Origin of the small-x resummation discrepancy

Understand the structural origin of the discrepancy between the exact four-loop pure-singlet quark-to-quark splitting function and the small-x resummation prediction in the subleading-color contribution proportional to \(\zeta_2\) at next-to-next-to-leading logarithmic level.

Background

The paper derives the four-loop pure-singlet contribution to the quark-to-quark splitting function in QCD analytically for arbitrary Mellin moments and examines its small-momentum-fraction behavior. Most terms in the small-$x$ expansion agree with an existing small-$x$ resummation prediction, but the authors identify a discrepancy in the coefficient proportional to $\zeta_2$ at next-to-next-to-leading logarithmic order.

The discrepancy occurs in a subleading-color contribution and persists despite agreement in the leading-color approximation. The paper explicitly leaves unresolved the structural explanation for why the exact fixed-order result and the resummation prediction differ in this term.