Subatomic Physics

14.3. The Form of the Pion–Nucleon Interaction 431
HπN = FπNσ·(�τ · ∇ � Φ(x)), (14.14)
where FπN is a coupling constant. This Hamiltonian describes a point interaction:
Pion and nucleon interact only if they are at the same point. However, the interaction
is known to occur over an extended region. To smear out the interaction, a
weighting (source) function ρ(x) is introduced; ρ(x) can, for instance, be taken to
represent the nucleon probability density, ρ = ψ ∗ ψ. The function ρ(x) falls rapidly
to zero beyond about 1 fm and is normalized so that
�
d 3 xρ(x) =1. (14.15)
The Hamiltonian between a pion and an extended nucleon fixed at the origin of the
coordinate system becomes
�
d 3 xρ(x)σ·(�τ · ∇� Φ(x)). (14.16)
HπN = FπN
This interaction is the simplest one that leads to single emission and absorption of
pions. It is not unique; additional terms such as F ′� 2 Φ may be present. Moreover,
it is nonrelativistic and therefore limited in its range of validity. However, at higher
energies, where Eq. (14.16) is no longer valid, other particles and processes complicate
the situation so that consideration of the pion–nucleon force alone becomes
meaningless anyway.
The integral in Eq. (14.16) vanishes for a spherical source function ρ(r) unless
the pion wave function describes a p wave (l = 1). This prediction is in agreement
with the experimental data described in the previous section.
The first successful description of pion–nucleon scattering and pion photoproduction
was due to Chew and Low, (8) who used the Hamiltonian (14.16). Because
of the angular momentum barrier present in the l = 1 state, the low-energy pion–
nucleon scattering cross section (below about 50 MeV) can be computed in perturbation
theory. At higher energies, the approach is more sophisticated, but it
can be shown that the Hamiltonian (14.16) leads to an attractive force in the state
I = 3
2
,J = 3
2 and can explain the observed resonance.(9) At still higher energies,
the nonrelativistic approach is no longer adequate.
The numerical value of the pion–nucleon coupling constant FπN is determined
by comparing the measured and computed values for the pion–nucleon scattering
cross section. It is customary not to quote FπN but rather the corresponding dimensionless
and rationalized coupling constant, fπNN. The dimension of FπN in
8 G. F. Chew, Phys. Rev. 95, 1669 (1954); G. F. Chew and F. E. Low, Phys. Rev. 101, 1570
(1956); G. C. Wick, Rev. Mod. Phys. 27, 339 (1955).
9 Detailed descriptions of the Chew-Low approach can be found in G. Källen, Elementary Particle
Physics, Addison-Wesley, Reading, Mass., 1964; E. M. Henley and W. Thirring, Elementary
Quantum Field Theory, McGraw-Hill, New York, 1962; and J. D. Bjorken and S. D. Drell,
Relativistic Quantum Mechanics, McGraw-Hill, New York, 1964. While these accounts are not
elementary, they contain more details than the original papers.