Detailed study of the $K^\pm \rightarrow \pi^0 \pi^0 e^\pm \nu $ ($K_{e4}^{00}$) decay properties

A sample of 65210 $K^\pm \rightarrow \pi^0 \pi^0 e^\pm \nu $ ($K_{e4}^{00}$) decay candidates with $1\%$ background contamination has been collected in 2003--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 $K^\pm \rightarrow \pi^0 \pi^0 \pi^\pm $ decay mode as normalization. It is measured to be BR$(K_{e4}) = (2.552 ± 0.010_{stat} \pm 0.010_{syst} \pm 0.032_{ext}) \times 10^{−5}$ which improves the current world average precision by an order of magnitude while the $1.4\%$ 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 ($K_{e4}^{+-}$) is also presented.