Analytic closed form for the leading high-temperature free-energy integral

Derive an analytic closed-form expression for the convergent integral representation of the leading high-temperature free energy of the large-\(N\) critical \(O(N)\) model with angular potential, given in equation (\ref{eq:physical-functional}).

Background

The authors reduce the leading high-temperature free energy per scalar component to an expression involving a single convergent integral over the angular-velocity parameter $x$ and the thermal mass $\tilde m\beta$. This representation is useful for deriving systematic expansions near zero angular potential and near the pole at $x=0$.

Although the integral can be evaluated numerically and is used in the subsequent asymptotic analysis, the paper does not obtain an analytic closed-form evaluation. Finding such a form remains an explicit unresolved calculation within the framework developed in the paper.