The NMSSM Close to the R-symmetry Limit and Naturalness in $h \to aa$   Decays for $m_a<2\mb$
  Dominant decay of a SM-like Higgs boson into particles beyond those contained in the minimal supersymmetric standard model has been identified as a natural scenario to avoid fine tuning in electroweak symmetry breaking while satisfying all LEP limits. In the simplest such extension, the next-to-minimal supersymmetric model, the lightest CP-even Higgs boson can decay into two pseudoscalars. In the scenario with least fine tuning the lightest CP-even Higgs boson has mass of order 100 GeV. In order to escape LEP limits it must decay to a pair of the lightest CP-odd Higgs bosons with Br(h -> aa)>.7 and m_a<2m_b (so that a -> \tau^+ \tau^- or light quarks and gluons). The mass of the lightest CP-odd Higgs boson is controlled by the soft-trilinear couplings, A_\lambda(m_Z) and A_\kappa(m_Z). We identify the region of parameter space where this situation occurs and discuss how natural this scenario is. It turns out that in order to achieve m_a < 2 m_b with A_\lambda(m_Z), A_\kappa(m_Z) of order the typical radiative corrections, the required tuning of trilinear couplings need not be larger than 5-10 %. Further, the necessity for this tuning can be eliminated in specific SUSY breaking scenarios. Quite interestingly, Br(h -> aa) is typically above 70 % in this region of parameter space and thus an appropriately large value requires no additional tuning. 
Radovan Dermisek (IAS) and John F. Gunion (U.C. Davis)