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HFAG-Tau Early 2012 Report

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2  Branching fractions fit

The measurements listed in Table 1 have been used in a minimum χ2 fit subject to the equality 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 2.2. The fitted quantities and the measurements are labelled using the PDG [91] 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 [31].

The fitted branching fractions consist on 40 “base nodes” and 45 derived branching fractions, described either as sum of base nodes (see Section 2.2) or as ratios of branching fractions (see Table 1). Furthermore, we define (see Section 2.2) ΓAll as the sum of all the base modes, which correspond to all non-overlapping tau decay modes, Γ998 = 1 −ΓAll and Γ110 = Xs− ντ, which is the total branching fraction of the tau to modes with the strangeness quantum number equal to one.

The fitted HFAG-Tau averages are reported in Table 1. The fit has χ2/d.o.f. = 143.5/118 , corresponding to a confidence level CL = 5.5% . We use a total of 157 measurements and 47 constraint equations to fit 86 quantities. The fit is statistically consistent with the unitarity constraint, but the unitarity constraint is not applied.

In several cases, when it is statistically equivalent within the HFAG-Tau fitting procedure, for historical reasons the statistical and systematic errors are added in quadrature and are reported in the above table in the location of the statistical error, reporting zero as systematic error. A scale factor of 5.44 (as in the former report [31]) has been applied in the fit to the quoted errors of the two inconsistent measurements of Γ96 = τ → KKKν by BaBar and Belle.

With respect to the end-of-2009 HFAG report [31], following comments by M. Davier [62], we have included 3 new modes:

     
 Γ49     
=  π− π0 K0 
K
0 ντ, 
   
 Γ804     =  π− KL0 KL0 ντ,    
 Γ805     =  a1− (→ π− γ) ντ    

along with the related measurements

     
 Γ46     
= π− K0 
K
0 ντ    
 = (0.1530 ± 0.0340 ± 0.0000) · 10−2  (ALEPH [47]),      
 Γ49     
= π− π0 K0 
K
0 ντ    
 = (3.1000 ± 2.3000 ± 0.0000) ·  10−4  (ALEPH [50]),      

the estimate

     
 Γ805     = a1− (→ π− γ) ντ     = (4.0000 ± 2.0000 ± 0.0000) · 10−4  (ALEPH [96]),      

and the constraint

     
  Γ46= Γ48 + Γ47 + Γ804 .          

Furthermore, the following new measurements were added:

     
 Γ128     = K− η ντ     = (1.4200 ± 0.1100 ± 0.0700) ·  10−4  (BaBar [24]),      
 Γ40     
= 
K
0 π− π0 ντ    
 = (0.3840 ± 0.0040 ± 0.0160) · 10−2  (Belle [95]),      
 Γ42     = K− π0 K0 ντ     = (0.1480 ± 0.0020 ± 0.0080) · 10−2  (Belle [95]).      

Finally, the constraint parameters (see Section 2.2) have been updated to the PDG 2011 results [91].

Table 1: HFAG Winter 2012 branching fractions fit results.
Tau lepton branching fractionValueExp.Ref.
 
