$\overline{p}p$ annihilation at rest into $K_L{K}^{\pm }{\pi }^{\mp }$

We present a measurement and partial wave analysis of the final state $K_L{K}^{\pm }{\pi }^{\mp }$ of $\overline{p}p$ annihilation at rest in liquid hydrogen. This reaction is important for the study of the $\overline{K}K$ decay mode of scalar resonances, in particular, the isovectors $a_0\mathrm{}\left(980\right)\mathrm{}$ and $a_0\mathrm{}\left(1450\right).$ The determination of the $a_0\mathrm{}\left(1450\right)\mathrm{}$ production also fixes the $\overline{K}K$ coupling of the isoscalar $f_0\mathrm{}\left(1500\right)\mathrm{}$ which is discussed as a glueball. We find $B\left(\overline{p}\overrightarrow{p}{a}_0\mathrm{}\right(1450\left)\mathrm{}\pi {;a}_0\mathrm{}\right(1450)\mathrm{}\to \overline{K}K)=(8.88\mathrm{}\pm 1.68)\times {10}^{-4}$ and hence $B\left(\overline{p}\overrightarrow{p}{f}_0\mathrm{}\right(1500\left)\mathrm{}\pi {;f}_0\mathrm{}\right(1500)\mathrm{}\to \overline{K}K)=(4.52\mathrm{}\pm 0.36)\times {10}^{-4}.$ The mass and width of $a_0\mathrm{}\left(1450\right)\mathrm{}$ are $m=1480\mathrm{}\pm 30\mathrm{MeV}{/c}^2$ and $\Gamma =265\mathrm{}\pm 15\mathrm{MeV}{/c}^2,$ respectively. For $a_0\mathrm{}\left(980\right)\mathrm{}$ we determine the relative ratio $B(\overline{p}\overrightarrow{p}{a}_0\pi {;a}_0\to \overline{K}K)/B(\overline{p}\overrightarrow{p}{a}_0\pi {;a}_0\to \pi \eta )$ =0.23±0.05 and its relative coupling, $\overline{K}K$ to πη, is $1.03\pm \mathrm{0.14.}$