Properties of scattering forms and their relation to associahedra

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 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

UV cancellations in gravity loop integrands

“…In parallel, there has been great progress in the effort to understand amplitudes as differential forms on various geometric spaces. This was first seen in the context of the planar N = 4 sYM theory where tree-level amplitudes and the loop integrands correspond to logarithmic forms on the Amplituhedron space [13,14], more recently similar structures have been seen for tree-level amplitudes in φ 3 theory via the Associahedron [15][16][17][18] and in cosmological correlation functions [19,20]. The important input in the Amplituhedron construction is the uniqueness of tree-level amplitudes and loop integrands -they are fully fixed by a list of certain homogenous conditions: logarithmic singularities, no poles at infinity, and absence of unphysical singularities.…”

Scattering forms and the positive geometry of kinematics, color and the worldsheet

“…Moreover, from general considerations of positive geometries we know that there should also be a "worldsheet canonical form" associated with this worldsheet associahedron, which turns out to be the famous "worldsheet Parke-Taylor form" [29] (for related discussions see e.g. [28,30]), an object whose importance has been highlighted in Nair's observation [31] and Witten's twistor string [4], and especially in the story of scattering equations and the CHY formulas for scattering amplitudes [5,6,15,32].…”

On positive geometries of quartic interactions: Stokes polytopes, lower forms on associahedra and world-sheet forms

“…14 , we define a 1-form on 12 In [2], the world-sheet form was obtained by replacing X ij → σ ij−1 and this was precisely the Parke-Taylor n − 3 form. We will come back to this in Section 6.3.…”

“…This space admits a real section, M 0,n (R) which is parametrized by an (equivalence class of) ordered set of points on a disk 1 . It was shown in [2,12] that there exists a compactification M 0,n (R) of M 0,n (R) which is also an associahedron. CHY scattering equations precisely generate a diffeomorphism between the world-sheet associahedron M 0,n (R) and the kinematic space associahedron A n such that the CHY formula for scattering amplitude can be understood in terms of pushforward of the canonical form between the two associahedra.…”