HFLAV-Tau Winter 2022 - Branching fraction fit

A fit of the available experimental measurements is used to determine the τ branching fractions, together with their uncertainties and correlations.

All relevant published statistical and systematic correlations among the measurements are used. In addition, for a selection of measurements, particularly the most precise and the most recent ones, the documented systematic uncertainty contributions are examined to consider systematic dependence on external parameters. We use the standard HFLAV procedures to account for the updated values and uncertainties of the external parameters and for the correlations induced on different measurements that have a systematic dependence on the same external parameter.

Both the measurements and the fitted quantities consist of either τ decay branching fractions, labelled B_i, or ratios of two τ decay branching fractions, labelled B_i/ B_j. Some branching fractions are sums of other branching fractions, for instance, B₈ = B(τ⁻→h⁻ ν_τ), which is the sum of B₉ = B(τ⁻→π⁻ ν_τ) and B₁₀ = B(τ⁻→K⁻ ν_τ), with the symbol h referring to a π or a K. The fit χ² is constructed following Eq. ‍(??) and minimized subject to a list of constraints on the fitted quantities:

The constraints are implemented with Lagrange multipliers (see Sec. ‍??. In some cases, constraints arise from approximate relations that nevertheless hold within the present experimental precision and are treated as exact. For instance, the constraint B(τ⁻→K⁻ K⁻ K⁺ ν_τ) = B(τ⁻→K⁻ φ ν_τ) × B(φ→ K⁺K⁻) is justified given the current experimental evidence. Section ‍2.6 lists all constraint equations relating the fitted quantities.

Following a convention established in the Review of Particle Physics, τ branching fractions are often labelled with the final state content of π^±, π⁰, K^±, γ, implicitly including decay chains that involve intermediate particles, e.g., K_S⁰→π⁺π⁻, and η, ω, φ decays. When measurements exclude the contribution of some or all the known intermediate particles, the branching fraction notation flags this information by adding, e.g., “ex.K⁰”.

2.1 Fit results

We use a total of 171 measurements to fit 135 quantities subject to 88 constraints. The fit has χ²/d.o.f. = 134/124, corresponding to a confidence level CL = 24.56%. The fitted quantity values and uncertainties are listed in Table ‍1. Although the fit treats all quantities in the same way, for the purpose of presenting correlations we select a set of 47 “basis quantities” from which all remaining quantities can be calculated using the definitions listed in Section ‍2.6. The off-diagonal correlation coefficients between the basis quantities are listed in Section ‍2.5.

Table ‍1 also reports B₁₁₀ = B(τ⁻ → X_s⁻ ν_τ) (see Section ‍2.6), the total measured branching fraction for τ decays to final states with strangeness 1. Also reported is the unitarity residual B₉₉₈ = 1 − B_All = (0.0684 ± 0.1068) · 10⁻², where 1 − B_All is the sum of the τ branching fractions into all measured final states. We find that B₉₉₈ is consistent with 0 to within the experimental uncertainty. A unitarity constraint forcing B₉₉₈ to be 0 is not applied.

In performing the fit, a scale factor of 5.44 was applied to the published uncertainties of the two severely inconsistent measurements of B₉₆ = τ → KKKν by BaBar and Belle. The scale-factor value was chosen using the PDG procedure, i.e., it is such that χ²/d.o.f.=1 when fitting just the two B₉₆ measurements.

2.2 Changes with respect to the previous report

In the previous HFLAV reports ‍[2, 3, 4, 5], information from the ALEPH Collaboration ‍[6] was used to compute inclusive τ branching fractions for final states with one or more hadron, where each hadron can be either a pion or a kaon. In the current report, we use Ref. ‍[6] for the branching fractions for exclusive final states containing one or more charged pions. The past choice granted some minor advantages, which are now dropped in the interest of simplicity. As a consequence, the τ branching fraction global fit reported here matches more closely the fit that we supply to the PDG, reported in Ref. ‍[7].

A set of preliminary BaBar results presented in 2018 ‍[8], used in the τ branching fraction fit in the previous HFLAV report ‍[5], are not used here since they have not yet been published. Therefore, we now use the older BaBar measurement of B₁₆ = B(τ⁻→K⁻ π⁰ ν_τ) in Ref. ‍[9]. This revision of input measurements causes a significant shift of the value of B(τ⁻→K⁻ 3π⁰ ν_τ (ex.K⁰,η)), which is however consistent with the large uncertainty on the sole direct measurement of this mode by ALEPH.

Since this edition, we use new improved calculations of the radiative corrections for the theory predictions of the τ decays to pseudoscalar mesons ‍[10]. The estimated uncertainties are increased but more reliable in comparison to the previous estimations ‍[11, 12, 13, 14]. There is a minor increase of the uncertainties on the lepton universality tests based on hadronic τ decays (section ‍3) and on |V_us| (section ‍5).

The parameters used to update the measurements’ systematic biases and the parameters appearing in the constraint equations in Section ‍2.6 have been updated to the PDG 2020 update ‍[7].

2.3 Differences between the HFLAV 2021 fit and the PDG 2021 fit

Our branching-fraction fit is different from that of the PDG ‍[7] in several ways.

The PDG fit enforces the unitarity constraint B₉₉₈=0, while the HFLAV 2021 fit does not.

As in our previous report ‍[5], we use the ALEPH ‍[7] estimate for B₈₀₅ = B(τ⁻→a₁⁻(π⁻ γ) ν_τ), which is not a direct measurement. By contrast, the PDG fit defines B₈₀₅ = B(a₁→πγ)× B(τ → 3π ν), using the PDG average of B(a₁→πγ) as a parameter in the fit. As a consequence, the PDG fit procedure does not take into account the large uncertainty on B(a₁→πγ). This results in an underestimated uncertainty on B₈₀₅, which is then properly adjusted with respect to the fit result in the PDG listings.

2.4 Branching fraction fit results and experimental inputs

Table ‍1 reports the experimental inputs to the τ branching-fraction fit and the fit results.

Table 1: HFLAV 2021 branching fractions fit results.

τ lepton branching fraction Experiment Reference

B₁ = particle⁻ ≥ 0  neutrals ≥ 0  K⁰  ν_τ

0.8518 ± 0.0011 average

B₂ = particle⁻ ≥ 0  neutrals ≥ 0  K_L⁰  ν_τ

0.8452 ± 0.0010 average

B₃ = µ⁻ ν_µν_τ

0.17387 ± 0.00040 average

0.17319 ± 0.00070 ± 0.00032 ALEPH [6]

0.17325 ± 0.00095 ± 0.00077 DELPHI [15]

0.17342 ± 0.00110 ± 0.00067 L3 [16]

0.17340 ± 0.00090 ± 0.00060 OPAL [17]

B₃

B₅

=
µ⁻ ν_µν_τ

e⁻ ν_e ν_τ

0.9762 ± 0.0028 average

0.9970 ± 0.0350 ± 0.0400 ARGUS [18]

0.9796 ± 0.0016 ± 0.0036 BaBar [19]

0.9777 ± 0.0063 ± 0.0087 CLEO [20]

B₅ = e⁻ ν_e ν_τ

0.17811 ± 0.00041 average

0.17837 ± 0.00072 ± 0.00036 ALEPH [6]

0.17760 ± 0.00060 ± 0.00170 CLEO [20]

0.17877 ± 0.00109 ± 0.00110 DELPHI [15]

0.17806 ± 0.00104 ± 0.00076 L3 [16]

0.17810 ± 0.00090 ± 0.00060 OPAL [21]

B₇ = h⁻ ≥ 0  K_L⁰  ν_τ

0.12020 ± 0.00055 average

0.12400 ± 0.00700 ± 0.00700 DELPHI [22]

0.12470 ± 0.00260 ± 0.00430 L3 [23]

0.12100 ± 0.00700 ± 0.00500 OPAL [24]

B₈ = h⁻ ν_τ

0.11504 ± 0.00054 average

0.11520 ± 0.00050 ± 0.00120 CLEO [20]

0.11571 ± 0.00120 ± 0.00114 DELPHI [25]

0.11980 ± 0.00130 ± 0.00160 OPAL [26]

B₈

B₅

=
h⁻ ν_τ

e⁻ ν_e ν_τ

0.6459 ± 0.0033 average

B₉ = π⁻ ν_τ

0.10808 ± 0.00053 average

0.10828 ± 0.00070 ± 0.00078 ALEPH [6]

B₉

B₅

=
π⁻ ν_τ

e⁻ ν_e ν_τ

0.6068 ± 0.0032 average

0.5945 ± 0.0014 ± 0.0061 BaBar [19]

B₁₀ = K⁻ ν_τ

(0.6957 ± 0.0096) · 10⁻² average

(0.6960 ± 0.0250 ± 0.0140) · 10⁻² ALEPH [27]

(0.6600 ± 0.0700 ± 0.0900) · 10⁻² CLEO [28]

