The $f_0(500)$ meson is often labeled a "non-Breit-Wigner" resonance due to the large background phase required for its description. By fitting elastic phase shifts of the $f_0(500)$, $\rho(770)$, and $\Delta(1232)$, we extract the effective background phase $\phi_\mathrm{ B}$ and uncover an empirical regularity within our formalism: $\phi_\mathrm{ B}$ is equal to the angle subtended by the pole and the threshold $\phi_0$. This implies the residue phase is geometrically constrained to be $\theta \approx 2\phi_0$. The anomalous nature of the $f_0(500)$ thus arises from the proximity of its pole to the $\pi\pi$ threshold. A unified scaling relation confirms the $f_0(500)$ behaves as a standard Breit-Wigner resonance.
Categories: hep-ph