Next-to-leading order QCD predictions for the hadronic $WH$+jet production

We calculate the next-to-leading order(NLO) QCD corrections to the $WH^0$ production in association with a jet at hadron colliders. We study the impacts of the complete NLO QCD radiative corrections to the integrated cross sections, the scale dependence of the cross sections, and the differential cross sections ($\frac{d \sigma}{d\cos\theta}$, $\frac{d \sigma}{dp_T}$) of the final $W$-, Higgs-boson and jet. We find that the corrections significantly modify the physical observables, and reduce the scale uncertainty of the LO cross section. Our results show that by applying the inclusive scheme with $p_{T,j}^{cut}=20 GeV$ and taking $m_H=120 GeV$, $\mu=\mu_0\equiv\frac{1}{2}(m_W+m_H)$, the K-factor is 1.15 for the process $p\bar p \to W^{\pm}H^0j+X$ at the Tevatron, while the K-factors for the processes $pp \to W^-H^0j+X$ and $pp \to W^+H^0j+X$ at the LHC are 1.12 and 1.08 respectively. We conclude that to understand the hadronic associated $WH^0$ production, it is necessary to study the NLO QCD corrections to $WH^0j$ production process which is part of the inclusive $WH^0$ production.