10.2 Problems 555
10.83 Classify the following semileptonic decays of theD+(1,869)=cdmeson
as Cabibbo-allowed, Cabibbo-suppressed or forbidden in lowest order weak
interactions.
(a)D+→K−+π++e++υe
(b)D+→π++π++e−+νe
10.84 Which of the following decays are allowed in lowest order weak interac-
tions?
(a)K−→π++π−+e−+νe
(b)Ξ^0 →Σ−+e++νe
(c)Ω−→Ξ−+π++π−
10.85 Which of the following decays are allowed and which are forbidden?
(a)K^0 →π−e+νe
(b)K^0 →π+e−νe
(c)K^0 →π+e−νe
(d)K^0 →π−e+νe10.86 A muon neutrino is generated at timet=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
Pμ(t)= 1 −sin^22 θsin^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 probabilityPe(t) that
at the same timet, the neutrino has oscillated into an electron neutrino.
(b) Derive the expression for the time at which the probabilitiesPμ(t) and
Pe(t) are first equal. Assuming thatmνe =2 eV andmνμ =3 eV and
θ= 340 ; find timetwhen beam energy is 1 GeV
10.88 Show how the following data prove the universality of the weak coupling
constant.τμ= 2. 197 × 10 −^6 s,ττ= 2. 91 × 10 −^13 s, the branching fraction
of the tauon, B(τ+→e+νeντ)= 0. 178 , mμ= 105 .658 MeV/c^2 ,mτ =
1,777 MeV. Note thatΓ(μ→eνeνμ)=G^2 m^5 μ
192 π^3in natural units and the
Fermi constantG ∝ g^2 , wheregis the weak charge also known as the
coupling amplitude.10.89 Consider the semi leptonic weak decays (a)Σ−→n+e−+νe(b)Σ+→
n+e++νeExplain why the reaction (a) is observed but (b) is not.
10.90 The D+meson (cd) decays via the weak interaction toK^0 μ+νμ
Alternatively the D+can decay toπ^0 μ+νμ. What are the predictions of
Cabibbo’s theory for the relative rates of the two decays?