Predictiveness and power counting

2.5 Predictiveness and power counting

The entire rationale of an effective Lagrangian is to incorporate the virtual effects of high-energy particles in low-energy processes, order-by-order in powers of the small ratio $r$ of these two scales (e.g., $r = E ∕m ℛ$ in the toy model). In order to use an effective Lagrangian it is therefore necessary to know which terms contribute to physical processes to any given order in $r$ .

This determination is explicitly possible if the low-energy degrees of freedom are weakly interacting, because in this case perturbation theory in the weak interactions may be analyzed graphically, permitting the use of power-counting arguments to systematically determine where powers of $r$ originate. Notice that the assumption of a weakly-interacting low-energy theory does not presuppose the underlying physics to be also weakly interacting. For instance, for the toy model the Goldstone boson of the low-energy theory is weakly interacting provided only that the $U (1)$ symmetry is spontaneously broken, since its interactions are all suppressed by powers of $r$ . Notice that this is true independent of the size of the coupling $λ$ of the underlying theory.

For example, in the toy model the effective Lagrangian takes the general form

where the sum is over interactions $𝒪id$ , involving $i$ powers of the dimensionless field $θ$ and $d$ derivatives. The power of $m R$ premultiplying each term is chosen to ensure that the coefficient $c id$ is dimensionless. (For instance, the interaction $μ 2 (∂μθ ∂ θ)$ has $i = d = 4$ .) There are three useful properties which all of the operators in this sum must satisfy:

$d$ must be even by virtue of Lorentz invariance.
Since the sum is only over interactions, it does not include the kinetic term, which is the unique term for which $d = i = 2$ .
The $U (1)$ symmetry implies every factor of $θ$ is differentiated at least once, and so $d ≥ i$ . Furthermore, any term linear in $θ$ must therefore be a total derivative, and so may be omitted, implying $i ≥ 2$ without loss.

2.5.1 Power-counting low-energy Feynman graphs

It is straightforward to track the powers of $v$ and $mR$ that interactions of the form (15 ) contribute to an $ℓ$ -loop contribution to the amplitude $𝒜 ℰ(E)$ for the scattering of $Ni$ initial Goldstone bosons into $Nf$ final Goldstone bosons at centre-of-mass energy $E$ . The label $ℰ = Ni + Nf$ here denotes the number of external lines in the corresponding graph. (The steps presented in this section closely follow the discussion of [28 ].)

With the desire of also being able to include later examples, consider the following slight generalization of the Lagrangian of Equation (15 ):

Here $φ$ denotes a generic boson field, $cn$ are again dimensionless coupling constants which we imagine to be at most $𝒪 (1)$ , and $f$ , $M$ , and $v$ are mass scales of the underlying problem. The sum is again over operators which are powers of the fields and their derivatives, and $dn$ is the dimension of the operator $𝒪n$ , in powers of mass. For example, in the toy-model application we have $2 f = v mR$ , $M = m ℛ$ , and we have written $θ = φ ∕v$ . In the toy-model example the sum over $n$ corresponds to the sum over $i$ and $d$ , and $dn = d$ .

Imagine using this Lagrangian to compute a scattering amplitude $𝒜 ℰ(E )$ involving the scattering of $ℰ$ relativistic particles whose energy and momenta are of order $E$ . We wish to focus on the contribution to $𝒜$ due to a Feynman graph having $I$ internal lines and $Vik$ vertices. The labels $i$ and $k$ here indicate two characteristics of the vertices: $i$ counts the number of lines which converge at the vertex, and $k$ counts the power of momentum which appears in the vertex. Equivalently, $i$ counts the number of powers of the fields $φ$ which appear in the corresponding interaction term in the Lagrangian, and $k$ counts the number of derivatives of these fields which appear there.

2.5.1.1 Some useful identities

The positive integers $I$ , $E$ , and $Vik$ , which characterize the Feynman graph in question are not all independent since they are related by the rules for constructing graphs from lines and vertices.

