Tree Amplitudes in Gauge Theory as Scalar MHV Diagrams

TL;DR

This paper extends the CSW scalar-MHV diagram framework to compute tree-level gauge theory amplitudes that include fermions, validating the approach by agreement with known results and enabling new compact expressions. It treats MHV amplitudes as scalar vertices connected by 1/p^2 propagators, with an off-shell continuation defined by a reference spinor, and shows a direct, universal way to obtain kinematic amplitudes A_n for various external content. The authors derive explicit closed-form expressions for non-MHV amplitudes with two and four fermions, and perform nontrivial checks against established results for n=4 and n=5, demonstrating the method’s reliability and potential for simplifying multi-parton computations. The work also discusses loop-level challenges and IR divergences, situating the scalar-graph approach within broader efforts linking gauge theory amplitudes to twistor string theory and suggesting directions for future developments in more general theories and colours.

Abstract

It was proposed in hep-th/0403047 that all tree amplitudes in pure Yang-Mills theory can be constructed from known MHV amplitudes. We apply this approach for calculating tree amplitudes of gauge fields and fermions and find agreement with known results.The formalism amounts to an effective scalar perturbation theory which offers a much simpler alternative to the usual Feynman diagrams in gauge theory and can be used for deriving new simple expressions for tree amplitudes. At tree level the formalism works in a generic gauge theory, with or without supersymmetry, and for a finite number of colours.

Figures (2)

• Figure 1: Tree diagrams with MHV vertices contributing to the $---+++\ldots ++$ amplitude with 2 fermions and $n-2$ gluons in Eq. \ref{['case1']}. Fermions are represented by dashed lines and gluons -- by solid lines.
• Figure 2: Diagrams contributing to the 4-fermion $n$-point amplitude \ref{['case2']}