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

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.