HFAG-Tau Summer 2016 - Branching fraction fit

A global fit of the available experimental measurements is used to determine the τ branching fractions, together with their uncertainties and statistical correlations. The τ branching fractions provide a test for theory predictions based on the Standard Model (SM) EW and QCD interactions and can be further elaborated to test the EW charged-current universality for leptons, to determine the CKM matrix coefficient |V_us| and the QCD coupling constant α_s at the τ mass.

The measurements used in the fit are listed in Table 1 and consist of either τ decay mode branching fractions, labelled as Γ_i, or ratios of two τ decay mode branching fractions, labelled as Γ_i/Γ_j. A minimum χ² fit is performed for all the measured quantities and for some additional branching fractions and ratios of branching fractions, and all ft results are listed in Table 1. Some fitted quantities are equal to the ratio of two other fitted quantities, as documented with the notation Γ_i/Γ_j in Table 1. Some fitted quantities are sum of other fitted quantities, for instance Γ₈ = B (τ→h⁻ ν_τ) is the sum of Γ₉ = B (τ→π⁻ ν_τ) and Γ₁₀ = B (τ→K⁻ ν_τ). The symbol h is used to mean either a π or K. Section 2.7 lists all equations relating one quantity to the sum of other quantities. In the following, we refer to both types of relations between fitted quantities collectively as constraint equations or constraints. The fit χ² is minimized subject to all these above mentioned constraints. The fit procedure is equivalent to the one employed in the former HFAG reports [1, 2, 3].

2.1 Technical implementation of the fit procedure

The fit computes the quantities q_i by minimizing a χ² while respecting a series of equality constraints on the q_i. The χ² is computed using the measurements m_i and their covariance matrix E_ij as χ² = (m_i − A_ikq_k)^t E_ij⁻¹ (m_j − A_jlq_l) where the model matrix A_ij is used to get the vector of the predicted measurements m_i^′ from the vector of the fit parameters q_j as m_i^′= A_ijq_j. In this particular implementation the measurements are grouped by the quantity that they measure, and all quantities with at least one measurement correspond to a fit parameter. Therefore, the matrix A_ij has one row per measurement m_i and one column per fitted quantity q_j, with unity coefficients for the rows and column that identify a measurement m_i of the quantity q_j, respectively. In summary, the fit requires:

where Eq. 2 defines the relations between the quantities q_i and its left term defines the respective constraint expressions. Using the method of Lagrange multipliers, a set of equations is obtained by taking the derivatives with respect to the fitted quantities q_k and the Lagrange multipliers λ_r of the sum of the χ² and the constraint expressions multiplied by the Lagrange multipliers λ_r, one for each constraint:

Eq. 4 defines a set of equations for the vector of the unknowns (q_k, λ_r), some of which may be non-linear, in case of non-linear constraints. An iterative minimization procedure approximates at each step the non-linear constraint expressions by their first order Taylor expansion around the current values of the fitted quantities, q_s:

where c_r^′ are the resulting constant known terms, independent of q_s at first order. After linearization, the differentiation by q_k and λ_r is trivial and leads to a set of linear equations

where u_j = (q_k, λ_r) and v_i is the vector of the known constant terms running over the index k and then r in the right terms of Eq. 7 and Eq. 8, respectively. Solving the equation set in Eq. 9 by matrix inversion gives the the fitted quantities and their variance and covariance matrix, using the measurements and their variance and covariance matrix. The fit procedure starts by computing the linear approximation of the non-linear constraint expressions around the quantities seed values. With an iterative procedure, the unknowns are updated at each step by solving the equations and the equations are then linearized around the updated values, until the variation of the fitted unknowns is reduced below a numerically small threshold.

2.2 Fit results

The fit output consists of 135 fitted quantities that correspond to either branching fractions or ratios of branching fractions. The fitted quantities values and undertainties are listed in Table 1. The off-diagonal statistical correlation terms between a subset of 47 “basis quantities” are listed in Section 2.6. All the remaining statistical correlation terms can be obtained using the constraint equations listed in Table 1 and Section 2.7.

The fit has χ²/d.o.f. = 137.3/123, corresponding to a confidence level CL = 17.84%. We use a total of 170 measurements to fit the above mentioned 135 quantities subjected to 88 constraints. Although the unitarity constraint is not applied, the fit is statistically consistent with unitarity, where the residual is Γ₉₉₈ = 1 − Γ_All = (0.0355 ± 0.1031) · 10⁻².

A scale factor of 5.44 (as in the three previous reports [1, 2, 3]) has been applied to the published uncertainties of the two severely inconsistent measurements of Γ₉₆ = τ → KKKν by BaBar and Belle. The scale factor has been determined using the PDG procedure, i.e. to the proper size in order to obtain a reduced χ² equal to 1 when fitting just the two Γ₉₆ measurements.

For several old results, for historical reasons, the table lists the statistical errors formed from the quadratic sum of the statistical and systematic errors, setting the systematics errors to zero: this does not affect the fit result as the systematic and statistical errors are treated in the same way.

2.3 Changes with respect to the previous report

The following changes have been introduced with respect to the previous HFAG report [3].

They where announced in 2008 and 2009, respectively, but have not been published yet.

In the 2014 report, for several BaBar and Belle experimental results we used more precise numerical values than the published ones, using internal information from the Collaborations. We revert to the published figures in this report, as the improvements in the fit results were negligible. In so doing, we use in this report the same values that are used in the PDG 2016 fit.

The Belle result on τ⁻ → K_S⁰ (particles)⁻ ν_τ [6] has been discarded, because it was determined that the published information does not permit a reliable determination of the correlations with the other results in the same paper. The correlations estimated for the HFAG 2014 report were inconsistent and made the covariance matrix of the results in the corresponding paper non positive-definite, as well as the overall correlation matrix for the branching ratio fit results. It has been found that the inconsistency had negligible impact on lepton universality and |V_us| measurements.

The ALEPH result on Γ₄₆ (τ⁻ → π⁻ K⁰ K⁰ ν_τ) [7] has been removed from the fit inputs, since it is the simply sum of twice Γ₄₇ = π⁻ K_S⁰ K_S⁰ ν_τ and Γ₄₈ = π⁻ K_S⁰ K_L⁰ ν_τ from the same paper, hence 100% correlated with them.

Several minor corrections have been applied to the constraints. The list of constraints included in the following fully documents the changes when compared with the same list in the 2014 edition. In some cases the relation equating one decay mode to a sum of modes included some minor terms that did not match the mode definitions. In other cases, the sum included modes with overlapping components. The effects on the 2014 fit results have been found to be modest with respect to the quoted uncertainties.

For instance, the definition of the total branching fraction has been updated as follows:

In the 2014 definition, the term Γ₇₈ = h⁻ h⁻ h⁺ 3π⁰ ν_τ included the contributions of Γ₅₀ = π⁻ π⁰ K_S⁰ K_S⁰ ν_τ and Γ₁₃₂ = π⁻ K⁰ η ν_τ, which were already included. In the present definition, Γ₇₈ has been replaced with modes whose sum corresponds to Γ₈₁₀ = 2π⁻ π⁺ 3π⁰ ν_τ (ex. K⁰). As in 2014, the total τ branching fraction Γ_All definition includes two modes that have overlapping final states, to a minor extent that we consider negligible:

Finally, we updated to the PDG 2015 results [8] all the parameters corresponding to the measurements systematic biases and uncertainties and all the parameters appearing in the constraint equations in Section 2.7 and Table 1.

2.4 Differences between the HFAG 2016 fit and the PDG 2016 fit

As is standard for the PDG branching fraction fits, the PDG 2016 τ branching fraction fit is unitarity constrained, while the HFAG 2016 fit is unconstrained.

The HFAG-Tau fit uses the ALEPH measurements of branching fractions defined according to the final state content of “hadrons” and kaons, where a “hadron” corresponds to either a pion or a kaon, since this set of results is closer to the actual experimental measurements and facilitates a more comprehensive treatment of the experimental results correlations [1]. The PDG 2016 fit on the other hand continues to use – as in the past editions – the ALEPH measurements of modes with pions and kaons, which correspond to the final set of published measurements of the collaboration. It is planned to eventually update the PDG fit to use the same ALEPH measurement set that is used by HFAG.

The HFAG 2016 fit, as in 2014, uses the ALEPH estimate for Γ₈₀₅ = B (τ→a₁⁻ (→ π⁻ γ) ν_τ), which is not a direct experimental measurement. The PDG 2016 fit uses the PDG average of B (a₁→πγ) as a parameter and defines Γ₈₀₅ = B (a₁→πγ)×B (τ → 3π ν). As a consequence, the PDG fit procedure does not take into account the large uncertainty on B (a₁→πγ), resulting in an underestimated fit uncertainty on Γ₈₀₅. Therefore, in this case an appropriate correction has to be applied after the fit.

2.5 Branching ratio fit results and experimental inputs

Table 1: HFAG Summer 2016 branching fractions fit results.

τ lepton branching fraction Fit value / Exp. HFAG Fit / Ref.

