HFLAV-Tau 2023 - Combination of upper limits on τ lepton-flavour-violating branching fractions

The Standard Model prediction for the τ lepton-flavour-violating (LFV) branching fractions are of order10⁻⁵⁴ [119, 120], which are too small to be measured experimentally. Recently, the contributions due to finite neutrino masses for τ⁻ → ℓ_i⁻ ℓ_j⁺ ℓ_a⁻ decays (where a = i or j, and i may or may not be equal to j), were re-scrutinized and also found to be in the range of [10⁻⁵⁶,10⁻⁵⁴] [121, 122]. However, these branching fractions are expected to be of the order of 10⁻¹⁰−10⁻⁹ in several new physics scenarios [123, 124, 125, 126, 127], which is just below the current experimental sensitivity. Several searches have been performed to observe neutrino-less τ decays to final states categorized as ℓ γ, ℓ P⁰, ℓ V⁰, ℓ S⁰, ℓ ℓ ℓ, ℓ h h, including decays that are also baryon-number-violating (BNV). In Table 7 and Figure 5, we report the observed upper limits on these branching fractions published by the B-factories BaBar and Belle and the LHC experiments. Weaker upper limits (mainly from CLEO) have been omitted from Table 7, but are shown in Figure 5, to document the historical progress. For channels where multiple results of similar strengths are available, we also report the combined upper limits, following the prescription described below. No preliminary results are included in this report.

We combine the reported upper limits by first averaging the central values of the corresponding branching-fraction measurements to obtain the average branching fraction B and its uncertainty δ B,

where B_i is the central value for the branching fraction of measurement i and w_i = 1 / δ B_i² is the inverse of square of its uncertainty. We then compute the 90% confidence level (CL) upper limit as B⁹⁰ = B + 1.645 δ B, assuming a Gaussian uncertainty. The majority of published limits do not have reported values for B_i and w_i. In such cases, we take the central value to be B_i = (N_obsⁱ−N_bkgⁱ)/(N_τⁱεⁱ), where N_obsⁱ is the number of observed events, N_bkgⁱ is the number of expected background events, N_τⁱ is the number of τ decays in the data sample, and εⁱ is the signal reconstruction efficiency in the signal region. Additional information for central value of B(τ⁻ → µ⁻ µ⁺ µ⁻) are obtained from the CMS [3] and LHCb [128] collaborations, respectively, and are listed in Table 7 for future reference.

The uncertainty δ B_i is computed assuming that the reported 90% CL limit corresponds to B_i⁹⁰ = B_i + 1.645 δ B_i. Using this combination procedure, the resulting upper limit may be weaker than the most stringent input limit, but always corresponds to a more precise branching fraction, obtained by averaging multiple measurements. Since there is negligible gain in combining limits of very different strength, the combinations do not include the limits that are more than 10 times larger than the strictest limit in the respective channel. This excludes limits of the CLEO and ATLAS searches and the LHCb searches for τ decays into proton and two muons. We report in Table 7 and Figure 5 the observed upper limits on the τ lepton-flavour-violating decays and their combinations.

Table 7: 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 combinations of upper limits are reported in the rows with label “Comb.” in the “Experiment / Reference” column. The measurements used to compute the combinations are tagged with an asterisk attached to their reference. Where available, additional information is reported about the number of τ decays used in the search, the efficiency of the selection, the expected background and the number of observed events, or the central value of branching fractions.

Decay mode
90% CL

Limit

Exp./Ref.
N_τ

(10⁶)

ε

(%)

N_bkg N_obs

ℓγ category

B(τ⁻ → e⁻ γ) 3.3 · 10⁻⁸ BaBar [129]* 963 3.90 ± 0.30 1.60 ± 0.40 0

5.6 · 10⁻⁸ Belle [130]* 1824 2.90 ± 0.19 5.50 ± 4.65 5

2.6 · 10⁻⁸ Comb.

B(τ⁻ → µ⁻ γ) 4.4 · 10⁻⁸ BaBar [129]* 963 6.10 ± 0.50 3.60 ± 0.70 2

4.2 · 10⁻⁸ Belle [130]* 1824 3.70 ± 0.21 5.30 ± 2.75 5

2.7 · 10⁻⁸ Comb.

