Auxiliary field method for the square root potential

TL;DR

This paper develops an analytical approach using the auxiliary field method to approximate eigenenergies of a two-particle system with a square root potential, relevant for hybrid mesons, demonstrating high accuracy.

Contribution

The paper introduces an analytical expression for eigenenergies of a specific square root potential using the auxiliary field method, applicable to hybrid meson models.

Findings

Asymptotic expressions are derived for the eigenenergies.

Approximate results closely match exact solutions.

The method shows excellent accuracy.

Abstract

Using the auxiliary field method, we give an analytical expression for the eigenenergies of a system composed of two non-relativistic particles interacting via a potential of type $\sqrt{a^2 r^2 + b}$. This situation is usual in the case of hybrid mesons in which the quark-antiquark pair evolves in an excited gluonic field. Asymptotic expressions are proposed and the approximate results are compared to the exact ones. It is shown that the accuracy is excellent.