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6  Combination of upper limits on τ lepton-flavour-violating branching fractions

The Standard Model predicts that the τ lepton-flavour-violating (LFV) branching fractions are too small to be measured with the available experimental precision. We report in Table 14 and Figure 2 the experimental upper limits on these branching fractions that have been published by the B-factories BaBar and Belle and later experiments. We omit previous weaker upper limits (mainly from CLEO) and all preliminary results older than a few years. Presently, no preliminary result is included.

Combining upper limits is a delicate issue, since there is no standard and generally agreed procedure. Furthermore, the τ LFV searches published limits are extracted from the data with a variety of methods, and cannot be directly combined with a uniform procedure. It is however possible to use a single and effective upper limit combination procedure for all modes by re-computing the published upper limits with just one extraction method, using the published information that documents the upper limit determination: number of observed candidates, expected background, signal efficiency and number of analyzed τ decays.

We chose to use the CLs method [99] to re-compute the τ LFV upper limits, since it is well known and widely used (see the Statistics review of PDG 2018 [6]), and since the limits computed with the CLs method can be combined in a straightforward way (see below). The CLs method is based on two hypotheses: signal plus background and background only. We calculate the observed confidence levels for the two hypotheses:

     
 
CLs+b = Ps+b(Q ≤ Qobs) = ∫
Qobs


− ∞
 
dPs+b
dQ
 dQ,   
            (10)
 
CLb = Pb(Q ≤ Qobs) = ∫
Qobs


− ∞
 
dPb
dQ
 dQ,  
            (11)

where CLs+b is the confidence level observed for the signal plus background hypotheses, CLb is the confidence level observed for the background only hypothesis, dPs+b/dQ and dPb/dQ are the probability distribution functions (PDFs) for the two corresponding hypothesis and Q is called the test statistic. The CLs value is defined as the ratio between the confidence level for the signal plus background hypothesis and the confidence level for the background hypothesis:

     
CLs = 
CLs+b
CLb
.
             (12)

When multiple results are combined, the PDFs in Eqs. (10) and (11) are the product of the individual PDFs,

     
CLs = 
N
∏
i=1
ni
∑
n=0
 
e−(si+bi) (si+bi)n
n!
 
N
∏
i=1
  
ni
∑
n=0
 
e−bi bin
n!
    
N
∏
j=1
 ⎡
⎣		siSi(xij)+biBi(xij)⎤
⎦
N
∏
j=1
Bi(xij)
 ,
             (13)

where N is the number of results (or channels), and, for each channel i, ni is the number of observed candidates, xij are the values of the discriminating variables (with index j), si and bi are the number of signal and background events and Si, Bi are the probability distribution functions of the discriminating variables. The discriminating variables xij are assumed to be uncorrelated. The expected signal si is related to the τ lepton branching fraction B(τ → fi) into the searched final state fi by si = Niєi B(τ → fi), where Ni is the number of produced τ leptons and єi is the detection efficiency for observing the decay τ→ fi. For e+ e− experiments, Ni = 2Liσττ, where Li is the integrated luminosity and σττ is the τ pair production cross section σ(e+ e− → τ+ τ−) [100]. In experiments where τ leptons are produced in more complex multiple reactions, the effective Ni is typically estimated with Monte Carlo simulations calibrated with related data yields.

The extraction of the upper limits is performed using the code provided by Tom Junk [101]. The systematic uncertainties are modeled in the Monte Carlo toy experiments by convolving the Si and Bi PDFs with Gaussian distributions corresponding to the nuisance parameters.

Table 14 reports the HFLAV combinations of the τ LFV limits. Since there is negligible gain in combining limits of very different strength, the combinations do not include the CLEO searches and do not include results where the single event sensitivity is more than a factor of 5 lower than the value for the search with the best limit.

Figure 3 reports a graphical representation of the τ LFV limits combinations listed in Table 14. The published information that has been used to obtain these limits is reported in Table 15. In the previous HFLAV reports, the determination of combined limit B183 = µ− µ+ µ− erroneously counted twice the systematic uncertainty of the LHCb limit. That has been fixed now, and the combination of the upper limits on B183 = µ− µ+ µ− has changed from < 1.2 · 10−8 to < 1.1 · 10−8.

