Subatomic Physics

14.6. Meson Theory of the Nucleon–Nucleon Force 443
present section we shall establish the expression for the interaction energy between
two nucleons.
We begin with the simplest
case, where the interaction
is mediated by the exchange
of a neutral scalar
meson. The emission and
absorption of such a meson
is described by an interaction
Hamiltonian. For the
pseudoscalar case, the corresponding
Hamiltonian HπN
has been discussed in Section
12.3.
Figure 14.16: Typical two pion-exchange potential diagrams.
The Hamiltonian, Hs, for the scalar interaction can be obtained by similar invariance
arguments: Φ is now a scalar in ordinary and in isospin space, and the simplest
expression for the energy of interaction between a scalar meson and a fixed nucleon
characterized by a source function ρ(x) is
Hs = g(�c) 3/2
�
d 3 xΦ(x)ρ(x). (14.39)
Between emission and absorption, the meson is free. The wave function of a free
spinless meson satisfies the Klein–Gordon equation, Eq. (14.26). In the timeindependent
case, it reads
�
∇ 2 −
�
mc
� �
2
Φ(x) =0. (14.40)
�
Together with Hamilton’s equations of motion, (25,29) Eqs. (14.39) and (14.40) lead
to �
∇ 2 −
�
mc
� �
2
�
Φ(x) = 4πgρ(x)
. (14.41)
(�c) 1/2
This expression is identical to Eq. (14.22). In Section 14.4, we constructed it by
starting from the corresponding one in electromagnetism and adding a mass term.
Here it follows logically from the wave equation for the scalar meson together with
the simplest form for the interaction Hamiltonian. The solution to Eq. (14.42) has
29 A brief derivation is given in W. Pauli, Meson Theory of Nuclear Forces, Wiley-Interscience,
New York, 1946. The elements of Lagrange and Hamiltonian mechanics can be found in most
texts on mechanics. The application to wave functions (fields) is described in E. M. Henley and
W. Thirring, Elementary Quantum Field Theory, McGraw-Hill, New York, 1962, p. 29, or F.
Mandl, Introduction to Quantum Field Theory, Wiley-Interscience, New York, 1959, chapter 2.