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

Balma Duch; Pere Masjuan

arxiv: 2512.20721 · v2 · submitted 2025-12-23 · ✦ hep-ph

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

Balma Duch , Pere Masjuan This is my paper

Pith reviewed 2026-05-16 20:27 UTC · model grok-4.3

classification ✦ hep-ph

keywords Padé approximantsf0(500) resonanceππ scatteringthreshold conditionsanalytic continuationpole extractionpartial-wave amplitude

0 comments

The pith

Imposing correct threshold behavior on admissible parametrizations of the scalar-isoscalar ππ partial wave improves determinations of the f0(500) pole position via Padé approximants.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the analytic continuation of the ππ-scattering amplitude to the complex plane using Padé approximants, focusing on how threshold conditions affect the extracted resonance pole. Authors take a class of admissible parametrizations for the scalar-isoscalar partial wave and enforce the proper threshold behavior required by unitarity and analyticity. This constraint yields tighter pole positions for the f0(500) than earlier unconstrained analyses. A sympathetic reader sees the result as showing that physical boundary conditions at low energy reduce extrapolation uncertainty and strengthen the Padé method as a practical tool for resonance extraction from data.

Core claim

Using Padé approximants to continue the ππ scattering amplitude analytically from the physical region, with input from admissible parametrizations of the scalar-isoscalar partial wave now constrained to the correct threshold behavior, produces improved determinations of the f0(500) resonance pole position and its uncertainty compared with prior work that omitted the threshold constraint.

What carries the argument

Padé approximants applied to threshold-constrained admissible parametrizations of the scalar-isoscalar ππ partial wave

If this is right

Uncertainties on the extracted f0(500) pole position decrease once threshold conditions are enforced.
The Padé method becomes more precise for resonance extraction when input amplitudes satisfy low-energy physical constraints.
Similar threshold constraints can be applied to other partial waves or scattering processes to reduce pole-extraction uncertainties.
The approach provides a simpler alternative to full model-dependent fits for locating resonance poles.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The improvement suggests that threshold conditions act as a general regularizer for analytic continuations of scattering data.
The method could be validated by applying it to synthetic amplitudes generated from known models whose poles are known exactly.
Extensions to multi-channel cases or to reactions involving heavier particles would test whether the same threshold enforcement remains effective.

Load-bearing premise

The selected class of admissible parametrizations continues to represent the true ππ amplitude faithfully in the physical region even after the threshold constraints are imposed.

What would settle it

An independent high-precision lattice QCD calculation of the f0(500) pole position that lies outside the uncertainty band obtained with threshold constraints but inside the band obtained without them would falsify the claimed improvement.

Figures

Figures reproduced from arXiv: 2512.20721 by Balma Duch, Pere Masjuan.

**Figure 2.** Figure 2: FIG. 2: Overlap of the 68% CL ellipses for the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6: Comparison of the 68% confidence level (CL) ellipses for the six parameterizations. The darker ellipses correspond to [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7: Comparison of the 68% confidence level (CL) ellipses for the six parameterizations. The darker ellipses correspond [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Adding the standard threshold constraint to their existing Padé setup gives a modest improvement on the f0(500) pole from Ref. [1], but the paper stays light on actual numbers.

read the letter

The main point is that forcing the input parametrizations of the pi pi S-wave to obey the usual q^2 threshold behavior shifts or tightens the extracted f0(500) pole relative to their prior work. It is a small, practical adjustment rather than a new technique. They take a class of admissible parametrizations, impose the kinematic threshold factor, and run the Padé continuation to the complex plane. This keeps the method simple and shows that the constraint helps the continuation without adding much machinery. The approach is direct and the claim is falsifiable by comparing the new pole values to the old ones. The soft spot is that the abstract gives no pole positions, error bars, or side-by-side tables, so the size of the improvement is stated but not shown in the summary. Without those numbers it is hard to judge whether the change is large enough to matter for most applications. The result also inherits whatever limitations the chosen parametrizations have in the physical region. This is for people already working with Padé or other analytic continuation tools in hadron spectroscopy. A reader who follows resonance extraction would pick up a useful check on threshold effects. I would send it to peer review as a short note; the scope is narrow and the central numerical comparison is straightforward to verify once the full tables are in front of a referee.

