Beyond the Daisy Chain: Running and the 3D EFT View of Supercooled Phase Transitions

TL;DR

The paper addresses predicting gravitational waves from supercooled first-order phase transitions in nearly conformal dark sectors by comparing finite-temperature potential schemes. It systematically contrasts 4D one-loop (Full and HT with Daisy), 4D HT with RG improvement, the one-parameter approximation, and 3D DR potentials up to two loops using DRalgo, emphasizing renormalisation-scale choices. The main finding is that RG-improved 4D HT at an optimal scale aligns well with two-loop DR results, while OPA and other partial higher-order schemes exhibit larger scale dependence and deviations; this yields robust predictions for $T_n$, $\alpha$, and $\beta/H$, hence for the gravitational-wave spectrum. The study provides a computationally efficient and theoretically sound framework for PTA-scale GW predictions from supercooled phase transitions, offering a concrete route to test scale-invariant dark sectors against NANOGrav data and guiding future complementary probes.

Abstract

Pulsar timing arrays have recently observed a stochastic gravitational wave background at nano-Hertz frequencies. This raises the question whether the signal can be of primordial origin. Supercooled first-order phase transitions are among the few early Universe scenarios that can successfully explain it. To further scrutinise this possibility, a precise theoretical understanding of the dynamics of the phase transition is required. Here we perform such an analysis for a dark sector with an Abelian Higgs model in the conformal limit, which is known to admit large supercooling. We compare simple analytic parametrisations of the bounce action, one-loop finite temperature calculations including Daisy resummation, and results of a dimensionally reduced (3D) effective theory including up to two-loop corrections using the DRalgo framework. Consistent renormalisation group evolution (RGE) of the couplings is essential for a meaningful interpretation of the results. We find that the 3D EFT with consistent expansion in the 4D parameters gives a significantly reduced scale dependence of the phase transition parameters. With a suitable choice of RGE scale, the 4D high temperature expanded effective potential yields results consistent with the 3D calculations, while the analytic parametrisation deviates significantly in the limit of large supercooling.

Figures (7)

• Figure 1: Left: Renormalisation-group evolution of the model parameters for $g(\mu_0)=0.6$, with input conditions $m^2(\mu_0)=0$ and $\lambda(\mu_0)=0$ at $\mu_0=1\GeV$ (see \ref{['tab:mu_scales']}). The gauge coupling $g$ (shown as $g^2/2\pi$) runs slowly, while the quartic coupling $\lambda$ decreases and becomes negative at intermediate scales, triggering radiative symmetry breaking. The mass parameter $m^2$ remains nearly constant, reflecting its vanishing initial condition. The VEV emerges around the scale where $\lambda$ crosses zero, marking the onset of spontaneous symmetry breaking. Right: The VEV as a function of $\mu$ for $g(\mu_0)=0.5-0.8$, $\lambda(\mu_0)=0$ and $m^2(\mu_0)=0$. The residual $\mu$-dependence is minimised for $\mu\sim\mu_0$, showing that the explicit scaling in \ref{['eq:vev']} is compensated once the running of the couplings is included.
• Figure 2: Comparison of different prescriptions for the effective potential at $g(\mu_0)=0.6$. Left: Comparison between the 4D Full potential, its 4D HT expansion, the OPA, and the 3D dimensionally reduced potentials at different loop orders at $T=0.39\, v$. Right: Residual dependence on the renormalisation scale at $T = 0.01\, v$. For the 4D HT potential (green band), the renormalisation scale is set to $\mu = \pi T$ and varied between $\mu = \pi T/4$ and $\mu = 4\pi T$ to illustrate the scale dependence. For the 3D two-loop (NLO) potential (purple band), the scale is instead fixed at $\mu_{\rm Match} = 2\pi T$ and varied analogously from $\mu_{\rm Match} = (2\pi T)/4$ to $\mu_{\rm Match} = 4(2\pi T)$. In each case, the dotted line shows the potential evaluated at the corresponding reference scale used in the corresponding approach. The dash-dotted green line corresponds to the 4D HT potential at $\mu=\mu_0$, whereas the orange dotted line depicts the OPA (without running).
• Figure 3: Comparison of the tunnelling action $S_3(T)/T$ computed with different effective potential schemes: 4D HT, one-parameter approximation (OPA), and dimensional reduction (DRalgo). Results are shown for $\mu_0=1$ and $\lambda(\mu_0) = 0$, at two benchmark values of the dark gauge coupling. Left:$g(\mu_0)=0.5$, where all schemes are in close agreement. Right:$g(\mu_0)=0.8$, where discrepancies between methods become more pronounced due to the increased sensitivity to thermal corrections at larger coupling.
• Figure 4: Nucleation temperature $T_n$ as a function of the dark gauge coupling $g$, normalised to the symmetry-breaking scale $v$. Different curves correspond to different computation schemes for the effective potential: 4D HT, OPA, and dimensional reduction.
• Figure 5: Left: Strength ($\alpha$) and Right: inverse duration ($\beta/H$) of the phase transition as a function of the dark gauge coupling $g$. Both panels are done for $\mu_0=1\GeV$ and $\lambda(\mu_0)= 0$. $\alpha$ and $\beta/H$ are computed for different effective potential schemes: 4D HT, OPA and DRalgo.