Comment on `Nucleon Structure Functions from Operator Product Expansion on the Lattice'

TL;DR

This paper critiques a proposed method for extracting nucleon structure functions from lattice QCD, highlighting issues with analytic continuation and operator mixing that challenge the validity of the approach.

Contribution

It identifies fundamental problems in relating Euclidean lattice correlators to Minkowski structure functions, emphasizing the need for careful treatment of analytic properties and operator mixing.

Findings

Euclidean correlators lack the necessary analyticity for dispersion relations.

Operator mixing introduces divergences that complicate the extraction of moments.

The proposed identification of correlators with structure function moments is problematic.

Abstract

It is suggested in the paper by A.J. Chambers {\it et al.} (Phys. Rev. Lett. 118, 242001 (2017), arXiv:1703.01153) that the time-ordered current-curent correlator in the nucleon calculated on the lattice is to be identified as the forward Compton amplitude so that it is related to the sum of the even moments of the structure function as in the Minkowski space in the continuum. We point out two problems with this identification. First of all, the current-current correlator defined in the Euclidean space is not analytic everywhere on the rest of the complex $\nu$ or $\omega$ plane, besides the cuts on the real axis. As such, there is no dispersion relation to relate it to its imaginary part and hence the moments of the structure function. On the lattice, there is an additional difficulty in that the higher dimensional local operators from the operator production expansion (OPE) of the current-current product can mix with lower dimensional higher-twist operators which leads to divergences in the powers of inverse lattice spacing. This mixing needs to be removed before their matrix elements can be identified as the moments of the structure function.