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:

where and (and and ) are the 4-momenta of the initial (and final) particles.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, , and the massless mass-shell condition . Something similar occurs for – scattering, which also vanishes due to a cancellation amongst the graphs of Figure 1 in the zero-momentum limit.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.

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