  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.
