Measurements of the form-factors slopes of KL -> pi mu nu decay with the KLOE Detector

We present a measurement of the K-$\pi$ form-factor parameters for the decay \klpmn. We use 328 pb^{-1} of data collected in 2001 and 2002, corresponding to $\sim $ 1.8 million \semil events. Measurements of semileptonic form factors provide information about the dynamics of the strong interaction and are necessary for evaluation of the phase-space integral $I^\mu_K$ needed to measure the CKM matrix element $|V_{us}|$ for \klpmn decays and to test lepton universality in kaon decays. Using a new parameterization for the vector and scalar form factors we find $\lambda_+$=\pt(25.6\pm 0.4_{\rm{stat.}}\pm 0.3_{\rm{syst.}}),-3, and $\lambda_0$=\pt(14.3\pm 1.7_{\rm{stat.}}\pm 1.1_{\rm{stat.}}),-3,. In the more usual quadratic expansion of the form factor the above result is corresponds to $\lambda'_+=\lambda_+$, $\lambda''_+=2\lambda^2_+$, $\lambda'_0=\lambda_0$ and $\lambda''_0=(\lambda^2_0+0.000416)/2$. Our results, together with recent lattice calculations of $f_\pi$, $f_K$ and $f(0)$, satisfy the Callan-Trieman relation.