10.2 Problems
555
10.83 Classify the following semileptonic decays of the D
+ (1,869) = cd meson
as Cabibbo-allowed, Cabibbo-suppressed or forbidden in lowest order weak
interactions.
(a) D
+
→ K
−
+ π
+
+ e
+
+ υ e
(b) D
+
→ π
+
+ π
+
+ e
−
+ ν e
10.84 Which of the following decays are allowed in lowest order weak interactions?
(a) K
−
→ π
+
+ π
−
+ e
−
+ ν e
(b) Ξ
0
→ Σ
−
+ e
+
+ ν e
(c) Ω
−
→ Ξ
−
+ π
+
+ π
−
10.85 Which of the following decays are allowed and which are forbidden ?
(a) K
0
→ π
− e
+
ν e
(b) K
0
→ π
+ e
−
ν e
(c) K 0 → π
+ e
−
ν e
(d) K 0 → π
− e
+
ν e
10.86 A muon neutrino is generated at time t = 0 at a particle accelerator. Show
that at a later time t the probability that it is still a muon neutrino is, in natural
units and in the neutrino rest frame
P μ (t) = 1 − sin
2 2θ sin
2
(E 2 − E 1 )t
2
[Courtesy D.H. Perkins, Introduction to High Energy Physics, University of
Cambridge Press]
10.87 (a) In Problem 10.86, write down an expression for the probability P e (t) that
at the same time t, the neutrino has oscillated into an electron neutrino.
(b) Derive the expression for the time at which the probabilities P μ (t) and
P e (t) are first equal. Assuming that m νe = 2 eV and m νμ = 3 eV and
θ = 34
0 ; find time t when beam energy is 1 GeV
10.88 Show how the following data prove the universality of the weak coupling
constant. τ μ = 2.197 × 10
−6 s, τ τ = 2.91 × 10
−13 s, the branching fraction
of the tauon, B(τ
+
→ e
+
ν e ν τ ) = 0.178, m μ = 105.658 MeV/c
2
, m τ =
1,777 MeV. Note that Γ(μ → eν e ν μ ) =
G
2 m
5
μ
192π 3 in natural units and the
Fermi constant G ∝ g
2 , where g is the weak charge also known as the
coupling amplitude.
10.89 Consider the semi leptonic weak decays (a) Σ
−
→ n + e
−
+ ν e (b) Σ
+
→
n + e
+
+ ν e Explain why the reaction (a) is observed but (b) is not.
10.90 The D
+ meson (cd) decays via the weak interaction to K 0 μ
+
ν μ
Alternatively the D
+ can decay to π
0
μ
+
ν μ . What are the predictions of
Cabibbo’s theory for the relative rates of the two decays?