Minimal Subtraction vs. Physical Factorisation Schemes in Small-x QCD

We investigate the relationship of “physical” parton densities defined by k-factorisation, to those in the minimal subtraction scheme, by comparing their small-x behaviour. We first summarize recent results on the above scheme change derived from the BFKL equation at NLx level, and we then propose a simple extension to the renormalisation-group improved (RGI) equation. In this way we are able the examine the difference between resummed gluon distributions in the Q0 and MS schemes and also to show MS scheme resummed results for Pgg and approximate ones for Pqg. We find that, due to the stability of the RGI approach, small-x resummation effects are not much affected by the scheme-change in the gluon channel, while they are relatively more sensitive for the quark–gluon mixing. Predictions of perturbative Quantum Chromodynamics for DGLAP [1] evolution kernels in hard processes have been substantially improved in the past few years, both by higher order calculations for any Bjorken x [2] and by resummation methods in the small-x region [3–14]. However, higher order splitting functions are factorisation-scheme dependent: while the NNLO results and standard parton densities [15, 16] are obtained in the minimal subtraction (MS) scheme, the resummed ones are in the so-called Q0-scheme, in which the gluon density is defined by k-factorisation of a physical process. Therefore, in order to compare theoretical results, or to exploit the small-x results in the analysis of data, we need a precise understanding of the relationship of physical schemes based on k-factorisation and of minimal subtraction ones, with sufficient accuracy. The starting point in this direction is the work of Catani, Hautmann and one of us (M.C.) [17, 18], who calculated the leading-log x (LLx) coefficient function R of the gluon density in the (dimensional) Q0-scheme 1 versus the minimal subtraction one, namely g0(t, ω) = R (