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6 |Vus| measurement
The CKM matrix element |Vus| is most precisely determined from kaon
decays [77], and its precision is limited by the
uncertainties of the lattice QCD estimates of f+Kπ(0) and fK/fπ.
Using the τ branching fractions, it is possible to determine |Vus| in an
alternative way [78] that does not depend on lattice QCD and
has small theory uncertainties (see Section 6.1).
Moreover, |Vus| can be determined using the τ branching fractions
similarly to the kaon case, using the same lattice QCD estimates, in order
to check the overall experimental consistency.
We have updated the CKM coefficient |Vus| determinations that we did in the
previous report using the updated data from HFAG 2014 and PDG 2013.
6.1 Inclusive τ partial width to strange
The τ hadronic partial width is the sum of the τ partial widths to
strange and to non-strange hadronic final states, Γhad =
Γs + ΓVA . Dividing any partial width Γx
by the electronic partial width, Γe, we obtain partial width ratios
Rx (which are equal to the respective branching fraction ratios
B x/B e) for which Rhad = Rs + RVA . In terms of
such ratios, |Vus| is measured as [78]
| |Vus| τ s | = | | |
Rs/ | ⎡
⎢
⎢
⎣ | | − δ Rtheory | ⎤
⎥
⎥
⎦ |
|
| ,
|
| | | | | | | | | |
|
where δ Rtheory can be determined in the context of low
energy QCD theory, partly relying on experimental low energy scattering
data. The literature reports several
calculations [78, 79, 80]. In this
report we use Ref. [78], whose estimated uncertainty size is
in between the two other ones. We use the information in that paper and the
PDG 2013 value for the s-quark mass ms = 93.50 ± 2.50 MeV [17]
to calculate δ Rtheory = 0.239 ± 0.030.
We proceed following the same procedure of the 2012 HFAG
report [3], using the universality improved
B euni = (17.814 ± 0.023)%
(see Section 5) to compute the Rx ratios, and
using the sum of the τ branching fractions to strange and
non-strange hadronic final states to compute Rs and RVA,
respectively.
Using the τ branching fraction fit results with their uncertainties
and correlations (Section 2), we compute B s =
(2.882 ± 0.047)% (see also Table 13) and
B VA = B hadrons − B s =
(61.81 ± 0.10)%, where B hadrons =
Γhadrons defined in section 5. PDG 2013 averages
are used for non-τ quantities, including |Vud| = 0.97425 ± 0.00022, which
comes from Ref. [81] like for the previous HFAG report.
We obtain |Vus| τ s = 0.2176 ± 0.0021, which
is 3.4σ lower than the unitarity CKM
prediction |Vus| uni = 0.22547 ± 0.00095, from (|Vus| uni)2 = 1 −
|Vud| 2. The |Vus| τ s uncertainty includes a systematic error
contribution of 0.44% from the theory uncertainty on
δ Rtheory. There is no significant change with respect to
the previous HFAG report.
Kim Maltman has computed an alternative theoretical estimate based on Fixed
Order Pertubation Theory and experimental inputs restricted to τ quantities [82].
The result is δ Rtheory = 0.254 ± 0.038.
The uncertainty includes an additional contribution to account
for the differences between using Fixed Order Pertubation Theory and Contour
Improved Perturbation Theory. With this alternative value, we would obtain
|Vus| τ s = 0.2181 ± 0.0022, which would be 3.1σ lower than
the unitarity CKM prediction.
