Gauge invariance in the effective action and potential

Abstract

The gauge dependence of the effective action $\ensuremath{\Gamma}$ and potential $V$ are studied in general gauge theories. Explicit expressions which manifest all the gauge dependences of $\ensuremath{\Gamma}$ and $V$ are obtained. From these equations, it is concluded that $\ensuremath{\Gamma}$ (or $V$) has gauge-invariant values for any solution of the Euler-Lagrange equation $\frac{\ensuremath{\delta}\ensuremath{\Gamma}}{\ensuremath{\delta}\ensuremath{\varphi}}=0$ (or $\frac{\ensuremath{\partial}V}{\ensuremath{\partial}\ensuremath{\varphi}}=0$) 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 $\ensuremath{\Gamma}$. A wide class of gauges are allowed for the effective potential $V$; for instance, in scalar QED the allowed gauges are $\ensuremath{-}(\frac{1}{2}\ensuremath{\alpha}){(\ensuremath{\partial}A)}^{2}$, $\ensuremath{-}(\frac{1}{2}\ensuremath{\alpha}){(\ensuremath{\partial}A\ensuremath{-}v{\ensuremath{\Phi}}_{2})}^{2}$ (where the direction of condensation is restricted to ${\ensuremath{\Phi}}_{1}$), the Coulomb gauge, and the axial gauge. In particular the ${R}_{\ensuremath{\xi}}$ gauge is also an allowed gauge.