Perturbative aspects of the electroweak phase transition with a complex singlet and implications for gravitational wave predictions

Abstract

We present a detailed analysis of strong first-order electroweak phase transitions within the extension of the Standard Model by a complex scalar singlet (cxSM). Focusing on the impact of renormalization scale and gauge dependence, we systematically compare commonly used perturbative frameworks for predicting thermodynamic observables that characterize the phase transition and the associated gravitational-wave (GW) spectrum. These include both the four-dimensional ($4D$) formalism and the dimensionally reduced three-dimensional effective field theory ($3D$ EFT) approach in different renormalization schemes. Within the $3D$ EFT, we compute the effective potential up to two-loop order in a general $R_ξ$ gauge, and demonstrate that applying the $\hbar$-expansion yields gauge-independent results in excellent agreement with those obtained from a direct minimization of the loop-corrected potential. In contrast, large discrepancies between the two methods persist in the $4D$ approaches. We find that, across most of the parameter space, the $3D$ EFT approach provides the most robust predictions for phase-transition parameters and GW spectra, reducing theoretical uncertainties in the GW peak amplitude by more than an order of magnitude compared to the $4D$ calculations. We point out, however, that the $3D$ EFT approach is subject to an additional theory uncertainty from truncating the EFT at finite operator dimension and show that higher-dimensional operators within the $3D$ EFT approach can substantially modify the predicted transition strength and GW signals. This indicates a potential breakdown of the high-temperature expansion precisely in the region with the lowest transition temperatures, where the strongest GW signals are expected and the detection prospects with LISA are most promising.

Figures (20)

• Figure 1: The pattern of a possible strong FOEWPT in the cxSM model explored in this work. The arrow points in the direction of decreasing temperature.
• Figure 2: Topologies of prototype diagrams with background field dependent propagators entering the effective potential calculation at the two-loop level.
• Figure 3: Illustration of the procedure for systematically integrating out relevant thermal scales within the dimensional reduction approach. Starting from the initial scale ($\mu\sim m_Z$), Lagrangian parameters---determined from physical input parameters (see \ref{['app:parameters']})---are evolved to the hard scale ($\mu\sim\pi T$), where the full theory is matched to the dimensionally reduced EFT. The effective couplings are then evolved to the soft scale ($\mu_{3D}^{\rm soft}\sim m_D$), where temporal vector modes are subsequently integrated out, yielding the final EFT used to compute thermodynamic observables at the ultrasoft scale, characterized by the gauge coupling ($\mu_{3D}\sim g_{3D}^2$).
• Figure 4: Two-loop ghost--ghost--scalar prototype diagrams that appear in the effective potential in $R_\xi\xspace$-gauge, but are absent in the Landau gauge. See \ref{['app:masseigenstates']} for the gauge-dependent mass eigenvalues used in their evaluation.
• Figure 5: Critical temperature (left) and latent heat (right) for $m_s=100\,\text{GeV}$, $\lambda_s=1$, $\lambda_{hs}\in[1,1.3]$ determined by direct potential minimization in the Landau gauge. For the $4D$ approaches, the shaded bands around the solid lines are obtained from a variation of the renormalization scale by a factor of 2, with the central value being fixed at $\mu = \pi T$ and $\mu = v$ for $\overline{\text{MS}}$ and $\widetilde{\text{OS}}$ renormalization schemes, respectively (see text for details). For the $3D$ approach, the shaded band results from a variation of the hard matching scale by a factor of two around the central value $\mu = \pi T$.