3 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. Starting with our 2014 report ‍[3], the precision of such tests was significantly improved due to use of the Belle τ lifetime measurement ‍[74], while improvements from the τ branching fraction fit are negligible. We perform the universality tests by using ratios of the partial widths of a heavier lepton α decaying to a lighter lepton β ‍[75],

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 ought to be 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 and R_γ^µ=1−42.4· 10⁻⁴ ‍[75] and M_W from PDG 2021 ‍[7]. We use HFLAV 2021 averages and PDG 2021 for the other quantities. Using pure leptonic processes we obtain the coupling ratios

Using the expressions for the τ hadronic partial widths, we obtain

where h = π or K. The radiative corrections δ R_τ/π and δ R_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)% ‍[10], respectively. We obtain:

The largest contributions to the uncertainties of the tests are the uncertainty on δ R_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 because 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 R_τ/π and δ R_τ/K uncertainties are uncorrelated as they are estimated to be with good approximation ‍[10]. Table ‍12 reports the correlation coefficients for the fitted coupling ratios.

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Table 12: Universality coupling ratios correlation coefficients (%).

( gτ/ge )τ	51
( gµ/ge )τ	-50	49
( gτ/gµ )π	16	18	1
( gτ/gµ )K	12	11	-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. Indeed, in the reported correlation matrix there is one eigenvalue that is consistent with zero within the numerical accuracy of the reported figures.