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

Abstract

We present a determination of the ratio of kaon and pion leptonic decay constants in isosymmetric QCD (isoQCD), $f_K/f_{\ddot{I}}€$, 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, $f_{\ddot{I}}{€}^{isoQCD}=130.4(2)$ MeV, at the isoQCD pion point, $M_{\ddot{I}}{€}^{isoQCD}=135.0(2)$ 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 $(f_K/f_{\ddot{I}}€{)}^{isoQCD}=1.1995(44)$, 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.

68 pages, 14 figures, 12 tables. Version to appear in PRD