# Flux Tube S-matrix Bootstrap

## TL;DR

This paper develops a bootstrap approach to constrain the S-matrix of massless particles in 2D quantum field theories, with implications for flux tube physics and bounds on effective action coefficients.

## Contribution

It introduces a novel bootstrap method for massless 2D S-matrices, constraining low energy expansions and related physical quantities, including flux tube spectra.

## Key findings

- Bounded the low energy S-matrix expansion terms.
- Identified a resonance matching the world-sheet axion.
- Provided bounds on flux tube ground state energy and level splitting.

## Abstract

We bootstrap the S-matrix of massless particles in unitary, relativistic two dimensional quantum field theories. We find that the low energy expansion of such S-matrices is strongly constrained by the existence of a UV completion. In the context of flux tube physics, this allows us to constrain several terms in the S-matrix low energy expansion or -- equivalently -- on Wilson coefficients of several irrelevant operators showing up in the flux tube effective action. These bounds have direct implications for other physical quantities; for instance, they allow us to further bound the ground state energy as well as the level splitting of degenerate energy levels of large flux tubes. We find that the S-matrices living at the boundary of the allowed space exhibit an intricate pattern of resonances with one sharper resonance whose quantum numbers, mass and width are precisely those of the world-sheet axion proposed in [1,2]. The general method proposed here should be extendable to massless S-matrices in higher dimensions and should lead to new quantitative bounds on irrelevant operators in theories of Goldstones and also in gauge and gravity theories.