Dynamics of Non-renormalizable Electroweak Symmetry Breaking

TL;DR

This work analyzes electroweak symmetry breaking in the Standard Model augmented by a dimension-six Higgs operator $H^6$ using a complete one-loop finite-temperature effective potential $V_{eff}( ext{f},T)$. It develops a careful treatment of infrared issues via ring (Daisy) resummation, matches renormalization to physical inputs $(v_0,m_h,f)$, and analyzes bubble nucleation and supercooling to identify regions that yield a strong first-order EWPT. It then estimates the gravitational-wave signal from the transition via the parameters $ ext{α}$ and $eta/H_n$ and concludes that detectable signals require fine-tuning of $(m_h,f)$ and are mainly within reach of future space-based detectors. Overall, the results connect beyond-SM Higgs self-interactions to cosmological phase-transition dynamics and potential gravitational-wave observables, highlighting both viable baryogenesis scenarios and challenging gravitational-wave prospects.

Abstract

We compute the complete one-loop finite temperature effective potential for electroweak symmetry breaking in the Standard Model with a Higgs potential supplemented by higher dimensional operators as generated for instance in composite Higgs and Little Higgs models. We detail the resolution of several issues that arise, such as the cancellation of infrared divergences at higher order and imaginary contributions to the potential. We follow the dynamics of the phase transition, including the nucleation of bubbles and the effects of supercooling. We characterize the region of parameter space consistent with a strong first-order phase transition which may be relevant to electroweak baryogenesis. Finally, we investigate the prospects of present and future gravity wave detectors to see the effects of a strong first-order electroweak phase transition.

Figures (8)

• Figure 1: Different potentials close to the critical temperature for $m_h=115$ GeV and $f=620$ GeV ($f$ is the decay constant of the strong sector the Higgs emerges from). The dashed curve is the potential of Grojean:2004xa which includes only the thermal mass term of the Higgs, while the solid and dotted ones represent the full one-loop potential with (solid) and without (dotted) the ring diagram contributions. In blue, we have also plotted the imaginary part of the full one-loop potential with the ring contributions (solid blue) as well as the imaginary part of the ring contributions alone (dashed blue). This illustrates the cancelation of the large imaginary parts between the ring and the one-loop contributions, while there still exists an additional and smaller imaginary part for some values of $\phi$ due to a negative quartic coupling (see the discussion in Section \ref{['sec:ImCancelation']} for details). An imaginary part of the potential can be interpreted as a decay rate of some quantum states of the scalar fields to some others but the imaginary part of the full potential is always tiny compared to the real part around the transition temperature and the system is stable enough throughout the entire time of the transition.
• Figure 2: Some generic examples of ring diagrams where each solid line may represent either a scalar, a fermion or a gauge field. The small loops correspond to thermal loops in the IR limit. They are all separately IR divergent, but their sum is IR finite.
• Figure 3: The left panel of this figure shows contours of the nucleation temperature $T_n$ in the allowed region for an EW symmetry-breaking first order phase transition ($f$ is the decay constant of the strong sector the Higgs emerges from, and $m_h$ is the physical Higgs mass). Below the red lower bound the EW symmetry remains intact in the vacuum while above the blue upper one the phase transition is second order or not even occurs. Within the red band, the universe is trapped in a metastable vacuum since no expanding bubble is nucleated and the transition never proceeds. The contours are from left to right for $T_n=\{50,100,150\}\, {\rm GeV}$. The right panel of this figure shows contours of the relative deviation of the nucleation temperature from the critical one: $\epsilon_T=(T_c-T_n)/T_c$. This measures the degree to which the phase transition is delayed by the overcooling effect. The contours are, from above, for $\epsilon_T=\{10^{-3},10^{-2},0.1,0.3\}$.
• Figure 4: Plot of the ratio $\xi_n=\langle\phi(T_n)\rangle/T_n$ characterizing the strength of the phase transition using the thermal mass approximation of Grojean:2004xa (left) and the complete one-loop potential (right). The contours are for $\xi_n=\{1,2,3,4\}$ from top to bottom. $f$ is the decay constant of the strong sector the Higgs emerges from, and $m_h$ is the physical Higgs mass.
• Figure 5: The panel on the left contains contours of the latent heat $\alpha=\{5.10^{-3},10^{-2},5.10^{-2},0.1,0.5\}$ from top to bottom. The panel on the right draws contours of the parameter, $\beta/H_n$, measuring the duration of the phase transition. From above one has $\beta/H_n=\{10^5,10^4,10^3,200\}$. $f$ is the decay constant of the strong sector the Higgs emerges from, and $m_h$ is the physical Higgs mass.