Comparisons of Exact Results for the Virtual Photon Contribution to Single Hard Bremsstrahlung in Radiative Return for Electron-Positron Annihilation

We compare fully differential exact results for the virtual photon correction to single hard photon bremsstrahlung obtained using independent calculations, both for $e^+e^-$ annihilation at high-energy colliders and for radiative returnapplications. The results are compared using Monte Carlo evaluations of the matrix elements as well as by direct analytical evaluation of certain critical limits. Special attention is given to the issues of numerical stability and the treatment of finite-mass corrections. It is found that agreement on the order of $10^{-5}$ or better is obtained over most of the range of hard photon energies, at CMS energies relevant to both high energy collisions and radiative return experiments.