A Spin One Bottomonium Study in the Functional Formalism in the Feynman Gauge

TL;DR

This paper solves the Bethe-Salpeter equation for spin-one bottomonium in a specific gauge using a lattice-extrapolated gluon propagator, introducing a regulator mass to manage infrared divergences, and achieves good agreement with experimental data.

Contribution

It presents a novel numerical solution for bottomonium states in the Feynman gauge using a lattice-based interaction kernel with a regulator mass.

Findings

Good agreement with experimental bottomonium data below the $BB$ threshold.

Introduction of a symmetry breaking regulator mass $m_L$ effectively manages infrared divergences.

Provides insights into the potential origin of the interaction used in the model.

Abstract

The Bethe-Salpeter equation for spin-one bottomonium coupled to quark and gluon propagators is solved in a class of non-trivial linear gauges. The interaction kernel is based on a known gluon propagator extrapolated from the lattice and contains only small amount of additional phenomenology. The first numerical results are obtained in the Feynman gauge where the associated problem with infrared divergences is circumvented by by introducing a symmetry breaking regulator mass $m_L$. The presence of this mass renders the bound state vertices of vector states finite but it is also necessary to prevent inconsistent solutions of the bound state equation. For $m_L\simeq \Lambda_{QCD}$ it gives us good agreement with the experimental data for vector and axial vector bottomonia below the $BB$ threshold. While the primary concern pertains to the physical ramifications of the proposed concept, a potential origin for the proposed interaction is also presented.