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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/


RVA
|Vud| 2
 −  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 hadronsB 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 fractionHFAG Summer 2014 fit
Γ10 = K ντ
(0.6955 ± 0.0096) · 10−2
Γ16 = K π0 ντ
(0.4331 ± 0.0149) · 10−2
Γ23 = K 2π0 ντ (ex. K0)
(0.0630 ± 0.0220) · 10−2
Γ28 = K 3π0 ντ (ex. K0,η)
(0.0419 ± 0.0216) · 10−2
Γ35 = π K0 ντ
(0.8378 ± 0.0123) · 10−2
Γ40 = π K0 π0 ντ
(0.3680 ± 0.0103) · 10−2
Γ44 = π K0 π0 π0 ντ (ex. K0)
(0.0124 ± 0.0204) · 10−2
Γ53 = K0 h h h+ ντ
(0.0222 ± 0.0202) · 10−2
Γ128 = K η ντ
(0.0155 ± 0.0008) · 10−2
Γ130 = K π0 η ντ
(0.0048 ± 0.0012) · 10−2
Γ132 = π K0 η ντ
(0.0093 ± 0.0015) · 10−2
Γ151 = K ω ντ
(0.0410 ± 0.0092) · 10−2
Γ801 = K φ ντ(φ → KK)
(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
Γ110 = Xs ντ
(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)
=
fK2 |Vus| 2
fπ2 |Vud| 2
 

1 − mK2/mτ2 
2

1 −  mπ2/mτ2 
2
rLD → Kντ)
rLD → πντ)
 .
         

We use fK/fπ= 1.1920 ± 0.0050 from the FLAG 2013 Lattice averages with Nf=2+1 [83]. We compute |Vus| τ K = 0.2232 ± 0.0019, 1.0σ below the CKM unitarity prediction.

We proceed like in 2012 also to determine |Vus| from the branching fraction BK ντ ) using

     
  B → Kντ) =
GF2 fK2 |Vus| 2 mτ3 ττ
16πℏ
 


1 − 
mK2
mτ2
 


2



 
 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


PNG format PDF format
Vus summary plot
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              
 
from  
1 − |Vud| 2
   (CKM unitarity)
 , 
   
 |Vus| τ s     = 0.2176     ± 0.0021        −3.4σ      from  Γ(τ → Xs ντ) ,    
 |Vus| τ K     = 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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