Measuring the ratio of $HWW$ and $HZZ$ couplings through $W^{+}{W}^{-}H$ production

For a generic Higgs boson, measuring the relative sign and magnitude of its couplings with the $W$ and $Z$ bosons is essential in determining its origin. Such a test is also indispensable for the 125-GeV Higgs boson. We propose that the ratio of the $HWW$ and $HZZ$ couplings ${\lambda }_{WZ}$ can be directly determined through the $W^{+}{W}^{-}H$ production, where $H$ denotes a generic Higgs boson, owing to the tree-level interference effect. While this is impractical at the LHC due to the limited sensitivity, it can be done at future $e^{+}{e}^{-}$ colliders, such as a 500-GeV ILC with the beam polarization $P(e^{-},e^{+})=(-0.8,+0.3)$ in the $jj{\mathrm{\ell }}^{\pm }bb$ and ${\mathrm{\ell }}^{\pm }{\mathrm{\ell }}^{\pm }{\mathrm{\ell }}^{\mp }jj$ channels. The discovery potential of a general ratio and the power to discriminate it from the SM value are studied in detail. Combining the cross section of $e^{+}{e}^{-}\to W^{+}{W}^{-}H$ with the measurements of $HZZ$ coupling at the HL-LHC, one can further improve the sensitivity of ${\lambda }_{WZ}$.