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Proof of the graviton MHV formula using Plebański's second heavenly equation

Noah Miller

TL;DR

The paper advances a non-twistor, non-BCFW derivation of the gravitational NSVW MHV amplitude by constructing a perturbiner expansion for Plebański's second heavenly equation, organized as marked tree graphs that generate self-dual spacetimes with arbitrary numbers of positive-helicity gravitons. It develops a detailed on-shell action approach, using Plebański's action plus a boundary term, to extract the MHV amplitude from boundary contributions, and enriches the perturbiner framework with both a linear and exponential formulation of perturbations, including a recursion operator. A novel NSVW generalization emerges from a graph-based reexpression of the generating function, which is subsequently shown to reduce to the classic NSVW form when auxiliary spinor choices are specialized. The work connects self-dual gravity, cluster-graph expansions, and on-shell action techniques, offering a coherent, first-principles route to gravitational MHV amplitudes that may extend to broader backgrounds and other theories, such as YM, with potential twistor-theoretic interpretations to be explored. Overall, the paper provides a self-contained, diagrammatic, and algebraic derivation of the NSVW formula, together with new structural insights and a complementary binary-tree expansion framework.

Abstract

Self-dual spacetimes can be thought of as spacetimes containing only positive helicity gravitons. In this work we give a perturbiner expansion for self-dual spacetimes based on Plebański's second heavenly equation. The expansion is naturally organized as a sum over ``marked tree graphs'' where each node corresponds to a positive helicity graviton and can have an arbitrary number of edges. Negative helicity gravitons must be added in by hand. We then use this perturbiner expansion to give a first principles derivation of the NSVW tree formula for the MHV amplitude in Einstein gravity. A unique feature of this proof is that it does not use BCFW recursion or twistor theory. It works by plugging the spacetime with arbitrarily many $+$ gravitons and two $-$ gravitons into the on-shell gravitational action and evaluating it. The action we use is the self-dual Plebański action plus an additional boundary term, and the amplitude itself comes entirely from the boundary term. Along the way, we also find an interesting new generalization of the NSVW formula which has not previously appeared in the literature. In the appendix we give another way to express the perturbiner expansion using binary tree graphs instead of marked tree graphs, and prove the equivalence of these two expansions diagrammatically. We also provide an introduction to self-dual gravity aimed at non-experts, as well as a proof of the Parke-Taylor formula in Yang Mills theory analogous to our proof of the NSVW formula in gravity.

Proof of the graviton MHV formula using Plebański's second heavenly equation

TL;DR

The paper advances a non-twistor, non-BCFW derivation of the gravitational NSVW MHV amplitude by constructing a perturbiner expansion for Plebański's second heavenly equation, organized as marked tree graphs that generate self-dual spacetimes with arbitrary numbers of positive-helicity gravitons. It develops a detailed on-shell action approach, using Plebański's action plus a boundary term, to extract the MHV amplitude from boundary contributions, and enriches the perturbiner framework with both a linear and exponential formulation of perturbations, including a recursion operator. A novel NSVW generalization emerges from a graph-based reexpression of the generating function, which is subsequently shown to reduce to the classic NSVW form when auxiliary spinor choices are specialized. The work connects self-dual gravity, cluster-graph expansions, and on-shell action techniques, offering a coherent, first-principles route to gravitational MHV amplitudes that may extend to broader backgrounds and other theories, such as YM, with potential twistor-theoretic interpretations to be explored. Overall, the paper provides a self-contained, diagrammatic, and algebraic derivation of the NSVW formula, together with new structural insights and a complementary binary-tree expansion framework.

Abstract

Self-dual spacetimes can be thought of as spacetimes containing only positive helicity gravitons. In this work we give a perturbiner expansion for self-dual spacetimes based on Plebański's second heavenly equation. The expansion is naturally organized as a sum over ``marked tree graphs'' where each node corresponds to a positive helicity graviton and can have an arbitrary number of edges. Negative helicity gravitons must be added in by hand. We then use this perturbiner expansion to give a first principles derivation of the NSVW tree formula for the MHV amplitude in Einstein gravity. A unique feature of this proof is that it does not use BCFW recursion or twistor theory. It works by plugging the spacetime with arbitrarily many gravitons and two gravitons into the on-shell gravitational action and evaluating it. The action we use is the self-dual Plebański action plus an additional boundary term, and the amplitude itself comes entirely from the boundary term. Along the way, we also find an interesting new generalization of the NSVW formula which has not previously appeared in the literature. In the appendix we give another way to express the perturbiner expansion using binary tree graphs instead of marked tree graphs, and prove the equivalence of these two expansions diagrammatically. We also provide an introduction to self-dual gravity aimed at non-experts, as well as a proof of the Parke-Taylor formula in Yang Mills theory analogous to our proof of the NSVW formula in gravity.
Paper Structure (21 sections, 6 theorems, 299 equations, 12 figures)

This paper contains 21 sections, 6 theorems, 299 equations, 12 figures.

Table of Contents

  1. Introduction
  2. A perturbiner expansion for Plebański's second heavenly equation
  3. Linear perturbations on a SD background
  4. Linear SD perturbations and the recursion operator
  5. Linear ASD perturbations
  6. Writing linear perturbations as an exponential
  7. Scattering amplitudes and on-shell actions
  8. The Gravitational MHV Amplitude
  9. MHV scattering and the Plebański Action
  10. Massaging the MHV generating function
  11. Proof of the NSVW formula
  12. Discussion
  13. A binary tree expansion for Plebański's second heavenly equation
  14. A crash course in self-dual gravity
  15. Spinor conventions
  16. ...and 6 more sections

Key Result

Theorem 2.1

The complete perturbiner expansion of the Plebański scalar is given by a sum over all marked tree graphs in $\mathcal{T}_N$ via

Figures (12)

  • Figure 1: Examples of graphs in $\mathcal{T}_9$.
  • Figure 2: A edge in a graph corresponds to $D^{ij}/z_{ij}$.
  • Figure 3: An example of a graph $\mathbf{t}$ and its corresponding $\phi_\mathbf{t}$.
  • Figure 4: A edge corresponding to $\Box(\phi_i \phi_j)$.
  • Figure 5: A edge corresponding to $D^{ij} \phi_i \phi_j$.
  • ...and 7 more figures

Theorems & Definitions (57)

  • Definition 1
  • Definition 2
  • Example
  • Definition 3
  • Example
  • Theorem 2.1
  • proof
  • Definition 4
  • Definition 5
  • Definition 6
  • ...and 47 more