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 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 $\sim 0.092\mathrm{fm}$ with linear lattice size up to $L\sim 5.5\mathrm{fm}$. The scale is set by the particle data group (PDG) value of the pion decay constant, $f_{\pi }^{\text{ISO}\mathrm{QCD}}=130.4(2)\mathrm{MeV}$, at the ISOQCD pion point, $M_{\pi }^{\text{ISO}\mathrm{QCD}}=135.0(2)\mathrm{MeV}$, obtaining for the gradient-flow scales the values $w_0=0.17383(63)\mathrm{fm}$, $\sqrt{t_0}=0.14436(61)\mathrm{fm}$ and $t_0/w_0=0.11969(62)\mathrm{fm}$. The data are analyzed within the framework of SU(2) chiral perturbation theory without resorting to the use of renormalized quark masses. At the ISOQCD kaon point $M_K^{\text{ISO}\mathrm{QCD}}=494.2(4)\mathrm{MeV}$ we get $(f_K/f_{\pi }{)}^{\text{ISO}\mathrm{QCD}}=1.1995(44)$, where the error includes both statistical and systematic uncertainties. Implications for the Cabibbo-Kobayashi-Maskawa matrix element $|V_{us}|$ and for the first-row Cabibbo-Kobayashi-Maskawa unitarity are discussed.

Figure 1

Values of the pion decay constant $w_0{f}_{\pi }$ (left panel) and of the quantity $w_0{X}_{\pi }=w_0(f_{\pi }{M}_{\pi }^4{)}^{1/5}$ (right panel) versus the squared pion mass $(w_0{M}_{\pi }{)}^2$ in units of the GF scale $w_0$. For the ensemble cA211.12.48 the corrected value of $f_{\pi }$ given by Eq. (18) is considered.

Figure 2

Values of the pion mass (left panel) and pion decay constant (right panel) in lattice units for the three ensembles cB211.25.XX of Table 1. The red circles represent the data versus $M_{\pi }L$. The expansion variable ${\xi }_{\pi }$ is given by Eq. (42). The blue squares correspond to the data corrected by the GL formula, while black diamonds represent the data corrected by the CDH formula, adopting for the NLO LECs the values ${\overline{\ell }}_1^{\mathrm{phys}}=-0.4$, ${\overline{\ell }}_2^{\mathrm{phys}}=4.3$, ${\overline{\ell }}_3^{\mathrm{phys}}=3.2$ and ${\overline{\ell }}_4^{\mathrm{phys}}=4.4$. The green triangles correspond to the CDH correction assuming ${\overline{\ell }}_2^{\mathrm{phys}}=3.3$. The horizontal dotted lines are the values of the pion mass and decay constant in the infinite volume limit.

Figure 3

The same as in Fig. 2, but adopting the alternative definition (43) for the expansion variable ${\xi }_{\pi }$ and assuming $f=122.5\mathrm{MeV}$ and $a=0.080\mathrm{fm}$.

Figure 4

Values of $w_0{X}_{\pi }$ versus $M_{\pi }L$ for the three ensembles cB211.25.XX differing only for the lattice size L. The dashed line indicates the simple exponential fit of the form $A[1+B{e}^{-M_{\pi }L}/(M_{\pi }L{)}^{3/2}]$.

Figure 5

Values of the pion decay constant $w_0{f}_{\pi }$ corrected for FVEs according to Eq. (47) (open markers) and compared to the results of the NLO ChPT fit corresponding to $A_2=0$ in Eq. (49) applied to all data points ($M_{\pi }\lesssim 350\mathrm{MeV}$). The solid line represents the results of the fit in the continuum limit, while the dashed lines correspond to the fit evaluated at each value of $\beta$. The cross represents the result at the physical pion point (4) corresponding to the value $w_0=0.1740(15)\mathrm{fm}$, obtained as described in the text.

Figure 6

Top panel: values of the quantity $w_0{X}_{\pi }$ (open markers) compared to the results of the NLO fit corresponding to $A_2^{\prime }=F_{\mathrm{FVE}}=0$ in Eq. (56) applied to all data points ($M_{\pi }\lesssim 350\mathrm{MeV}$). The solid line represents the results of the fit in the continuum limit, while the dashed lines correspond to the fit evaluated at each value of $\beta$. The cross represents the result at the physical pion point (4) corresponding to the value $w_0=0.17394(58)\mathrm{fm}$. Bottom panel: the quantity $w_0{X}_{\pi }$ after subtraction of its extrapolation to the continuum limit. The discretization terms proportional both to $a^2$ and to $a^2\xi$ present in Eq. (56) are clearly visible.

Figure 7

Values of the ratio $f_K/f_{\pi }$ interpolated at the kaon reference mass (72) versus the squared pion mass. The vertical dotted line indicates the location of the physical ISOQCD point (4). For the ensemble cA211.12.48 the corrected value of the ratio $f_K/f_{\pi }$, obtained using Eqs. (18) and (69), is considered.

Figure 8

Values of the ratio $f_K/f_{\pi }$ corrected for FVEs according to Eq. (76) (open markers) compared to the results of the NLO fit corresponding to $R_2={\tilde{D}}_1=0$ in Eq. (77) applied to all pion masses ($M_{\pi }\lesssim 350\mathrm{MeV}$). The solid line represents the results of the fit in the continuum limit, while the dashed lines correspond to the fit evaluated at each value of $\beta$. The cross represents the result at the physical pion point (4).

Figure 9

The plot compares the information for $V_{ud}$ and $V_{us}$ obtained in the FLAG-4 review for $N_f=2+1+1$ [18] and in this work by using Eq. (80) and the semileptonic result of Ref. [37]. The determinations of $V_{ud}$ obtained from superallowed nuclear $\beta$ transitions obtained in Refs. [8] and [21] are also shown as green and orange bands, labeled, respectively, $HT$ and $SGPR$. The dotted line indicates the correlation between $V_{ud}$ and $V_{us}$ that follows assuming the unitarity of the CKM matrix. The ellipses represent 68% likelihood contours.

Figure 10

Light quark-mass dependence and extrapolations to the physical point of the GF scales $s_0\equiv \sqrt{t_0}$, $w_0$, $t_0/w_0$, normalized by their values at the physical point for ease of comparing the results at different lattice spacings.

Figure 11

Continuum extrapolations of the dimensionless ratio of GF scales $s_0/w_0$ at the physical point in terms of the lattice spacing in units of the GF scales $s_0^{\mathrm{phys}}$, $w_0^{\mathrm{phys}}$ and $t_0^{\mathrm{phys}}/w_0^{\mathrm{phys}}$ at the physical point.

Figure 12

Global fit of the lattice spacing and light quark-mass dependence of the dimensionless ratio $s_0/w_0\equiv \sqrt{t_0}/w_0$. The black line with the error band at the bottom shows the fit result in the continuum, while the data points on this line represent the data corrected by the lattice artifacts as described by the global fit.

Figure 13

The same as in Fig. 6, but in the case of the GF scales $\sqrt{t_0}$ (left panel) and $t_0/w_0$ (right panel).

Figure 14

Ratio of the lattice spacing $a$ obtained from the GF scales $\sqrt{t_0}$ (red circles) and $t_0/w_0$ (blue squares) with the one determined from the GF scale $w_0$ (see Table 12). The solid and dashed lines are linear fits.