On the Inverse Problem in Effective Field Theory

We show that the tree-level spectrum of heavy particles can be directly extracted from the Wilson coefficients of the corresponding effective field theory at low energies. This procedure is exact when the number of resonances is finite, and otherwise approximate. Our results are derived from a new class of analytic dispersion relations that depend nonlinearly on the scattering amplitude and apply to an exceedingly broad class of theories and kinematics.