Ratio of kaon and pion leptonic decay constants with $N_f = 2 + 1 + 1$ Wilson-clover twisted-mass fermions

We present a determination of the ratio of kaon and pion leptonic decay constants in isosymmetric QCD (isoQCD), $f_K / f_\pi$, making use of the gauge ensembles produced by the Extended Twisted Mass Collaboration (ETMC) with $N_f = 2 + 1 + 1$ flavors of Wilson-clover twisted-mass quarks, including configurations close to the physical point for all dynamical flavors. The simulations are carried out at three values of the lattice spacing ranging from $\sim 0.068$ to $\sim 0.092$ fm with linear lattice size up to $L \sim 5.5$~fm. The scale is set by the PDG value of the pion decay constant, $f_\pi^{isoQCD} = 130.4~(2)$ MeV, at the isoQCD pion point, $M_\pi^{isoQCD} = 135.0~(2)$ MeV, obtaining for the gradient-flow (GF) scales the values $w_0 = 0.17383~(63)$ fm, $\sqrt{t_0} = 0.14436~(61)$ fm and $t_0 / w_0 = 0.11969~(62)$ fm. The data are analyzed within the framework of SU(2) Chiral Perturbation Theory (ChPT) without resorting to the use of renormalized quark masses. At the isoQCD kaon point $M_K^{isoQCD} = 494.2~(4)$ MeV we get $(f_K / f_\pi)^{isoQCD} = 1.1995~(44)$, where the error includes both statistical and systematic uncertainties. Implications for the Cabibbo-Kobayashi-Maskawa (CKM) matrix element $|V_{us}|$ and for the first-row CKM unitarity are discussed.