5 Measurements of |V_us|

The CKM matrix element magnitude |V_us| is most precisely determined from kaon decays ‍[79] (see Figure ‍1), and its precision is limited by the uncertainties of the lattice QCD estimates of the meson form factor f₊^Kπ(0) and decay constant in f_K±/f_π±. Using the τ branching fractions, it is possible to determine |V_us| in an alternative way ‍[80, 81] 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 lattice QCD predictions for the meson decay constants.

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 satisfy R_had = R_s + R_VA . In terms of such ratios, |V_us| can be measured as ‍[80, 81]

where δ R_theory can be determined using perturbative QCD and partly relying on experimental low energy scattering data ‍[82, 83, 84]. The calculations in the first two references have been criticized for falling short in dealing with the biases and uncertainties in the low-energy regime of QCD ‍[85], but are still supported by the authors. In order to obtain smaller and more reliable QCD uncertainties, alternative procedures for computing |V_us| using τ decays have been proposed, which involve the τ spectral functions ‍[85] and lattice QCD methods ‍[86].

In the following, we compute |V_us| using the τ branching fraction fit results according to eq. ‍16, since the complexity of the other proposed procedures and the effort that is required to reproduce them exceed the scope of this report. We use Ref. ‍[82] and the s-quark mass m_s = 93.00 ± 8.54 MeV  ‍[7] to calculate δ R_theory = 0.238 ± 0.033, since that reference quotes uncertainties that are intermediate with respect to the other two assessments in the above mentioned existing literature.

We proceed following the same procedure of the 2012 HFLAV report ‍[2]. We sum the strange and non-strange hadronic τ branching fractions B_s and B_VA , and use the universality-improved B_e^uni (see Section ‍4) to compute the R_s and R_VA ratios. In past determinations of |V_us|, such as the 2009 HFLAV report ‍[87], the total hadronic branching fraction was computed using unitarity as B_had ^uni = 1 − B_e − B_µ, and B_VA was obtained from B_had ^uni − B_s . Here we use the 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 offset by increased correlations in the expressions (1− B_e − B_µ)/ B_e^uni and B_s/( B_had− B_s) used in the |V_us| calculation. Second, if there are unobserved τ hadronic decay modes, they will 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 = (2.908 ± 0.048)% (see Table ‍13) and B_VA = B_had − B_s = (61.83 ± 0.10)%. PDG 2021 averages ‍[7] are used for quantities other than the results of the HFLAV τ branching fractions fit; |V_ud| = 0.97373 ± 0.00031 is taken from a 2020 updated determination ‍[88]. We obtain |V_us| _{τ s} = 0.2184 ± 0.0021, where the uncertainty includes a systematic error contribution of 0.0011 from the theory uncertainty on δ R_theory. This value is 3.7σ lower than the value |V_us| _uni = 0.2277 ± 0.0013 predicted from the CKM unitarity relation (|V_us| _uni)² = 1 − |V_ud| ² − |V_ub| ². We also compute (|V_us|/|V_ud|)_{τ s} = 0.2243 ± 0.0022.

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Table 13: HFLAV 2021 τ branching fractions to strange final states.

Branching fraction	HFLAV 2021 fit (%)
K− ντ	0.6957 ± 0.0096
K− π0 ντ	0.4322 ± 0.0148
K− 2π0 ντ (ex.K0)	0.0634 ± 0.0219
K− 3π0 ντ (ex.K0,η)	0.0465 ± 0.0213
π− K0 ντ	0.8375 ± 0.0139
π− K0 π0 ντ	0.3810 ± 0.0129
π− K0 2π0 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(K+ K−) ντ	0.0022 ± 0.0008
K− φ(KS0 KL0) ντ	0.0015 ± 0.0006
K− π− π+ ντ (ex.K0,ω)	0.2924 ± 0.0068
K− π− π+ π0 ντ (ex.K0,ω,η)	0.0387 ± 0.0142
K− 2π− 2π+ ντ (ex.K0)	0.0001 ± 0.0001
K− 2π− 2π+ π0 ντ (ex.K0)	0.0001 ± 0.0001
Xs− ντ	2.9076 ± 0.0478

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5.2 |V_us| from B(τ → Kν) / B(τ → πν)

We compute |V_us|/|V_ud| from the ratio of branching fractions B(τ → K⁻ ν_τ) / B(τ → π⁻ ν_τ) = (6.437 ± 0.092) · 10⁻² using the equation ‍[89]:

and we get |V_us|/|V_ud| = 0.2289 ± 0.0019, using the ratio of decay constants f_K±/f_π± = 1.1932 ± 0.0021 from the FLAG 2019 lattice QCD averages with N_f=2+1+1 ‍[90, 91, 92, , 93] and δ R_{τ K/τπ} = (0.10 ± 0.80)% ‍[10].

By using |V_ud| ‍[88] we compute |V_us| _{τ K/π} = 0.2229 ± 0.0019, 2.1σ 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 ‍[90, 92, 94, ], S_EW = 1.02320 ± 0.00030 ‍[95] and δ R_{τ K} = (−0.15 ± 0.57)% ‍[10]. We obtain |V_us| _{τ K} = 0.2219 ± 0.0017, which is 2.6σ below the CKM unitarity prediction. The physical constants G_F and ℏ are taken from CODATA 2018 ‍[96]. This edition fixes a transcription error on the physical constants taken from PDG 2018 that caused an incorrect shift of the |V_us| _{τ K} determination by about +0.5 σ in the previous HFLAV report ‍[5].

5.4 Summary of |V_us| from τ decays

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Figure 1: |Vus| determinations. In the CKM-unitarity evaluation, |Vud| is taken from a 2020 experimental update ‍[88]. The value |Vus| Kℓ3 from K−→ π0 µ−νµ decays is taken from a 2021 update ‍[97]. The value |Vus| Kℓ2 from K−→ µ−νµ decays is taken from Ref. ‍[7].

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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 two |V_us| determinations that rely on exclusive τ branching fractions, we obtain:

Averaging the τ inclusive and exclusive |V_us| determinations, we obtain:

In calculating the averages, the correlation between f_K± and f_K±/f_π± is taken to be zero, in absence of public information. Taking it to be ± 100% varies the |V_us| central value by about 8% of its uncertainty and the |V_us| uncertainty by about 1% relative. From the purpose of estimating the correlations between δ R_{τ K}, δ R_τπ and δ R_{τ K/τπ} we use the information ‍[10] that the uncertainties on δ R_{τ K} and δ R_τπ are uncorrelated to a good approximation and that δ R_{τ K/τπ} = δ R_{τ K} − δ R_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 ‍[76, 98, 99].

Alternative determinations of |V_us| from B(τ → X_sν) ‍[85, 86], based on partially different sets of experimental inputs, report |V_us| values consistent with the unitarity determination.

Figure ‍1 reports our |V_us| determinations using the τ branching fractions, compared to two determinations based on kaon data ‍[7] and to the value obtained from |V_ud| with CKM-matrix unitarity ‍[7].