2. Analytical e ± Spectra from Three-Body Cascade Decays Appendix C: Fitting Positron Spectra In this Appendix, we present the positron spectra got through the above procedure. Similarly, we can calculate the anti-proton spectra, although we will not present the details here. The basic method is to simulate the positron and anti-proton spectra in the rest frame of the sparticles and Higgs bosons via PYTHIA (each state ( I, F ) must be simulated separately and then summed over). We will not pay much attention to the detail fitting but present the results directly. Notation is consistent with the one given above, but we change the variable to energy, as in the original form from simulation. The spectra are divided into three types: the type of sleptons, the type of Higgs bosons and the type of Higgsinos. For sleptons, the spectra from the left- and right-handed sparticles 22 The correction term is complicated but important since without it we cannot have the right order. This expression is valid for the energy fraction 2 ϵ$_{µ}$η$_{2}$ ≤ x S e ≤ η$_{2}$R max µ . The DM three-body cascade decay produces a harder spectrum than the two-body decay since its intermediate particle produces a hard spectrum. This is clearly shown in the muon case. dN dx S e = ∫ R max µ x S $_{e}$/η 2 dx$_{µ}$ [ 5 3 − 3 ( x S e x$_{µ}$ ) 2 + 4 3 ( x S e x$_{µ}$ ) 3 ] × ( 1 2 − ϵ 2 I 1 − x$_{µ}$ ) x 2 µ ≈ 5 18 − 3 2 ( x S e ) 2 + 11 9 x S e 3 − 2 3 x S e ) $^{3}$log x S e + ϵ 2 I 3 [ 5 − 5 x S e + 2 ( x S e ) 2 − 2 ( x S e ) 3 + 4 ( x S e ) $^{3}$log x S e +2 ( 5 − 9 ( x S e ) 2 + 4 ( x S e ) 3 )] . (B17) If we are only interested in the positrons with energy higher than 1 GeV, we obtain R − ≈ x S $_{e}$/x$_{µ}$ . Integrating over x µ e and x$_{µ}$ , we have where the approximated spectrum is calculated according to Eq. (B5). dN dx µ e = 2( x µ $_{e}$) $^{2}$(3 − 2 x µ $_{e}$) , d ˜ N dx$_{µ}$ ≈ ( 1 2 − ϵ 2 I 1 − x$_{µ}$ ) x 2 µ , (B16) As an application, we use the above formulae to calculate the e ± spectrum functions from S → µ + fermion + scalar followed by µ three-body decay. Notice that ϵ$_{µ}$ ≪ ϵ$_{e}$ ≪ 1 and η$_{1}$ ≤ r ≤ η$_{2}$ , we obtain 2 η$_{1}$ϵ$_{µ}$ ≤ x S e ≤ R max µ η$_{2}$ . The spectrum functions dN/dx µ e and d ˜ N/dx$_{µ}$ are respectively given by