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. A global constrained fit of the available experimental measurements is used to obtain the averages of the τ branching fractions to a complete set of the observed final states, together with their uncertainties and statistical correlations: these data summarize the experimental information for further elaborations. The fit procedure is functionally equivalent to the one employed in the former HFAG reports [1, 3] and consists in a minimum χ2 fit subject to linear and non-linear constraints.
The measurements listed in Table 1 have been used in a minimum χ2 fit subject to the constraints that are listed either in the same table (where some fitted quantities and experimental measurements are expressed as ratios of fit quantities) or in Section 2.4. The fitted quantities and the measurements are labelled using the PDG Γn notation, where n is an integer number, which matches the PDG notation for n<800. We use n≥ 800 to denote some additional branching fractions, as documented in the former HFAG report [3]. The PDG Γn notation does not maintain the same numbers across editions. We continue using the PDG 2010 [4] numbers with the aim to eventually switch to a different stable notation, probably based on PDG identifiers [5].
The fit output consists in 103 quantities, which correspond to either branching fractions or ratios of linear combinations of branching fractions. Although the fit treats all quantities in the same way, for the purpose of describing the results we divide the above quantities in a set of 47 “base nodes” that permit the definition of all the remaining ones as either a sum of “base nodes” (56 quantities, see Section 2.4) or as ratios of linear combinations of “base nodes” (the remaining quantities, see Table 1).
Furthermore we define (see Section 2.4) Γ110 = B (τ− → Xs− ντ), the total branching fraction of the τ decays to final states with the strangeness quantum number equal to one, and ΓAll, the branching fraction of the τ into any measured final state, which is supposed to be equal to 1 within the experimental uncertainty. We define the unitarity residual as Γ998 = 1 −ΓAll.
The fitted HFAG-Tau averages are reported in Table 1. The fit has χ2/d.o.f. = 142.5/127, corresponding to a confidence level CL = 16.45%. We use a total of 174 measurements to fit the above mentioned 103 quantities. Although the unitarity constraint is not applied, the fit is statistically consistent with unitarity, and the unitarity residual is Γ998 = 1 − ΓAll = (9.902 ± 9.850) · 10−4.
A scale factor of 5.44 (as in the two previous reports [1, 3]) has been applied to the published uncertainties of the two severely inconsistent measurements of Γ96 = τ → KKKν by BaBar and Belle, following the same procedure as the PDG.
For several old results, for historical reasons, the table reports as statistical errors the sum in quadrature of the statistical and systematic errors and as systematic errors zero: this does not affect the fit results since the systematic errors are treated exactly like the statistical ones.
The following additions and changes have been done with respect to the previous HFAG report [3].
Published results from two BaBar papers and one Belle paper have been added. The following results have been used from the BaBar high multiplicity decay τ branching fractions paper [6]:
Γ811 = π− 2π0 ω ντ (ex. K0) ( 7.3 ± 1.2 ± 1.2 ) · 10 −5
Γ812 = 2π− π+ 3π0 ντ (ex. K0, η, ω, f1) ( 0.1 ± 0.08 ± 0.30 ) · 10 −4
Γ821 = 3π− 2π+ ντ (ex. K0, ω, f1) ( 7.68 ± 0.04 ± 0.40 ) · 10 −4
Γ822 = K− 2π− 2π+ ντ (ex. K0) ( 0.6 ± 0.5 ± 1.1 ) · 10 −6
Γ831 = 2π− π+ ω ντ (ex. K0) ( 8.4 ± 0.4 ± 0.6 ) · 10 −5
Γ832 = 3π− 2π+ π0 ντ (ex. K0, η, ω, f1) ( 0.36 ± 0.03 ± 0.09 ) · 10 −4
Γ833 = K− 2π− 2π+ π0 ντ (ex. K0) ( 1.1 ± 0.4 ± 0.4 ) · 10 −6
Γ910 = 2π− π+ η ντ (η → 3π0) (ex. K0) ( 8.27 ± 0.88 ± 0.81 ) · 10 −5
Γ911 = π− 2π0 η ντ (η → π+ π− π0) (ex. K0) ( 4.57 ± 0.77 ± 0.50 ) · 10 −5
Γ920 = π− f1 ντ (f1 → 2π− 2π+) ( 5.20 ± 0.31 ± 0.37 ) · 10 −5
Γ930 = 2π− π+ η ντ (η → π+π−π0) (ex. K0) ( 5.39 ± 0.27 ± 0.41 ) · 10 −5
Γ944 = 2π− π+ η ντ (η → γγ) (ex. K0) ( 8.26 ± 0.35 ± 0.51 ) · 10 −5 .
These results supersede the previous BaBar results Γ136 = τ− → π− π− π+ η ντ (ex. K0) [7] and Γ103 = τ− → 3h− 2h+ ντ (ex. K0) [8]. The following results have been used from the BaBar paper on the τ branching fractions with two KS [9]:
Γ47 = π− KS0 KS0 ντ ( 2.31 ± 0.04 ± 0.08 ) · 10 −4
Γ50 = π− π0 KS0 KS0 ντ ( 1.60 ± 0.20 ± 0.22 ) · 10 −5 .
The following results have been used from the Belle paper on the KS final states [10]:
Γ33 = KS0 (particles)− ντ ( 9.15 ± 0.01 ± 0.15 ) · 10 −3
Γ35 = π−
K 0 ντ ( 8.32 ± 0.02 ± 0.16 ) · 10 −3
Γ37 = K− K0 ντ ( 14.8 ± 0.14 ± 0.54 ) · 10 −4
Γ40 = π−
K 0 π0 ντ ( 3.86 ± 0.04 ± 0.14 ) · 10 −3
Γ42 = K− π0 K0 ντ ( 14.96 ± 0.20 ± 0.74 ) · 10 −4
Γ47 = π− KS0 KS0 ντ ( 2.33 ± 0.03 ± 0.09 ) · 10 −4
Γ50 = π− π0 KS0 KS0 ντ ( 2.00 ± 0.22 ± 0.20 ) · 10 −5 .
