Single-scale Renormalisation Group Improvement of Multi-scale Effective Potentials

TL;DR

This work develops a single-scale RG-improvement framework for multi-scale effective potentials by solving the RG equation with boundary data on a tree-level hypersurface where one-loop corrections vanish. The resulting RG-improved potential takes the form of the tree-level potential evaluated at a field-dependent scale, enabling resummation of dominant logarithmic contributions without introducing multiple renormalisation scales. The method is shown to reproduce pivot-logarithm resummations in regimes with a dominant pivot and is demonstrated across several models, including SU(2)cSM, Higgs–Yukawa, and massless O(N) $\phi^4$, where stability analyses and large-field behavior are clarified. It offers a practical, numerically straightforward alternative to multi-scale RG methods while preserving gauge-independence at fixed-loop orders and extending applicability to complex scalar sectors. The approach hinges on perturbativity at the chosen scale and provides a coherent framework for analyzing radiative symmetry breaking and vacuum stability across energy scales.

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.

Figures (9)

• Figure 1: The tree-level hypersurface is symbolically represented by $\Sigma_*$. If we start from point $A$ and travel a distance $t'_*$ along the characteristic curve to reach point $B$, then we return to A by starting from $B$ and by travelling the same distance in the opposite direction.
• Figure 2: The one-loop unimproved effective potential $V_1$ (dashed lines) and the one-loop improved effective potential $V$ (solid lines) for two values of renormalisation scale $\mu$, $\mu=246\,\mathrm{GeV}$ (green) and $\mu=M_P$ (dark blue). In the left panel the value of $h$ is fixed to $246\,\mathrm{GeV}$ and the potentials are plotted as functions of $\varphi$. In the right panel $\varphi=600\,\mathrm{GeV}$ and the potentials are plotted along the $h$ direction. Note that three of the curves overlap, that is why it is hard to see the dark blue curves.
• Figure 3: Contour plots of $t_*^{(0)}(\mu, h, \varphi)$ for $h=246\,\mathrm{GeV}$ in different ranges of $\varphi$. In the left panel also the hypersurfaces $\mathbb{B}=0$ (red solid line) and $V^{(1)}=0$ (black dashed line) are displayed.
• Figure 4: Left panel: The one-loop unimproved effective potential $V_1$ (dashed lines) and the one-loop improved effective potential $V$ (solid lines) for two values of renormalisation scale $\mu$, $\mu=246\,\mathrm{GeV}$ (green) and $\mu=800\,\mathrm{GeV}$ (dark blue). The value of $h$ is fixed to $246\,\mathrm{GeV}$ at the scale $\mu=800\,\mathrm{GeV}$ (the value at $\mu=246\,\mathrm{GeV}$ is obtained as a value of the running field at this scale). Right panel: The one-loop improved (with $t_*^{(0)}$ approximation) and unimproved potential for $\mu=800\,\mathrm{GeV}$ showed in the left panel (dark blue) superimposed with the one-loop improved potential (with full $t_*$) (orange).
• Figure 5: The running coupling constant $\lambda$ (left panel) and the running mass term $m$ (right panel) evaluated at different field-dependent scales: $t_*^{(0)}$, $\log \frac{\phi}{\tilde{\mu}_0}$ and $\log\frac{\tilde{\mu}_*}{\tilde{\mu}_0}$, where $\tilde{\mu}_0=\textrm{exp}(3/4)\cdot 10 \textrm{TeV}$. The curves for $\log\frac{\tilde{\mu}_*}{\tilde{\mu}_0}$ are reproduced from ref. Casas.