Γ3 = µ− νµντ
(17.392 ± 0.040) · 10−2HFAGWinter 2012 fit
 (17.319 ± 0.077 ± 0.000) · 10−2ALEPH[96]
 (17.325 ± 0.122 ± 0.000) · 10−2DELPHI[10]
 (17.342 ± 0.129 ± 0.000) · 10−2L3[13]
 (17.340 ± 0.108 ± 0.000) · 10−2OPAL[5]
Γ3
Γ5
 = 
µ− νµντ
e− νe ντ
0.9761 ± 0.0028HFAGWinter 2012 fit
 0.9970 ± 0.0532 ± 0.0000ARGUS[22]
 0.9796 ± 0.0039 ± 0.0005BaBar [42]
 0.9777 ± 0.0107 ± 0.0000CLEO[25]
Γ5 = e− νe ντ
(17.818 ± 0.041) · 10−2HFAGWinter 2012 fit
 (17.837 ± 0.080 ± 0.000) · 10−2ALEPH[96]
 (17.760 ± 0.180 ± 0.000) · 10−2CLEO[25]
 (17.877 ± 0.155 ± 0.000) · 10−2DELPHI[10]
 (17.806 ± 0.129 ± 0.000) · 10−2L3[13]
 (17.810 ± 0.108 ± 0.000) · 10−2OPAL[1]
Γ7 = h− ≥0 KL0 ντ
(12.020 ± 0.055) · 10−2HFAGWinter 2012 fit
 (12.400 ± 0.990 ± 0.000) · 10−2DELPHI[8]
 (12.470 ± 0.502 ± 0.000) · 10−2L3[11]
 (12.100 ± 0.860 ± 0.000) · 10−2OPAL[23]
Γ8 = h− ντ
(11.507 ± 0.054) · 10−2HFAGWinter 2012 fit
 (11.524 ± 0.105 ± 0.000) · 10−2ALEPH[96]
 (11.520 ± 0.130 ± 0.000) · 10−2CLEO[25]
 (11.571 ± 0.166 ± 0.000) · 10−2DELPHI[7]
 (11.980 ± 0.206 ± 0.000) · 10−2OPAL[15]
Γ9 = π− ντ
(10.811 ± 0.053) · 10−2HFAGWinter 2012 fit
Γ9
Γ5
 = 
π− ντ
e− νe ντ
(60.675 ± 0.321) · 10−2HFAGWinter 2012 fit
 (59.450 ± 0.574 ± 0.248) · 10−2BaBar [42]
Γ10 = K− ντ
(0.6955 ± 0.0096) · 10−2HFAGWinter 2012 fit
 (0.6960 ± 0.0287 ± 0.0000) · 10−2ALEPH[49]
 (0.6600 ± 0.1140 ± 0.0000) · 10−2CLEO[52]
 (0.8500 ± 0.1800 ± 0.0000) · 10−2DELPHI[9]
 (0.6580 ± 0.0396 ± 0.0000) · 10−2OPAL[4]
Γ10
Γ5
 = 
K− ντ
e− νe ντ
(3.9031 ± 0.0543) · 10−2HFAGWinter 2012 fit
 (3.8820 ± 0.0630 ± 0.0174) · 10−2BaBar [42]
Γ13 = h− π0 ντ
(25.936 ± 0.090) · 10−2HFAGWinter 2012 fit
 (25.924 ± 0.129 ± 0.000) · 10−2ALEPH[96]
 (25.670 ± 0.010 ± 0.390) · 10−2Belle[69]
 (25.870 ± 0.437 ± 0.000) · 10−2CLEO[30]
 (25.740 ± 0.244 ± 0.000) · 10−2DELPHI[7]
 (25.050 ± 0.610 ± 0.000) · 10−2L3[11]
 (25.890 ± 0.336 ± 0.000) · 10−2OPAL[15]
Γ14 = π− π0 ντ
(25.504 ± 0.092) · 10−2HFAGWinter 2012 fit
Γ16 = K− π0 ντ
(0.4322 ± 0.0149) · 10−2HFAGWinter 2012 fit
 (0.4440 ± 0.0354 ± 0.0000) · 10−2ALEPH[49]
 (0.4160 ± 0.0030 ± 0.0180) · 10−2BaBar [34]
 (0.5100 ± 0.1221 ± 0.0000) · 10−2CLEO[52]
 (0.4710 ± 0.0633 ± 0.0000) · 10−2OPAL[6]
Γ17 = h− ≥2 π0 ντ
(10.803 ± 0.095) · 10−2HFAGWinter 2012 fit
 ( 9.910 ± 0.411 ± 0.000) · 10−2OPAL[15]
Γ19 = h− 2π0 ντ (ex. K0)
(9.3044 ± 0.0972) · 10−2HFAGWinter 2012 fit
 (9.2950 ± 0.1217 ± 0.0000) · 10−2ALEPH[96]
 (9.4980 ± 0.4219 ± 0.0000) · 10−2DELPHI[7]
 (8.8800 ± 0.5597 ± 0.0000) · 10−2L3[11]
Γ19
Γ13
 = 
h− 2π0 ντ (ex. K0)
h− π0 ντ
(35.874 ± 0.442) · 10−2HFAGWinter 2012 fit