These results supersede the preliminary ones [11] and the former published result on τ− → π− K0 ντ [12]. In order to profit from the measurements of branching fractions with two KS in the final state, we discard the inclusive ALEPH measurement on τ− → π− K0 K0 π0 ντ [13] and we add the exclusive ALEPH measurement on τ− → π− π0 KS0 KL0 ντ [13].
The CLEO result on τ− → π− π− π+ η ντ (ex. K0) [14] has been discarded since it is very correlated with the branching fractions into six pions measured in the same paper, which are dominated by the η and ω resonances.
We added the τ− → K− η ντ measurements by CLEO [15] and ALEPH [16] to complete the list of useful experimental inputs for this mode.
In order to best integrate the above new results in the global fit, the following constraints have been added:
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> + Γ801·Γφ→ KS KL/(Γφ→ K+K−+Γφ→ KS KL) |
Γ49 = | Γ50 + Γ51 + Γ806 |
Γ78 = | Γ810 + Γ50·2·ΓKS→π+π−·ΓKS→π0π0 + Γ132·(Γ<K0|KS>·ΓKS→π+π−·Γη→3π0) |
Γ103 = | Γ820 + Γ822 + Γ831·Γω→π+π− |
Γ104 = | Γ830 + Γ833 |
Γ806 = | Γ50 · (Γ<K0|KL>·Γ<K0|KL>) / (Γ<K0|KS>·Γ<K0|KS>) |
Γ810 = | Γ910 + Γ911 + Γ811·Γω→π+π−π0 + Γ812 |
Γ820 = | Γ920 + Γ821 |
Γ830 = | Γ930 + Γ831·Γω→π+π−π0 + Γ832 |
Γ910 = | Γ136·Γη→3π0 |
Γ930 = | Γ136·Γη→π+π−π0 |
Γ944 = | Γ136·Γη→γγ . |
It was realised that the Γ44 fit quantity, which is exclusively determined by one single ALEPH result [13], does actually exclude the contribution from K0→π0π0, so we renamed it to Γ44 = π− K0 π0 π0 ντ (ex. K0). The definition of the total branching fraction and of the inclusive branching fraction of the τ lepton into a strange final states have been updated as follows:
Γ110 = | Γ10 + Γ16 + Γ23 + Γ28 + Γ35 + Γ40 + Γ128 + Γ802 + Γ803 + Γ151 + Γ130 + Γ132 + Γ44 + Γ53 + Γ801 + Γ822 + Γ833 |
ΓAll = | Γ3 + Γ5 + Γ9 + Γ10 + Γ14 + Γ16 + Γ20 + Γ23 + Γ27 + Γ28 + Γ30 + Γ35 + Γ37 + Γ40 + Γ42 + Γ47 + Γ48 + Γ804 + Γ62 + Γ70 + Γ77 + Γ78 + Γ93 + Γ94 + Γ104 + Γ126 + Γ128 + Γ802 + Γ803 + Γ800 + Γ151 + Γ130 + Γ132 + Γ44 + Γ53 + Γ50 + Γ51 + Γ806 + Γ805 + Γ801 + Γ152 + Γ103 . |
The total τ branching fraction ΓAll definition includes two modes that have overlapping final states, to a minor extent:
|
The amount of overlap cannot be disentangled with the presently available measurements, however we consider it negligible since the involved branching fractions are small and their overlap is conceivably minor. An inaccurate constraint used in the two previous reports has been removed:
Γ136 = | Γ104· Γη → π+π−π0 + Γ78 · Γη → 3π0 . |
The inaccurate constraint had negligible effect on the global fit in the previous HFAG reports, except for the involved specific branching ratios. In particular, the effects on the lepton universality tests and in the |Vus| determination were negligible.
Finally, the constraint parameters (see Section 2.4) have been updated to the PDG 2013 results [17].
Table 1 reports the τ branching ratio fit results and experimental inputs.
τ lepton branching fraction Fit value / Exp. HFAG Fit / Ref.
Γ3 = µ− νµντ 0.17391 ± 0.00040 HFAG Summer 2014 fit 0.17319 ± 0.00077 ± 0.00000 ALEPH [18] 0.17325 ± 0.00122 ± 0.00000 DELPHI [19] 0.17342 ± 0.00129 ± 0.00000 L3 [20] 0.17340 ± 0.00108 ± 0.00000 OPAL [21]
