1000 Solved Problems in Modern Physics

10.3 Solutions 573

10.45
Fig. 10.8Chain decay of
B-meson

The decays are weak at each stage as the quantum numbersb, candS

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 mec^2

9 π

(1)

(a) By analogy the QCD formula for charmonium (ccbound state) would be

Γ(3g)=

(2π^2 −9)α^6 smcc^2

9 π

(2)

whereαsis the coupling constant for strong interaction and mcc^2 ≈

1 .55 GeV is the constituent mass of the c-quark. SubstitutingΓ( 3 g)=

8 × 10 −^5 GeV in (2), we findαs= 0. 307

(b) For the radiative decay of thebbbound state the analogous formula

would be

Γ(ggγ)=

2(π^2 −9)α^4 sα^2 mbc^2

9 π

SubstitutingΓ(ggγ)= 1. 6 × 10 −^7 GeV,α= 1 / 137 ,mbc^2 = 4 .72 GeV,

we findαs= 0. 319

10.47 The ratio of the cross sections

R=

3 σ(e+e−→QQ)

σ(e+e−→μ+μ−)

=

3

∑

i

(qi/e)^2

1

(1)

whereqi/eis the charge of the quark in terms of the charge of the electron.

The charges and masses for the quarks are tabulated below

q/eM(GeV/c^2 )

u-quark + 2 /30.31

d-quark − 1 /30.31

s-quark − 1 /30.5

c-quark + 2 /31.6

b-quark − 1 /34.6

t-quark + 2 /3 175