Abstract
Recently, there has been renewed interest in a crossing-symmetric dispersion relation from the 1970s due to its implications for both regular quantum field theory and conformal field theory. However, this dispersion relation introduces nonlocal spurious singularities and requires additional locality constraints for their removal, a process that presents considerable technical challenges. In this Letter, we address this issue by deriving a new crossing-symmetric dispersion relation free of spurious singularities. Our formulation offers a compact and nonperturbative representation of the local block expansion, effectively resumming both Witten (in conformal field theory) and Feynman (in quantum field theory) diagrams. Consequently, we explicitly derive all contact terms in relation to the corresponding perturbative expansion. Our results establish a solid foundation for the Polyakov-Mellin bootstrap in conformal field theories and the crossing-symmetry S-matrix bootstrap in quantum field theories.