Basics of Generalized Unitarity

TL;DR

<3-5 sentence high-level summary>Generalized unitarity provides a powerful, tree-based framework for constructing loop amplitudes directly from on-shell tree amplitudes across supersymmetric and non-supersymmetric theories, including non-planar contributions. The paper reviews both four-dimensional and six-dimensional on-shell formalisms, detailing cut construction, cut-merging strategies, and the use of ansatzes to determine integrands, with regularization addressed via six-dimensional helicity and Higgs-like regulators. It then demonstrates how tree-level properties such as dual conformal symmetry and color–kinematics duality can be carried over to loops, through explicit examples in ${\cal N}=4$ SYM, and discusses extensions to gravity via the double-copy construction. The work highlights practical techniques for computing multi-loop amplitudes and provides a foundation for exploring deeper symmetries and regularization schemes in gauge and gravity theories.

Abstract

We review generalized unitarity as a means for obtaining loop amplitudes from on-shell tree amplitudes. The method is generally applicable to both supersymmetric and non-supersymmetric amplitudes, including non-planar contributions. Here we focus mainly on N=4 Yang-Mills theory, in the context of on-shell superspaces. Given the need for regularization at loop level, we also review a six-dimensional helicity-based superspace formalism and its application to dimensional and massive regularizations. An important feature of the unitarity method is that it offers a means for carrying over any identified tree-level property of on-shell amplitudes to loop level, though sometimes in a modified form. We illustrate this with examples of dual conformal symmetry and a recently discovered duality between color and kinematics.

Figures (10)

• Figure 1: The $s$ and $t$ channel two-particle cuts of the one-loop four-point amplitude.
• Figure 2: The procedure for obtaining a spanning set of cuts for a planar two-loop four-point amplitude. Only topologically distinct diagrams are shown. The exposed intermediate lines are all placed on shell.
• Figure 3: The double-box three-particle cuts. The cut contributions (a) and (b) are two distinct cut contributions of the same horizontal double-box integral. The vertical double box has only a single contribution to the three-particle cut.
• Figure 4: An example showing how cuts are combined to obtain contributions with fewer cut conditions. In this equation the momentum labels of the different cut contributions need to be aligned, before being combined. On the right hand side, the only remaining cut conditions
• Figure 5: A three-particle supercut for the four-point amplitude. This cut contribution contains one MHV and one $\overline{\hbox{MHV}}$ superamplitude. The "$+$" indicates the MHV amplitude, while the "$-$" indicates the $\overline{\hbox{MHV}}$ one.