Heavy-Quark Production in DIS: Mellin Moments for Phenomenology

Heavy-quark production in deep-inelastic scattering probes the gluon distribution, and the precision of current and future data requires predictions beyond next-to-leading order. We calculate the even Mellin moments $N=2$ to $22$ of the next-to-leading-order heavy-quark coefficient functions for $F_2$ and $F_L$, retaining the full heavy-quark mass dependence. This includes the gluon channel, for which no analytic result is available. They provide a test of the parametrisations used in phenomenology: at a representative kinematic point these are accurate to better than one per cent in the gluon channel, whereas in the quark channels we find deviations of up to $20\mathrm{\%}$ over a wide range of Mellin moments, signalling significant differences in the corresponding $x$-space distributions. As a new result at next-to-next-to-leading order, the lowest moment $N=2$ has been determined exactly.