(0.8500 ± 0.1800 ± 0.0000) · 10⁻² DELPHI [29]

(0.6580 ± 0.0270 ± 0.0290) · 10⁻² OPAL [30]

B₁₀

B₅

=
K⁻ ν_τ

e⁻ ν_e ν_τ

(3.906 ± 0.054) · 10⁻² average

(3.882 ± 0.032 ± 0.057) · 10⁻² BaBar [19]

B₁₀

B₉

=
K⁻ ν_τ

π⁻ ν_τ

(6.437 ± 0.092) · 10⁻² average

B₁₁ = h⁻ ≥ 1  neutrals  ν_τ

0.36977 ± 0.00098 average

B₁₂ = h⁻ ≥ 1  π⁰  ν_τ (ex.K⁰)

0.36477 ± 0.00098 average

B₁₃ = h⁻ π⁰ ν_τ

0.25918 ± 0.00090 average

0.25670 ± 0.00010 ± 0.00390 Belle [31]

0.25870 ± 0.00120 ± 0.00420 CLEO [32]

0.25740 ± 0.00201 ± 0.00138 DELPHI [25]

0.25050 ± 0.00350 ± 0.00500 L3 [23]

0.25890 ± 0.00170 ± 0.00290 OPAL [26]

B₁₄ = π⁻ π⁰ ν_τ

0.25486 ± 0.00090 average

0.25471 ± 0.00097 ± 0.00085 ALEPH [6]

B₁₆ = K⁻ π⁰ ν_τ

(0.4322 ± 0.0148) · 10⁻² average

(0.4440 ± 0.0260 ± 0.0240) · 10⁻² ALEPH [27]

(0.4160 ± 0.0030 ± 0.0180) · 10⁻² BaBar [9]

(0.5100 ± 0.1000 ± 0.0700) · 10⁻² CLEO [28]

(0.4710 ± 0.0590 ± 0.0230) · 10⁻² OPAL [33]

B₁₇ = h⁻ ≥ 2  π⁰  ν_τ

0.10794 ± 0.00097 average

0.09910 ± 0.00310 ± 0.00270 OPAL [26]

B₁₈ = h⁻ 2π⁰ ν_τ

(9.460 ± 0.100) · 10⁻² average

B₁₉ = h⁻ 2π⁰ ν_τ (ex.K⁰)

(9.309 ± 0.100) · 10⁻² average

(9.498 ± 0.320 ± 0.275) · 10⁻² DELPHI [25]

(8.880 ± 0.370 ± 0.420) · 10⁻² L3 [23]

B₁₉

B₁₃

=
h⁻ 2π⁰ ν_τ (ex.K⁰)

h⁻ π⁰ ν_τ

0.3592 ± 0.0045 average

0.3420 ± 0.0060 ± 0.0160 CLEO [34]

B₂₀ = π⁻ 2π⁰ ν_τ (ex.K⁰)

(9.245 ± 0.099) · 10⁻² average

(9.239 ± 0.086 ± 0.090) · 10⁻² ALEPH [6]

B₂₃ = K⁻ 2π⁰ ν_τ (ex.K⁰)

(0.0634 ± 0.0219) · 10⁻² average

(0.0560 ± 0.0200 ± 0.0150) · 10⁻² ALEPH [27]

(0.0900 ± 0.1000 ± 0.0300) · 10⁻² CLEO [28]

B₂₄ = h⁻ ≥ 3  π⁰  ν_τ

(1.335 ± 0.066) · 10⁻² average

B₂₅ = h⁻ ≥ 3  π⁰  ν_τ (ex.K⁰)

(1.250 ± 0.066) · 10⁻² average

(1.403 ± 0.214 ± 0.224) · 10⁻² DELPHI [25]

B₂₆ = h⁻ 3π⁰ ν_τ

(1.173 ± 0.072) · 10⁻² average

(1.700 ± 0.240 ± 0.380) · 10⁻² L3 [23]

B₂₆

B₁₃

=
h⁻ 3π⁰ ν_τ

h⁻ π⁰ ν_τ

(4.526 ± 0.278) · 10⁻² average

(4.400 ± 0.300 ± 0.500) · 10⁻² CLEO [34]

B₂₇ = π⁻ 3π⁰ ν_τ (ex.K⁰)

(1.040 ± 0.071) · 10⁻² average

(0.977 ± 0.069 ± 0.058) · 10⁻² ALEPH [6]

B₂₈ = K⁻ 3π⁰ ν_τ (ex.K⁰,η)

(4.648 ± 2.131) · 10⁻⁴ average

(3.700 ± 2.100 ± 1.100) · 10⁻⁴ ALEPH [27]

B₂₉ = h⁻ 4π⁰ ν_τ (ex.K⁰)

(0.1587 ± 0.0391) · 10⁻² average

(0.1600 ± 0.0500 ± 0.0500) · 10⁻² CLEO [34]

B₃₀ = h⁻ 4π⁰ ν_τ (ex.K⁰,η)

(0.1118 ± 0.0391) · 10⁻² average

(0.1120 ± 0.0370 ± 0.0350) · 10⁻² ALEPH [6]

B₃₁ = K⁻ ≥ 0  π⁰ ≥ 0  K⁰ ≥ 0  γ ν_τ

(1.548 ± 0.029) · 10⁻² average

(1.700 ± 0.120 ± 0.190) · 10⁻² CLEO [28]

(1.540 ± 0.240 ± 0.000) · 10⁻² DELPHI [29]

(1.528 ± 0.039 ± 0.040) · 10⁻² OPAL [30]

B₃₂ = K⁻ ≥ 1  (π⁰ or K⁰ or γ) ν_τ

(0.8556 ± 0.0282) · 10⁻² average

B₃₃ = K_S⁰ (particles)⁻ ν_τ

(0.9370 ± 0.0292) · 10⁻² average

(0.9700 ± 0.0580 ± 0.0620) · 10⁻² ALEPH [35]

(0.9700 ± 0.0900 ± 0.0600) · 10⁻² OPAL [36]

B₃₄ = h⁻ K⁰ ν_τ

(0.9861 ± 0.0138) · 10⁻² average

(0.8550 ± 0.0360 ± 0.0730) · 10⁻² CLEO [37]

B₃₅ = π⁻ K⁰ ν_τ

(0.8375 ± 0.0139) · 10⁻² average

(0.9280 ± 0.0450 ± 0.0340) · 10⁻² ALEPH [27]

(0.8320 ± 0.0025 ± 0.0150) · 10⁻² Belle [38]

(0.9500 ± 0.1500 ± 0.0600) · 10⁻² L3 [39]

(0.9330 ± 0.0680 ± 0.0490) · 10⁻² OPAL [40]

B₃₇ = K⁻ K⁰ ν_τ

(0.1486 ± 0.0034) · 10⁻² average

(0.1580 ± 0.0420 ± 0.0170) · 10⁻² ALEPH [35]

(0.1620 ± 0.0210 ± 0.0110) · 10⁻² ALEPH [27]

(0.1478 ± 0.0022 ± 0.0040) · 10⁻² BaBar [41]

(0.1480 ± 0.0013 ± 0.0055) · 10⁻² Belle [38]

(0.1510 ± 0.0210 ± 0.0220) · 10⁻² CLEO [37]

B₃₈ = K⁻ K⁰ ≥ 0  π⁰  ν_τ

(0.2985 ± 0.0073) · 10⁻² average

(0.3300 ± 0.0550 ± 0.0390) · 10⁻² OPAL [40]

B₃₉ = h⁻ K⁰ π⁰ ν_τ

(0.5310 ± 0.0134) · 10⁻² average

(0.5620 ± 0.0500 ± 0.0480) · 10⁻² CLEO [37]

B₄₀ = π⁻ K⁰ π⁰ ν_τ

(0.3810 ± 0.0129) · 10⁻² average

(0.2940 ± 0.0730 ± 0.0370) · 10⁻² ALEPH [35]

(0.3470 ± 0.0530 ± 0.0370) · 10⁻² ALEPH [27]

(0.3860 ± 0.0031 ± 0.0135) · 10⁻² Belle [38]

(0.4100 ± 0.1200 ± 0.0300) · 10⁻² L3 [39]

B₄₂ = K⁻ K⁰ π⁰ ν_τ

(0.1499 ± 0.0070) · 10⁻² average

(0.1520 ± 0.0760 ± 0.0210) · 10⁻² ALEPH [35]

(0.1430 ± 0.0250 ± 0.0150) · 10⁻² ALEPH [27]

(0.1496 ± 0.0019 ± 0.0073) · 10⁻² Belle [38]

(0.1450 ± 0.0360 ± 0.0200) · 10⁻² CLEO [37]

B₄₃ = π⁻ K⁰ ≥ 1  π⁰  ν_τ

(0.4045 ± 0.0260) · 10⁻² average

(0.3240 ± 0.0740 ± 0.0660) · 10⁻² OPAL [40]

