Scalar and Fermion Interactions

A few local interactions already reveal several central features of QFT: connected scattering, scale-dependent couplings, and the dependence of excitations on the chosen vacuum. These profiles compare scalar self-interactions with scalar–fermion coupling. They distinguish classical potential analysis from conclusions about a quantum phase or a continuum limit.

Enter this chapter

Begin with Scalar φ⁴ Theory to relate one interaction term to a four-particle vertex and to the shape of a classical potential. The O(N) Vector Model then shows what changes when a discrete reflection becomes a continuous internal symmetry.

Topics

1. Scalar φ⁴ Theory — Determine the quartic vertex, inspect the classical minima, and track the coupling’s engineering dimension.
2. O(N) Vector Model — Separate radial and angular fluctuations and identify how symmetry and component number organize approximations.
3. Yukawa Theory — Couple a scalar to fermions and calculate a decay rate with its mass threshold and spin sum.

How the ideas fit together

The same polynomial interaction can play different roles depending on dimension and state. Power counting measures how its coupling scales near a Gaussian theory. A classical potential identifies candidate backgrounds. Expanding about one of those backgrounds determines the masses and vertices used in perturbation theory. These operations answer different questions and must be combined in that order when a vacuum shift changes the interaction terms.

For example, expanding a quartic potential about a nonzero minimum also produces a cubic interaction for the fluctuation. Its tree-level scattering is therefore not exhausted by the quartic contact term. The symmetry-breaking expansion is developed in Srednicki 2007, §30.

In an O(N) theory, rotations of the field direction connect classically degenerate minima, while a radial fluctuation changes the field magnitude. A nonlinear sigma model removes that radial degree of freedom from its defining variables; it is not merely another name for the linear vector model. Quantum fluctuations and the order of infinite-volume and zero-source limits remain essential when interpreting symmetry breaking.

Where to go next

Continue to Gauge Theories to compare these global symmetries with gauge redundancy. For the geometric counterpart of a vector model, see the O(N) Nonlinear Sigma Model.

References

• Srednicki, Mark. Quantum Field Theory. Cambridge University Press, 2007, §30. DOI.