E$. The “beam constrained" invariant mass of the $B$ candidates, $M_B$, is computed from the formula $$M_B^2=E_{beam}^2-(\sum_i\overrightarrow{p_i})^2~~~.$$

To further reduce backgrounds we define $$\label{eq:chisq}
\chi_b^2=\left({{\Delta M_{D^*}}\over {\sigma(\Delta 
M_{D^*})}}\right)^2 + 
\left({{\Delta M_{D}}\over {\sigma(\Delta M_{D})}}\right)^2 + 
\sum_{n(\pi^o)}\left({{\Delta M_{\pi^o}}\over {\sigma(\Delta 
M_{\pi^o})}}\right)^2~~~,$$ where $\Delta M_{D^*}$ is the computed $D^*-D^o$ mass difference minus the nominal value, $\Delta M_{D}$ is the invariant candidate $D^o$ mass minus the known $D^o$ mass and $\Delta M_{\pi^o}$ is the measured $\gamma\gamma$ invariant mass minus the known $\pi^o$ mass. All $\pi^o$’s in the final state are included in the sum. The $\sigma$’s are the measurement errors. We select candidate events in each mode requiring that $\chi^2_b < C_n$, where $C_n$ varies for each decay $D^o$ decay mode. For the $Kn\pi$ decay modes we use $C_n=~12,$ 8, and 6, respectively.

Branching Fraction and $(4\pi)^-$ Mass Spectrum
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We start with the $D^o\to K^-\pi^+$ decay mode. We show the candidate $B$ mass distribution, $M_B$, for $\Delta E$ in the side-bands from -5.0 to -3.0$\sigma$ and 5.0 to 3.0$\sigma$ on Fig. \[bm\_4pi\_kpi\](a). The $\Delta E$ resolution is 18 MeV ($\sigma).$ This gives a good representation of the background in the signal region. We fit this distribution with a shape given as $$back(r)=p_1 r\sqrt{1-r^2}e^{-p_2(1-r^2)}~~~,$$ where 