Quartic interaction

In quantum field theory, a quartic interaction or φ<sup>4</sup> theory is a type of self-interaction in a scalar field. Other types of quartic interactions may be found under the topic of four-fermion interactions. A classical free scalar field <math>\varphi</math> satisfies the Klein–Gordon equation. If a scalar field is denoted <math>\varphi</math>, a quartic interaction is represented by adding an interaction energy term <math>({\lambda}/{4!}) \varphi^4</math> to the Lagrangian density. The coupling constant <math>\lambda</math> is dimensionless in 4-dimensional spacetime.

This article uses the <math>(+ - - -)</math> metric signature for Minkowski space.

Lagrangian for a massive, real scalar field

The Lagrangian density for a massive, real scalar field with a quartic interaction is

:<math>\mathcal{L}(\varphi)=\frac{1}{2} [\partial^\mu \varphi \partial_\mu \varphi -m^2 \varphi^2] -\frac{\lambda}{4!} \varphi^4.</math>

The first term between the brackets is the energy related to the four-momentum of the particle, the second term describes its restmass energy.

This Lagrangian has a global Z<sub>2</sub> symmetry mapping <math>\varphi\to-\varphi</math>.

Lagrangian for a complex scalar field

The Lagrangian for a complex scalar field can be motivated as follows. For two scalar fields <math>\varphi_1</math> and <math>\varphi_2</math> the Lagrangian has the form

:<math> \mathcal{L}(\varphi_1,\varphi_2) =

\frac{1}{2} [ \partial_\mu \varphi_1 \partial^\mu \varphi_1 - m^2 \varphi_1^2]

+ \frac{1}{2} [ \partial_\mu \varphi_2 \partial^\mu \varphi_2 - m^2 \varphi_2^2]

- \frac{1}{4} \lambda (\varphi_1^2 + \varphi_2^2)^2,

</math>

which can be written more concisely introducing a complex scalar field <math>\phi</math> defined as

:<math> \phi \equiv \frac{1}{\sqrt{2 (\varphi_1 + i \varphi_2), </math>

:<math> \phi^* \equiv \frac{1}{\sqrt{2 (\varphi_1 - i \varphi_2). </math>

Expressed in terms of this complex scalar field, the above Lagrangian becomes

:<math>\mathcal{L}(\phi)=\partial^\mu \phi^* \partial_\mu \phi -m^2 \phi^* \phi -\lambda (\phi^* \phi)^2,</math>

which is thus equivalent to the SO(2) model of real scalar fields <math>\varphi_1, \varphi_2</math>, as can be seen by expanding the complex field <math>\phi</math> in real and imaginary parts.

With <math>N</math> real scalar fields, we can have a <math>\varphi^4</math> model with a global SO(N) symmetry given by the Lagrangian

:<math>\mathcal{L}(\varphi_1,...,\varphi_N)=\frac{1}{2} [\partial^\mu \varphi_a \partial_\mu \varphi_a - m^2 \varphi_a \varphi_a] -\frac{1}{4} \lambda (\varphi_a \varphi_a)^2, \quad a=1,...,N.</math>

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 <math>\lambda</math> 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, <math>\phi^4</math> theories have a Landau pole. This means that without a cut-off on the high-energy scale, renormalization would render the theory trivial.

The <math>\phi^4</math> model belongs to the Griffiths-Simon class, meaning that it can be represented also as the weak limit of an Ising model on a certain type of graph. The triviality of both the <math>\phi^4</math> model and the Ising model in <math>d\geq 4</math> can be shown via a graphical representation known as the random current expansion.

