Quartic interaction

, a quartic interaction is represented by adding a potential energy term

the Lagrangian has the form which can be written more concisely introducing a complex scalar field

defined as Expressed in terms of this complex scalar field, the above Lagrangian becomes which is thus equivalent to the SO(2) model of real scalar fields

model with a global SO(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.

This means that without a cut-off on the high-energy scale, renormalization would render the theory trivial.

can be shown via a graphical representation known as the random current expansion.

[3] The time-ordered vacuum expectation values of polynomials in φ, known as the n-particle Green's functions, are constructed by integrating over all possible fields, normalized by the vacuum expectation value with no external fields, All of these Green's functions may be obtained by expanding the exponential in J(x)φ(x) in the generating function A Wick rotation may be applied to make time imaginary.

Changing the signature to (++++) then gives a φ4 statistical mechanics integral over a 4-dimensional Euclidean space, Normally, this is applied to the scattering of particles with fixed momenta, in which case, a Fourier transform is useful, giving instead where

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: 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 + i ε), where the ε term represents the small Wick rotation needed to make the Minkowski-space Gaussian integral converge.

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.

[4] 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.

[5] 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.

[6] The simplest relativistic system in which we can see spontaneous symmetry breaking is one with a single scalar field

leads to We now expand the field around this minimum writing and substituting in the lagrangian we get where we notice that the scalar

Thinking in terms of vacuum expectation values lets us understand what happens to a symmetry when it is spontaneously broken.

are 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 now the dispersion relation Finally, for the case of a symmetry breaking one has being

and the following dispersion relation holds These wave solutions are interesting as, notwithstanding we started with an equation with a wrong mass sign, the dispersion relation has the right one.

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