Application of Britto-Cachazo-Feng-Witten recursion relations and the Feynman-tree theorem to the four-gluon amplitude with all plus helicities

M Maniatis

doi:10.1103/physrevd.100.096022

Abstract

Recently, it has been shown that in gauge theories amplitudes to any perturbation order can be obtained by gluing together simple three-point on-shell amplitudes. These three-point amplitudes, in turn, are fixed by locality and Lorentz invariance. This factorization into three-point on-shell amplitudes follows from the Britto-Cachazo-Feng-Witten recursion relations and the Feynman-tree theorem. In an explicit example, that is, the four-gluon amplitude with all plus helicities, we illustrate the method. In a conventional calculation, this amplitude corresponds to one-loop box diagrams.

Highlights

A milestone in the understanding of scattering amplitudes are the Britto-Cachazo-Feng-Witten (BCFW) recursion relations [1,2]: By an analytic continuation of the external momenta, tree amplitudes factorize into elementary building blocks of three-point amplitudes in any gauge theory or in gravity
Even though the BCFW recursion relations are very powerful, they are limited to tree diagrams which, form an unphysical subset of diagrams to a certain perturbation order
The application of the Feynman-tree theorem followed by the BCFW recursion relations can be reversed, and the amplitudes be constructed by gluing together elementary on-shell three-point amplitudes

Summary

INTRODUCTION

A milestone in the understanding of scattering amplitudes are the Britto-Cachazo-Feng-Witten (BCFW) recursion relations [1,2]: By an analytic continuation of the external momenta, tree amplitudes factorize into elementary building blocks of three-point amplitudes in any gauge theory or in gravity. The Feynman-tree theorem recursively opens all loops: Any loop is expressed in terms of generalized cut diagrams which reduce the loop order about at least one unit. It has been shown that only a subset of possible generalized cuts contribute in multiloop diagrams It has been shown [32,33,34] that the application of the Feynman-tree theorem followed by the BCFW recursion allows one to factorize amplitudes at any loop order in terms of elementary three-point amplitudes in gauge theories. Following the Feynman-tree theorem, in this process particle-antiparticle pairs in the forward limit have to be taken into account These pairs are unobservable but contribute, in general, to the corresponding perturbation order. This amplitude provides an excellent framework to study the methods, since it is rather simple but reveals the main steps of the calculation

GLUING TOGETHER ON-SHELL SUBAMPLITUDES

DISCUSSION

CONCLUSION