HFLAV-Tau 2023 - Branching fraction fit

A fit of the available experimental measurements is used to determine the τ branching fractions. 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 documentation on the systematic uncertainties’ contributions is examined to determine how they depend on parameters such as other τ branching fractions or other external parameters, e.g., η and ω meson branching fractions and the cross-section σ(e⁺e⁻ → τ⁺τ⁻). We use the standard HFLAV procedures [24] to adjust the measurements’ values and uncertainties to account for the updated values and uncertainties of the external parameters, which are taken from the Review of Particle Physics [8]. The measurements’ correlations are adjusted to account for the dependence of different measurement on the same parameters.

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”.

Each measurement is modeled with one fit parameter that is either a τ decay branching fraction, labelled B_i, or a ratio of two τ decay branching fractions, labelled B_i/j = B_i/ B_j. Some branching fractions are linear combinations of other branching fractions: for instance B₈ = B(τ⁻ → h⁻ ν_τ) 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. 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 according to the current experimental evidence. The constraint equations depend on uncertain quantities, such as B(φ→ K⁺K⁻) in this example. The uncertainties associated with these constraints are typically smaller in relative value than the uncertainties on the τ branching fractions. Therefore, we neglect all of them except for the poorly known branching fraction B₉₈₀ = B(a₁⁻ → π⁻ γ), which is modeled with a nuisance fit parameter, based on the value and uncertainty estimated by the ALEPH collaboration [25]. All the constraints are listed in Section 4.4.

where m_i is a measurement result, q_k is a fit parameter, V_ij is the covariance matrix that describes the measurements’ uncertainties and correlations, M_ik is a model matrix that specifies the linear combination of the fit parameters corresponding to the measurement m_i, p_r is a nuisance fit parameter, and n_r and σ_{n_r} are, respectively, the value and uncertainty that summarize the existing prior knowledge on p_r. In the current fit, the second sum includes just one nuisance fit parameter.

The χ² is minimized varying all fit parameters while respecting all constraints, using the technique of the Lagrange multipliers .

In performing the fit, a scale factor of 5.44 was applied to the published uncertainties of the two severely inconsistent measurements of B₉₆ = B(τ⁻ → K⁻ K⁻ K⁺ ν_τ) 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.

4.1 Changes with respect to the previous report

The 2021 HFLAV report [24] used the ALEPH 2006 estimate of B₈₀₅ = B(τ⁻ → a₁⁻(π⁻ γ) ν_τ) [25] and treated it as an experimental measurement, therefore B₈₀₅ = B(τ⁻ → a₁⁻(π⁻ γ) ν_τ) was determined using B₉₈₀ = B(a₁⁻ → π⁻ γ) and the values of B₆₂ = B(τ⁻ → π⁻ π⁺ π⁻ ν_τ (ex.K⁰,ω)) and B₂₀ = B(τ⁻ → π⁻ 2π⁰ ν_τ (ex.K⁰)) as known at the time of the ALEPH 2006 publication. In this edition, we use an estimate of B₉₈₀ in a constraint equation that defines B₈₀₅, and B₉₈₀ is optimized in the τ branching fraction fit as nuisance fit parameter. With the current procedure, B₈₀₅ is determined using B₉₈₀ and the fit values of B₆₂ and B₂₀. The fit returns B₈₀₅ = (0.38 ± 0.15)· 10⁻³, while the HFLAV 2021 report [24] had B₈₀₅ = (0.40± 0.20)·10⁻³ (the ALEPH estimate [25]). The other τ branching fractions averages are not affected by this modification of the fit procedure.

We include in the τ branching fraction fit measurements the recently published measurement of the ratio of the τ leptonic branching fractions:

by the Belle II experiment [2]. Since this result is the most precise measurement from a single experiment to date, the precision of the fitted leptonic branching fractions is also improved.

We also report the results from a unitarity-constrained fit, along with the results from the non-unitarity-constrained fit. The results from the unitarity-constrained fit are required as inputs for tau-decay Monte Carlo event generators, such as Tauola [26], and were not included in the previous reports.

The parameters used to update the measurements’ systematic biases and the parameters appearing in the constraint equations in Section 4.4 have been updated to values reported in the Review of Particle Physics 2022 and 2023 update [8].

4.2 Differences with respect to the PDG fit

The τ branching fraction fit provided by HFLAV for the 2024 edition of the Review of Particle Physics reports as in the past editions only the unitarity-constrained fit results, and includes all the updates of this report, except the recent Belle II measurement of the ratio of the τ leptonic branching fractions, which was published later than the new literature publication deadline.

4.3 Fit results

We use a total of 171 τ branching fractions and ratios measurements and 1 nuisance fit parameter measurement to optimize 137 fit parameters and 1 nuisance fit parameter, while respecting 91 equality constraints. When considering just the τ branching fractions, omitting the nuisance fit parameter, the fit has χ²/d.o.f. = 138/125, corresponding to a confidence level CL = 20.2%. The fitted parameters’ values and uncertainties corresponding to τ branching fractions measurements are listed in Table 3, for both the universality-constrained fit and the unconstrained fit. The table also includes:

When no unitarity constraint is applied, we find that B_ur is consistent with 0. The nuisance fit parameter’s value and uncertainty are listed in Table 4. All resulting fit parameters, their input measurements, and their correlation matrix are available at [9].

Table 3: Fit averages and measurements of τ branching fractions and ratios of branching fractions. The label “u.c.” indicates the results from the unitarity-constrained fit.

Fit parameter Experiment Reference

B₁ = B(τ⁻ → particle⁻ ≥ 0  neutrals ≥ 0  K⁰  ν_τ)

0.8518 ± 0.0011 average

0.8524 ± 0.0006 average with u.c.

B₂ = B(τ⁻ → particle⁻ ≥ 0  neutrals ≥ 0  K_L⁰  ν_τ)

0.8452 ± 0.0010 average

0.8458 ± 0.0006 average with u.c.

B₃ = B(τ⁻ → µ⁻ ν_µν_τ)

0.17360 ± 0.00037 average

0.17366 ± 0.00036 average with u.c.

