The toy model revisited

2.3 The toy model revisited

The key to understanding this model’s low-energy limit is to recognize that the low-energy suppression of scattering amplitudes (as well as the exact masslessness of the light particle) is a consequence of the theory’s $U (1 )$ symmetry. (The massless state has these properties because it is this symmetry’s Nambu–Goldstone boson. The earliest general formulation of non-Abelian Goldstone-boson interactions arose through the study of low-energy pion interactions [144 , 145 , 32 , 72 ]; for reviews of Goldstone boson properties see [83 , 106 ]; see also [147

, 28

].) The simplicity of the low-energy behaviour is therefore best displayed by

making the symmetry explicit for the low-energy degrees of freedom, and
performing the low-energy approximation as early as possible.

2.3.1 Exhibiting the symmetry

The $U (1 )$ symmetry can be made to act exclusively on the field which represents the light particle by parameterizing the theory using a different set of variables than $ℐ$ and $ℛ$ . To this end imagine instead using polar coordinates in field space

In terms of $θ$ and $χ$ the action of the $U(1)$ symmetry is simply $θ → θ + ω$ , and the model’s Lagrangian becomes

The semiclassical spectrum of this theory is found by expanding $ℒ$ in powers of the canonically-normalized fluctuations, $χ′ = √2-(χ − v)$ and $θ′ = √2-v θ$ , about the vacuum $χ = v$ , revealing that $χ ′$ describes the mass- $mR$ particle while $′ θ$ represents the massless particle.

With the $U (1)$ symmetry realized purely on the massless field $θ$ , we may expect good things to happen if we identify the low-energy dynamics.

2.3.2 Timely performance the low-energy approximation

To properly exploit the symmetry of the low-energy limit we integrate out all of the high-energy degrees of freedom as the very first step, leaving the inclusion of the low-energy degrees of freedom to last. This is done most efficiently by computing the following low-energy effective (or Wilson) action.

A conceptually simple (but cumbersome in practice) way to split degrees of freedom into ‘heavy’ and ‘light’ categories is to classify all field modes in momentum space as heavy if (in Euclidean signature) they satisfy $2 2 2 p + m > Λ$ ,where $m$ is the corresponding particle mass and $Λ$ is an appropriately chosen cutoff.

Light modes are then all of those which are not heavy. The cutoff $Λ$ , which defines the boundary between these two kinds of modes,is chosen to lie well below the high-energy scale (i.e., well below $m R$ in the toy model),but is also chosen to lie well above the low-energy scale of ultimate interest (like the centre-of-mass energies $E$ of low-energy scattering amplitudes). Notice that in the toy model the heavy degrees of freedom defined by this split include all modes of the field $χ′$ , as well as the high-frequency components of the massless field $θ′$ .

If $h$ and $ℓ$ schematically denote the fields which are, respectively, heavy or light in this characterization, then the influence of heavy fields on light-particle scattering at low energies is completely encoded in the following effective Lagrangian:

The $Λ$ -dependence which is introduced by the low-energy/high-energy split of the integration measure is indicated explicitly in this equation.

Physical observables at low energies are now computed by performing the remaining path integral over the light degrees of freedom only. By virtue of its definition, each configuration in the integration over light fields is weighted by a factor of $exp [i∫ d4x ℒeff(ℓ)]$ implying that the effective Lagrangian weights the low-energy amplitudes in precisely the same way as the classical Lagrangian does for the integral over both heavy and light degrees of freedom. In detail, the effects of virtual contributions of heavy states appear within the low-energy theory through the contributions of new effective interactions, such as are considered in detail for the toy model in some of the next sections (see, e.g., Sections 2.3.3, 2.3.4, and 2.5.2).

Although this kind of low-energy/high-energy split in terms of cutoffs most simply illustrates the conceptual points of interest, in practical calculations it is usually dimensional regularization which is more useful. This is particularly true for theories (like general relativity) involving gauge symmetries, which can be conveniently kept manifest using dimensional regularization. We therefore return to this point in subsequent sections to explain how dimensional regularization can be used with an effective field theory.

