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 ${\bar{\mathcal{M}}}_{0,n}\mathrm{\backslash }{\mathcal{M}}_{0,n}$. 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.