HFLAV-Tau 2018 - 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 dependencies from 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 with a systematic dependence from the same external parameter.

Both the measurements and the fitted quantities consist of either τ decay branching fractions, labelled as B_i, or ratios of two τ decay branching fractions, labelled as B_i/ B_j. Some branching fractions are sums 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). The symbol h is used to mean either a π or a K. The fit χ² is minimized while respecting a list of constraints on the fitted quantities:

In some cases, constraints describe approximate relations that nevertheless hold within the present experimental precision. For instance, the constraint B(τ→K⁻ K⁻ K⁺ ν_τ) = B(τ→K⁻ φ ν_τ) × B(φ→ K⁺K⁻) is justified within the current experimental evidence. Section 2.7 lists all equations relating one quantity to other quantities.

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 x_i and their covariance matrix V_ij as

where the model matrix A_ij is used to get the vector of the predicted measurements x_i^′ from the vector of the fit parameters q_j as x_i^′= A_ijq_j. In this particular implementation, the measurements are grouped according to the measured quantity, and all quantities with at least one measurement correspond to a fit parameter. Therefore, the matrix A_ij has one row per measurement x_i and one column per fitted quantity q_j, with unity coefficients for the rows and column that identify a measurement x_i of the quantity q_j. In summary, the χ² given in Eq. (1) is minimized subject to the constraints

where Eq. (2) corresponds to the constraint equations, written as a set of “constraint expressions” that are equated to zero. 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:

Equation (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). Solving the equation set in Eq. (9) gives the fitted quantities and their covariance matrix, using the measurements and their 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 RMS average of relative variation of the fitted unknowns is reduced below 10⁻¹².

2.2 Fit results

Although the fit treats all quantities in the same way, for the purpose of describing the results we select a set of 47 “basis quantities” from which all remaining quantities can be calculated using the definitions listed in Section 2.7.

The fit output consists of 136 fitted quantities that correspond to either branching fractions or ratios of branching fractions. The fitted quantities values and uncertainties are listed in Table 1. The off-diagonal correlation terms between the basis quantities are listed in Section 2.6.

Furthermore we define (see Section 2.7) B₁₁₀ = B(τ⁻ → X_s⁻ ν_τ), the total branching fraction of the τ decays to final states with the strangeness quantum number equal to one, and B_All, the branching fraction of the τ into any measured final state, which should be equal to 1 within the experimental uncertainty. We define the unitarity residual as B₉₉₈ = 1 − B_All.

The fit has χ²/d.o.f. = 142/129, corresponding to a confidence level CL = 20.13%. We use a total of 176 measurements to fit the above mentioned 136 quantities subjected to 89 constraints. Although the unitarity constraint is not applied, the fit is statistically consistent with unitarity, where the residual is B₉₉₈ = 1 − B_All = (0.0274 ± 0.1026) · 10⁻².

A scale factor of 5.44 has been applied to the published uncertainties of the two severely inconsistent measurements of B₉₆ = 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 B₉₆ measurements.

2.3 Changes with respect to the previous report

The following changes have been introduced with respect to the previous HFLAV report [2].

The parameters used to update the measurements’ systematic biases and the parameters appearing in the constraint equations in Section 2.7 have been updated to the PDG 2018 averages [6].

2.4 Differences between the HFLAV 2018 fit and the PDG 2018 fit

As is standard for the PDG branching fraction fits, the PDG 2018 τ branching fraction fit is unitarity constrained, while the HFLAV 2018 fit is unconstrained.

The HFLAV-Tau fit uses an elaboration of the measurements reported on the main ALEPH paper on τ branching fractions [7] to obtain branching fractions to inclusive final states with “hadrons” (where a hadron is either a pion or a kaon), since this set of results is closer to the actual experimental measurements and facilitates a more appropriate and comprehensive treatment of the experimental results correlations. The PDG 2018 fit on the other hand continues to use – as in the past editions – the published ALEPH measurements of branching fractions to esclusive final states with pions [7].

As in 2016, HFLAV uses the ALEPH estimate for B₈₀₅ = B(τ→a₁⁻ (→ π⁻ γ) ν_τ), which is not a direct measurement, and the PDG 2018 fit uses the PDG average of B(a₁→πγ) as a parameter and defines B₈₀₅ = 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 B₈₀₅. Therefore, in this case an appropriate correction has been applied after the fit.

Finally, the HFLAV 2018 τ branching fraction fit includes measurements that appeared after the deadline for inclusion in the PDG, and preliminary measurements that are not included in the PDG.

2.5 Branching ratio fit results and experimental inputs

Table 1: HFLAV 2018 branching fractions fit results.