Γ3 Γ5 =
µ− νµντ e− νe ντ 0.97610 ± 0.00278 HFAG Summer 2014 fit 0.99700 ± 0.05315 ± 0.00000 ARGUS [22] 0.97960 ± 0.00390 ± 0.00053 BaBar [23] 0.97770 ± 0.01074 ± 0.00000 CLEO [24]
Γ5 = e− νe ντ 0.17817 ± 0.00041 HFAG Summer 2014 fit 0.17837 ± 0.00080 ± 0.00000 ALEPH [18] 0.17760 ± 0.00180 ± 0.00000 CLEO [24] 0.17877 ± 0.00155 ± 0.00000 DELPHI [19] 0.17806 ± 0.00129 ± 0.00000 L3 [20] 0.17810 ± 0.00108 ± 0.00000 OPAL [25]
Γ7 = h− ≥0 KL0 ντ 0.12026 ± 0.00054 HFAG Summer 2014 fit 0.12400 ± 0.00990 ± 0.00000 DELPHI [26] 0.12470 ± 0.00502 ± 0.00000 L3 [27] 0.12100 ± 0.00860 ± 0.00000 OPAL [28]
Γ8 = h− ντ 0.11509 ± 0.00054 HFAG Summer 2014 fit 0.11524 ± 0.00105 ± 0.00000 ALEPH [18] 0.11520 ± 0.00130 ± 0.00000 CLEO [24] 0.11571 ± 0.00166 ± 0.00000 DELPHI [29] 0.11980 ± 0.00206 ± 0.00000 OPAL [30]
Γ9 = π− ντ 0.10813 ± 0.00053 HFAG Summer 2014 fit
Γ9 Γ5 =
π− ντ e− νe ντ 0.6069 ± 0.0032 HFAG Summer 2014 fit 0.5945 ± 0.0057 ± 0.0025 BaBar [23]
Γ10 = K− ντ (0.6955 ± 0.0096) · 10−2 HFAG Summer 2014 fit (0.6960 ± 0.0287 ± 0.0000) · 10−2 ALEPH [31] (0.6600 ± 0.1140 ± 0.0000) · 10−2 CLEO [32] (0.8500 ± 0.1800 ± 0.0000) · 10−2 DELPHI [33] (0.6580 ± 0.0396 ± 0.0000) · 10−2 OPAL [34]
Γ10 Γ5 =
K− ντ e− νe ντ (3.903 ± 0.054) · 10−2 HFAG Summer 2014 fit (3.882 ± 0.063 ± 0.017) · 10−2 BaBar [23]
Γ13 = h− π0 ντ 0.25936 ± 0.00090 HFAG Summer 2014 fit 0.25924 ± 0.00129 ± 0.00000 ALEPH [18] 0.25670 ± 0.00010 ± 0.00390 Belle [35] 0.25870 ± 0.00437 ± 0.00000 CLEO [36] 0.25740 ± 0.00244 ± 0.00000 DELPHI [29] 0.25050 ± 0.00610 ± 0.00000 L3 [27] 0.25890 ± 0.00336 ± 0.00000 OPAL [30]
Γ14 = π− π0 ντ 0.25502 ± 0.00092 HFAG Summer 2014 fit
Γ16 = K− π0 ντ (0.4331 ± 0.0149) · 10−2 HFAG Summer 2014 fit (0.4440 ± 0.0354 ± 0.0000) · 10−2 ALEPH [31] (0.4160 ± 0.0030 ± 0.0180) · 10−2 BaBar [37] (0.5100 ± 0.1221 ± 0.0000) · 10−2 CLEO [32] (0.4710 ± 0.0633 ± 0.0000) · 10−2 OPAL [38]
Γ17 = h− ≥2 π0 ντ 0.10804 ± 0.00095 HFAG Summer 2014 fit 0.09910 ± 0.00411 ± 0.00000 OPAL [30]
Γ19 = h− 2π0 ντ (ex. K0) (9.303 ± 0.097) · 10−2 HFAG Summer 2014 fit (9.295 ± 0.122 ± 0.000) · 10−2 ALEPH [18] (9.498 ± 0.422 ± 0.000) · 10−2 DELPHI [29] (8.880 ± 0.560 ± 0.000) · 10−2 L3 [27]
Γ19 Γ13 =
h− 2π0 ντ (ex. K0) h− π0 ντ 0.3587 ± 0.0044 HFAG Summer 2014 fit 0.3420 ± 0.0171 ± 0.0000 CLEO [39]
Γ20 = π− 2π0 ντ (ex. K0) (9.240 ± 0.100) · 10−2 HFAG Summer 2014 fit
Γ23 = K− 2π0 ντ (ex. K0) (0.0630 ± 0.0220) · 10−2 HFAG Summer 2014 fit (0.0560 ± 0.0250 ± 0.0000) · 10−2 ALEPH [31] (0.0900 ± 0.1044 ± 0.0000) · 10−2 CLEO [32]
Γ25 = h− ≥ 3π0 ντ (ex. K0) (1.233 ± 0.065) · 10−2 HFAG Summer 2014 fit (1.403 ± 0.310 ± 0.000) · 10−2 DELPHI [29]
Γ26 = h− 3π0 ντ (1.157 ± 0.072) · 10−2 HFAG Summer 2014 fit (1.082 ± 0.093 ± 0.000) · 10−2 ALEPH [18] (1.700 ± 0.449 ± 0.000) · 10−2 L3 [27]
Γ26 Γ13 =
h− 3π0 ντ h− π0 ντ (4.460 ± 0.277) · 10−2 HFAG Summer 2014 fit (4.400 ± 0.583 ± 0.000) · 10−2 CLEO [39]
Γ27 = π− 3π0 ντ (ex. K0) (1.030 ± 0.075) · 10−2 HFAG Summer 2014 fit
Γ28 = K− 3π0 ντ (ex. K0,η) (4.190 ± 2.160) · 10−4 HFAG Summer 2014 fit (3.700 ± 2.371 ± 0.000) · 10−4 ALEPH [31]
Γ29 = h− 4π0 ντ (ex. K0) (0.1566 ± 0.0391) · 10−2 HFAG Summer 2014 fit (0.1600 ± 0.0707 ± 0.0000) · 10−2 CLEO [39]