Γ₁ = (particles)⁻ ≥ 0  neutrals ≥ 0   K⁰  ν_τ

0.8519 ± 0.0011 HFAG Summer 2016 fit

Γ₂ = (particles)⁻ ≥ 0  neutrals ≥ 0   K_L⁰  ν_τ

0.8453 ± 0.0010 HFAG Summer 2016 fit

Γ₃ = µ⁻ ν_µν_τ

0.17392 ± 0.00040 HFAG Summer 2016 fit

0.17319 ± 0.00077 ± 0.00000 ALEPH [9]

0.17325 ± 0.00095 ± 0.00077 DELPHI [10]

0.17342 ± 0.00110 ± 0.00067 L3 [11]

0.17340 ± 0.00090 ± 0.00060 OPAL [12]

Γ₃

Γ₅

=
µ⁻ ν_µν_τ

e⁻ ν_e ν_τ

0.9762 ± 0.0028 HFAG Summer 2016 fit

0.9970 ± 0.0350 ± 0.0400 ARGUS [13]

0.9796 ± 0.0016 ± 0.0036 BaBar [14]

0.9777 ± 0.0063 ± 0.0087 CLEO [15]

Γ₅ = e⁻ ν_e ν_τ

0.17816 ± 0.00041 HFAG Summer 2016 fit

0.17837 ± 0.00080 ± 0.00000 ALEPH [9]

0.17760 ± 0.00060 ± 0.00170 CLEO [15]

0.17877 ± 0.00109 ± 0.00110 DELPHI [10]

0.17806 ± 0.00104 ± 0.00076 L3 [11]

0.17810 ± 0.00090 ± 0.00060 OPAL [16]

Γ₇ = h⁻ ≥ 0   K_L⁰  ν_τ

0.12023 ± 0.00054 HFAG Summer 2016 fit

0.12400 ± 0.00700 ± 0.00700 DELPHI [17]

0.12470 ± 0.00260 ± 0.00430 L3 [18]

0.12100 ± 0.00700 ± 0.00500 OPAL [19]

Γ₈ = h⁻ ν_τ

0.11506 ± 0.00054 HFAG Summer 2016 fit

0.11524 ± 0.00105 ± 0.00000 ALEPH [9]

0.11520 ± 0.00050 ± 0.00120 CLEO [15]

0.11571 ± 0.00120 ± 0.00114 DELPHI [20]

0.11980 ± 0.00130 ± 0.00160 OPAL [21]

Γ₈

Γ₅

=
h⁻ ν_τ

e⁻ ν_e ν_τ

0.6458 ± 0.0033 HFAG Summer 2016 fit

Γ₉ = π⁻ ν_τ

0.10810 ± 0.00053 HFAG Summer 2016 fit

Γ₉

Γ₅

=
π⁻ ν_τ

e⁻ ν_e ν_τ

0.6068 ± 0.0032 HFAG Summer 2016 fit

0.5945 ± 0.0014 ± 0.0061 BaBar [14]

Γ₁₀ = K⁻ ν_τ

(0.6960 ± 0.0096) · 10⁻² HFAG Summer 2016 fit

(0.6960 ± 0.0287 ± 0.0000) · 10⁻² ALEPH [22]

(0.6600 ± 0.0700 ± 0.0900) · 10⁻² CLEO [23]

(0.8500 ± 0.1800 ± 0.0000) · 10⁻² DELPHI [24]

(0.6580 ± 0.0270 ± 0.0290) · 10⁻² OPAL [25]

Γ₁₀

Γ₅

=
K⁻ ν_τ

e⁻ ν_e ν_τ

(3.906 ± 0.054) · 10⁻² HFAG Summer 2016 fit

(3.882 ± 0.032 ± 0.057) · 10⁻² BaBar [14]

Γ₁₀

Γ₉

=
K⁻ ν_τ

π⁻ ν_τ

(6.438 ± 0.094) · 10⁻² HFAG Summer 2016 fit

Γ₁₁ = h⁻ ≥ 1   neutrals  ν_τ

0.36973 ± 0.00097 HFAG Summer 2016 fit

Γ₁₂ = h⁻ ≥ 1  π⁰  ν_τ (ex.  K⁰)

0.36475 ± 0.00097 HFAG Summer 2016 fit

Γ₁₃ = h⁻ π⁰ ν_τ

0.25935 ± 0.00091 HFAG Summer 2016 fit

0.25924 ± 0.00129 ± 0.00000 ALEPH [9]

0.25670 ± 0.00010 ± 0.00390 Belle [26]

0.25870 ± 0.00120 ± 0.00420 CLEO [27]

0.25740 ± 0.00201 ± 0.00138 DELPHI [20]

0.25050 ± 0.00350 ± 0.00500 L3 [18]

0.25890 ± 0.00170 ± 0.00290 OPAL [21]

Γ₁₄ = π⁻ π⁰ ν_τ

0.25502 ± 0.00092 HFAG Summer 2016 fit

Γ₁₆ = K⁻ π⁰ ν_τ

(0.4327 ± 0.0149) · 10⁻² HFAG Summer 2016 fit

(0.4440 ± 0.0354 ± 0.0000) · 10⁻² ALEPH [22]

(0.4160 ± 0.0030 ± 0.0180) · 10⁻² BaBar [28]

(0.5100 ± 0.1000 ± 0.0700) · 10⁻² CLEO [23]

(0.4710 ± 0.0590 ± 0.0230) · 10⁻² OPAL [29]

Γ₁₇ = h⁻ ≥ 2   π⁰  ν_τ

0.10775 ± 0.00095 HFAG Summer 2016 fit

0.09910 ± 0.00310 ± 0.00270 OPAL [21]

Γ₁₈ = h⁻ 2π⁰ ν_τ

(9.458 ± 0.097) · 10⁻² HFAG Summer 2016 fit

Γ₁₉ = h⁻ 2π⁰ ν_τ (ex.  K⁰)

(9.306 ± 0.097) · 10⁻² HFAG Summer 2016 fit

(9.295 ± 0.122 ± 0.000) · 10⁻² ALEPH [9]

(9.498 ± 0.320 ± 0.275) · 10⁻² DELPHI [20]

(8.880 ± 0.370 ± 0.420) · 10⁻² L3 [18]

Γ₁₉

Γ₁₃

=
h⁻ 2π⁰ ν_τ (ex.  K⁰)

h⁻ π⁰ ν_τ

0.3588 ± 0.0044 HFAG Summer 2016 fit

0.3420 ± 0.0060 ± 0.0160 CLEO [30]

Γ₂₀ = π⁻ 2π⁰ ν_τ (ex. K⁰)

(9.242 ± 0.100) · 10⁻² HFAG Summer 2016 fit

Γ₂₃ = K⁻ 2π⁰ ν_τ (ex. K⁰)

(0.0640 ± 0.0220) · 10⁻² HFAG Summer 2016 fit

(0.0560 ± 0.0250 ± 0.0000) · 10⁻² ALEPH [22]

(0.0900 ± 0.1000 ± 0.0300) · 10⁻² CLEO [23]

Γ₂₄ = h⁻ ≥ 3  π⁰  ν_τ

(1.318 ± 0.065) · 10⁻² HFAG Summer 2016 fit

Γ₂₅ = h⁻ ≥ 3  π⁰  ν_τ (ex.  K⁰)

(1.233 ± 0.065) · 10⁻² HFAG Summer 2016 fit

(1.403 ± 0.214 ± 0.224) · 10⁻² DELPHI [20]

Γ₂₆ = h⁻ 3π⁰ ν_τ

(1.158 ± 0.072) · 10⁻² HFAG Summer 2016 fit

(1.082 ± 0.093 ± 0.000) · 10⁻² ALEPH [9]

(1.700 ± 0.240 ± 0.380) · 10⁻² L3 [18]

Γ₂₆

Γ₁₃

=
h⁻ 3π⁰ ν_τ

h⁻ π⁰ ν_τ

(4.465 ± 0.277) · 10⁻² HFAG Summer 2016 fit

(4.400 ± 0.300 ± 0.500) · 10⁻² CLEO [30]

Γ₂₇ = π⁻ 3π⁰ ν_τ (ex. K⁰)

(1.029 ± 0.075) · 10⁻² HFAG Summer 2016 fit

Γ₂₈ = K⁻ 3π⁰ ν_τ (ex. K⁰,η)

(4.283 ± 2.161) · 10⁻⁴ HFAG Summer 2016 fit

(3.700 ± 2.371 ± 0.000) · 10⁻⁴ ALEPH [22]

Γ₂₉ = h⁻ 4π⁰ ν_τ (ex.  K⁰)

(0.1568 ± 0.0391) · 10⁻² HFAG Summer 2016 fit

(0.1600 ± 0.0500 ± 0.0500) · 10⁻² CLEO [30]

Γ₃₀ = h⁻ 4π⁰ ν_τ (ex. K⁰,η)

(0.1099 ± 0.0391) · 10⁻² HFAG Summer 2016 fit

(0.1120 ± 0.0509 ± 0.0000) · 10⁻² ALEPH [9]

Γ₃₁ = K⁻ ≥ 0  π⁰ ≥ 0  K⁰ ≥ 0  γ ν_τ

(1.545 ± 0.030) · 10⁻² HFAG Summer 2016 fit

(1.700 ± 0.120 ± 0.190) · 10⁻² CLEO [23]

(1.540 ± 0.240 ± 0.000) · 10⁻² DELPHI [24]

(1.528 ± 0.039 ± 0.040) · 10⁻² OPAL [25]

Γ₃₂ = K⁻ ≥ 1  (π⁰ or K⁰ or γ) ν_τ

(0.8528 ± 0.0286) · 10⁻² HFAG Summer 2016 fit

Γ₃₃ = K_S⁰ (particles)⁻ ν_τ

(0.9372 ± 0.0292) · 10⁻² HFAG Summer 2016 fit

(0.9700 ± 0.0849 ± 0.0000) · 10⁻² ALEPH [7]

(0.9700 ± 0.0900 ± 0.0600) · 10⁻² OPAL [31]

