A toy example

2.2 A toy example

In order to make the discussion as concrete as possible, consider the following model for a single complex scalar field $φ$ :

with

This theory enjoys a continuous $U (1)$ symmetry of the form $φ → eiωφ$ , where the parameter $ω$ is a constant. The two parameters of the model are $λ$ and $v$ . Since $v$ is the only dimensionful quantity it sets the model’s overall energy scale.

The semiclassical approximation is justified if the dimensionless quantity $λ$ 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 $U (1 )$ transformation) by $φ = v$ . For small $λ$ the spectrum consists of two weakly-interacting particle types described by the fields $ℛ$ and $ℐ$ , where $( ) φ = v + 1√-ℛ + √i-ℐ 2 2$ . To leading order in $λ$ the particle masses are $m ℐ = 0$ and $m ℛ = λv$ .

The low-energy regime in this model is $E ≪ m ℛ$ . The masslessness of $ℐ$ ensures the existence of degrees of freedom in this regime, with the potential for nontrivial low-energy interactions, which we next explore.

2.2.1 Massless-particle scattering

The interactions amongst the particles in this model are given by the scalar potential:

Figure 1:

The Feynman graphs responsible for tree-level $ℛ – ℐ$ scattering in the toy model. Here solid lines denote $ℛ$ particles and dashed lines represent $ℐ$ particles.

Imagine using the potential of Equation (3 ) to calculate the amplitude for the scattering of $ℐ$ particles at low energies to lowest-order in $λ$ . For example, the Feynman graphs describing tree-level $ℐ$ – $ℛ$ scattering are given in Figure 1 . The $S$ -matrix obtained by evaluating the analogous tree-level diagrams for $ℐ$ self-scattering is proportional to the following invariant amplitude:

where $sμ$ and $rμ$ (and $s′μ$ and $r′μ$ ) 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

[ ( )2] ( )2 3λ2- -3-- λ2v- -2-- λ2v- ′ ′ 𝒜 = − 2 + m2 √2-- + m4 √2-- [− r ⋅ s + r ⋅ r + r ⋅ s ] + 𝒪 (quartic in momenta ) ℛ ℛ = 0 + 𝒪 (quartic in momenta ). (5 )

The last equality uses conservation of 4-momentum, $sμ + rμ = s′μ + r′μ$ , and the massless mass-shell condition $2 r = 0$ . Something similar occurs for $ℛ$ – $ℐ$ 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 $r = E ∕mR$ . The real size of the scattering rate might depend crucially on the relative size of $r$ and $λ2$ , should the vanishing of the leading low-energy terms turn out to be an artifact of leading-order perturbation theory.

If $ℐ$ 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 $𝒜$ vanish order-by-order in perturbation theory. Furthermore, a similar suppression turns out also to hold for all other amplitudes involving $ℐ$ particles, with the $n$ -point amplitude for $ℐ$ scattering being suppressed by $n$ powers of $r$ .

Clearly the hard way to understand these low-energy results is to first compute to all orders in $λ$ and then expand the result in powers of $r$ . A much more efficient approach exploits the simplicity of small $r$ before calculating scattering amplitudes.