Γ30 = h− 4π0 ντ (ex. K0,η) (0.1097 ± 0.0391) · 10−2 HFAG Summer 2014 fit (0.1120 ± 0.0509 ± 0.0000) · 10−2 ALEPH [18]
Γ31 = K− ≥0 π0 ≥0 K0 ≥0 γ ντ (1.548 ± 0.030) · 10−2 HFAG Summer 2014 fit (1.700 ± 0.225 ± 0.000) · 10−2 CLEO [32] (1.540 ± 0.240 ± 0.000) · 10−2 DELPHI [33] (1.528 ± 0.056 ± 0.000) · 10−2 OPAL [34]
Γ33 = KS0 (particles)− ντ (0.9019 ± 0.0081) · 10−2 HFAG Summer 2014 fit (0.9700 ± 0.0849 ± 0.0000) · 10−2 ALEPH [40] (0.9150 ± 0.0010 ± 0.0150) · 10−2 Belle [10] (0.9700 ± 0.1082 ± 0.0000) · 10−2 OPAL [41]
Γ34 = h− K0 ντ (0.9878 ± 0.0119) · 10−2 HFAG Summer 2014 fit (0.8550 ± 0.0814 ± 0.0000) · 10−2 CLEO [42]
Γ35 = π− K0 ντ (0.8378 ± 0.0123) · 10−2 HFAG Summer 2014 fit (0.9280 ± 0.0564 ± 0.0000) · 10−2 ALEPH [31] (0.8400 ± 0.0040 ± 0.0230) · 10−2 BaBar [43] (0.8320 ± 0.0020 ± 0.0160) · 10−2 Belle [10] (0.9500 ± 0.1616 ± 0.0000) · 10−2 L3 [44] (0.9330 ± 0.0838 ± 0.0000) · 10−2 OPAL [45]
Γ37 = K− K0 ντ (0.1500 ± 0.0050) · 10−2 HFAG Summer 2014 fit (0.1580 ± 0.0453 ± 0.0000) · 10−2 ALEPH [40] (0.1620 ± 0.0237 ± 0.0000) · 10−2 ALEPH [31] (0.1480 ± 0.0014 ± 0.0054) · 10−2 Belle [10] (0.1510 ± 0.0304 ± 0.0000) · 10−2 CLEO [42]
Γ38 = K− K0 ≥0 π0 ντ (0.3029 ± 0.0074) · 10−2 HFAG Summer 2014 fit (0.3300 ± 0.0674 ± 0.0000) · 10−2 OPAL [45]
Γ39 = h− K0 π0 ντ (0.5209 ± 0.0114) · 10−2 HFAG Summer 2014 fit (0.5620 ± 0.0693 ± 0.0000) · 10−2 CLEO [42]
Γ40 = π− K0 π0 ντ (0.3680 ± 0.0103) · 10−2 HFAG Summer 2014 fit (0.2940 ± 0.0818 ± 0.0000) · 10−2 ALEPH [40] (0.3470 ± 0.0646 ± 0.0000) · 10−2 ALEPH [31] (0.3420 ± 0.0060 ± 0.0150) · 10−2 BaBar [46] (0.3860 ± 0.0040 ± 0.0140) · 10−2 Belle [10] (0.4100 ± 0.1237 ± 0.0000) · 10−2 L3 [44]
Γ42 = K− π0 K0 ντ (0.1528 ± 0.0070) · 10−2 HFAG Summer 2014 fit (0.1520 ± 0.0789 ± 0.0000) · 10−2 ALEPH [40] (0.1430 ± 0.0291 ± 0.0000) · 10−2 ALEPH [31] (0.1496 ± 0.0020 ± 0.0074) · 10−2 Belle [10] (0.1450 ± 0.0412 ± 0.0000) · 10−2 CLEO [42]
Γ43 = π− K0 ≥1 π0 ντ (0.3805 ± 0.0229) · 10−2 HFAG Summer 2014 fit (0.3240 ± 0.0992 ± 0.0000) · 10−2 OPAL [45]
Γ44 = π− K0 π0 π0 ντ (ex. K0) (1.245 ± 2.043) · 10−4 HFAG Summer 2014 fit (2.600 ± 2.400 ± 0.000) · 10−4 ALEPH [13]
Γ46 = π− K0 K0 ντ (0.1329 ± 0.0110) · 10−2 HFAG Summer 2014 fit (0.1530 ± 0.0340 ± 0.0000) · 10−2 ALEPH [40]
Γ47 = π− KS0 KS0 ντ (2.359 ± 0.061) · 10−4 HFAG Summer 2014 fit (2.600 ± 1.118 ± 0.000) · 10−4 ALEPH [40] (2.310 ± 0.040 ± 0.080) · 10−4 BaBar [9] (2.330 ± 0.030 ± 0.090) · 10−4 Belle [10] (2.300 ± 0.583 ± 0.000) · 10−4 CLEO [42]
Γ48 = π− KS0 KL0 ντ (0.0857 ± 0.0104) · 10−2 HFAG Summer 2014 fit (0.1010 ± 0.0264 ± 0.0000) · 10−2 ALEPH [40]
Γ49 = π− K0 K0 π0 ντ (2.896 ± 1.051) · 10−4 HFAG Summer 2014 fit
Γ50 = π− π0 KS0 KS0 ντ (1.845 ± 0.206) · 10−5 HFAG Summer 2014 fit (1.600 ± 0.200 ± 0.220) · 10−5 BaBar [9] (2.000 ± 0.220 ± 0.200) · 10−5 Belle [10]
Γ51 = π− π0 KS0 KL0 ντ (2.527 ± 1.047) · 10−4 HFAG Summer 2014 fit (3.100 ± 1.100 ± 0.500) · 10−4 ALEPH [40]
Γ53 = K0 h− h− h+ ντ (2.221 ± 2.024) · 10−4 HFAG Summer 2014 fit (2.300 ± 2.025 ± 0.000) · 10−4 ALEPH [40]
Γ54 = h− h− h+ ≥0 neutrals ≥0 KL0 ντ 0.15201 ± 0.00059 HFAG Summer 2014 fit 0.15000 ± 0.00500 ± 0.00000 CELLO [47] 0.14400 ± 0.00671 ± 0.00000 L3 [48] 0.15100 ± 0.01000 ± 0.00000 TPC [49]
