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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:
ℒ = − ∂ φ∗∂μ φ − V(φ ∗φ), (1 ) μ
2 V = λ--(φ∗φ − v2)2 . (2 ) 4
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:

2( ) λ-- √ -- 2 2 2 V = 16 2 2v ℛ + ℛ + ℐ . (3 )
View Image

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 (3View Equation) 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 ℐℛ scattering are given in Figure 1View Image. The S-matrix obtained by evaluating the analogous tree-level diagrams for ℐ self-scattering is proportional to the following invariant amplitude:

2 ( 2 )2 [ ] 𝒜 = − 3λ--+ λ√-v- ---------1---------+ ---------1--------- + ---------1--------- , (4 ) 2 2 (s + r)2 + m2ℛ − iε (r − r′)2 + m2ℛ − iε (r − s′)2 + m2ℛ − iε
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 ℛℐ scattering, which also vanishes due to a cancellation amongst the graphs of Figure 1View Image 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 (3View Equation), 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.

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