# Anomalous dimension of the gauge invariant canonical decomposition for proton momentum with the background field method

## TL;DR

This paper investigates the anomalous dimension of the gauge-invariant canonical decomposition of the proton's energy-momentum tensor using the background field method, revealing inconsistencies and gauge-parameter dependence.

## Contribution

It identifies an inconsistency in the gauge-invariant decomposition's renormalization within the background field method and discusses the persistent gauge-parameter dependence.

## Key findings

- Naive gauge-invariant decomposition is inconsistent with background field renormalization.
- Gauge-parameter dependence remains in the anomalous dimension calculation.
- Results suggest the need for extended analysis in gauge-invariant spin decompositions.

## Abstract

The anomalous dimension for the gauge-invariant-canonical decomposition of the energy-momentum tensor for quarks and gluons is studied by the background field method. In particular, the consistency between the background field method and the renormalization in the gluonic sectors is investigated. The analysis shows that the naive gauge-invariant-decomposition has an inconsistency between its definition and the renormalization in the background field method. Although we try to consider a trick to overcome this inconsistency in computing the anomalous dimension, the gauge-parameter dependence remains in the final result. This result should be extended to the problems on the gauge-invariant-canonical-spin decomposition.