0.17319 ± 0.00070 ± 0.00032 ALEPH [25]

0.17325 ± 0.00095 ± 0.00077 DELPHI [27]

0.17342 ± 0.00110 ± 0.00067 L3 [28]

0.17340 ± 0.00090 ± 0.00060 OPAL [29]

B_3 / 5 =
B(τ⁻ → µ⁻ ν_µν_τ)

B(τ⁻ → e⁻ ν_e ν_τ)

0.9730 ± 0.0022 average

0.9730 ± 0.0022 average with u.c.

0.9970 ± 0.0350 ± 0.0400 ARGUS [30]

0.9796 ± 0.0016 ± 0.0036 BaBar [31]

0.9675 ± 0.0007 ± 0.0036 Belle II [2]

0.9777 ± 0.0063 ± 0.0087 CLEO [15]

B₅ = B(τ⁻ → e⁻ ν_e ν_τ)

0.1784 ± 0.0004 average

0.1785 ± 0.0004 average with u.c.

0.1784 ± 0.0007 ± 0.0004 ALEPH [25]

0.1776 ± 0.0006 ± 0.0017 CLEO [15]

0.1788 ± 0.0011 ± 0.0011 DELPHI [27]

0.1781 ± 0.0010 ± 0.0008 L3 [28]

0.1781 ± 0.0009 ± 0.0006 OPAL [32]

B₇ = B(τ⁻ → h⁻ ≥ 0  K_L⁰  ν_τ)

0.1202 ± 0.0005 average

0.1203 ± 0.0005 average with u.c.

0.1240 ± 0.0070 ± 0.0070 DELPHI [33]

0.1247 ± 0.0026 ± 0.0043 L3 [34]

0.1210 ± 0.0070 ± 0.0050 OPAL [35]

B₈ = B(τ⁻ → h⁻ ν_τ)

0.1150 ± 0.0005 average

0.1151 ± 0.0005 average with u.c.

0.1152 ± 0.0005 ± 0.0012 CLEO [15]

0.1157 ± 0.0012 ± 0.0011 DELPHI [36]

0.1198 ± 0.0013 ± 0.0016 OPAL [37]

B_8 / 5 =
B(τ⁻ → h⁻ ν_τ)

B(τ⁻ → e⁻ ν_e ν_τ)

0.6448 ± 0.0032 average

0.6450 ± 0.0032 average with u.c.

B₉ = B(τ⁻ → π⁻ ν_τ)

0.1081 ± 0.0005 average

0.1082 ± 0.0005 average with u.c.

0.1083 ± 0.0007 ± 0.0008 ALEPH [25]

B_9 / 5 =
B(τ⁻ → π⁻ ν_τ)

B(τ⁻ → e⁻ ν_e ν_τ)

0.6058 ± 0.0032 average

0.6060 ± 0.0031 average with u.c.

0.5945 ± 0.0014 ± 0.0061 BaBar [31]

B₁₀ = B(τ⁻ → K⁻ ν_τ)

0.00696 ± 0.00010 average

0.00697 ± 0.00010 average with u.c.

0.00696 ± 0.00025 ± 0.00014 ALEPH [38]

0.00660 ± 0.00070 ± 0.00090 CLEO [39]

0.00850 ± 0.00180 ± 0.00000 DELPHI [40]

0.00658 ± 0.00027 ± 0.00029 OPAL [41]

B_10 / 5 =
B(τ⁻ → K⁻ ν_τ)

B(τ⁻ → e⁻ ν_e ν_τ)

0.03901 ± 0.00054 average

0.03903 ± 0.00054 average with u.c.

0.03882 ± 0.00032 ± 0.00057 BaBar [31]

B_10 / 9 =
B(τ⁻ → K⁻ ν_τ)

B(τ⁻ → π⁻ ν_τ)

0.0644 ± 0.0009 average

0.0644 ± 0.0009 average with u.c.

B₁₁ = B(τ⁻ → h⁻ ≥ 1  neutrals  ν_τ)

0.3698 ± 0.0010 average

0.3700 ± 0.0009 average with u.c.

B₁₂ = B(τ⁻ → h⁻ ≥ 1  π⁰  ν_τ (ex.K⁰))

0.3648 ± 0.0010 average

0.3650 ± 0.0009 average with u.c.

B₁₃ = B(τ⁻ → h⁻ π⁰ ν_τ)

0.25917 ± 0.00090 average

0.25926 ± 0.00089 average with u.c.

0.25670 ± 0.00010 ± 0.00390 Belle [42]

0.25870 ± 0.00120 ± 0.00420 CLEO [43]

0.25740 ± 0.00201 ± 0.00138 DELPHI [36]

0.25050 ± 0.00350 ± 0.00500 L3 [34]

0.25890 ± 0.00170 ± 0.00290 OPAL [37]

B₁₄ = B(τ⁻ → π⁻ π⁰ ν_τ)

0.2548 ± 0.0009 average

0.2549 ± 0.0009 average with u.c.

0.2547 ± 0.0010 ± 0.0008 ALEPH [25]

B₁₆ = B(τ⁻ → K⁻ π⁰ ν_τ)

0.004321 ± 0.000148 average

0.004328 ± 0.000148 average with u.c.

0.004440 ± 0.000260 ± 0.000240 ALEPH [38]

0.004160 ± 0.000030 ± 0.000180 BaBar [44]

0.005100 ± 0.001000 ± 0.000700 CLEO [39]

0.004710 ± 0.000590 ± 0.000230 OPAL [45]

B₁₇ = B(τ⁻ → h⁻ ≥ 2  π⁰  ν_τ)

0.1079 ± 0.0010 average

0.1081 ± 0.0009 average with u.c.

0.0991 ± 0.0031 ± 0.0027 OPAL [37]

B₁₈ = B(τ⁻ → h⁻ 2π⁰ ν_τ)

0.0946 ± 0.0010 average

0.0948 ± 0.0010 average with u.c.

B₁₉ = B(τ⁻ → h⁻ 2π⁰ ν_τ (ex.K⁰))

0.0931 ± 0.0010 average

0.0932 ± 0.0010 average with u.c.

0.0950 ± 0.0032 ± 0.0027 DELPHI [36]

0.0888 ± 0.0037 ± 0.0042 L3 [34]

B_19 / 13 =
B(τ⁻ → h⁻ 2π⁰ ν_τ (ex.K⁰))

B(τ⁻ → h⁻ π⁰ ν_τ)

0.359 ± 0.005 average

0.360 ± 0.004 average with u.c.

