# What's the difference and connection between symmetry breaking and anomaly?

I'm just wondering what's the difference between symmetry breaking and anomaly.

From my understanding, symmetry breaking means: there is a symmetry in the action, but in the ground state of the theory(minimum of the action), the symmetry becomes smaller, which is a subgroup of the original symmetry.

While anomaly means that there is a symmetry in the classical theory ,but it's no longer a symmetry after quantization.

I think there should be some connections between them, but I don't know yet.

• Anomalous symmetry The current is conserved classically, but due to a property of the Dirac sea, develops a non-vanishing divergence proportional to $$\hbar$$ computable via fermion loop corrections to it. It is a property of the fermion sea/measure, not the vacuum altered by the hamiltonian. Globally, as in flavor chiral ones, they control important physics processes, like neutral pion decay, and are generally summarized by WZWN actions. Locally, they invalidate gauge invariance and render gauge theories inconsistent, so are avoided/cancelled (by the addition of suitable fermions) with extreme prejudice.