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

Auxiliary field method for the square root potential