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independent U(1) symmetries for both the left and right handed symmetries. These phases always cancel out since the fermions are always bilinear. The presence of the non-zero VEV (1.33) also breaks this classical U (1)$_{L}$ ⊗ U (1)$_{R}$ symmetry to leave a single U (1)$_{V}$ symmetry. And once again, Goldstone’s theorem applies: a spontaneously broken symmetry implies the presence of a massless boson. Here the broken symmetry has one only one generator, so we’d expect one massless boson. A look at the known QCD mass spectrum shows that the only suitable candidate, in terms of quantum numbers, is the η ′ (958) meson. However it has a disappointingly high mass. Even allowing for the small masses of the quarks, we’d expect our boson to have a mass similar to that of the pions. 

The explanation of this dichotomy puzzled physicists for a long while, and wasn’t explained until 1986 [31]. It is quite easy to show [14, 32] that due to the necessity of regularization and renormalization (in particular triangle diagrams such as figure 1.6) the classical axial symmetry of QCD ( ∂$_{µ}$j µ 5 = 0) is broken by quantum effects to 

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The obvious question to ask is what is the source of such an anomaly, and the answer lies in the complicated field of instantons [31–34]. There are 

