The semiclassical approximation is justified if
the dimensionless quantity should be sufficiently small.
In this approximation the vacuum field configuration is found by
minimizing the system’s energy density, and so is given (up to a
transformation) by
. For small
the spectrum consists of two weakly-interacting particle types
described by the fields
and
, where
. To leading order in
the particle masses
are
and
.
The low-energy regime in this model is . The masslessness of
ensures the
existence of degrees of freedom in this regime, with the potential
for nontrivial low-energy interactions, which we next explore.
The interactions amongst the particles in this model are given by the scalar potential:
|
Imagine using the potential of
Equation (3) to calculate the
amplitude for the scattering of
particles at low energies to
lowest-order in
. For example, the Feynman graphs
describing tree-level
-
scattering are given in
Figure 1
. The
-matrix obtained by evaluating the analogous
tree-level diagrams for
self-scattering is
proportional to the following invariant amplitude:
An interesting feature of this amplitude is that when it is expanded in powers of external four-momenta, both its leading and next-to-leading terms vanish. That is
The last equality uses conservation of 4-momentum,Clearly the low-energy particles interact more
weakly than would be expected given a cursory inspection of the
scalar potential, Equation (3), since at tree level
the low-energy scattering rate is suppressed by at least eight
powers of the small energy ratio
. The real
size of the scattering rate might depend crucially on the relative
size of
and
, should the vanishing of the leading
low-energy terms turn out to be an artifact of leading-order
perturbation theory.
If scattering were of direct experimental
interest, one can imagine considerable effort being invested in
obtaining higher-order corrections to this low-energy result. And
the final result proves to be quite interesting: As may be verified
by explicit calculation, the first two terms in the low-energy
expansion of
vanish order-by-order in perturbation
theory. Furthermore, a similar suppression turns out also to hold
for all other amplitudes involving
particles, with the
-point amplitude for
scattering being suppressed
by
powers of
.
Clearly the hard way to understand these
low-energy results is to first compute to all orders in
and then expand the result in powers of
. A much more efficient
approach exploits the simplicity of small
before calculating scattering amplitudes.