A power-counting estimate for the -loop contribution to the
-point
graviton-scattering amplitude
, which involves
vertices involving
derivatives and the
emission or absorption of
gravitons, may be found by
arguments identical to those used previously for the toy model. The
main difference from the toy-model analysis is the existence for
gravity of interactions involving two derivatives, which all come
from the Einstein-Hilbert term in
. (Such terms also
arise for Goldstone bosons for symmetry-breaking patterns involving
non-Abelian groups and are easily incorporated into the analysis.)
The resulting estimate for
turns out to be of order
Equations (31) and (32
) share many of the
noteworthy features of Equations (26
) and (27
). Again the weakness
of the graviton’s coupling follows only from the low-energy
approximations,
and
. When written as
in Equation (32
), it is clear that
even though the ratio
is potentially much larger
than
, it does not actually arise in
unless contributions from at least curvature-cubed
interactions are included (for which
).
These expressions permit a determination of the
dominant low-energy contributions to scattering amplitudes. The
minimum suppression comes when and
, and so is given by arbitrary tree graphs
constructed purely from the Einstein-Hilbert action. We are led in
this way to what we are in any case inclined to believe: It is
classical general relativity which governs the low-energy dynamics
of gravitational waves!
But the next-to-leading contributions are also
quite interesting. These arise in one of two ways, either (i) and
for any
, or (ii)
,
,
is arbitrary, and all other
vanish. That is, the next to leading contribution is
obtained by computing the one-loop corrections using only Einstein
gravity, or by working to tree level and including precisely one
curvature-squared interaction in addition to any number of
interactions from the Einstein-Hilbert term. Both are suppressed
compared to the leading term by a factor of
, and the one-loop contribution carries an additional
factor of
.
For instance, for 2-body graviton scattering we
have , and so the above arguments imply the leading
behaviour is
, where the numbers
and
have been explicitly calculated. At tree level all
of the amplitudes turn out to vanish except for those which are
related by crossing symmetry to the amplitude for which all
graviton helicities have the same sign, and this is given
by [52]:
The one-loop correction to this result has also
been computed [64], and is
infrared divergent. These infrared divergences cancel in the usual
way with tree-level bremsstrahlung diagrams [144], leading to a
finite result [62], which is
suppressed as expected relative to the tree contribution by terms
of order , up to logarithmic corrections.
The observables of most practical interest for experimental purposes involve the gravitational interactions of various kinds of matter. It is therefore useful to generalize the previous arguments to include matter and gravity coupled to one another. In most situations this generalization is reasonably straightforward, but somewhat paradoxically it is more difficult to treat the interactions of non-relativistic matter than of relativistic matter. This section describes the reasons for this difference.
Particular kinds of higher-derivative terms
involving the matter fields may also be included equally trivially,
provided the mass scales which appear in these terms appear in the
same way as they did for the graviton. For instance, scalar
functions built from arbitrary powers of and its
derivatives
can be included, along the lines of
Similar estimates also apply if a mass for the scalar field is included, provided that this
mass is not larger than the energy flowing through the external
lines:
. This kind of mass does not change the
power-counting result appreciably for observables which are
infrared finite (which may require, as mentioned above summing over
an indeterminate number of soft final gravitons). They do not
change the result because infrared-finite quantities are at most
logarithmically singular as
[150], and so
their expansion in
simply adds terms for which factors of
are replaced by smaller factors of
. But the above discussion can change dramatically if
, since an important ingredient in the dimensional
estimate is the assumption that the largest scale in the graph is
the external energy
. Consequently the power-counting given
above only directly applies to relativistic particles.
The case of non-relativistic particles is also of
real practical interest for the applications of effective field
theories in other branches of physics. This is so, even though one
might think that an effective theory should contain only particles
which are very light. Non-relativistic particles can nevertheless
arise in practice within an effective field theory, even particles
having masses which are large compared to those of the particles
which were integrated out to produce the effective field theory in
the first place. Such massive particles can appear consistently in
a low-energy theory provided they are stable (or extremely
long-lived), and so cannot decay and release enough energy to
invalidate the low-energy approximation. Some well-known examples
of this include the low-energy nuclear interactions of nucleons (as
described within chiral perturbation theory [153, 154, 101, 110]), the
interactions of heavy fermions like the and
quark (as described by heavy-quark effective theory
(HQET) [91, 92, 109]), and the
interactions of electrons and nuclei in atomic physics (as
described by non-relativistic quantum electrodynamics
(NRQED) [34
, 112
, 104
, 105
, 128
, 111
]).
