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HFAG-Tau Summer 2016 Report

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2  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 |Vus| and the QCD coupling constant αs 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 Γi, or ratios of two τ decay mode branching fractions, labelled as Γij. 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 Γij in Table 1. Some fitted quantities are sum of other fitted quantities, for instance Γ8 = B (τ→h ντ) is the sum of Γ9 = B (τ→π ντ) and Γ10 = B (τ→K ντ). 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 qi by minimizing a χ2 while respecting a series of equality constraints on the qi. The χ2 is computed using the measurements mi and their covariance matrix Eij as χ2 = (miAikqk)t Eij−1 (mjAjlql) where the model matrix Aij is used to get the vector of the predicted measurements mi from the vector of the fit parameters qj as mi= Aijqj. 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 Aij has one row per measurement mi and one column per fitted quantity qj, with unity coefficients for the rows and column that identify a measurement mi of the quantity qj, respectively. In summary, the fit requires:

     
min (mi − Aikqk)t Eij−1 (mj − Ajl ql),                (1)
     
subjected to   fr(qs) − cr = 0,                (2)

where Eq. 2 defines the relations between the quantities qi 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 qk and the Lagrange multipliers λr of the sum of the χ2 and the constraint expressions multiplied by the Lagrange multipliers λr, one for each constraint:

     
min 
(Aikqk − mi)t Eij−1 (Ajlql − mj) + 2λr(fr(qs) − cr
 
             (3)
     
(∂/∂ qk, ∂/∂ λr) [expression above] = 0                (4)

Eq. 4 defines a set of equations for the vector of the unknowns (qk, λr), 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, qs:

     
fr(qs) − cr = fr(
q
s) + 
∂ fr(qs)
∂ qs



 



q
s
 (qs − 
q
s) − cr 
             (5)

which can be written as

     
Brs qs − cr               (6)

where cr are the resulting constant known terms, independent of qs at first order. After linearization, the differentiation by qk and λr is trivial and leads to a set of linear equations

     
Akit Eij−1 Ajl ql + Bkrt λr =  Akit Eij−1 mj                (7)
     
Brs qs = cr               (8)

which can be expressed as:

     
Fij uj = vi                (9)

where uj = (qk, λr) and vi 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 − ΓAll = (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].

Two old preliminary results have been removed:

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 τKS0 (particles) ντ [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 |Vus| measurements.

The ALEPH result on Γ46 → π K0 K0 ντ) [7] has been removed from the fit inputs, since it is the simply sum of twice Γ47 = π KS0 KS0 ντ and Γ48 = π KS0 KL0 ντ 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:

ΓAll =Γ3 + Γ5 + Γ9 + Γ10 + Γ14 + Γ16 + Γ20 + Γ23 + Γ27 + Γ28 + Γ30 + Γ35 + Γ37 + Γ40 + Γ42 + Γ47·(1 + ((Γ<K0|KL>·Γ<K0|KL>) / (Γ<K0|KS>·Γ<K0|KS>))) + Γ48 + Γ62 + Γ70 + Γ77 + Γ811 + Γ812 + Γ93 + Γ94 + Γ832 + Γ833 + Γ126 + Γ128 + Γ802 + Γ803 + Γ800 + Γ151 + Γ130 + Γ132 + Γ44 + Γ53 + Γ50·(1 + ((Γ<K0|KL>·Γ<K0|KL>) / (Γ<K0|KS>·Γ<K0|KS>))) + Γ51 + Γ167·(Γφ→ K+Kφ→ KS KL) + Γ152 + Γ920 + Γ821 + Γ822 + Γ831 + Γ136 + Γ945 + Γ805 .

In the 2014 definition, the term Γ78 = h h h+0 ντ included the contributions of Γ50 = π π0 KS0 KS0 ντ and Γ132 = π K0 η ντ, which were already included. In the present definition, Γ78 has been replaced with modes whose sum corresponds to Γ810 = 2π π+0 ντ (ex. K0). As in 2014, the total τ branching fraction ΓAll definition includes two modes that have overlapping final states, to a minor extent that we consider negligible:

     
 Γ50 =  π π0 KS0 KS0 ντ          
 
Γ132 =  π 
K
0 η ντ .
         

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) ντ), 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 reports the τ branching ratio fit results and experimental inputs.


Table 1: HFAG Summer 2016 branching fractions fit results.
τ lepton branching fractionFit value / Exp.HFAG Fit / Ref.
 
