Second Order Electroweak Phase Transition

• The topic is defined by a continuous evolution of the Higgs field's vacuum expectation value without latent heat or bubble nucleation.
• It provides a framework where a slow-rolling Higgs field links inflationary dynamics, reheating processes, and a deformed equation of state.
• Observable imprints include modifications to the stochastic gravitational wave spectrum, offering a probe into early universe phase transition dynamics.

The second order electroweak phase transition refers to a scenario in which the symmetry-breaking transition associated with the Standard Model (SM) Higgs sector occurs continuously, without a potential barrier and hence without latent heat or bubble nucleation. Historically, early calculations in the SM indicated a second-order or crossover transition for realistic Higgs masses. Recent studies have clarified the cosmological and phenomenological consequences of such a transition, emphasizing its impact on stochastic gravitational wave backgrounds, phase transition dynamics, and connections to inflationary cosmology.

1. Definition and Physical Features of Second Order Electroweak Phase Transition

The second order electroweak phase transition is characterized by the absence of a discontinuity in the order parameter, i.e., the vacuum expectation value (VEV) of the Higgs field evolves smoothly as the Universe cools through the critical temperature $T_c$. The finite-temperature effective potential $V(\phi, T)$ does not develop a barrier between the symmetric phase ($\phi = 0$) and the broken symmetry phase ($\phi \neq 0$). Instead, the Higgs field continuously rolls down its potential as the effective mass term changes sign. The absence of latent heat implies no macroscopic bubble nucleation, distinguishing second order transitions from first order scenarios central to baryogenesis mechanisms.

In SM physics, for a Higgs mass $m_H \sim 125$ GeV, both lattice calculations and perturbative results indicate a very weak first order or crossover transition with

$v_c / T_c \simeq 0.121$

where $v_c$ is the Higgs VEV at the critical temperature $T_c \sim 150$ GeV. This value lies well below the threshold required for strong first order transitions ($v_c / T_c \gtrsim 0.6 - 1.4$), justifying classification as second order or crossover (Oikonomou, 19 Oct 2025).

2. Higgs Inflation and Reheating in a Second Order Framework

Within models that invoke a non-minimally coupled Higgs field as the inflaton, the scalar potential in the Einstein frame takes the form

$$S_E = \int d^4x \sqrt{-\tilde{g}} \left[ \frac{M_P^2}{2}\tilde{R} - \frac{1}{2}\tilde{g}^{\mu\nu}\partial_\mu\phi \partial_\nu\phi - V(\phi) \right]$$

where

$$V(\phi) \simeq V_0 \left(1 - \exp\left(-\sqrt{\frac{2}{3}}\frac{\phi}{M_P}\right)\right)^{-2}$$

with $V_0 = \lambda M_P^4 / (4\xi^2)$ and $\xi$ a non-minimal coupling parameter (Oikonomou, 19 Oct 2025). This formalism yields inflationary dynamics compatible with Planck data.

Reheating proceeds at a temperature

$$T_{\text{re}} \sim \left( \frac{45 V_{\text{fin}}}{\pi^2 g_{\text{re}}} \right)^{1/4} \exp\left[ -\frac{3(1 + w_{\text{re}}) N_{\text{re}}}{4} \right]$$

where $g_{\text{re}}$ counts relativistic degrees of freedom and $w_{\text{re}}$ is the equation of state during reheating.

3. Slow-Rolling Higgs Field and Deformation of the Equation of State

As the Universe cools to $T \sim 150$ GeV, the second order phase transition allows the Higgs field to slow-roll toward the new minimum. Unlike first order transitions, the kinetic energy of the field remains subdominant and the equation of state (EoS) for the Higgs sector approaches $w_H \simeq -1$ during this period (Oikonomou, 19 Oct 2025). This results in a temporary deviation of the total EoS from pure radiation ($w = 1/3$), with effective values such as $w_{\text{eff}} \simeq 0.25$ or as low as $0.15$. The timescale for this deformation is set by the slow-roll dynamics and can be long compared to other microphysical processes.

