Relations for Einstein-Yang-Mills amplitudes from the CHY representation

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.Comment: 13 pages, version to be publishe

Properties of scattering forms and their relation to associahedra

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 $\overline{\mathcal M}_{0,n} \backslash {\mathcal M}_{0,n}$. 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.Comment: 40 pages, version to be publishe

Proof of the fundamental BCJ relations for QCD amplitudes

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 $2 p_i p_j$. The BCJ-relations are well established for pure gluonic amplitudes as well as for amplitudes in ${\mathcal N}=4$ 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.Comment: 24 pages, version to be publishe

Asymptotic behavior of global entropy solutions for nonstrictly hyperbolic systems with linear damping

In this paper we investigate the large time behavior of the global weak entropy solutions to the symmetric Keyftiz-Kranzer system with linear damping. It is proved that as t tends to infinite the entropy solutions tend to zero in the L p nor

Classical off-shell currents

We consider tree-level off-shell currents of two massive particles and $n$ massless bosons in the classical limit, which can be fused into the classical limit of $n+2$ scattering amplitudes. We show that dressing up the current with coherent wave-functions associated with the massive particles leads to the recently proposed Worldline Quantum Field Theory (WQFT) path integral. The currents thus constructed encode solutions of classical equations of motion so they can be applied to contexts where the classical limit is relevant, including hard thermal loops. We give several examples of these currents in scalar, gauge and gravitational theories.Comment: 28 pages, v2: clarifications in Sec.2, references added, typos fixed, matches published versio