# Gauge dependence in the anomalous dimension of the gauge invariant canonical decomposition for proton momentum

## TL;DR

This paper investigates the gauge dependence of the anomalous dimension of the gauge-invariant canonical energy-momentum tensor for protons, revealing issues with gauge dependence and suggesting the need for improved treatment of certain conditions.

## Contribution

It identifies the gauge dependence in the background field method calculation and proposes a more covariant approach to address this problem at higher orders.

## Key findings

- Naive calculations lack counter terms in gluonic sectors.
- The gauge dependence arises from the treatment of the condition $F^{

u}_{pure}=0$.
- Further improvements are needed for gauge independence and eigenvalue analysis.

## Abstract

The gauge dependence in the anomalous dimension of the gauge-invariant-canonical-energy-momentum tensor for proton is studied by the background field method. The naive calculation shows the problem, the absence of the counter term in the gluonic sectors. The analysis shows that the result [Chen et al., Phys. Rev. Lett. 103, 062001 (2009)] is derived from the background field method after we introduced a trick to avoid the problem except for the gluon-to-gluon sector; it is gauge dependent. The possible reason of this gauge-dependent result comes from the nontrivial treatment of the condition $F^{\mu\nu}_{pure}=0$ at a higher order. This result shows that one needs a further improvement in treating this condition with a covariant way at a higher order by the background field method. In particular, we have to focus on two checkpoints, the gauge independence and zero eigenvalue in the anomalous-dimension matrix, in order to test the validity of the gauge-invariant-canonical-energy-momentum tensor.