1000 Solved Problems in Modern Physics

580 10 Particle Physics – II

Reaction (3) has relative weightage of 3 because of three colours. Thus

the branching fraction of (1) is predicted to be 1/5 or 0.2, a value which is in

agreement with the experimental value of 0.18.

10.69 According to Sargent’s law,Γ∝Q^5

Decay Q (MeV) τ(s) Q^5 τ(MeV^5 −s)

(a) μ+→e++νe+νμ 105.15 2. 197 × 10 −^62. 82 × 104

(b) n→p+e−+νe 0.782 900 2. 63 × 102

(c) τ+→e++νe+ντ 1784 3. 4 × 10 −^136. 14 × 103

(d) π+→π^0 +e++νe 4.08 2.56 2. 89 × 103

(e)^14 O→^14 N∗+e++νe 1.81 198 3. 85 × 103

The numbers in the last column vary only over one order of magnitude

which is small compared to the lifetimes which span over 15 orders of mag-

nitude, thereby conforming to the Sargents’ law (τ∝ 1 /T^5 max) and therefore

favoring the same coupling constant for the weak decays.

10.70 (a) Forbidden because lepton number is not conserved, left side hasLμ=−^1
while right side hasLμ=+ 1
(b) Forbidden because charge is not conserved.
(c) Allowed becauseB, Lμ, Qetc are conserved. Also the selection rule,
ΔQ=ΔS=±1 is obeyed.
(d) Allowed for reason stated in (c).

10.71 Introduce the spurion, (Sp) a hypothetical particle of spin^1 / 2 and isospin 1/2
and convert the weak decay into a strong interaction

Sp+Ξ−→Λ+π−

I^1 / 2 1 / 2 01

I 3 − 1 / 2 − 1 / 20 − 1

The reaction must occur in pureI =1 state. Looking up the Clebsch –

Gordon coefficients for^1 / 2 ×^1 / 2 , in Table 3.3,

∣

∣Λ,π−

〉

=a 1 | 1 ,− 1 〉

For the second decay, we associate again a spurion and consider the reaction

Sp+Ξ^0 →Λ+π^0

I^1 / 2 1 / 2 01

I 3 − 1 / 2 + 1 / 200