Gauge invariance in the effective action and potential

Abstract

The gauge dependence of the effective action $\Gamma$ and potential $V$ are studied in general gauge theories. Explicit expressions which manifest all the gauge dependences of $\Gamma$ and $V$ are obtained. From these equations, it is concluded that $\Gamma$ (or $V$) has gauge-invariant values for any solution of the Euler-Lagrange equation $\frac{\delta \Gamma }{\delta \phi }=0\mathrm{}$ (or $\frac{\partial V}{\partial \phi }=0\mathrm{}$) Introduction of a certain concept about the categories of gauge conditions resolves the appearance of a gauge-dependent unphysical "vacuum." Any gauge can be used to calculate $\Gamma$. A wide class of gauges are allowed for the effective potential $V$; for instance, in scalar QED the allowed gauges are $-\left(\frac{1}{2}\alpha \right){\left(\partial A\right)}^2$, $-\left(\frac{1}{2}\alpha \right){(\partial A-v{\Phi }_2)}^2$ (where the direction of condensation is restricted to ${\Phi }_1$), the Coulomb gauge, and the axial gauge. In particular the $R_{\xi }$ gauge is also an allowed gauge.