We consider a novel scenario for Vector Strongly Interacting Massive Particle (VSIMP) dark
matter with local

Article funded by SCOAP3

\frac{1}{2}$, the dark-charged gauge boson $X$ can be the lightest gauge boson in the dark sector, becoming a candidate for non-abelian dark matter. On the other hand, once taking into consideration the VEV of $S$, $Z'$ gauge boson can decouple from the $X$ spectrum due to the contribution of $v_S$ to its mass.
To maintain the thermal equilibrium of $X_\mu$ with the Standard Model bath, one needs to consider first its coupling to $Z'$.
Indeed, in order to communicate between non-abelian dark matter and SM by renormalizable couplings, it is sufficient to consider the dark Higgs $H_X$ with nonzero $Z'$ charge and take a sizable $g_{Z'}$.
Then, the mass matrix for neutral gauge bosons in the basis \mbox{($Z'_\mu$, $X_{3\mu}$)}, receives a correction term violating the dark custodial symmetry, as follows,
\begin{align}
M^2_{2\times 2}=m^2_{X_3}\left(\begin{array}{cc} \alpha s^2_X & -s_X c_X \\ -s_X c_X & c^2_X \end{array} \right) \label{mass220}
\end{align}
where $m^2_{X_3}\equiv (g^2_X+g^2_{Z'}) I^2 v^2_I=m^2_{X_3,0}+g^2_{Z'} I^2 v^2_I$,
$c_X\equiv \cos\theta_X$ and $s_X\equiv \sin\theta_X$, with $\sin\theta_X=g_{Z'}/\sqrt{g^2_X+g^2_{Z'}}$, and
\begin{align}
\alpha\equiv 1+\frac{q^2_S v^2_S}{I^2 v^2_I}.
\end{align}
Unlike in the SM, there exists a singlet scalar $S$ contributing to the $Z'$ mass.
The most general dark gauge boson masses with VEVs of Higgs fields in arbitrary representations are given in the appendix~\ref{TypSec_A}.
In the absence of the gauge kinetic mixing, the mass matrix~(\ref{mass220}) can be diagonalized explicitly by introducing a dark Weinberg angle as in the SM\@.
Performing a rotation of dark gauge fields to mass eigenstates,\footnote{In the presence of a gauge kinetic mixing between $Z'$ and hypercharge gauge boson, mass eigenstates as well as mass eigenvalues of neutral gauge bosons including the $Z$-boson are modified, but we consider the case where mass corrections are negligible but new interactions of extra gauge bosons to the SM are kept in the leading order in the gauge kinetic mixing parameter, as will be discussed in the next section.} ${\tilde Z}'_\mu, {\tilde X}_{3\mu}$, as
\begin{align}
\left(\begin{array}{c} Z'_\mu \\ X_{3\mu} \end{array} \right)= \left(\begin{array}{cc} \cos\theta'_X & -\sin\theta'_X \\ \sin\theta'_X & \cos\theta'_X \end{array} \right) \left(\begin{array}{c} {\tilde Z}'_\mu \\ {\tilde X}_{3\mu} \end{array} \right)
\end{align}
with
\begin{align}
\tan(2\, \theta'_X)= \frac{2c_X s_X}{c^2_X-\alpha\, s^2_X}, \label{gaugemix0}
\end{align}
we obtain the mass eigenvalues for dark gauge bosons,
\begin{align}
m^2_{ {\tilde Z}'} & = m^2_{X_3}c^2_X (1-\cot\theta'_X\, \tan\theta_X),
\label{mass20} \\
m^2_{{\tilde X}_{3}} & = m^2_{X_3}c^2_X(1+\tan\theta'_X\, \tan\theta_X). \label{mass30}
\end{align}
Therefore, from the results in eqs.~(\ref{mass10})--(\ref{mass30}), we can keep the hierarchy of masses,
\begin{align}
m^2_X< m^2_{{\tilde X}_3} < m^2_{{\tilde Z}'}\, ,
\end{align}
as far as
\begin{align}
\tan\theta'_X<0, \quad \quad |\tan\theta'_X|< \frac{1}{2\tan\theta_X}.
\end{align}
In this case, the dark charged gauge boson is still the lightest gauge boson in the dark sector, so that the $2\rightarrow 2$ annihilation of $X_\mu, X^\dagger_\mu$ is forbidden while $3\rightarrow 2$ processes with gauge self-interactions become dominant for determining the relic density of $X_\mu, X^\dagger_\mu$.
\relax
We note that if $m_X>m_{{\tilde X}_3}$, the $2\rightarrow 2$ annihilation of $X_\mu, X^\dagger_\mu$ into a ${\tilde X}_3$ pair is open, dominating the relic abundance calculation and leading to an interesting possibility for WIMP dark matter. However, in this work, we are interested in the production of light dark matter below sub-GeV scale, so henceforth we focus on the case with $m_X

