Virtuous trees at five- and six-point levels for Yang-Mills theory and gravity

Abstract

We present a particularly nice D-dimensional graph-based representation of the full color-dressed five-point tree-level gluon amplitude. It possesses the following virtues: 1) it satisfies the colorkinematic correspondence, and thus trivially generates the associated five-point graviton amplitude, 2) all external state information is encoded in color-ordered partial amplitudes, and 3) one function determines the kinematic contribution of all graphs in the Yang-Mills amplitude, so the associated gravity amplitude is manifestly permutation symmetric. The third virtue, while shared among all known loop-level correspondence-satisfying representations, is novel for tree-level representations sharing the first two virtues. This new D-dimensional representation makes contact with the recently found multiloop five-point representations, suggesting all-loop, all-multiplicity ramifications through unitarity. Additionally we present a slightly less virtuous representation of the six-point MHV and MHV amplitudes which holds only in four dimensions.

Note on symmetric BCJ numerator

“…A technical issue lies with this approach is that because of the complexity involved, along with the ambiguity introduced by generalized gauge invariance, it is practically difficult to write down an analytic expression for generic numerator. Nevertheless at tree level, explicit numerators were worked out by Broedel and Carrasco [42] at 4 and 5-points, and 6-points in the case of four dimensions, which satisfy the following three properties:…”

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Abstract: We present an algorithm that leads to BCJ numerators satisfying manifestly the three properties proposed by Broedel and Carrasco in [42]. We explicitly calculate the numerators at 4, 5 and 6-points and show that the relabeling property is generically satisfied.

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Simplifying multiloop integrands and ultraviolet divergences of gauge theory and gravity amplitudes

“…There is by now substantial evidence in favor of the duality, especially at tree level (L = 0) [43][44][45][46], where explicit representations of the numerators in terms of partial amplitudes are known for any number of external legs [47]. A consequence of this duality is the existence of nontrivial relations between the color-ordered partial tree amplitudes of gauge theory [29], which have been proven both from field theory [48] and string theory [49] perspectives.…”

“…The third constraint is clearly specific to the four-point amplitude, and should be modified for higher-point amplitudes because they have a more complicated structure. For the five-point case, however, a simple generalization has been found [33], involving pre-factors that are proportional [44] to linear combinations of five-point tree-amplitudes. For amplitudes in less supersymmetric theories, the first and second conditions should be relaxed (in addition to the third one), because one-loop triangle and bubble subgraphs are known to appear in such theories.…”

Algebras for amplitudes

“…The fact that this representation is possible for gauge theory amplitudes (and for amplitudes of the closely related theories to be discussed next) implies linear relations among color-ordered amplitudes, the BCJ relations. The color-kinematics duality has been discussed extensively in recent work, see [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38].…”