content_image/334521.jpg content_image/334532.jpg content_image/334531.jpg content_image/334530.jpg
<img src='content_image/334521.jpg'>

where the integration upper limit l$_{max}$ = $^{√}$r 2 MW − r 2 $_{⊙}$sin $^{2}$φ ′ + r$_{⊙}$ cos φ ′ and the DM halo size r$_{MW}$ ≈ 30 kpc. 

The muon neutrinos interact with the earth rock to produce the upgoing muon flux, which can be detected by the water Cherenkov detector Super-Kamiokande (SK) [33, 34]. The neutrino induced muon flux is give by [35] 

<img src='content_image/334532.jpg'>

where L ( E$_{µ}$ ) is the range of a muon with energy E$_{µ}$ until its energy drops below the SK threshold E$_{thr}$ = 1 . 6 GeV: 

<img src='content_image/334531.jpg'>

where α$_{µ}$ = 2 × 10 − $^{3}$g − $^{1}$GeV cm2 and β$_{µ}$ = 4 . 2 × 10 − $^{6}$g − $^{1}$cm$^{2}$. ρ is the density of the material in g cm − 3 and n$_{a}$ ≈ r$_{a}$ρ/m$_{p}$ are the number densities of neutrons and protons with r$_{p}$ ≈ 5 / 9 and r$_{n}$ ≈ 4 / 9. For the detection cross section, we use 

<img src='content_image/334530.jpg'>

where A n,p ν$_{µ}$ = 0 . 25 , 0 . 15, B n,p ν$_{µ}$ = 0 . 06 , 0 . 04 and A n,p ¯ ν$_{µ}$ = B p,n ν$_{µ}$ , B n,p ¯ ν$_{µ}$ = A p,n ν$_{µ}$ . 

As mentioned in Sec II, an important feature of this model is that the Yukawa couplings for δ ++ L ℓ $^{−}$ℓ$^{−}$, δ + $_{L}$ℓ $^{−}$ν$_{ℓ}$ and δ 0 $_{L}$ν$_{ℓ}$ ν$_{ℓ}$ are equal. In the DM particle decay processes, the final state 

