Theoretical description of the $\boldsymbol{J/\psi \to \eta (\eta') h_1(1380)}$, $\boldsymbol{J/\psi \to \eta (\eta') h_1(1170)}$ and $\boldsymbol{J/\psi \to \pi^0 b_1(1235)^0}$ reactions

We have made a study of the $J/\psi \to \eta' h_1, \eta h_1$ (with $h_1$ being $h_1(1170)$ and $h_1(1380)$) and $J/\psi \to \pi^0 b_1(1235)^0$ assuming the axial vector mesons to be dynamically generated from the pseudoscalar-vector meson interaction. We have taken the needed input from previous studies of the $J/\psi \to \phi \pi \pi, \omega \pi \pi$ reactions. We obtain fair agreement with experimental data and provide an explanation on why the recent experiment on $J/\psi \to \eta' h_1(1380), h_1(1380) \to K^{*+} K^- +c.c.$ observed in the $K^+ K^- \pi^0$ mode observes the peak of the $h_1(1380)$ at a higher energy than its nominal mass.


I. INTRODUCTION
The axial vector mesons of low energy are emerging as a powerful source of information on hadron dynamics. In quark models [1,2], they are not so well reproduced as the corresponding vector mesons. Subsequent studies in the context of QCD, large N c behavior, combined with phenomenology, have shown that the vector mesons are largely qq objects [3]. However, this is not the case following the same investigation, for the low-lying axial vector mesons [4]. Actually these axial vector mesons have been studied within the context of the chiral unitary approach and have proved to be well described from the interaction of pseudoscalar and vector mesons [5][6][7][8] using potentials provided by the chiral Lagrangians [9]. Radiative decays of the axial vector mesons have been largely studied from different points of view [10][11][12], and more concretely from the point of view as being dynamically generated from the pseudoscalar-vector interaction [13,14]. One of the topics where axial vector mesons are shown to play an important role is in τ − decays. The τ − → ν τ π + π − π − decay has been shown to be dominated by the formation of the a 1 (1260) using different models [15][16][17], and concretely assuming the resonance to be largely made from the ρπ interaction [8]. Another τ − decay more recently studied is the τ − → ν τ π − f 1 (1285) which is investigated in Ref. [18] from the perspective of the Nambu-Jona-Lasinio model and in Ref. [19] from the perspective of the f 1 (1285) being dynamically generated from the K * K interaction. A generalization of this reaction to the general case of τ − → ν τ P A (P for pseudoscalar and A for axial vector meson) has been done in Ref. [20] and estimates based on vector meson dominance have been done before [21]. Other weak decays, as the B 0 s → J/ψf 1 (1285), have also been studied [22].
Strong decays of axial vector mesons have also been investigated. The three pion decay of the a 1 (1260) is studied in Ref. [23], the strong decays of the b 1 (1235) in Ref. [12], and the decay of f 1 (1285) → ρ 0 π + π − has been investigated in Ref. [24]. The f 1 (1285) is a particular case in chiral theory since it is generated from the single channel K * K − c.c. in s-wave [6,7].
J/ψ decays have also contributed to this field and studies have been done for the J/ψ → φKK * and J/ψ → φf 1 (1285) decays [38]. Another interesting reaction is the J/ψ → ηK * 0K * 0 reaction measured by the BES collaboration in Ref. [39], where the peak observed in the K * 0K * 0 invariant mass around the K * 0K * 0 threshold was interpreted in Ref. [40] as a manifestation of the h 1 (1830) axial vector meson predicted by Ref. [41]. The behavior of a 1 (1260) resonance in a nuclear medium from the perspective of its ρπ main component was also studied in Ref. [42].
Very recently the BESIII collaboration has reported the observation of the h 1 (1380) in the J/ψ → η ′ KKπ decay [43]. Selecting η or η ′ in the final state is a good filter for isospin and C-parity such that the extra meson produced is a I G (J P C ) = 0 − (1 +− ) state, assuming s-wave production. This corresponds to the quantum numbers of the h 1 axial vector meson.
