$J/\psi \to \gamma \pi\pi$, $\gamma\pi^0\eta$ reactions and the $f_0(980)$ and $a_0(980)$ resonances

X iv :1 90 9. 08 88 8v 1 [ he pph ] 1 9 Se p 20 19 The J/ψ → γππ, γπη reactions and the f0(980) and a0(980) resonances S. Sakai, 2, ∗ Wei-Hong Liang, 3, † G. Toledo, 5 and E. Oset 4, ‡ Department of Physics, Guangxi Normal University, Guilin 541004, China Institute of Theoretical Physics, CAS, Zhong Guan Cun East Street 55 100190 Beijing, China Guangxi Key Laboratory of Nuclear Physics and Technology, Guangxi Normal University, Guilin 541004, China Departamento de F́ısica Teórica and IFIC, Centro Mixto Universidad de Valencia CSIC, Institutos de Investigación de Paterna, Aptdo. 22085, 46071 Valencia, Spain Instituto de Fisica, Universidad Nacional Autonoma de Mexico, AP20-364, Ciudad de Mexico 01000, Mexico We study the J/ψ → γππ, γπη reactions from the perspective that they come from the J/ψ → φ(ω)ππ, ρπη reactions where the ρ, ω, and φ get converted into a photon via vector meson dominance. Using models successfully used previously to study the J/ψ → ω(φ)ππ reactions, we make determinations of the invariant mass distributions for ππ in the regions of the f0(500), f0(980) and for π η in the region of the a0(980). The integrated differential widths lead to branching ratios below present upper bounds, but they are sufficiently large for future check in updated facilities.

The extension of the J/ψ → φ(ω)f 0 (980) reaction to the J/ψ → γf 0 (980) should be equally clarifying concerning the nature of the scalar resonances. Actually, since the γ does not have given isospin, now the J/ψ → γa 0 (980) reaction is equally allowed and the comparison of the production rates introduces new elements to test productions from this molecular picture of scalar mesons.
Experimentally there are no data in the PDG [33] for these decay modes, and only upper limits exist, but same spectra that can be associated to f 0 (500) and f 0 (980) production is available in Ref. [34]. Theoretically there is already one paper making predictions [35]. The model used is the one of Ref. [24], where J/ψ → KK and the photon is emitted from the Kaons, together with related contact terms. The φ → KK coupling is substituted by the J/ψ → K + K − which is taken from experiment.
In the present approach we take a different point of view. We rely upon the models of Refs. [26,27] for J/ψ → φ(ω)ππ, which proved rather successful to interpret experimental data [36][37][38][39], and implement vector meson dominance (VMD) with φ(ω) → γ conversion [40,41]. Yet, the model has to be extended to include J/ψ → ρ 0 P P → γP P (with P the pseudoscalar meson) and we relate J/ψ → ρ 0 P P to J/ψ → φ(ω)P P implementing SU(3) symmetry in the primary production vertex J/ψ → V P P , assuming the J/ψ(cc) as an SU(3) singlet, in the same way as an ss state is assumed to be an isospin singlet. In this way, we find direct ρ 0 π 0 η production and ρ 0 P P with P P in isospin I = 1, which upon final state interaction produces the a 0 (980). The f 0 (980) and a 0 (980) are, thus, produced without isospin violation, given the fact that the photon carries no determined isospin. The rates obtained are below the upper experimental bounds but reachable in future experiments.

A. Primary vector-pseudoscalar-pseudoscalar production
In the J/ψ → φ(ω)ππ reaction studied in Ref. [26], a dominant OZI conserving and a subleading OZI violating terms were considered. An equivalent reformulation of the problem classifying the structures in terms of singlet and octet operators was given in Ref. [27] and continued in Ref. [42]. In the study of the J/ψ → η(η ′ )h 1 (1380) reaction done in Ref. [43], the same primary production vertex was assumed, yet, with a different, more intuitive and practical formulation. One starts assuming that J/ψ(cc) is a singlet of SU(3), in the same way that an ss state is a singlet of isospin SU (2). There are then several structures that are SU(3) singlet with two pseudoscalars and one vector: with ... standing for the SU(3) trace, where P and V stand for the SU(3) pseudoscalar and vector matrices corresponding to qq. This classification was already introduced in the study of the χ c1 → ηπ + π − and η c → ηπ + π − reactions [44,45], with a primary production of P P P . The P 3 structure was found completely off from data [46] and the P P P was the dominant one. Consequently, in the study of J/ψ → η(η ′ )h 1 (1380), the V P P , V P P structures were assumed, with the V P P being the dominant one, and a good agreement with data [47] was found.
The formalism was found equivalent to those of Refs. [26,27] and the weight of the V P P and V P P structures were found compatible with the results obtained in Refs. [26,27]. This finding by itself is important since in Refs. [26,27] the P P pair was allowed to propagate to generate the f 0 (980) state, while in Ref. [43] it was a P V pair that was allowed to propagate to generate the h 1 (1380) state, that, within the chiral unitary approach, is dynamically generated from the V P interaction [48][49][50]. That both processes are well described starting from the same primary V P P production, allowing either the P P to interact to form the f 0 (980), of the V P to produce the h 1 (1380), speaks much in favor of the dynamically generated nature of these resonances from the meson-meson interaction.
The P and V SU(3) matrices corresponding to qq are given by where in P the η-η ′ mixing of Ref. [51] has been assumed.
The terms going into V P P are given below.
From these terms we select those that have a ω, φ, ρ 0 to implement VMD and we get the weights, h i , for primary production of one vector and two pseudoscalars, h ρ 0 π 0 π 0 = 0; where we have multiplied by √ 2 the weights appearing directly from Eqs.(3)-(5) for π 0 π 0 , ηη production because we take into account the factor 2! for production of two identical particles and also use for convenience the unitary Concerning the V P P structure we have from where we get the weights and we have ignored the η ′ terms that do not play any relevant role in f 0 , a 0 production because of its big mass.
If we use the production vertex the production weights will be In the study of Ref. [43], two solutions for A and β were found, with one of them preferred, and consistent with Refs. [26,27]. We take this solution here corresponding to A = −g;g = 0.032; β = 0.0927.
As to the spin structure, following Refs. [26,27] we assume it to be of the type which was found consistent with the experimental information.

