Assessing the role of threshold conditions in the determination of uncertainties in pole extractions using Pad\'e approximants

In this letter, we discuss the determination of the $f_0(500)$ resonance by analytic continuation through Pad\'e approximants of the $\pi\pi$-scattering amplitude from the physical region to the pole in the complex energy plane. Using as input a class of admissible parametrizations of the scalar-isoscalar $\pi\pi$ partial wave and imposing now the correct threshold behavior of the partial amplitude, we improve on the determinations of pole positions obtained in Ref. [1], thus empowering the Pad\'e method as a simple and precise tool for extracting resonance poles from amplitudes.