where the
's are tree-level amplitudes in a gravity theory, the
's are color-stripped tree-level amplitudes in a gauge theory,
and
. In these equations the polarization and momentum labels are
suppressed, but the label ``
'' is kept to distinguish the external legs. The coupling
constants have been removed from the amplitudes, but are
reinserted below in Eqs. (12
) and (13
). An explicit generalization to
n
-point field theory gravity amplitudes may be found in
Appendix A of Ref. [23
]. The KLT relations before the field theory limit is taken may,
of course, be found in the original paper [85
].
The KLT equations generically hold for any closed string
states, using their Fock space factorization into pairs of open
string states. Although not obvious, the gravity
amplitudes (10) and (11
) have all the required symmetry under interchanges of identical
particles. (This is easiest to demonstrate in string theory by
making use of an
SL
(2,
Z) symmetry on the string world sheet.)
In the field theory limit the KLT equations must hold in any dimension, because the gauge theory amplitudes appearing on the right-hand side have no explicit dependence on the space-time dimension; the only dependence is implicit in the number of components of momenta or polarizations. Moreover, if the equations hold in, say, ten dimensions, they must also hold in all lower dimensions since one can truncate the theory to a lower-dimensional subspace.
The amplitudes on the left-hand side of Eqs. (10) and (11
) are exactly the scattering amplitudes that one obtains via
standard gravity Feynman rules [44,
45,
138]. The gauge theory amplitudes on the right-hand side may be
computed via standard Feynman rules available in any modern
textbook on quantum field theory [107,
141]. After computing the full gauge theory amplitude, the
color-stripped partial amplitudes
appearing in the KLT relations (10
) and (11
), may then be obtained by expressing the full amplitudes in a
color trace basis [8,
90,
98,
99
,
48
]:
where the sum runs over the set of all permutations, but with
cyclic rotations removed and
g
as the gauge theory coupling constant. The
partial amplitudes that appear in the KLT relations are defined
as the coefficients of each of the independent color traces. In
this formula, the
are fundamental representation matrices for the Yang-Mills gauge
group
, normalized so that
. Note that the
are completely independent of the color and are the same for any
value of
. Eq. (12
) is quite similar to the way full open string amplitudes are
expressed in terms of the string partial amplitudes by dressing
them with Chan-Paton color factors [106].
Instead, it is somewhat more convenient to use color-ordered
Feynman rules [99,
48,
20
] since they directly give the
color-stripped gauge theory amplitudes appearing in the KLT
equations. These Feynman rules are depicted in Fig.
6
. When obtaining the partial amplitudes from these Feynman rules
it is essential to order the external legs following the order
appearing in the corresponding color trace. One may view the
color-ordered gauge theory rules as a new set of Feynman rules
for gravity theories at tree-level, since the KLT relations allow
one to convert the obtained diagrams to tree-level gravity
amplitudes [14
] as shown in Fig.
6
.
To obtain the full amplitudes from the KLT relations in
Eqs. (10), (11
) and their
n
-point generalization, the couplings need to be reinserted. In
particular, when all states couple gravitationally, the full
gravity amplitudes including the gravitational coupling constant
are:
where
expresses the coupling
in terms of Newton's constant
. In general, the precise combination of coupling constants
depends on how many of the interactions are gauge or other
interactions and how many are gravitational.
For the case of four space-time dimensions, it is very
convenient to use helicity representation for the physical
states [36,
87
,
142
]. With helicity amplitudes the scattering amplitudes in either
gauge or gravity theories are, in general, remarkably compact,
when compared with expressions where formal polarization vectors
or tensors are used. For each helicity, the graviton polarization
tensors satisfy a simple relation to gluon polarization
vectors:
The
are essentially ordinary circular polarization vectors
associated with, for example, circularly polarized light. The
graviton polarization tensors defined in this way automatically
are traceless,
, because the gluon helicity polarization vectors satisfy
. They are also transverse,
, because the gluon polarization vectors satisfy
, where
is the four momentum of either the graviton or gluon.
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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 |