This Lagrangian has a global Z2 symmetry mapping to .
For two scalar fields and the Lagrangian has the form
which can be written more concisely introducing a complex scalar field defined as
Expressed in terms of this scalar field, the above Lagrangian becomes
which is thus equivalent to the SO(2) model of real scalar fields , as can be seen expanding the complex field in real and imaginary parts.
With real scalar fields, we can have a model with a globalSO(N) symmetry given by the Lagrangian
Expanding the complex field in real and imaginary parts shows that it is equivalent to the SO(2) model of real scalar fields.
In all of the models above, the coupling constant must be positive, since, otherwise, the potential would be unbounded below, and there would be no stable vacuum. Also, the Feynman path integral discussed below would be ill-defined. In 4 dimensions, theories have a Landau pole. This means that without a cut-off on the high-energy scale, renormalization would render the theory trivial.
The standard trick to evaluate this functional integral is to write it as a product of exponential factors, schematically,
The second two exponential factors can be expanded as power series, and the combinatorics of this expansion can be represented graphically. The integral with λ = 0 can be treated as a product of infinitely many elementary Gaussian integrals, and the result may be expressed as a sum of Feynman diagrams, calculated using the following Feynman rules:
Each field in the n-point Euclidean Green's function is represented by an external line (half-edge) in the graph, and associated with momentum p.
Each vertex is represented by a factor -λ.
At a given order λk, all diagrams with n external lines and k vertices are constructed such that the momenta flowing into each vertex is zero. Each internal line is represented by a factor 1/(q2 + m2), where q is the momentum flowing through that line.
Any unconstrained momenta are integrated over all values.
The result is divided by a symmetry factor, which is the number of ways the lines and vertices of the graph can be rearranged without changing its connectivity.
Do not include graphs containing "vacuum bubbles", connected subgraphs with no external lines.
The last rule takes into account the effect of dividing by . The Minkowski-space Feynman rules are similar, except that each vertex is represented by , while each internal line is represented by a factor i/(q2-m2 + iε), where the ε term represents the small Wick rotation needed to make the Minkowski-space Gaussian integral converge.
The integrals over unconstrained momenta, called "loop integrals", in the Feynman graphs typically diverge. This is normally handled by renormalization, which is a procedure of adding divergent counter-terms to the Lagrangian in such a way that the diagrams constructed from the original Lagrangian and counterterms are finite. A renormalization scale must be introduced in the process, and the coupling constant and mass become dependent upon it. It is this dependence that leads to the Landau pole 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.
An interesting feature can occur if m2 happens to be negative, but with λ 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) theory. A continuous broken symmetry leads to a Goldstone boson. This type of spontaneous symmetry breaking is the essential component of the Higgs mechanism.
The simplest relativistic system in which we can see spontaneous symmetry breaking is one with a single scalar field with Lagrangian
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.
with a Jacobi elliptic function and two integration constants, provided 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
These wave solutions are interesting as, notwithstanding we started with an equation with a wrong mass sign, the dispersion relation has the right one. Besides, Jacobi function has no real zeros and so the field is never zero but moves around a given constant value that is initially chosen describing a spontaneous breaking of symmetry.
A proof of uniqueness can be provided if we note that the solution can be sought in the form being . Then, the partial differential equation becomes an ordinary differential equation that is the one defining the Jacobi elliptic function with satisfying the proper dispersion relation.