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# BCF anomaly and higher-group structure in the low energy effective theories of mesons

Dec 2022

We discuss the BCF anomaly of massless QCD-like theories, first obtained byAnber and Poppitz, from the viewpoint of the low energy effective theories. Weassume that the QCD-like theories exhibit spontaneous chiral symmetry breakingdue to a quark bilinear condensate. Using the 't Hooft anomaly matchingcondition for the BCF anomaly, we find that the low energy effective action iscomposed of a chiral Lagrangian and a Wess-Zumino-Witten term together with aninteraction term of the $\eta^\prime$ meson with the background gauge field fora discrete one-form symmetry. It is shown that the low energy effective actioncancels the quantum inconsistencies associated with $\eta^\prime$ due to anambiguity of how to uplift the action to a five-dimensional spacetime with aboundary. The $\eta^\prime$ term plays a substantial role in exploring theemergent higher-group structure at low energies.

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