Gluon Amplitudes as 2d Conformal Correlators (1706.03917v1)

Abstract: Recently, spin-one wavefunctions in four dimensions that are conformal primaries of the Lorentz group SL(2,C) were constructed. We compute low-point, tree-level gluon scattering amplitudes in the space of these conformal primary wavefunctions. The answers have the same conformal covariance as correlators of spin-one primaries in a 2d CFT. The BCFW recursion relation between three- and four-point gluon amplitudes is recast into this conformal basis.

• The paper establishes a conformal basis for gluon scattering amplitudes by using spin-one primary wavefunctions and the Mellin transform.
• It adapts BCFW recursion relations to the conformal framework, confirming the preservation of traditional amplitude techniques.
• The research deepens the connection between 4d quantum field theory and 2d conformal field theory, offering potential insights into holographic quantum gravity.

Overview of "Gluon Amplitudes as $2d$ Conformal Correlators"

This paper explores the connection between four-dimensional (4d) massless quantum field theory (QFT) scattering amplitudes and two-dimensional (2d) conformal field theories (CFTs), with a focus on gluon scattering amplitudes. The authors, Pasterski, Shao, and Strominger, investigate the representation of gluon amplitudes in a novel basis comprised of conformal primary wavefunctions, which transform under the 4d Lorentz group as $SL(2,\mathbb{C})$ primaries. This work is motivated by the broader question of whether 4d QFT scattering amplitudes can be described by some approximations of 2d CFTs, potentially contributing to the holographic understanding of quantum gravity in flat space.

Key Findings and Methodology

1. Conformal Primary Wavefunctions: The authors construct spin-one conformal primary wavefunctions in 4d that are primaries of the Lorentz group $SL(2,\mathbb{C})$. These wavefunctions are used to express tree-level gluon scattering amplitudes in a manner that highlights their conformal covariance, akin to 2d CFT correlators.
2. Mellin Transform: A crucial technical tool employed is the Mellin transform, which effectively recasts the scattering amplitudes from the traditional momentum space into a conformal basis. This facilitates the representation of amplitudes as 2d conformal correlators and emphasizes the connection to CFT.
3. Three- and Four-Point Amplitudes: The paper provides explicit calculations of Mellin-transformed three- and four-point gluon amplitudes. These computations reveal that the transformed amplitudes exhibit the expected conformal covariance properties. Importantly, the four-point amplitudes display a constraint that mirrors the structure of CFT correlators.
4. BCFW Recursion Relations: The authors extend the BCFW recursion relations, a powerful technique in calculating scattering amplitudes, to the conformal basis. They demonstrate that the recursion relations maintain their validity in this new setting, suggesting that traditional amplitude techniques can be adapted to the conformal basis.

Implications and Speculations

The implications of this work are significant both from practical and theoretical standpoints. By establishing a conformal representation for gluon amplitudes, the research opens pathways for utilizing CFT techniques in analyzing scattering processes in 4d QFT. On a theoretical level, this work supports the notion of a deeper connection between 4d field theories and 2d CFTs, potentially offering insights into holographic duality and quantum gravity.

Looking forward, future research could aim to further generalize these findings to include graviton amplitudes, where the 4d-2d connection might be even more pronounced given the infinite-dimensional local symmetries. Further exploration into the role of soft theorems and their representation as Ward identities in this framework could enhance our understanding of symmetry principles in quantum gravity.

In sum, this paper makes a compelling case for the power of using a 2d conformal approach to understand gluon amplitudes, setting a foundation for further research into the interplay between 4d QFT and 2d CFTs and their potential holographic implications.