Induced Electroweak Walls in Extended Higgs Models

• Induced electroweak walls are defined as localized interfaces generated by non-Higgs scalar configurations that render the Higgs sector symmetric in the wall core.
• The mechanism utilizes scalar portal couplings (e.g., in N2HDM) to create a spatially dependent Higgs mass profile, leading to unsuppressed sphaleron activity within a quasi-two-dimensional region.
• This phenomenon provides a natural setting for baryogenesis and gravitational wave production, distinguishing induced walls from traditional first-order electroweak bubble walls.

Searching arXiv for the specified paper and closely related work on induced electroweak walls, domain walls, and electroweak symmetry restoration. Induced electroweak wall denotes a spatially localized interface generated by a non-Higgs scalar configuration—typically a domain wall, but in some constructions also a shock wave—that renders the Higgs sector electroweak-symmetric in or near its core while the ambient medium remains electroweak-broken. In this setting, the wall does not merely separate pre-existing electroweak phases produced by a first-order electroweak phase transition; rather, it induces a local sign change or suppression in the Higgs mass parameter through scalar portal couplings, thereby creating a quasi-two-dimensional electroweak-symmetric region with unsuppressed weak sphaleron transitions. In the N2HDM realization of Sassi and Moortgat-Pick, the mechanism is implemented by a singlet-field domain wall associated with an exact discrete symmetry, and electroweak symmetry restoration occurs in the vicinity of the wall because the singlet-dependent contributions to the Higgs-doublet masses vanish in the wall core (Sassi et al., 2024). Closely related realizations have been developed in singlet-extended Standard Model frameworks, embedded-wall constructions, and axion-like setups that use wall motion to source baryogenesis (Azzola et al., 2024, Schröder et al., 2024, Vanvlasselaer et al., 22 Apr 2026).

1. Definition and conceptual scope

The defining feature of an induced electroweak wall is that the interface is sourced by a field other than the electroweak order parameter itself. The wall profile belongs to an auxiliary scalar sector, while the Higgs expectation value responds to it through portal terms. This distinguishes induced electroweak walls from conventional electroweak bubble walls in first-order transitions, where the Higgs condensate is the primary order parameter and the wall separates thermodynamically competing minima of the finite-temperature Higgs effective potential (0903.4099, Friedlander et al., 2020).

In the N2HDM realization, the relevant auxiliary field is a real gauge singlet $S\equiv \Phi_s$ odd under an exact $Z_2$ symmetry, $S\to -S$. The spontaneous breaking of this discrete symmetry produces two degenerate vacua $S=\pm v_s$, so a domain wall forms between them. The portal interactions $\lambda_7 S^2|\Phi_1|^2/2$ and $\lambda_8 S^2|\Phi_2|^2/2$ then make the Higgs-doublet effective masses spatially dependent across the wall. Because $S(z)\to 0$ in the wall core, the singlet-induced mass contributions vanish there, and the electroweak vacuum expectation values can be driven to zero if the remaining mass parameters are positive in that region (Sassi et al., 2024).

This general pattern recurs across several models. In the minimal singlet extension with approximate $Z_2$, electroweak-symmetric wall cores arise when the Higgs mass term $\mu_H^2+\lambda_{HS}S^2/2$ becomes positive at the wall center, again because $S(0)=0$ (Azzola et al., 2024). In the axion-like induced-wall mechanism, the Higgs mass parameter is taken to depend directly on a wall-forming scalar $Z_2$0 as $Z_2$1, so one side of the wall is electroweak-symmetric and the other broken (Vanvlasselaer et al., 22 Apr 2026). These examples support a broad usage of the term “induced electroweak wall” for any wall-like scalar background that locally creates an electroweak phase boundary.

2. N2HDM realization: scalar structure and discrete symmetry

The N2HDM contains two $Z_2$2 doublets $Z_2$3 with hypercharge $Z_2$4, together with a real gauge singlet $Z_2$5. The discrete symmetry relevant for the induced electroweak wall is $Z_2$6, with all other fields neutral. This symmetry ensures two degenerate singlet vacua and therefore the possibility of a domain-wall solution (Sassi et al., 2024).

