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 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 symmetry is spontaneously broken, since its interactions are all suppressed by powers of . Notice that this is true independent of the size of the coupling of the underlying theory.
For example, in the toy model the effective Lagrangian takes the general form
It is straightforward to track the powers of and that interactions of the form (15) contribute to an -loop contribution to the amplitude for the scattering of initial Goldstone bosons into final Goldstone bosons at centre-of-mass energy . The label here denotes the number of external lines in the corresponding graph. (The steps presented in this section closely follow the discussion of .)
With the desire of also being able to include later examples, consider the following slight generalization of the Lagrangian of Equation (15):
Imagine using this Lagrangian to compute a scattering amplitude involving the scattering of relativistic particles whose energy and momenta are of order . We wish to focus on the contribution to due to a Feynman graph having internal lines and vertices. The labels and here indicate two characteristics of the vertices: counts the number of lines which converge at the vertex, and counts the power of momentum which appears in the vertex. Equivalently, counts the number of powers of the fields which appear in the corresponding interaction term in the Lagrangian, and counts the number of derivatives of these fields which appear there.
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: . On the other hand, there are two ends for each internal line, and one end for each external line in the graph: . Equating these gives the identity which expresses the ‘conservation of ends’:
A second useful identity gives the number of loops for each (connected) graph:defining the number of loops.
As usual for a connected graph, all but one of the momentum-conserving -functions in Equation (19) can be used to perform one of the momentum integrals in Equation (20). The one remaining -function which is left after doing so depends only on the external momenta , 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 by
The number of four-momentum integrations which are left after having used all of the momentum-conserving -functions is then . This last equality uses the definition, Equation (18), of the number of loops .
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 are comparable to (or larger than) the masses of the light particles whose scattering is being calculated, then 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
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 :
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 . As a result it permits the determination of which interactions in the effective Lagrangian are required to reproduce any given order in 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 . (Notice also that the factor in Equation (24) implies, all other things being equal, that the scale cannot be taken to be systematically smaller than without ruining the validity of the loop expansion in the effective low-energy theory.)
We now apply this power-counting estimate to the toy model discussed earlier. Using the relations and , we have
Equations (26) and (27) have several noteworthy features:
To see how Equations (26) and (27) are used, consider the first few powers of in the toy model. For any the leading contributions for small come from tree graphs, . The tree graphs that dominate are those for which takes the smallest possible value. For example, for 2-particle scattering , and so precisely one tree graph is possible for which , corresponding to and all other . This identifies the single graph which dominates the 4-point function at low energies, and shows that the resulting leading energy dependence is .
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 contributions are achieved if and only if either (i) and with all others zero, or (ii) and . Since there are no interactions, no one-loop graphs having 4 external lines can be built using precisely one vertex, and so only tree graphs can contribute. Of these, the only two choices allowed by at order are therefore the choices or . Both of these contribute a result of order .
© Max Planck Society and the author(s)