B₄₄ = π⁻ K⁰ 2π⁰ ν_τ (ex.K⁰)

(2.342 ± 2.306) · 10⁻⁴ average

(2.600 ± 2.400 ± 0.000) · 10⁻⁴ ALEPH [42]

B₄₆ = π⁻ K⁰ K⁰ ν_τ

(0.1517 ± 0.0247) · 10⁻² average

B₄₇ = π⁻ K_S⁰ K_S⁰ ν_τ

(2.349 ± 0.065) · 10⁻⁴ average

(2.600 ± 1.000 ± 0.500) · 10⁻⁴ ALEPH [35]

(2.310 ± 0.040 ± 0.080) · 10⁻⁴ BaBar [43]

(2.330 ± 0.033 ± 0.093) · 10⁻⁴ Belle [38]

(2.300 ± 0.500 ± 0.300) · 10⁻⁴ CLEO [37]

B₄₈ = π⁻ K_S⁰ K_L⁰ ν_τ

(0.1048 ± 0.0247) · 10⁻² average

(0.1010 ± 0.0230 ± 0.0130) · 10⁻² ALEPH [35]

B₄₉ = π⁻ π⁰ K⁰ K⁰ ν_τ

(3.543 ± 1.193) · 10⁻⁴ average

B₅₀ = π⁻ K_S⁰ K_S⁰ π⁰ ν_τ

(1.820 ± 0.207) · 10⁻⁵ average

(1.600 ± 0.200 ± 0.220) · 10⁻⁵ BaBar [43]

(2.000 ± 0.216 ± 0.202) · 10⁻⁵ Belle [38]

B₅₁ = π⁻ K_S⁰ K_L⁰ π⁰ ν_τ

(3.179 ± 1.192) · 10⁻⁴ average

(3.100 ± 1.100 ± 0.500) · 10⁻⁴ ALEPH [35]

B₅₃ = K⁰ h⁻ h⁻ h⁺ ν_τ

(2.223 ± 2.024) · 10⁻⁴ average

(2.300 ± 1.900 ± 0.700) · 10⁻⁴ ALEPH [35]

B₅₄ = h⁻ h⁻ h⁺ ≥ 0  neutrals ≥ 0  K_L⁰  ν_τ

0.15193 ± 0.00063 average

0.15000 ± 0.00400 ± 0.00300 CELLO [44]

0.14400 ± 0.00600 ± 0.00300 L3 [45]

0.15100 ± 0.00800 ± 0.00600 TPC [46]

B₅₅ = h⁻ h⁻ h⁺ ≥ 0  neutrals  ν_τ (ex.K⁰)

0.14545 ± 0.00058 average

0.14556 ± 0.00105 ± 0.00076 L3 [47]

0.14960 ± 0.00090 ± 0.00220 OPAL [48]

B₅₆ = h⁻ h⁻ h⁺ ν_τ

(9.790 ± 0.055) · 10⁻² average

B₅₇ = h⁻ h⁻ h⁺ ν_τ (ex.K⁰)

(9.449 ± 0.054) · 10⁻² average

(9.510 ± 0.070 ± 0.200) · 10⁻² CLEO [49]

(9.317 ± 0.090 ± 0.082) · 10⁻² DELPHI [25]

B₅₇

B₅₅

=
h⁻ h⁻ h⁺ ν_τ (ex.K⁰)

h⁻ h⁻ h⁺ ≥ 0  neutrals  ν_τ (ex.K⁰)

0.6496 ± 0.0031 average

0.6600 ± 0.0040 ± 0.0140 OPAL [48]

B₅₈ = h⁻ h⁻ h⁺ ν_τ (ex.K⁰,ω)

(9.419 ± 0.054) · 10⁻² average

B₅₉ = π⁻ π⁺ π⁻ ν_τ

(9.300 ± 0.052) · 10⁻² average

B₆₀ = π⁻ π⁺ π⁻ ν_τ (ex.K⁰)

(9.010 ± 0.052) · 10⁻² average

(8.830 ± 0.010 ± 0.130) · 10⁻² BaBar [50]

(8.420 ± 0.000 _−0.250^+0.260) · 10⁻² Belle [51]

(9.130 ± 0.050 ± 0.460) · 10⁻² CLEO3 [52]

B₆₂ = π⁻ π⁺ π⁻ ν_τ (ex.K⁰,ω)

(8.981 ± 0.052) · 10⁻² average

(9.041 ± 0.060 ± 0.076) · 10⁻² ALEPH [6]

B₆₃ = h⁻ h⁻ h⁺ ≥ 1  neutrals  ν_τ

(5.293 ± 0.052) · 10⁻² average

B₆₄ = h⁻ h⁻ h⁺ ≥ 1  π⁰  ν_τ (ex.K⁰)

(5.088 ± 0.052) · 10⁻² average

B₆₅ = h⁻ h⁻ h⁺ π⁰ ν_τ

(4.757 ± 0.054) · 10⁻² average

B₆₆ = h⁻ h⁻ h⁺ π⁰ ν_τ (ex.K⁰)

(4.573 ± 0.054) · 10⁻² average

(4.230 ± 0.060 ± 0.220) · 10⁻² CLEO [49]

(4.545 ± 0.106 ± 0.103) · 10⁻² DELPHI [25]

B₆₇ = h⁻ h⁻ h⁺ π⁰ ν_τ (ex.K⁰,ω)

(2.791 ± 0.071) · 10⁻² average

B₆₈ = π⁻ π⁺ π⁻ π⁰ ν_τ

(4.620 ± 0.054) · 10⁻² average

B₆₉ = π⁻ π⁺ π⁻ π⁰ ν_τ (ex.K⁰)

(4.488 ± 0.054) · 10⁻² average

(4.598 ± 0.057 ± 0.064) · 10⁻² ALEPH [6]

(4.190 ± 0.100 ± 0.210) · 10⁻² CLEO [53]

B₇₀ = π⁻ π⁺ π⁻ π⁰ ν_τ (ex.K⁰,ω)

(2.743 ± 0.071) · 10⁻² average

B₇₄ = h⁻ h⁻ h⁺ ≥ 2  π⁰  ν_τ (ex.K⁰)

(0.5154 ± 0.0312) · 10⁻² average

(0.5610 ± 0.0680 ± 0.0950) · 10⁻² DELPHI [25]

B₇₅ = h⁻ h⁻ h⁺ 2π⁰ ν_τ

(0.5041 ± 0.0311) · 10⁻² average

B₇₆ = h⁻ h⁻ h⁺ 2π⁰ ν_τ (ex.K⁰)

(0.4942 ± 0.0311) · 10⁻² average

(0.4350 ± 0.0300 ± 0.0350) · 10⁻² ALEPH [6]

B₇₆

B₅₄

=
h⁻ h⁻ h⁺ 2π⁰ ν_τ (ex.K⁰)

h⁻ h⁻ h⁺ ≥ 0  neutrals ≥ 0  K_L⁰  ν_τ

(3.252 ± 0.203) · 10⁻² average

(3.400 ± 0.200 ± 0.300) · 10⁻² CLEO [54]

B₇₇ = h⁻ h⁻ h⁺ 2π⁰ ν_τ (ex.K⁰,ω,η)

(9.790 ± 3.562) · 10⁻⁴ average

B₇₈ = h⁻ h⁻ h⁺ 3π⁰ ν_τ

(2.124 ± 0.299) · 10⁻⁴ average

(2.200 ± 0.300 ± 0.400) · 10⁻⁴ CLEO [55]

B₇₉ = K⁻ h⁻ h⁺ ≥ 0  neutrals  ν_τ

(0.6276 ± 0.0140) · 10⁻² average

B₈₀ = K⁻ π⁻ h⁺ ν_τ (ex.K⁰)

(0.4364 ± 0.0073) · 10⁻² average

B₈₀

B₆₀

=
K⁻ π⁻ h⁺ ν_τ (ex.K⁰)

π⁻ π⁺ π⁻ ν_τ (ex.K⁰)

(4.843 ± 0.079) · 10⁻² average

(5.440 ± 0.210 ± 0.530) · 10⁻² CLEO [56]

B₈₁ = K⁻ π⁻ h⁺ π⁰ ν_τ (ex.K⁰)

(8.498 ± 1.169) · 10⁻⁴ average

B₈₁

B₆₉

=
K⁻ π⁻ h⁺ π⁰ ν_τ (ex.K⁰)

π⁻ π⁺ π⁻ π⁰ ν_τ (ex.K⁰)

(1.893 ± 0.264) · 10⁻² average

(2.610 ± 0.450 ± 0.420) · 10⁻² CLEO [56]

B₈₂ = K⁻ π⁻ π⁺ ≥ 0  neutrals  ν_τ

(0.4759 ± 0.0136) · 10⁻² average

(0.5800 _−0.1300^+0.1500 ± 0.1200) · 10⁻² TPC [57]

