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HFLAV-Tau Spring 2017 Report
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8 Combination of upper limits on τ lepton-flavour-violating 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 [110] to re-compute the
τ LFV upper limits, since it is well known and widely used (see the
Statistics review of PDG 2013 [2]), 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) = | ∫ | | | | dQ,
|
| | | | | | | | | (10) |
| | | CLb = Pb(Q ≤ Qobs) = | ∫ | | | | 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:
When multiple results are combined, the PDFs in
Eqs. (10) and (11) are the
product of the individual PDFs,
|
| | CLs = | | | | | ⎡
⎣ | siSi(xij)+biBi(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 = 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−) [111].
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 [112]. 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 15 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 limits in Table 15.
The published information that has been used to obtain these limits is
reported in Table 16.
| Table 15: Combinations of upper limits on lepton flavour violating τ decay
modes. The modes are grouped according to the properties of their final
states. Modes with baryon number violation are labelled with “BNV”.
|
|
| Decay mode | Category | | Refs. |
|
| |
| Γ156 = e− γ | ℓγ | 5.4 · 10−8 | [93, 92] |
| Γ157 = µ− γ | | 5.0 · 10−8 | [93, 92] |
|
| Γ158 = e− π0 | ℓ P0 | 4.9 · 10−8 | [95, 94] |
| Γ159 = µ− π0 | | 3.6 · 10−8 | [95, 94] |
| Γ160 = e− KS0 | | 1.4 · 10−8 | [97, 96] |
| Γ161 = µ− KS0 | | 1.5 · 10−8 | [97, 96] |
| Γ162 = e− η | | 5.5 · 10−8 | [95, 94] |
| Γ163 = µ− η | | 3.8 · 10−8 | [95, 94] |
| Γ172 = e− η′(958) | | 9.9 · 10−8 | [95, 94] |
| Γ173 = µ− η′(958) | | 6.3 · 10−8 | [95, 94] |
|
| Γ164 = e− ρ0 | ℓ V0 | 1.5 · 10−8 | [99, 98] |
| Γ165 = µ− ρ0 | | 1.5 · 10−8 | [99, 98] |
| Γ166 = e− ω | | 3.3 · 10−8 | [99, 100] |
| Γ167 = µ− ω | | 4.0 · 10−8 | [99, 100] |
| Γ168 = e− K*(892)0 | | 2.3 · 10−8 | [99, 98] |
| Γ169 = µ− K*(892)0 | | 6.0 · 10−8 | [99, 98] |
| Γ170 = e− K*(892)0 | | 2.2 · 10−8 | [99, 98] |
| Γ171 = µ− K*(892)0 | | 4.2 · 10−8 | [99, 98] |
| Γ176 = e− φ | | 2.0 · 10−8 | [99, 98] |
| Γ177 = µ− φ | | 6.8 · 10−8 | [99, 98] |
|
| Γ178 = e− e+ e− | ℓℓℓ | 1.4 · 10−8 | [103, 102] |
| Γ179 = e− µ+ µ− | | 1.6 · 10−8 | [103, 102] |
| Γ180 = µ− e+ µ− | | 9.8 · 10−9 | [103, 102] |
| Γ181 = µ− e+ e− | | 1.1 · 10−8 | [103, 102] |
| Γ182 = e− µ+ e− | | 8.4 · 10−9 | [103, 102] |
| Γ183 = µ− µ+ µ− | | 1.2 · 10−8 | [103, 102, 105] |
|
| |
| 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 limit used in the
combinations (from LHCb) has been originally determined with the CLs method.
|
|
| Decay mode | Exp. | Ref. | | | Nbkg | Nobs |
|
| |
| Γ156 = e− γ | BaBar | [92] | 963 | 3.90 ± 0.30 | 1.60 ± 0.40 | 0 |
| Γ156 = e− γ | Belle | [93] | 983 | 3.00 ± 0.10 | 5.14 ± 3.30 | 5 |
| Γ157 = µ− γ | BaBar | [92] | 963 | 6.10 ± 0.50 | 3.60 ± 0.70 | 2 |
| Γ157 = µ− γ | Belle | [93] | 983 | 5.07 ± 0.20 | 13.90 ± 5.00 | 10 |
| Γ158 = e− π0 | BaBar | [94] | 339 | 2.83 ± 0.25 | 0.17 ± 0.04 | 0 |
| Γ158 = e− π0 | Belle | [95] | 401 | 3.93 ± 0.18 | 0.20 ± 0.20 | 0 |
| Γ159 = µ− π0 | BaBar | [94] | 339 | 4.75 ± 0.37 | 1.33 ± 0.15 | 1 |
