Multiparticle production in weak coupling theories in the Lipatov approach

We study the amplitudes with a large number of external legs in gφ⁴ theory within the Lipatov approach. This approach has been originally suggested to determine how the amplitudes with a fixed number of external lines depend on the order of perturbation theory, at large orders. We discuss the generalization of this method to amplitudes with a large number of external legs, N, and few or no loops at all (tree-level multiparticle amplitudes). We show that the N-dependence in this limit is under theoretical control and we reproduce the factorial growth discussed in the literature. We elucidate the relation between the alleged onset of the strong-coupling regime in multiparticle production and the well-known problem of the divergence of perturbation theory at large orders. We argue that if the number of particles produced exceeds a critical value the cross sections are bounded by exp(-const./g).