3.3 Tree-level applications

Using a helicity representation [3688142], Berends, Giele, and Kuijf (BGK) [10Jump To The Next Citation Point] 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 [105990] 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

Atree(1− ,2−,3+ ,4+) = s12, (15 ) 4 g g g g s23
tree − − + + s12 A 4 (1g ,2g ,4g ,3g ) = s24, (16 )
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. (15View Equation) may be computed using the color-ordered Feynman diagrams depicted in Figure 7View Image. The diagrams for the partial amplitude in Eq. (16View Equation) 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.
View Image

Figure 7: The three color-ordered Feynman diagrams contributing to the QCD partial amplitude in Eq. (15View Equation).

The corresponding four-graviton amplitude follows from the KLT equation (10View Equation). After including the coupling from Eq. (13View Equation), the four-graviton amplitude is:

( ) tree − − + + κ- 2 s12 s12 ℳ 4 (1 h,2h,3h ,4h) = 2 s12 × s23 × s24, (17 )
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 (17View Equation) 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. [14Jump To The Next Citation Point]. For example, by applying the Feynman rules in Figure 6View Image, one can obtain amplitudes for gluon amplitudes dressed with gravitons. A sampling of these, to leading order in the graviton coupling, is

M4 (1+g ,2+g ,3 −h,4−h) = 0, | | − + − + ( κ)2 || s523 ||1∕2 M4 (1 g ,2g ,3 h,4h) = 2- ||s--s2-|| , 13 12 ( ) || 4 ||1∕2 (18 ) M (1+,2+ ,3+,4 −,5−) = g2 κ- ||----s45----|| , 5 g g g g h 2 |s12s23s34s41| M5 (1+g ,2+g ,3+h,4 −h,5−h) = 0
for the coefficients of the color traces a1 am Tr [T ⋅⋅⋅T ] following the ordering of the gluon legs. Again, for simplicity, overall phases are eliminated from the amplitudes. (In Ref. [14Jump To The Next Citation Point] 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 [14Jump To The Next Citation Point]. 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. k → 0 i) or collinear (i.e. k i parallel to k j), 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 Figure 6View Image, without any modifications. For example, in a supergravity theory, the scattering of a gravitino by a graviton is

( )2 ℳtree (1−,2− ,3+,4+) = κ- s A (1−,2− ,3+,4+ )A (1− ,2−,4+ ,3+) 4 &tidle;h h h &tidle;h 2 12 4 g g g g 4 g&tidle; g &tidle;g g ( κ)2 s12 ∘ s12- = -- s12 × ---× ---, (19 ) 2 s23 s24
where the subscript &tidle;h signifies a spin 3/2 gravitino and &tidle;g 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:
ℳex (1−i1,2+¯ı2,3−a3,4+a4) 4 q ¯q g (g )2 = κ- s A (1i1,2¯ı2,3−a3,4+a4) A (1− ,2+,4+ ,3−) 2 12 4 s s Q Q¯ 4 q ¯q Q¯ Q ( )2 ∘ --3---- = κ- --|s13s23|δ ¯ı2δa3a4, (20 ) 2 s12 i1
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 (i1,¯ı2,a3,a4) 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. [14Jump To The Next Citation Point], for both amplitudes (19View Equation) and (20View Equation), 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.


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