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Concept# Quartic interaction

Summary

In quantum field theory, a quartic interaction 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 satisfies the Klein–Gordon equation. If a scalar field is denoted , a quartic interaction is represented by adding a potential energy term to the Lagrangian density. The coupling constant is dimensionless in 4-dimensional spacetime.
This article uses the metric signature for Minkowski space.
The Lagrangian density for a real scalar field with a quartic interaction is
This Lagrangian has a global Z2 symmetry mapping .
The Lagrangian for a complex scalar field can be motivated as follows. 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 complex scalar field, the above Lagrangian becomes
which is thus equivalent to the SO(2) model of real scalar fields , as can be seen by expanding the complex field in real and imaginary parts.
With real scalar fields, we can have a 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.
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 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 model and the Ising model in can be shown via a graphical representation known as the random current expansion.
Path integral formulation
The Feynman diagram expansion may be obtained also from the Feynman path integral formulation.

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