Single-scale renormalisation group improvement of multi-scale effective potentials

Abstract

We present a new method for renormalisation group improvement of the effective potential of a quantum field theory with an arbitrary number of scalar fields. The method amounts to solving the renormalisation group equation for the effective potential with the boundary conditions chosen on the hypersurface where quantum corrections vanish. This hypersurface is defined through a suitable choice of a field-dependent value for the renormalisation scale. The method can be applied to any order in perturbation theory and it is a generalisation of the standard procedure valid for the one-field case. In our method, however, the choice of the renormalisation scale does not eliminate individual logarithmic terms but rather the entire loop corrections to the effective potential. It allows us to evaluate the improved effective potential for arbitrary values of the scalar fields using the tree-level potential with running coupling constants as long as they remain perturbative. This opens the possibility of studying various applications which require an analysis of multi-field effective potentials across different energy scales. In particular, the issue of stability of the scalar potential can be easily studied beyond tree level.

Gravitational waves from conformal symmetry breaking

“…As explained in refs. [85,90], if the potential beyond tree level is considered, these constrains should be imposed on the running couplings evaluated at some high energy scale (e.g. the Planck scale).…”

Vacuum stability of Froggatt-Nielsen models

“…In general, there is no straightforward way to resum all the leading logs using the usual RG methods (not even in the SM, but large logs from light particles are not expected to contribute strongly to the potential, since this would lead to IR divergences in the limit of vanishing masses). Several papers in the literature have proposed solutions to this problem that rely on a generalized MS multiscale approach [10,11,12,13,14]. We will follow a related but slightly different route here.…”

“…In the current study, we have the additional complication that we have to deal with two scalar VEVs. There are several proposals in the literature how to obtain a RG-improved effective potential in this circumstance [10,11,12,13,14]. We will see that we do not require these techniques here and discuss in detail how to resum the leading logs in our context.…”

Construction of a renormalization group improved effective potential in a two real scalar system

“…In ref. [30], a new method is suggested. The guiding principle for the method is to choose the renormalization scale so that the total loop correction vanishes.…”

“…[31] the RG improved effective potential in classical conformal theory is analyzed based on the method of ref. [30]. In the present paper, we also approach the problem for the RG improvement of the effective potential.…”