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9  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 limits 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 [103] to re-compute the τ LFV upper limits, since it is well known and widely used (see the Statistics review of PDG 2013 [17]), 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,   
            (1)
 
CLb = Pb(Q ≤ Qobs) = ∫
Qobs


− ∞
 
dPb
dQ
 dQ,  
            (2)

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
.
             (3)

When multiple results are combined, the PDFs in Equations 1 and 2 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)
 ,
             (4)

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−) [104]. 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 [105]. 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 re-computed limits for the B factories results as well as the corresponding HFAG combination. 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 4 reports the re-computed τ LFV searches upper limits and their combination.

Table 15: Combinations of upper limits on lepton flavor violating τ decay modes. The table includes, for each experimental result, the published information that has been used to re-compute the limit with the CLs method, i.e. the integrated luminosity, the cross-section for τ lepton pairs production, the detection efficiency, the number of expected background events and the number of observed events. Since the LHCb collaboration published a limit determined with the CLs method, in this case only the published limit is reported. The table finally reports the combined limit that is obtained by combining the re-computed limits. The modes are grouped according to the particle content of their final states. Modes with lepton number violation are labeled with “(L)”, modes with baryon number violation are labeled with “(BNV)”.
Decay modeCat.
90% CL
Limit
Exp.
L
(fb−1)
σττ
(nb)
efficiency
(%)
NbkgNobs
 
Γ156 =  e− γ 
lγ <  5.4 · 10−8HFAG
    Belle5350.9193.00 ± 0.105.14 ± 3.305
    BaBar 5240.9193.90 ± 0.301.60 ± 0.400
Γ157 =  µ− γ 
  <  5.0 · 10−8HFAG
    Belle5350.9195.07 ± 0.2013.90 ± 5.0010
    BaBar 5240.9196.10 ± 0.503.60 ± 0.702
Γ160 =  e− KS0 
lP0 <  1.4 · 10−8HFAG
    Belle6710.91910.20 ± 0.670.18 ± 0.180
    BaBar 4690.9199.10 ± 1.730.59 ± 0.251
Γ161 =  µ− KS0 
  <  1.5 · 10−8HFAG
    Belle6710.91910.70 ± 0.730.35 ± 0.210
    BaBar 4690.9196.14 ± 0.200.30 ± 0.181
Γ164 =  e− ρ0 
l V0 <  1.5 · 10−8HFAG
    Belle8540.9197.58 ± 0.410.29 ± 0.150
    BaBar 4510.9197.31 ± 0.201.32 ± 0.171
Γ165 =  µ− ρ0 
  <  1.5 · 10−8HFAG
    Belle8540.9197.09 ± 0.371.48 ± 0.350
    BaBar 4510.9194.52 ± 0.402.04 ± 0.190
Γ168 =  e− K*(892)0 
  <  2.3 · 10−8HFAG
    Belle8540.9194.37 ± 0.240.29 ± 0.140
    BaBar 4510.9198.00 ± 0.201.65 ± 0.232
Γ169 =  µ− K*(892)0 
  <  6.0 · 10−8HFAG
    Belle8540.9193.39 ± 0.190.53 ± 0.201
    BaBar 4510.9194.60 ± 0.401.79 ± 0.214
Γ170 =  
e− 
K
*(892)0
 
  <  2.2 · 10−8HFAG
    Belle8540.9194.41 ± 0.250.08 ± 0.080
    BaBar 4510.9197.80 ± 0.202.76 ± 0.282
Γ171 =  
µ− 
K
*(892)0
 
  <  4.2 · 10−8HFAG
    Belle8540.9193.60 ± 0.200.45 ± 0.171
    BaBar 4510.9194.10 ± 0.301.72 ± 0.171
Γ176 =  e− φ 
  <  2.0 · 10−8HFAG
    Belle8540.9194.18 ± 0.250.47 ± 0.190
    BaBar 4510.9196.40 ± 0.200.68 ± 0.120
Γ177 =  µ− φ 
  <  6.8 · 10−8HFAG
    Belle8540.9193.21 ± 0.190.06 ± 0.061
    BaBar 4510.9195.20 ± 0.302.76 ± 0.166
Γ166 =  e− ω 
  <  3.3 · 10−8HFAG
    Belle8540.9192.92 ± 0.180.30 ± 0.140
    BaBar 4510.9192.96 ± 0.130.35 ± 0.060
Γ167 =  µ− ω 
  <  4.0 · 10−8HFAG
    Belle8540.9192.38 ± 0.140.72 ± 0.180
    BaBar 4510.9192.56 ± 0.160.73 ± 0.030
Γ178 =  e− e+ e− 
lll <  1.4 · 10−8HFAG
    Belle7820.9196.00 ± 0.590.21 ± 0.150
    BaBar 4720.9198.60 ± 0.200.12 ± 0.020
Γ181 =  µ− e+ e− 
  <  1.1 · 10−8HFAG
    Belle7820.9199.30 ± 0.730.04 ± 0.040
    BaBar 4720.9198.80 ± 0.500.64 ± 0.190
Γ179 =  e− µ+ µ− 
  <  1.6 · 10−8HFAG
    Belle7820.9196.10 ± 0.580.10 ± 0.040
    BaBar 4720.9196.40 ± 0.400.54 ± 0.140
Γ183 =  µ− µ+ µ− 
  <  1.2 · 10−8HFAG
    Belle7820.9197.60 ± 0.560.13 ± 0.200
    BaBar 4720.9196.60 ± 0.600.44 ± 0.170
   <  4.6 · 10−8LHCb
Γ182 =  e− µ+ e− 
  <  8.4 · 10−9HFAG
    Belle7820.91911.50 ± 0.890.01 ± 0.010
    BaBar 4720.91912.70 ± 0.700.34 ± 0.120
Γ180 =  µ− e+ µ− 
  <  9.8 · 10−9HFAG
    Belle7820.91910.10 ± 0.770.02 ± 0.020
    BaBar 4720.91910.20 ± 0.600.03 ± 0.020
Γ211 =   π− Λ  
BNV <  1.9 · 10−8HFAG
    Belle9060.9194.39 ± 0.360.31 ± 0.180
    BaBar 2370.91912.20 ± 8.500.56 ± 0.560
Γ212 =  
 π− 
Λ
 
  <  1.8 · 10−9HFAG
    Belle9060.9194.80 ± 0.390.21 ± 0.150
    BaBar 2370.91912.28 ± 8.500.42 ± 0.420
Γ213 =   K− Λ  
  <  3.7 · 10−9HFAG
    Belle9060.9193.16 ± 0.270.42 ± 0.190
    BaBar 2370.9199.47 ± 0.660.12 ± 0.121
Γ214 =  
 K− 
Λ
 
  <  2.0 · 10−9HFAG
    Belle9060.9194.11 ± 0.350.31 ± 0.140
    BaBar 2370.91910.63 ± 0.740.26 ± 0.260
 
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