] were the first to exploit the KLT relations to obtain
amplitudes in Einstein gravity. In quantum chromodynamics (QCD)
an infinite set of helicity amplitudes known as the Parke-Taylor
amplitudes [105,
9,
89] were already known. These maximally helicity violating (MHV)
amplitudes describe the tree-level scattering of
n
gluons when all gluons but two have the same helicity, treating
all particles as outgoing. (The tree amplitudes in which all or
all but one of the helicities are identical vanish.) BGK used the
KLT relations to directly obtain graviton amplitudes in pure
Einstein gravity, using the known QCD results as input.
Remarkably, they also were able to obtain a compact formula for
n
-graviton scattering with the special helicity configuration in
which two legs are of opposite helicity from the remaining ones.
As a particularly simple example, the color-stripped four-gluon tree amplitude with two minus helicities and two positive helicities in QCD is given by
where the
g
subscripts signify that the legs are gluons and the
superscripts signify the helicities. With the conventions used
here, helicities are assigned by treating all particles as
outgoing. (This differs from another common choice which is to
keep track of which particles are incoming and which are
outgoing.) In these amplitudes, for simplicity, overall phases
have been removed. The gauge theory partial amplitude in
Eq. (15
) may be computed using the color-ordered Feynman diagrams
depicted in Fig.
7
. The diagrams for the partial amplitude in Eq. (16) are similar except that the labels for legs 3 and 4 are
interchanged. Although QCD contains fermion quarks, they do not
contribute to tree amplitudes with only external gluon legs
because of fermion number conservation; for these amplitudes QCD
is entirely equivalent to pure Yang-Mills theory.
).
The corresponding four-graviton amplitude follows from the KLT
equation (10). After including the coupling from Eq. (13), the four-graviton amplitude is:
where the subscript
h
signifies that the particles are gravitons and, as with the
gluon amplitudes, overall phases are removed. As for the case of
gluons, the
superscripts signify the helicity of the graviton. This
amplitude necessarily must be identical to the result for pure
Einstein gravity with no other fields present, because any other
states, such as an anti-symmetric tensor, dilaton, or fermion, do
not contribute to
n
-graviton tree amplitudes. The reason is similar to the reason
why the quarks do not contribute to pure glue tree amplitudes in
QCD. These other physical states contribute only when they appear
as an external state, because they couple only in pairs to the
graviton. Indeed, the amplitude (17) is in complete agreement with the result for this helicity
amplitude obtained by direct diagrammatic calculation using the
pure gravity Einstein-Hilbert action as the starting point [7] (and taking into account the different conventions for
helicity).
The KLT relations are not limited to pure gravity amplitudes.
Cases of gauge theory coupled to gravity have also been discussed
in Ref. [14]. For example, by applying the Feynman rules in Fig.
6, one can obtain amplitudes for gluon amplitudes dressed with
gravitons. A sampling of these, to leading order in the graviton
coupling, is
for the coefficients of the color traces
following the ordering of the gluon legs. Again, for simplicity,
overall phases are eliminated from the amplitudes. (In
Ref. [14] mixed graviton matter amplitudes including the phases may be
found.)
These formulae have been generalized to infinite sequences of
maximally helicity-violating tree amplitudes for gluon amplitudes
dressed by external gravitons. The first of these were obtained
by Selivanov using a generating function technique [121]. Another set was obtained using the KLT relations to find the
pattern for an arbitrary number of legs [14]. In doing this, it is extremely helpful to make use of the
analytic properties of amplitudes as the momenta of various
external legs become soft (i.e.
) or collinear (i.e.
parallel to
), as discussed in the next subsection.
Cases involving fermions have not been systematically studied, but at least for the case with a single fermion pair the KLT equations can be directly applied using the Feynman rules in Fig. 6, without any modifications. For example, in a supergravity theory, the scattering of a gravitino by a graviton is
where the subscript
signifies a spin 3/2 gravitino and
signifies a spin 1/2 gluino. As a more subtle example, the
scattering of fundamental representation quarks by gluons via
graviton exchange also has a KLT factorization:
where
q
and
Q
are distinct massless fermions. In this equation, the gluons are
factorized into products of fermions. On the right-hand side the
group theory indices
are interpreted as global flavor indices but on the left-hand
side they should be interpreted as color indices of local gauge
symmetry. As a check, in Ref. [14], for both amplitudes (19) and (20), ordinary gravity Feynman rules were used to explicitly verify
that the expressions for the amplitudes are correct.
Cases with multiple fermion pairs are more involved. In particular, for the KLT factorization to work in general, auxiliary rules for assigning global charges in the color-ordered amplitudes appear to be necessary. This is presumably related to the intricacies associated with fermions in string theory [62].
When an underlying string theory does exist, such as for the case of maximal supergravity discussed in Section 7, then the KLT equations necessarily must hold for all amplitudes in the field theory limit. The above examples, however, demonstrate that the KLT factorization of amplitudes is not restricted only to the cases where an underlying string theory exists.
|
Perturbative Quantum Gravity and its Relation to Gauge
Theory
Zvi Bern http://www.livingreviews.org/lrr-2002-5 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |
