Measurements of the total and differential Higgs boson production cross-sections, via $WH$ and $ZH$ associated production using $H\rightarrow WW^\ast\rightarrow\ellν\ellν$ and $H\rightarrow WW^\ast\rightarrow\ellνjj$ decays, are presented. The analysis uses proton-proton events delivered by the Large Hadron Collider at a centre-of-mass energy of 13 TeV and recorded by the ATLAS detector between 2015 and 2018. The data correspond to an integrated luminosity of 140 fb$^{-1}$. The sum of the $WH$ and $ZH$ cross-sections times the $H\rightarrow WW^\ast$ branching fraction is measured to be $0.44^{+0.10}_{-0.09}$ (stat.) $^{+0.06}_{-0.05}$ (syst.) pb, in agreement with the Standard Model prediction. Higgs boson production is further characterised through measurements of the differential cross-section as a function of the transverse momentum of the vector boson and in the framework of Simplified Template Cross-Sections.
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