$|{\mathrm{V}}_{us}|$ from $K_{\ell 3}$ decay and four-flavor lattice QCD

Using highly improved staggered quark (HISQ) $N_f=2+1+1$ MILC ensembles with five different values of the lattice spacing, including four ensembles with physical quark masses, we perform the most precise computation to date of the $K\to \pi \ell \nu$ vector form factor at zero momentum transfer, $f_{+}^{K^0{\pi }^{-}}(0)=0.9696(15{)}_{\mathrm{stat}}(12{)}_{\mathrm{syst}}$. This is the first calculation that includes the dominant finite-volume effects, as calculated in chiral perturbation theory at next-to-leading order. Our result for the form factor provides a direct determination of the Cabibbo-Kobayashi-Maskawa (CKM) matrix element $|V_{us}|=0.22333(44{)}_{f_{+}(0)}(42{)}_{\exp }$, with a theory error that is, for the first time, at the same level as the experimental error. The uncertainty of the semileptonic determination is now similar to that from leptonic decays and the ratio $f_{K^{+}}/f_{{\pi }^{+}}$, which uses $|V_{ud}|$ as input. Our value of $|V_{us}|$ is in tension at the $2–2.6\sigma$ level both with the determinations from leptonic decays and with the unitarity of the CKM matrix. In the test of CKM unitarity in the first row, the current limiting factor is the error in $|V_{ud}|$, although a recent determination of the nucleus-independent radiative corrections to superallowed nuclear $\beta$ decays could reduce the $|V_{ud}{|}^2$ uncertainty nearly to that of $|V_{us}{|}^2$. Alternative unitarity tests using only kaon decays, for which improvements in the theory and experimental inputs are likely in the next few years, reveal similar tensions and could be further improved by taking correlations between the theory inputs. As part of our analysis, we calculated the correction to $f_{+}^{K\pi }(0)$ due to nonequilibrated topological charge at leading order in chiral perturbation theory, for both the full-QCD and the partially quenched cases. We also obtain the combination of low-energy constants in the chiral effective Lagrangian $[C_{12}^r+C_{34}^r-(L_5^r{)}^2](M_{\rho })=(2.92\pm 0.31)\times {10}^{-6}$.

Figure 1

Gauge-field ensembles analyzed in this work (parameters of these gauge-field ensembles are listed in Table 1). The area of each disk is proportional to the statistical sample size $N_{\mathrm{conf}}\times N_{\mathrm{src}}$. Ensembles on which we have increased the statistics or we have added since our earlier work in Ref. [24] are indicated with black outlines. The three disks with $a\approx 0.12\mathrm{fm}$ and $m_{\pi }\approx 200\mathrm{MeV}$ correspond to the three ensembles with $m_l/m_s^{\mathrm{sea}}=0.1$ and different volumes (smaller to larger from top to bottom) in Table 1. The yellow disk (smallest volume) is not included in the final analysis but used only to study finite-volume effects.

Figure 2

Comparison of data and fit results for the rescaled three-point functions defined in Eq. (2.8) on the $a\approx 0.09\mathrm{fm}$ ensemble with physical quark masses. Green squares are the data points and orange circles are obtained from the fit posteriors. The fit includes the three three-point functions shown, with $T=23$, 27, 32, in the fit ranges shown by the orange lines. Correlators with $T=27$, 23 are given a vertical offset, different for each $T$, so results for the three correlators do not lie on top of each other. Errors are statistical only.

Figure 3

Variation of the fit result for $f_{+}(0)$ with $t_{\min }$ (left panel) and $N_{\exp }$ (right panel) for the ensemble with $a\approx 0.09\mathrm{fm}$ and physical light-quark masses. The errors are statistical, generated with a 500-bootstrap distribution. The black point on each figure corresponds to our preferred fit with $t_{\min }=8$ and $N_{\exp }=3+3$.

Figure 4

Variation of $f_{+}(0)$ with the block size for the ensemble with $a\approx 0.15\mathrm{fm}$ and physical quark masses. The errors are statistical, generated with a 500-bootstrap distribution. The black point corresponds to our preferred fit with $N_{\text{block}}=4$.

Figure 5

Form factor $f^{K^0{\pi }^{-}}(0)$ vs light-quark mass. The data points are the raw results listed in Table 3 before applying the corrections described in Sec. 3. Errors shown are statistical only, obtained from 500 bootstrap resamples. Different symbols and colors denote different lattice spacings. Data at the same light-quark mass but different lattice spacing are offset horizontally. The open orange circle corresponds to the smallest volume ensemble with $a\approx 0.12\mathrm{fm}$ and $m_l/m_s=0.1$.