Table 14: Experimental upper limits on lepton flavour violating τ decays. The modes are grouped according to the properties of their final states. Modes with baryon number violation are labelled with “BNV”. The experiment “HFLAV” denotes the combinations of upper limits computed by HFLAV. The references associated with the combination list what upper limits have been used.
Decay modeCategory
90% CL
Limit
ExperimentReferences
 
B156 = e− γℓγ3.3 · 10−8BaBar[102]
  1.2 · 10−7Belle[103]
  5.4 · 10−8HFLAV[103, 102]
B157 = µ− γ 4.4 · 10−8BaBar[102]
  4.5 · 10−8Belle[103]
  5.0 · 10−8HFLAV[103, 102]
B158 = e− π0ℓ P01.3 · 10−7BaBar[104]
  8.0 · 10−8Belle[105]
  4.9 · 10−8HFLAV[105, 104]
B159 = µ− π0 1.1 · 10−7BaBar[104]
  1.2 · 10−7Belle[105]
  3.6 · 10−8HFLAV[105, 104]
B160 = e− KS0 3.3 · 10−8BaBar[106]
  2.6 · 10−8Belle[107]
  1.4 · 10−8HFLAV[107, 106]
B161 = µ− KS0 4.0 · 10−8BaBar[106]
  2.3 · 10−8Belle[107]
  1.5 · 10−8HFLAV[107, 106]
B162 = e− η 1.6 · 10−7BaBar[104]
  9.2 · 10−8Belle[105]
  5.5 · 10−8HFLAV[105, 104]
B163 = µ− η 1.5 · 10−7BaBar[104]
  6.5 · 10−8Belle[105]
  3.8 · 10−8HFLAV[105, 104]
B172 = e− η′(958) 2.4 · 10−7BaBar[104]
  1.6 · 10−7Belle[105]
  9.9 · 10−8HFLAV[105, 104]
B173 = µ− η′(958) 1.4 · 10−7BaBar[104]
  1.3 · 10−7Belle[105]
  6.3 · 10−8HFLAV[105, 104]
B164 = e− ρ0ℓ V04.6 · 10−8BaBar[108]
  1.8 · 10−8Belle[109]
  1.5 · 10−8HFLAV[109, 108]
B165 = µ− ρ0 2.6 · 10−8BaBar[108]
  1.2 · 10−8Belle[109]
  1.5 · 10−8HFLAV[109, 108]
B166 = e− ω 1.1 · 10−7BaBar[110]
  4.8 · 10−8Belle[109]
  3.3 · 10−8HFLAV[109, 110]
B167 = µ− ω 1.0 · 10−7BaBar[110]
  4.7 · 10−8Belle[109]
  4.0 · 10−8HFLAV[109, 110]
B168 = e− K*(892) 5.9 · 10−8BaBar[108]
  3.2 · 10−8Belle[109]
  2.3 · 10−8HFLAV[109, 108]
B169 = µ− K*(892) 1.7 · 10−7BaBar[108]
  7.2 · 10−8Belle[109]
  6.0 · 10−8HFLAV[109, 108]
B170 = e− K*(892) 4.6 · 10−8BaBar[108]
  3.4 · 10−8Belle[109]
  2.2 · 10−8HFLAV[109, 108]
B171 = µ− K*(892) 7.3 · 10−8BaBar[108]
