Microscopic quark study of the eta and eta' masses

TL;DR

This paper demonstrates that accurately modeling the eta and eta' meson masses requires going beyond the simple rainbow-ladder approximation, by including dominant diagrams in the Bethe-Salpeter equation and analyzing various flavor scenarios.

Contribution

It introduces a self-consistent method to incorporate dominant diagrams beyond the rainbow-ladder approximation in the Bethe-Salpeter equation for meson mass calculations.

Findings

Identified key diagrams necessary for eta and eta' mass splitting.

Developed a numerical approach for improved Bethe-Salpeter equation solutions.

Analyzed flavor scenarios relevant to lattice QCD and realistic quark masses.

Abstract

We show that it is necessary to go beyond the BCS (rainbow-ladder) approximation to split the $\eta$ and $\eta'$ masses from the $\pi$ and K masses. We determine the self-consistent set of one-quark-loop diagrams both for the Schwinger-Dyson quark mass gap equation and for the Bethe-Salpeter quark-antiquark boundstate equation. We identify the dominant diagrams, and we focus on the boundstate equation. We detail the Bethe-Salpeter equation, adding the dominant new diagram to the BCS kernel. The relevant numerical techniques are also discussed. The ideal cases of one, two and three light flavors, relevant to lattice QCD are also explored, together with the case of realistic current quark masses.