where
R
is the usual scalar curvature and
is the Yang-Mills field strength. An inspection of these two
Lagrangians does not reveal any obvious factorization property
that might explain the KLT relations. Indeed, one might be
tempted to conclude that the KLT equations could not possibly
hold in pure Einstein gravity. However, although somewhat
obscure, the Einstein-Hilbert Lagrangian can in fact be
rearranged into a form that is compatible with the KLT relations
(as argued in this section). Of course, there should be such a
rearrangement, given that in the low energy limit pure graviton
tree amplitudes in string theory should match those of Einstein
gravity. All other string states either decouple or cannot enter
as intermediate states in pure graviton amplitudes because of
conservation laws. Indeed, explicit calculations using ordinary
gravity Feynman rules confirm this to be true [117,
26
,
14]. (In loops, any state of the string that survives in the low
energy limit will in fact contribute, but in this section only
tree amplitudes are being considered.)
One of the key properties exhibited by the KLT
relations (10
) and (11) is the separation of graviton space-time indices into `left'
and `right' sets. This is a direct consequence of the
factorization properties of closed strings into open strings.
Consider the graviton field,
. We define the
index to be a ``left'' index and the
index to be a ``right'' one. In string theory, the ``left''
space-time indices would arise from the world-sheet left-mover
oscillator and the ``right'' ones from the right-mover
oscillators. Of course, since
is a symmetric tensor it does not matter which index is assigned
to the left or to the right. In the KLT relations each of the two
indices of a graviton are associated with two distinct gauge
theories. For convenience, we similarly call one of the gauge
theories the ``left'' one and the other the ``right'' one. Since
the indices from each gauge theory can never contract with the
indices of the other gauge theory, it must be possible to
separate all the indices appearing in a gravity amplitude into
left and right classes such that the ones in the left class only
contract with left ones and the ones in the right class only with
right ones.
This was first noted by Siegel, who observed that it should be
possible to construct a complete field theory formalism that
naturally reflects the left-right string theory factorization of
space-time indices. In a set of remarkable papers [124,
123,
125], he constructed exactly such a formalism. With appropriate
gauge choices, indices separate exactly into ``right'' and
``left'' categories, which do not contract with each other. This
does not provide a complete explanation of the KLT relations,
since one would still need to demonstrate that the gravity
amplitudes can be expressed directly in terms of gauge theory
ones. Nevertheless, this formalism is clearly a sensible starting
point for trying to derive the KLT relations directly from
Einstein gravity. Hopefully, this will be the subject of future
studies, since it may lead to a deeper understanding of the
relationship of gravity to gauge theory. A Lagrangian with the
desired properties could, for example, lead to more general
relations between gravity and gauge theory classical
solutions.
Here we outline a more straightforward order-by-order
rearrangement of the Einstein-Hilbert Lagrangian, making it
compatible with the KLT relations [26]. A useful side-benefit is that this provides a direct
verification of the KLT relations up to five points starting from
the Einstein-Hilbert Lagrangian in its usual form. This is a
rather non-trivial direct verification of the KLT relations,
given the algebraic complexity of the gravity Feynman rules.
In conventional gauges, the difficulty of factorizing the Einstein-Hilbert Lagrangian into left and right parts is already apparent in the kinetic terms. In de Donder gauge, for example, the quadratic part of the Lagrangian is
so that the propagator is the one given in Eq. (2). Although the first term is acceptable since left and right
indices do not contract into each other, the appearance of the
trace
in Eq. (32) is problematic since it contracts a left graviton index with a
right one. (The indices are raised and lowered using the flat
space metric
and its inverse.)
In order for the kinematic term (32) to be consistent with the KLT equations, all terms which
contract a ``left'' space-time index with a ``right'' one need to
be eliminated. A useful trick for doing so is to introduce a
``dilaton'' scalar field that can be used to remove the graviton
trace from the quadratic terms in the Lagrangian. The appearance
of the dilaton as an auxiliary field to help rearrange the
Lagrangian is motivated by string theory, which requires the
presence of such a field. Following the discussion of
Refs. [25,
26], consider instead a Lagrangian for gravity coupled to a
scalar:
Since the auxiliary field
is quadratic in the Lagrangian, it does not appear in any tree
diagrams involving only external gravitons [26]. It therefore does not alter the tree
S
-matrix of purely external gravitons. (For theories containing
dilatons one can allow the dilaton to be an external physical
state.) In de Donder gauge, for example, taking
, the quadratic part of the Lagrangian including the dilaton
is
The term involving
can be eliminated with the field redefinitions
and
yielding
One might be concerned that the field redefinition might alter
gravity scattering amplitudes. However, because this field
redefinition does not alter the trace-free part of the graviton
field it cannot change the scattering amplitudes of traceless
gravitons [26].
Of course, the rearrangement of the quadratic terms is only the first step. In order to make the Einstein-Hilbert Lagrangian consistent with the KLT factorization, a set of field variables should exist where all space-time indices can be separated into ``left'' and ``right'' classes. To do so, all terms of the form
need to be eliminated since they contract left indices with
right ones. A field redefinition that accomplishes this is [26]:
This field redefinition was explicitly checked in Ref. [26] through
, to eliminate all terms of the type in Eq. (38), before gauge fixing. However, currently there is no formal
understanding of why this field variable choice eliminates terms
that necessarily contract left and right indices.
It turns out that one can do better by performing further
field redefinitions and choosing a particular non-linear gauge.
The explicit forms of these are a bit complicated and may be
found in Ref. [26]. With a particular gauge choice it is possible to express the
off-shell three-graviton vertex in terms of Yang-Mills three
vertices:
where
is the color-ordered Gervais-Neveu [65] gauge Yang-Mills three-vertex, from which the color factor has
been stripped. This is not the only possible reorganization of
the three-vertex that respects the KLT factorization. It just
happens to be a particularly simple form of the vertex. For
example, another gauge that has a three-vertex that factorizes
into products of color-stripped Yang-Mills three-vertices is the
background-field [130,
46,
1] version of de Donder gauge for gravity and Feynman gauge
for QCD. (However, background field gauges are meant for loop
effective actions and not for tree-level
S
-matrix elements.) Interestingly, these gauge choices have a
close connection to string theory [65,
24].
The above ideas represent some initial steps in reorganizing the Einstein-Hilbert Lagrangian so that it respects the KLT relations. An important missing ingredient is a derivation of the KLT equations starting from the Einstein-Hilbert Lagrangian (and also when matter fields are present).
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Perturbative Quantum Gravity and its Relation to Gauge
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