τ lepton branching fraction Experiment Reference

B₁ = (particles)⁻ ≥ 0  neutrals ≥ 0   K⁰  ν_τ

0.8521 ± 0.0011 average

B₂ = (particles)⁻ ≥ 0  neutrals ≥ 0   K_L⁰  ν_τ

0.8455 ± 0.0010 average

B₃ = µ⁻ ν_µν_τ

0.17392 ± 0.00039 average

0.17319 ± 0.00070 ± 0.00032 ALEPH [7]

0.17325 ± 0.00095 ± 0.00077 DELPHI [8]

0.17342 ± 0.00110 ± 0.00067 L3 [9]

0.17340 ± 0.00090 ± 0.00060 OPAL [10]

B₃

B₅

=
µ⁻ ν_µν_τ

e⁻ ν_e ν_τ

0.9761 ± 0.0028 average

0.9970 ± 0.0350 ± 0.0400 ARGUS [11]

0.9796 ± 0.0016 ± 0.0036 BaBar [12]

0.9777 ± 0.0063 ± 0.0087 CLEO [13]

B₅ = e⁻ ν_e ν_τ

0.17817 ± 0.00041 average

0.17837 ± 0.00072 ± 0.00036 ALEPH [7]

0.17760 ± 0.00060 ± 0.00170 CLEO [13]

0.17877 ± 0.00109 ± 0.00110 DELPHI [8]

0.17806 ± 0.00104 ± 0.00076 L3 [9]

0.17810 ± 0.00090 ± 0.00060 OPAL [14]

B₇ = h⁻ ≥ 0   K_L⁰  ν_τ

0.12019 ± 0.00053 average

0.12400 ± 0.00700 ± 0.00700 DELPHI [15]

0.12470 ± 0.00260 ± 0.00430 L3 [16]

0.12100 ± 0.00700 ± 0.00500 OPAL [17]

B₈ = h⁻ ν_τ

0.11502 ± 0.00053 average

0.11524 ± 0.00070 ± 0.00078 ALEPH [7]

0.11520 ± 0.00050 ± 0.00120 CLEO [13]

0.11571 ± 0.00120 ± 0.00114 DELPHI [18]

0.11980 ± 0.00130 ± 0.00160 OPAL [19]

B₈

B₅

=
h⁻ ν_τ

e⁻ ν_e ν_τ

0.6456 ± 0.0033 average

B₉ = π⁻ ν_τ

0.10804 ± 0.00052 average

B₉

B₅

=
π⁻ ν_τ

e⁻ ν_e ν_τ

0.6064 ± 0.0032 average

0.5945 ± 0.0014 ± 0.0061 BaBar [12]

B₁₀ = K⁻ ν_τ

(0.6986 ± 0.0085) · 10⁻² average

(0.6960 ± 0.0250 ± 0.0140) · 10⁻² ALEPH [20]

(0.7170 ± 0.0031 ± 0.0210) · 10⁻² BaBar [4]

(0.6600 ± 0.0700 ± 0.0900) · 10⁻² CLEO [21]

(0.8500 ± 0.1800 ± 0.0000) · 10⁻² DELPHI [22]

(0.6580 ± 0.0270 ± 0.0290) · 10⁻² OPAL [23]

B₁₀

B₅

=
K⁻ ν_τ

e⁻ ν_e ν_τ

(3.921 ± 0.048) · 10⁻² average

(3.882 ± 0.032 ± 0.057) · 10⁻² BaBar [12]

B₁₀

B₉

=
K⁻ ν_τ

π⁻ ν_τ

(6.467 ± 0.084) · 10⁻² average

B₁₁ = h⁻ ≥ 1   neutrals  ν_τ

0.36996 ± 0.00094 average

B₁₂ = h⁻ ≥ 1  π⁰  ν_τ (ex.  K⁰)

0.36495 ± 0.00094 average

B₁₃ = h⁻ π⁰ ν_τ

0.25938 ± 0.00090 average

0.25924 ± 0.00097 ± 0.00085 ALEPH [7]

0.25670 ± 0.00010 ± 0.00390 Belle [24]

0.25870 ± 0.00120 ± 0.00420 CLEO [25]

0.25740 ± 0.00201 ± 0.00138 DELPHI [18]

0.25050 ± 0.00350 ± 0.00500 L3 [16]

0.25890 ± 0.00170 ± 0.00290 OPAL [19]

B₁₄ = π⁻ π⁰ ν_τ

0.25447 ± 0.00091 average

B₁₆ = K⁻ π⁰ ν_τ

(0.4904 ± 0.0092) · 10⁻² average

(0.4440 ± 0.0260 ± 0.0240) · 10⁻² ALEPH [20]

(0.5050 ± 0.0020 ± 0.0150) · 10⁻² BaBar [4]

(0.5100 ± 0.1000 ± 0.0700) · 10⁻² CLEO [21]

(0.4710 ± 0.0590 ± 0.0230) · 10⁻² OPAL [26]

B₁₇ = h⁻ ≥ 2   π⁰  ν_τ

0.10793 ± 0.00091 average

0.09910 ± 0.00310 ± 0.00270 OPAL [19]

B₁₈ = h⁻ 2π⁰ ν_τ

(9.421 ± 0.092) · 10⁻² average

B₁₉ = h⁻ 2π⁰ ν_τ (ex.  K⁰)

(9.270 ± 0.092) · 10⁻² average

(9.295 ± 0.084 ± 0.088) · 10⁻² ALEPH [7]

(9.498 ± 0.320 ± 0.275) · 10⁻² DELPHI [18]

(8.880 ± 0.370 ± 0.420) · 10⁻² L3 [16]

B₁₉

B₁₃

=
h⁻ 2π⁰ ν_τ (ex.  K⁰)

h⁻ π⁰ ν_τ

0.3574 ± 0.0042 average

0.3420 ± 0.0060 ± 0.0160 CLEO [27]

