Initial conditions for electron and photon structure and fragmentation functions

In the computation of short-distance cross sections initiated by electrons and photons one can adopt the so-called structure-function approach, in which these particles play formally the same roles as hadrons do in QCD factorisation theorems, and must thus be associated with PDFs (equivalently known as structure functions in this context). At variance with their QCD counterparts, such PDFs are entirely calculable in QED. In this paper we present the results, at the next-to-leading order in the QED coupling constant $\alpha$, for the initial conditions of the unpolarised electron and photon PDFs, which are a necessary ingredient for their eventual collinear evolution at the next-to-leading logarithmic accuracy. We also compute the analogous final-state quantities, namely the initial conditions for fragmentation functions into electrons and photons.