Suppose we have some complicated fundamental theory of fermions and gauge fields which involves global chiral symmetry and global anomalies which breaks some subgroup of this symmetr. At some energy scale spontaneously symmetry breaking happens, and goldstone bosons arise. We may then write chiral effective lagrangian (usual kinetic terms plus Wess-Zumino term). The problem of such description is that it doesn't contain information about global chiral anomalies. Thus theory have to be consistent if we gauge global subgroups of chiral group which are breaked by anomalies. For doing that we must introduce background gauge fields.

In QCD roles of gauge bosons is taken by vector and axial-vector mesons with spin one.

Why we can treat vector mesons as background gauge fields if they vanish on $\text{GeV}$ scale? What is the motivation of such treatment?


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