Why does the $\tilde\chi^0_2 \,\tilde\chi^\pm_1$ cross section increase in the focus point region?

Question

The focus point is an interesting region of the cMSSM parameter space at high $m_0$ and low $m_{1/2}$. Features are high scalar masses (> 1-2 TeV), light charginos / neutralinos (which are higgsino-like), and fairly low fine-tuning. One can also achieve correct dark matter relic densities and the correct higgs mass.

There seems to be a curve in the $m_0, m_{1/2}$ plane in this region, where if you approach it, the $\tilde\chi^0_2 \,\tilde\chi^\pm_1$ cross section goes up radically (maybe even diverges, spectrum generators can be finicky in that region). Why is that so? You can also rephrase the question, why do the chargino and neutralino masses go down there? I'd expect to first order a simple relationship between $m(\chi)$ and $m_{1/2}$, but around the focus point the masses seem very sensitive to $m_0$.

I guess this is due to the renormalization flow of the particle masses. The slope of the running masses seems a bit steeper than usual in focus point SUSY. There seem to be some interesting cancellations going on. It would be great if someone could elaborate on this.

Answer 1

Electroweak symmetry breaking requires that $\partial V/\partial H_{u,d} = 0$. Combining the two expressions, we find the superpotential bilinear $\mu$, $$ |\mu|^2 = \frac12 \left[\frac{|m_{H_d}^2 - m_{H_u}^2|}{\cos 2\beta} - m_{H_u}^2 - m_{H_d}^2 - M_Z^2\right]_{\text{EW-scale}} $$

In the focus-point region $|m_{H_u}^2|$ and $|m_{H_d}^2|$ are "focused" to $\lesssim M_Z^2$ at the electroweak scale; this is insensitive to their values at the high-scale.

Careful with the sign of $m_{H_u}^2$, though; it is parameter unto itself, rather than the square of a real parameter, and it is typically negative at the electroweak scale.

As a result $|\mu| \lesssim M_Z$, and certainly $|\mu| \ll M_1, M_2$. (Note that in minimal models like the CMSSM, $\mu$ is calculated in this way, but in more general models, we can trade e.g. a free parameter $m_{H_d}^2$ for $\mu$, and have $\mu$ as an input parameter and $m_{H_d}^2$ calculated.) The lightest neutralino/chargino is therefore Higgsino-like, with $$ m_{\chi_{1,2}} \approx |\mu|,\\ m_{\chi_{3}} \approx M_1,\\ m_{\chi_{4}} \approx M_2,\\ m_{\chi^\pm_{1}} \approx |\mu|,\\ m_{\chi^\pm_{2}} \approx M_2. $$