The KLT relations in field theory

3.2 The KLT relations in field theory

The fact that the KLT relations hold for the extensive variety of compactified string models [102 , 103 , 49 , 50 , 87 , 3 ] implies that they should also be generally true in field theories of gravity. For the cases of four- and five-particle scattering amplitudes, in the field theory limit the KLT relations [86

] reduce to:

where the $Mn$ ’s are tree-level amplitudes in a gravity theory, the $An$ ’s are color-stripped tree-level amplitudes in a gauge theory, and $2 sij ≡ (ki + kj)$ . In these equations the polarization and momentum labels are suppressed, but the label “ $j = 1,...,n$ ” 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 [86

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 $An$ appearing in the KLT relations (10 ) and (11 ), may then be obtained by expressing the full amplitudes in a color trace basis [8 , 91 , 100 , 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 $An$ partial amplitudes that appear in the KLT relations are defined as the coefficients of each of the independent color traces. In this formula, the $T ai$ are fundamental representation matrices for the Yang–Mills gauge group $SU (Nc )$ , normalized so that $Tr (TaT b) = δab$ . Note that the $An$ are completely independent of the color and are the same for any value of $Nc$ . 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 $An$ color-stripped gauge theory amplitudes appearing in the KLT equations. These Feynman rules are depicted in Figure 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 Figure 6 .

Figure 6: The color-ordered gauge theory Feynman rules for obtaining tree-level scattering amplitudes for gravity coupled to matter, via the KLT equations. The Greek indices are space-time ones and the Latin ones are group theory ones. The curly lines are vectors, dotted ones scalars, and the solid ones fermions.

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 $κ2 = 32πGN$ expresses the coupling $κ$ in terms of Newton’s constant $GN$ . 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 , 88 , 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 $± 𝜀μ$ are essentially ordinary circular polarization vectors associated with, for example, circularly polarized light. The graviton polarization tensors defined in this way automatically are traceless, $𝜀±μ μ= 0$ , because the gluon helicity polarization vectors satisfy $𝜀± ⋅ 𝜀± = 0$ . They are also transverse, $𝜀± k ν = 𝜀± kμ = 0 μν μν$ , because the gluon polarization vectors satisfy $𝜀± ⋅ k = 0$ , where $kμ$ is the four momentum of either the graviton or gluon.