B. Vector meson dominance
Next we implement VMD by converting ρ 0 , ω, φ into a photon. For this we use the conversion Lagrangian V → γ [41] and the J/ψ → γP P starting state is depicted in Fig. 1.
Considering the V propagator in Fig. 1 and the conversion Lagrangian of Eq. (16), we find that VMD is implemented with the change where V is any of the ρ 0 , ω, φ vectors and

C. Final state interaction
Final state interaction is implemented letting the P P pair interact. This proceeds diagrammatically as shown in Figs. 2, 3, and 4.
Analytically we have where M inv is the π + π − invariant mass, G i are the loop functions of the intermediate states, i = π 0 π 0 , π + π − , K + K − , K 0K 0 , ηη, and t i,π + π − the scattering matrices of the chiral unitary approach [10]. With respect to Ref. [10] we introduce the ηη coupled channel and we find that in the cut-off regularization method q max = 600 MeV is required to fit phenomenology [53]. Similarly with i = ηη, K + K − , K 0K 0 .
Analogously, we have with The amplitude t i,π 0 η occurs now in I = 1 and the corresponding matrices, following Ref. [10], were evaluated in Ref. [54], again with a cut off q max = 600 MeV, which provides the clear cusp lineshape of the a 0 (980) as shown in the χ c1 → ηπ + π − reaction, both experimentally [46] and theoretically [44].
Taking P µ , K µ the momenta of the J/ψ and the photon, respectively, the most general structure for the reaction amplitude is given by with T µν = a g µν + b P µ P ν + c P µ K ν + d P ν K µ + e K µ K ν .
Gauge invariance (T µν K ν = 0) forces b = 0 and requires d = −a/(P · K). Furthermore the c and e terms do not contribute to t since ǫ ν (γ) K ν = 0. Hence only the a and d terms are needed and they are related by the former relationship. The former arguments are also used in Refs. [24,35] and the d coefficient is explicitly evaluated there. Here, we have explicitly calculated the a term, since our amplitudes go as ǫ µ (J/ψ) ǫ ν (γ)g µν . We can trivially incorporate the d structure of Eq. (25) into our framework, but it is unnecessary if we work in the Coulomb gauge (ǫ 0 (γ) = 0, ǫ · K = 0), which explicitly works with transverse photons. In this gauge the d term vanishes in the rest frame of the J/ψ, P = 0, since ǫ(γ) · P = 0. Thus, we evaluate the decay width in this frame and we only have to consider the terms from VMD that we have evaluated, with the condition that pol. Thus

