Role of the $f_1(1285)$ state in the $J/\psi \to \phi \bar{K} K^*$ and $J/\psi \to \phi f_1(1285)$ decays

We study the role of the $f_1(1285)$ resonance in the decays of $J/\psi \to \phi \bar{K} K^*$ and $J/\psi \to \phi f_1(1285)$. The theoretical approach is based on the results of chiral unitary theory where the $f_1(1285)$ resonance is dynamically generated from the $K^* \bar{K} - c.c.$ interaction. In order to further test the dynamical nature of the $f_1(1285)$ state, we investigate the $J/\psi \to \phi \bar{K} K^*$ decay close to the $\bar{K} K^*$ threshold and make predictions for the ratio of the invariant mass distributions of the $J/\psi \to \phi \bar{K} K^*$ decay and the $J/\psi \to \phi f_1(1285)$ partial decay width with all the parameters of the mechanism fixed in previous studies. The results can be tested in future experiments and therefore offer new clues on the nature of the $f_1(1285)$ state.


I. INTRODUCTION
The f 1 (1285) resonance [I G (J P C ) = 0 + (1 ++ )] is an axial-vector state with mass M f1 = 1281.9±0.5 MeV and total decay width Γ f1 = 24.2 ± 1.1 MeV [1]. This state is described as a qq state within the quark model [2][3][4][5][6][7]. On the other hand, the f 1 (1285) is also suggested to be a dynamically generated state made from the single chan-nelKK * interaction in the chiral unitary approach [8]. As shown in Ref. [8], because the f 1 (1285) resonance has positive G parity, it cannot couple to other pseudoscalarvector channels. For reasons of parity it can also not decay into two pseudoscalar mesons. Thus, since the resonance is located below theKK * mass threshold, its observation is difficult in two body decays. Indeed, the main decay channels of the f 1 (1285) are 4π (branching ratio = 33%), ηππ (52%), and πKK (9%).
While Nature is probably more complicated and the f 1 (1285) state might have components of either type (see discussions in Ref. [9]), two comments are in order. First, the fact that states of different nature are possible does not mean that there should be a duplication of states with the same quantum numbers corresponding to each type of structure. The different structures mix and at the end it is a particular mixture what gives rise to the observed states. These features were well described in Refs. [10][11][12][13] for the σ (f 0 (500)) meson. One starts with a seed of qq and lets it couple to ππ components respecting unitarity of the ππ interaction. At the end, a physical state develops in which the original seed has been eaten up by the meson cloud, which becomes the dominant compo-nent of the wave function. The other comment is that, depending on the reaction, one or the other component will evidence itself more clearly, and in the present case, where we have aKK * produced at the end, it is quite clear that it is this component the one which will show up.
In Refs. [9,14], the decays of f 1 (1285) → ηπ 0 π 0 and f 1 (1285) → πKK were studied using the picture in which the f 1 (1285) is dynamically generated from the single channelKK * interaction. The theoretical predictions are compatible with the experimental measurements. Very recently, the production of the f 1 (1285) resonance in the reaction K − p → f 1 (1285)Λ within an effective Lagrangian approach was studied in Ref. [15] based on the results obtained in chiral unitary theory. The theoretical calculations are in agreement with the experimental data which provides further support for the molecular structure of the f 1 (1285) state.
In the present work, following the formalism of Ref. [8], we study the decays of J/ψ → φKK * and J/ψ → φf 1 (1285) with the picture that the f 1 (1285) resonance is dynamically generated from the single channelKK * interaction. This paper is organized as follows. In Sec. II, we discuss the formalism and the main ingredients of the model. In Sec. III, we present our main results and, finally, a short summary and conclusions are given in Sec. IV.

