Finite-temperature stability from doublet inflation field with right-handed neutrinos

TL;DR

This work extends the Standard Model by adding a Z2-odd inert SU(2) doublet and right-handed neutrinos to address inflation, dark matter, neutrino masses, and electroweak phase transition dynamics. Inflation is driven by the inert doublet through a non-minimal gravity coupling, yielding $\lambda_2/\zeta_2^2 \approx 4.4\times 10^{-10}$ and a reheating temperature near $10^{14}$ GeV; the RG running and two-loop $\beta$-functions constrain couplings to preserve perturbativity up to high scales. The electroweak phase transition is enhanced by the inert doublet, with strong FOPT requiring $\frac{\phi_+(T_c)}{T_c} \ge 1.0$ which fixes $m_{22}$ (e.g., $\sim 400$ GeV for $Y_N=0.01$ under GUT-scale unitarity), while Planck-scale unitarity relaxes this to $\frac{\phi_+(T_c)}{T_c} \ge 0.6$ but restricts $m_{22}$ to $\lesssim 70$ GeV for the same $Y_N$ and becomes incompatible for $Y_N=0.4$. Finite-temperature analysis shows thermal corrections increase tunneling probabilities, tightening stability bounds, though larger inert-doublet quartics can sustain EW vacuum stability up to the Planck scale for modest $Y_N$; a TeV-scale DM candidate arises from the inert doublet, with relic-density-consistent regions. Overall, the framework links early-universe inflation, dark matter, and electroweak cosmology within a perturbative and thermally consistent setup.

Abstract

We study the augmentation of the Standard Model (SM) with another $SU(2)$ Higgs doublet and right-handed neutrinos. The second Higgs doublet ($Φ_2$) is defined to be odd under the $Z_2$ symmetry, and hence, the lightest stable neutral particle from the additional doublet becomes the cold dark matter candidate. The right-handed neutrino field coupled to the Higgs field provides non-zero mass for the neutrinos. The inert doublet field coupled non-minimally to gravity as $ζ_2 Φ_2^\dagger Φ_2 R$ also acts as an inflaton field. The inflationary bounds restrict the interaction couplings as $λ_2/ζ_2^2 \approx 4\times 10^{-10}$. After inflation ends, the scalar bosonic degrees of freedom from the inert doublet can contribute to the electroweak phase transition. The strongly first-order phase transition bound, i.e., $\frac{φ_{+}(T_c)}{T_c} \geq 1.0$ restricts the bare mass parameter of the additional doublet to $m_{22}=400.0$ GeV, demanding GUT scale perturbative unitarity for $Y_N=0.01$. The increase in $Y_N$ reduces the strength of phase transition, and it is no longer satisfied even for vanishing bare mass parameter. The Planck scale perturbative unitarity allows for the first-order phase transition, $\frac{φ_{+}(T_c)}{T_c} \geq 0.6$, until $m_{22}=70.0$ GeV for $Y_N=0.01$, and none of the mass values satisfies the first-order phase transition for $Y_N=0.4$. The thermal corrections also affect the probability of tunneling from the false vacuum to the true vacuum, and hence, the finite temperature stability of the electroweak vacuum has been studied, including the finite-temperature effects.

Figures (13)

• Figure 1: (a) denotes the DM mass $M_{H^0}$ variation in GeV with the interaction quartic coupling $\lambda_L=(\lambda_3+\lambda_4+\lambda_5)$ and the mass splitting between the DM mass $M_{H^0}$ and $M_{A^0}$ is given in (b). The green points describes the allowed parameter space from strongly first-order phase transition. The running of the couplings is considered from the EW scale $m_t$, to $\mu=246$ GeV and the couplings are constrained to be perturbative till the scale $\mu=246$ GeV. The SM input parameters at two-loop level chosen at the EW scale are given in \ref{['tab:SMint']}.
• Figure 2: (a) denotes the variation of the interaction coupling $\lambda_3$ with the strength of phase transition for different values of the doublet mass parameter $(m_{22})$. The perturbative scale is chosen till $\mu=10^{15}$ GeV and the dashed pink line correspond to the maximum possible allowed value of the interaction quartic coupling $\lambda_3$ at the EW scale for $Y_N=0.01$ and $Y_N=0.4$ in \ref{['fig:06']}(a) and (b), respectively. The upper dashed black line correspond to the criteria for strongly first-order phase transition, i.e, $\phi_{+}(T_c)/T_c \sim 1$ and the lower one denotes the first-order phase transition, i.e, $\phi_{+}(T_c)/T_c \sim 0.6$.
• Figure 3: (a) Variation of the interaction quartic coupling $\lambda_3$ with the strength of phase transition $\phi_{+}(T_c)/T_c$ for $Y_N=0.01$ and $Y_N=0.4$ in (b). The perturbative scale is chosen to be $\mu=10^{19}$ GeV and the mass parameter is constrained to $m_{22}\mathrel{\mathop{ \hbox{$<$}} \hbox{$\sim$}}70$ GeV from the first-order phase transition as shown by (a).
• Figure 4: The variation of the field, $h_1$ with the euclidean distance, $r$ for quartic couplings $\lambda_i=0.01$ with $i \in \{2,3,4,5\}$ and $Y_N=0.01$. The field $h_1$, and the euclidean action are both rescaled with the Planck mass, $M_P=1.22 \times 10^{19}$ GeV. The value of the bounce solution at the center comes out to be $h_{1B}(0)=9.638 \times 10^{18}$ GeV.
• Figure 5: The variation of the action with temperature $T$ in GeV.