mentioned earlier, and requires that the cutoff be kept finite. Alternatively, if the cutoff is allowed to go to infinity, the Landau pole can be avoided only if the renormalized coupling runs to zero, rendering the theory trivial. Spontaneous symmetry breaking An interesting feature can occur if m2 turns negative, but with λ still positive. In this case, the vacuum consists of two lowest-energy states, each of which spontaneously breaks the Z2 global symmetry of the original theory. This leads to the appearance of interesting collective states like domain walls. In the O(2) theory, the vacua would lie on a circle, and the choice of one would spontaneously break the O(2) symmetry. A continuous broken symmetry leads to a Goldstone boson. This type of spontaneous symmetry breaking is the essential component of the Higgs mechanism. Spontaneous breaking of discrete symmetries The simplest relativistic system in which we can see spontaneous symmetry breaking is one with a single scalar field with Lagrangian where and Minimizing the potential with respect to leads to We now expand the field around this minimum writing and substituting in the lagrangian we get where we notice that the scalar has now a positive mass term. Thinking in terms of vacuum expectation values lets us understand what happens to a symmetry when it is spontaneously broken. The original Lagrangian was invariant under the symmetry . Since are both minima, there must be two different vacua: with Since the symmetry takes , it must take as well. The two possible vacua for the theory are equivalent, but one has to be chosen. Although it seems that in the new Lagrangian the symmetry has disappeared, it is still there, but it now acts as This is a general feature of spontaneously broken symmetries: the vacuum breaks them, but they are not actually broken in the Lagrangian, just hidden, and often realized only in a nonlinear way. Exact solutions There exists a set of exact classical solutions to the equation of motion of the theory written in the form that can be written for the massless, case as with a Jacobi elliptic function and two integration constants, provided the following dispersion relation holds The interesting point is that we started with a massless equation but the exact solution describes a wave with a dispersion relation proper to a massive solution. When the mass term is not zero one gets being