Step 4a is required when the low-energy expansion is being used as an efficient means to accurately calculating observables in a well-understood theory. It is the option of choosing instead Step 4b, however, which introduces much of the versatility of effective-Lagrangian methods. Step 4b is useful both when the underlying theory is not known (such as when searching for physics beyond the Standard Model) and when the underlying physics is known but complicated (like when describing the low-energy interactions of pions in quantum chromodynamics).
The effective Lagrangian is in this way seen to
be predictive even though it is not renormalizable in the usual
sense. In fact, renormalizable theories are simply the special case
of Step 4b
where one stops at order , and so are the ones which
dominate in the limit that the light and heavy scales are very
widely separated. We see in this way why renormalizable interactions play
ubiquitous roles through physics! These observations have important
conceptual implications for the quantum behaviour of other
non-renormalizable theories, such as gravity, to which we return in
the next Section 3.
The effective Lagrangian of the toy model seems
to carry much more information when is used to represent
the light particles than it would if
were used. How can
physics depend on the fields which are used to parameterize the
theory?
Physical quantities do not depend on what
variables are used to describe them, and the low-energy scattering
amplitude is suppressed by the same power of
in the toy model regardless of whether it is the effective
Lagrangian for
or
which is used at an
intermediate stage of the calculation.
The final result would nevertheless appear quite
mysterious if were used as the low-energy variable,
since it would emerge as a cancellation only at the end of the
calculation. With
the result is instead manifest at every
step. Although the physics does not depend on the variables in
terms of which it is expressed, it nevertheless pays mortal
physicists to use those variables which make manifest the
symmetries of the underlying system.
The definition of appears to depend
on lots of calculational details, like the value of
(or, in dimensional regularization, the matching
scale) and the minutae of how the cutoff is implemented. Why
doesn’t
depend on all of these details?
generally does depend on all of the regularizational
details. But these details all must cancel in final expressions for
physical quantities. Thus, some
-dependence enters into
scattering amplitudes through the explicit dependence which is
carried by the couplings of
(beyond tree level). But
also potentially enters scattering amplitudes
because loops over all light degrees of freedom must be cut off at
in the effective theory, by definition. The
cancellation of these two sources of cutoff-dependence is
guaranteed by the observation that
enters only as a
bookmark, keeping track of the light and heavy degrees of freedom
at intermediate steps of the calculation.
This cancellation of in all physical
quantities ensures that we are free to make any choice of cutoff
which makes the calculation convenient. After all, although all
regularization schemes for
give the same answers, more
work is required for some schemes than for others. Again, mere
mortal physicists use an inconvenient scheme at their own peril!
In practice this is not a problem, so long as the
effective interactions are chosen to properly reproduce the
dimensionally-regularized scattering amplitudes of the full theory
(order-by-order in ). This is possible ultimately because
the difference between the cutoff- and dimensionally-regularized
low-energy theory can itself be parameterized by appropriate local
effective couplings within the low-energy theory. Consequently, any
regularization-dependent properties will necessarily drop out of
final physical results, once the (renormalized) effective couplings
are traded for physical observables.
In practice this means that one does not
construct a dimensionally-regularized effective theory by
explicitly performing a path integral over successively
higher-energy momentum modes of all fields in the underlying
theory. Instead one defines effective dimensionally regularized
theories for which heavy fields are completely removed. For
instance, suppose it is the low-energy influence of a heavy
particle having mass
which is of interest. Then
the high-energy theory consists of a dimensionally-regularized
collection of light fields
and
, while the effective theory is a
dimensionally-regularized theory of the light fields
only. The effective couplings of the low-energy
theory are obtained by performing a matching calculation, whereby the couplings
of the low-energy effective theory are chosen to reproduce
scattering amplitudes or Green’s functions of the underlying theory
order-by-order in powers of the inverse heavy scale
. Once the couplings of the effective theory are
determined in this way in terms of those of the underlying
fundamental theory, they may be used to compute any purely
low-energy observable.
An important technical point arises if calculations are being done to one-loop accuracy (or more) using dimensional regularization. For these calculations it is convenient to trade the usual minimal-subtraction (or modified-minimal-subtraction) renormalization scheme, for a slightly modified decoupling subtraction scheme [149, 124, 125]. In this scheme couplings are defined using minimal (or modified-minimal) subtraction between successive particle threshholds, with the couplings matched from the underlying theory to the effective theory as each heavy particle is successively integrated out. This results in a renormalization group evolution of effective couplings which is almost as simple as for minimal subtraction, but with the advantage that the implications of heavy particles in running couplings are explicitly decoupled as one passes to energies below the heavy particle mass. Some textbooks which describe effective Lagrangians are [74, 59]; some reviews articles which treat low-energy effective field theories (mostly focussing on pion interactions) are [117, 108, 114, 133, 127, 100, 75].
A great advantage of the
dimensionally-regularized effective theory is the absence of the
cutoff scale , which implies that the only
dimensionful scales which arise are physical particle masses. This
was implicitly used in the power-counting arguments given earlier,
wherein integrals over loop momenta were replaced by powers of
heavy masses on dimensional grounds. This gives a sufficiently
accurate estimate despite the ultraviolet divergences in these
integrals, provided the integrals are dimensionally regularized.
For effective theories it is powers of the arbitrary cutoff scale
which would arise in these estimates, and because
cancels out of physical quantities, this just
obscures how heavy physical masses appear in the final results.