m_X$, which means $I>\frac{1}{2}$, that is, at least a triplet Higgs field with nonzero VEV must be introduced.
Moreover, in order for SIMP processes $XXX\rightarrow X{\tilde X}_3$
(which should be the dominant one in the relic abundance calculation) to be kinematically allowed, we also require $m_{{\tilde X}_{3}}<2m_X$, which means $I\leq 2$.
Thus, we need $m_X

0$, we can have ${\tilde Z}'$ decoupled. This case is possible for a nonzero VEV of the Higgs field with $I=1, \frac{3}{2}, 2$. Then, all the non-abelian gauge bosons of $\SU(2)$ can participate in the full $3\rightarrow 2$ processes, without Boltzmann suppression. But, in this case, the mixing angle between dark neutral gauge bosons become suppressed due to $|\tan\theta'_X|\ll 1$. Therefore, DM-SM elastic scattering through the kinetic mixing would be suppressed, so we need to rely on Higgs portal coupling for kinetic equilibrium. If one considers multiple Higgs fields with different isospins, in particular, doublet $\Phi$ and triplet $T$, from eq.~(\ref{genmassdiff}), we get the approximate mass difference in the limit of a large ${\tilde Z}'$ mass: \begin{align} m^2_{{\tilde X}_3}-m^2_X\approx \frac{1}{2} g^2_X v^2_T - \frac{1}{\beta}\, g^2_X \Big(\frac{1}{4}v^2_\Phi+v^2_T\Big) \end{align} with \begin{align} \beta= 1+\frac{q^2_S v^2_S}{\frac{1}{2} v^2_\Phi+ v^2_T}\gg 1. \end{align} Then, we can tune dark Higgs VEVs, $v_\Phi$ and $v_T$ such that $m^2_{{\tilde X}_3}\approx m^2_X$. In this case, as discussed in the appendix, the mixing angle between dark neutral gauge bosons becomes \begin{align} \tan\theta'_X\approx -\frac{1}{\beta\tan\theta_X}. \end{align} Therefore, even in this case, the mixing angle in the dark gauge sector is small so Higgs portal coupling would be more relevant for kinetic equilibrium as in the previous case. ]]>

2m_X$, whereas the $2\rightarrow 2$ annihilations such as $X_+ X_-\rightarrow {\tilde X}_3{\tilde X}_3$ are open and dominant in the lower green region. For comparison, the mass relation, $m_{{\tilde X}_3}\approx \sqrt{3} m_X$, is satisfied along the blue dashed line. Below the blue dashed line, the forbidden channels tend to contribute to the DM annihilations dominantly. But, above the blue dashed line, the SIMP channels are dominant, and the relic density is saturated close to $m_{{\tilde X}_3}\sim 2m_X$, due to the $t$-channel diagrams. This behavior is regularized by a relatively large velocity of dark matter during freeze-out. \boldmath ]]>

