Complete Classification of Domain Wall Solutions in the $\mathbb{Z}_2$-symmetric 2HDM

TL;DR

This work classifies all domain wall solutions in the Z2-symmetric 2HDM by reducing the field content to six active components and identifying four interconnected solution classes: standard, superconducting, CP-violating, and a combination of superconducting and CP-violating. It combines a six-field reduction with mass-dependent stability analyses to map the parameter space, explaining why dynamical simulations generate wall configurations that deviate from the previously known minimum-energy state. The authors demonstrate current-carrying (superconducting) walls, show how CP-violating walls can localize CP violation along the wall (CP1-type), and reveal a two-dimensional CP-violating composite wall, all supported by detailed numerical simulations and semi-analytic arguments. These results have cosmological relevance, offering mechanisms for gravitational wave production, baryogenesis, and the possible realization of Kinky Vortons in a tractable lower-dimensional setting. The work thus provides a predictive framework linking wall structure to scalar mass hierarchies and EW rotations, with clear avenues for three-dimensional extensions and phenomenological exploration.

Abstract

We present a complete classification of domain wall solutions in the two-Higgs Doublet Model (2HDM) with a global $\mathbb{Z}_2$ symmetry, categorised as superconducting, CP-violating, or neither, depending on the scalar particle masses and the ratio of the two Higgs doublets' vacuum expectation values. We demonstrate that any domain wall solution can be reduced to depend on only six of the eight general field components, with further field reductions possible within different regions of the parameter space. Furthermore, we show that the superconducting solutions can be used to construct stable, current-carrying domain walls in two spatial dimensions. Similarly, the CP-violating solutions allow for two-dimensional configurations where CP symmetry is locally broken on the $\mathbb{Z}_2$-symmetric wall, which could provide an out-of-equilibrium environment for CP-violating processes to occur.

Figures (19)

• Figure 1: $(2+1)$-dimensional simulations of the 2HDM with a $\mathbb{Z}_2$-symmetric potential for different mass parametrisations (A, B, C and D). Shown are the bi-linear components $R^1$ (top), $R^2$ (middle), and $R^\mu R_\mu$ (bottom). Colour mappings of each individual component are normalized across all parameter sets, with each component type having a separate scale.
• Figure 2: Values of $R^2$ and $\sqrt{R^\mu R_\mu}$ at the centre of fixed boundary domain wall solutions, and the total energy, E, as functions of $\gamma_1$ and $\theta_1$ in parameter set A, for relatively EW rotated vacua. The rotations were varied in increments of $\pi/128$ in the range $[0, 4\pi]$, such that $512^2$ different solutions were computed.
• Figure 3: Values of $R^4$ and $R^5$ at the centre of fixed boundary domain wall solutions, as functions of $\gamma_1$ and $\theta_1$ in parameter set A for relatively EW rotated vacua. The rotations were varied in increments of $\pi/8$ in the range $[0, 4\pi]$, such that $32^3$ different solutions were computed.
• Figure 4: Relatively EW rotated domain wall solutions where the solution exhibits either non-zero $R^2$ (top) and $R^4$ (bottom) in parameter set A. Solutions where the value at the centre of the wall is positive are depicted on the left and those of equal magnitude but negative on the right. Within the space of ($\gamma_1, \theta_1, \theta_2$) presented are solutions (a):($0,\pi,0)$, (b):($2\pi,\pi,0$), (c):($3\pi,0,0$), (d):($\pi,0,0$). In (a) and (b) the field components $f_+,\, \chi,\, \gamma_1,\, \gamma_3$ are identically zero for all $x$, in (c) and (d) $\xi,\, \chi,\, \gamma_2,\, \gamma_3$ are identically zero for all $x$.
• Figure 5: (a) $\sqrt{R^\mu R_\mu}$ at the centre of the domain wall for naturally bounded solutions of the field configuration (\ref{['eq:gen_rep']}) with the field restriction $\gamma_1 = \eta_1 = \eta_2 = \eta_3 = 0$, and parameter restriction $M_A > M_H^\pm$. Scalar masses are given in GeV, and we set $M_A$ to be $100 \text{ GeV}$ greater than $M_{H^\pm}$, however the magnitude of this mass difference does not affect the solutions. There is a clear dark-shaded region where there exist solutions with a stable condensate. The dashed line represents $M_H = M_{H^\pm}$. (b) Example solution for parameter set D, where a condensate forms in $f_+$.