0.342 ± 0.006 ± 0.016 CLEO [46]

B₂₀ = B(τ⁻ → π⁻ 2π⁰ ν_τ (ex.K⁰))

0.0925 ± 0.0010 average

0.0926 ± 0.0010 average with u.c.

0.0924 ± 0.0009 ± 0.0009 ALEPH [25]

B₂₃ = B(τ⁻ → K⁻ 2π⁰ ν_τ (ex.K⁰))

0.00063 ± 0.00022 average

0.00065 ± 0.00022 average with u.c.

0.00056 ± 0.00020 ± 0.00015 ALEPH [38]

0.00090 ± 0.00100 ± 0.00030 CLEO [39]

B₂₄ = B(τ⁻ → h⁻ ≥ 3  π⁰  ν_τ)

0.0134 ± 0.0007 average

0.0134 ± 0.0007 average with u.c.

B₂₅ = B(τ⁻ → h⁻ ≥ 3  π⁰  ν_τ (ex.K⁰))

0.0125 ± 0.0007 average

0.0126 ± 0.0007 average with u.c.

0.0140 ± 0.0021 ± 0.0022 DELPHI [36]

B₂₆ = B(τ⁻ → h⁻ 3π⁰ ν_τ)

0.0117 ± 0.0007 average

0.0118 ± 0.0007 average with u.c.

0.0170 ± 0.0024 ± 0.0038 L3 [34]

B_26 / 13 =
B(τ⁻ → h⁻ 3π⁰ ν_τ)

B(τ⁻ → h⁻ π⁰ ν_τ)

0.0453 ± 0.0028 average

0.0454 ± 0.0028 average with u.c.

0.0440 ± 0.0030 ± 0.0050 CLEO [46]

B₂₇ = B(τ⁻ → π⁻ 3π⁰ ν_τ (ex.K⁰))

0.0104 ± 0.0007 average

0.0104 ± 0.0007 average with u.c.

0.0098 ± 0.0007 ± 0.0006 ALEPH [25]

B₂₈ = B(τ⁻ → K⁻ 3π⁰ ν_τ (ex.K⁰,η))

(0.46 ± 0.21)· 10⁻³ average

(0.48 ± 0.21)· 10⁻³ average with u.c.

(0.37 ± 0.21 ± 0.11)· 10⁻³ ALEPH [38]

B₂₉ = B(τ⁻ → h⁻ 4π⁰ ν_τ (ex.K⁰))

0.0016 ± 0.0004 average

0.0016 ± 0.0004 average with u.c.

0.0016 ± 0.0005 ± 0.0005 CLEO [46]

B₃₀ = B(τ⁻ → h⁻ 4π⁰ ν_τ (ex.K⁰,η))

0.00112 ± 0.00039 average

0.00112 ± 0.00039 average with u.c.

0.00112 ± 0.00037 ± 0.00035 ALEPH [25]

B₃₁ = B(τ⁻ → K⁻ ≥ 0  π⁰ ≥ 0  K⁰ ≥ 0  γ ν_τ)

0.01548 ± 0.00029 average

0.01552 ± 0.00029 average with u.c.

0.01700 ± 0.00120 ± 0.00190 CLEO [39]

0.01540 ± 0.00240 ± 0.00000 DELPHI [40]

0.01528 ± 0.00039 ± 0.00040 OPAL [41]

B₃₂ = B(τ⁻ → K⁻ ≥ 1  (π⁰ or K⁰ or γ) ν_τ)

0.00856 ± 0.00028 average

0.00859 ± 0.00028 average with u.c.

B₃₃ = B(τ⁻ → K_S⁰ (particles)⁻ ν_τ)

0.00937 ± 0.00029 average

0.00943 ± 0.00028 average with u.c.

0.00970 ± 0.00058 ± 0.00062 ALEPH [47]

0.00970 ± 0.00090 ± 0.00060 OPAL [48]

B₃₄ = B(τ⁻ → h⁻ K⁰ ν_τ)

0.00986 ± 0.00014 average

0.00987 ± 0.00014 average with u.c.

0.00855 ± 0.00036 ± 0.00073 CLEO [49]

B₃₅ = B(τ⁻ → π⁻ K⁰ ν_τ)

0.008375 ± 0.000139 average

0.008384 ± 0.000138 average with u.c.

0.009280 ± 0.000450 ± 0.000340 ALEPH [38]

0.008320 ± 0.000025 ± 0.000150 Belle [50]

0.009500 ± 0.001500 ± 0.000600 L3 [51]

0.009330 ± 0.000680 ± 0.000490 OPAL [52]

B₃₇ = B(τ⁻ → K⁻ K⁰ ν_τ)

0.001486 ± 0.000034 average

0.001486 ± 0.000034 average with u.c.

0.001580 ± 0.000420 ± 0.000170 ALEPH [47]

0.001620 ± 0.000210 ± 0.000110 ALEPH [38]

0.001478 ± 0.000022 ± 0.000040 BaBar [53]

0.001480 ± 0.000013 ± 0.000055 Belle [50]

0.001510 ± 0.000210 ± 0.000220 CLEO [49]

B₃₈ = B(τ⁻ → K⁻ K⁰ ≥ 0  π⁰  ν_τ)

0.00298 ± 0.00007 average

0.00299 ± 0.00007 average with u.c.

0.00330 ± 0.00055 ± 0.00039 OPAL [52]

B₃₉ = B(τ⁻ → h⁻ K⁰ π⁰ ν_τ)

0.00531 ± 0.00013 average

0.00532 ± 0.00013 average with u.c.

0.00562 ± 0.00050 ± 0.00048 CLEO [49]

B₄₀ = B(τ⁻ → π⁻ K⁰ π⁰ ν_τ)

0.003810 ± 0.000129 average

0.003817 ± 0.000129 average with u.c.

0.002940 ± 0.000730 ± 0.000370 ALEPH [47]

0.003470 ± 0.000530 ± 0.000370 ALEPH [38]

0.003860 ± 0.000031 ± 0.000135 Belle [50]

0.004100 ± 0.001200 ± 0.000300 L3 [51]

B₄₂ = B(τ⁻ → K⁻ K⁰ π⁰ ν_τ)

0.001499 ± 0.000070 average

0.001500 ± 0.000070 average with u.c.

