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

We compute two quantities that are used for further tests involving the τ branching fractions:

• the “universality-improved” value B_e^uni of B_e = B(τ → e ν ν), determined with 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^uni, which is a measure of the ratio Γ(τ → had) / Γ(τ→ eνν) of the respective partial widths.

Following Ref. ‍[1], we obtain the improved value B_e^uni 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) ‍,                   (10)

• 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  ·  mτ5 mµ5  · f( me2 mτ2 )/f( me2 mµ2 ) · (RγτRWτ)/(RγµRWµ) ‍,                   (11)

  where B(µ → e ν_e ν_µ) = 1.

Accounting for correlations, we obtain

We use B_e^uni to obtain the ratio

where B_had is the sum of all measured branching fractions to hadrons. An alternative definition of B_had uses the unitarity of the sum of all branching fractions, B_had^uni = 1 − B_e − B_µ= (64.80 ± 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) = (64.87 ± 0.04)%, yielding

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.