The first such relation can be obtained by equating two equivalent ways of counting how internal and external lines can end in a graph. On the one hand, since all lines end at a vertex, the number of ends is given by summing over all of the ends which appear in all of the vertices: $∑ ik iVik$ . On the other hand, there are two ends for each internal line, and one end for each external line in the graph: $2I + E$ . Equating these gives the identity which expresses the ‘conservation of ends’:

A second useful identity gives the number of loops $L$ for each (connected) graph:

For simple planar graphs, this last equation agrees with the intuitive notion of what the number of loops in a graph means, since it expresses a topological invariant which states how the Euler number for a disc can be expressed in terms of the number of edges, corners, and faces of the triangles in one of its triangularization. For graphs which do not have the topology of a plane, Equation (18

) should instead be read as defining the number of loops.

2.5.1.2 Dimensional estimates

We now collect the dependence of $𝒜 ℰ(a)$ on the parameters appearing in $ℒe ff$ . Reading the Feynman rules from the Lagrangian of Equation (16

) shows that the vertices in the Feynman graph contribute the following factor:

where $p$ generically denotes the various momenta running through the vertex. Similarly, there are $I$ internal lines in the graph, each of which contributes the additional factor:

where, again, $p$ denotes the generic momentum flowing through the line. $m$ generically denotes the mass of any light particles which appear in the effective theory, and it is assumed that the kinetic terms which define their propagation are those terms in $ℒeff$ involving two derivatives and two powers of the fields $φ$ .

As usual for a connected graph, all but one of the momentum-conserving $delta$ -functions in Equation (19 ) can be used to perform one of the momentum integrals in Equation (20 ). The one remaining $delta$ -function which is left after doing so depends only on the external momenta $4 δ (q)$ , and expresses the overall conservation of four-momentum for the process. Future formulae are less cluttered if this factor is extracted once and for all, by defining the reduced amplitude $𝒜&tidle;$ by

Here $q$ generically represents the external four-momenta of the process, whose components are of order $E$ in size.

The number of four-momentum integrations which are left after having used all of the momentum-conserving $delta$ -functions is then $∑ I − ik Vik + 1 = L$ . This last equality uses the definition, Equation (18 ), of the number of loops $L$ .

We now estimate the result of performing the integration over the internal momenta. To do so it is most convenient to regulate the ultraviolet divergences which arise using dimensional regularization² . For dimensionally-regularized integrals, the key observation is that the size of the result is set on dimensional grounds by the light masses or external momenta of the theory. That is, if all external energies $q$ are comparable to (or larger than) the masses $m$ of the light particles whose scattering is being calculated, then $q$ is the light scale controlling the size of the momentum integrations, so dimensional analysis implies that an estimate of the size of the momentum integrations is

with a dimensionless pre-factor which depends on the dimension $n$ of spacetime, and which may be singular in the limit that $n → 4$ . Notice that the assumption that $q$ is the largest relevant scale in the low-energy theory explicitly excludes the case of the scattering of non-relativistic particles.

One might worry whether such a simple dimensional argument can really capture the asymptotic dependence of a complicated multi-dimensional integral whose integrand is rife with potential singularities. The ultimate justification for this estimate lies with general results like Weinberg’s theorem [142 , 84 , 128 , 78 ], which underly the power-counting analyses of renormalizability. These theorems ensure that the simple dimensional estimates capture the correct behaviour up to logarithms of the ratios of high- and low-energy mass scales.

With this estimate for the size of the momentum integrations, we find the following contribution to the amplitude $&tidle; 𝒜 ℰ(E)$ :

where $∑ X = ik kVik$ and $∑ Y = 4L − 2I + ik kVik$ . Liberal use of the identities (17

) and (18

), and taking $q ∼ E$ , allows this to be rewritten as the following estimate:

with $P = 2 + 2L + ∑ (k − 2)Vik ik$ . Equivalently, if we group terms depending on $L$ , Equation (24

) may also be written as

with $P ′ = 2 + ∑ (k − 2)Vik ik$ .