Γ55 = h− h− h+ ≥0 neutrals ντ (ex. K0) 0.14573 ± 0.00056 HFAG Summer 2014 fit 0.14556 ± 0.00130 ± 0.00000 L3 [50] 0.14960 ± 0.00238 ± 0.00000 OPAL [51]
Γ57 = h− h− h+ ντ (ex. K0) (9.448 ± 0.053) · 10−2 HFAG Summer 2014 fit (9.510 ± 0.212 ± 0.000) · 10−2 CLEO [52] (9.317 ± 0.122 ± 0.000) · 10−2 DELPHI [29]
Γ57 Γ55 =
h− h− h+ ντ (ex. K0) h− h− h+ ≥0 neutrals ντ (ex. K0) 0.6483 ± 0.0029 HFAG Summer 2014 fit 0.6600 ± 0.0146 ± 0.0000 OPAL [51]
Γ58 = h− h− h+ ντ (ex. K0, ω) (9.418 ± 0.053) · 10−2 HFAG Summer 2014 fit (9.469 ± 0.096 ± 0.000) · 10−2 ALEPH [18]
Γ60 = π− π− π+ ντ (ex. K0) (9.0097 ± 0.0510) · 10−2 HFAG Summer 2014 fit (8.8337 ± 0.0074 ± 0.1267) · 10−2 BaBar [53] (8.4200 ± 0.0033 ± 0.2588) · 10−2 Belle [54] (9.1300 ± 0.4627 ± 0.0000) · 10−2 CLEO3 [55]
Γ62 = π− π− π+ ντ (ex. K0,ω) (8.980 ± 0.051) · 10−2 HFAG Summer 2014 fit
Γ66 = h− h− h+ π0 ντ (ex. K0) (4.603 ± 0.051) · 10−2 HFAG Summer 2014 fit (4.734 ± 0.077 ± 0.000) · 10−2 ALEPH [18] (4.230 ± 0.228 ± 0.000) · 10−2 CLEO [52] (4.545 ± 0.148 ± 0.000) · 10−2 DELPHI [29]
Γ69 = π− π− π+ π0 ντ (ex. K0) (4.516 ± 0.052) · 10−2 HFAG Summer 2014 fit (4.190 ± 0.233 ± 0.000) · 10−2 CLEO [56]
Γ70 = π− π− π+ π0 ντ (ex. K0,ω) (2.767 ± 0.071) · 10−2 HFAG Summer 2014 fit
Γ74 = h− h− h+ ≥ 2π0 ντ (ex. K0) (0.5130 ± 0.0311) · 10−2 HFAG Summer 2014 fit (0.5610 ± 0.1168 ± 0.0000) · 10−2 DELPHI [29]
Γ76 = h− h− h+ 2π0 ντ (ex. K0) (0.4919 ± 0.0310) · 10−2 HFAG Summer 2014 fit (0.4350 ± 0.0461 ± 0.0000) · 10−2 ALEPH [18]
Γ76 Γ54 =
h− h− h+ 2π0 ντ (ex. K0) h− h− h+ ≥0 neutrals ≥0 KL0 ντ (3.236 ± 0.202) · 10−2 HFAG Summer 2014 fit (3.400 ± 0.361 ± 0.000) · 10−2 CLEO [57]
Γ77 = h− h− h+ 2π0 ντ (ex. K0,ω,η) (9.734 ± 3.546) · 10−4 HFAG Summer 2014 fit
Γ78 = h− h− h+ 3π0 ντ (2.109 ± 0.299) · 10−4 HFAG Summer 2014 fit (2.200 ± 0.500 ± 0.000) · 10−4 CLEO [14]
Γ80 Γ60 =
K− π− h+ ντ (ex. K0) π− π− π+ ντ (ex. K0) (4.845 ± 0.081) · 10−2 HFAG Summer 2014 fit (5.440 ± 0.570 ± 0.000) · 10−2 CLEO [58]
Γ81 Γ69 =
K− π− h+ π0 ντ (ex. K0) π− π− π+ π0 ντ (ex. K0) (1.932 ± 0.266) · 10−2 HFAG Summer 2014 fit (2.610 ± 0.615 ± 0.000) · 10−2 CLEO [58]
Γ82 = K− π− π+ ≥0 neutrals ντ (0.4796 ± 0.0138) · 10−2 HFAG Summer 2014 fit (0.5800 ± 0.1845 ± 0.0000) · 10−2 TPC [59]
Γ85 = K− π− π+ ντ (ex. K0) (0.2929 ± 0.0068) · 10−2 HFAG Summer 2014 fit (0.2140 ± 0.0470 ± 0.0000) · 10−2 ALEPH [60] (0.2726 ± 0.0018 ± 0.0092) · 10−2 BaBar [53] (0.3300 ± 0.0013 ± 0.0166) · 10−2 Belle [54] (0.3840 ± 0.0405 ± 0.0000) · 10−2 CLEO3 [55] (0.4150 ± 0.0664 ± 0.0000) · 10−2 OPAL [38]
Γ88 = K− π− π+ π0 ντ (ex. K0) (8.113 ± 1.168) · 10−4 HFAG Summer 2014 fit (6.100 ± 4.295 ± 0.000) · 10−4 ALEPH [60] (7.400 ± 1.360 ± 0.000) · 10−4 CLEO3 [61]
Γ92 = π− K− K+ ≥0 neutrals ντ (0.1497 ± 0.0033) · 10−2 HFAG Summer 2014 fit (0.1590 ± 0.0566 ± 0.0000) · 10−2 OPAL [62] (0.1500 ± 0.0855 ± 0.0000) · 10−2 TPC [59]
Γ93 = π− K− K+ ντ (0.14363 ± 0.00274) · 10−2 HFAG Summer 2014 fit (0.16300 ± 0.02702 ± 0.00000) · 10−2 ALEPH [60] (0.13461 ± 0.00100 ± 0.00364) · 10−2 BaBar [53] (0.15500 ± 0.00066 ± 0.00556) · 10−2 Belle [54] (0.15500 ± 0.01082 ± 0.00000) · 10−2 CLEO3 [55]
Γ93 Γ60 =
π− K− K+ ντ π− π− π+ ντ (ex. K0) (1.594 ± 0.030) · 10−2 HFAG Summer 2014 fit (1.600 ± 0.335 ± 0.000) · 10−2 CLEO [58]
