mass of the intermediate states, which also provide a possibility to probe the masses of the triplets. The dark matter cascade decay processes S$_{D}$ → δ$_{L}$δ$_{L}$ → 4 ℓ , 2 ℓ 2 ν and 4 ν$_{ℓ}$ contribute to new sources of primary positrons, neutrinos and gamma-rays in our galaxy. Besides the mass and lifetime of S$_{D}$ , the final energy spectra of the cosmic-ray particles depend on the mass ratio between S$_{D}$ and the triplets. In this section, we consider two typical mass hierarchies: the small mass hierarchy case (SH) , in which we take the mass ratio r$_{m}$ ≡ 2 m$_{δ}$$_{L}$/m$_{D}$ = 0 . 8, and the large hierarchy case (LH) in which r$_{m}$ = 0 . 1. Note that in the two extreme cases with r$_{m}$ =1 and 0, the shape of the final energy spectra should reduce to that of the two- body and four-body decays respectively. The relative strengths of the Yukawa couplings also play important roles, as they determine the corresponding branching ratios of different decay modes. In the basis in which the left-handed weak gauge interaction is diagonal, the leptonic Yukawa couplings Y$_{ij}$ ( i, j = e, µ, τ ) are related to y$_{ij}$ by Y = U T · y · U , where U is the mixing matrix for charged lepton. For the Yukawa couplings, we are going to consider four representative cases. In each of the first three cases, the Yukawa couplings are assumed to be dominant by one of the three generation leptons, while in the last case the couplings are assumed to be the same for all the generations, i.e., I ) | Y$_{ee}$ | ≫ | Y$_{µµ}$ | , | Y$_{ττ}$ | , the triplet scalars decay mainly into 4 e , 2 e 2 ν$_{e}$ and 4 ν$_{e}$ ; II ) | Y$_{µµ}$ | ≫ | Y$_{ee}$ | , | Y$_{ττ}$ | , the triplets decay mainly into 4 µ , 2 µ 2 ν$_{µ}$ and 4 ν$_{µ}$ ; III ) | Y$_{ττ}$ | ≫ | Y$_{ee}$ | , | Y$_{µµ}$ | , the triplets decay mainly into 4 τ , 2 τ 2 ν$_{τ}$ and 4 ν$_{τ}$ ; and IV ) | Y$_{ee}$ | ≈ | Y$_{µµ}$ | ≈ | Y$_{ττ}$ | , the triplet scalars decay to all leptons with the same branching ratio. In fact, the case IV will give the same results for the flavor democratic case in which all the Yukawa matrix elements are nearly identical. For simplicity we have assumed that the off-diagonal Yukawa couplings Y$_{ij}$ ( i = j ) are all small such that no lepton flavor violating process are considered. For each case, we consider various values of the mass and lifetime of S$_{D}$ with paying attention to the case that can reproduce the current data. Note that in this model, the large mass hierarchy r$_{m}$ = 0 . 1 is unlikely for the cases LH-I and LH-II unless S$_{D}$ is much heavier than 10 TeV, since the lower bounds of the triplet masses are expected to be around TeV scale. However, we still keep them for the sake of completeness as such light intermediate states may be possible in other models. III. ENERGY SPECTRA OF THE COSMIC-RAY PARTICLES 9