A Multi-scale Subtraction Scheme and Partial Renormalization Group Equations in the $O(N)$-symmetric $ϕ^4$-theory

Abstract

To resum large logarithms in multi-scale problems a generalization of $\MS$ is introduced allowing for as many renormalization scales as there are generic scales in the problem. In the new \lq\lq minimal multi-scale subtraction scheme&#39;&#39; standard perturbative boundary conditions become applicable. However, the multi-loop beta functions depend on the various renormalization scale ratios and a large logarithms resummation has to be performed on them. Using these improved beta functions the \lq\lq partial&#39;&#39; renormalization group equations corresponding to the renormalization point independence of physical quantities allows one to resum the logarithms. As an application the leading and next-to-leading order two-scale analysis of the effective potential in the $O(N)$-symmetric $ϕ^4$-theory is performed. This calculation indicates that there is no stable vacuum in the broken phase of the theory for $1<N\leq4$.