A complete ${\cal O}(\alpha \alpha_s)$ calculation of the Photon + 1 Jet rate in $e^+e^-$ annihilation

We present a complete calculation of the photon +~1 jet rate in $e^+e^-$ annihilation up to ${\cal O}(\alpha \alpha_{s})$. Although formally of next-to-leading order in perturbation theory, this calculation contains several ingredients appropriate to a next-to-next-to-leading order calculation of jet observables. In particular, we describe a generalization of the commonly used phase space slicing method to isolate the singularities present when more than one particle is unresolved. Within this approach, we analytically evaluate the singularities associated with the following double unresolved regions; triple collinear, soft/collinear and double single collinear configurations as well as those from the collinear limit of virtual graphs. By comparing the results of our calculation with the existing data on the photon +~1 jet rate from the ALEPH Collaboration at CERN, we make a next-to-leading order determination of the process-independent non-perturbative quark-to-photon fragmentation function $D_{q \to \gamma}(z,\mu_{F})$ at ${\cal O}(\alpha \alpha_{s})$. As a first application of this measurement allied with our improved perturbative calculation, we determine the dependence of the isolated photon +~1 jet cross section in a democratic clustering approach on the jet resolution parameter $\ycut$ at next-to-leading order. The next-to-leading order corrections to this observable are moderate but improve the agreement between theoretical prediction and experimental data.