0.001520 ± 0.000760 ± 0.000210 ALEPH [47]

0.001430 ± 0.000250 ± 0.000150 ALEPH [38]

0.001496 ± 0.000019 ± 0.000073 Belle [50]

0.001450 ± 0.000360 ± 0.000200 CLEO [49]

B₄₃ = B(τ⁻ → π⁻ K⁰ ≥ 1  π⁰  ν_τ)

0.00404 ± 0.00026 average

0.00408 ± 0.00025 average with u.c.

0.00324 ± 0.00074 ± 0.00066 OPAL [52]

B₄₄ = B(τ⁻ → π⁻ K⁰ 2π⁰ ν_τ (ex.K⁰))

(0.23 ± 0.23)· 10⁻³ average

(0.26 ± 0.23)· 10⁻³ average with u.c.

(0.26 ± 0.24 ± 0.00)· 10⁻³ ALEPH [54]

B₄₆ = B(τ⁻ → π⁻ K⁰ K⁰ ν_τ)

0.00152 ± 0.00025 average

0.00155 ± 0.00024 average with u.c.

B₄₇ = B(τ⁻ → π⁻ K_S⁰ K_S⁰ ν_τ)

(234.8 ± 6.5)· 10⁻⁶ average

(234.9 ± 6.5)· 10⁻⁶ average with u.c.

(260.0 ± 100.0 ± 50.0)· 10⁻⁶ ALEPH [47]

(231.0 ± 4.0 ± 8.0)· 10⁻⁶ BaBar [55]

(233.0 ± 3.3 ± 9.3)· 10⁻⁶ Belle [50]

(230.0 ± 50.0 ± 30.0)· 10⁻⁶ CLEO [49]

B₄₈ = B(τ⁻ → π⁻ K_S⁰ K_L⁰ ν_τ)

0.00105 ± 0.00025 average

0.00108 ± 0.00024 average with u.c.

0.00101 ± 0.00023 ± 0.00013 ALEPH [47]

B₄₉ = B(τ⁻ → π⁻ π⁰ K⁰ K⁰ ν_τ)

(0.35 ± 0.12)· 10⁻³ average

(0.36 ± 0.12)· 10⁻³ average with u.c.

B₅₀ = B(τ⁻ → π⁻ K_S⁰ K_S⁰ π⁰ ν_τ)

(18.2 ± 2.1)· 10⁻⁶ average

(18.2 ± 2.1)· 10⁻⁶ average with u.c.

(16.0 ± 2.0 ± 2.2)· 10⁻⁶ BaBar [55]

(20.0 ± 2.2 ± 2.0)· 10⁻⁶ Belle [50]

B₅₁ = B(τ⁻ → π⁻ K_S⁰ K_L⁰ π⁰ ν_τ)

(0.32 ± 0.12)· 10⁻³ average

(0.32 ± 0.12)· 10⁻³ average with u.c.

(0.31 ± 0.11 ± 0.05)· 10⁻³ ALEPH [47]

B₅₃ = B(τ⁻ → K⁰ h⁻ h⁻ h⁺ ν_τ)

(0.22 ± 0.20)· 10⁻³ average

(0.25 ± 0.20)· 10⁻³ average with u.c.

(0.23 ± 0.19 ± 0.07)· 10⁻³ ALEPH [47]

B₅₄ = B(τ⁻ → h⁻ h⁻ h⁺ ≥ 0  neutrals ≥ 0  K_L⁰  ν_τ)

0.1519 ± 0.0006 average

0.1520 ± 0.0006 average with u.c.

0.1500 ± 0.0040 ± 0.0030 CELLO [56]

0.1440 ± 0.0060 ± 0.0030 L3 [57]

0.1510 ± 0.0080 ± 0.0060 TPC [58]

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

0.1454 ± 0.0006 average

0.1455 ± 0.0006 average with u.c.

0.1456 ± 0.0010 ± 0.0008 L3 [59]

0.1496 ± 0.0009 ± 0.0022 OPAL [60]

B₅₆ = B(τ⁻ → h⁻ h⁻ h⁺ ν_τ)

0.0979 ± 0.0005 average

0.0980 ± 0.0005 average with u.c.

B₅₇ = B(τ⁻ → h⁻ h⁻ h⁺ ν_τ (ex.K⁰))

0.0945 ± 0.0005 average

0.0945 ± 0.0005 average with u.c.

0.0951 ± 0.0007 ± 0.0020 CLEO [61]

0.0932 ± 0.0009 ± 0.0008 DELPHI [36]

B_57 / 55 =
B(τ⁻ → h⁻ h⁻ h⁺ ν_τ (ex.K⁰))

B(τ⁻ → h⁻ h⁻ h⁺ ≥ 0  neutrals  ν_τ (ex.K⁰))

0.6496 ± 0.0031 average

0.6498 ± 0.0031 average with u.c.

0.6600 ± 0.0040 ± 0.0140 OPAL [60]

B₅₈ = B(τ⁻ → h⁻ h⁻ h⁺ ν_τ (ex.K⁰,ω))

0.0942 ± 0.0005 average

0.0942 ± 0.0005 average with u.c.

B₅₉ = B(τ⁻ → π⁻ π⁺ π⁻ ν_τ)

0.0930 ± 0.0005 average

0.0931 ± 0.0005 average with u.c.

B₆₀ = B(τ⁻ → π⁻ π⁺ π⁻ ν_τ (ex.K⁰))

0.09010 ± 0.00052 average

0.09016 ± 0.00051 average with u.c.

0.08830 ± 0.00010 ± 0.00130 BaBar [62]

0.08420 ± 0.00000 _−0.00250^+0.00260 Belle [63]

0.09130 ± 0.00050 ± 0.00460 CLEO3 [64]

B₆₂ = B(τ⁻ → π⁻ π⁺ π⁻ ν_τ (ex.K⁰,ω))

0.0898 ± 0.0005 average

0.0899 ± 0.0005 average with u.c.

0.0904 ± 0.0006 ± 0.0008 ALEPH [25]

B₆₃ = B(τ⁻ → h⁻ h⁻ h⁺ ≥ 1  neutrals  ν_τ)

0.0529 ± 0.0005 average

0.0529 ± 0.0005 average with u.c.