Γ1 = (particles) ≥ 0  neutrals ≥ 0   K0  ντ
0.8519 ± 0.0011HFAG Summer 2016 fit
Γ2 = (particles) ≥ 0  neutrals ≥ 0   KL0  ντ
0.8453 ± 0.0010HFAG Summer 2016 fit
Γ3 = µ νµντ
0.17392 ± 0.00040HFAG Summer 2016 fit
0.17319 ± 0.00077 ± 0.00000ALEPH[9]
0.17325 ± 0.00095 ± 0.00077DELPHI[10]
0.17342 ± 0.00110 ± 0.00067L3[11]
0.17340 ± 0.00090 ± 0.00060OPAL[12]
Γ3
Γ5
 = 
µ νµντ
e νe ντ
0.9762 ± 0.0028HFAG Summer 2016 fit
0.9970 ± 0.0350 ± 0.0400ARGUS[13]
0.9796 ± 0.0016 ± 0.0036BaBar[14]
0.9777 ± 0.0063 ± 0.0087CLEO[15]
Γ5 = e νe ντ
0.17816 ± 0.00041HFAG Summer 2016 fit
0.17837 ± 0.00080 ± 0.00000ALEPH[9]
0.17760 ± 0.00060 ± 0.00170CLEO[15]
0.17877 ± 0.00109 ± 0.00110DELPHI[10]
0.17806 ± 0.00104 ± 0.00076L3[11]
0.17810 ± 0.00090 ± 0.00060OPAL[16]
Γ7 = h ≥ 0   KL0  ντ
0.12023 ± 0.00054HFAG Summer 2016 fit
0.12400 ± 0.00700 ± 0.00700DELPHI[17]
0.12470 ± 0.00260 ± 0.00430L3[18]
0.12100 ± 0.00700 ± 0.00500OPAL[19]
Γ8 = h ντ
0.11506 ± 0.00054HFAG Summer 2016 fit
0.11524 ± 0.00105 ± 0.00000ALEPH[9]
0.11520 ± 0.00050 ± 0.00120CLEO[15]
0.11571 ± 0.00120 ± 0.00114DELPHI[20]
0.11980 ± 0.00130 ± 0.00160OPAL[21]
Γ8
Γ5
 = 
h ντ
e νe ντ
0.6458 ± 0.0033HFAG Summer 2016 fit
Γ9 = π ντ
0.10810 ± 0.00053HFAG Summer 2016 fit
Γ9
Γ5
 = 
π ντ
e νe ντ
0.6068 ± 0.0032HFAG Summer 2016 fit
0.5945 ± 0.0014 ± 0.0061BaBar[14]
Γ10 = K ντ
(0.6960 ± 0.0096) · 10−2HFAG 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 ντ
e νe ντ
(3.906 ± 0.054) · 10−2HFAG Summer 2016 fit
(3.882 ± 0.032 ± 0.057) · 10−2 BaBar[14]
Γ10
Γ9
 = 
K ντ
π ντ
(6.438 ± 0.094) · 10−2HFAG Summer 2016 fit
Γ11 = h ≥ 1   neutrals  ντ
0.36973 ± 0.00097HFAG Summer 2016 fit
Γ12 = h ≥ 1  π0  ντ (ex.  K0)
0.36475 ± 0.00097HFAG Summer 2016 fit
Γ13 = h π0 ντ
0.25935 ± 0.00091HFAG Summer 2016 fit
0.25924 ± 0.00129 ± 0.00000ALEPH[9]
0.25670 ± 0.00010 ± 0.00390Belle[26]
0.25870 ± 0.00120 ± 0.00420CLEO[27]
0.25740 ± 0.00201 ± 0.00138DELPHI[20]
0.25050 ± 0.00350 ± 0.00500L3[18]
0.25890 ± 0.00170 ± 0.00290OPAL[21]
Γ14 = π π0 ντ
0.25502 ± 0.00092HFAG Summer 2016 fit
Γ16 = K π0 ντ
(0.4327 ± 0.0149) · 10−2HFAG 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  ντ
0.10775 ± 0.00095HFAG Summer 2016 fit
0.09910 ± 0.00310 ± 0.00270OPAL[21]
Γ18 = h 2π0 ντ
(9.458 ± 0.097) · 10−2HFAG Summer 2016 fit
Γ19 = h 2π0 ντ (ex.  K0)
(9.306 ± 0.097) · 10−2HFAG 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 ντ (ex.  K0)
h π0 ντ
0.3588 ± 0.0044HFAG Summer 2016 fit
0.3420 ± 0.0060 ± 0.0160CLEO[30]
Γ20 = π 2π0 ντ (ex. K0)
(9.242 ± 0.100) · 10−2HFAG Summer 2016 fit
Γ23 = K 2π0 ντ (ex. K0)
(0.0640 ± 0.0220) · 10−2HFAG 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  ντ
(1.318 ± 0.065) · 10−2HFAG Summer 2016 fit
Γ25 = h ≥ 3  π0  ντ (ex.  K0)
(1.233 ± 0.065) · 10−2HFAG Summer 2016 fit
(1.403 ± 0.214 ± 0.224) · 10−2 DELPHI[20]
Γ26 = h 3π0 ντ
(1.158 ± 0.072) · 10−2HFAG 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 ντ
h π0 ντ
(4.465 ± 0.277) · 10−2HFAG Summer 2016 fit
(4.400 ± 0.300 ± 0.500) · 10−2 CLEO[30]
Γ27 = π 3π0 ντ (ex. K0)
(1.029 ± 0.075) · 10−2HFAG Summer 2016 fit
Γ28 = K 3π0 ντ (ex. K0,η)
(4.283 ± 2.161) · 10−4HFAG Summer 2016 fit
(3.700 ± 2.371 ± 0.000) · 10−4 ALEPH[22]
Γ29 = h 4π0 ντ (ex.  K0)
(0.1568 ± 0.0391) · 10−2HFAG Summer 2016 fit
(0.1600 ± 0.0500 ± 0.0500) · 10−2 CLEO[30]
Γ30 = h 4π0 ντ (ex. K0,η)
(0.1099 ± 0.0391) · 10−2HFAG Summer 2016 fit
(0.1120 ± 0.0509 ± 0.0000) · 10−2 ALEPH[9]
Γ31 = K ≥ 0  π0 ≥ 0  K0 ≥ 0  γ ντ
(1.545 ± 0.030) · 10−2HFAG 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) ντ
(0.8528 ± 0.0286) · 10−2HFAG Summer 2016 fit
Γ33 = KS0 (particles) ντ
(0.9372 ± 0.0292) · 10−2HFAG 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 ντ
(0.9865 ± 0.0139) · 10−2HFAG Summer 2016 fit
(0.8550 ± 0.0360 ± 0.0730) · 10−2 CLEO[32]
Γ35 = π K0 ντ
(0.8386 ± 0.0141) · 10−2HFAG 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 ντ
(0.1479 ± 0.0053) · 10−2HFAG 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  ντ
(0.2982 ± 0.0079) · 10−2HFAG Summer 2016 fit
(0.3300 ± 0.0550 ± 0.0390) · 10−2 OPAL[34]
Γ39 = h K0 π0 ντ
(0.5314 ± 0.0134) · 10−2HFAG Summer 2016 fit
(0.5620 ± 0.0500 ± 0.0480) · 10−2 CLEO[32]
Γ40 = π K0 π0 ντ
(0.3812 ± 0.0129) · 10−2HFAG 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 ντ
(0.1502 ± 0.0071) · 10−2HFAG 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  ντ
(0.4046 ± 0.0260) · 10−2HFAG Summer 2016 fit
