Nakanishi integral representation of pseudoscalar fermion-antifermion bound states in the Bethe--Salpeter equation

Abstract

We propose a method to solve for the pseudoscalar bound state amplitude of a fermion and an antifermion from its Bethe--Salpeter equation (BSE) in the Minkowski space applying spectral representations. With the dressing of the fermion propagators specified by their spectral functions, we first derive the explicit expressions for the Nakanishi spectral functions of the Bethe--Salpeter wave function (BSWF) in terms of these propagators and the Bethe--Salpeter amplitude (BSA) directly from the definition in the momentum space. Applying the rainbow-ladder truncation in the covariant gauge, we then convert the BSE in the momentum space into identities that calculate the spectral functions of the BSA from those of the gauge-boson propagator and of the BSWF. These relations form a closed set of linear functionals for the Nakanishi spectral functions with analytical kernels, laying the foundation for the numerical solution of the BSE in the Minkowski space.