Photon Fragmentation at LEP

The production of final state photons in hadronic $Z$-boson decays can be used to study the quark-to-photon fragmentation function $D_{q\to \gamma}(z,\mu_{F})$. Currently, two different observables are used at LEP to probe this function: the `photon' +~1 jet rate and the inclusive photon energy distribution. We outline the results of a calculation of the `photon' +~1 jet rate at fixed ${\cal O}(\alpha \alpha_{s})$, which yield a next-to-leading order determination of the quark-to-photon fragmentation function $D_{q\to \gamma}(z,\mu_{F})$. The resulting predictions for the isolated photon rate and the inclusive photon spectrum at the same, fixed order, are found to be in good agreement with experimental data. Furthermore, we outline the main features of conventional approaches using parameterizations of the resummed solutions of the evolution equation and point out deficiencies of these currently available parameterizations in the large $z$-region. We finally demonstrate that the ALEPH data on the `photon' +~1 jet rate are able to discriminate between different parameterizations of the quark-to-photon fragmentation function, which are equally allowed by the OPAL photon energy distribution data.