Energy Dependence of the Form Factor inKe3+Decay

In a spark chamber and counter experiment at the Princeton-Pennsylvania Accelerator we have investigated the decay $K^{+}\to e^{+}+{\pi }^0+\nu$. A Dalitz plot of 1393 ${K_{e3\mathrm{}}}^{+}$ events agrees well with vector coupling. The ${\pi }^0$ energy spectrum was investigated to determine the energy dependence of the strong-interaction form factor $f_{+}$ which is a function of $q^2={M_K}^2+{M_{\pi }}^2-2\mathrm{}{M}_K{E}_{\pi }$ for $K^{+}$ decays at rest. For a linear expansion $f_{+}\left(q^2\right)=f_{+}\mathrm{}\left(0\right)(1+\frac{\lambda q^2}{{M_{\pi }}^2})$, we obtain $\lambda =0.016\mathrm{}\pm 0.016$, which agrees well with previous measurements of $\lambda$ for ${K_{e3\mathrm{}}}^{+}$. Assuming the interaction is dominated by a $J=1^{-}$, $I=\frac{1}{2}$ intermediate state of mass $M$, such as the $K^{*}$ resonance at 890 MeV, we obtain $M=\left({{1180}_{-387\mathrm{}}}^{+\infty }\right)$ MeV. This result is consistent with $K^{*}$ dominance but also with no energy dependence for $f_{+}$. The value of $\lambda$ for ${K_{e3\mathrm{}}}^{+}$ obtained in this experiment agrees with the weighted mean value $〈{\lambda }^0〉=0.010\mathrm{}\pm 0.018$ obtained from four ${K_{e3\mathrm{}}}^0$ experiments and thus offers no evidence for violation of the leptonic $\left|\Delta \mathrm{I}\right|=\frac{1}{2}$ rule.