Crossing Symmetric Dispersion Relations for Mellin Amplitudes.

Area of Science:

• Theoretical Physics
• High Energy Physics
• String Theory

Background:

• Conformal field theories (CFTs) describe critical phenomena and are key to understanding quantum field theories.
• Mellin amplitudes provide a powerful tool for analyzing CFT correlators, especially in higher dimensions.
• Nonperturbative methods like the Polyakov bootstrap are essential for studying strongly coupled CFTs.

Purpose of the Study:

• To establish a rigorous, nonperturbative framework for CFTs in Mellin space.
• To resolve ambiguities in crossing symmetric blocks using a novel dispersion relation approach.
• To connect the CFT Mellin space formalism with geometric descriptions in anti-de Sitter space.

Main Methods:

• Development of manifestly crossing symmetric dispersion relations for Mellin amplitudes.
• Introduction of "locality" constraints as an alternative to traditional crossing symmetry requirements.
• Demonstration of the equivalence between sum rules from two-channel and the new dispersion relations.

Main Results:

• A firm foundation for the nonperturbative Polyakov bootstrap in Mellin space.
• Resolution of contact term ambiguities in crossing symmetric blocks.
• Identification of Polyakov blocks with Witten diagrams in anti-de Sitter (AdS) space.
• Derivation of two-sided bounds for Wilson coefficients in AdS effective field theories.

Conclusions:

• The new dispersion relation method provides a robust framework for CFT analysis in Mellin space.
• This approach unifies different aspects of CFT calculations and their holographic duals.
• The results offer new tools for studying effective field theories in curved spacetimes.