B₂₀ = π⁻ 2π⁰ ν_τ (ex. K⁰)

(9.211 ± 0.092) · 10⁻² average

B₂₃ = K⁻ 2π⁰ ν_τ (ex. K⁰)

(0.0585 ± 0.0027) · 10⁻² average

(0.0560 ± 0.0200 ± 0.0150) · 10⁻² ALEPH [20]

(0.0615 ± 0.0012 ± 0.0034) · 10⁻² BaBar [4]

(0.0900 ± 0.1000 ± 0.0300) · 10⁻² CLEO [21]

B₂₄ = h⁻ ≥ 3  π⁰  ν_τ

(1.372 ± 0.034) · 10⁻² average

B₂₅ = h⁻ ≥ 3  π⁰  ν_τ (ex.  K⁰)

(1.288 ± 0.034) · 10⁻² average

(1.403 ± 0.214 ± 0.224) · 10⁻² DELPHI [18]

B₂₆ = h⁻ 3π⁰ ν_τ

(1.236 ± 0.030) · 10⁻² average

(1.082 ± 0.071 ± 0.059) · 10⁻² ALEPH [7]

(1.700 ± 0.240 ± 0.380) · 10⁻² L3 [16]

B₂₆

B₁₃

=
h⁻ 3π⁰ ν_τ

h⁻ π⁰ ν_τ

(4.764 ± 0.118) · 10⁻² average

(4.400 ± 0.300 ± 0.500) · 10⁻² CLEO [27]

B₂₇ = π⁻ 3π⁰ ν_τ (ex. K⁰)

(1.1381 ± 0.0292) · 10⁻² average

(1.1680 ± 0.0060 ± 0.0380) · 10⁻² BaBar [4]

B₂₈ = K⁻ 3π⁰ ν_τ (ex. K⁰,η)

(1.127 ± 0.263) · 10⁻⁴ average

(3.700 ± 2.100 ± 1.100) · 10⁻⁴ ALEPH [20]

(1.250 ± 0.160 ± 0.240) · 10⁻⁴ BaBar [4]

B₂₉ = h⁻ 4π⁰ ν_τ (ex.  K⁰)

(0.1333 ± 0.0071) · 10⁻² average

(0.1600 ± 0.0500 ± 0.0500) · 10⁻² CLEO [27]

B₃₀ = h⁻ 4π⁰ ν_τ (ex. K⁰,η)

(0.0864 ± 0.0067) · 10⁻² average

(0.1120 ± 0.0370 ± 0.0350) · 10⁻² ALEPH [7]

B₃₁ = K⁻ ≥ 0  π⁰ ≥ 0  K⁰ ≥ 0  γ ν_τ

(1.568 ± 0.018) · 10⁻² average

(1.700 ± 0.120 ± 0.190) · 10⁻² CLEO [21]

(1.540 ± 0.240 ± 0.000) · 10⁻² DELPHI [22]

(1.528 ± 0.039 ± 0.040) · 10⁻² OPAL [23]

B₃₂ = K⁻ ≥ 1  (π⁰ or K⁰ or γ) ν_τ

(0.8729 ± 0.0141) · 10⁻² average

B₃₃ = K_S⁰ (particles)⁻ ν_τ

(0.9366 ± 0.0292) · 10⁻² average

(0.9700 ± 0.0580 ± 0.0620) · 10⁻² ALEPH [28]

(0.9700 ± 0.0900 ± 0.0600) · 10⁻² OPAL [29]

B₃₄ = h⁻ K⁰ ν_τ

(0.9860 ± 0.0138) · 10⁻² average

(0.8550 ± 0.0360 ± 0.0730) · 10⁻² CLEO [30]

B₃₅ = π⁻ K⁰ ν_τ

(0.8378 ± 0.0139) · 10⁻² average

(0.9280 ± 0.0450 ± 0.0340) · 10⁻² ALEPH [20]

(0.8320 ± 0.0025 ± 0.0150) · 10⁻² Belle [31]

(0.9500 ± 0.1500 ± 0.0600) · 10⁻² L3 [32]

(0.9330 ± 0.0680 ± 0.0490) · 10⁻² OPAL [33]

B₃₇ = K⁻ K⁰ ν_τ

(0.1483 ± 0.0034) · 10⁻² average

(0.1580 ± 0.0420 ± 0.0170) · 10⁻² ALEPH [28]

(0.1620 ± 0.0210 ± 0.0110) · 10⁻² ALEPH [20]

(0.1478 ± 0.0022 ± 0.0040) · 10⁻² BaBar [3]

(0.1480 ± 0.0013 ± 0.0055) · 10⁻² Belle [31]

(0.1510 ± 0.0210 ± 0.0220) · 10⁻² CLEO [30]

B₃₈ = K⁻ K⁰ ≥ 0   π⁰  ν_τ

(0.2977 ± 0.0073) · 10⁻² average

(0.3300 ± 0.0550 ± 0.0390) · 10⁻² OPAL [33]

B₃₉ = h⁻ K⁰ π⁰ ν_τ

(0.5302 ± 0.0134) · 10⁻² average

(0.5620 ± 0.0500 ± 0.0480) · 10⁻² CLEO [30]

