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Structure of the MHV-rules Lagrangian

James H. Ettle, Tim R. Morris

TL;DR

The paper provides an explicit all-orders solution to the canonical transformation that maps Yang–Mills into an MHV-rules Lagrangian, yielding holomorphic expansion coefficients and CSW-compatible vertices that reproduce MHV amplitudes at tree level. It verifies the 3-, 4-, and 5-point vertices match the CSW off-shell continuations and derives recursive formulas for the A and \bar{A} expansions, including their holomorphic structure. It then addresses quantum aspects by analyzing ET matching factors at one loop and showing they vanish in dimensional regularisation for massless theories, supporting the consistency of MHV rules within standard QFT. The results underpin a Lagrangian basis for CSW/MHV methods and suggest pathways to regularised CSW approaches and possible hybrid calculation strategies for amplitudes.

Abstract

Recently, a canonical change of field variables was proposed that converts the Yang-Mills Lagrangian into an MHV-rules Lagrangian, i.e. one whose tree level Feynman diagram expansion generates CSW rules. We solve the relations defining the canonical transformation, to all orders of expansion in the new fields, yielding simple explicit holomorphic expressions for the expansion coefficients. We use these to confirm explicitly that the three, four and five point vertices are proportional to MHV amplitudes with the correct coefficient, as expected. We point out several consequences of this framework, and initiate a study of its implications for MHV rules at the quantum level. In particular, we investigate the wavefunction matching factors implied by the Equivalence Theorem at one loop, and show that they may be taken to vanish in dimensional regularisation.

Structure of the MHV-rules Lagrangian

TL;DR

The paper provides an explicit all-orders solution to the canonical transformation that maps Yang–Mills into an MHV-rules Lagrangian, yielding holomorphic expansion coefficients and CSW-compatible vertices that reproduce MHV amplitudes at tree level. It verifies the 3-, 4-, and 5-point vertices match the CSW off-shell continuations and derives recursive formulas for the A and \bar{A} expansions, including their holomorphic structure. It then addresses quantum aspects by analyzing ET matching factors at one loop and showing they vanish in dimensional regularisation for massless theories, supporting the consistency of MHV rules within standard QFT. The results underpin a Lagrangian basis for CSW/MHV methods and suggest pathways to regularised CSW approaches and possible hybrid calculation strategies for amplitudes.

Abstract

Recently, a canonical change of field variables was proposed that converts the Yang-Mills Lagrangian into an MHV-rules Lagrangian, i.e. one whose tree level Feynman diagram expansion generates CSW rules. We solve the relations defining the canonical transformation, to all orders of expansion in the new fields, yielding simple explicit holomorphic expressions for the expansion coefficients. We use these to confirm explicitly that the three, four and five point vertices are proportional to MHV amplitudes with the correct coefficient, as expected. We point out several consequences of this framework, and initiate a study of its implications for MHV rules at the quantum level. In particular, we investigate the wavefunction matching factors implied by the Equivalence Theorem at one loop, and show that they may be taken to vanish in dimensional regularisation.

Paper Structure

This paper contains 14 sections, 73 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and Transformation
  3. Preliminaries
  4. The transformation
  5. The precise correspondence
  6. Explicit Expansion
  7. $A$ expansion coefficients to all orders
  8. ${\bar{A}}$ expansion coefficients to all orders
  9. Three-point vertex
  10. Four-point vertices
  11. Five-point vertices
  12. Equivalence Theorem Matching Factors
  13. Conclusions
  14. Off-shell matching factors

Figures (1)

  • Figure 1: Topology of the one-loop ET matching factors, with \ref{['oneloop-a']} the contribution to negative helicity ${\bar{B}}$ legs; and \ref{['oneloop-b']} the contribution to positive helicity $B$ legs. Wavy lines denote $A$ and ${\bar{A}}$ fields, straight lines $B$ and ${\bar{B}}$.