Γ94 = π− K− K+ π0 ντ (0.611 ± 0.183) · 10−4 HFAG Summer 2014 fit (7.500 ± 3.265 ± 0.000) · 10−4 ALEPH [60] (0.550 ± 0.184 ± 0.000) · 10−4 CLEO3 [61]
Γ94 Γ69 =
π− K− K+ π0 ντ π− π− π+ π0 ντ (ex. K0) (0.1353 ± 0.0406) · 10−2 HFAG Summer 2014 fit (0.7900 ± 0.4682 ± 0.0000) · 10−2 CLEO [58]
Γ96 = K− K− K+ ντ (2.156 ± 0.800) · 10−5 HFAG Summer 2014 fit (1.578 ± 0.130 ± 0.123) · 10−5 BaBar [53] (3.290 ± 0.169 ± 0.196) · 10−5 Belle [54]
Γ102 = 3h− 2h+ ≥0 neutrals ντ (ex. K0) (0.0986 ± 0.0037) · 10−2 HFAG Summer 2014 fit (0.0970 ± 0.0121 ± 0.0000) · 10−2 CLEO [63] (0.1020 ± 0.0290 ± 0.0000) · 10−2 HRS [64] (0.1700 ± 0.0341 ± 0.0000) · 10−2 L3 [50]
Γ103 = 3h− 2h+ ντ (ex. K0) (8.224 ± 0.315) · 10−4 HFAG Summer 2014 fit (7.200 ± 1.500 ± 0.000) · 10−4 ALEPH [18] (6.400 ± 2.508 ± 0.000) · 10−4 ARGUS [65] (7.700 ± 1.030 ± 0.000) · 10−4 CLEO [63] (9.700 ± 1.581 ± 0.000) · 10−4 DELPHI [29] (5.100 ± 2.000 ± 0.000) · 10−4 HRS [64] (9.100 ± 1.523 ± 0.000) · 10−4 OPAL [66]
Γ104 = 3h− 2h+ π0 ντ (ex. K0) (1.637 ± 0.113) · 10−4 HFAG Summer 2014 fit (2.100 ± 0.922 ± 0.000) · 10−4 ALEPH [18] (1.700 ± 0.283 ± 0.000) · 10−4 CLEO [14] (1.600 ± 1.342 ± 0.000) · 10−4 DELPHI [29] (2.700 ± 2.012 ± 0.000) · 10−4 OPAL [66]
Γ110 = Xs− ντ (2.882 ± 0.047) · 10−2 HFAG Summer 2014 fit
Γ126 = π− π0 η ντ (0.1387 ± 0.0072) · 10−2 HFAG Summer 2014 fit (0.1800 ± 0.0447 ± 0.0000) · 10−2 ALEPH [16] (0.1350 ± 0.0030 ± 0.0070) · 10−2 Belle [67] (0.1700 ± 0.0283 ± 0.0000) · 10−2 CLEO [68]
Γ128 = K− η ντ (1.548 ± 0.080) · 10−4 HFAG Summer 2014 fit (2.900 −1.200+1.300 ± 0.700) · 10−4 ALEPH [16] (1.420 ± 0.110 ± 0.070) · 10−4 BaBar [69] (1.580 ± 0.050 ± 0.090) · 10−4 Belle [67] (2.600 ± 0.500 ± 0.500) · 10−4 CLEO [15]
Γ130 = K− π0 η ντ (0.483 ± 0.116) · 10−4 HFAG Summer 2014 fit (0.460 ± 0.110 ± 0.040) · 10−4 Belle [67] (1.770 ± 0.904 ± 0.000) · 10−4 CLEO [70]
Γ132 = π− K0 η ντ (0.934 ± 0.149) · 10−4 HFAG Summer 2014 fit (0.880 ± 0.140 ± 0.060) · 10−4 Belle [67] (2.200 ± 0.734 ± 0.000) · 10−4 CLEO [70]
Γ136 = π− π− π+ η ντ (ex. K0) (2.186 ± 0.129) · 10−4 HFAG Summer 2014 fit
Γ150 = h− ω ντ (1.995 ± 0.064) · 10−2 HFAG Summer 2014 fit (1.910 ± 0.092 ± 0.000) · 10−2 ALEPH [16] (1.600 ± 0.491 ± 0.000) · 10−2 CLEO [71]
Γ150 Γ66 =
h− ω ντ h− h− h+ π0 ντ (ex. K0) 0.4333 ± 0.0139 HFAG Summer 2014 fit 0.4310 ± 0.0330 ± 0.0000 ALEPH [72] 0.4640 ± 0.0233 ± 0.0000 CLEO [52]
Γ151 = K− ω ντ (4.100 ± 0.922) · 10−4 HFAG Summer 2014 fit (4.100 ± 0.922 ± 0.000) · 10−4 CLEO3 [61]
Γ152 = h− π0 ω ντ (0.4054 ± 0.0418) · 10−2 HFAG Summer 2014 fit (0.4300 ± 0.0781 ± 0.0000) · 10−2 ALEPH [16]
Γ152 Γ76 =
h− ω π0 ντ h− h− h+ 2π0 ντ (ex. K0) 0.8243 ± 0.0757 HFAG Summer 2014 fit 0.8100 ± 0.0848 ± 0.0000 CLEO [57]
Γ800 = π− ω ντ (1.954 ± 0.065) · 10−2 HFAG Summer 2014 fit
Γ801 = K− φ ντ(φ → KK) (3.664 ± 1.360) · 10−5 HFAG Summer 2014 fit
Γ802 = K− π− π+ ντ (ex. K0,ω) (0.2922 ± 0.0068) · 10−2 HFAG Summer 2014 fit
Γ803 = K− π− π+ π0 ντ (ex. K0,ω,η) (4.101 ± 1.429) · 10−4 HFAG Summer 2014 fit
Γ804 = π− KL0 KL0 ντ (2.359 ± 0.061) · 10−4 HFAG Summer 2014 fit
Γ805 = a1− (→ π− γ) ντ (4.000 ± 2.000) · 10−4 HFAG Summer 2014 fit (4.000 ± 2.000 ± 0.000) · 10−4 ALEPH [18]
Γ806 = π− π0 KL0 KL0 ντ (1.845 ± 0.206) · 10−5 HFAG Summer 2014 fit