Figure 6

Diagrams contributing to $⟨\pi |\rho |K⟩$ at tree level, where $\rho$ is either the vector current $V^{\mu }$ or the scalar density $\tilde{S}$. The squares are weak vertices with the insertion of the current or density and the black dot is a strong vertex.

Figure 7

Form factor $f^{K^0{\pi }^{-}}(0)$ vs light-quark mass. The data points correspond to the results in Table 3 and are corrected for the one-loop finite-volume effects also listed in that table. Different symbols and colors denote different lattice spacings. The data point at the $a\approx 0.042\mathrm{fm}$ ensemble includes the correction given in Sec. 3b. The error bars on the data points are statistical only, obtained from 500 bootstrap resamples. Data points at the same light-quark mass but different lattice spacings are offset horizontally. The gray continuous line shows the continuum extrapolation in the isospin limit as a function of the light-quark mass, and the yellow star is the continuum result interpolated to the physical light-quark masses. The cyan error band, as well as the error bar on the physical point, is the statistical chiral-continuum fit error (obtained using 500 bootstrap resamples), which includes discretization and higher-order chiral errors, as well as the uncertainty from some of the input parameters, as discussed in the text. The continuum extrapolation line is obtained by setting the valence and sea light-quark masses equal, setting $m_s$ to its physical value, turning off all discretization effects, and considering $m_u=m_d$, i.e., without isospin-breaking effects. On the other hand, the yellow star is the interpolation to the physical masses and includes strong isospin-breaking effects at NNLO.

Figure 8

Stability of the continuum extrapolation and chiral interpolation with respect to the choice of fit function. The blue band corresponds to our preferred fit function (labeled “base”). Notice that the analytical parametrization is applied only at NNLO and beyond. The PQSChPT expression is used at NLO, including isospin corrections. The $Q$ value for each fit is shown in the right-hand-side plot. The red point on that plot corresponds to a fit with ${\chi }^2/\mathrm{dof}<0.05$. See the text for the explanation of the different tests performed.

Figure 9

Comparison of $f_{+}^{K^0{\pi }^{-}}(0)$ from this analysis with previous lattice results entering in the FLAG averages [6] together with those averages for $N_f=2+1+1$ and $N_f=2+1$, as well as nonlattice determinations based on ChPT. The beige band corresponds to our result. The references and numerical results for all determinations are given in Table 7.

Figure 10

Summary of recent $|V_{us}|$ determinations. The semileptonic determinations, labeled $K_{l3}$, use inputs for $f_{+}^{K\pi }(0)$ from the most recent lattice calculations in Refs. [23, 25, 26], respectively. The leptonic determinations, labeled $K_{l2}$, use as inputs the $2+1$-flavor lattice-QCD average $f_K$ from FLAG [6], which only includes calculations where the lattice scale is set from physical inputs other than $f_{\pi }$, and the most recent and precise determination of $f_{K^{\pm }}/f_{{\pi }^{\pm }}$ from Ref. [46]. The inclusive hadronic $\tau$-decay determinations are the most recent ones, from Boyle et al. 2018 [36] and Hudspith et al. 2017 [35]. The second value from Ref. [36] comes from relating the $\tau \to K\ell \nu$ branching fraction to the $K_{\mu 2}$ branching fraction to get the experimental contribution from the $K$ pole. The two values in Ref. [35] correspond to using the normalization for $\tau$ decays into $K\pi$ modes as obtained in Ref. [37] or as given by HFLAV [39]. For the exclusive $\tau$ determination we follow the calculation by the HFLAV group [39], but we update the value of the ratio $f_{K^{\pm }}/f_{{\pi }^{\pm }}$ to that in Ref. [46]. The unitarity value is taken to be $|V_{us}|=\sqrt{1-|V_{ud}{|}^2}$ with $|V_{ud}|$ from Ref. [2]. RC stands for radiative corrections. The dotted magenta vertical lines correspond to this unitarity value. The gray vertical band corresponds to our result in Eq. (7.1).

Figure 11

Constraints on $|V_{ud}|$ and $|V_{us}|$ from our results ($K_{l3}$), kaon leptonic decays ($K_{l2}$), superallowed nuclear $\beta$ decays, unitarity, and $|V_{cd}|$, as discussed in the text. Blue ellipses correspond to the allowed region from $K_{l3}$ and one of the other two constraints with a 68% probability. Both regions have no overlap with unitarity (black line). Correlations between $K_{l2}$ and $K_{l3}$ are not taken into account. The orange horizontal line in the yellow region corresponds to the central value for $|V_{us}|$ as extracted from $|V_{cd}|$.

Figure 12

Comparison of the unitarity point using $|V_{ud}|=0.97366(15)$ with the results in this work, and with the unitarity point corresponding to $|V_{ud}|=0.97420(21)$. RC stands for radiative corrections.