The double copy suggests that the basis of the dynamics of general relativity is Yang-Mills theory. Motivated by the importance of the relativistic two-body problem, we study the classical dynamics of colour-charged particle scattering from the perspective of amplitudes, rather than equations of motion. We explain how to compute the change of colour, and the radiation of colour, during a classical collision. We apply our formalism at next-to-leading order for the colour change and at leading order for colour radiation.

We show that a recently discovered relation, which expresses tree-level single trace Einstein-Yang-Mills amplitudes with one graviton and (n − 1) gauge bosons as a linear combination of pure Yang-Mills tree amplitudes with n gauge bosons, can be derived from the CHY representation. In addition we show that there is a generalisation, which expresses tree-level single trace Einstein-Yang-Mills amplitudes with r gravitons and (n − r) gauge bosons as a linear combination of pure Yang-Mills tree amplitudes with n gauge bosons. We present a general formula for this case.

Abstract:The fundamental BCJ-relation is a linear relation between primitive tree amplitudes with different cyclic orderings. The cyclic orderings differ by the insertion place of one gluon. The coefficients of the fundamental BCJ-relation are linear in the Lorentz invariants 2p i p j . The BCJ-relations are well established for pure gluonic amplitudes as well as for amplitudes in N = 4 super-Yang-Mills theory. Recently, it has been conjectured that the BCJ-relations hold also for QCD amplitudes. In this paper we give a proof of this conjecture. The proof is valid for massless and massive quarks.

We show that the half-integrands in the CHY representation of tree amplitudes give rise to the definition of differential forms -the scattering forms -on the moduli space of a Riemann sphere with n marked points. These differential forms have some remarkable properties. We show that all singularities are on the divisor M 0,n \M 0,n . Each singularity is logarithmic and the residue factorises into two differential forms of lower points. In order for this to work, we provide a threefold generalisation of the CHY polarisation factor (also known as reduced Pfaffian) towards off-shell momenta, unphysical polarisations and away from the solutions of the scattering equations. We discuss explicitly the cases of bi-adjoint scalar amplitudes, Yang-Mills amplitudes and gravity amplitudes.In this article we bring three things together, which really should be viewed together: (i) the Cachazo-He-Yuan (CHY) representation of tree-level n-point scattering amplitudes [1-3], (ii) the moduli space M 0,n of n marked points on a Riemann surface of genus zero and (iii) "positive" geometries / "canonical" forms, as recently discussed by Bai and Lam [4]. The integrand of the CHY representation for bi-adjoint scalar amplitudes, Yang-Mills amplitudes and gravity amplitudes is constructed from two factors, a cyclic factor (or Parke-Taylor factor) and a polarisation factor (also known as reduced Pfaffian). We show that the cyclic factor and the polarisation factor lead to differential (n − 3)-forms Ω cyclic scattering and Ω pol scattering , respectively, on the compactification M 0,n of M 0,n , such that the only singularities of the differential forms Ω scattering are on the divisor M 0,n \M 0,n . Each singularity is logarithmic and the residue factorises into two differential forms of lower points. These scattering forms figure prominently in the recent work by Mizera [5,6]. The scattering forms are cocycles and Mizera has shown that the amplitudes are intersection numbers of these cocycles, twisted by a one-form derived from the scattering equations.We put "positive" geometry into quotes. The reason is the following: The solutions of the scattering equations are in general complex and correspond to points in M 0,n . Only for very special external momenta p are the solutions of the scattering equations real [7][8][9]. If the solutions are real, we may limit ourselves to the space of real points M 0,n (R). This is a positive space in the sense of Arkani-Hamed, Bai and Lam [4], with boundary M 0,n (R)\M 0,n (R). However, we are interested in the general situation. This forces us to work throughout the paper with the complex numbers C instead of the real numbers R. For simplicity we write M 0,n instead of M 0,n (C). We find that the notion of "positivity" is not essential, what is essential is the structure of the divisor

Inspired by the recent progress in the field of scattering amplitudes, we discuss hypothetical particles which can be characterised as the double copies of fermions -in the same way gravitons can be viewed as double copies of gauge bosons. As the gravitons, these hypothetical particles interact only through gravitational interactions. We present two equivalent methods for the computation of the relevant scattering amplitudes. The hypothetical particles can be massive and non-relativistic.