Inverse Electroweak Phase Transition

• Inverse electroweak phase transition is a non-standard scenario where the Higgs vacuum expectation value follows a reversed or nonmonotonic thermal history due to extensions beyond the Standard Model.
• It arises from mechanisms such as extended scalar sectors, negative thermal mass effects, and topological influences that induce high-temperature symmetry breaking and multi-step transitions.
• This phenomenon impacts cosmological observables including gravitational waves, baryogenesis, and dark matter signatures, offering new avenues for testing beyond Standard Model physics.

An inverse electroweak phase transition is a scenario in which the usual thermal history of electroweak symmetry breaking is reversed or rendered nonmonotonic, such that the Higgs vacuum expectation value (VEV) transitions from broken back to symmetric, typically at high temperatures, or exhibits intermediate phases not present in the Standard Model. This concept encompasses a class of non-standard thermal or topological phase histories in which the broken phase appears at unexpectedly high temperatures, can be temporarily restored, or is even induced by quantum effects associated with spacetime topology, rather than by the usual thermal mechanisms.

1. Definition and Physical Context

The standard electroweak phase transition (EWPT) refers to the process in the early Universe where the Higgs field acquires a nonzero VEV as the temperature drops below the electroweak scale, spontaneously breaking the $SU(2)_L \times U(1)_Y$ gauge symmetry to $U(1)_{\rm EM}$. In an "inverse" EWPT, the order or nature of this transition is inverted or modified compared to this conventional scenario:

• The broken phase may exist at high temperature, with the Universe transitioning to the symmetric phase as it cools ("unrestored symmetry" or "non-restoration") (Meade et al., 2018).
• There may be intermediate, multi-step, or nonmonotonic thermal histories, including reentrant or restorative transitions at lower temperature, often enabled by extended Higgs sectors, singlet fields, or modified topological or cosmological backgrounds (Chen et al., 1 Mar 2025, Jarvinen, 2010).
• In some models, quantum effects associated with spacetime topology, rather than temperature, control when the symmetry breaking occurs (Oikonomou, 1 Apr 2025).

This behavior is typically associated with extended scalar sectors, strong portal couplings, specific multi-field dynamics, or vacuum structures influenced by additional symmetries or spacetime properties.

2. Mechanisms Realizing Inverse Electroweak Phase Transitions

2.1. Thermal Inverse Symmetry Breaking via Extended Scalars

In models with scalar fields beyond the Standard Model Higgs, thermal corrections to the effective potential can invert the conventional pattern of symmetry restoration:

• Scalar Singlet Models and Portal Couplings: A real scalar $S$ with a $\mathbb{Z}_2$ symmetry and significant Higgs-portal coupling can develop a nonzero VEV at high temperatures due to negative thermal mass-squared terms. The high-temperature effective potential can have its minimum at $(h=0, s=w)$ with $w\ne 0$, breaking $\mathbb{Z}_2$, before restoring the symmetry at lower temperatures as the Universe cools (Chen et al., 1 Mar 2025).
• Large Negative Mixed Quartics: Negative mixed quartic couplings between SM Higgs and many singlets can lead to symmetry non-restoration: the Higgs VEV persists (or is even enhanced) at high temperatures (Meade et al., 2018).

The essential mathematical mechanism involves the finite-temperature effective mass squared for the scalar direction: $m^2_{\rm eff}(T) = m^2 + c T^2$ where $c$ can become negative and large due to the interplay of couplings and the number of singlet degrees of freedom, reversing the expected symmetry restoration.

2.2. Topological and Nonthermal Inverse Transitions

Nonthermal inverse transitions arise from the global properties of spacetime rather than temperature:

• Compactification/Topology-Induced EWPT: On $S^1 \times \mathbb{R}^3$, the SM effective potential develops an additional minimum as the compact radius $L$ increases. For $L < L_c$ only the symmetric vacuum exists; for $L > L_c$ a broken-phase minimum appears and overtakes the origin, with a barrier separating the two. The tunneling (first-order) transition proceeds as space expands and $L \uparrow$ rather than by cooling (Oikonomou, 1 Apr 2025).

2.3. Multi-Step and Rotational Transitions in Extended Higgs Sectors

In certain supersymmetric or multi-singlet models, the field space may admit more complex intermediate vacua:

• Rotations in Singlet Space: The $\mu\nu$SSM features three right-handed sneutrino singlets. In some regions of parameter space, thermal evolution leads to multistep transitions in which singlet VEVs acquire nonzero values in intermediate phases with vanishing Higgs VEV, followed by a rotational transition to the final EWSB vacuum. The nontrivial rotation of singlet field VEVs is unique to models with an enlarged singlet vector space and can produce inverse or nonmonotonic EWPT trajectories (Chung et al., 2010).

2.4. Inverse EWPT in Strongly-Coupled and Composite Models

Technicolor or composite Higgs frameworks featuring multiple strongly coupled sectors can yield scenarios with multiple (and possibly inverse) phase transitions:

• Multiple or Reentrant Transitions: With two matter sectors (e.g., Ultra Minimal Technicolor), the EW symmetry can be broken, restored, and broken again at distinct temperature intervals due to the interplay of condensates in different sectors. This allows temporarily restored symmetry at intermediate temperatures—an "inverse" transition (Jarvinen, 2010).

