content_image/333212.jpg content_image/333213.jpg content_image/333210.jpg content_image/333211.jpg
when the squarks are heavy. This upper bound is largely a consequence of the LEP bounds on the chargino masses which force the mixing ∼ m$_{Z}$/µ to be less than one. Eq. (29) provides a good analytic understanding of this number – it comes within approximately 10% of this value. The deviation is due to mixing effects that occur away from the large M$_{1}$ limit. 

In many models of SUSY breaking the relation M$_{1}$/α$_{1}$ = M$_{2}$/α$_{2}$ = M$_{3}$/α$_{3}$ holds. We refer to this condition as unified gaugino masses. Because this is equivalent to M$_{2}$ ≈ 2 M$_{1}$ at the weak scale, the LSP is mostly Bino and Higgsino. In this case, 

<img src='content_image/333212.jpg'>

<img src='content_image/333213.jpg'>

Finally, for m$_{χ}$ > m$_{W}$ , a thermal relic density within ± 3 σ of the WMAP measurement implies an upper limit on the amount of Higgsino in the DM particle. Therefore, 

<img src='content_image/333210.jpg'>

<img src='content_image/333211.jpg'>

This result holds for the case with unified gaugino masses as well. Note that Eqs. (31), (33), and (35) all occur for a DM mass of O (80 GeV). 

To saturate the above bound ( i.e. maximize σ$_{SD}$ for thermal, well-tempered DM) requires a Bino/Higgsino mixture (recall that d$_{q}$ vanishes for a pure Higgsino), with a negligible Wino contribution. The largest values of SD DD occur when the DM has the largest Bino/Higgsino mixing which happens for the lowest values of the DM mass. As the mass of the DM increases, a larger component of Higgsino or Wino is needed for the DM to efficiently annihilate down to the correct relic density, which in turn typically leads to a decrease in σ$_{SD}$ . 

As shown in Fig. 2, there is a tight correlation between the SD cross section and the DM mass, in the decoupling limit when there is gaugino mass unification and a thermal relic abundance. 

For low masses, the neutralino is well-tempered for low masses and as m$_{χ}$ → O (1 TeV) the neutralino approaches a pure Higgsino. Examining Fig. 2, except for when the annihilation channel χχ → t ¯ opens, σ$_{SD}$ is a smooth, monotonically decreasing curve. An experiment sensitive to cross sections of O (10 − $^{4}$pb) will probe m$_{χ}$ ≲ 200 GeV. There is a spread in the points in this figure from the liberal range taken on the relic density constraint. For masses approaching O (1 TeV), there is additional extent from the variation in the Bino content of the neutralino and from contributions from squark exchange. For masses at 1 TeV, σ$_{SD}$ goes from 10 − $^{6}$pb → 0 for M$_{1}$ from 1300 GeV → ∞ . Note that the projected reach of a 1 ton version of COUPP is O (10 − $^{6}$) pb for m$_{χ}$ = 1 TeV [17], which would probe the entire range of SD cross sections for neutralinos excepting a nearly pure TeV Higgsino . 