Γ₃₄ = h⁻ K⁰ ν_τ

(0.9865 ± 0.0139) · 10⁻² HFAG Summer 2016 fit

(0.8550 ± 0.0360 ± 0.0730) · 10⁻² CLEO [32]

Γ₃₅ = π⁻ K⁰ ν_τ

(0.8386 ± 0.0141) · 10⁻² HFAG Summer 2016 fit

(0.9280 ± 0.0564 ± 0.0000) · 10⁻² ALEPH [22]

(0.8320 ± 0.0025 ± 0.0150) · 10⁻² Belle [6]

(0.9500 ± 0.1500 ± 0.0600) · 10⁻² L3 [33]

(0.9330 ± 0.0680 ± 0.0490) · 10⁻² OPAL [34]

Γ₃₇ = K⁻ K⁰ ν_τ

(0.1479 ± 0.0053) · 10⁻² HFAG Summer 2016 fit

(0.1580 ± 0.0453 ± 0.0000) · 10⁻² ALEPH [7]

(0.1620 ± 0.0237 ± 0.0000) · 10⁻² ALEPH [22]

(0.1480 ± 0.0013 ± 0.0055) · 10⁻² Belle [6]

(0.1510 ± 0.0210 ± 0.0220) · 10⁻² CLEO [32]

Γ₃₈ = K⁻ K⁰ ≥ 0   π⁰  ν_τ

(0.2982 ± 0.0079) · 10⁻² HFAG Summer 2016 fit

(0.3300 ± 0.0550 ± 0.0390) · 10⁻² OPAL [34]

Γ₃₉ = h⁻ K⁰ π⁰ ν_τ

(0.5314 ± 0.0134) · 10⁻² HFAG Summer 2016 fit

(0.5620 ± 0.0500 ± 0.0480) · 10⁻² CLEO [32]

Γ₄₀ = π⁻ K⁰ π⁰ ν_τ

(0.3812 ± 0.0129) · 10⁻² HFAG Summer 2016 fit

(0.2940 ± 0.0818 ± 0.0000) · 10⁻² ALEPH [7]

(0.3470 ± 0.0646 ± 0.0000) · 10⁻² ALEPH [22]

(0.3860 ± 0.0031 ± 0.0135) · 10⁻² Belle [6]

(0.4100 ± 0.1200 ± 0.0300) · 10⁻² L3 [33]

Γ₄₂ = K⁻ π⁰ K⁰ ν_τ

(0.1502 ± 0.0071) · 10⁻² HFAG Summer 2016 fit

(0.1520 ± 0.0789 ± 0.0000) · 10⁻² ALEPH [7]

(0.1430 ± 0.0291 ± 0.0000) · 10⁻² ALEPH [22]

(0.1496 ± 0.0019 ± 0.0073) · 10⁻² Belle [6]

(0.1450 ± 0.0360 ± 0.0200) · 10⁻² CLEO [32]

Γ₄₃ = π⁻ K⁰ ≥ 1   π⁰  ν_τ

(0.4046 ± 0.0260) · 10⁻² HFAG Summer 2016 fit

(0.3240 ± 0.0740 ± 0.0660) · 10⁻² OPAL [34]

Γ₄₄ = π⁻ K⁰ π⁰ π⁰ ν_τ (ex. K⁰)

(2.340 ± 2.306) · 10⁻⁴ HFAG Summer 2016 fit

(2.600 ± 2.400 ± 0.000) · 10⁻⁴ ALEPH [35]

Γ₄₆ = π⁻ K⁰ K⁰ ν_τ

(0.1513 ± 0.0247) · 10⁻² HFAG Summer 2016 fit

Γ₄₇ = π⁻ K_S⁰ K_S⁰ ν_τ

(2.332 ± 0.065) · 10⁻⁴ HFAG Summer 2016 fit

(2.600 ± 1.118 ± 0.000) · 10⁻⁴ ALEPH [7]

(2.310 ± 0.040 ± 0.080) · 10⁻⁴ BaBar [36]

(2.330 ± 0.033 ± 0.093) · 10⁻⁴ Belle [6]

(2.300 ± 0.500 ± 0.300) · 10⁻⁴ CLEO [32]

Γ₄₈ = π⁻ K_S⁰ K_L⁰ ν_τ

(0.1047 ± 0.0247) · 10⁻² HFAG Summer 2016 fit

(0.1010 ± 0.0264 ± 0.0000) · 10⁻² ALEPH [7]

Γ₄₉ = π⁻ K⁰ K⁰ π⁰ ν_τ

(3.540 ± 1.193) · 10⁻⁴ HFAG Summer 2016 fit

Γ₅₀ = π⁻ π⁰ K_S⁰ K_S⁰ ν_τ

(1.815 ± 0.207) · 10⁻⁵ HFAG Summer 2016 fit

(1.600 ± 0.200 ± 0.220) · 10⁻⁵ BaBar [36]

(2.000 ± 0.216 ± 0.202) · 10⁻⁵ Belle [6]

Γ₅₁ = π⁻ π⁰ K_S⁰ K_L⁰ ν_τ

(3.177 ± 1.192) · 10⁻⁴ HFAG Summer 2016 fit

(3.100 ± 1.100 ± 0.500) · 10⁻⁴ ALEPH [7]

Γ₅₃ = K⁰ h⁻ h⁻ h⁺ ν_τ

(2.218 ± 2.024) · 10⁻⁴ HFAG Summer 2016 fit

(2.300 ± 2.025 ± 0.000) · 10⁻⁴ ALEPH [7]

Γ₅₄ = h⁻ h⁻ h⁺ ≥ 0  neutrals ≥ 0   K_L⁰  ν_τ

0.15215 ± 0.00061 HFAG Summer 2016 fit

0.15000 ± 0.00400 ± 0.00300 CELLO [37]

0.14400 ± 0.00600 ± 0.00300 L3 [38]

0.15100 ± 0.00800 ± 0.00600 TPC [39]

Γ₅₅ = h⁻ h⁻ h⁺ ≥ 0   neutrals  ν_τ (ex.  K⁰)

0.14567 ± 0.00057 HFAG Summer 2016 fit

0.14556 ± 0.00105 ± 0.00076 L3 [40]

0.14960 ± 0.00090 ± 0.00220 OPAL [41]

Γ₅₆ = h⁻ h⁻ h⁺ ν_τ

(9.780 ± 0.054) · 10⁻² HFAG Summer 2016 fit

Γ₅₇ = h⁻ h⁻ h⁺ ν_τ (ex.  K⁰)

(9.439 ± 0.053) · 10⁻² HFAG Summer 2016 fit

(9.510 ± 0.070 ± 0.200) · 10⁻² CLEO [42]

(9.317 ± 0.090 ± 0.082) · 10⁻² DELPHI [20]

Γ₅₇

Γ₅₅

=
h⁻ h⁻ h⁺ ν_τ (ex.  K⁰)

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

0.6480 ± 0.0030 HFAG Summer 2016 fit

0.6600 ± 0.0040 ± 0.0140 OPAL [41]

Γ₅₈ = h⁻ h⁻ h⁺ ν_τ (ex.  K⁰, ω)

(9.408 ± 0.053) · 10⁻² HFAG Summer 2016 fit

(9.469 ± 0.096 ± 0.000) · 10⁻² ALEPH [9]

Γ₅₉ = π⁻ π⁺ π⁻ ν_τ

(9.290 ± 0.052) · 10⁻² HFAG Summer 2016 fit

Γ₆₀ = π⁻ π⁺ π⁻ ν_τ (ex.  K⁰)

(9.000 ± 0.051) · 10⁻² HFAG Summer 2016 fit

(8.830 ± 0.010 ± 0.130) · 10⁻² BaBar [43]

(8.420 ± 0.000 _−0.250^+0.260) · 10⁻² Belle [44]

(9.130 ± 0.050 ± 0.460) · 10⁻² CLEO3 [45]

Γ₆₂ = π⁻ π⁻ π⁺ ν_τ (ex. K⁰,ω)

(8.970 ± 0.052) · 10⁻² HFAG Summer 2016 fit

Γ₆₃ = h⁻ h⁻ h⁺ ≥ 1   neutrals  ν_τ

(5.325 ± 0.050) · 10⁻² HFAG Summer 2016 fit

Γ₆₄ = h⁻ h⁻ h⁺ ≥ 1   π⁰  ν_τ (ex.  K⁰)

(5.120 ± 0.049) · 10⁻² HFAG Summer 2016 fit

Γ₆₅ = h⁻ h⁻ h⁺ π⁰ ν_τ

(4.790 ± 0.052) · 10⁻² HFAG Summer 2016 fit

Γ₆₆ = h⁻ h⁻ h⁺ π⁰ ν_τ (ex.  K⁰)

(4.606 ± 0.051) · 10⁻² HFAG Summer 2016 fit

(4.734 ± 0.077 ± 0.000) · 10⁻² ALEPH [9]

(4.230 ± 0.060 ± 0.220) · 10⁻² CLEO [42]

(4.545 ± 0.106 ± 0.103) · 10⁻² DELPHI [20]

Γ₆₇ = h⁻ h⁻ h⁺ π⁰ ν_τ (ex.  K⁰, ω)

(2.820 ± 0.070) · 10⁻² HFAG Summer 2016 fit

Γ₆₈ = π⁻ π⁺ π⁻ π⁰ ν_τ

(4.651 ± 0.053) · 10⁻² HFAG Summer 2016 fit

Γ₆₉ = π⁻ π⁺ π⁻ π⁰ ν_τ (ex.  K⁰)

(4.519 ± 0.052) · 10⁻² HFAG Summer 2016 fit

(4.190 ± 0.100 ± 0.210) · 10⁻² CLEO [46]

