Mellin Representation for the Heavy Flavor Contributions to Deep Inelastic Structure Functions

TL;DR

The paper tackles the inclusion of heavy-flavor contributions in deep inelastic scattering by developing semi-analytic Mellin-N representations for the heavy-quark Wilson coefficients at LO and NLO, expressed as meromorphic functions in complex $N$. The authors employ a MINIMAX polynomial approach in the scaling variable $z$ and transform the result with Riemann-Liouville fractional integrals to obtain efficient, accurate Mellin-space representations across a wide range of $\xi=Q^2/m^2$. These parameterizations enable fast, scheme-invariant QCD evolution in Mellin space, with a comprehensive $\xi$-grid stored for precise inverse transforms and practical inclusion in global fits of parton distributions. The work provides publicly accessible parameterizations and demonstrates high-precision accuracy, facilitating precise determinations of $\alpha_s$ and refined heavy-flavor treatments in DIS analyses.

Abstract

We derive semi--analytic expressions for the analytic continuation of the Mellin transforms of the heavy flavor QCD coefficient functions for neutral current deep inelastic scattering in leading and next-to-leading order to complex values of the Mellin variable $N$. These representations are used in Mellin--space QCD evolution programs to provide fast evaluations of the heavy flavor contributions to the structure functions $F_2(x,Q^2), F_L(x,Q^2)$ and $g_1(x,Q^2)$.