Closed-form angular integration in the orthogonal characteristic function

Determine whether the remaining integral over the center angular variable \(\Phi\) in the orthogonal-symmetry characteristic function for time-reversal-invariance-breaking scattering-matrix elements can be evaluated in closed form.

Background

In the orthogonal-symmetry calculation, the authors reduce the saddle-point-manifold integral to a four-fold integral over angular and radial variables. After transforming the angular variables to a center variable $\Phi$ and a difference variable, the difference-variable integral can be expressed using modified Bessel functions.

The remaining $\Phi$-integral is not obtained in closed form. Although a power-series expansion would technically permit termwise phase integrations, the resulting resummation produces complicated sums without a meaningful closed expression. A closed-form evaluation would simplify the exact characteristic function for arbitrary time-reversal-invariance-breaking strength.