Dimensional regularization and PCAC

In a space-time of arbitrary even dimension neither the mass term nor the kinetic-energy term of the free spinor Lagrangian is chirally invariant. We construct a generalized sigma-model whereby the divergence of the axial current is converted into the sum of two distinct pion fields. The first pion couples in the conventional manner with the fermion whereas the second pion couples through the anomalous kineticenergy interaction ψ¯∂↔JΓ[JΓKΓLΓMΓN]τψ\bar \psi \overleftrightarrow \partial ^J \Gamma _{[J} \Gamma _K \Gamma _L \Gamma _M \Gamma _{N]} \tau \psi . This second coupling disappears from all tree graphs as we descend to four dimensions, but through closed fermion loops is responsible for all the anomalous meson interactions. The 〈AAA〉 and 〈VAAA〉 anomalies are thereby calculated in terms of the 〈VVA〉 anomaly.