Summing Radiative Corrections to the Effective Potential

Abstract

When one uses the Coleman-Weinberg renormalization condition, the effective potential $V$ in the massless $\ddot{I}{•}_4^4$ theory with O(N) symmetry is completely determined by the renormalization group functions. It has been shown how the $(p+1)$ order renormalization group function determine the sum of all the N$^{\mbox{\scriptsize p}}$LL order contribution to $V$ to all orders in the loop expansion. We discuss here how, in addition to fixing the N$^{\mbox{\scriptsize p}}$LL contribution to $V$, the $(p+1)$ order renormalization group functions also can be used to determine portions of the N$^{\mbox{\scriptsize p+n}}$LL contributions to $V$. When these contributions are summed to all orders, the singularity structure of \mcv is altered. An alternate rearrangement of the contributions to $V$ in powers of $\ln \ddot{I}•$, when the extremum condition $V^{\prime }(\ddot{I}•=v)=0$ is combined with the renormalization group equation, show that either $v=0$ or $V$ is independent of $\ddot{I}•$. This conclusion is supported by showing the LL, $\cdots$, N${}^4$LL contributions to $V$ become progressively less dependent on $\ddot{I}•$.

16 pages; added 2 figures and 2 tables; references revised