Global Determination of $|V_{us}|$

TL;DR

This work presents a first global determination of $|V_{us}|$ by simultaneously analyzing $K\to \ell\nu$, $K\to \pi\ell\nu$, $\tau\to K\nu$, and $\tau\to K\pi\nu$ decays within a novel dispersively-bounded parametrization of the $K\pi$ form factors. The method enforces unitarity, Callan-Treiman constraints, and resonance structure, enabling a unified extraction of $|V_{us}|$ and hadronic form-factor parameters without relying on lattice inputs for the normalization at $q^2=0$. The results yield $f_+(0)$ in agreement with lattice determinations and a global $|V_{us}|=0.2244(5)$, while uncovering a sustained tension with CKM unitarity when combined with $|V_{ud}|$; no compelling evidence for a right-handed current is observed within the current uncertainties. The analysis also provides new predictions for $f_0$ at $q^2=m_D^2$ relevant to non-leptonic $D$ decays and demonstrates the potential for future refinements if long-distance EM effects and angular observables are incorporated. Overall, this pilot study establishes a consistent framework for a more precise global determination of $|V_{us}|$ and related hadronic quantities.

Abstract

In light of ongoing issues with the first-row unitarity test of the CKM matrix, we showcase a global fit to measurements of $K\to \ell ν$, $K\to π\ell ν$, $τ\to Kν$, and $τ\to Kπν$ decays for the first time. Fitting the semileptonic and 3-body $τ$ decay data simultaneously becomes computationally feasible because we employ a simple form factor parametrisation for the $Kπ$ form factor that manifestly connects the semileptonic and pair-production regions. We find good agreement with the data and use our analyses to infer $|V_{us}|$ and the parameters for the $Kπ$ form factors. Our result for $|V_{us}|$ sits between the results extracted from exclusive $K_{\ell 2}$ and $K_{\ell 3}$ decays and somewhat above the inclusive $τ$ decay result. Moreover, our results for the $K\to π$ vector form factor at zero momentum transfer are compatible with lattice QCD determinations, despite not using lattice QCD inputs for this quantity. We are further able to determine some of the pole parameters of the individual scalar and vector $Kπ$ resonances with masses below the $τ$ mass. We caution that our results account only for short-distance electromagnetic corrections but not long-distance contributions; this is due to the lack of a consistent description of long-distance corrections to the $Kπ$ matrix elements both above and below threshold for an arbitrary model of the form factors.

Figures (4)

• Figure 1: Illustration of the relevant phase space regions and locations of the resonance poles in the complex $q^2$ (left) and $z$ (right) planes for the $f_+$ form factor. The phase space of $\tau \to K \pi \nu$ is illustrated as purple lines and curves. The phase space of $K\to \pi \ell^-\bar{\nu}$ is illustrated as brown lines. The resonance poles are marked with coloured crosses ($K^*(896)$: light blue, $K^*(1410)$: yellow, $K^*(1680)$: red).
• Figure 2: Normalised differential tau decay data from Belle Belle:2007goc (blue error bars), along with the posterior prediction and 68 probability envelope of our nominal model fit (orange band).
• Figure 3: Our result for $|V_{us}^{}|\xspace$ shown as the shaded vertical band, in comparison to results in the literature. We show four independent results obtained from inclusive $\tau$ decays, corresponding to results from Refs. Gamiz:2002nuGamiz:2004ar (top-most data point), Refs. Maltman:2015xwaMaltman:2019xeh (second from top), Refs. RBC:2018uykMaltman:2019xeh (second from bottom), and Ref. ExtendedTwistedMass:2024myu (bottom). We refer the interested reader to discussions in the HFLAV report HeavyFlavorAveragingGroupHFLAV:2024ctgHFLAV:tau-vus-web-report for details of the differences between these determinations. The $K_{l3}$, hyperon, and $\beta$ decay + unitarity results are taken from Refs. ParticleDataGroup:2024cfkPDG:Vud_Vus_review, while the global CKM fit is obtained by the CKMfitter collaboration Charles:2004jdCKMfitter:summer2023.
• Figure 4: Joint distribution of $|V_{us}^{}|\xspace$, $f_{K}$, and $f_+(0)$ in our CKM fit model, along with a comparison to the $f_+(0)$ posterior in our FF fit model.