  We extend the HAL QCD approach, with which potentials between two hadrons can
be obtained in QCD at energy below inelastic thresholds, to inelastic and
multi-particle scatterings. We first derive asymptotic behaviors of the
Nambu-Bethe-Salpeter (NBS) wave function at large space separations for systems
with more than 2 particles, in terms of the one-shell $T$-matrix consrainted by
the unitarity of quantum field theories. We show that its asymptotic behavior
contains phase shifts and mixing angles of $n$ particle scatterings. This
property is one of the essential ingredients of the HAL QCD scheme to define
"potential" from the NBS wave function in quantum field theories such as QCD.
We next construct energy independent but non-local potentials above inelastic
thresholds, in terms of these NBS wave functions. We demonstrate an existence
of energy-independent coupled channel potentials with a non-relativistic
approximation, where momenta of all particles are small compared with their own
masses. Combining these two results, we can employ the HAL QCD approach also to
investigate inelastic and multi-particle scatterings.

High Energy Physics - Lattice;High Energy Physics - Phenomenology;Nuclear Theory