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Global parametrizations of $\pi\pi$ scattering with dispersive constraints: Beyond the S0 wave

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper provides analytic global fits for the S2, P, D0, D2, F, G0, and G2 partial waves of pion-pion scattering, valid up to at least 1.8 GeV, and shows that, together with a slightly updated S0 wave, they satisfy nine dispersion…

desk verdict Useful global pi-pi parametrizations with real constraints, but the claimed FDR fulfillment up to 1.6 GeV rests on an untested L=4 truncation. read the letter →

arxiv 2412.15327 v2 pith:PO4UE6Y5 submitted 2024-12-19 hep-ph hep-exhep-latnucl-th

J. R. Peláez , P. Rabán , J. Ruiz de Elvira This is my paper · ORCID
classification hep-phhep-exhep-latnucl-th PACS 13.75.Lb11.55.Fv
keywords pion-pionscatteringpartial-waveanalysisdispersionrelationsRoyequationsGKPYinelasticityReggeparametrizationglobalfits
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper aims to make the dispersive description of pion-pion scattering practical by packaging it into analytic, ready-to-use parametrizations. It supplies differentiable expressions for seven partial waves (S2, P, D0, D2, F, G0, G2) from threshold to at least 1.8 GeV, with the S0 wave updated only slightly. Unlike earlier piecewise or purely numerical outputs, these fits can be evaluated anywhere on the real axis and used directly in other hadronic calculations. The paper's central demonstration is that the fits satisfy nine dispersion relations—three forward relations, three Roy equations, and three GKPY equations—with the forward relations involving the P wave now tested up to 1.6 GeV, about 200 MeV higher than before. If correct, a single set of simple expressions reproduces the content of dispersive partial-wave analyses while remaining convenient for phenomenology.

What carries the argument

The central device is a hybrid analytic parametrization: conformal-mapping expansions in variables like $w(s)=(\sqrt{s}-\alpha\sqrt{s_0-s})/(\sqrt{s}+\alpha\sqrt{s_0-s})$ describe the low-energy elastic region, Chebyshev polynomials in a rescaled variable describe the region above 1.4 GeV, and inelasticity is switched on smoothly by factors with the correct angular-momentum threshold behavior, such as the $\bar{J}_{\pi\omega}$ function for the P wave and Blatt-Weisskopf barrier factors for the F and G resonances. The full fit is constrained by penalty functions that quantify the distance between direct evaluations and dispersion-relation integrals, with a Regge parametrization supplying the high-energy part of the integrals.

What would settle it

Evaluate the forward amplitudes at 1.5 GeV with the published G-wave set and then with a trial H-wave (angular momentum five) built from the same Regge input; if the second evaluation moves the direct-versus-dispersive difference by more than the stated uncertainty band, the claimed dispersion-relation fulfillment depends on truncating the partial-wave expansion rather than on the data.

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Extended reading notes

Core claim

The central claim is that one can write the pion-pion partial-wave amplitudes (equivalently phase shifts and elasticities) as relatively simple analytic functions that are continuous and differentiable from threshold up to 1.8–2.1 GeV depending on the wave, and that these functions satisfy dispersive constraints. Below roughly 0.9 GeV the new parametrizations mimic the previous constrained fits to data; above that energy they are free fits to the main scattering data sets. The authors then require the fits to satisfy, within uncertainties, three forward dispersion relations, three Roy equations, and three GKPY equations. The notable improvements over earlier S0–P global fits are that the P-wave inelasticity now opens at the pion-omega threshold, the P-wave elastic input comes from a recent pion vector form factor analysis, the F and G waves are parametrized for the first time, and the forward relations involving the P wave are imposed up to 1.6 GeV with a cleaner matching to the Regge regime. For the region above 1.4 GeV the authors provide three solution variants reflecting three mutually incompatible data sets; all three satisfy the dispersion relations, with the first variant somewhat favored.

Load-bearing premise

The load-bearing assumption is that the partial-wave sum up to G waves (angular momentum four) is accurate below 1.6 GeV, the energy at which the paper switches to a high-energy Regge description; the authors themselves warn that near 1.7 GeV the omitted higher partial waves are as large as the retained ones, so hidden contributions below 1.6 GeV would bias all nine dispersion-relation tests.