B₄₀ = π⁻ K⁰ π⁰ ν_τ

(0.3807 ± 0.0129) · 10⁻² average

(0.2940 ± 0.0730 ± 0.0370) · 10⁻² ALEPH [28]

(0.3470 ± 0.0530 ± 0.0370) · 10⁻² ALEPH [20]

(0.3860 ± 0.0031 ± 0.0135) · 10⁻² Belle [31]

(0.4100 ± 0.1200 ± 0.0300) · 10⁻² L3 [32]

B₄₂ = K⁻ π⁰ K⁰ ν_τ

(0.1494 ± 0.0070) · 10⁻² average

(0.1520 ± 0.0760 ± 0.0210) · 10⁻² ALEPH [28]

(0.1430 ± 0.0250 ± 0.0150) · 10⁻² ALEPH [20]

(0.1496 ± 0.0019 ± 0.0073) · 10⁻² Belle [31]

(0.1450 ± 0.0360 ± 0.0200) · 10⁻² CLEO [30]

B₄₃ = π⁻ K⁰ ≥ 1   π⁰  ν_τ

(0.4042 ± 0.0260) · 10⁻² average

(0.3240 ± 0.0740 ± 0.0660) · 10⁻² OPAL [33]

B₄₄ = π⁻ K⁰ 2π⁰ ν_τ (ex. K⁰)

(2.346 ± 2.306) · 10⁻⁴ average

(2.600 ± 2.400 ± 0.000) · 10⁻⁴ ALEPH [34]

B₄₆ = π⁻ K⁰ K⁰ ν_τ

(0.1516 ± 0.0247) · 10⁻² average

B₄₇ = π⁻ K_S⁰ K_S⁰ ν_τ

(2.342 ± 0.065) · 10⁻⁴ average

(2.600 ± 1.000 ± 0.500) · 10⁻⁴ ALEPH [28]

(2.310 ± 0.040 ± 0.080) · 10⁻⁴ BaBar [35]

(2.330 ± 0.033 ± 0.093) · 10⁻⁴ Belle [31]

(2.300 ± 0.500 ± 0.300) · 10⁻⁴ CLEO [30]

B₄₈ = π⁻ K_S⁰ K_L⁰ ν_τ

(0.1048 ± 0.0247) · 10⁻² average

(0.1010 ± 0.0230 ± 0.0130) · 10⁻² ALEPH [28]

B₄₉ = π⁻ K⁰ K⁰ π⁰ ν_τ

(3.543 ± 1.193) · 10⁻⁴ average

B₅₀ = π⁻ π⁰ K_S⁰ K_S⁰ ν_τ

(1.816 ± 0.207) · 10⁻⁵ average

(1.600 ± 0.200 ± 0.220) · 10⁻⁵ BaBar [35]

(2.000 ± 0.216 ± 0.202) · 10⁻⁵ Belle [31]

B₅₁ = π⁻ π⁰ K_S⁰ K_L⁰ ν_τ

(3.179 ± 1.192) · 10⁻⁴ average

(3.100 ± 1.100 ± 0.500) · 10⁻⁴ ALEPH [28]

B₅₃ = K⁰ h⁻ h⁻ h⁺ ν_τ

(2.220 ± 2.024) · 10⁻⁴ average

(2.300 ± 1.900 ± 0.700) · 10⁻⁴ ALEPH [28]

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

0.15206 ± 0.00061 average

0.15000 ± 0.00400 ± 0.00300 CELLO [36]

0.14400 ± 0.00600 ± 0.00300 L3 [37]

0.15100 ± 0.00800 ± 0.00600 TPC [38]

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

0.14558 ± 0.00056 average

0.14556 ± 0.00105 ± 0.00076 L3 [39]

0.14960 ± 0.00090 ± 0.00220 OPAL [40]

B₅₆ = h⁻ h⁻ h⁺ ν_τ

(9.769 ± 0.053) · 10⁻² average

B₅₇ = h⁻ h⁻ h⁺ ν_τ (ex.  K⁰)

(9.428 ± 0.053) · 10⁻² average

(9.510 ± 0.070 ± 0.200) · 10⁻² CLEO [41]

(9.317 ± 0.090 ± 0.082) · 10⁻² DELPHI [18]

B₅₇

B₅₅

=
h⁻ h⁻ h⁺ ν_τ (ex.  K⁰)

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

0.6476 ± 0.0029 average

0.6600 ± 0.0040 ± 0.0140 OPAL [40]

B₅₈ = h⁻ h⁻ h⁺ ν_τ (ex.  K⁰, ω)

(9.397 ± 0.053) · 10⁻² average

(9.469 ± 0.062 ± 0.073) · 10⁻² ALEPH [7]

B₅₉ = π⁻ π⁺ π⁻ ν_τ

(9.279 ± 0.051) · 10⁻² average

B₆₀ = π⁻ π⁺ π⁻ ν_τ (ex.  K⁰)

(8.990 ± 0.051) · 10⁻² average

(8.830 ± 0.010 ± 0.130) · 10⁻² BaBar [42]

(8.420 ± 0.000 _−0.250^+0.260) · 10⁻² Belle [43]