Feynman integral quantization

The Feynman diagram expansion may be obtained also from the Feynman path integral formulation. 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,

:<math>\langle\Omega|\mathcal{T}\{\int \mathcal{D}\phi e^{i\int d^4x \left({1\over 2}\partial^\mu \phi \partial_\mu \phi -{m^2 \over 2}\phi^2-{\lambda\over 4!}\phi^4\right).</math>

All of these Green's functions may be obtained by expanding the exponential in J(x)φ(x) in the generating function

:<math>Z[J] =\int \mathcal{D}\phi e^{i\int d^4x \left({1\over 2}\partial^\mu \phi \partial_\mu \phi -{m^2 \over 2}\phi^2-{\lambda\over 4!}\phi^4+J\phi\right)} = Z[0] \sum_{n=0}^{\infty} \frac{1}{n!} \langle\Omega|\mathcal{T}\_{\text{unimportant constant

+ \underbrace{\frac{1}{2} [( \partial \sigma)^2 - (\sqrt{2}\mu)^2 \sigma^2 ]}_{\text{massive scalar field

+ \underbrace{ (-\lambda v \sigma^3 - \frac{\lambda}{4} \sigma^4) }_{\text{self-interactions. </math>

where we notice that the scalar <math>\sigma</math> 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 <math>Z_2</math> symmetry <math> \varphi \rightarrow -\varphi</math>. Since

:<math> \langle \Omega | \varphi | \Omega \rangle = \pm \sqrt{ \frac{6\mu^2}{\lambda} }</math>

are both minima, there must be two different vacua: <math>|\Omega_\pm \rangle</math> with

:<math> \langle \Omega_\pm | \varphi | \Omega_\pm \rangle = \pm \sqrt{ \frac{6\mu^2}{\lambda} }. </math>

Since the <math>Z_2</math> symmetry takes <math> \varphi \rightarrow -\varphi</math>, it must take <math> | \Omega_+ \rangle \leftrightarrow | \Omega_- \rangle </math> 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 <math>Z_2</math> symmetry has disappeared, it is still there, but it now acts as

<math> \sigma \rightarrow -\sigma - 2v. </math>

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

:<math> \partial^2\varphi+\mu_0^2\varphi+\lambda\varphi^3=0</math>

that can be written for the massless, <math>\mu_0=0</math>, case as

:<math>\varphi(x) = \pm\mu\left(\frac{2}{\lambda}\right)^{1\over 4}{\rm sn}(p\cdot x+\theta,i),</math>

where <math>\, \rm sn\!</math> is the Jacobi elliptic sine function and <math>\,\mu,\theta</math> are two integration constants, provided the following dispersion relation holds

:<math>p^2=\mu^2\left(\frac{\lambda}{2}\right)^{1\over 2}.</math>

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

:<math>\varphi(x) = \pm\sqrt{\frac{2\mu^4}{\mu_0^2 + \sqrt{\mu_0^4 + 2\lambda\mu^4}{\rm sn}\left(p\cdot x+\theta,\sqrt{\frac{-\mu_0^2 + \sqrt{\mu_0^4 + 2\lambda\mu^4{-\mu_0^2 -

\sqrt{\mu_0^4 + 2\lambda\mu^4}\right)</math>

being now the dispersion relation

:<math>p^2=\mu_0^2+\frac{\lambda\mu^4}{\mu_0^2+\sqrt{\mu_0^4+2\lambda\mu^4.</math>

Finally, for the case of a symmetry breaking one has

:<math>\varphi(x) =\pm v\cdot {\rm dn}(p\cdot x+\theta,i),</math>

being <math>v=\sqrt{\frac{2\mu_0^2}{3\lambda</math> and the following dispersion relation holds

:<math>p^2=\frac{\lambda v^2}{2}.</math>

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, the Jacobi function dn 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 <math>\varphi=\varphi(\xi)</math> being <math>\xi=p\cdot x</math>. Then, the partial differential equation becomes an ordinary differential equation that is the one defining the Jacobi elliptic function with <math>p</math> satisfying the proper dispersion relation.

Quartic interaction

Lagrangian for a massive, real scalar field

Lagrangian for a complex scalar field

Feynman integral quantization

Exact solutions

See also

References

Further reading