Abstract
We propose a new formalism to analyse the extremum structure of scale-invariant effective potentials. The problem is stated in a compact matrix form, used to derive general expressions for the stationary point equation and the mass matrix of a multi-field RG-improved effective potential. Our method improves on (but is not limited to) the Gildener-Weinberg approximation and identifies a set of conditions that signal the presence of a radiative minimum. When the conditions are satisfied at different scales, or in different subspaces of the field space, the effective potential has more than one radiative minimum. We illustrate the method through simple examples and study in detail a Standard-Model-like scenario where the potential admits two radiative minima. Whereas we mostly concentrate on biquadratic potentials, our results carry over to the general case by using tensor algebra.
Highlights
The emergence of radiative minima in scale-invariant potentials is relevant to many problems in contemporary physics
We propose a new formalism to analyse the extremum structure of scaleinvariant effective potentials
We have considered the minimisation of effective potentials with multiple scalar fields by way of a novel matrix formalism in which calculations are presented in a compact and intuitive manner
Summary
The emergence of radiative minima in scale-invariant potentials is relevant to many problems in contemporary physics. Because observables cannot depend on an arbitrary parameter, the explicit dependence of a process on this quantity must be compensated by an implicit dependence of the involved physical parameters This is the essence of the Callan-Symanzik equation, which determines the evolution of couplings, masses and fields with the renormalisation scale, resulting in the loss of scale symmetry at the quantum level. That quantum fluctuations generally violate scale invariance is demonstrated by the mechanism of dimensional transmutation [55] In this case, the quantum corrections induce the spontaneous symmetry breaking of the theory by producing a radiatively generated minimum in an otherwise trivial scalar potential. The computation of the minima of the effective potential is generally a cumbersome task: on top of the dimensionality of the problem that scales with the number of scalar fields, the analysis is complicated by the nature of the radiative corrections.
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