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Based on data samples collected with the BESIII detector at the BEPCII collider at 9 center-of-mass energies from 4.21 to 4.42 GeV, we search for the production of $e^+e^-\to \omega\chi_{cJ}$ ($J$=0, 1, 2).
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The upper limit is calculated by using a frequentist method with unbounded profile likelihood treatment of systematic uncertainties, which is implemented by a C++ class trolke
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