The KLT relation from the tree formula and permutohedron

Qu Cao; Liang Zhang

doi:10.1140/epjc/s10052-022-11168-1

2023 Impact factor 4.2

Particles and Fields

Open Access

Eur. Phys. J. C (2023) 83: 78
https://doi.org/10.1140/epjc/s10052-022-11168-1

Regular Article - Theoretical Physics

The KLT relation from the tree formula and permutohedron

Qu Cao^a and Liang Zhang

Department of Physics, Zhejiang Institute of Modern Physics, Zhejiang University, 310027, Hangzhou, China

^a qucao@zju.edu.cn

Received: 13 September 2022
Accepted: 23 December 2022
Published online: 27 January 2023

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.

In their paper [1], they admit that the formula they present is known from the older work by Bern, Dixon, Perelstein, and Rozowsky [2]. So the tree formula should be called the BDPR formula. In our paper, we call the tree formula for convenience.

© The Author(s) 2023

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