Scattering Amplitudes (1308.1697v2)

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.

• The paper introduces on-shell techniques that significantly reduce the complexity of high-point scattering amplitude calculations in QCD and supersymmetric theories.
• It employs spinor helicity, BCFW recursion, and color-ordering methods to bridge traditional Feynman diagrams with modern amplitude formulations.
• The review reveals deep hidden symmetries, such as dual conformal invariance, and suggests future non-perturbative approaches in quantum gravity research.

Overview of "Scattering Amplitudes" by Henriette Elvang and Yu-tin Huang

The paper "Scattering Amplitudes" by Henriette Elvang and Yu-tin Huang presents a comprehensive review of the developments in on-shell scattering amplitudes in quantum field theory, particularly focusing on gauge theories such as quantum chromodynamics (QCD) and supersymmetric theories like $\mathcal{N}=4$ super Yang-Mills (SYM) theory.

Key Concepts and Techniques

The paper bridges the gap between standard quantum field theory courses and modern amplitude techniques. It starts with Feynman rules in simple processes and progresses to advanced concepts like:

1. Spinor Helicity Formalism: This formalism simplifies the representation of scattering amplitudes in four-dimensional space by exploiting the properties of Weyl spinors. It becomes particularly powerful for massless particles, allowing a clear depiction of helicity states and simplifying the algebra involved in amplitude calculations.
2. On-Shell Recursion Relations: Initially developed by Britto, Cachazo, Feng, and Witten (BCFW), these relations allow the construction of higher-point amplitudes from lower-point ones using complex deformation of momenta. This approach significantly reduces computational complexity compared to traditional Feynman diagram techniques.
3. Supersymmetric Amplitudes: The paper reviews how supersymmetry (SUSY) provides powerful constraints on amplitudes, leading to relations among different scattering processes via SUSY Ward identities. This is particularly elaborated in the context of $\mathcal{N}=4$ SYM, where all tree-level amplitudes can be organized into superamplitudes that respect the full $\mathcal{N}=4$ supersymmetry.
4. Color-Ordering and Gauge Invariance: The techniques employed also extend to organizing calculations based on color orbits in non-abelian gauge theories, leading to significant simplifications. It introduces color-ordered partial amplitudes, which reveal the underlying gauge symmetry more transparently.
5. Dual Conformal Symmetry and the Grassmannian: This symmetry emerges in planar $\mathcal{N}=4$ SYM and is a powerful tool in the formulation of scattering amplitudes. The authors discuss its implications and how it leads to the concept of the Grassmannian, a geometrical structure underlying scattering amplitudes.
6. Twistor and Momentum Twistor Variables: These are allied variables that linearize the action of the conformal group on scattering amplitudes, thereby exposing hidden structures like dual superconformal symmetry.

Numerical Results and Theoretical Implications

The paper provides strong numerical and conceptual evidence for the simplification of processes involved in high multi-leg calculations, particularly in the context of MHV (Maximally Helicity Violating) and NMHV (Next-to-Maximally Helicity Violating) amplitudes. It showcases the simplicity that arises from using modern techniques and how these techniques uncover deeper symmetries in quantum field theories.

Future Directions and Speculations

Henriette Elvang and Yu-tin Huang propose further exploration into non-perturbative approaches and the geometric underpinnings of amplitudes, specifically with the aim of gaining insights into unsolved problems in theoretical physics, such as quantum gravity. The potential unification of techniques across different dimensions and extensions beyond supersymmetry are points of interest highlighted by the authors.

Conclusion

This review encapsulates the transition from traditional diagrammatic methods to modern amplitude techniques, revealing new dimensions of understanding and efficiency in the computation of scattering processes. The interplay of symmetries, computational techniques, and physical insights makes it a pivotal contribution to the field of high-energy theoretical physics, emphasizing the elegance and power of on-shell methods in quantum field theory.