Γ810 = 2π− π+ 3π0 ντ (ex. K0) (1.925 ± 0.298) · 10−4 HFAG Summer 2014 fit
Γ811 = π− 2π0 ω ντ (ex. K0) (7.110 ± 1.586) · 10−5 HFAG Summer 2014 fit (7.300 ± 1.200 ± 1.200) · 10−5 BaBar [6]
Γ812 = 2π− π+ 3π0 ντ (ex. K0, η, ω, f1) (1.336 ± 2.682) · 10−5 HFAG Summer 2014 fit (1.000 ± 0.800 ± 3.000) · 10−5 BaBar [6]
Γ820 = 3π− 2π+ ντ (ex. K0, ω) (8.205 ± 0.315) · 10−4 HFAG Summer 2014 fit
Γ821 = 3π− 2π+ ντ (ex. K0, ω, f1) (7.685 ± 0.296) · 10−4 HFAG Summer 2014 fit (7.680 ± 0.040 ± 0.400) · 10−4 BaBar [6]
Γ822 = K− 2π− 2π+ ντ (ex. K0) (0.596 ± 1.208) · 10−6 HFAG Summer 2014 fit (0.600 ± 0.500 ± 1.100) · 10−6 BaBar [6]
Γ830 = 3π− 2π+ π0 ντ (ex. K0) (1.626 ± 0.113) · 10−4 HFAG Summer 2014 fit
Γ831 = 2π− π+ ω ντ (ex. K0) (8.370 ± 0.624) · 10−5 HFAG Summer 2014 fit (8.400 ± 0.400 ± 0.600) · 10−5 BaBar [6]
Γ832 = 3π− 2π+ π0 ντ (ex. K0, η, ω, f1) (3.783 ± 0.873) · 10−5 HFAG Summer 2014 fit (3.600 ± 0.300 ± 0.900) · 10−5 BaBar [6]
Γ833 = K− 2π− 2π+ π0 ντ (ex. K0) (1.108 ± 0.566) · 10−6 HFAG Summer 2014 fit (1.100 ± 0.400 ± 0.400) · 10−6 BaBar [6]
Γ910 = 2π− π+ η ντ (η → 3π0) (ex. K0) (7.144 ± 0.423) · 10−5 HFAG Summer 2014 fit (8.270 ± 0.880 ± 0.810) · 10−5 BaBar [6]
Γ911 = π− 2π0 η ντ (η → π+ π− π0) (ex. K0) (4.424 ± 0.867) · 10−5 HFAG Summer 2014 fit (4.570 ± 0.770 ± 0.500) · 10−5 BaBar [6]
Γ920 = π− f1 ντ (f1 → 2π− 2π+) (5.202 ± 0.444) · 10−5 HFAG Summer 2014 fit (5.200 ± 0.310 ± 0.370) · 10−5 BaBar [6]
Γ930 = 2π− π+ η ντ (η → π+π−π0) (ex. K0) (5.010 ± 0.297) · 10−5 HFAG Summer 2014 fit (5.390 ± 0.270 ± 0.410) · 10−5 BaBar [6]
Γ944 = 2π− π+ η ντ (η → γγ) (ex. K0) (8.615 ± 0.510) · 10−5 HFAG Summer 2014 fit (8.260 ± 0.350 ± 0.510) · 10−5 BaBar [6]
Γ998 = 1 − ΓAll (9.902 ± 9.850) · 10−4 HFAG Summer 2014 fit
The following tables report the correlation coefficients between base nodes, in percent.
Γ5 23 Γ9 7 5 Γ10 3 6 1 Γ14 -13 -14 -13 -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 1 0 -0 2 -1 1 -0 1 -0 Γ37 0 0 -0 -0 0 -2 1 -2 1 -2 0 -29 Γ40 -0 -1 1 0 -0 2 -0 1 -2 1 -0 -10 -1 Γ3 Γ5 Γ9 Γ10 Γ14 Γ16 Γ20 Γ23 Γ27 Γ28 Γ30 Γ35 Γ37 Γ40
Γ42 -0 0 -0 -0 1 -3 1 -5 0 -5 0 -3 -27 -17 Γ44 -0 -0 0 -0 -0 -0 0 -1 0 -1 0 13 16 1 Γ47 0 0 -0 0 -0 1 -0 1 -0 1 -0 -9 -9 -6 Γ48 -0 -0 1 -0 0 -1 1 -3 0 -3 -0 40 51 12 Γ50 0 0 -0 -0 0 -0 0 -0 0 -0 -0 -3 3 -1 Γ51 -0 -0 0 -0 0 -0 0 -1 0 -1 -0 13 17 4 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 3 -1 2 Γ70 -6 -6 -7 -1 -9 -1 -1 0 -1 0 3 -1 0 -1 Γ77 -1 -0 -3 -1 -2 -0 -0 0 2 0 2 -0 0 -0 Γ93 -1 -1 2 0 -1 2 -1 -0 -1 -0 -1 1 -0 1 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 1 -0 0 Γ3 Γ5 Γ9 Γ10 Γ14 Γ16 Γ20 Γ23 Γ27 Γ28 Γ30 Γ35 Γ37 Γ40
Γ130 0 0 0 0 0 0 -0 -0 0 -0 -0 0 -0 0 Γ132 0 0 0 0 0 0 -0 -0 0 -0 -0 1 1 0 Γ136 0 0 0 0 -0 0 -0 -0 -0 -0 -1 0 -0 0 Γ151 0 0 0 -0 0 0 0 -0 0 -0 0 -0 -0 -0 Γ152 -1 -0 -3 -1 -2 -0 -1 0 2 0 2 0 0 0 Γ800 -2 -2 -2 -0 -3 -0 -0 0 -0 0 1 -0 0 -0 Γ801 -0 -0 0 -0 -0 -0 0 -0 0 -0 -0 1 1 0 Γ802 -1 -1 0 0 -1 -1 -2 0 -2 0 -1 -0 0 -0 Γ803 -0 -0 -0 -0 -0 -0 -0 0 -0 0 0 -0 -0 -0 Γ805 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Γ811 0 0 0 0 0 0 -0 -0 -0 -0 -0 0 -0 0 Γ812 0 1 0 0 0 -0 0 0 -0 0 -0 -0 -0 -0 Γ821 0 0 0 0 0 0 -1 -0 -0 -0 -1 0 -0 0 Γ822 0 0 0 0 0 -0 -0 0 -0 0 -0 -0 0 -0 Γ3 Γ5 Γ9 Γ10 Γ14 Γ16 Γ20 Γ23 Γ27 Γ28 Γ30 Γ35 Γ37 Γ40
