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Description
Contrary to the general notion that the background part of resonant scattering amplitude must be large and complicated for broad resonances, we show that the size of background depends only on the geometry, more precisely, on the angle at which the resonant pole is "seen" from the threshold. We use an approximation which treats the threshold as a simple zero instead of a branch point [1,2], and it works perfectly not only for mostly elastic meson and baryon resonances, but also for the highly inelastic Z boson. It enables us to connect the background to the complex residue phase, an elusive resonant parameter from the world of nucleon resonances, and it equals twice the background phase shift. To include the Z boson in this picture we study two separate definitions of resonant mass and width: as pole, and Breit-Wigner parameters. We clarify that the two different definitions come from two ways of combining the total phase shift from resonant and background terms: the sum of tangents, or the tangent of the sum. Using all this, we show that the infamous non-Breit-Wigner meson resonance f_0(500) has the same shape as any broad Breit-Wigner resonance with the mass close to the threshold [3].
[1] Model-Independent Extraction of the Pole and Breit-Wigner Resonance Parameters, S. Ceci, M. Korolija, and B. Zauner, Phys. Rev. Lett. 111, 112004 (2013)
[2] Fundamental properties of resonances, S. Ceci, M. Hadzimehmedovic, H. Osmanovic, A. Percan & B. Zauner
Scientific Reports volume 7, Article number: 45246 (2017)
[3] The strangest non-strange meson is not so strange after all, Sasa Ceci, Marin Vuksic, and Branimir Zauner, https://arxiv.org/abs/2005.11564v3
