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## 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 

<img src='content_image/336924.jpg'>

<img src='content_image/336921.jpg'>

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. 

## Appendix B: Fragmentation Function 

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 

