Contributions of order $\mathcal{O}(m_{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{k}}^2)$ to $K_{\mathrm{\ell }3}$ form factors and unitarity of the CKM matrix

The form factors for the $K_{\mathrm{\ell }3}$ semileptonic decay are computed to order $O(p^4)$ in generalized chiral perturbation theory. The main difference with the standard $O(p^4)$ expressions consists in contributions quadratic in quark masses, which are described by a single divergence-free low-energy constant, $A_3$. A new simultaneous analysis is presented for the CKM matrix element $V_{us}$, the ratio $F_K\mathrm{/}{F}_{\pi }$, $K_{\mathrm{\ell }3}$ decay rates and the scalar form factor slope ${\lambda }_0$. This framework easily accommodates the precise value for $V_{ud}$ deduced from superallowed nuclear $\beta$-decays.