Speaker
Description
In the context of continuum Schwinger methods, multiplicative renormalization of the Schwinger-Dyson equations is a highly nontrivial problem. In particular, a naive truncation of the tower of equations can spoil the cancellations among divergent subdiagrams required for multiplicative renormalizability. For the quark gap equation, this problem becomes especially acute as one moves beyond the widely used Rainbow-Ladder truncation by incorporating the dressed quark-gluon vertex. In this work, we apply to the quark gap equation a method originally developed in the formal renormalization of QED, through which the vertex renormalization constants that multiply the quark self-energy are eliminated in favor of dressed quark-gluon vertices, together with additional diagrams that prevent overcounting. As a result, all nonperturbative divergent subdiagrams are formally eliminated, yielding a manifestly finite system of equations. The validity of this procedure relies on the symmetry of the kernel of the vertex Schwinger-Dyson equation under a specific exchange of its internal and external legs. In particular, this condition is valid for the equations derived from the 3PI effective action. Using this truncation, we compute the quark propagator and quark-gluon vertex, obtaining excellent agreement with lattice results. We demonstrate analytically that the solutions exhibit the correct ultraviolet behavior dictated by the renormalization group, and confirm this behavior numerically. Furthermore, we argue that, although the vertex renormalization constant is divergent in perturbation theory, it becomes finite nonperturbatively; in fact, it vanishes in the limit when the ultraviolet regulator is lifted. Beyond their intrinsic interest, our results provide valuable constraints on truncations of the quark Schwinger-Dyson equations, which play a central role in hadron phenomenology, and offer new insights into nonperturbative renormalization in other contexts, such as the gauge sector.