10.2 Problems
551
10.51 (a) What are the quark constituents of the states Δ
−
, Δ
0
, Δ
+
, Δ
++ ?
(b) Assuming the quarks are in states of zero angular momentum, what fundamental difficulty appears to be associated with the Δ states, which have
I = 3/2 and how is it resolved?
(c) How do you explain the occurrence of excited states of the nucleons with
the higher values of J. What parities would the higher states have?
10.52 Use the quark model to determine the quark composition of (a) Σ
+
, Σ
− , n
and p (b) K
+
, K
−
, π
+
, π
− mesons.
10.53 (a) What are the three particles described by taking the three identical quarks?
(b) What are the quantum numbers of the b quark?
10.54 Draw the quark flow diagrams for the decays (a) ϕ → K
+ K
− (b) ω →
π
+
π
−
π
0 (c) Show that the decay ϕ → π
+
π
−
π
0 is suppressed.
10.55 At a beam energy of 60 GeV, σ (π
+ p) ∼ = σ (π
− p) = 25 mb while σ (pp) ∼ =
σ (pn) = 38 mb. Show that the ratio of cross sections σ (π N )/σ (NN ) can be
explained by simple quark model.
10.56 The production of a leptonic pair in a pion–nucleon collision is explained by
the Drell–Yan mechanism which consists of the annihilation of the anti-quark
from the pion with a quark from the nucleon, producing a virtual photon that
transforms to a muon pair. Show that the cross section in π12 C collisions
away from heavy meson resonances is predicted as
σ (π
− C)/σ (π
+ C) = 4 : 1
[Courtesy D.H. Perkins, Introduction to High Energy Physics, University of
Cambridge Press]
10.57 Below the production threshold of charm particles, the cross-section for the
reaction e
+ e
−
→ μ
+
μ
− is 20 nb. Estimate the cross-section for hadron production
10.58 In the quark model, a meson is described as a bound quark-antiquark state.
It is usual to represent the potential energy between q and q by V (r ) =
−
A
r
+ Br where A and B are positive constants. At r > few fermis, A is
negligible. Use the method of variation, with the trial function ψ(r ) = e
−r/a
to show that the ground state energy is given by E 0 = 2.96
B
2
2
mq
1 / 3 , where
m q is the quark mass.
10.2.4 Electromagnetic Interactions
10.59 The reaction e
+ e
−
→ μ
+
μ
− is studied using colliding beams each of energy
10 GeV and at these energies the reaction is predominantly electromagnetic.
Draw the lowest order Feynman diagram. The differential cross-section is
given by
551
10.51 (a) What are the quark constituents of the states Δ
−
, Δ
0
, Δ
+
, Δ
++ ?
(b) Assuming the quarks are in states of zero angular momentum, what fundamental difficulty appears to be associated with the Δ states, which have
I = 3/2 and how is it resolved?
(c) How do you explain the occurrence of excited states of the nucleons with
the higher values of J. What parities would the higher states have?
10.52 Use the quark model to determine the quark composition of (a) Σ
+
, Σ
− , n
and p (b) K
+
, K
−
, π
+
, π
− mesons.
10.53 (a) What are the three particles described by taking the three identical quarks?
(b) What are the quantum numbers of the b quark?
10.54 Draw the quark flow diagrams for the decays (a) ϕ → K
+ K
− (b) ω →
π
+
π
−
π
0 (c) Show that the decay ϕ → π
+
π
−
π
0 is suppressed.
10.55 At a beam energy of 60 GeV, σ (π
+ p) ∼ = σ (π
− p) = 25 mb while σ (pp) ∼ =
σ (pn) = 38 mb. Show that the ratio of cross sections σ (π N )/σ (NN ) can be
explained by simple quark model.
10.56 The production of a leptonic pair in a pion–nucleon collision is explained by
the Drell–Yan mechanism which consists of the annihilation of the anti-quark
from the pion with a quark from the nucleon, producing a virtual photon that
transforms to a muon pair. Show that the cross section in π12 C collisions
away from heavy meson resonances is predicted as
σ (π
− C)/σ (π
+ C) = 4 : 1
[Courtesy D.H. Perkins, Introduction to High Energy Physics, University of
Cambridge Press]
10.57 Below the production threshold of charm particles, the cross-section for the
reaction e
+ e
−
→ μ
+
μ
− is 20 nb. Estimate the cross-section for hadron production
10.58 In the quark model, a meson is described as a bound quark-antiquark state.
It is usual to represent the potential energy between q and q by V (r ) =
−
A
r
+ Br where A and B are positive constants. At r > few fermis, A is
negligible. Use the method of variation, with the trial function ψ(r ) = e
−r/a
to show that the ground state energy is given by E 0 = 2.96
B
2
2
mq
1 / 3 , where
m q is the quark mass.
10.2.4 Electromagnetic Interactions
10.59 The reaction e
+ e
−
→ μ
+
μ
− is studied using colliding beams each of energy
10 GeV and at these energies the reaction is predominantly electromagnetic.
Draw the lowest order Feynman diagram. The differential cross-section is
given by