Finite Energy Instantons in the O(3) Non-linear Sigma Model

We consider winding number transitions in the two dimensional O(3) non-linear sigma model, modified by a suitable conformal symmetry breaking term. We discuss the general properties of the relevant instanton solutions which dominate the transition amplitudes at finite energy, and find the solutions numerically. The Euclidean period of the solution increases with energy, contrary to the behavior found in the abelian Higgs model or simple one dimensional systems. This indicates that there is a sharp crossover from instanton dominated tunneling to sphaleron dominated thermal activation at a certain critical temperature in this model. We argue that the electroweak theory in four dimensions should exhibit a similar behavior. e-mail: [email protected] [email protected] [email protected] Gauge theories of the strong and electroweak interactions are characterized by a multiple vacuum structure. Tunneling transitions between different vacua are responsible for physically interesting effects, such as baryon number violation in the electroweak theory. At zero temperature and energy these winding number transitions are dominated by the familiar zero energy instanton solutions of the Euclidean field equations, with vacuum boundary conditions. At finite temperatures thermal activation over the potential barrier separating the multiple vacua can occur in addition to quantum tunneling. The static classical solution whose energy is equal to the top of this barrier between neighboring vacua is the sphaleron solution. At sufficiently high temperatures transitions between different winding number sectors are dominated by classical thermal activation with a rate controlled by the energy of the sphaleron. In simple one dimensional systems there is a smooth crossover from the zero energy instanton dominated tunneling transition to the high temperature sphaleron dominated regime. The corresponding classical solutions interpolating between these two situations are known as periodic instantons with turning points at finite Euclidean time β, which give a non-pertubative contribution to the partition function and transition rate at temperature β. One would expect similar considerations to apply in quantum field theory, although the situation is much less well explored and very few classical solutions of this kind are known. Periodic instantons appear naturally also in the context of zero temperature field theory, if one considers the transition probability between different winding number sectors at fixed energy, E. This probability may be expressed in the form,