10.2 Problems 553
10.68 Use lepton universality and lepton-quark symmetry to estimate the branching
fraction for the decayτ−→e−+νe+ντ. Ignore final states that are cabibbo-
suppressed 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++νe+νμ 106 2. 19 × 10 −^6
(b)n→p+e−+νe 0.78 900
(c)τ+→e++νe+ντ 1776 3. 4 × 10 −^13
(d)π+→π^0 +e++νe 4.1 2.56
(e)^14 O→^14 N∗+e++νe 1.81 198Apply 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)νμ+p→μ++n
(b)νe+p→e−+π++p
(c)K+→π^0 +μ++νμ
(d)Λ→π++e−+νe
10.71 Introduce a fictitious particle called Spurion to calculate the ratio of decay
ratesΞ−→Λ+π−/Ξ^0 →Λ+π^0
10.72 Consider the following decays ofΣhyperons
Σ+→n+π+
Σ−→n+π−
Σ+→p+π^0
Using theΔI=^1 / 2 rule, show that the triangle relation for the amplitudes is
a++√
2a 0 =a−10.73Λhyperon can decay predominantly through the non-leptonic modesΛ→
p+π−andΛ→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 × 108 km. Assume that the total energy output of
the sun isLΘ= 3. 83 × 1026 Js−^1 , and that eachα-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