Properties of scattering forms and their relation to associahedra

TL;DR

This work shows that CHY half-integrands define differential forms on the compactified moduli space \overline{\mathcal{M}}_{0,n}, with all singularities on the boundary being logarithmic and residues factorizing into two lower-point forms, thereby linking CHY data to a geometric, associahedron-like structure. It introduces a threefold generalization of the CHY polarization factor to handle off-shell momenta and unphysical polarisations, enabling the construction of polarization forms that share the same favorable analytic properties as the cyclic ones. The amplitudes in bi-adjoint scalar theory, Yang-Mills, and gravity arise as twisted intersection numbers of these scattering forms with a twist η encoding the scattering equations, situating the construction within Mizera’s framework. Collectively, the paper provides a concrete geometric realization of tree-level amplitudes via scattering forms on \overline{M}_{0,n}, connects to positive geometry ideas, and suggests directions for extending these structures beyond tree level and to other theories.

Abstract

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.

Figures (9)

• Figure 1: The moduli space ${\mathcal{M}}_{0,5}({\mathbb R})$ (left). The region $X$ is bounded by $z_2=0$, $z_3=1$ and $z_2=z_3$. The right figure shows $\overline{\mathcal{M}}_{0,5}({\mathbb R})$, obtained from ${\mathcal{M}}_{0,5}({\mathbb R})$ by blowing up the points $(z_2,z_3)=(0,0)$, $(z_2,z_3)=(1,1)$ and $(z_2,z_3)=(\infty,\infty)$.
• Figure 2: A hexagon, where the edges are labelled by the cyclic ordered variables $(z_1,z_2,...,z_6)$ (left picture). The middle picture shows the chord $(2,5)$. Right picture: A chord divides the hexagon into two lower $n$-gons, in this case two quadrangles.
• Figure 3: We may visualise a scattering form on $\mathcal{M}_{0,n}^\pi$ as shown in the left figure: The dihedral structure defines an $n$-gon, each external line of the scattering process crosses one edge of the $n$-gon. In the limit $u_{i_0,n}\rightarrow 0$ the residue factorises into two scattering forms of lower points, as shown in the right figure.
• Figure 4: A hexagon with a triangulation, given by the dashed lines. The dual graph is shown in red.
• Figure 5: A hexagon with the dihedral structure $\pi=(1,2,3,4,5,6)$. The dashed line shows the chord $(3,6)$. The dotted red lines show the bonds corresponding to the cyclic factor $C(\sigma,z)$ for $\sigma=(1,3,2,4,5,6)$. The bonds $(z_2-z_4)$ and $(z_6-z_1)$ cross the chord $(3,6)$.