Equation (24 ) is the principal result of this section. Its utility lies in the fact that it links the contributions of the various effective interactions in the effective Lagrangian (16 ) with the dependence of observables on small energy ratios such as $r = E ∕M$ . As a result it permits the determination of which interactions in the effective Lagrangian are required to reproduce any given order in $E ∕M$ in physical observables.

Most importantly, Equation (24 ) shows how to calculate using non-renormalizable theories. It implies that even though the Lagrangian can contain arbitrarily many terms, and so potentially arbitrarily many coupling constants, it is nonetheless predictive so long as its predictions are only made for low-energy processes, for which $E ∕M ≪ 1$ . (Notice also that the factor $(M ∕f )4L$ in Equation (24 ) implies, all other things being equal, that the scale $f$ cannot be taken to be systematically smaller than $M$ without ruining the validity of the loop expansion in the effective low-energy theory.)

2.5.2 Application to the toy model

We now apply this power-counting estimate to the toy model discussed earlier. Using the relations $2 f = vm ℛ$ and $M = m ℛ$ , we have

where $∑ P = 2 + 2L + id(d − 2 )Vid$ . As above, $Vid$ counts the number of times an interaction involving $i$ powers of fields and $d$ derivatives appears in the amplitude. An equivalent form for this expression is

Equations (26 ) and (27 ) have several noteworthy features:

The condition $d ≥ i ≥ 2$ ensures that all of the powers of $E$ appearing in $𝒜 ℰ$ are positive. Furthermore, since $d = 2$ only occurs for the kinetic term, $d ≥ 4$ for all interactions, implying $𝒜 ℰ$ is suppressed by at least 4 powers of $E$ , and higher loops cost additional powers of $E$ . Furthermore, it is straightforward to identify the graphs which contribute to $𝒜 ℰ$ to any fixed power of $E$ .
As is most clear from Equation (27 ), perturbation theory in the effective theory relies only on the assumptions $E ≪ 4 πv$ and $E ≪ m ℛ$ . In particular, it does not rely on the ratio $m ℛ∕(4πv )$ being small. Since $m ℛ = λv$ in the underlying toy model, this ratio is simply of order $λ∕(4π)$ , showing how weak coupling for the Goldstone boson is completely independent of the strength of the couplings in the underlying theory.

To see how Equations (26 ) and (27 ) are used, consider the first few powers of $E$ in the toy model. For any $ℰ$ the leading contributions for small $E$ come from tree graphs, $L = 0$ . The tree graphs that dominate are those for which $∑ ′ id(d − 2)Vid$ takes the smallest possible value. For example, for 2-particle scattering $ℰ = 4$ , and so precisely one tree graph is possible for which $∑ ′ id(d − 2)Vid = 2$ , corresponding to $V44 = 1$ and all other $Vid = 0$ . This identifies the single graph which dominates the 4-point function at low energies, and shows that the resulting leading energy dependence is $4 2 2 𝒜4 (E ) ∼ E ∕(v m R)$ .

The utility of power-counting really becomes clear when subleading behaviour is computed, so consider the size of the leading corrections to the 4-point scattering amplitude. Order $E6$ contributions are achieved if and only if either (i) $L = 1$ and $Vi4 = 1$ with all others zero, or (ii) $L = 0$ and $∑ (4V + 2V ) = 4 i i6 i4$ . Since there are no $d = 2$ interactions, no one-loop graphs having 4 external lines can be built using precisely one $d = 4$ vertex, and so only tree graphs can contribute. Of these, the only two choices allowed by $ℰ = 4$ at order $6 E$ are therefore the choices $V46 = 1$ or $V34 = 2$ . Both of these contribute a result of order $𝒜4 (E ) ∼ E6∕(v2 m4ℛ)$ .