ℓ P⁰ category

B(τ⁻ → e⁻ π⁰) 1.3 · 10⁻⁷ BaBar [131]* 623 2.80 ± 0.25 0.17 ± 0.04 0

8.0 · 10⁻⁸ Belle [132]* 737 3.88 ± 0.17 0.20 ± 0.20 0

6.4 · 10⁻⁸ Comb.

B(τ⁻ → µ⁻ π⁰) 1.1 · 10⁻⁷ BaBar [131]* 623 4.69 ± 0.37 1.33 ± 0.15 1

1.2 · 10⁻⁷ Belle [132]* 737 4.48 ± 0.20 0.58 ± 0.34 1

8.1 · 10⁻⁸ Comb.

B(τ⁻ → e⁻ K_S⁰) 3.3 · 10⁻⁸ BaBar [133]* 862 9.40 ± 0.20 1.00 ± 0.40 1

2.6 · 10⁻⁸ Belle [134]* 1234 7.06 ± 0.47 0.18 ± 0.18 0

2.0 · 10⁻⁸ Comb.

B(τ⁻ → µ⁻ K_S⁰) 4.0 · 10⁻⁸ BaBar [133]* 862 7.00 ± 0.39 5.30 ± 2.20 2

2.3 · 10⁻⁸ Belle [134]* 1234 7.40 ± 0.50 0.35 ± 0.21 0

1.8 · 10⁻⁸ Comb.

B(τ⁻ → e⁻ η) 1.6 · 10⁻⁷ BaBar [131]* 623 2.12 ± 0.20 0.22 ± 0.05 0

9.2 · 10⁻⁸ Belle [132]* 737 2.87 ± 0.14 0.78 ± 0.59 0

7.4 · 10⁻⁸ Comb.

B(τ⁻ → µ⁻ η) 1.5 · 10⁻⁷ BaBar [131]* 623 3.59 ± 0.41 0.75 ± 0.08 1

6.5 · 10⁻⁸ Belle [132]* 737 4.08 ± 0.20 0.64 ± 0.38 0

6.1 · 10⁻⁸ Comb.

B(τ⁻ → e⁻ η^′(958)) 2.4 · 10⁻⁷ BaBar [131]* 623 1.53 ± 0.16 0.12 ± 0.03 0

1.6 · 10⁻⁷ Belle [132]* 737 2.15 ± 0.11 0.57 ± 0.41 0

1.3 · 10⁻⁷ Comb.

B(τ⁻ → µ⁻ η^′(958)) 1.4 · 10⁻⁷ BaBar [131]* 623 2.18 ± 0.26 0.49 ± 0.04 0

1.3 · 10⁻⁷ Belle [132]* 737 2.45 ± 0.13 0.46 ± 0.33 0

8.6 · 10⁻⁸ Comb.

ℓ S⁰ category

B(τ⁻ → e⁻ f₀(980)) 6.8 · 10⁻⁸ Belle [135] 1233 2.90 ± 0.33 0.10 ± 0.07 0

B(τ⁻ → µ⁻ f₀(980)) 6.4 · 10⁻⁸ Belle [135] 1233 3.01 ± 0.33 0.11 ± 0.08 0

ℓ V⁰ category

B(τ⁻ → e⁻ ρ⁰) 4.6 · 10⁻⁸ BaBar [136]* 829 7.31 ± 0.18 1.32 ± 0.17 1

2.2 · 10⁻⁸ Belle [137]* 1796 8.49 ± 0.37 0.80 ± 0.27 1

2.0 · 10⁻⁸ Comb.

B(τ⁻ → µ⁻ ρ⁰) 2.6 · 10⁻⁸ BaBar [136]* 829 4.52 ± 0.41 2.04 ± 0.19 0

1.7 · 10⁻⁸ Belle [137]* 1796 7.78 ± 0.36 0.95 ± 0.25 0

1.3 · 10⁻⁸ Comb.

