## C. Large SI and Small SD 

There are points which have large SI and SD with a nearly maximal gaugino fraction. If one relaxes the requirement of large SD, then the gaugino fraction can be pushed to nearly 100% while keeping the product Z$_{B,W}$ Z$_{H}$$_{u,d}$ approximately fixed, which in turn keeps the SI cross section constant. The relic density constraint can still be satisfied since both Winos and Higgsinos annihilate to W ± bosons with approximately the same rate. 

There is another way to have small SD while allowing large SI. In the context of the SDM, one can take λ = λ $^{′}$, i.e. t$_{β}$ = 1 in the MSSM. From the SDM mass matrix (see Appendix B), one can see that mixing between S and D$_{−}$ will vanish. Since the SD cross section is proportional to this mixing factor, Z$_{D}$$_{−}$ , it will be zero as well. This effect accounts for the empty region in Figs. 3 and 4 since we restricted t$_{β}$ > 5 in our numerical scans. 

For tan β ∼ > 1 . 5, we find that for σ$_{SI}$ ∼ 5 × 10 − 9 pb the smallest cross section for SD is σ$_{SD}$ ∼ 10 − 6 pb. If one allows σ$_{SI}$ < 5 × 10 − 9 pb, then as | Z$_{H}$$_{u,d}$ | → 0, σ$_{SD}$/σ$_{SI}$ → | Z$_{H}$$_{u,d}$ | 2 → 0. Hence, SD falls off faster than SI. However, this is the µ → ∞ limit which leads to fine-tuning as described above. 

## D. Small SI and Large SD 

Large SD requires a well-tempered neutralino, which naively also leads to large SI DD. In this section we will enumerate the various options one has for suppressing SI signals. We will argue that all options require fine-tuning or numerical coincidences $^{1}$. 

Here are the options for minimizing σ$_{SI}$ : 

1. One can make m$_{h}$ and m$_{H}$ heavy; however m$_{h}$ ≈ 115 GeV in the MSSM in the absence of large fine-tunings. Even in SplitSUSY, m$_{h}$ ∼ < 160 GeV. 

2. Since c$_{u,d}$ ∼ ( Z$_{W}$ − t$_{w}$ Z$_{B}$ ), i.e the Higgs couples to the Zino, one could attempt to restrict the DM to only be a photino-Higgsino admixture. In Appendix C, we show that this is impossible when one restricts M$_{2}$ by the LEP bound. 

3. One can tune $^{(}$f ( N ) Tu + 2 2 27 f ( N ) TG ) c$_{u}$ m$_{u}$ against $^{(}$f ( N ) Td + f ( N ) Ts + 2 27 f ( N ) TG ) c$_{d}$ m$_{d}$ by tuning the contribution from H against that from h . As we will discuss below, it is not possible 

