# 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. Sep 24, 2020 at 22:15

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.
• Are you saying we can consider QCD with $U_A(1)$ symmetry simply by adding the $i\bar\psi \gamma^5 \psi$ term (a term that should be there any way by EFT logic). If that is true why don't we write it? Is it due to experimental data? Nov 18, 2021 at 19:28