Γ₇₀ = π⁻ π⁻ π⁺ π⁰ ν_τ (ex. K⁰,ω)

(2.769 ± 0.071) · 10⁻² HFAG Summer 2016 fit

Γ₇₄ = h⁻ h⁻ h⁺ ≥ 2  π⁰  ν_τ (ex.  K⁰)

(0.5135 ± 0.0312) · 10⁻² HFAG Summer 2016 fit

(0.5610 ± 0.0680 ± 0.0950) · 10⁻² DELPHI [20]

Γ₇₅ = h⁻ h⁻ h⁺ 2π⁰ ν_τ

(0.5024 ± 0.0310) · 10⁻² HFAG Summer 2016 fit

Γ₇₆ = h⁻ h⁻ h⁺ 2π⁰ ν_τ (ex.  K⁰)

(0.4925 ± 0.0310) · 10⁻² HFAG Summer 2016 fit

(0.4350 ± 0.0461 ± 0.0000) · 10⁻² ALEPH [9]

Γ₇₆

Γ₅₄

=
h⁻ h⁻ h⁺ 2π⁰ ν_τ (ex.  K⁰)

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

(3.237 ± 0.202) · 10⁻² HFAG Summer 2016 fit

(3.400 ± 0.200 ± 0.300) · 10⁻² CLEO [47]

Γ₇₇ = h⁻ h⁻ h⁺ 2π⁰ ν_τ (ex. K⁰,ω,η)

(9.759 ± 3.550) · 10⁻⁴ HFAG Summer 2016 fit

Γ₇₈ = h⁻ h⁻ h⁺ 3π⁰ ν_τ

(2.107 ± 0.299) · 10⁻⁴ HFAG Summer 2016 fit

(2.200 ± 0.300 ± 0.400) · 10⁻⁴ CLEO [48]

Γ₇₉ = K⁻ h⁻ h⁺ ≥ 0   neutrals  ν_τ

(0.6297 ± 0.0141) · 10⁻² HFAG Summer 2016 fit

Γ₈₀ = K⁻ π⁻ h⁺ ν_τ (ex.  K⁰)

(0.4363 ± 0.0073) · 10⁻² HFAG Summer 2016 fit

Γ₈₀

Γ₆₀

=
K⁻ π⁻ h⁺ ν_τ (ex.  K⁰)

π⁻ π⁺ π⁻ ν_τ (ex.  K⁰)

(4.847 ± 0.080) · 10⁻² HFAG Summer 2016 fit

(5.440 ± 0.210 ± 0.530) · 10⁻² CLEO [49]

Γ₈₁ = K⁻ π⁻ h⁺ π⁰ ν_τ (ex.  K⁰)

(8.726 ± 1.177) · 10⁻⁴ HFAG Summer 2016 fit

Γ₈₁

Γ₆₉

=
K⁻ π⁻ h⁺ π⁰ ν_τ (ex.  K⁰)

π⁻ π⁺ π⁻ π⁰ ν_τ (ex.  K⁰)

(1.931 ± 0.266) · 10⁻² HFAG Summer 2016 fit

(2.610 ± 0.450 ± 0.420) · 10⁻² CLEO [49]

Γ₈₂ = K⁻ π⁻ π⁺ ≥ 0   neutrals  ν_τ

(0.4780 ± 0.0137) · 10⁻² HFAG Summer 2016 fit

(0.5800 _−0.1300^+0.1500 ± 0.1200) · 10⁻² TPC [50]

Γ₈₃ = K⁻ π⁻ π⁺ ≥ 0   π⁰  ν_τ (ex.  K⁰)

(0.3741 ± 0.0135) · 10⁻² HFAG Summer 2016 fit

Γ₈₄ = K⁻ π⁻ π⁺ ν_τ

(0.3441 ± 0.0070) · 10⁻² HFAG Summer 2016 fit

Γ₈₅ = K⁻ π⁺ π⁻ ν_τ (ex.  K⁰)

(0.2929 ± 0.0067) · 10⁻² HFAG Summer 2016 fit

(0.2140 ± 0.0470 ± 0.0000) · 10⁻² ALEPH [51]

(0.2730 ± 0.0020 ± 0.0090) · 10⁻² BaBar [43]

(0.3300 ± 0.0010 _−0.0170^+0.0160) · 10⁻² Belle [44]

(0.3840 ± 0.0140 ± 0.0380) · 10⁻² CLEO3 [45]

(0.4150 ± 0.0530 ± 0.0400) · 10⁻² OPAL [29]

Γ₈₅

Γ₆₀

=
K⁻ π⁺ π⁻ ν_τ (ex. K⁰)

π⁻ π⁺ π⁻ ν_τ (ex. K⁰)

(3.254 ± 0.074) · 10⁻² HFAG Summer 2016 fit

Γ₈₇ = K⁻ π⁻ π⁺ π⁰ ν_τ

(0.1331 ± 0.0119) · 10⁻² HFAG Summer 2016 fit

Γ₈₈ = K⁻ π⁻ π⁺ π⁰ ν_τ (ex.  K⁰)

(8.115 ± 1.168) · 10⁻⁴ HFAG Summer 2016 fit

(6.100 ± 4.295 ± 0.000) · 10⁻⁴ ALEPH [51]

(7.400 ± 0.800 ± 1.100) · 10⁻⁴ CLEO3 [52]

Γ₈₉ = K⁻ π⁻ π⁺ π⁰ ν_τ (ex.  K⁰, η)

(7.761 ± 1.168) · 10⁻⁴ HFAG Summer 2016 fit

Γ₉₂ = π⁻ K⁻ K⁺ ≥ 0   neutrals  ν_τ

(0.1495 ± 0.0033) · 10⁻² HFAG Summer 2016 fit

(0.1590 ± 0.0530 ± 0.0200) · 10⁻² OPAL [53]

(0.1500 _−0.0700^+0.0900 ± 0.0300) · 10⁻² TPC [50]

Γ₉₃ = π⁻ K⁻ K⁺ ν_τ

(0.1434 ± 0.0027) · 10⁻² HFAG Summer 2016 fit

(0.1630 ± 0.0270 ± 0.0000) · 10⁻² ALEPH [51]

(0.1346 ± 0.0010 ± 0.0036) · 10⁻² BaBar [43]

(0.1550 ± 0.0010 _−0.0050^+0.0060) · 10⁻² Belle [44]

(0.1550 ± 0.0060 ± 0.0090) · 10⁻² CLEO3 [45]

Γ₉₃

Γ₆₀

=
π⁻ K⁻ K⁺ ν_τ

π⁻ π⁺ π⁻ ν_τ (ex.  K⁰)

(1.593 ± 0.030) · 10⁻² HFAG Summer 2016 fit

(1.600 ± 0.150 ± 0.300) · 10⁻² CLEO [49]

Γ₉₄ = π⁻ K⁻ K⁺ π⁰ ν_τ

(0.611 ± 0.183) · 10⁻⁴ HFAG Summer 2016 fit

(7.500 ± 3.265 ± 0.000) · 10⁻⁴ ALEPH [51]

(0.550 ± 0.140 ± 0.120) · 10⁻⁴ CLEO3 [52]

Γ₉₄

Γ₆₉

=
π⁻ K⁻ K⁺ π⁰ ν_τ

π⁻ π⁺ π⁻ π⁰ ν_τ (ex.  K⁰)

(0.1353 ± 0.0405) · 10⁻² HFAG Summer 2016 fit

(0.7900 ± 0.4400 ± 0.1600) · 10⁻² CLEO [49]

Γ₉₆ = K⁻ K⁻ K⁺ ν_τ

(2.174 ± 0.800) · 10⁻⁵ HFAG Summer 2016 fit

(1.578 ± 0.130 ± 0.123) · 10⁻⁵ BaBar [43]

(3.290 ± 0.170 _−0.200^+0.190) · 10⁻⁵ Belle [44]

Γ₁₀₂ = 3h⁻ 2h⁺ ≥ 0   neutrals  ν_τ (ex.  K⁰)

(0.0985 ± 0.0037) · 10⁻² HFAG Summer 2016 fit

(0.0970 ± 0.0050 ± 0.0110) · 10⁻² CLEO [54]

(0.1020 ± 0.0290 ± 0.0000) · 10⁻² HRS [55]

(0.1700 ± 0.0220 ± 0.0260) · 10⁻² L3 [40]

Γ₁₀₃ = 3h⁻ 2h⁺ ν_τ (ex. K⁰)

(8.216 ± 0.316) · 10⁻⁴ HFAG Summer 2016 fit

(7.200 ± 1.500 ± 0.000) · 10⁻⁴ ALEPH [9]

(6.400 ± 2.300 ± 1.000) · 10⁻⁴ ARGUS [56]

(7.700 ± 0.500 ± 0.900) · 10⁻⁴ CLEO [54]

(9.700 ± 1.500 ± 0.500) · 10⁻⁴ DELPHI [20]

(5.100 ± 2.000 ± 0.000) · 10⁻⁴ HRS [55]

(9.100 ± 1.400 ± 0.600) · 10⁻⁴ OPAL [57]

Γ₁₀₄ = 3h⁻ 2h⁺ π⁰ ν_τ (ex. K⁰)

(1.634 ± 0.114) · 10⁻⁴ HFAG Summer 2016 fit

(2.100 ± 0.700 ± 0.900) · 10⁻⁴ ALEPH [9]

(1.700 ± 0.200 ± 0.200) · 10⁻⁴ CLEO [48]

