A Proof of the Explicit Minimal-basis Expansion of Tree Amplitudes in Gauge Field Theory

TL;DR

This work addresses the problem of proving the explicit minimal-basis expansion for color-ordered gauge-theory tree amplitudes, demonstrating that all amplitudes can be expressed in a $(n-3)!$-element basis. It combines a field-theory derivation of the general BCJ relations via BCFW recursion with an inductive proof of an explicit expansion formula using a specially defined function $\\mathcal{F}$. The main contributions are a rigorous field-theory proof of the general BCJ relation and a constructive, closed-form minimal-basis expansion formula, thereby solidifying the connection between BCJ, KK, and KLT relations and enabling direct amplitude computations. The results have significant impact for simplifying gauge-theory amplitudes and for understanding the structure underlying gravity-gauge theory relations in a purely field-theoretic framework.

Abstract

In last couple years, an important relation (BCJ relation) between color-ordered tree-level scattering amplitudes of gauge theory has inspired many studies. This relation implies that the minimal basis for the color-ordered tree-level amplitudes is $(n-3)!$ and other amplitudes can be expanded into a particular chosen basis. In this paper we will prove the conjectured explicit minimal basis expansion. For this purpose we will write down general BCJ relation of gauge theory by taking the field theory limit of BCJ relation in string theory. Then we prove these general BCJ relations using BCFW on-shell recursion relation. Using these general BCJ relations, we prove the conjectured explicit minimal-basis expansion of gauge theory tree amplitudes inductively.