Scalar φ⁴ Theory

Scalar phi-four theory adds the simplest stable polynomial self-interaction to a real scalar field with a reflection symmetry. It is a laboratory for scattering, renormalization, phase transitions, and the difference between a classical potential and a quantum theory.

Quartic scalar interaction

In $d$ spacetime dimensions, consider

$$\mathcal L=\frac12(\partial\phi)^2 -\frac12m^2\phi^2-\frac{\lambda}{4!}\phi^4, \qquad \lambda>0.$$

The reflection $\phi\mapsto-\phi$ forbids odd powers. The field dimension is $(d-2)/2$, so $[\lambda]=4-d$. The coupling is relevant by Gaussian power counting below four dimensions, marginal in four, and irrelevant above four. “Marginal” is a classical classification; quantum running decides its subsequent behavior.

The equation of motion is $(\Box+m^2)\phi+\lambda\phi^3/6=0$. Unlike a free Gaussian theory, it connects different Fourier modes and permits connected scattering.

Scattering and the classical vacuum

For scattering about the symmetric vacuum with $m^2>0$, expand the interaction exponential once. Contracting four labeled external fields with $\phi^4$ gives $4!$ assignments, cancelling the denominator. The four-point vertex is therefore $-i\lambda$, and the tree-level $2\to2$ invariant amplitude is $\mathcal M=-\lambda$ in the convention that the connected scattering matrix contains $i\mathcal M$. At this order it has no angular dependence. Higher orders introduce momentum dependence and renormalization. In a broken vacuum, shifting the field also produces cubic vertices, whose exchange diagrams contribute at tree level. See Srednicki 2007, §§9–10.

The classical potential gives a different useful check. For $m^2>0$, its minimum is $\phi=0$. For $m^2<0$,

$$v^2=-\frac{6m^2}{\lambda}, \qquad V''(\pm v)=-2m^2>0.$$

The two minima have equal energy and are interchanged by the reflection. The fluctuation mass squared about either minimum is positive. This is a classical result; quantum symmetry breaking also depends on dimension, volume, and the order of limits.

What this model can establish

Connected correlators, scattering amplitudes, correlation lengths, and order-parameter response are natural observables. Perturbation theory is controlled only when renormalized couplings and relevant logarithms are small. Near a critical point in $d<4$, an interacting fixed point can require an epsilon expansion, a large-component generalization, or nonperturbative methods.

A positive quartic term stabilizes the classical potential but does not by itself prove an ultraviolet-complete interacting continuum theory in four dimensions. A regulated effective theory with a stated cutoff is a separate, useful construction.

Develop the calculation

Renormalization and Effective Field Theory develops regulator removal and running couplings. What an Interacting Lagrangian Does and Does Not Specify explains the additional defining data. Compare the continuous symmetry and additional limits in the O(N) Vector Model.

References

• Srednicki, Mark. Quantum Field Theory. Cambridge University Press, 2007, §§9–10, 28, and 30. DOI.