Editorial extensions

If this is right

  • Anyone who needs pion-pion input for a hadronic-process calculation can use one analytic expression per wave from threshold to 1.8 GeV instead of gluing together piecewise tables.
  • The two forward dispersion relations that involve the P wave now constrain fits up to 1.6 GeV, so the inelastic region above the rho(770) tail is no longer arbitrary.
  • The three solution variants bracket the ambiguity in the older scattering data above 1.4 GeV, giving users a quantitative handle on systematic uncertainty.
  • Because the fits are analytic on the real axis, derivatives and integrals needed in phenomenological applications are well-defined, while the fits should not be extrapolated beyond their stated maximum energies.
  • The new G-wave parametrizations, although built with no direct scattering data, now enter the forward dispersion relations, so future G-wave measurements can be checked against the dispersive bands.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the truncation at angular momentum four hides a real H-wave contribution below 1.6 GeV, the excellent dispersion-relation agreement could be a coincidence of the truncation; adding a trial H wave built from the same Regge input would reveal the shift.
  • The same conformal-plus-Chebyshev construction could be applied to pion-kaon or pion-pion to kaon-antikaon amplitudes, where global parametrizations satisfying Roy-like constraints would be equally valuable.
  • The authors' warning that naive extrapolation to the complex plane is model-dependent implies that extracting resonance poles from these global fits requires analytic continuation via dispersion relations; the fits themselves should be used on the real axis.
  • The marked reduction in the P-wave uncertainty band suggests that future high-precision measurements of the pion vector form factor, such as the data the authors deliberately set aside, would immediately translate into sharper tests of these fits.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. This paper constructs global analytic parametrizations of the pi pi -> pi pi partial waves S2, P, D0, D2, F, G0, and G2, with a slightly updated S0 wave, covering energies up to approximately 1.8-2.1 GeV depending on the wave. The parametrizations are fit to scattering data and to the constrained fits of Garcia-Martin et al. [54], and the fits are then modified by imposing, as penalty terms, three forward dispersion relations (FDRs) up to 1.4 or 1.6 GeV, together with three Roy and three GKPY equations up to 1.1 GeV. The paper reports that the resulting Global Fits describe the data and satisfy the dispersion relations within uncertainties, with average quadratic distances dbar_i^2 <= 1 in the main energy regions, and it highlights an improved treatment of the P-wave inelasticity starting at the pi omega threshold and a better matching to the Regge regime.

Significance. If the claims hold, this is a valuable phenomenological resource: it provides relatively simple analytic expressions for seven partial waves with realistic uncertainty bands over a much wider energy range than the previous global parametrizations in [68], and it extends the FDR constraints to 1.6 GeV for the two relations that involve the P wave. The before-and-after comparison in Fig. 14 is a concrete and useful demonstration that imposing the dispersion relations changes the fits substantially, and the improvement over the unconstrained fits is clear. However, the central quantitative claim of FDR fulfillment up to 1.6 GeV depends on assumptions about the partial-wave truncation and on fits that use the same relations as penalties; these assumptions need quantitative scrutiny before the claim can be taken at face value.

