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What is the meaning of the restoration at finite temperature of a symmetry that is "broken" by the presence of an anomaly. If the symmetry is not there why is it restored at finite temperature? Is there some quantitative way to deal with this where there are something like a $\frac{1}{T}$ corrections that quantify the breaking of the symmetry? Is the story equivalent to normal explicit symmetry breaking? (Say if we turn on a magnetic field in the Ising model breaking the $Z_2$ symmetry. Then at high temperature I think the symmetry will never be exactly restored but will be approximately restored for any observable measured at high temperature.)

The restoration of the $U_A(1)$ axial symmetry is suggested for example in "Remarks on the chiral phase transition in chromodynamics" (1984) by Pisarski and Wilczek. (Or the same thing is discussed more recently in a review paper https://arxiv.org/abs/2111.00569 for a more recent example.)

Is there any other example where an axial symmetry is restored (and where it subsequently plays a role in spontaneous symmetry breaking).

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    $\begingroup$ A 2021 review by Azcoiti, (Axial $U_A(1)$ Anomaly: A New Mechanism to Generate Massless Bosons, link to pdf), says this: "despite the great effort devoted to investigating the fate of the axial anomaly in the chirally symmetric [high-temperature] phase of QCD, we still do not have a clear answer..." Also, the newer paper you linked cites hep-ph/9310253, which uses the phrases "practically restored" and "to some accuracy level" for the anomalous $U(1)_A$ at high temperature. $\endgroup$ Nov 19, 2021 at 4:48
  • $\begingroup$ You seem to be "warm" in visualizing anomaly breaking as some idiosyncratic explicit breaking, washed out in high temperatures... $\endgroup$ Nov 19, 2021 at 17:22

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