Local Integrals for Planar Scattering Amplitudes (1012.6032v1)

Abstract: Recently, an explicit, recursive formula for the all-loop integrand of planar scattering amplitudes in N=4 SYM has been described, generalizing the BCFW formula for tree amplitudes, and making manifest the Yangian symmetry of the theory. This has made it possible to easily study the structure of multi-loop amplitudes in the theory. In this paper we describe a remarkable fact revealed by these investigations: the integrand can be expressed in an amazingly simple and manifestly local form when represented in momentum-twistor space using a set of chiral integrals with unit leading singularities. As examples, we present very-concise expressions for all 2- and 3-loop MHV integrands, as well as all 2-loop NMHV integrands. We also describe a natural set of manifestly IR-finite integrals that can be used to express IR-safe objects such as the ratio function. Along the way we give a pedagogical introduction to the foundations of the subject. The new local forms of the integrand are closely connected to leading singularities---matching only a small subset of all leading singularities remarkably suffices to determine the full integrand. These results strongly suggest the existence of a theory for the integrand directly yielding these local expressions, allowing for a more direct understanding of the emergence of local spacetime physics.

• The paper introduces a novel representation of multi-loop integrands via a recursive formula that highlights Yangian symmetry in N=4 SYM.
• It employs momentum-twistor space to recast integrals into chiral forms with unit leading singularities, simplifying complex amplitude computations.
• The study establishes finite, IR-safe integrals, paving the way for more efficient calculations and deeper insights into the structure of scattering amplitudes.

Insights into "Local Integrals for Planar Scattering Amplitudes"

The paper presented provides a detailed examination of local integrals in the context of planar scattering amplitudes within maximally supersymmetric Yang-Mills theory ($\mathcal{N}=4$ SYM). The central contribution lies in a novel representation of the integrand for multi-loop scattering amplitudes that illuminates both the structure and the simplicity of the theory's underlying mathematical framework.

Recursive Formula and Yangian Symmetry

The authors explore a recursive formula generalizing the Britto-Cachazo-Feng-Witten (BCFW) recursion for tree amplitudes to all-loop integrands. This approach hinges on the Yangian symmetry of the theory, a prominent feature of $\mathcal{N}=4$ SYM, which the recursive construction makes manifest. By using momentum-twistor space, the integrands are recast into forms that highlight their natural cyclicity and dual conformal invariance.

Exploiting Momentum-Twistor Space

One of the remarkable aspects of this research is the effective utilization of momentum-twistor space, introduced by Andrew Hodges, which offers a more intuitive geometric perspective on scattering amplitudes. In this space, the integrals are represented by chiral integrands characterized by unit leading singularities, facilitating an unprecedented simplicity in the expressions for amplitudes. The authors focus on planar theories, where dual conformal symmetry allows for the reduction of multiloop integrals into a coherent form that simplifies calculations and exposes the symmetry inherent in these integrals.

Simplification of Multi-Loop Integrands

The authors extend the formalism to multi-loop amplitudes, illustrating the application of their methods to 2-loop and 3-loop amplitudes for both MHV and NMHV sectors. The main achievement here is representing these sophisticated integrals as strikingly simple sums over specific momentum-twistor configurations. These representations effectively capture all leading singularities of the amplitudes, hypothesized to be fundamental units of the theory's scattering data.

Finite and IR-Safe Integrals

Another significant aspect explored is the construction of IR-finite integrals. By carefully choosing chiral integrands, the authors define new local forms that remain finite upon integration, precluding the need for regularization. This feature underscores the potential of these forms for simplifying calculations in complex quantum field theoretical contexts.

Future Directions and Implications

The insights garnered from this paper not only impact practical calculations in $\mathcal{N}=4$ SYM but also hint at a deeper theoretical structure governing scattering amplitudes. The expression of amplitudes in terms of local, purely chiral integrands suggests a potential for further reducing the complexity of higher-loop computations and uncovering more profound connections between scattering theory and geometric structures such as polytopes. Additionally, the techniques and results may extend to less symmetric contexts or to the exploration of simlar constructs in string theory.

This paper lays a foundation for continued investigation into integrable structures in scattering amplitudes, addressing both practical computation needs and theoretical questions about the relationship between amplitudes and emergent geometric and algebraic objects. As such, it invites speculations on the evolution of amplitude theory, potentially leading toward a more comprehensive understanding of quantum field theories.