B₈₃ = K⁻ π⁻ π⁺ ≥ 0  π⁰  ν_τ (ex.K⁰)

(0.3719 ± 0.0134) · 10⁻² average

B₈₄ = K⁻ π⁻ π⁺ ν_τ

(0.3444 ± 0.0069) · 10⁻² average

B₈₅ = K⁻ π⁺ π⁻ ν_τ (ex.K⁰)

(0.2930 ± 0.0068) · 10⁻² average

(0.2140 ± 0.0370 ± 0.0290) · 10⁻² ALEPH [58]

(0.2730 ± 0.0020 ± 0.0090) · 10⁻² BaBar [50]

(0.3300 ± 0.0010 _−0.0170^+0.0160) · 10⁻² Belle [51]

(0.3840 ± 0.0140 ± 0.0380) · 10⁻² CLEO3 [52]

(0.4150 ± 0.0530 ± 0.0400) · 10⁻² OPAL [33]

B₈₅

B₆₀

=
K⁻ π⁺ π⁻ ν_τ (ex.K⁰)

π⁻ π⁺ π⁻ ν_τ (ex.K⁰)

(3.252 ± 0.074) · 10⁻² average

B₈₇ = K⁻ π⁻ π⁺ π⁰ ν_τ

(0.1308 ± 0.0119) · 10⁻² average

B₈₈ = K⁻ π⁻ π⁺ π⁰ ν_τ (ex.K⁰)

(7.891 ± 1.161) · 10⁻⁴ average

(6.100 ± 3.900 ± 1.800) · 10⁻⁴ ALEPH [58]

(7.400 ± 0.800 ± 1.100) · 10⁻⁴ CLEO3 [59]

B₈₉ = K⁻ π⁻ π⁺ π⁰ ν_τ (ex.K⁰,η)

(7.536 ± 1.161) · 10⁻⁴ average

B₉₂ = π⁻ K⁻ K⁺ ≥ 0  neutrals  ν_τ

(0.1495 ± 0.0033) · 10⁻² average

(0.1590 ± 0.0530 ± 0.0200) · 10⁻² OPAL [60]

(0.1500 _−0.0700^+0.0900 ± 0.0300) · 10⁻² TPC [57]

B₉₃ = π⁻ K⁻ K⁺ ν_τ

(0.1434 ± 0.0027) · 10⁻² average

(0.1630 ± 0.0210 ± 0.0170) · 10⁻² ALEPH [58]

(0.1346 ± 0.0010 ± 0.0036) · 10⁻² BaBar [50]

(0.1550 ± 0.0010 _−0.0050^+0.0060) · 10⁻² Belle [51]

(0.1550 ± 0.0060 ± 0.0090) · 10⁻² CLEO3 [52]

B₉₃

B₆₀

=
π⁻ K⁻ K⁺ ν_τ

π⁻ π⁺ π⁻ ν_τ (ex.K⁰)

(1.592 ± 0.030) · 10⁻² average

(1.600 ± 0.150 ± 0.300) · 10⁻² CLEO [56]

B₉₄ = π⁻ K⁻ K⁺ π⁰ ν_τ

(0.607 ± 0.183) · 10⁻⁴ average

(7.500 ± 2.900 ± 1.500) · 10⁻⁴ ALEPH [58]

(0.550 ± 0.140 ± 0.120) · 10⁻⁴ CLEO3 [59]

B₉₄

B₆₉

=
π⁻ K⁻ K⁺ π⁰ ν_τ

π⁻ π⁺ π⁻ π⁰ ν_τ (ex.K⁰)

(0.1352 ± 0.0408) · 10⁻² average

(0.7900 ± 0.4400 ± 0.1600) · 10⁻² CLEO [56]

B₉₆ = K⁻ K⁻ K⁺ ν_τ

(2.169 ± 0.800) · 10⁻⁵ average

(1.578 ± 0.130 ± 0.123) · 10⁻⁵ BaBar [50]

(3.290 ± 0.170 _−0.200^+0.190) · 10⁻⁵ Belle [51]

B₁₀₂ = 3h⁻ 2h⁺ ≥ 0  neutrals  ν_τ (ex.K⁰)

(0.0993 ± 0.0037) · 10⁻² average

(0.0970 ± 0.0050 ± 0.0110) · 10⁻² CLEO [61]

(0.1020 ± 0.0290 ± 0.0000) · 10⁻² HRS [62]

(0.1700 ± 0.0220 ± 0.0260) · 10⁻² L3 [47]

B₁₀₃ = 3h⁻ 2h⁺ ν_τ (ex.K⁰)

(8.281 ± 0.314) · 10⁻⁴ average

(7.200 ± 0.900 ± 1.200) · 10⁻⁴ ALEPH [6]

(6.400 ± 2.300 ± 1.000) · 10⁻⁴ ARGUS [63]

(7.700 ± 0.500 ± 0.900) · 10⁻⁴ CLEO [61]

(9.700 ± 1.500 ± 0.500) · 10⁻⁴ DELPHI [25]

(5.100 ± 2.000 ± 0.000) · 10⁻⁴ HRS [62]

(9.100 ± 1.400 ± 0.600) · 10⁻⁴ OPAL [64]

B₁₀₄ = 3h⁻ 2h⁺ π⁰ ν_τ (ex.K⁰)

(1.645 ± 0.114) · 10⁻⁴ average

(2.100 ± 0.700 ± 0.900) · 10⁻⁴ ALEPH [6]

(1.700 ± 0.200 ± 0.200) · 10⁻⁴ CLEO [55]

(1.600 ± 1.200 ± 0.600) · 10⁻⁴ DELPHI [25]

(2.700 ± 1.800 ± 0.900) · 10⁻⁴ OPAL [64]

B₁₀₆ = (5π)⁻ ν_τ

(0.7793 ± 0.0534) · 10⁻² average

B₁₁₀ = X_s⁻ ν_τ

(2.908 ± 0.048) · 10⁻² average

B₁₂₆ = π⁻ π⁰ η ν_τ

(0.1386 ± 0.0072) · 10⁻² average

(0.1800 ± 0.0400 ± 0.0200) · 10⁻² ALEPH [65]

(0.1350 ± 0.0030 ± 0.0070) · 10⁻² Belle [66]

(0.1700 ± 0.0200 ± 0.0200) · 10⁻² CLEO [67]

B₁₂₈ = K⁻ η ν_τ

(1.547 ± 0.080) · 10⁻⁴ average

(2.900 _−1.200^+1.300 ± 0.700) · 10⁻⁴ ALEPH [65]

(1.420 ± 0.110 ± 0.070) · 10⁻⁴ BaBar [68]

(1.580 ± 0.050 ± 0.090) · 10⁻⁴ Belle [66]

(2.600 ± 0.500 ± 0.500) · 10⁻⁴ CLEO [69]

B₁₃₀ = K⁻ π⁰ η ν_τ

(0.483 ± 0.116) · 10⁻⁴ average

(0.460 ± 0.110 ± 0.040) · 10⁻⁴ Belle [66]

(1.770 ± 0.560 ± 0.710) · 10⁻⁴ CLEO [70]

B₁₃₂ = π⁻ K⁰ η ν_τ

(0.937 ± 0.149) · 10⁻⁴ average

(0.880 ± 0.140 ± 0.060) · 10⁻⁴ Belle [66]

(2.200 ± 0.700 ± 0.220) · 10⁻⁴ CLEO [70]

B₁₃₆ = π⁻ π⁺ π⁻ η ν_τ (ex.K⁰)

(2.202 ± 0.129) · 10⁻⁴ average

B₁₄₉ = h⁻ ω ≥ 0  neutrals  ν_τ

(2.395 ± 0.075) · 10⁻² average

B₁₅₀ = h⁻ ω ν_τ

(1.988 ± 0.064) · 10⁻² average

(1.910 ± 0.070 ± 0.060) · 10⁻² ALEPH [65]

(1.600 ± 0.270 ± 0.410) · 10⁻² CLEO [71]

B₁₅₀

B₆₆

=
h⁻ ω ν_τ

h⁻ h⁻ h⁺ π⁰ ν_τ (ex.K⁰)

0.4348 ± 0.0140 average

0.4310 ± 0.0330 ± 0.0000 ALEPH [72]

0.4640 ± 0.0160 ± 0.0170 CLEO [49]

B₁₅₁ = K⁻ ω ν_τ

(4.101 ± 0.922) · 10⁻⁴ average

(4.100 ± 0.600 ± 0.700) · 10⁻⁴ CLEO3 [59]

B₁₅₂ = h⁻ π⁰ ω ν_τ

(0.4069 ± 0.0419) · 10⁻² average

(0.4300 ± 0.0600 ± 0.0500) · 10⁻² ALEPH [65]

