5  |V_us| measurement

The CKM matrix element |V_us| is most precisely determined from kaon decays [71] (see also Figure 1), and its precision is limited by the uncertainties of the lattice QCD estimates of f₊^Kπ(0) and f_K/f_π. Using the τ branching fractions, it is possible to determine |V_us| in an alternative way [72] 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 lattice QCD estimates, in order to check the overall experimental consistency.

In the following Sections 5.1, 5.2 and 5.3 we update the CKM coefficient |V_us| determinations that were shown in the previous report using the 2015 determination of |V_ud| [73] and the updated averages from HFAG 2016 and PDG 2015 for the other quantities.

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| is measured as [72]

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 [72, 74, 75]. In this report we use Ref. [72], whose estimated uncertainty size is in between the two other ones. We use the information in that paper and the PDG 2015 value for the s-quark mass m_s = 95.00 ± 5.00 MeV [8] to calculate δ R_theory = 0.242 ± 0.032.

We proceed following the same procedure of the 2012 HFAG report [2], using the universality improved B _e^uni = (17.815 ± 0.023)% (see Section 4) to compute the R_x ratios, and using the sum of the τ branching fractions to strange and non-strange hadronic final states to compute R_s and R_VA, respectively.

Using the τ branching fraction fit results with their uncertainties and correlations (Section 2), we compute B _s = (2.909 ± 0.048)% (see also Table 13) and B _VA = B _hadrons − B _s = (61.85 ± 0.10)%, where B _hadrons is equal to Γ_hadrons defined in section 4. PDG 2015 averages are used for non-τ quantities, including |V_ud| = 0.97417 ± 0.00021, which comes from Ref. [76] like for the previous HFAG report.

We obtain |V_us| _{τ s} = 0.2186 ± 0.0021, which is 3.1σ lower than the unitarity CKM prediction |V_us| _uni = 0.22582 ± 0.00091, from (|V_us| _uni)² = 1 − |V_ud| ² [77]. The |V_us| _{τ s} uncertainty includes a systematic error contribution of 0.47% from the theory uncertainty on δ R_theory. There is no significant change with respect to the previous HFAG report.

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Table 13: HFAG Summer 2016 τ branching fractions to strange final states.

Branching fraction	HFAG Summer 2016 fit (%)
K− ντ	0.6960 ± 0.0096
K− π0 ντ	0.4327 ± 0.0149
K− 2π0 ντ (ex. K0)	0.0640 ± 0.0220
K− 3π0 ντ (ex. K0,η)	0.0428 ± 0.0216
π− K0 ντ	0.8386 ± 0.0141
π− K0 π0 ντ	0.3812 ± 0.0129
π− K0 π0 π0 ντ (ex. K0)	0.0234 ± 0.0231
K0 h− h− h+ ντ	0.0222 ± 0.0202
K− η ντ	0.0155 ± 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	2.9087 ± 0.0482

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

We follow the same procedure of the HFAG 2012 report to compute |V_us| from the ratio of branching fractions B (τ → K⁻ ν_τ) / B (τ → π⁻ ν_τ) = (6.438 ± 0.094) · 10⁻² from the equation

We use f_K/f_π= 1.1930 ± 0.0030 from the FLAG 2016 Lattice averages with N_f=2+1+1 [78].

The ratio of radiative corrections R_{τ K/τπ} is estimated as

where

and

We compute |V_us| _{τ K/π} = 0.2231 ± 0.0018, 1.3σ 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 = 155.6 ± 0.4 MeV from FLAG 2016 with N_f=2+1+1 [78] and the radiative correction S_EW = 1.02010 ± 0.00030 [82]. We obtain |V_us| _{τ K} = 0.2223 ± 0.0016, which is 1.9σ below the CKM unitarity prediction. The physical constants have been taken from PDG 2015 (which uses CODATA 2014 [83]).

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 (taking into account all correlations due to the usage of the fitted τ branching fractions and the other mentioned inputs) we obtain:

We could not find a published estimate of the correlation of the uncertainties on f_K and f_K/f_π, but even if we assume ± 100% correlation, the uncertainty on |V_us| _τ does not change more than about ± 5%.

Recent studies [84, 85] indicate that the currently used theory uncertainties for the |V_us| determination from inclusive τ → X_s ν appear to be underestimated. This may explain the measured discrepancy with respect to |V_us| determined from kaon decays and from |V_ud| and the CKM matrix unitarity. The same studies propose an alternative determination of |V_us| that uses the τ spectral functions in addition to the τ branching fractions. The resulting value of |V_us| is consistent with the other |V_us| determinations.

Figure 1 summarizes the |V_us| results, reporting also recent determinations of |V_us| from kaon decays [77], CKM matrix unitarity [77] and the above mentioned determination of |V_us| from inclusive τ → X_s ν decays [85].