Bootstrapping Massive Quantum Field Theories

TL;DR

This paper introduces a novel non-perturbative approach to analyze UV complete, unitary quantum field theories with a mass gap, utilizing unitarity constraints expressed through positive semi-definite matrices involving scattering data.

Contribution

It develops a new method based on unitarity and positive semi-definiteness to bound properties of quantum field theories, tested in two dimensions with promising results.

Findings

Numerical bounds on the central charge are obtained.

Some bounds are saturated by known integrable models.

The method is applicable to a broad class of theories.

Abstract

We propose a new non-perturbative method for studying UV complete unitary quantum field theories (QFTs) with a mass gap in general number of spacetime dimensions. The method relies on unitarity formulated as positive semi-definiteness of the matrix of inner products between asymptotic states (in and out) and states created by the action of local operators on the vacuum. The corresponding matrix elements involve scattering amplitudes, form factors and spectral densities of local operators. We test this method in two-dimensional QFTs by setting up a linear optimization problem that gives a lower bound on the central charge of the UV CFT associated to a QFT with a given mass spectrum of stable particles (and couplings between them). Some of our numerical bounds are saturated by known form factors in integrable theories like the sine-Gordon, E8 and O(N) models.