φ4(phi-four) theory is a simple example of a quantum field theory (QFT) that exhibits many of the general properties of such theories, and is thus often used as a "toy model" for teaching QFT basics or for theoretical studies. It is renormalizable but it is not a gauge theory.
φ4 theory got its name from the φ4 interaction term in the Lagrangian.
One real, scalar field φ≡φ(x).
L(φ, ∂μφ) = 1/2 (∂μφ)(∂μφ) - 1/2 m2φ2 - g/4! φ4
Equation of motion
∂μ∂μφ + m2φ + g/3! φ3 = 0.
For a free field (g = 0), this is the Klein-Gordon equation.