Towards Feynman rules for Mellin amplitudes in AdS/CFT (1107.1504v3)

Abstract: We investigate the use of the embedding formalism and the Mellin transform in the calculation of tree-level conformal correlation functions in $AdS$/CFT. We evaluate 5- and 6-point Mellin amplitudes in $\phi^3$ theory and even a 12-pt diagram in $\phi^4$ theory, enabling us to conjecture a set of Feynman rules for scalar Mellin amplitudes. The general vertices are given in terms of Lauricella generalized hypergeometric functions. We also show how to use the same combination of Mellin transform and embedding formalism for amplitudes involving fields with spin. The complicated tensor structures which usually arise can be written as certain operators acting as projectors on much simpler index structures - essentially the same ones appearing in a flat space amplitude. Using these methods we are able to evaluate a four-point current diagram with current exchange in Yang-Mills theory.

Towards Feynman Rules for Mellin Amplitudes in AdS/CFT

The paper "Towards Feynman rules for Mellin amplitudes in AdS/CFT" by Miguel F. Paulos endeavors to create a framework to simplify computations of correlation functions in the AdS/CFT correspondence by employing Mellin amplitudes. The motivation stems from the intricate and computationally burdensome nature of traditional Witten diagrams when calculating these correlation functions.

Main Contributions

1. Use of Mellin Transform: The paper proposes utilizing the Mellin transform as a more efficient basis for these calculations. This approach is advantageous due to its conformity with the symmetries inherent in conformal field theories (CFTs). For instance, the Mellin amplitudes simplify the analytic structure of correlation functions, providing a more straightforward path to elucidate factorization properties.
2. Formulation of Feynman Rules: By evaluating specific scalar theories, such as $\phi^3$ and $\phi^4$, the paper ventures to establish a set of Feynman-like rules for Mellin amplitudes. These theoretical rules offer a systematic method to calculate n-point amplitudes by gluing together lower-point amplitudes, which inherently simplifies the computation by focusing on three-point functions.
3. Lauricella Functions: The vertices in the proposed Feynman rules are expressed using generalized Lauricella hypergeometric functions. This substitution displays a novel and efficient analytic methodology for managing descendant fields in AdS/CFT, representing a step forward in calculating the Mellin amplitudes for higher-point functions.
4. Extension to Spin Fields: The paper extends these methodologies to amplitudes involving fields with spin, demonstrating that complex tensor structures typically required for these calculations can be efficiently handled using projector operators, thus reducing the computational burden.
5. Case Studies on Specific Scenarios: Calculations are performed for scenarios involving five, six, and even twelve-point diagrams, corroborating the proposed framework. These examples validate the feasibility of the theorized Feynman rules in simplifying and tractably representing higher-order diagrams in Mellin space.

Implications and Future Directions

The implications of this research are substantial as it provides a promising direction for rendering previously cumbersome calculations in AdS/CFT more manageable. The introduction of Mellin amplitudes as a complement to the embedding formalism paves the way for potential advancements in theoretical physics, especially regarding complex quantum field theories in curved spacetime contexts.

Future developments could include the extension of these Feynman rules to include fields of various spins beyond scalars, and potentially, to the computations involving loop amplitudes. Moreover, the analogies drawn to flat-space scattering amplitudes suggest intriguing connections that could further bridge concepts from different areas in theoretical physics. Finally, exploring applications of these concepts to practical problems in quantum gravity and high-energy physics could provide additional insights into the structure of the universe at fundamental levels.

In conclusion, this paper contributes a firm foundation towards developing a systematic method for dealing with Mellin amplitudes in AdS/CFT, marking a significant stride towards more efficient calculations within the field of theoretical physics.