Referee Report

2 major / 1 minor

Summary. The paper claims that imposing the correct threshold behavior (the standard q^{2L} factor with L=0) on a class of admissible parametrizations of the scalar-isoscalar ππ partial wave improves the f0(500) resonance pole positions extracted via Padé approximants relative to the results of Ref. [1]. The work positions this constraint as a simple kinematic input that enhances the precision and reliability of the Padé method for resonance pole determination from scattering amplitudes.

Significance. If the reported numerical improvement in pole-position uncertainties holds under the stated parametrization class, the result would provide a practical, falsifiable demonstration that standard threshold constraints reduce extrapolation errors in analytic continuation techniques. This could modestly strengthen the case for using Padé approximants as a lightweight tool in hadron spectroscopy, particularly when input amplitudes are already constrained by low-energy kinematics.

major comments (2)

[Results section] Results section: the central claim of improvement over Ref. [1] requires an explicit side-by-side numerical comparison (pole real/imaginary parts and uncertainties) with and without the threshold constraint; without this table or set of values the magnitude of the improvement cannot be assessed.
[§2] §2 (parametrization class): the manuscript must specify how the admissible parametrizations are selected and validated against data once the threshold factor is imposed, to confirm that the constrained class remains representative of the physical ππ amplitude in the fitting region.

minor comments (1)

The abstract would be strengthened by quoting the numerical shift in pole position and uncertainty reduction achieved by the threshold condition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the recommendation for minor revision. We address each major comment below and will update the manuscript accordingly.

read point-by-point responses

Referee: [Results section] Results section: the central claim of improvement over Ref. [1] requires an explicit side-by-side numerical comparison (pole real/imaginary parts and uncertainties) with and without the threshold constraint; without this table or set of values the magnitude of the improvement cannot be assessed.

Authors: We agree that an explicit side-by-side comparison is required to quantify the improvement. In the revised manuscript we will add a table in the Results section that reports the real and imaginary parts of the f0(500) pole together with their uncertainties, obtained from the same set of admissible parametrizations both with and without the imposed threshold factor. This will make the magnitude of the improvement directly assessable. revision: yes
Referee: [§2] §2 (parametrization class): the manuscript must specify how the admissible parametrizations are selected and validated against data once the threshold factor is imposed, to confirm that the constrained class remains representative of the physical ππ amplitude in the fitting region.

Authors: We will expand §2 to describe the precise selection criteria for the admissible parametrizations, the manner in which the q^{2L} (L=0) threshold factor is imposed, the fitting interval used, and the validation procedure against experimental data that confirms the constrained class remains representative of the physical amplitude. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper improves pole extractions for the f0(500) by applying Padé approximants to a class of admissible parametrizations of the ππ S-wave after imposing the standard kinematic threshold condition (the q^{2L} factor with L=0). This constraint is an external physical requirement drawn from partial-wave analyticity and is independent of the extracted pole position. The self-citation to Ref. [1] supplies only the baseline parametrizations and numerical results for comparison; the new work demonstrates a direct numerical improvement without redefining any fitted quantity in terms of the output pole or invoking a uniqueness theorem that reduces to prior self-work. No step equates a prediction to its own input by construction, and the method remains falsifiable against external amplitude data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of a well-defined class of admissible parametrizations that can be further constrained by threshold behavior without losing consistency with data; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption The scalar-isoscalar ππ partial wave must obey the correct threshold behavior near the two-pion threshold.
Imposed as an additional constraint on the input parametrizations to improve pole extraction.

pith-pipeline@v0.9.0 · 5395 in / 1271 out tokens · 47715 ms · 2026-05-16T20:27:03.321211+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

From QCD-Based Descriptions to Direct Fits: A Unified Study of Nucleon Electromagnetic Form Factors
hep-ph 2025-12 unverdicted novelty 4.0

A combined GPD and vector-meson model fitted to data produces global Padé parametrizations for nucleon electromagnetic form factors in the spacelike region.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages · cited by 1 Pith paper

[1]

