Precise measurement of the Decay $K^\pm \to \pi^0\pi^0\mu^\pm\nu$

We report the first observation of the decay $K^{\pm} \rightarrow \pi^{0} \pi^{0} \mu^{\pm} \nu$ ($K^{00}_{\mu4}$) by the NA48/2 experiment at the CERN SPS. From 2437 detected signal candidates with a S/B ratio of about 6, the branching ratio of the decay is determined with high precision. In the region of squared dilepton mass above $0.03$~GeV$^2/c^4$, the branching ratio is found to be $BR(K^{00}_{\mu 4},S_l>0.03) = (0.65 \pm 0.03) \times 10^{-6}$. The full phase space result $BR(K^{00}_{\mu 4}) = (3.4 \pm 0.2) \times 10^{-6}$, depending on the decay model extrapolation, is in a reasonable agreement with the R form factor prediction from 1-loop Chiral Perturbation Theory.