7 Measurements of |V_us|

The CKM matrix element magnitude |V_us| is most precisely determined from kaon decays [93] (see Figure 4), and its precision is limited by the uncertainties of the lattice QCD estimates of the meson form factor f₊^Kπ(0) and decay-constant ratio f_K±/f_π±. The following sections report alternative determinations of |V_us| based on the experimental measurements of the τ branching fractions.

7.1 |V_us| from B(τ → X_sν)

|V_us| is computed using the inclusive τ branching fraction to strange hadronic final states as [94, 95]

where R_us = B_us / B_e^uni = 0.1632 ± 0.0027, B_us = 2.908 ± 0.048 is the inclusive τ branching fraction to strange hadronic final states (see also Table 6), R_ud = R_had ^uni − R_us = 3.470 ± 0.008, and |V_ud| = 0.97384 ± 0.00026 is taken from a recent average of measurements of superallowed nuclear β decays, neutron decay measurements [96], and the correction δ R_{τ-SU(3)-break} accounts for the SU(3)-breaking effects due to the difference between the strange, up and down quark masses. δ R_{τ-SU(3)-break} is determined using perturbative QCD and experimental low energy scattering measurements [97, 98, 99]. In this report, we compute δ R_{τ-SU(3)-break} = 0.239 ± 0.032 using Ref. [97] and the s-quark mass m_s = 93 ± 7 MeV  [8]. Ref. [97] estimates a δ R_{τ-SU(3)-break} uncertainty that is intermediate with respect to the other two assessments [98, 99]. The uncertainty on δ R_{τ-SU(3)-break} contributes a 0.47% systematic contribution on |V_us| _τ-OPE-1. The above-mentioned δ R_{τ-SU(3)-break} calculations have been criticized for falling short in dealing with the biases and uncertainties in the low-energy regime of QCD [100]. However, the community has not reached yet a consensus on the matter. In order to obtain more reliable QCD uncertainties, alternative procedures for computing |V_us| using τ decays have been proposed, which involve the τ spectral functions [100] and lattice QCD methods [101]. Because of the complexity of these two last proposed procedures, the effort that is required to use them to compute |V_us| using the current HFLAV-Tau branching fractions exceeds the scope of this report. Therefore, we just report the most recent updates:

We also use a recently reported lattice QCD calculation of the inclusive τ hadronic decay rate [104] to compute an additional determination of |V_us|,

where (R_us / |V_us| ²)_latt-incl = 3.407 ± 0.022 [104]. It is worth noting that all calculations of the τ inclusive decay rate to hadrons are not presently including the long-distance isospin-breaking corrections [104].

In several of the past determinations of |V_us|, such as the 2009 HFLAV report [105], the total hadronic branching fraction was computed using unitarity as B_had ^uni = 1 − B_e − B_µ, and B_ud was obtained from B_had ^uni − B_us . Here we rather use the direct experimental determination of B_ud for two reasons. First, both methods result in comparable uncertainties on |V_us|, since the better precision on B_had ^uni = 1 − B_e − B_µ with respect to B_had equal to the sum of all τ hadronic branching fractions is offset by increased correlations when considering R_had ^uni = B_had/ B_e^uni and similar expressions in the |V_us| calculation. Second, if there are unobserved τ hadronic decay modes, they will affect B_ud and B_us in a more asymmetric way when using B_had ^uni.

The values of |V_us| _τ-OPE-1 and |V_us| _τ-latt-incl are respectively 3.6σ and 3.7σ lower than the value |V_us| _uni = 0.2272 ± 0.0011 predicted from the CKM unitarity relation (|V_us| _uni)² = 1 − |V_ud| ² − |V_ub| ², with |V_ub| = 0.00382 ± 0.00020 [8]. Using Eqs. 21 and 24 we also compute |V_us|/|V_ud| _τ-OPE-1 = 0.2243 ± 0.0022 and |V_us|/|V_ud| _τ-latt-incl = 0.2248 ± 0.0020.

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Table 6: A summary of the averaged τ branching fractions to strange final states.

Branching fraction	HFLAV 2023 fit (%)
B(τ− → K− ντ)	0.6959 ± 0.0096
B(τ− → K− π0 ντ)	0.4321 ± 0.0148
B(τ− → K− 2π0 ντ (ex.K0))	0.0634 ± 0.0219
B(τ− → K− 3π0 ντ (ex.K0,η))	0.0465 ± 0.0213
B(τ− → π− K0 ντ)	0.8375 ± 0.0139
B(τ− → π− K0 π0 ντ)	0.3810 ± 0.0129
B(τ− → π− K0 2π0 ντ (ex.K0))	0.0234 ± 0.0231
B(τ− → K0 h− h− h+ ντ)	0.0222 ± 0.0202
B(τ− → K− η ντ)	0.0155 ± 0.0008
B(τ− → K− π0 η ντ)	0.0048 ± 0.0012
B(τ− → π− K0 η ντ)	0.0094 ± 0.0015
B(τ− → K− ω ντ)	0.0410 ± 0.0092
B(τ− → K− φ(K+ K−) ντ)	0.0022 ± 0.0008
B(τ− → K− φ(KS0 KL0) ντ)	0.0015 ± 0.0006
B(τ− → K− π− π+ ντ (ex.K0,ω))	0.2924 ± 0.0068
B(τ− → K− π− π+ π0 ντ (ex.K0,ω,η))	0.0388 ± 0.0142
B(τ− → K− 2π− 2π+ ντ (ex.K0))	0.0001 ± 0.0001
B(τ− → K− 2π− 2π+ π0 ντ (ex.K0))	0.0001 ± 0.0001
B(τ− → Xs− ντ)	2.9078 ± 0.0478

