Summing Radiative Corrections to the Effective Potential

When one uses the Coleman-Weinberg renormalization condition, the effective potential \(V\) in the massless \(\phi_4^4\) 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 \(\ln \phi\), when the extremum condition \(V^\prime (\phi = v) = 0\) is combined with the renormalization group equation, show that either \(v = 0\) or \(V\) is independent of \(\phi\). This conclusion is supported by showing the LL, \(\cdots\), N\(^4\)LL contributions to \(V\) become progressively less dependent on \(\phi\).