586 10 Particle Physics – II
10.87 (a)Pe(t)= 1 −Pμ(t)=sin^22 θsin^2
[
(E 2 −E 1 )
2 t]
(b)Pμ(t)=Pe(t)1 −sin^22 θsin^2[
(E 2 −E 1 )
2
t]
=sin^22 θsin^2[
(E 2 −E 1 )
2
t]
2sin^22 θsin^2[
(E 2 −E 1 )
2
t]
= 1
Restoring to practical units the above equation becomes2sin^22 θsin^2[
Δm^2 c^4
2 Et]
= 1
whereΔm^2 =m 22 −m 12 andθ= 340
Ifm 1 andm 2 are in eV/c^2 , andEin MeV andLthe distance from the source,
then the last equation becomes2sin^22 θsin^2(
1. 27 Δm^2 .L
E)
= 1
Insertingθ = 340 ,Δm^2 = 52 − 32 =16 andE =1,000 MeV, we find
L=426 m, givingt=L/C= 1. 42 × 10 −^6 s.10.88τ(μ+→e+νeντ)=
G^2
(c)^6m^5 μ
192 π^3(1)
G^2 ∼g^2/
Mw^2 , where Mwis the mass of W-boson.
From theτlepton lifetime and formula (1) for the dependence of parent
particle mass, we can test the universality of the couplingsgμandgτto the
W – boson
(
gτ
gμ) 4
=B
(
τ+→e+νeντ)
(
mμ
mτ) 5 (
τμ
ττ)
InsertingB= 0. 178 ,mμ= 105 .658 MeV/c^2 ,mτ= 1777 .0MeV/c^2 ,τμ=
2. 197 × 10 −^6 s andττ= 2. 91 × 10 −^13 s, we find
gτ
gμ= 0. 987
Comment: From the branching fractions forτ+ → e+νeντandτ+ →
μ+νμντthe ratiogμ/ge = 1 .001. A similar result is obtained from the
branching ratio ofπ→eνandπ→μe, proving thereby different flavours
of leptons have identical couplings to the W±bosons.
The principle of universality is equally valid for theZ^0 coupling. Thus,
the branching fractions are predicted as
Z^0 →e+e−:μ+μ−:τ+τ−=1:1:1
in agreement with the experimental ratios. Formula (1) affords the most accu-
rate determination of G, the Fermi constant because the mass and lifetime of
muon are precisely known by experiment.