5  |V_us| measurement

The CKM matrix element magnitude |V_us| is most precisely determined from kaon decays [78] (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 [79, 80] 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 [79, 80]

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 [81, 82, 83]. In this report we use Ref. [81], whose estimated uncertainty size is intermediate 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 [9] to calculate δ R_theory = 0.242 ± 0.032.

We proceed following the same procedure of the 2012 HFLAV report [3]. 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 [1], 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 vanified by increased statistical 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 = (2.909 ± 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, and |V_ud| = 0.97417 ± 0.00021 [84].

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 ± 0.00089, from (|V_us| _uni)² = 1 − |V_ud| ². 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 HFLAV report.

---

Table 13: HFLAV Spring 2017 τ branching fractions to strange final states.

Branching fraction	HFLAV Spring 2017 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

---

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) = (6.438 ± 0.094) · 10⁻² from the equation [70]:

We use f_K/f_π= 1.1930 ± 0.0030 from the FLAG 2016 Lattice averages with N_f=2+1+1 [85],

We compute |V_us| _{τ K/π} = 0.2236 ± 0.0018, 1.1σ below the CKM unitarity prediction.

5.3  |V_us| from τ summary

---

PNG format	PDF format

Figure 1: |Vus| averages.

---

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 two 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:

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 [70, 88, 89].

A recent determination of |V_us| [90, 91] that relies on the τ spectral functions in addition to the inclusive τ → X_s ν branching fraction reports a |V_us| value about 1σ lower than the CKM-unitarity determination. This determination uses inputs that are partially different from the ones used in this report. Specifically, the HFLAV average of B (τ→K⁻ ν_τ) has been replaced with the SM prediction based on the measured B (K⁻ → µ⁻ ν_µ) and the HFLAV average of B (τ→K⁻ π⁰ ν_τ) has been replaced with an in-progress BaBar measurement that is published in a PhD thesis. Both changes increase the resulting τ → X_s ν inclusive branching fraction. This study claims that the newly proposed |V_us| calculation has a more stable and reliable theory uncertainty, which could possibly have been underestimated in former studies, which are used for the HFLAV |V_us| average.

In previous editions of the HFLAV report, we also computed |V_us| using the branching fraction B (τ → Kν) and without taking the ratio with B (τ → πν). We do not report this additional determination because it did not include the long-distance radiative corrections in addition to the short-distance contribution, and because it had a negligible effect on the overall precision of the |V_us| calculation with τ data.

Figure 1 reports the HFLAV |V_us| determinations that use the τ branching fractions, compared to two |V_us| determinations based on kaon data [2] and to |V_us| obtained from |V_ud| and the CKM matrix unitarity [2].