particle. But the positron production is not clear in the course of fragmentation, so we cannot make sure the value of I $_{e}$. However, for m$_{e}$ ≪ m$_{I,F}$ , we have I e = 1 [44]. For three-body decays, the upper limit R max I,F can be found in Eq. (B2). In the rest frame of the ( I, F ) particle, θ$_{I,F}$ is the angle between the electron spatial momentum direction and the boost axis of the ( I, F ) particle. The δ − function simply indicates that for some fixed boosted energy x S $_{e}$, one should integrate over all possible configurations for ( x I,F e , cos θ$_{I,F}$ ). In other words, Eq. (B12) makes the Lorentze boost of the energy spectrum: dN/dx I,F e → d ˜ N/dx S $_{e}$, and then convoluted by the distribution dN/dx$_{I,F}$ that denotes the differential probability of S decay to ( I, F ) particle in the rest frame of S . Finally, we get the total positron energy spectrum. For isotropic decay of the ( I, F ) particle, the distribution function is independent on angle, which can be integrated out ∫ dN dx S e dx S e = multi$_{I,F}$ , (B14) R $_{±}$( x$_{I,F}$ ) = 1 2 ( x S e ± √ 1 − 4 ϵ 2 I,F √ ( x S $_{e}$) 2 − 4( ϵ$_{e}$ϵ$_{I,F}$ x$_{I,F}$ ) 2 ) ϵ − 2 $_{I,F}$x − 1 $_{I,F}$. (B15) dN dx S e = Γ$_{eff}$ Γ$_{total}$ ∑ I,F C 2 $_{I,F}$R$_{I,F}$ √ 1 − 4 ϵ 2 I,F ∫ R max I,F 2 ϵ$_{I,F}$ dx$_{I,F}$ 1 R$_{I,F}$ d ˜ N dx$_{I,F}$ ∫ max ( x$_{I,F}$ ) min ( x$_{I,F}$ ) dx I,F e 1 √ ( x I,F e ) 2 − 4 ϵ 2 e dN dx I,F e . (B13) where multi$_{I,F}$ denotes how many electrons are produced by the fragmentation of each ( I, F ) particle. The integrand region in Eq. (B13) is given by min( x$_{I,F}$ ) = R $_{−}$( x$_{I,F}$ ) and max( x$_{I,F}$ ) = min( I $_{e}$, R $_{+}$) with If R − > I $_{e}$, the integral is zero. Then the constrained R $_{±}$, which depends only on the ratio r = x S $_{e}$/x$_{I,F}$ , in turn constrains the integral region for x$_{I,F}$ . If ϵ$_{I,F}$ ≪ 1 is satisfied, then R$_{+}$ ≈ x S $_{e}$/ ( x$_{I}$ϵ 2 $_{I}$) ≫ 1. So the upper limit always takes I $_{e}$. For example, let us choose ( I, F ) = µ . But in this paper, this is not always true since the sparticles and Higgs bosons are heavy. Consequently there are some values of r which gives R + > I $_{e}$. For the later case, one can show that R ± is monotonically increase with r , and the lower limit smaller than I e gives the upper bound for r ≤ η$_{2}$ , which is the larger root of R − = 0. However, in the former case, R − > I e gives bound η$_{1}$ ≤ r ≤ η$_{2}$ . In both cases, for fixed x S $_{e}$, we must have x$_{I}$ ≥ x S $_{e}$/η$_{2}$ . 21 The normalization condition of dN/dx S e in each decay channel is