58. τ Branching Fractions

158. τ Branching Fractions
Revised April 2024 by Sw. Banerjee (Louisville U.) and A. Lusiani (SNS, Pisa; INFN, Pisa).

58.1 τ Branching Fractions
The τ Listings contains 252 entries that correspond to either a τ partial decay fraction intoa specific decay mode (branching fraction) or a ratio of two τ partial decay fractions (branchingratio). Experimental measurements provide values for 148 of these quantities, upper limits for 67
branching fractions to Lepton Family number, Lepton number, or Baryon number violating modes,and 37 additional upper limits for other modes. A total of 129 τ branching fractions and branchingratios are determined with a fit of 170 measurements. 85 quantities have at least one measurementin the fit.

58.2 The constrained fit to τ branching fractions
The τ branching fractions fit uses the reported values, uncertanties and statistical correlationsof the τ branching fractions and branching ratios measurements. Asymmetric uncertainties are
2 + σ 2 )/2. If only a few measurements are correlated, the correla2= (σ+symmetrized as σsymm−tion coefficients are listed in the footnote for each measurement (see for example Γ (particle− ≥0 neutrals ≥ 0 K 0 ντ (“1-prong”))/Γtotal ). If a large number of measurements are correlated, thenthe full correlation matrix is listed in the footnote to the measurement that first appears in the τListings. Footnotes to the other measurements refer to the first one. For example, the large correlation matrices for the branching fraction or ratio measurements contained in Refs. [1] [2] are listedin Footnotes to the Γ (e− ν e ντ )/Γtotal and Γ (h− ντ )/Γtotal measurements respectively. Additionally,
the most precise experimental inputs are treated according to how they depend on external parameters on the basis of their documentation [3]. The τ measurements may depend on parameterssuch as the τ pair production cross-section in e+ e− annihilations at the Υ (4S) peak. In some cases,measurements reported in different papers by the same collaboration may depend on common parameters like the estimate of the integrated luminosity or of particle identification efficiencies. For
all the significant detected dependencies, the τ measurements and their uncertainties are updated
to account for the updated values of the external parameters. The dependencies on common systematic effects are also determined in size and sign, and all the common systematic dependencies ofdifferent measurements are used together with the published statistical and systematic uncertainties and correlations in order to compute an updated all-inclusive variance and covariance matrixof the experimental inputs of the fit.
The fit procedure parameters correspond to all measured τ branching fractions and ratios, tosome non-measured branching fractions and ratios like for instance B(τ − → π − KL0 KL0 ντ ) and to
one nuisance variable. When discussing the fit results in the following, the fit χ2 , the number ofdegrees of freedom, the residuals and pulls all refer to the subset of fit parameters that correspondto τ branching fractions and ratios, excluding nuisance variables. The fit parameters are optimized
while respecting relations described by a series of constraint equations. All the experimental inputs
and all the constraint equations are reported in the τ Listings section that follows this review. In
some cases, constraints describe approximate relations that nevertheless hold within the present
experimental precision. For instance, the constraint B(τ − → K − K − K + ντ ) = B(τ − → K − φντ ) ×
B(φ → K + K − ) is justified within the current experimental evidence. The constraint equations
between the τ branching fractions and ratios include quantities other than τ branching fractions
and branching ratios, like for instance the branching fractions of the η and ω mesons. We neglect the−uncertainties on these values for all quantities except B(a−1 → π γ), whose value and uncertainty−2were estimated by ALEPH [1] to be (0.21 ± 0.08)·10 , relying on a measurement of Γ (a−1 →S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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58. τ Branching Fractions