H(T_\text{F}) \Big(\frac{m_X}{T_\text{F}} \Big)^2 \end{align} where $T_\text{F}\simeq m_X/15$ is the freeze-out temperature and $H(T_\text{F})$ is the Hubble parameter at freeze-out. On the other hand, since dark-neutral gauge boson ${\tilde X}_3$ has a similar mass as dark matter in our model, its abundance is not that suppressed around freeze-out temperature. Thus, ${\tilde X}_3$ plays an important role of keeping dark matter in thermal equilibrium through the SM by the decays into electron or muon pairs, the kinetic equilibrium for dark matter can be also achieved by the elastic scattering between dark matter and ${\tilde X}_3$. Thus, in this case, we require \begin{equation} n_{\tilde{X}_3}^{\rm eq} \Gamma_{\tilde{X}_3} > H(T_f)n_X^{\rm eq}. \end{equation} We find that kinetic equilibrium for dark matter can be easily achieved by the elastic scattering between dark matter and dark-neutral gauge boson ${\tilde X}_3$ as far as the latter remains in kinetic equilibrium for a tiny gauge kinetic mixing $\epsilon\sim 10^{-6}$ in the parameter space of our interest. In figure~\ref{epsilon1}, we showed the parameter space for $\varepsilon\approx c_W \xi$ vs $m_X$ with various constraints coming from the model consistency and experiments. We have assumed that the VEV of a quadruplet dark Higgs determines $\SU(2)_X$ gauge boson masses and chosen different values for the DM self-coupling $\alpha_X=1, 0.5$ in the top and bottom panels, respectively, varying $m_{{\tilde Z}'}$. We have similar plots for $\varepsilon$ vs $m_{{\tilde Z}'}$ in figure~\ref{epsilon2}. \begin{figure} \centering \includegraphics[height=0.295\textwidth]{figure/fig10a.pdf}\hfill \includegraphics[height=0.295\textwidth]{figure/fig10b.pdf}\hfill \includegraphics[height=0.295\textwidth]{figure/fig10c.pdf} ]]>

m^2_X$, we need $I>\frac{1}{2}$, namely, at least a triplet dark Higgs with nonzero VEV. For instance, ignoring the mass splitting due to the dark Weinberg mixing and keeping only one Higgs representation with $I=\frac{1}{2}, 1, \frac{3}{2}, 2$, we get \begin{align} m^2_{{\tilde X}_{3}} & \approx m^2_X, \quad 2m^2_X, \quad 3 m^2_X, \quad 4m^2_X. \end{align} Then, we get $\Delta\equiv (m_{{\tilde X}_{3}}-m_X)/m_X$ for $I=\frac{1}{2}, 1, \frac{3}{2}, 2$, as follows, \begin{align} \Delta=0, \quad \sqrt{2}-1, \quad \sqrt{3}-1, \quad 1. \end{align} But, for general VEVs of all Higgs representations with $\frac{1}{2}\leq I\leq 2$, we can cover the entire range of the mass splitting continuously for $0\leq \Delta\leq 2$. ]]>