III. RESULTS
We evaluate the differential width for where we introduce the normalization factor A of Eq. (13) and p γ is the photon momentum in the J/ψ rest frame andp π the π momentum in the π + π − rest frame, with t given by Eq. (19), and For π 0 π 0 in the final state the result is 1 2 of that of π + π − , assuming isospin symmetry, as we have done. Analogously for π 0 η in the final state we have the same differential width substitutingp π bỹ and t from Eq. (22), Let us recall that when we use the production amplitude of Eq. (12), V P P + β V P P , in Eqs. (20), (21) and (23) we must substitute h i by h i + βh ′ i . In this case in t φ of Eq. (21) we must also include π + π − , π 0 π 0 in the i sum over intermediate states.
Let us first look at π + π − production. In Fig. 5 we show dΓ/dM inv (π + π − ) as a function of the π + π − invariant mass. We find a neat peak for the f 0 (980), but we also find a sizeable strength in the region of the f 0 (500). In   Ref. [35] this region is not investigated and the ππ invariant mass distribution for γπ + π − is plotted from 700 MeV on. One can envisage a small contribution from the f 0 (500) since the KK channel considered in Ref. [35] couples strongly to f 0 (980) but weakly to f 0 (500). Conversely, the f 0 (500) couples strongly to ππ, ηη and weakly to KK. In our approach we have π + π − production in the f 0 (500) region through the coefficients h ωπ 0 π 0 , h ωπ + π − , h ωηη , h ′ ωπ 0 π 0 , h ′ ωπ + π − , h ′ ωηη , h ′ φπ 0 π 0 , h φπ + π − and h ′ φηη , and we obtain a sizeable contribution of π + π − production in the f 0 (500) region. It is interesting to compare the results of Fig. 5 with those of BESIII [34] 0 ++ mode of π 0 π 0 production. The results look qualitatively similar. Although our f 0 (980) peak is more prominent than the one in Ref. [34], we should note that we would have to take into account the experimental resolution to compare, that would flatten our peak. The best comparison is the ratio of areas below the peak of the broad f 0 (500) region and the f 0 (980). We However, we should note that the fraction of 0 ++ around the f 0 (980) is very small compared to the total, more than three orders of magnitude smaller, and these numbers should have necessarily large uncertainties.
one can think of it as responsible for a certain fraction of the J/ψ → γπ 0 η, non a 0 (980), decay reported in Ref. [57].
We next proceed to eliminate the tree level contributions to obtain what can be better compared with the experimental resonance contribution and with Ref. [35]. The results are shown in Fig. 7. We can see that the strength at the peak of the f 0 (980) is reduced by about a factor of 2. The strength of the f 0 (500) does not change much but the shape changes appreciably and the interference effect that made the strength zero around 940 MeV is no longer present. It is interesting to see that a similar interference shows up also in the experimental analysis of Ref. [34]. As to the a 0 (980) the strength at the peak is also reduced by a factor 2.2, but an apparent background from 700 MeV up to 950 MeV disappears.
To facilitate the comparison with Ref. [35], driven by KK production, we also take zero all h i except h V KK . The results are shown in Fig. 8. We can see that the peak of the a 0 (980) is further reduced by about a factor 2.4, and the  one of the f 0 (980) by about a factor 1.2. However, the most striking thing is that the f 0 (500) strength disappears totally. This indicates the relevance of the non KK original channels in producing the f 0 (500) strength. This is reminiscent of the picture found in the B 0 → J/ψπ + π − and B 0 s → J/ψπ + π − reactions [58], where the first reaction shows the clear f 0 (500) production and very small f 0 (980) production, while the second one produces clearly the f 0 (980) and no sign of the f 0 (500). This was naturally interpreted in Ref. [53], since after hadronization of a dd pair the first reaction produces mostly ππ but no KK, while the second produces mostly KK and no ππ after the hadronization of an ss pair.

IV. CONCLUSIONS
We have tackled the J/ψ → γπ + π − , γπ 0 η reactions by taking advantage of previous work on J/ψ → φ(ω)ππ(f 0 (980)) and J/ψ → η(η ′ )h 1 (1380). These decay modes might look disconnected but we proved that one can make predictions for one of them using experimental data from the other. The link stems from the fact that in a first step both reactions proceed creating one vector and two pseudoscalars. In the first case the two pseudoscalars interact to produce the f 0 (980) state and in the second case a vector and a pseudoscalar interact to produce the axial vector h 1 (1380). The interaction of these pairs of mesons is done using the chiral unitary approach, and the success of these studies gives further strength to the picture where these resonances are dynamically generated from the interaction of pairs of mesons.
As a next step we take the common picture of the V P P primary production in J/ψ → φ(ω)P P and J/ψ → ηV P , and by means of vector meson dominance we convert the J/ψ → ω(φ, ρ 0 )P P production into J/ψ → γP P production.
The next step consist in taking into account the P P final state interaction to produce π + π − , π 0 π 0 or π 0 η at the end, that produce peaks around the f 0 (500) and f 0 (980)(π 0 π 0 , π + π − ) and around the a 0 (980) (π 0 η). We find a distinct signal for both the a 0 (980) and f 0 (980) production and a broad distribution in the region of the f 0 (500). The a 0 (980) signal is found much larger than that of the f 0 (980) but still lower than the experimental upper bound. Yet, it is not much smaller than this bound, which gives us hopes that in future updates these decay modes will be observed.
We also mentioned that the f 0 (980) mode was apparently observed, but the small fraction of the total γπ 0 π 0 in this mode, together with the need to separate the different multipole contributions, makes advisable further looks with improved statistics and methods.
Having precise measurements of these decay modes will be an important complement to the φ → γππ, γπ 0 η reactions, helping us gain insight into the nature of the low mass scalar mesons.