b. Annihilation into ν$_{s}$, ν$_{s}$ (heavy sterile-like neutrinos) |M| 2 = 32 ∣ g 2 $^{B}$−$_{L}$q$_{f}$ q$_{N}$ s − M 2 Z ′ + iM$_{Z}$ ′ Γ$_{Z}$ ′ ∣ ∣$^{2}$$^{(}$$^{s}$ − 4 m 2 $_{N}$) ( 3 8 s − 1 2 ( s 2 + m 2 ν$_{s}$ ) + 1 2 ( s 4 + m 2 ν$_{s}$ ) cos 2 θ ) +4 λ 2 $_{N}$λ 2 ν$_{s}$ ∣ ∣$^{∂}$$^{Ψ}$ ∂h i s − M 2 h + iM$_{h}$ Γ$_{h}$ ∂ Ψ ∂h + ∂ Ψ ∂H i s − M 2 H + iM$_{H}$ Γ$_{H}$ ∂ Ψ $_{∂H}$∣ ∣ ∣$^{2}$$^{(}$$^{s}$ − 4 m 2 $_{N}$)( s − 4 m 2 $_{ν$_{s}$}$) . (B3) |M| 2 = 8 λ 2 N ( 1 2 g $^{2}$v ) 2 ∣ ∣$^{∂}$$^{Ψ}$ ∂h 1 s − M 2 h + iM$_{h}$ Γ$_{h}$ ∂φ ∂h + ∂ Ψ ∂H 1 s − M 2 H + iM$_{H}$ Γ$_{H}$ ∂φ $_{∂H}$∣ ∣ ∣$^{2}$ ( s − 4 m 2 $_{N}$) ( 1 + 1 2 M 4 W ( s 2 − M 2 W ) 2 ) . (B4) |M| 2 = 8 λ 2 N ( 1 4 ( g 2 + g ′ $^{2}$) v ) 2 ∣ ∣$^{∂}$$^{Ψ}$ ∂h 1 s − M 2 h + iM$_{h}$ Γ$_{h}$ ∂φ ∂h + ∂ Ψ ∂H 1 s − M 2 H + iM$_{H}$ Γ$_{H}$ ∂φ $_{∂H}$∣ ∣ ∣$^{2}$ ( s − 4 m 2 $_{N}$) ( 1 + 1 2 M 4 Z ( s 2 − M 2 Z ) 2 ) . (B5) ∣ ∂h i s − M 2 h + iM$_{h}$ Γ$_{h}$ iλ$_{hhh}$ + ∂ Ψ ∂H i s − M 2 H + iM$_{H}$ Γ$_{H}$ iλ$_{Hhh}$ ∣ , (B7) |M| 2 = |M 1 + M 2 | $^{2}$, (B6) |M 1 | 2 = λ 2 N ( s 2 − 2 m 2 N ) 11 4. Annihilation into ZZ 3. Annihilation into W $^{+}$W − 5. Annihilation into hh M 1 denotes the amplitude by s -channel Higgs bosons h and H exchange, while M 2 does that for t ( u )-channel N exchange diagram. The formulas for NN → hH and HH can be obtained by appropriate replacement of the vertexes, e.g., λ$_{hhh}$ → λ$_{hhH}$ .