Search for CP violation in the decay of tagged $\overline{K}^0$ and $K^0$ to $\pi^{0}\pi^{0}\pi^{0}$

The CPLEAR experiment measured the eigentime-dependent asymmetry in the rates of initially tagged \kzerob\ and \kzero\ decaying to \threepizero\ in order to study the interference between the CP-violating \kshort\ and the CP-conserving \klong\ decay amplitudes. Without assuming CPT invariance, we obtain for the CP-violation parameter \etazero\ the values $\reetazero = 0.18 \pm 0.14_{\mbox{stat.}} \pm 0.06_{\mbox{syst.}} $ and $\imetazero = 0.15 \pm 0.20_{\mbox{stat.}} \pm 0.03_{\mbox{syst.}} .$ Requiring \reetazero\ to be equal to \ree\ we obtain $\imetazero = -0.05 \pm 0.12_{\mbox{stat.}} \pm 0.05_{\mbox{syst.}} .$ The corresponding upper limit for the branching ratio of the $\kshort \ra \threepizero$ decay is deduced to be $B_{\kshort\ra\threepizero} < 1.9 \times 10^{-5}$ at the 90\% confidence level.