5  Universality improved B (τ → e ν ν) and R_had

We compute two quantities that are used in this report and have been traditionally used for further elaborations and tests involving the τ branching fractions: the “universality improved” experimental determination of B _e = B (τ → e ν ν), which relies on assuming that the Standard Model and lepton universality hold, and the ratio R_had between the total branching fraction of the τ to hadrons and the universality improved B _e, which is the same as the ratio of the two respective partial widths.

Following Ref. [76], we obtain a more precise experimental determination of B _e using the τ branching fraction to muon and the τ lifetime. We average:

• the B _e fit value Γ₅,
• the B _e determination from the B _µ= B (τ → µ ν ν) fit value Γ₃ assuming that g_µ/g_e = 1 hence (see also Section 4) B _e = B _µ· f(m_e²/m_τ²)/f(m_µ²/m_τ²),
• the B _e determination from the τ lifetime assuming that g_τ/g_µ=1 hence B _e = B (µ → e ν_e ν_µ)· (τ_τ/ τ_µ) · (m_τ/m_µ)⁵ · f(m_e²/m_τ²)/f(m_e²/m_µ²) · (δ_γ^τδ_W^τ)/(δ_γ^µδ_W^µ) where B (µ → e ν_e ν_µ) = 1.

Accounting for statistical correlations, we obtain

The recent Belle τ lifetime measurement has brought a significant improvement. We use B _e^uni to obtain the ratio

where Γ(τ → hadrons) and Γ(τ→ eνν) indicate the partial widths and Γ_hadrons is the total branching fraction of the τ to hadrons, or the total branching fraction in any measured final state minus the leptonic branching fractions, i.e. with our notation Γ_hadrons = Γ_All − Γ₃ − Γ₅ = (64.69 ± 0.10)% (see Section 2 and Table 1 for the definitions of Γ_All, Γ₃, Γ₅). We underline that this report’s definition of Γ_hadrons corresponds to summing all τ hadronic decay modes, like in the previous report, rather than – as done elsewhere – subtracting the leptonic branching fractions from unity, i.e. Γ_hadrons = 1 − Γ₃ − Γ₅.