Domain wall constraints on two-Higgs-doublet models with $Z_2$ symmetry

The two-Higgs-doublet model (2HDM) with spontaneously broken $Z_2$ symmetry predicts a production of domain walls at the electroweak scale. We derive cosmological constraints on model parameters for both type-I and type-II 2HDMs from the requirement that domain walls do not dominate the Universe by the present day. For type-I 2HDMs, we deduce the lower bound on the key parameter $\tan \beta >{10}^5$ for a wide range of Higgs-boson masses $\sim 100\mathrm{GeV}$ or greater, close to the Standard Model alignment limit. In addition, we perform numerical simulations of the 2HDM with an approximate as well as an exact $Z_2$ symmetry but biased initial conditions. In both cases, we find that domain wall networks are unstable and, hence, do not survive at late times. The domain walls experience an exponential suppression of scaling in these models, which can help ameliorate the stringent constraints found in the case of an exact discrete symmetry. For a 2HDM with softly broken $Z_2$ symmetry, we relate the size of this exponential suppression to the soft-breaking bilinear parameter $m_{12}$, allowing limits to be placed on this parameter of order $\mu \mathrm{eV}$, such that domain wall domination can be avoided. In particular, for type-II 2HDMs, we obtain a corresponding lower limit on the $CP$-odd phase $\theta$ generated by QCD instantons, $\theta \gtrsim {10}^{-11}/(\sin \beta \cos \beta )$, which is in some tension with the upper limit of $\theta \lesssim {10}^{-11}–{10}^{-10}$, as derived from the nonobservation of a nonzero neutron electric dipole moment. For a $Z_2$-symmetric 2HDM with biased initial conditions, we are able to relate the size of the exponential suppression to a biasing parameter $ϵ$ so as to avoid domain wall domination.

Figure 1

Variation of dimensionless energy per unit area of minimum energy kink solutions with $\tan \beta$ (left) and the SM alignment parameter, $\cos (\alpha -\beta )$ (right), for benchmark values of the $CP$-even scalar mass, $M_H$. In all cases, the energy per unit area of kink solutions goes to zero in the limit of large $\tan \beta$.

Figure 2

Variation of the dimensionless energy per unit area in the $Z_2$-symmetric 2HDM with SM alignment limit as a function of the $CP$-even scalar mass, $M_H$, and $CP$-odd mixing angle, $\tan \beta =v_2/v_1$, for minimum energy kink solutions obtained via gradient flow. Contour regions indicate the order of magnitude of the energy per unit area for a given set of physical parameters.

Figure 3

Evolution of domain walls in a 2HDM with approximate $Z_2$ symmetry in ($2+1$) dimensions for dimensionless soft-breaking parameter, ${\widehat{m}}^2=2\times {10}^{-3}$, $4\times {10}^{-3}$, $5\times {10}^{-3}$, and $7\times {10}^{-3}$ top-to-bottom. Remaining parameters are common to all sets of simulations and were chosen as $M_H=M_A=M_{H^{\pm }}=200\mathrm{GeV}$, $\tan \beta =0.85$, $\cos (\alpha -\beta )=1.0$. Simulations were run for time, $t=448$ with temporal grid spacing, $\mathrm{\Delta }t=0.2$ and spatial grid size, $P=4096$ with spacing, $\mathrm{\Delta }x=0.9$. Each set of plots progress in time left-to-right, and each plot is at double the time step of the previous. Each panel is a binary color map, indicating which of the two vacua the field lies in throughout the space.

Figure 4

Evolution of the number of domain walls in 2D 2HDM simulations with approximate $Z_2$ symmetry averaged over ten realizations for various values of the dimensionless soft breaking parameter, ${\widehat{m}}^2$. Remaining parameters chosen were $M_H=M_A=M_{H^{\pm }}=200\mathrm{GeV}$, $\tan \beta =0.85$, and $\cos (\alpha -\beta )=1.0$. Simulations were run for time, $t=1260$ with temporal grid spacing, $\mathrm{\Delta }t=0.2$ and spatial grid size, $P=4096$ with spacing, $\mathrm{\Delta }x=0.9$. Error bars, which are the standard deviation amongst the realizations, illustrate the numerical scatter. Also shown are nonlinear least squares fittings for an exponentially suppressed power law for $n=1$ fixed (top) and allowing $n$ to vary (bottom).

Figure 5

2D simulations of the evolution of domain walls in a 2HDM with $Z_2$ symmetry from increasingly biased initial conditions top-to-bottom. Parameters chosen were $M_H=M_A=M_{H^{\pm }}=200\mathrm{GeV}$, $\tan \beta =0.85$, and $\cos (\alpha -\beta )=1.0$. Simulation was run for time, $t=448$ with temporal grid spacing, $\mathrm{\Delta }t=0.2$, and spatial grid size, $P=4096$ with spacing, $\mathrm{\Delta }x=0.9$. Each set of plots progress in time left-to-right and each plot is at double the time step of the previous.

Figure 6

Evolution of the number of domain walls in 2D 2HDM simulations with $Z_2$ symmetry from biased initial conditions (i.e., normally distributed about $ϵ$) averaged over ten realizations. Also plotted is the standard power law scaling for a domain wall network $\propto t^{-1}$. Parameters chosen were $M_H=M_A=M_{H^{\pm }}=200\mathrm{GeV}$, $\tan \beta =0.85$, and $\cos (\alpha -\beta )=1.0$. Simulations were run for time, $t=448$ with temporal grid spacing, $\mathrm{\Delta }t=0.2$ and spatial grid size, $P=4096$ with spacing, $\mathrm{\Delta }x=0.9$. Error bars, which are the standard deviation amongst the realizations, illustrate the numerical scatter.