  7.0 · 10−8Belle[109]
  4.2 · 10−8HFLAV[109, 108]
B176 = e− φ 3.1 · 10−8BaBar[108]
  3.1 · 10−8Belle[109]
  2.0 · 10−8HFLAV[109, 108]
B177 = µ− φ 1.9 · 10−7BaBar[108]
  8.4 · 10−8Belle[109]
  6.8 · 10−8HFLAV[109, 108]
B174 = e− f0(980)ℓ S03.2 · 10−8Belle[111]
B175 = µ− f0(980) 3.4 · 10−8Belle[111]
B178 = e− e+ e−ℓℓℓ2.9 · 10−8BaBar[112]
  2.7 · 10−8Belle[113]
  1.4 · 10−8HFLAV[113, 112]
B179 = e− µ+ µ− 3.2 · 10−8BaBar[112]
  2.7 · 10−8Belle[113]
  1.6 · 10−8HFLAV[113, 112]
B180 = µ− e+ µ− 2.6 · 10−8BaBar[112]
  1.7 · 10−8Belle[113]
  9.8 · 10−9HFLAV[113, 112]
B181 = µ− e+ e− 2.2 · 10−8BaBar[112]
  1.8 · 10−8Belle[113]
  1.1 · 10−8HFLAV[113, 112]
B182 = e− µ+ e− 1.8 · 10−8BaBar[112]
  1.5 · 10−8Belle[113]
  8.4 · 10−9HFLAV[113, 112]
B183 = µ− µ+ µ− 3.8 · 10−7ATLAS[114]
  3.3 · 10−8BaBar[112]
  2.1 · 10−8Belle[113]
  4.6 · 10−8LHCb[115]
  1.1 · 10−8HFLAV[113, 112, 115]
B184 = e− π+ π−ℓ hh1.2 · 10−7BaBar[116]
  2.3 · 10−8Belle[117]
B185 = e+ π− π− 2.7 · 10−7BaBar[116]
  2.0 · 10−8Belle[117]
B186 = µ− π+ π− 2.9 · 10−7BaBar[116]
  2.1 · 10−8Belle[117]
B187 = µ+ π− π− 7.0 · 10−8BaBar[116]
  3.9 · 10−8Belle[117]
B188 = e− π+ K− 3.2 · 10−7BaBar[116]
  3.7 · 10−8Belle[117]
B189 = e− K+ π− 1.7 · 10−7BaBar[116]
  3.1 · 10−8Belle[117]
B190 = e+ π− K− 1.8 · 10−7BaBar[116]
  3.2 · 10−8Belle[117]
B191 = e− KS0 KS0 7.1 · 10−8Belle[107]
B192 = e− K+ K− 1.4 · 10−7BaBar[116]
  3.4 · 10−8Belle[117]
B193 = e+ K− K− 1.5 · 10−7BaBar[116]
  3.3 · 10−8Belle[117]
B194 = µ− π+ K− 2.6 · 10−7BaBar[116]
  8.6 · 10−8Belle[117]
B195 = µ− K+ π− 3.2 · 10−7BaBar[116]
  4.5 · 10−8Belle[117]
B196 = µ+ π− K− 2.2 · 10−7BaBar[116]
  4.8 · 10−8Belle[117]
B197 = µ− KS0 KS0 8.0 · 10−8Belle[107]
B198 = µ− K+ K− 2.5 · 10−7BaBar[116]
  4.4 · 10−8Belle[117]
B199 = µ+ K− K− 4.8 · 10−7BaBar[116]
  4.7 · 10−8Belle[117]
B211 = π− ΛBNV7.2 · 10−8Belle[118]
B212 = π− Λ 1.4 · 10−7Belle[118]
B215 = p µ− µ− 4.4 · 10−7LHCb[119]
B216 = p µ+ µ− 3.3 · 10−7LHCb[119]
 
