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

TL;DR

This paper proves the explicit minimal-basis expansion of gauge theory tree amplitudes, confirming the conjecture that all such amplitudes can be expressed in terms of a basis of size (n-3)! using BCJ relations and BCFW recursion.

Contribution

It provides a rigorous proof of the conjectured minimal-basis expansion of gauge theory tree amplitudes using BCJ relations derived from string theory and BCFW recursion.

Findings

Confirmed the (n-3)! basis size for gauge theory amplitudes

Derived general BCJ relations from string theory limits

Proved the minimal-basis expansion inductively

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.