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# System prepended metadata

title: Untitled

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1. The sub-decay branching fraction $\mathcal{B}(K_S^0 \rightarrow \pi^+ \pi^-)$ is taken care in the efficiency. I calculate efficiency as: 
$\epsilon = \frac{\text{# of }K_S^0 (\to π^+ π^-) π^0\text{ cand}}{\text{# of generated }K_S^0 π0}$
    > It would be better I change the eff value by your convention (?).

2. I have consider the efficiency correction on the final state KS0 π0:
\begin{equation}
\begin{aligned}
\xi_{\rm CF} &= \prod_{K_S^0, π^0} \frac{\epsilon_{\rm data}}{\epsilon_{\rm MC}}\\
\xi_{\rm CF}(K_S^0) &= \frac{\left(\frac{N(D^0 \rightarrow K_S^0 \pi^+ \pi-)}{N(D^0 \rightarrow K^- \pi^+ \pi^+ \pi-)}\right)_{\rm data}}{\left(\frac{N(D^0 \rightarrow K_S^0 \pi^+ \pi-)}{N(D^0 \rightarrow K^- \pi^+ \pi^+ \pi-)}\right)_{\rm MC}} = 1.043 \pm 0.072 \\
\xi_{\rm CF}(\pi^0) &= \frac{\epsilon_{\rm data}}{\epsilon_{\rm MC}} = \frac{0.480 \pm 0.031}{0.466 \pm 0.022} = 1.03 \pm 0.082\\
\epsilon' &= \epsilon \xi_{\rm CF} (\epsilon' \text{ is the final efficiency used in determined BF on data})\\
\end{aligned}
\end{equation}

3. N(BB) I just use $\int Ldt \cdot σ$, it is not very accurate since I haven't considered $f^{00}$.
    > should I refine, this would make result go from 10.9 -> 11.0?
 
4. Final BF
\begin{equation}
\begin{aligned}
\mathcal{B}(B^0 \rightarrow K^0 \pi^0) &= \frac{N_{\rm sig}}{\int Ldt \cdot σ \cdot 0.5 \cdot \epsilon' }\\
 &= \frac{34.9}{34.58 \times 10^{15} \times 1.1 \times 10^{-9} \times 0.5 \times 2 \times 0.5 \times 0.157  \times 1.043 \times 1.03} \sim 10.9 \times 10^{-6}
\end{aligned}
\end{equation}



5. No $\xi_{\rm CF}$, refined cross-section
\begin{equation}
\begin{aligned}
\mathcal{B}(B^0 \rightarrow K^0 \pi^0) &= \frac{N_{\rm sig}}{N(B\bar{B}) \cdot 2 \cdot 0.5 \cdot \epsilon }\\
 &= \frac{34.9_{-8.5}^{+9.2}}{34.58 \times 10^{15} \times 1.11 \times 10^{-9} \times 0.487 \times 2  \times 0.5 \times 0.157} \sim (11.9_{-2.9}^{+3.1}) \times 10^{-6}
\end{aligned}
\end{equation}


