Truncated Mellin moments: Useful relations and implications for the spin structure function $g_2$

TL;DR

This paper reviews the use of truncated Mellin moments in QCD, deriving evolution equations and sum rules for spin structure functions, and demonstrating their application to the analysis of $g_2$.

Contribution

It introduces new relations and sum rules for truncated Mellin moments and generalizes the Wandzura-Wilczek relation in this context.

Findings

Derived the evolution equation for double truncated moments.

Generalized the Wandzura-Wilczek relation for truncated moments.

Numerically solved the DGLAP-like evolution equation for $g_2$.

Abstract

We review our previous studies of truncated Mellin moments of parton distributions. We show in detail the derivation of the evolution equation for double truncated moments. The obtained splitting function has the same rescaled form as in a case of the single truncated moments. We apply the truncated moments formalism to QCD analyses of the spin structure functions of the nucleon, $g_1$ and $g_2$. We generalize the Wandzura-Wilczek relation in terms of the truncated moments and find new sum rules. We derive the DGLAP-like evolution equation for the twist-2 part of $g_2$ and solve it numerically. We find also useful relations between the truncated and untruncated moments.