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where n$_{p}$ ( n$_{n}$ ) is the number density of protons (neutrons) in matter around the detector, the muon range R ( E$_{µ}$ ) is the distance that a muon could travel in matter before its energy drops be- low the detector’s threshold energy E$_{th}$ which is given by 

with α, β the parameters describing the energy loss of muons in matter. 

The cross sections of deep inelastic neutrino- nucleon scattering processes are given by [28] 

where y ≡ 1 − E$_{ℓ}$/E$_{ν}$ . We can see that the neu- trino detector is more powerful to observe neutrino signals in the high energy region. Because the 

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neutrino-nucleon cross section and muon range in- crease as E$_{ν}$ , while the atmospheric neutrino back- ground decrease as E 3 $_{ν}$. 

To obtain limits on the cross section per nucleon times the branching ratio σ$_{n}$ · Br$_{i}$ , we use the upper bound on high energy up-going muon flux in the direction of Sun given by Super-K group [29]. The Super-K detector is located in a mine with 1000 m rock overburden which contains 50,000 ton water. The detector can detect up-going muons with mea- sured path length of at least 7 m which is equiva- lent to the detector energy threshold 1.6 GeV. So the Super-K detector has a wide detecting range for high energy neutrino. Super-K group analyzed the data in 1679.6 days of detector live time, and no events induced by DM annihilation/decay are confirmed in the directions of the Sun, the center of Earth, and the Galactic Center. A stringent limit for up-going muon flux O (10 − $^{15}$) cm − 2 s − 1 con- strains the DM model. Here we use the constraint in Ref. [29] for all annihilation channels. 

The Super-K limits on σ$_{n}$ · Br$_{i}$ to different an- nihilation channels for iSI and iSD DM with dif- ferent mass splitting δ are presented in FIG. 3, and 4 respectively. For a typical DM annihilation cross section of 10 − 4 pb (or higher) that explains DAMA in the iSI DM case, the limits are quite constraining on hard channels ( W $^{+}$W $^{−}$, ZZ , t ¯ t , τ $^{+}$τ $^{−}$), are lessened in softer channels like c ¯ c and b ¯ b and are quite loose in the light quark channels. In the iSD DM case, the typical DM annihilation cross section (or higher) to explain DAMA is en- hanced to 10 pb [12]. Interestingly, even though 