2

number of measurements / 0.16π − γ) [4]. This quantity is included in the fit parameters as nuisance variable, with a χ2 term−− + −
0corresponding to its estimate. We assume that B(τ − → a−1 ντ ) = B(τ → π π π ντ (ex.K , ω)) +−B(τ − → π − 2π 0 ντ (ex.K 0 ))+B(τ − → a1 (π − γ)ντ ), neglecting the observed but negligible branching−− → a− ν ) · B(a− → π − γ). The
fractions to other modes, and that B(τ − → a−
1 (π γ)ντ ) = B(τ1 τ
1values of all other quantities in the constraint equations are taken from the 2023 edition of theReview of Particle Physics.
In the fit, uncertainty scale factors are applied to the published uncertainties of measurements
only if significant inconsistency between different measurements remain after accounting for allrelevant uncertainties and correlations. When performing the fit with no scale factors, the twomeasurements of B(τ − → K − K − K + ντ ) have pulls exceeding 5 σ from the fit values. There are 170pulls, one per measurement. They are partially correlated, and the effective number of independent
pulls is equal to the number of degrees of freedom of the fit, 125. The probability of getting pulls
equal or larger than either one of the two very large pulls in a sample of 125 is smaller that theprobability of a 3 σ deviation for a Normal variable. Therefore, it has been decided to apply an
uncertainty scale factor of 5.4 on all measurements of B(τ − → K − K − K + ντ ) (one by BaBar and oneby Belle). The scale factor has been computed according to the standard PDG procedure. After
applying the scale factor, the pull distribution of the measurements in figure 58.1 is reasonably
Normal and the pull probability distribution in figure 58.2 is reasonably flat.
141210864
20−3.5 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 2.5

pull
Figure 58.1: Pulls of individual measurements against the respective fitted quantity.
Considering only the residuals with respect to τ branching fractions and ratios measurements,
the constrained fit has a χ2 of 135 for 125 degrees of freedom, corresponding to a χ2 probabilityof 26.0%. We use 170 measurements and 84 constraints on the branching fractions and ratios
to determine 129 quantities, consisting of 112 branching fractions and 17 branching ratios. Theconstraints include the unitarity constraint on the sum of all the exclusive τ decay modes, Ball =1. If the unitarity constraint is released, the fit result for Ball is consistent with unitarity with
1 − Ball = (0.07 ± 0.11)%.
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58. τ Branching Fractions

number of measurements / 0.04

3

12
10
86
4
2
0
0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

probability
Figure 58.2: Probability of individual measurement pulls against the respective fitted quantity.
For the convenience of summarizing the fit results, we list in the following the values anduncertainties for a set of 46 “basis” decay modes, from which all remaining branching fractions
and ratios can be obtained using the constraints. The basis decay modes are not intended tosum up to 1. Since some basis quantities represent multiple branching fractions that are related
by constraint equations, the unitarity constraint corresponds to a linear combination of branching
fractions, with the coefficients listed in the following. The coefficients of B(τ − → π − 2π 0 ντ (ex.K 0 ))and B(τ − → π − π + π − ντ (ex.K 0 , ω)) depend on the fitted value of the nuisance variable, B(a−1 →π − γ) = 0.0022 ± 0.0008. The correlation matrix between the basis modes is reported in the τListings.
In defining the fit constraints and in selecting the modes that sum up to one we made someassumptions and choices. We assume that some channels, like τ − → π − K + π − ≥ 0π 0 ντ andτ − → π + K − K − ≥ 0π 0 ντ , have negligible branching fractions as expected from the StandardModel, even if the experimental limits for these branching fractions are not very stringent. The 95%
confidence level upper limits are B(τ − → π − K + π − ≥ 0π 0 ντ ) < 0.25% and B(τ − → π + K − K − ≥0π 0 ντ ) < 0.09%, values not so different from measured branching fractions for allowed 3-prong
modes containing charged kaons. For decays to final states containing one neutral kaon we assume
that the branching fraction with the KL0 are the same as the corresponding one with a KS0 . On
decays with two neutral kaons we assume that the branching fractions with KL0 KL0 are the same asthe ones with KS0 KS0 .

58.3 BaBar and Belle measure on average lower branching fractions and ratios.
We compare the BaBar and Belle measurements with the results of a fit where all the B-factories
measurements have been excluded. We restrict the comparison to the measurements that are usedin the fit, omitting two measurements that are superseeded with other results, and one measurementthat is fully correlated with other measurements. We find that BaBar and Belle measure on averagelower τ branching fractions and ratios than the other experiments. Figures 58.3 and 58.4 show
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4

