Partial-wave amplitudes and multiparticle production

Recursion relations for integrals of amplitudes over the phase space, i.e. for partial wave amplitudes, are introduced. In their simplest form these integrals are proportional to the s-wave amplitudes and represent rigorous lower bounds on the total cross sections. The connection with classical field equations in D dimensions is established. Previous results on multiparticle amplitudes are easily reproduced. CERN-TH.6971/93 August 1993 The high-multiplicity limit of processes involving scalar particles has been studied in several contexts [1, 2, 3, 4, 5, 6, 7]. Ordinary perturbation theory for the amplitude A(∞ → \) predicts a factorial growth, n!, which is inconsistent with the unitarity limits. Higher-order corrections at threshold [8, 9, 10] exhibit an even faster growth with n, and the whole perturbative expansion breaks down for sufficiently high multiplicities. On the other hand, the nullification phenomenon, i.e. the nullification of A(∈ → \) amplitudes at threshold in certain theories [8, 11, 12, 13], including the Standard Model with specific relations among the different couplings [14, 15], might be the signal that, under certain conditions, the perturbative expansion is still consistent with the unitarity limits: delicate cancellations among different graphs contributing to the amplitude might soften the n! growth below the limiting case of the √ n! one, so that the cross sections still satisfy unitarity bounds. The case of sinh-Gordon interactions in two dimensions is a quite interesting example of such cancellations, where the nullification survives at any energy and/or at any order of perturbation theory [16]. In this paper we are introducing some new recursion relations involving the integrals of the amplitudes over the phase space, a(n, w) = ∫ n