(0.3240 ± 0.0740 ± 0.0660) · 10−2 OPAL[34]
Γ44 = π K0 π0 π0 ντ (ex. K0)
(2.340 ± 2.306) · 10−4HFAG Summer 2016 fit
(2.600 ± 2.400 ± 0.000) · 10−4 ALEPH[35]
Γ46 = π K0 K0 ντ
(0.1513 ± 0.0247) · 10−2HFAG Summer 2016 fit
Γ47 = π KS0 KS0 ντ
(2.332 ± 0.065) · 10−4HFAG 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 KS0 KL0 ντ
(0.1047 ± 0.0247) · 10−2HFAG Summer 2016 fit
(0.1010 ± 0.0264 ± 0.0000) · 10−2 ALEPH[7]
Γ49 = π K0 K0 π0 ντ
(3.540 ± 1.193) · 10−4HFAG Summer 2016 fit
Γ50 = π π0 KS0 KS0 ντ
(1.815 ± 0.207) · 10−5HFAG 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 KS0 KL0 ντ
(3.177 ± 1.192) · 10−4HFAG Summer 2016 fit
(3.100 ± 1.100 ± 0.500) · 10−4 ALEPH[7]
Γ53 = K0 h h h+ ντ
(2.218 ± 2.024) · 10−4HFAG Summer 2016 fit
(2.300 ± 2.025 ± 0.000) · 10−4 ALEPH[7]
Γ54 = h h h+ ≥ 0  neutrals ≥ 0   KL0  ντ
0.15215 ± 0.00061HFAG Summer 2016 fit
0.15000 ± 0.00400 ± 0.00300CELLO[37]
0.14400 ± 0.00600 ± 0.00300L3[38]
0.15100 ± 0.00800 ± 0.00600TPC[39]
Γ55 = h h h+ ≥ 0   neutrals  ντ (ex.  K0)
0.14567 ± 0.00057HFAG Summer 2016 fit
0.14556 ± 0.00105 ± 0.00076L3[40]
0.14960 ± 0.00090 ± 0.00220OPAL[41]
Γ56 = h h h+ ντ
(9.780 ± 0.054) · 10−2HFAG Summer 2016 fit
Γ57 = h h h+ ντ (ex.  K0)
(9.439 ± 0.053) · 10−2HFAG 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+ ντ (ex.  K0)
h h h+ ≥ 0   neutrals  ντ (ex.  K0)
0.6480 ± 0.0030HFAG Summer 2016 fit
0.6600 ± 0.0040 ± 0.0140OPAL[41]
Γ58 = h h h+ ντ (ex.  K0, ω)
(9.408 ± 0.053) · 10−2HFAG Summer 2016 fit
(9.469 ± 0.096 ± 0.000) · 10−2 ALEPH[9]
Γ59 = π π+ π ντ
(9.290 ± 0.052) · 10−2HFAG Summer 2016 fit
Γ60 = π π+ π ντ (ex.  K0)
(9.000 ± 0.051) · 10−2HFAG 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+ ντ (ex. K0,ω)
(8.970 ± 0.052) · 10−2HFAG Summer 2016 fit
Γ63 = h h h+ ≥ 1   neutrals  ντ
(5.325 ± 0.050) · 10−2HFAG Summer 2016 fit
Γ64 = h h h+ ≥ 1   π0  ντ (ex.  K0)
(5.120 ± 0.049) · 10−2HFAG Summer 2016 fit
Γ65 = h h h+ π0 ντ
(4.790 ± 0.052) · 10−2HFAG Summer 2016 fit
Γ66 = h h h+ π0 ντ (ex.  K0)
(4.606 ± 0.051) · 10−2HFAG 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 ντ (ex.  K0, ω)
(2.820 ± 0.070) · 10−2HFAG Summer 2016 fit
Γ68 = π π+ π π0 ντ
(4.651 ± 0.053) · 10−2HFAG Summer 2016 fit
Γ69 = π π+ π π0 ντ (ex.  K0)
(4.519 ± 0.052) · 10−2HFAG Summer 2016 fit
(4.190 ± 0.100 ± 0.210) · 10−2 CLEO[46]
Γ70 = π π π+ π0 ντ (ex. K0,ω)
(2.769 ± 0.071) · 10−2HFAG Summer 2016 fit
Γ74 = h h h+ ≥ 2  π0  ντ (ex.  K0)
(0.5135 ± 0.0312) · 10−2HFAG Summer 2016 fit
(0.5610 ± 0.0680 ± 0.0950) · 10−2 DELPHI[20]
Γ75 = h h h+ 2π0 ντ
(0.5024 ± 0.0310) · 10−2HFAG Summer 2016 fit
Γ76 = h h h+ 2π0 ντ (ex.  K0)
(0.4925 ± 0.0310) · 10−2HFAG Summer 2016 fit
(0.4350 ± 0.0461 ± 0.0000) · 10−2 ALEPH[9]
Γ76
Γ54
 = 
h h h+ 2π0 ντ (ex.  K0)
h h h+ ≥ 0  neutrals ≥ 0   KL0  ντ
(3.237 ± 0.202) · 10−2HFAG Summer 2016 fit
(3.400 ± 0.200 ± 0.300) · 10−2 CLEO[47]
Γ77 = h h h+ 2π0 ντ (ex. K0,ω,η)
(9.759 ± 3.550) · 10−4HFAG Summer 2016 fit
Γ78 = h h h+ 3π0 ντ
(2.107 ± 0.299) · 10−4HFAG Summer 2016 fit
(2.200 ± 0.300 ± 0.400) · 10−4 CLEO[48]
Γ79 = K h h+ ≥ 0   neutrals  ντ
(0.6297 ± 0.0141) · 10−2HFAG Summer 2016 fit
Γ80 = K π h+ ντ (ex.  K0)
(0.4363 ± 0.0073) · 10−2HFAG Summer 2016 fit
Γ80
Γ60
 = 
K π h+ ντ (ex.  K0)
π π+ π ντ (ex.  K0)
(4.847 ± 0.080) · 10−2HFAG Summer 2016 fit
(5.440 ± 0.210 ± 0.530) · 10−2 CLEO[49]
Γ81 = K π h+ π0 ντ (ex.  K0)
(8.726 ± 1.177) · 10−4HFAG Summer 2016 fit
Γ81
Γ69
 = 
K π h+ π0 ντ (ex.  K0)
π π+ π π0 ντ (ex.  K0)
(1.931 ± 0.266) · 10−2HFAG Summer 2016 fit
(2.610 ± 0.450 ± 0.420) · 10−2 CLEO[49]
Γ82 = K π π+ ≥ 0   neutrals  ντ
(0.4780 ± 0.0137) · 10−2HFAG Summer 2016 fit
(0.5800 −0.1300+0.1500 ± 0.1200) · 10−2 TPC[50]
Γ83 = K π π+ ≥ 0   π0  ντ (ex.  K0)
(0.3741 ± 0.0135) · 10−2HFAG Summer 2016 fit
Γ84 = K π π+ ντ
(0.3441 ± 0.0070) · 10−2HFAG Summer 2016 fit
Γ85 = K π+ π ντ (ex.  K0)
(0.2929 ± 0.0067) · 10−2HFAG 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 ντ (ex. K0)
π π+ π ντ (ex. K0)
(3.254 ± 0.074) · 10−2HFAG Summer 2016 fit
Γ87 = K π π+ π0 ντ
(0.1331 ± 0.0119) · 10−2HFAG Summer 2016 fit
Γ88 = K π π+ π0 ντ (ex.  K0)
(8.115 ± 1.168) · 10−4HFAG 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 ντ (ex.  K0, η)
(7.761 ± 1.168) · 10−4HFAG Summer 2016 fit
Γ92 = π K K+ ≥ 0   neutrals  ντ
(0.1495 ± 0.0033) · 10−2HFAG 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+ ντ
(0.1434 ± 0.0027) · 10−2HFAG 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+ ντ
