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

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.

💡 Research Summary

The authors address five outstanding shortcomings of the Standard Model—absence of inflation, a viable dark‑matter candidate, non‑zero neutrino masses, a strongly first‑order electroweak phase transition (EWPT), and the long‑term stability of the electroweak vacuum—by introducing a minimal extension that adds a second SU(2) scalar doublet Φ₂ and three right‑handed neutrinos Nᵢ. A discrete Z₂ symmetry is imposed under which Φ₂ is odd while all other fields are even. Consequently, Φ₂ does not acquire a vacuum expectation value, and the lightest neutral component (chosen to be H⁰) is stable and serves as a cold dark‑matter particle.

The scalar potential is the usual inert‑doublet form with quartic couplings λ₁…λ₅ and a bilinear mass term m₂₂² for Φ₂. The right‑handed neutrinos couple to the SM Higgs doublet Φ₁ via Yukawa matrix Y_N, providing Dirac masses for the active neutrinos after electroweak symmetry breaking.

A crucial feature of the model is a non‑minimal coupling of Φ₂ to gravity, ζ₂ Φ₂†Φ₂ R. During inflation the dynamics are dominated by the neutral component φ of Φ₂, while Φ₁ is set to zero. After a Weyl transformation to the Einstein frame the effective inflaton potential becomes

U(φ)= λ₂ φ⁴ /