# What's the real resolution of the $U(1)_A$ problem?

To recap the problem, consider QCD with three massless quark flavors. There is a symmetry $$SU(3)_L \times SU(3)_R \times U(1)_L \times U(1)_R$$ corresponding to independent rotations of the left-chiral and right-chiral quark fields. Vector symmetries are the subset $$SU(3)_V \times U(1)_V$$ which rotate the left-chiral and right-chiral quark fields the same way, while the axial symmetry $U(1)_A$ rotates the fields in opposite directions. Finally, we define $SU(3)_A$ by $$SU(3)_A = SU(3)_L \times SU(3)_R / SU(3)_V$$ and conventionally call it an "axial symmetry group", though it's merely a coset. All of these symmetries except $U(1)_V$ are explicitly broken by the quark masses, but we can treat this as a small effect and ignore it below.

The formation of the chiral condensate spontaneously breaks the symmetry to $SU(3)_V \times U(1)_V$, so we should have $8 + 1$ Goldstone bosons due to $SU(3)_A$ and $U(1)_A$. The $U(1)_A$ problem is the fact that there is no Goldstone boson corresponding to it. The candidate is the $\eta'$, which is much heavier than the $8$ others.

According to most textbooks, the resolution of the $U(1)_A$ problem is that the $U(1)_A$ symmetry is anomalous by a $U(1)_A SU(3)^2$ triangle diagram, and hence not a true symmetry of the quantum field theory. Since it's not a symmetry, it can't be spontaneously broken.

I don't buy this. The problem is that $SU(3)_A$ is also anomalous, by about the same amount. For example, the $U(1)$ subgroup of $SU(3)_A$ corresponding to the pion $\pi^0$ has a $U(1) U(1)_{\text{EM}}^2$ anomaly which accounts for the fast decay $\pi^0 \to \gamma \gamma$. This is important because it's how anomalies were discovered in the first place. So by this reasoning the pion should be heavy as well, but it isn't.

What distinguishes $U(1)_A$ here? Is the anomaly alone really the solution to the $U(1)_A$ problem?

• Indeed, no EM configurations exist to "seize the vacuum" and give the neutral pion extra mass, as QCD instantons give the $\eta '$ extra mass and also. But it is a longish story, relying on the topological structure of the vacuum... There may be related questions with bits and pieces of it. Commented Jul 25, 2018 at 18:44
• If you only consider the symmetries that you mention at the beginning of your post, then $SU(3)_A$ is not anomalous. The problem arises when you add electromagnetic interactions that break the $SU(3)_L \times SU(3)_R$ symmetry. And indeed, there is a mass difference between the neutral and charged pions precisely due to electromagnetic interactions. This splitting is small compared to the mass of the $\eta'$ because electromagnetism is much weaker than the strong interaction.
– M.Jo
Commented Aug 21, 2018 at 10:16
• Did you ever find a good resolution to this problem? It seems to me as though M.Jo is on the right track: $U(1)_A$ is anomalous in pure QCD, whereas $SU(3)_A$ is not. Only when we couple to QED do we see an anomaly in $SU(3)_A$ (specifically, the $\lambda_3$ and $\lambda_8$ generators), and these effects are relatively smaller on account that $e < g_s$. Commented Apr 8, 2019 at 16:27
• See arxiv.org/abs/1810.05338 for an explanation using more modern language. Commented Aug 1, 2019 at 20:05

We are trying to understand why the $$\eta'$$ acquires a mass in pure QCD (no external fields). This is indeed explained by the $$U(1)_A\times [SU(3)]^2$$ anomaly. Note that the $$SU(3)$$ is the QCD gauge symmetry, so we cannot turn these fields off. The anomaly in the $$U(1)_A$$ current is proportional to $$G_{\mu\nu}\tilde{G}^{\mu\nu}$$, which is a total divergence. This is not an impediment, because QCD has topological $$|n\rangle$$ sectors, and the partition function in the presence of a topological $$\theta$$-term $${\cal L}_\theta\sim\theta G_{\mu\nu}\tilde{G}^{\mu\nu}$$ has $$\theta$$ dependence. The topological term has a vanishing vacuum expectation value (QCD does not spontaneously break CP), but the $$\eta'$$ mass is controlled by the associated susceptibility (with a caveat, explained by Witten and Veneziano) $$\chi_{top} \sim \frac{1}{V} \int d^4x\, \langle G_{\mu\nu}\tilde{G}^{\mu\nu}(0)G_{\alpha\beta}\tilde{G}^{\alpha\beta}(x)\rangle .$$ This quanitity is of order $$\Lambda_{QCD}^4$$, and as a result the $$\eta'$$ acquires a mass comparable to other non-Goldstone bosons.
Pure QCD has a $$SU(3)_L\times SU(3)_R$$ flavor symmetry, which I can try to (weakly) gauge. We find that this symmetry is anomalous, and in the chiral broken phase this anomaly can be represented by a Wess-Zumino term, which (among other things) reproduces the $$\pi^0\to 2\gamma$$ decay. I can ask whether this anomaly contributes to the mass of the $$\pi^0$$. Note that 1) EM certainly does contribute to the mass of charged Goldstone bosons, 2) the anomaly is again proportional to $$F_{\mu\nu}\tilde{F}^{\mu\nu}$$, which is a surface term (but QED does not have topological sectors). Finally, in the standard model $$SU(2)_L\times U(1)_Y$$ is gauged, but the QCD anomaly (represented by the Wess-Zumino term) is cancelled by leptons.
P.S.: An explanation of why the $$\pi^0$$ does not acquire a mass (in more modern language) is also given here.