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CHY formula and MHV amplitudes

Yi-jian Du, Fei Teng, Yong-Shi Wu

TL;DR

This work establishes a precise link between the CHY formulation of scattering amplitudes and 4D MHV amplitudes in Yang-Mills and gravity, proving that only a single rational solution of the scattering equations yields the MHV sector. By developing a manifest Möbius covariant formalism and two structural properties, the authors show that the CHY integral localized at Weinzierl's special solution reproduces both the Parke-Taylor expression for color-ordered YM MHV amplitudes and the Hodges determinant form for gravity MHV amplitudes, for any number of external legs. They further demonstrate that all other solutions do not contribute to MHV amplitudes and analytically confirm that the other Weinzierl rational solution supports anti-MHV amplitudes. The results uncover a surprising feature of CHY: the external helicity configuration is encoded in the choice of scattering-equation solutions, opening avenues to classify solution spaces and extend analyses to non-MHV sectors, with connections to Kumar–Kleiss–Kuijf relations and Eulerian-number patterns.

Abstract

In this paper, we study the relation between the Cachazo-He-Yuan (CHY) formula and the maximal-helicity-violating (MHV) amplitudes of Yang-Mills and gravity in four dimensions. We prove that only one special rational solution of the scattering equations found by Weinzierl support the MHV amplitudes. Namely, localized at this solution, the integrated CHY formula reproduces the Parke-Taylor formula for Yang-Mills amplitudes as well as the Hodges formula for gravitational amplitudes. This is achieved by developing techniques, in a manifestly Möbius covariant formalism, to explicitly compute relevant reduced Pfaffians/determinants. We observe and prove two interesting properties (or identities), which facilitate the computations. We also check that all the other $(n-3)!-1$ solutions to the scattering equations do not support the MHV amplitudes, and prove analytically that this is indeed true for the other special rational solution proposed by Weinzierl, that actually supports the anti-MHV amplitudes.

CHY formula and MHV amplitudes

TL;DR

This work establishes a precise link between the CHY formulation of scattering amplitudes and 4D MHV amplitudes in Yang-Mills and gravity, proving that only a single rational solution of the scattering equations yields the MHV sector. By developing a manifest Möbius covariant formalism and two structural properties, the authors show that the CHY integral localized at Weinzierl's special solution reproduces both the Parke-Taylor expression for color-ordered YM MHV amplitudes and the Hodges determinant form for gravity MHV amplitudes, for any number of external legs. They further demonstrate that all other solutions do not contribute to MHV amplitudes and analytically confirm that the other Weinzierl rational solution supports anti-MHV amplitudes. The results uncover a surprising feature of CHY: the external helicity configuration is encoded in the choice of scattering-equation solutions, opening avenues to classify solution spaces and extend analyses to non-MHV sectors, with connections to Kumar–Kleiss–Kuijf relations and Eulerian-number patterns.

Abstract

In this paper, we study the relation between the Cachazo-He-Yuan (CHY) formula and the maximal-helicity-violating (MHV) amplitudes of Yang-Mills and gravity in four dimensions. We prove that only one special rational solution of the scattering equations found by Weinzierl support the MHV amplitudes. Namely, localized at this solution, the integrated CHY formula reproduces the Parke-Taylor formula for Yang-Mills amplitudes as well as the Hodges formula for gravitational amplitudes. This is achieved by developing techniques, in a manifestly Möbius covariant formalism, to explicitly compute relevant reduced Pfaffians/determinants. We observe and prove two interesting properties (or identities), which facilitate the computations. We also check that all the other solutions to the scattering equations do not support the MHV amplitudes, and prove analytically that this is indeed true for the other special rational solution proposed by Weinzierl, that actually supports the anti-MHV amplitudes.

Paper Structure

This paper contains 25 sections, 60 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preparation: a review of the CHY, Parke-Taylor and Hodges formula
  3. CHY formula
  4. Parke-Taylor formula
  5. Hodges formula
  6. MHV Yang-Mills and gravity amplitudes from CHY formula
  7. $A_{n}(1^{-}2^{-}\ldots)$ and $\mathcal{M}_{n}(1^{-}2^{-}\ldots)$ from the integrated CHY formula with solution (\ref{['eq:sl1']})
  8. General MHV amplitudes
  9. Manifest Möbius invariance
  10. Summary
  11. Other solutions at MHV and Non-MHV
  12. Independent set of characteristic equations
  13. Non-MHV solutions and Non-MHV amplitudes
  14. Conclusions and discussions
  15. Spinor-helicity formalism
  16. ...and 10 more sections

Figures (1)

  • Figure 1: The structure of the matrix $\Psi$ after we fixed the gauge. Only the shaded regions are generally nonzero.