High Statistics Measurement of theK+→π0e+ν(Ke3+) Branching Ratio

E865 at the Brookhaven National Laboratory AGS collected about 70 000 $K_{e3}^{+}$ events to measure the $K_{e3}^{+}$ branching ratio relative to the observed $K^{+}\to {\pi }^{+}{\pi }^0$, $K^{+}\to {\pi }^0{\mu }^{+}\nu$, and $K^{+}\to {\pi }^{+}{\pi }^0{\pi }^0$ decays. The ${\pi }^0$ in all the decays was detected using the $e^{+}{e}^{-}$ pair from ${\pi }^0\to e^{+}{e}^{-}\gamma$ decay and no photons were required. Using the 2002 Particle Data Group branching ratios for the normalization decays, we obtain $\mathrm{B}\mathrm{R}(K_{e3(\gamma )}^{+})=(5.13\pm {0.02}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}\pm {0.09}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}}\pm {0.04}_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}})\%$, where $K_{e3(\gamma )}^{+}$ includes the effect of virtual and real photons. This result is $\approx 2.3\sigma$ higher than the current Particle Data Group value. Implications for the $V_{us}$ element of the CKM matrix, and the matrix’s unitarity are discussed.