General solutions for tunneling of scalar fields with quartic potentials

For the theory of a single scalar field $\varphi$ with a quartic potential $V\left(\varphi \right)$, we find semianalytic expressions for the Euclidean action in both four and three dimensions. The action in four dimensions determines the quantum tunneling rate at zero temperature from a false vacuum state to the true vacuum state; similarly, the action in three dimensions determines the thermal tunneling rate for a finite temperature theory. We show that for all quartic potentials the action can be obtained from a one-parameter family of instanton solutions corresponding to a one-parameter family of differential equations. We find the solutions numerically and use polynomial fitting formulas to obtain expressions for the Euclidean action. These results allow one to calculate tunneling rates for the entire possible range of quartic potentials, from the thin-wall (nearly degenerate) limit to the opposite limit of vanishing barrier height. We also present a similar calculation for potentials containing ${\varphi }^4\ln {\varphi }^2$ terms, which arise in the one-loop approximation to the effective potential in electroweak theory.