A systematic approach to sketch Bethe-Salpeter equation

TL;DR

This paper presents a systematic approach to solving the Bethe-Salpeter equation for mesons by incorporating non-perturbative gluon mass effects and symmetry constraints, improving the understanding of meson properties.

Contribution

It introduces a method to solve the coupled gap and Bethe-Salpeter equations using Ward-Green-Takahashi identities, ensuring symmetry preservation and incorporating a non-perturbative gluon mass scale.

Findings

Derived a nontrivial quark-gluon vertex structure.

Linked the Bethe-Salpeter kernel to the dressed quark-gluon vertex.

Provided a symmetry-consistent truncation scheme for meson studies.

Abstract

To study meson properties, one needs to solve the gap equation for the quark propagator and the Bethe-Salpeter (BS) equation for the meson wavefunction, self-consistently. The gluon propagator, the quark-gluon vertex, and the quark--anti-quark scattering kernel are key pieces to solve those equations. Predicted by lattice-QCD and Dyson-Schwinger analyses of QCD's gauge sector, gluons are non-perturbatively massive. In the matter sector, the modeled gluon propagator which can produce a veracious description of meson properties needs to possess a mass scale, accordingly. Solving the well-known longitudinal Ward-Green-Takahashi identities (WGTIs) and the less-known transverse counterparts together, one obtains a nontrivial solution which can shed light on the structure of the quark-gluon vertex. It is highlighted that the phenomenologically proposed anomalous chromomagnetic moment (ACM) vertex originates from the QCD Lagrangian symmetries and its strength is proportional to the magnitude of dynamical chiral symmetry breaking (DCSB). The color-singlet vector and axial-vector WGTIs can relate the BS kernel and the dressed quark-gluon vertex to each other. Using the relation, one can truncate the gap equation and the BS equation, systematically, without violating crucial symmetries, e.g., gauge symmetry and chiral symmetry.