Non-local anomaly of the axial-vector current for bound states

We demonstrate that the amplitude $<\rho\gamma|\partial_\nu (\bar q\gamma_\nu \gamma_5 q)|0>$ does not vanish in the limit of zero quark masses. This represents a new kind of violation of the classical equation of motion for the axial current and should be interpreted as the axial anomaly for bound states. The anomaly emerges in spite of the fact that the one loop integrals are ultraviolet-finite as guaranteed by the presence of the bound-state wave function. As a result, the amplitude behaves like $\sim 1\mathrm{/}{p}^2$ in the limit of a large momentum $p$ of the current. This is to be compared with the amplitude $<\gamma \gamma \mathrm{\mid }{\mathrm{\partial }}_{\nu }\bar{q}{\gamma }_{\nu }{\gamma }_{5}q\mathrm{\mid }0>$ which remains finite in the limit $p^2\to \mathrm{\infty }$. The observed effect leads to the modification of the classical equation of motion of the axial-vector current in terms of the non-local operator and can be formulated as a non-local axial anomaly for bound states.