Bound states and the effective potential in field theories

Abstract

We calculate the leading contribution to the effective potential, $V\left(\phi \right)$ under the assumptions that there is a deep bound state and the corresponding pole dominates the one-particle irreducible vertices for spin-zero field theories. We find that $V$ has a $\frac{3}{2}$-power branch point in the $\phi$ plane on the real axis. We show that the $\frac{3}{2}$ power is due to the fact that $V^{\prime }=\frac{\mathrm{dV}}{d\phi }$ satisfies an algebraic equation quadratic and cubic in $V^{\prime }$ for the cases considered. In the domain of $\phi$ for which $V$ is real, the leading pole contribution is negative, allowing for the possibility of an instability in the normal vacuum.