Positronium decays with a dark $Z$ and fermionic dark matter

We investigate the invisible decay of positronium to probe the fermionic light dark matter mediated by the dark $Z$ boson. The invisible decay rate of positronium through weak interaction in the standard model is too small to be detected in the experiment. We show that it can be enhanced to be observed in the future if the dark matter is lighter than the electron in the dark $Z$ model. We also compute the relic abundance of such light dark matter and discuss the big bang nucleosynthesis constraint with an alternative thermal history scenario.

Figure 1

Exclusion regions by the positronium invisible decays in the plane of the model parameter $(m_{Z^{\prime }},|\sin {\theta }_X|)$. The magenta region is excluded by the current limit $\mathrm{Br}(\mathrm{o}\text{−}\mathrm{Ps}\to \mathrm{invisible})<4.2\times {10}^{-7}$, the green region by the future limit $<{10}^{-9}$, and the blue region by the future limit $<{10}^{-11}$. Electroweak constraints of $\mathrm{\Delta }\rho$ is the red region (overlapped by the Ps exclusion regions) and the APV constraints above the red line. The black line is the $1\text{−}\sigma$ exclusion line by the anomalous magnetic moment of the electron.

Figure 2

Behaviors of relic density for a few benchmark points, as functions of the interaction strength between DM pairs and $Z^{\prime }$ boson.

Figure 3

Branching ratios of the o-Ps invisible decay for points that accommodate the observed relic density of DM within a $3\sigma$ range as a function of $Z^{\prime }$ mass in unit of MeV. The points which satisfy the ‘Bullet Cluster’ constraint are represented by large squares.

Figure 4

Branching ratios of o-Ps and p-Ps decay into a photon and a $Z^{\prime }$ boson for points that accommodate the observed relic density of DM within $3\sigma$ range as a function of the $Z^{\prime }$ mass in unit of MeV. They appear as the conversion of positronium to single photon. The points which satisfy the ‘Bullet Cluster’ constraint are represented by large squares.