Strong First-Order Electroweak Phase Transition

• Strong First-Order Electroweak Phase Transition is a process marked by bubble nucleation and a large order parameter-to-temperature ratio crucial for electroweak baryogenesis.
• The mechanism is exemplified in warped extra-dimensional frameworks like the Goldberger–Wise stabilized Randall–Sundrum model, where radion dynamics dictate the phase transition.
• Supercooling and delayed bubble nucleation in SFOEWPT yield distinct cosmological predictions, including testable gravitational wave signatures and reheating constraints.

A strong first-order electroweak phase transition (SFOEWPT) is a phase transition in which the Universe’s thermal evolution through electroweak symmetry breaking at finite temperature proceeds via bubble nucleation, accompanied by a substantial potential barrier between symmetric and broken vacua and a large order parameter-to-temperature ratio. This phenomenon is a central requirement for models of electroweak baryogenesis and represents a sharp departure from the Standard Model prediction of a smooth crossover for the measured Higgs mass. SFOEWPTs are of particular interest in scenarios where the mechanism for baryon asymmetry and testable cosmological signals—such as stochastic gravitational waves—are both present. The theoretical framework is exemplified by warped extra-dimensional models, such as the Goldberger–Wise stabilized Randall–Sundrum (RS) scenario, which realizes a highly supercooled, confining, strong first-order electroweak phase transition with distinct cosmological and phenomenological implications (0706.3388).

1. Theoretical Structure: RS Geometry, Phases, and the Role of the Radion

The RS framework features a five-dimensional, warped anti-de Sitter spacetime truncated by a UV and an IR brane. At finite temperature, the system admits two gravitational backgrounds:

• Confined (RS) phase: Both branes are present; holographically, this is a confining phase in the dual CFT, and the IR brane is at a physical scale μ (interpreted as the radion field), where the Higgs sector resides.
• Deconfined (AdS–Schwarzschild) phase: The IR brane disappears behind a black hole horizon; the field theory is in a high-temperature, deconfined phase and the symmetry is restored.

The radion field μ encodes the position of the IR brane and is massless (a flat direction) in the pure RS setup, corresponding to a modulus of the extra-dimensional volume. Absence of a radion potential obstructs any finite-temperature first-order transition.

Goldberger–Wise stabilization introduces a bulk scalar field, Φ, with tailored boundary conditions that yield a potential for the radion:

$$V_{GW}(\mu) = \epsilon v_0^2 k^4 + \mu^4\left[(4 + 2\epsilon)(v_1 - v_0 (\mu/k)^\epsilon)^2 - \epsilon v_1^2 + \delta T_1\right]$$

where parameters encode brane and bulk field/deformation data, and $\epsilon$ quantifies deviation from scaling.

This potential provides a shallow barrier, critical for supercooled, strongly first-order transitions by introducing a new, low critical temperature for the phase change.

2. Finite-Temperature Effective Potential and Phase Transition Dynamics

The full tree-level effective potential for the system, incorporating both the Goldberger–Wise stabilized radion and the IR-brane-localized Higgs, reads:

$$V_{RS}(\mu, \phi_c) = V_{GW}(\mu) + \left(\frac{\mu}{\mu_{TeV}}\right)^4 \frac{\lambda}{4}(\phi_c^2 - v^2)^2$$

Here, $\phi_c$ is the classical background Higgs field. The $(\mu/\mu_{TeV})^4$ factor ensures redshift of the Higgs potential, making the radion contribution dominant and enforcing electroweak symmetry breaking to occur only at extremely low temperature.

Because $V_{GW}(\mu)$ remains "flat" over a large range, the system is trapped in the deconfined phase deep below the critical temperature, experiencing extreme supercooling. The critical temperature $T_c$ is defined as the point where the free energies of the two phases coincide:

$F_d(T_c) = F_c(\mu=\mu_{TeV}, \phi_c = v)$

Actual nucleation occurs at $T_n \ll T_c$ (where tunneling action $S(T_n) = B \simeq 150$), triggering rapid expansion of the new phase.

