An example of semiclassical instanton-like scattering: (1 + 1)-dimensional sigma model

Abstract

A solution to the classical field equations in the massless (1 + 1)-dimensional O(3) sigma model is found, which describes a multiparticle instanton-like transition at high energy. In the limit of a small number of initial particles, the number of final particles is shown to be also small, and the probability of the transition is suppressed by exp(−2 S 0), where S 0 is the instanton action. This solution, however, does not correspond to the maximum transition probability among all states with given number of incoming particles and energy. Unless the limit g 2 n initial → 0 is exponentially sensitive to the structure of the initial state, our results imply that well above the sphaleron energy, the instanton-induced cross section becomes again suppressed by the instanton exponent, and the number of final particles is again small.