The condensate <tr(A_{\mu}^{2})> in commutative and noncommutative theories

TL;DR

This paper investigates the gauge invariance and independence of the vacuum condensate of the operator A_{}^2 in both commutative and noncommutative gauge theories, revealing that gauge invariance does not imply gauge independence of the condensate.

Contribution

It demonstrates that gauge invariance of a specific operator in noncommutative gauge theory does not ensure its vacuum condensate's gauge independence, and derives generalized Ward identities for related Green's functions.

Findings

Gauge invariance does not guarantee gauge independence of the condensate.

Generalized Ward identities are derived for Green's functions involving the condensate operator.

The study applies to both commutative and noncommutative gauge theories.

Abstract

It is shown that gauge invariance of the operator \int dx tr(A_{\mu}^{2}-\frac{2}{g \xi} x^{\nu} \theta_{\mu\nu} A^{\mu}) in noncommutative gauge theory does not lead to gauge independence of its vacuum condensate. Generalized Ward identities are obtained for Green's functions involving operator \underset{\Omega \to \infty}{lim}\frac{1}{\Omega} \int\limits_{\Omega} dx tr(A_{\mu}^{2}) in noncommutative and commutative gauge theories.