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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) = | ∫ | | | | dQ,
|
| | | | | | | | | (1) |
| CLb = Pb(Q ≤ Qobs) = | ∫ | | | | 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:
When multiple results are combined, the PDFs in
Equations 1 and 2 are the
product of the individual PDFs,
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 mode | Cat. | | Exp. | | | | Nbkg | Nobs |
|
|
| lγ | < | 5.4 · 10−8 | HFAG |
| | | | Belle | 535 | 0.919 | 3.00 ± 0.10 | 5.14 ± 3.30 | 5 |
| | | | BaBar | 524 | 0.919 | 3.90 ± 0.30 | 1.60 ± 0.40 | 0 |
| | < | 5.0 · 10−8 | HFAG |
| | | | Belle | 535 | 0.919 | 5.07 ± 0.20 | 13.90 ± 5.00 | 10 |
| | | | BaBar | 524 | 0.919 | 6.10 ± 0.50 | 3.60 ± 0.70 | 2 |
|
| lP0 | < | 1.4 · 10−8 | HFAG |
| | | | Belle | 671 | 0.919 | 10.20 ± 0.67 | 0.18 ± 0.18 | 0 |
| | | | BaBar | 469 | 0.919 | 9.10 ± 1.73 | 0.59 ± 0.25 | 1 |
| | < | 1.5 · 10−8 | HFAG |
| | | | Belle | 671 | 0.919 | 10.70 ± 0.73 | 0.35 ± 0.21 | 0 |
| | | | BaBar | 469 | 0.919 | 6.14 ± 0.20 | 0.30 ± 0.18 | 1 |
|
| l V0 | < | 1.5 · 10−8 | HFAG |
| | | | Belle | 854 | 0.919 | 7.58 ± 0.41 | 0.29 ± 0.15 | 0 |
| | | | BaBar | 451 | 0.919 | 7.31 ± 0.20 | 1.32 ± 0.17 | 1 |
| | < | 1.5 · 10−8 | HFAG |
| | | | Belle | 854 | 0.919 | 7.09 ± 0.37 | 1.48 ± 0.35 | 0 |
| | | | BaBar | 451 | 0.919 | 4.52 ± 0.40 | 2.04 ± 0.19 | 0 |
| | < | 2.3 · 10−8 | HFAG |
| | | | Belle | 854 | 0.919 | 4.37 ± 0.24 | 0.29 ± 0.14 | 0 |
| | | | BaBar | 451 | 0.919 | 8.00 ± 0.20 | 1.65 ± 0.23 | 2 |
| | < | 6.0 · 10−8 | HFAG |
| | | | Belle | 854 | 0.919 | 3.39 ± 0.19 | 0.53 ± 0.20 | 1 |
| | | | BaBar | 451 | 0.919 | 4.60 ± 0.40 | 1.79 ± 0.21 | 4 |
| | < | 2.2 · 10−8 | HFAG |
| | | | Belle | 854 | 0.919 | 4.41 ± 0.25 | 0.08 ± 0.08 | 0 |
| | | | BaBar | 451 | 0.919 | 7.80 ± 0.20 | 2.76 ± 0.28 | 2 |
| | < | 4.2 · 10−8 | HFAG |
| | | | Belle | 854 | 0.919 | 3.60 ± 0.20 | 0.45 ± 0.17 | 1 |
| | | | BaBar | 451 | 0.919 | 4.10 ± 0.30 | 1.72 ± 0.17 | 1 |
| | < | 2.0 · 10−8 | HFAG |
| | | | Belle | 854 | 0.919 | 4.18 ± 0.25 | 0.47 ± 0.19 | 0 |
| | | | BaBar | 451 | 0.919 | 6.40 ± 0.20 | 0.68 ± 0.12 | 0 |