π π+ π ντ (ex.  K0)
(1.593 ± 0.030) · 10−2HFAG Summer 2016 fit
(1.600 ± 0.150 ± 0.300) · 10−2 CLEO[49]
Γ94 = π K K+ π0 ντ
(0.611 ± 0.183) · 10−4HFAG 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 ντ
π π+ π π0 ντ (ex.  K0)
(0.1353 ± 0.0405) · 10−2HFAG Summer 2016 fit
(0.7900 ± 0.4400 ± 0.1600) · 10−2 CLEO[49]
Γ96 = K K K+ ντ
(2.174 ± 0.800) · 10−5HFAG 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  ντ (ex.  K0)
(0.0985 ± 0.0037) · 10−2HFAG 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+ ντ (ex. K0)
(8.216 ± 0.316) · 10−4HFAG 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 ντ (ex. K0)
(1.634 ± 0.114) · 10−4HFAG 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) ντ
(0.7748 ± 0.0534) · 10−2HFAG Summer 2016 fit
Γ110 = Xs ντ
(2.909 ± 0.048) · 10−2HFAG Summer 2016 fit
Γ126 = π π0 η ντ
(0.1386 ± 0.0072) · 10−2HFAG 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 ντ
(1.547 ± 0.080) · 10−4HFAG 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 ντ
(0.483 ± 0.116) · 10−4HFAG 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 ντ
(0.937 ± 0.149) · 10−4HFAG 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 ντ (ex.  K0)
(2.184 ± 0.130) · 10−4HFAG Summer 2016 fit
Γ149 = h ω ≥ 0   neutrals  ντ
(2.401 ± 0.075) · 10−2HFAG Summer 2016 fit
Γ150 = h ω ντ
(1.995 ± 0.064) · 10−2HFAG 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 ντ
h h h+ π0 ντ (ex.  K0)
0.4332 ± 0.0139HFAG Summer 2016 fit
0.4310 ± 0.0330 ± 0.0000ALEPH[65]
0.4640 ± 0.0160 ± 0.0170CLEO[42]
Γ151 = K ω ντ
(4.100 ± 0.922) · 10−4HFAG Summer 2016 fit
(4.100 ± 0.600 ± 0.700) · 10−4 CLEO3[52]
Γ152 = h π0 ω ντ
(0.4058 ± 0.0419) · 10−2HFAG Summer 2016 fit
(0.4300 ± 0.0781 ± 0.0000) · 10−2 ALEPH[58]
Γ152
Γ54
 = 
h ω π0 ντ
h h h+ ≥ 0  neutrals ≥ 0   KL0  ντ
(2.667 ± 0.275) · 10−2HFAG Summer 2016 fit
Γ152
Γ76
 = 
h ω π0 ντ
h h h+ 2π0 ντ (ex.  K0)
0.8241 ± 0.0757HFAG Summer 2016 fit
0.8100 ± 0.0600 ± 0.0600CLEO[47]
Γ167 = K φ ντ
(4.445 ± 1.636) · 10−5HFAG Summer 2016 fit
Γ168 = K φ ντ (φ → K+ K)
(2.174 ± 0.800) · 10−5HFAG Summer 2016 fit
Γ169 = K φ ντ (φ → KS0 KL0)
(1.520 ± 0.560) · 10−5HFAG Summer 2016 fit
Γ800 = π ω ντ
(1.954 ± 0.065) · 10−2HFAG Summer 2016 fit
Γ802 = K π π+ ντ (ex. K0,ω)
(0.2923 ± 0.0067) · 10−2HFAG Summer 2016 fit
Γ803 = K π π+ π0 ντ (ex. K0,ω,η)
(4.103 ± 1.429) · 10−4HFAG Summer 2016 fit
Γ804 = π KL0 KL0 ντ
(2.332 ± 0.065) · 10−4HFAG Summer 2016 fit
Γ805 = a1 (→ π γ) ντ
(4.000 ± 2.000) · 10−4HFAG Summer 2016 fit
(4.000 ± 2.000 ± 0.000) · 10−4 ALEPH[9]
Γ806 = π π0 KL0 KL0 ντ
(1.815 ± 0.207) · 10−5HFAG Summer 2016 fit
Γ810 = 2π π+ 3π0 ντ (ex. K0)
(1.924 ± 0.298) · 10−4HFAG Summer 2016 fit
Γ811 = π 2π0 ω ντ (ex. K0)
(7.105 ± 1.586) · 10−5HFAG Summer 2016 fit
(7.300 ± 1.200 ± 1.200) · 10−5 BaBar[66]
Γ812 = 2π π+ 3π0 ντ (ex. K0, η, ω, f1)
(1.344 ± 2.683) · 10−5HFAG Summer 2016 fit
(1.000 ± 0.800 ± 3.000) · 10−5 BaBar[66]
Γ820 = 3π 2π+ ντ (ex. K0, ω)
(8.197 ± 0.315) · 10−4HFAG Summer 2016 fit
Γ821 = 3π 2π+ ντ (ex. K0, ω, f1)
(7.677 ± 0.297) · 10−4HFAG Summer 2016 fit
(7.680 ± 0.040 ± 0.400) · 10−4 BaBar[66]
Γ822 = K 2π 2π+ ντ (ex. K0)
(0.596 ± 1.208) · 10−6HFAG Summer 2016 fit
(0.600 ± 0.500 ± 1.100) · 10−6 BaBar[66]
Γ830 = 3π 2π+ π0 ντ (ex. K0)
(1.623 ± 0.114) · 10−4HFAG Summer 2016 fit
Γ831 = 2π π+ ω ντ (ex. K0)
(8.359 ± 0.626) · 10−5HFAG Summer 2016 fit
(8.400 ± 0.400 ± 0.600) · 10−5 BaBar[66]
Γ832 = 3π 2π+ π0 ντ (ex. K0, η, ω, f1)
(3.771 ± 0.875) · 10−5HFAG Summer 2016 fit
(3.600 ± 0.300 ± 0.900) · 10−5 BaBar[66]
Γ833 = K 2π 2π+ π0 ντ (ex. K0)
(1.108 ± 0.566) · 10−6HFAG Summer 2016 fit
(1.100 ± 0.400 ± 0.400) · 10−6 BaBar[66]
Γ910 = 2π π+ η ντ (η → 3π0)  (ex. K0)
(7.136 ± 0.424) · 10−5HFAG Summer 2016 fit
(8.270 ± 0.880 ± 0.810) · 10−5 BaBar[66]
Γ911 = π 2π0 η ντ (η → π+ π π0)  (ex. K0)
(4.420 ± 0.867) · 10−5HFAG Summer 2016 fit
(4.570 ± 0.770 ± 0.500) · 10−5 BaBar[66]
Γ920 = π f1 ντ (f1 → 2π 2π+)
(5.197 ± 0.444) · 10−5HFAG Summer 2016 fit
(5.200 ± 0.310 ± 0.370) · 10−5 BaBar[66]
Γ930 = 2π π+ η ντ (η → π+ππ0)  (ex. K0)
(5.005 ± 0.297) · 10−5HFAG Summer 2016 fit
(5.390 ± 0.270 ± 0.410) · 10−5 BaBar[66]
Γ944 = 2π π+ η ντ (η → γγ)  (ex. K0)
(8.606 ± 0.511) · 10−5HFAG Summer 2016 fit
(8.260 ± 0.350 ± 0.510) · 10−5 BaBar[66]
Γ945 = π 2π0 η ντ
(1.929 ± 0.378) · 10−4HFAG Summer 2016 fit
Γ998 = 1 − ΓAll
(0.0355 ± 0.1031) · 10−2HFAG Summer 2016 fit
 