Key consequence: Large order parameter $\phi_c(T_n)/T_n \gg 1$ is generically achieved. This ensures the phase transition is not only first-order, but “strongly” so.

3. Sphaleron Processes, Baryon Number Preservation, and Cosmological Constraints

For electroweak baryogenesis viability, sphaleron transitions (which violate $B+L$) must be suppressed in the broken phase to preserve any generated baryon asymmetry after the transition. The sphaleron rate inside the bubbles is

$\Gamma_{sph} \sim \kappa \alpha_W^4 T^4$

where the exponential suppression depends on $\phi_c(T_n)/T_n$, readily satisfied due to supercooling.

Important constraints arise:

Constraint	Quantitative Formulation	Physical Significance
Sphaleron suppression inside	$\phi_c(T_n)/T_n \gg 1$	Baryon number preservation in bubbles
Sphaleron active outside	$N_e \lesssim 26$ (max e-folds)	Avoid excessive expansion; baryogenesis must be possible in symmetric phase; ties to horizon problem
Post-reheating sphaleron off	$T_R < T_{sph-decouple}$	Prevents reactivation and baryon washout; $T_{sph-decouple}$ increases for heavier Higgs

Thus, a heavier Higgs is favored in this realization, as it relaxes the upper bound on the reheating temperature after the phase transition, in contrast to conventional scenarios.

4. Supercooling, Bubble Nucleation, and Inflationary Aspects

Due to the extreme flatness of $V_{GW}$, bubble nucleation is delayed until $T_n$ drops far below $T_c$. The reduction in $T$ is logarithmically slow with $S(T)$, and only a small number of e-folds ($N_e \sim \mathcal{O}(1)$–$26$) of inflation are accommodated before successful tunneling. This is restrictive compared to requirements for “solving” the cosmological horizon problem (which typically demands $N_e \gtrsim 30$), and suggests a potential tension for low-scale inflationary resolutions within this setup.

The transition dynamics entail:

• Bubbles of the RS (IR-brane present, Higgs-condensed) phase nucleating within the AdS–S (deconfined) background
• Percolation and reheating to $T_R$, which must respect the constraints described above to ensure baryonic asymmetry is preserved and baryogenesis is not washed out

5. Mathematical Summary: Critical Conditions and Order Parameters

The phase transition and baryogenesis constraints are succinctly encoded through the following relations:

• Tree-level RS potential:

$$V_{RS}(\mu, \phi_c) = V_{GW}(\mu) + \left(\frac{\mu}{\mu_{TeV}}\right)^4 \frac{\lambda}{4}(\phi_c^2 - v^2)^2$$

• Condition for nucleation:

$S(T_n) = B \sim 150$

• Sphaleron suppression:

$\phi_c(T_n)/T_n \gg 1$

• Max e-folds for sphaleron activity outside:

$N_e \lesssim 26$

• Post-reheating washout avoidance:

$T_R < T_{sph-decouple}(m_H)$

with $T_{sph-decouple}$ rising with heavier $m_H$.

6. Implications for Model Building and Electroweak Baryogenesis

The confining SFOEWPT in the Goldberger–Wise–stabilized RS model exemplifies the following principles:

• Supercooled transitions can vastly enhance the order parameter, ensuring strong first order EWPTs without special tuning in the Higgs sector or light exotic states.
• The radion (modulus of IR brane position) assumes the central dynamic role.
• Viable baryogenesis depends not only on achieving $\phi_c/T_n \gg 1$ but also on careful control of reheating and on the expansion rate, as only a finite number of e-folds are compatible with sustained out-of-bubble sphaleron transitions.
• A phenomenologically heavier Higgs is favored to relax reheating contraints, a counterintuitive result versus typical SM and minimal extensions.

This construction highlights the relevance of extra-dimensional stabilization mechanisms and strongly-coupled composite dynamics as robust providers of SFOEWPT, with distinctive predictions for the cosmological, gravitational wave, and possibly collider phenomenology. Any successful implementation also presupposes an additional source of CP violation to satisfy the remaining Sakharov conditions for baryogenesis.