  The influence of a two-hadron threshold is studied for the hadron mass
scaling with respect to some quantum chromodynamics parameters. A quantum
mechanical model is introduced to describe the system with a one-body bare
state coupled with a single elastic two-body scattering. The general behavior
of the energy of the bound and resonance state near the two-body threshold for
a local potential is derived from the expansion of the Jost function around the
threshold. It is shown that the same scaling holds for the nonlocal potential
induced by the coupling to a bare state. In p or higher partial waves, the
scaling law of the stable bound state continues across the threshold describing
the real part of the resonance energy. In contrast, the leading contribution of
the scaling is forbidden by the nonperturbative dynamics near the s-wave
threshold. As a consequence, the bound state energy is not continuously
connected to the real part of the resonance energy. This universal behavior
originates in the vanishing of the field renormalization constant of the
zero-energy resonance in the s wave. A proof is given for the vanishing of the
field renormalization constant, together with a detailed discussion.
