Finite Size Corrections in Massive Thirring Model

We calculate for the first time the finite size corrections in the massive Thirring model. This is done by numerically solving the equations of periodic boundary conditions of the Bethe ansatz solution. It is found that the corresponding central charge extracted from the 1/L term is around 0.4 for the coupling constant of g0 = − π 4 and decreases down to zero when g0 = − π 3 . This is quite different from the predicted central charge of the sine-Gordon model. PACS numbers: 11.10.Lm e-mail: [email protected] e-mail: [email protected] 1 In two dimensional field theory, there is a remarkable correspondence between the fermionic and bosonic field theories. This was first recognized by Coleman [1], and he proved that the sine-Gordon field theory and the massive Thirring model are equivalent to each other in that the arbitrary order of the correlation functions turn out to be the same. Recently, however, Klassen and Melzer [2] argue that the equivalence between the sine-Gordon and the massive Thirring models may be violated at the finite size correction. They proved by using the perturbed conformal field theory that these two models are different in finite-volume energy levels, for example. In this paper, we calculate the finite size corrections to the ground state energy. We solve numerically the equations of the periodic boundary condition in the Bethe ansatz solutions of the massive Thirring model [3-5]. The ground state energy can be expressed as Ev = E0L− πc̃ 6L + ... (1) where L denotes the box size. c̃ corresponds to a central charge at the massless limit [6,7]. The present calculation shows that the corresponding central charge c̃ in the negative coupling constant regions (no bound states) is around 0.4 for g0 = − π 4 and that it becomes zero when g0 = − π 3 . These values can be compared with those calculated for the sine-Gordon field theory [8,9]. The central charge for the sine-Gordon field theory with the massless limit can be expressed as c = 1− 6 p(p+ 1) (2) where p is an integer and is related to the coupling constant g0 as g0 = − π 2 (1− 1 p ). (3) In fig.1, we summarize the calculated central charge as the function of the coupling constant for the sine-Gordon model by Itoyama and Moxhay, and for the massive 2 Thirring model by the present calculations. One can see that the values of the central charge predicted for the two models are very different from each other. It is, however, not very clear to us whether this difference may be related to a possible violation of the equivalence between the sine-Gordon field theory and the massive Thirring model at the finite volume energy as suggested by Klassen and Melzer. Here, we briefly review the massive Thirring model whose lagrangian density can be written as [10] L = ψ̄(iγμ∂ μ −m0)ψ − 1 2 g0j jμ (4) with the fermion current jμ =: ψ̄γμψ :. Choosing a basis where γ5 is diagonal, we write the hamiltonian as