Explicit Bosonization of the Massive Thirring Model in 3+1 Dimensions

We bosonize the Massive Thirring Model in 3+1D for small coupling constant and arbitrary mass. The bosonized action is explicitly obtained both in terms of a Kalb-Ramond tensor field as well as in terms of a dual vector field. An exact bosonization formula for the current is derived. The small and large mass limits of the bosonized theory are examined in both the direct and dual forms. We finally obtain the exact bosonization of the free fermion with an arbitrary mass. ∗ 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] The method of bosonization has proven to be a very powerful tool for investigating two-dimensional theories allowing, for instance, the obtainment of exact solutions of nonlinear theories like Quantum Electrodynamics and SineGordon model [1]. A lot of effort has been made in order to generalize this method to higher dimensions [2, 3] but explicit results are only available in 2+1D [4, 5, 6, 7, 8]. In the present work we aim to provide an explicit bosonization of the Massive Thirring Model (MTM) in 3+1D. Using the functional methods developed in [2] we show that the MTM for arbitrary mass and coupling can be bosonized either in terms of a second rank pseudotensor Kalb-Ramond gauge field or in terms of a dual vector gauge field. An exact bosonization formula for the current is derived both in terms of the tensor and vector fields. We then perform a small coupling expansion, obtaining thereby concrete expressions for the bosonized lagrangian and its dual for arbitrary mass. The small and large mass limits are analyzed in detail. The leading contribution in the small mass limit behaves as a Proca theory, either in the tensor or vector cases. In the large mass limit, however, the leading contribution is nonlocal. This is to be compared with the corresponding analysis in 2+1D [4, 5, 6, 7, 8], where the roles of the two limits are reversed. We also 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 explicit results for the bosonized lagrangian and its dual. Interestingly, the tensor field lagrangian that appears in the small mass limit has been shown to be connected with QED [9]. Let us consider the master lagrangian [2] LM = 1 6 FμναF μνα + ψ̄(i 6 ∂ −m− λ 6 B)ψ + Bμ∂νAαβ (1) where Aαβ is the second rank antisymmetric Kalb-Ramond tensor field and F μνα = ∂ A (2) is the corresponding field intensity tensor. Bμ is an external vector field and λ is its coupling constant to the fermions. This lagrangian is invariant under independent gauge transformations on the vector and tensor fields. Consider the euclidean