B₆₄ = B(τ⁻ → h⁻ h⁻ h⁺ ≥ 1  π⁰  ν_τ (ex.K⁰))

0.0509 ± 0.0005 average

0.0509 ± 0.0005 average with u.c.

B₆₅ = B(τ⁻ → h⁻ h⁻ h⁺ π⁰ ν_τ)

0.0476 ± 0.0005 average

0.0476 ± 0.0005 average with u.c.

B₆₆ = B(τ⁻ → h⁻ h⁻ h⁺ π⁰ ν_τ (ex.K⁰))

0.0457 ± 0.0005 average

0.0457 ± 0.0005 average with u.c.

0.0423 ± 0.0006 ± 0.0022 CLEO [61]

0.0454 ± 0.0011 ± 0.0010 DELPHI [36]

B₆₇ = B(τ⁻ → h⁻ h⁻ h⁺ π⁰ ν_τ (ex.K⁰,ω))

0.0279 ± 0.0007 average

0.0279 ± 0.0007 average with u.c.

B₆₈ = B(τ⁻ → π⁻ π⁺ π⁻ π⁰ ν_τ)

0.0462 ± 0.0005 average

0.0462 ± 0.0005 average with u.c.

B₆₉ = B(τ⁻ → π⁻ π⁺ π⁻ π⁰ ν_τ (ex.K⁰))

0.0449 ± 0.0005 average

0.0449 ± 0.0005 average with u.c.

0.0460 ± 0.0006 ± 0.0006 ALEPH [25]

0.0419 ± 0.0010 ± 0.0021 CLEO [65]

B₇₀ = B(τ⁻ → π⁻ π⁺ π⁻ π⁰ ν_τ (ex.K⁰,ω))

0.0274 ± 0.0007 average

0.0274 ± 0.0007 average with u.c.

B₇₄ = B(τ⁻ → h⁻ h⁻ h⁺ ≥ 2  π⁰  ν_τ (ex.K⁰))

0.00515 ± 0.00031 average

0.00517 ± 0.00031 average with u.c.

0.00561 ± 0.00068 ± 0.00095 DELPHI [36]

B₇₅ = B(τ⁻ → h⁻ h⁻ h⁺ 2π⁰ ν_τ)

0.00504 ± 0.00031 average

0.00505 ± 0.00031 average with u.c.

B₇₆ = B(τ⁻ → h⁻ h⁻ h⁺ 2π⁰ ν_τ (ex.K⁰))

0.00494 ± 0.00031 average

0.00495 ± 0.00031 average with u.c.

0.00435 ± 0.00030 ± 0.00035 ALEPH [25]

B_76 / 54 =
B(τ⁻ → h⁻ h⁻ h⁺ 2π⁰ ν_τ (ex.K⁰))

B(τ⁻ → h⁻ h⁻ h⁺ ≥ 0  neutrals ≥ 0  K_L⁰  ν_τ)

0.0325 ± 0.0020 average

0.0326 ± 0.0020 average with u.c.

0.0340 ± 0.0020 ± 0.0030 CLEO [66]

B₇₇ = B(τ⁻ → h⁻ h⁻ h⁺ 2π⁰ ν_τ (ex.K⁰,ω,η))

0.0010 ± 0.0004 average

0.0010 ± 0.0004 average with u.c.

B₇₈ = B(τ⁻ → h⁻ h⁻ h⁺ 3π⁰ ν_τ)

(212 ± 30)· 10⁻⁶ average

(213 ± 30)· 10⁻⁶ average with u.c.

(220 ± 30 ± 40)· 10⁻⁶ CLEO [67]

B₇₉ = B(τ⁻ → K⁻ h⁻ h⁺ ≥ 0  neutrals  ν_τ)

0.00628 ± 0.00014 average

0.00629 ± 0.00014 average with u.c.

B₈₀ = B(τ⁻ → K⁻ π⁻ h⁺ ν_τ (ex.K⁰))

0.00436 ± 0.00007 average

0.00437 ± 0.00007 average with u.c.

B_80 / 60 =
B(τ⁻ → K⁻ π⁻ h⁺ ν_τ (ex.K⁰))

B(τ⁻ → π⁻ π⁺ π⁻ ν_τ (ex.K⁰))

0.0484 ± 0.0008 average

0.0485 ± 0.0008 average with u.c.

0.0544 ± 0.0021 ± 0.0053 CLEO [68]

B₈₁ = B(τ⁻ → K⁻ π⁻ h⁺ π⁰ ν_τ (ex.K⁰))

(0.85 ± 0.12)· 10⁻³ average

(0.86 ± 0.12)· 10⁻³ average with u.c.

B_81 / 69 =
B(τ⁻ → K⁻ π⁻ h⁺ π⁰ ν_τ (ex.K⁰))

B(τ⁻ → π⁻ π⁺ π⁻ π⁰ ν_τ (ex.K⁰))

0.0189 ± 0.0026 average

0.0191 ± 0.0026 average with u.c.

0.0261 ± 0.0045 ± 0.0042 CLEO [68]

B₈₂ = B(τ⁻ → K⁻ π⁻ π⁺ ≥ 0  neutrals  ν_τ)

0.00476 ± 0.00014 average

0.00477 ± 0.00014 average with u.c.

0.00580 _−0.00130^+0.00150 ± 0.00120 TPC [69]

B₈₃ = B(τ⁻ → K⁻ π⁻ π⁺ ≥ 0  π⁰  ν_τ (ex.K⁰))

0.00372 ± 0.00013 average

0.00373 ± 0.00013 average with u.c.

B₈₄ = B(τ⁻ → K⁻ π⁻ π⁺ ν_τ)

0.00344 ± 0.00007 average

0.00345 ± 0.00007 average with u.c.

B₈₅ = B(τ⁻ → K⁻ π⁺ π⁻ ν_τ (ex.K⁰))

0.002930 ± 0.000068 average

0.002933 ± 0.000068 average with u.c.

