Symmetry preserving truncations of the gap and Bethe-Salpeter equations

TL;DR

This paper develops a symmetry-consistent method to determine the Bethe-Salpeter kernel in hadron physics, revealing the limitations of ladder truncations and the importance of two-loop H-diagrams for accurate meson descriptions.

Contribution

It introduces a novel representation of the gluon-quark vertex to derive the unique, symmetry-preserving Bethe-Salpeter kernel, highlighting the role of H-diagrams and limitations of ladder truncations.

Findings

H-diagrams originate from two-loop contributions involving the three-gluon vertex.

Ladder-like truncations cannot preserve Ward-Green-Takahashi identities.

Adding crossed-box diagrams does not resolve the symmetry preservation issue.

Abstract

Ward-Green-Takahashi (WGT) identities play a crucial role in hadron physics, e.g. imposing stringent relationships between the kernels of the one- and two-body problems, which must be preserved in any veracious treatment of mesons as bound-states. In this connection, one may view the dressed gluon-quark vertex, $\Gamma_\mu^a$, as fundamental. We use a novel representation of $\Gamma_\mu^a$, in terms of the gluon-quark scattering matrix, to develop a method capable of elucidating the unique quark-antiquark Bethe-Salpeter kernel, $K$, that is symmetry-consistent with a given quark gap equation. A strength of the scheme is its ability to expose and capitalise on graphic symmetries within the kernels. This is displayed in an analysis that reveals the origin of $H$-diagrams in $K$, which are two-particle-irreducible contributions, generated as two-loop diagrams involving the three-gluon vertex, that cannot be absorbed as a dressing of $\Gamma_\mu^a$ in a Bethe-Salpeter kernel nor expressed as a member of the class of crossed-box diagrams. Thus, there are no general circumstances under which the WGT identities essential for a valid description of mesons can be preserved by a Bethe-Salpeter kernel obtained simply by dressing both gluon-quark vertices in a ladder-like truncation; and, moreover, adding any number of similarly-dressed crossed-box diagrams cannot improve the situation.