A model of the dark sector where $O(\text{few}\mathrm{GeV})$ mass dark matter particles $\chi$ couple to a lighter dark force mediator $V$, $m_V\ll m_{\chi }$, is motivated by the recently discovered mismatch between simulated and observed shapes of galactic halos. Such models, in general, provide a challenge for direct detection efforts and collider searches. We show that for a large range of coupling constants and masses, the production and decay of the bound states of $\chi$, such as $0^{-+}$ and $1^{--}$ states, ${\eta }_D$ and ${\mathrm{\Upsilon }}_D$, is an important search channel. We show that $e^{+}{e}^{-}\to {\eta }_D+V$ or ${\mathrm{\Upsilon }}_D+\gamma$ production at $B$ factories for ${\alpha }_D>0.1$ is sufficiently strong to result in multiple pairs of charged leptons and pions via ${\eta }_D\to 2V\to 2(l^{+}{l}^{-})$ and ${\mathrm{\Upsilon }}_D\to 3V\to 3(l^{+}{l}^{-})$ $(l=e,\mu ,\pi )$. The absence of such final states in the existing searches performed at BABAR and Belle sets new constraints on the parameter space of the model. We also show that a search for multiple bremsstrahlung of dark force mediators, $e^{+}{e}^{-}\to \chi \overline{\chi }+nV$, resulting in missing energy and multiple leptons, will further improve the sensitivity to self-interacting dark matter.

• Received 29 October 2015

DOI:https://doi.org/10.1103/PhysRevLett.116.151801