The behavior of multi-point Green functions at zero energy and momentum carried by all external particles is studied in the limit of large number of particles in a scalar theory with and without spontaneous breaking of a discrete symmetry. It is found that in the theory without the symmetry breaking the exact asymptotic behavior coincides with that given by the tree graphs, so that the loop corrections are not essential. In contrast, in the case of spontaneously broken symmetry the number of essential loops grows proportionally to the number of external particles.
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It is my pleasure to acknowledge valuable as well as critical discussions with L. McLerran, M. Shifman and A. Vainshtein. This work is supported in part by DOE grant DE-ACO2-83ER40105.