The devil in the transition: NLO nucleation and the particle physics behind the PTA signal

We quantify how the treatment of a strongly supercooled phase transition affects the particle-physics interpretation of a nanohertz gravitational-wave signal. For the classically conformal Abelian Higgs model, we calculate the nucleation rate at NLO in the dimensionally reduced theory and test the gradient expansion against explicit fluctuation determinants. A calibrated correction reproduces the determinant exponent with a mean absolute relative difference of $0.8\mathrm{\%}$ and allows this information to be included throughout the parameter scan. We propagate the corrected rate through percolation, reheating and gravitational-wave production, and confront the resulting spectra with the NANOGrav 15-year and IPTA DR2 data. The corrected NLO calculation shifts the preferred gauge coupling by $9\mathrm{\%}$ and $13\mathrm{\%}$, respectively, and the input scale by $18\mathrm{\%}$ for NANOGrav. After the determinant calibration, the largest theoretical uncertainties arise from uncertainties in the gravitational-wave spectra and the relation between $R_{\star }$ and $\beta \mathrm{/}H$.