6 Universality-improved values of B(τ → e ν_e ν_τ) and R_had

Following Ref. [1], we compute the experimental value of B_e = B(τ → e ν_e ν_τ) assuming the Standard Model lepton universality by averaging:

• the B_e fit value B₅,
• the B_e determination from the B_µ= B(τ → µ ν_µν_τ) fit value B₃ assuming that g_µ/g_e = 1 (Eq. 9),
• the B_e determination from the τ lifetime assuming that g_τ/g_µ=1 (Eq. 10) .

Accounting for correlations, we obtain

We compute the sum of all measured branching fractions to hadrons and the experimental value of the ratio between the τ hadronic and electronic widths assuming the Standard Model lepton universality:

An alternative definition of B_had uses the unitarity of the sum of all branching fractions, resulting in:

A third definition of B_had uses the unitarity of the sum of all τ branching fractions, 1 = B_e + B_µ+ B_had, the Standard Model prediction of B_µ from B_e in Eq. 9 and B_e^uni to define:

Although B_had^{uni, lept} and B_had^{uni, SM} are more precise than B_had^uni, the precision of R_had ^{uni, lept} and R_had ^{uni, SM} is comparable to the one of R_had ^uni because there are larger correlations between B_had^{uni, lept}, B_had^{uni, SM} and B_e^uni than between B_had and B_e^uni.