  The worldsheet axion plays a crucial role in the dynamics of the Yang-Mills
confining flux tubes. According to the lattice measurements, its mass is of
order the string tension and its coupling is close to a certain critical value.
Using the S-matrix Bootstrap, we construct non-perturbative $2 \to 2$ branon
scattering amplitudes which also feature a weakly coupled axion resonance with
these properties. We study the extremal bootstrap amplitudes in detail and show
that the axion plays a dominant role in their UV completion in two distinct
regimes, in one of which it cannot be considered a parametrically light
particle. We conjecture that the actual flux tube amplitudes exhibit a similar
behavior.