(9.130 ± 0.050 ± 0.460) · 10⁻² CLEO3 [44]

B₆₂ = π⁻ π⁻ π⁺ ν_τ (ex. K⁰,ω)

(8.960 ± 0.051) · 10⁻² average

B₆₃ = h⁻ h⁻ h⁺ ≥ 1   neutrals  ν_τ

(5.327 ± 0.049) · 10⁻² average

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

(5.122 ± 0.049) · 10⁻² average

B₆₅ = h⁻ h⁻ h⁺ π⁰ ν_τ

(4.791 ± 0.052) · 10⁻² average

B₆₆ = h⁻ h⁻ h⁺ π⁰ ν_τ (ex.  K⁰)

(4.607 ± 0.051) · 10⁻² average

(4.734 ± 0.059 ± 0.049) · 10⁻² ALEPH [7]

(4.230 ± 0.060 ± 0.220) · 10⁻² CLEO [41]

(4.545 ± 0.106 ± 0.103) · 10⁻² DELPHI [18]

B₆₇ = h⁻ h⁻ h⁺ π⁰ ν_τ (ex.  K⁰, ω)

(2.821 ± 0.070) · 10⁻² average

B₆₈ = π⁻ π⁺ π⁻ π⁰ ν_τ

(4.652 ± 0.053) · 10⁻² average

B₆₉ = π⁻ π⁺ π⁻ π⁰ ν_τ (ex.  K⁰)

(4.520 ± 0.052) · 10⁻² average

(4.190 ± 0.100 ± 0.210) · 10⁻² CLEO [45]

B₇₀ = π⁻ π⁻ π⁺ π⁰ ν_τ (ex. K⁰,ω)

(2.770 ± 0.071) · 10⁻² average

B₇₄ = h⁻ h⁻ h⁺ ≥ 2  π⁰  ν_τ (ex.  K⁰)

(0.5148 ± 0.0311) · 10⁻² average

(0.5610 ± 0.0680 ± 0.0950) · 10⁻² DELPHI [18]

B₇₅ = h⁻ h⁻ h⁺ 2π⁰ ν_τ

(0.5037 ± 0.0309) · 10⁻² average

B₇₆ = h⁻ h⁻ h⁺ 2π⁰ ν_τ (ex.  K⁰)

(0.4937 ± 0.0309) · 10⁻² average

(0.4350 ± 0.0300 ± 0.0350) · 10⁻² ALEPH [7]

B₇₆

B₅₄

=
h⁻ h⁻ h⁺ 2π⁰ ν_τ (ex.  K⁰)

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

(3.247 ± 0.202) · 10⁻² average

(3.400 ± 0.200 ± 0.300) · 10⁻² CLEO [46]

B₇₇ = h⁻ h⁻ h⁺ 2π⁰ ν_τ (ex. K⁰,ω,η)

(9.812 ± 3.555) · 10⁻⁴ average

B₇₈ = h⁻ h⁻ h⁺ 3π⁰ ν_τ

(2.114 ± 0.299) · 10⁻⁴ average

(2.200 ± 0.300 ± 0.400) · 10⁻⁴ CLEO [47]

B₇₉ = K⁻ h⁻ h⁺ ≥ 0   neutrals  ν_τ

(0.6293 ± 0.0140) · 10⁻² average

B₈₀ = K⁻ π⁻ h⁺ ν_τ (ex.  K⁰)

(0.4361 ± 0.0072) · 10⁻² average

B₈₀

B₆₀

=
K⁻ π⁻ h⁺ ν_τ (ex.  K⁰)

π⁻ π⁺ π⁻ ν_τ (ex.  K⁰)

(4.851 ± 0.080) · 10⁻² average

(5.440 ± 0.210 ± 0.530) · 10⁻² CLEO [48]

B₈₁ = K⁻ π⁻ h⁺ π⁰ ν_τ (ex.  K⁰)

(8.727 ± 1.177) · 10⁻⁴ average

B₈₁

B₆₉

=
K⁻ π⁻ h⁺ π⁰ ν_τ (ex.  K⁰)

π⁻ π⁺ π⁻ π⁰ ν_τ (ex.  K⁰)

(1.931 ± 0.266) · 10⁻² average

(2.610 ± 0.450 ± 0.420) · 10⁻² CLEO [48]

B₈₂ = K⁻ π⁻ π⁺ ≥ 0   neutrals  ν_τ

(0.4779 ± 0.0137) · 10⁻² average

(0.5800 _−0.1300^+0.1500 ± 0.1200) · 10⁻² TPC [49]

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

(0.3741 ± 0.0135) · 10⁻² average

B₈₄ = K⁻ π⁻ π⁺ ν_τ

(0.3442 ± 0.0068) · 10⁻² average

B₈₅ = K⁻ π⁺ π⁻ ν_τ (ex.  K⁰)

(0.2929 ± 0.0067) · 10⁻² average

(0.2140 ± 0.0370 ± 0.0290) · 10⁻² ALEPH [50]

(0.2730 ± 0.0020 ± 0.0090) · 10⁻² BaBar [42]

(0.3300 ± 0.0010 _−0.0170^+0.0160) · 10⁻² Belle [43]

(0.3840 ± 0.0140 ± 0.0380) · 10⁻² CLEO3 [44]

