Comment on `Controversy concerning the definition of quark and gluon angular momentum' by Elliot Leader (arXiv:1101.5956, PRD 83, 096012 (2011))

TL;DR

This paper discusses the correct definition of quark momentum and angular momentum in nucleons, proposing a gauge-invariant approach using partial derivatives and testing it through lattice QCD calculations.

Contribution

It introduces a gauge-invariant lattice QCD method for evaluating quark momentum and angular momentum using the canonical form with partial derivatives.

Findings

Ratios of three-point to two-point functions are zero within errors for u and d quarks.

Replacing gauge links with unity affects the matrix elements, supporting the proposed definition.

The results challenge the conventional gauge-invariant definitions of quark angular momentum.

Abstract

It is argued by the author that the canonical form of the quark energy-momentum tensor with a partial derivative instead of the covariant derivative is the correct definition for the quark momentum and angular momentum fraction of the nucleon in covariant quantization. Although it is not manifestly gauge invariant, its matrix elements in the nucleon will be non-vanishing and are gauge invariant. We test this idea in the path-integral quantization by calculating correlation functions on the lattice with a gauge-invariant nucleon interpolation field and replacing the gauge link in the quark lattice momentum operator with unity, which corresponds to the partial derivative in the continuum. We find that the ratios of three-point to two-point functions are zero within errors for both the u and d quarks, contrary to the case without setting the gauge links to unity.