On higher-order flavour-singlet splitting and coefficient functions at large x

TL;DR

The paper investigates the large-${x}$ behaviour of flavour-singlet QCD quantities, focusing on the off-diagonal splitting functions $P_{\rm qg}$ and $P_{\rm gq}$ and the gluon coefficient functions $C_{2,\rm g}$ and $C_{\rm L, g}$. By analyzing physical evolution kernels for pairs of structure functions, it shows that these kernels are single-log enhanced at large ${x}$ and derives explicit leading-log predictions, including all-order LL expressions via unfactorized amplitudes and mass factorization. The work connects four-loop off-diagonal behaviour to known three-loop results and demonstrates cancellations of the leading double-log terms, providing nontrivial cross-checks and a framework for extending LL resummations to flavour-singlet quantities. It also outlines a pathway toward phenomenological parametrizations and potential extensions to other processes like Higgs and Z decays, SCET approaches, and path-integral formulations. Overall, it advances the understanding of high-${x}$ structure in QCD and establishes a foundation for all-order LL resummations in flavour-singlet channels.

Abstract

We discuss the large-x behaviour of the splitting functions P_qg and P_gq and of flavour-singlet coefficient functions, such as the gluon contributions C_2,g and C_L,g to the structure functions F_2,L, in massless perturbative QCD. These quantities are suppressed by one or two powers of 1-x with respect to the 1/(1-x) terms which are the subject of the well-known threshold exponentiation. We show that the double-logarithmic contributions to P_qg, P_gq and C_L at order alpha_s^4 can be predicted from known third-order results and present, as a first step towards a full all-order generalization, the leading-logarithmic large-x behaviour of P_qg, P_gq and C_2,g at all orders in alpha_s.

Figures (2)

• Figure 1: The kinematics and perturbative QCD factorization of photon-exchange DIS.
• Figure 2: The function ${\cal B}_{0}(x)$ in Eq. (\ref{['B0']}), evaluated using its defining Taylor expansion.