Applying this condition allows the use of one-order higher PAs without modifying the input data or parameters of Ref

by explicitly enforcing the physical condition that the imaginarypartoftheamplitudevanishesatthepion-pion threshold in the construction of the PAs. Applying this condition allows the use of one-order higher PAs without modifying the input data or parameters of Ref. [1]. For the set of parameterizations, we report the central values of the input parameters...

work page
[2]

The study coversarangeofexpansionpointss 0 withintheelasticre- gion, fromtheππthresholdto0.85GeV.Theconvergence of the sequences is illustrated in Figs

andP N 2 (up toP 3 2), both sequences chosen to allow a direct com- parison with the previous analysis [1, 18, 19]. The study coversarangeofexpansionpointss 0 withintheelasticre- gion, fromtheππthresholdto0.85GeV.Theconvergence of the sequences is illustrated in Figs. 1 and 2, showing that, within each sequence, the most stable determina- tions are obtain...

work page
[3]

The orange cross is the reference numbersp = (457+14 −13 −i279 +11 −7 )

Panels a) to e) correspond to parameterizationsv1 −v 5. The orange cross is the reference numbersp = (457+14 −13 −i279 +11 −7 ). pole term, which slows convergence and increases sen- sitivity to small variations. By employingM= 2, one pole accurately locates the resonance, while the other pole effectively parametrizes the background and the in- fluence of...

work page
[4]

ForP 3 2, the mass uncertainty decreases by approximately 24%, while the half-width increases slightly by 14%

Although thev 6 parametrization carries larger individual uncertainties, its inclusion still improves the overall determination of the pole parameters [1, 18, 19]. ForP 3 2, the mass uncertainty decreases by approximately 24%, while the half-width increases slightly by 14%. In the case ofP 4 1, both the mass and half-width uncer- tainties are reduced by r...

work page 2021
[5]

Uncertainty estimates of theσ- pole determination by padé approximants.Physical Re- view D, 93(7):076004, April 2016

Irinel Caprini, Pere Masjuan, Jacobo Ruiz de Elvira, and Juan José Sanz-Cillero. Uncertainty estimates of theσ- pole determination by padé approximants.Physical Re- view D, 93(7):076004, April 2016

work page 2016
[6]

José R. Peláez. From controversy to precision on the sigma meson: A review on the status of the non-ordinary σ(500)resonance.Physics Reports, 658:1–111, November 2016

work page 2016
[7]

Nuclear Physics B, 603(1–2):125–179, June 2001

G.Colangelo, J.Gasser, andH.Leutwyler.ππscattering. Nuclear Physics B, 603(1–2):125–179, June 2001

work page 2001
[8]

Ananthanarayan, G

B. Ananthanarayan, G. Colangelo, J. Gasser, and H. Leutwyler. Roy equation analysis ofππscattering. Physics Reports, 353(4):207–279, November 2001

work page 2001
[9]

S.M. Roy. Exact integral equation for pion-pion scatter- ing involving only physical region partial waves.Physics Letters B, 36(4):353–356, 1971

work page 1971
[10]

Garcia-Martin, R

R. Garcia-Martin, R. Kaminski, J. R. Pelaez, J. Ruiz de Elvira, and F. J. Yndurain. The Pion-pion scattering am- plitude. IV: Improved analysis with once subtracted Roy- like equations up to 1100 MeV.Phys. Rev. D, 83:074004, 2011

work page 2011
[11]

Descotes-Genon, and B

Paul Buettiker, S. Descotes-Genon, and B. Moussallam. A new analysis ofπK scattering from Roy and Steiner type equations.Eur. Phys. J. C, 33:409–432, 2004

work page 2004
[12]

Mass and width of the lowest resonance in QCD.Phys

Irinel Caprini, Gilberto Colangelo, and Heinrich Leutwyler. Mass and width of the lowest resonance in QCD.Phys. Rev. Lett., 96:132001, 2006

work page 2006
[13]

Garcia-Martin, R

R. Garcia-Martin, R. Kaminski, J. R. Pelaez, and 7 FIG. 6: Comparison of the 68% confidence level (CL) ellipses for the six parameterizations. The darker ellipses correspond to the results obtained from Ref. [1] forP 3 1, while the lighter ellipses show the updated results presented in this work forP4 1. Colors indicate the different parameter sets: red f...

