Summing Radiative Corrections to the Effective Potential

TL;DR

This paper explores how renormalization group functions can sum radiative corrections to the effective potential in massless scalar theory, revealing conditions under which the potential becomes independent of the field or vanishes.

Contribution

It demonstrates how higher-order renormalization group functions can determine and sum contributions to the effective potential, affecting its singularity structure and dependence on the field.

Findings

Summation of all orders alters the potential's singularity structure.

The extremum condition implies either a zero vacuum expectation value or a field-independent potential.

Higher-order contributions become less dependent on the field as order increases.

Abstract

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 $\phi$.