The renormalization group improved effective potential in massless models

The effective potential $V$ is considered in massless $\lambda {\varphi }_4^4$ theory. The expansion of $V$ in powers of the coupling $\lambda$ and of the logarithm of the background field $\varphi$ is reorganized in two ways; first as a series in $\lambda$ alone, then as a series in $\ln \varphi$ alone. By applying the renormalization group (RG) equation to $V$, these expansions can be summed. Using the condition $V^{\mathrm{\prime }}(v)=0$ (where $v$ is the vacuum expectation value of $\varphi$) in conjunction with the expansion of $V$ in powers of $\ln \varphi$ fixes $V$ provided $v\mathrm{\ne }0$. In this case, the dependence of $V$ on $\varphi$ drops out and $V$ is not analytic in $\lambda$. Massless scalar electrodynamics is considered using the same approach.