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The rare decay K0(L) --> pi0 pi0 nu nu-bar was studied with the E391adetector at the KEK 12-GeV proton synchrotron. Based on 9.4 x 10^9 K0L decays,an upper limit of 8.1 x 10^{-7} was obtained for the branching fraction at 90%confidence level. We also set a limit on the K0(L) --> pi0 pi0 X (X -->invisible particles) process; the limit on the branching fraction varied from7.0 x 10^{-7} to 4.0 x 10^{-5} for the mass of X ranging from 50 MeV/c^2 to 200MeV/c^2.

Study of the K0(L) --> pi0 pi0 nu nu-bar decay