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It is shown that the technique recently suggested by Brown for summing the tree graphs at threshold can be extended to calculate the loop effects. An explicit result is derived for the sum of one-loop graphs for the amplitude of threshold production of n on-mass-shell particles by one virtual particle in the unbroken h44 theory. It is also found that the tree-level amplitude of production of n particles by two incoming on-mass-shell particles vanishes at the threshold for n > 4. PACS numberk): 1 l.lO.Jj, 11.1O.EfThe problem of calculating amplitudes of processes with many weakly interacting particles has recently attracted a considerable interest, initially triggered by the observation [l-31 that such processes in particular are associated with a possible baryon-and lepton-number violation in high-energy electroweak interactions. Cornwall [4] and Goldberg [5] have pointed out that in perturbative amplitudes with many external particles the weak coupling may get compensated by a large number of diagrams. This is a manifestation of the old-standing problem of the factorial growth of the coefficients in the perturbation theory [6]. Since the perturbative expansion for multiparticle amplitudes starts from a high order in the coupling constant, for a sufficiently large number n of particles the factorial growth of the coefficients in the series invalidates the perturbative calculation of such amplitudes. Given the lack of a better approach it seems useful to quantify and study the problem within the perturbation theory itself. A simple model example in which the problem arises with full strength is the amplitude A , where a virtual particle of a real scalar field 4 produces a large number n of on-mass-shell 4 particles in the h 4 4 theory. It has been recently found that the sum of all tree graphs for this amplitude in the threshold limit, i.e., when all the produced particles are at rest, can be calculated exactly for arbitrary n both in the case of unbroken symmetry [7] and in the case of theory with spontaneous breaking of the symmetry under the reflection 4 --4 [81.Originally the calculation [7] was done by directly solving a recursion relation for the tree graphs. Argyres, Kleiss, and Papadopoulous [8] applied a regular method of solving the recursion relations based on a generating function for which the recursion relation for the amplitudes A , is equivalent to second-order nonlinear differential equation. Most recently Brown [9] has shown that the generating function is nothing else than the classical field 4 , ( t ) generated by an external source p=poeim' which field is a complex solution of the Euler-Lagrange classical equation satisfying the condition that it has only the positive frequency part. The equation and their solutions in both approaches are related by a simple change of variable. Thus Brown has reproduced the previous results [7,8] in a simple and elegant way.The purpose of this paper is to show that Brown's technique can be extended to calculate the loop contributions to the amplitudes A , as...

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