Bootstrapping Pion Form Factors at Large $N$

Abstract

We initiate a bootstrap study of pion form factors in large $N$ QCD. We consider the mixed system of the vector-current two-point function, the pion vector form factor, and the pion scattering amplitude in the chiral limit. At large $N$ these observables are meromorphic, with spectral data constrained by unitarity, crossing symmetry, and Regge boundedness. We obtain bounds of two kinds. The first are rigorous and universal: from analyticity, unitarity and the asymptotic Brodsky-Farrar scaling, we constrain low-energy form-factor coefficients. The second are more phenomenological, of the Shifman-Vainshtein-Zakharov type: feeding in the perturbative ultraviolet behavior at a finite scale lets us bound the pion decay constant, convert a large $N$ lattice measurement into a lower bound on the scale at which asymptotic freedom sets in, and constrain the pion charge radius. Combining these inputs, the space of allowed chiral Lagrangians shrinks toward the region where large $N$ QCD is expected to sit. Our results illustrate how local gauge-invariant probes provide a canonical bridge between the hadronic bootstrap and the microscopic QCD Lagrangian.