Bootstrapping QCD Using Pion Scattering Amplitudes

We consider the S-matrix bootstrap of four-dimensional scattering amplitudes with O(3) symmetry and no bound states. We explore the allowed space of scattering lengths which parametrize the interaction strength at threshold of the various scattering channels. Next we consider an application of this formalism to pion physics. A signature of pions is that they are derivatively coupled leading to (chiral) zeros in their scattering amplitudes. In this work we explore the multidimensional space of chiral zeros positions, scattering length values, and resonance mass values. Interestingly, we encounter lakes, peninsulas, and kinks depending on which sections of this intricate multidimensional space we consider. We discuss the remarkable location where QCD seems to lie in these plots, based on various experimental and theoretical expectations.