Hints of the J PC = 0 −− and 1 −− K ∗ ¯ K 1 (1270) Molecules in the J/ψ → ϕηη ′ Decay

The primary objective of this study is to investigate hadronic molecules of K ∗ ¯ K 1 (1270) using a one-boson-exchange model, which incorporates exchanges of vector and pseudoscalar mesons in the t -channel, as well as the pion exchange in the u -channel. Additionally, careful consideration is given to the three-body effects resulting from the on-shell pion originating from K 1 (1270) → K ∗ π . Then the BESIII data of the J/ψ → ϕηη ′ process is fitted using the K ∗ ¯ K 1 (1270) scattering amplitude with J PC = 0 −− or 1 −− . The analysis reveals that both the J PC = 0 −− and 1 −− assumptions for K ∗ ¯ K 1 (1270) scattering provide good descriptions of the data, with similar fit qualities. Notably, the parameters obtained from the best fits indicate the existence of K ∗ ¯ K 1 (1270) bound states, denoted by ϕ (2100) and ϕ 0 (2100) for the 1 −− and 0 −− states, respectively. The current experimental data, including the η polar angular distribution, cannot distinguish which K ∗ ¯ K 1 (1270) bound state contributes to the J/ψ → ϕηη ′ process, or if both are involved. Therefore, we propose further explorations of this process, as well as other processes, in upcoming experiments with many more J/ψ events to disentangle the different possibilities.


I. INTRODUCTION
Exotic hadrons, which lie beyond the conventional quark model [1,2], have gained significant attention in the past two decades due to the observation of numerous exotic states or their candidates in experiments.Despite of extensive research on the structures and properties of these exotic states, many of them remain subjects of debate.We refer to Refs.[3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] for recent reviews on the experimental and theoretical status of exotic hadrons.One intriguing observation is that many of the observed peaks are located very close to the thresholds of hadron pairs that they can couple to.This proximity can be attributed to the S-wave attraction between the relevant hadron pair, as discussed in Ref. [21].Consequently, a natural interpretation for these states is the formation of hadronic molecules, as extensively reviewed in Refs.[3,8,14,17,19,20].
In Ref. [50], a total of 1.3×10 9 J/ψ events were used to investigate the decay process J/ψ → ϕη ′ η.Notably, an enhancement around 2.1 GeV was observed in the final states involving ϕη ′ .By incorporating a Breit-Wigner (BW) resonance with J P = 1 + or 1 − , the invariant mass distribution of ϕη ′ was well described, while the possibil-ity of J P = 0 − was ruled out based on the distribution of the η polar angle, which represents the angle between the outgoing η meson and the incoming e + e − beams in the rest frame of the J/ψ.However, we will later explain that the current data do not provide conclusive evidence to exclude the J P = 0 − possibility due to the significant contribution of the phase space (PHSP) processes derived from the experimental Monte Carlo simulations.
In the Review of Particle Physics (RPP) [51], there are two K 1 particles, namely K 1 (1270) and K 1 (1400).Given that the observed enhancement in J/ψ → ϕη ′ η is slightly below the threshold of K * K1 (1270), it is reasonable to investigate whether the ϕη ′ invariant mass distribution can be explained by the presence of K * K1 (1270) molecular states.In the following analysis, we will use K 1 to refer to K 1 (1270) unless otherwise specified.
The flavor wave function of the K * K1 state with specific J P C can be expressed as where J 1 represents the spin of K * , J 2 represents the spin of K 1 , and C refers to the charge conjugation operator.Using the following phase conventions for the charge conjugation transformation, we have In order to assess the exchanges of the vector meson (V ) and pseudoscalar meson (P ) between K * and K1 in the t-channel, the Lagrangian of K * K * V /P coupling is needed.From the hidden local symmetry formalism, the relevant Lagrangians can be constructed as [52][53][54] where and ⟨• • • ⟩ means the trace in flavor space.The coupling constant g is expressed as g = m V /(2F π ) where m V represents the mass of the vector meson ρ and F π = 92.4MeV is the pion decay constant.The coupling constant G ′ is expressed as Expanding Eqs. ( 5) and ( 6), we obtain the following K * K * V /P couplings, with and τ are the Pauli matrices in the isospin space.
