A geometric link between resonance parameters
The central claim is a geometric one: elastic unitarity—the condition imposed on an elastic reaction in this model—links the reaction threshold, the S-matrix pole, its residue phase and the observable peak. The link is expressed as β = α, and equating those angles yields a unitary Breit–Wigner peak position. The result is a proposed way to connect these resonance parameters.
To build this geometry, the amplitude contains a resonance pole, a zero at the reaction threshold and a constant background. The analysis imposes elastic S-matrix unitarity on that construction and tests the resulting relation across six orders of magnitude in energy.
Near the pole, however, the model is only a first-order Laurent approximation—a local mathematical approximation—so α and β may differ in practice; the analysis treats β as the more robust of the two. In canonical elastic nuclear and hadron cases, the departure metric ξE was near zero and β was close to α, with agreement to available reference phases.
From peaks to predictions
That pattern leads to predictions for three named states. A residue phase is the angle assigned to the pole’s residue in the model. The formalism gives −44(12)° for 5 He, −24(7)° for Σ(1385)+ and −7(13)° for Ξ(1530)0. The parenthetical values are the reported uncertainties.
The same formalism can also be inverted. For the Υ(4S), the reconstruction estimates a complex pole of 10575(1) − i 8.3(13) MeV and a residue phase of −52(6)°. The pole’s real and imaginary components are represented together in that complex notation.
For W and Z bosons, exact complex-pole calculations produced reference phases matching Höhler’s rule. In the Higgs analysis, the reported result was no peak-mass shift and a zero residue phase.
Where the picture bends
The broad-scalar cases expose the model’s boundary. When empirical Breit–Wigner masses from the PDG are used, f0 (500) and K0∗ (700) show strong departures from the elastic-unitarity constraint, with ξE much greater than zero. Yet the unitary construction gives residue-phase predictions of −115(5)° for f0 and −174(3)° for K0∗, close to dispersive reference values of −118(12)° and −180(8)°, respectively.
The contrast is central to the paper’s caution: a close match between a unitary phase prediction and a dispersive reference does not remove the strong departure found when the empirical Breit–Wigner masses are used.
An additional reconstruction check for Δ(1232) yields a residue-phase estimate of −42(1)°.
A conditional estimator, not a final measurement
The construction is conditional on its starting assumptions: a pole, a threshold zero, a constant background and elastic unitarity. Since the model is a first-order approximation near the pole, α and β may differ in practice, and β is treated as more robust.
Independent analyses remain needed to test the three predicted phases and the Υ(4S) inverse reconstruction. The supplied document is an arXiv version 1 preprint dated 26 Aug 2026. Its end matter reports no funding, conflict-of-interest, data, code or supplementary-material statement.
Paper data and sources
Original title: How elastic unitarity governs resonance peaks and residue phases
Authors: S. Ceci, R. Omerović, H. Osmanović et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text