work page 2011
[14]

Work- man

Alfred Švarc, Mirza Hadzimehmedovic, Hedim Osman- ovic, Jugoslav Stahov, Lothar Tiator, and Ron L. Work- man. Introducing the Pietarinen expansion method into the single-channel pole extraction problem.Phys. Rev. C, 88(3):035206, 2013

work page 2013
[15]

Meissner and J

Ulf-G. Meissner and J. A. Oller. Chiral unitary meson baryon dynamics in the presence of resonances: Elas- tic pion nucleon scattering.Nucl. Phys. A, 673:311–334, 2000

work page 2000
[16]

Dispersive determination of resonances from ππscattering data.https://arxiv.org/abs/2512.09033, December 2025

José Ramón Peláez, Pablo Rabán, and Jacobo Ruiz de Elvira. Dispersive determination of resonances from ππscattering data.https://arxiv.org/abs/2512.09033, December 2025

work page arXiv 2025
[17]

Baker and Peter Graves-Morris.Padé Ap- proximants

George A. Baker and Peter Graves-Morris.Padé Ap- proximants. Encyclopedia of Mathematics and its Appli- cations. Cambridge University Press, 2 edition, 1996

work page 1996
[18]

Masjuan, J.J

P. Masjuan, J.J. Sanz-Cillero, and J. Virto. Some re- marks on the padé unitarization of low-energy ampli- tudes.Physics Letters B, 668(1):14–19, September 2008

work page 2008
[19]

Miguel Salg, Fernando Romero-López, and William I. Jay. Bayesian analysis and analytic continuation of scattering amplitudes from lattice qcd.Phys. Rev. D, 112(11):114502, 2025

work page 2025
[20]

A Rational approach to resonance saturation in large-N(c) QCD.JHEP, 05:040, 2007

Pere Masjuan and Santiago Peris. A Rational approach to resonance saturation in large-N(c) QCD.JHEP, 05:040, 2007

work page 2007
[21]

A Rational approxi- mation to⟨V V−AA⟩and its O(p 6) low-energy constant

Pere Masjuan and Santiago Peris. A Rational approxi- mation to⟨V V−AA⟩and its O(p 6) low-energy constant. Phys. Lett. B, 663:61–65, 2008

work page 2008
[22]

Padé approx- imants and resonance poles.Eur

Pere Masjuan and Juan José Sanz-Cillero. Padé approx- imants and resonance poles.Eur. Phys. J. C, 73:2594, 2013

work page 2013
[23]

Precise determination of resonance pole parameters through padé approximants.Phys

Pere Masjuan, Jacobo Ruiz de Elvira, and Juan José Sanz-Cillero. Precise determination of resonance pole parameters through padé approximants.Phys. Rev. D, 90(9):097901, 2014

work page 2014
[24]

de Montessus de Ballore

R. de Montessus de Ballore. Sur les fractions contin- ues algébriques.Bulletin de la Société Mathématique de France, 30:28–36, 1902

work page 1902
[25]

Finding the sigma pole by analytic ex- trapolation ofππscattering data.Physical Review D, 77(11):114019, June 2008

Irinel Caprini. Finding the sigma pole by analytic ex- trapolation ofππscattering data.Physical Review D, 77(11):114019, June 2008

work page 2008
[26]

García-Martín, J

R. García-Martín, J. R. Peláez, and F. J. Ynduráin. Ex- perimental status of theππisoscalars-wave at low en- ergy:f 0(600)pole and scattering length.Physical Review D, 76(7):074034, October 2007

work page 2007
[27]

J. R. Peláez, A. Rodas, and J. Ruiz de Elvira. Global parameterization ofππscattering up to 2 gev.The Eu- ropean Physical Journal C, 79(12):7509, December 2019

work page 2019
[28]

J. R. Peláez, P. Rabán, and J. Ruiz de Elvira. Global parametrizations ofππscattering with dispersive con- straints: Beyond the s0 wave.Physical Review D, 111(7):074003, April 2025

work page 2025

[1] [1]

