Determinations of the form factors of semileptonic \(D\rightarrow K\) decays and leptoquark constraints

By analyzing all existing measurements for \( D\rightarrow K \ell ^+ \nu _{\ell } \) (\(\ell =e,\ \mu \)) decays, we find that the determinations of both the vector form factor \(f_+^K(q^2)\) and scalar form factor \(f_0^K(q^2)\) for semileptonic \(D\rightarrow K\) decays from these measurements are feasible. By taking the parameterization of the one order series expansion of the \(f_+^K(q^2)\) and \(f_0^K(q^2)\), \(f_+^K(0)|V_{cs}|\) is determined to be \(0.7182\pm 0.0029\), and the shape parameters of \(f_+^K(q^2)\) and \(f_0^K(q^2)\) are \(r_{+1}=-2.16\pm 0.007\) and \(r_{01}=0.89\pm 3.27\), respectively. Combining with the average \(f_+^K(0)\) of \(N_f=2+1\) and \(N_f=2+1+1\) lattice calculaltion, the \(|V_{cs}|\) is extracted to be \(0.964\pm 0.004\pm 0.019\) where the first error is experimental and the second theoretical. Alternatively, the \(f_+^K(0)\) is extracted to be \(0.7377\pm 0.003\pm 0.000\) by taking the \(|V_{cs}|\) as the value from the global fit with the unitarity constraint of the CKM matrix. Moreover, using the obtained form factors by \(N_f=2+1+1\) lattice QCD, we re-analyze these measurements in the context of new physics. Constraints on scalar leptoquarks are obtained for different final states of semileptonic \(D \rightarrow K\) decays.