lpetrich said:
There's a simpler hypothesis: both particles' masses are associated with electroweak symmetry breaking, and that gives them similar masses.
Well the electron and top quark both get masses from EWSB but their masses are not similar at all.
In any case, it can only be for a small region of MSSM parameter space that the higgsino has a mass comparable to the lightest Higgs, or for the higgsino to even be the lightest superpartner (LSP). Martin,
http://arxiv.org/abs/hep-ph/9709356, is a very nice review of the MSSM and contains details that I'll leave out.
First of all, the lightest Higgs has a tree-level mass
$$ m_{h^0} \sim m_Z \cos (2\beta) + \mathcal{O}(m_Z/|\mu|), $$
where ##m_Z## is the mass of the Z, ##\beta## is the angle defined by the Higgs vevs, and ##\mu## is the mass term in the MSSM superpotential. This formula also gets corrections from Higgs masses in the soft-breaking potential, as well as a significantly large 1-loop contribution from top and stop loops.
The approximation used above was that ##m_Z \ll |\mu|## and the soft-breaking mass parameters. This is a reasonable assumption given our biases about the SUSY breaking scale and current detection limits.
The neutralinos mix to give mass eigenstates that are linear combinations of the bino, zino, and higginos. The tree-level mass matrix is described by the chargino masses ##M_{1,2}##, ##\mu##, ##m_Z##, and the angles ##\beta,\theta_W##. The chargino masses come from soft-breaking terms, while as before ##\mu## comes from the superpotential.
Once again it's useful to expand for ##m_Z \ll |\mu|,M_{1,2}##, in which case the mass eigenvalues are
$$\begin{split}
& m_1 \sim M_1 - \frac{m_Z^2}{|\mu|} C_1, \\
& m_2 \sim M_2 - \frac{m_Z^2}{|\mu|} C_2, \\
& m_{3,4} \sim |\mu| - \frac{m_Z^2}{|\mu|} C_{3,4}, \\
\end{split}$$
where the ##C_i## are of order one in most of the parameter space, though they can get large when there are coincidences like ##M_{1,2}\sim |\mu|##. Away from these special points, these have been ordered so that the corresponding eigenvectors are in order: "mostly bino", "mostly zino", and two "mostly higgsino" states.
The first thing we should note is that the lightest higgsino-like mass eigenstate has a mass that has little in common with that of the lightest Higgs. To find them to be close in value would probably be a complete coincidence given the arbitrariness of the MSSM parameters.
It's also very unlikely that a higgsino-like particle could be the LSP, or even as light as the lightest Higgs boson. Recall that ##\mu## has it's own fine-tuning/hierarchy problem: since it is protected from corrections above the SUSY breaking scale, there's no good reason for it to be so much smaller than the GUT or Planck scales.
It is more reasonable to assume that the soft-breaking masses ##M_{1,2}## are naturally of order the SUSY breaking scale. In models with gauge coupling unification at some higher GUT scale, ##M_1 \sim 0.5 M_2##, so it's typically assumed that ##M_1 < M_2 \ll |\mu|##, which is consistent with the labeling of states we made above. Such a hierarchy also suggests that the bino is usually the LSP.
It's possible to say more, but nothing less speculative. I think from the above it's clear that, while not impossible, a higgsino-like LSP with a mass similar to that of the lightest Higgs would be quite unusual.
