10.3 Solutions
573
10.45
Fig. 10.8 Chain decay of
B-meson
The decays are weak at each stage as the quantum numbers b, c and S
change by one unit and is evidenced by narrow decay widths characteristic
of weak interactions. The route for the chain decays is in accordance with the
Cabibbo scheme as explained in the Sect. 10.1.
10.46 The QED formula is
Γ(3γ ) =
(2π
2
− 9)α
6 m e c
2
9π
(1)
(a) By analogy the QCD formula for charmonium (cc bound state) would be
Γ(3g) =
(2π
2
− 9)α
6
s m c c
2
9π
(2)
where α s is the coupling constant for strong interaction and m c c
2
≈
1.55 GeV is the constituent mass of the c-quark. Substituting Γ (3g) =
8 × 10
−5 GeV in (2), we find α s = 0.307
(b) For the radiative decay of the bb bound state the analogous formula
would be
Γ(ggγ ) =
2(π
2
− 9)α
4
s α
2 m b c
2
9π
Substituting Γ(ggγ ) = 1.6 × 10
−7 GeV, α = 1/137, m b c
2
= 4.72 GeV,
we find α s = 0.319
10.47 The ratio of the cross sections
R =
3σ (e
+ e
−
→ Q Q)
σ (e + e − → μ + μ − )
=
3
i
(q i /e)
2
1
(1)
where q i /e is the charge of the quark in terms of the charge of the electron.
The charges and masses for the quarks are tabulated below
q/e
M (GeV/c
2 )
u-quark +2/3 0.31
d-quark −1/3 0.31
s-quark −1/3 0.5
c-quark +2/3 1.6
b-quark −1/3 4.6
t-quark +2/3 175