Generalized Unitarity and Six-Dimensional Helicity

TL;DR

The work develops a six-dimensional spinor-helicity framework to extend unitarity-based loop calculations, enabling dimensionally regulated amplitudes and revealing regulator-compatible structures. It demonstrates concrete results in both QCD (one-loop four-point) and maximally supersymmetric theories (two- and four-loop four-point amplitudes, including nonplanar pieces), while linking methods to the Higgs regulator and exploring higher-dimensional dual conformal properties. The approach leverages six-dimensional BCFW recursion, a chiral–conjugate factorization, and an on-shell superspace to efficiently construct and validate loop integrands across dimensions. The findings support the universality of the unitarity method in higher dimensions and point to promising extensions to phenomenology and deeper symmetry structures such as six-dimensional dual conformal invariance. Overall, the paper provides a practical toolkit and conceptual framework for applying six-dimensional helicity to complex loop calculations and regulator-aware analyses.

Abstract

We combine the unitarity method with the six-dimensional helicity formalism of Cheung and O'Connell to construct loop-level scattering amplitudes. As a first example, we construct dimensionally regularized QCD one-loop four-point amplitudes. As a nontrivial multiloop example, we confirm that the recently constructed four-loop four-point amplitude of N=4 super-Yang-Mills theory, including nonplanar contributions, is valid for dimensions less than or equal to six. We comment on the connection of our approach to the recently discussed Higgs infrared regulator and on dual conformal properties in six dimensions.

Figures (7)

• Figure 1: Cuts used to obtain a one-loop four-point amplitude. We display the triple cuts and ordinary two-particle cut in the $s_{14}$ channel. In general, we must also evaluate $s_{12}$ channel cuts.
• Figure 2: The quadruple cut. The cuts are indicated by dashed (red) lines. Here two pairs of three-point amplitudes are grouped together to form two easier-to-use four-point tree amplitudes, indicated by the (blue) ovals. The cut propagators of the four-point tree amplitudes are canceled by multiplying by inverse propagators prior to imposing cut conditions.
• Figure 3: Two categories of BCFW diagrams. (A) contains a three-point subamplitude, which does not have the full supermomentum delta function $\delta^4(q)\delta^4(\tilde{q})$. (B) contains no three-point subamplitude, thus all delta functions are of degree four.
• Figure 4: A sample cut of the four-loop four-point amplitude.
• Figure 5: The three-particle cut of a two-loop four-point amplitude.