Reheating after the Supercooled Phase Transitions with Radiative Symmetry Breaking

Theories with radiative symmetry breaking (RSB) lead to first-order phase transitions and the production of gravitational waves as well as primordial black holes if the supercooling period lasted long enough. Here we explain how to efficiently reheat the universe after such period in the above-mentioned class of theories. Two cases are possible, depending on whether the RSB scale is much larger than the electroweak (EW) symmetry breaking scale or not. When it is, the dominant reheating mechanism can be the decays of the field responsible for RSB in the Standard Model (SM) sector. We point out that in a similar way dark matter (DM) can be produced and we analyze in some detail the case of a sterile-neutrino, finding that the full DM abundance is reproduced when this particle is at the $10^2$ MeV scale in a well-motivated SM completion. When the RSB scale is not much larger than the EW symmetry breaking scale, we find that efficient reheating always occurs when the energy density of the false vacuum is first entirely transferred to a dark photon and then to SM fermions via dark-photon decays.

💡 Research Summary

The paper investigates how the Universe can be efficiently reheated after a prolonged super‑cooled first‑order phase transition (PT) that occurs in theories where symmetry breaking is driven radiatively (Radiative Symmetry Breaking, RSB). In RSB the scalar field χ develops a flat direction at a particular renormalization scale, and quantum corrections generate a new scale χ₀ via dimensional transmutation. The χ field acquires a mass mχ = √β̄ χ₀, where β̄ is the one‑loop beta‑function of the quartic coupling along the flat direction.

Two qualitatively different regimes are identified, depending on the hierarchy between the RSB scale χ₀ and the electroweak (EW) scale vEW.

1. χ₀ ≫ vEW (SM‑connected RSB).
In this case the Standard Model (SM) is embedded in the RSB sector, so χ couples directly to SM particles through gauge and Yukawa interactions. The authors write the most general no‑scale Lagrangian, diagonalise the scalar, vector and fermion mass matrices in unitary gauge, and derive the interaction terms that allow χ to decay into SM scalars, vectors, and fermions. They compute the two‑body decay widths Γ₂S, Γ₂F and Γ₂V (Eqs. 4.3‑4.5) and also the three‑body channels χ → ϕϕϕ and χ → ϕ B B (Eqs. 4.6‑4.7). The rates are expressed in terms of β̄ and the dimensionless ratios ζ = 4 m²_prod / m²χ, which measure how light the decay products are relative to χ. When ζ ≪ 1 (the usual situation because SM particles are much lighter than χ), the two‑body decays dominate and the total inclusive width simplifies to Γ_tot ≈ β̄³ᐟ² ζ_tot χ₀ / (32π).

The reheating temperature follows the standard relation T_RH ≈ (90/π²g_*)¹ᐟ⁴ √(Γ_tot M_P). For typical β̄ ∼ 10⁻³–10⁻² and χ₀ ∼ 10⁸–10¹⁰ GeV, one obtains T_RH well above the MeV scale, ensuring compatibility with Big‑Bang Nucleosynthesis.

A particularly interesting application is the production of sterile‑neutrino dark matter (DM). The χ decay can generate right‑handed neutrinos N₁ with a tiny Yukawa coupling y_N₁ ≈ 10⁻⁸. If m_{N₁} ≈ 10² MeV, the non‑thermal population from χ decay reproduces the observed DM relic density Ω_DM ≈ 0.26. Thus, in this regime the same super‑cooled PT that sources gravitational waves (GWs) and possibly primordial black holes (PBHs) can also account for the particle DM component.

2. χ₀ ≲ vEW (Hidden‑sector RSB).
When the RSB scale is comparable to or below the EW scale, experimental bounds force the couplings between χ and SM fields to be extremely small. The authors therefore consider a dark sector gauge group (e.g. a hidden U(1)D) under which χ is charged. The dominant decay channel is χ → A′ A′, where A′ is a massive dark photon with mass m{A′}=g_D χ₀. The corresponding width scales as Γ_{χ→A′A′} ∝ g_D⁴ χ₀/(32π).

The dark photons subsequently decay into SM fermion pairs via kinetic mixing ε F_{μν}F′^{μν}. For ε ≈ 10⁻⁶–10⁻⁴ and m_{A′} > 2 m_f, the decay A′ → f \bar f proceeds efficiently, transferring the vacuum energy to the SM plasma. This two‑step reheating (χ → A′ A′ → SM) can be viewed as a form of preheating driven by the time‑dependent background χ field, leading to parametric resonance production of A′ quanta. The resulting reheating temperature can reach 10 MeV–GeV, again safely above the BBN threshold.

Bubble dynamics and reheating time scales.
The paper compares the reheating time τ_RH ≈ 1/Γ_tot with the Hubble time H⁻¹ and the bubble expansion time (bubble wall velocity ≈ c). Efficient reheating requires τ_RH ≪ H⁻¹ so that the energy stored in the false vacuum is transferred to radiation before the bubbles percolate completely. This condition also guarantees that the gravitational‑wave spectrum generated by the PT is not significantly altered by late‑time entropy injection.

Three‑body decays and model‑dependent effects.
While three‑body channels can be comparable to two‑body ones if the couplings G_{Nc}·λ′_{abc} are large, the authors argue that in most realistic RSB constructions the two‑body processes dominate. They also discuss possible additional contributions such as particle production from bubble collisions, but these are left for future model‑specific studies.

Conclusions and outlook.
The authors provide a unified framework for reheating after super‑cooled RSB‑driven PTs. In the SM‑connected case, χ decays directly into SM particles and can simultaneously generate sterile‑neutrino DM around the 100 MeV scale. In the hidden‑sector case, reheating proceeds via dark photons that later decay into SM fermions, with kinetic mixing controlling the efficiency. Both mechanisms yield reheating temperatures well above the BBN bound and are compatible with the generation of observable GW signals and PBH formation. The results open avenues for connecting upcoming PTA GW observations, PBH searches, and DM experiments (especially those targeting sterile neutrinos or dark photons) to the underlying radiative symmetry‑breaking dynamics.