major comments (4)
  1. [§II, §IV.A.5, Eqs. (56)-(57)] The extension of the F0+ and It=1 FDRs to 1.6 GeV rests on a partial-wave series truncated at angular momentum l=4 with Regge input only above 1.62 GeV, while Section II itself warns that 'already at 1.7 GeV the F wave is as large as the P wave, the D0 as the S0, and the D2 is larger than the S2.' The FDR integrands are not suppressed for s' near the external s, so an omitted l=5 (H) wave between roughly 1.4 and 1.6 GeV enters the principal-value integral directly. The paper provides no numerical estimate of this omitted contribution, and the reported dbar_i^2 <= 1 therefore demonstrates consistency between two representations that share the same l<=4 truncation rather than establishing that higher partial waves are negligible. Please quantify the H-wave contribution, for example by adding a Regge- or ChPT-motivated l=5 input below the matching point or by showing the sensitivity of Delta_i(s) to a conservative H-wave estimate.
  2. [§IV.A.3, Eq. (59)] Because the FDR constraints are inserted as penalty functions in the same minimization, the post-fit dbar_i^2 values are not an independent validation of the dispersion relations; they quantify how well the penalty has been satisfied. The paper should state this limitation explicitly in the summary and abstract, and ideally provide an out-of-sample check, such as FDR fulfillment in an energy region or linear combination not used in the penalty, or a comparison with the independent Roy/GKPY output. In addition, the statement that after the constrained minimization 'we keep their uncertainties delta p_k = delta p_k^U' means that the error bands entering dbar_i^2 and the figures are not the covariance of the constrained fit; this should either be corrected by propagating the constrained uncertainties or be clearly flagged as a conservative approximation.
  3. [Table VIII, Fig. 16] The abstract and Section V claim without qualification that the new parametrizations improve their fulfillment of forward dispersion relations, but Table VIII shows that Global Fits II and III have dbar_It=1^2 = 1.81 and 1.56 in [0.93,1.06] GeV and 1.04 and 1.47 in [1.46,1.56] GeV, i.e., they do not satisfy the It=1 FDR within the paper's own dbar^2<=1 criterion in these subregions. The global averages are below one only because the violations are localized. Please qualify the summary claims, or revise Fits II and III (the authors themselves suggest that moving the D0 inelasticity onset below K anti-K would help) so that all three fits meet the stated criterion uniformly.
  4. [§III.D, §IV.A, Tables IV and X] The G0 wave is entirely built from an educated guess: no scattering data exist, and the input is the RPP mass and width of the f4(2050), a sum-rule scattering length, and a BWBW form, with the input width uncertainty as large as 80 MeV. Since the G0 wave now contributes to the FDR integrands up to 1.6 GeV, the claimed dbar_i^2<=1 fulfillment may be sensitive to this model-dependent input. Please quantify this sensitivity, for example by repeating the constrained fit with a different f4(2050) width or with the G0 uncertainties enlarged, and by reporting how dbar_i^2 changes when the G0 wave is omitted or varied within its input errors.
minor comments (5)
  1. [§IV.A.3, Eq. (59)] The description of which parameters enter the k-sum as p_k^U and which are instead refit through the q^exp_m data is hard to follow; please list explicitly, for each wave, which parameter sets are kept fixed and which are varied in the constrained minimization.
  2. [Eq. (34) and Table III] The parameter x_rho3 appears in the BWBW expression and in Table III but is not defined in the text; please state its role (e.g., an inelasticity or peak-normalization parameter) and its allowed range.
  3. [Tables IV and X] For the G waves, Table X reports input values from [73] with uncertainties multiplied by five, while the 'Best values' column appears to use the original uncertainties from [73]; this distinction is mentioned in the table caption but should also be stated in the main text where the G-wave inputs are introduced.
  4. [§III.A.2 and §III.A.3] The text says that some parameters 'are constrained indirectly by refitting' the experimental data, but it does not give an example or indicate how large the resulting shifts are; one sentence with a concrete example (e.g., the P-wave K_i coefficients) would improve clarity.
  5. [General] Since the parametrizations are intended for practical phenomenological use, a machine-readable file with the final parameter values and their correlation/covariance information would be a useful addition; the paper currently provides only the parameter tables.

Circularity Check

2 steps flagged · score 6.0 of 10

FDR 'fulfillment' and the low-energy 'reproduction' of earlier analyses are built into the minimization, though the global parametrizations themselves rest on independent experimental input.

  1. fitted input called prediction [Sec. IV.A.3, Eqs. (58)-(59), Fig. 14 (right panels)]
    "In particular, we divide the calculation of the ¯d2_i into different energy regions, assigning them a different weight to ensure a ¯d2_i ≤ 1 in all of them. The ¯d2_i are often called penalty functions in the literature. [...] and we now show that they also satisfy FDRs within uncertainties up to their maximum applicability region."

    The metric used to claim FDR fulfillment, dbar^2_i defined in Eq. (58), is the same quantity that is minimized as a penalty in Eq. (59): the pseudo-chi2 contains sum_i W_i^2 dbar^2_i. The weights W_i are explicitly chosen so that dbar^2_i <= 1 in each energy region. Therefore the later statement that the constrained Global Fits 'satisfy FDRs within uncertainties' is not an independent test or prediction; it reports the value of the objective function that was minimized. The genuine non-circular content is the pre-constraint check (left panels of Fig. 14), where unconstrained fits fail with dbar^2 ~ 3-11; the post-constraint agreement is enforced by construction.

  2. self definitional [Abstract; Sec. II; Sec. V; Appendix B]
    "With earlier S0-wave parametrizations, slightly updated here, they reproduce previous partial wave dispersion analyses up to the πω threshold. [...] up to 0.9 GeV, we have basically mimicked them."

