Kinky vortons in the 2HDM

Abstract

We construct and analyse two-dimensional, current-carrying ring solutions, known as kinky vortons, in the $\mathbb{Z}_2$-symmetric global two-Higgs-doublet model (2HDM). We demonstrate the existence of multiple dynamically stable configurations that persist under non-axially symmetric perturbations. These solutions are described with high accuracy by the thin string approximation and elastic string formalism, which correctly capture both their equilibrium radii and dynamical oscillation frequencies. Kinky vortons in the $\mathbb{Z}_2$-symmetric theory establish the viability of vorton solutions in a phenomenologically motivated extension of the Standard Model, and should provide a computationally tractable proxy for vortons in the $U(1)$-symmetric 2HDM. In addition, we identify a composite domain wall configuration in which localized condensates are supported on secondary domain walls existing on a $\mathbb{Z}_2$ wall, suggesting a mechanism by which kinky-vorton-like defects could arise in a three dimensional setting.

Figures (14)

• Figure 1: Example energy-minimizing kink solution for parameter set A in the $\mathbb{Z}_2$-symmetric global 2HDM, with $\kappa = -0.60$. Shown are the field profiles in the (a) general representation, (b) $g_i$ representation, and (c) bi-linear representation. Here, $R^4 = -\sqrt{R^\mu R_\mu}$ and hence $R^5 = 0$ globally. As $R^\mu R_\mu \neq 0$ at the centre of the kink, the $U(1)_{\rm EM}$ symmetry is broken locally.
• Figure 2: Predicted radius per winding number, $R_*/N$ as a function of $\kappa$ for $\kappa \in [-0.700, -0.001]$ for the parameter sets in Table \ref{['tab:param_sets']}, using kink solutions computed with $n_x=4000,\, \Delta x=0.05,\, \delta=10^{-7}$ (see Appendix \ref{['sec:numericals-kinks']} for definitions). Clear asymptotic behaviour is observed as $\kappa\to 0$.
• Figure 3: Effective mass of the condensate field, $f_+$, as $r \to \infty$, as a function of $\kappa$ for $\kappa\in[-0.700,-0.001]$ for the parameter sets in Table \ref{['tab:param_sets']}, using kink solutions computed with $n_x=4000$, $\Delta x=0.05$, and $\delta=10^{-7}$ (see Appendix \ref{['sec:numericals-kinks']} for definitions). Vacuum stability provides a clear lower bound on the $\kappa$-range in which kinky vortons may exist. For parameter set A, where the physical masses are larger than in sets B--D, this bound occurs at substantially larger $|\kappa|$. The corresponding upper bounds on $\omega$ are $\omega_A \le 1.961$, $\omega_B \le 0.970$, $\omega_C \le 0.944$, and $\omega_D \le 1.219$.
• Figure 4: Longitudinal and transverse perturbation propagation speeds, $c_L^2$ (solid) and $c_T^2$ (dashed), for current-carrying walls in the 2HDM as a function of $\kappa$ for $\kappa\in[-0.700,-0.001]$ for the parameter sets in Table \ref{['tab:param_sets']}. Kink solutions were computed with $n_x=4000$, $\Delta x=0.05$, and $\delta=10^{-7}$ (see Appendix \ref{['sec:numericals-kinks']} for definitions). A second-order finite-difference scheme is used to evaluate $\Sigma_2^\prime$ for the calculations of $c_L^2$ and $c_T^2$. We see non-standard behaviour in $c_L^2$ as $\kappa \to 0$, as Eq. \ref{['eq:speeds']} would naively suggest that $c_L^2 \to 1$.
• Figure 5: Intervals of instability for $\kappa\in[-0.700,-0.001]$ for different modes of oscillation ranging from $m=2$ to $m=50$. The coloured regions show where the discriminant of the cubic is less than zero, and as such values of $\kappa$ for which kinky vortons would be expected to be unstable to a given vibrational mode. Shaded areas of the plots correspond to regions of vacuum instability. There exist regions of stability in parameter sets A and B, when their vacuum instability is also considered, parameter set C is always unstable and, despite appearances, parameter set D does not have regions of stability because $c_L^2$ is always negative, see Fig. \ref{['fig:prop_speeds']}.