2.6  Correlation terms between basis branching fractions uncertainties

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


Table 2: Basis nodes correlation coefficients in percent, section 1.
Γ5 23             
Γ9 75            
Γ10 351           
Γ14 -13-14-12-3          
Γ16 0-12-1-16         
Γ20 -5-5-7-1-402        
Γ23 000-22-13-22       
Γ27 -4-3-8-103-366      
Γ28 000-22-135-21-29     
Γ30 -5-4-11-2-9060-420    
Γ35 00000001010   
Γ37 00000-21-31-30-22  
Γ40 00000101-210-124 
  Γ3 Γ5 Γ9 Γ10 Γ14 Γ16 Γ20 Γ23 Γ27 Γ28 Γ30 Γ35 Γ37 Γ40


Table 3: Basis nodes correlation coefficients in percent, section 2.
Γ42 00001-31-51-502-21-20
Γ44 00000000000-10-4
Γ47 00000000000-11-4
Γ48 00000000000-30-2
Γ50 0000000-10-10070
Γ51 00000000000-10-1
Γ53 00000000000000
Γ62 -3-580-45-7-1-5-1-5000
Γ70 -6-6-7-1-8-1-10-103000
Γ77 -10-3-1-2000202000
Γ93 -1-130-12-10-10-1000
Γ94 00000000000000
Γ126 000000-1000-2000
Γ128 0010010-10-10000
  Γ3 Γ5 Γ9 Γ10 Γ14 Γ16 Γ20 Γ23 Γ27 Γ28 Γ30 Γ35 Γ37 Γ40


Table 4: Basis nodes correlation coefficients in percent, section 3.
Γ130 00000000000000
Γ132 00000000000000
Γ136 00000000000000
Γ151 00000000000000
Γ152 -10-3-1-20-10202000
Γ167 00000000000000
Γ800 -2-2-20-3000001000
Γ802 -1-100-10-20-20-1000
Γ803 00000000000000
Γ805 00000000000000
Γ811 00000000000000
Γ812 01000000000000
Γ821 001000-1000-1000
Γ822 00000000000000
  Γ3 Γ5 Γ9 Γ10 Γ14 Γ16 Γ20 Γ23 Γ27 Γ28 Γ30 Γ35 Γ37 Γ40


Table 5: Basis nodes correlation coefficients in percent, section 4.
Γ831 00000000000000
Γ832 00000000000000
Γ833 00000000000000
Γ920 00000000000000
Γ945 00000000000000
  Γ3 Γ5 Γ9 Γ10 Γ14 Γ16 Γ20 Γ23 Γ27 Γ28 Γ30 Γ35 Γ37 Γ40


Table 6: Basis nodes correlation coefficients in percent, section 5.
Γ44 0             
Γ47 10            
Γ48 -1-60           
Γ50 50-70          
Γ51 0-30-60         
Γ53 000000        
Γ62 0010000       
Γ70 0000000-20      
Γ77 0000000-1-7     
Γ93 000000014-40    
Γ94 00000000-200   
Γ126 001000010-500  
Γ128 0010000200104 
  Γ42 Γ44 Γ47 Γ48 Γ50 Γ51 Γ53 Γ62 Γ70 Γ77 Γ93 Γ94 Γ126 Γ128


