We present a search for new phenomena in events with two reconstructed $Z$ bosons and large missing transverse momentum, sensitive to processes $pbar{p}rightarrow X_2X_2 rightarrow Z Z X_1 X_1$, where $X_2$ is an unstable particle decaying as $X_2rightarrow ZX_1$ and $X_1$ is undetected. The particles $X_1$ and $X_2$ may be, among other possibilities, fourth generation neutrinos or supersymmetric particles. We study the final state in which one $Z$ boson decays to two charged leptons and the second decays hadronically. In data corresponding to an integrated luminosity of 4.2 fb$^{-1}$ from proton-antiproton collisions recorded by the CDF II detector at the Tevatron, with center-of-mass energy of 1.96 TeV, we find agreement between data and standard-model backgrounds. We calculate 95% confidence level upper limits on the cross section of the process $pbar{p}rightarrow X_2X_2 rightarrow Z Z X_1 X_1$ ranging from 50 fb to 1 pb, depending on the masses of $X_1$ and $X_2$.
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