    The abstract's claim that the parametrizations 'reproduce previous partial wave dispersion analyses up to the πω threshold' is a restatement of the construction: below 0.9 GeV the fits are deliberately built to mimic the CFD of [54], and the S0 wave is carried over from [68]. Agreement with those earlier analyses in that energy region is therefore an input condition, not an output result. The paper itself acknowledges this by saying it 'mimics' the CFD. This is a transparent, mild definitional circularity in the framing; it does not affect the new content above 0.9 GeV.

full rationale

The central product of the paper is a set of analytic global fits to pi pi scattering data, which is a data-driven construction rather than a first-principles derivation. The most important circularity is that the claimed satisfaction of the forward dispersion relations is not an independent check: the dbar^2_i measures defined in Eq. (58) are used as penalty functions in Eq. (59), with weights chosen to force dbar^2_i <= 1 in each region. Reporting dbar^2_i <= 1 afterwards therefore reduces to the minimization target. A second, milder circularity is that the agreement with earlier dispersive analyses below 0.9 GeV is by design, since the CFD of [54] is used as input and the S0 wave is inherited from [68]. Both circularities are partly acknowledged in the text ('penalty functions', 'mimicked'), so they are transparency issues rather than hidden derivations. The paper does not rely on a load-bearing self-citation chain, a uniqueness theorem, or a smuggled ansatz: the unconstrained fits to actual experimental data, the updated pion vector form factor input from [77], and the Regge matching provide independent content. The score is 6 rather than 8 or 10 because the parametrizations themselves are anchored to external data and the circularity is confined to the 'dispersive fulfillment' claims and the reproduced low-energy region; the beyond-1.6 GeV parts are explicitly unconstrained fits to data.

Assumptions & free parameters 10 free parameters · 6 assumptions · 0 invented entities

The parametrizations rest on standard dispersion machinery plus several data and modeling choices. The most fragile assumptions are the partial-wave truncation at 1.62 GeV and the reliability of the selected 1970s data sets. No new particles, forces, or conserved quantities are introduced; the free parameters are the many coefficients of the analytic functions.

free parameters (10)
  • P-wave coefficients, Global Fit I (Table I) = 18 coefficients; see Table I
    All coefficients in the conformal, inelastic, and high-energy P-wave parametrizations are fitted to phase shift, elasticity, and vector form factor input.
  • D0-wave coefficients, Global Fit I (Table II) = 12 coefficients plus inelasticity onset at 0.9 GeV
    Fitted to CFD input below 0.9 GeV, Hyams data above, and dispersion relation penalties; the onset energy is a phenomenological choice.
  • F-wave coefficients (Table III) = 6 coefficients: B2, B3, m_rho3, Gamma_rho3, x_rho3, R_rho3
    Parameters of the conformal low-energy piece and the Blatt-Weisskopf Breit-Wigner resonance piece, fitted to scattering data and constraints.
  • G0-wave coefficients (Table IV) = m_f4, Gamma_f4, x_f4, R_f4
    Resonance parameters for the f4(2050)-like shape, partially fixed by the scattering length input and matching conditions.
  • S2-wave coefficients (Table V) = 6 coefficients plus inelasticity onset at 0.915 GeV
    Fitted to CFD input, I=2 scattering data, and dispersion constraints; the onset is an empirical choice.
  • D2-wave coefficients (Table VI) = 6 coefficients: B0, B1, B2, Delta, Bh2, Bh3
    Fitted to CFD threshold input and I=2 data; the wave is kept elastic.
  • G2-wave coefficients (Table VII) = 6 coefficients: B0, B1, B2, Delta, Bh2, Bh3
    Fitted to sparse I=2 data and the scattering length from sum rules.
  • S0-wave updated coefficients (Tables XI-XIII) = S0 parameters from reference [68], slightly updated
    Changes in other waves induce small shifts in the S0 parameters when dispersion relations are imposed.
  • FDR penalty weights W_i in Eq. (59) = Chosen so that dbar^2_i <= 1
    Hand-tuned weights set the strength of the dispersion relation penalties and directly control the reported fulfillment metrics.
  • Regge matching points = 1.42, 1.62, and 1.62 GeV
    Chosen at round energies near where the partial-wave series and Regge description match within uncertainties; they set the maximum FDR constraint energies.
assumptions (6)
  • standard math Forward dispersion relations, Roy equations, and GKPY equations are valid for pion-pion amplitudes and can be used as constraints in the stated energy ranges.
    These relations follow from analyticity, crossing, and unitarity; the paper cites standard references and applies them in their conventional range.
  • domain assumption The partial-wave series truncated at angular momentum 4 is sufficient to reconstruct the imaginary parts used in the forward dispersion relation integrands up to 1.62 GeV.
    The paper itself notes in Section II that partial-wave convergence is questionable at 1.7 GeV, yet the fits use only S through G waves inside the FDR integrals.
  • domain assumption The Regge parametrization from references [99, 54] is a valid semi-local description of the high-energy amplitudes above the matching point.
    Regge theory is treated as an average description; the matching point is chosen near where partial-wave and Regge curves agree within uncertainties.
  • domain assumption The selected experimental data sets, Solutions I, II, and III and the chosen I=2 data, are unbiased samples of the true pion-pion amplitude.
    The data sets are mutually inconsistent, and the paper provides three separate fits, meaning the analysis carries a real data-selection ambiguity.
  • domain assumption The pion vector form factor dispersive analysis of Colangelo, Hoferichter, and Stoffer provides the P-wave phase shift below the pi-omega threshold.
    The P-wave elastic input is taken from a form factor analysis rather than from direct scattering data, relying on the Watson final-state phase relation.
  • domain assumption Isospin symmetry and neglect of electromagnetic corrections are valid for the amplitudes under study.
    The parametrizations are written in the isospin limit and compare with data that include isospin-breaking effects only implicitly.

