Partial wave analysis of ππ scattering below 2 GeV

All resonances included in multi-channel D (ππ, 4π, KK̄ and ηη) and F (ππ, 4π, ωπ and KK̄) waves are analyzed. Results of the analysis are based on a set of well determined ππ scattering amplitudes for S F waves in a unitary multi-channel approach. The amplitudes are refined and re-fitted to the dispersion relations up to 1.1 GeV, and to the experimental data in the effective two pion mass from the threshold to 2.7 GeV and 1.9 GeV for D and F waves, respectively. Many existing parameterizations of the ππ scattering amplitudes still significantly do not satisfy the crossing symmetry condition or do not describe the ππ threshold region. Since they are used in some analyses, having a correct determination of ππ scattering amplitudes is essential to achieve a proper result. In our previous work [1] we presented a general method of refining the ππ amplitudes by fitting them to the GKPY equations [2, 3] which are once-subtracted dispersion relations with imposed crossing symmetry condition. The method was demonstrated on refining of the unitary multi-channel S (ππ, KK̄ and ηη) and P (ππ, ρ2π, and ρσ) wave amplitudes from [4] mostly below 1.1 GeV with no change in their original mathematical structure. In Ref. [5] we included the multi-channel D (ππ, effective (2π)(2π), KK̄ and ηη) and F (ππ, effective (2π)(2π), ωπ and KK̄) wave amplitudes from [6]. The S -F amplitudes were refined and then re-fitted to the available experimental data and to the GKPY equations [2, 3]. In Ref. [5] we presented some part of the results of analysis for the D and F wave amplitudes especially, details on formalism of the analysis and precise determination of ππ scattering amplitudes for D and F waves. In this paper we discuss on the resonances included in the analysis and show their importance separately in the corresponding partial waves. The method of analysis is generally the same as in Ref. [1]. To construct the D0 and F1 waves, we have used the same formalism as in Refs. [4, 6] and updated the list of contributing resonant states for the D0 wave according to the latest issue of PDG [7]. Then, a correct behavior of the phase shift near the threshold was controlled by the pseudo data (see Secs.II in [5]). The initial amplitudes for the S 0 and P1 waves were from [1], both with added polynomial near the threshold (the “extended” amplitudes in [1]). Since there is no any resonance for S 2 and D2 waves, the amplitudes for these isotensor waves were taken from [2] and kept unchanged during the analysis. In the modification of the original amplitudes only those parameters that were expected to contribute significantly to the dispersive integrals, i.e. the low-energy resonances and the ππ background were allowed to change. The results provided precise determination of D0 and F1 partial wave amplitudes which satisfy the crossing symmetry condition and describe the experimental data very well. e-mail: vnazari@ifj.edu.pl DOI: 10.1051/ , 07016 (2016) 130 EPJ Web of Conferences 13007016 MESON 2016 epjconf/2016 © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). In the construction of the partial waves, apart from background part which is relatively small, the resonance contribution described by the multi-channel Breit-Wigner form determines the main structure of the amplitudes. Concerning the dominant, important and insignificant resonances of the D0 wave in the multi-channel analysis approach, various fits have been performed. Results show that not all first four lightest states, i.e. f2(1270), f2(1430), f2(1525), and f2(1640), are the most important states in the analysis, as it was expected at the beginning. For this, Table 1 demonstrates the significance of each resonance in the analysis separately. It shows the χ2 for the fits when one particular resonance was omitted. The χ2 was only composed of the χData for D0 wave in the Eq. (16) in [8] and χDR for all partial waves in the Eq. (17) in [8]. Nevertheless, conditions are the same for all fits and they just differ by omitted resonances. Values of the χ2 achieved from fits 9, 11 and 12 are almost equal to the value in the fit 1 where all resonance states are included. Accordingly, from the point of view of the χ2 one can omit the f2(2010), f2(2300) and f2(2340) resonances