Isolated Prompt Photon Production in Hadronic Final States of $e^+e^-$ Annihilation

We provide complete analytic expressions for the isolated prompt photon production cross section in $e^+e^-$ annihilation reactions through one-loop order in quantum chromodynamics (QCD) perturbation theory. Functional dependences on the isolation cone size $\delta$ and isolation energy parameter $\epsilon$ are derived. The energy dependence as well as the full angular dependence of the cross section on $\theta_\gamma$ are displayed, where $\theta_\gamma$ specifies the direction of the photon with respect to the $e^+e^-$ collision axis. We point out that conventional perturbative QCD factorization breaks down for isolated photon production in $e^+e^-$ annihilation reactions in a specific region of phase space. We discuss the implications of this breakdown for the extraction of fragmentation functions from $e^+e^-$ annihilation data and for computations of prompt photon production in hadron-hadron reactions.