Coupled propagator and vertex Dyson-Schwinger equations

W. Detmold

Dyson-Schwinger equations (DSEs) are the quantum equations of motion of a field theory and in principle contain all information in the theory. However, they form an infinite tower of coupled equations (the $n^{\rm th}$ depending on the $(n+1)^{\rm th}$) that can so far only be solved by truncation or in the weak coupling limit where they reduce to perturbation theory.

Many different truncations have been studied over the years. Typically one might study the DSE for the quark propagator, which depends not only on the quark propagator but also on the gluon propagator and the quark-gluon vertex. These unknown functions can either be modeled or (eventually) be obtained from lattice QCD. However, in the simple model defined by the Munczek-Nemirovsky gluon propagator it has been possible to solve the coupled DSEs for the quark propagator and the quark-gluon vertex without further truncation. This was carried out in a scalar model [37] and the extensions to a QCD-like theory containing fermions and applications beyond the Munczek-Nemirovsky model are under investigation.