The Evolution of Unpolarized Singlet Structure Functions at Small x

TL;DR

This work analyzes how all-order small-$x$ resummations in the singlet sector affect the QCD evolution of unpolarized DIS structure functions and the photon structure, using a comprehensive Mellin-$N$ space framework. It contrasts fixed-order and resummed evolution, examines leading and next-to-leading small-$x$ contributions to anomalous dimensions and coefficient functions, and investigates less singular terms via momentum-sum-rule inspired prescriptions. The results show that NL$x$ quark contributions can dominate the small-$x$ rise of the singlet density, while gluonic terms can yield large, sometimes opposing, corrections; the longitudinal structure function $F_L$ is especially sensitive to resummation in its coefficient functions. The photon structure function and its inhomogeneous evolution provide a complementary testing ground, but complete conclusions require higher-order and subleading corrections, underscoring the need for continued development of all-order small-$x$ resummation and fixed-order matching.

Abstract

A systematic study is performed of the impact of the various resummed small-$x$ contributions to the anomalous dimensions and coefficient functions on the evolution of unpolarized structure functions in deep-inelastic scattering. The proton structure functions $F_2^p$ and $F_L^p$ as well as the photon structure function $F_2^γ$ are considered together with the corresponding parton densities. The general analytic solution of the evolution equations in Mellin-$N$ space is derived, and different approximate solutions are compared. Potential effects of less singular small-$x$ terms in the anomalous dimension and coefficient functions are discussed.