Hard thermal contributions to phase transition observables at NNLO

TL;DR

This work develops a high-temperature effective field theory for gauge–Higgs systems up to $\mathcal{O}(g^6)$ using dimensional reduction and three-loop matching, with a detailed treatment of the Abelian Higgs model as a testbed for higher-dimensional operators and loop corrections on phase-transition thermodynamics relevant to gravitational waves. It provides three-loop scalar and Debye masses and two-loop quartic couplings, demonstrates gauge independence of physical quantities, and identifies a master-integral mismatch that points to missing contributions in the reduction basis. The analysis reveals that dimension-6 operators can dominate the thermodynamics in the small-$x$ (strong transition) regime, while loop corrections dominate at larger $x$, establishing a quantitative transition around $x_{\text{LO}} \sim 1$ and informing GW and primordial black hole phenomenology. Overall, the results complete the NNLO soft-scale EFT parameter determination and set the stage for future nonequilibrium studies, including bubble nucleation rates and extensions to SU($N$) theories.

Abstract

To construct the high-temperature effective field theory of gauge-Higgs models up to $\mathcal{O}(g^6)$ in the gauge coupling, we integrate out hard modes to three-loop level and use the next-to-next-to-leading order effective potential. For the Abelian Higgs model, we quantify the impact of both higher-dimensional operators and higher-loop corrections on thermodynamic parameters relevant for gravitational-wave observables, finding that one-loop dimension-six effects typically dominate over two- and three-loop corrections to super-renormalizable parameters for the strongest transitions. We derive the three-loop scalar and Debye masses for the ${\rm U(1)}$ and ${\rm SU}(N)$ gauge-Higgs models, as well as the two-loop quartic couplings for the Abelian case, show gauge independence of physical parameters, and demonstrate that no new master integrals are required for the matching, while consistency of 4d and 3d renormalizability points to previously missing contributions in these master integrals.

Figures (5)

• Figure 1: Three-loop contributions to the bare two-point functions in the Abelian Higgs model at the soft scale. Dashed directed lines denote scalars ($\phi$) and wiggly lines denote gauge fields ($B_\mu$).
• Figure 2: Two-loop contributions to the bare four-point functions in the Abelian Higgs model at the soft scale including all possible permutations of external legs. Dashed directed lines denote scalars ($\phi$) and wiggly lines denote gauge fields ($B_\mu$).
• Figure 3: Debye (left panels) and scalar (right panels) masses as functions of temperature (top row) and matching scale $\bar{\Lambda}$ at $T=100$ GeV (bottom row). The 4d couplings are fixed at \ref{['eq:BM1']}. The 3d renormalization scale is set to the optimized value $\bar{\Lambda}_{\hbox{\scriptsize 3d,opt}}=2.85\,T$Stevenson:1981vjLaine:2005aiGhisoiu:2015uza, and we assume $X=0$. Setting $X$ to slightly different values (for instance, $X=-2/5$) leads to only imperceptible deviations in the results. In the upper panels, the matching scale is varied in the range $\bar{\Lambda} \in [2^{-3},2^{3}]\,\pi T$. The $n$-loop curves show the $n$-loop matching results for the Debye and scalar masses, with the 4d parameters evolved using two-loop running \ref{['4d running']}. The plots illustrate that the dependence on the matching scale is further suppressed at higher orders when two-loop running and three-loop matching contributions are included.
• Figure 4: Relative ratios of the thermodynamic quantities $\tilde{y}_{\rm c}$, $\Delta\langle\phi^\dagger\phi\rangle_{\rm c}$, and $\Delta\langle(\phi^\dagger\phi)^2\rangle_{\rm c}$ with (dashed) and without (solid) the dimension-6 contribution $c_6$, computed at $\mathcal{O}(g^4)$ (red) and $\mathcal{O}(g^6)$ (blue) matching orders. For all plots, $T_{\rm c}=1.2$ GeV, $g^2(\overline{T}_{\rm c})=0.3$, $\bar{\Lambda}=\overline{T}_{\rm c}\equiv 4\pi e^{\gamma_{\hbox{\tiny\rm{E}}}} T$, and $\bar{\Lambda}_{\hbox{\scriptsize 3d,opt}}=2.85\,T_{\rm c}$Ghisoiu:2015uza.
• Figure 5: Left: Transition strength $\alpha_{\rm c}$ at the critical temperature \ref{['eq:transition_strength']}, computed using $\mathcal{O}(g^4)$ matching while neglecting dimension-6 operators. Right: Ratio of corrections from higher-dimensional operators ($c_6 \neq 0$) to those from $\mathcal{O}(g^6)$ higher-loop matching as defined in eq. \ref{['eq:ratio_transition_strength']}. The four-dimensional couplings are fixed as in \ref{['eq:BM1']}.