, Calcutta, India. Work supported in part by CNPq-Brazilian National Research Council. E-Mail addresses: [email protected]; [email protected] Bosonization in spacetime dimensions higher than two has attracted a lot of attention in the past few years [1, 2, 3, 4, 5]. Even though a direct Mandelstam-like operator bosonization has only been found in the case of the free massless Dirac fermion field in 2+1D [1], bosonized expressions for the fermion lagrangian and the current have been obtained in different mass regimes of free theories [5]. For interacting theories, on the other hand, only the large mass limit has been explored [2, 3, 4]. General bosonized expressions for the current and lagrangian in such interacting theories are still lacking for an arbitrary value of the fermion mass. Furthermore, a new question which arises in this framework and which has not been discussed in the literature is the issue of renormalizability of the fermionic theory when compared with its bosonized counterpart. This is a feature which is absent in 1+1D. In this work, we present the explicit bosonization of the Massive Thirring Model in three dimensions (MTM) in the weak coupling regime, but for arbitrary mass. Contrary to previous approaches, we develop a new method of bosonization that is based on the direct comparison of current correlation functions. We derive explicit bosonized forms for the current and lagrangian for arbitrary values of the fermion mass. It is shown that the MTM for weak coupling but with arbitrary mass is mapped into a generalized free gauge theory. This suggests that at least in this regime we can make sense out of the MTM in spite of the fact that it is perturbatively nonrenormalizable. Interestingly, the bosonized version of the MTM turns out to be nonlocal for all values of the fermion mass m, except m→∞. We also define a dual current which has identical correlation functions as the usual current. We finally consider the case of a free fermion with an arbitrary mass as a limiting situation of the MTM when the coupling constant vanishes. In this case, we get exact results for the bosonized lagrangian and current. In the zero mass limit these expressions reduce to the ones obtained in [1]. In the m → ∞ regime, we derive current algebra relations directly from the bosonic expressions and find a divergent Schwinger term as a result of the infinite mass scale. Let us consider the euclidean generating functional of the MTM in three dimen-