II. FORMALISM
We want to study the role of the f 1 (1285) state, which is dynamically generated by theK and K * interaction, in the J/ψ → φKK * decay. In the chiral unitary approach of Ref. [8], the f 1 (1285) resonance was obtained by solving the Bethe-Salpeter equation in theKK * channel to obtain the scattering amplitude where v is theKK * →KK * transition potential and G is the loop function for the propagators of theK and K * mesons given in Ref. [8].
The v and G depend on the invariant mass M inv of theKK * system, and hence the scattering amplitude t is also dependent on M inv . The loop function G is divergent, and it can be regularized both with a cutoff prescription or with dimensional regularization in terms of a subtraction constant [19]. In this work we will make use of the cutoff regularization scheme, which introduces a cutoff parameter q max . The cut off is tuned to get a pole of the t matrix at the mass (1281.3 MeV) of the f 1 (1285). This provides the coupling g f1 = 7555 MeV of the resonance to theKK * channel (see more details in Ref. [9]). With the explicit expressions for v and G taken from Ref. [8], we obtain a good description of the f 1 (1285) resonance using a cutoff q max = 990 MeV, as in Ref. [8].
For J/ψ → φKK * , the decay mechanism is shown in Fig. 1. To take into account the final state interaction of theKK * pair, we have to consider the resummation of the diagrams shown in the figure.
According to the diagrams in Fig. 1, the transition matrix for the process J/ψ → φKK * can be given by where the last equality follows from Eq. (1). The V P and C s are the bare production vertex and the spin structure (the spin of K * together with the one of the φ must give the spin of J/ψ: 1 1 → 1) factor for J/ψ → φKK * . We assume that this bare vertex is of a short range nature, i.e., just a coupling constant in the field theory language.
The spin structure of the J/ψ, K * , and φ coupling can be written as Summing and averaging C 2 s over final and initial polarizations of the vector mesons we find where p φ and and p K * are the φ and K * momenta in the J/ψ rest frame, respectively, where M φK is the invariant mass of φK system, and λ(x, y, z) is the Kählen or triangle function. We can easily get theKK * invariant mass spectrum for the J/ψ → φKK * as [20][21][22]: For a given value of M inv , the range of M φK is defined as, inv are the energies ofK and φ in theKK * rest frame.
On the other hand, if we are interested in the production of the f 1 (1285) resonance, the relevant mechanism is depicted diagrammatically in Fig. 2 and we have where the spin factor C ′ s is easily obtained. We must recall that the coupling of f 1 (1285) toKK * −c.c. is given by g f1 ε i (f 1 )ε i (K * ). Contracting the two ε(K * ) in the K * propagator in Fig. 2 we have Then, the partial decay width of J/ψ → φf 1 (1285) is given by with and p ′ φ is the φ meson momentum obtained in the J/ψ rest frame which is The chiral theory cannot provide the value of the constant V P in Eqs. (7) and (10), however, if we divide dΓ/dM inv by Γ J/ψ→φf1(1285) the constant V P is cancelled, and we can make precise predictions for the ratio R Γ as, This ratio is relevant because it has no free parameters (all the parameters are fixed by previous works) and, thus, it is a prediction of the theory. The shape, as well as the absolute values of the ratio R Γ for theKK * mass distribution, can be compared with the experimental measurements.

III. NUMERICAL RESULTS AND DISCUSSION
In Fig. 3, the numerical results of R Γ as a function of the invariant mass M inv of theKK * system are shown. The solid curve stands for the theory prediction and the dotted curve stands for the phase space. For evaluating the contributions of the phase space, we replace t(M inv )/v(M inv ) of Eq. (2) by a constant, thus removing any effect of the M inv dependence of the f 1 (1285) resonance. Then we tune this constant such that the M inv integrated R Γ in the range of energies from theKK * threshold to 1.7 GeV is the same as the one evaluated with the explicit resonance formalism.
In addition, in Fig. 3 we also show the results which are obtained without considering the spin structure factor by the dashed curve in Fig. 3. We see that the structure factor gives a small effect to our predictions and could be neglected. We see a clear threshold enhancement in Fig. 3 which is caused by the contributions of the f 1 (1285) state below threshold, which is dynamically generated by theKK * interaction. The theoretical predictions can be tested by future experiments.
Actually, the range of the invariant mass ofKK * in the decay of J/ψ → φKK * is from the threshold ofKK * up to 2.077 GeV (M J/ψ − m φ = 2077 MeV), however, we cannot go so far because the chiral theory works well about 200 − 300 MeV from the threshold, hence we con-sider only the range of 300 MeV above theKK * threshold as shown in Fig. 3.
One might think we should compare our theoretical result, R = M J/ψ −m φ mK +m K * R Γ dM inv , to the experimental result in Eq. (16), but, as discussed before, we take thē KK * →KK * scattering amplitude t(M inv ) from the chiral unitary approach, and we can not go too far from thē KK * threshold. Furthermore, there could be also other contributions from higher mass states with spin-parity J P = 1 + and 2 + at higher invariant mass region ofKK * . These higher states will not contribute too much to the lower energy region and hence will not affect our predictions here. On the other hand, note that the experimental results of Ref. [17] were obtained in the 1980s and only few signal events were observed. Further improvement can be done in the future at BESIII or BelleII. The future experimental observation of the mass distribution R Γ would provide very valuable information on the mechanism of the J/ψ → φKK * decay.

IV. SUMMARY
In summary, we have studied the decays of J/ψ → φKK * and J/ψ → φf 1 (1285) with the theoretical approach which is based on results of chiral unitary theory where the f 1 (1285) resonance is dynamically generated from the K * K − c.c. interaction. The ratio R Γ = dΓ J/ψ→φKK * /dMinv Γ J/ψ→φf 1 (1285) as a function of invariant mass M inv ofKK * is predicted. A clear threshold enhancement in Fig. 3 compared with the phase space appears, which is caused by the presence of the f 1 (1285) state below threshold . The experimental observation of this mass distribution would then provide very valuable information to check our predictions and the basic nature of the f 1 (1285) resonance.