7 with 〈 σv 〉 $^{6}$Li the thermal average of the cross section times the relative velocity for this process [38], and n$_{D}$ the number density of deuterium. By using (24), we see that the additional $^{6}$Li production is constrained to be Δ Y $^{6}$Li < O (10 − $^{21}$). Numerical value of 〈 σv 〉 $^{6}$Li gives 〈 σv 〉 6 $_{Li}$n$_{D}$ /H ∼ O (10 − $^{6}$) at T ∼ 10 keV. Then from (27) it is found that the upper bound on the abundance of stau should be Y$_{˜}$ τ − ≲ 10 − $^{15}$. Because the bound state ( $^{4}$He˜ τ $^{−}$) forms at T ≲ 10 keV, this process is strongly constrained for τ$_{˜}$ τ − ≳ 10 4 with τ$_{˜}$ τ − being the stau life- time, which corresponds to δm ≲ 100 MeV. Note that the ratio 〈 σv 〉 6 $_{Li}$n$_{D}$ /H rapidly decreases as the cosmic temperature decreases, and this nonthermal production of $^{6}$Li is much more effective just after formation of the bound state. This is the reason why we can estimate (27) at around 10 keV. On the other hand, the rates of $^{7}$Be and $^{7}$Li destruc- tion through the internal conversion [14–16] could be nearly equal to the formation rates of the bound sate ( $^{7}$Be˜ τ $^{−}$) and ( $^{7}$Li˜ τ $^{−}$), respectively. This is because the timescale of the destruction through the internal conver- sion is much faster than that of any other nuclear reaction rates and the Hubble expansion rate. Then the destroyed amount of $^{7}$Be (or $^{7}$Li after its electron capture) is ap- proximately represented by Further constraints come from the relic density of the DM, which can be stated in terms of the stau relic density. It is calculated as shown in Fig. 4 for the present values of parameters. Applying all the constraints, we are led to the allowed interval shown by the thick line in the figure. D. Constraint on parameter space of stau Δ Y $^{7}$Be ∼ 〈 σv 〉 bnd , $_{7}$n $^{7}$Be H Y$_{˜}$ τ − , (28) where 〈 σv 〉 bnd , 7 ∼ 10 − $^{2}$GeV − $^{2}$( T/ 30keV) − 1 / $^{2}$( Z/ 4) 2 × ( A/ 7) − 3 / $^{2}$( E$_{b}$ 7 $_{Be}$/ 1350keV) is the thermally-averaged cross section times the relative velocity of the bound- state formation for ( $^{7}$Be˜ τ $^{−}$) [16, 17]. We request Δ Y $^{7}$Be to become ∼ O (10 − $^{20}$) to reduce the abundance of $^{7}$Be to fit the observational data. Then the abundance of ˜ τ − should be the order of Δ Y 7 $_{Be}$( 〈 σv 〉 bnd , $_{7}$n 7 $_{Be}$/H ) − 1 ∼ O (10 − $^{12}$) with 〈 σv 〉 bnd , $_{7}$n 7 $_{Be}$/H ∼ 10 − 8 at T = 30 keV. Because 〈 σv 〉 bnd , $_{7}$n 7 $_{Be}$/H decreases as the cosmic tem- perature decreases ( ∝ T 1 / $^{2}$), the destruction is more ef- fective just after the formation of The bound state. This validates that we have estimated (28) at 30 keV. There- fore the parameter region at around Y$_{˜}$ τ − ∼ 10 − 12 and δm ≲ 130 MeV is allowed by the observational $^{7}$Li. Here δm ≲ 130 MeV corresponds to τ$_{˜}$ τ ≳ 10 3 s. The case for the destruction of ( $^{7}$Li˜ τ $^{−}$) through the internal conver- sion is also similar to that of ( $^{7}$Be˜ τ $^{−}$) [14, 15]. with a conservative systematic error (+0.106) [35]. For the $^{7}$Li abundance, we adopt two observational values of the n $^{7}$Li to n$_{H}$ ratio. Recently it has been reported to be We have adopted following observatinal abundances of $^{6}$Li and $^{7}$Li. Throughout this subsection, n$_{i}$ denotes the number density of a particle “ i ”, and observational errors are given at 1 σ . For the n $^{6}$Li to n $^{7}$Li ratio, we use the upper bound [34], FIG. 4: Parameter regions that are consistent with the ob- served abundances of the DM and the light elements. Con- straints from the observed $^{7}$Li abundances are due to Bonifa- cio et al. [36] and Melendez and Ramilez [37]. The calculated relic number density of stau with the indicated parameter is also shown. The top left region is excluded from $^{6}$Li overpro- duction for Y$_{˜}$ τ − ≳ 10 $_{−}$15 and δm ≲ 100 MeV. Allowed 10 -11 10 -12 10 -13 10 -14 10 -16 10 -15 10 100 Relic Dens i ty of the Stau Y τ ~ Stau-Neutralino Mass Difference δ m /[MeV] 3 σ  CL m $_{stau}$ = 350 GeV sin θ$_{τ}$  = 0.8 γ$_{τ}$  = 0 η  = 6.225 10 –10 Lo wer B ound  from Me lende z-Ramil e z Uppe r Boun d from Me lendez-Ra mil e z Upper  Bound  from  Bonifacio  et al . Overproduction of Dark Matter Overproduction of $^{6}$Li Low er B ound  from  Bon ifacio  et al. C a l c u l a t e d R e l i c S t a u D e n s i t y ( n 6 $_{Li}$/n 7 $_{Li}$)$_{p}$ < 0 . 046 ± 0 . 022 + 0 . 106 , (24) log$_{10}$( n 7 $_{Li}$/n$_{H}$ )$_{p}$ = − 9 . 90 ± 0 . 09 , (25) by Ref. [36], and on the other hand, a milder one was also given by Ref. [37], In the current scenario $^{6}$Li can be overproduced by the scattering of the bound state ( $^{4}$He˜ τ $^{−}$) off the background deuterium through ( $^{4}$He˜ τ $^{−}$) + D → 6 Li + ˜ τ − [4]. The abundance of the nonthermally-produced $^{6}$Li through this process is approximately represented by Finally, we show in Fig. 5 the parameter space in which the calculated abundances of the DM and that of the light elements are consistent with their values from the observations. Here, based on the discussion in previous Δ Y $^{6}$Li ∼ 〈 σv 〉 6 $_{Li}$n$_{D}$ H Y$_{˜}$ τ − , (27) log$_{10}$( n 7 $_{Li}$/n$_{H}$ )$_{p}$ = − 9 . 63 ± 0 . 06 . (26)