Restrictions on the growth of the mass-shell elastic scattering amplitude following from the short range character of the interaction

Bounds of Greenberg–Low type and analyticlty in the squared momentum transfer in a certain ellipse are proved for the elastic scattering amplitude $F(s,\cos\theta)$ for a large class of field theories, (including nonlocalizable theories [10]). It is shown that the conditions that are sufficient for the proof of the existence of the $S$-matrix by the Haag–Ruelle methodare sufficient for the proof of the bounds on the partial amplitudes. The proof is based on the cluster property of the vacuum expectation values. These properties reflect the shortrange character of the interaction forces between the particles.