Systematic analysis of radiative symmetry breaking in models with extended scalar sector

TL;DR

The paper investigates radiative symmetry breaking in models with extended scalar sectors, addressing how perturbative schemes and renormalization-scale choices affect the existence and location of radiatively generated minima. It surveys single-field and multifield cases, delineating when tree-level terms balance with loop corrections and how scale dependence emerges in multi-VEV settings, proposing RG-improved potentials to mitigate these issues. The authors apply these insights to the conformal Standard Model extended by an SU(2)_X doublet (SU(2)cSM), performing detailed one-loop and RG-improved analyses and comparing with the Gildener–Weinberg approach. They demonstrate that RG improvement reduces scale sensitivity and yields more reliable predictions for minima, masses, and couplings, supporting the viability of SU(2)cSM up to the Planck scale and underscoring implications for dark-sector dynamics and early-universe phenomena.

Abstract

Radiative symmetry breaking (RSB) is a theoretically appealing framework for the generation of mass scales through quantum effects. It can be successfully implemented in models with extended scalar and gauge sectors. We provide a systematic analysis of RSB in such models: we review the common approximative methods of studying RSB, emphasising their limits of applicability and discuss the relevance of the relative magnitudes of tree-level and loop contributions as well as the dependence of the results on the renormalisation scale. The general considerations are exemplified within the context of the conformal Standard Model extended with a scalar doublet of a new SU(2)$_X$ gauge group, the so-called SU(2)cSM. We show that various perturbative methods of studying RSB may yield significantly different results due to renormalisation-scale dependence. Implementing the renormalisation-group (RG) improvement method recently developed in arXiv:1801.05258, which is well-suited for multi-scale models, we argue that the use of the RG improved effective potential can alleviate this scale dependence providing more reliable results.

Figures (8)

• Figure 1: Results of the scan of the $(\lambda_2, g_{X})$ parameter space for the mass hierarchy $M_H<M_S$. Upper panel: contour plots of the values of the mass of the extra scalar $M_S$ (left) and of the new gauge bosons $M_X$ (right). The orange short-dashed lines show the decimal logarithm of the energy scale at which a Landau pole appears, the thick black line represents the boundary of the region where the potential is bounded from below at the Planck scale (to the right of the curve). The long-dashed red lines represent the contours of constant values of $\cos\theta$, see eq. \ref{['eq:mixing']}. Middle panel: values of the VEV of the $\varphi$ field, $w$. Lower panel: values of $\lambda_1$ and $\lambda_3$. The dot-dashed lines correspond to contours of constant values of the ratio given in eq. \ref{['eq:scalar-approx']}. The white dot represents the benchmark point from table \ref{['tab:BM']}.
• Figure 2: Results of the scan of the $(\lambda_2, g_{X})$ parameter space for the mass hierarchy $M_H>M_S$. Upper panel: contour plots of the values of the mass of the extra scalar $M_S$ (left) and of the new gauge bosons $M_X$ (right). The thick black line shows the boundary of the region where the potential is bounded from below (above the curve), the long-dashed red lines represent the contours of constant values of $|\sin\theta|$, see eq. \ref{['eq:mixing']}. Middle panel: the values of the VEV of the $\varphi$ field, $w$. Lower panel: the values of $\lambda_1$ and $\lambda_3$. The red dot-dashed lines in the lower panel correspond to contours of constant values of the ratio given in eq. \ref{['eq:scalar-approx']}. The grey region is excluded since there is no stable minimum in this region.
• Figure 3: Relative differences between masses and scalar VEVs computed using the GW method and the one-loop potential as described in section \ref{['sec:method-numerical']}. The relative differences are defined in eq. \ref{['eq:rel-dif']}. The thick black line shows the boundary of the stable region. The grey region is exluded.
• Figure 4: The scale $\mu_{\textrm{GW}}$ at which the GW determinant vanishes. Left panel: $M_H<M_S$, right panel: $M_H>M_S$. The light grey shaded regions are excluded since there at least one of the scalar masses becomes complex (the GW method does not find a minimum). The black lines denote the boundary of the region where potential is stable up to the Planck scale (in the left panel to the right of the curve, in the right panel above the curve). The grey region is excluded, the points there do not correspond to stable minima.
• Figure 5: The relative differences between the VEVs of the scalar fields, computed using the one-loop CW potential at $\mu_{\textrm{GW}}$ and the GW method (defined as in eq. \ref{['eq:rel-dif']} with the change that $v_{\textrm{CW}}$ and $w_{\textrm{CW}}$ are defined at $\mu_{\textrm{GW}}$). The black curve is the boundary of the region where the potential is stable up to the Planck scale, the shaded grey region is not viable.