Effect of the $\eta\eta$ channel and interference phenomena in the two-pion transitions of charmonia and bottomonia

The basic shape of di-pion mass spectra in the two-pion transitions of both charmonia and bottomonia states is explained by an unified mechanism based on contributions of the $\pi\pi$, $K\overline{K}$ and $\eta\eta$ coupled channels including their interference. The role of the individual $f_0$ resonances in shaping the di-pion mass distributions in the charmonia and bottomonia decays is considered.


I. INTRODUCTION
In this contribution we report on new results in continuation of our study of scalar meson properties analyzing jointly the data on the isoscalar S-wave processes ππ → ππ, KK, ηη and on the two-pion transitions of heavy mesons, when it is reasonable to consider that the two-pion pair is produced in the S-wave state and the final meson remains a spectator. We analyzed data on the charmonium decay processes -J/ψ → φππ, ψ(2S) → J/ψππ -from the Crystal Ball, DM2, Mark II, Mark III, and BES II Collaborations and 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). Moreover, the contribution of multi-channel ππ scattering (namely, ππ → ππ, KK, ηη) in the final-state interactions is considered. The multi-channel ππ scattering is described in our model-independent approach based on analyticity and unitarity and using an uniformization procedure. A novel feature in the analysis is accounting for effects from the ηη channel in the indicated two-pion transitions which is assumed not only kinematically, i.e., via including the channel threshold in the uniformizing variable, but also by adding the ππ → ηη amplitude in the corresponding formulas for the decays.
We showed that the experimentally observed interesting (even mysterious) behavior of the ππ spectra of the Υfamily decays, which starts to be apparent in the second radial excitation and is also seen for the higher states,a bell-shaped form in the near-ππ-threshold region, smooth dips at about 0.6 GeV in the Υ(4S, 5S) → Υ(1S)π + π − , about 0.45 GeV in the Υ(4S, 5S) → Υ(2S)π + π − , and at about 0.7 GeV in the Υ(3S) → Υ(1S)(π + π − , π 0 π 0 ), and also sharp dips near 1 GeV in the Υ(4S, 5S) → Υ(1S)π + π − -can be explained by the interference between the ππ scattering amplitudes, KK → ππ and ηη → ππ, in the final-state re-scattering (by the constructive interference in the near-ππ-threshold region and by the destructive one in the dip regions). Note that in a number of works (see, e.g., [1] and the references therein) various assumptions were made to explain this observed behavior of the di-meson mass distributions.
The di-meson mass distributions in the quarkonia decays are calculated using a formalism analogous to that proposed in Ref. [12] for the decays J/ψ → φ(ππ, KK) and V ′ → V ππ (V = ψ, Υ) which is extended with allowing for amplitudes of transitions between the ππ, KK and ηη channels in decay formulas. It is assumed that the pion pairs in the final state have zero isospin and spin. Only these pairs of pions undergo the final state interactions whereas the final Υ(nS) meson (n < m) remains as a spectator. The decay amplitudes are related with the scattering amplitudes F Υ(mS) → Υ(nS)ππ = e (mn) 1 where s; indices m and n correspond to Υ(mS) and Υ(nS), respectively. The free parameters α 2 , depend on the couplings of J/ψ, ψ(2S), and 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 where ρ i = 1 − s i /s and s i is the reaction threshold. The S-matrix elements are taken as the products where S res represents the contribution of resonances, S B is the background part. The S res -matrix elements are parameterized on the uniformization plane of the ππ-scattering S-matrix element 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 [13] 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. Resonance representations on the Riemann surface are obtained using formulas in Table I [ 14], expressing analytic continuations of the S-matrix elements to all sheets in terms of those on the physical (I) sheet that have only the resonances zeros (beyond the real axis), at least, around the physical region. In Table I the Roman numerals denote the Riemann-surface sheets, the superscript I is omitted to simplify the notation, det S is the determinant of the 3 × 3 S-matrix on sheet I, D αβ is the minor of the element S αβ , that is, 12 , D 12 = S 12 S 33 − S 13 S 23 , D 23 = S 11 S 23 − S 12 S 13 , etc. These formulas show how singularities and resonance poles and zeros are transferred from the matrix element S 11 to matrix elements of coupled processes.