decay mode

fit result coefficientB(τ − → µ− ν̄µ ντ )0.1739 ± 0.0004
B(τ − → e− ν̄e ντ )0.1782 ± 0.0004−−
B(τ → π ντ )0.1082 ± 0.0005
B(τ − → K − ντ )0.00696 ± 0.00010
B(τ − → π − π 0 ντ )0.2549 ± 0.0009
B(τ − → K − π 0 ντ )0.00433 ± 0.00015
B(τ − → π − 2π 0 ντ (ex.K 0 ))0.0926 ± 0.0010−−00
B(τ → K 2π ντ (ex.K ))(0.65 ± 0.22) · 10−3−−00
B(τ → π 3π ντ (ex.K ))0.0104 ± 0.0007−−00
B(τ → K 3π ντ (ex.K , η))(0.48 ± 0.21) · 10−3−−00
B(τ → h 4π ντ (ex.K , η))
0.0011 ± 0.0004
B(τ − → π − K̄ 0 ντ )0.00838 ± 0.00014
B(τ − → K − K 0 ντ )0.001486 ± 0.000034
B(τ − → π − K̄ 0 π 0 ντ )0.00382 ± 0.00013−−00
B(τ → K K π ντ )0.00150 ± 0.00007
B(τ − → π − K̄ 0 2π 0 ντ (ex.K 0 ))(0.26 ± 0.23) · 10−3B(τ − → π − KS0 KS0 ντ )(235 ± 6) · 10−6
B(τ − → π − KS0 KL0 ντ )0.00108 ± 0.00024
B(τ − → π − KS0 KS0 π 0 ντ )(18.2 ± 2.1) · 10−6
B(τ − → π − KS0 KL0 π 0 ντ )(0.32 ± 0.12) · 10−3−0−−+B(τ → K̄ h h h ντ )(0.25 ± 0.20) · 10−3−−+−0B(τ → π π π ντ (ex.K , ω))
0.0899 ± 0.0005−−+−00B(τ → π π π π ντ (ex.K , ω))0.0274 ± 0.0007
B(τ − → h− h− h+ 2π 0 ντ (ex.K 0 , ω, η))
0.0010 ± 0.0004−−−+B(τ → π K K ντ )0.001435 ± 0.000027
B(τ − → π − K − K + π 0 ντ )(61 ± 18) · 10−6−−0B(τ → π π ηντ )0.00139 ± 0.00007−−
B(τ → K ηντ )(155 ± 8) · 10−6−−0B(τ → K π ηντ )(48 ± 12) · 10−6B(τ − → π − K̄ 0 ηντ )(94 ± 15) · 10−6−−+−0B(τ → π π π ηντ (ex.K ))(220 ± 13) · 10−6−−
B(τ → K ωντ )(0.41 ± 0.09) · 10−3B(τ − → h− π 0 ωντ )0.0041 ± 0.0004
B(τ − → K − φντ )(44 ± 16) · 10−6−−B(τ → π ωντ )0.0195 ± 0.0006−−−+0
B(τ → K π π ντ (ex.K , ω))
0.00293 ± 0.00007
B(τ − → K − π − π + π 0 ντ (ex.K 0 , ω, η))(0.39 ± 0.14) · 10−3B(τ − → π − 2π 0 ωντ )(72 ± 16) · 10−6−−+00
B(τ → 2π π 3π ντ (ex.K , η, ω, f1 ))(14 ± 27) · 10−6−−+0
B(τ → 3π 2π ντ (ex.K , ω, f1 ))(775 ± 30) · 10−6B(τ − → K − 2π − 2π + ντ (ex.K 0 ))(0.6 ± 1.2) · 10−6−−+0
B(τ → 2π π ωντ (ex.K ))(84 ± 6) · 10−6B(τ − → 3π − 2π + π 0 ντ (ex.K 0 , η, ω, f1 ))(38 ± 9) · 10−6−−−+00
B(τ → K 2π 2π π ντ (ex.K ))(1.1 ± 0.6) · 10−6−−−+B(τ → π f1 (2π 2π )ντ )(52 ± 4) · 10−6
B(τ − → π − 2π 0 ηντ (ex.K 0 ))(0.19 ± 0.04) · 10−3
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1.00001.00001.00001.00001.00001.00001.00221.00001.00001.00001.00001.00001.00001.00001.00001.00002.00001.00002.00001.00001.00001.00221.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00000.83001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.000058. τ Branching Fractions

5number of measurements / 0.5

histograms of the 25 pulls of the differences between B-factory measurements and the respective
non-B-factory fit results. The average pull between the two sets of measurements is -0.8σ (-0.7σ
for the 14 Belle measurements and -0.9σ for the 11 BaBar measurements).

4BaBar3210−3−2−10123

standard deviations

number of measurements / 0.5

Figure 58.3: Distribution of the normalized difference between 11 measurements of branching
fractions and ratios published by the BaBar collaboration and the respective averages computed
using only non-B-factory measurements.

