Study of WIMP annihilations into a pair of on-shell scalar mediators

In this article, we focus on a new scalar $\phi$ mediated scalar/vectorial WIMPs (weakly interacting massive particles) with $\phi$'s mass slightly below the WIMP mass. To explain the Galactic center 1 - 3 GeV gamma-ray excess, here we consider the case that a WIMP pair predominantly annihilates into an on-shell $\phi \phi$ pair with $\phi$ mainly decaying to $\tau \bar{\tau}$. The masses of WIMPs are in a range about 14 - 22 GeV, and the annihilations of WIMPs are phase space suppressed today. In this annihilation scheme, the couplings of the $\phi$ - standard model (SM) particles are almost arbitrary small, and the WIMP-nucleus spin-independent scattering can be tolerant by the present dark matter (DM) direct detections. A scalar mediator-Higgs field mixing is introduced, which is small and available. The lower limit on the couplings of the $\phi$-SM particles set by the thermal equilibrium in the early universe is derived, and this constraint is above the neutrino background for scalar DM in direct detections. The WIMPs may be detectable at the upgraded DM direct detection experiment in the next few years, and the exotic decay $h \rightarrow \phi \phi$, the production of $\phi$ may be observable at future high-luminosity $e^+ e^-$ collider.