 (34.200 ± 1.709 ± 0.000) · 10−2CLEO[93]
Γ20 = π− 2π0 ντ (ex. K0)
(9.2414 ± 0.0997) · 10−2HFAGWinter 2012 fit
Γ23 = K− 2π0 ντ (ex. K0)
(0.0630 ± 0.0222) · 10−2HFAGWinter 2012 fit
 (0.0560 ± 0.0250 ± 0.0000) · 10−2ALEPH[49]
 (0.0900 ± 0.1044 ± 0.0000) · 10−2CLEO[52]
Γ25 = h− ≥ 3π0 ντ (ex. K0)
(1.2349 ± 0.0650) · 10−2HFAGWinter 2012 fit
 (1.4030 ± 0.3098 ± 0.0000) · 10−2DELPHI[7]
Γ26 = h− 3π0 ντ
(1.1573 ± 0.0717) · 10−2HFAGWinter 2012 fit
 (1.0820 ± 0.0926 ± 0.0000) · 10−2ALEPH[96]
 (1.7000 ± 0.4494 ± 0.0000) · 10−2L3[11]
Γ26
Γ13
 = 
h− 3π0 ντ
h− π0 ντ
(4.4622 ± 0.2767) · 10−2HFAGWinter 2012 fit
 (4.4000 ± 0.5831 ± 0.0000) · 10−2CLEO[93]
Γ27 = π− 3π0 ντ (ex. K0)
(1.0322 ± 0.0749) · 10−2HFAGWinter 2012 fit
Γ28 = K− 3π0 ντ (ex. K0,η)
(4.1870 ± 2.1761) · 10−4HFAGWinter 2012 fit
 (3.7000 ± 2.3710 ± 0.0000) · 10−4ALEPH[49]
Γ29 = h− 4π0 ντ (ex. K0)
(0.1558 ± 0.0391) · 10−2HFAGWinter 2012 fit
 (0.1600 ± 0.0707 ± 0.0000) · 10−2CLEO[93]
Γ30 = h− 4π0 ντ (ex. K0,η)
(0.1091 ± 0.0391) · 10−2HFAGWinter 2012 fit
 (0.1120 ± 0.0509 ± 0.0000) · 10−2ALEPH[96]
Γ31 = K− ≥0 π0 ≥0 K0 ≥0 γ ντ
(1.5481 ± 0.0310) · 10−2HFAGWinter 2012 fit
 (1.7000 ± 0.2247 ± 0.0000) · 10−2CLEO[52]
 (1.5400 ± 0.2400 ± 0.0000) · 10−2DELPHI[9]
 (1.5280 ± 0.0559 ± 0.0000) · 10−2OPAL[4]
Γ33 = KS0 (particles)− ντ
(0.8953 ± 0.0255) · 10−2HFAGWinter 2012 fit
 (0.9700 ± 0.0849 ± 0.0000) · 10−2ALEPH[47]
 (0.9700 ± 0.1082 ± 0.0000) · 10−2OPAL[19]
Γ34 = h− K0 ντ
(0.9797 ± 0.0233) · 10−2HFAGWinter 2012 fit
 (0.8550 ± 0.0814 ± 0.0000) · 10−2CLEO[61]
Γ35 = π− K0 ντ
(0.8206 ± 0.0182) · 10−2HFAGWinter 2012 fit
 (0.9280 ± 0.0564 ± 0.0000) · 10−2ALEPH[49]
 (0.8400 ± 0.0040 ± 0.0230) · 10−2BaBar [40]
 (0.8080 ± 0.0040 ± 0.0260) · 10−2Belle[67]
 (0.9500 ± 0.1616 ± 0.0000) · 10−2L3[12]
 (0.9330 ± 0.0838 ± 0.0000) · 10−2OPAL[3]
Γ37 = K− K0 ντ
(0.1591 ± 0.0157) · 10−2HFAGWinter 2012 fit
 (0.1580 ± 0.0453 ± 0.0000) · 10−2ALEPH[47]
 (0.1620 ± 0.0237 ± 0.0000) · 10−2ALEPH[49]
 (0.1510 ± 0.0304 ± 0.0000) · 10−2CLEO[61]
Γ38 = K− K0 ≥0 π0 ντ
(0.3041 ± 0.0168) · 10−2HFAGWinter 2012 fit
 (0.3300 ± 0.0674 ± 0.0000) · 10−2OPAL[3]
Γ39 = h− K0 π0 ντ
(0.5099 ± 0.0146) · 10−2HFAGWinter 2012 fit
 (0.5620 ± 0.0693 ± 0.0000) · 10−2CLEO[61]
Γ40 = π− K0 π0 ντ
(0.3649 ± 0.0108) · 10−2HFAGWinter 2012 fit
 (0.2940 ± 0.0818 ± 0.0000) · 10−2ALEPH[47]
 (0.3470 ± 0.0646 ± 0.0000) · 10−2ALEPH[49]
 (0.3420 ± 0.0060 ± 0.0150) · 10−2BaBar [92]
 (0.3840 ± 0.0040 ± 0.0160) · 10−2Belle[95]
 (0.4100 ± 0.1237 ± 0.0000) · 10−2L3[12]
Γ42 = K− π0 K0 ντ
(0.1450 ± 0.0071) · 10−2HFAGWinter 2012 fit
 (0.1520 ± 0.0789 ± 0.0000) · 10−2ALEPH[47]
