Analyticity and the Holographic S-Matrix (1111.6972v3)

Abstract: We derive a simple relation between the Mellin amplitude for AdS/CFT correlation functions and the bulk S-Matrix in the flat spacetime limit, proving a conjecture of Penedones. As a consequence of the Operator Product Expansion, the Mellin amplitude for any unitary CFT must be a meromorphic function with simple poles on the real axis. This provides a powerful and suggestive handle on the locality vis-a-vis analyticity properties of the S-Matrix. We begin to explore analyticity by showing how the familiar poles and branch cuts of scattering amplitudes arise from the holographic description. For this purpose we compute examples of Mellin amplitudes corresponding to 1-loop and 2-loop Witten diagrams in AdS. We also examine the flat spacetime limit of conformal blocks, implicitly relating the S-Matrix program to the Bootstrap program for CFTs. We use this connection to show how the existence of small black holes in AdS leads to a universal prediction for the conformal block decomposition of the dual CFT.

• The paper demonstrates that Mellin amplitudes in large N CFTs correspond to the flat-space S-Matrix via a specific integral transform.
• It details derivation techniques using one- and two-loop Witten diagrams to reveal poles and branch cuts that mirror Feynman diagram factorization.
• It connects conformal block decomposition to scattering amplitudes, offering insights into holographic dualities and black hole thermodynamics.

Analyticity and the Holographic S-Matrix: A Review

The paper "Analyticity and the Holographic S-Matrix" provides a detailed exploration of the relationship between Mellin amplitudes in AdS/CFT correspondence and the S-Matrix of a quantum field theory in the flat spacetime limit. This work proves an existing conjecture proposed by Penedones, which hypothesizes a correspondence between Mellin representation of conformal field theory (CFT) correlation functions and the S-Matrix in flat space. The paper achieves this by establishing a methodology to convert Mellin amplitudes into effective bulk S-Matrix descriptions as the curvature radius of Anti-de Sitter space (AdS) approaches infinity.

Key Findings and Methodological Advances

1. Mellin Representation and S-Matrix Equivalence: The authors demonstrate that the Mellin amplitudes of a large $N$ CFT, when taken in the flat spacetime limit, correspond directly to the bulk S-Matrix. This correspondence is articulated through a specific integral transform, proving Penedones’ conjecture for scalar fields and one-loop diagrams. The paper illustrates how Mellin amplitudes encode information about the locality and unitarity properties of the holographic S-Matrix through simple poles on the real axis, providing insights into the analyticity of scattering amplitudes.
2. Derivation Techniques: By analyzing one-loop and two-loop Witten diagrams, this research elucidates the emergence of familiar poles and branch cuts in scattering amplitudes from the holographic dual. These calculations reveal how Mellin amplitudes decompose into lower-point functions, with poles manifesting multi-particle factorization that parallels usual Feynman diagrams at tree level in flat spacetime.
3. Conformal Block Decomposition: The paper aligns the paper of Mellin amplitudes with the conformal bootstrap approach by relating the conformal block decomposition of CFT 4-point functions to the flat-space S-Matrix. This connection is articulated by calculating the flat-space limit of conformal blocks, showing that they transform into delta functions in center-of-mass energy with angular dependencies described by Gegenbauer polynomials.
4. Insights on Black Hole Thermodynamics: The research suggests that high-energy scattering amplitudes, characterized by the emergence of black holes, can inform the conformal block expansions in CFT with gravity duals. The intriguing proposal is that, at trans-Planckian energies, scattering amplitudes diminish due to the entropy associated with black hole formations.

Implications and Future Directions

The implications of this research extend to both theoretical and phenomenological domains. Practically, the results provide techniques for computing loop diagrams in AdS which are crucial for understanding holographic duals of strongly coupled gauge theories. Theoretically, this work sets the stage for evaluating whether all consistent S-Matrices in quantum gravity can be derived from the AdS/CFT framework—a question pivotal to understanding quantum gravity in flat spacetime.

This paper opens avenues for exploring non-perturbative aspects of quantum gravity via the analytic structure of Mellin amplitudes. Future research can extend these methodologies to include fields with spin or investigate the interplay between conformal bootstrap techniques and the S-Matrix program in greater detail. Additionally, a deeper understanding of how black hole physics manifests in the dual CFTs could elucidate the microscopic underpinnings of Hawking radiation and black hole entropy, addressing some of the most fundamental questions in theoretical physics.

Overall, "Analyticity and the Holographic S-Matrix" bridges significant conceptual gaps in holographic duality, providing a foundation for deeper insights into the intersection of conformal field theories and quantum gravity.