Measurement of $K^{0} _{\mu 3}$ form factors

This paper reports on a new high precision measurement of the form factors of the $K_{L}\rightarrow~\pi^{\pm}~\mu^{\mp}~\nu_{\mu}$} decay. The data sample of about $2.3 \times 10^{6}$ events was recorded in 1999 by the NA48 experiment at CERN. Studying the Dalitz plot density we measured a linear, $\lambda^{'}_{+} = (16.8\pm 2.3_{stat} \pm 2.4_{syst})\times 10^{-3}$, and a quadratic, $\lambda^{''}_{+} = (4.0\pm 1.0_{stat} \pm 1.0_{syst})\times 10^{-3}$ term in the power expansion of the vector form factor. No evidence was found for a second order term for the scalar form factor; the linear slope was determined to be $\lambda_{0} = (9.1\pm 1.1_{stat} \pm 0.8_{syst})\times 10^{-3}$. \noindent Using a linear fit our results were: $\lambda_{+} = (26.1\pm 0.6_{stat} \pm 0.8_{syst} )\times 10^{-3}$ and, $\lambda_{0} = (12.6\pm 0.7_{stat} \pm 1.0_{syst} )\times 10^{-3}$. A pole fit of the form factors yields: $m_V = ( 915 \pm 10_{stat} \pm 17_{syst} )$ MeV/c$^2$ and $m_S = (1348 \pm 43_{stat} \pm 53_{syst} )$ MeV/c$^2$.