(0.4150 ± 0.0530 ± 0.0400) · 10⁻² OPAL [26]

B₈₅

B₆₀

=
K⁻ π⁺ π⁻ ν_τ (ex. K⁰)

π⁻ π⁺ π⁻ ν_τ (ex. K⁰)

(3.258 ± 0.074) · 10⁻² average

B₈₇ = K⁻ π⁻ π⁺ π⁰ ν_τ

(0.1329 ± 0.0119) · 10⁻² average

B₈₈ = K⁻ π⁻ π⁺ π⁰ ν_τ (ex.  K⁰)

(8.116 ± 1.168) · 10⁻⁴ average

(6.100 ± 3.900 ± 1.800) · 10⁻⁴ ALEPH [50]

(7.400 ± 0.800 ± 1.100) · 10⁻⁴ CLEO3 [51]

B₈₉ = K⁻ π⁻ π⁺ π⁰ ν_τ (ex.  K⁰, η)

(7.762 ± 1.168) · 10⁻⁴ average

B₉₂ = π⁻ K⁻ K⁺ ≥ 0   neutrals  ν_τ

(0.1493 ± 0.0033) · 10⁻² average

(0.1590 ± 0.0530 ± 0.0200) · 10⁻² OPAL [52]

(0.1500 _−0.0700^+0.0900 ± 0.0300) · 10⁻² TPC [49]

B₉₃ = π⁻ K⁻ K⁺ ν_τ

(0.1431 ± 0.0027) · 10⁻² average

(0.1630 ± 0.0210 ± 0.0170) · 10⁻² ALEPH [50]

(0.1346 ± 0.0010 ± 0.0036) · 10⁻² BaBar [42]

(0.1550 ± 0.0010 _−0.0050^+0.0060) · 10⁻² Belle [43]

(0.1550 ± 0.0060 ± 0.0090) · 10⁻² CLEO3 [44]

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 [48]

B₉₄ = π⁻ K⁻ K⁺ π⁰ ν_τ

(0.611 ± 0.183) · 10⁻⁴ average

(7.500 ± 2.900 ± 1.500) · 10⁻⁴ ALEPH [50]

(0.550 ± 0.140 ± 0.120) · 10⁻⁴ CLEO3 [51]

B₉₄

B₆₉

=
π⁻ K⁻ K⁺ π⁰ ν_τ

π⁻ π⁺ π⁻ π⁰ ν_τ (ex.  K⁰)

(0.1353 ± 0.0405) · 10⁻² average

(0.7900 ± 0.4400 ± 0.1600) · 10⁻² CLEO [48]

B₉₆ = K⁻ K⁻ K⁺ ν_τ

(2.169 ± 0.800) · 10⁻⁵ average

(1.578 ± 0.130 ± 0.123) · 10⁻⁵ BaBar [42]

(3.290 ± 0.170 _−0.200^+0.190) · 10⁻⁵ Belle [43]

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

(0.0990 ± 0.0037) · 10⁻² average

(0.0970 ± 0.0050 ± 0.0110) · 10⁻² CLEO [53]

(0.1020 ± 0.0290 ± 0.0000) · 10⁻² HRS [54]

(0.1700 ± 0.0220 ± 0.0260) · 10⁻² L3 [39]

B₁₀₃ = 3h⁻ 2h⁺ ν_τ (ex. K⁰)

(8.260 ± 0.314) · 10⁻⁴ average

(7.200 ± 0.900 ± 1.200) · 10⁻⁴ ALEPH [7]

(6.400 ± 2.300 ± 1.000) · 10⁻⁴ ARGUS [55]

(7.700 ± 0.500 ± 0.900) · 10⁻⁴ CLEO [53]

(9.700 ± 1.500 ± 0.500) · 10⁻⁴ DELPHI [18]

(5.100 ± 2.000 ± 0.000) · 10⁻⁴ HRS [54]

(9.100 ± 1.400 ± 0.600) · 10⁻⁴ OPAL [56]

B₁₀₄ = 3h⁻ 2h⁺ π⁰ ν_τ (ex. K⁰)

(1.641 ± 0.114) · 10⁻⁴ average

(2.100 ± 0.700 ± 0.900) · 10⁻⁴ ALEPH [7]

(1.700 ± 0.200 ± 0.200) · 10⁻⁴ CLEO [47]

(1.600 ± 1.200 ± 0.600) · 10⁻⁴ DELPHI [18]

(2.700 ± 1.800 ± 0.900) · 10⁻⁴ OPAL [56]

B₁₀₆ = (5π)⁻ ν_τ

(0.7532 ± 0.0356) · 10⁻² average

B₁₁₀ = X_s⁻ ν_τ

(2.931 ± 0.041) · 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 [57]

(0.1350 ± 0.0030 ± 0.0070) · 10⁻² Belle [58]

(0.1700 ± 0.0200 ± 0.0200) · 10⁻² CLEO [59]

B₁₂₈ = K⁻ η ν_τ

(1.543 ± 0.080) · 10⁻⁴ average

(2.900 _−1.200^+1.300 ± 0.700) · 10⁻⁴ ALEPH [57]

(1.420 ± 0.110 ± 0.070) · 10⁻⁴ BaBar [60]

