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A super MHV vertex expansion for N=4 SYM theory

Michael Kiermaier, Stephen G. Naculich

TL;DR

The authors formulate a super MHV vertex expansion for all tree-level amplitudes in N=4 SYM by introducing four reference Grassmann parameters alongside a reference spinor. This framework enables significant diagrammatic simplifications via careful choices of eta_{X a}, and it is derived from and consistent with holomorphic all-line supershift recursion relations. They establish 1/z^k falloff for N^kMHV generating functions under all-line supershifts, with enhanced 1/z^{k+4} falloff for anti-MHV under special shifts, and show how these structures extend to all N^kMHV amplitudes. The work provides practical counting rules for diagram elimination, explicit NMHV/N^2MHV generating functions, and a unifying recursion-based derivation of the entire super MHV expansion, with potential implications for N=8 supergravity via KLT relations.

Abstract

We present a supersymmetric generalization of the MHV vertex expansion for all tree amplitudes in N=4 SYM theory. In addition to the choice of a reference spinor, this super MHV vertex expansion also depends on four reference Grassmann parameters. We demonstrate that a significant fraction of diagrams in the expansion vanishes for a judicious choice of these Grassmann parameters, which simplifies the computation of amplitudes. Even pure-gluon amplitudes require fewer diagrams than in the ordinary MHV vertex expansion. We show that the super MHV vertex expansion arises from the recursion relation associated with a holomorphic all-line supershift. This is a supersymmetric generalization of the holomorphic all-line shift recently introduced in arXiv:0811.3624. We study the large-z behavior of generating functions under these all-line supershifts, and find that they generically provide 1/z^k falloff at (Next-to)^k MHV level. In the case of anti-MHV generating functions, we find that a careful choice of shift parameters guarantees a stronger 1/z^(k+4) falloff. These particular all-line supershifts may therefore play an important role in extending the super MHV vertex expansion to N=8 supergravity.

A super MHV vertex expansion for N=4 SYM theory

TL;DR

The authors formulate a super MHV vertex expansion for all tree-level amplitudes in N=4 SYM by introducing four reference Grassmann parameters alongside a reference spinor. This framework enables significant diagrammatic simplifications via careful choices of eta_{X a}, and it is derived from and consistent with holomorphic all-line supershift recursion relations. They establish 1/z^k falloff for N^kMHV generating functions under all-line supershifts, with enhanced 1/z^{k+4} falloff for anti-MHV under special shifts, and show how these structures extend to all N^kMHV amplitudes. The work provides practical counting rules for diagram elimination, explicit NMHV/N^2MHV generating functions, and a unifying recursion-based derivation of the entire super MHV expansion, with potential implications for N=8 supergravity via KLT relations.

Abstract

We present a supersymmetric generalization of the MHV vertex expansion for all tree amplitudes in N=4 SYM theory. In addition to the choice of a reference spinor, this super MHV vertex expansion also depends on four reference Grassmann parameters. We demonstrate that a significant fraction of diagrams in the expansion vanishes for a judicious choice of these Grassmann parameters, which simplifies the computation of amplitudes. Even pure-gluon amplitudes require fewer diagrams than in the ordinary MHV vertex expansion. We show that the super MHV vertex expansion arises from the recursion relation associated with a holomorphic all-line supershift. This is a supersymmetric generalization of the holomorphic all-line shift recently introduced in arXiv:0811.3624. We study the large-z behavior of generating functions under these all-line supershifts, and find that they generically provide 1/z^k falloff at (Next-to)^k MHV level. In the case of anti-MHV generating functions, we find that a careful choice of shift parameters guarantees a stronger 1/z^(k+4) falloff. These particular all-line supershifts may therefore play an important role in extending the super MHV vertex expansion to N=8 supergravity.

Paper Structure

This paper contains 19 sections, 87 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Review
  3. The super MHV vertex expansion
  4. NMHV amplitudes
  5. Simplified NMHV amplitude computations
  6. All tree amplitudes
  7. Simplification of general amplitude computations
  8. All-line supershifts
  9. Anti-MHV generating functions
  10. N$^k$MHV generating functions under all-line supershifts
  11. The super MHV vertex expansion from all-line supershifts
  12. All-line supershift recursion relations
  13. NMHV generating function
  14. N$^2$MHV generating function
  15. All tree amplitudes
  16. ...and 4 more sections

Figures (3)

  • Figure 1: The diagrams of the MHV vertex expansion at the (a) NMHV, (b) N$^2$MHV, and (c) N$^3$MHV level.
  • Figure 2: (a) The five diagrams contributing to the ordinary MHV vertex expansion of $5$-point NMHV amplitudes. (b) The single remaining diagram that contributes to the super MHV vertex expansion.
  • Figure 3: A diagram $\alpha$ of the all-line supershift recursion relation of an N$^2$MHV generating function. The super MHV vertex expansion is substituted for the NMHV subamplitude, and the internal line $P_\beta$ of the super MHV vertex diagram is evaluated at shifted momenta.