 (0.1430 ± 0.0291 ± 0.0000) · 10−2ALEPH[49]
 (0.1480 ± 0.0020 ± 0.0080) · 10−2Belle[95]
 (0.1450 ± 0.0412 ± 0.0000) · 10−2CLEO[61]
Γ43 = π− K0 ≥1 π0 ντ
(0.3917 ± 0.0250) · 10−2HFAGWinter 2012 fit
 (0.3240 ± 0.0992 ± 0.0000) · 10−2OPAL[3]
Γ44 = π− K0 π0 π0 ντ
(2.6854 ± 2.3037) · 10−4HFAGWinter 2012 fit
 (2.6000 ± 2.4000 ± 0.0000) · 10−4ALEPH[50]
Γ46 = π− K0 K0 ντ
(0.1562 ± 0.0209) · 10−2HFAGWinter 2012 fit
 (0.1530 ± 0.0340 ± 0.0000) · 10−2ALEPH[47]
Γ47 = π− KS0 KS0 ντ
(2.3957 ± 0.5026) · 10−4HFAGWinter 2012 fit
 (2.6000 ± 1.1180 ± 0.0000) · 10−4ALEPH[47]
 (2.3000 ± 0.5831 ± 0.0000) · 10−4CLEO[61]
Γ48 = π− KS0 KL0 ντ
(0.1082 ± 0.0203) · 10−2HFAGWinter 2012 fit
 (0.1010 ± 0.0264 ± 0.0000) · 10−2ALEPH[47]
Γ49 = π− K0 K0 π0 ντ
(3.1000 ± 2.3000) · 10−4HFAGWinter 2012 fit
 (3.1000 ± 2.3000 ± 0.0000) · 10−4ALEPH[50]
Γ53 = K0 h− h− h+ ντ
(2.2224 ± 2.0236) · 10−4HFAGWinter 2012 fit
 (2.3000 ± 2.0248 ± 0.0000) · 10−4ALEPH[47]
Γ54 = h− h− h+ ≥0 neutrals ≥0 KL0 ντ
(15.192 ± 0.060) · 10−2HFAGWinter 2012 fit
 (15.000 ± 0.500 ± 0.000) · 10−2CELLO[54]
 (14.400 ± 0.671 ± 0.000) · 10−2L3[17]
 (15.100 ± 1.000 ± 0.000) · 10−2TPC[18]
Γ55 = h− h− h+ ≥0 neutrals ντ (ex. K0)
(14.574 ± 0.056) · 10−2HFAGWinter 2012 fit
 (14.556 ± 0.130 ± 0.000) · 10−2L3[14]
 (14.960 ± 0.238 ± 0.000) · 10−2OPAL[20]
Γ57 = h− h− h+ ντ (ex. K0)
(9.4404 ± 0.0530) · 10−2HFAGWinter 2012 fit
 (9.5100 ± 0.2119 ± 0.0000) · 10−2CLEO[44]
 (9.3170 ± 0.1218 ± 0.0000) · 10−2DELPHI[7]
Γ57
Γ55
 = 
h− h− h+ ντ (ex. K0)
h− h− h+ ≥0 neutrals ντ (ex. K0)
(64.776 ± 0.294) · 10−2HFAGWinter 2012 fit
 (66.000 ± 1.456 ± 0.000) · 10−2OPAL[20]
Γ58 = h− h− h+ ντ (ex. K0, ω)
(9.4099 ± 0.0531) · 10−2HFAGWinter 2012 fit
 (9.4690 ± 0.0958 ± 0.0000) · 10−2ALEPH[96]
Γ60 = π− π− π+ ντ (ex. K0)
(9.0018 ± 0.0510) · 10−2HFAGWinter 2012 fit
 (8.8337 ± 0.0074 ± 0.1267) · 10−2BaBar [36]
 (8.4200 ± 0.0033 ± 0.2588) · 10−2Belle[81]
 (9.1300 ± 0.4627 ± 0.0000) · 10−2CLEO3[57]
Γ62 = π− π− π+ ντ (ex. K0,ω)
(8.9719 ± 0.0511) · 10−2HFAGWinter 2012 fit
Γ66 = h− h− h+ π0 ντ (ex. K0)
(4.6019 ± 0.0513) · 10−2HFAGWinter 2012 fit
 (4.7340 ± 0.0767 ± 0.0000) · 10−2ALEPH[96]
 (4.2300 ± 0.2280 ± 0.0000) · 10−2CLEO[44]
 (4.5450 ± 0.1478 ± 0.0000) · 10−2DELPHI[7]
Γ69 = π− π− π+ π0 ντ (ex. K0)
(4.5146 ± 0.0524) · 10−2HFAGWinter 2012 fit
 (4.1900 ± 0.2326 ± 0.0000) · 10−2CLEO[66]
Γ70 = π− π− π+ π0 ντ (ex. K0,ω)
(2.7659 ± 0.0710) · 10−2HFAGWinter 2012 fit
Γ74 = h− h− h+ ≥ 2π0 ντ (ex. K0)
(0.5231 ± 0.0311) · 10−2HFAGWinter 2012 fit
 (0.5610 ± 0.1168 ± 0.0000) · 10−2DELPHI[7]
Γ76 = h− h− h+ 2π0 ντ (ex. K0)
