Subatomic Physics

428 Strong Interactions
gives a mass of 1232 MeV/c 2 for both processes, and it is appealing to assume
that they represent the same resonance. The discovery of this resonance, called
∆(1232), was already discussed in Section 5.12. The cross sections in Figs. 14.5 and
14.6 show that the interaction of pions with nucleons at energies below about 500
MeV is dominated by this resonance.
Figure 14.7: Pion scattering
and pion photoproduction at
low energies are dominated
by the formation of an excited
nucleon, N ∗ , usually called
∆(1232).
Isospin and spin of ∆(1232) can be established by simple
arguments. Pion (I = 1) and nucleon (I = 1
2 )canform
states with I = 1
2
and I = 3
2
. If ∆(1232) had I = 1
2 ,only
two charge states of the resonance would occur. According
to the Gell-Mann–Nishijima relation, Eq. (8.30), they
would have the same electric charges as the nucleons,
namely 0 and 1. These two resonances, ∆ 0 (1232) and
∆ + (1232), are indeed observed. In addition, however,
the ∆ ++ (1232) appears in the process π + p → π + p,and
∆consequentlymusthaveI = 3
2 . The fourth member
of the isospin multiplet, ∆− (1232), cannot be observed
with proton targets; deuteron targets permit the investigation
of the reaction π−n → π−n,where∆−shows up. To establish the spin of ∆(1232), we note that the
maximum cross section for the scattering of unpolarized
particles is given by (6)
σmax =4πλ 2
2J +1
(2Jπ + 1)(2JN +1) =4πλ2
�
J + 1
�
.
2
(14.7)
J, Jπ, andJNare the spins of the resonance and of the colliding particles, and
λ is the c.m. reduced pion wavelength at resonance. 4πλ 2 at 155 MeV is almost
100 mb, and σmax is about 200 mb, so that J + 1
3
2 ≈ 2orJ = 2 . To form a state with
spin 3
2 in pion–nucleon scattering, the incoming pions must carry one unit of orbital
angular momentum. Pion–nucleon scattering at low energies occurs predominantly
in p waves.
• Thefactthatpion–nucleonscattering at low energies occurs predominantly
in the state J = 3 3
2 ,I = 2 (the so-called 3–3 resonance) can be verified by a spin–
isospin phase-shift analysis. We shall not present the complete analysis here, but
we shall outline its isospin part because it provides an example for the use of isospin
invariance. We first note that experimental states are prepared with well-defined
charges. Theoretically, however, it is more appropriate to use well-defined values of
the total isospin. It is therefore necessary to express the experimentally prepared
6 The maximum cross section for the scattering of spinless particles with zero orbital angular
momentum is given by 4πλ 2 . A particle with spin J is (2J + 1)-fold-degenerate. By assuming that
the above cross section holds for each substate, Eq. (14.7) follows.