The prospective Higgs factories, exemplified by ILC, FCC-ee and CEPC, plan to conduct the precision Higgs measurements at the e + e − center-of-mass energy around 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) electroweak corrections appears far from sufficient to match the unprecedented experimental precision. In this work, we make an important advancement toward this direction by investigating the mixed electroweak-QCD corrections to e + e − → HZ at next-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-order results, thereby must be incorporated in the future confrontation with the data.

Within the nonrelativistic QCD (NRQCD) factorization framework, we have computed the O(α 2 s) corrections to the exclusive production of P-wave spin-triplet charmonia χ cJ (J = 0, 1, 2) accompanied with a hard photon at B factory. For the first time, we have explicitly verified the validity of NRQCD factorization for exclusive Pwave quarkonium production to two-loop order. Unlike the χ cJ electromagnetic decays, the O(α 2 s) corrections are found to be smaller than the O(α s) corrections in all three channels e + e − → χ c0,1,2 + γ. In particular, the O(α 2 s) corrections appear moderate for χ c1 + γ case, and marginal for χ c0 + γ. Moreover, the predictions in next-to-next-to-leading order (NNLO) accuracy for the production rates of χ c0,1 + γ are insensitive to the renormalization and factorization scales. All of these features may indicate that perturbative expansion in these two channels exhibits a decent convergence behavior. By contrast, both the O(α s) and O(α 2 s) corrections to the χ c2 + γ production rate are sizable, which reduces the Born order cross section by one order of magnitude after including the NNLO perturbative corrections. Taking the values of the long-distance NRQCD matrix elements from nonrelativistic potential model, our prediction to χ c1 + γ production rate is consistent with the recent Belle measurement. The NNLO predictions to the χ c0,2 + γ production rates are much smaller than that for χ c1 + γ, which seems to naturally explain why the e + e − → χ c0,2 + γ channels have escaped experimental detection to date.