5 FIG. 2: Total abundance of staus and neutralinos, which cor- responds to the relic abundance of DM. Each line shows the total abundance, and attached value represents the stau mass. Yellow band represents the allowed region from the WMAP observation at the 3 σ level , and the region inside the horizon- tal dotted lines corresponds to allowed region at the 2 σ level [1]. In the left side of vertical lines, the LSP is left-handed sneutrino. Three lines correspond to m ˜ τ = 400 , 350 , 300 GeV from left to right. III. NUMERICAL RESULTS We obtain the relic density of stau at the BBN era by integrating these equations from T$_{f}$ to the temperature for beginning the BBN under the initial condition of the total number density of stau and neutralino. These equa- tions make it clear that if the tau number density is out of the equilibrium, the ratio between those of stau and neutralino does not satisfy the Eq.(13). Here g$_{∗}$ ( T ) is the relativistic degrees of freedom, and we use s = 2 π 2 45 g$_{∗}$ ( T ) T 3 , H = 1 . 66 g 1 / 2 ∗ T 2 m$_{pl}$ . (21) dY$_{˜}$ χ dT = [ 3 HT g$_{∗}$ ( T ) ] − 1 [ 3 g$_{∗}$ ( T ) + T dg$_{∗}$ ( T ) dT ] s × { 〈 σv 〉 ˜ χτ − → ˜ τ − $_{γ}$Y$_{˜}$ $_{χ}$Y$_{τ}$ − − 〈 σv 〉 ˜ τ − γ → ˜ χτ − Y$_{˜}$ τ − Y$_{γ}$ + 〈 σv 〉 ˜ χγ → ˜ τ − τ + Y$_{˜}$ $_{χ}$Y$_{γ}$ − 〈 σv 〉 ˜ τ − τ $^{+}$→ ˜ $_{χγ}$Y$_{˜}$ τ − Y$_{τ}$ + + σv ˜ χτ $^{+}$→ ˜ τ + $_{γ}$Y$_{˜}$ $_{χ}$Y$_{τ}$ + σv 〉 ˜ τ $^{+}$γ → ˜ χτ + Y$_{˜}$ τ + Y$_{γ}$ + 〈 σv 〉 ˜ χγ → ˜ τ $^{+}$τ − Y$_{˜}$ $_{χ}$Y$_{γ}$ − 〈 σv 〉 ˜ τ $^{+}$τ − → ˜ $_{χγ}$Y$_{˜}$ τ + Y$_{τ}$ − } . (20)  0.07  0.08  0.09  0.1  0.11  0.12  0.13  0.14  0.15  0.4  0.5  0.6  0.7  0.8  0.9  1 GLYPH<C5>GLYPH<CX>GLYPH<DC>GLYPH<CX>GLYPH<D2>GLYPH<CV> GLYPH<CP>GLYPH<D2>GLYPH<CV>GLYPH<D0>GLYPH<CT> GLYPH<D3>GLYPH<CU> GLYPH<D0>GLYPH<CT>GLYPH<CU>GLYPH<D8>GLYPH<B9> GLYPH<CP>GLYPH<D2>GLYPH<CS> GLYPH<D6>GLYPH<CX>GLYPH<CV>GLYPH<CW> GLYPH<D8>GLYPH<B9>GLYPH<CW>GLYPH<CP>GLYPH<D2>GLYPH<CS>GLYPH<CT>GLYPH<CS> GLYPH<D7>GLYPH<D8>GLYPH<CP>GLYPH<D9> GLYPH<D7>GLYPH<CX>GLYPH<D2> GLYPH<AI> GLYPH<AS> GLYPH<CC> GLYPH<D3>GLYPH<D8>GLYPH<CP>GLYPH<D0> GLYPH<D6>GLYPH<CT>GLYPH<D0>GLYPH<CX>  GLYPH<CP>GLYPH<CQ>GLYPH<D9>GLYPH<D2>GLYPH<CS>GLYPH<CP>GLYPH<D2> GLYPH<CT>$^{GLYPH<AA>}$ GLYPH<BW> GLYPH<C5> GLYPH<CW> GLYPH<BE> GLYPH<CF>GLYPH<C5>GLYPH<BT>GLYPH<C8> GLYPH<BG>GLYPH<BC>GLYPH<BC>GLYPH<BZ>GLYPH<CT>GLYPH<CE> GLYPH<BF>GLYPH<BH>GLYPH<BC>GLYPH<BZ>GLYPH<CT>GLYPH<CE> GLYPH<BF>GLYPH<BC>GLYPH<BC>GLYPH<BZ>GLYPH<CT>GLYPH<CE> GLYPH<BE> GLYPH<AR> GLYPH<BE> GLYPH<AR> sin θ$_{τ}$ ≃ 0 . 8. The increase of the abundance can be un- derstood by the fact that the annihilation cross section of ˜ τ + ˜ τ → τ + τ becomes smaller as the heavier stau mixes. The increase is gradually compensated by two annihila- tion processes, ˜ τ +˜ τ ∗ → W $^{+}$+ W − and ˜ χ 0 $_{1}$+˜ τ → W $^{−}$+ ν$_{τ}$ , as the left-right mixing becomes large. The latter process can not be ignored because left-handed sneutrino is de- generate to stau and neutralino in the present parameter set. These processes become significant for sin θ$_{τ}$ < 0 . 