(1.580 ± 0.050 ± 0.090) · 10⁻⁴ Belle [58]

(2.600 ± 0.500 ± 0.500) · 10⁻⁴ CLEO [61]

B₁₃₀ = K⁻ π⁰ η ν_τ

(0.483 ± 0.116) · 10⁻⁴ average

(0.460 ± 0.110 ± 0.040) · 10⁻⁴ Belle [58]

(1.770 ± 0.560 ± 0.710) · 10⁻⁴ CLEO [62]

B₁₃₂ = π⁻ K⁰ η ν_τ

(0.936 ± 0.149) · 10⁻⁴ average

(0.880 ± 0.140 ± 0.060) · 10⁻⁴ Belle [58]

(2.200 ± 0.700 ± 0.220) · 10⁻⁴ CLEO [62]

B₁₃₆ = π⁻ π⁺ π⁻ η ν_τ (ex.  K⁰)

(2.196 ± 0.129) · 10⁻⁴ average

B₁₄₉ = h⁻ ω ≥ 0   neutrals  ν_τ

(2.402 ± 0.075) · 10⁻² average

B₁₅₀ = h⁻ ω ν_τ

(1.996 ± 0.064) · 10⁻² average

(1.910 ± 0.070 ± 0.060) · 10⁻² ALEPH [57]

(1.600 ± 0.270 ± 0.410) · 10⁻² CLEO [63]

B₁₅₀

B₆₆

=
h⁻ ω ν_τ

h⁻ h⁻ h⁺ π⁰ ν_τ (ex.  K⁰)

0.4331 ± 0.0139 average

0.4310 ± 0.0330 ± 0.0000 ALEPH [64]

0.4640 ± 0.0160 ± 0.0170 CLEO [41]

B₁₅₁ = K⁻ ω ν_τ

(4.100 ± 0.922) · 10⁻⁴ average

(4.100 ± 0.600 ± 0.700) · 10⁻⁴ CLEO3 [51]

B₁₅₂ = h⁻ π⁰ ω ν_τ

(0.4066 ± 0.0419) · 10⁻² average

(0.4300 ± 0.0600 ± 0.0500) · 10⁻² ALEPH [57]

B₁₅₂

B₅₄

=
h⁻ ω π⁰ ν_τ

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

(2.674 ± 0.275) · 10⁻² average

B₁₅₂

B₇₆

=
h⁻ ω π⁰ ν_τ

h⁻ h⁻ h⁺ 2π⁰ ν_τ (ex.  K⁰)

0.8236 ± 0.0757 average

0.8100 ± 0.0600 ± 0.0600 CLEO [46]

B₁₆₇ = K⁻ φ ν_τ

(4.409 ± 1.626) · 10⁻⁵ average

B₁₆₈ = K⁻ φ ν_τ (φ → K⁺ K⁻)

(2.169 ± 0.800) · 10⁻⁵ average

B₁₆₉ = K⁻ φ ν_τ (φ → K_S⁰ K_L⁰)

(1.499 ± 0.553) · 10⁻⁵ average

B₈₀₀ = π⁻ ω ν_τ

(1.955 ± 0.065) · 10⁻² average

B₈₀₂ = K⁻ π⁻ π⁺ ν_τ (ex. K⁰,ω)

(0.2923 ± 0.0067) · 10⁻² average

B₈₀₃ = K⁻ π⁻ π⁺ π⁰ ν_τ (ex. K⁰,ω,η)

(4.105 ± 1.429) · 10⁻⁴ average

B₈₀₄ = π⁻ K_L⁰ K_L⁰ ν_τ

(2.342 ± 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 [7]

B₈₀₆ = π⁻ π⁰ K_L⁰ K_L⁰ ν_τ

(1.816 ± 0.207) · 10⁻⁵ average

B₈₀₉ = π⁻ 4π⁰ ν_τ (ex.  K⁰, η)

(8.640 ± 0.670) · 10⁻⁴ average

(9.020 ± 0.400 ± 0.650) · 10⁻⁴ BaBar [4]

B₈₁₀ = 2π⁻ π⁺ 3π⁰ ν_τ (ex. K⁰)

(1.931 ± 0.298) · 10⁻⁴ average

B₈₁₁ = π⁻ 2π⁰ ω ν_τ (ex. K⁰)

(7.139 ± 1.586) · 10⁻⁵ average

(7.300 ± 1.200 ± 1.200) · 10⁻⁵ BaBar [65]

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

(1.325 ± 2.682) · 10⁻⁵ average

(1.000 ± 0.800 ± 3.000) · 10⁻⁵ BaBar [65]

B₈₂₀ = 3π⁻ 2π⁺ ν_τ (ex. K⁰, ω)

(8.242 ± 0.313) · 10⁻⁴ average

B₈₂₁ = 3π⁻ 2π⁺ ν_τ (ex. K⁰, ω, f₁)

(7.719 ± 0.295) · 10⁻⁴ average

(7.680 ± 0.040 ± 0.400) · 10⁻⁴ BaBar [65]

B₈₂₂ = K⁻ 2π⁻ 2π⁺ ν_τ (ex. K⁰)

