($_{−}$ 4 + s − 4 m N + A 2 b ln ∣ A 2 $_{b}$∣ ∣ ) , (B9) ∫ d cos θ 2 |M 2 | 2 = λ 4 N ( ∂ Ψ ∂h ) 4 ($_{−}$ 8 − I$_{22}$ + J$_{22}$ ln ∣ ∣$^{A}$$^{+ 2}$$^{b}$ A − 2 $_{b}$∣ ∣ ) , (B8) ∫ d cos θ 2 M 1 M ∗ 2 = 4 m$_{N}$λ 3 N ( ∂ Ψ ∂h ) 2 ( ∂ Ψ ∂h i s − M 2 h + iM$_{h}$ Γ$_{h}$ iλ$_{hhh}$ + ∂ Ψ ∂H i s − M 2 H + iM$_{H}$ Γ$_{H}$ iλ$_{Hhh}$ ) I$_{22}$ ( s ) ≡ 4 ( A + 2 a ) 2 − 2( s + 4 m 2 $_{N}$) A − s ( A + m 2 $_{N}$) − 3 m 2 $_{N}$( s − 4 m 2 $_{N}$) A 2 4 b 2 , (B10) b ( s, m$_{N}$, m$_{h}$ ) ≡ √ s 4 − m 2 h √ s 4 − m 2 $_{N}$. (B13) J$_{22}$ ( s, m$_{h}$ ) ≡ 1 Ab ( 2 A ( A + 2 a ) − A ( s + 4 m 2 $_{N}$) + A 2 − 4 a 2 − ( s − 2 m 2 $_{N}$)( m 2 N − m 2 $_{h}$) +3 m 2 $_{N}$( s − 4 m 2 $_{N}$) ) , (B11) s A ( s, m$_{h}$ ) ≡ − 2 + m 2 $_{h}$, (B12) 〈 σv 〉 = 1 m 2 N [ w ( s ) − 3 2 ( 2 w ( s ) − 4 m 2 N dw ds ) T m$_{N}$ ]∣ ∣ s =4 m 2 N (C1) = 6 dw $_{ds}$∣ ∣ s =4 m 2 N T m$_{N}$ , (C2) 4 w ( s ) ≡ ∫ d LIPS ∑ |M| 2 = 1 8 $_{π}$√ s − 4 m 2 final s ∫ d cos θ 2 ∑ |M| $^{2}$, (C3) where θ is the scattering angle in the center of mass frame. The auxiliary functions appear above are defined as In partial wave expansion, the thermal averaged cross section is given by with where m$_{final}$ is the mass of final state particle. Appendix C: Thermal averaged annihilation cross section [1] T. Yanagida, in Proceedings of Workshop on the Unified Theory and the Baryon Number in the Universe , Tsukuba, Japan, edited by A. Sawada and A. Sugamoto (KEK, Tsukuba, 1979), p 95; M. Gell-Mann, P. Ramond, and R. Slansky, in Supergravity , Proceedings of Workshop, 12