Resummation of small-x double logarithms in QCD: inclusive deep-inelastic scattering

Abstract

Abstract We present a comprehensive study of high-energy double logarithms in inclusive DIS. They appear parametrically as $$ {\alpha}_s^n $$ α s n ln2n−kx at the n-th order in perturbation theory in the splitting functions for the parton evolution and the coefficient functions for the hard scattering process, and represent the leading corrections at small x in the flavour non-singlet case. We perform their resummation, in terms of modified Bessel functions, to all orders in full QCD up to NNLL accuracy, and partly to N3LL and beyond in the large-nc limit, and provide fixed-order expansions up to five loops. In the flavour-singlet sector, where these double logarithms are sub-dominant at small x compared to single-logarithmic $$ {\alpha}_s^n $$ α s n x−1 lnn−kx BFKL contributions, we construct fixed-order expansions up to five loops at NNLL accuracy in full QCD. The results elucidate the analytic small-x structure underlying inclusive DIS results in fixed-order perturbation theory and provide important information for present and future numerical and analytic calculations of these quantities.