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Pith. "Pith review of Global parametrizations of $\pi\pi$ scattering with dispersive constraints: Beyond the S0 wave." pith.science (2026). https://pith.science/paper/PO4UE6Y5

@misc{pith2026241215327,
  author       = {Pith},
  title        = {Pith review of: Global parametrizations of $\pi\pi$ scattering with dispersive constraints: Beyond the S0 wave},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PO4UE6Y5}},
  note         = {Machine review of arXiv:2412.15327}
}
abstract

We provide new global parametrizations of $\pi\pi \to \pi\pi$ scattering for the S2, P, D, F, and G partial waves up to at least 1.8 GeV, easy to implement for phenomenological use. With earlier S0-wave parametrizations, slightly updated here, they reproduce previous partial wave dispersion analyses up to the $\pi\omega$ threshold. In addition, these new parametrizations have improved their description of recent P-wave data, the inelasticity in various waves, and their fulfillment of Roy-like and forward dispersion relations. The latter now test very high partial waves and have an improved matching with the Regge regime, extending those with P-wave contributions up to 1.6 GeV. Above 1.6 GeV and up to 1.8 GeV, or sometimes somewhat beyond, the parametrizations are simple unconstrained fits to data.

Figures

Figures reproduced from arXiv: 2412.15327 by the authors.

Figure 1
Figure 1. FIG. 1. We show our unconstrained and dispersively con [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Comparison among the P-wave Global Fits I, II, [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. D0-wave phase shift (top) and elasticity (bottom). [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (17 more)
Figure 4
Figure 4. Figure 4: Finally, in these two solutions, the elasticity was [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 4
Figure 4. Figure 4: FIG. 4. Comparison among Global Fits I, II, and III for the [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. F-wave phase shift (top) and elasticity (bottom). We [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Comparison among Global Fits I, II, and III for the F [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. G0-wave phase shift (top) and elasticity (bottom). [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. S2-wave phase shift (top) and elasticity (bot [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Comparison among Global Fits I, II, and III for the [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. D2-wave phase shift. We show our unconstrained [PITH_FULL_IMAGE:figures/full_fig_p022_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Comparison among Global Fits I, II, and III for the [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. G2-wave phase shift. We show our unconstrained [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Comparison among Global Fits I, II, and III for the [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Fulfillment of forward dispersion relations before (left column) and after (right column), they have been imposed as [PITH_FULL_IMAGE:figures/full_fig_p026_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Comparison between the FDR error bands when [PITH_FULL_IMAGE:figures/full_fig_p027_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Detail of the fulfillment of the [PITH_FULL_IMAGE:figures/full_fig_p028_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. Im [PITH_FULL_IMAGE:figures/full_fig_p029_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. Fulfillment of Roy and GKPY equations (left and right columns, respectively) by the constrained Global Fit I. Note [PITH_FULL_IMAGE:figures/full_fig_p031_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19. S0-wave phase shift (top) and elasticity (bottom). [PITH_FULL_IMAGE:figures/full_fig_p035_19.png]

