$K \to π$ semileptonic form factors with $N_f=2+1+1$ Twisted Mass fermions
We present a lattice QCD determination of the vector and scalar form factors of the semileptonic $K \to π\ell ν$ decay which are relevant for the extraction of the CKM matrix element $|V_{us}|$ from experimental data. Our results are based on the gauge configurations produced by the European Twisted Mass Collaboration with $N_f = 2+1+1$ dynamical fermions, which include in the sea, besides two light mass degenerate quarks, also the strange and the charm quarks. We use data simulated at three different values of the lattice spacing and with pion masses as small as $210$ MeV. Our final result for the vector form factor at zero momentum transfer is $f_+(0) = 0.9709 (46)$, where the uncertainty is both statistical and systematic combined in quadrature. Using the latest experimental value of $f_+(0) |V_{us}|$ from $K_{\ell 3}$ decays, we obtain $|V_{us}| = 0.2230 (11)$, which allows to test the unitarity constraint of the Standard Model below the permille level once the determination of $|V_{ud}|$ from superallowed nuclear $β$ decays is adopted. A slight tension with unitarity at the level of $\sim 2$ standard deviations is observed. Moreover we present our results for the semileptonic scalar $f_0(q^2)$ and vector $f_+(q^2)$ form factors in the whole range of values of the squared four-momentum transfer $q^2$ measured in $K_{\ell 3}$ decays, obtaining a very good agreement with the momentum dependence of the experimental data. We provide a set of synthetic data points representing our results for the vector and scalar form factors at the physical point for several selected values of $q^2$.