Table 7: Basis nodes correlation coefficients in percent, section 6.
Γ130 000000000-10011
Γ132 00000000000021
Γ136 00000000-100000
Γ151 000000001200000
Γ152 0000000-1-11-640000
Γ167 0000000-1001000
Γ800 0000000-8-69-2-1000
Γ802 000000016-600000
Γ803 0000000-1-1900-20-1
Γ805 00000000000000
Γ811 00000000000000
Γ812 0000-1000-100000
Γ821 00000000-100000
Γ822 00000000000000
  Γ42 Γ44 Γ47 Γ48 Γ50 Γ51 Γ53 Γ62 Γ70 Γ77 Γ93 Γ94 Γ126 Γ128


Table 8: Basis nodes correlation coefficients in percent, section 7.
Γ831 00000000000000
Γ832 00000000000000
Γ833 00000000000000
Γ920 00000000000000
Γ945 00000000000000
  Γ42 Γ44 Γ47 Γ48 Γ50 Γ51 Γ53 Γ62 Γ70 Γ77 Γ93 Γ94 Γ126 Γ128


Table 9: Basis nodes correlation coefficients in percent, section 8.
Γ132 0             
Γ136 00            
Γ151 000           
Γ152 0000          
Γ167 00000         
Γ800 000-14-30        
Γ802 000-201-1       
Γ803 000-580091      
Γ805 000000000     
Γ811 0-1200000000    
Γ812 0-2-80000000-16   
Γ821 004700000008-4  
Γ822 00-1000000000-1 
  Γ130 Γ132 Γ136 Γ151 Γ152 Γ167 Γ800 Γ802 Γ803 Γ805 Γ811 Γ812 Γ821 Γ822


Table 10: Basis nodes correlation coefficients in percent, section 9.
Γ831 0039000000014-439-1
Γ832 00300000002030
Γ833 00-1000000000-10
Γ920 002100000003-235-1
Γ945 0-125000000010-11100
  Γ130 Γ132 Γ136 Γ151 Γ152 Γ167 Γ800 Γ802 Γ803 Γ805 Γ811 Γ812 Γ821 Γ822


Table 11: Basis nodes correlation coefficients in percent, section 10.
Γ832 -2    
Γ833 -1-1   
Γ920 1710  
Γ945 17204 
  Γ831 Γ832 Γ833 Γ920 Γ945

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 Γn labels. The equations include as coefficients the values of some non-tau branching fractions, denoted e.g. with the self-describing notation ΓKS → π0π0. Some coefficients are probabilities corresponding to modulus square amplitudes describing quantum mixtures of states such as K0, K0, KS, KL, denoted with e.g. Γ<K0|KS> = |<K0|KS>|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·Γη→neutral           
   + Γ128·Γη→neutral + Γ130·Γη→neutral + Γ132·Γη→neutral           
   + Γ800·Γω→π0γ + Γ151·Γω→π0γ + Γ152·Γω→π0γ           
   + Γ167·Γφ→ KS KL          
     
Γ2 = Γ3 + Γ5 + Γ9 + Γ10 + Γ14 + Γ16           
   + Γ20 + Γ23 + Γ27 + Γ28 + Γ30 + Γ35·(Γ<K0|KS>·ΓKS→π0π0          
   +Γ<K0|KL>) + Γ40·(Γ<K0|KS>·ΓKS→π0π0<K0|KL>) + Γ44·(Γ<K0|KS>·ΓKS→π0π0          
   +Γ<K0|KL>) + Γ37·(Γ<K0|KS>·ΓKS→π0π0<K0|KL>) + Γ42·(Γ<K0|KS>·ΓKS→π0π0          
   +Γ<K0|KL>) + Γ47·(ΓKS→π0π0·ΓKS→π0π0) + Γ48·ΓKS→π0π0           
   + Γ804 + Γ50·(ΓKS→π0π0·ΓKS→π0π0) + Γ51·ΓKS→π0π0           
   + Γ806 + Γ126·Γη→neutral + Γ128·Γη→neutral + Γ130·Γη→neutral           
   + Γ132·(Γη→neutral·(Γ<K0|KS>·ΓKS→π0π0<K0|KL>)) + Γ800·Γω→π0γ           
   + Γ151·Γω→π0γ + Γ152·Γω→π0γ + Γ167·(Γφ→ KS KL·ΓKS→π0π0)          
     
Γ7 = Γ35·Γ<K0|KL> + Γ9 + Γ804 + Γ37·Γ<K0|KL>           
   + Γ10          
     
Γ8 = Γ9 + Γ10          
     
Γ11 = Γ14 + Γ16 + Γ20 + Γ23 + Γ27 + Γ28           
   + Γ30 + Γ35·(Γ<K0|KS>·ΓKS→π0π0) + Γ37·(Γ<K0|KS>·ΓKS→π0π0)           
   + Γ40·(Γ<K0|KS>·ΓKS→π0π0) + Γ42·(Γ<K0|KS>·ΓKS→π0π0)           
   + Γ47·(ΓKS→π0π0·ΓKS→π0π0) + Γ50·(ΓKS→π0π0·ΓKS→π0π0)           
   + Γ126·Γη→neutral + Γ128·Γη→neutral + Γ130·Γη→neutral           
   + Γ132·(Γ<K0|KS>·ΓKS→π0π0·Γη→neutral) + Γ151·Γω→π0γ           
   + Γ152·Γω→π0γ + Γ800·Γω→π0γ          
     
Γ12 = Γ128·Γη→3π0 + Γ30 + Γ23 + Γ28 + Γ14           
   + Γ16 + Γ20 + Γ27 + Γ126·Γη→3π0 + Γ130·Γη→3π0          
     
Γ13 = Γ14 + Γ16          
     
Γ17 = Γ128·Γη→3π0 + Γ30 + Γ23 + Γ28 + Γ35·(Γ<K0|KS>·ΓKS→π0π0)           
   + Γ40·(Γ<K0|KS>·ΓKS→π0π0) + Γ42·(Γ<K0|KS>·ΓKS→π0π0)           
   + Γ20 + Γ27 + Γ47·(ΓKS→π0π0·ΓKS→π0π0) + Γ50·(ΓKS→π0π0·ΓKS→π0π0)           
   + Γ126·Γη→3π0 + Γ37·(Γ<K0|KS>·ΓKS→π0π0) + Γ130·Γη→3π0          
     