Applying this condition allows the use of one-order higher PAs without modifying the input data or parameters of Ref

by explicitly enforcing the physical condition that the imaginarypartoftheamplitudevanishesatthepion-pion threshold in the construction of the PAs. Applying this condition allows the use of one-order higher PAs without modifying the input data or parameters of Ref. [1]. For the set of parameterizations, we report the central values of the input parameters...

work page

[2] [2]

The study coversarangeofexpansionpointss 0 withintheelasticre- gion, fromtheππthresholdto0.85GeV.Theconvergence of the sequences is illustrated in Figs

andP N 2 (up toP 3 2), both sequences chosen to allow a direct com- parison with the previous analysis [1, 18, 19]. The study coversarangeofexpansionpointss 0 withintheelasticre- gion, fromtheππthresholdto0.85GeV.Theconvergence of the sequences is illustrated in Figs. 1 and 2, showing that, within each sequence, the most stable determina- tions are obtain...

work page

[3] [3]

The orange cross is the reference numbersp = (457+14 −13 −i279 +11 −7 )

Panels a) to e) correspond to parameterizationsv1 −v 5. The orange cross is the reference numbersp = (457+14 −13 −i279 +11 −7 ). pole term, which slows convergence and increases sen- sitivity to small variations. By employingM= 2, one pole accurately locates the resonance, while the other pole effectively parametrizes the background and the in- fluence of...

work page

[4] [4]

ForP 3 2, the mass uncertainty decreases by approximately 24%, while the half-width increases slightly by 14%

Although thev 6 parametrization carries larger individual uncertainties, its inclusion still improves the overall determination of the pole parameters [1, 18, 19]. ForP 3 2, the mass uncertainty decreases by approximately 24%, while the half-width increases slightly by 14%. In the case ofP 4 1, both the mass and half-width uncer- tainties are reduced by r...

work page 2021

[5] [5]

Uncertainty estimates of theσ- pole determination by padé approximants.Physical Re- view D, 93(7):076004, April 2016

Irinel Caprini, Pere Masjuan, Jacobo Ruiz de Elvira, and Juan José Sanz-Cillero. Uncertainty estimates of theσ- pole determination by padé approximants.Physical Re- view D, 93(7):076004, April 2016

work page 2016

[6] [6]

José R. Peláez. From controversy to precision on the sigma meson: A review on the status of the non-ordinary σ(500)resonance.Physics Reports, 658:1–111, November 2016

work page 2016

[7] [7]

Nuclear Physics B, 603(1–2):125–179, June 2001

G.Colangelo, J.Gasser, andH.Leutwyler.ππscattering. Nuclear Physics B, 603(1–2):125–179, June 2001

work page 2001

[8] [8]

Ananthanarayan, G

B. Ananthanarayan, G. Colangelo, J. Gasser, and H. Leutwyler. Roy equation analysis ofππscattering. Physics Reports, 353(4):207–279, November 2001

work page 2001

[9] [9]

S.M. Roy. Exact integral equation for pion-pion scatter- ing involving only physical region partial waves.Physics Letters B, 36(4):353–356, 1971

work page 1971

[10] [10]

Garcia-Martin, R

R. Garcia-Martin, R. Kaminski, J. R. Pelaez, J. Ruiz de Elvira, and F. J. Yndurain. The Pion-pion scattering am- plitude. IV: Improved analysis with once subtracted Roy- like equations up to 1100 MeV.Phys. Rev. D, 83:074004, 2011

work page 2011

[11] [11]

Descotes-Genon, and B

Paul Buettiker, S. Descotes-Genon, and B. Moussallam. A new analysis ofπK scattering from Roy and Steiner type equations.Eur. Phys. J. C, 33:409–432, 2004

work page 2004

[12] [12]

Mass and width of the lowest resonance in QCD.Phys

Irinel Caprini, Gilberto Colangelo, and Heinrich Leutwyler. Mass and width of the lowest resonance in QCD.Phys. Rev. Lett., 96:132001, 2006

work page 2006

[13] [13]

Garcia-Martin, R

R. Garcia-Martin, R. Kaminski, J. R. Pelaez, and 7 FIG. 6: Comparison of the 68% confidence level (CL) ellipses for the six parameterizations. The darker ellipses correspond to the results obtained from Ref. [1] forP 3 1, while the lighter ellipses show the updated results presented in this work forP4 1. Colors indicate the different parameter sets: red f...

