Summing One-Loop Graphs at Multi-Particle Threshold

It is shown that the technique recently suggested by Lowell Brown for summing the tree graphs at threshold can be extended to calculate the loop effects. 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 in the unbroken $\lambda \phi^4$ 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 $n > 4$.