Nonperturbative analysis of the gravitational waves from a first-order electroweak phase transition

TL;DR

The paper develops an end-to-end nonperturbative framework for predicting gravitational waves from a thermal first-order EWPT by leveraging dimensional reduction to a 3-d SM-like EFT and pre-existing lattice results. It shows that any BSM theory reducible to this infrared EFT yields GW signals that are below detectability, making collider experiments the primary probe of the EWPT phase structure. As a concrete test, it analyzes the real singlet-extended SM (xSM), mapping its phase diagram nonperturbatively and benchmarking against perturbative methods, finding GW signatures remain weak in the analyzed regions. The results validate perturbative approaches in certain regimes but emphasize the necessity of including higher-dimension operators or light BSM fields for potentially observable GW signals, motivating dedicated nonperturbative studies and highlighting collider experiments as complementary probes of EWPT physics.

Abstract

We present the first end-to-end nonperturbative analysis of the gravitational wave power spectrum from a thermal first-order electroweak phase transition (EWPT), using the framework of dimensionally reduced effective field theory and pre-existing nonperturbative simulation results. We are able to show that a first-order EWPT in any beyond the Standard Model (BSM) scenario that can be described by a Standard Model-like effective theory at long distances will produce gravitational wave signatures too weak to be observed at existing and planned detectors. This implies that colliders are likely to provide the best chance of exploring the phase structure of such theories, while transitions strong enough to be detected at gravitational wave experiments require either previously neglected higher-dimension operators or light BSM fields to be included in the dimensionally reduced effective theory and therefore necessitate dedicated nonperturbative studies. As a concrete application, we analyze the real singlet-extended Standard Model and identify regions of parameter space with single-step first-order transitions, comparing our findings to those obtained using a fully perturbative method. We discuss the prospects for exploring the electroweak phase diagram in this model at collider and gravitational wave experiments in light of our nonperturbative results.

Figures (6)

• Figure 1: A schematic illustration of the steps of dimensional reduction in the presence of superheavy and heavy BSM scalar fields $S$ and $s$, respectively. In the first step of DR, all fermionic modes, hard bosonic modes and all $S$ modes are integrated out, resulting in an effective 3-d theory of purely spatial zero-modes. In the second step of DR, the heavy scalars $A_0$ and $s_3$ are integrated out, resulting in a simplified EFT of a doublet $\bar{\phi}_3$ and a gauge field $\bar{A}_i$ describing the long distance physics relevant for the phase transition.
• Figure 2: The entire region of gravitational wave parameter space onto which the 3-d Standard Model-like effective field theory can be mapped, along with prospects for detection. In panel (a), the corresponding region is shaded gray, along with explanations of the various limits to the effective field theory's validity. Also shown are LISA $\mathrm{SNR}=10$ sensitivity curves Audley:2017drz (assuming a five year mission duration) for two cases: one where the sound source is on until the shock formation time ($t_{\rm end}=t_{\rm sh}$), and the other where the sound source lasts the full Hubble time ($t_{\rm end}=H^{-1}_*$). In panel (b), the parameter space mapping to the 3-d SM-like EFT is shown in light blue, while the black line shows the ranges of $\alpha$ and $\beta/H_*$ that can be determined nonperturbatively from the results of Ref. Moore:2000jw. Additional LISA SNR curves are shown in (b), assuming that the sound source lasts for a Hubble time (the dashed contours give the shock formation time, and only in the shaded region is the shock formation time longer than a Hubble time). Note that other gravitational wave experiments are not expected to provide additional sensitivity to the SM-like 3-d EFT regions. Finally, the dark blue region in (b) is the area of relevance for the real singlet model (xSM), as discussed in Sec. \ref{['sec:pheno']}. The points shown correspond to the benchmark comparison of our nonperturbative analysis and the predictions of 3-d and 4-d perturbation theory. The sensitivity curves assume $v_w =1$ and $T_*=140$ GeV. The latter is the appropriate choice for transitions described by the 3-d SM-like theory, as the magnitude of $T_*$ is primarily set by the Higgs mass absent other light degrees of freedom or large higher-dimension operator effects. This figure shows that the 3-d SM-like EFT and existing lattice results cannot be reliably used to study electroweak phase transitions predicting an observable gravitational wave signal.
• Figure 3: Phase structure of the xSM in the $Z_2$ symmetric limit. Our nonperturbative approach predicts a first-order electroweak phase transition in the light green region. The darker green shaded region features $x<0$ so that the higher dimension operator $(\phi^\dagger \phi)^3$ must be kept in the dimensionally reduced theory to resolve one-step transitions. Furthermore, above the dotted green contour, the dimension-6 operator coefficient, $c_6$, becomes larger than 10 times the corresponding SM value, $c_{6, {\rm SM}}$, and the neglected $(\phi^\dagger \phi)^3$ term can already have significant effects. The gray shaded region corresponds to where the singlet mass parameter $\mu^2_{\sigma}<0$ so that the superheavy dimensional reduction completely breaks down. The parameter space in which 4-d perturbation theory predicts a first-order transition with $v_c/T_c = 0.3-0.6$ is shaded orange. We also show contours indicating the size of the singlet $\overline{\text{MS}}$ mass parameter $\mu_\sigma$. When $\mu_\sigma \lessapprox g T \approx 100$ GeV, the superheavy singlet approximation becomes compromised. A first-order EWPT is robustly excluded for small values of $\lambda_{221}$, where our nonperturbative treatment is well-justified.
• Figure 4: Parameter space of the xSM predicting a first-order electroweak phase transition for various values of $m_2$ and $\sin \theta$ with $b_3=0$, in both the 4-d perturbative and 3-d nonperturbative approaches. The light green regions features a first-order electroweak phase transition as predicted by existing lattice results. The darker shaded region features $x<0$ so that the higher dimension operator $(\phi^\dagger \phi)^3$ must be kept in the dimensionally reduced theory to resolve one-step electroweak transitions. Furthermore, to the right of the dotted green contour, the dimension-6 operator coefficient, $c_6$, becomes larger than 10 times the corresponding SM value, $c_{6, {\rm SM}}$, and the neglected $(\phi^\dagger \phi)^3$ term can already have significant effects. We also show contours indicating the size of the singlet $\overline{\text{MS}}$ mass parameter $\mu_\sigma$. When $\mu_\sigma \lessapprox g T \approx 100$ GeV, the superheavy singlet approximation becomes compromised. The gray shaded regions correspond to where the singlet mass parameter $\mu^2_{\sigma}<0$ so that the superheavy dimensional reduction completely breaks down. The parameter space in which 4-d perturbation theory predicts a first-order transition with $v_c/T_c = 0.3-0.6$ is shaded orange, and matches up well with the nonperturbative first-order regions especially in regions where the $(\phi^\dagger \phi)^3$ contribution is unimportant.
• Figure 5: Parameter space of the xSM predicting a first-order phase transition for various values of $m_2$ and $\sin \theta$ away from the $b_3=0$ limit. The shading and contours are as in Figs. \ref{['fig:Z2']}-\ref{['fig:boomerang_compare_3-d_and_4-d']}, with additional contours and gray shaded regions indicating where the electroweak vacuum is metastable in a 1-loop analysis. Our nonperturbative predictions for the first-order electroweak phase transition regions correspond roughly to the parameter space in which 4-d perturbation theory predict a first-order transition with $v_c/T_c = 0.3-0.6$.