(0.4911 ± 0.0310) · 10−2HFAGWinter 2012 fit
 (0.4350 ± 0.0461 ± 0.0000) · 10−2ALEPH[96]
Γ76
Γ54
 = 
h− h− h+ 2π0 ντ (ex. K0)
h− h− h+ ≥0 neutrals ≥0 KL0 ντ
(3.2326 ± 0.2024) · 10−2HFAGWinter 2012 fit
 (3.4000 ± 0.3606 ± 0.0000) · 10−2CLEO[56]
Γ77 = h− h− h+ 2π0 ντ (ex. K0,ω,η)
(9.7301 ± 3.5416) · 10−4HFAGWinter 2012 fit
Γ78 = h− h− h+ 3π0 ντ
(3.1986 ± 0.3124) · 10−4HFAGWinter 2012 fit
 (2.2000 ± 0.5000 ± 0.0000) · 10−4CLEO[26]
Γ80
Γ60
 = 
K− π− h+ ντ (ex. K0)
π− π− π+ ντ (ex. K0)
(4.8482 ± 0.0808) · 10−2HFAGWinter 2012 fit
 (5.4400 ± 0.5701 ± 0.0000) · 10−2CLEO[94]
Γ81
Γ69
 = 
K− π− h+ π0 ντ (ex. K0)
π− π− π+ π0 ντ (ex. K0)
(1.9323 ± 0.2660) · 10−2HFAGWinter 2012 fit
 (2.6100 ± 0.6155 ± 0.0000) · 10−2CLEO[94]
Γ82 = K− π− π+ ≥0 neutrals ντ
(0.4801 ± 0.0147) · 10−2HFAGWinter 2012 fit
 (0.5800 ± 0.1845 ± 0.0000) · 10−2TPC[53]
Γ85 = K− π− π+ ντ (ex. K0)
(0.2929 ± 0.0068) · 10−2HFAGWinter 2012 fit
 (0.2140 ± 0.0470 ± 0.0000) · 10−2ALEPH[48]
 (0.2726 ± 0.0018 ± 0.0092) · 10−2BaBar [36]
 (0.3300 ± 0.0013 ± 0.0166) · 10−2Belle[81]
 (0.3840 ± 0.0405 ± 0.0000) · 10−2CLEO3[57]
 (0.4150 ± 0.0664 ± 0.0000) · 10−2OPAL[6]
Γ88 = K− π− π+ π0 ντ (ex. K0)
(8.1122 ± 1.1680) · 10−4HFAGWinter 2012 fit
 (6.1000 ± 4.2950 ± 0.0000) · 10−4ALEPH[48]
 (7.4000 ± 1.3600 ± 0.0000) · 10−4CLEO3[28]
Γ92 = π− K− K+ ≥0 neutrals ντ
(0.1496 ± 0.0033) · 10−2HFAGWinter 2012 fit
 (0.1590 ± 0.0566 ± 0.0000) · 10−2OPAL[2]
 (0.1500 ± 0.0855 ± 0.0000) · 10−2TPC[53]
Γ93 = π− K− K+ ντ
(0.1435 ± 0.0027) · 10−2HFAGWinter 2012 fit
 (0.1630 ± 0.0270 ± 0.0000) · 10−2ALEPH[48]
 (0.1346 ± 0.0010 ± 0.0036) · 10−2BaBar [36]
 (0.1550 ± 0.0007 ± 0.0056) · 10−2Belle[81]
 (0.1550 ± 0.0108 ± 0.0000) · 10−2CLEO3[57]
Γ93
Γ60
 = 
π− K− K+ ντ
π− π− π+ ντ (ex. K0)
(1.5940 ± 0.0305) · 10−2HFAGWinter 2012 fit
 (1.6000 ± 0.3354 ± 0.0000) · 10−2CLEO[94]
Γ94 = π− K− K+ π0 ντ
(0.6113 ± 0.1829) · 10−4HFAGWinter 2012 fit
 (7.5000 ± 3.2650 ± 0.0000) · 10−4ALEPH[48]
 (0.5500 ± 0.1844 ± 0.0000) · 10−4CLEO3[28]
Γ94
Γ69
 = 
π− K− K+ π0 ντ
π− π− π+ π0 ντ (ex. K0)
(0.1354 ± 0.0406) · 10−2HFAGWinter 2012 fit
 (0.7900 ± 0.4682 ± 0.0000) · 10−2CLEO[94]
Γ96 = K− K− K+ ντ
(2.1774 ± 0.8005) · 10−5HFAGWinter 2012 fit
 (1.5777 ± 0.1300 ± 0.1231) · 10−5BaBar [36]
 (3.2900 ± 0.1694 ± 0.1962) · 10−5Belle[81]
Γ102 = 3h− 2h+ ≥0 neutrals ντ (ex. K0)
(0.1022 ± 0.0037) · 10−2HFAGWinter 2012 fit
 (0.0970 ± 0.0121 ± 0.0000) · 10−2CLEO[72]
 (0.1020 ± 0.0290 ± 0.0000) · 10−2HRS[60]
 (0.1700 ± 0.0341 ± 0.0000) · 10−2L3[14]
Γ103 = 3h− 2h+ ντ (ex. K0)
(8.2349 ± 0.3060) · 10−4HFAGWinter 2012 fit
 (7.2000 ± 1.5000 ± 0.0000) · 10−4ALEPH[96]
