Lattice Operators for Moments of the Structure Functions and their Transformation under the Hypercubic Group

TL;DR

This work addresses the symmetry classification challenge for lattice operators used to compute moments of nucleon structure functions in lattice QCD by exploiting the hypercubic group $H(4)$. It develops a systematic method to decompose lattice tensor operators into $H(4)$-irreducible subspaces, constructing explicit, orthonormal bases for tensors up to rank $4$ and for three operator families $O^0$, $O^5$, and $O^g$, with careful treatment of charge conjugation and trace properties. The authors combine GL(4) embedding with Young-frame techniques and $S_4$ symmetries to obtain a comprehensive set of bases, and they discuss axial operators and mixing both conceptually and with examples, including 1-loop perturbative results. The results enable symmetry-consistent, controlled operator construction and mixing assessment for lattice determinations of structure function moments, highlighting the need to avoid lower-dimensional mixing while managing same-dimension mixing in practical calculations.

Abstract

For lattice operators that are relevant to the calculation of moments of nucleon structure functions we investigate the transformation properties under the hypercubic group. We give explicit bases of irreducible subspaces for tensors of rank up to 4.