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The prospective Higgs factories, exemplified by ILC, FCC-ee and CEPC, plan toconduct the precision Higgs measurements at the $e^+e^-$ center-of-mass energyaround 250 GeV. The cross sections for the dominant Higgs production channel,the Higgsstrahlung process, can be measured to a (sub-) percent accuracy.Merely incorporating the well-known next-to-leading order (NLO) electroweakcorrections appears far from sufficient to match the unprecedented experimentalprecision. In this work, we make an important advancement toward this directionby investigating the mixed electroweak-QCD corrections to $e^+e^-\to HZ$ atnext-to-next-to-leading order (NNLO) for both unpolarized and polarized $Z$boson. The corrections turn out to reach one percent level of the Born-orderresults, thereby must be incorporated in the future confrontation with thedata.

Mixed electroweak-QCD corrections to $e^+e^-\to HZ$ at Higgs factory