Appendix A: The DM S Three-body Decay In this Appendix, we develop some useful approximation formulae for the three-body DM S decay. First, let us explain the convention. The decay width is calculated in the rest frame of the heavy parent particle S (Fermionic or scalar component depends on the choice.). In a decay channel, we always have the order of the final state masses as m$_{3}$ > m$_{2}$ > m$_{1}$ . In this case, in general m$_{3}$ is the mass of a slepton or Higgs bonson, with m$_{3}$/m$_{D}$ ≲ 0 . 1. And m$_{2}$ denotes the mass of Higgsino, m$_{2}$/m$_{D}$ ≲ 0 . 05. Finally, the lightest particle is either muon or neutrino, whose mass m$_{1}$ = m$_{µ,ν}$ < 10 − $^{4}$m$_{D}$ is ignorable. To simplify the calculations we set m$_{1}$ = 0. First, we consider a general three-body decay process described by a heavy complex scalar S with mass M that decays into two fermions ψ$_{1}$$_{,}$$_{2}$ plus a scalar φ$_{3}$ . The Feynman diagram is given in Fig. 5. It is not difficult to get the corresponding amplitude square with the final state spin summed over where C is the coupling constant with mass dimension − 1, and C I is the possible mixing factor arising from converting the interaction state into the mass eigenstate. The mass m$_{1}$ is still kept in the momentum, and then will be set to zero finally. Fragmentation function dN/dx$_{I}$ is analytically calculable for three-body decay Y → a + b + c . For the isotropic decay it is very convenient [43] to write the phase space integral according to the scaled variables x$_{I}$ = 2 P$_{I}$ · $_{Y/}$√$_{Y}$ $^{2}$, where the invariant mass is defined as Y 2 = ( a + b + c ) $^{2}$. Here, the particles and their four-momenta are labeled with the same symbols. In the rest frame of Y , we have Y = m$_{Y}$ and x$_{I}$ = 2 E$_{I}$/m$_{Y}$ . The three-body phase Appendix B: Fragmentation Function | i M| 2 =4 C $^{2}$C2 $_{I}$( p$_{1}$ · p$_{2}$ + m$_{1}$m$_{2}$ ) , (A1) 18 FIG. 5: The DM S three-body decays into the Higgs bosons and leptons, or the sleptons, Higgsinos as well as leptons. S ψ 1 ψ 2 φ 3 S ∗ ψ † 1 ψ † 2 φ ∗ 3