and conclude that these resonances are insignificant in the description. Value of the χ2 in the fits 3, 5, 7 and 8 show that the f2(1430), f2(1640), f2(1910) and f2(1950) don’t play a very important role on the results but one should keep them. Additionally, the f2(1525), f2(1810), and f2(2150) resonances which strongly influence behavior of the amplitude and therefore value of the χ2 are the most important resonances in the D0 wave and obviously the dominant resonance which dominate the behaviour of the amplitude is f2(1270) resonance. From two resonance states ρ3(1690) and ρ3(1990) listed in PDG [7] for the isovector F1 wave only the ρ3(1690) resonance state has been considered in the analysis. For the ρ3(1990) resonance state, the partial widths are not known. In Refs. [5, 8] we have found that enough is only one resonance state ρ3(1690) which can satisfactorily describe the available experimental data in [9, 10] and give a reasonable description in the threshold region. Table 1. Values of χ2 after fitting when some specific resonance states are omitted in the D0 wave. Fit No. χ2/n.d. f . 1 All stated are included 326.21/330=0.9885 Omitted Resonances 2 f2(1270) 5838.8/330=17.69 3 f2(1430) 327.39/330=0.9921 4 f2(1525) 336.97/330=1.0211 5 f2(1640) 327.76/330=0.9932 6 f2(1810) 348.07/330=1.0548 7 f2(1910) 327.36/330=0.9920 8 f2(1950) 327.26/330=0.9917 9 f2(2010) 326.68/330=0.9899 10 f2(2150) 368.81/330=1.1176 11 f2(2300) 326.54/330=0.9895 12 f2(2340) 326.64/330=0.9898 A figurative comparison of the phase shift of the resonances in ππ → ππ channel is presented in the Figs. 1-3 for D0 and F1 waves individually. The D0 wave consists of 11 resonances ( f2(1270), f2(1430), f2(1525), f2(1640), f2(1810), f2(1910), f2(1950), f2(2010), f2(2150), f2(2300), f2(2340)) from which the f2(1270) is the most dominant one. This can be seen in the Fig. 1 where the f2(1270) determines structure of the full phase shift. Effect of other phase shifts are comparably small and influence of the background part on the full phase shift notably appears above ∼1600 MeV which describe the data very well. Removing the full phase shift of the D0 wave (Full phase), f2(1270), DOI: 10.1051/ , 07016 (2016) 130 EPJ Web of Conferences 13007016 MESON 2016 epjconf/2016

Many existing parameterizations of the ππ scattering amplitudes still significantly do not satisfy the crossing symmetry condition or do not describe the ππ threshold region.Since they are used in some analyses, having a correct determination of ππ scattering amplitudes is essential to achieve a proper result.
In our previous work [1] we presented a general method of refining the ππ amplitudes by fitting them to the GKPY equations [2,3] which are once-subtracted dispersion relations with imposed crossing symmetry condition.The method was demonstrated on refining of the unitary multi-channel S (ππ, K K and ηη) and P (ππ, ρ2π, and ρσ) wave amplitudes from [4] mostly below 1.1 GeV with no change in their original mathematical structure.In Ref. [5] we included the multi-channel D (ππ, effective (2π)(2π), K K and ηη) and F (ππ, effective (2π)(2π), ωπ and K K) wave amplitudes from [6].The S -F amplitudes were refined and then re-fitted to the available experimental data and to the GKPY equations [2,3].In Ref. [5] we presented some part of the results of analysis for the D and F wave amplitudes especially, details on formalism of the analysis and precise determination of ππ scattering amplitudes for D and F waves.In this paper we discuss on the resonances included in the analysis and show their importance separately in the corresponding partial waves.
The method of analysis is generally the same as in Ref. [1].To construct the D0 and F1 waves, we have used the same formalism as in Refs.[4,6] and updated the list of contributing resonant states for the D0 wave according to the latest issue of PDG [7].Then, a correct behavior of the phase shift near the threshold was controlled by the pseudo data (see Secs.II in [5]).The initial amplitudes for the S 0 and P1 waves were from [1], both with added polynomial near the threshold (the "extended" amplitudes in [1]).Since there is no any resonance for S 2 and D2 waves, the amplitudes for these isotensor waves were taken from [2] and kept unchanged during the analysis.
