Probing next-to-minimal-supersymmetric models with minimal fine tuning by searching for decays of theΥto a lightCP-odd Higgs boson

Completely natural electroweak symmetry breaking is easily achieved in supersymmetric models if there is a SM-like Higgs boson, $h$, with $m_h\lesssim 100\mathrm{GeV}$. In the minimal supersymmetric model, such an $h$ decays mainly to $b\overline{b}$ and is ruled out by LEP constraints. However, if the MSSM Higgs sector is expanded so that $h$ decays mainly to still lighter Higgs bosons, e.g. $h\to aa$, with $\mathrm{Br}(h\to aa)>0.7$, and if $m_a<2m_b$, then the LEP constraints are satisfied even if $m_h\lesssim 100$ GeV. In this paper, we show that in the next-to-minimal supersymmetric model the above $h$ and $a$ properties (for the lightest $CP$-even and $CP$-odd Higgs bosons, respectively) imply a lower bound on $\mathrm{Br}({\rm Y}\to \gamma a)$ that dedicated runs at present (and future) $B$ factories can explore.