(1.600 ± 1.200 ± 0.600) · 10⁻⁴ DELPHI [20]

(2.700 ± 1.800 ± 0.900) · 10⁻⁴ OPAL [57]

Γ₁₀₆ = (5π)⁻ ν_τ

(0.7748 ± 0.0534) · 10⁻² HFAG Summer 2016 fit

Γ₁₁₀ = X_s⁻ ν_τ

(2.909 ± 0.048) · 10⁻² HFAG Summer 2016 fit

Γ₁₂₆ = π⁻ π⁰ η ν_τ

(0.1386 ± 0.0072) · 10⁻² HFAG Summer 2016 fit

(0.1800 ± 0.0447 ± 0.0000) · 10⁻² ALEPH [58]

(0.1350 ± 0.0030 ± 0.0070) · 10⁻² Belle [59]

(0.1700 ± 0.0200 ± 0.0200) · 10⁻² CLEO [60]

Γ₁₂₈ = K⁻ η ν_τ

(1.547 ± 0.080) · 10⁻⁴ HFAG Summer 2016 fit

(2.900 _−1.200^+1.300 ± 0.700) · 10⁻⁴ ALEPH [58]

(1.420 ± 0.110 ± 0.070) · 10⁻⁴ BaBar [61]

(1.580 ± 0.050 ± 0.090) · 10⁻⁴ Belle [59]

(2.600 ± 0.500 ± 0.500) · 10⁻⁴ CLEO [62]

Γ₁₃₀ = K⁻ π⁰ η ν_τ

(0.483 ± 0.116) · 10⁻⁴ HFAG Summer 2016 fit

(0.460 ± 0.110 ± 0.040) · 10⁻⁴ Belle [59]

(1.770 ± 0.560 ± 0.710) · 10⁻⁴ CLEO [63]

Γ₁₃₂ = π⁻ K⁰ η ν_τ

(0.937 ± 0.149) · 10⁻⁴ HFAG Summer 2016 fit

(0.880 ± 0.140 ± 0.060) · 10⁻⁴ Belle [59]

(2.200 ± 0.700 ± 0.220) · 10⁻⁴ CLEO [63]

Γ₁₃₆ = π⁻ π⁺ π⁻ η ν_τ (ex.  K⁰)

(2.184 ± 0.130) · 10⁻⁴ HFAG Summer 2016 fit

Γ₁₄₉ = h⁻ ω ≥ 0   neutrals  ν_τ

(2.401 ± 0.075) · 10⁻² HFAG Summer 2016 fit

Γ₁₅₀ = h⁻ ω ν_τ

(1.995 ± 0.064) · 10⁻² HFAG Summer 2016 fit

(1.910 ± 0.092 ± 0.000) · 10⁻² ALEPH [58]

(1.600 ± 0.270 ± 0.410) · 10⁻² CLEO [64]

Γ₁₅₀

Γ₆₆

=
h⁻ ω ν_τ

h⁻ h⁻ h⁺ π⁰ ν_τ (ex.  K⁰)

0.4332 ± 0.0139 HFAG Summer 2016 fit

0.4310 ± 0.0330 ± 0.0000 ALEPH [65]

0.4640 ± 0.0160 ± 0.0170 CLEO [42]

Γ₁₅₁ = K⁻ ω ν_τ

(4.100 ± 0.922) · 10⁻⁴ HFAG Summer 2016 fit

(4.100 ± 0.600 ± 0.700) · 10⁻⁴ CLEO3 [52]

Γ₁₅₂ = h⁻ π⁰ ω ν_τ

(0.4058 ± 0.0419) · 10⁻² HFAG Summer 2016 fit

(0.4300 ± 0.0781 ± 0.0000) · 10⁻² ALEPH [58]

Γ₁₅₂

Γ₅₄

=
h⁻ ω π⁰ ν_τ

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

(2.667 ± 0.275) · 10⁻² HFAG Summer 2016 fit

Γ₁₅₂

Γ₇₆

=
h⁻ ω π⁰ ν_τ

h⁻ h⁻ h⁺ 2π⁰ ν_τ (ex.  K⁰)

0.8241 ± 0.0757 HFAG Summer 2016 fit

0.8100 ± 0.0600 ± 0.0600 CLEO [47]

Γ₁₆₇ = K⁻ φ ν_τ

(4.445 ± 1.636) · 10⁻⁵ HFAG Summer 2016 fit

Γ₁₆₈ = K⁻ φ ν_τ (φ → K⁺ K⁻)

(2.174 ± 0.800) · 10⁻⁵ HFAG Summer 2016 fit

Γ₁₆₉ = K⁻ φ ν_τ (φ → K_S⁰ K_L⁰)

(1.520 ± 0.560) · 10⁻⁵ HFAG Summer 2016 fit

Γ₈₀₀ = π⁻ ω ν_τ

(1.954 ± 0.065) · 10⁻² HFAG Summer 2016 fit

Γ₈₀₂ = K⁻ π⁻ π⁺ ν_τ (ex. K⁰,ω)

(0.2923 ± 0.0067) · 10⁻² HFAG Summer 2016 fit

Γ₈₀₃ = K⁻ π⁻ π⁺ π⁰ ν_τ (ex. K⁰,ω,η)

(4.103 ± 1.429) · 10⁻⁴ HFAG Summer 2016 fit

Γ₈₀₄ = π⁻ K_L⁰ K_L⁰ ν_τ

(2.332 ± 0.065) · 10⁻⁴ HFAG Summer 2016 fit

Γ₈₀₅ = a₁⁻ (→ π⁻ γ) ν_τ

(4.000 ± 2.000) · 10⁻⁴ HFAG Summer 2016 fit

(4.000 ± 2.000 ± 0.000) · 10⁻⁴ ALEPH [9]

Γ₈₀₆ = π⁻ π⁰ K_L⁰ K_L⁰ ν_τ

(1.815 ± 0.207) · 10⁻⁵ HFAG Summer 2016 fit

Γ₈₁₀ = 2π⁻ π⁺ 3π⁰ ν_τ (ex. K⁰)

(1.924 ± 0.298) · 10⁻⁴ HFAG Summer 2016 fit

Γ₈₁₁ = π⁻ 2π⁰ ω ν_τ (ex. K⁰)

(7.105 ± 1.586) · 10⁻⁵ HFAG Summer 2016 fit

(7.300 ± 1.200 ± 1.200) · 10⁻⁵ BaBar [66]

Γ₈₁₂ = 2π⁻ π⁺ 3π⁰ ν_τ (ex. K⁰, η, ω, f₁)

(1.344 ± 2.683) · 10⁻⁵ HFAG Summer 2016 fit

(1.000 ± 0.800 ± 3.000) · 10⁻⁵ BaBar [66]

Γ₈₂₀ = 3π⁻ 2π⁺ ν_τ (ex. K⁰, ω)

(8.197 ± 0.315) · 10⁻⁴ HFAG Summer 2016 fit

Γ₈₂₁ = 3π⁻ 2π⁺ ν_τ (ex. K⁰, ω, f₁)

(7.677 ± 0.297) · 10⁻⁴ HFAG Summer 2016 fit

(7.680 ± 0.040 ± 0.400) · 10⁻⁴ BaBar [66]

Γ₈₂₂ = K⁻ 2π⁻ 2π⁺ ν_τ (ex. K⁰)

(0.596 ± 1.208) · 10⁻⁶ HFAG Summer 2016 fit

(0.600 ± 0.500 ± 1.100) · 10⁻⁶ BaBar [66]

Γ₈₃₀ = 3π⁻ 2π⁺ π⁰ ν_τ (ex. K⁰)

(1.623 ± 0.114) · 10⁻⁴ HFAG Summer 2016 fit

Γ₈₃₁ = 2π⁻ π⁺ ω ν_τ (ex. K⁰)

(8.359 ± 0.626) · 10⁻⁵ HFAG Summer 2016 fit

(8.400 ± 0.400 ± 0.600) · 10⁻⁵ BaBar [66]

Γ₈₃₂ = 3π⁻ 2π⁺ π⁰ ν_τ (ex. K⁰, η, ω, f₁)

(3.771 ± 0.875) · 10⁻⁵ HFAG Summer 2016 fit

(3.600 ± 0.300 ± 0.900) · 10⁻⁵ BaBar [66]

Γ₈₃₃ = K⁻ 2π⁻ 2π⁺ π⁰ ν_τ (ex. K⁰)

(1.108 ± 0.566) · 10⁻⁶ HFAG Summer 2016 fit

(1.100 ± 0.400 ± 0.400) · 10⁻⁶ BaBar [66]

Γ₉₁₀ = 2π⁻ π⁺ η ν_τ (η → 3π⁰)  (ex. K⁰)

(7.136 ± 0.424) · 10⁻⁵ HFAG Summer 2016 fit

(8.270 ± 0.880 ± 0.810) · 10⁻⁵ BaBar [66]

Γ₉₁₁ = π⁻ 2π⁰ η ν_τ (η → π⁺ π⁻ π⁰)  (ex. K⁰)

(4.420 ± 0.867) · 10⁻⁵ HFAG Summer 2016 fit

(4.570 ± 0.770 ± 0.500) · 10⁻⁵ BaBar [66]

Γ₉₂₀ = π⁻ f₁ ν_τ (f₁ → 2π⁻ 2π⁺)

(5.197 ± 0.444) · 10⁻⁵ HFAG Summer 2016 fit

(5.200 ± 0.310 ± 0.370) · 10⁻⁵ BaBar [66]

Γ₉₃₀ = 2π⁻ π⁺ η ν_τ (η → π⁺π⁻π⁰)  (ex. K⁰)

(5.005 ± 0.297) · 10⁻⁵ HFAG Summer 2016 fit

(5.390 ± 0.270 ± 0.410) · 10⁻⁵ BaBar [66]

Γ₉₄₄ = 2π⁻ π⁺ η ν_τ (η → γγ)  (ex. K⁰)

(8.606 ± 0.511) · 10⁻⁵ HFAG Summer 2016 fit

(8.260 ± 0.350 ± 0.510) · 10⁻⁵ BaBar [66]

Γ₉₄₅ = π⁻ 2π⁰ η ν_τ

(1.929 ± 0.378) · 10⁻⁴ HFAG Summer 2016 fit

Γ₉₉₈ = 1 − Γ_All

(0.0355 ± 0.1031) · 10⁻² HFAG Summer 2016 fit

2.6 Correlation terms between basis branching fractions uncertainties

The following tables report the correlation coefficients between basis nodes, in percent.

