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MHV Lagrangian for N=4 Super Yang-Mills

Haidong Feng, Yu-tin Huang

TL;DR

The paper addresses realizing CSW MHV perturbation theory within a Lagrangian for N=4 SYM by employing a light-cone superspace formulation and two successive field redefinitions. The canonical (equal-time) redefinition is shown to produce a CSW-compatible MHV Lagrangian, and explicit calculations demonstrate that the on-shell 4-point MHV amplitude matches the known form $\langle12\rangle^2/(\langle34\rangle\langle41\rangle)$, with off-shell CSW continuation consistent up to an overall factor that cancels in amplitudes. The one-loop equivalence theorem is established by showing self-energy diagrams are scaleless and vanish, supporting the action-level realization of CSW in this supersymmetric setting. Overall, the work provides a supersymmetric, light-cone action framework in which only MHV vertices appear, clarifying the role of currents and canonical constraints in connecting the Lagrangian to CSW perturbation theory and its loop structure.

Abstract

Here we formulate two field redefinitions for N=4 Super Yang-Mills in light cone superspace that generates only MHV vertices in the new Lagrangian. After careful consideration of the S-matrix equivalence theorem, we see that only the canonical transformation gives the MHV Lagrangian that would correspond to the CSW expansion. Being in superspace, it is easier to analyse the equivalence theorem at loop level. We calculate the on shell amplitude for 4pt $(\barΛ\bar{\rm A}Λ{\rm A})$ MHV in the new lagrangian and show that it reproduces the previously known form. We also briefly discuss the relationship with the off-shell continuation prescription of CSW.

MHV Lagrangian for N=4 Super Yang-Mills

TL;DR

The paper addresses realizing CSW MHV perturbation theory within a Lagrangian for N=4 SYM by employing a light-cone superspace formulation and two successive field redefinitions. The canonical (equal-time) redefinition is shown to produce a CSW-compatible MHV Lagrangian, and explicit calculations demonstrate that the on-shell 4-point MHV amplitude matches the known form , with off-shell CSW continuation consistent up to an overall factor that cancels in amplitudes. The one-loop equivalence theorem is established by showing self-energy diagrams are scaleless and vanish, supporting the action-level realization of CSW in this supersymmetric setting. Overall, the work provides a supersymmetric, light-cone action framework in which only MHV vertices appear, clarifying the role of currents and canonical constraints in connecting the Lagrangian to CSW perturbation theory and its loop structure.

Abstract

Here we formulate two field redefinitions for N=4 Super Yang-Mills in light cone superspace that generates only MHV vertices in the new Lagrangian. After careful consideration of the S-matrix equivalence theorem, we see that only the canonical transformation gives the MHV Lagrangian that would correspond to the CSW expansion. Being in superspace, it is easier to analyse the equivalence theorem at loop level. We calculate the on shell amplitude for 4pt MHV in the new lagrangian and show that it reproduces the previously known form. We also briefly discuss the relationship with the off-shell continuation prescription of CSW.

Paper Structure

This paper contains 11 sections, 58 equations, 2 figures.

Table of Contents

  1. Introduction
  2. N=4 Light-Cone Superspace
  3. The Field Redefinition
  4. Field redefinition I $\Phi(\chi)$
  5. Field redefinition I for YM
  6. Field redefinition II (canonical redefinition)
  7. Explicit Calculation for MHV amplitude $\bar{\Lambda}\bar{{\rm A}}\Lambda {\rm A}$
  8. CSW-off-shell continuation
  9. Equivalence Theorem at one-loop
  10. Discussion
  11. Proof of field redefinition

Figures (2)

  • Figure 1: In this figure we show how the field redefinition may contribute to tree graphs from the modification of coupling to the source current. The solid circle indicate the $(--+)$ vertex while the empty circle indicates contraction with the currents. Due to new terms in coupling, the $C\bar{J}_3BB$ term, one can actually construct contribution to the $(--++)$ amplitude by using this term, denoted by the larger empty circle, as a vertex.
  • Figure 2: These are the two relevant contribution to the one-loop self-energy diagram. For simplicity we only denote the positions of $d^4$ and $\bar{d}^4$ to indicate which legs of the vertex was used for the loop contraction.