B(τ⁻ → e⁻ ω) 1.1 · 10⁻⁷ BaBar [138]* 706 2.61 ± 0.11 0.35 ± 0.06 0

2.4 · 10⁻⁸ Belle [137]* 1796 5.41 ± 0.24 0.74 ± 0.43 0

2.2 · 10⁻⁸ Comb.

B(τ⁻ → µ⁻ ω) 1.0 · 10⁻⁷ BaBar [138]* 706 2.26 ± 0.14 0.73 ± 0.03 0

3.9 · 10⁻⁸ Belle [137]* 1796 3.27 ± 0.16 0.32 ± 0.30 0

3.4 · 10⁻⁸ Comb.

B(τ⁻ → e⁻ K^*(892)) 5.9 · 10⁻⁸ BaBar [136]* 829 8.00 ± 0.19 1.65 ± 0.23 2

1.9 · 10⁻⁸ Belle [137]* 1796 6.94 ± 0.28 0.54 ± 0.26 0

1.9 · 10⁻⁸ Comb.

B(τ⁻ → µ⁻ K^*(892)) 1.7 · 10⁻⁷ BaBar [136]* 829 4.57 ± 0.36 1.79 ± 0.21 4

2.9 · 10⁻⁸ Belle [137]* 1796 4.52 ± 0.19 0.84 ± 0.40 0

3.3 · 10⁻⁸ Comb.

B(τ⁻ → e⁻ K^*(892)) 4.6 · 10⁻⁸ BaBar [136]* 829 7.76 ± 0.18 2.76 ± 0.28 2

1.7 · 10⁻⁸ Belle [137]* 1796 7.45 ± 0.31 0.25 ± 0.11 0

1.5 · 10⁻⁸ Comb.

B(τ⁻ → µ⁻ K^*(892)) 7.3 · 10⁻⁸ BaBar [136]* 829 4.11 ± 0.32 1.72 ± 0.17 1

4.3 · 10⁻⁸ Belle [137]* 1796 4.58 ± 0.20 0.58 ± 0.21 1

3.7 · 10⁻⁸ Comb.

B(τ⁻ → e⁻ φ) 3.1 · 10⁻⁸ BaBar [136]* 829 6.43 ± 0.16 0.68 ± 0.12 0

2.0 · 10⁻⁸ Belle [137]* 1796 6.45 ± 0.29 0.38 ± 0.21 0

1.6 · 10⁻⁸ Comb.

B(τ⁻ → µ⁻ φ) 1.9 · 10⁻⁷ BaBar [136]* 829 5.18 ± 0.27 2.76 ± 0.16 6

2.3 · 10⁻⁸ Belle [137]* 1796 5.59 ± 0.27 0.47 ± 0.16 0

2.6 · 10⁻⁸ Comb.

ℓℓℓ category

B(τ⁻ → e⁻ e⁺ e⁻) 2.9 · 10⁻⁸ BaBar [139]* 860 8.60 ± 0.20 0.12 ± 0.02 0

2.7 · 10⁻⁸ Belle [140]* 1437 6.00 ± 0.59 0.21 ± 0.15 0

1.9 · 10⁻⁸ Comb.

B(τ⁻ → e⁻ µ⁺ µ⁻) 3.2 · 10⁻⁸ BaBar [139]* 860 6.40 ± 0.40 0.54 ± 0.14 0

2.7 · 10⁻⁸ Belle [140]* 1437 6.10 ± 0.58 0.10 ± 0.04 0

2.0 · 10⁻⁸ Comb.

B(τ⁻ → µ⁻ e⁺ µ⁻) 2.6 · 10⁻⁸ BaBar [139]* 860 10.20 ± 0.60 0.03 ± 0.02 0

1.7 · 10⁻⁸ Belle [140]* 1437 10.10 ± 0.77 0.02 ± 0.02 0

1.4 · 10⁻⁸ Comb.

B(τ⁻ → µ⁻ e⁺ e⁻) 2.2 · 10⁻⁸ BaBar [139]* 860 8.80 ± 0.50 0.64 ± 0.19 0

1.8 · 10⁻⁸ Belle [140]* 1437 9.30 ± 0.73 0.04 ± 0.04 0

1.3 · 10⁻⁸ Comb.

