Detailed study of the ${\mathrm{K}}^{\pm }\to {\pi }^0{\pi }^0{\mathrm{e}}^{\pm }\nu$ (${\mathrm{K}}_{\mathrm{e}4}^{00}$) decay properties

A sample of 65210 $${\mathrm{K}}^{\pm }$ \rightarrow \pi^{0} \pi^{0} ${\mathrm{e}}^{\pm }$ \nu$($$\mathrm{K}$_{$\mathrm{e}$4}^{00}$)decaycandidateswith$1%$backgroundcontaminationhasbeencollectedin2003$-$2004 by the NA48/2 collaboration at the CERN SPS. A study of the differential rate provides the first measurement of the hadronic form factor variation in the plane $(M_{\pi \pi }^2,M_{e\nu }^2)$ and brings evidence for a cusp-like structure in the distribution of the squared ${\pi }^0{\pi }^0$ invariant mass around $4m_{{\pi }^{+}}^2$. Exploiting a model independent description of this form factor, the branching ratio, inclusive of radiative decays, is obtained using the ${\mathrm{K}}^{\pm }\to {\pi }^0{\pi }^0{\pi }^{\pm }$ decay mode as normalization. It is measured to be BR(${\mathrm{K}}_{\mathrm{e}4}^{00}$) = $(2.552\pm {0.010}_{stat}\pm {0.010}_{syst}\pm {0.032}_{ext})\times {10}^{-5}$, which improves the current world average precision by an order of magnitude while the $1.4\mathrm{\%}$ relative precision is dominated by the external uncertainty from the normalization mode. A comparison with the properties of the corresponding mode involving a ${\pi }^{+}{\pi }^{-}$ pair (${\mathrm{K}}_{\mathrm{e}4}^{+-}$) is also presented.