Virtual Particles

We define particles or resonances which do not correspond to the usual Breit-Wigner type poles in the partial wave amplitudes. They are described in terms of the trajectory of the poles of the $S$ matrix in the complex angular-momentum plane as a function of energy, when this trajectory does not quite reach a physical value of $J$. Range and scattering parameters are used to determine the trajectories near the threshold, which in turn are related to high-energy cross sections for processes in which these particles are exchanged in the cross channels. The trajectories for the two definitely known examples, namely the $I=0\mathrm{}$, $\pi -\pi$, $S$-wave "virtual state", and the single $n-p$ $S$-wave "virtual state", as well as the triplet $n-p$ trajectory, are determined.