Table 15: Published information that has been used to re-compute upper limits with the CLs method, i.e. the number of τ leptons produced, the signal detection efficiency and its uncertainty, the number of expected background events and its uncertainty, and the number of observed events. The uncertainty on the efficiency includes the minor uncertainty contribution on the number of τ leptons (typically originating on the uncertainties on the integrated luminosity and on the production cross-section). The additional limit used in the combinations (from LHCb) has been originally determined with the CLs method.
Decay modeExp.Ref.
Nτ
(millions)
efficiency
(%)
NbkgNobs
 
B156 = e− γBaBar[102]9633.90 ± 0.301.60 ± 0.400
B156 = e− γBelle[103]9833.00 ± 0.105.14 ± 3.305
B157 = µ− γBaBar[102]9636.10 ± 0.503.60 ± 0.702
B157 = µ− γBelle[103]9835.07 ± 0.2013.90 ± 5.0010
B158 = e− π0BaBar[104]3392.83 ± 0.250.17 ± 0.040
B158 = e− π0Belle[105]4013.93 ± 0.180.20 ± 0.200
B159 = µ− π0BaBar[104]3394.75 ± 0.371.33 ± 0.151
B159 = µ− π0Belle[105]4014.53 ± 0.200.58 ± 0.341
B160 = e− KS0BaBar[106]8629.10 ± 1.730.59 ± 0.251
B160 = e− KS0Belle[107]127410.20 ± 0.670.18 ± 0.180
B161 = µ− KS0BaBar[106]8626.14 ± 0.200.30 ± 0.181
B161 = µ− KS0Belle[107]127410.70 ± 0.730.35 ± 0.210
B162 = e− ηBaBar[104]3392.12 ± 0.200.22 ± 0.050
B162 = e− ηBelle[105]4012.87 ± 0.200.78 ± 0.780
B163 = µ− ηBaBar[104]3393.59 ± 0.410.75 ± 0.081
B163 = µ− ηBelle[105]4014.08 ± 0.280.64 ± 0.040
B172 = e− η′(958)BaBar[104]3391.53 ± 0.160.12 ± 0.030
B172 = e− η′(958)Belle[105]4011.59 ± 0.130.01 ± 0.410
B173 = µ− η′(958)BaBar[104]3392.18 ± 0.260.49 ± 0.260
B173 = µ− η′(958)Belle[105]4012.47 ± 0.200.23 ± 0.460
B164 = e− ρ0BaBar[108]8297.31 ± 0.201.32 ± 0.171
B164 = e− ρ0Belle[109]15547.58 ± 0.410.29 ± 0.150
B165 = µ− ρ0BaBar[108]8294.52 ± 0.402.04 ± 0.190
B165 = µ− ρ0Belle[109]15547.09 ± 0.371.48 ± 0.350
B166 = e− ωBaBar[110]8292.96 ± 0.130.35 ± 0.060
B166 = e− ωBelle[109]15542.92 ± 0.180.30 ± 0.140
B167 = µ− ωBaBar[110]8292.56 ± 0.160.73 ± 0.030
B167 = µ− ωBelle[109]15542.38 ± 0.140.72 ± 0.180
B168 = e− K*(892)BaBar[108]8298.00 ± 0.201.65 ± 0.232
B168 = e− K*(892)Belle[109]15544.37 ± 0.240.29 ± 0.140
B169 = µ− K*(892)BaBar[108]8294.60 ± 0.401.79 ± 0.214
B169 = µ− K*(892)Belle[109]15543.39 ± 0.190.53 ± 0.201
B170 = e− K*(892)BaBar[108]8297.80 ± 0.202.76 ± 0.282
B170 = e− K*(892)Belle[109]15544.41 ± 0.250.08 ± 0.080
B171 = µ− K*(892)BaBar[108]8294.10 ± 0.301.72 ± 0.171
B171 = µ− K*(892)Belle[109]15543.60 ± 0.200.45 ± 0.171
B176 = e− φBaBar[108]8296.40 ± 0.200.68 ± 0.120
B176 = e− φBelle[109]15544.18 ± 0.250.47 ± 0.190
B177 = µ− φBaBar[108]8295.20 ± 0.302.76 ± 0.166
B177 = µ− φBelle[109]15543.21 ± 0.190.06 ± 0.061
B178 = e− e+ e−BaBar[112]8688.60 ± 0.200.12 ± 0.020
B178 = e− e+ e−Belle[113]14376.00 ± 0.590.21 ± 0.150
B179 = e− µ+ µ−BaBar[112]8686.40 ± 0.400.54 ± 0.140
B179 = e− µ+ µ−Belle[113]14376.10 ± 0.580.10 ± 0.040
B180 = µ− e+ µ−BaBar[112]86810.20 ± 0.600.03 ± 0.020
B180 = µ− e+ µ−Belle[113]143710.10 ± 0.770.02 ± 0.020
B181 = µ− e+ e−BaBar[112]8688.80 ± 0.500.64 ± 0.190
B181 = µ− e+ e−Belle[113]14379.30 ± 0.730.04 ± 0.040
B182 = e− µ+ e−BaBar[112]86812.70 ± 0.700.34 ± 0.120
B182 = e− µ+ e−Belle[113]143711.50 ± 0.890.01 ± 0.010
B183 = µ− µ+ µ−BaBar[112]8686.60 ± 0.600.44 ± 0.170
B183 = µ− µ+ µ−Belle[113]14377.60 ± 0.560.13 ± 0.200
 
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