On the Structure of Supersymmetric Sums in Multi-Loop Unitarity Cuts

TL;DR

This work develops two synergistic frameworks to systematize the sum over on-shell supermultiplet states crossing generalized unitarity cuts in four-dimensional multi-loop amplitudes. The first is a linear-algebraic approach that reduces fermionic integrals to determinants derived from the coefficients of a system of equations, while the second is a diagrammatic index-diagram method that tracks SU(4) R-symmetry flows and translates diagrams into spinor expressions via explicit sign rules. Together, they yield compact, highly organized expressions for supersymmetric sums in ${\cal N}=4$ SYM, with explicit multi-loop (including four-loop) examples and clear pathways to theories with fewer supersymmetries and to ${\cal N}=8$ supergravity through KLT. The results expose universal structures, such as the fourth-power factorization over SU(4) indices and the ability to replace complicated state sums with momentum-based factorization identities, aiding both analytic understanding and computational efficiency. The methods provide a robust toolkit for constructing and validating multi-loop amplitudes across supersymmetric theories, with implications for UV behavior and gravity/gauge dualities.

Abstract

In this paper we describe algebraic and diagrammatic methods, related to the MHV generating function method, for evaluating and exposing the structure of supersymmetric sums over the states crossing generalized unitarity cuts of multi-loop amplitudes in four dimensions. We focus mainly on cuts of maximally supersymmetric Yang-Mills amplitudes. We provide various concrete examples, some of which are directly relevant for the calculation of four-loop amplitudes. Additionally, we discuss some cases with less than maximal supersymmetry. The results of these constructions carry over to generalized cuts of multi-loop supergravity amplitudes through use of the Kawai-Lewellen-Tye relations between gravity and gauge-theory tree amplitudes.

Figures (22)

• Figure 1: For an MHV amplitude the shaded (blue) "index line" (a) connecting leg $i$ to leg $j$ represents $\langle q_i^a \, q_j^a\rangle$. The two endpoints (and line) carry the same $SU(4)$ index. A solid (black) line (b) without endpoint dots represents a spinor product in the denominator.
• Figure 2: Examples of $SU(4)$ index diagrams for specifying numerator factors of MHV tree amplitudes. The diagrams (a), (b) and (c) correspond to the amplitudes in eq. (\ref{['SuperTreeExamples']}). The shaded (blue) line connecting leg $i$ to leg $j$ represents a factor of $\eta^a_i\left\langle i\,j\right\rangle\eta^a_j$ respectively, and solid (black) lines represents $\left\langle i\,j\right\rangle^{-1}$. The white "$+$" label on black background indicates that the amplitude is holomorphic, or MHV.
• Figure 3: The same amplitudes as in fig. \ref{['SampleMHVTreesFigure']}, now in the $\overline{\hbox{MHV}}$ representation. The shaded (blue) line connecting leg $i$ to leg $j$ represents a factor of $\widetilde{\eta}_{ia}\left[i\,j\right]\widetilde{\eta}_{ja}$ respectively, and solid (black) lines represents $\left[i\,j\right]^{-1}$. The white "$-$" label on black background indicates that the amplitude is anti-holomorphic, or $\overline{\hbox{MHV}}$.
• Figure 4: The MHV vertex construction builds non-MHV superamplitudes from MHV superamplitudes. The blobs are MHV superamplitudes, and the dots signify an arbitrary number of external legs, of which a few are drawn explicitly.
• Figure 5: The index lines for two out of the nine diagrams of the MHV vertex expansion for the amplitude $\langle g_-^{abcd}(1) g_+(2) f_-^{abc}(3)f_+^d(4) s^{ab}(5) g_+(6) s^{cd}(7)\rangle$. The dashed vertical (red) line signifies that the intermediate state is on-shell. The integration $\int d^4\eta$ will force exactly four $SU(4)$ index lines to end (or start) on the intermediate on-shell state.