Γ18 = Γ23 + Γ35·(Γ<K0|KS>·ΓKS→π0π0) + Γ20 + Γ37·(Γ<K0|KS>·ΓKS→π0π0)          
     
Γ19 = Γ23 + Γ20          
     
Γ24 = Γ27 + Γ28 + Γ30 + Γ40·(Γ<K0|KS>·ΓKS→π0π0)           
   + Γ42·(Γ<K0|KS>·ΓKS→π0π0) + Γ47·(ΓKS→π0π0·ΓKS→π0π0)           
   + Γ50·(ΓKS→π0π0·ΓKS→π0π0) + Γ126·Γη→3π0 + Γ128·Γη→3π0           
   + Γ130·Γη→3π0 + Γ132·(Γ<K0|KS>·ΓKS→π0π0·Γη→3π0)          
     
Γ25 = Γ128·Γη→3π0 + Γ30 + Γ28 + Γ27 + Γ126·Γη→3π0           
   + Γ130·Γη→3π0          
     
Γ26 = Γ128·Γη→3π0 + Γ28 + Γ40·(Γ<K0|KS>·ΓKS→π0π0)           
   + Γ42·(Γ<K0|KS>·ΓKS→π0π0) + Γ27          
     
Γ29 = Γ30 + Γ126·Γη→3π0 + Γ130·Γη→3π0          
     
Γ31 = Γ128·Γη→neutral + Γ23 + Γ28 + Γ42 + Γ16           
   + Γ37 + Γ10 + Γ167·(Γφ→ KS KL·ΓKS→π0π0)          
     
Γ32 = Γ16 + Γ23 + Γ28 + Γ37 + Γ42 + Γ128·Γη→neutral           
   + Γ130·Γη→neutral + Γ167·(Γφ→ KS KL·ΓKS→π0π0)          
     
Γ33 = Γ35·Γ<K0|KS> + Γ40·Γ<K0|KS> + Γ42·Γ<K0|KS>           
   + Γ47 + Γ48 + Γ50 + Γ51 + Γ37·Γ<K0|KS>           
   + Γ132·(Γ<K0|KS>·Γη→neutral) + Γ44·Γ<K0|KS> + Γ167·Γφ→ KS KL          
     
Γ34 = Γ35 + Γ37          
     
Γ38 = Γ42 + Γ37          
     
Γ39 = Γ40 + Γ42          
     
Γ43 = Γ40 + Γ44          
     
Γ46 = Γ48 + Γ47 + Γ804          
     
Γ49 = Γ50 + Γ51 + Γ806          
     
Γ54 = Γ35·(Γ<K0|KS>·ΓKS→π+π) + Γ37·(Γ<K0|KS>·ΓKS→π+π)           
   + Γ40·(Γ<K0|KS>·ΓKS→π+π) + Γ42·(Γ<K0|KS>·ΓKS→π+π)           
   + Γ47·(2·ΓKS→π+π·ΓKS→π0π0) + Γ48·ΓKS→π+π           
   + Γ50·(2·ΓKS→π+π·ΓKS→π0π0) + Γ51·ΓKS→π+π           
   + Γ53·(Γ<K0|KS>·ΓKS→π0π0<K0|KL>) + Γ62 + Γ70           
   + Γ77 + Γ78 + Γ93 + Γ94 + Γ126·Γη→charged           
   + Γ128·Γη→charged + Γ130·Γη→charged + Γ132·(Γ<K0|KL>·Γη→π+ππ0           
   + Γ<K0|KS>·ΓKS→π0π0·Γη→π+ππ0 + Γ<K0|KS>·ΓKS→π+π·Γη→3π0)           
   + Γ151·(Γω→π+ππ0ω→π+π) + Γ152·(Γω→π+ππ0ω→π+π)           
   + Γ167·(Γφ→ K+K + Γφ→ KS KL·ΓKS→π+π) + Γ802 + Γ803           
   + Γ800·(Γω→π+ππ0ω→π+π)          
     
Γ55 = Γ128·Γη→charged + Γ152·(Γω→π+ππ0ω→π+π) + Γ78           
   + Γ77 + Γ94 + Γ62 + Γ70 + Γ93 + Γ126·Γη→charged           
   + Γ802 + Γ803 + Γ800·(Γω→π+ππ0ω→π+π) + Γ151·(Γω→π+ππ0          
   +Γω→π+π) + Γ130·Γη→charged + Γ168          
     
Γ56 = Γ35·(Γ<K0|KS>·ΓKS→π+π) + Γ62 + Γ93 + Γ37·(Γ<K0|KS>·ΓKS→π+π)           
   + Γ802 + Γ800·Γω→π+π + Γ151·Γω→π+π + Γ168          
     
Γ57 = Γ62 + Γ93 + Γ802 + Γ800·Γω→π+π + Γ151·Γω→π+π           
   + Γ167·Γφ→ K+K          
     
Γ58 = Γ62 + Γ93 + Γ802 + Γ167·Γφ→ K+K          
     
Γ59 = Γ35·(Γ<K0|KS>·ΓKS→π+π) + Γ62 + Γ800·Γω→π+π          
     
Γ60 = Γ62 + Γ800·Γω→π+π          
     
Γ63 = Γ40·(Γ<K0|KS>·ΓKS→π+π) + Γ42·(Γ<K0|KS>·ΓKS→π+π)           
   + Γ47·(2·ΓKS→π+π·ΓKS→π0π0) + Γ50·(2·ΓKS→π+π·ΓKS→π0π0)           
   + Γ70 + Γ77 + Γ78 + Γ94 + Γ126·Γη→charged           
   + Γ128·Γη→charged + Γ130·Γη→charged + Γ132·(Γ<K0|KS>·ΓKS→π+π·Γη→neutral           
   + Γ<K0|KS>·ΓKS→π0π0·Γη→charged) + Γ151·Γω→π+ππ0 + Γ152·(Γω→π+ππ0          
   +Γω→π+π) + Γ800·Γω→π+ππ0 + Γ803          
     
