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Dark Matter and Electroweak Baryogenesis with Spontaneous $CP$ Violation in the Early Universe

Subhojit Roy

Abstract

Dark matter (DM) and the baryon asymmetry of the universe (BAU) are among the most compelling indications of physics beyond the Standard Model. We revisit the inelastic Higgs-portal complex singlet, a minimal framework in which a complex scalar splits into two nearly degenerate real states, with an off-diagonal Higgs-portal interaction that drives coannihilation to set the relic density, while the elastic DM-Higgs coupling can be tuned small enough to evade direct-detection limits. This setup naturally supports a strong first-order electroweak phase transition (SFOEWPT) and can account for the long-standing Galactic Center gamma-ray excess (GCE) via present-day DM annihilation into Higgs pairs. In this work, we show that the same framework, extended by a $Z_2$-symmetric dimension-6 $CP$-violating top Yukawa operator, can also generate the BAU via the electroweak baryogenesis (EWBG) mechanism. The cosmological history involves a two-step electroweak phase transition: first, the singlet fields acquire nonzero vacuum expectation values (vevs); then a strongly first-order transition occurs in which the Higgs develops its nonzero vev while the singlet vevs vanish. During this second step, both fields remain nonzero only within the advancing bubble wall, generating wall-localized $CP$ violation that biases sphaleron transitions and enables EWBG. After the phase transition, $CP$ and $Z_2$ symmetries are restored: the lightest singlet state becomes a stable DM candidate, while the vanishing singlet vevs allow the model to naturally satisfy the stringent constraints on $CP$ violation. We delineate the SFOEWPT-favored parameter space, identifying the criteria for the two-step phase transition region that simultaneously yields the observed BAU and relic density, explains the GCE, and predicts gravitational wave spectra accessible to next-generation space-based detectors.

Dark Matter and Electroweak Baryogenesis with Spontaneous $CP$ Violation in the Early Universe

Abstract

Dark matter (DM) and the baryon asymmetry of the universe (BAU) are among the most compelling indications of physics beyond the Standard Model. We revisit the inelastic Higgs-portal complex singlet, a minimal framework in which a complex scalar splits into two nearly degenerate real states, with an off-diagonal Higgs-portal interaction that drives coannihilation to set the relic density, while the elastic DM-Higgs coupling can be tuned small enough to evade direct-detection limits. This setup naturally supports a strong first-order electroweak phase transition (SFOEWPT) and can account for the long-standing Galactic Center gamma-ray excess (GCE) via present-day DM annihilation into Higgs pairs. In this work, we show that the same framework, extended by a -symmetric dimension-6 -violating top Yukawa operator, can also generate the BAU via the electroweak baryogenesis (EWBG) mechanism. The cosmological history involves a two-step electroweak phase transition: first, the singlet fields acquire nonzero vacuum expectation values (vevs); then a strongly first-order transition occurs in which the Higgs develops its nonzero vev while the singlet vevs vanish. During this second step, both fields remain nonzero only within the advancing bubble wall, generating wall-localized violation that biases sphaleron transitions and enables EWBG. After the phase transition, and symmetries are restored: the lightest singlet state becomes a stable DM candidate, while the vanishing singlet vevs allow the model to naturally satisfy the stringent constraints on violation. We delineate the SFOEWPT-favored parameter space, identifying the criteria for the two-step phase transition region that simultaneously yields the observed BAU and relic density, explains the GCE, and predicts gravitational wave spectra accessible to next-generation space-based detectors.

Paper Structure

This paper contains 14 sections, 75 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Theoretical Framework
  3. The Inelastic Higgs-Portal Dark Matter Model
  4. Finite Temperature Corrections to the Effective Potential
  5. Quantum Corrections at Zero Temperature
  6. Thermal Corrections
  7. Daisy Resummation
  8. Scalar and Gauge Boson Masses at Finite Temperature
  9. Generalities of Electroweak Baryogenesis
  10. Gravitational Wave Signatures from a First-Order Phase Transition
  11. Dark Matter Phenomenology
  12. FOEWPT-favored parameter space
  13. Spontaneous $CP$ violation and the generated baryon asymmetry
  14. Conclusions

Figures (6)

  • Figure 1: Feynman diagrams for DM annihilation channels at $f_1 \to 0$. Top row: (a) $\phi_1\phi_1\!\to hh$ (via $\phi_2$ exchange), (b) $\phi_1\phi_2\!\to \mathrm{SM}\,\mathrm{SM}$ (via $h$), (c) $\phi_1\phi_2\!\to hh$ (contact $\phi_1\phi_2 h^2$), (d) $\phi_1\phi_2\!\to hh$ (via $\phi_2$ exchange). Bottom row: (e) $\phi_2\phi_2\!\to hh$ (contact $\phi_2^2 h^2$), (f) $\phi_2\phi_2\!\to \mathrm{SM}\,\mathrm{SM}$ (via $h$), (g) $\phi_2\phi_2\!\to hh$ (via $\phi_1$ exchange), (h) $\phi_2\phi_2\!\to hh$ (via $\phi_2$ exchange). Dashed lines denote scalars ($\phi_i$, $h$); solid lines denote the SM fields. Here, the corresponding $u$-channel contributions, analogous to the $t$-channel diagrams, are not displayed.
  • Figure 2: Relic abundance $\Omega_{\phi_1} h^2$ as a function of $m_{\phi_1}$ for representative benchmark sets of $(\Delta m,\,g,\,f_2)$. The coupling $f_1$ is fixed at $10^{-5}$, chosen small enough to ensure that elastic Higgs-mediated scattering remains well below current direct-detection limits Hooper:2025fdaLZCollaboration:2024lux. The horizontal dashed line indicates the Planck measurement of the DM relic abundance Planck:2018vyg.
  • Figure 3: Four one-loop diagrams contributing to spin-independent scattering of $\phi_1$ on a nucleon $n$. Dashed lines: scalars $(\phi_{1}, \phi_2, h)$; solid lines: nucleon $n$.
  • Figure 4: [Left]: Scan in the $m_{\phi_2}$–$f_2$ plane showing points that realize an FOEWPT. We vary $m_{\phi_2}\in[100,250]~\mathrm{GeV}$ and $\Delta m \in[10,20]~\mathrm{GeV}$, while holding $\lambda_{1,2,12} \sim 0.5$ and $g \sim 0.05$ fixed. [Right]: Peak amplitude $\left(\Omega_{\mathrm{GW}} h^2\right)_{\mathrm{peak}}$ versus peak frequency $f_{\mathrm{peak}}$ of the predicted stochastic GW signal from FOEWPT-allowed points. Sensitivity curves for prospective GW observatories LISA, Taiji, ALIA, BBO, and UDECIGO are overlaid for comparison. In both panels, the color scale encodes the transition strength $\xi_n \equiv v_n/T_n$.
  • Figure 5: Profiles of $h(z)$, $\phi_1(z)$, and $\phi_2(z)$ across the bubble wall (left plot), and the left-handed baryon chemical potential, $\mu_{B_L}$, obtained from solving the transport equations (right plot). Both are shown as functions of the coordinate $z$, transverse to the bubble wall, at $T = T_n$ for the benchmark point given in Tab. \ref{['bptable']}.
  • ...and 1 more figures