2.7 Equality constraints

We list in the following the equality constraints that relate a branching fraction to a sum of branching fractions, which have been introduced in Section 2. In the constraint equations, the τ branching fractions are denoted with Γ_n labels. The equations include as coefficients the values of some non-tau branching fractions, denoted e.g. with the self-describing notation Γ_{K_S → π⁰π⁰}. Some coefficients are probabilities corresponding to modulus square amplitudes describing quantum mixtures of states such as K⁰, K⁰, K_S, K_L, denoted with e.g. Γ_{<K⁰|K_S>} = |<K⁰|K_S>|². All non-tau quantities are taken from the PDG 2015 [8] fits (when available) or averages, and are used without accounting for their uncertainties, which are however in general small with respect to the uncertainties on the τ branching fractions.

The following list does not include the constraints already introduced in Section 2, and listed in Table 1, where some measured ratios of branching fractions are expressed as ratios of two branching fractions.

Γ₁ =	Γ₃ + Γ₅ + Γ₉ + Γ₁₀ + Γ₁₄ + Γ₁₆
	+ Γ₂₀ + Γ₂₃ + Γ₂₇ + Γ₂₈ + Γ₃₀ + Γ₃₅
	+ Γ₄₀ + Γ₄₄ + Γ₃₇ + Γ₄₂ + Γ₄₇ + Γ₄₈
	+ Γ₈₀₄ + Γ₅₀ + Γ₅₁ + Γ₈₀₆ + Γ₁₂₆·Γ_η→neutral
	+ Γ₁₂₈·Γ_η→neutral + Γ₁₃₀·Γ_η→neutral + Γ₁₃₂·Γ_η→neutral
	+ Γ₈₀₀·Γ_ω→π⁰γ + Γ₁₅₁·Γ_ω→π⁰γ + Γ₁₅₂·Γ_ω→π⁰γ
	+ Γ₁₆₇·Γ_{φ→ K_S K_L}

Γ₂ =	Γ₃ + Γ₅ + Γ₉ + Γ₁₀ + Γ₁₄ + Γ₁₆
	+ Γ₂₀ + Γ₂₃ + Γ₂₇ + Γ₂₈ + Γ₃₀ + Γ₃₅·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁰π⁰}
	+Γ_{<K⁰\|K_L>}) + Γ₄₀·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁰π⁰}+Γ_{<K⁰\|K_L>}) + Γ₄₄·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁰π⁰}
	+Γ_{<K⁰\|K_L>}) + Γ₃₇·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁰π⁰}+Γ_{<K⁰\|K_L>}) + Γ₄₂·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁰π⁰}
	+Γ_{<K⁰\|K_L>}) + Γ₄₇·(Γ_{K_S→π⁰π⁰}·Γ_{K_S→π⁰π⁰}) + Γ₄₈·Γ_{K_S→π⁰π⁰}
	+ Γ₈₀₄ + Γ₅₀·(Γ_{K_S→π⁰π⁰}·Γ_{K_S→π⁰π⁰}) + Γ₅₁·Γ_{K_S→π⁰π⁰}
	+ Γ₈₀₆ + Γ₁₂₆·Γ_η→neutral + Γ₁₂₈·Γ_η→neutral + Γ₁₃₀·Γ_η→neutral
	+ Γ₁₃₂·(Γ_η→neutral·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁰π⁰}+Γ_{<K⁰\|K_L>})) + Γ₈₀₀·Γ_ω→π⁰γ
	+ Γ₁₅₁·Γ_ω→π⁰γ + Γ₁₅₂·Γ_ω→π⁰γ + Γ₁₆₇·(Γ_{φ→ K_S K_L}·Γ_{K_S→π⁰π⁰})

Γ₇ =	Γ₃₅·Γ_{<K⁰\|K_L>} + Γ₉ + Γ₈₀₄ + Γ₃₇·Γ_{<K⁰\|K_L>}
	+ Γ₁₀

Γ₈ =

Γ₉ + Γ₁₀

Γ₁₁ =	Γ₁₄ + Γ₁₆ + Γ₂₀ + Γ₂₃ + Γ₂₇ + Γ₂₈
	+ Γ₃₀ + Γ₃₅·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁰π⁰}) + Γ₃₇·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁰π⁰})
	+ Γ₄₀·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁰π⁰}) + Γ₄₂·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁰π⁰})
	+ Γ₄₇·(Γ_{K_S→π⁰π⁰}·Γ_{K_S→π⁰π⁰}) + Γ₅₀·(Γ_{K_S→π⁰π⁰}·Γ_{K_S→π⁰π⁰})
	+ Γ₁₂₆·Γ_η→neutral + Γ₁₂₈·Γ_η→neutral + Γ₁₃₀·Γ_η→neutral
	+ Γ₁₃₂·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁰π⁰}·Γ_η→neutral) + Γ₁₅₁·Γ_ω→π⁰γ
	+ Γ₁₅₂·Γ_ω→π⁰γ + Γ₈₀₀·Γ_ω→π⁰γ

Γ₁₂ =	Γ₁₂₈·Γ_η→3π⁰ + Γ₃₀ + Γ₂₃ + Γ₂₈ + Γ₁₄
	+ Γ₁₆ + Γ₂₀ + Γ₂₇ + Γ₁₂₆·Γ_η→3π⁰ + Γ₁₃₀·Γ_η→3π⁰

Γ₁₃ =

Γ₁₄ + Γ₁₆

Γ₁₇ =	Γ₁₂₈·Γ_η→3π⁰ + Γ₃₀ + Γ₂₃ + Γ₂₈ + Γ₃₅·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁰π⁰})
	+ Γ₄₀·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁰π⁰}) + Γ₄₂·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁰π⁰})
	+ Γ₂₀ + Γ₂₇ + Γ₄₇·(Γ_{K_S→π⁰π⁰}·Γ_{K_S→π⁰π⁰}) + Γ₅₀·(Γ_{K_S→π⁰π⁰}·Γ_{K_S→π⁰π⁰})
	+ Γ₁₂₆·Γ_η→3π⁰ + Γ₃₇·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁰π⁰}) + Γ₁₃₀·Γ_η→3π⁰

Γ₁₈ =

Γ₂₃ + Γ₃₅·(Γ_{<K⁰|K_S>}·Γ_{K_S→π⁰π⁰}) + Γ₂₀ + Γ₃₇·(Γ_{<K⁰|K_S>}·Γ_{K_S→π⁰π⁰})

Γ₁₉ =

Γ₂₃ + Γ₂₀

Γ₂₄ =	Γ₂₇ + Γ₂₈ + Γ₃₀ + Γ₄₀·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁰π⁰})
	+ Γ₄₂·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁰π⁰}) + Γ₄₇·(Γ_{K_S→π⁰π⁰}·Γ_{K_S→π⁰π⁰})
	+ Γ₅₀·(Γ_{K_S→π⁰π⁰}·Γ_{K_S→π⁰π⁰}) + Γ₁₂₆·Γ_η→3π⁰ + Γ₁₂₈·Γ_η→3π⁰
	+ Γ₁₃₀·Γ_η→3π⁰ + Γ₁₃₂·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁰π⁰}·Γ_η→3π⁰)

Γ₂₅ =	Γ₁₂₈·Γ_η→3π⁰ + Γ₃₀ + Γ₂₈ + Γ₂₇ + Γ₁₂₆·Γ_η→3π⁰
	+ Γ₁₃₀·Γ_η→3π⁰

Γ₂₆ =	Γ₁₂₈·Γ_η→3π⁰ + Γ₂₈ + Γ₄₀·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁰π⁰})
	+ Γ₄₂·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁰π⁰}) + Γ₂₇

Γ₂₉ =

Γ₃₀ + Γ₁₂₆·Γ_η→3π⁰ + Γ₁₃₀·Γ_η→3π⁰

Γ₃₁ =	Γ₁₂₈·Γ_η→neutral + Γ₂₃ + Γ₂₈ + Γ₄₂ + Γ₁₆
	+ Γ₃₇ + Γ₁₀ + Γ₁₆₇·(Γ_{φ→ K_S K_L}·Γ_{K_S→π⁰π⁰})

Γ₃₂ =	Γ₁₆ + Γ₂₃ + Γ₂₈ + Γ₃₇ + Γ₄₂ + Γ₁₂₈·Γ_η→neutral
	+ Γ₁₃₀·Γ_η→neutral + Γ₁₆₇·(Γ_{φ→ K_S K_L}·Γ_{K_S→π⁰π⁰})