(0.594 ± 1.208) · 10⁻⁶ average

(0.600 ± 0.500 ± 1.100) · 10⁻⁶ BaBar [65]

B₈₃₀ = 3π⁻ 2π⁺ π⁰ ν_τ (ex. K⁰)

(1.630 ± 0.113) · 10⁻⁴ average

B₈₃₁ = 2π⁻ π⁺ ω ν_τ (ex. K⁰)

(8.400 ± 0.624) · 10⁻⁵ average

(8.400 ± 0.400 ± 0.600) · 10⁻⁵ BaBar [65]

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

(3.775 ± 0.874) · 10⁻⁵ average

(3.600 ± 0.300 ± 0.900) · 10⁻⁵ BaBar [65]

B₈₃₃ = K⁻ 2π⁻ 2π⁺ π⁰ ν_τ (ex. K⁰)

(1.108 ± 0.566) · 10⁻⁶ average

(1.100 ± 0.400 ± 0.400) · 10⁻⁶ BaBar [65]

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

(7.176 ± 0.422) · 10⁻⁵ average

(8.270 ± 0.880 ± 0.810) · 10⁻⁵ BaBar [65]

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

(4.444 ± 0.867) · 10⁻⁵ average

(4.570 ± 0.770 ± 0.500) · 10⁻⁵ BaBar [65]

B₉₂₀ = π⁻ f₁ ν_τ (f₁ → 2π⁻ 2π⁺)

(5.225 ± 0.444) · 10⁻⁵ average

(5.200 ± 0.310 ± 0.370) · 10⁻⁵ BaBar [65]

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

(5.033 ± 0.296) · 10⁻⁵ average

(5.390 ± 0.270 ± 0.410) · 10⁻⁵ BaBar [65]

B₉₄₄ = 2π⁻ π⁺ η ν_τ (η → γγ)  (ex. K⁰)

(8.654 ± 0.509) · 10⁻⁵ average

(8.260 ± 0.350 ± 0.510) · 10⁻⁵ BaBar [65]

B₉₄₅ = π⁻ 2π⁰ η ν_τ

(1.939 ± 0.378) · 10⁻⁴ average

B₉₉₈ = 1 −  B_All

(0.0274 ± 0.1026) · 10⁻² average

2.6 Correlation terms 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.

2.7 Equality constraints

The constraints on the τ branching fractions fitted quantities are listed in the following. The constraint equations include as coefficients the values of some non-tau branching fractions, denoted e.g., with the self-describing notation B_{K_S → π⁰π⁰}. Some coefficients are probabilities corresponding to the modulus square of amplitudes describing quantum mixtures of states such as K⁰, K⁰, K_S, K_L, denoted with e.g., B_{<K⁰|K_S>} = |<K⁰|K_S>|². All non-tau quantities are taken from the PDG 2018 [6] averages. The fit procedure does not account for their uncertainties, which are generally small with respect to the uncertainties on the τ branching fractions. Please note that, in the following table, when a quantity like B₃/ B₅ appears on the left side of the equation, it represents a fitted quantity, and when it appears on the right side it represents the ratio of two separate fitted quantities.

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>}))

B₈₀₉ =

B₃₀

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


B₄₂	0	0	0	-2	1	-5	1	-4	-4	-2	-2	-1	-15	-20
B₄₄	0	0	0	0	0	0	0	0	0	0	0	-1	0	-4
B₄₇	0	-1	2	1	-1	2	-1	1	1	0	0	-1	2	-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	-4	-5	6	2	-4	1	-11	-1	-2	-2	-3	-1	3	0
B₇₀	-5	-6	-7	-2	-8	-1	-1	-1	-1	0	0	0	-1	0
B₇₇	0	0	-2	0	-2	1	0	1	2	1	1	0	0	0
B₉₃	-1	-1	2	1	-1	1	-2	0	0	0	0	0	1	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	0	0	0	0
B₁₂₈	0	0	1	0	0	0	0	0	0	0	0	0	1	0
	B₃	B₅	B₉	B₁₀	B₁₄	B₁₆	B₂₀	B₂₃	B₂₇	B₂₈	B₃₀	B₃₅	B₃₇	B₄₀

HFLAV-Tau 2018 Report

2 Branching fraction fit

2.1 Technical implementation of the fit procedure

2.2 Fit results

2.3 Changes with respect to the previous report

2.4 Differences between the HFLAV 2018 fit and the PDG 2018 fit

2.5 Branching ratio fit results and experimental inputs

2.6 Correlation terms between basis branching fractions uncertainties

2.7 Equality constraints

HFLAV-Tau 2018 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	1	0	1	-1	0	0	0	0	0	1	0
B₁₅₁	0	0	0	0	0	0	0	0	0	0	0	0	0	0
B₁₅₂	0	0	-3	0	-2	1	0	1	2	1	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	0	0	0	0
B₈₀₂	-1	-1	0	0	-1	-1	-3	-1	-2	-1	-1	0	0	0
B₈₀₃	0	0	0	0	0	0	0	0	0	0	0	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	1	1	0	0	0	0	0	0	0	0	0	0	0	0
B₈₂₁	0	0	2	1	0	1	-2	0	0	0	0	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	0	-1	0	0	0	0	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	0	-1	0	0	0	0	0	0	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