Data-driven dispersive analysis of the $\ensuremath{\pi}\ensuremath{\pi}$ and $\ensuremath{\pi}K$ scattering

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Data-driven dispersive analysis of the $π π$ and $π K$ scattering

Igor Danilkin, Oleksandra Deineka, and Marc Vanderhaeghen

Phys. Rev. D 103, 114023 – Published 22 June 2021

Abstract

We present a data-driven analysis of the resonant $S$ -wave $π π \to π π$ and $π K \to π K$ reactions using the partial-wave dispersion relation. The contributions from the left-hand cuts are accounted for using the Taylor expansion in a suitably constructed conformal variable. The fits are performed to experimental and lattice data as well as Roy analyses. For the $π π$ scattering we present both a single- and a coupled-channel analysis by including additionally the $K \bar{K}$ channel. For the latter the central result is the Omnès matrix, which is consistent with the most recent Roy and Roy-Steiner results on $π π \to π π$ and $π π \to K \bar{K}$ , respectively. By the analytic continuation to the complex plane, we found poles associated with the lightest scalar resonances $σ / f_{0} (500)$ , $f_{0} (980)$ , and $κ / K_{0}^{*} (700)$ for the physical pion mass value and in the case of $σ / f_{0} (500)$ , $κ / K_{0}^{*} (700)$ also for unphysical pion mass values.

Received 3 January 2021
Accepted 22 April 2021

DOI:https://doi.org/10.1103/PhysRevD.103.114023

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Hadron-hadron interactions Strong interaction

Kaons Mesons Pions

Particles & Fields

Authors & Affiliations

Igor Danilkin, Oleksandra Deineka , and Marc Vanderhaeghen

Institut für Kernphysik & PRISMA+ Cluster of Excellence, Johannes Gutenberg Universität, D-55099 Mainz, Germany

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Vol. 103, Iss. 11 — 1 June 2021

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Figure 1
Left-hand cut singularities (solid black curves) in the complex $s$ -plane for the (a) $π π \to π π$ and (b) $π K \to π K$ scattering. In the plot we schematically show the position of the closest left-hand cut singularity ( $s_{L}$ ), Adler zero ( $s_{A}$ ), threshold ( $s_{t h}$ ), and expansion point ( $s_{E}$ ). Dashed lines determine the specific form of the conformal map and subsequently the domain of convergence of the conformal expansion in Eq. (13).
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Figure 2
Results for the $π π \to π π$ scattering with $J = 0$ , $I = 0$ in the single-channel case. Top, central, and bottom panels correspond to $m_{π} = physical$ , 236, 391 MeV, respectively. Left panels show the convergence of the conformal expansion in Eq. (13), central panels show the comparison with the data, and right panels show the corresponding Omnès functions. In the phase shift plot for the physical pion mass two curves are shown: fit to the experimental data [49, 50] (dashed curve) and fit to the pseudodata from Roy analysis [6, 7] (thick curve). Note that for the sake of comparison with the coupled-channel case (see Fig. 4), we adopted for this case $s_{E}$ based on $\sqrt{s_{\max}} = 1.2 GeV$ , as discussed in the text.
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Figure 3
Comparison between the coupled-channel $N / D$ fits and the Roy-like solution from [7]. The dashed curves are the fit solely to the experimental data, while the solid curves take advantage of both the experimental data and the results of Roy (Roy-Steiner) analyses on $π π \to π π$ ( $π π \to K \bar{K}$ ).
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Figure 4
Results for the $π π \to π π, K \bar{K}$ scattering with $J = 0$ , $I = 0$ in the coupled-channel case. Top, central, and bottom panels correspond to 11,12, and 22 matrix elements, respectively, with $1 = π π$ and $2 = K \bar{K}$ . Left panels show the convergence of the conformal expansion in Eq. (13), central panels show the comparison with the data, and right panels show the elements of the Omnès matrix. In the central plots two curves are shown: fit to the experimental data [49, 50] (dashed curve) and fit to the pseudodata from Roy analyses [6, 7] (thick curve).
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Figure 5
Left panel: two-photon decay width of $σ / f_{0} (500)$ and $f_{0} (980)$ compared to the recent dispersive estimations from [24, 64, 81, 85, 86]. Right panel: total cross section of $γ γ \to π^{0} π^{0}$ $(| \cos θ | < 0.8)$ from [73] with updated $I = 0$ , $J = 0$ contribution. The data for the cross section are taken from [87].
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Figure 6
Results for the $π K \to π K$ scattering with $J = 0, I = 1 / 2$ in the single-channel approximation. Top and bottom panels correspond to $m_{π} = physical, 239 MeV$ , respectively. Left panels show the convergence of the conformal expansion in Eq. (13), central panels show the comparison with the data, and right panels show the corresponding Omnès functions. In the phase shift plot for the physical pion mass two curves are shown: fit to the experimental data [51, 52] (dashed curve) and fit to the pseudodata from Roy-Steiner analysis [11] (thick curve).
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Physical Review D

covering particles, fields, gravitation, and cosmology

Data-driven dispersive analysis of the $π π$ and $π K$ scattering

Igor Danilkin, Oleksandra Deineka, and Marc Vanderhaeghen

Phys. Rev. D 103, 114023 – Published 22 June 2021

Abstract

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Physical Review D

covering particles, fields, gravitation, and cosmology

Data-driven dispersive analysis of the ππ and πK scattering

Igor Danilkin, Oleksandra Deineka, and Marc Vanderhaeghen

Phys. Rev. D 103, 114023 – Published 22 June 2021

Abstract

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Data-driven dispersive analysis of the $π π$ and $π K$ scattering