The signature-changing soft-gluon term as a possible source of non-planar $ζ_2$ terms at small $x$ in the non-singlet quark-plus-antiquark splitting function

Abstract

A relation underlying resummations at small momentum fraction $x$ predicts the highest logarithms of quark-plus-antiquark flavor-difference splitting functions from lower orders. Complete three- and four-loop results violate it at third-highest logarithms by pi-squared terms suppressed for many colors, of unexplained origin and color structure. At four loops Moch and Vogt suggested a squared imaginary phase. We retain the signature-changing soft-gluon term also in positive signature in Kirschner and Lipatov's double-logarithmic equations for forward quark-antiquark scattering, without derivation, with their printed negative-signature sign and adjoint Born term. We show that the proposed equations reproduce both violations in full color for quark-plus-antiquark flavor differences at zero power of $x$, fixed coupling and first order in pi squared, for defining-representation quarks of special unitary groups with at least three colors. The proposed equations at third-highest logarithms use one wave per exchanged color and signature, the retained term at lowest-order strength as the positive-signature color-neutral channel's only new source of these violations, no such terms from gluons across the softest exchanged quark-antiquark pair at comparable or larger transverse momentum, and unchanged quark-pair normalization in the splitting function. This agreement gives three independent tests of one system with no continuous coefficient adjusted to the violations, although corresponding non-quartic quark-minus-antiquark terms are missed. In the proposed equations, the retained term weighted by this phase is a possible origin of both violations. The equations predict these full-color five-loop quark-plus-antiquark terms if the relation otherwise holds and no new source first acts at five loops. Confirmation would support the retained term as the origin of the violations through five loops.