B(τ⁻ → e⁻ µ⁺ e⁻) 1.8 · 10⁻⁸ BaBar [139]* 860 12.70 ± 0.70 0.34 ± 0.12 0

1.5 · 10⁻⁸ Belle [140]* 1437 11.50 ± 0.89 0.01 ± 0.01 0

1.1 · 10⁻⁸ Comb.

B(τ⁻ → µ⁻ µ⁺ µ⁻) 3.8 · 10⁻⁷ ATLAS [141]

3.3 · 10⁻⁸ BaBar [139]* 860 6.60 ± 0.60 0.44 ± 0.17 0

2.1 · 10⁻⁸ Belle [140]* 1437 7.60 ± 0.56 0.13 ± 0.20 0

1.9 · 10⁻⁸ Belle II [4]* 777 20.42 ± 0.93 0.70 ± 0.56 1

2.9 · 10⁻⁸ CMS [3]* B = 5.1 · 10⁻⁹

4.6 · 10⁻⁸ LHCb [128]* B = 2.3 · 10⁻⁸

1.6 · 10⁻⁸ Comb.

ℓ hh category

B(τ⁻ → e⁻ π⁺ π⁻) 1.2 · 10⁻⁷ BaBar [142]* 407 3.30 ± 0.15 0.81 ± 0.13 0

2.3 · 10⁻⁸ Belle [143]* 1564 5.45 ± 0.31 0.55 ± 0.23 0

2.1 · 10⁻⁸ Comb.

B(τ⁻ → e⁺ π⁻ π⁻) 2.7 · 10⁻⁷ BaBar [142] 407 3.40 ± 0.15 0.41 ± 0.10 1

2.0 · 10⁻⁸ Belle [143] 1564 6.56 ± 0.36 0.37 ± 0.19 0

B(τ⁻ → µ⁻ π⁺ π⁻) 2.9 · 10⁻⁷ BaBar [142] 407 3.40 ± 0.19 2.99 ± 0.41 3

2.1 · 10⁻⁸ Belle [143] 1564 5.83 ± 0.33 0.63 ± 0.23 0

B(τ⁻ → µ⁺ π⁻ π⁻) 7.0 · 10⁻⁸ BaBar [142]* 407 3.30 ± 0.18 1.46 ± 0.27 0

3.9 · 10⁻⁸ Belle [143]* 1564 6.55 ± 0.37 0.33 ± 0.16 1

3.5 · 10⁻⁸ Comb.

B(τ⁻ → e⁻ π⁺ K⁻) 3.2 · 10⁻⁷ BaBar [142]* 407 3.10 ± 0.13 0.14 ± 0.06 1

3.7 · 10⁻⁸ Belle [143]* 1564 3.97 ± 0.25 0.18 ± 0.13 0

3.8 · 10⁻⁸ Comb.

B(τ⁻ → e⁻ K⁺ π⁻) 1.7 · 10⁻⁷ BaBar [142]* 407 3.08 ± 0.13 0.32 ± 0.08 0

3.1 · 10⁻⁸ Belle [143]* 1564 4.07 ± 0.25 0.55 ± 0.31 0

2.9 · 10⁻⁸ Comb.

B(τ⁻ → e⁺ π⁻ K⁻) 1.8 · 10⁻⁷ BaBar [142]* 407 3.19 ± 0.14 0.16 ± 0.06 0

3.2 · 10⁻⁸ Belle [143]* 1564 4.00 ± 0.25 0.46 ± 0.21 0

3.1 · 10⁻⁸ Comb.

B(τ⁻ → e⁻ K_S⁰ K_S⁰) 7.1 · 10⁻⁸ Belle [134] 1234 2.79 ± 0.31 0.07 ± 0.07 0

B(τ⁻ → e⁻ K⁺ K⁻) 1.4 · 10⁻⁷ BaBar [142]* 407 3.77 ± 0.16 0.22 ± 0.06 0

3.4 · 10⁻⁸ Belle [143]* 1564 4.29 ± 0.29 0.17 ± 0.10 0

3.3 · 10⁻⁸ Comb.