 (6.4000 ± 2.5080 ± 0.0000) · 10−4ARGUS[21]
 (8.5600 ± 0.0500 ± 0.4200) · 10−4BaBar [33]
 (7.7000 ± 1.0300 ± 0.0000) · 10−4CLEO[72]
 (9.7000 ± 1.5810 ± 0.0000) · 10−4DELPHI[7]
 (5.1000 ± 2.0000 ± 0.0000) · 10−4HRS[60]
 (9.1000 ± 1.5230 ± 0.0000) · 10−4OPAL[16]
Γ104 = 3h− 2h+ π0 ντ (ex. K0)
(1.9801 ± 0.2437) · 10−4HFAGWinter 2012 fit
 (2.1000 ± 0.9220 ± 0.0000) · 10−4ALEPH[96]
 (1.7000 ± 0.2828 ± 0.0000) · 10−4CLEO[26]
 (1.6000 ± 1.3420 ± 0.0000) · 10−4DELPHI[7]
 (2.7000 ± 2.0120 ± 0.0000) · 10−4OPAL[16]
Γ110 = Xs− ντ
(2.8746 ± 0.0498) · 10−2HFAGWinter 2012 fit
Γ126 = π− π0 η ντ
(0.1386 ± 0.0072) · 10−2HFAGWinter 2012 fit
 (0.1800 ± 0.0447 ± 0.0000) · 10−2ALEPH[59]
 (0.1350 ± 0.0030 ± 0.0070) · 10−2Belle[78]
 (0.1700 ± 0.0283 ± 0.0000) · 10−2CLEO[29]
Γ128 = K− η ντ
(1.5285 ± 0.0808) · 10−4HFAGWinter 2012 fit
 (1.4200 ± 0.1100 ± 0.0700) · 10−4BaBar [24]
 (1.5800 ± 0.0500 ± 0.0900) · 10−4Belle[78]
Γ130 = K− π0 η ντ
(0.4825 ± 0.1161) · 10−4HFAGWinter 2012 fit
 (0.4600 ± 0.1100 ± 0.0400) · 10−4Belle[78]
 (1.7700 ± 0.9043 ± 0.0000) · 10−4CLEO[55]
Γ132 = π− K0 η ντ
(0.9364 ± 0.1491) · 10−4HFAGWinter 2012 fit
 (0.8800 ± 0.1400 ± 0.0600) · 10−4Belle[78]
 (2.2000 ± 0.7338 ± 0.0000) · 10−4CLEO[55]
Γ136 = π− π− π+ η ντ (ex. K0)
(1.4921 ± 0.0968) · 10−4HFAGWinter 2012 fit
 (1.6000 ± 0.0500 ± 0.1100) · 10−4BaBar [37]
 (2.3000 ± 0.5000 ± 0.0000) · 10−4CLEO[26]
Γ150 = h− ω ντ
(1.9945 ± 0.0641) · 10−2HFAGWinter 2012 fit
 (1.9100 ± 0.0922 ± 0.0000) · 10−2ALEPH[59]
 (1.6000 ± 0.4909 ± 0.0000) · 10−2CLEO[51]
Γ150
Γ66
 = 
h− ω ντ
h− h− h+ π0 ντ (ex. K0)
(43.340 ± 1.389) · 10−2HFAGWinter 2012 fit
 (43.100 ± 3.300 ± 0.000) · 10−2ALEPH[58]
 (46.400 ± 2.335 ± 0.000) · 10−2CLEO[44]
Γ151 = K− ω ντ
(4.1000 ± 0.9220) · 10−4HFAGWinter 2012 fit
 (4.1000 ± 0.9220 ± 0.0000) · 10−4CLEO3[28]
Γ152 = h− π0 ω ντ
(0.4049 ± 0.0418) · 10−2HFAGWinter 2012 fit
 (0.4300 ± 0.0781 ± 0.0000) · 10−2ALEPH[59]
Γ152
Γ76
 = 
h− ω π0 ντ
h− h− h+ 2π0 ντ (ex. K0)
(82.453 ± 7.575) · 10−2HFAGWinter 2012 fit
 (81.000 ± 8.485 ± 0.000) · 10−2CLEO[56]
Γ800 = π− ω ντ
(1.9535 ± 0.0647) · 10−2HFAGWinter 2012 fit
Γ801 = K− φ ντ(φ → KK)
(3.7002 ± 1.3604) · 10−5HFAGWinter 2012 fit
Γ802 = K− π− π+ ντ (ex. K0,ω)
(0.2923 ± 0.0068) · 10−2HFAGWinter 2012 fit
Γ803 = K− π− π+ π0 ντ (ex. K0,ω,η)
(4.1074 ± 1.4286) · 10−4HFAGWinter 2012 fit
Γ804 = π− KL0 KL0 ντ
(2.3957 ± 0.5026) · 10−4HFAGWinter 2012 fit
Γ805 = a1− (→ π− γ) ντ
(4.0000 ± 2.0000) · 10−4HFAGWinter 2012 fit
 (4.0000 ± 2.0000 ± 0.0000) · 10−4ALEPH[96]
Γ998 = 1 − ΓAll
(0.0704 ± 0.1060) · 10−2HFAGWinter 2012 fit
 

