Crossing Symmetric Dispersion Relations for Mellin Amplitudes
Rajesh Gopakumar, Aninda Sinha, Ahmadullah Zahed
TL;DR
The paper develops a manifestly crossing-symmetric dispersion framework for Mellin amplitudes of identical-scalar CFT four-point functions, introducing locality constraints to fix contact-term ambiguities and enabling a Witten-diagram expansion. It demonstrates an equivalence between the resulting sum rules and existing two-channel dispersion results (joaopaper), while connecting Polyakov blocks to AdS/Witten diagrams. Comprehensive checks include 2d and 3d Ising model tests, epsilon-expansion analyses, and explicit ℓ=4 contact-term expressions, establishing convergence and consistency of the approach. Additionally, it derives two-sided positivity bounds for AdS EFT Wilson coefficients, highlighting practical implications for EFTs in AdS/CFT and offering a robust non-perturbative bootstrap tool in Mellin space.
Abstract
We consider manifestly crossing symmetric dispersion relations for Mellin amplitudes of scalar four point correlators in conformal field theories (CFTs). This allows us to set up the non-perturbative Polyakov bootstrap for CFTs in Mellin space on a firm foundation, thereby fixing the contact term ambiguities in the crossing symmetric blocks. Our new approach employs certain "locality" constraints replacing the requirement of crossing symmetry in the usual fixed-$t$ dispersion relation. Using these constraints we show that the sum rules based on the two channel dispersion relations and the present dispersion relations are identical. Our framework allows us to connect with the conceptually rich picture of the Polyakov blocks being Witten diagrams in anti-de Sitter (AdS) space. We also give two sided bounds for Wilson coefficients for effective field theories in AdS space.