Large $N$ hadron scattering bootstrap: massive pions and glueballs

We extend the large $N$ bootstrap of pion scattering to nonzero pion mass, and apply the same formalism to glueball scattering. Our bounds apply to Taylor coefficients and on-shell couplings of the exact planar amplitude. They do not rely on a truncated chiral expansion, so we can follow them as the quark mass varies, and apply them to glueballs, for which no low-energy effective theory exists. For pions, we bound the $O(p^4)$ coefficients and the $\pi \pi f_2$ coupling. As in the chiral limit, the bound on the $\pi \pi f_2$ coupling features a kink, where the extracted spectrum closely resembles the real-world meson spectrum. For glueballs, we bound the couplings of the lightest scalar glueball to heavier glueballs of higher spin, using the large $N$ lattice spectrum as input, and find extremal solutions with linear Regge trajectories.