Heavy quark coefficient functions at asymptotic values $Q^2 \gg m^2$

TL;DR

The paper derives analytic asymptotic expressions for heavy-quark coefficient functions in deep-inelastic scattering in the limit $Q^2 \gg m^2$, at next-to-leading order in $\alpha_s$, using the operator product expansion and finite operator matrix elements. It shows how mass factorization transfers large logarithms $\ln(Q^2/m^2)$ into transition functions and provides explicit forms for heavy-quark contributions in gluon- and quark-initiated channels, including Bethe-Heitler and Compton processes. The results serve as cross-checks against exact, numerically-intensive calculations and inform the applicability of the variable-flavour scheme for charm production at HERA. The study quantifies the kinematic domain where the asymptotic expressions reliably approximate full NLO results, enabling efficient phenomenology of charm production and heavy-flavour evolution.

Abstract

In this paper we present the analytic form of the heavy-quark coefficient functions for deep-inelastic lepton-hadron scattering in the kinematical regime $Q^2 \gg m^2$ . Here $Q^2$ and $m^2$ stand for the masses squared of the virtual photon and heavy quark respectively. The calculations have been performed up to next-to-leading order in the strong coupling constant $α_s$ using operator product expansion techniques. Apart from a check on earlier calculations, which however are only accessible via large computer programs, the asymptotic forms of the coefficient functions are useful for charm production at HERA when the condition $Q^2 \gg m_c^2$ is satisfied. Furthermore the analytical expressions can also be used when one applies the variable heavy flavour scheme up to next-to-leading order in $α_s$.