5  |V_us| measurement

The CKM coefficient |V_us| can be measured in several ways from the comparison of tau partial widths to strange and non-strange final states.

5.1  Inclusive tau partial width to strange

The tau hadronic partial width is the sum of the tau partial width to strange and to non-strange hadronic final states, Γ_had = Γ_s + Γ_VA . Dividing by the partial width to electron, Γ_e, we obtain partial width ratios (which are equal to the respective branching fraction ratios) for which R_had = R_s + R_VA . In terms of such ratios, |V_us| is measured as

where δ R_theory can be determined in the context of low energy QCD theory, partly relying on experimental low energy scattering data. We use δ R_theory = 0.240 ± 0.032  [70], which induces a systematic error on |V_us| that lies between two more recent estimates [71, 83].

In the following, we use the universality improved B _e^uni (see Section 4) to compute the R ratios. The most direct experimental determination of R_s and R_VA = R_had−R_s come from the tau inclusive branching fractions to hadronic and strange hadronic states, B _had and B _s . However often the total hadronic branching fraction has been replaced by the indirect but more precise expression B _had ^uni = 1 − B _e −B _µ (or similar expressions based on B _e^uni), using unitarity, see for example the 2009 HFAG report [31]. We depart from this choice here, and we use the most direct determination of R_had, for two reasons: first there is no significant statistical gain in the final errors, because of statistical correlations in the R_had expression (1−B _e −B _µ)/B _e^univ, and second the indirect determination of R_VA = R_had ^uni − R_s would absorb the effect of possible unobserved hadronic states entirely in R_VA, while they could also be strange final states.

With the above choices, using |V_ud| = 0.97425 ± 0.00022  [73], using HFAG values of this report, including the above-mentioned B _e^univ, B _s = (2.872 ± 0.050)% (see also Table 9), B _VA = (61.85 ± 0.11)%) and the PDG 2011 averages, we obtain |V_us| _{τ s} = 0.2172 ± 0.0022 , which is 3.4 σ lower than the unitarity CKM prediction |V_us| _uni = 0.2255 ± 0.0010 , from (|V_us| _uni)² = 1 − |V_ud| ². The |V_us| _{τ s} uncertainty includes a systematic error contribution of 0.0010 from the theory uncertainty on δ R_theory,

If we use the alternative above mentioned definitions of B _had, the mismatch remains 3.4σ. Using a unitarity-constrained tau branching fraction fit, the mismatch remains 3.4σ. The 3.4 σ discrepancy is close to the unconstrained fit result of the 2009 HFAG report, 3.6σ [31], and also to the 3.3σ from the HFAG-Tau 2011 intermediate document [46], based on a unitarity-constrained fit.

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Table 9: HFAG Winter 2012 Tau branching fractions to strange final states.

Branching fraction	HFAG Winter 2012 fit
Γ10 = K− ντ	(0.6955 ± 0.0096) · 10−2
Γ16 = K− π0 ντ	(0.4322 ± 0.0149) · 10−2
Γ23 = K− 2π0 ντ (ex. K0)	(0.0630 ± 0.0222) · 10−2
Γ28 = K− 3π0 ντ (ex. K0,η)	(0.0419 ± 0.0218) · 10−2
Γ35 = π− K0 ντ	(0.8206 ± 0.0182) · 10−2
Γ40 = π− K0 π0 ντ	(0.3649 ± 0.0108) · 10−2
Γ44 = π− K0 π0 π0 ντ	(0.0269 ± 0.0230) · 10−2
Γ53 = K0 h− h− h+ ντ	(0.0222 ± 0.0202) · 10−2
Γ128 = K− η ντ	(0.0153 ± 0.0008) · 10−2
Γ130 = K− π0 η ντ	(0.0048 ± 0.0012) · 10−2
Γ132 = π− K0 η ντ	(0.0094 ± 0.0015) · 10−2
Γ151 = K− ω ντ	(0.0410 ± 0.0092) · 10−2
Γ801 = K− φ ντ(φ → KK)	(0.0037 ± 0.0014) · 10−2
Γ802 = K− π− π+ ντ (ex. K0,ω)	(0.2923 ± 0.0068) · 10−2
Γ803 = K− π− π+ π0 ντ (ex. K0,ω,η)	(0.0411 ± 0.0143) · 10−2
Γ110 = Xs− ντ	(2.8746 ± 0.0498) · 10−2

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

We use the ratio of branching fractions B (τ⁻ → K⁻ ν_τ )/B (τ⁻ → π⁻ ν_τ ) = 0.0643 ± 0.0009 to measure |V_us| from the equation

In this ratio, the short-distance radiative corrections cancel. The term r_LD(p) = 1 + δ_LD(p) corresponds to the long-distance electroweak radiative correction factor for the process p. Following Ref. [45], the ratio of radiative correction factors is estimated as r_LD^Kπ = r_LD(τ⁻ → K⁻ν/K⁻ → µ⁻ν) / r_LD(τ⁻ → π⁻ν/π⁻ → µ⁻ν) · r_LD(K⁻ → µ⁻ν)/r_LD(π⁻ → µ⁻ν), where the first ratio is [1+ (0.90 ± 0.22)%]/[1+ (0.16 ± 0.14)%] [65] and the second ratio is (0.9930 ± 0.0035)% [85], hence assuming independent errors r_LD^Kπ = 1.0003 ± 0.0044 . The ratio f_K/f_π is estimated in lattice QCD to be 1.1936 ± 0.0053 [80]. We measure |V_us| _{τ K/π} = 0.2229 ± 0.0020 , 1.2 σ below the CKM unitarity prediction.

We use the branching fraction B (τ⁻ → K⁻ ν_τ ) to measure |V_us| from the equation

where f_K = 156.1 ± 1.1  MeV  [80] is the kaon decay constant estimated with lattice QCD, and S_EW = 1.0201 ± 0.0003  [68] accounts for the radiative corrections. We obtain |V_us| _{τ K} = 0.2214 ± 0.0022 , wich is 1.7 σ below the CKM unitarity prediction. CODATA 2006 results [90] and PDG 2011 have been used for the physics constants.

5.3  |V_us| from tau summary

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Figure 1: |Vus| averages of this document compared with the FlaviaNet results [27].

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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:

Thanks to the improved lattice QCD determination of f_K [80], the uncertainty on |V_us| _{τ K} has been significantly reduced with respect to the previous HFAG report. Averaging the three above |V_us| determinations 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%. Figure 1 summarizes the |V_us| results.