  The bootstrap approach (demanding consistency conditions to scattering
amplitudes) has shown to be quite powerful to tightly constrain gauge theories
at large $N_c$. We extend previous analysis to scattering amplitudes involving
pions and external gauge bosons. These amplitudes allow us to access the chiral
anomaly and connect low-energy physical quantities to UV properties of the
theory. In particular, we are able to obtain an analytic bound on the chiral
anomaly coefficient as a function of the pion dipole polarizabilities. This
bound can be useful for holographic models whose dual UV completions are not
known, and provide a consistency condition to lattice simulations.