Γ831 -0 0 0 0 -0 0 -0 -0 -0 -0 -1 0 -0 0 Γ832 -0 -0 -0 -0 -0 -0 0 0 0 0 -0 0 0 -0 Γ833 -0 -0 -0 -0 -0 -0 0 0 0 0 -0 -0 0 -0 Γ920 0 0 0 0 0 0 -0 -0 -0 -0 -0 0 -0 0 Γ3 Γ5 Γ9 Γ10 Γ14 Γ16 Γ20 Γ23 Γ27 Γ28 Γ30 Γ35 Γ37 Γ40
Γ44 7 Γ47 -4 17 Γ48 20 -96 48 Γ50 3 6 -12 20 Γ51 7 -32 17 -100 7 Γ53 -0 0 -0 -0 -0 -0 Γ62 -1 1 0 1 0 0 -0 Γ70 0 0 -0 -0 -0 -0 -0 -19 Γ77 0 -0 0 0 0 0 0 -1 -7 Γ93 -0 0 0 1 0 0 -0 14 -4 -0 Γ94 0 0 -0 -0 -0 -0 -0 -0 -2 -0 -0 Γ126 -0 0 1 0 0 0 -0 1 -0 -5 0 -0 Γ128 -0 0 0 0 0 0 -0 2 -0 -0 1 -0 4 Γ42 Γ44 Γ47 Γ48 Γ50 Γ51 Γ53 Γ62 Γ70 Γ77 Γ93 Γ94 Γ126 Γ128
Γ130 -0 0 0 0 0 0 -0 0 -0 -1 0 -0 1 1 Γ132 0 -1 1 -4 0 -1 -0 0 -0 -0 0 -0 2 1 Γ136 -0 0 0 0 -0 0 -0 -0 -1 0 0 -0 0 0 Γ151 -0 0 -0 0 -0 -0 -0 0 12 -0 0 0 -0 -0 Γ152 0 -0 0 0 0 0 0 -1 -11 -64 -0 -0 -0 -0 Γ800 0 0 -0 -0 -0 -0 -0 -8 -69 -2 -1 0 -0 -0 Γ801 0 -1 1 -4 0 -1 -0 -1 -0 -0 1 -0 0 0 Γ802 0 -0 -0 -0 -0 -0 -0 17 -6 -0 -0 -0 -0 -0 Γ803 -0 -0 -0 -0 -0 -0 -0 -1 -19 -0 -0 -2 -0 -1 Γ805 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Γ811 -0 -0 0 -0 -0 -0 -0 -0 -0 -0 -0 -0 0 0 Γ812 -0 -0 0 -0 -1 -0 -0 -0 -1 -0 -0 -0 -0 -0 Γ821 -0 0 0 0 0 0 0 0 -1 0 0 -0 0 0 Γ822 0 -0 -0 -0 0 -0 0 -0 -0 0 -0 -0 -0 -0 Γ42 Γ44 Γ47 Γ48 Γ50 Γ51 Γ53 Γ62 Γ70 Γ77 Γ93 Γ94 Γ126 Γ128
Γ831 -0 0 0 0 -0 0 0 -0 -0 0 0 -0 0 0 Γ832 -0 -0 0 -0 -0 -0 0 -0 -0 0 -0 -0 0 0 Γ833 0 -0 -0 -0 0 -0 0 -0 0 0 -0 0 -0 -0 Γ920 -0 0 0 0 0 0 0 0 -0 0 0 -0 0 0 Γ42 Γ44 Γ47 Γ48 Γ50 Γ51 Γ53 Γ62 Γ70 Γ77 Γ93 Γ94 Γ126 Γ128
Γ132 0 Γ136 0 -0 Γ151 -0 -0 -0 Γ152 -0 -0 0 -0 Γ800 -0 -0 -0 -14 -3 Γ801 0 -0 0 -0 -0 -0 Γ802 -0 -0 -0 -2 -0 -1 1 Γ803 -0 -0 -0 -58 -0 9 -0 1 Γ805 0 0 0 0 0 0 0 0 0 Γ811 0 -1 20 -0 -0 -0 -0 -0 -0 0 Γ812 -0 -2 -8 -0 -0 -0 -0 -0 -0 0 -16 Γ821 0 -0 47 -0 0 -0 0 -0 -0 0 8 -4 Γ822 -0 0 -1 -0 0 -0 -0 0 -0 0 -0 0 -1 Γ130 Γ132 Γ136 Γ151 Γ152 Γ800 Γ801 Γ802 Γ803 Γ805 Γ811 Γ812 Γ821 Γ822
Γ831 0 -0 39 -0 0 -0 0 -0 -0 0 14 -4 39 -1 Γ832 0 -0 3 -0 0 -0 -0 -0 -0 0 2 -0 3 -0 Γ833 -0 0 -1 -0 0 0 -0 -0 0 0 -0 0 -1 0 Γ920 0 -0 20 -0 0 -0 0 -0 -0 0 3 -2 35 -1 Γ130 Γ132 Γ136 Γ151 Γ152 Γ800 Γ801 Γ802 Γ803 Γ805 Γ811 Γ812 Γ821 Γ822
Γ832 -2 Γ833 -1 -1 Γ920 17 1 -0 Γ831 Γ832 Γ833 Γ920
We use equality constraints that relate a branching fraction to a sum of branching fractions. As mentioned above, the τ branching fractions are denoted with Γn labels. In the constraint relations we use the values of some non-tau branching fractions, denoted e.g. with the self-describing notation ΓKS → π0π0. We also use 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. In the fit, all non-tau quantities are taken from the PDG 2013 [17] 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 τ branching fractions are illustrated in Table 1. The equations in the following permit the computation of the values and uncertainties for branching fractions that are not listed in Table 1, once they are expressed as function of the quantities that are listed there. The following list does not include the (non-linear) constraints already introduced in Section 2, and illustrated in Table 1, where some measured branching fractions are expressed as ratios of “base” branching fractions.
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