First-principles quantitative viability of electroweak baryogenesis from transient spontaneous CP violation in the 2HDM+a (bubble-wall velocity, dark-matter interplay, and the radiative origin of the CP-breaking phase)

Electroweak baryogenesis driven by a transient period of spontaneous CP violation — realized in a two-Higgs-doublet model extended by a singlet pseudoscalar $a$ (2HDM+ $a$ ), where $\langle a\rangle\neq 0$ in the early Universe and CP is restored after the electroweak phase transition — evades electric-dipole-moment bounds because CP is (approximately) conserved today. The original 2023 analysis established the viable parameter region only at tree level plus leading $\mathcal{O}(T^2)$ thermal corrections, with an analytic semiclassical estimate of the baryon asymmetry and an assumed bubble-wall velocity. The authors' 2025 follow-up upgraded the computation to the full one-loop finite-temperature effective potential with daisy resummation and all-wall-velocity transport equations, finding the baryon asymmetry suppressed by roughly an order of magnitude and requiring larger singlet–doublet pseudoscalar mixing ( $\sin\theta\gtrsim 0.15$ ). What remains open is the fully first-principles completion: computing the bubble-wall velocity from the model's own scalar–plasma dynamics and redoing the baryon-asymmetry prediction with it; mapping the simultaneous requirements of baryogenesis and thermal-relic dark matter in the same model; deriving the tiny explicit CP-breaking phase $\delta_{12}$ on which the mechanism relies; and extending the scan beyond the alignment limit and to high scalar masses — jointly settling whether this predictive, collider-testable scenario survives.

Background

The Universe contains matter but essentially no antimatter, quantified by the baryon asymmetry $n_B/s\simeq 8.7\times 10^{-11}$ (baryon number per entropy). Sakharov's conditions for dynamically generating it require baryon-number violation, C and CP violation, and departure from thermal equilibrium. In electroweak baryogenesis (EWBG) all three arise around the electroweak scale: sphalerons — the thermal tunneling configurations that violate baryon number in the Standard Model (SM) — are active in the symmetric phase; CP-violating interactions of particles traversing the walls of bubbles of the broken phase separate chiral charge, which sphalerons convert to baryon number; and departure from equilibrium is provided by a first-order electroweak phase transition (EWPT), in which Higgs-field bubbles nucleate and expand through the hot plasma. Successful EWBG requires the transition to be strong enough that sphalerons are switched off inside bubbles, and enough new CP violation outside the SM, which is typically constrained by stringent electric-dipole-moment (EDM) limits on CP-violating beyond-the-SM physics today. The scenario studied here (Huber, Mimasu and No, Phys. Rev. D 107 (2023) 075042) realizes EWBG in a non-minimal Higgs sector of two Higgs doublets plus a singlet pseudoscalar $a$ (the 2HDM+ $a$ ): a finite-temperature minimum with $\langle a\rangle\neq 0$ spontaneously breaks CP in the early Universe, and the electroweak phase transition later restores CP by driving $\langle a\rangle\to 0$ , so present-day EDMs are barely affected. The baryon asymmetry of the Universe (BAU) is computed by solving transport equations — coupled Boltzmann equations for the perturbations of particle densities near the bubble wall, with CP-violating source terms from the varying Higgs background — feeding the net chiral charge to sphaleron washout; the result depends sensitively on the wall thickness $L_W$ and wall velocity $v_W$ , which are themselves determined by the friction of the plasma on the expanding wall. Predictions require the finite-temperature effective potential, computed at one loop with daisy (ring) resummation — the resummation of thermal mass insertions that tames infrared-divergent Matsubara zero modes — in a fixed (Landau) gauge. Because the transition is (or approaches) first order, it can also source a stochastic gravitational-wave (GW) background, potentially detectable at space-based detectors such as LISA. The 2023 analysis necessarily approximated each of these ingredients: tree-level potential plus $\mathcal{O}(T^2)$ corrections, an analytic semiclassical BAU estimate in an assumed wall-thickness regime with bracketed $v_W$ , dark-matter (DM) interplay deferred (the model naturally extends to a Dirac fermion DM candidate $\chi$ coupled to $a$ ), no GW study, and a merely argued radiative origin for the tiny explicit CP phase $\delta_{12}$ (a bias $\delta_{12}\gtrsim 10^{-15}$ is needed so that domains of opposite CP sign do not annihilate the generated asymmetry, while EDM limits demand it stay extremely small). Prior work by Cline and Kainulainen (Phys. Rev. D 101 (2020) 063525) derived fluid transport equations valid for arbitrary wall velocity, showing smooth BAU suppression with $v_W$ , and Ai, Laurent and van de Vis (JHEP 02 (2025) 119) derived collision-term-independent upper (local-thermal-equilibrium) and lower (ballistic) bounds on $v_W$ — machinery that is available but has not been replaced by a true first-principles wall-velocity solution for this model. A recent review (van de Vis, de Vries and Postma, Prog. Part. Nucl. Phys. 2026) documents wall velocity, transport treatment, and sphaleron rates as the field-wide precision frontier of EWBG.

Problem Statement

Within the 2HDM+ $a$ realization of electroweak baryogenesis from transient spontaneous CP violation, complete the quantitative first-principles determination of whether the observed baryon asymmetry $n_B/s\simeq 8.7\times 10^{-11}$ can be generated in a region of parameter space consistent with all theoretical and experimental constraints. The common objective uniting the requested results is a fully controlled prediction of the baryon asymmetry and its correlated experimental signatures — or the exclusion of the scenario. Specifically: (1) compute the bubble-wall velocity $v_W$ (and thickness $L_W$ ) from the model's own microphysics in place of ballistic/LTE bracketing, and determine how the viable baryogenesis region in the $(m_{a_1},\sin\theta,M,\tan\beta)$ planes (where $m_{a_1}$ is the lightest CP-odd scalar mass, $\sin\theta$ the singlet–doublet pseudoscalar mixing angle, $M$ the common mass scale of the heavier two-Higgs-doublet scalars, and $\tan\beta$ the ratio of the two Higgs-doublet vacuum expectation values), established with the full one-loop daisy-resummed finite-temperature potential, shifts once $v_W$ is predicted rather than scanned; (2) map the simultaneous requirements of successful baryogenesis and thermal-relic dark matter in the same model, i.e. how the coupling $y_\chi$ fixed by the observed relic density of the Dirac fermion $\chi$ modifies the CP-violating transition temperature $T_S$ and the viable region, and how the opened decay $a_1\to\chi\bar\chi$ reshapes the collider testability; (3) determine the radiative origin and magnitude of the tiny explicit CP-breaking phase $\delta_{12}$ in concrete UV embeddings — determining whether the generated $\sin\delta_{12}$ exceeds the domain-bias threshold $\sim 10^{-15}$ while remaining consistent with EDM bounds; and (4) extend the finite-temperature analysis beyond the alignment limit (the decoupling regime in which the observed 125 GeV Higgs boson is the purely Standard-Model-like CP-even scalar) and to common 2HDM mass scales above $\sim 700$ GeV (requiring more stable numerics for the bounce action — the Euclidean tunneling action whose extremum determines the bubble-nucleation rate), establishing the correlated predictions — exotic Higgs decays, flavour observables, EDMs, and the stochastic gravitational-wave signal of the viable region — or excluding the scenario. The scientific objective is the joint resolution of items (1)–(4).

References

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T. Gent, S. Huber, K. Mimasu, J. M. No, "Towards precise baryogenesis in the 2HDM+ $a$ ", arXiv:2512.22081, https://arxiv.org/abs/2512.22081
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