Weak gauge-invariance of dimension two condensate in Yang-Mills theory

TL;DR

This paper proves that the vacuum condensate of mass dimension two in Yang-Mills theory is gauge-invariant in a weak sense, suggesting its numerical value is independent of gauge-fixing conditions, with considerations on Gribov copies.

Contribution

It provides a formal proof of the weak gauge-invariance of the dimension two condensate in Yang-Mills theory under certain conditions.

Findings

The condensate's vacuum expectation value is gauge-invariant in the weak sense.

Gauge-fixing condition independence is shown for small deformations from specific gauges.

Discussion on modifications needed when Gribov copies are present.

Abstract

We give a formal proof that the space-time average of the vacuum condensate of mass dimension two, i.e., the vacuum expectation value of the squared potential $\mathscr{A}_\mu^2$, is gauge invariant in the weak sense that it is independent of the gauge-fixing condition adopted in quantizing the Yang-Mills theory. This is shown at least for the small deformation from the generalized Lorentz and the modified Maximal Abelian gauge in the naive continuum formulation neglecting Gribov copies. This suggests that the numerical value of the condensate could be the same no matter what gauge-fixing conditions for choosing the representative from the gauge orbit are adopted to measure it. Finally, we discuss how this argument should be modified when the Gribov copies exist.