A comprehensive approach to the physics of mesons

We develop an inclusive approach for the self-consistent solution of coupled sets of Bethe-Salpeter and Schwinger–Dyson equations in QCD. This framework allows us to maintain the axial Ward-Takahashi identities of the theory within advanced approximation schemes, such as the skeleton or three-particle irreducible expansions. For this purpose we reformulate the Schwinger-Dyson equation of the axial-vector vertex such that the bulk of its quantum corrections is expressed in terms of an effective vertex, containing an additional gluon. Crucially, this vertex satisfies a symmetry-induced relation of its own, which involves the full quark-gluon vertex. As a result, the Schwinger-Dyson equation reproduces the standard Ward-Takahashi identity satisfied by the axial-vector vertex. Consequently, the known relation between the quark mass function and the wave function of the pion in the chiral limit is duly fulfilled. The present approach offers valuable insights into the interplay between symmetry and dynamics, and provides a practical path towards computations of hadron physics within sophisticated approximations. In particular, the one-loop dressed truncation of the key dynamical equations, including that of the quark-gluon vertex, is shown to be completely compatible with the required symmetry relations. Further extensions and potential phenomenological applications of the developed framework are briefly discussed.