# Why do we think that the $U(1)$ problem is solved by instantons?

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?

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.