∫ dcos θ
2 |M2|2 = λ4
N
( ∂Ψ
∂h
) 4 (
−8 − I22 + J22 ln
⏐
⏐
⏐
⏐
A+ 2b
A− 2b
⏐
⏐
⏐
⏐
)
, (B8)
∫ dcos θ
2 M1M∗
2 = 4 mN λ3
N
( ∂Ψ
∂h
) 2 ( ∂Ψ
∂h
i
s− M2
h + iMhΓh
iλhhh + ∂Ψ
∂H
i
s− M2
H + iMH ΓH
iλHhh
)
(
−4 + s− 4m2
N+ A
2b ln
⏐
⏐
⏐
⏐
A+ 2b
A− 2b
⏐
⏐
⏐
⏐
)
, (B9)
where θ is the scattering angle in the center of mass frame. The auxiliary fun ctions appear
above are deﬁned as
I22(s) ≡ 4(A+ 2a)2 − 2(s+ 4m2
N)A− s(A+ m2
N) − 3m2
N(s− 4m2
N)
A2 − 4b2 , (B10)
J22(s,mh) ≡ 1
Ab
(
2A(A+ 2a) − A(s+ 4m2
N) + A2 − 4a2 − (s− 2m2
N)(m2
N− m2
h)
+3m2
N(s− 4m2
N)
)
, (B11)
A(s,mh) ≡ − s
2 + m2
h, (B12)
b(s,mN ,mh) ≡
√
s
4 − m2
h
√
s
4 − m2
N. (B13)
Appendix C: Thermal averaged annihilation cross section
In partial wave expansion, the thermal averaged cross section isgiven by
⟨σv⟩ = 1
m2
N
[
w(s) − 3
2
(
2w(s) − 4m2
N
dw
ds
) T
mN
] ⏐
⏐
⏐
⏐
s=4m2
N
(C1)
= 6 dw
ds
⏐
⏐
⏐
⏐
s=4m2
N
T
mN
, (C2)
with
4w(s) ≡
∫
dLIPS
∑
|M|2 = 1
8π
√
s− 4m2
ﬁnal
s
∫ dcos θ
2
∑
|M|2, (C3)
where mﬁnal is the mass of ﬁnal state particle.
[1] T. Yanagida, inProceedings of Workshop on the Uniﬁed Theory and the Baryon Nu mber 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, inSupergravity, Proceedings of Workshop,
12