We assume that the K 1 K 1 V /P couplings have the same form as the K * K * V /P couplings, with We further assume that the coupling constants g 1 and G ′ 1 should be of the same order as g and G ′ , respectively.As a result, we opt to set g 1 = g and G ′ 1 = G ′ in the following calculations.This would be the case in the massive Yang-Mills model for vector mesons [52].We have verified that any deviation of approximately 20% in g 1 and G ′ 1 can be adequately accommodated by varying the cutoff to be introduced later.
Using the above Lagrangian, we obtain the potentials from t-channel meson exchanges in momentum space, V V (0) where A (0) and µ is the reduced mass of K * K1 and q = k − k ′ is the three-momentum of the exchanged π with k and k ′ the three-momenta of the incoming and outgoing particles in the center-of-mass (c.m.) frame, respectively.The superscripts (1) and (0) represent the results in the 1 −− and 0 −− cases, respectively.The flavor factors are

The constant components of the potentials, A
(1/0) V /P , will be rewritten as two scale-dependent parameters [44,56], P , (30) P , (31) which will serve as counterterms to absorb the cutoff (Λ) dependence as will be explained later.We will take C (1)  and C (0) as free parameters to be fitted.
The S-wave K 1 K * π coupling can be expressed as1 where the coupling constant g S = 3.4 is determined by the partial decay width of K 1 → K * π.Note that we have ignored a possible D-wave contribution.Utilizing the Lagrangian in Eq. ( 32), we obtain the potential for the u-channel π exchange as where q represents the four-momentum of the exchanged pion.

B. Lippmann-Schwinger Equation
The scattering amplitude can be obtained by solving the Lippmann-Schwinger Equation (LSE), where k and k ′ are the three-momenta of the initial and final states in the c.m. frame, in order, µ is the reduced mass of K * K1 , and E is the energy relative to the threshold.The energy-dependent width Γ(E; l) is the sum of the widths of K * and K 1 .The integral is ultraviolet divergent and it is regularized by introducing a Gaussian form factor, where Λ is the cutoff parameter.The effects of the variation of Λ can be absorbed by adjusting the value of C (1)  or C (0) introduced in Eqs.(30,31).
After the S-wave projection, the LSE in Eq. ( 35) is reduced to with k, k ′ and l the magnitudes of the corresponding three-momenta.We would like to emphasize that the S-wave projection of the potential is nontrivial, and it will result in additional cuts to the scattering amplitude [44,57].This introduces significant complexity, particularly for the u-channel π exchange.Further details can be found in Appendix A. The K * dominantly decays into Kπ with a decay width around Γ K * = 50 MeV [51].The total decay width of the K 1 is (90 ± 20) MeV and the branching ratio of K 1 → K * π is (21 ± 10)% [51]. 2 For simplicity, in Γ(E; l) we only include the energy dependence of the partial width of K 1 → K * π since this process contribute to the pion exchange between K * and K1 in the u-channel.Explicitly, we have where Γ cons K1 = 71 MeV is the decay width of K 1 apart from the K * π channel, is the invariant mass of K * π from the K 1 decay, and q eff (E; l) is the momentum of the π in the rest frame of K 1 , determined by 2 The central values are used in the following calculations.The function yields the momentum of m 1 in the rest frame of M in the decay process of M → m 1 m 2 and λ(x, y, z) = x 2 + y 2 + z 2 − 2xy − 2yz − 2zx is the Källén triangle function.The K * π loop in the K 1 propagator introduces an additional cut, which is represented by a three-body cut extending from the K * K * π threshold to infinity.To ensure a smooth crossing of this cut when searching for poles in the complex energy plane, the cut of the square root function in Eq. ( 42) is defined along the negative imaginary axis [44,57,58].