B₁₅₂

B₅₄

=
h⁻ ω π⁰ ν_τ

h⁻ h⁻ h⁺ ≥ 0  neutrals ≥ 0  K_L⁰  ν_τ

(2.678 ± 0.275) · 10⁻² average

B₁₅₂

B₇₆

=
h⁻ ω π⁰ ν_τ

h⁻ h⁻ h⁺ 2π⁰ ν_τ (ex.K⁰)

0.8235 ± 0.0757 average

0.8100 ± 0.0600 ± 0.0600 CLEO [54]

B₁₆₇ = K⁻ φ ν_τ

(4.408 ± 1.626) · 10⁻⁵ average

B₁₆₈ = K⁻ φ(K⁺ K⁻) ν_τ

(2.169 ± 0.800) · 10⁻⁵ average

B₁₆₉ = K⁻ φ(K_S⁰ K_L⁰) ν_τ

(1.499 ± 0.553) · 10⁻⁵ average

B₈₀₀ = π⁻ ω ν_τ

(1.947 ± 0.065) · 10⁻² average

B₈₀₂ = K⁻ π⁻ π⁺ ν_τ (ex.K⁰,ω)

(0.2924 ± 0.0068) · 10⁻² average

B₈₀₃ = K⁻ π⁻ π⁺ π⁰ ν_τ (ex.K⁰,ω,η)

(3.874 ± 1.423) · 10⁻⁴ average

B₈₀₄ = π⁻ K_L⁰ K_L⁰ ν_τ

(2.349 ± 0.065) · 10⁻⁴ average

B₈₀₅ = a₁⁻(π⁻ γ) ν_τ

(4.000 ± 2.000) · 10⁻⁴ average

(4.000 ± 2.000 ± 0.000) · 10⁻⁴ ALEPH [6]

B₈₀₆ = π⁻ K_L⁰ K_L⁰ π⁰ ν_τ

(1.820 ± 0.207) · 10⁻⁵ average

B₈₁₀ = 2π⁻ π⁺ 3π⁰ ν_τ (ex.K⁰)

(1.940 ± 0.298) · 10⁻⁴ average

B₈₁₁ = π⁻ 2π⁰ ω ν_τ

(7.164 ± 1.586) · 10⁻⁵ average

(7.300 ± 1.200 ± 1.200) · 10⁻⁵ BaBar [73]

B₈₁₂ = 2π⁻ π⁺ 3π⁰ ν_τ (ex.K⁰,η,ω,f₁)

(1.353 ± 2.683) · 10⁻⁵ average

(1.000 ± 0.800 ± 3.000) · 10⁻⁵ BaBar [73]

B₈₂₀ = 3π⁻ 2π⁺ ν_τ (ex.K⁰,ω)

(8.262 ± 0.313) · 10⁻⁴ average

B₈₂₁ = 3π⁻ 2π⁺ ν_τ (ex.K⁰,ω,f₁)

(7.738 ± 0.295) · 10⁻⁴ average

(7.680 ± 0.040 ± 0.400) · 10⁻⁴ BaBar [73]

B₈₂₂ = K⁻ 2π⁻ 2π⁺ ν_τ (ex.K⁰)

(0.593 ± 1.208) · 10⁻⁶ average

(0.600 ± 0.500 ± 1.100) · 10⁻⁶ BaBar [73]

B₈₃₀ = 3π⁻ 2π⁺ π⁰ ν_τ (ex.K⁰)

(1.633 ± 0.113) · 10⁻⁴ average

B₈₃₁ = 2π⁻ π⁺ ω ν_τ (ex.K⁰)

(8.417 ± 0.624) · 10⁻⁵ average

(8.400 ± 0.400 ± 0.600) · 10⁻⁵ BaBar [73]

B₈₃₂ = 3π⁻ 2π⁺ π⁰ ν_τ (ex.K⁰,η,ω,f₁)

(3.772 ± 0.874) · 10⁻⁵ average

(3.600 ± 0.300 ± 0.900) · 10⁻⁵ BaBar [73]

B₈₃₃ = K⁻ 2π⁻ 2π⁺ π⁰ ν_τ (ex.K⁰)

(1.107 ± 0.566) · 10⁻⁶ average

(1.100 ± 0.400 ± 0.400) · 10⁻⁶ BaBar [73]

B₉₁₀ = 2π⁻ π⁺ η(3π⁰) ν_τ (ex.K⁰)

(7.195 ± 0.422) · 10⁻⁵ average

(8.270 ± 0.880 ± 0.810) · 10⁻⁵ BaBar [73]

B₉₁₁ = π⁻ 2π⁰ η(π⁺ π⁻ π⁰) ν_τ (ex.K⁰)

(4.457 ± 0.867) · 10⁻⁵ average

(4.570 ± 0.770 ± 0.500) · 10⁻⁵ BaBar [73]

B₉₂₀ = π⁻ f₁(2π⁻ 2π⁺) ν_τ

(5.237 ± 0.444) · 10⁻⁵ average

(5.200 ± 0.310 ± 0.370) · 10⁻⁵ BaBar [73]

B₉₃₀ = 2π⁻ π⁺ η(π⁺ π⁻ π⁰) ν_τ (ex.K⁰)

(5.046 ± 0.296) · 10⁻⁵ average

(5.390 ± 0.270 ± 0.410) · 10⁻⁵ BaBar [73]

B₉₄₄ = 2π⁻ π⁺ η(γ γ) ν_τ (ex.K⁰)

(8.676 ± 0.509) · 10⁻⁵ average

(8.260 ± 0.350 ± 0.510) · 10⁻⁵ BaBar [73]

B₉₄₅ = π⁻ 2π⁰ η ν_τ (ex.K⁰)

(1.945 ± 0.378) · 10⁻⁴ average

B₉₉₈ = 1 − B_All

(0.0684 ± 0.1068) · 10⁻² average

2.5 Correlation coefficients between basis branching fractions uncertainties

The following tables report the correlation coefficients between basis quantities that were obtained from the τ branching fractions fit, in percent.

2.6 Equality constraints

The constraints on the τ branching-fractions fit quantities are listed in the following equations. When a quantity such as B₃/ B₅ appears on the left side of the equation it represents a fitted quantity, while when it appears on the right side it represents the ratio of two separate fitted quantities.

The equations include coefficients that arise from non-τ branching fractions, denoted, e.g., with the self-describing notation B_{K_S → π⁰π⁰}. Some coefficients are probabilities corresponding to the squared moduli of amplitudes describing quantum state mixtures, such as K⁰, K⁰, K_S, K_L. These are denoted with, e.g., B_{<K⁰|K_S>} = |<K⁰|K_S>|². The values of all non-τ quantities are taken from the PDG 2021 ‍[7] averages. The fit procedure does not account for their uncertainties, which are generally small with respect to the uncertainties on the τ branching fractions.

B₁ =	B₃ + B₅ + B₉ + B₁₀ + B₁₄ + B₁₆
	+ B₂₀ + B₂₃ + B₂₇ + B₂₈ + B₃₀ + B₃₅
	+ B₄₀ + B₄₄ + B₃₇ + B₄₂ + B₄₇ + B₄₈
	+ B₈₀₄ + B₅₀ + B₅₁ + B₈₀₆ + B₁₂₆· B_η→neutral
	+ B₁₂₈· B_η→neutral + B₁₃₀· B_η→neutral + B₁₃₂· B_η→neutral
	+ B₈₀₀· B_ω→π⁰γ + B₁₅₁· B_ω→π⁰γ + B₁₅₂· B_ω→π⁰γ
	+ B₁₆₇· B_{φ→ K_S K_L}

B₂ =	B₃ + B₅ + B₉ + B₁₀ + B₁₄ + B₁₆
	+ B₂₀ + B₂₃ + B₂₇ + B₂₈ + B₃₀ + B₃₅·( B_{<K⁰\|K_S>}· B_{K_S→π⁰π⁰}
	+ B_{<K⁰\|K_L>}) + B₄₀·( B_{<K⁰\|K_S>}· B_{K_S→π⁰π⁰}+ B_{<K⁰\|K_L>}) + B₄₄·( B_{<K⁰\|K_S>}· B_{K_S→π⁰π⁰}
	+ B_{<K⁰\|K_L>}) + B₃₇·( B_{<K⁰\|K_S>}· B_{K_S→π⁰π⁰}+ B_{<K⁰\|K_L>}) + B₄₂·( B_{<K⁰\|K_S>}· B_{K_S→π⁰π⁰}
	+ B_{<K⁰\|K_L>}) + B₄₇·( B_{K_S→π⁰π⁰}· B_{K_S→π⁰π⁰}) + B₄₈· B_{K_S→π⁰π⁰}
	+ B₈₀₄ + B₅₀·( B_{K_S→π⁰π⁰}· B_{K_S→π⁰π⁰}) + B₅₁· B_{K_S→π⁰π⁰}
	+ B₈₀₆ + B₁₂₆· B_η→neutral + B₁₂₈· B_η→neutral + B₁₃₀· B_η→neutral
	+ B₁₃₂·( B_η→neutral·( B_{<K⁰\|K_S>}· B_{K_S→π⁰π⁰}+ B_{<K⁰\|K_L>})) + B₈₀₀· B_ω→π⁰γ
	+ B₁₅₁· B_ω→π⁰γ + B₁₅₂· B_ω→π⁰γ + B₁₆₇·( B_{φ→ K_S K_L}· B_{K_S→π⁰π⁰})

