The strangest non-strange meson is not so strange: Phase shift analysis reveals the geometric origin of the $f_0(500)$ residue phase

Abstract

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 $f_0(500)$, $ρ(770)$, and $Δ(1232)$, we extract the effective background phase $φ_\mathrm{B}$ and uncover an empirical regularity within our formalism: $φ_\mathrm{B}$ is equal to the angle subtended by the pole and the threshold $φ_0$. This implies the residue phase is geometrically constrained to be $θ\approx 2φ_0$. The anomalous nature of the $f_0(500)$ thus arises from the proximity of its pole to the $ππ$ threshold. A unified scaling relation confirms the $f_0(500)$ behaves as a standard Breit-Wigner resonance.

Figures (3)

• Figure 1: Partial fits of our model to the meson phase shift data. The $\pi \pi$ elastic phase shifts are from Protopopescu Pro73 (blue squares), Ishida Ish97 (black disks), and Estabrooks Est74 (magenta diamonds). The black solid line is the full phase $\phi$, red dashed line is resonant phase, and background phase $\phi_\mathrm{B}$ is the constant difference between the two lines.
• Figure 2: Partial fits of our model to the data. The phase of the scattering amplitude calculated from the GWU partial-wave single-energy solutions for the $\pi N$ elastic scattering ArndtSAID. The black solid line is the full phase $\phi$, red dashed line is resonant phase, and background phase $\phi_\mathrm{B}$ is the constant difference between the two lines. Fits are done in the proximity of the resonant pole, yet the agreement is excellent even with the non-fitted data towards the threshold.
• Figure 3: The self-consistency plot. Each "*" is at the position of the fitted $M$ and $\Gamma$, while for each "+" we calculate $\Gamma$ using Eq. (\ref{['Phi0']}) from fitted values of $\phi_\mathrm{B}$ and $M$ (with $\phi_0=\phi_\mathrm{B}$).