Coefficient of the four-flavor fractional determinant

Determine the coefficient exponent \(\alpha_2\) multiplying the \(j=2\) fractional anomalous determinant for four degenerate flavors and three colors in the syncretic chiral effective Lagrangian.

Background

For four flavors and three colors, the paper writes the leading fractional anomalous terms using coefficient functions assumed, for illustration, to depend only on det⁡Φ†Φ\det\Phi^\dagger\Phi. The exponent α2\alpha_2 controls the prefactor of the term proportional to (det⁡Φ)2/3(\det\Phi)^{2/3}.

Different requirements for the small-Φ\Phi behavior give different values: requiring the term to scale as Φ3\Phi^3 yields α2=1/12\alpha_2=1/12, while requiring Φ4\Phi^4 yields α2=1/6\alpha_2=1/6. The paper therefore leaves the physically appropriate exponent unresolved and notes that a more general dependence on additional invariants is likely necessary.

References

For example, assuming that the ${\cal F}_2$ is a function only of $\det \Phi\dagger \Phi$, the value of the coefficient $\alpha_2$ is unclear.

— Fractional anomalous determinants and the chiral phase transition  (2610.00472 - Pisarski, 30 Sep 2026) in Section 6.4, “Four flavors”