Kaon semileptonic vector form factor with $N_f=2+1+1$ Twisted Mass fermions

We investigate the vector form factor relevant for the $K_{\mathrm{\ell }3}$ semileptonic decay using maximally twisted-mass fermions with 4 dynamical flavours ($N_f=2+1+1$). Our simulations feature pion masses ranging from $210$ MeV to approximately $450$ MeV and lattice spacing values as small as $0.06$fm. Our main result for the vector form factor at zero 4-momentum transfer is $f_{+}(0)=0.9683(65)$ where the uncertainty is both statistical and systematic. By combining our result with the experimental value of $f_{+}(0)\mathrm{\mid }{V}_{us}\mathrm{\mid }$ we obtain $\mathrm{\mid }{V}_{us}\mathrm{\mid }=0.2234(16)$, which satisfies the unitarity constraint of the Standard Model at the permille level.