3  Tests of lepton universality

Lepton universality tests probe the Standard Model prediction that the charged weak current interaction has the same coupling for all lepton generations. The precision of such tests has been significantly improved since the 2014 edition by the addition of the Belle τ lifetime measurement [66], while improvements from the τ branching fraction fit are negligible. We compute the universality tests by using ratios of the partial widths of a heavier lepton λ decaying to a lighter lepton ρ [67],

where

We use R_γ^τ=1−43.2· 10⁻⁴ and R_γ^µ=1−42.4· 10⁻⁴ [67] and M_W from PDG 2018 [6]. We use HFLAV 2018 averages and PDG 2018 for the other quantities. Using pure leptonic processes we obtain

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

where h = π or K and the radiative corrections are δ R_τ/π = (0.16 ± 0.14)% and δ R_τ/K = (0.90 ± 0.22)% [71, 72, 73, 74]. We measure:

Similar tests could be performed with decays to electrons, however they are less precise because the hadron two body decays to electrons are helicity-suppressed. Averaging the three g_τ/g_µ ratios we obtain

accounting for correlations. Table 12 reports the correlation coefficients for the fitted coupling ratios.

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

( gτ/ge )	51
( gµ/ge )	-50	49
( gτ/gµ )π	23	25	2
( gτ/gµ )K	11	10	-1	6
	( gτ/gµ )	( gτ/ge )	( gµ/ge )	( gτ/gµ )π

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Since there is 100% correlation between g_τ/g_µ, g_τ/g_e and g_µ/g_e, the correlation matrix is expected to be positive semi-definite, with one eigenvalue equal to zero. Due to numerical inaccuracies, one eigenvalue is expected to be close to zero rather than exactly zero.