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4  Tests of lepton universality

The lepton universality tests probe the Standard Model prediction that the charged weak current interaction has the same coupling for all leptons. The precision of such tests has been significantly improved by the addition of the recent Belle τ lifetime measurement [2], while improvements from the τ branching fraction fit are negligible. We compute the universality tests like in the previous report by using proper ratios of the partial widths of a heavier lepton L decaying to a lighter lepton ℓ [74],

     
  Γ(L → νL ℓ 
ν
 (γ)) =
B (L → νL ℓ 
ν
)
τL
 =
GL G mL5
192 π3
  f


m2
mL2



rWL rγL ,
          

where

     
 G
g2
2
 MW2
 ,    
       f(x)= 1 −8x +8x3 −x4 −12x2 lnx ,  
rWL
= 1 + 
3
5
 
mL2
MW2
 ,    
         rγL
= 1+
α(mL)
 


25
4
−π2


 .
 

We use rγτ=1−43.2· 10−4 and rγµ=1−42.4· 10−4 [74] and MW from PDG 2013 [17]. We use HFAG 2014 averages and PDG 2013 for the other quantities. Using pure leptonic processes we obtain

     
  


gτ
gµ
 


= 1.0011 ± 0.0015 ,
    
 


gτ
ge
 


= 1.0029 ± 0.0015 ,
    
 


gµ
ge
 


= 1.0018 ± 0.0014 .
      

Using semi-hadronic processes

     
  


gτ
gµ
 


2



 
 =
B (τ → h ντ)
B (
h → µ 
ν
µ
)
2mh mµ2τh
(1+δh)mτ3ττ



1−mµ2/mh2
1−mh2/mτ2
 


2



 
 ,
          

where h = π or K and the radiative corrections are δπ = (0.16 ± 0.14)% and δK = (0.90 ± 0.22)% [75], we measure:

     
  


gτ
gµ
 


 



π
= 0.9963 ± 0.0027 ,    
 


gτ
gµ
 


 



K
 = 0.9858 ± 0.0071 .
        

Similar tests could be performed with decays to electrons, however they are less precise because the hadron two body decays to electrons are helicity-suppressed. Averaging the three gτ/gµ ratios we obtain

     
  


gτ
gµ
 


 



τ+π+K
= 1.0001 ± 0.0014 ,          

accounting for statistical correlations. Table 12 reports the statistical correlation coefficients for the fitted coupling ratios:


Table 12: Universality coupling ratios correlation coefficients (%).
( gτ/ge )53
( gµ/ge )-4948
( gτ/gµ )π24262
( gτ/gµ )K1210-25
 ( gτ/gµ )( gτ/ge )( gµ/ge )( gτ/gµ )π


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