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It is usually thought that the $U(1)$ problem is solved when 't Hooft realized that instantons induce additional symmetry breaking of the $U(1)_A$ symmetry aside from the non-vanishing quark masses. This additional symmetry breaking split the mass of $\eta'$ from other pseudo-Goldstone bosons ($\pi, K, \eta$).

This looks strange to me since much earlier than the discovery of BPST instantons, Adler, Bell and Jackiw already discovered that the $U(1)_A$ current is anomalous which by itself already means that $U(1)_A$ has additional breaking due to the anomaly. Why then people do not think that it is simply the ABJ anomaly that solves the U(1) problem?

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I agree that it is not so much instantons that are relevent, but the anomaly and $\theta_{QCD}$ term. By the early 1970's, when I became a grad student, the anomaly was well understood to be the source of the $e^2/\pi$ mass for the fermion/photon mode in the 1+1 dimensional Schwinger model. Several people (Kogut-Susskind, and also myself in my thesis) wrote about the analogy between the Schwinger effect and the $\eta-\eta'$ splitting in 1976. I forget whther it was t'Hooft or Witten who related the 3+1 dimensional $\eta'$ mass to the topological susceptabilty $\partial^2 {\mathcal E}_{vac}/\partial \theta^2$. That was the key contribution surely? --- rather than the minimal action instanton field configuation itself. Of course, without instantons, the $\theta_{QCD}$ parameter (also introduced in 1976 I think) would not have been thought of.

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