Primal Bootstrap for Pion Scattering at Large-N

Abstract

We introduce a basis for tree-level meromorphic scattering amplitudes suitable for describing pion scattering in the large-N limit. The basis is constructed as linear combinations of Lovelace-Shapiro-like amplitudes with varying Regge slopes and intercepts. The resulting amplitudes satisfy by construction the fundamental requirements of analyticity, crossing symmetry, and Regge behavior. We analyze their behavior in specific kinematical regimes, including the high-energy fixed-angle limit. We also show that finite linear combinations of our basis elements need not violate unitarity. Nonetheless, because unitarity is not imposed by construction, we enforce it a posteriori by requiring positivity of the partial-wave decomposition. This condition can be formulated as an optimization problem and solved numerically. The solutions to this primal bootstrap problem yield meromorphic amplitudes that satisfy all the aforementioned constraints. We compare several observables with the bounds obtained from the dual positivity conditions and show that our family of amplitudes spans the full allowed parameter space. With appropriate modifications, this method can be extended to construct amplitude families for broader applications.