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We search for di-muon decays of a low-mass Higgs boson ($A^0$) in the fullyreconstructed decay chain of $\Upsilon(2S,3S) \to \pipi\Upsilon(1S)$,$\Upsilon(1S) \to \g A^0$, $A^0 \to \mumu$. The $A^0$ is predicted by severalextensions of the Standard Model (SM), including the Next-to-MinimalSupersymmetric Standard Model (NMSSM). NMSSM introduces a \CP-odd light Higgsboson whose mass could be less than 10 \gevcc. The data samples used in thisanalysis contain $92.8 \times 10^6$ $\Upsilon(2S)$ and $116.8 \times 10^6$$\Upsilon(3S)$ events collected by the \babar\ detector. The $\Upsilon(1S)$sample is selected by tagging the pion pair in the $\Upsilon(2S, 3S) \to\pi^+\pi^-\Upsilon(1S)$ transitions. We find no evidence for $A^0$ productionand set 90% confidence level (C.L.) upper limits on the product branchingfraction $\mathcal{B}(\Upsilon(1S) \to \g A^0) \times \mathcal{B}(A^0 \to\mumu)$ in the range of $(0.28 - 9.7)\times 10^{-6}$ for $0.212 \le m_{A^0} \le9.20$ \gevcc. We also combine our results with previous \babar\ results of$\Upsilon(2S,3S) \to \g A^0$, $A^0 \to \mumu$ to set limits on the effectivecoupling ($f_{\Upsilon}$) of the \b-quark to the $A^0$, $f_{\Upsilon}^2 \times\mathcal{B}(A^0 \to \mumu)$, at the level of $(0.29 - 40)\times 10^{-6}$ for$0.212 \le m_{A^0} \le 9.2$ GeV/$c^2$.

Search for di-muon decays of a light scalar Higgs boson in radiative Y(1S) decays