8
Elementary Processes in Scalar and Spinor
Electrodynamics
8.1 Coulomb scattering of charged spin-0 particles
We begin our study of electromagnetic interactions by considering the simplest case, that of the scattering of a (hypothetical) positively charged spin-0
particle ‘s
+ ’ by a fixed Coulomb potential, treated as a classical field. This
will lead us to the relativistic generalization of the Rutherford formula for
the cross section. We shall use this example as an exercise to gain familiarity
with the quantum field-theoretic approach of chapter 6, since it can also be
done straightforwardly using the ‘wavefunction’ approach familiar from nonrelativistic quantum mechanics, when supplemented by the work of chapter 3.
We shall also look at ‘s
− ’ Coulomb scattering, to test the antiparticle prescriptions of chapter 3. Incidentally, we call these scalar particles s
± to emphasize
that they are not to be identified with, for instance, the physical pions π
± ,
since the latter are composite (q¯ q) systems, and hence their interactions are
more complicated than those of our hypothetical ‘point-like’ s
± (as we shall
see in section 8.4). No point-like charged scalar particles have been discovered,
as yet.
8.1.1 Coulomb scattering of s + (wavefunction approach)
Consider the scattering of a spin-0 particle of charge e and mass M , the ‘s
+ ’, in
an electromagnetic field described by the classical potential A
μ . The process
we are considering is
s
+ (p) → s
+ (p
′ )
(8.1)
′
as shown in figure 8.1, where p and p are the initial and final 4-momenta
respectively. The appropriate potential for use in the KG equation has been
given in section 3.5:
V ˆ KG = ie(∂ μ A
μ + A
μ ∂ μ ) − e
2 A
2 .
(8.2)
As we shall see in more detail as we go along, the parameter characterizing
each order of perturbation theory based on this potential is found to be e
2 /4π.
221
Elementary Processes in Scalar and Spinor
Electrodynamics
8.1 Coulomb scattering of charged spin-0 particles
We begin our study of electromagnetic interactions by considering the simplest case, that of the scattering of a (hypothetical) positively charged spin-0
particle ‘s
+ ’ by a fixed Coulomb potential, treated as a classical field. This
will lead us to the relativistic generalization of the Rutherford formula for
the cross section. We shall use this example as an exercise to gain familiarity
with the quantum field-theoretic approach of chapter 6, since it can also be
done straightforwardly using the ‘wavefunction’ approach familiar from nonrelativistic quantum mechanics, when supplemented by the work of chapter 3.
We shall also look at ‘s
− ’ Coulomb scattering, to test the antiparticle prescriptions of chapter 3. Incidentally, we call these scalar particles s
± to emphasize
that they are not to be identified with, for instance, the physical pions π
± ,
since the latter are composite (q¯ q) systems, and hence their interactions are
more complicated than those of our hypothetical ‘point-like’ s
± (as we shall
see in section 8.4). No point-like charged scalar particles have been discovered,
as yet.
8.1.1 Coulomb scattering of s + (wavefunction approach)
Consider the scattering of a spin-0 particle of charge e and mass M , the ‘s
+ ’, in
an electromagnetic field described by the classical potential A
μ . The process
we are considering is
s
+ (p) → s
+ (p
′ )
(8.1)
′
as shown in figure 8.1, where p and p are the initial and final 4-momenta
respectively. The appropriate potential for use in the KG equation has been
given in section 3.5:
V ˆ KG = ie(∂ μ A
μ + A
μ ∂ μ ) − e
2 A
2 .
(8.2)
As we shall see in more detail as we go along, the parameter characterizing
each order of perturbation theory based on this potential is found to be e
2 /4π.
221