In the expression of annihilation cross section, we used the following notations :
∂Φ
∂h = 1√
2 cos θ,
∂Φ
∂H = 1√
2
sin θ,
∂Ψ
∂h = − 1√
2 sin θ,
∂Ψ
∂H = 1√
2
cos θ. (A6)
Appendix B: Amplitude
We give explicit formulas of the invariant amplitude squared for the pair annihilation
processes of the RH neutrinos.
1. Annihilation into charged fermions
|M|2 =
32
⏐
⏐
⏐
⏐
g2
B−Lqf qN
s− M2
Z′ + iMZ′ ΓZ′
⏐
⏐
⏐
⏐
2
(s− 4m2
N)
( 3
8s− 1
2
( s
2 − m2
f
)
+ 1
2
( s
4 − m2
f
)
cos2 θ
)
+16λ2
N
⏐
⏐
⏐
⏐yf
( ∂Φ
∂h
i
s− M2
h + iMhΓh
∂Ψ
∂h + ∂Φ
∂H
i
s− M2
H + iMH ΓH
∂Ψ
∂H
) ⏐
⏐
⏐
⏐
2
(s− 4m2
N)
( s
4 − m2
f
)
. (B1)
2. Annihilation into neutrinos
a. Annihilation intoνa, ν a (light active-like neutrinos)
|M|2 =
32
⏐
⏐
⏐
⏐
g2
B−Lqf qN
s− M2
Z′ + iMZ′ ΓZ′
⏐
⏐
⏐
⏐
2
(s− 4m2
N)
( 3
8s− 1
2
( s
2 + m2
νa
)
+ 1
2
( s
4 + m2
νa
)
cos2 θ
)
.(B2)
10