The KLT Relation from the Tree formula and Permutohedron

TL;DR

This paper extends the NSVW tree formula to all helicity sectors, revealing a deep connection between the Permutohedron and KLT relations, and providing new insights into gauge and gravity amplitude mappings.

Contribution

It introduces a generalized tree formula for all helicity sectors, linking Permutohedron structures to KLT relations and gravity amplitudes.

Findings

Established a mapping between Permutohedron and KLT relations.

Derived gravity amplitudes from a determinant via the matrix-tree theorem.

Re-derived soft and collinear limits of amplitudes.

Abstract

In this paper, we generalize the Nguyen-Spradlin-Volovich-Wen (NSVW) tree formula from the MHV sector to any helicity sector. We find a close connection between the Permutohedron and the KLT relation, and construct a non-trivial mapping between them, linking the amplitudes in the gauge and gravity theories. The gravity amplitude can also be mapped from a determinant followed from the matrix-tree theorem. Besides, we use the binary tree graphs to manifest its Lie structure. In our tree formula, there is an evident Hopf algebra of the permutation group behind the gravity amplitudes. Using the tree formula, we can directly re-derive the soft/collinear limit of the amplitudes.