4. Imprints on the Stochastic Gravitational Wave Background

The time-dependent EoS induced by slow-rolling during the second order electroweak phase transition alters the expansion rate relevant for primordial gravitational wave (GW) modes entering the horizon at $T \sim 150$ GeV and below. The GW spectrum $\Omega_{\text{gw}}(f)$ receives a multiplicative correction:

$$\Omega_{\text{gw}}(f) = S_k(f) \times \frac{k^2}{12 H_0^2} r \mathcal{P}_\zeta(k_{\text{ref}}) \left( \frac{k}{k_{\text{ref}}} \right)^{n_T} T_1^2(x_{\text{eq}}) T_2^2(x_R)$$

where

$$S_k(f) = \left( \frac{k}{k_s} \right)^{r_s} \quad \text{with} \quad r_s = -2 \frac{1-3w}{1+3w}$$

Here, $k_s$ is the wavenumber of modes reentering near $T \sim 150$ GeV; $n_T$ is the tensor tilt; $T_1$ and $T_2$ are transfer functions accounting for changes in effective relativistic degrees of freedom and reheating, respectively. The spectral deformation, most prominent for $w_{\text{eff}} < 1/3$, produces observable departures from scale invariance in $\Omega_{\text{gw}}(f)$ for $f$ corresponding to the electroweak epoch and below.

Current and future GW observatories (e.g., LiteBIRD) are being designed with the sensitivity to probe such features. A successful detection of deviations consistent with $S_k(f)$ would provide direct information on the expansion history and phase transition dynamics in the early Universe (Oikonomou, 19 Oct 2025).

5. Cosmological and Phenomenological Consequences

The absence of a strong first order electroweak phase transition impacts several cosmological processes:

• Baryogenesis: Second order transitions lack bubble walls and the associated out-of-equilibrium conditions crucial for electroweak baryogenesis, requiring alternate mechanisms (e.g., leptogenesis or physics beyond the SM).
• Topological Defects and Dark Matter: Smooth transitions generally do not generate ’t Hooft–Polyakov monopoles; however, model-dependent details may alter this conclusion in extended frameworks (Niemi et al., 2020).
• GW Signatures as Probes: The characteristic imprints of a deformed EoS during the electroweak crossover provide a unique observational avenue, distinct from the bubble collision signals of first order transitions.
• Inflationary Consistency: Higgs-driven inflation models naturally accommodate a slow-roll second order phase transition—linking cosmic microwave background (CMB) observables (e.g., $n_s$, $r$) with GW spectra and phase transition physics.

6. Theoretical Tools and Model Extensions

Analysis of the second order electroweak phase transition employs:

• One-loop and finite-temperature effective potentials with daisy resummation for SM-like sectors (Oikonomou, 19 Oct 2025).
• Dimensional reduction and 3d EFTs for more complicated scalar sectors, suited for lattice studies and for precision calculation of transition order and strength (Niemi et al., 2018, Niemi et al., 2021, Niemi et al., 2020).
• Non-minimal coupling to gravity for Higgs inflation scenarios (Oikonomou, 19 Oct 2025), integrating reheating and symmetry breaking in a unified framework.

Extension to models with singlet scalars, higher-dimensional operators, or modifications to the Higgs potential can “unlock” strong first order transitions in previously excluded regions, but these are technically first order phenomena, not applicable to pure second order transitions in the SM parameter regime (Oikonomou et al., 3 Mar 2024).

7. Distinction Between Crossover, Second Order, and Weak First Order

The SM with $m_H \sim 125$ GeV sits near the boundary between weak first order and crossover transitions. The phase transition order becomes ambiguous for extremely shallow potentials, but physically the absence of bubble nucleation and the continuous change in the Higgs VEV justifies classification as second order (or crossover). Quantitatively, if $v_c/T_c \lesssim 0.2$, the transition is too weak to fulfill baryogenesis requirements and produces continuous thermodynamic evolution (Oikonomou, 19 Oct 2025).

A plausible implication is that the SM electroweak phase transition is a second order (or crossover) event for phenomenologically realistic Higgs masses, and that observable consequences (e.g., GW spectral features) arise primarily from the EoS deformation rather than bubble dynamics.

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In summary, the second order electroweak phase transition describes a smooth, continuous breaking of electroweak symmetry with no latent heat or bubble nucleation, characteristic of the SM for experimentally viable Higgs mass. Its cosmological signatures are imprinted in gravitational wave backgrounds via the deformation of the equation of state during slow-roll Higgs evolution. These effects are distinguishable from those arising in strong first order transitions and are a target for precision GW cosmology and studies of early universe dynamics (Oikonomou, 19 Oct 2025).