Scattering Amplitudes

TL;DR

The review surveys modern on-shell methods for scattering amplitudes, starting from spinor-helicity formalism and color-ordering and culminating in unitarity and on-shell recursion. It develops a coherent framework—incorporating SUSY, twistors, dual conformal symmetry, and momentum twistors—that connects tree-level amplitudes to loop-level structure via unitarity, BCFW, and leading singularities. Key contributions include the systematic construction of tree amplitudes in N=4 SYM (including MHV/NMHV via super-BCFW and R-invariants), the emergence of dual conformal symmetry and momentum twistor space, and the unitarity-based assembly of loop amplitudes with a focus on planar theories and their geometric organization. The work underscores the power of on-shell, symmetry-guided approaches to reveal deep mathematical structures (Grassmannians, polytopes, Yangian symmetry) underlying perturbative quantum field theory and perturbative gravity, with broad implications for computations and conceptual understanding. Practically, these methods offer compact, efficient representations of amplitudes and pave the way for further insights into perturbative dynamics and potential non-perturbative connections.

Abstract

The purpose of this review is to bridge the gap between a standard course in quantum field theory and recent fascinating developments in the studies of on-shell scattering amplitudes. We build up the subject from basic quantum field theory, starting with Feynman rules for simple processes in Yukawa theory and QED. The material covered includes spinor helicity formalism, on-shell recursion relations, superamplitudes and their symmetries, twistors and momentum twistors, loops and integrands, Grassmannians, polytopes, and amplitudes in perturbative supergravity as well as 3d Chern-Simons-matter theories. Multiple examples and exercises are included.

Figures (11)

• Figure 1: A graphical representation of the map between dual space $y^\mu$ and momentum twistor space $Z_i^I=(|i\rangle, [\mu_i|)$. The lefthand figure illustrate the incidence relations (\ref{['incidence']}): a null line in dual space, defined by the two points $y_i$ and $y_{i+1}$, corresponds to a point $Z_i^I=(|i\rangle, [\mu_i| )$ in momentum twistor space. The righthand figure shows how a point $y_i$ in dual space maps to a line in momentum twistor space via the relation (\ref{['inversemap']}).
• Figure 2: The sum of residues from all Feynman diagrams with propagators $\ell^2$ and $(\ell-p_1-p_2)^2$ on-shell must give the product of two tree-amplitudes.
• Figure 3: 1-loop box diagram with $K^{(i)}_1=p_1+\cdots+p_i$, $K^{(i)}_2=p_{i+1}+\cdots+p_{j}$, $K^{(i)}_3=p_{j+1}+\cdots+p_{k}$ and $K^{(i)}_2=p_{k+1}+\cdots+p_{n}$. The corresponding box coefficient $C_4^{(i)}$ in (\ref{['BoxCoef']}) is the product of the four tree amplitudes at each corner.
• Figure 4: The integrands of $\mathcal{N}=4$ SYM 4-point amplitude to 3-loop order. These are the unique scalar integrands that are dual conformal invariant.
• Figure 7: The geometry of (\ref{['ihat']}) and (\ref{['ZI']}): (a) shows that the on-shell condition $\hat{P}_I^2=0$ fixes the shifted momentum twistor $\hat{Z}_i$ to be at the intersection of the line $(i,i+1)$ with the plane $(i-1,j,j-1)$. In (b) the momentum twistor $Z_I$ is located at the intersection of line $(j, j-1)$ and the plane $(i-1,i,i+1)$.