Measurement of the branching ratios of the decays $\Xi^{0}\rightarrow \sigma^{+} e^{-} \overline{\nu}_{e}$ and $\overline{\Xi^{0}}\rightarrow \overline{\sigma^{+}} e^{+} \nu_{e}$

From 56 days of data taking in 2002, the NA48/1 experiment observed 6316 $\Xi^{0}\rightarrow \Sigma^{+} e^{-} \overline{\nu}_{e}$ candidates (with the subsequent $\Sigma^{+} \rightarrow p \pi^{0}$ decay) and 555 $\overline{\Xi^{0}}\rightarrow \overline{\Sigma^{+}} e^{+} \nu_{e}$ candidates with background contamination of $215 \pm 44$ and $136 \pm 8$ events, respectively. From these samples, the branching ratios BR($\Xi^{0}\rightarrow \Sigma^{+} e^{-} \overline{\nu}_{e})=(2.51 \pm0.03_{\mathrm{stat}}\pm 0.09_{\mathrm{syst}})\times10^{-4}$ and BR($\overline{\Xi^{0}}\rightarrow \overline{\Sigma^{+}} e^{+} \nu_{e})= (2.55 \pm 0.12_{\mathrm{stat}}\pm{0.10}_{\mathrm{syst}}) \times 10^{-4}$ were measured allowing the determination of the CKM matrix element $|V_{\mathrm{us}}| = 0.209^{+0.023}_{-0.028}$. Using the Particle Data Group average for $|V_{\mathrm{us}}|$ obtained in semileptonic kaon decays, we measured the ratio $g_1/f_1 = 1.20 \pm 0.05$ of the axial-vector to vector form factors.