Γ₃₃ =	Γ₃₅·Γ_{<K⁰\|K_S>} + Γ₄₀·Γ_{<K⁰\|K_S>} + Γ₄₂·Γ_{<K⁰\|K_S>}
	+ Γ₄₇ + Γ₄₈ + Γ₅₀ + Γ₅₁ + Γ₃₇·Γ_{<K⁰\|K_S>}
	+ Γ₁₃₂·(Γ_{<K⁰\|K_S>}·Γ_η→neutral) + Γ₄₄·Γ_{<K⁰\|K_S>} + Γ₁₆₇·Γ_{φ→ K_S K_L}

Γ₃₄ =

Γ₃₅ + Γ₃₇

Γ₃₈ =

Γ₄₂ + Γ₃₇

Γ₃₉ =

Γ₄₀ + Γ₄₂

Γ₄₃ =

Γ₄₀ + Γ₄₄

Γ₄₆ =

Γ₄₈ + Γ₄₇ + Γ₈₀₄

Γ₄₉ =

Γ₅₀ + Γ₅₁ + Γ₈₀₆

Γ₅₄ =	Γ₃₅·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁺π⁻}) + Γ₃₇·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁺π⁻})
	+ Γ₄₀·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁺π⁻}) + Γ₄₂·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁺π⁻})
	+ Γ₄₇·(2·Γ_{K_S→π⁺π⁻}·Γ_{K_S→π⁰π⁰}) + Γ₄₈·Γ_{K_S→π⁺π⁻}
	+ Γ₅₀·(2·Γ_{K_S→π⁺π⁻}·Γ_{K_S→π⁰π⁰}) + Γ₅₁·Γ_{K_S→π⁺π⁻}
	+ Γ₅₃·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁰π⁰}+Γ_{<K⁰\|K_L>}) + Γ₆₂ + Γ₇₀
	+ Γ₇₇ + Γ₇₈ + Γ₉₃ + Γ₉₄ + Γ₁₂₆·Γ_η→charged
	+ Γ₁₂₈·Γ_η→charged + Γ₁₃₀·Γ_η→charged + Γ₁₃₂·(Γ_{<K⁰\|K_L>}·Γ_{η→π⁺π⁻π⁰}
	+ Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁰π⁰}·Γ_{η→π⁺π⁻π⁰} + Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁺π⁻}·Γ_η→3π⁰)
	+ Γ₁₅₁·(Γ_{ω→π⁺π⁻π⁰}+Γ_{ω→π⁺π⁻}) + Γ₁₅₂·(Γ_{ω→π⁺π⁻π⁰}+Γ_{ω→π⁺π⁻})
	+ Γ₁₆₇·(Γ_{φ→ K⁺K⁻} + Γ_{φ→ K_S K_L}·Γ_{K_S→π⁺π⁻}) + Γ₈₀₂ + Γ₈₀₃
	+ Γ₈₀₀·(Γ_{ω→π⁺π⁻π⁰}+Γ_{ω→π⁺π⁻})

Γ₅₅ =	Γ₁₂₈·Γ_η→charged + Γ₁₅₂·(Γ_{ω→π⁺π⁻π⁰}+Γ_{ω→π⁺π⁻}) + Γ₇₈
	+ Γ₇₇ + Γ₉₄ + Γ₆₂ + Γ₇₀ + Γ₉₃ + Γ₁₂₆·Γ_η→charged
	+ Γ₈₀₂ + Γ₈₀₃ + Γ₈₀₀·(Γ_{ω→π⁺π⁻π⁰}+Γ_{ω→π⁺π⁻}) + Γ₁₅₁·(Γ_{ω→π⁺π⁻π⁰}
	+Γ_{ω→π⁺π⁻}) + Γ₁₃₀·Γ_η→charged + Γ₁₆₈

Γ₅₆ =	Γ₃₅·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁺π⁻}) + Γ₆₂ + Γ₉₃ + Γ₃₇·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁺π⁻})
	+ Γ₈₀₂ + Γ₈₀₀·Γ_{ω→π⁺π⁻} + Γ₁₅₁·Γ_{ω→π⁺π⁻} + Γ₁₆₈

Γ₅₇ =	Γ₆₂ + Γ₉₃ + Γ₈₀₂ + Γ₈₀₀·Γ_{ω→π⁺π⁻} + Γ₁₅₁·Γ_{ω→π⁺π⁻}
	+ Γ₁₆₇·Γ_{φ→ K⁺K⁻}

Γ₅₈ =

Γ₆₂ + Γ₉₃ + Γ₈₀₂ + Γ₁₆₇·Γ_{φ→ K⁺K⁻}

Γ₅₉ =

Γ₃₅·(Γ_{<K⁰|K_S>}·Γ_{K_S→π⁺π⁻}) + Γ₆₂ + Γ₈₀₀·Γ_{ω→π⁺π⁻}

Γ₆₀ =

Γ₆₂ + Γ₈₀₀·Γ_{ω→π⁺π⁻}

Γ₆₃ =	Γ₄₀·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁺π⁻}) + Γ₄₂·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁺π⁻})
	+ Γ₄₇·(2·Γ_{K_S→π⁺π⁻}·Γ_{K_S→π⁰π⁰}) + Γ₅₀·(2·Γ_{K_S→π⁺π⁻}·Γ_{K_S→π⁰π⁰})
	+ Γ₇₀ + Γ₇₇ + Γ₇₈ + Γ₉₄ + Γ₁₂₆·Γ_η→charged
	+ Γ₁₂₈·Γ_η→charged + Γ₁₃₀·Γ_η→charged + Γ₁₃₂·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁺π⁻}·Γ_η→neutral
	+ Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁰π⁰}·Γ_η→charged) + Γ₁₅₁·Γ_{ω→π⁺π⁻π⁰} + Γ₁₅₂·(Γ_{ω→π⁺π⁻π⁰}
	+Γ_{ω→π⁺π⁻}) + Γ₈₀₀·Γ_{ω→π⁺π⁻π⁰} + Γ₈₀₃

Γ₆₄ =	Γ₇₈ + Γ₇₇ + Γ₉₄ + Γ₇₀ + Γ₁₂₆·Γ_{η→π⁺π⁻π⁰}
	+ Γ₁₂₈·Γ_{η→π⁺π⁻π⁰} + Γ₁₃₀·Γ_{η→π⁺π⁻π⁰} + Γ₈₀₀·Γ_{ω→π⁺π⁻π⁰}
	+ Γ₁₅₁·Γ_{ω→π⁺π⁻π⁰} + Γ₁₅₂·(Γ_{ω→π⁺π⁻π⁰}+Γ_{ω→π⁺π⁻}) + Γ₈₀₃

Γ₆₅ =	Γ₄₀·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁺π⁻}) + Γ₄₂·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁺π⁻})
	+ Γ₇₀ + Γ₉₄ + Γ₁₂₈·Γ_{η→π⁺π⁻π⁰} + Γ₁₅₁·Γ_{ω→π⁺π⁻π⁰}
	+ Γ₁₅₂·Γ_{ω→π⁺π⁻} + Γ₈₀₀·Γ_{ω→π⁺π⁻π⁰} + Γ₈₀₃

Γ₆₆ =	Γ₇₀ + Γ₉₄ + Γ₁₂₈·Γ_{η→π⁺π⁻π⁰} + Γ₁₅₁·Γ_{ω→π⁺π⁻π⁰}
	+ Γ₁₅₂·Γ_{ω→π⁺π⁻} + Γ₈₀₀·Γ_{ω→π⁺π⁻π⁰} + Γ₈₀₃

Γ₆₇ =

Γ₇₀ + Γ₉₄ + Γ₁₂₈·Γ_{η→π⁺π⁻π⁰} + Γ₈₀₃

Γ₆₈ =	Γ₄₀·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁺π⁻}) + Γ₇₀ + Γ₁₅₂·Γ_{ω→π⁺π⁻}
	+ Γ₈₀₀·Γ_{ω→π⁺π⁻π⁰}

Γ₆₉ =

Γ₁₅₂·Γ_{ω→π⁺π⁻} + Γ₇₀ + Γ₈₀₀·Γ_{ω→π⁺π⁻π⁰}

Γ₇₄ =	Γ₁₅₂·Γ_{ω→π⁺π⁻π⁰} + Γ₇₈ + Γ₇₇ + Γ₁₂₆·Γ_{η→π⁺π⁻π⁰}
	+ Γ₁₃₀·Γ_{η→π⁺π⁻π⁰}

Γ₇₅ =	Γ₁₅₂·Γ_{ω→π⁺π⁻π⁰} + Γ₄₇·(2·Γ_{K_S→π⁺π⁻}·Γ_{K_S→π⁰π⁰})
	+ Γ₇₇ + Γ₁₂₆·Γ_{η→π⁺π⁻π⁰} + Γ₁₃₀·Γ_{η→π⁺π⁻π⁰}

Γ₇₆ =

Γ₁₅₂·Γ_{ω→π⁺π⁻π⁰} + Γ₇₇ + Γ₁₂₆·Γ_{η→π⁺π⁻π⁰} + Γ₁₃₀·Γ_{η→π⁺π⁻π⁰}

Γ₇₈ =

Γ₈₁₀ + Γ₅₀·(2·Γ_{K_S→π⁺π⁻}·Γ_{K_S→π⁰π⁰}) + Γ₁₃₂·(Γ_{<K⁰|K_S>}·Γ_{K_S→π⁺π⁻}·Γ_η→3π⁰)

Γ₇₉ =	Γ₃₇·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁺π⁻}) + Γ₄₂·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁺π⁻})
	+ Γ₉₃ + Γ₉₄ + Γ₁₂₈·Γ_η→charged + Γ₁₅₁·(Γ_{ω→π⁺π⁻π⁰}
	+Γ_{ω→π⁺π⁻}) + Γ₁₆₈ + Γ₈₀₂ + Γ₈₀₃

