# Does the Coleman-Weinberg mechanism belong to the dynamical symmetry breaking or the anomaly?

We know that a massless $\phi^4$ theory $$S=\int d^4x \left[\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{\lambda}{4!}\phi^4\right],$$ has conformal invariance at the classical level. But within the Coleman-Weinberg mechanism, at the one-loop level, quantum fluctuations will generate a vacuum expected value for $\phi$, introducing a mass scale and breaking the conformal invariance. Is this phenomenon a dynamical symmetry breaking or an anomaly? How can we distinguish between them?

Just $\phi$ acquiring a VEV would not mean an anomaly, that would just be ordinary spontaneous symmetry breaking. However, the appearance of the $\phi^2$ term in the effective potential also means that we have an anomaly, i.e. the quantum effective action is not invariant under the classical symmetry - this is a clear case of a quantum anomaly. That is, in this case, the Coleman-Weinberg mechanism leads to both spontaneous symmetry breaking and a quantum anomaly, but it is perfectly conceivable to have one without the other - they are completely distinct things. It might be debatable whether we want to speak of spontaneously "breaking" a symmetry that became anomalous to begin with, though.
Strictly speaking, the phase dynamical symmetry breaking, although essentially a kind of spontaneous symmetry breaking proces, means something slightly different. While spontaneous symmetry breaking usually refers to cases where an elementary scalar field (such as $\phi$) acquires a VEV, dynamical symmetry breaking refers to cases where the scalar field that acquires a VEV is a composite field. In this sense the term dynamical refers to a force that is strong enough to bind fields so strongly that the composite acquires a VEV. Chiral symmetry breaking is an example of dynamical symmetry breaking. Technicolor theories are attempts to describe the electroweak symmetry breaking in terms of a dynamical symmetry breaking process.