In the present paper we study this reaction from the theoretical point of view. In the picture two h 1 states, corresponding to the h 1 (1170) and h 1 (1380), together with their companions f 1 , a 1 , b 1 axial vector mesons, are dynamically generated from the pseudoscalar-vector meson interaction. To produce η ′ h 1 (1380) we first produce η ′ plus an extra pseudoscalar and a vector meson, letting them propagate, using the chiral unitary approach to describe it, and the interaction generates the resonance. Thus, we have a primary transition J/ψ going to one vector and two pseudoscalars and one needs a theory for it. Lacking a description for such a complicated dynamics, a symmetry is invoked, considering the J/ψ to be an SU (3) flavor singlet. The two reduced matrix elements needed are taken from the former study of the J/ψ → ωππ and J/ψ → φππ reactions in Refs. [44,45]. As we shall see, we are able to provide reasonable rates for this reaction and for J/ψ → π 0 b 1 (1235) 0 , and at the same time we make predictions for the rates of the J/ψ → η ′ h 1 (1170) and J/ψ → ηh 1 (1170) (ηh 1 (1380)) reactions, which constitute extra test for the dynamical origin of the axial vector mesons and we hope are measured in the near future.

II. FORMALISM
In Ref. [6] the h 1 (1170) and h 1 (1380) resonances were generated, albeit with somewhat different mass, from the interaction of the channels 1 √ 2 (K * K −K * K), φη, ωη, ρπ. The η ′ V (η ′ -vector) states have too large mass and were not included in Ref. [6]. We shall not take the theoretical masses of Ref. [6], but use the experimental ones to be accurate in the phase space calculations. However, we shall take the couplings of the resonance to the different channels obtained in Ref. [6] which are rather stable with respect to changes in the input of the model. Given the nature assumed for these resonances as dynamically generated from the interaction of these channels, the mechanism to produce them is depicted in Fig. 1.
The first thing we need is to describe the J/ψ coupling to two pseudoscalars and one vector (P P V ). For this we rely on the work done in Ref. [46], where the experimental data on the χ c1 → ηπ + π − [47] were studied and the a 0 (980) and f 0 (980) signals were well reproduced in the ηπ and π + π − mass distributions, respectively. The basic assumption done in Ref. [46] was that J/ψ was an SU(3) flavor singlet state and that it coupled to the structure Trace(P P P ). Since this is not the only SU(3) singlet structure, in Ref. [48], where the η c → ηπ + π − reaction was studied, it was shown that other possible structures lead to results in flagrant conflict with experiment. Based on this finding we shall also take as the first step the structure where A 1 is a constant, P and V are the SU(3) qq matrices written in terms of the pseu- In the pseudoscalar-meson matrix P , the mixing of η and η ′ is taken into account following Ref. [49], and the ideal mixing is assumed for ω and φ in the vector-meson matrix V . From there we obtain where If we isolate the terms with η ′ , we obtain the coefficients W i for the different channels that we show in Table I.
In Table II we show the coefficients W i for the terms involving η from Trace(P P V ).
We shall also study the J/ψ → π 0 b 1 (1235) 0 reaction. According to Ref. [6] the b 1 (1235) couples to 1 √ 2 |K * K I=1 + |K * K I=1 , φπ, ωπ, ρη and the terms of the trace of P P V where a π 0 appears have weights which we show in Table III. It is interesting to see that with the coefficients W i of Table I for η ′ K * K or η ′K * K, we find the same coupling for the terms of the combination which is the K * K ,K * K combination with isospin I = 0 and C-parity negative that we obtain (K − = −|1/2, −1/2 , K * − = −|1/2, −1/2 , CK * + = −K * − ), as it corresponds to the h 1 state. We also find a common coupling to this state in Table II for channels with η, but Tables I and II we also see a common coupling to the terms of the combination which is our ρπ state with For the terms that contain a π 0 in Table III we find a common coupling to the terms of the combination which has I = 1 and C-parity negative, as it corresponds to the b 1 state.
