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

We provide complete analytic expressions for the inclusive prompt photon production cross section in hadronic final states of $e^+e^-$ annihilation reactions through one-loop order in quantum chromodynamics perturbation theory. Computed explicitly are direct photon production through first order in the electromagnetic strength $\alpha_{em}$ and the quark-to-photon and gluon-to-photon fragmentation contributions through first order in the strong coupling $\alpha_s$. The full angular dependence of the cross sections is displayed, separated into transverse $(1 +\cos ^2\theta _\gamma)$ and longitudinal $(\sin ^2\theta_\gamma)$ components, where $\theta_\gamma$ specifies the direction of the photon with respect to the $e^+e^-$ collision axis. We discuss extraction of fragmentation functions from $e^+e^-$ data.