Assessing the role of threshold conditions in the determination of uncertainties in pole extractions using Padé approximants

TL;DR

This work addresses the challenge of extracting the $f_0(500)$ resonance pole from elastic $\pi\pi$ scattering by analytic continuation using Padé Approximants. It employs 2-point Padé approximants that merge an expansion around a reference point $s_0$ with the physical threshold, enforcing the correct threshold behavior to reduce theoretical uncertainties. Using multiple parameterizations of the $\pi\pi$ scalar-isoscalar phase shift and sequences $P_1^N$, $P_2^N$, the study demonstrates improved convergence and more precise pole determinations, with updated results around $s_p \approx (457 - i\,279)$ MeV and $M \approx 459$–$466$ MeV, $\Gamma/2 \approx 295$ MeV. The incorporation of threshold constraints significantly tightens the uncertainties, validating Padé-based analytic continuation as a simple, robust method for resonance pole extraction in hadron spectroscopy.

Abstract

In this letter, we discuss the determination of the $f_0(500)$ resonance by analytic continuation through Padé approximants of the $ππ$-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 $ππ$ 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 Padé method as a simple and precise tool for extracting resonance poles from amplitudes.

Figures (7)

• Figure 1: Overlap of the 68% CL ellipses for the $P_{1}^{N}$'s sequence pole position: red for $P_{1}^{2}$, orange for $P_{1}^{3}$, and green for $P_{1}^{4}$. Panels a) to e) correspond to parameterizations $v_1$-$v_5$. The orange cross is the reference number $s_p=(457^{+14}_{-13} - i 279^{+11}_{-7})$.
• Figure 2: Overlap of the 68% CL ellipses for the $P_{2}^{N}$'s sequence pole position: red for $P_{2}^{1}$, green for $P_{2}^{2}$, and blue for $P_{2}^{3}$. Panels a) to e) correspond to parameterizations $v_1-v_5$. The orange cross is the reference number $s_p=(457^{+14}_{-13} - i 279^{+11}_{-7})$.
• Figure 3: Illustration of the analytic continuation's sensitivity of the PA method by showing the 68% confidence level (CL) regions for the pole positions obtained from the different parameterizations. The colors correspond to the five parameterizations as follows: red for $v_{1}$, green for $v_{2}$, blue for $v_{3}$, purple for $v_{4}$, and yellow for $v_{5}$. The orange cross is the reference number $s_p=(457^{+14}_{-13} - i 279^{+11}_{-7})$.
• Figure 4: Overlap of the 68% CL ellipses for the pole positions derived from the $P_1^N$ ($P_1^2$–$P_1^4$), upper panel, and $P_2^N$ ($P_2^1$–$P_2^3$), lower panel. Red, green, and blue colors correspond to increasing PA order, respectively. The orange cross is the reference number $s_p=(457^{+14}_{-13} - i 279^{+11}_{-7})$.
• Figure 5: Overlap of the 68% CL ellipses for $P^4_1$ (upper panel) and $P^3_2$ (lower panel). The colors correspond to the six parameterizations as follows: red for $v_{1}$, green for $v_{2}$, blue for $v_{3}$, purple for $v_{4}$, yellow for $v_{5}$ and gray for $v_{6}$. The orange cross represents the reference value $s_p = (457^{+14}_{-13} - i 279^{+11}_{-7})$.