The Einstein–Hilbert Lagrangian and Gauge Theory

4 The Einstein–Hilbert Lagrangian and Gauge Theory

Consider the Einstein–Hilbert and Yang–Mills Lagrangians,

where $R$ is the usual scalar curvature and $Fμaν$ 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, $hμν$ . We define the $μ$ index to be a “left” index and the $ν$ 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 $h μν$ 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 $hμμ$ 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 $η μν$ 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 $ϕ$ 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 $gμν = ημν + κh μν$ , the quadratic part of the Lagrangian including the dilaton is

The term involving $μ hμ$ 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 $𝒪 (h6)$ , 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:

iG μ1ν1,μ2ν2,μ3ν3(k ,k ,k ) = ( ) 1 2 3 − i- κ- [V μ1μ2μ3(k ,k ,k ) × V ν1ν2ν3(k ,k ,k ) 2 2 GN 1 2 3 GN 1 2 3 +V μG2Nμ1μ3(k2,k1,k3) × V μG2Nμ1μ3(k2,k1,k3)], (40 )

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).