Determination of theKL0→π−μ+νμform factorξ(q2)by muon polarization measurements

The polarization of the muon in the decay $K_L^0\to {\pi }^{-}{\mu }^{+}{\nu }_{\mu } \left(K_{\mu 3}\right)$ was measured as a function of $q^2$, the four-momentum transferred to the lepton pair, by using a precession polarimeter with a double-armed spectrometer at the Bevatron. A sample of 207 260 events collected with a vertical precession field was used to determine the $K_{\mu 3}$ form factor $\xi \left(q^2\right)$, assuming $\mathrm{Im}\xi \left(q^2\right) = 0.0$. If one parametrizes the $q^2$ dependence of $\xi$ by $\xi \left(q^2\right) = \xi \mathrm{}\left(0\right)\mathrm{}+\frac{\Lambda q^2}{{m_{\pi }}^2}$, then $\xi \mathrm{}\left(0\right)\mathrm{} = 0.178\pm 0.105-3.80\Lambda$. A sample of 55 604 events collected with a horizontal precession field was used to determine $\mathrm{Im}\xi \left(q^2\right)$. If one assumes that $\mathrm{Im}\xi \left(q^2\right)$ has no dependence on $q^2$, then $\mathrm{Im}\xi \mathrm{}\left(0\right)\mathrm{} = 0.35\pm 0.30+0.21\mathrm{Re}\xi \mathrm{}\left(0\right)\mathrm{}$.