Tree-Level Formalism

TL;DR

The paper surveys two powerful, on-shell methods for tree-level scattering amplitudes: the MHV diagram (CSW) approach, which builds general amplitudes from MHV vertices connected by scalar propagators, and the BCFW recursion relations, which reconstruct amplitudes from complex-analytic shifts and factorization. It develops both formalisms in detail, including a supersymmetric, on-shell spacetime formulation for ${\cal N}=4$ SYM, a Lagrangian derivation of MHV rules, and extensions to gravity and massive particles. The authors discuss loop-related aspects, rational terms, and various regularisation schemes, highlighting the deep connections to twistor space, momentum twistor formalisms, and the broader S-matrix program. Together, these techniques provide covariant, efficient means to derive amplitudes across gauge theories and gravity, with broad theoretical and computational implications.

Abstract

We review two novel techniques used to calculate tree-level scattering amplitudes efficiently: MHV diagrams, and on-shell recursion relations. For the MHV diagrams, we consider applications to tree-level amplitudes and focus in particular on the N=4 supersymmetric formulation. We also briefly describe the derivation of loop amplitudes using MHV diagrams. For the recursion relations, after presenting their general proof, we discuss several applications to massless theories with and without supersymmetry, to theories with massive particles, and to graviton amplitudes in General Relativity. This article is an invited review for a special issue of Journal of Physics A devoted to "Scattering Amplitudes in Gauge Theories".

Figures (8)

• Figure 1: The two MHV diagrams contributing to $\langle 1^+ 2^- 3^- 4^-\rangle$
• Figure 2: On the left we represent a two-mass easy box function $F_{n:r;i}^{2m\,e}$. $n$ is the total number of external legs, the labels $i$ and $r$ are defined in the figure. On the right we depict the same box function in a slightly simplified notation, used later in \ref{['boxcsw22']}. The massless legs are called $p_{i-1} := p$ and $p_{i+r} := q$.
• Figure 3: One-loop MHV Feynman diagram, using MHV amplitudes as interaction vertices, with the CSW off-shell prescription.
• Figure 4: The super-MHV diagrams contributing to the NMHV superamplitude.
• Figure 5: One of the recursive diagrams contributing to the BCFW recursion relation for a colour-ordered amplitude $A(1, \ldots , n)$. The particles with shifted momenta are adjacent -- namely 1 and 2. Therefore, there is a single sum in the recursion relation, labeled by $j$.