Path-integral formulation of backward and UMKLAPP scattering for one-dimensional spinless fermions

We present a $(1+1)$-dimensional fermionic quantum field theory with nonlocal couplings between currents. This model describes an ensemble of spinless fermions interacting through forward, backward, and UMKLAPP scattering processes. We express the vacuum-to-vacuum functional in terms of a nontrivial fermionic determinant. Using path-integral methods, we find a bosonic representation for this determinant and an effective action depending on three scalar fields, of which two correspond to the physical collective excitations and one is an auxiliary field, which can be integrated out using an approximation technique.