Supersums for all supersymmetric amplitudes

TL;DR

The paper develops an on-shell diagrammatic framework for superamplitudes in $\mathcal{N}=4$ super Yang–Mills and extends it to $\mathcal{N}<4$ via a $\Phi$-$\Psi$ truncation scheme. It provides an $\mathcal{N}$-dependent MHV generating function and a graphical index-diagram representation that translates directly to analytic expressions for tree-level and one-loop amplitudes. A key result is the organization of supersymmetric sums in unitarity cuts into compact factorized forms that depend on $\mathcal{N}$, with extensions to non-MHV amplitudes via MHV-vertex constructions. The approach offers a practical, scalable method for computing supersymmetric amplitudes and points to rich futures directions in multiloop and matter-coupled theories.

Abstract

We present an on-shell graphical framework for superamplitudes in super Yang-Mills theory with arbitrary supersymmetry. Our diagrammatic procedure is derived through manipulations of the full N = 4 superamplitude and illustrated by a number of explicit examples.

Figures (5)

• Figure 1: The individual terms in the spinor string in the denominator are diagrammatrically represented by a curved line without endpoints. The blue index line with endpoints translates into a spinor product of supermomenta of the corresponding individual legs. For $\mathcal{N} < 4$, states of the $\Psi$-sector must be connected with a solid green sector line without end points. Identical graphs exist in the $\overline{\mathrm{MHV}}$ picture with obvious continued expressions.
• Figure 2: The analytic expressions for the $\mathcal{N} = 4$ amplitudes \ref{['SIMPLEMHV1']}-\ref{['SIMPLEMHV3']} are neatly captured by these three simple index diagrams. If reinterpreted in the $\overline{\mathrm{MHV}}$ picture the first diagram would have four index lines between the two positive helicity gluons for instance.
• Figure 3: Many amplitudes exist for several values of $\mathcal{N}$. Here \ref{['SIMPLEMHV1']}-\ref{['SIMPLEMHV3']} are shown for $\mathcal{N} = 2$. In general, if a diagram has at most $\Lambda$ grouped index lines in $\mathcal{N} = 4$, then it can be non-zero for reduced supersymmetry only provided $\Lambda \geq 4-\mathcal{N}$.
• Figure 4: $\mathcal{N} = 3$ and $\mathcal{N} = 2$ MHV tree-level examples as suggested by the number of index lines.
• Figure 5: The left and right index diagrams should respectively illustrate internal gluon and fermion contributions in a unitarity cut of the four-point one-loop amplitude. The cut marked by the dashed red line splits the amplitude into $\mathrm{MHV}$ and $\overline{\mathrm{MHV}}$ parts. Again the green sector line accounts for reduced supersymmetry. Horizontal flips of these diagrams and two additional diagrams representing internal scalars are very easy to draw, but are left out here.