4  Universality-improved B(τ → e ν ν) and R_had

We compute two quantities that are used in this report and that 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 the assumption that the Standard Model and lepton universality hold;
• the ratio R_had between the total branching fraction of the τ to hadrons, B_had and the universality-improved B_e, which is the same as the ratio of the two respective partial widths, Γ(τ → had) and Γ(τ→ eνν).

Following Ref. [1], we obtain a more precise experimental determination of B_e using the τ branching fraction to µ ν ν, B_µ, and the τ lifetime. We average:

• the B_e fit value B₅,
• the B_e determination from the B_µ= B(τ → µ ν ν) fit value B₃ assuming that g_µ/g_e = 1, hence (see also Section 3)

  Be =  Bµ· f(me2/mτ2)/f(mµ2/mτ2) ,

• the B_e determination from the τ lifetime assuming that g_τ/g_µ=1, hence

  Be =  B(µ → e

  ν

  e νµ)· (ττ/ τµ) · (mτ/mµ)5 · f(me2/mτ2)/f(me2/mµ2) · (RγτRWτ)/(RγµRWµ) ,

  where B(µ → e ν_e ν_µ) = 1.

Accounting for correlations, we obtain

We use B_e^uni to obtain the ratio

We define B_had as the sum of all measured branching fractions to hadrons, which corresponds to the sum of all branching fractions minus the leptonic branching fractions, B_had = B_All − B_e − B_µ= (64.76 ± 0.10)% (see Section 2 and Table 1 for more details on the definition of B_All). An alternative definition of B_had uses the unitarity of the sum of all branching fractions, B_had^uni = 1 − B_e − B_µ= (64.79 ± 0.06)%, and results in:

A third definition of B_had uses the unitarity of the sum of all branching fractions, the Standard Model prediction B_µ= B_e · f(m_µ²/m_τ²)/f(m_e²/m_τ²) and B_e^uni to define B_had^{uni, SM} = 1 − B_e^uni − B_e^uni · f(m_µ²/m_τ²)/f(m_e²/m_τ²) = (64.86 ± 0.04)%, and to compute

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