New structures in scattering amplitudes: a review

TL;DR

This review advocates reframing perturbative quantum field theory in terms of on-shell data, using analyticity, unitarity, and locality as guiding principles to reveal the underlying structure of scattering amplitudes. It surveys how three-particle amplitudes serve as fundamental building blocks, with BCFW and related momentum-space deformations enabling recursive construction of higher-point tree amplitudes, including supersymmetric extensions. It then connects tree-level structures to loop-level physics through generalized unitarity, quadruple/triple/double cuts, and the forward-limit insights that emerge most clearly in planar N=4 SYM, while introducing on-shell diagrammatics and the amplituhedron as geometric incarnations of positivity and locality. The framework highlights how locality and unitarity can appear as emergent properties and how all-loop descriptions in special theories can be captured by on-shell, gauge-invariant building blocks, offering a complementary perspective to conventional Feynman diagrammatics with potential for broad applicability across field theories.

Abstract

We review some recent developments in the understanding of field theories in the perturbative regime. In particular, we discuss the notions of analyticity, unitarity and locality, and therefore the singularity structure of scattering amplitudes in general interacting theories. We describe their tree-level structure and their on-shell representations, as well as the links between the tree-level structure itself and the structure of the loop amplitudes. Finally, we describe the on-shell diagrammatics recently proposed both on general grounds and in the remarkable example of planar supersymmetric theories. This review is partially based on lectures given at: Dipartimento di Fisica and INFN, Università di Bologna; Departamento de Fısica de Partıculas, Universidade de Santiago de Compostela; and as part of the program Strings@ar Lectures on Advanced Topics of High Energy Physics held at the IAFE

Figures (15)

• Figure 1: One-loop integral expansion. The minimal integral basis on which a one-loop amplitude in four dimensions can be expanded shows at most box-integrals, and then the lower point ones. This is due to the fact that in four dimensions and in the complexified momentum space it is possible to send on-shell simultaneously at most four internal propagators
• Figure 2: Four-particle cut. It is obtained by sending on-shell four internal propagators. This is the highest codimension singularity in four-dimensions. Performing this class of cuts, one obtains the coefficients of the box-integrals as product of four tree-level amplitudes.
• Figure 3: Three-particle cut. It is obtained by sending on-shell three internal propagators.
• Figure 4: Double cut. It is obtained by sending on-shell two internal propagators.
• Figure 5: Three-particle amplitudes. The black three-point vertex represents the holomorphic three-particle amplitudes, while the white vertex the anti-holomorphic one. The directions of the arrows represent the sign of the helicity of a given state: if the arrow is incoming, the helicity is negative, if it is outgoing the helicity of the given state is positive.