B₃

B₅

B₃

B₅

B₇ =	B₃₅· B_{<K⁰\|K_L>} + B₉ + B₈₀₄ + B₃₇· B_{<K⁰\|K_L>}
	+ B₁₀

B₈ =

B₉ + B₁₀

B₈

B₅

B₈

B₅

B₉

B₅

B₉

B₅

B₁₀

B₅

B₁₀

B₅

B₁₀

B₉

B₁₀

B₉

B₁₁ =	B₁₄ + B₁₆ + B₂₀ + B₂₃ + B₂₇ + B₂₈
	+ B₃₀ + B₃₅·( B_{<K⁰\|K_S>}· B_{K_S→π⁰π⁰}) + B₃₇·( B_{<K⁰\|K_S>}· B_{K_S→π⁰π⁰})
	+ B₄₀·( B_{<K⁰\|K_S>}· B_{K_S→π⁰π⁰}) + B₄₂·( B_{<K⁰\|K_S>}· B_{K_S→π⁰π⁰})
	+ B₄₇·( B_{K_S→π⁰π⁰}· B_{K_S→π⁰π⁰}) + B₅₀·( B_{K_S→π⁰π⁰}· B_{K_S→π⁰π⁰})
	+ B₁₂₆· B_η→neutral + B₁₂₈· B_η→neutral + B₁₃₀· B_η→neutral
	+ B₁₃₂·( B_{<K⁰\|K_S>}· B_{K_S→π⁰π⁰}· B_η→neutral) + B₁₅₁· B_ω→π⁰γ
	+ B₁₅₂· B_ω→π⁰γ + B₈₀₀· B_ω→π⁰γ

B₁₂ =	B₁₂₈· B_η→3π⁰ + B₃₀ + B₂₃ + B₂₈ + B₁₄
	+ B₁₆ + B₂₀ + B₂₇ + B₁₂₆· B_η→3π⁰ + B₁₃₀· B_η→3π⁰

B₁₃ =

B₁₄ + B₁₆

B₁₇ =	B₁₂₈· B_η→3π⁰ + B₃₀ + B₂₃ + B₂₈ + B₃₅·( B_{<K⁰\|K_S>}· B_{K_S→π⁰π⁰})
	+ B₄₀·( B_{<K⁰\|K_S>}· B_{K_S→π⁰π⁰}) + B₄₂·( B_{<K⁰\|K_S>}· B_{K_S→π⁰π⁰})
	+ B₂₀ + B₂₇ + B₄₇·( B_{K_S→π⁰π⁰}· B_{K_S→π⁰π⁰}) + B₅₀·( B_{K_S→π⁰π⁰}· B_{K_S→π⁰π⁰})
	+ B₁₂₆· B_η→3π⁰ + B₃₇·( B_{<K⁰\|K_S>}· B_{K_S→π⁰π⁰}) + B₁₃₀· B_η→3π⁰

B₁₈ =

B₂₃ + B₃₅·( B_{<K⁰|K_S>}· B_{K_S→π⁰π⁰}) + B₂₀ + B₃₇·( B_{<K⁰|K_S>}· B_{K_S→π⁰π⁰})

B₁₉ =

B₂₃ + B₂₀

B₁₉

B₁₃

B₁₉

B₁₃

B₂₄ =	B₂₇ + B₂₈ + B₃₀ + B₄₀·( B_{<K⁰\|K_S>}· B_{K_S→π⁰π⁰})
	+ B₄₂·( B_{<K⁰\|K_S>}· B_{K_S→π⁰π⁰}) + B₄₇·( B_{K_S→π⁰π⁰}· B_{K_S→π⁰π⁰})
	+ B₅₀·( B_{K_S→π⁰π⁰}· B_{K_S→π⁰π⁰}) + B₁₂₆· B_η→3π⁰ + B₁₂₈· B_η→3π⁰
	+ B₁₃₀· B_η→3π⁰ + B₁₃₂·( B_{<K⁰\|K_S>}· B_{K_S→π⁰π⁰}· B_η→3π⁰)

B₂₅ =	B₁₂₈· B_η→3π⁰ + B₃₀ + B₂₈ + B₂₇ + B₁₂₆· B_η→3π⁰
	+ B₁₃₀· B_η→3π⁰

B₂₆ =	B₁₂₈· B_η→3π⁰ + B₂₈ + B₄₀·( B_{<K⁰\|K_S>}· B_{K_S→π⁰π⁰})
	+ B₄₂·( B_{<K⁰\|K_S>}· B_{K_S→π⁰π⁰}) + B₂₇

B₂₆

B₁₃

B₂₆

B₁₃

B₂₉ =

B₃₀ + B₁₂₆· B_η→3π⁰ + B₁₃₀· B_η→3π⁰

B₃₁ =	B₁₂₈· B_η→neutral + B₂₃ + B₂₈ + B₄₂ + B₁₆
	+ B₃₇ + B₁₀ + B₁₆₇·( B_{φ→ K_S K_L}· B_{K_S→π⁰π⁰})

B₃₂ =	B₁₆ + B₂₃ + B₂₈ + B₃₇ + B₄₂ + B₁₂₈· B_η→neutral
	+ B₁₃₀· B_η→neutral + B₁₆₇·( B_{φ→ K_S K_L}· B_{K_S→π⁰π⁰})

B₃₃ =	B₃₅· B_{<K⁰\|K_S>} + B₄₀· B_{<K⁰\|K_S>} + B₄₂· B_{<K⁰\|K_S>}
	+ B₄₇ + B₄₈ + B₅₀ + B₅₁ + B₃₇· B_{<K⁰\|K_S>}
	+ B₁₃₂·( B_{<K⁰\|K_S>}· B_η→neutral) + B₄₄· B_{<K⁰\|K_S>} + B₁₆₇· B_{φ→ K_S K_L}

B₃₄ =

B₃₅ + B₃₇

B₃₈ =

B₄₂ + B₃₇

B₃₉ =

B₄₀ + B₄₂

B₄₃ =

B₄₀ + B₄₄

B₄₆ =

B₄₈ + B₄₇ + B₈₀₄

B₄₉ =

B₅₀ + B₅₁ + B₈₀₆

B₅₄ =	B₃₅·( B_{<K⁰\|K_S>}· B_{K_S→π⁺π⁻}) + B₃₇·( B_{<K⁰\|K_S>}· B_{K_S→π⁺π⁻})
	+ B₄₀·( B_{<K⁰\|K_S>}· B_{K_S→π⁺π⁻}) + B₄₂·( B_{<K⁰\|K_S>}· B_{K_S→π⁺π⁻})
	+ B₄₇·(2· B_{K_S→π⁺π⁻}· B_{K_S→π⁰π⁰}) + B₄₈· B_{K_S→π⁺π⁻}
	+ B₅₀·(2· B_{K_S→π⁺π⁻}· B_{K_S→π⁰π⁰}) + B₅₁· B_{K_S→π⁺π⁻}
	+ B₅₃·( B_{<K⁰\|K_S>}· B_{K_S→π⁰π⁰}+ B_{<K⁰\|K_L>}) + B₆₂ + B₇₀
	+ B₇₇ + B₇₈ + B₉₃ + B₉₄ + B₁₂₆· B_η→charged
	+ B₁₂₈· B_η→charged + B₁₃₀· B_η→charged + B₁₃₂·( B_{<K⁰\|K_L>}· B_{η→π⁺π⁻π⁰}
	+ B_{<K⁰\|K_S>}· B_{K_S→π⁰π⁰}· B_{η→π⁺π⁻π⁰} + B_{<K⁰\|K_S>}· B_{K_S→π⁺π⁻}· B_η→3π⁰)
	+ B₁₅₁·( B_{ω→π⁺π⁻π⁰}+ B_{ω→π⁺π⁻}) + B₁₅₂·( B_{ω→π⁺π⁻π⁰}+ B_{ω→π⁺π⁻})
	+ B₁₆₇·( B_{φ→ K⁺K⁻} + B_{φ→ K_S K_L}· B_{K_S→π⁺π⁻}) + B₈₀₂ + B₈₀₃
	+ B₈₀₀·( B_{ω→π⁺π⁻π⁰}+ B_{ω→π⁺π⁻})

