Subatomic Physics

462 Strong Interactions
texts include J. Gribbin, The Search for Superstrings, Symnmetry, and the Theory
of Everything, Little, Brown and Co., New York, 1998; B. Greene,TheFabricofthe
Cosmos, A.A. Knopf, New York, 2004; D. Falk, Universe on a T-Shirt: The Quest
for the Theory of Everything, Arcade Publ. Co., New York, 2004.
Problems
14.1. (a) List 10 possible pion–nucleon scattering processes, with, at most, one
pion and one nucleon.
(b) Which of these processes are related by time-reversal invariance?
(c) Express all cross sections in terms of M 3/2 and M 1/2.
14.2. ∗ Sketch an experimental arrangement used to study pion–nucleon scattering.
(a) How is the total cross section observed?
(b) How is the charge-exchange reaction cross section determined?
14.3. Use the observed cross sections to show that the peaks of the first resonance
in pion–nucleon and in photonucleon reactions occur at the same mass of the
∆. Take recoil into account.
14.4. Treat the pion–nucleon scattering at the first resonance classically: Compute
the classical distance from the center of the nucleon at which a pion with
angular momentum l =0, 1, 2, 3 (in units of �) will strike. Which partial
waves will contribute significantly according to this argument? Use a parity
argument to rule out the values l =0andl =2.
14.5. Justify Eq. (14.7) by a crude (nonrigorous) argument.
14.6. Verify the expansions (14.8).
14.7. Consider HπN, Eq. (14.16). Assume a spherical source function ρ(r). Assume
the pion wave function to be a plane wave. Show that only the p-wave part
of this plane wave leads to a nonvanishing integral.
14.8. Consider Fig. 5.35. The second and third resonances in the π − p system have
no counterpart in the π + p system. What is the isospin of these resonances?
14.9. (a) Do conservation laws permit terms in the pion–nucleon interaction that
are quadratic in the pion wave function � Φ? If so, give an example.
(b) Repeat part (a) for terms cubic in � Φ. If your answer is yes, give an
example.