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3  Tests of lepton universality

Lepton universality tests probe the Standard Model prediction that the charged weak current interaction has the same coupling for all lepton generations. The precision of such tests has been significantly improved since the 2014 edition by the addition of the Belle τ lifetime measurement [67], while improvements from the τ branching fraction fit are negligible. We compute the universality tests as in the previous report by using ratios of the partial widths of a heavier lepton λ decaying to a lighter lepton ρ [68],

     
  Γ(λ → νλ ρ 
ν
ρ (γ)) =
B (λ → νλ ρ 
ν
ρ)
τλ
 =
Gλ Gρ mλ5
192 π3
  f


mρ2
mλ2



RWλ Rγλ ,
          

where

     
 Gρ
gρ2
2
 MW2
 ,    
       f(x)= 1 −8x +8x3 −x4 −12x2 lnx ,  
RWλ
= 1 + 
3
5
 
mλ2
MW2
 ,    
         Rγλ
= 1+
α(mλ)
 


25
4
−π2


 .
 

We use Rγτ=1−43.2· 10−4 and Rγµ=1−42.4· 10−4 [68] and MW from PDG 2015 [8]. We use HFAG 2014 averages and PDG 2015 for the other quantities. Using pure leptonic processes we obtain

     
  


gτ
gµ
 


= 1.0010 ± 0.0015 ,
    
 


gτ
ge
 


= 1.0029 ± 0.0015 ,
    
 


gµ
ge
 


= 1.0019 ± 0.0014 .
      

Using the expressions for the τ semi-hadronic partial widths [69] we obtain

     
  


gτ
gµ
 


2



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



1−mµ2/mh2
1−mh2/mτ2
 


2



 
 ,
          

where h = π or K and the radiative corrections are δ Rτ/π = (0.16 ± 0.12)% and δ Rτ/K = (0.90 ± 0.22)% [69], we measure:

     
  


gτ
gµ
 


 



π
= 0.9961 ± 0.0027 ,    
 


gτ
gµ
 


 



K
 = 0.9860 ± 0.0070 .
        

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
= 1.0000 ± 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 )K1110-16
 ( gτ/gµ )( gτ/ge )( gµ/ge )( gτ/gµ )π

Since there is 100% correlation between gτ/ge, gµ/ge and gµ/ge, the correlation matrix is expected to be positive semi-definite, with one eigenvalue equal to zero. Due to numerical inaccuracies, one eigenvalue is expected to be close to zero rather than exactly zero.


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