Abstract
The duality between color and kinematics was originally observed for purely adjoint massless gauge theories, and later found to hold even after introducing massive fermionic and scalar matter in arbitrary gauge-group representations. Such a generalization was critical for obtaining both loop amplitudes in pure Einstein gravity and realistic gravitational matter from the double copy. In this paper we elaborate on the double copy that yields amplitudes in gravitational theories coupled to flavored massive matter with spin, which is relevant to the problems of black-hole scattering and gravitational waves. Our construction benefits from making the little group explicit for the massive particles, as shown on lower-point examples. For concreteness, we focus on the double copy of QCD with massive quarks, for which we work out the gravitational Lagrangian up to quartic scalar and vector-scalar couplings. We find new gauge-invariant double-copy formulae for tree-level amplitudes with two distinct-flavor pairs of matter and any number of gravitons. These are similar to, but inherently different from, the well-known Kawai-Lewellen-Tye formulae, since the latter only hold for the double copy of purely adjoint gauge theories.
Highlights
A number of impressive results and techniques relevant to black-hole scattering [3,4,5,6,7,8,9,10,11,12,13]
The framework is based on the results and lessons learned from the extension of the color-kinematics duality to fundamental matter [20], and in particular to quantum chromodynamics (QCD) with massive quarks [21]
We expect it to be more generally applicable to gravitational amplitudes obtained from a pair of gauge theories that obey color-kinematics duality and that have massive matter transforming in non-adjoint representations of the gauge group
Summary
We review the external wavefunctions for massive particles using the massive spinor-helicity formalism of ref. [78].2 As in the purely massless case, the idea of this formalism is to build up all of the scattering kinematics from basic SL(2, C) spinors such that all resulting formulae are covariant with respect to the little group of the each physical particle. Any scattering amplitude can be represented [78] as having 2s symmetric indices for each s-spin massive state with momentum p, as carried by the helicity spinors λpαa and λαpa. We follow the textbook convention in that we normalize these spinors to 2m, upaubp = vpavpb = 2mδab This identity illustrates the general fact that the upper and lower little-group indices are related by complex conjugation, which will be evident from numerous equations below. Recall that the little-group indices a = 1, 2 permit SO(3) rotations of the spin quantization axis. This axis can be set to the three-momentum of the particle, which corresponds to the definite-helicity spinor parametrization detailed in refs. The translation from QCD to its self-conjugate version [74] is straightforward
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