work page 2011

[14] [14]

Work- man

Alfred Švarc, Mirza Hadzimehmedovic, Hedim Osman- ovic, Jugoslav Stahov, Lothar Tiator, and Ron L. Work- man. Introducing the Pietarinen expansion method into the single-channel pole extraction problem.Phys. Rev. C, 88(3):035206, 2013

work page 2013

[15] [15]

Meissner and J

Ulf-G. Meissner and J. A. Oller. Chiral unitary meson baryon dynamics in the presence of resonances: Elas- tic pion nucleon scattering.Nucl. Phys. A, 673:311–334, 2000

work page 2000

[16] [16]

Dispersive determination of resonances from ππscattering data.https://arxiv.org/abs/2512.09033, December 2025

José Ramón Peláez, Pablo Rabán, and Jacobo Ruiz de Elvira. Dispersive determination of resonances from ππscattering data.https://arxiv.org/abs/2512.09033, December 2025

work page arXiv 2025

[17] [17]

Baker and Peter Graves-Morris.Padé Ap- proximants

George A. Baker and Peter Graves-Morris.Padé Ap- proximants. Encyclopedia of Mathematics and its Appli- cations. Cambridge University Press, 2 edition, 1996

work page 1996

[18] [18]

Masjuan, J.J

P. Masjuan, J.J. Sanz-Cillero, and J. Virto. Some re- marks on the padé unitarization of low-energy ampli- tudes.Physics Letters B, 668(1):14–19, September 2008

work page 2008

[19] [19]

Miguel Salg, Fernando Romero-López, and William I. Jay. Bayesian analysis and analytic continuation of scattering amplitudes from lattice qcd.Phys. Rev. D, 112(11):114502, 2025

work page 2025

[20] [20]

A Rational approach to resonance saturation in large-N(c) QCD.JHEP, 05:040, 2007

Pere Masjuan and Santiago Peris. A Rational approach to resonance saturation in large-N(c) QCD.JHEP, 05:040, 2007

work page 2007

[21] [21]

A Rational approxi- mation to⟨V V−AA⟩and its O(p 6) low-energy constant

Pere Masjuan and Santiago Peris. A Rational approxi- mation to⟨V V−AA⟩and its O(p 6) low-energy constant. Phys. Lett. B, 663:61–65, 2008

work page 2008

[22] [22]

Padé approx- imants and resonance poles.Eur

Pere Masjuan and Juan José Sanz-Cillero. Padé approx- imants and resonance poles.Eur. Phys. J. C, 73:2594, 2013

work page 2013

[23] [23]

Precise determination of resonance pole parameters through padé approximants.Phys

Pere Masjuan, Jacobo Ruiz de Elvira, and Juan José Sanz-Cillero. Precise determination of resonance pole parameters through padé approximants.Phys. Rev. D, 90(9):097901, 2014

work page 2014

[24] [24]

de Montessus de Ballore

R. de Montessus de Ballore. Sur les fractions contin- ues algébriques.Bulletin de la Société Mathématique de France, 30:28–36, 1902

work page 1902

[25] [25]

Finding the sigma pole by analytic ex- trapolation ofππscattering data.Physical Review D, 77(11):114019, June 2008

Irinel Caprini. Finding the sigma pole by analytic ex- trapolation ofππscattering data.Physical Review D, 77(11):114019, June 2008

work page 2008

[26] [26]

García-Martín, J

R. García-Martín, J. R. Peláez, and F. J. Ynduráin. Ex- perimental status of theππisoscalars-wave at low en- ergy:f 0(600)pole and scattering length.Physical Review D, 76(7):074034, October 2007

work page 2007

[27] [27]

J. R. Peláez, A. Rodas, and J. Ruiz de Elvira. Global parameterization ofππscattering up to 2 gev.The Eu- ropean Physical Journal C, 79(12):7509, December 2019

work page 2019

[28] [28]

J. R. Peláez, P. Rabán, and J. Ruiz de Elvira. Global parametrizations ofππscattering with dispersive con- straints: Beyond the s0 wave.Physical Review D, 111(7):074003, April 2025

work page 2025