0.002140 ± 0.000370 ± 0.000290 ALEPH [70]

0.002730 ± 0.000020 ± 0.000090 BaBar [62]

0.003300 ± 0.000010 _−0.000170^+0.000160 Belle [63]

0.003840 ± 0.000140 ± 0.000380 CLEO3 [64]

0.004150 ± 0.000530 ± 0.000400 OPAL [45]

B_85 / 60 =
B(τ⁻ → K⁻ π⁺ π⁻ ν_τ (ex.K⁰))

B(τ⁻ → π⁻ π⁺ π⁻ ν_τ (ex.K⁰))

0.0325 ± 0.0007 average

0.0325 ± 0.0007 average with u.c.

B₈₇ = B(τ⁻ → K⁻ π⁻ π⁺ π⁰ ν_τ)

0.00131 ± 0.00012 average

0.00131 ± 0.00012 average with u.c.

B₈₈ = B(τ⁻ → K⁻ π⁻ π⁺ π⁰ ν_τ (ex.K⁰))

(0.79 ± 0.12)· 10⁻³ average

(0.80 ± 0.12)· 10⁻³ average with u.c.

(0.61 ± 0.39 ± 0.18)· 10⁻³ ALEPH [70]

(0.74 ± 0.08 ± 0.11)· 10⁻³ CLEO3 [71]

B₈₉ = B(τ⁻ → K⁻ π⁻ π⁺ π⁰ ν_τ (ex.K⁰,η))

(0.75 ± 0.12)· 10⁻³ average

(0.76 ± 0.12)· 10⁻³ average with u.c.

B₉₂ = B(τ⁻ → π⁻ K⁻ K⁺ ≥ 0  neutrals  ν_τ)

0.001495 ± 0.000033 average

0.001496 ± 0.000033 average with u.c.

0.001590 ± 0.000530 ± 0.000200 OPAL [72]

0.001500 _−0.000700^+0.000900 ± 0.000300 TPC [69]

B₉₃ = B(τ⁻ → π⁻ K⁻ K⁺ ν_τ)

0.001434 ± 0.000027 average

0.001435 ± 0.000027 average with u.c.

0.001630 ± 0.000210 ± 0.000170 ALEPH [70]

0.001346 ± 0.000010 ± 0.000036 BaBar [62]

0.001550 ± 0.000010 _−0.000050^+0.000060 Belle [63]

0.001550 ± 0.000060 ± 0.000090 CLEO3 [64]

B_93 / 60 =
B(τ⁻ → π⁻ K⁻ K⁺ ν_τ)

B(τ⁻ → π⁻ π⁺ π⁻ ν_τ (ex.K⁰))

0.01592 ± 0.00030 average

0.01592 ± 0.00030 average with u.c.

0.01600 ± 0.00150 ± 0.00300 CLEO [68]

B₉₄ = B(τ⁻ → π⁻ K⁻ K⁺ π⁰ ν_τ)

(61 ± 18)· 10⁻⁶ average

(61 ± 18)· 10⁻⁶ average with u.c.

(750 ± 290 ± 150)· 10⁻⁶ ALEPH [70]

(55 ± 14 ± 12)· 10⁻⁶ CLEO3 [71]

B_94 / 69 =
B(τ⁻ → π⁻ K⁻ K⁺ π⁰ ν_τ)

B(τ⁻ → π⁻ π⁺ π⁻ π⁰ ν_τ (ex.K⁰))

0.0014 ± 0.0004 average

0.0014 ± 0.0004 average with u.c.

0.0079 ± 0.0044 ± 0.0016 CLEO [68]

B₉₆ = B(τ⁻ → K⁻ K⁻ K⁺ ν_τ)

(21.7 ± 8.0)· 10⁻⁶ average

(21.7 ± 8.0)· 10⁻⁶ average with u.c.

(15.8 ± 1.3 ± 1.2)· 10⁻⁶ BaBar [62]

(32.9 ± 1.7 _−2.0^+1.9)· 10⁻⁶ Belle [63]

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

0.00099 ± 0.00004 average

0.00099 ± 0.00004 average with u.c.

0.00097 ± 0.00005 ± 0.00011 CLEO [73]

0.00102 ± 0.00029 ± 0.00000 HRS [74]

0.00170 ± 0.00022 ± 0.00026 L3 [59]

B₁₀₃ = B(τ⁻ → 3h⁻ 2h⁺ ν_τ (ex.K⁰))

(828 ± 31)· 10⁻⁶ average

(829 ± 31)· 10⁻⁶ average with u.c.

(720 ± 90 ± 120)· 10⁻⁶ ALEPH [25]

(640 ± 230 ± 100)· 10⁻⁶ ARGUS [75]

(770 ± 50 ± 90)· 10⁻⁶ CLEO [73]

(970 ± 150 ± 50)· 10⁻⁶ DELPHI [36]

(510 ± 200 ± 0)· 10⁻⁶ HRS [74]

(910 ± 140 ± 60)· 10⁻⁶ OPAL [76]

B₁₀₄ = B(τ⁻ → 3h⁻ 2h⁺ π⁰ ν_τ (ex.K⁰))

(165 ± 11)· 10⁻⁶ average

(165 ± 11)· 10⁻⁶ average with u.c.

(210 ± 70 ± 90)· 10⁻⁶ ALEPH [25]

(170 ± 20 ± 20)· 10⁻⁶ CLEO [67]

(160 ± 120 ± 60)· 10⁻⁶ DELPHI [36]

(270 ± 180 ± 90)· 10⁻⁶ OPAL [76]

B₁₀₆ = B(τ⁻ → (5π)⁻ ν_τ)

0.0078 ± 0.0005 average

0.0078 ± 0.0005 average with u.c.

B₁₁₀ = B(τ⁻ → X_s⁻ ν_τ)

0.0291 ± 0.0005 average

0.0292 ± 0.0004 average with u.c.

B₁₂₆ = B(τ⁻ → π⁻ π⁰ η ν_τ)

0.001386 ± 0.000072 average

0.001389 ± 0.000072 average with u.c.

0.001800 ± 0.000400 ± 0.000200 ALEPH [77]

0.001350 ± 0.000030 ± 0.000070 Belle [78]

0.001700 ± 0.000200 ± 0.000200 CLEO [79]

B₁₂₈ = B(τ⁻ → K⁻ η ν_τ)

(155 ± 8)· 10⁻⁶ average

(155 ± 8)· 10⁻⁶ average with u.c.

