Baryon number at singular linear-sigma configurations

Determine how the baryon-number operator constructed from the unitary projection \(\widetilde U=\Phi(\Phi^\dagger\Phi)^{-1/2}\) behaves when \(\det\Phi=0\), where the linear chiral field \(\Phi\) is non-invertible and the baryon number may change discontinuously.

Background

The paper constructs a baryon-number operator for the linear sigma model by projecting the complex matrix field $\Phi$ onto its unitary part, $\widetilde U$. This construction is valid only when $\Phi$ is invertible, equivalently when $\det\Phi\neq0$, because the inverse square root of $\Phi^\dagger\Phi$ is otherwise singular or undefined.

For invertible configurations, the projected field lies in the same relevant homotopy class as the nonlinear sigma-model field and yields an integer-valued winding number. The unresolved issue is the behavior at configurations where $\det\Phi$ vanishes: the paper notes that the baryon number can jump there, but does not determine the precise operator or its dynamics at these singular points.