5  |V_us| measurement

The CKM matrix element magnitude |V_us| is most precisely determined from kaon decays [75] (see Figure 1), and its precision is limited by the uncertainties of the lattice QCD estimates of the meson decay constants f₊^Kπ(0) and f_K±/f_π±. Using the τ branching fractions, it is possible to determine |V_us| in an alternative way [76, 77] that does not depend on lattice QCD and has small theory uncertainties (as discussed in Section 5.1). Moreover, |V_us| can be determined using the τ branching fractions similarly to the kaon case, using the same meson decay constants from lattice QCD.

5.1  |V_us| from B(τ → X_sν)

The τ hadronic partial width is the sum of the τ partial widths to strange and to non-strange hadronic final states, Γ_had = Γ_s + Γ_VA . The suffix “VA” traditionally denotes the sum of the τ partial widths to non-strange final states, which proceed through either vector or axial-vector currents.

Dividing any partial width Γ_x by the electronic partial width, Γ_e, we obtain partial width ratios R_x (which are equal to the respective branching fraction ratios B_x/ B_e) for which R_had = R_s + R_VA . In terms of such ratios, |V_us| can be measured as [76, 77]

where δ R_theory can be determined in the context of low energy QCD theory, partly relying on experimental low energy scattering data. The literature reports several calculations [78, 79, 80]. In this report we use Ref. [78], whose estimated uncertainty size is intermediate between the two other ones. We use the information in that paper and the PDG 2018 value for the s-quark mass m_s = 95.00 ± 6.70 MeV [6] to calculate δ R_theory = 0.242 ± 0.033.

We proceed following the same procedure of the 2012 HFLAV report [81]. We sum the relevant τ branching fractions to compute B_VA and B_s and we use the universality-improved B_e^uni (see Section 4) to compute the R_VA and R_s ratios. In past determinations of |V_us|, for example in the 2009 HFLAV report [82], the total hadronic branching fraction has been computed using unitarity as B_had ^uni = 1 − B_e − B_µ, obtaining then B_s from the sum of the strange branching fractions and B_VA from B_had ^uni − B_s . We prefer to use the more direct experimental determination of B_VA 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 is counterbalanced by increased correlations in the expressions (1− B_e − B_µ)/ B_e^univ and B_s/( B_had− B_s) in the |V_us| calculation. Second, if there are unobserved τ hadronic decay modes, they would affect B_VA and B_s in a more asymmetric way when using unitarity.

Using the τ branching fraction fit results with their uncertainties and correlations (Section 2), we compute B_s = (2.931 ± 0.041)% (see also Table 13) and B_VA = B_had − B_s = (61.83 ± 0.10)%, where B_had has been defined in section 4. PDG 2018 averages are used for non-τ quantities; |V_ud| = 0.97420 ± 0.00021 [83, 84].

We obtain |V_us| _{τ s} = 0.2195 ± 0.0019, which is 2.9σ lower than the unitarity CKM prediction |V_us| _uni = 0.22565 ± 0.00089, from (|V_us| _uni)² = 1 − |V_ud| ² − |V_ub| ². The |V_us| _{τ s} uncertainty includes a systematic error contribution of 0.0011 from the theory uncertainty on δ R_theory. The 2018 BaBar preliminary results improved the |V_us| precision by about 10% and reduced the discrepancy by about 6.5%.

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Table 13: HFLAV 2018 τ branching fractions to strange final states.

Branching fraction	HFLAV 2018 fit (%)
K− ντ	0.6986 ± 0.0085
K− π0 ντ	0.4904 ± 0.0092
K− 2π0 ντ (ex. K0)	0.0585 ± 0.0027
K− 3π0 ντ (ex. K0,η)	0.0113 ± 0.0026
π− K0 ντ	0.8378 ± 0.0139
π− K0 π0 ντ	0.3807 ± 0.0129
π− K0 2π0 ντ (ex. K0)	0.0235 ± 0.0231
K0 h− h− h+ ντ	0.0222 ± 0.0202
K− η ντ	0.0154 ± 0.0008
K− π0 η ντ	0.0048 ± 0.0012
π− K0 η ντ	0.0094 ± 0.0015
K− ω ντ	0.0410 ± 0.0092
K− φ ντ (φ → K+ K−)	0.0022 ± 0.0008
K− φ ντ (φ → KS0 KL0)	0.0015 ± 0.0006
K− π− π+ ντ (ex. K0,ω)	0.2923 ± 0.0067
K− π− π+ π0 ντ (ex. K0,ω,η)	0.0410 ± 0.0143
K− 2π− 2π+ ντ (ex. K0)	0.0001 ± 0.0001
K− 2π− 2π+ π0 ντ (ex. K0)	0.0001 ± 0.0001
Xs− ντ	2.9308 ± 0.0412

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5.2  |V_us| from B(τ → Kν) / B(τ → πν)

We compute |V_us| from the ratio of branching fractions B(τ → K⁻ ν_τ) / B(τ → π⁻ ν_τ) = (6.467 ± 0.084) · 10⁻² from the equation [85]:

We use f_K±/f_π± = 1.1932 ± 0.0019 from the FLAG 2019 lattice QCD averages with N_f=2+1+1 [86, 87, 88, 89],

The value of δ R_K/π in the Spring 2017 HFLAV-Tau report [2] incorrectly included a strong isospin-breaking correction that is not needed when using f_K±/f_π± rather than its isospin-limit variant. We compute |V_us| _{τ K/π} = 0.2236 ± 0.0015, 1.2σ below the CKM unitarity prediction.

5.3  |V_us| from B(τ → Kν)

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

We use f_K± = 155.7 ± 0.3 MeV from the FLAG 2019 lattice QCD averages with N_f=2+1+1 [86, 87, 92, 88], δ R_τ/K = (0.90 ± 0.22)% [71, 72, 73, 74] and δ R_Kµ2 = (1.07 ± 0.21)% [90, 6, 93], which includes short and long-distance radiative corrections. We obtain |V_us| _{τ K} = 0.2234 ± 0.0015, which is 1.3σ below the CKM unitarity prediction. The physical constants have been taken from PDG 2018 (which uses CODATA 2014 [94]).

5.4  |V_us| from τ summary

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Figure 1: |Vus| averages.

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We summarize the |V_us| results reporting the values, the discrepancy with respect to the |V_us| determination from CKM unitarity, and an illustration of the measurement method:

Averaging the three above |V_us| determinations that rely on the τ branching fractions (taking into account all correlations due to the τ HFLAV and other mentioned inputs) we obtain, for |V_us| and its discrepancy:

The correlation between f_K± and f_K±/f_π± has been assumed to be zero. Even assuming ± 100% correlation, the |V_us| uncertainty varies by less than 10%.

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 major τ branching fractions to kaons [ B(τ→K⁻ ν_τ), B(τ→K⁻ π⁰ ν_τ) and B(τ→π⁻ K⁰ ν_τ)] are lower than their determinations from the kaon branching fractions into final states with leptons within the SM [68, 95, 96].

Alternative deterrminations of |V_us| from B(τ → X_sν) [97, 98], based on partially different sets of experimental inputs, report |V_us| values consistent with the unitarity determination.

Figure 1 reports the HFLAV |V_us| determinations that use the τ branching fractions, compared to two |V_us| determinations based on kaon data [6] and to |V_us| obtained from |V_ud| and the CKM matrix unitarity [6].