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?
Answer
First, dynamical symmetry breaking (which I take to be either synonymous with or a subset of spontaneous symmetry breaking) and anomalies are two completely different things. An anomaly is when a symmetry group acquires a central extension, due to some obstruction in the process of representing it in our theory. Such obstructions can exist purely classical, or they can arise in the course of quantization, but they are crucially features of the whole theory. For more information on anomalies, see this excellent answer by DavidBarMoshe. In contrast, in spontaneous symmetry breaking, the theory retains the symmetry, just its vacuum state does not, which leads to the symmetry being non-linearly realized on the natural perturbative degrees of freedom (being "broken").
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.
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