Signals of the electroweak phase transition at colliders and gravitational wave observatories

TL;DR

This work investigates the electroweak phase transition (EWPT) within an effective field theory extended to dimension eight, focusing on the operators $\mathcal{O}_6$ and $\mathcal{O}_8$ to realize a strongly first-order transition. It computes the finite-temperature potential, performs mean-field and full numerical analyses to determine nucleation temperatures and GW parameters, and assesses the reach of LISA for detecting related gravitational waves. The study then matches concrete UV completions, including weakly-coupled scalars and custodial quadruplets, highlighting that many UV scenarios inevitably generate additional operators beyond $\mathcal{O}_6$ and $\mathcal{O}_8$, while some custodial-symmetric setups can isolate effects in the Higgs potential. A key conclusion is that LISA has the potential to probe a significant portion of the EWPT parameter space earlier than collider experiments, while collider Higgs self-coupling measurements provide complementary constraints and, in some cases, are less sensitive to the EWPT dynamics. Overall, the results emphasize gravitational-wave observations as a crucial, early probe of EWPT physics and the structure of possible UV completions.

Abstract

If the electroweak phase transition (EWPT) is of strongly first order due to higher dimensional operators, the scale of new physics generating them is at the TeV scale or below. In this case the effective-field theory (EFT) neglecting operators of dimension higher than six may overlook terms that are relevant for the EWPT analysis. In this article we study the EWPT in the EFT to dimension eight. We estimate the reach of the future gravitational wave observatory LISA for probing the region in which the EWPT is strongly first order and compare it with the capabilities of the Higgs measurements via double-Higgs production at current and future colliders. We also match different UV models to the previously mentioned dimension-eight EFT and demonstrate that, from the top-down point of view, the double-Higgs production is not the best signal to explore these scenarios.

Figures (6)

• Figure 1: The potentials $V_{\rm tree}$ (black solid curve), $V_{1\ell}$ at $T=0$ (orange dashed curve) and $V_{1\ell}$ at $T=T_x$ (green dashed curve) for the choices of $c_6/f^2$ and $c_8/f^4$ indicated in each panel. In the left panel, there exist two vacua already at zero temperature ($\mu^2\simeq -3100\,GeV^2$, $\lambda\simeq-0.23$, $T_x = T_c = 35\,GeV$). In the central panel, the existence of two vacua arises only at finite temperature ($\mu^2=1900\,GeV^2$, $\lambda=-0.06$, $T_x=T_c=82\,GeV$). In the right panel, the potential is unbounded from below, but the instability scale is above the cutoff $f=1\,$TeV ($\mu^2=3000\,GeV^2$, $\lambda=-0.03$, $T_x = T_n=99\,GeV$). $T_c$ is the critical temperature obtained in the mean-field approximation.
• Figure 2: $c_{6}/f^2-c_{8}/f^4$ of parameter space for a SFOEWPT in the mean-field approximation. Left) The filled region shows the allowed values for $c_{6}$ and $c_{8}$ such that at $T = 0$ the deepest minimum is at $v$. In the darker areas there is a second minimum above the one at $v$. For negative $c_{8}$, we cut off the potential at $1$ TeV and demand that $V(1 \text{TeV})>V(v)$ to ensure that the global minimum is at $v$. Superimposed are shades of yellow to green to show the strength of the phase transition, $v_{T_c}/T_{c}$, based on the critical temperature. Right) Zoomed version on the black rectangle of the left panel (note the different axis ranges). Lines of constant $T_c$ are depicted.
• Figure 3: Values of $T_n$ (top left), $v_{T_n}/T_n$ (top right), $\alpha$ (bottom left) and $\beta/H$ (bottom right) characterising the SFOEWPT in the plane $c_6/f^2$--$c_8/f^4$. The labels of $T_n$ and $T_c$ are in GeV units. On the right of the grey area the condition in Eq. \ref{['eq:c6_upper']} is violated. In the gray area to the right of the dashed line, the lifetime of the EW symmetric vacuum is longer that the age of the Universe, whereas on the left the transition results too weak for our purposes, i.e. $v_{T_n}/T_n<0.7$. Below the grey area, the EW vacuum at zero temperature is not the global minimum at scales below the cutoff $f=1\,$TeV. In orange the parameter region LISA is sensitive to.
• Figure 4: Left panel) The values of $(T_c-T_n)/T_c$ in the plane $c_6/f^2$--$c_8/f^4$ (dotted lines). The rest stands as in Fig. \ref{['fig:ewpt']}. Right panel) The values of $\beta/H$ and $10^6\times\alpha$ as a function of $c_6/f^2$ for $c_8/f^4=5\,{\rm TeV}^{-4}$ (dotted curves), $c_8/f^4=2\,{\rm TeV}^{-4}$ (dashed curves) and $c_8/f^4=0$ (solid curves).
• Figure 5: Left panel) Region of Fig. \ref{['fig:c6c8']} where the SFOEWPT is achieved accordingly to the criterion $v_{T_n}\gtrsim T_n$ instead of $v_{T_c}\gtrsim T_c$. The reaches of FCC-ee DiVita:2017vrr and LISA Caprini:2015zlo are also displayed. Right panel) Allowed region from the left panel translated to the $\lambda_{3}/\lambda_{3,\text{SM}}$--$\lambda_{4}/\lambda_{4, \text{SM}}$ plane together with the future experimental sensitivities Papaefstathiou:2015paa.