2.3.3 Implications for the low-energy limit

Now comes the main point. When applied to the toy model, the condition of symmetry and the restriction to the low-energy limit together have strong implications for $ℒe ff(θ)$ . Specifically:

Invariance of $ℒeff(θ)$ under the symmetry $θ → θ + ω$ implies $ℒeff$ can depend on $θ$ only through the invariant quantity $∂ θ μ$ .
Interest in the low-energy limit permits the expansion of $ℒe ff$ in powers of derivatives of $θ$ . Because only low-energy functional integrals remain to be performed, higher powers of $∂μθ$ correspond in a calculable way to higher suppression of observables by powers of $E∕m ℛ$ .

Combining these two observations leads to the following form for $ℒeff$ :

where the ellipses represent terms which involve more than six derivatives, and so more than two inverse powers of $m ℛ$ .

A straightforward calculation confirms the form (9 ) in perturbation theory, but with the additional information

In this formulation it is clear that each additional factor of $θ$ is always accompanied by a derivative, and so implies an additional power of $r$ in its contribution to all light-particle scattering amplitudes. Because Equation (9 ) is derived assuming only general properties of the low energy effective Lagrangian, its consequences (such as the suppression by $rn$ of low-energy $n$ -point amplitudes) are insensitive of the details of the underlying model. They apply, in particular, to all orders in $λ$ .

Conversely, the details of the underlying physics only enter through specific predictions, such as Equations (10 ), for the low-energy coefficients $a$ , $b$ , and $c$ . Different models having a $U(1)$ Goldstone boson in their low-energy spectrum can differ in the low-energy self-interactions of this particle only through the values they predict for these coefficients.

2.3.4 Redundant interactions

The effective Lagrangian (9 ) does not contain all possible polynomials of $∂ μθ$ . For example, two terms involving 4 derivatives which are not written are

where $d$ and $e$ are arbitrary real constants. These terms are omitted because their inclusion would not alter any of the predictions of $ℒ eff$ . Because of this, interactions such as those in Equation (11

) are known as redundant interactions.

There are two reasons why such terms do not contribute to physical observables. The first reason is the old saw that states that total derivatives may be dropped from an action. More precisely, such terms may be integrated to give either topological contributions or surface terms evaluated at the system’s boundary. They may therefore be dropped provided that none of the physics of interest depends on the topology or what happens on the system’s boundaries. (See, however, [2 ] and references therein for a concrete example where boundary effects play an important role within an effective field theory.) Certainly boundary terms are irrelevant to the form of the classical field equations far from the boundary. They also do not contribute perturbatively to scattering amplitudes, as may be seen from the Feynman rules which are obtained from a simple total derivative interaction like

since these are proportional to

This shows that the two interactions of Equation (11

) are not independent, since we can integrate by parts to replace the couplings $(d,e)$ with $(d′,e′) = (d − e,0)$ .

The second reason why interactions might be physically irrelevant (and so redundant) is if they may be removed by performing a field redefinition. For instance under the infinitesimal redefinition $δθ = A □ θ$ , the leading term in the low-energy action transforms to

This redefinition can be used to set the effective coupling $e$ to zero, simply by choosing $2 2Av = e$ . This argument can be repeated order-by-order in powers of $1∕mR$ to remove more and more terms in $ℒeff$ without affecting physical observables.

Since the variation of the lowest-order action is always proportional to its equations of motion, it is possible to remove in this way any interaction which vanishes when evaluated at the solution to the lower-order equations of motion. Of course, a certain amount of care must be used when so doing. For instance, if our interest is in how the $θ$ -field affects the interaction energy of classical sources, we must add a source coupling $Δℒ = J μ∂μθ$ to the Lagrangian. Once this is done the lowest-order equations of motion become $2v2□ θ = ∂μJ μ$ , and so an effective interaction like $□ θ□ θ$ is no longer completely redundant. It is instead equivalent to the contact interactions like $μ 2 4 (∂μJ ) ∕(4v )$ .