Analytical CFND for radially isotropic bivariate Cauchy distributions

Derive a closed-form analytical expression for the continuous Frobenius norm of the difference between two radially isotropic bivariate Cauchy probability density functions with common scale parameter γ whose centroids are located at the same radial distance μ from the origin and differ by a rotation angle θ; equivalently, compute the L2 distance over R^2 between the two rotated isotropic Cauchy densities to obtain CFND(θ) in closed form.

Background

The paper defines the continuous Frobenius norm of the difference (CFND) as the L2 distance between two continuous 2D distributions and derives a closed-form expression for Gaussian models. To explore non-Gaussian behavior, the authors consider Cauchy (Lorentzian) models.

While they provide an analytical CFND expression for a separated bivariate Cauchy (product of independent 1D Cauchy distributions) and show its first-order absolute-sine behavior, they report being unable to obtain a closed-form result for the radially isotropic bivariate Cauchy case. This leaves the exact CFND for the isotropic Cauchy model unresolved.

References

The integral of the radially isotropic bivariate Cauchy distribution is analytically too difficult to solve because the expansion of the product of two bivariate Cauchy distributions is too high order—we do not yet know the analytical solution.

— Algorithm to extract direction in 2D discrete distributions and a continuous Frobenius norm (2506.17360 - Yepez et al., 20 Jun 2025) in Subsubsection "First-order Cauchy approximation," within Section 3 (Direction Algorithm)