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
where the sum is over interactions
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 [29].)
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:
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 [143, 85, 129, 79], 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
.