Consistent Use of Effective Potentials
It is well known that effective potentials can be gauge-dependent while their values at extrema should be gauge-invariant. Unfortunately, establishing this invariance in perturbation theory is not straightforward, since contributions from arbitrarily high- order loops can be of the same size. We show in massless scalar QED that an infinite class of loops can be summed (and must be summed) to give a gauge invariant value for the potential at its minimum. In addition, we show that the exact potential depends on both the scale at which it is calculated and the normalization of the fields, but the vacuum energy does not. Using these insights, we propose a method to extract some physical quantities from effective potentials which is self-consistent order-by-order in perturbation theory, including improvement with the renormalization group.