In Tables IV and V  As one can see in Table IV, the couplings of h 1 (1170) to the φη and ωη channels are small. Then, we omit the contribution from these channels in the calculation of J/ψ → In view of Eqs. (8), (9) and (10), we can write the coupling of the resonance to an 1 Note that the couplings of Ref. [6] are calculated for 1 for h 1 and As a consequence, the couplings of h 1 to (K * K ) I=0 in Table IV have opposite sign to those in Ref. [6] while the coupling of b 1 to (K * K ) I=1 in Table V has the same sign.
individual channel as Then, apart from a factor A 1 ǫ J/ψ · ǫ V , the transition matrices are given by for h i , either of the two h 1 states, where the extra factor two in the ωη and φη channels in Eq. (13) comes from the identity of the two η particles in the channels. Finally, with R standing for b 1 . Prefactors 2 in front of W π 0 φπ 0 and W π 0 ωπ 0 appear from the identity of two π 0 .
In Eqs. (12), (13), and (14), G i , with i being K * K , φη, ωη, ρπ, φπ, ωπ, ρη, is the mesonmeson loop function. Here, G i is regularized with dimensional regularization as in Ref. [6], and for G i with K * or ρ the vector meson mass is smeared with the spectral function (see e.g. Ref. [50] for details), and the K * K loop function is averaged over the isospin after the With the amplitudes Eqs. (12), (13), and (14), the decay width of J/ψ → P R (P R = η (′) h 1 (1170), η (′) h 1 (1380), and π 0 b 1 (1235) 0 ) is obtained by with φηη ωηη π 0 φπ 0 π 0 ωπ 0 One attractive thing of the J/ψP P V Lagrangian considered is that it automatically fulfills the Okubo-Zweig-Iizuka (OZI) rule. As we can see in (P P V ) 33 , Eq. (7), the φππ channel does not appear. The φ, implicitly assumed there as a ss state, only couples to kaons or η, η ′ that contain s quarks. Actually the J/ψ → φππ decay is suppressed with respect to ωππ by one order of magnitude, but it is not zero. In Refs. [44,45] an extra Lagrangian was used, that in our new formalism can be cast as where Trace(P P ) =(P P ) 11 + (P P ) 22 + (P P ) 33 , with We shall call W ′ i the weights for the different channels discussed above. In Table VI, we show the weights W ′ i which appear in our calculation. To include these new terms we only need minor changes: i) Changes in Eq. (12): Ref. [45] Present work ii) Changes in Eq. (13): It is easy to establish the correspondence of A 1 , β to the parametersg and ν of Ref. [45] or the λ φ parameter of Ref. [44]. For this we choose the J/ψ → ωπ + π − , J/ψ → φπ + π − , J/ψ → ωK + K − , J/ψ → φK + K − transitions and take the weights from the Lagrangian of Eqs. (1) and (16) and compare them with the results of Ref. [45] (see in Eq. (18) of that work the term without rescattering, the term 1 in (1 + Gt)). We show these terms in Table VII, from where In addition, in Ref. [45] the relationship of ν to the parameter λ φ of Ref. [44] was found as Note that ν = 1 corresponds to β = 0 and is the case with the OZI respecting Lagrangian.

IV. RESULTS
In Table VIII we show the branching ratios that we obtain for the different reactions.
Let us see how we approach these issues.
In Table X, we show the different channels in K * K + c.c. and their decay modes, together with the weight of each mode given by the square of the corresponding isospin Clebsch-Gordan (CG) coefficient. In K * + K − + K * − K + measured in K + K − π 0 mode the weight is 1 2 · 1 3 of the weight for the decay in all channels. In K * K + c.c. measured in the K 0 S K ± π ∓ one counts 1 2 · 2 3 of the sum of the rates since the K 0 S K ± π ∓ is selected in the K * ± mode and there is an extra factor 1/2 reduction for measuring K 0 S rather than K 0 orK 0 . Thus the two η ] h 1 (1380) rates should be equivalent assuming isospin symmetry as indicated in Ref. [43]. Actually, within errors the two experimental rates are compatible, although some isospin violation is claimed in Ref. [43]. We will stick to the isospin symmetry for the vertices, but some isospin violation necessarily appears in the final results due to differences in the phase space due to different masses of the particles. However, in the present study we do not enter the issue of the isospin violation, and focus only on the K + K − π 0 mode.
