A matrix formulation for small-x singlet evolution

TL;DR

This paper introduces a matrix evolution equation in (x,kt)-space that unifies DGLAP and BFKL formalisms for small-x singlet evolution, incorporating NLO and NLx effects with improved theoretical consistency.

Contribution

It presents a novel matrix formulation that generalizes existing evolution equations, including resummation effects and scheme constraints, bridging DGLAP and BFKL approaches.

Findings

Resummation effects delay down to x=0.0001

Shallow dip in gluon-gluon splitting functions around x=0.001

Matrix formulation satisfies NLO constraints and highlights NNLO limitations.

Abstract

We propose a matrix evolution equation in (x,kt)-space for flavour singlet, unintegrated quark and gluon densities, which generalizes DGLAP and BFKL equations in the relevant limits. The matrix evolution kernel is constructed so as to satisfy renormalization group constraints in both the ordered and antiordered regions of exchanged momenta kt, and incorporates the known NLO anomalous dimensions in the MSbar scheme as well as the NLx BFKL kernel. We provide a hard Pomeron exponent and effective eigenvalue functions that include the n_f-dependence, and give also the matrix of resummed DGLAP splitting functions. The results connect smoothly with those of the single-channel approach. The novel P_{qa} splitting functions show resummation effects delayed down to x=0.0001, while both P_{ga} entries show a shallow dip around x=0.001, similarly to the gluon-gluon single-channel results. We remark that the matrix formulation poses further constraints on the consistency of a BFKL framework with the MSbar scheme, which are satisfied at NLO, but marginally violated by small n_f/N_c^2-suppressed terms at NNLO.