8. Another process which reduces the total abundance due to the left-right mixing is ˜ τ +˜ τ ∗ → t + ¯ t through s-channel exchange of the heavy Higgses. This annihilation process becomes significant as the mixing reaches to π/ 4 and the masses of two staus are split. In fig. 2, the mass differ- ence between the lighter and the heavier stau is fixed to be 30 GeV to maximize the DM abundance, but the same result can be obtained by changing the stau mass for an- other values of the mass difference. As shown in Fig. 2, the total abundance also strongly depends on m$_{˜}$ $_{τ}$. This is understood as follows. In the non-relativistic limit, since the relic number density of relic species is proportional to ( m$_{relic}$ 〈 σv 〉 $_{sum}$) − 1 and 〈 σv 〉 sum (Eq. (8)) is proportional to 1 /m 2 relic [30], the total number density N is propor- tional to m$_{˜}$ $_{τ}$, As a first step for the calculation of the relic number density of stau, based on the discussion in section II B, we calculate the total abundance of stau and neutralino with micrOMEGAs [28]. Fig. 2 shows the total abundance, which corresponds to the relic abundance of DM, as a function of sin θ$_{τ}$ , where θ$_{τ}$ is the mixing angle between left and right-handed stau. Each curved line shows the total abundance for each stau mass, and horizontal band represents the allowed region from the WMAP observa- tion at the 3 σ level (0 . 0913 ≤ Ω$_{DM}$ h 2 ≤ 0 . 1285), and the region inside the horizontal dotted lines corresponds to allowed region at the 2 σ level (0 . 0963 ≤ Ω$_{DM}$ h 2 ≤ 0 . 1213) [1]. In the left side of vertical lines, the LSP is the left-handed sneutrino. Three lines correspond to m$_{˜}$ τ = 400 , 350 , 300 GeV from left to right. Since the left-handed sneutrino DM has been ruled out by con- straints from the direct detection experiments [29], we focus on the right-side region. Here we took γ$_{τ}$ = 0 and δm = 100 MeV. The total abundance increases first as the heavier stau mixes to the lighter stau, then turns to decrease at and the total abundance is given by Ω$_{DM}$ h 2 ∼ m$_{˜}$ $_{τ}$N . Thus the total abundance is proportinal to m 2 ˜ $_{τ}$, and it is consistent with the result in Fig. 2. In this section, we will first show the evolution of the stau number density, and then study the relation between the relic density of stau and the modification of nucle- osynthesis. Finally, we study a solution of the $^{7}$Li prob- lem with long-lived stau in the coannihilation scenario based on Ref. [14–16]. A. Total abundance N ∝ 1 m$_{˜}$ τ 〈 σv 〉 sum ∝ 1 1 /m$_{˜}$ τ = m$_{˜}$ τ , (22)