The exercise done above indicates that we get a reduction of a factor of 6 from the rate obtained for J/ψ → η ′ h 1 (1380) in Table VIII, assuming it to be dominated by the K * K + c.c. channel. This makes the new rate much closer to experiment, but still larger by about a factor 2-2.5. We then conduct the investigation forward and perform two different evaluations: 1) J/ψ → η ′ K * K + c.c.: Here we consider the K * K in the final state explicitly as shown in Fig. 2. The differential width for this channel is given by with where we will take the masses for K * + K − .
The t matrix in Eq. (33) is given by witht with t J/ψ,η ′ h 1 given by Eq. (12) together with Eq. (22) and g R,(K * K ) I=0 taken from Table IV. We also take the mass and width of the h 1 (1380) from the PDG [26], M h 1 = 1407 MeV and Γ h 1 = 89 MeV. By taking the coupling g R,(K * K ) I=0 we are automatically summing all four final K * K +K * K channels. The ǫ J/ψ · ǫ K * coupling stems from the primary vertex ǫ J/ψ · ǫ V together with the ǫ V · ǫ h 1 from the h 1 → P V vertex [6], and the final ǫ h 1 · ǫ K * vertex, after summing over the polarizations of the intermediate V and h 1 . The sum and average over polarizations of ǫ J/ψ · ǫ K * in Eq. (36) gives unity.
Integrating Eq. (33) over the K * K invariant mass we should get a factor 6 times bigger rate than the one obtained projecting over the measured channels in Ref. [43]. Hence, we should divide the results obtained by a factor 6 for comparison with the experimental numbers.
2) J/ψ → η ′ h 1 (1380), h 1 (1380) → K * + K − , K * + → K + π 0 : We take into account this channel by looking at the mechanism of Fig. 3. Here we take into account explicitly the h 1 (1380) and K * + propagators, and hence, automatically the mass distribution of these two states. We have now the double differential mass distribution [51] where and t ′ is now given by wheret is given by Eq. (37) and the effective coupling g K * ,Kπ is defined below calculating the K * width, and where, once again, the ǫ J/ψ · p K factor appears after summing over the polarization of the internal vector and axial vector mesons. The factor 1/2 accounts for the coupling of the resonance to K * + K − , −g R,(K * K ) I=0 /2, and the (−) 1 √ 3 the CG coefficient of K * + → K + π 0 . Taking into account that the full width of the K * is given by and that withp K + given by Eq. (41), and Finally, let us recall that we have calculated J/ψ → η ′ K * + K − and then projected to K + K − π 0 , but to match the experiment we have to add the J/ψ → η ′ K * − K + projected to K + K − π 0 , which gives the same contribution. Thus, we have to multiply by a factor two the results obtained by means of Eq. (45).
In Fig. 4, we show the results for d 2 Γ/dM inv (K + K − π 0 )dM inv (K + π 0 ) as a function of    Next we integrate this differential cross section over M inv (K + π 0 ) in the interval [M K * − 2Γ 0 K * , M K * + 2Γ 0 K * ] and plot it in Fig. 5. What we see in Fig. 5 is that the peak of dΓ/dM inv (K + K − π 0 ) is shifted forwards higher energies than the nominal mass of the h 1 (1380). The reason for it is that in Eq. (45) there are two factors competing. One is the h 1 (1380) propagator that makes the distribution peak at the nominal mass of the h 1 (1380), and the other one is the K * propagator that becomes more on shell as the energy of K * + K − increases. The combination of these factors makes the peak of the distribution to appear at higher energies. This explains what is observed in the experiment, but the mass of the h 1 (1380) is the one appearing in its propagator and the measured peak in Ref. [43] should not be taken as the mass of the h 1 (1380). In other words, our fair reproduction of the mass distribution of Ref. [43] should be seen as a factor supporting the mass of the h 1 (1380) at the nominal PDG mass of 1407 MeV, which can be observed in other decay channels.