Γ64 = Γ78 + Γ77 + Γ94 + Γ70 + Γ126·Γη→π+ππ0           
   + Γ128·Γη→π+ππ0 + Γ130·Γη→π+ππ0 + Γ800·Γω→π+ππ0           
   + Γ151·Γω→π+ππ0 + Γ152·(Γω→π+ππ0ω→π+π) + Γ803          
     
Γ65 = Γ40·(Γ<K0|KS>·ΓKS→π+π) + Γ42·(Γ<K0|KS>·ΓKS→π+π)           
   + Γ70 + Γ94 + Γ128·Γη→π+ππ0 + Γ151·Γω→π+ππ0           
   + Γ152·Γω→π+π + Γ800·Γω→π+ππ0 + Γ803          
     
Γ66 = Γ70 + Γ94 + Γ128·Γη→π+ππ0 + Γ151·Γω→π+ππ0           
   + Γ152·Γω→π+π + Γ800·Γω→π+ππ0 + Γ803          
     
Γ67 = Γ70 + Γ94 + Γ128·Γη→π+ππ0 + Γ803          
     
Γ68 = Γ40·(Γ<K0|KS>·ΓKS→π+π) + Γ70 + Γ152·Γω→π+π           
   + Γ800·Γω→π+ππ0          
     
Γ69 = Γ152·Γω→π+π + Γ70 + Γ800·Γω→π+ππ0          
     
Γ74 = Γ152·Γω→π+ππ0 + Γ78 + Γ77 + Γ126·Γη→π+ππ0           
   + Γ130·Γη→π+ππ0          
     
Γ75 = Γ152·Γω→π+ππ0 + Γ47·(2·ΓKS→π+π·ΓKS→π0π0)           
   + Γ77 + Γ126·Γη→π+ππ0 + Γ130·Γη→π+ππ0          
     
Γ76 = Γ152·Γω→π+ππ0 + Γ77 + Γ126·Γη→π+ππ0 + Γ130·Γη→π+ππ0          
     
Γ78 = Γ810 + Γ50·(2·ΓKS→π+π·ΓKS→π0π0) + Γ132·(Γ<K0|KS>·ΓKS→π+π·Γη→3π0)          
     
Γ79 = Γ37·(Γ<K0|KS>·ΓKS→π+π) + Γ42·(Γ<K0|KS>·ΓKS→π+π)           
   + Γ93 + Γ94 + Γ128·Γη→charged + Γ151·(Γω→π+ππ0          
   +Γω→π+π) + Γ168 + Γ802 + Γ803          
     
Γ80 = Γ93 + Γ802 + Γ151·Γω→π+π          
     
Γ81 = Γ128·Γη→π+ππ0 + Γ94 + Γ803 + Γ151·Γω→π+ππ0          
     
Γ82 = Γ128·Γη→charged + Γ42·(Γ<K0|KS>·ΓKS→π+π) + Γ802           
   + Γ803 + Γ151·(Γω→π+ππ0ω→π+π) + Γ37·(Γ<K0|KS>·ΓKS→π+π)          
     
Γ83 = Γ128·Γη→π+ππ0 + Γ802 + Γ803 + Γ151·(Γω→π+ππ0          
   +Γω→π+π)          
     
Γ84 = Γ802 + Γ151·Γω→π+π + Γ37·(Γ<K0|KS>·ΓKS→π+π)          
     
Γ85 = Γ802 + Γ151·Γω→π+π          
     
Γ87 = Γ42·(Γ<K0|KS>·ΓKS→π+π) + Γ128·Γη→π+ππ0 + Γ151·Γω→π+ππ0           
   + Γ803          
     
Γ88 = Γ128·Γη→π+ππ0 + Γ803 + Γ151·Γω→π+ππ0          
     
Γ89 = Γ803 + Γ151·Γω→π+ππ0          
     
Γ92 = Γ94 + Γ93          
     
Γ96 = Γ167·Γφ→ K+K          
     
Γ102 = Γ103 + Γ104          
     
Γ103 = Γ820 + Γ822 + Γ831·Γω→π+π          
     
Γ104 = Γ830 + Γ833          
     
Γ106 = Γ30 + Γ44·Γ<K0|KS> + Γ47 + Γ53·Γ<K0|KS>           
   + Γ77 + Γ103 + Γ126·(Γη→3π0η→π+ππ0) + Γ152·Γω→π+ππ0          
     
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·Γφ→ K+K          
     
Γ169 = Γ167·Γφ→ KS KL          
     
Γ804 = Γ47 · ((Γ<K0|KL>·Γ<K0|KL>) / (Γ<K0|KS>·Γ<K0|KS>))          
     
Γ806 = Γ50 · ((Γ<K0|KL>·Γ<K0|KL>) / (Γ<K0|KS>·Γ<K0|KS>))          
     
Γ810 = Γ910 + Γ911 + Γ811·Γω→π+ππ0 + Γ812          
     
Γ820 = Γ920 + Γ821          
     
Γ830 = Γ930 + Γ831·Γω→π+ππ0 + Γ832          
     
Γ910 = Γ136·Γη→3π0          
     
Γ911 = Γ945·Γη→π+ππ0          
     
Γ930 = Γ136·Γη→π+ππ0          
     
Γ944 = Γ136·Γη→γγ          
     
ΓAll = Γ3 + Γ5 + Γ9 + Γ10 + Γ14 + Γ16           
   + Γ20 + Γ23 + Γ27 + Γ28 + Γ30 + Γ35           
   + Γ37 + Γ40 + Γ42 + Γ47·(1 + ((Γ<K0|KL>·Γ<K0|KL>) / (Γ<K0|KS>·Γ<K0|KS>)))           
   + Γ48 + Γ62 + Γ70 + Γ77 + Γ811 + Γ812           
   + Γ93 + Γ94 + Γ832 + Γ833 + Γ126 + Γ128           
   + Γ802 + Γ803 + Γ800 + Γ151 + Γ130 + Γ132           
   + Γ44 + Γ53 + Γ50·(1 + ((Γ<K0|KL>·Γ<K0|KL>) / (Γ<K0|KS>·Γ<K0|KS>)))           
   + Γ51 + Γ167·(Γφ→ K+Kφ→ KS KL) + Γ152 + Γ920           
   + Γ821 + Γ822 + Γ831 + Γ136 + Γ945 + Γ805          

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HFAG-Tau Summer 2016 Report

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