$f_K/f_π$ in iso-symmetric QCD and the CKM matrix unitarity

TL;DR

This work delivers a first-principles lattice QCD determination of $f_K/f_\pi$ in the iso-symmetric limit with $N_f=2+1$, using a hybrid Wilson unitary and mixed-action setup to tighten the continuum extrapolation. It extends the analysis to include strong isospin-breaking and QED corrections to extract $|V_{us}|/|V_{ud}|$ and perform a CKM first-row unitarity test, yielding $|V_{us}|/|V_{ud}|=0.2330(11)(17)(5)(2)(3)$ and a unitarity-satisfying sum $|V_{ud}|^2+|V_{us}|^2=0.9995(6)(7)(2)(7)(1)$. The scale is set via $f_\pi$, avoiding uncertainties tied to other theory scales, and the two-regulation strategy enhances control over lattice artifacts. The dominant limitation is the lattice error in $f_K/f_\pi$, suggesting that future work should target increased statistics and NNLO $ chi$PT effects to reduce systematic uncertainties. Overall, the results provide a precise isoQCD prediction and a CKM unitarity check consistent with the Standard Model.

Abstract

We present lattice results for $f_K/f_π$ in the iso-symmetric limit of pure QCD (isoQCD) with $N_f=2+1$ flavours, along with a determination of $|V_{us}|/|V_{ud}|$ and a study on the unitarity of the first row of the Cabibbo-Kobayashi-Maskawa (CKM) matrix after introducing strong isospin-breaking and QED effects. The results obtained are based on a combination of a Wilson unitary action and the mixed-action setup introduced in arXiv:2309.14154, arXiv:2510.20450. The combination of the two regularisations enables a more precise control over the continuum-limit extrapolation.

Figures (6)

• Figure 1: Measured values of $\rho_{2},\;\rho_{4}$, defined in \ref{['eq:rho2']} and \ref{['eq:rho4']}, for the set of CLS ensembles employed in this work, following the ${\rm tr}(M_{q})=2m_{l}+m_{s}\approx{\rm const.}$ chiral trajectory. Empty points correspond to the Wilson unitary setup, while filled ones correspond to the Wtm mixed action. The difference in $\rho_2$ and $\rho_4$ between the Wilson and Wtm mixed action for any given ensemble is due to different cutoff effects in $f_{\pi}$ for each regularisation, as the pion and kaon masses -- in lattice units -- are matched to be the same in both. The physical point, marked by a black cross, is reached by interpolating both in $\rho_2$ and $\rho_4$.
• Figure 2: $R_X$ for $X=m_{\pi},f_{\pi},f_K$ for the set of ensembles considered in this work in the Wilson unitary setup, as given by \ref{['eq:FVE']} (blue crosses), together with the expected dependence on $m_{\pi}L$ for each of these ensembles (grey dashed lines). It can be seen that the finite-volume correction is of a few per mille at most, well below the statistical precision of the lattice data (the red horizontal line shows a representative example of the relative statistical precision). $R_{f_K}-R_{f_{\pi}}$ in the bottom right plot is the relevant correction factor for the ratio $f_K/f_{\pi}$.
• Figure 3: Model exploration for the chiral-continuum extrapolation of the ratio $f_K/f_{\pi}$. In the first bracket, labels $[\chi \rm PT]$ and $[\chi \rm PT\;ratio]$ refer to the use of \ref{['eq:ratio_expanded']} and \ref{['eq:ratio']} respectively, with cutoff effects added according to \ref{['eq:cutoff_parameters']}, while $f_{\pi K}$ or $f_{\pi}$ refers to what was used for the scale $\mu$ inside the chiral logarithms in \ref{['eq:Lpi', 'eq:LK', 'eq:Leta']}. [Tay constr] refers to \ref{['eq:Taylor_constrained']}, and [Tay unconstr] to \ref{['eq:Taylor_un']}. The second label indicates which data are included: [$-$] means all data are used, $[\beta>3.40]$ that the coarsest lattice spacing is removed, $[m_{\pi}<360{\rm\;MeV}]$ that only ensembles with pion masses smaller than $360$ MeV are included in the analysis, and $[m_{\pi}L\geq4.2]$ the same for lattices with the volume satisfying this condition. The horizontal band in the top panel represents the final quoted result in \ref{['eq:fKfpi_ph']}, with statistical and systematic uncertainties added in quadrature.
• Figure 4: Contribution of the different lattice ensembles to the statistical error squared in \ref{['eq:fKfpi_ph']}.
• Figure 5: Left column: chiral-continuum extrapolation employing \ref{['eq:ratio_expanded']}, using $\mu=f_{\pi}$ in the definition of the chiral logarithms, and adding cutoff effects according to \ref{['eq:cutoff_parameters']} with $\Gamma_{i}=0$, removing $m_{\pi}>360$ MeV. Right column: chiral-continuum extrapolation employing \ref{['eq:ratio']}, using $\mu=f_{\pi}$, and adding cutoff effects according to \ref{['eq:cutoff_parameters']} with $\Gamma_{i}=0$, removing the coarsest lattice spacing $\beta=3.40$. In the two top plots, the points in the $y$-axis are projected to $\rho_4^{\rm ph}$ and vanishing lattice spacing, while in the two middle plots they are projected to $\rho_2^{\rm ph}$ and vanishing lattice spacing. The vertical dashed line marks the position of the physical point, and the bands correspond to the mass-dependence in the continuum. Finally, in the two bottom plots the data in the $y$-axis are projected to both $\rho_2^{\rm ph},\;\rho_4^{\rm ph}$. The two bands in each plot correspond to the cutoff dependence for both the Wilson unitary and Wtm mixed-actions setups. Empty points are computed with the Wilson unitary action, while filled ones are with the mixed action. The two models shown correspond to the ones giving the lowest and highest results that enter in the model average (cf. Fig. \ref{['fig:model_av']}), and thus define the width of our error band in \ref{['eq:fKfpi_ph']}, corresponding to the black star point in the two bottom plots.