Bootstrapping Pions at Large $N$. Part II: Background Gauge Fields and the Chiral Anomaly

We continue the program [1] of carving out the space of large $N$ confining gauge theories by modern S-matrix bootstrap methods, with the ultimate goal of cornering large $N$ QCD. In this paper, we focus on the effective field theory of massless pions coupled to background electromagnetic fields. We derive the full set of positivity constraints encoded in the system of 2 $\to$ 2 scattering amplitudes of pions and photons. This system probes a larger set of intermediate meson states, and is thus sensitive to intricate large $N$ selection rules, especially when supplemented with expectations from Regge theory. It also has access to the coefficient of the chiral anomaly. We find novel numerical bounds on several ratios of Wilson coefficients, in units of the rho mass. By matching the chiral anomaly with the microscopic theory, we also derive bounds that contain an explicit $N$ dependence.