  We derive asymptotic behaviors of the Nambu-Bethe-Salpeter (NBS) wave
function at large space separations for systems with more than 2 particles in
quantum field theories. To deal with $n$-particles in the center of mass flame
coherently, we introduce the Jacob coordinates of $n$ particles and then
combine their $3(n-1)$ coordinates into the one spherical coordinate in
$D=3(n-1)$ dimensions. We parametrize on-shell $T$-matrix for $n$-particle
system of scalar fields at low energy, using the unitarity constraint of the
$S$-matrix. We then express asymptotic behaviors of the NBS wave function for
$n$ particles at low energy, in terms of parameters of $T$-matrix, and show
that the NBS wave function carry the information of $T$-matrix such as phase
shifts and mixing angles of the $n$-particle system in its own asymptotic
behavior, so that the NBS wave function can be considered as the scattering
wave of $n$-particles in quantum mechanics. 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. Our results, together with
an extension to systems with spin 1/2 particles, justify the HAL QCD's
definition of potentials for 3 or more nucleons(baryons) in terms the NBS wave
functions.