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Reference graph

Works this paper leans on

115 extracted references · 21 canonical work pages · cited by 3 Pith papers

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    Gomez Nicola and J

    A. Gomez Nicola and J. R. Pelaez, Phys. Rev. D65, 054009 (2002), arXiv:hep-ph/0109056 [hep-ph]

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    Garc ´ ıa-Mart ´ ın, R

    R. Garc ´ ıa-Mart ´ ın, R. Kami´ nski, J. R. Pel´ aez, J. Ruiz de Elvira, and F. J. Yndur´ ain, Phys.Rev. D83, 074004 (2011), arXiv:1102.2183 [hep-ph]

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    G0-wave Global Fit and parameters As commented before, there are no scattering data for this wave. Thus our input consists of the a(0) 4 scattering length provided in Table X in Appendix A, along with the mass and width of the f4(2050) resonance taken from the weighted average of their determinations listed in the RPP [90] with two pions in the final stat...

  4. [2]

    The three G0-wave Global Fits In practice, as seen in Table IV, our three G0-wave Global Fits share the same parameters up to the pre- cision that we give them. The fact that the three of them share as input the same value for the scattering length with its small uncertainty, fixes the parameters within that precision, even after imposing the dispersive c...

    2011

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    8, we show the phase shift and elasticity of our constrained Global Fit I, together with all available scattering data

    S2-wave Global Fit and parameters In Fig. 8, we show the phase shift and elasticity of our constrained Global Fit I, together with all available scattering data. However, the input we use in our fit is the CFD below 0.9 GeV, the data represented by solid symbols above 0.9 GeV and the a(2) 0 and b(2) 0 threshold parameters in Table X in Appendix A. Regardi...

  6. [4]

    The three S2-wave Global Fits In contrast to the S0, P, D0, and F waves, each I = 2 wave has only one data set to fit from the start. There- fore, we expect the three constrained Global Fits to the I = 2 waves to differ very little among themselves since their separation is an effect induced indirectly from the other waves used as input in the dispersion ...

  7. [5]

    In addition, we fit the CFD results below 0.915 GeV, as well as the data on the phase shift that we show in Fig

    D2-wave Global Fit and parameters For this wave, we include in the fit the values of the threshold parameters a(2) 2 and b(2) 2 obtained from the CFD parametrization in [54], given in Table X here. In addition, we fit the CFD results below 0.915 GeV, as well as the data on the phase shift that we show in Fig. 10. It should be noted that this was the wave ...

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    The three D2-wave Global Fits The three Global Fits are very compatible, almost identical up to 1.25 GeV, as shown in Fig. 11. Their parameters, provided in Table VI are very compatible too. 0.50 0.75 1.00 1.25 1.50 1.75 2.00 s (GeV) 7 6 5 4 3 2 1 0 (2) 2 (s) ( ) Global I Global II Global III Cohen et al. Durusoy et al. OPEDP Durusoy et al. OPE Losty et a...

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    [23], Durusoy et al

    G2-wave Global Fit and parameters Three different experimental collaborations, Cohen et al. [23], Durusoy et al. [21], and Losty et al. [22] provide measurements of the phase shift for this wave, shown in Fig. 12. The set from Cohen et al. [23] only has two data points at 1.2 ...

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    13 and in Table VII, the three Global Fits I, II, and III are remarkably compatible and fully consistent within uncertainties

    The three G2-wave Global Fits As seen in Fig. 13 and in Table VII, the three Global Fits I, II, and III are remarkably compatible and fully consistent within uncertainties. 0.50 0.75 1.00 1.25 1.50 1.75 2.00 s (GeV) 7 6 5 4 3 2 1 0 (2) 4 (s) ( ) Global I Global II Global III C...

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    They are very relevant for three reasons

    Definitions Let us first discuss the fixed t = 0 dispersion relations, also known as forward dispersion relations. They are very relevant for three reasons. First, their applicability can be extended, in principle, to any value of s. Second, the optical theorem relates the ima...

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    dispersive

    FDRs as checks In [54, 70–73] the three FDRs in Eqs. (56) and (57) were imposed as constraints of phenomenological fits to data, to be satisfied within uncertainties to obtain the CFD parametrizations. Moreover, in [54] Roy and GKPY equations—to be explained below—were imposed...

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