4Belle32

10−3−2−10123

standard deviations
Figure 58.4: Distribution of the normalized difference between 14 measurements of branching
fractions and ratios published by the Belle collaboration and the respective averages computed
using only non-B-factory measurements.

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58. τ Branching Fractions

658.4 Overconsistency of Leptonic Branching Fraction Measurements.
As observed in the previous editions of this review, measurements of the leptonic branching
fractions are more consistent with each other than expected from the quoted uncertainties on theindividual measurements. The χ2 /n.d.o.f. are 0.34/4 for Be and 0.08/4 for Bµ . The probability ofgetting a smaller χ2 is 1.3% for Be and 0.08% for Bµ .

58.5 Technical implementation of the fit
The fit computes a set of quantities denoted with qi by minimizing a χ2 while respecting aseries of equality constraints on the qi . The quantities qi represent τ branching fractions andbranching ratios, and nuisance variables. The fit minimization procedure is equivalent to choosinga set of basis fit variables, using functions of these basis variables to predict measurements, anddetermining these basis variables by minimizing the measurements’ χ2 . The χ2 is computed using−1
the measurements mi and their covariance matrix Eij as χ2 = (mi − Aik qk )t Eij(mj − Ajl ql ), wherethe model matrix Aij is used to get the vector of the predicted measurements m0i from the vectorof the fit parameters qj as m0i = Aij qj . There is one fit variable for each of the modes that havemeasurements, therefore Aij = 1 when qj corresponds to the τ branching fraction or branchingratio of the measurement mi , and Aij = 0 otherwise. The constraints are equations involving thefit parameters. The fit does not impose limitations on the functional form of the constraints. In
summary, the fit requires:hiih−1(mj − Ajl ql ) ,min χ2 (qk ) = min (mi − Aik qk )t Eij

subjected tofr (qs ) − cr = 0 ,(58.1)(58.2)

where the left term of Eq. 58.2 defines the constraint expressions. Using the method of Lagrange
multipliers, a set of equations is obtained by taking the derivatives with respect to the fitted
quantities qk and the Lagrange multipliers λr of the sum of the χ2 and the constraint expressions
multiplied by the Lagrange multipliers λr , one for each constraint:
ih−1(Ajl ql −mj ) + 2λr (fr (qs ) − cr ) =min (Aik qk −mi )t Eijhi= min χ̃2 (qk , λr ) ,hi(∂/∂qk , ∂/∂λr ) χ̃2 (qk , λr ) = 0 .

(58.3)

Eq. 58.3 defines a set of equations for the vector of the unknowns (qk , λr ), some of which may be
non-linear, in case of non-linear constraints. An iterative minimization procedure approximates at
each step the non-linear constraint expressions by their first order Taylor expansion around thecurrent values of the fitted quantities, q̄s :fr (qs ) − cr = fr (q̄s ) +∂fr (qs )(qs − q̄s ) − cr ,∂qs q̄swhich can be written asBrs qs − c0r ,where c0r are the resulting constant known terms, independent of qs at first order. After linearization,
the differentiation by qk and λr is trivial and leads to a set of linear equations−1−1tAtki EijAjl ql + Bkr
λr = Atki Eij
mj ,(58.4)

Brs qs = c0r ,(58.5)

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58. τ Branching Fractions

which can be expressed as
Fij uj = vi ,(58.6)

where uj = (qk , λr ) and vi is the vector of the known constant terms running over the index k andthen r in the right terms of Eq. 58.4 and Eq. 58.5, respectively. Solving the equation set in Eq. 58.6
by matrix inversion gives the the fitted quantities and their variance and covariance matrix, using
the measurements and their variance and covariance matrix. The fit procedure starts by computing
the linear approximation of the non-linear constraint expressions around the quantities seed values.
With an iterative procedure, the unknowns are updated at each step by solving the equations
and the equations are then linearized around the updated values, until the variation of the fitted
unknowns is reduced below a numerically small threshold.
References[1] S. Schael et al. (ALEPH), Phys. Rept. 421, 191 (2005), [hep-ex/0506072].
[2] J. Abdallah et al. (DELPHI), Eur. Phys. J. C46, 1 (2006), [hep-ex/0603044].
[3] D. Asner et al. (Heavy Flavor Averaging Group) (2010), [arXiv:1010.1589].
[4] M. Zielinski et al., Phys. Rev. Lett. 52, 1195 (1984).

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