Non-MHV Tree Amplitudes in Gauge Theory

TL;DR

This paper demonstrates that non-MHV (NMHV) tree-level amplitudes in gauge theories with up to N=4 supersymmetry can be obtained directly from the CSW scalar graph framework built from MHV vertices. By employing off-shell continuation with a fixed reference spinor, the intermediate diagrams are free of unphysical poles and the final amplitudes are Lorentz- and gauge-invariant, enabling straightforward numerical usage without extra helicity-spinor algebra. The authors explicitly derive NMHV amplitudes with three negative helicities for cases with two or four fermions and show how supersymmetric Ward identities relate purely gluonic NMHV amplitudes to fermionic ones, enabling compact closed-form expressions. They further develop an analytic, superspace-based approach using Nair's analytic supervertex and explore iterative constructions with two analytic vertices, outlining a path to systematically generate all NMHV amplitudes and suggesting extensions to loop calculations via unitarity methods.

Abstract

We show how all non-MHV tree-level amplitudes in 0 =< N =< 4 gauge theories can be obtained directly from the known MHV amplitudes using the scalar graph approach of Cachazo, Svrcek and Witten. Generic amplitudes are given by sums of inequivalent scalar diagrams with MHV vertices. The novel feature of our method is that after the `Feynman rules' for scalar diagrams are used, together with a particular choice of the reference spinor, no further helicity-spinor algebra is required to convert the results into a numerically usable form. Expressions for all relevant individual diagrams are free of singularities at generic phase space points, and amplitudes are manifestly Lorentz- (and gauge-) invariant. To illustrate the method, we derive expressions for n-point amplitudes with three negative helicities carried by fermions and/or gluons. We also write down a supersymmetric expression based on Nair's supervertex which gives rise to all such amplitudes in 0 =< N =< 4 gauge theories.

Figures (6)

• Figure 1: Tree diagrams with MHV vertices contributing to the amplitude $A_n(\Lambda_{m_1}^-,g_{m_2}^-,g_{m_3}^-,\Lambda_k^+)$. Fermions, $\Lambda^+$ and $\Lambda^-,$ are represented by dashed lines and negative helicity gluons, $g^-,$ by solid lines. Positive helicity gluons $g^+$ emitted from each vertex are indicated by dotted semicircles with labels showing the bounding $g^+$ lines in each MHV vertex.
• Figure 2: Tree diagrams with MHV vertices contributing to the amplitude $A_n(\Lambda_{m_1}^-,g_{m_2}^-,\Lambda_k^+,g_{m_3}^-)$.
• Figure 3: Tree diagrams with MHV vertices contributing to the amplitude $A_n(\Lambda_{m_1}^-,\Lambda_k^+,g_{m_2}^-,g_{m_3}^-)$ .
• Figure 4: Tree diagrams with MHV vertices contributing to the four fermion amplitude $A_n(g_1^-,\Lambda_{m_2}^-,\Lambda_{m_3}^-,\Lambda_{m_p}^+,\Lambda_{m_q}^+)$.
• Figure 5: Tree diagrams with MHV vertices contributing to the four fermion amplitude $A_n(g_1^-,\Lambda_{m_2}^-,\Lambda_{m_p}^+,\Lambda_{m_3}^-,\Lambda_{m_q}^+)$.