A lot of topics in this thread.
1)
Anomalies and the chiral anomaly. As they say, an anomaly occurs when a symmetry of the classical theory is no longer true of the quantum theory. I usually think of this as happening because the integration measure in the sum over histories does not respect the symmetry, but there are a variety of ways of deriving an anomaly.
A chiral anomaly exists when the amount of handedness is not conserved (i.e. the difference in the number of left-handed and right-handed particles).
David Tong's QFT lecture notes, which have acquired a very good reputation, discuss the chiral anomaly in chapters 3 and 5. In chapter 3 he derives the chiral anomaly for massless fermions in several ways, in chapter 5 he does it for QCD and the neutral pion decay.
The simplest-sounding explanation is on
page 125, figure 25. This is in 1+1 dimensions, where the chiralities are "moving to the left" and "moving to the right", and there is a rightward-pointing electric field that boosts all the right-movers to higher energy levels. If there were only finitely many energy levels, the occupation of new higher levels would be matched by a vacation of lower energy levels, and the number of right-movers would stay the same. But because the Dirac sea is infinitely deep, it amounts to a raising of the sea of right-movers and lowering of the sea of left-movers, changing the relative number of right-movers and left-movers. Then on the next page, he makes an analogous argument for chiral states in 3+1 dimensions.
Ideally, I think one would be able to switch between this kind of explanation, the measure-theoretic explanation, and the triangle-diagram perturbative calculations that ABJ originally used. I haven't attained this flexibility myself, but all three explanations are there, side by side, in Tong's notes.
2)
Chiral anomaly and neutral pion decay. The theory of this is in Tong's chapter 5. There are several new ingredients, e.g. we're talking about pions, and we don't even know how to calculate their behavior from QCD first principles (except on the lattice), so instead we'll the effect is derived in the context of Weinberg's chiral perturbation theory.
I don't have anything simple to say about this, so instead I will provide some references for the theoretical calculation of neutral pion decay, courtesy of
this 2018 thesis. See pages 9-10, which first discusses the theoretical calculation assuming up and down quarks are massless, then various more sophisticated calculations that take the masses into account, and compares them all to the measured decay rate. The measured width is 7.74 +/- 0.46 eV, the calculation which treats quarks as massless predicts 7.76 +/- 0.04 eV, the corrected calculations which include the light quark masses give results around 8 eV (near the upper bound on the measured width).
3)
Generalized symmetries and non-invertible symmetries. This is a new take on symmetries in QFT. I first tried to understand it a year ago, my grasp of it is still very weak, but here we go.
First we have the generalized global symmetries. You start with an ordinary quantum field theory, you take a submanifold of space-time (this can be a point, a line, a surface, or a hyperplane), and you associate a "topological operator", labelled by an element of a group, with that submanifold. Mathematically, this corresponds to a deformation of the original QFT, in which you tensor it with a topological QFT associated with the submanifold. Presumably the Hilbert space of the original theory has to be enlarged to include topological quantum numbers characterizing the special submanifold.
It is said that the usual global symmetries of QFT correspond in this scheme to a situation where the submanifold is just a point. I've seen the Gauss law referenced here, as an example of a conservation law with a topological character - you can deform the closed surface through which the electric flux is passing, and the law is still valid. And to mention another QFT concept that may be familiar, when a Wilson line is present, the submanifold is the line... However, I haven't understood how any of this actually works in any detail.
This all started ten years ago, with a paper by Gaiotto et al,
"Generalized Global Symmetries". Originally, the algebra of labels of the topological operators were always groups. However, in the last few years, people are also labeling them with "fusion algebras" or "fusion categories" which aren't groups. In some cases this algebra involves relations mapping a single submanifold to multiple submanifolds or vice versa, I guess that's the "fusion". And some of these algebras have non-invertible relations, and those define the "non-invertible symmetries".
I mentioned above that the chiral anomaly of the pion has been formulated in multiple ways. Apparently another way has been figured out, involving a non-invertible symmetry. I have no idea what the special submanifold is - maybe the center-of-mass worldline? (which would include the world-line of the pion until it decays). I hope it makes sense to me at some point.
4)
@arivero's plot of decay widths. I thought about these observations of his a few years back, specifically why Z boson decay would resemble pion decay. At the time I thought it might be evidence that the Higgs is a top-antitop bound state. My reasoning is that the spin-0 component of the Z (which gives the Z its mass) would then be a kind of neutral top-pion, and so maybe the same anomalous mechanism that applies to the neutral pion, might also apply to it. But I never really got to the bottom of that.
