Functional form of the fractional-anomaly coefficients

Determine the functions \({\cal F}_j(\Phi^\dagger\Phi)\) governing the fractional anomalous determinant terms in the broken-phase chiral effective Lagrangian, including their dependence on the independent invariants of \(\Phi^\dagger\Phi\).

Background

The proposed broken-phase anomalous Lagrangian is an infinite sum involving coefficient functions ${\cal F}_j(\Phi^\dagger\Phi)$ multiplying fractional powers $(\det\Phi)^{j/N_c}$. The paper imposes symmetry and matching conditions on these functions but does not derive their explicit form.

The author emphasizes that the functions may depend on invariants beyond $\det(\Phi^\dagger\Phi)$, such as traces of powers of $\Phi^\dagger\Phi$. Their determination is important for making quantitative predictions and for testing the proposed fractional-instanton description of the chiral transition.