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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 f:
L = - @ f*@mf - V(f*f), (1) m
with
2 V = c--(f*f - v2)2 . (2) 4
This theory enjoys a continuous U (1) symmetry of the form f --> eiwf, where the parameter w is a constant. The two parameters of the model are c 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 c 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 f = v. For small c the spectrum consists of two weakly-interacting particle types described by the fields R and I, where ( ) f = v + 1 V~ -R + V~ i-I 2 2. To leading order in c the particle masses are mI = 0 and mR = cv.

The low-energy regime in this model is E « mR. 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( ) c-- V~ -- 2 2 2 V = 16 2 2vR + 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 c. For example, the Feynman graphs describing tree-level I-R 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 = - 3c--+ c V~ -v- ---------1---------+ ---------1--------- + ---------1--------- , (4) 2 2 (s + r)2 + m2R - ie (r - r')2 + m2R - ie (r - s')2 + m2R - ie
where sm and rm (and s'm and r'm) 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 3c2- -3-- c2v- -2-- c2v- ' ' A = - 2 + m2 V~ 2- + m4 V~ 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, sm + rm = s'm + r'm, 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 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 c2, 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 c 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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