Subatomic Physics

432 Strong Interactions
Eq. (14.16) depends on the normalization of the pion wave function � Φ. Since the
pion should be treated relativistically, the probability density is normalized not to
unity, but to E −1 ,whereE is the energy of the state. This normalization gives the
probability density the correct Lorentz transformation properties; the probability
density is not a relativistic scalar, but transforms like the zeroth component of a
four-vector. With this normalization, � Φ has the dimension of E −1/2 L −3/2 and the
dimensionless rationalized coupling constant has the value (10)
f 2 πNN = m2 π
4π� 5 c F 2 πN ≈ 0.08. (14.17)
When the pion was the only known meson, the subject of the pion–nucleon
interaction played a dominant role in theoretical and experimental investigations.
It was felt that a complete knowledge of this interaction would be the clue to
a complete understanding of strong physics. However, attempts to explain, for
instance, the nucleon–nucleon force and the nucleon structure in terms of the pion
alone were never successful. Other mesons were postulated, and these and some
unexpected ones were found. It became clear that the pion–nucleon interaction is
not the only problem of interest and that an interaction-by-interaction approach
would not necessarily solve the entire problem. At present, in this energy domain
the field is very complicated and far beyond a brief and low-brow description. Our
discussion here is therefore limited; we shall not treat other interactions but shall
turn to the nucleon–nucleon force because it plays an important role in nuclear and
particle physics.
14.4 The Yukawa Theory of Nuclear Forces
We have stated at the beginning of Section 14.2 that Yukawa introduced a heavy
boson for the explanation of nuclear forces in 1934. The fundamental idea thus
antedates the discovery of the pion by years. The role of mesons in nuclear physics
was not discovered experimentally; it was predicted through a brilliant theoretical
speculation. For this reason we shall first sketch the basic idea of Yukawa’s theory
before expounding the experimental facts. We shall introduce the Yukawa potential
in its simplest form by analogy with the electromagnetic interaction.
The interaction of a charged particle with a Coulomb potential has been discussed
in chapter 10. The scalar potential A0 produced by a charge distribution
qρ(x ′ ) satisfies the wave equation (11)
∇ 2 A0 − 1
c 2
∂2A0 = −4πqρ. (14.18)
∂t2 10 O. Dumbrajs et al., Nucl. Phys. B216, 277 (1983).
11 The inhomogeneous wave equation can be found in most texts on electrodynamics, for instance,
in Jackson, Eq. (6.73). As in chapter 10, our notation differs slightly from Jackson; here ρ is not
a charge but a probability distribution.