The background is introduced to the S B -matrix elements 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. First, this confirms well our assumption (5). Second, this shows that the representation of multi-channel resonances by the pole and zeros on the uniformization plane given in Table 1 is good and quite sufficient. This result is also a criterion for the correctness of the approach.
In Table II we show the poles corresponding to f 0 resonances, obtained in the analysis. Generally, the wide multichannel states are most adequately represented by poles, because the poles give the main model-independent effect of resonances and are rather stable characteristics for various models, whereas masses and total widths are very  model-dependent for wide resonances [16]. The masses, widths, and the coupling constants of resonances should be calculated using the poles on sheets II, IV and VIII, because only on these sheets the analytic continuations have the forms (see Table I): ∝ 1/S I 11 , ∝ 1/S I 22 and ∝ 1/S I 33 , respectively, i.e., the pole positions of resonances are at the same points of the complex-energy plane, as the resonance zeros on the physical sheet, and are not shifted due to the coupling of channels.
Further, since studying the decays of charmonia and bottomonia, we investigated the role of the individual f 0 resonances in contributing to the shape of the di-pion mass distributions in these decays, firstly we studied their role in forming the energy dependence of amplitudes of reactions ππ → ππ, KK, ηη. 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 (maybe, with elements of the pseudo-background) to have the more-or-less acceptable description of the multi-channel ππ scattering. Below we therefore considered description of the multi-channel ππ scattering for two more cases: • first, when leaving out a minimal set of the f 0 mesons consisting of the f 0 (500), f 0 (980), and f ′ 0 (1500), which is sufficient to achieve a description of the processes ππ → ππ, KK, ηη with a total χ 2 /ndf ≈ 1.20.
• Second, from the above-indicated three mesons only the f 0 (500) can be omitted while still obtaining a reasonable description of multi-channel ππ scattering (though with appearance of a pseudo-background) with the total χ 2 /ndf ≈ 1.43.
A satisfactory description of all considered processes (including ππ → ππ, KK, ηη) was obtained with the total χ 2 /ndf = 736.457/(710 − 118) ≈ 1.24; for the ππ scattering, χ 2 /ndf ≈ 1.15. Results for the distributions are shown in Figs. 2-4 with the same notation as in Fig. 1. Here the effects of omitting some resonance are more apparent than in Fig. 1.
It is shown that the di-pion 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 [2,3] 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.
Accounting for the effect of the ηη channel in the considered decays, both kinematically (i.e. via the uniformizing variable) and also by adding the ππ → ηη amplitude in the formulas for the decays, permits us to eliminate nonphysical (i.e. those related with no channel thresholds) non-regularities in some ππ distributions, which are present without this extension of the description [15]. We obtained a reasonable and satisfactory description of all considered ππ spectra in the two-pion transitions of charmonium and bottomonium.
It was also very useful to consider the role of individual f 0 resonances in contributions to the di-pion mass distributions in the indicated decays. For example, it is seen that the sharp dips near 1 GeV in the Υ(4S, 5S) → Υ(1S)π + π − decays are related with the f 0 (500) contribution to the interfering amplitudes of ππ scattering and KK, ηη → ππ processes. Namely consideration of this role of the f 0 (500) allows us to make a conclusion on existence of the sharp dip at about 1 GeV in the di-pion mass spectrum of the Υ(4S) → Υ(1S)π + π − decay where, unlike Υ(5S) → Υ(1S)π + π − ,  the scarce data do not permit to draw such conclusions yet. Also, a manifestation of the f 0 (1370) turned out to be interesting and unexpected. First, in the satisfactory description of the ππ spectrum of decay J/ψ → φππ, the second large peak in the 1.4-GeV region can be naively explained as the contribution of the f 0 (1370). We have shown that this is not right -the constructive interference between the contributions of the ηη and ππ and KK channels plays the main role in formation of the 1.4-GeV peak. This is quite in agreement with our earlier conclusion that the f 0 (1370) has a dominant ss component [2].
On the other hand, it turned out that the f 0 (1370) contributes considerably in the near-ππ-threshold region of many di-pion mass distributions, especially making the threshold bell-shaped form of the di-pion spectra in the decays Υ(mS) → Υ(nS)ππ (m > n, m = 3, 4, 5, n = 1, 2, 3). This fact confirms, first, the existence of the f 0 (1370)