3. Bubble Nucleation Dynamics and Vacuum Trapping

In scenarios where the potential is sufficiently flat or the barrier particularly high, the phase transition can proceed through:

• Inverse $s$-Bubble Nucleation: Bubbles with nonzero singlet VEV nucleate but may immediately collapse if the surface tension is large relative to the vacuum energy difference ($\Delta V$), i.e., if the critical radius $R_c = 2\sigma/\Delta V$ is not achieved. This leads to "vacuum trapping," where the Universe remains in the high-temperature broken phase (Chen et al., 1 Mar 2025).
• No Nucleation in Extremely Strong PTs: If the nucleation rate is too low, the phase transition never completes, and the cosmological history is altered. These failed transitions leave distinctive signatures, such as the suppression of gravitational-wave production.

The nucleation and bubble-wall structure are governed by the bounce equation: $$\frac{d^2 \phi}{dr^2} + \frac{2}{r} \frac{d\phi}{dr} = \frac{\partial V_{\rm eff}(\phi, T)}{\partial \phi}$$ with the action $S_3(T)$ determining the nucleation rate.

4. Theoretical Formulation and Quantitative Criteria

Typical features are encapsulated in the finite-temperature effective potential, generalized for multi-field scenarios: $$V_{\rm eff}(h, s; T) = V_0(h, s) + V_{\text{1-loop}}(h, s) + V_T(h, s; T) + V_{\text{daisy}}(h, s; T)$$ Critical temperatures, bubble nucleation rates, and transition strengths are all computed from this ensemble.

Symmetry breaking (or restoration) at high temperature is diagnosed via the (temperature-dependent) second derivative of the potential at the origin: $$\left. \frac{\partial^2 V_{\rm eff}}{\partial s^2} \right|_{h=0, \, s=0}$$ A negative value at high $T$ for $s$ yields high-$T$ symmetry breaking (inverse breaking).

The strength of first-order transitions is characterized by $v_c/T_c$ or, in multi-field models, $\sqrt{2}v(T_c)/T_c$, with values exceeding 1–1.3 being indicative of a transition strong enough for baryogenesis.

5. Cosmological and Experimental Implications

Gravitational Waves

The dynamics of inverse or multi-step EWPTs can directly impact the stochastic gravitational-wave background. If the transition fails (due to vacuum trapping or collapsing bubbles), GW signals are suppressed or absent. Multi-step or inverse transitions can yield complex GW spectra with multiple peaks, corresponding to sequential or overlapping transitions, potentially observable by detectors such as LISA or BBO (Chen et al., 1 Mar 2025, Jarvinen, 2010).

Baryogenesis and Sphaleron Processes

Inverse EWPTs alter the standard narrative for baryogenesis:

• If the transition is delayed, or occurs via symmetric to broken to symmetric again, the window for out-of-equilibrium sphaleron transitions is shifted.
• In multi-step or rotational transitions, CP violation may be enhanced at bubble walls traversing nontrivial field-space trajectories (e.g., by sourcing extra CP-phases in the singlet sector) (Chung et al., 2010).
• Failed completion of the transition may preclude the out-of-equilibrium conditions needed for baryogenesis.

Dark Matter Phenomenology

In models where the extended scalar is a dark matter candidate (e.g., the $\mathbb{Z}_2$-odd real singlet), inverse symmetry breaking and associated vacuum trapping modify the relic density and freeze-out conditions. The VEV pattern for the singlet across the Universe's history impacts couplings and annihilation cross-sections relevant for indirect detection.

Collider Signatures

Parameter regions leading to inverse EWPTs typically require sizable scalar self-couplings or portal couplings, which can be probed via modifications to the Higgs self-coupling, invisible decay width, or in exotic decay channels at future colliders.

6. Comparison with Standard Electroweak Thermal Histories

The inverse electroweak phase transition paradigm departs from the standard high-$T$ symmetric, low-$T$ broken narrative in several ways:

• Thermal ISB and Nonmonotonic Orders: The order parameter trajectory (Higgs VEV vs. $T$) is nonmonotonic and may include broken-to-symmetric or symmetric-to-broken-to-symmetric transitions.
• Topological Nonthermal Transitions: In topologically nontrivial spaces, even with $T = 0$, the expansion of the Universe (i.e., increasing compactification radius) can induce symmetry breaking through quantum corrections to the effective potential, leading to phase transitions disjoint from the thermal history (Oikonomou, 1 Apr 2025).
• Distinct Bubble and Wall Dynamics: Nucleation, collapse, and coexistence of bubble profiles become more diverse, with bubbles tracing nontrivial directions in scalar field space or failing to grow to the critical radius necessary for percolation.

7. Theoretical and Phenomenological Outlook

The concept of inverse or nonmonotonic electroweak phase transitions is now well established as a generic possibility in a wide range of beyond-the-Standard-Model scenarios, particularly those featuring extended scalar sectors, strong portal couplings, or nontrivial spacetime structure. These phenomena challenge the traditional thermal history assumptions, introduce rich dynamical behavior during the epoch of electroweak symmetry breaking, and motivate new experimental and observational strategies, both for collider searches and cosmological probes such as gravitational wave experiments. They also offer alternative mechanisms for baryogenesis and dark matter genesis, highlighting the interplay between particle physics, cosmology, and global properties of spacetime.