Recursion Relations, Generating Functions, and Unitarity Sums in N=4 SYM Theory

TL;DR

This paper develops a comprehensive on-shell framework for ${\cal N}=4$ SYM amplitudes by proving the existence of valid 3-line shifts for NMHV amplitudes and establishing a universal NMHV generating function that encodes all external-state dependences. It shows how MHV generating functions and NMHV diagrams combine to yield a shift-independent MHV-vertex expansion, and it extends these ideas to loop-level spin sums using Grassmann integration, producing compact generating functions for unitarity cuts across 1–4 loops. The authors also construct anti-MHV and anti-NMHV generating functions via conjugation and Grassmann-Fourier transforms, enabling complete spin sums in mixed NMHV/anti-NMHV cuts. A central outcome is the proof of valid 2-line shifts for any ${\cal N}=4$ SYM tree amplitude, implying universal BCFW recursion for all amplitudes, with caveats for ${\cal N}=8$ supergravity. The work provides a powerful, unifying toolkit for exact amplitude computations and unitarity cuts in highly supersymmetric gauge theories.

Abstract

We prove that the MHV vertex expansion is valid for any NMHV tree amplitude of N=4 SYM. The proof uses induction to show that there always exists a complex deformation of three external momenta such that the amplitude falls off at least as fast as 1/z for large z. This validates the generating function for n-point NMHV tree amplitudes. We also develop generating functions for anti-MHV and anti-NMHV amplitudes. As an application, we use these generating functions to evaluate several examples of intermediate state sums on unitarity cuts of 1-, 2-, 3- and 4-loop amplitudes. In a separate analysis, we extend the recent results of arXiv:0808.0504 to prove that there exists a valid 2-line shift for any n-point tree amplitude of N=4 SYM. This implies that there is a BCFW recursion relation for any tree amplitude of the theory.

Figures (7)

• Figure 1: Diagrammatic expansion of an amplitude $A_n(1^-,\dots,x,\dots,n)$ under a 2-line shift $[1^-,x\rangle$.
• Figure 2: A generic MHV vertex diagram of an NMHV amplitude $A_n(m_1,\dots,m_2,\dots,m_3,\dots)$, arising from a 3-line shift $[m_1,m_2,m_3|$. The set of lines $\hat{m}_i$, $\hat{m}_j$, $\hat{m}_k$ is a cyclic permutation of $\hat{m}_1$, $\hat{m}_2$, $\hat{m}_3$.
• Figure 3: N$^k$MHV loop amplitude evaluated by a unitarity cut of $(L+1)$-lines. The sum over intermediate states involves all subamplitudes $I$ and $J$ with $\eta$-counts $r_I$ and $r_J$ such that $r_I + r_J=4(k+L+3)$. (For $L=1$ we assume that $I$ and $J$ each have more than one external leg, so that 3-point anti-MHV does not occur in the spin sum.)
• Figure 4: Triple cut of NMHV 1-loop amplitude gives MHV subamplitudes $I$, $J$, and $K$.
• Figure 5: 4-point $L$-loop MHV amplitude with $(L+1)$-line cut.