2.1  Correlation between base nodes uncertainties

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

Table 2: Base nodes correlation coefficients in percent, section 1
Γ5 23             
Γ9 75            
Γ10 361           
Γ14 -13-14-12-3          
Γ16 -0-12-1-16         
Γ20 -5-5-7-1-402        
Γ23 00-0-22-12-22       
Γ27 -4-3-8-103-366      
Γ28 00-0-12-124-19-29     
Γ30 -5-4-11-2-9-060-420    
Γ35 -0-110-02-11-01-0   
Γ37 00-1-11-83-124-120-6  
Γ40 -0-11-0-001-2-2-2-00-3 
  Γ3 Γ5 Γ9 Γ10 Γ14 Γ16 Γ20 Γ23 Γ27 Γ28 Γ30 Γ35 Γ37 Γ40
Table 3: Base nodes correlation coefficients in percent, section 2
Γ42 -0-00-00-31-5-1-50-0-730
Γ44 00-00-00-00000-2-2-4
Γ47 -0-0-0-0-0000000-0-0-0
Γ48 000000-01-00-0-4-3-3
Γ53 00000-000000-0-0-0
Γ62 -3-580-45-7-1-5-1-54-13
Γ70 -6-6-7-1-9-1-10-103-10-1
Γ77 -1-0-3-1-2-0-00202-00-0
Γ78 112011-0-0-0-001-01
Γ93 -1-120-12-1-0-1-0-12-01
Γ94 -0-0-0-0-0-0-00-000-00-0
Γ103 002001-1-0-0-0-11-01
Γ104 -1-1-1-0-100-00-0-10-00
Γ126 000000-1-00-0-20-00
  Γ3 Γ5 Γ9 Γ10 Γ14 Γ16 Γ20 Γ23 Γ27 Γ28 Γ30 Γ35 Γ37 Γ40
Table 4: Base nodes correlation coefficients in percent, section 3
Γ128 -0-01-0-01-0-1-0-1-01-01
Γ130 000000-0-00-0-00-00
Γ132 00-00-00-0-00-0-00-00
Γ151 -0-0-0-0-00-0-0-0-00-0-0-0
Γ152 -1-0-3-1-2-0-10202-000
Γ800 -2-2-2-0-3-0-00-001-00-0
Γ801 -0-00-0-00-0-00-0-0-0-0-0
Γ802 -1-100-1-1-20-20-1-1-0-0
Γ803 -0-0-0-0-0-0-00-000-0-0-0
Γ805 00000000000000
  Γ3 Γ5 Γ9 Γ10 Γ14 Γ16 Γ20 Γ23 Γ27 Γ28 Γ30 Γ35 Γ37 Γ40
Table 5: Base nodes correlation coefficients in percent, section 4
Γ44 -2             
Γ47 -0-0            
Γ48 -2-5-19           
Γ53 -000-0          
Γ62 1-0-0-0-0         
Γ70 -000-0-0-19        
Γ77 0-0-000-1-7       
Γ78 0-0-0-0-02-2-1      
Γ93 0-0-0-0-014-4-01     
Γ94 000-0-0-0-2-0-0-0    
Γ103 0-0-0-0-03-1-041-0   
Γ104 -0-0000-001-3600-11  
Γ126 0-0-0-0-01-0-500-000 
  Γ42 Γ44 Γ47 Γ48 Γ53 Γ62 Γ70 Γ77 Γ78 Γ93 Γ94 Γ103 Γ104 Γ126
Table 6: Base nodes correlation coefficients in percent, section 5
Γ128 0-0-0-0-02-0-001-0104
Γ130 0-0-0-0-00-0-100-0001
Γ132 -0-0-0-0-00-0-000-00-02
Γ151 000-0-0012000-0000
Γ152 0-0-000-1-11-64-1-0-0-01-0
Γ800 -000-0-0-8-69-2-0-10-00-0
Γ801 -0-0-0-0-0-1-0-001-0000
Γ802 -000-0-017-6-0-0-0-0-0-0-0
Γ803 -0000-0-1-19-0-0-0-2-00-0
Γ805 00000000000000
  Γ42 Γ44 Γ47 Γ48 Γ53 Γ62 Γ70 Γ77 Γ78 Γ93 Γ94 Γ103 Γ104 Γ126
Table 7: Base nodes correlation coefficients in percent, section 6
Γ130 1         
Γ132 10        
Γ151 00-0       
Γ152 -0-000      
Γ800 -0-0-0-14-3     
Γ801 00-0-0-0-0    
Γ802 -0-0-0-2-0-11   
Γ803 -1-0-0-58-09-01  
Γ805 000000000 
  Γ128 Γ130 Γ132 Γ151 Γ152 Γ800 Γ801 Γ802 Γ803 Γ805

2.2  Equality constraints

We use equality constraints that relate a branching fraction to a sum of branching fractions. As mentioned above, the tau 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 2011 [91] 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 tau branching fractions. The tau 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.

     
  Γ7 = Γ35·Γ<K0|KL> + Γ9 + Γ804 + Γ37·Γ<K0|KL> + Γ10           
     
  Γ8 = Γ9 + Γ10           
     
  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) + Γ48·ΓKS→π0π0 + Γ126·Γη→3π0 + Γ37·(Γ<K0|KS>·ΓKS→π0π0)          
  + Γ130·Γη→3π0           
     
  Γ19 = Γ23 + Γ20           
     
  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          
  + Γ801·(Γφ→ KS KL·ΓKS→π0π0)/ (Γφ→ K+K− +Γφ→ KS KL)           
     
  Γ33 = Γ35·Γ<K0|KS> + Γ40·Γ<K0|KS> + Γ42·Γ<K0|KS> + Γ47 + Γ48 + Γ37·Γ<K0|KS>          
  + Γ132·(Γ<K0|KS>·Γη→neutral) + Γ44·Γ<K0|KS> + Γ801·Γφ→ KS KL/(Γφ→ K+K− +Γφ→ KS KL)           
     
