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.
Highlights
A basic tool to study quantum corrections of a given quantum field theory is the effective action
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
If the mass scales are very different, it is not possible to choose a value of the renormalisation scale such that all the ratios are of the same order and all the logarithmic terms do not grow large
Summary
We begin by considering massless φ4-theory to illustrate our method of RG-improvement in a simple setting. Running along a characteristic curve can improve the validity of the perturbative expansion of the effective potential at a given loop order. With t log μ(t,μ) μ we can interpret λ(t, λ) as the value of the coupling parameter when the renormalisation scale is chosen to be μ, which is related to the corresponding value at the arbitrary reference scale μ by a resummation of terms of all orders in λ log μμ This observation is the key to regaining perturbativity in the case of large logarithms. We will argue that it is possible to improve the effective potential by solving the RG equation with the boundary given at the hypersurface of vanishing quantum corrections To put it differently, the appropriate generalisation to the multi-scale case consists of choosing the single renormalisation scale to be μ = μ∗, which is the scale at which V (1) = 0. The method presented is generalised to an arbitrary loop order in appendix A
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