A Duality For The S Matrix (0907.5418v1)

Abstract: We propose a dual formulation for the S Matrix of N = 4 SYM. The dual provides a basis for the "leading singularities" of scattering amplitudes to all orders in perturbation theory, which are sharply defined, IR safe data that uniquely determine the full amplitudes at tree level and 1-loop, and are conjectured to do so at all loop orders. The scattering amplitude for n particles in the sector with k negative helicity gluons is associated with a simple integral over the space of k planes in n dimensions, with the action of parity and cyclic symmetries manifest. The residues of the integrand compute a basis for the leading singularities. A given leading singularity is associated with a particular choice of integration contour, which we explicitly identify at tree level and 1-loop for all NMHV amplitudes as well as the 8 particle NNMHV amplitude. We also identify a number of 2-loop leading singularities for up to 8 particles. There are a large number of relations among residues which follow from the multi-variable generalization of Cauchy's theorem known as the "global residue theorem". These relations imply highly non-trivial identities guaranteeing the equivalence of many different representations of the same amplitude. They also enforce the cancellation of non-local poles as well as consistent infrared structure at loop level. Our conjecture connects the physics of scattering amplitudes to a particular subvariety in a Grassmannian; space-time locality is reflected in the topological properties of this space.

• The paper introduces a dual reformulation of scattering amplitudes in N=4 SYM, positing that leading singularities capture complete tree-level and loop-level behavior.
• It constructs amplitudes as integrals over k-planes, embedding cyclic and parity symmetries via Grassmannian geometry.
• Numerical results from 6- and 7-particle NMHV cases support the framework through precise residue calculations and the use of the global residue theorem.

Analysis of "A Duality For The S Matrix"

The paper "A Duality For The S Matrix" presents an intriguing exploration into reformulating the S Matrix, particularly within the context of ${\cal N} = 4$ supersymmetric Yang-Mills (SYM) theory. It proposes a duality that aims to redefine how scattering amplitudes—vital components of quantum field theory—are conceptualized and calculated. This dual formulation is concerned primarily with the leading singularities of scattering amplitudes across all perturbative orders, positing a novel mathematical foundation that purports to encapsulate the complete amplitudes at tree level and conjecturally at all loop orders.

Key Concepts and Methodologies

Central to the authors’ proposal is the construction of scattering amplitudes from leading singularities, which are defined as sharply delineated and infrared-safe data. The authors associate each scattering amplitude with a simple integral over the space of $k$-planes in $n$ dimensions, embedding the influence of parity and cyclic symmetries directly into the formulation of the amplitude. The residues of this integrand compute a basis for these leading singularities, linking each singularity with specific integration contours.

The paper expounds on the mathematical and physical implications of adopting a duality framework, aligning the behavior of scattering amplitudes within the geometry of Grassmannians. Grassmannians provide the mathematical structure underlying the new formulation, suggesting a geometric perspective on amplitude calculations. The authors take care to align their findings with existing methodologies, such as the Britto-Cachazo-Feng-Witten (BCFW) recursion relations which already enhance computational techniques in field theory.

Numerical Results and Theorems

Concrete numerical results highlight the prowess of this duality framework. The researchers achieve significant results by calculating the residues that align with known loop-level phenomena—providing supporting evidence for their conjecture. In particular, they develop heuristic evidence that supports their theses by examining cases like the 6-particle NMHV and 7-particle NMHV amplitudes. Each residue aligns precisely with known data from one-loop computations.

Several theorems, notably the global residue theorem, underlie the mathematical rigor of the approach. These theorems assist in guaranteeing the validity of the amplitude calculations proposed by the duality framework. The global residue theorem plays a crucial role by accounting for relations that naturally eliminate inconsistencies arising from various representations of the same amplitude and mandating the absence of unphysical poles.

Future Directions and Implications

The proposed dual perspective on the S Matrix fundamentally changes how theoretical physicists might approach amplitude calculations in quantum field theory. It primes several future avenues of research, both theoretically and computationally. The authors speculate on the theoretical implications of simplifying the treatment of non-local poles and doubly logarithmic infrared divergences at loop levels.

Furthermore, the adoption of Grassmannian geometries opens up interesting theoretical pursuits related to the holographic mappings of scattering amplitudes, perhaps offering insights that are transferrable to quantum gravity and string theory. This new conceptual framework might foster fresh computational algorithms which are intrinsically orderly, compared to the tangled web of conventional Feynman diagrams.

In summary, "A Duality For The S Matrix" offers compelling evidence and a well-motivated conjecture for rethinking the S Matrix through the lens of duality and geometry, setting the stage for a potential recalibration of strategies in theoretical and computational particle physics. As the researchers continue to refine this framework, the field might edge closer to resolving some of the outstanding complexities that have long challenged high-energy physicists.