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The associated production of Higgs boson with a muon pair, $e^+e^-\to\mu^+\mu^- H$, is one of the golden channels to pin down the properties of theHiggs boson in the prospective Higgs factories exemplified by \textsf{CEPC}.The projected accuracy of the corresponding cross section measurement is aboutper cent level at CEPC. In this work, we investigate both ${\mathcalO}(\alpha)$ weak correction and the ${\mathcal O}(\alpha\alpha_s)$ mixedelectroweak-QCD corrections for this channel, appropriately taking into accountthe effect of finite $Z^0$ width. The $\mu^+\mu^-$ invariant mass spectrum isalso predicted. The mixed electroweak-QCD correction turns out to reach 1.5\%of the Born-order result, and thereby must be included in future confrontationwith the data. We also observe that, after including higher-order corrections,the simplified prediction for the integrated cross section employing thenarrow-width-approximation may deviate from our full result by a few per cents.

Mixed electroweak-QCD corrections to $e^+e^-\to \mu^+\mu^- H$ at CEPC with finite-width effect