HFLAV $\tau$ branching fractions fit and measurements of Vus with $\tau$ lepton data

We report the status of the Heavy Flavour Averaging Group (HFLAV) averages of the $\tau$ lepton measurements We then update the latest published HFLAV global fit of the $\tau$ lepton branching fractions (Spring 2017) with recent results by BABAR. We use the fit results to update the Cabibbo-Kobayashi-Maskawa (CKM) matrix element Vus measurements with the $\tau$ branching fractions. We combine the direct $\tau$ branching fraction measurements with indirect predictions using kaon branching fractions measurements to improve the determination of Vus using $\tau$ branching fractions. The Vus determinations based on the inclusive branching fraction of $\tau$ to strange final states are about $3\sigma$ lower than the Vus determination from the CKM matrix unitarity.


Introduction
The τ subgroup of the Heavy Flavour Averaging Group (HFLAV) provides a global fit of the τ branching fractions, the lepton universality tests and the |V us | determination based on τ measurements. The latest published report for the τ lepton is labelled "Spring 2017" [1]. A version of the HFLAV τ branching fractions fit with unitarity constraint is published on the Review of Particle Physics [2] (RPP). There are additional minor differences between the two fits [1,3]. The fit results are used to test lepton universality and to compute |V us | [1].
The HFLAV-Tau group collects and combines also a list of upper limits set by searches of lepton-flavour-violating τ decays [1].
In the following, we update the HFLAV-Tau global fit inputs with two BABAR measurements that became public in 2018 [4,5] and we update the |V us | determinations based on τ data. The new results have a negligible effect on the lepton universality tests.
Finally, we add to the fit input measurements of three τ branching fractions that are indirectly determined using measurements of kaon branching fractions [6], in order to improve the precision on |V us |.

New τ branching fraction measurements
Since the last HFLAV report, BABAR published [4] a measurement of and presented [5] preliminary measurements of

|V us | determination including the 2018 BABAR results
We add the measurements listed in the previous section to the HFLAV-Tau global fit, removing a former BABAR measurement of B(τ − → K − π 0 ν τ ) [7] that has been superseded [5]. The new measurements of the branching fractions τ decaying to a kaon and 0, 1, 2, 3 π 0 's improve the experimental resolution on several modes that most contribute to the uncertainty on |V us |.
We compute also where f K± /f π± = 1.193 ± 0.003 from the FLAG 2016 Lattice averages with N f = 2 + 1 + 1 [10][11][12][13] (the same value persists in the FLAG 2017 web update). The radiative correction terms are [18][19][20]. The third value differs from the one quoted in the Spring 2017 HFLAV-Tau report [1], which incorrectly included a strong isospin-breaking correction that is not needed when using f K± /f π± rather than its isospin-limit variant. The other parameters are taken from the Review of Particle Physics (RPP) 2018 [2]. Averaging the two above |V us | determinations, we obtain |V us | τ = 0.2220 ± 0.0014.

τ branching fraction predictions from kaon measurements
Assuming the validity of the Standard Model (SM), three τ branching fractions have been computed using the precisely measured K 2 and K 3 branching fractions and the measured τ − → (Kπ) − ν τ spectra [6]: The uncertainties on the last two results are fully correlated. It has been observed [6,18] that all the above indirect values are higher than the corresponding directly measured τ branching fractions. If the indirect values replace the direct ones, |V us | = 0.2207 ± 0.027 [6]. We add the kaon-indirect determinations of the three above τ branching fractions to the data set used in the previous section in order to obtain improved calculations of |V us | τ s = 0.2202 ± 0.0018, |V us | τ K/π = 0.22546 ± 0.00097 and their average |V us | τ = 0.22439 ± 0.00088.

Consistency of |V us | with the CKM matrix unitarity
Assuming the CKM matrix unitarity, using |V ud | = 0.97420 ± 0.00021 [9] and |V ub | = (0.3940 ± 0.0360) · 10 −2 [2]. Table 1 summarizes the residuals, expressed as numbers of standard deviations, of the above mentioned |V us | determinations with respect to the |V us | computation from the CKM matrix unitarity. |V us | computed with the τ -inclusive method is significantly lower, but the significance of the discrepancy is mildly reduced alongside a mild progress in the experimental resolution.  Figure 1 reports the |V us | τ s determinations described above, a determination of |V us | τ s obtained replacing some τ branching fractions measurements with the indirect predictions based on kaon branching fractions [6], and other more complex determinations that use the τ spectral functions [21] and Lattice QCD techniques [22]. Updates on the last two determinations have been presented at the Tau 2018 workshop [23]. The last four determinations use an older and in some cases partial set of experimental τ branching fractions measurements. The τ based |V us | determinations use the |V ud | measurements as input. The dependence on |V ud | is however very small, and there is just a small correlation between |V us | and |V ud | when doing a simultaneous fit. Figure 2 shows the results of a |V ud |-|V us | simultaneous fit on the τ measurements corresponding to the HFLAV Spring 2017 fit and the BABAR 2018 results. The fit results are: |V ud | = 0.97420 ± 0.00021 |V us | = 0.2223 ± 0.0014 |V ud | -|V us | correlation = 0.035 Tables 2 and 3 report the contributions to the |V us | τ s uncertainty before and after the BABAR 2018 results. The largest contributions come from the τ branching fractions to strange final states and from the theory. The BABAR 2018 measurements reduced significantly several large contributions. High multiplicity τ decays to strange final states dominate the |V us | τ s uncertainty. The Belle II super flavour factory will offer the opportunity to improve the experimental precision on the τ strange branching fractions. More precise τ branching fractions and spectral function measurements will help improving also the theory uncertainty.  Figure 1: |V us | τ s determinations obtained in this document, from the top: |V us | uni , |V us | τ s with the HFLAV Spring 2017 fit, after adding the BABAR 2018 data, after adding both the BABAR 2018 and the kaon indirect predictions, from Ref. [6], from Ref. [21], and two determinations from Ref. [22]. : Results of a |V ud |-|V us | simultaneous fit. The bands describe the constraints corresponding to the |V ud | measurement, the |V us | τ s and the |V us | τ K/π determinations that use the τ measurements. The oblique line corresponds to the CKM matrix unitarity constraint. The ellipse corresponds to 1σ uncertainty on the |V ud | and |V us | fit results.