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

We discuss the BCF anomaly of massless QCD-like theories, first obtained by Anber and Poppitz, from the viewpoint of the low energy effective theories. We assume that the QCD-like theories exhibit spontaneous chiral symmetry breaking due to a quark bilinear condensate. Using the 't Hooft anomaly matching condition for the BCF anomaly, we find that the low energy effective action is composed of a chiral Lagrangian and a Wess-Zumino-Witten term together with an interaction term of the $\eta^\prime$ meson with the background gauge field for a discrete one-form symmetry. It is shown that the low energy effective action cancels the quantum inconsistencies associated with $\eta^\prime$ due to an ambiguity of how to uplift the action to a five-dimensional spacetime with a boundary. The $\eta^\prime$ term plays a substantial role in exploring the emergent higher-group structure at low energies.