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4 Universality-improved B(τ → e ν ν) and Rhad
We compute two quantities that are used for further
tests involving the τ branching fractions:
- the “universality-improved” value Beuni of Be = B(τ → e ν ν), determined with the assumption
that the Standard Model and lepton universality hold;
- the ratio Rhad between the
total branching fraction of the τ to hadrons, Bhad and the universality-improved Beuni, 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 Beuni using the
τ branching fraction to µ ν ν, Bµ, and the τ lifetime. We average:
- the Be fit value B5,
- the Be determination from the Bµ= B(τ → µ ν
ν) fit value B3 assuming that gµ/ge = 1,
hence (see also Section 3)
| Be = Bµ· f(me2/mτ2)/f(mµ2/mτ2) ,
| | | | | | | | | | (10) |
|
- the Be determination from the τ lifetime assuming that
gτ/gµ=1, hence
| Be = B(µ → e | | e
νµ)· | | · | | ·
f( | | )/f( | | ) · (RγτRWτ)/(RγµRWµ) ,
|
| | | | | | | | | | (11) |
|
where B(µ → e
νe νµ) = 1.
Accounting for correlations, we obtain
| Beuni = (17.812 ± 0.022)%.
| | | | | | | | | | (12) |
|
We use Beuni to obtain the ratio
| Rhad = | | = | | =
3.6343 ± 0.0082 ,
|
| | | | | | | | | | (13) |
|
where Bhad is the sum of all measured
branching fractions to hadrons.
An alternative definition of Bhad uses the unitarity of
the sum of all branching fractions,
Bhaduni = 1 − Be − Bµ=
(64.80 ± 0.06)%, and results in:
| Rhad uni = | | =
3.6381 ± 0.0075 .
|
| | | | | | | | | | (14) |
|
A third definition of Bhad uses the unitarity of
the sum of all branching fractions, the Standard Model prediction
Bµ= Be · f(mµ2/mτ2)/f(me2/mτ2) and Beuni to define
Bhaduni, SM = 1 − Beuni −
Beuni · f(mµ2/mτ2)/f(me2/mτ2) =
(64.87 ± 0.04)%, yielding
| Rhad uni, SM = | 1 − Beuni −
Beuni· f(mµ2/mτ2)/f(me2/mτ2) |
|
Beuni |
| =
3.6417 ± 0.0070 .
|
| | | | | | | | | | (15) |
|
Although Bhaduni and
Bhaduni, SM are more precise than
Bhad, the precision of Rhad uni and
Rhad uni, SM is just slightly better than the one of Rhad because there are larger correlations between
Bhaduni, Bhaduni, SM and
Beuni than between Bhad and Beuni.
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