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The gauge dependence in the anomalous dimension of thegauge-invariant-canonical-energy-momentum tensor for proton is studied by thebackground field method. The naive calculation shows the problem, the absenceof 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 thebackground field method after we introduced a trick to avoid the problem exceptfor the gluon-to-gluon sector; it is gauge dependent. The possible reason ofthis gauge-dependent result comes from the nontrivial treatment of thecondition $F^{\mu\nu}_{pure}=0$ at a higher order. This result shows that oneneeds a further improvement in treating this condition with a covariant way ata higher order by the background field method. In particular, we have to focuson two checkpoints, the gauge independence and zero eigenvalue in theanomalous-dimension matrix, in order to test the validity of thegauge-invariant-canonical-energy-momentum tensor.

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