Future space interferometers such as LISA may one day see a background of gravitational waves produced when the early universe changed phase suddenly. This paper shows that for a common class of particle models that predict very strong “supercooled” transitions, the ability to work backwards from a detected gravitational-wave signal to the underlying particle physics depends strongly on how theorists treat quantum and thermal corrections. In one widely used approximation the theoretical error is larger than the experimental error expected from LISA, which undermines attempts to reconstruct model parameters with confidence.

The authors study a simple representative model: a classically conformal U(1)_X extension of the Standard Model. “Classically conformal” means there is no explicit mass scale in the tree-level potential, and masses arise through quantum effects (the Coleman–Weinberg mechanism). In these setups a thermal barrier separating the symmetric and broken phases can persist down to very low temperatures. The transition then happens after strong supercooling, bubbles of the new phase are unusually thick, and the nucleation physics is sensitive to thermal fluctuations. These features make the prediction of a gravitational-wave spectrum delicate and dependent on how thermal effects are resummed.

The paper compares two practical ways to do that resummation. The first is the common four-dimensional “daisy” resummation, where certain thermal loops are summed and the nucleation-rate prefactor is estimated on dimensional grounds. The second is a more systematic high-temperature dimensional-reduction approach that builds a three-dimensional effective field theory matched at two-loop order, includes next-to-leading-order corrections to the tunneling (bounce) action, and uses the full one-loop functional determinants for the nucleation prefactor. The latter reduces sensitivity to the renormalization scale and includes higher-order thermal effects.