Subatomic Physics

444 Strong Interactions
already been given in Section 14.4. In particular, for a point nucleon at position
x = 0, it is the Yukawa potential, Eq. (14.24). The nucleon acts as a source of the
meson field and
g
Φ(x) =−
(�c) 1/2 mc
exp(−kr), r = |x|, k = , (14.42)
r �
is the field produced at x by a point nucleon sitting at the origin. The interaction
energy between this and a second point nucleon at position x is found by inserting
Eq. (14.42) into Eq. (14.39) and by using the fact that ρ(x) now also describes a
point nucleon. The interaction energy then becomes
Vs = −g 2 �c exp(−kr)
. (14.43)
r
The negative sign means attraction and two nucleons consequently attract each
other if the force is produced by a neutral scalar meson.
Pions are pseudoscalar and not scalar particles, although the latter appear in
the meson exchange potentials used to fit data. (25) As the next step, we therefore
consider the contribution from a neutral pseudoscalar meson. Looking through
the list at PDG indicates that η, with a mass of 549 MeV/c2 , is such a particle.
The interaction Hamiltonian is very similar to the one given in Eq. (14.16); for an
isoscalar particle, this relation simplifies to
�
Hp = F d 3 xρ(x)σ · ∇Φ. (14.44)
The free pseudoscalar meson is also described by the Klein–Gordon equation, Eq.
(14.44), because it is not possible to distinguish between free scalar and pseudoscalar
particles. For the meson field in the presence of a nucleon, Eqs. (14.44) and (14.40)
together yield �
∇ 2 −
�
mc
� �
2
�
Φ=− 4π
�2 F σ ·∇ρ(x). (14.45)
c2 This equation is solved as in Section 14.4. Inserting the solution into Eq. (14.44)
then gives, for the potential energy due to the exchange of the neutral pseudoscalar
meson between point nucleons A and B,
2 F
Vp =
�2c2 (σA ·∇)(σB ·∇) exp(−kr)
. (14.46)
r
The differentiations can be performed, and the final result is (25,30)
2 F
Vp =
�2c2 �
1
3 σA
�
1 1 1
· σB + SAB + +
3 kr (kr) 2
��
2 exp(−kr)
× k , (14.47)
r
30Details can be found in L. R. B. Elton, Introductory Nuclear Theory, 2nd ed, Saunders,
Philadelphia, 1966, Section 10.3. Vp, as given in Eq. (14.47), is not complete; a term proportional
to δ(r) is missing. The omission is unimportant because the short-range repulsion between nucleons
makes the term ineffective.