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7.2 |V_us| from B(τ → Kν) / B(τ → πν)

We also compute |V_us|/|V_ud| from the ratio of branching fractions B_{10 / 9} = B(τ⁻ → K⁻ ν_τ) / B(τ⁻ → π⁻ ν_τ) = 0.0644 ± 0.0009 (Table 3) using the equation [106]:

where δ_{τ K/τπ} = (0.10 ± 0.80)% [92] is the radiative correction for the ratio of partial widths Γ(τ→ Kν_τ[γ]) / Γ(τ→ πν_τ[γ]). Using the ratio of decay constants f_K±/f_π± = 1.1934 ± 0.0019 from the 2023 web update of the FLAG 2021 lattice QCD averages with N_f=2+1+1 [107, 108, 109, 110, 111, 112, 113], we obtain |V_us|/|V_ud| = 0.2289 ± 0.0019. By using the above-mentioned |V_ud| value, we compute |V_us| _{τ K/π} = 0.2229 ± 0.0019, which is 2.0σ below the CKM unitarity prediction.

7.3 |V_us| from B(τ → Kν)

We also determine |V_us| from the branching fraction B(τ⁻ → K⁻ ν_τ ) using

We use f_K± = 155.7 ± 0.3 MeV from the 2023 web update of the FLAG 2021 lattice QCD averages with N_f=2+1+1 [107, 108, 109, 111, 114, 112]. The universal short-distance electroweak correction for τ hadronic decays is S_EW^{τ h} = S_EW^{Rτ h} · S_EW^sub,lep = 1.01910 ± 0.00030, where the radiative correction for the tau spectral functions is S_EW^{Rτ h} = 1.02350 ± 0.00030 [115, 88, 116] and the sub-leading universal short-distance correction for the τ leptonic decays is S_EW^sub,lep = 0.9957 [115]. The long-distance radiative correction for B(τ⁻ → K⁻ ν_τ ) is δ_{τ K} = (−0.15 ± 0.57)% [92]. The physical constants G_F and ℏ are taken from CODATA 2018 [91]. We obtain |V_us| _{τ K} = 0.2224 ± 0.0017, which is 2.3σ below the CKM unitarity prediction.

7.4 Summary of |V_us| from τ decays

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Figure 4: |Vus| determinations. The values of |Vus| Kℓ3, |Vus| Kℓ2 and the expected |Vus| from the CKM matrix unitarity are taked from Ref. [96]. The other reported |Vus| values are documented in the text. When two uncertainties are reported, the first one accounts for the uncertainties of the HFLAV-Tau fit results, and the second one accounts for the uncertainties of the theory and the other inputs that are used for the |Vus| determinations.

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We summarize the |V_us| determinations that we updated using the present HFLAV-Tau averages by reporting the values, the discrepancy with respect to the |V_us| determination from CKM unitarity, and an illustration of the measurement method:

Averaging the two |V_us| determinations that rely on exclusive τ branching fractions (Eqs. 32, 33), we obtain:

Averaging the τ inclusive and exclusive |V_us| determinations (Eqs. 30, 31, 32, 33), we obtain:

In calculating the averages, all correlations arising from using the τ branching fractions fit results and common external inputs are accounted for. The systematic uncertainties of |V_us| _τ-OPE-1 and |V_us| _τ-latt-incl have been assumed to be uncorrelated. The correlation between f_K± and f_K±/f_π± is conservatively set to 100%. For the purpose of estimating the correlations between δ_{τ K}, δ_τπ and δ_{τ K/τπ}, we use the information [92] that the uncertainties on δ _{τ K} and δ_τπ are uncorrelated to a good approximation, and that δ_{τ K/τπ} = δ_{τ K} − δ_τπ.

All |V_us| determinations based on measured τ branching fractions are lower than both the kaon and the CKM-unitarity determinations. This is correlated with the fact that the direct measurements of the three largest τ branching fractions to kaons [ B(τ⁻ → K⁻ ν_τ), B(τ⁻ → K⁻ π⁰ ν_τ) and B(τ⁻ → π⁻ K⁰ ν_τ)] yield lower values than their SM predictions based on the branching fractions of leptonic kaon decays [88, 117, 118].

Figure 4 reports our |V_us| determinations using the present HFLAV-Tau averages, two additional |V_us| determinations using partially different τ measurements inputs [102, 101, 103], two determinations based on kaon data [96], and the |V_us| value obtained from |V_ud| using the CKM-matrix unitarity [96].