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Coupled-channel effects are taken into account for the study of scalar formfactors in semi-leptonic $D\to\pi\bar{\ell}\nu_\ell$ and $D\to\bar{K}\bar{\ell}\nu_\ell$ decays, by solving the Muskhelishvili-Omn\`esintegral equations. As inputs, we employ the unitarized amplitudes taken fromchiral effective theory for the region not far away from thresholds, while, athigher energies of the Goldstone bosons, proper asymptotic conditions areemployed. Within Muskhelishvili-Omn\`es formalism, the scalar form factors arerepresented in terms of Omn\`es matrix multiplied by a vector of polynomials.We reduce the number of subtraction constants by matching to the scalar formfactors derived in chiral perturbation theory up to next-to-leading order. Therecent lattice QCD data by ETM collaboration for ${D\to\pi}$ and${D\to\bar{K}}$ scalar form factors are simultaneously well described. Thescalar form factors for $D\to\eta$, $D_s\to \bar{K}$ and $D_s\to \eta$transitions are predicted in their whole kinematical regions. Using our fittingparameters, we also extract the following Cabibbo-Kobayashi-Maskawa elements:$|V_{cd}|=0.243(_{-12}^{+11})_{\rm sta.}({3})_{\rm sys.}(3)_{\rm exp.}$ and$|V_{cs}|=0.950(_{-40}^{+39})_{\rm sta.}(1)_{\rm sys.}(7)_{\rm exp.}$. Theapproach used in this work can be straightforwardly extended to thesemileptonic $B$ decays.

Scalar form factors of semi-leptonic $D\to\pi/ \bar{K}$ transitions with coupled-channel effects