B₅₅ =	B₁₂₈· B_η→charged + B₁₅₂·( B_{ω→π⁺π⁻π⁰}+ B_{ω→π⁺π⁻}) + B₇₈
	+ B₇₇ + B₉₄ + B₆₂ + B₇₀ + B₉₃ + B₁₂₆· B_η→charged
	+ B₈₀₂ + B₈₀₃ + B₈₀₀·( B_{ω→π⁺π⁻π⁰}+ B_{ω→π⁺π⁻}) + B₁₅₁·( B_{ω→π⁺π⁻π⁰}
	+ B_{ω→π⁺π⁻}) + B₁₃₀· B_η→charged + B₁₆₈

B₅₆ =	B₃₅·( B_{<K⁰\|K_S>}· B_{K_S→π⁺π⁻}) + B₆₂ + B₉₃ + B₃₇·( B_{<K⁰\|K_S>}· B_{K_S→π⁺π⁻})
	+ B₈₀₂ + B₈₀₀· B_{ω→π⁺π⁻} + B₁₅₁· B_{ω→π⁺π⁻} + B₁₆₈

B₅₇ =	B₆₂ + B₉₃ + B₈₀₂ + B₈₀₀· B_{ω→π⁺π⁻} + B₁₅₁· B_{ω→π⁺π⁻}
	+ B₁₆₇· B_{φ→ K⁺K⁻}

B₅₇

B₅₅

B₅₇

B₅₅

B₅₈ =

B₆₂ + B₉₃ + B₈₀₂ + B₁₆₇· B_{φ→ K⁺K⁻}

B₅₉ =

B₃₅·( B_{<K⁰|K_S>}· B_{K_S→π⁺π⁻}) + B₆₂ + B₈₀₀· B_{ω→π⁺π⁻}

B₆₀ =

B₆₂ + B₈₀₀· B_{ω→π⁺π⁻}

B₆₃ =	B₄₀·( B_{<K⁰\|K_S>}· B_{K_S→π⁺π⁻}) + B₄₂·( B_{<K⁰\|K_S>}· B_{K_S→π⁺π⁻})
	+ B₄₇·(2· B_{K_S→π⁺π⁻}· B_{K_S→π⁰π⁰}) + B₅₀·(2· B_{K_S→π⁺π⁻}· B_{K_S→π⁰π⁰})
	+ B₇₀ + B₇₇ + B₇₈ + B₉₄ + B₁₂₆· B_η→charged
	+ B₁₂₈· B_η→charged + B₁₃₀· B_η→charged + B₁₃₂·( B_{<K⁰\|K_S>}· B_{K_S→π⁺π⁻}· B_η→neutral
	+ B_{<K⁰\|K_S>}· B_{K_S→π⁰π⁰}· B_η→charged) + B₁₅₁· B_{ω→π⁺π⁻π⁰} + B₁₅₂·( B_{ω→π⁺π⁻π⁰}
	+ B_{ω→π⁺π⁻}) + B₈₀₀· B_{ω→π⁺π⁻π⁰} + B₈₀₃

B₆₄ =	B₇₈ + B₇₇ + B₉₄ + B₇₀ + B₁₂₆· B_{η→π⁺π⁻π⁰}
	+ B₁₂₈· B_{η→π⁺π⁻π⁰} + B₁₃₀· B_{η→π⁺π⁻π⁰} + B₈₀₀· B_{ω→π⁺π⁻π⁰}
	+ B₁₅₁· B_{ω→π⁺π⁻π⁰} + B₁₅₂·( B_{ω→π⁺π⁻π⁰}+ B_{ω→π⁺π⁻}) + B₈₀₃

B₆₅ =	B₄₀·( B_{<K⁰\|K_S>}· B_{K_S→π⁺π⁻}) + B₄₂·( B_{<K⁰\|K_S>}· B_{K_S→π⁺π⁻})
	+ B₇₀ + B₉₄ + B₁₂₈· B_{η→π⁺π⁻π⁰} + B₁₅₁· B_{ω→π⁺π⁻π⁰}
	+ B₁₅₂· B_{ω→π⁺π⁻} + B₈₀₀· B_{ω→π⁺π⁻π⁰} + B₈₀₃

B₆₆ =	B₇₀ + B₉₄ + B₁₂₈· B_{η→π⁺π⁻π⁰} + B₁₅₁· B_{ω→π⁺π⁻π⁰}
	+ B₁₅₂· B_{ω→π⁺π⁻} + B₈₀₀· B_{ω→π⁺π⁻π⁰} + B₈₀₃

B₆₇ =

B₇₀ + B₉₄ + B₁₂₈· B_{η→π⁺π⁻π⁰} + B₈₀₃

B₆₈ =	B₄₀·( B_{<K⁰\|K_S>}· B_{K_S→π⁺π⁻}) + B₇₀ + B₁₅₂· B_{ω→π⁺π⁻}
	+ B₈₀₀· B_{ω→π⁺π⁻π⁰}

B₆₉ =

B₁₅₂· B_{ω→π⁺π⁻} + B₇₀ + B₈₀₀· B_{ω→π⁺π⁻π⁰}

B₇₄ =	B₁₅₂· B_{ω→π⁺π⁻π⁰} + B₇₈ + B₇₇ + B₁₂₆· B_{η→π⁺π⁻π⁰}
	+ B₁₃₀· B_{η→π⁺π⁻π⁰}

B₇₅ =	B₁₅₂· B_{ω→π⁺π⁻π⁰} + B₄₇·(2· B_{K_S→π⁺π⁻}· B_{K_S→π⁰π⁰})
	+ B₇₇ + B₁₂₆· B_{η→π⁺π⁻π⁰} + B₁₃₀· B_{η→π⁺π⁻π⁰}

B₇₆ =

B₁₅₂· B_{ω→π⁺π⁻π⁰} + B₇₇ + B₁₂₆· B_{η→π⁺π⁻π⁰} + B₁₃₀· B_{η→π⁺π⁻π⁰}

B₇₆

B₅₄

B₇₆

B₅₄

B₇₈ =

B₈₁₀ + B₅₀·(2· B_{K_S→π⁺π⁻}· B_{K_S→π⁰π⁰}) + B₁₃₂·( B_{<K⁰|K_S>}· B_{K_S→π⁺π⁻}· B_η→3π⁰)

B₇₉ =	B₃₇·( B_{<K⁰\|K_S>}· B_{K_S→π⁺π⁻}) + B₄₂·( B_{<K⁰\|K_S>}· B_{K_S→π⁺π⁻})
	+ B₉₃ + B₉₄ + B₁₂₈· B_η→charged + B₁₅₁·( B_{ω→π⁺π⁻π⁰}
	+ B_{ω→π⁺π⁻}) + B₁₆₈ + B₈₀₂ + B₈₀₃

B₈₀ =

B₉₃ + B₈₀₂ + B₁₅₁· B_{ω→π⁺π⁻}

B₈₀

B₆₀

B₈₀

B₆₀

B₈₁ =

B₁₂₈· B_{η→π⁺π⁻π⁰} + B₉₄ + B₈₀₃ + B₁₅₁· B_{ω→π⁺π⁻π⁰}

B₈₁

B₆₉

B₈₁

B₆₉

B₈₂ =	B₁₂₈· B_η→charged + B₄₂·( B_{<K⁰\|K_S>}· B_{K_S→π⁺π⁻}) + B₈₀₂
	+ B₈₀₃ + B₁₅₁·( B_{ω→π⁺π⁻π⁰}+ B_{ω→π⁺π⁻}) + B₃₇·( B_{<K⁰\|K_S>}· B_{K_S→π⁺π⁻})

B₈₃ =	B₁₂₈· B_{η→π⁺π⁻π⁰} + B₈₀₂ + B₈₀₃ + B₁₅₁·( B_{ω→π⁺π⁻π⁰}
	+ B_{ω→π⁺π⁻})

B₈₄ =

B₈₀₂ + B₁₅₁· B_{ω→π⁺π⁻} + B₃₇·( B_{<K⁰|K_S>}· B_{K_S→π⁺π⁻})

B₈₅ =

B₈₀₂ + B₁₅₁· B_{ω→π⁺π⁻}

B₈₅

B₆₀

B₈₅

B₆₀

B₈₇ =	B₄₂·( B_{<K⁰\|K_S>}· B_{K_S→π⁺π⁻}) + B₁₂₈· B_{η→π⁺π⁻π⁰} + B₁₅₁· B_{ω→π⁺π⁻π⁰}
	+ B₈₀₃

