A Renormalisation Group Equation Study of the Scalar Sector of the Minimal B-L Extension of the Standard Model

TL;DR

This work derives the complete set of one-loop Renormalisation Group Equations for the minimal gauged $U(1)_{B-L}$ extension of the Standard Model, incorporating the extended scalar sector and three right-handed neutrinos. Focusing on the pure $B-L$ case, it maps the scalar potential to physical observables through the masses $m_{h_1}$, $m_{h_2}$ and mixing angle $\alpha$, and analyzes triviality and vacuum stability up to a high scale $Q$, with boundary conditions at the electroweak scale. The results delineate regions of parameter space where the model remains perturbative and vacuum-stable up to scales beyond the LHC, highlighting the roles of the singlet vev $x$, heavy-neutrino Yukawas $y^M$, and $Z'$ phenomenology in shaping viable Higgs-sector signatures. Overall, the study constrains Higgs-sector physics in the minimal $B-L$ extension and links it to neutrino and gauge-boson sectors, guiding future collider tests and theoretical consistency checks.

Abstract

We present the complete set of Renormalisation Group Equations (RGEs) at one loop for the non-exotic minimal U(1) extension of the Standard Model (SM). It includes all models that are anomaly-free with the SM fermion content augmented by one Right-Handed (RH) neutrino per generation. We then pursue the numerical study of the pure B-L model, deriving the triviality and vacuum stability bounds on an enlarged scalar sector comprising one additional Higgs singlet field with respect to the SM.

Figures (8)

• Figure 1: Maximum allowed values by eq. (\ref{['cond_g']}) for $g'_1(Q_{EW})$ in the $B-L$ model as a function of the scale $Q$.
• Figure 2: The $95\%$ C.L. upper bound on $\xi=g_{HZZ}/g^{SM}_{HZZ}$Barate:2003sz. In the $B-L$ model, $\xi=\cos{\alpha}(\sin{\alpha})$ for $H=h_1(h_2)$.
• Figure 3: Allowed values in the $m_{h_1}$ vs. $m_{h_2}$ space in the $B-L$ model by eqs. (\ref{['cond_1']}) and (\ref{['cond_2']}), for (\ref{['mh1_mh2_a0']}) $\alpha =0$, (\ref{['mh1_mh2_a01']}) $\alpha =0.1$, (\ref{['mh1_mh2_api4']}) $\alpha =\pi /4$ and (\ref{['mh1_mh2_api3']}) $\alpha =\pi /3$. Colours refer to different values of $Q/$GeV: blue ($10^{3}$), red ($10^{7}$), green ($10^{10}$), purple ($10^{15}$) and cyan ($10^{19}$). The shaded black region is forbidden by our convention $m_{h_2} > m_{h_1}$, while the shaded red region refers to the values of of the scalar masses forbidden by LEP. Here: $x=7.5$ TeV, $m_{\nu_h}=200$ GeV.
• Figure 4: Allowed values (that are those between the same colour lines) for $m_{h_2}$ as a function of the scale $Q$ in the $B-L$ model by eqs. (\ref{['cond_1']}) and (\ref{['cond_2']}), for several values of $m_{h_1}$ and (\ref{['mh2_vs_Q_a0']}) $\alpha =0$, (\ref{['mh2_vs_Q_a01']}) $\alpha =0.1$, (\ref{['mh2_vs_Q_pi8']}) $\alpha =\pi /8$ and (\ref{['mh2_vs_Q_api4']}) $\alpha = \pi /4$. Also, $x=3.5$ TeV and $m_{\nu_h}=200$ GeV. Only the allowed values by our convention $m_{h_2} > m_{h_1}$ are shown.
• Figure 5: Allowed values in the $m_{h_2}$ vs. $\alpha$ space in the $B-L$ model by eqs. (\ref{['cond_1']}) and (\ref{['cond_2']}), for (\ref{['mh2_a_mh1-100']}) $m_{h_1}=100$ GeV, (\ref{['mh2_a_mh1-100']}) $m_{h_1}=120$ GeV, (\ref{['mh2_a_mh1-100']}) $m_{h_1}=160$ GeV and (\ref{['mh2_a_mh1-100']}) $m_{h_1}=180$ GeV. Colours refer to different values of $Q/$GeV: blue ($10^{3}$), red ($10^{7}$), green ($10^{10}$), purple ($10^{15}$) and cyan ($10^{19}$). The plots already encode our convention $m_{h_2} > m_{h_1}$ and the shaded red region refers to the values of $\alpha$ forbidden by LEP. Here: $x=3.5$ TeV, $m_{\nu_h}=200$ GeV.