| Γ159 = µ− π0 | Belle | [95] | 401 | 4.53 ± 0.20 | 0.58 ± 0.34 | 1 |
| Γ160 = e− KS0 | BaBar | [96] | 862 | 9.10 ± 1.73 | 0.59 ± 0.25 | 1 |
| Γ160 = e− KS0 | Belle | [97] | 1274 | 10.20 ± 0.67 | 0.18 ± 0.18 | 0 |
| Γ161 = µ− KS0 | BaBar | [96] | 862 | 6.14 ± 0.20 | 0.30 ± 0.18 | 1 |
| Γ161 = µ− KS0 | Belle | [97] | 1274 | 10.70 ± 0.73 | 0.35 ± 0.21 | 0 |
| Γ162 = e− η | BaBar | [94] | 339 | 2.12 ± 0.20 | 0.22 ± 0.05 | 0 |
| Γ162 = e− η | Belle | [95] | 401 | 2.87 ± 0.20 | 0.78 ± 0.78 | 0 |
| Γ163 = µ− η | BaBar | [94] | 339 | 3.59 ± 0.41 | 0.75 ± 0.08 | 1 |
| Γ163 = µ− η | Belle | [95] | 401 | 4.08 ± 0.28 | 0.64 ± 0.04 | 0 |
| Γ172 = e− η′(958) | BaBar | [94] | 339 | 1.53 ± 0.16 | 0.12 ± 0.03 | 0 |
| Γ172 = e− η′(958) | Belle | [95] | 401 | 1.59 ± 0.13 | 0.01 ± 0.41 | 0 |
| Γ173 = µ− η′(958) | BaBar | [94] | 339 | 2.18 ± 0.26 | 0.49 ± 0.26 | 0 |
| Γ173 = µ− η′(958) | Belle | [95] | 401 | 2.47 ± 0.20 | 0.23 ± 0.46 | 0 |
| Γ164 = e− ρ0 | BaBar | [98] | 829 | 7.31 ± 0.20 | 1.32 ± 0.17 | 1 |
| Γ164 = e− ρ0 | Belle | [99] | 1554 | 7.58 ± 0.41 | 0.29 ± 0.15 | 0 |
| Γ165 = µ− ρ0 | BaBar | [98] | 829 | 4.52 ± 0.40 | 2.04 ± 0.19 | 0 |
| Γ165 = µ− ρ0 | Belle | [99] | 1554 | 7.09 ± 0.37 | 1.48 ± 0.35 | 0 |
| Γ166 = e− ω | BaBar | [100] | 829 | 2.96 ± 0.13 | 0.35 ± 0.06 | 0 |
| Γ166 = e− ω | Belle | [99] | 1554 | 2.92 ± 0.18 | 0.30 ± 0.14 | 0 |
| Γ167 = µ− ω | BaBar | [100] | 829 | 2.56 ± 0.16 | 0.73 ± 0.03 | 0 |
| Γ167 = µ− ω | Belle | [99] | 1554 | 2.38 ± 0.14 | 0.72 ± 0.18 | 0 |
| Γ168 = e− K*(892)0 | BaBar | [98] | 829 | 8.00 ± 0.20 | 1.65 ± 0.23 | 2 |
| Γ168 = e− K*(892)0 | Belle | [99] | 1554 | 4.37 ± 0.24 | 0.29 ± 0.14 | 0 |
| Γ169 = µ− K*(892)0 | BaBar | [98] | 829 | 4.60 ± 0.40 | 1.79 ± 0.21 | 4 |
| Γ169 = µ− K*(892)0 | Belle | [99] | 1554 | 3.39 ± 0.19 | 0.53 ± 0.20 | 1 |
| Γ170 = e− K*(892)0 | BaBar | [98] | 829 | 7.80 ± 0.20 | 2.76 ± 0.28 | 2 |
| Γ170 = e− K*(892)0 | Belle | [99] | 1554 | 4.41 ± 0.25 | 0.08 ± 0.08 | 0 |
| Γ171 = µ− K*(892)0 | BaBar | [98] | 829 | 4.10 ± 0.30 | 1.72 ± 0.17 | 1 |
| Γ171 = µ− K*(892)0 | Belle | [99] | 1554 | 3.60 ± 0.20 | 0.45 ± 0.17 | 1 |
| Γ176 = e− φ | BaBar | [98] | 829 | 6.40 ± 0.20 | 0.68 ± 0.12 | 0 |
| Γ176 = e− φ | Belle | [99] | 1554 | 4.18 ± 0.25 | 0.47 ± 0.19 | 0 |
| Γ177 = µ− φ | BaBar | [98] | 829 | 5.20 ± 0.30 | 2.76 ± 0.16 | 6 |
| Γ177 = µ− φ | Belle | [99] | 1554 | 3.21 ± 0.19 | 0.06 ± 0.06 | 1 |
| Γ178 = e− e+ e− | BaBar | [102] | 868 | 8.60 ± 0.20 | 0.12 ± 0.02 | 0 |
| Γ178 = e− e+ e− | Belle | [103] | 1437 | 6.00 ± 0.59 | 0.21 ± 0.15 | 0 |
| Γ179 = e− µ+ µ− | BaBar | [102] | 868 | 6.40 ± 0.40 | 0.54 ± 0.14 | 0 |
| Γ179 = e− µ+ µ− | Belle | [103] | 1437 | 6.10 ± 0.58 | 0.10 ± 0.04 | 0 |
| Γ180 = µ− e+ µ− | BaBar | [102] | 868 | 10.20 ± 0.60 | 0.03 ± 0.02 | 0 |
| Γ180 = µ− e+ µ− | Belle | [103] | 1437 | 10.10 ± 0.77 | 0.02 ± 0.02 | 0 |
| Γ181 = µ− e+ e− | BaBar | [102] | 868 | 8.80 ± 0.50 | 0.64 ± 0.19 | 0 |
| Γ181 = µ− e+ e− | Belle | [103] | 1437 | 9.30 ± 0.73 | 0.04 ± 0.04 | 0 |
| Γ182 = e− µ+ e− | BaBar | [102] | 868 | 12.70 ± 0.70 | 0.34 ± 0.12 | 0 |
| Γ182 = e− µ+ e− | Belle | [103] | 1437 | 11.50 ± 0.89 | 0.01 ± 0.01 | 0 |
| Γ183 = µ− µ+ µ− | BaBar | [102] | 868 | 6.60 ± 0.60 | 0.44 ± 0.17 | 0 |
| Γ183 = µ− µ+ µ− | Belle | [103] | 1437 | 7.60 ± 0.56 | 0.13 ± 0.20 | 0 |
|
| |
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