A proposal to use CPLEAR as antigravitometer

The reconstructed kinematics of the isotropically emitted $K^0\overline{K}^0$ and their decay vertex following a $p\overline{p}$ annihilation at rest allows to select events with specific neutral kaon topology {\it and} energy. Thus, from the accumulated events with CPLEAR one can define high statistics subgroups of any gravitationally oriented neutral kaons with $\sim \pm 60$ cm maximum vertical displacement, in order to search for (anti)gravity effects. The suggested new data analysis allows to combine for the first time upward moving $K^0$'s with downward emitted $overline{K}^0$'s, and {\it vice versa}, in order to search for any directional dependence of the $CP$ parameters. The achieved limit for the relative $K^0-overline{K}^0$ mass difference ($\sim 10^{-18}$) may be indicative of a probably unique sensitivity for the proposed antigravity search. Existing anomalous energy dependent $CP$ results along with the used non-horizontal kaon beam (having in fact an equal vertical kaon displacement per $\tau_S$ as in CPLEAR) are in favour of this proposal whatever the theoretical interpretation.