At tree level, the scalar potential is

$Z_2$7

For neutral vacua without charge or CP breaking on the wall boundaries, the vacuum ansatz is

$Z_2$8

with

$Z_2$9

The tree-level minimization conditions quoted for this setup are

$S\to -S$0

$S\to -S$1

$S\to -S$2

Within this structure, the induced electroweak wall is not an independent object added to the model. It is a dynamical consequence of the $S\to -S$3-odd singlet domain wall and the portal-induced spatial dependence of the doublet masses (Sassi et al., 2024). A plausible implication is that the mechanism is especially natural in extended Higgs sectors where discrete symmetries and scalar portals already play a structural role, rather than being introduced solely for baryogenesis.

3. Domain-wall solution and induced Higgs profiles

The planar wall is obtained as a static, $S\to -S$4-dependent kink connecting the two singlet vacua,

$S\to -S$5

with the Higgs doublet expectation values approaching their broken values far from the wall. The one-dimensional Euler–Lagrange equations are

$S\to -S$6

subject to

$S\to -S$7

These equations are solved numerically, for example by gradient flow, to obtain the full wall profiles (Sassi et al., 2024).

The central dynamical quantity is the spatially varying Higgs-doublet effective mass. Because of the couplings $S\to -S$8, one has, for example,

$S\to -S$9

and similarly for the second doublet. Since $S=\pm v_s$0 changes sharply across the domain wall, these masses can change sign across the profile (Sassi et al., 2024).

A typical numerical result quoted for the N2HDM wall gives a thickness

$S=\pm v_s$1

and a tension

$S=\pm v_s$2

The paper summary further states that sufficiently large negative ratios $S=\pm v_s$3 can extend the region over which $S=\pm v_s$4 over several $S=\pm v_s$5, so that the Higgs vacuum expectation values remain zero in a slab of thickness $S=\pm v_s$6 around the wall (Sassi et al., 2024). This enlarged electroweak-symmetric slab is an important refinement: the induced electroweak wall is not necessarily limited to the microscopic defect core itself, but can include a parametrically wider environment whose existence follows from the scalar mass hierarchy and portal couplings.

4. Electroweak symmetry restoration and sphaleron activity

Electroweak symmetry restoration in the induced-wall mechanism occurs because the wall core drives the doublet vacuum expectation values to zero. In the N2HDM description, when $S=\pm v_s$7 in the core, the singlet-induced mass terms $S=\pm v_s$8 vanish. If $S=\pm v_s$9 or $\lambda_7 S^2|\Phi_1|^2/2$0, the Higgs-doublet mass parameters become positive in the core,

$\lambda_7 S^2|\Phi_1|^2/2$1

The wall therefore interpolates between electroweak-broken bulk regions and a locally electroweak-symmetric interior (Sassi et al., 2024).

The cosmological significance of this restoration is governed by the weak sphaleron rate. Outside the wall, in the broken phase, sphalerons are exponentially suppressed: $\lambda_7 S^2|\Phi_1|^2/2$2 Inside the wall, by contrast, $\lambda_7 S^2|\Phi_1|^2/2$3, so $\lambda_7 S^2|\Phi_1|^2/2$4 and sphaleron transitions become unsuppressed: $\lambda_7 S^2|\Phi_1|^2/2$5 with the parametric enhancement

$\lambda_7 S^2|\Phi_1|^2/2$6

The condition for unsuppressed sphalerons is $\lambda_7 S^2|\Phi_1|^2/2$7 throughout the core, and since $\lambda_7 S^2|\Phi_1|^2/2$8 inside, this is automatic in the quoted setup (Sassi et al., 2024).

This induced-wall logic parallels several other constructions. The minimal singlet domain-wall baryogenesis scenario likewise requires $\lambda_7 S^2|\Phi_1|^2/2$9, ensuring sphaleron activity in the core and suppression outside (Azzola et al., 2024). Embedded walls are engineered so that the Higgs vanishes in the wall interior, restoring $\lambda_8 S^2|\Phi_2|^2/2$0 there and allowing sphalerons whose size satisfies $\lambda_8 S^2|\Phi_2|^2/2$1 to fit inside the wall (Schröder et al., 2024). The repeated appearance of the same condition across otherwise distinct models indicates that the induced electroweak wall is best understood as a mechanism for localizing baryon-number violation to a moving or persistent lower-dimensional defect.

