A More-Effective Potential

Abstract

In theories with spontaneous symmetry breaking, the loop expansion of the effective potential is awkward. In such theories, the exact effective potential $V(\phi_c,T)$ is real and convex (as a function of the classical field $\phi_c$), but its perturbative series can be complex with a real part that is concave. These flaws limit the utility of the effective potential, particularly in studies of the early universe. A generalization of the effective potential is available that is real, that has no obvious convexity problems, and that can be computed in perturbation theory. For the theory with classical potential $V(\phi) = (\lambda/4)(\phi^2 - \sigma^2)^2$, this more-effective potential closely tracks the usual effective potential where the latter is real $|\phi_c| \geq \sigma/\sqrt{3}$ and naturally extends it to $\phi_c = 0$, revealing that the critical temperature at the one-loop level runs from $T_C \approx 1.81 \sigma$ for $\lambda = 0.1$ to $T_C \approx 1.74 \sigma$ for $\lambda = 1$.