Explicit BCJ Numerators from Pure Spinors (1104.5224v1)

Abstract: We derive local kinematic numerators for gauge theory tree amplitudes which manifestly satisfy Jacobi identities analogous to color factors. They naturally emerge from the low energy limit of superstring amplitudes computed with the pure spinor formalism. The manifestation of the color--kinematics duality is a consequence of the superstring computation involving no more than (n-2)! kinematic factors for the full color dressed n-point amplitude. The bosonic part of these results describe gluon scattering independent on the number of supersymmetries and captures any N^kMHV helicity configuration after dimensional reduction to D=4 dimensions.

• The paper presents a systematic method for constructing local, Jacobi-satisfying BCJ numerators using pure spinor techniques in gauge theory tree amplitudes.
• It leverages the low-energy limits of superstring amplitudes to avoid ambiguities typical in traditional field theory BCJ constructions.
• The derived numerators facilitate double copy constructions, simplifying loop corrections in both Yang-Mills and N=8 supergravity theories.

Overview of "Explicit BCJ Numerators from Pure Spinors"

The paper "Explicit BCJ Numerators from Pure Spinors" authored by Carlos R. Mafra, Oliver Schlotter, and Stephan Stieberger, advances the paper of gauge theory scattering amplitudes through the lens of the newly established color-kinematics duality. The core contribution of this work is the derivation of local kinematic numerators for gauge theory tree amplitudes that satisfy Jacobi identities. These numerators reveal an underlying duality with color factors, which reflects a hidden symmetry within scattering amplitudes.

Key Contributions

1. Color-Kinematics Duality: The paper builds on the work of Bern, Carrasco, and Johansson (BCJ) by developing a parametrization scheme for gauge theory amplitudes. This scheme ensures that kinematic factors adhere to relations analogous to the Jacobi identity—a foundational concept in the treatment of color factors in gauge theories.
2. Pure Spinor Formalism: The low-energy limits of superstring amplitudes, as computed using the pure spinor formalism, naturally suggest the numerators derived in this work. The paper uses this formalism to avoid the inherent ambiguities that arise in directly constructing BCJ numerators within field theory, particularly those related to contact terms.
3. Construction of BCJ Numerators: The authors present a systematic method to construct BCJ numerators explicitly. By analyzing tree-level diagrams with cubic vertices, the numerators are defined in a manner that is manifestly local and supersymmetric. These constructions remain valid across different spacetime dimensions and helicity configurations, which highlights their general applicability.
4. Applications to Supergravity and SYM: Importantly, the resulting numerators are applicable not only to Yang-Mills theories but also to N=8 supergravity through the double copy construction. This demonstrates the utility of their approach for examining the ultraviolet characteristics of gravity theories at higher loops.

Numerical Results and Implications

The manuscript provides strong numerical evidence supporting the viability of their numerators for various N-point amplitudes. For instance, the authors show how to derive four-point and five-point amplitudes using the numerators, preserving the required Jacobi-like identities across all channels and ensuring consistency with existing theoretical frameworks.

These implicit numerators drastically reduce the complexity involved in calculating loop corrections in supergravity compared to traditional methods. The success in simplifying these calculations significantly impacts our understanding of gravity's perturbative aspects.

Speculative Implications and Future Directions

The work opens avenues for further refinements in amplitude calculations in both gauge theories and gravity. There are specific indications that the methods developed could streamline computations in both perturbative and non-perturbative sectors. Additionally, potential applications in exploring quantum gravity's ultraviolet behavior remain an attractive line of inquiry, particularly when considering the convergence of string theory and gauge field methods.

The future of this research may involve extending these methods to higher-point calculations and further exploring the crossing symmetry lost in current constructions. The prospect of achieving a unified framework that maintains both crossing symmetry and duality could fundamentally simplify theoretical computations and deepen our understanding of underlying physical processes.

Conclusion

The paper makes substantial contributions to the use of pure spinor formalism for understanding the dual nature of color and kinematics in gauge theory amplitudes. By bypassing traditional obstacles and establishing a kinematic basis, the authors provide a robust framework for constructing gauge amplitudes that could potentially unlock new insights in both gauge theory and gravitational studies. This work represents a significant step in harmonizing concepts across various theoretical physics domains.