Multi-step phase transition and gravitational wave from general $\mathbb{Z}_2$ scalar extensions

TL;DR

This work analyzes a Z2-symmetric N-plet scalar extension of the Standard Model to realize a two-step electroweak phase transition. It develops the finite-temperature one-loop effective potential, derives analytic conditions for a two-step path, and identifies narrow but viable regions in parameter space where the second step is strongly first-order and produces a detectable gravitational-wave signal. The study finds that the second-step barrier is often tree-level in origin, favoring GW frequencies below the sub-Hz range, while the first step may also be first-order via thermal loops but is typically less accessible to detection. Collider constraints, especially on h to gamma gamma, are shown to be compatible with the two-step scenario, and future GW experiments such as LISA and BBO can probe significant portions of the viable parameter space, potentially revealing the restoration of the discrete symmetry after electroweak breaking.

Abstract

Multi-step phase transition provides a paradigm in which a broken symmetry during phase transition can be restored, enriching the phenomena of both dark matter and baryon asymmetry. We study the dynamics of the multi-step phase transition in the standard model extension with additional isospin $N$-plet scalar field $Φ_2$ under a discrete $\mathbb{Z}_2$ symmetry. We find that the multi-step phase transition could be triggered if there is a moderately large coupling between the Higgs and the $Φ_2$ and this coupling is required to be larger as the mass of the $Φ_2$ and/or isospin increase. The first-order phase transition at the first (second) step can be realized by the thermal loop (tree-level barrier) effects. Thus it is more likely that a detectable spectrum of gravitational waves can be produced at the second step of the phase transition.

Figures (12)

• Figure 1: The path of phase transitions in the model with two classical fields $\left\langle \Phi_1 \right\rangle$ and $\left\langle \Phi_2 \right\rangle$. The blue cross marks represent the global minimum point of the potential at zero temperature. The value of $\left\langle \Phi_2 \right\rangle$ is zero in (a) and (b) phase transitions, while it is nonzero in (c)--(e) phase transitions, respectively.
• Figure 2: Extrema in the tree-level potential.
• Figure 3: Parameter regions that satisfies the conditions of Eqs. \ref{['eq:con1']}, \ref{['eq:con2']} and \ref{['eq:con3']}, with $(I_2, Y_2) = (1/2, 1/2)$ and $m_H = 100, 200, 300 \,{\text{GeV}}$. The region enclosed by blue, yellow, and green dashed lines corresponds to each condition, and the red is the union region. It could be found that the allowed regions for $\lambda_{12}$ are narrow belts scaling as $m_H$, and the upper bounds are set by the 'global minimum' condition. Also, the lower bounds from 'local minimum' and $T_2 > T_1$ are quite close, indicating that the phase transition goes through two steps given the condition that there are two local minima.
• Figure 4: The potential as the temperature varieties. The left column shows the effective potential at $T = 150~{\text{GeV}}$, $118~{\text{GeV}}$ and $T = 0~{\text{GeV}}$ respectively with $\lambda_2 = \lambda_{12} = 2.0,~ m_H = 200 ~{\text{GeV}}$ and $(I_2, Y_2) = (1/2, 1/2)$, while the right is $T = 150 ~{\text{GeV}}$, $140 {\text{GeV}}$ and $T = 0 ~{\text{GeV}}$ respectively with $\lambda_2 = 0.25,~ \lambda_{12} = 3.02,~ m_H = 300~ {\text{GeV}}$ and $(I_2, Y_2) = (1, 1)$. Only the left case goes through a two-step phase transition.
• Figure 5: Polar coordinates in the effective potential. $z$ and $\gamma$ are a radius and an angle in the coordinates.