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Exact Multiparticle Amplitudes at Threshold in Large-N Component phi^4 Theory

I derive the set of recurrence relations between the amplitudes of multiparticle production at threshold in the standard large-N limit of the O(N)-symmetric phi^4$ theory which sums all relevant diagrams with arbitrary number of loops. I find an exact solution to the recurrence relations using the Gelfand--Dikii representation of the diagonal resolvent of the Schrodinger operator. The result coincides with the tree amplitudes while the effect of loops is the renormalization of the coupling constant and mass. The form of the solution is due to the fact that the exact amplitude of the process 2->n at n>2 vanishes on mass shell when averaged over the O(N)-indices of incoming particles for dynamical reasons because of the cancellation between diagrams. I discuss some possible applications of large-N amplitudes, in particular, for the renormalon problem.