The key to understanding the effective field theory for very massive, stable particles at low energies lies in the recognition that their anti-particles need not be included since they would have already been integrated out to obtain the effective field theory of interest. As a result heavy-particle lines within the Feynman graphs of the effective theory only directly connect external lines, and never arise as closed loops.
The most direct approach to estimating the size
of quantum corrections in this case is to power-count as before,
subject to the restriction that graphs including internal closed
loops of heavy particles are to be excluded. Donoghue and
Torma [61] have performed such
a power-counting analysis along these lines, and shows that quantum
effects remain suppressed by powers of light-particle energies (or small momentum
transfers) divided by
through the first few
nontrivial orders of perturbation theory. Although heavy-particle
momenta can be large,
, they only arise in physical
quantities through the small relativistic parameter
rather than through
, extending the
suppression of quantum effects obtained earlier to non-relativistic
problems.
Unfortunately, if a calculation is performed
within a covariant gauge, individual Feynman graphs can depend on large powers like , even though these all cancel in physical
amplitudes. For this reason an all-orders inductive proof of the
above power-counting remains elusive. As Donoghue and
Torma [61
] also make clear,
progress towards such an all-orders power-counting result is likely
to be easiest within a physical, non-covariant gauge, since such a
gauge allows powers of small quantities like
to be most easily followed.
Consider, then, a complex massive scalar field (we take a complex field to ensure low-energy conservation of heavy-particle number) having action
for which the conserved current for heavy-particle number is Our interest is in exhibiting the leading couplings of this field to gravity, organized in inverse powers ofWhen treating non-relativistic particles it is
convenient to remove the rest mass of the heavy particle from the
energy, since (by assumption) this energy is not available to other
particles in the low-energy theory. For the observers just
described this can be done by extracting a time-dependent phase
from the heavy-particle field according to .
is
chosen for later convenience, to ensure a conventional
normalization for the field
. With this choice we have
, and the extra
-dependence introduced this way has the effect of
making the large-
limit of the positive-frequency part of
a relativistic action easier to follow.
With these variables the action for the scalar field becomes
and the conserved current for heavy-particle number becomes HereNotice that for Minkowski space, where , the first term in
vanishes, leaving a result which is finite in the
limit. Furthermore - keeping in mind
that the leading time and space derivatives are of the same order
of magnitude (
) - the leading large-
part of
is equivalent to the usual
non-relativistic Schrödinger Lagrangian density,
. In the same limit the
density of
particles also takes the standard
Schrödinger form
.
The next step consists of integrating out the
anti-particles, which (by assumption) cannot be produced by the
low-energy physics of interest. In principle, this can be done by
splitting the relativistic field into its positive- and
negative-frequency parts
, and performing the
functional integral over the negative-frequency part
. (To leading order this often simply corresponds to
setting the negative-frequency part to zero.) Once this has been
done the fields describing the heavy particles have the
non-relativistic expansion
Writing the heavy-particle action in this way
extends the standard parameterized post-Newtonian (PPN)
expansion [69, 67, 68, 147] to the effective
quantum theory, and so forms the natural setting for an all-orders
power-counting analysis which keeps track of both quantum and
relativistic effects. For instance, for weak gravitational fields
having , the leading gravitational
coupling for large
may be read off from Equation (39
) to be
The real power of the effective theory lies in
identifying the subdominant contributions in powers of , however, and the above discussion shows that
different components of the metric couple to matter at different
orders in this small quantity. Once
is shifted by the
static non-relativistic Newtonian potential, however, the remaining
contributions are seen to couple with a strength which is
suppressed by negative powers of
, rather than
positive powers. A full power-counting analysis using such an
effective theory, along the lines of the analogous electromagnetic
problems [34, 112, 104, 105, 128, 111], would be very
instructive.