5 Tests of lepton universality

Lepton universality tests probe the Standard-Model prediction that the weak charged-current interaction has the same coupling for all lepton generations. Beginning with our 2014 report [86], the precision of such tests was significantly improved thanks to the Belle τ lifetime measurement [19]. The most significant improvement of the τ branching fraction fit results comes from the recent Belle II measurement [2]. We perform the universality tests by using ratios of the partial widths of a heavier lepton L decaying to a lighter lepton ℓ [87],

where

The equation holds at leading perturbative order (with some corrections being computed at next-to-leading order) for branching fractions to final states that include a soft photon, as detailed in the notation. The inclusion of soft photons is not explicitly mentioned in the branching fractions notation used in this chapter, but is implicitly assumed, since experimental measurements do include soft photons. For most measurements of τ branching fractions, soft photons are not experimentally reconstructed but accounted for in the simulations used to estimate the experimental efficiency. We use R_γ^τ=1−43.2· 10⁻⁴, R_γ^µ=1−42.4· 10⁻⁴ [87] and take the averages for the τ mass, lifetime and branching fractions from this report. We use the CODATA 2018 report [91] for the electron and muon masses, and PDG 2022 and 2023 update [8] for the muon lifetime, the π^± and K^± meson masses and lifetimes, and the W boson mass.

Using pure leptonic processes we obtain the coupling ratios

Under the assumption that the muon and electron charged weak couplings are equal, we average B_e = B(τ → e ν_e ν_τ) and its prediction from B_µ= B(τ → µν_µν_τ),

to obtain B_e^′, a measurement of the τ electronic branching fraction that summarizes the experimental information on the electron and the muon couplings from the measurements of B_e and B_µ. We test the universality of the couplings of the lighter leptons with respect to the τ by comparing B_e^′ to the predicted τ electronic branching fraction from the measurement of the τ lifetime:

In Figure 3, the plot shows the measured B_e^′, the measured τ lifetime, and a band corresponding to B_e^τ_τ as function of the τ lifetime, whose width depends primarily on the uncertainty on the τ mass, which has been reduced thanks to the recent Belle measurement [6].

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Figure 3: Test of the universality of the couplings of the lighter leptons e and µ, assumed to be the same, and of the τ lepton. The plot shows values and uncertainties of the τ lifetime and of Be′, a determination of the electronic τ branching fractions obtained from the direct measurements of the electronic and the muonic τ branching fractions. The contour delimits the 68% CL region. The oblique band shows the Standard Model prediction of Be′ as a function of the τ lifetime, and its width is mainly determined by the uncertainty on the τ mass.

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An additional universality test is obtained using the measurements of semileptonic branching fractions of pseudoscalar mesons, using [88]:

where h = π or K. The radiative corrections δ_τ/π and δ_τ/K have been recently updated with an improved estimation of their uncertainties and their values are (0.18 ± 0.57)% and (0.97 ± 0.58)% [92], respectively. Using B(π → µ ν_µ) and B(K → µ ν_µ) from the Review of Particle Physics 2022 edition and 2023 update [8], we obtain:

The largest contribution to the uncertainties of these tests are the uncertainty on δ_τ/π for (g_τ/g_µ)_π and the uncertainty on the τ branching fraction for (g_τ/g_µ)_K. Similar tests can be performed using measurements of decay modes with electrons, but are less precise, since the meson decays to electrons are helicity suppressed and have less precise experimental measurements. Averaging the three g_τ/g_µ ratios we obtain

accounting for correlations and assuming that the δ_τ/π and δ_τ/K uncertainties are uncorrelated, as they are estimated to be with good approximation [92]. Table 5 reports the correlation coefficients for the fitted coupling ratios.

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Table 5: Correlation coefficients (%) of the coupling ratios.

( gτ/ge )τ	67
( gµ/ge )τ	-41	40
( gτ/gµ )π	18	19	1
( gτ/gµ )K	12	12	-1	7
	( gτ/gµ )τ	( gτ/ge )τ	( gµ/ge )τ	( gτ/gµ )π

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Since (g_τ/g_µ)_τ= (g_τ/g_e)_τ/ (g_µ/g_e)_τ, the correlation matrix is expected to be positive semi-definite, with one eigenvalue equal to zero. A numerical calculation of the eigenvalues returns the values 1.804, 1.329, 0.959, 0.907, 2.887·10⁻¹⁵.