Geometric Constraint on Residue Phases: Resolving the N(2190) Anomaly and Diagnosing Exotic States

Abstract

We derive a parameter-free geometric constraint on residue phases dictated by the pole-threshold angle. Using the N(2190) anomaly as a test case, this constraint reveals a sign ambiguity in prior data; correcting it yields a phase of $-28^\circ\pm10^\circ$, matching our prediction. This consistency validates the method as a model-independent diagnostic for distinguishing compact from molecular states, offering a rigorous tool for exotic spectroscopy.

Figures (4)

• Figure 1: Complex phase of the scattering amplitude $T$. (Top) The unitary toy model, connecting the physical (upper half-plane) and non-physical (lower half-plane) Riemann sheets. (Bottom) Our effective model [Eq. (\ref{['eq:our_amplitude']})]. Note the remarkable similarity near the pole position ($M-i\Gamma/2$, white circle) and the Breit-Wigner mass ($M_\mathrm{BW}$, bullseye marker). Deviations are visible primarily near the threshold ($E_0$, black circle).
• Figure 2: A comparison (not a fit) of our prediction, based on the PDG estimates of masses and widths, to the $\pi N$ elastic WI08 single energy data WI08. Evidently, our prediction is not getting worse with higher partial waves. The real part departs somewhat closer to the threshold, but the imaginary part is rather good there. Notably, with exception of N(2190), higher partial waves have increasingly better agreement between our residue phase prediction ($\theta_\mathrm{OM}$) and the PDG estimate ($\theta_\mathrm{PDG}$).
• Figure 3: The $\pi N$ elastic residue phase for the lightest resonances in partial waves. Black circles are PDG estimates, red squares show the value of twice the threshold angle $\alpha$, blue diamonds show twice the Manley angle $\beta$, and purple stars (the average between the two) are our predictions. Based on agreement with our model, first resonances are classified in types Ia and Ib as introduced in Ref. Ceci26. The N(2190) resonance is the only one not fitting this scheme.
• Figure 4: Resolution of the N(2190) anomaly. Amplitude prediction calculated with the statistical averages of the PDG parameters falls to the data, and the predicted (OM) and measured (PDG) residue phase are fully consistent.