(290 ₋₁₂₀⁺¹³⁰ ± 70)· 10⁻⁶ ALEPH [77]

(142 ± 11 ± 7)· 10⁻⁶ BaBar [80]

(158 ± 5 ± 9)· 10⁻⁶ Belle [78]

(260 ± 50 ± 50)· 10⁻⁶ CLEO [81]

B₁₃₀ = B(τ⁻ → K⁻ π⁰ η ν_τ)

(48 ± 12)· 10⁻⁶ average

(48 ± 12)· 10⁻⁶ average with u.c.

(46 ± 11 ± 4)· 10⁻⁶ Belle [78]

(177 ± 56 ± 71)· 10⁻⁶ CLEO [82]

B₁₃₂ = B(τ⁻ → π⁻ K⁰ η ν_τ)

(94 ± 15)· 10⁻⁶ average

(94 ± 15)· 10⁻⁶ average with u.c.

(88 ± 14 ± 6)· 10⁻⁶ Belle [78]

(220 ± 70 ± 22)· 10⁻⁶ CLEO [82]

B₁₃₆ = B(τ⁻ → π⁻ π⁺ π⁻ η ν_τ (ex.K⁰))

(220 ± 13)· 10⁻⁶ average

(220 ± 13)· 10⁻⁶ average with u.c.

B₁₄₉ = B(τ⁻ → h⁻ ω ≥ 0  neutrals  ν_τ)

0.0240 ± 0.0008 average

0.0240 ± 0.0008 average with u.c.

B₁₅₀ = B(τ⁻ → h⁻ ω ν_τ)

0.0199 ± 0.0006 average

0.0199 ± 0.0006 average with u.c.

0.0191 ± 0.0007 ± 0.0006 ALEPH [77]

0.0160 ± 0.0027 ± 0.0041 CLEO [83]

B_150 / 66 =
B(τ⁻ → h⁻ ω ν_τ)

B(τ⁻ → h⁻ h⁻ h⁺ π⁰ ν_τ (ex.K⁰))

0.435 ± 0.014 average

0.435 ± 0.014 average with u.c.

0.431 ± 0.033 ± 0.000 ALEPH [84]

0.464 ± 0.016 ± 0.017 CLEO [61]

B₁₅₁ = B(τ⁻ → K⁻ ω ν_τ)

(0.41 ± 0.09)· 10⁻³ average

(0.41 ± 0.09)· 10⁻³ average with u.c.

(0.41 ± 0.06 ± 0.07)· 10⁻³ CLEO3 [71]

B₁₅₂ = B(τ⁻ → h⁻ π⁰ ω ν_τ)

0.0041 ± 0.0004 average

0.0041 ± 0.0004 average with u.c.

0.0043 ± 0.0006 ± 0.0005 ALEPH [77]

B_152 / 54 =
B(τ⁻ → h⁻ ω π⁰ ν_τ)

B(τ⁻ → h⁻ h⁻ h⁺ ≥ 0  neutrals ≥ 0  K_L⁰  ν_τ)

0.0268 ± 0.0028 average

0.0269 ± 0.0028 average with u.c.

B_152 / 76 =
B(τ⁻ → h⁻ ω π⁰ ν_τ)

B(τ⁻ → h⁻ h⁻ h⁺ 2π⁰ ν_τ (ex.K⁰))

0.82 ± 0.08 average

0.82 ± 0.08 average with u.c.

0.81 ± 0.06 ± 0.06 CLEO [66]

B₁₆₇ = B(τ⁻ → K⁻ φ ν_τ)

(44 ± 16)· 10⁻⁶ average

(44 ± 16)· 10⁻⁶ average with u.c.

B₁₆₈ = B(τ⁻ → K⁻ φ(K⁺ K⁻) ν_τ)

(22 ± 8)· 10⁻⁶ average

(22 ± 8)· 10⁻⁶ average with u.c.

B₁₆₉ = B(τ⁻ → K⁻ φ(K_S⁰ K_L⁰) ν_τ)

(15 ± 6)· 10⁻⁶ average

(15 ± 6)· 10⁻⁶ average with u.c.

B₈₀₀ = B(τ⁻ → π⁻ ω ν_τ)

0.0195 ± 0.0006 average

0.0195 ± 0.0006 average with u.c.

B₈₀₂ = B(τ⁻ → K⁻ π⁻ π⁺ ν_τ (ex.K⁰,ω))

0.00292 ± 0.00007 average

0.00293 ± 0.00007 average with u.c.

B₈₀₃ = B(τ⁻ → K⁻ π⁻ π⁺ π⁰ ν_τ (ex.K⁰,ω,η))

(0.39 ± 0.14)· 10⁻³ average

(0.39 ± 0.14)· 10⁻³ average with u.c.

B₈₀₄ = B(τ⁻ → π⁻ K_L⁰ K_L⁰ ν_τ)

(235 ± 6)· 10⁻⁶ average

(235 ± 6)· 10⁻⁶ average with u.c.

B₈₀₅ = B(τ⁻ → a₁⁻(π⁻ γ) ν_τ)

(0.38 ± 0.15)· 10⁻³ average

(0.40 ± 0.15)· 10⁻³ average with u.c.

B₈₀₆ = B(τ⁻ → π⁻ K_L⁰ K_L⁰ π⁰ ν_τ)

(18.2 ± 2.1)· 10⁻⁶ average

(18.2 ± 2.1)· 10⁻⁶ average with u.c.

B₈₁₀ = B(τ⁻ → 2π⁻ π⁺ 3π⁰ ν_τ (ex.K⁰))

(194 ± 30)· 10⁻⁶ average

(194 ± 30)· 10⁻⁶ average with u.c.

B₈₁₁ = B(τ⁻ → π⁻ 2π⁰ ω ν_τ)

(72 ± 16)· 10⁻⁶ average

(72 ± 16)· 10⁻⁶ average with u.c.

(73 ± 12 ± 12)· 10⁻⁶ BaBar [85]

B₈₁₂ = B(τ⁻ → 2π⁻ π⁺ 3π⁰ ν_τ (ex.K⁰,η,ω,f₁))

(14 ± 27)· 10⁻⁶ average

(14 ± 27)· 10⁻⁶ average with u.c.

(10 ± 8 ± 30)· 10⁻⁶ BaBar [85]

B₈₂₀ = B(τ⁻ → 3π⁻ 2π⁺ ν_τ (ex.K⁰,ω))

(826 ± 31)· 10⁻⁶ average

(827 ± 31)· 10⁻⁶ average with u.c.

