Proof of the MHV vertex expansion for all tree amplitudes in N=4 SYM theory

TL;DR

This paper proves the validity of the MHV vertex expansion for all tree-level amplitudes in N=4 SYM theory, establishing decay properties under momentum shifts and deriving compact generating functions.

Contribution

It provides a rigorous proof of the MHV vertex expansion for all tree amplitudes in N=4 SYM and introduces improved generating functions and sum rules.

Findings

All N^kMHV tree amplitudes fall off as 1/z^k or faster under the shift.

Derived compact and efficient generating functions for all N^kMHV amplitudes.

Established sum rules for diagrams in the MHV vertex expansion.

Abstract

We prove the MHV vertex expansion for all tree amplitudes of N=4 SYM theory. The proof uses a shift acting on all external momenta, and we show that every N^kMHV tree amplitude falls off as 1/z^k, or faster, for large z under this shift. The MHV vertex expansion allows us to derive compact and efficient generating functions for all N^kMHV tree amplitudes of the theory. We also derive an improved form of the anti-NMHV generating function. The proof leads to a curious set of sum rules for the diagrams of the MHV vertex expansion.