Complex Scalar Singlet Model: Electroweak Phase Transition and Gravitational Waves

TL;DR

The paper investigates electroweak phase transitions in the cxSM, a minimal SM extension with a complex singlet scalar. It develops the most general renormalizable scalar potential, computes the one-loop finite-temperature effective potential with daisy resummation, and conducts a comprehensive parameter scan under theoretical and experimental constraints to identify strong first-order phase transition regions. It demonstrates both single-step and multi-step SFOPTs and presents benchmark points with distinct field-direction dynamics, predicting gravitational-wave spectra that can be probed by next-generation detectors such as LISA, BBO, DECIGO, and U-DECIGO. The study highlights a compelling multimessenger picture where electroweak baryogenesis can be accompanied by detectable stochastic gravitational waves, while also outlining future work on CP-violation mechanisms and collider probes to further constrain the model.

Abstract

The Standard Model (SM) cannot explain the observed baryon asymmetry of the Universe (BAU), thus driving the need for physics beyond the SM, which can generate electroweak baryogenesis through a strong first-order electroweak phase transition (SFOPT). We extend the SM with a complex singlet scalar (cxSM) and examine the phase transition behavior using a fully general renormalizable scalar potential that permits a complex vacuum expectation value for the singlet and coupled dynamics among multiple scalar fields. Employing the one-loop thermal effective potential with daisy resummation and appropriate counter terms, we conduct an extensive scan of the parameter space, enforcing both theoretical and experimental limits on the scalar sector. This analysis reveals viable domains yielding SFOPT. From these regions, we select representative benchmark scenarios demonstrating multi-stage transitions, producing stochastic gravitational wave signals via bubble nucleation dynamics. The resulting spectra lie within the projected sensitivity of next-generation observatories, including LISA, BBO, DECIGO, and U-DECIGO. Thus, the cxSM offers a compelling setting for electroweak baryogenesis, enriched by correlated gravitational-wave and collider phenomenology.

Figures (6)

• Figure 1: Parameter space satisfying SFOPT in the cxSM. The gray points are allowed by all the constraints, while the colored points exhibit SFOPT. The colored markers denote the benchmark points for which the details are presented in Table \ref{['tab_bp_details']}.
• Figure 2: Upper panel (\ref{['fig_bp1_vev_evolve']}) shows the evolution of all the fields as a function of temperature for BP1, where the different phases are denoted via solid and dashed lines. The vertical dashed gray and purple lines denote the critical and nucleation temperatures, respectively. Lower panel (\ref{['fig_bp1_phase_diagram']}) depicts the phase diagram for BP1 in the three different field spaces, while temperature has been shown parametrically in the color bar with the arrows denoting the discontinuity of fields at the critical temperature. The numerical values of all the fields and temperature shown in the figures are in the units of GeV.
• Figure 3: Same as Fig. \ref{['fig_bp1_vev_evolve']} for BP2 (left panel) and BP3 (right panel).
• Figure 4: Same as Fig. \ref{['fig_bp1_vev_evolve']} for BP4 (left panel) and BP5 (right panel).
• Figure 5: The stochastic GW spectrum for BP1 (left panel), BP4 (middle panel) and BP5 (right panel). Individual contributions for each benchmark point from sound wave, magnetohydrodynamic turbulence, and bubble collisions are shown in red, green, and blue solid lines corresponding to Eq. \ref{['eq: omega_sw']}, \ref{['eq: omega_turb']}, and \ref{['eq: omega_col']} respectively. The dashed gray, violet, yellow, and light blue lines correspond to the sensitivity curves for LISA, BBO, DECIGO, and U-DECIGO respectively.