A Note on Polytopes for Scattering Amplitudes (1012.6030v1)

Abstract: In this note we continue the exploration of the polytope picture for scattering amplitudes, where amplitudes are associated with the volumes of polytopes in generalized momentum-twistor spaces. After a quick warm-up example illustrating the essential ideas with the elementary geometry of polygons in CP^2, we interpret the 1-loop MHV integrand as the volume of a polytope in CP^3x CP^3, which can be thought of as the space obtained by taking the geometric dual of the Wilson loop in each CP^3 of the product. We then review the polytope picture for the NMHV tree amplitude and give it a more direct and intrinsic definition as the geometric dual of a canonical "square" of the Wilson-Loop polygon, living in a certain extension of momentum-twistor space into CP^4. In both cases, one natural class of triangulations of the polytope produces the BCFW/CSW representations of the amplitudes; another class of triangulations leads to a striking new form, which is both remarkably simple as well as manifestly cyclic and local.

• The paper represents perturbative scattering amplitudes geometrically as volumes of specific polytopes in momentum-twistor space.
• Scattering amplitudes can be derived from triangulations of these polytopes, offering novel representations emphasizing cyclicity and locality.
• This geometric framework provides new insights into amplitude structures and suggests alternative computational methods for gauge theories.

Polytopes and Scattering Amplitudes: A Geometric Interpretation

The paper "A Note on Polytopes for Scattering Amplitudes" by Arkani-Hamed et al. provides an intriguing perspective on perturbative scattering amplitudes, particularly within ${\cal N}=4$ Supersymmetric Yang-Mills (SYM) theory, using geometric polytope concepts. This work explores the connections between scattering amplitudes and the geometrical properties of polytopes in complex projective spaces, offering potential insights both algebraically and geometrically.

Summary of Results and Definitions

The authors approach scattering amplitudes by associating them with volumes of polytopes located in generalized momentum-twistor spaces. These polytopes originate from extensions of the BCFW recursion relations, formulated naturally in momentum-twistor space. One of the significant results involves expressing scattering amplitudes via triangulations of these polytopes, revealing forms that emphasize cyclicity and locality distinctly compared to usual representations.

Key Components

1. Momentum-Twistor Space: The paper adopts momentum-twistor space as the optimal backdrop for formulating scattering amplitudes, making the symmetry structures such as Yangian symmetry more apparent.
2. Polytope Interpretation: A notable conceptual leap is interpreting amplitudes as polytopes' volumes, specifically in ${\mathbb{CP}^3 \times \mathbb{CP}^3}$ and their geometric dualities. This mapping provides an innovative way to understand amplitude properties, like cyclic invariance and locality, through simple volume integrals over polytopes.
3. Triangulation Methods: Various triangulation methods of the polytopes lead to representations of amplitudes. Traditional BCFW triangulations offer compact expressions but can introduce spurious poles, whereas alternative triangulations proposed here yield expressions that are simpler and exhibit manifest symmetries.

Detailed Investigations

• One-Loop MHV Integrand: By mapping this integrand onto certain polytope volumes, the paper presents a novel form that emphasizes the manifest cyclicity and locality, and derives alternate forms that eliminate spurious singularities conventionally present in other representations.
• NMHV Tree Amplitudes: The paper conjectures these amplitudes as the volume of geometric duals of specific polytopes. Previously reported identities among $R$-invariants are reimagined through additive simplicial structures, enhancing understanding regarding amplitude representation and transformation properties.

Implications and Future Directions

The implications are profound, especially regarding how scattering amplitudes might be fundamentally represented. This geometric and algebraic framework lends itself to generalizations beyond simple perturbative calculations, inviting further explorations in non-supersymmetric theories.

Future developments may include:

• Extending these polytope methods to higher loop orders and amplitudes beyond MHV and NMHV.
• Investigating connections between these geometric representations and potential physical or phenomenological insights into gauge theories in non-supersymmetric regimes.
• Using these insights to streamline amplitude calculations in scenarios that maximize symmetries while ensuring physical interpretability.

Conclusion

This paper suggests a compelling blend of geometry and algebra in understanding the fabric of scattering amplitudes within gauge theories. The polytope and momentum-twistor approaches promise advances in deciphering the algebraic structure underlying physical amplitudes while suggesting novel computational techniques. This may lead to more streamlined representations both encapsulating inherent symmetries and practical for extensive multiparticle processes. The methodological shift from purely algebraic manipulations to spatial-geometric comprehension marks a significant step toward unifying various representations of quantum field theoretic phenomena.