In the modification of the original amplitudes only those parameters that were expected to contribute significantly to the dispersive integrals, i.e. the low-energy resonances and the ππ background were allowed to change.The results provided precise determination of D0 and F1 partial wave amplitudes which satisfy the crossing symmetry condition and describe the experimental data very well.
In the construction of the partial waves, apart from background part which is relatively small, the resonance contribution described by the multi-channel Breit-Wigner form determines the main structure of the amplitudes.Concerning the dominant, important and insignificant resonances of the D0 wave in the multi-channel analysis approach, various fits have been performed.Results show that not all first four lightest states, i.e. f 2 (1270), f 2 (1430), f 2 (1525), and f 2 (1640), are the most important states in the analysis, as it was expected at the beginning.For this, Table 1 demonstrates the significance of each resonance in the analysis separately.It shows the χ 2 for the fits when one particular resonance was omitted.The χ 2 was only composed of the χ 2 Data for D0 wave in the Eq. ( 16) in [8] and χ 2 DR for all partial waves in the Eq. ( 17) in [8].Nevertheless, conditions are the same for all fits and they just differ by omitted resonances.Values of the χ 2 achieved from fits 9, 11 and 12 are almost equal to the value in the fit 1 where all resonance states are included.Accordingly, from the point of view of the χ 2 one can omit the f 2 (2010), f 2 (2300) and f 2 (2340) resonances and conclude that these resonances are insignificant in the description.Value of the χ 2 in the fits 3, 5, 7 and 8 show that the f 2 (1430), f 2 (1640), f 2 (1910) and f 2 (1950) don't play a very important role on the results but one should keep them.Additionally, the f 2 (1525), f 2 (1810), and f 2 (2150) resonances which strongly influence behavior of the amplitude and therefore value of the χ 2 are the most important resonances in the D0 wave and obviously the dominant resonance which dominate the behaviour of the amplitude is f 2 (1270) resonance.
From two resonance states ρ 3 (1690) and ρ 3 (1990) listed in PDG [7] for the isovector F1 wave only the ρ 3 (1690) resonance state has been considered in the analysis.For the ρ 3 (1990) resonance state, the partial widths are not known.In Refs.[5,8] we have found that enough is only one resonance state ρ 3 (1690) which can satisfactorily describe the available experimental data in [9,10] and give a reasonable description in the threshold region.
from which the f 2 (1270) is the most dominant one.This can be seen in the Fig. 1 where the f 2 (1270) determines structure of the full phase shift.Effect of other phase shifts are comparably small and influence of the background part on the full phase shift notably appears above ∼1600 MeV which describe the data very well.Removing the full phase shift of the D0 wave (Full phase), f 2 (1270), Finally for the F1 wave which has only one resonance (ρ 3 (1690)), description of the phase shift is much simpler than D0 wave. Figure 3 shows the influence of the resonance and background parts on the full phase shift.Due to the scattered data below ∼1700 MeV one can omit the background part, since only the ρ 3 (1690) phase shift is enough to describe the data.Effect of the background part is seen above ∼1700 MeV to describe the data.

Figure 3 .
Figure 3. Illustrative comparison of the resonance and background (Bkg.) with the full phase shift of the F1 wave (Full phase) and experimental data [9, 10].

Table 1 .
Values of χ 2 after fitting when some specific resonance states are omitted in the D0 wave. of the phase shift of the resonances in ππ → ππ channel is presented in the Figs.1-3 for D0 and F1 waves individually.The D0 wave consists of 11 resonances ( f 2 (1270), Bkg.) and experimental data in the Fig.1one can see the differences between other phase shifts in the Fig.2.Influence of these phases on the full phase shift is less than 1 degree.However, one can conclude that effect of the f 2 (1640), f 2 (1910), f 2 (1950), f 2 (2010) and f 2 (2150) phase shifts is almost negligible.Note that the scarce data above ∼1600 MeV indicate almost no resonant structure.Illustrative comparison of the phase shifts of the resonances in the D0 wave.The f 2 (1270) resonance is overlapped by sum of all resonances (Sum of all Res.) when background part (Bkg.) is omitted.Except f 2 (1270) contributions of other resonances are relatively very small (less than 1 degree).