5. Baryogenesis mechanisms associated with induced walls

The N2HDM wall construction directly furnishes one of the Sakharov ingredients—unsuppressed baryon-number violation—by restoring electroweak symmetry in the wall core. The remaining necessary ingredient is CP violation localized on or near the wall. The N2HDM summary states that if CP-violating phases are encoded, for instance, in a $\lambda_8 S^2|\Phi_2|^2/2$2-dependent Goldstone angle $\lambda_8 S^2|\Phi_2|^2/2$3 along the wall, chiral fermion currents can be biased into the symmetric region, where sphalerons convert the chiral asymmetry into net baryon number that is then frozen in when the fermions return to the broken region (Sassi et al., 2024). A full study of this domain-wall baryogenesis scenario is described there as in progress.

Related papers make the baryogenesis logic more explicit. In the N2HDM electroweak symmetry restoration study, a localized Higgs-doublet phase $\lambda_8 S^2|\Phi_2|^2/2$4 may be induced near the wall even if $\lambda_8 S^2|\Phi_2|^2/2$5, and the corresponding top mass term $\lambda_8 S^2|\Phi_2|^2/2$6 supplies a CP-asymmetric reflection source (Sassi et al., 2024). In the minimal singlet model, explicit CP-violating top couplings of $\lambda_8 S^2|\Phi_2|^2/2$7 generate semiclassical sources

$\lambda_8 S^2|\Phi_2|^2/2$8

with distinct behavior for linear and quadratic couplings under $\lambda_8 S^2|\Phi_2|^2/2$9; the quadratic source is even and therefore survives averaging over walls of opposite orientation (Azzola et al., 2024). In embedded-wall baryogenesis, diffusion-mediated transport from the wall edges produces a left-handed density that weak sphalerons convert into baryon number, with quoted estimates reaching the observed range under the parameter choices summarized in that work (Schröder et al., 2024).

A distinct but conceptually adjacent route is spontaneous baryogenesis on induced electroweak walls. In the axion-like setup of Vanvlasselaer and Yin, the wall-forming scalar couples both to the Higgs mass parameter and to the $S(z)\to 0$0 Chern–Simons density. The moving wall then generates a local effective $S(z)\to 0$1 chemical potential,

$S(z)\to 0$2

while unsuppressed sphalerons operate in front of the wall (Vanvlasselaer et al., 22 Apr 2026). This does not rely on CP-violating reflection in the conventional electroweak baryogenesis sense; instead, the anomalous derivative coupling biases baryon-number violation directly. A plausible implication is that “induced electroweak wall” has become an umbrella concept spanning both transport-based electroweak baryogenesis and spontaneous baryogenesis, provided the wall itself creates the required electroweak-symmetric region.

6. Relation to conventional electroweak bubble walls and seeded transitions

Induced electroweak walls should be sharply distinguished from standard electroweak bubble walls. In a conventional first-order transition, the wall is the phase interface of the Higgs background itself, and its properties are controlled by the finite-temperature effective potential, plasma friction, hydrodynamic matching, and possible runaway behavior (0903.4099, Bodeker et al., 2017, Friedlander et al., 2020). Such walls can be subsonic, terminal-velocity, or nearly luminal depending on the balance between vacuum pressure and friction, with important consequences for baryogenesis and gravitational waves (Friedlander et al., 2020, Bodeker et al., 2017).

In contrast, induced electroweak walls can arise without a first-order Higgs-driven phase transition. The out-of-equilibrium ingredient is supplied by the motion or dynamics of a wall in another scalar sector. This is explicit in the N2HDM discussion, where the wall is a $S(z)\to 0$3-domain wall in the singlet field and the electroweak-symmetric region is induced around it (Sassi et al., 2024). It is even more explicit in the minimal singlet domain-wall framework, where electroweak baryogenesis proceeds below the temperature of electroweak symmetry breaking as walls with restored or weakly broken electroweak cores sweep through space (Azzola et al., 2024).

