Perturbative re-construction of the massive derivative coupling model in 2 dimensions

Abstract

The Schroer model [1] is a 2-dimensional model built to discuss the structural characteristics of Quantum Electrodynamics (QED), namely the Hilbert space and dynamic particularities due to the infraparticle character of the electron. The interaction there is set through a massless boson φ. In this case, the fields do not live in the Fock space of free fermions due to IR divergences, and so one has to define the model through the Wightman functions and then reconstruct the Hilbert space. The model we studied contains the boson φ of mass m that is free, in the sense of obeying Klein-Gordon equation, and a fermion ψq of mass M, which are coupled through the equation of motion (i∂/ − M)ψq = −q(∂φ/ )ψq, where q is the coupling constant. The non-perturbative solution is the dressed Dirac field ψq(x) .=: e iqφ(x) : ψ(x) from [1], where φ is the free boson and ψ is the free Dirac field. We will call this the massive Schroer model. The IR divergences do not appear in the massive case. We suggest how the massive Schroer model arise from the free Dirac field with the interaction L = ∂µφjµ in the context of Epstein-Glaser perturbation theory, with φ being the massive boson and j µ the Dirac current. This model is renormalizable, with an infinite number of graphs to be normalized. We impose certain normalization conditions, which among others are the extended Ward identities. For tree graphs, these normalization conditions are automatically satisfied, while loop graphs are uniquely fixed by the respective normalization. This turns the model superrenormalizable. We show that, in the adiabatic limit, the S-matrix equals the unity, the interacting observables j µ , ∂µφ become free fields, and the interacting version of the free Dirac field coincides with the free dressed Dirac field ψq mentioned above.