Resummation of the Two Distinct Large Logarithms in the Broken $O(N)$-symmetric $\phi^4$-model

TL;DR

This paper introduces a new two-scale subtraction scheme, $ MS$, to systematically resum large logarithms in the effective potential of the $O(N)$-symmetric $^4$-model, revealing the absence of a stable vacuum for certain N values.

Contribution

A novel minimal two-scale subtraction scheme $ MS$ is developed for systematic resummation of large logarithms in the $O(N)$-symmetric $^4$-model.

Findings

No stable vacuum exists in the broken phase for 1<N≤4.

The $ MS$ scheme effectively resums large logarithms in the effective potential.

Leading logarithmic $ MS$ effective potential computed using standard boundary conditions.

Abstract

The loop-expansion of the effective potential in the $O(N)$-symmetric $\phi^4$-model contains generically two types of large logarithms. To resum those systematically a new minimal two-scale subtraction scheme $\tMS$ is introduced in an $O(N)$-invariant generalization of $\MS$. As the $\tMS$ beta functions depend on the renormalization scale-ratio a large logarithms resummation is performed on them. Two partial $\tMS$ renormalization group equations are derived to turn the beta functions into $\tMS$ running parameters. With the use of standard perturbative boundary conditions, which become applicable in $\tMS$, the leading logarithmic $\tMS$ effective potential is computed. The calculation indicates that there is no stable vacuum in the broken phase of the theory for $1<N\leq 4$.