B₈₂₁ = B(τ⁻ → 3π⁻ 2π⁺ ν_τ (ex.K⁰,ω,f₁))

(774 ± 30)· 10⁻⁶ average

(775 ± 30)· 10⁻⁶ average with u.c.

(768 ± 4 ± 40)· 10⁻⁶ BaBar [85]

B₈₂₂ = B(τ⁻ → K⁻ 2π⁻ 2π⁺ ν_τ (ex.K⁰))

(0.6 ± 1.2)· 10⁻⁶ average

(0.6 ± 1.2)· 10⁻⁶ average with u.c.

(0.6 ± 0.5 ± 1.1)· 10⁻⁶ BaBar [85]

B₈₃₀ = B(τ⁻ → 3π⁻ 2π⁺ π⁰ ν_τ (ex.K⁰))

(163 ± 11)· 10⁻⁶ average

(164 ± 11)· 10⁻⁶ average with u.c.

B₈₃₁ = B(τ⁻ → 2π⁻ π⁺ ω ν_τ (ex.K⁰))

(84 ± 6)· 10⁻⁶ average

(84 ± 6)· 10⁻⁶ average with u.c.

(84 ± 4 ± 6)· 10⁻⁶ BaBar [85]

B₈₃₂ = B(τ⁻ → 3π⁻ 2π⁺ π⁰ ν_τ (ex.K⁰,η,ω,f₁))

(37.7 ± 8.7)· 10⁻⁶ average

(37.7 ± 8.7)· 10⁻⁶ average with u.c.

(36.0 ± 3.0 ± 9.0)· 10⁻⁶ BaBar [85]

B₈₃₃ = B(τ⁻ → K⁻ 2π⁻ 2π⁺ π⁰ ν_τ (ex.K⁰))

(1.1 ± 0.6)· 10⁻⁶ average

(1.1 ± 0.6)· 10⁻⁶ average with u.c.

(1.1 ± 0.4 ± 0.4)· 10⁻⁶ BaBar [85]

B₉₁₀ = B(τ⁻ → 2π⁻ π⁺ η(3π⁰) ν_τ (ex.K⁰))

(72 ± 4)· 10⁻⁶ average

(72 ± 4)· 10⁻⁶ average with u.c.

(83 ± 9 ± 8)· 10⁻⁶ BaBar [85]

B₉₁₁ = B(τ⁻ → π⁻ 2π⁰ η(π⁺ π⁻ π⁰) ν_τ (ex.K⁰))

(45 ± 9)· 10⁻⁶ average

(45 ± 9)· 10⁻⁶ average with u.c.

(46 ± 8 ± 5)· 10⁻⁶ BaBar [85]

B₉₂₀ = B(τ⁻ → π⁻ f₁(2π⁻ 2π⁺) ν_τ)

(52.4 ± 4.4)· 10⁻⁶ average

(52.4 ± 4.4)· 10⁻⁶ average with u.c.

(52.0 ± 3.1 ± 3.7)· 10⁻⁶ BaBar [85]

B₉₃₀ = B(τ⁻ → 2π⁻ π⁺ η(π⁺ π⁻ π⁰) ν_τ (ex.K⁰))

(50.7 ± 3.0)· 10⁻⁶ average

(50.7 ± 3.0)· 10⁻⁶ average with u.c.

(53.9 ± 2.7 ± 4.1)· 10⁻⁶ BaBar [85]

B₉₄₄ = B(τ⁻ → 2π⁻ π⁺ η(γ γ) ν_τ (ex.K⁰))

(86.6 ± 5.1)· 10⁻⁶ average

(86.8 ± 5.1)· 10⁻⁶ average with u.c.

(82.6 ± 3.5 ± 5.1)· 10⁻⁶ BaBar [85]

B₉₄₅ = B(τ⁻ → π⁻ 2π⁰ η ν_τ (ex.K⁰))

(0.19 ± 0.04)· 10⁻³ average

(0.19 ± 0.04)· 10⁻³ average with u.c.

B_ur = B(τ⁻ unitarity residual)

0.0007 ± 0.0011 average

0.0 average with u.c.

B_all = B(τ⁻ → all modes)

0.9993 ± 0.0011 average

1.0 average with u.c.

Table 4: Nuisance fit parameter of the τ branching fractions fit.

Nuisance fit parameter Experiment Reference

B₉₈₀ = B(a₁⁻ → π⁻ γ)

0.0021 ± 0.0008 average

0.0021 ± 0.0008 ± 0.0000 ALEPH [25]

4.4 Equality constraints

The constraints on the τ branching-fractions used in the fit are listed in the following equations.

The equations involve fit parameters associated to τ branching fractions, nuisance fit parameters, external parameters corresponding to non-τ branching fractions (like B_{K_S → π⁰π⁰}), probabilities corresponding to squared moduli of amplitudes of quantum state mixtures, such as K⁰, K⁰, K_S, K_L. The probabilities 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 2022 and 2023 update [8] averages.

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})

B_3 / 5 =

B₃

B₅

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

B₈ =

B₉ + B₁₀

B_8 / 5 =

B₈

B₅

B_9 / 5 =

B₉

B₅

B_10 / 5 =

B₁₀

B₅

B_10 / 9 =

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

B_19 / 13 =

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

B_26 / 13 =

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⁻}

B_57 / 55 =

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}

B_76 / 54 =

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⁻}

B_80 / 60 =

B₈₀

B₆₀

B₈₁ =

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

B_81 / 69 =

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⁻}

B_85 / 60 =

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

B_93 / 60 =

B₉₃

B₆₀

B_94 / 69 =

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

B_150 / 66 =

B₁₅₀

B₆₆

B_152 / 54 =

B₁₅₂

B₅₄

B_152 / 76 =

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>}))

B₈₀₅ =

B₉₈₀/(1− B₉₈₀)·( B₂₀ + B₆₂)

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_ur =

1 − B_all

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

HFLAV-Tau 2023 Report

4 Branching fraction fit

4.1 Changes with respect to the previous report

4.2 Differences with respect to the PDG fit

4.3 Fit results

4.4 Equality constraints

HFLAV-Tau 2023 Report