| | < | 6.8 · 10−8 | HFAG |
| | | | Belle | 854 | 0.919 | 3.21 ± 0.19 | 0.06 ± 0.06 | 1 |
| | | | BaBar | 451 | 0.919 | 5.20 ± 0.30 | 2.76 ± 0.16 | 6 |
| | < | 3.3 · 10−8 | HFAG |
| | | | Belle | 854 | 0.919 | 2.92 ± 0.18 | 0.30 ± 0.14 | 0 |
| | | | BaBar | 451 | 0.919 | 2.96 ± 0.13 | 0.35 ± 0.06 | 0 |
| | < | 4.0 · 10−8 | HFAG |
| | | | Belle | 854 | 0.919 | 2.38 ± 0.14 | 0.72 ± 0.18 | 0 |
| | | | BaBar | 451 | 0.919 | 2.56 ± 0.16 | 0.73 ± 0.03 | 0 |
|
| lll | < | 1.4 · 10−8 | HFAG |
| | | | Belle | 782 | 0.919 | 6.00 ± 0.59 | 0.21 ± 0.15 | 0 |
| | | | BaBar | 472 | 0.919 | 8.60 ± 0.20 | 0.12 ± 0.02 | 0 |
| | < | 1.1 · 10−8 | HFAG |
| | | | Belle | 782 | 0.919 | 9.30 ± 0.73 | 0.04 ± 0.04 | 0 |
| | | | BaBar | 472 | 0.919 | 8.80 ± 0.50 | 0.64 ± 0.19 | 0 |
| | < | 1.6 · 10−8 | HFAG |
| | | | Belle | 782 | 0.919 | 6.10 ± 0.58 | 0.10 ± 0.04 | 0 |
| | | | BaBar | 472 | 0.919 | 6.40 ± 0.40 | 0.54 ± 0.14 | 0 |
| | < | 1.2 · 10−8 | HFAG |
| | | | Belle | 782 | 0.919 | 7.60 ± 0.56 | 0.13 ± 0.20 | 0 |
| | | | BaBar | 472 | 0.919 | 6.60 ± 0.60 | 0.44 ± 0.17 | 0 |
| | < | 4.6 · 10−8 | LHCb |
| | < | 8.4 · 10−9 | HFAG |
| | | | Belle | 782 | 0.919 | 11.50 ± 0.89 | 0.01 ± 0.01 | 0 |
| | | | BaBar | 472 | 0.919 | 12.70 ± 0.70 | 0.34 ± 0.12 | 0 |
| | < | 9.8 · 10−9 | HFAG |
| | | | Belle | 782 | 0.919 | 10.10 ± 0.77 | 0.02 ± 0.02 | 0 |
| | | | BaBar | 472 | 0.919 | 10.20 ± 0.60 | 0.03 ± 0.02 | 0 |
|
| BNV | < | 1.9 · 10−8 | HFAG |
| | | | Belle | 906 | 0.919 | 4.39 ± 0.36 | 0.31 ± 0.18 | 0 |
| | | | BaBar | 237 | 0.919 | 12.20 ± 8.50 | 0.56 ± 0.56 | 0 |
| | < | 1.8 · 10−9 | HFAG |
| | | | Belle | 906 | 0.919 | 4.80 ± 0.39 | 0.21 ± 0.15 | 0 |
| | | | BaBar | 237 | 0.919 | 12.28 ± 8.50 | 0.42 ± 0.42 | 0 |
| | < | 3.7 · 10−9 | HFAG |
| | | | Belle | 906 | 0.919 | 3.16 ± 0.27 | 0.42 ± 0.19 | 0 |
| | | | BaBar | 237 | 0.919 | 9.47 ± 0.66 | 0.12 ± 0.12 | 1 |
| | < | 2.0 · 10−9 | HFAG |
| | | | Belle | 906 | 0.919 | 4.11 ± 0.35 | 0.31 ± 0.14 | 0 |
| | | | BaBar | 237 | 0.919 | 10.63 ± 0.74 | 0.26 ± 0.26 | 0 |
|
|
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