Γ₈₀ =

Γ₉₃ + Γ₈₀₂ + Γ₁₅₁·Γ_{ω→π⁺π⁻}

Γ₈₁ =

Γ₁₂₈·Γ_{η→π⁺π⁻π⁰} + Γ₉₄ + Γ₈₀₃ + Γ₁₅₁·Γ_{ω→π⁺π⁻π⁰}

Γ₈₂ =	Γ₁₂₈·Γ_η→charged + Γ₄₂·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁺π⁻}) + Γ₈₀₂
	+ Γ₈₀₃ + Γ₁₅₁·(Γ_{ω→π⁺π⁻π⁰}+Γ_{ω→π⁺π⁻}) + Γ₃₇·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁺π⁻})

Γ₈₃ =	Γ₁₂₈·Γ_{η→π⁺π⁻π⁰} + Γ₈₀₂ + Γ₈₀₃ + Γ₁₅₁·(Γ_{ω→π⁺π⁻π⁰}
	+Γ_{ω→π⁺π⁻})

Γ₈₄ =

Γ₈₀₂ + Γ₁₅₁·Γ_{ω→π⁺π⁻} + Γ₃₇·(Γ_{<K⁰|K_S>}·Γ_{K_S→π⁺π⁻})

Γ₈₅ =

Γ₈₀₂ + Γ₁₅₁·Γ_{ω→π⁺π⁻}

Γ₈₇ =	Γ₄₂·(Γ_{<K⁰\|K_S>}·Γ_{K_S→π⁺π⁻}) + Γ₁₂₈·Γ_{η→π⁺π⁻π⁰} + Γ₁₅₁·Γ_{ω→π⁺π⁻π⁰}
	+ Γ₈₀₃

Γ₈₈ =

Γ₁₂₈·Γ_{η→π⁺π⁻π⁰} + Γ₈₀₃ + Γ₁₅₁·Γ_{ω→π⁺π⁻π⁰}

Γ₈₉ =

Γ₈₀₃ + Γ₁₅₁·Γ_{ω→π⁺π⁻π⁰}

Γ₉₂ =

Γ₉₄ + Γ₉₃

Γ₉₆ =

Γ₁₆₇·Γ_{φ→ K⁺K⁻}

Γ₁₀₂ =

Γ₁₀₃ + Γ₁₀₄

Γ₁₀₃ =

Γ₈₂₀ + Γ₈₂₂ + Γ₈₃₁·Γ_{ω→π⁺π⁻}

Γ₁₀₄ =

Γ₈₃₀ + Γ₈₃₃

Γ₁₀₆ =	Γ₃₀ + Γ₄₄·Γ_{<K⁰\|K_S>} + Γ₄₇ + Γ₅₃·Γ_{<K⁰\|K_S>}
	+ Γ₇₇ + Γ₁₀₃ + Γ₁₂₆·(Γ_η→3π⁰+Γ_{η→π⁺π⁻π⁰}) + Γ₁₅₂·Γ_{ω→π⁺π⁻π⁰}

Γ₁₁₀ =	Γ₁₀ + Γ₁₆ + Γ₂₃ + Γ₂₈ + Γ₃₅ + Γ₄₀
	+ Γ₁₂₈ + Γ₈₀₂ + Γ₈₀₃ + Γ₁₅₁ + Γ₁₃₀ + Γ₁₃₂
	+ Γ₄₄ + Γ₅₃ + Γ₁₆₈ + Γ₁₆₉ + Γ₈₂₂ + Γ₈₃₃

Γ₁₄₉ =

Γ₁₅₂ + Γ₈₀₀ + Γ₁₅₁

Γ₁₅₀ =

Γ₈₀₀ + Γ₁₅₁

Γ₁₆₈ =

Γ₁₆₇·Γ_{φ→ K⁺K⁻}

Γ₁₆₉ =

Γ₁₆₇·Γ_{φ→ K_S K_L}

Γ₈₀₄ =

Γ₄₇ · ((Γ_{<K⁰|K_L>}·Γ_{<K⁰|K_L>}) / (Γ_{<K⁰|K_S>}·Γ_{<K⁰|K_S>}))

Γ₈₀₆ =

Γ₅₀ · ((Γ_{<K⁰|K_L>}·Γ_{<K⁰|K_L>}) / (Γ_{<K⁰|K_S>}·Γ_{<K⁰|K_S>}))

Γ₈₁₀ =

Γ₉₁₀ + Γ₉₁₁ + Γ₈₁₁·Γ_{ω→π⁺π⁻π⁰} + Γ₈₁₂

Γ₈₂₀ =

Γ₉₂₀ + Γ₈₂₁

Γ₈₃₀ =

Γ₉₃₀ + Γ₈₃₁·Γ_{ω→π⁺π⁻π⁰} + Γ₈₃₂

Γ₉₁₀ =

Γ₁₃₆·Γ_η→3π⁰

Γ₉₁₁ =

Γ₉₄₅·Γ_{η→π⁺π⁻π⁰}

Γ₉₃₀ =

Γ₁₃₆·Γ_{η→π⁺π⁻π⁰}

Γ₉₄₄ =

Γ₁₃₆·Γ_η→γγ

Γ_All =	Γ₃ + Γ₅ + Γ₉ + Γ₁₀ + Γ₁₄ + Γ₁₆
	+ Γ₂₀ + Γ₂₃ + Γ₂₇ + Γ₂₈ + Γ₃₀ + Γ₃₅
	+ Γ₃₇ + Γ₄₀ + Γ₄₂ + Γ₄₇·(1 + ((Γ_{<K⁰\|K_L>}·Γ_{<K⁰\|K_L>}) / (Γ_{<K⁰\|K_S>}·Γ_{<K⁰\|K_S>})))
	+ Γ₄₈ + Γ₆₂ + Γ₇₀ + Γ₇₇ + Γ₈₁₁ + Γ₈₁₂
	+ Γ₉₃ + Γ₉₄ + Γ₈₃₂ + Γ₈₃₃ + Γ₁₂₆ + Γ₁₂₈
	+ Γ₈₀₂ + Γ₈₀₃ + Γ₈₀₀ + Γ₁₅₁ + Γ₁₃₀ + Γ₁₃₂
	+ Γ₄₄ + Γ₅₃ + Γ₅₀·(1 + ((Γ_{<K⁰\|K_L>}·Γ_{<K⁰\|K_L>}) / (Γ_{<K⁰\|K_S>}·Γ_{<K⁰\|K_S>})))
	+ Γ₅₁ + Γ₁₆₇·(Γ_{φ→ K⁺K⁻}+Γ_{φ→ K_S K_L}) + Γ₁₅₂ + Γ₉₂₀
	+ Γ₈₂₁ + Γ₈₂₂ + Γ₈₃₁ + Γ₁₃₆ + Γ₉₄₅ + Γ₈₀₅


Γ₅	23
Γ₉	7	5
Γ₁₀	3	5	1
Γ₁₄	-13	-14	-12	-3
Γ₁₆	0	-1	2	-1	-16
Γ₂₀	-5	-5	-7	-1	-40	2
Γ₂₃	0	0	0	-2	2	-13	-22
Γ₂₇	-4	-3	-8	-1	0	3	-36	6
Γ₂₈	0	0	0	-2	2	-13	5	-21	-29
Γ₃₀	-5	-4	-11	-2	-9	0	6	0	-42	0
Γ₃₅	0	0	0	0	0	0	0	1	0	1	0
Γ₃₇	0	0	0	0	0	-2	1	-3	1	-3	0	-22
Γ₄₀	0	0	0	0	0	1	0	1	-2	1	0	-12	4
	Γ₃	Γ₅	Γ₉	Γ₁₀	Γ₁₄	Γ₁₆	Γ₂₀	Γ₂₃	Γ₂₇	Γ₂₈	Γ₃₀	Γ₃₅	Γ₃₇	Γ₄₀


Γ₄₂	0	0	0	0	1	-3	1	-5	1	-5	0	2	-21	-20
Γ₄₄	0	0	0	0	0	0	0	0	0	0	0	-1	0	-4
Γ₄₇	0	0	0	0	0	0	0	0	0	0	0	-1	1	-4
Γ₄₈	0	0	0	0	0	0	0	0	0	0	0	-3	0	-2
Γ₅₀	0	0	0	0	0	0	0	-1	0	-1	0	0	7	0
Γ₅₁	0	0	0	0	0	0	0	0	0	0	0	-1	0	-1
Γ₅₃	0	0	0	0	0	0	0	0	0	0	0	0	0	0
Γ₆₂	-3	-5	8	0	-4	5	-7	-1	-5	-1	-5	0	0	0
Γ₇₀	-6	-6	-7	-1	-8	-1	-1	0	-1	0	3	0	0	0
Γ₇₇	-1	0	-3	-1	-2	0	0	0	2	0	2	0	0	0
Γ₉₃	-1	-1	3	0	-1	2	-1	0	-1	0	-1	0	0	0
Γ₉₄	0	0	0	0	0	0	0	0	0	0	0	0	0	0
Γ₁₂₆	0	0	0	0	0	0	-1	0	0	0	-2	0	0	0
Γ₁₂₈	0	0	1	0	0	1	0	-1	0	-1	0	0	0	0
	Γ₃	Γ₅	Γ₉	Γ₁₀	Γ₁₄	Γ₁₆	Γ₂₀	Γ₂₃	Γ₂₇	Γ₂₈	Γ₃₀	Γ₃₅	Γ₃₇	Γ₄₀

HFAG-Tau Summer 2016 Report

2 Branching fraction fit