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

Henriette Elvang, Daniel Z. Freedman, Michael Kiermaier

TL;DR

This work provides a rigorous proof that the MHV vertex (CSW) expansion applies to all tree-level ${ m N^kMHV}$ amplitudes in ${ m ext{N}=4}$ SYM, using all-line shifts to establish the necessary large-$z$ falloff. It simultaneously constructs generating functions ${ m F}^{ m N^kMHV}_n$ that package the entire ${ m N^kMHV}$ sector and demonstrates how to extract any amplitude via Grassmann differentiation. It also introduces a refined anti-$ m NMHV$ generating function with new sum rules, and shows that the entire framework yields compact, efficient expressions for amplitudes and supports practical unitarity calculations. The results solidify the CSW expansion as a robust tool for tree-level computations and provide a foundation for loop-level applications in ${ m N}=4$ SYM and related theories.

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.

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

TL;DR

This work provides a rigorous proof that the MHV vertex (CSW) expansion applies to all tree-level amplitudes in SYM, using all-line shifts to establish the necessary large- falloff. It simultaneously constructs generating functions that package the entire sector and demonstrates how to extract any amplitude via Grassmann differentiation. It also introduces a refined anti- generating function with new sum rules, and shows that the entire framework yields compact, efficient expressions for amplitudes and supports practical unitarity calculations. The results solidify the CSW expansion as a robust tool for tree-level computations and provide a foundation for loop-level applications in SYM and related theories.

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.

Paper Structure

This paper contains 23 sections, 130 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Notation and review
  3. The MHV vertex expansion
  4. Structure of the MHV vertex expansion
  5. $|X]$-independence and examples
  6. Generating functions for N$^k$MHV amplitudes
  7. The MHV vertex expansion from all-line shifts
  8. All-line shifts and their recursion relations
  9. N$^2$MHV amplitudes
  10. The MHV vertex expansion for all ${\cal N}=4$ SYM amplitudes
  11. Large $z$ behavior under all-line shifts
  12. BCFW representation of N$^k$MHV amplitudes
  13. Kinematics of the shifts
  14. Proof that ${\cal A}^\text{N$^k$MHV}_n(z) \sim z^{-k}$ as $z \rightarrow \infty$ under all-line shifts
  15. ${\cal N}=4$ SYM amplitudes under square spinor shifts
  16. ...and 8 more sections

Figures (7)

  • Figure 1: The 16 skeletons contributing to the MHV vertex expansion of $8$-point amplitudes at the N$^2$MHV level. Two diagrams carry a symmetry factor of $\frac{1}{2}$ because the sum over cyclic assignments of external particles to the legs of these skeletons overcounts their contribution. In fact, these two diagrams are invariant under the permutation $i\rightarrow i+4$ of external legs.
  • Figure 2: The two vertex diagrams that contribute to the MHV vertex expansion of the seven-point N$^{3}$MHV amplitude $\langle A^{12}(1)A^{1234}(2)A^{23}(3)A^{234}(4)A^{134}(5)A^{134}(6)A^{124}(7)\rangle$.
  • Figure 3: A typical diagram in the MHV vertex expansion of an N$^{4}$MHV amplitude.
  • Figure 4: (a) A diagram in the expansion (\ref{['rewritecommonindexRR']}) of an N$^k$MHV amplitude under the all-line shift. (b) A diagram in the expansion of an N$^2$MHV amplitude under the all-line shift.
  • Figure 5: The MHV vertex expansion is applied to the subamplitudes in the all-line recursion relation. As the arrows indicate, channels $\beta_A$ and $\gamma_B$ are chosen to include external states only. The dots in the figure represent the remaining parts of the MHV vertex diagrams.
  • ...and 2 more figures