A related but distinct phenomenon is domain-wall seeding of the electroweak phase transition. In the $S(z)\to 0$4-odd singlet model of Blasi and Mariotti, domain walls formed during an earlier step of symmetry breaking act as nucleation sites for the later electroweak transition, catalyzing the second step in a two-step thermal history (Blasi et al., 2022). There the wall is not necessarily itself an electroweak-symmetric slab in a broken ambient phase; instead, it reduces the bounce action for electroweak bubble nucleation on its surface. The seeded-transition mechanism is therefore adjacent to, but conceptually distinct from, the induced-wall mechanism.

This distinction matters because phenomenology is controlled by different time scales and geometric structures. Standard bubble walls are transient interfaces in a thermodynamic phase conversion. Seeded walls accelerate the onset of that conversion. Induced electroweak walls, by contrast, can be persistent defects or collapsing defects whose own cores host the active baryon-number-violating region. This suggests that induced electroweak walls occupy an intermediate category between topological-defect cosmology and electroweak phase-transition dynamics.

7. Cosmological implications, signals, and limitations

The most immediate cosmological consequence of an induced electroweak wall is the coexistence of electroweak-broken bulk regions with localized electroweak-symmetric slabs. This creates a spatially structured sphaleron network: baryon-number violation is inactive in the bulk but active on the wall. In the N2HDM summary, this motivates domain-wall baryogenesis in which CP-biased chiral transport into the wall core is converted into baryon number and subsequently frozen in outside the wall (Sassi et al., 2024).

Gravitational-wave production is another recurring implication. For the N2HDM domain-wall network, late annihilation is expected to emit stochastic gravitational radiation, with quoted typical ranges $S(z)\to 0$5 and $S(z)\to 0$6 (Sassi et al., 2024). In the axion-like induced-wall spontaneous-baryogenesis setup, domain-wall collapse and the attached induced electroweak bubble can yield signals spanning the nHz to mHz bands, with the nHz band identified there as especially promising (Vanvlasselaer et al., 22 Apr 2026). In collapsing-domain-wall baryogenesis with an axion-like field, the gravitational-wave spectrum is described as distinct from standard electroweak-scale first-order-transition spectra because the effective inverse time scale of wall collapse is $S(z)\to 0$7 rather than $S(z)\to 0$8 (Bai et al., 30 Apr 2026).

Several limitations and potential misconceptions also emerge from the literature. One misconception is that any electroweakly coupled wall automatically implies a first-order electroweak phase transition; induced-wall scenarios show that this is not required (Azzola et al., 2024, Schröder et al., 2024). Another is that all electroweak walls relevant to baryogenesis should behave like bubble walls with the same velocity constraints; induced walls may instead belong to scaling or collapse regimes characteristic of domain-wall networks, with RMS velocity $S(z)\to 0$9 in one regime and $Z_2$0 in another (Azzola et al., 2024). A further issue is model dependence of CP-odd sources: for example, linear CPV sources may average to zero across equally abundant wall orientations, whereas quadratic sources do not (Azzola et al., 2024).

Finally, induced electroweak walls should not be conflated with all electroweakly induced domain-wall claims in the literature. In Majoron models, for example, a purported electroweak instanton-induced domain-wall problem is argued to disappear unless explicit $Z_2$1 breaking is present, because the would-be anomaly-induced potential can be rotated away (Berbig, 3 Jun 2025). This illustrates that not every proposed connection between electroweak dynamics and wall formation survives detailed symmetry analysis.

Taken together, current work presents induced electroweak walls as a class of nonstandard electroweak interfaces generated by auxiliary scalar defects. In the N2HDM and related singlet or axion-like extensions, they realize local electroweak symmetry restoration, unsuppressed sphalerons, and potentially viable baryogenesis channels, while simultaneously opening a phenomenology that connects topological defects, extended Higgs sectors, and low-frequency gravitational-wave observables (Sassi et al., 2024, Azzola et al., 2024, Vanvlasselaer et al., 22 Apr 2026).