Two-point Functions and Bootstrap Applications in Quantum Field Theories

TL;DR

This work develops a comprehensive framework connecting two-point functions of conserved currents and the stress-tensor in UV-complete QFTs to their central charges, across generic spacetime dimensions. It builds real-time and Euclidean formalisms (Wightman, Källén-Lehmann, and Lehmann representations), derives explicit spectral densities in Lorentzian CFTs, and clarifies how UV/IR central charges govern asymptotics and 2d sum rules. A central achievement is showing how parity-odd sectors, unitarity, and conformal data feed into concrete bootstrap constraints, including stress-tensor form factors and their positivity bounds, enabling higher-dimensional numerical bootstrap explorations for theories with a mass gap. The results offer both analytic sum rules (notably in 2d) and a clear bootstrap program that links spectral data to UV/IR CFT data, with practical avenues for constraining QFTs via unitarity and positivity.

Abstract

We study two-point functions of local operators and their spectral representation in UV complete quantum field theories in generic dimensions focusing on conserved currents and the stress-tensor. We establish the connection with the central charges of the UV and IR fixed points. We re-derive c-theorems in 2d and show the absence of their direct analogs in higher dimensions. We conclude by focusing on quantum field theories with a mass gap. We study the stress tensor two-particle form factor, derive implications of unitarity and define concrete bootstrap problems in generic dimensions.