In the limit of large M$_{1}$ ( M$_{2}$ ) the neutralino is dominantly a Wino/Higgsino (Bino/Higgsino) admixture. We can explore this effective 3 state system using the SDM defined as (see Eq. (27) above): The resulting lightest eigenstate ( χ ) is specified by Following [9], it is useful to write this system in a basis defined by S and D$_{±}$ ≡ 1 √ 2 ( D ± ¯ D ). Note that the labels ± have nothing to do with electric charge. The mass matrix is then, in the ( S, D$_{+}$, D$_{−}$ ) basis, with the resulting lightest eigenstate, Since we are interested in the SD DD cross section, our goal is to extract the coupling of χ to the Z $^{0}$. The coefficient of the operator O SD q of Eq. (11) is given by Note that | 2 Z$_{D}$$_{+}$ Z$_{D}$$_{−}$ | ≡ | Z$_{D}$ | 2 − | Z ¯ $_{D}$| $^{2}$. One can find analytic expressions for the mass eigenstates and the combination | 2 Z$_{D}$$_{+}$ Z$_{D}$$_{−}$ | in various useful limits. To second order in v , for | µ$_{D}$ | , | µ$_{S}$ | , ( | µ$_{D}$ | − | µ$_{S}$ | ) ≫ λ v, λ ′ v and for | µ$_{D}$ | = | µ$_{S}$ | ≫ λ v, λ ′ v , m$_{χ}$ = µ$_{S}$ − 1 √ 2 | λ + λ $^{′}$|  | v | + ( λ − λ $^{′}$) 2 v 2 8 µ$_{S}$ (B.7) | 2 Z$_{D}$$_{+}$ Z$_{D}$$_{−}$ | = ( λ ′ − λ ) v 2 √ 2 | µ$_{S}$ | + ( λ ′ 2 − λ $^{2}$) v 2 8 µ 2 S . (B.8) | 2 Z$_{D}$$_{+}$ Z$_{D}$$_{−}$ | = ( λ ′ 2 − λ $^{2}$) v 2 µ 2 D − µ 2 S , (B.6) m$_{χ}$ = µ$_{S}$ − 2 λ λ ′ v 2 µ$_{D}$ − ( λ 2 + λ ′ $^{2}$) v 2 µ$_{S}$ µ 2 D (B.5) d$_{q}$ = − g 2 4 m 2 Z c 2 w | 2 Z$_{D}$$_{+}$ Z$_{D}$$_{−}$ | 2 T q $_{3}$. (B.4) χ ≡ Z$_{S}$ S + Z$_{D}$$_{+}$ D$_{+}$ + Z$_{D}$$_{−}$ D$_{−}$. (B.3) M$_{SDM}$ =     µ$_{S}$ 1 √ 2 ( λ + λ $^{′}$) v 1 √ 2 ( λ − λ $^{′}$) v 1 √ 2 ( λ + λ $^{′}$) v µ$_{D}$ 0 1 √ 2 ( λ − λ $^{′}$) v 0 − µ$_{D}$     , χ ≡ Z$_{S}$ S + Z$_{D}$ D + Z ¯ D ¯ D. (B.2) 27 L$_{SDM}$ ∋ µ$_{D}$ D ¯ D + λ h S D + λ $^{′}$h ∗ S ¯ D + µ$_{S}$ 2 S $^{2}$. (B.1) Appendix B: The Bino/Higgsino and Wino/Higgsino Limits