  Γ34 = Γ35 + Γ37           
     
  Γ38 = Γ42 + Γ37           
     
  Γ39 = Γ40 + Γ42           
     
  Γ43 = Γ40 + Γ44           
     
  Γ46 = Γ48 + Γ47 + Γ804           
     
  Γ54 = Γ128·Γη→charged + Γ152·(Γω→π+π−π0+Γω→π+π−) + Γ35·(Γ<K0|KS>·ΓKS→π+π−)          
  + Γ40·(Γ<K0|KS>·ΓKS→π+π−) + Γ42·(Γ<K0|KS>·ΓKS→π+π−) + Γ78          
   + Γ47·(2·ΓKS→π+π−·ΓKS→π0π0) + Γ77 + Γ48·ΓKS→π+π− + Γ94 + Γ62 + Γ70 + Γ93          
  + Γ126·Γη→charged + Γ37·(Γ<K0|KS>·ΓKS→π+π−) + Γ802 + Γ803          
  + Γ800·(Γω→π+π−π0+Γω→π+π−) + Γ151·(Γω→π+π−π0+Γω→π+π−) + Γ130·Γη→charged          
  + Γ132·(Γ<K0|KL>·Γη→π+π−π0          
  + Γ<K0|KS>·ΓKS→π0π0·Γη→π+π−π0 + Γ<K0|KS>·ΓKS→π+π−·Γη→3π0)          
  + Γ53·(Γ<K0|KS>·ΓKS→π0π0+Γ<K0|KL>)          
  + Γ801·(Γφ→ K+K− + Γφ→ KS KL·ΓKS→π+π−)/(Γφ→ K+K− +Γφ→ KS KL)           
     
  Γ55 = Γ128·Γη→charged + Γ152·(Γω→π+π−π0+Γω→π+π−) + Γ78 + Γ77 + Γ94 + Γ62 + Γ70 + Γ93          
  + Γ126·Γη→charged + Γ802 + Γ803 + Γ800·(Γω→π+π−π0+Γω→π+π−)          
  + Γ151·(Γω→π+π−π0 + Γω→π+π−) + Γ130·Γη→charged          
  + Γ801·Γφ→ K+K−/(Γφ→ K+K− + Γφ→ KS KL)           
     
  Γ57 = Γ62 + Γ93 + Γ802 + Γ800·Γω→π+π− + Γ151·Γω→π+π− + Γ801·Γφ→ K+K−/(Γφ→ K+K−+Γφ→ KS KL)           
     
  Γ58 = Γ62 + Γ93 + Γ802 + Γ801·Γφ→ K+K−/(Γφ→ K+K− +Γφ→ KS KL)           
     
  Γ60 = Γ62 + Γ800·Γω→π+π−           
     
  Γ66 = Γ128·Γη→π+π−π0 + Γ152·Γω→π+π− + Γ94 + Γ70 + Γ803 + Γ800·Γω→π+π−π0 + Γ151·Γω→π+π−π0           
     
  Γ68 = Γ152·Γω→π+π− + Γ40·(Γ<K0|KS>·ΓKS→π+π−) + Γ70 + Γ800·Γω→π+π−π0           
     
  Γ69 = Γ152·Γω→π+π− + Γ70 + Γ800·Γω→π+π−π0           
     
  Γ74 = Γ152·Γω→π+π−π0 + Γ78 + Γ77 + Γ126·Γη→π+π−π0 + Γ130·Γη→π+π−π0           
     
  Γ76 = Γ152·Γω→π+π−π0 + Γ77 + Γ126·Γη→π+π−π0 + Γ130·Γη→π+π−π0           
     
  Γ82 = Γ128·Γη→charged + Γ42·(Γ<K0|KS>·ΓKS→π+π−) + Γ802 + Γ803 + Γ151·(Γω→π+π−π0 +Γω→π+π−)          
  + Γ37·(Γ<K0|KS>·ΓKS→π+π−)           
     
  Γ85 = Γ802 + Γ151·Γω→π+π−           
     
  Γ88 = Γ128·Γη→π+π−π0 + Γ803 + Γ151·Γω→π+π−π0           
     
  Γ92 = Γ94 + Γ93           
     
  Γ96 = Γ801·Γφ→ K+K− / (Γφ→ K+K− + Γφ→ KS KL)           
     
  Γ102 = Γ103 + Γ104           
     
  Γ110 = Γ10 + Γ16 + Γ23 + Γ28 + Γ35 + Γ40 + Γ128 + Γ802 + Γ803 + Γ151 + Γ130 + Γ132 + Γ44 + Γ53 + Γ801           
     
  Γ136 = Γ104·Γη→π+π−π0 + Γ78·Γη→3π0           
     
  Γ150 = Γ800 + Γ151           
     
  Γ804 = Γ47·(Γ<K0|KL>·Γ<K0|KL>)/(Γ<K0|KS>·Γ<K0|KS>)          
     
  ΓAll = Γ3 + Γ5 + Γ9 + Γ10 + Γ14 + Γ16 + Γ20 + Γ23 + Γ27 + Γ28 + Γ30 + Γ35 + Γ37 + Γ40 + Γ42 + Γ47          
  + Γ48 + Γ62 + Γ70 + Γ77 + Γ78 + Γ93 + Γ94 + Γ104 + Γ126 + Γ128 + Γ802 + Γ803 + Γ800 + Γ151          
  + Γ130 + Γ132 + Γ44 + Γ53 + Γ49 + Γ804 + Γ805 + Γ801 + Γ152 + Γ103           

2.3  Fit procedure

The fit procedure is functionally equivalent to the one employed in the former HFAG report [31] and consists in a minimum χ2 fit subject to linear and non-linear constraints. The fit code has been improved to automatize the treatment of non-linear constraints, which are iteratively Taylor-expanded to obtain numerically approximate linear constraints, which permit an analytical solution for the χ2 minimization when, as it happens in this case, the χ2 is a quadratic function of the fitted quantities.

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