Significance of soft-scale breaking on primordial black hole production in Coleman-Weinberg type supercooling-phase transition

Abstract

Ultra-supercooling phase transitions can generate large overdensities in the Universe, potentially leading to the formation of primordial black holes (PBHs), which can also be a dark matter candidate. In this work, we focus on the supercooling phase transition for the scale symmetry breaking based on the effective potential of the Coleman-Weinberg (CW) type. We investigate the effect on the PBH production in the presence of an additional mass term for the CW scalar field, what we call a soft-scale breaking term, which serves as the extra explicit-scale breaking term other than the quantum scale anomaly induced by the CW mechanism. We demonstrate that even a small size of the soft-scale breaking term can significantly affect the PBH production depending on its sign: a positive term slows down the phase transition, thereby enhancing the PBH abundance and improving the model's ability to account for dark matter; in contrast, a negative term suppresses the PBH formation. The inclusion of such soft-scale breaking terms broadens the viable parameter space and increases the flexibility of the framework. We further illustrate our results through two ultraviolet-complete realizations: i) a many-flavor QCD-inspired model as a reference model which can dynamically induce a positive-soft scale breaking; ii) a Higgs portal model with a $B-L$ scalar as the benchmark for the case where a negative-soft scale breaking is induced. Our study would provide a new testable link between PBH dark matter and gravitational wave signatures in the CW-type scenario.

Figures (4)

• Figure 1: Left: Plots of the bounce action $S_3(T)/T$ with a negative (denoted as "Neg.") and positive (denoted as "Pos.") soft-scale breaking terms, as a function of $T$, for reference values of $g_D$ and $v_\phi$. Right: Variations of $\beta_H$ with respect to the coupling $g_D$ for different mass choices and a fixed $v_\phi$ consistently with the PBH-dark matter interpretation (See Eq. (\ref{['vphi-range']})).
• Figure 2: Plots of $S(T)\equiv S_3(T)/T - 4\ln T$ as a function of $T$ for two benchmark parameter sets. The blue, red, and orange curves respectively represent the result without any approximation on the nucleation, with linear approximation, and the second-order approximation. The vertical dashed lines (from right to left) indicate the nucleation temperature $T_n$, percolation temperature $T_p$, and PBH formation time $T_{\rm max}^{\rm PBH}$. Also have been displayed the relative error $\epsilon_\beta$ in Eq. (\ref{['linearity']}) (in unit of percent), which have been drwan by the dashed-light blue curves.
• Figure 3: Left: Plots of $S_3(T)/T$ as a function of $T$ for two benchmark parameter sets, BP A (blue) and BP A$'$ (Orange). The vertical dashed lines (from right to left) indicate the nucleation temperature $T_n$, percolation temperature $T_p$, and PBH formation time $T_{\rm max}^{\rm PBH}$ for BP A$'$. Right: Stochastic GW spectra in the cases with (red) and without (blue) the soft-scale breaking term. For each case, the solid and dashed curves correspond to the spectra sourced from sound waves and bubble wall collisions, respectively. The GW signals are compared with future prospected detector sensitivities Moore:2014lgaSchmitz:2020syl.
• Figure 4: PBH abundances as a function of mass for the benchmark points A$'$, B, and B$'$ given in Tabs. \ref{['tab:walking']} and \ref{['tab:B-L']}. The red shaded regions indicate current constraints from Hawking evaporation, including EGRB Carr:2009jm, 511 keV Laha:2019ssq, CMB Clark:2016nst, EDGES 21cm Mittal:2021egv, Voyager I Boudaud:2018hqb, INTEGRAL Laha:2020ivk, Dwarf heating Kim:2020ngi, Super K Dasgupta:2019cae, COMPTEL Coogan:2020tuf, AMS-02 Su:2024hrp, X-ray BG Tan:2024nbx, and Lyman-$\alpha$Khan:2025kag. The blue shaded regions correspond to existing microlensing limits from Hyper Suprime-Cam (HSC) Niikura:2017zjdSmyth:2019whbCroon:2020ouk, Kepler (K) Griest:2013esaGriest:2013aaa, EROS EROS-2:2006ryy, and OGLE Niikura:2019kqiMroz:2024mseMroz:2024wagMroz:2017mvf. This figure has been created with the publicly available Python code PBHboundsKavanagh:2024lgq.