10.2 Problems
553
10.68 Use lepton universality and lepton-quark symmetry to estimate the branching
fraction for the decay τ
−
→ e
−
+ν e +ν τ . Ignore final states that are cabibbosuppressed relative to the lepton modes.
10.69 The following decays are all ascribed to the weak interaction, resulting in
three final state particles. For each process, the available energy Q in the
decay is given as well as the mean lifetime.
Q(MeV)
τ (s)
(a) μ
+ → e
+ + ν e + ν μ
106
2.19 × 10
−6
(b) n → p + e
−
+ ν e
0.78
900
(c) τ
+
→ e
+
+ ν e + ν τ
1776
3.4 × 10
−13
(d) π
+
→ π
0
+ e
+
+ ν e
4.1
2.56
(e)
14 O →
14 N
∗
+ e
+
+ ν e
1.81
198
Apply Sergent’s law of radioactivity to show that the Q-values and mean
lifetimes are compatible with the same weak coupling.
10.70 Which of the following reactions are allowed and which forbidden as under
weak interactions?
(a) ν μ + p → μ
+
+ n
(b) ν e + p → e
−
+ π
+
+ p
(c) K
+
→ π
0
+ μ
+
+ ν μ
(d) Λ → π
+
+ e
−
+ ν e
10.71 Introduce a fictitious particle called Spurion to calculate the ratio of decay
rates Ξ
−
→ Λ + π
−
/Ξ
0
→ Λ + π
0
10.72 Consider the following decays of Σ hyperons
Σ
+
→ n + π
+
Σ
−
→ n + π
−
Σ
+
→ p + π
0
Using the ΔI = 1 / 2 rule, show that the triangle relation for the amplitudes is
a + +
√
2a 0 = a −
10.73 Λ hyperon can decay predominantly through the non-leptonic modes Λ →
p + π
− and Λ → n + π
0 . Introduce a fictitious particle called spurion of
isospin 1 / 2 to convert the decay into a reaction and determine the branching
ratios for these modes.
10.74 Assuming that the entire energy resulting from the p–p chain reaction escapes
from the sun’s surface, calculate the flux of neutrinos received on earth. Take
the earth-sun distance as 1.5×10
8 km. Assume that the total energy output of
the sun is L Θ = 3.83 × 10
26 Js
−1 , and that each α-particle produced implies
the generation of 26.72 MeV
10.75 The observation of neutrinos emitted by the supernova SN 1987A 170,000
years ago provided a rough estimate of neutrino’s mass. Assume that the
553
10.68 Use lepton universality and lepton-quark symmetry to estimate the branching
fraction for the decay τ
−
→ e
−
+ν e +ν τ . Ignore final states that are cabibbosuppressed relative to the lepton modes.
10.69 The following decays are all ascribed to the weak interaction, resulting in
three final state particles. For each process, the available energy Q in the
decay is given as well as the mean lifetime.
Q(MeV)
τ (s)
(a) μ
+ → e
+ + ν e + ν μ
106
2.19 × 10
−6
(b) n → p + e
−
+ ν e
0.78
900
(c) τ
+
→ e
+
+ ν e + ν τ
1776
3.4 × 10
−13
(d) π
+
→ π
0
+ e
+
+ ν e
4.1
2.56
(e)
14 O →
14 N
∗
+ e
+
+ ν e
1.81
198
Apply Sergent’s law of radioactivity to show that the Q-values and mean
lifetimes are compatible with the same weak coupling.
10.70 Which of the following reactions are allowed and which forbidden as under
weak interactions?
(a) ν μ + p → μ
+
+ n
(b) ν e + p → e
−
+ π
+
+ p
(c) K
+
→ π
0
+ μ
+
+ ν μ
(d) Λ → π
+
+ e
−
+ ν e
10.71 Introduce a fictitious particle called Spurion to calculate the ratio of decay
rates Ξ
−
→ Λ + π
−
/Ξ
0
→ Λ + π
0
10.72 Consider the following decays of Σ hyperons
Σ
+
→ n + π
+
Σ
−
→ n + π
−
Σ
+
→ p + π
0
Using the ΔI = 1 / 2 rule, show that the triangle relation for the amplitudes is
a + +
√
2a 0 = a −
10.73 Λ hyperon can decay predominantly through the non-leptonic modes Λ →
p + π
− and Λ → n + π
0 . Introduce a fictitious particle called spurion of
isospin 1 / 2 to convert the decay into a reaction and determine the branching
ratios for these modes.
10.74 Assuming that the entire energy resulting from the p–p chain reaction escapes
from the sun’s surface, calculate the flux of neutrinos received on earth. Take
the earth-sun distance as 1.5×10
8 km. Assume that the total energy output of
the sun is L Θ = 3.83 × 10
26 Js
−1 , and that each α-particle produced implies
the generation of 26.72 MeV
10.75 The observation of neutrinos emitted by the supernova SN 1987A 170,000
years ago provided a rough estimate of neutrino’s mass. Assume that the