The interference effects of multi-channel pion-pion scattering contributions to the final states of $\Psi$- and $\Upsilon$-meson family decays

It is shown that the basic shape of dipion mass distributions in the two-pion transitions of both charmonia and bottomonia states are explained by an unified mechanism based on the contribution of the $\pi\pi$, $K\overline{K}$ and $\eta\eta$ coupled channels including their interference.


I. INTRODUCTION
In the analysis of practically all available data on two-pion transitions of the Υ mesons from the ARGUS, CLEO, CUSB, Crystal Ball, Belle, and BaBar Collaborations -Υ(mS) → Υ(nS)ππ (m > n, m = 2, 3, 4, 5, n = 1, 2, 3) -the contribution of multi-channel ππ scattering in the final-state interactions is considered. The analysis, which is aimed at studying the scalar mesons, is performed jointly considering the above bottomonia decays, the isoscalar S-wave processes ππ → ππ, KK, ηη, which are described in our model-independent approach based on analyticity and unitarity and using an uniformization procedure, and the charmonium decay processes -J/ψ → φππ, ψ(2S) → J/ψππ -with data from the Crystal Ball, DM2, Mark II, Mark III, and BES II Collaborations.
The used formalism for calculating the di-meson mass distributions in the quarkonia decays is analogous to the one proposed in Ref. [11] for the decays J/ψ → φ(ππ, KK) and V ′ → V ππ (V = ψ, Υ) but with allowing for also amplitudes of transitions between the ππ, KK and ηη channels in decay formulas. There was assumed that the pion pairs in the final state have zero isospin and spin. Only these pairs of pions undergo final state interactions whereas the final Υ(nS) meson (n < m) remains a spectator. The amplitudes of decays are related with the scattering amplitudes s; indices m and n correspond to Υ(mS) and Υ(nS), respectively. The free parameters depend on the couplings of J/ψ, ψ(2S) and the Υ(mS) to the channels ππ, KK and ηη. The pole term in eq.(1) in front of T 21 is an approximation of possible φK states, not forbidden by OZI rules.
The amplitudes T ij are expressed through the S-matrix elements S ij = δ ij + 2i √ ρ 1 ρ 2 T ij where ρ i = 1 − s i /s and s i is the reaction threshold. The S-matrix elements are parameterized on the uniformization plane of the ππ scattering amplitude by poles and zeros which represent resonances. The uniformization plane is obtained by a conformal map of the 8-sheeted Riemann surface, on which the three-channel S matrix is determined, onto the plane. In the uniformizing variable used we have neglected the ππ-threshold branch point and allowed for the KK-and ηη-threshold branch points and left-hand branch point at s = 0 related to the crossed channels. The background is introduced to the amplitudes in a natural way: on the threshold of each important channel there appears generally speaking a complex phase shift. It is important that we have obtained practically zero background of the ππ scattering in the scalar-isoscalar channel. It confirms well our representation of resonances.
Studying the decays of charmonia and bottomonia, we investigated the role of the individual f 0 resonances in contributing to the shape of the dipion mass distributions. In this case we switched off only those resonances [f 0 (500), f 0 (1370), f 0 (1500) and f 0 (1710)], removal of which can be somehow compensated by correcting the background to have the more-or-less acceptable description of the multichannel ππ scattering. First, when leaving out before-mentioned resonances, a minimal set of the f 0 mesons consisting of the f 0 (500), f 0 (980), and f ′ 0 (1500) is sufficient to achieve a description of the processes ππ → ππ, KK, ηη with a total χ 2 /ndf ≈ 1.20. Second, from these three mesons only the f 0 (500) can be switched off while still obtaining a reasonable description of multi-channel ππ-scattering (though with an appearance of the pseudo-background) with a total χ 2 /ndf ≈ 1.43. In figures 1-3 the solid lines correspond to contribution of all relevant f 0 -resonances; the dotted, of the f 0 (500), f 0 (980), and f ′ 0 (1500); the dashed, of the f 0 (980) and f ′ 0 (1500).
It is shown that the dipion mass spectra in the above-indicated decays of charmonia and bottomonia are explained by the unified mechanism which is based on our previous conclusions on wide resonances [1,2] and is related to contributions of the ππ, KK and ηη coupled channels including their interference. It is shown that in the final states of these decays (except ππ scattering) the contribution of coupled processes, e.g., KK, ηη → ππ, is important even if these processes are energetically forbidden.