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8  Combination of upper limits on τ LFV branching fractions

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 [105] to re-compute the τ LFV upper limits, since it is well known and widely used (see the Statistics review of PDG 2013 [77]), 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 hypothesis, 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 statistics. The CLs value is defined as the ratio between the confidence level for the signal plus background hypothesis to the confidence level for the background hypothesis:

     
CLs = 
CLs+b
CLb
.
             (12)

When multiple results are combined, the PDFs in Equations 10 and 11 are the product of the individual PDFs,

     
CLs = 
Nchan
∏
i=1
ni
∑
n=0
 
e−(si+bi) (si+bi)n
n!
 
nchan
∏
i=1
  
ni
∑
n=0
 
e−bi bin
n!
    
n
∏
j=1
 siSi(xij)+biBi(xij)
ni
∏
j=1
Bi(xij)
 ,
             (13)

where Nchan 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 expected signal si is related to the τ lepton branching fraction B (τ → fi) into the searched final state fi by si = NiєiB (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−) [106]. 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 [107]. The systematic uncertainties are modeled in the Monte Carlo toy experiments by convolving the Si and Bi PDFs with with Gaussian distributions corresponding to the nuisance parameters.

Table 15 reports the HFAG combinations of the τ LFV limits, together with the published limits, excluding the older and weaker CLEO limits. Since there is negligible gain in combining limits of very different strength, the combinations do not include the CLEO searches and we do not combine results for modes where the best limit is more than an order of magnitude better than the other limits. Figure 3 reports a graphical representation of the limits in Table 15.

Table 15: Combinations of upper limits on lepton flavour violating τ decay modes. The modes are grouped according to the particle content of their final states. Modes with baryon number violation are labelled with “BNV”.
Decay modeCategory
90% CL
Limit
 
Γ156 = e− γℓγ5.4 · 10−8
Γ157 = µ− γ 5.0 · 10−8
Γ160 = e− KS0ℓ P01.4 · 10−8
Γ161 = µ− KS0 1.5 · 10−8
Γ164 = e− ρ0ℓ V01.5 · 10−8
Γ165 = µ− ρ0 1.5 · 10−8
Γ166 = e− ω 3.3 · 10−8
Γ167 = µ− ω 4.0 · 10−8
Γ168 = e− K*(892)0 2.3 · 10−8
Γ169 = µ− K*(892)0 6.0 · 10−8
Γ170 = e− K*(892)0 2.2 · 10−8
Γ171 = µ− K*(892)0 4.2 · 10−8
Γ176 = e− φ 2.0 · 10−8
Γ177 = µ− φ 6.8 · 10−8
Γ178 = e− e+ e−ℓℓℓ1.4 · 10−8
Γ179 = e− µ+ µ− 1.6 · 10−8
Γ180 = µ− e+ µ− 9.8 · 10−9
Γ181 = µ− e+ e− 1.1 · 10−8
Γ182 = e− µ+ e− 8.4 · 10−9
Γ183 = µ− µ+ µ− 1.2 · 10−8
Γ211 = π− ΛBNV1.9 · 10−8
Γ212 = π− Λ 1.8 · 10−8
Γ213 = K− Λ 3.7 · 10−8
Γ214 = K− Λ 2.0 · 10−8
 
Table 16: 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 limits used in the combinations (one from LHCb) have been determined with the CLs method already in their publication.
Decay modeExp.Ref.
Nτ
(millions)
efficiency
(%)
NbkgNobs
 