{\tilde \lambda}_T$, ensures the positive squared masses for neutral dark Higgs boson, $h_1\approx h_T$, for a small mixing quartic coupling. In order to ensure the consistency of the dark vacuum, we require $m_{h^{++}}^2>0$ or \begin{equation} \lambda_{TH}<\dfrac{4v_T^2 \tilde{\lambda}_T}{v^2}, \label{consistent} \end{equation} \looseness=-1 which naturally pushes the value of the quartic mixing to be $ \lambda_{TH} \lesssim 10^{-5}$ for $v_T \sim\text{GeV}$, implying the mixing angle $\tan 2\theta \lesssim 10^{-5}$ to be below the bound from the Higgs invisible decay. Moreover, from the kinetic terms of the triplet complex field in eq.~(\ref{eq:Vhiggs}), we also derive the following interactions between dark Higgs and gauge bosons, {\rdmathspace \begin{align} \mathcal{L}_\text{DH-DG} &= i h^{++} \overset{\leftrightarrow}{\partial}_\mu (h^{++})^\dagger \Big( g_X X_3^\mu+ g_{Z^\prime} Z^{\prime \mu} \Big) +g_X^2\Big(\dfrac{v_T+h_T }{\sqrt{2}} \Big) \Big( (h^{++})^\dagger X^\mu X_\mu+h^{++} X^\dagger_\mu X^{\dagger \mu} \Big) \nonumber \\ & \quad +|h^{++}|^2 \Big( g_{Z^\prime}^2 Z^{\prime \mu} Z^{\prime}_{\mu}+2g_X g_{Z^\prime} Z^{\prime \mu}X_{3 \mu}+g_X^2 \left( X_3^\mu X_{3 \mu} +X^{\dagger \mu} X_\mu \right) \Big) \nonumber \\ & \quad +\Big(\dfrac{1}{2}h_T^2+v_T h_T \Big) \Big( g_{Z^\prime}^2 Z^{\prime \mu} Z^{\prime}_{\mu}-2 g_X g_{Z^\prime} Z^{\prime \mu}X_{3 \mu} +g_X^2 \left( X_3^\mu X_{3 \mu} +X^{\dagger \mu} X_\mu \right) \Big). \label{higgs-gauge} \end{align}}\relax \paragraph{Quadruplet Higgs bosons.} For the quadruplet Higgs $Q_4 \!=\! \big(h^{(3)},h^{(2)},0,\frac{1}{\sqrt{2}}(v_{Q_4} \!+\! h_{Q_4})\big)^T$ with \begin{equation} v_{Q_4} = \frac{2m_{Q_4}}{\sqrt{4\lambda_{Q_4} - 9\tilde{\lambda}_{Q_4}}}, \end{equation} the masses of dark-charged and neutral Higgs bosons are \begin{align} m_{h^{(2)}}^2 & = 3 \tilde{\lambda}_{Q_4} v_{Q_4}^2 - \frac{1}{2} \lambda_{Q_4H}v^2\\ m_{h^{(3)}}^2 & = \frac{9}{2} \tilde{\lambda}_{Q_4} v_{Q_4}^2- \frac{1}{2} \lambda_{Q_4 H}v^2,\\ m_{h_{Q_4}}^2 & = \bigg(2\lambda_{Q_4} - \frac{9}{2}\tilde{\lambda}_{Q_4} \bigg)v_{Q_4}^2 - \frac{1}{2}\lambda_{Q_4 H}v^2 . \end{align} As a result, there are similar consistent conditions on the mixing quartic couplings for $m_{h^{(2)}}^2>0$ and $m_{h^{(2)}}^3>0$, as in eq.~(\ref{consistent}). We can ignore the dark charged Higgs contributions in the later discussion, if they are heavy enough for $\lambda_{Q_4} v_{Q_4}^2\sim \tilde{\lambda}_{Q_4} v_{Q_4}^2 \gg m^2_X$. Moerover, a similar vacuum stability bound, $\lambda_{Q_4}> \frac{9}{4}{\tilde \lambda}_{Q_4}$, ensures the positive squared masses for neutral dark Higgs boson $h_{Q_4}$, for a small mixing quartic coupling. \paragraph{Quintuplet Higgs bosons.} For the quintuplet Higgs \scalebox{.92}{$Q_5 \!=\! \big(h^{(4)}\!,h^{(3)}\!,h^{(2)}\!,0,\frac{1}{\sqrt{2}}(v_{Q_5} \!+\! h_{Q_5})\big)^T$} with \begin{equation} v_{Q_5}=\frac{m_{Q_5}}{\sqrt{\lambda_{Q_5} - 4\tilde{\lambda}_{Q_5}}}, \end{equation} the masses of dark-charged and neutral Higgs are \begin{align} m_{h^{(2)}}^2 & = 4 \tilde{\lambda}_{Q_5} v_{Q_5}^2 - \frac{1}{2} \lambda_{Q_5H}v^2,\\ m_{h^{(3)}}^2 & = 6\tilde{\lambda}_{Q_5} v_{Q_5}^2 - \frac{1}{2} \lambda_{Q_5 H}v^2,\\ m_{h^{(4)}}^2 & = 8 \tilde{\lambda}_{Q_5} v_{Q_5}^2 - \frac{1}{2} \lambda_{Q_5 H}v^2,\\ m_{h_{Q_5}}^2 & = 2\bigg(\lambda_{Q_5} - 4\tilde{\lambda}_{Q_5} \bigg)v_{Q_5}^2 - \frac{1}{2} \lambda_{Q_5H}v^2. \end{align} As a result, there are similar consistent conditions on the mixing quartic couplings for $m_{h^{(a)}}^2>0$ with $a=1,2,3$, as in eq.~(\ref{consistent}). Similarly, he dark charged Higgs contributions can be neglected in the later discussion, when $ \lambda_{Q_5} v_{Q_5}^2\sim \tilde{\lambda}_{Q_5} v_{Q_5}^2\gg m^2_X$. Moerover, a similar vacuum stability bound, $\lambda_{Q_5}> 4{\tilde \lambda}_{Q_5}$, ensures the positive squared masses for neutral dark Higgs boson $h_{Q_5}$, for a small mixing quartic coupling. ]]>