Table 13: HFAG Summer 2014 τ branching fractions to strange final states. |
|
Branching fraction | HFAG Summer 2014 fit |
|
| (0.6955 ± 0.0096) · 10−2 |
| (0.4331 ± 0.0149) · 10−2 |
| (0.0630 ± 0.0220) · 10−2 |
Γ28 = K− 3π0 ντ (ex. K0,η)
|
| (0.0419 ± 0.0216) · 10−2 |
| (0.8378 ± 0.0123) · 10−2 |
| (0.3680 ± 0.0103) · 10−2 |
Γ44 = π− K0 π0 π0 ντ (ex. K0)
|
| (0.0124 ± 0.0204) · 10−2 |
| (0.0222 ± 0.0202) · 10−2 |
| (0.0155 ± 0.0008) · 10−2 |
| (0.0048 ± 0.0012) · 10−2 |
| (0.0093 ± 0.0015) · 10−2 |
| (0.0410 ± 0.0092) · 10−2 |
| (0.0037 ± 0.0014) · 10−2 |
Γ802 = K− π− π+ ντ (ex. K0,ω)
|
| (0.2922 ± 0.0068) · 10−2 |
Γ803 = K− π− π+ π0 ντ (ex. K0,ω,η)
|
| (0.0410 ± 0.0143) · 10−2 |
Γ822 = K− 2π− 2π+ ντ (ex. K0)
|
| (0.0001 ± 0.0001) · 10−2 |
Γ833 = K− 2π− 2π+ π0 ντ (ex. K0)
|
| (0.0001 ± 0.0001) · 10−2
|
|
| (2.8817 ± 0.0470) · 10−2
|
|
6.2 |Vus| from B (τ → Kν) / B (τ → πν) and from B (τ → Kν)
We follow the same procedure of the HFAG 2012 report to compute |Vus| from
the ratio of branching fractions B (τ → K− ντ) / B (τ → π− ντ) =
(6.431 ± 0.094) · 10−2
[the wrong value corresponding to B (τ → K− ντ) / B (τ →
π− ντ) = (3.903 ± 0.054) · 10−2 has incorrectly been quoted
until 25 June 2015]
from the equation
| B (τ → K− ντ) |
|
B (τ → π− ντ) |
| =
| | | |
| rLD(τ− → K−ντ) |
|
rLD(τ− → π−ντ) |
| .
|
| | | | | | | | | |
|
We use fK/fπ= 1.1920 ± 0.0050 from the
FLAG 2013 Lattice averages with Nf=2+1 [83].
We compute |Vus| τ K/π = 0.2232 ± 0.0019,
1.0σ below the CKM unitarity prediction.
We proceed like in 2012 also to determine |Vus| from the branching fraction
B (τ− → K− ντ ) using
| B (τ− → K−ντ) =
| GF2 fK2 |Vus| 2 mτ3 ττ |
|
16πℏ |
| | ⎛
⎜
⎜
⎝ | 1 − | | | ⎞
⎟
⎟
⎠ | | SEW .
|
| | | | | | | | | | |
|
We use fK = 156.3 ± 0.9 MeV from FLAG 2013 with
Nf=2+1 [83]. We obtain |Vus| τ K = 0.2212 ± 0.0020,
which is 1.9σ below
the CKM unitarity prediction. CODATA 2010 results [84] and
PDG 2013 have been used for the physics constants.
6.3 |Vus| from τ summary
Figure 2: |Vus| averages of this document compared with the FlaviaNet results [77].
|
We summarize the |Vus| results reporting the values, the discrepancy with
respect to the |Vus| determination from CKM unitarity, and an illustration
of the measurement method:
| | |Vus| uni | | = 0.22547 | | ± 0.00095 | | | | | | | |
| |Vus| τ s | | = 0.2176 | | ± 0.0021 | | −3.4σ | | from Γ(τ− → Xs− ντ) , | | | |
| |Vus| τ K/π | | = 0.2232 | | ± 0.0019 | | −1.0σ | | from Γ(τ− → K− ντ )/Γ(τ− → π− ντ ) , | | | |
| |Vus| τ K | | = 0.2212 | | ± 0.0020 | | −1.9σ | | from Γ(τ− → K− ντ ) .
| | | |
|
Averaging the three above |Vus| determinations (taking into account all
correlations due to the usage of the fitted τ branching fractions and
the other mentioned inputs) we obtain:
| | |Vus| τ | | = 0.2204 ± 0.0014 | | −2.9σ | average of 3 |Vus| τ measurements.
| | | | | | |
|
We could not find a published estimate of the correlation of the
uncertainties on fK and fK/fπ, but even if we assume ±
100% correlation, the uncertainty on |Vus| τ does not change
more than about ± 5%. Figure 2 summarizes the
|Vus| results.
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