Γ156 = e− γBaBar[87]963.23.90 ± 0.301.60 ± 0.400
Γ156 = e− γBelle[86]983.43.00 ± 0.105.14 ± 3.305
Γ157 = µ− γBaBar[87]963.26.10 ± 0.503.60 ± 0.702
Γ157 = µ− γBelle[86]983.45.07 ± 0.2013.90 ± 5.0010
Γ160 = e− KS0BaBar[91]8629.10 ± 1.730.59 ± 0.251
Γ160 = e− KS0Belle[90]1273.610.20 ± 0.670.18 ± 0.180
Γ161 = µ− KS0BaBar[91]8626.14 ± 0.200.30 ± 0.181
Γ161 = µ− KS0Belle[90]1273.610.70 ± 0.730.35 ± 0.210
Γ164 = e− ρ0BaBar[94]828.87.31 ± 0.201.32 ± 0.171
Γ164 = e− ρ0Belle[93]1554.27.58 ± 0.410.29 ± 0.150
Γ165 = µ− ρ0BaBar[94]828.84.52 ± 0.402.04 ± 0.190
Γ165 = µ− ρ0Belle[93]1554.27.09 ± 0.371.48 ± 0.350
Γ166 = e− ωBaBar[95]828.82.96 ± 0.130.35 ± 0.060
Γ166 = e− ωBelle[93]1554.22.92 ± 0.180.30 ± 0.140
Γ167 = µ− ωBaBar[95]828.82.56 ± 0.160.73 ± 0.030
Γ167 = µ− ωBelle[93]1554.22.38 ± 0.140.72 ± 0.180
Γ168 = e− K*(892)0BaBar[94]828.88.00 ± 0.201.65 ± 0.232
Γ168 = e− K*(892)0Belle[93]1554.24.37 ± 0.240.29 ± 0.140
Γ169 = µ− K*(892)0BaBar[94]828.84.60 ± 0.401.79 ± 0.214
Γ169 = µ− K*(892)0Belle[93]1554.23.39 ± 0.190.53 ± 0.201
Γ170 = e− K*(892)0BaBar[94]828.87.80 ± 0.202.76 ± 0.282
Γ170 = e− K*(892)0Belle[93]1554.24.41 ± 0.250.08 ± 0.080
Γ171 = µ− K*(892)0BaBar[94]828.84.10 ± 0.301.72 ± 0.171
Γ171 = µ− K*(892)0Belle[93]1554.23.60 ± 0.200.45 ± 0.171
Γ176 = e− φBaBar[94]828.86.40 ± 0.200.68 ± 0.120
Γ176 = e− φBelle[93]1554.24.18 ± 0.250.47 ± 0.190
Γ177 = µ− φBaBar[94]828.85.20 ± 0.302.76 ± 0.166
Γ177 = µ− φBelle[93]1554.23.21 ± 0.190.06 ± 0.061
Γ178 = e− e+ e−BaBar[97]867.68.60 ± 0.200.12 ± 0.020
Γ178 = e− e+ e−Belle[96]1437.46.00 ± 0.590.21 ± 0.150
Γ179 = e− µ+ µ−BaBar[97]867.66.40 ± 0.400.54 ± 0.140
Γ179 = e− µ+ µ−Belle[96]1437.46.10 ± 0.580.10 ± 0.040
Γ180 = µ− e+ µ−BaBar[97]867.610.20 ± 0.600.03 ± 0.020
Γ180 = µ− e+ µ−Belle[96]1437.410.10 ± 0.770.02 ± 0.020
Γ181 = µ− e+ e−BaBar[97]867.68.80 ± 0.500.64 ± 0.190
Γ181 = µ− e+ e−Belle[96]1437.49.30 ± 0.730.04 ± 0.040
Γ182 = e− µ+ e−BaBar[97]867.612.70 ± 0.700.34 ± 0.120
Γ182 = e− µ+ e−Belle[96]1437.411.50 ± 0.890.01 ± 0.010
Γ183 = µ− µ+ µ−BaBar[97]867.66.60 ± 0.600.44 ± 0.170
Γ183 = µ− µ+ µ−Belle[96]1437.47.60 ± 0.560.13 ± 0.200
Γ211 = π− ΛBaBar[103]435.612.20 ± 8.500.56 ± 0.560
Γ211 = π− ΛBelle[102]1665.24.39 ± 0.360.31 ± 0.180
Γ212 = π− ΛBaBar[103]435.612.28 ± 8.500.42 ± 0.420
Γ212 = π− ΛBelle[102]1665.24.80 ± 0.390.21 ± 0.150
Γ213 = K− ΛBaBar[103]435.69.47 ± 0.660.12 ± 0.121
Γ213 = K− ΛBelle[102]1665.23.16 ± 0.270.42 ± 0.190
Γ214 = K− ΛBaBar[103]435.610.63 ± 0.740.26 ± 0.260
Γ214 = K− ΛBelle[102]1665.24.11 ± 0.350.31 ± 0.140
 
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