# Anomalous global symmetry in non-gauge theories

I’m a bit confused on the effects of anomalous global symmetries. So take for instance the following theory $$\mathscr{L}=\partial_\mu\phi\partial^\mu\phi^*+i\bar{\psi}\gamma_\mu\partial^\mu\psi-y \phi\bar{\psi}\psi+\text{h.c}-V(\phi)$$ with $$V(\phi)=m^2|\phi|^2+\lambda |\phi|^4$$ It has two global symmetries $$U_V(1)$$ with $$\psi\to e^{i\theta}\psi$$ and $$U_A(1)$$ with $$\psi\to e^{I\gamma_5\theta}\psi$$ and $$\phi\to e^{-2 i\theta}\phi$$.

These symmetries have significant physical consequences; of course $$U_A(1)$$ forbids a mass for $$\psi$$,also the interplay of $$U_V(1)$$ and $$U_A(1)$$ forbid $$\phi$$ from decaying since decaying into two fermions is forbidden by helicity consideration, and other decays are forbidden by either $$U_V(1)$$ or $$U_A(1)$$.

However we would usually consider the $$U_A(1)$$ to be anomalous; certainly it can’t be gauged. But it's unclear to me what physical affect this anomaly actually has. If $$U_V(1)$$ was gauged, then we would have $$\partial_\mu J^A_\mu=-\frac{g^2}{16\pi^2}F_{\mu\nu}\tilde{F}^{\mu\nu},$$ which would allow violations of $$U_A(1)$$.

However, when $$U_V(1)$$ is just a global symmetry, it seems like there is no physical consequence of the “anomaly”.

So my question is: are $$U_A(1)$$ and $$U_V(1)$$ good symmetries of the theory I described? If not, what observable consequences does this have? I understand that anomalies come from regularisation ambiguities so perhaps a different way to phase my question is: Is there a regularisation scheme that respects both $$U_A(1)$$ and $$U_V(1)$$ and if not what observables are ambiguous?

• Write down the currents you believe are classically conserved. This will specify what you believe your scalar actually does. – Cosmas Zachos Sep 24 '20 at 22:15

## 1 Answer

There is no anomaly problem with this system --- except that as written it does not have a continuous $$U_A(1)$$ symmetry. You need to include a term $$i\bar\psi \gamma^5 \psi$$ term in addition to the $$\bar\psi\psi$$ term. With that included it is a simple model that can be be used for illustrating chiral symmetry breaking.