B₈₈ =

B₁₂₈· B_{η→π⁺π⁻π⁰} + B₈₀₃ + B₁₅₁· B_{ω→π⁺π⁻π⁰}

B₈₉ =

B₈₀₃ + B₁₅₁· B_{ω→π⁺π⁻π⁰}

B₉₂ =

B₉₄ + B₉₃

B₉₃

B₆₀

B₉₃

B₆₀

B₉₄

B₆₉

B₉₄

B₆₉

B₉₆ =

B₁₆₇· B_{φ→ K⁺K⁻}

B₁₀₂ =

B₁₀₃ + B₁₀₄

B₁₀₃ =

B₈₂₀ + B₈₂₂ + B₈₃₁· B_{ω→π⁺π⁻}

B₁₀₄ =

B₈₃₀ + B₈₃₃

B₁₀₆ =	B₃₀ + B₄₄· B_{<K⁰\|K_S>} + B₄₇ + B₅₃· B_{<K⁰\|K_S>}
	+ B₇₇ + B₁₀₃ + B₁₂₆·( B_η→3π⁰+ B_{η→π⁺π⁻π⁰}) + B₁₅₂· B_{ω→π⁺π⁻π⁰}

B₁₁₀ =	B₁₀ + B₁₆ + B₂₃ + B₂₈ + B₃₅ + B₄₀
	+ B₁₂₈ + B₈₀₂ + B₈₀₃ + B₁₅₁ + B₁₃₀ + B₁₃₂
	+ B₄₄ + B₅₃ + B₁₆₈ + B₁₆₉ + B₈₂₂ + B₈₃₃

B₁₄₉ =

B₁₅₂ + B₈₀₀ + B₁₅₁

B₁₅₀ =

B₈₀₀ + B₁₅₁

B₁₅₀

B₆₆

B₁₅₀

B₆₆

B₁₅₂

B₅₄

B₁₅₂

B₅₄

B₁₅₂

B₇₆

B₁₅₂

B₇₆

B₁₆₈ =

B₁₆₇· B_{φ→ K⁺K⁻}

B₁₆₉ =

B₁₆₇· B_{φ→ K_S K_L}

B₈₀₄ =

B₄₇ · (( B_{<K⁰|K_L>}· B_{<K⁰|K_L>}) / ( B_{<K⁰|K_S>}· B_{<K⁰|K_S>}))

B₈₀₆ =

B₅₀ · (( B_{<K⁰|K_L>}· B_{<K⁰|K_L>}) / ( B_{<K⁰|K_S>}· B_{<K⁰|K_S>}))

B₈₁₀ =

B₉₁₀ + B₉₁₁ + B₈₁₁· B_{ω→π⁺π⁻π⁰} + B₈₁₂

B₈₂₀ =

B₉₂₀ + B₈₂₁

B₈₃₀ =

B₉₃₀ + B₈₃₁· B_{ω→π⁺π⁻π⁰} + B₈₃₂

B₉₁₀ =

B₁₃₆· B_η→3π⁰

B₉₁₁ =

B₉₄₅· B_{η→π⁺π⁻π⁰}

B₉₃₀ =

B₁₃₆· B_{η→π⁺π⁻π⁰}

B₉₄₄ =

B₁₃₆· B_η→γγ

B_All =	B₃ + B₅ + B₉ + B₁₀ + B₁₄ + B₁₆
	+ B₂₀ + B₂₃ + B₂₇ + B₂₈ + B₃₀ + B₃₅
	+ B₃₇ + B₄₀ + B₄₂ + B₄₇·(1 + (( B_{<K⁰\|K_L>}· B_{<K⁰\|K_L>}) / ( B_{<K⁰\|K_S>}· B_{<K⁰\|K_S>})))
	+ B₄₈ + B₆₂ + B₇₀ + B₇₇ + B₈₁₁ + B₈₁₂
	+ B₉₃ + B₉₄ + B₈₃₂ + B₈₃₃ + B₁₂₆ + B₁₂₈
	+ B₈₀₂ + B₈₀₃ + B₈₀₀ + B₁₅₁ + B₁₃₀ + B₁₃₂
	+ B₄₄ + B₅₃ + B₅₀·(1 + (( B_{<K⁰\|K_L>}· B_{<K⁰\|K_L>}) / ( B_{<K⁰\|K_S>}· B_{<K⁰\|K_S>})))
	+ B₅₁ + B₁₆₇·( B_{φ→ K⁺K⁻}+ B_{φ→ K_S K_L}) + B₁₅₂ + B₉₂₀
	+ B₈₂₁ + B₈₂₂ + B₈₃₁ + B₁₃₆ + B₉₄₅ + B₈₀₅


B₅	23
B₉	8	5
B₁₀	5	7	7
B₁₄	-14	-15	-13	-3
B₁₆	1	1	2	-1	-8
B₂₀	-5	-5	-8	-1	-41	1
B₂₃	2	2	0	-2	0	-13	-7
B₂₇	-5	-4	-8	-1	1	1	-36	1
B₂₈	2	2	1	-1	1	-13	-1	-22	-10
B₃₀	-4	-3	-10	-1	-8	0	6	-2	-44	2
B₃₅	0	0	0	0	0	0	0	0	0	0	0
B₃₇	0	-1	1	0	0	0	0	-2	0	-2	0	-15
B₄₀	0	0	0	0	0	1	0	1	-1	1	0	-12	2
	B₃	B₅	B₉	B₁₀	B₁₄	B₁₆	B₂₀	B₂₃	B₂₇	B₂₈	B₃₀	B₃₅	B₃₇	B₄₀


B₄₂	0	0	0	0	0	-3	1	-5	0	-5	0	-1	-14	-20
B₄₄	0	0	0	0	0	0	0	0	0	0	0	-1	0	-4
B₄₇	0	-1	2	0	0	2	0	0	0	0	0	-1	3	-4
B₄₈	0	0	0	0	0	0	0	0	0	0	0	-3	0	-2
B₅₀	0	0	0	0	0	0	0	0	0	0	0	1	5	0
B₅₁	0	0	0	0	0	0	0	0	0	0	0	-1	0	-1
B₅₃	0	0	0	0	0	0	0	0	0	0	0	0	0	0
B₆₂	-2	-4	8	0	-3	4	-7	0	-6	0	-5	-1	3	0
B₇₀	-6	-6	-7	-1	-10	0	-1	0	-1	0	3	0	-1	0
B₇₇	-1	0	-3	0	-2	0	0	0	2	0	2	0	0	0
B₉₃	0	-1	3	0	-1	2	-1	0	-1	0	-1	0	2	0
B₉₄	0	0	0	0	0	0	0	0	0	0	0	0	0	0
B₁₂₆	0	0	0	0	0	0	-1	0	0	0	-2	0	0	0
B₁₂₈	0	0	1	0	0	1	0	-1	0	-1	0	0	1	0
	B₃	B₅	B₉	B₁₀	B₁₄	B₁₆	B₂₀	B₂₃	B₂₇	B₂₈	B₃₀	B₃₅	B₃₇	B₄₀

HFLAV-Tau Winter 2022 Report

2 Branching fraction fit

2.1 Fit results

2.2 Changes with respect to the previous report

2.3 Differences between the HFLAV 2021 fit and the PDG 2021 fit

2.4 Branching fraction fit results and experimental inputs

2.5 Correlation coefficients between basis branching fractions uncertainties

2.6 Equality constraints

HFLAV-Tau Winter 2022 Report


B₁₃₀	0	0	0	0	0	0	0	0	0	0	0	0	0	0
B₁₃₂	0	0	0	0	0	0	0	0	0	0	0	0	0	0
B₁₃₆	0	0	1	0	0	1	0	0	0	0	-1	0	1	0
B₁₅₁	0	0	0	0	0	0	0	0	0	0	0	0	0	0
B₁₅₂	-1	-1	-3	0	-2	0	-1	0	2	0	2	0	0	0
B₁₆₇	0	0	0	0	0	0	0	0	0	0	0	0	0	0
B₈₀₀	-1	-1	-2	0	-3	0	0	0	0	0	1	0	0	0
B₈₀₂	1	0	1	0	0	0	-2	0	-1	0	-1	0	0	0
B₈₀₃	2	2	1	0	2	0	0	0	0	0	-1	0	0	0
B₈₀₅	0	0	0	0	0	0	0	0	0	0	0	0	0	0
B₈₁₁	0	0	0	0	0	0	0	0	0	0	0	0	0	0
B₈₁₂	0	1	0	0	0	0	0	0	0	0	0	0	0	0
B₈₂₁	0	0	2	0	0	1	-1	0	-1	0	-1	0	1	0
B₈₂₂	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	B₃	B₅	B₉	B₁₀	B₁₄	B₁₆	B₂₀	B₂₃	B₂₇	B₂₈	B₃₀	B₃₅	B₃₇	B₄₀


B₈₃₁	0	0	1	0	0	1	0	0	0	0	-1	0	1	0
B₈₃₂	0	0	0	0	0	0	0	0	0	0	0	0	0	0
B₈₃₃	0	0	0	0	0	0	0	0	0	0	0	0	0	0
B₉₂₀	0	0	1	0	0	1	0	0	0	0	-1	0	1	0
B₉₄₅	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	B₃	B₅	B₉	B₁₀	B₁₄	B₁₆	B₂₀	B₂₃	B₂₇	B₂₈	B₃₀	B₃₅	B₃₇	B₄₀