B(τ⁻ → e⁺ K⁻ K⁻) 1.5 · 10⁻⁷ BaBar [142]* 407 3.85 ± 0.16 0.04 ± 0.04 0

3.3 · 10⁻⁸ Belle [143]* 1564 4.64 ± 0.30 0.06 ± 0.06 0

3.2 · 10⁻⁸ Comb.

B(τ⁻ → µ⁻ π⁺ K⁻) 2.6 · 10⁻⁷ BaBar [142]* 407 2.87 ± 0.16 1.04 ± 0.18 1

8.6 · 10⁻⁸ Belle [143]* 1564 2.72 ± 0.17 0.72 ± 0.28 1

8.2 · 10⁻⁸ Comb.

B(τ⁻ → µ⁻ K⁺ π⁻) 3.2 · 10⁻⁷ BaBar [142]* 407 2.97 ± 0.16 1.67 ± 0.29 2

4.5 · 10⁻⁸ Belle [143]* 1564 2.62 ± 0.15 0.64 ± 0.23 0

4.6 · 10⁻⁸ Comb.

B(τ⁻ → µ⁺ π⁻ K⁻) 2.2 · 10⁻⁷ BaBar [142]* 407 2.85 ± 0.16 1.54 ± 0.25 1

4.8 · 10⁻⁸ Belle [143]* 1564 2.55 ± 0.16 0.56 ± 0.21 0

4.5 · 10⁻⁸ Comb.

B(τ⁻ → µ⁻ K_S⁰ K_S⁰) 8.0 · 10⁻⁸ Belle [134] 1234 2.43 ± 0.27 0.12 ± 0.08 0

B(τ⁻ → µ⁻ K⁺ K⁻) 2.5 · 10⁻⁷ BaBar [142]* 407 2.16 ± 0.12 0.24 ± 0.07 0

4.4 · 10⁻⁸ Belle [143]* 1564 2.85 ± 0.17 0.51 ± 0.19 0

4.2 · 10⁻⁸ Comb.

B(τ⁻ → µ⁺ K⁻ K⁻) 4.8 · 10⁻⁷ BaBar [142] 407 2.06 ± 0.11 0.07 ± 0.10 1

4.7 · 10⁻⁸ Belle [143] 1564 2.98 ± 0.18 0.25 ± 0.13 0

BNV category

B(τ⁻ → π⁻ Λ) 7.2 · 10⁻⁸ Belle [144]* 283 7.48 ± 0.68 1.70 ± 0.80 0

4.7 · 10⁻⁸ Belle II [5]* 669 9.50 ± 0.26 1.00 ± 1.20 0

3.0 · 10⁻⁸ Comb.

B(τ⁻ → π⁻ Λ) 1.4 · 10⁻⁷ Belle [144]* 283 7.54 ± 0.69 1.70 ± 0.80 1

4.3 · 10⁻⁸ Belle II [5]* 669 9.90 ± 0.28 0.50 ± 0.60 0

3.8 · 10⁻⁸ Comb.

B(τ⁻ → p e⁺ e⁻) 3.0 · 10⁻⁸ Belle [145] 1682 7.80 ± 0.39 0.50 ± 0.35 1

B(τ⁻ → p e⁻ e⁻) 3.0 · 10⁻⁸ Belle [145] 1682 8.00 ± 0.40 0.23 ± 0.07 1

B(τ⁻ → p e⁺ µ⁻) 2.0 · 10⁻⁸ Belle [145] 1682 6.50 ± 0.33 0.22 ± 0.06 0

B(τ⁻ → p e⁻ µ⁺) 1.8 · 10⁻⁸ Belle [145] 1682 6.90 ± 0.34 0.40 ± 0.28 0

B(τ⁻ → p µ⁻ µ⁻) 4.0 · 10⁻⁸ Belle [145] 1682 4.60 ± 0.23 1.30 ± 0.46 1

4.4 · 10⁻⁷ LHCb [146]

B(τ⁻ → p µ⁺ µ⁻) 1.8 · 10⁻⁸ Belle [145] 1682 5.00 ± 0.25 1.14 ± 0.43 0

3.3 · 10⁻⁷ LHCb [146]

HFLAV-Tau 2023 Report

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

HFLAV-Tau 2023 Report