Abstract
We present a general construction of two types of differential forms, based on any (n−3)-dimensional subspace in the kinematic space of n massless particles. The first type is the so-called projective, scattering forms in kinematic space, while the second is defined in the moduli space of n-punctured Riemann spheres which we call worldsheet forms. We show that the pushforward of worldsheet forms, by summing over solutions of scattering equations, gives the corresponding scattering forms, which generalizes the results of [1]. The pullback of scattering forms to subspaces can have natural interpretations as amplitudes in terms of Bern-Carrasco-Johansson double-copy construction or Cachazo-He-Yuan formula. As an application of our formalism, we construct in this way a large class of d log scattering forms and worldsheet forms, which are in one-to-one correspondence with non-planar MHV leading singularities in mathcal{N}=4 super-Yang-Mills. For every leading singularity function, we present a new determinant formula in moduli space, as well as a (combinatoric) polytope and associated scattering form in kinematic space. These include the so-called Cayley cases, where in each case the scattering form is the canonical forms of a convex polytope in the subspace, and scattering equations admit elegant rewritings as a map from the moduli space to the subspace.
Highlights
The primary example for canonical forms is the planar scattering form, Ωφ3(1, 2, · · ·, n), which is a d log scattering form from summing over planar cubic trees respecting the ordering; it represents a bi-adjoint φ3 amplitude with color stripped for one of the two groups
We present a general construction of two types of differential forms, based on any (n−3)-dimensional subspace in the kinematic space of n massless particles
As an application of our formalism, we construct in this way a large class of d log scattering forms and worldsheet forms, which are in one-to-one correspondence with non-planar MHV leading singularities in N = 4 super-Yang-Mills
Summary
In large enough spacetime dimensions, the kinematic space of n massless particles, Kn, can be spanned by all independent sab’s, it has dimension d. The meaning of such generalized scattering forms was discussed in [1]: they are the dual of color-dressed amplitudes in certain theories, where Wg’s are dual to color factors (the dual of Jacobi identities are given by (2.4)), and Ng’s the so-called Bern-Carassco-Johansson (BCJ) numerators that satisfy Jacobi identities It is an important open question how to find hyperplanes (or hypersurfaces) H such that the Ng’s become BCJ numerators of a given theory, such as Yang-Mills theory (YM) or non-linear sigma model (NLSM) [16,17,18,19]; equivalently, one can try to find H such that, on the support of scattering equations, the Jacobian JH equals the reduced Pfaffian, Pf Ψn( , k) for YM, or det An(s) for NLSM [20]
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