Next we plot in Fig. 6 the results for dΓ/dM inv (K * K ) divided by 6 obtained by means of Eq. (33) and we compare it with the results of Fig. 5 multiplied by two. Then we integrate dΓ/dM inv (K * K ) and compare these results with those of the previous method and the experiment of Ref. [43]. Figure 6 is instructive because there we do not have the K * propagator, only the h 1 one. In spite of that we observe that the peak of the K * K distribution also appears around 1425 MeV as in Fig. 5. Here, the reason for the displacement of the peak versus the nominal mass 1407 MeV must be found for the reduced phase space for the K * K decay since the threshold for K * K appears at 1385 MeV. It is then this threshold close to the nominal mass which makes the peak appear at higher energies than the nominal h 1 (1380) mass. Quantitatively we can see that the strength of the mass distribution of Fig. 5 multiplied by two is a bit smaller than that of Fig. 6, and it stretches at lower K * + K − invariant mass as a consequence of considering explicitly the mass distribution of the K * . It is rewarding that the two methods give very similar result but we should consider the method that takes into account the K * mass distribution as more accurate.
When we integrate the distributions over , we obtain the following branching ratios As we can see, we obtain with Method 2 a magnitude of 3.31 × 10 −4 a bit bigger than the experimental number, but this should be considered a reasonable success in view that we have not used any free parameters, and that we admit having uncertainties of the order of 10 − 20 % in the couplings that revert into uncertainties of 20 − 49 % in the branching ratios.
As mentioned before, in Table IX we made predictions for other decay models, one of them, the J/ψ → ηh 1 (1380), with a rate substantially smaller than for the other modes and which is tied directly to the assumption made of a dynamically generated h 1 (1380) resonance.
Certainly, the measurements of these decay modes will bring relevant information concerning the nature of the axial vector mesons.

VII. CONCLUSIONS
We have made a study of the J/ψ → η ′ h 1 and J/ψ → ηh 1 decays, with h 1 being h 1 (1170) and h 1 (1380), together with J/ψ → π 0 b 1 (1235) 0 . For this we have assumed that the axial vector mesons are dynamically generated from the pseudoscalar-vector meson interaction.
Using SU(3) symmetry with a small OZI violating term, which was determined previously in the study of J/ψ → φππ and J/ψ → ωππ reactions, we are able to obtain rates for these decays, which we compare with experiment. The comparison with the J/ψ → π 0 b 1 (1235) 0 data is fair. On the other hand, the comparison from the recent BESIII experiment [43] requires projection over the K * K channels, and then selecting K * + K − , K * − K + and furthermore looking into the K + K − π 0 or K 0 S K ± π ∓ modes. We have done that for the K + K − π 0 mode using two methods, one where we take the nominal mass of the K * assuming it as an elementary particle, and another one where the K * propagator, accounting for the K * mass distribution, is explicitly taken into account. The results obtained with both methods are similar, but the one accounting for the K * mass distribution is more accurate, and both produce a K * K distribution peaking at higher energies than the nominal h 1 (1380) mass, which justifies the results obtained in Ref. [43]. Once the integrations over the invariant masses are made, the final results for the branching ratios are in fair agreement with experiment considering errors. We also make predictions for modes not yet observed. We should stress that, apart from some uncertainty in the input used, from the study of the J/ψ → φππ, ωππ reactions, where no axial vector mesons were produced, we have no freedom in our approach. The predictions we made are tied to the nature assumed for the axial vector meson as dynamically generated from the pseudoscalar-vector meson interaction, and agreement with experiment should be seen as a factor in favor of this hypothesis. The investigation of the modes studied here, not yet measured, should be encouraged in this context.