The S-wave K * K1 system can couple to both the ϕη ′ and ϕη final states.In the following analysis, we consider the interaction between the ϕη, ϕη ′ and K * K1 coupled channels, which are labeled as channel 1, 2 and 3, respectively.The scattering amplitudes are described by the coupled-channel LSE, where G kk represents the loop function of the twoparticle propagators of channel k.The term V 33 is the K * K1 potential obtained in Sect.II A. Since the J P C of the system is either 1 −− or 0 −− , the ϕη (′) must be in P -wave.Therefore we neglect the interaction between ϕη (′) which is not expected to alter the existence of the K * K1 molecular states.Similarly, we expect V 31 and V 32 to be small and treat them in perturbation theory.Consequently, the potential matrix reads where v 31 and v 32 are constants, qη (′) represents the three-momentum of η (′) in the ϕη (′) c.m. frame.We thus have Upon disregarding the O V 2 31 , V 2 32 terms, it becomes apparent that T 33 corresponds to the single-channel K * K1 scattering amplitude, which has been derived in Eq. (37).The process of K * K1 → ϕη (′) inelastic scattering can be approximated as Here, both T 33 and V 33 are known, and the constants v 31 and v 32 can be absorbed into the normalization constant of the experimental data during the fitting process.
From the results obtained in Ref. [50], the contribution of the f 0 (1500) in ηη ′ can be considered negligible.Consequently, the amplitude of J/ψ → ϕη ′ η can be represented as follows, where q η ′ denotes the three-momentum of the η ′ in the J/ψ rest frame in Fig. 1(b), whereas q η denotes the threemomentum of the η in the J/ψ rest frame in Fig. 1(c).P a , P b and P c are constants that represent the production parameters of ϕη ′ η, K * K1 η ′ and K * K1 η in the decay of J/ψ, respectively.Note that we introduce the additional momentum q η ( ′ ) due to the fact that the η ′ in Fig. 1(b) is in P -wave, so is the η in Fig. 1(c).The P a term represents the production of the P -wave η and ϕη ′ , which is in fact a higher order term.The leading contribution from the S-wave ϕη ′ η production leads to a constant contact term and is covered by the background to be introduced in Eq. (50).The loop propagator of the K * K1 channel reads where the cutoff Λ 1 is fixed to 1 GeV.The Λ 1 -dependence of the physical results will be absorbed by the production parameters.
The differential decay width of J/ψ is now expressed as We have introduced a noninterfering background term αf bg (M ϕη ′ ), where f bg (M ϕη ′ ) mimics the lineshape of the PHSP process determined by the Monte Carlo simulation in Ref. [50].The parameter α represents the magnitude that needs to be fitted.It can come from a purely Swave production term, which does not interfere with the P -wave ones in Eq. (48).
Only the invariant mass distribution of ϕη ′ is published in Ref. [50] where the enhancement near 2.1 GeV was described by a BW resonance.Since the ϕη invariant mass distribution was not reported, we will first try to fit the data in Ref. [50] by considering only the K * K1 rescattering in the ϕη ′ channel, i.e., P b is fixed to 0. The reconstruction of the η ′ in Ref. [50] involves two modes, where the η ′ is reconstructed by γπ + π − and ηπ + π − , respectively.These two data sets are simultaneously fitted using the same differential decay width, as shown in Eq. ( 50).However, to take into account the efficiency difference between these two modes, a normalization factor β is introduced.In total, there are 5 free parameters: P a , P c , α, β and C (1) or C (0) .The parameters obtained from the best fits are listed in Table I with P b = 0 fixed and the fitting results are shown in Figs. 2 and 3. We can see that both assumptions, J P C = 1 −− and 0 −− , provide a satisfactory description of the data.With C (1) and C (0) from the best fits, the pole positions of the K * K1 molecules are determined to be (2079 − 68i) MeV for 1 −− , denoted by ϕ(2100) and (2091 − 61i) MeV for 0 −− , denoted by ϕ 0 (2100).
The quantum numbers of the introduced resonance were analyzed in Ref. [50] by examining the η polar angular distribution.If the quantum numbers J P of the introduced resonance in the ϕη ′ channel are 1 + , 1 − , or 0 − , the η polar angular distribution is proportional to 1, 1 + cos 2 θ, or sin 2 θ, respectively.It is found in Ref. [50] that both the assumptions of J P = 1 + and 1 − for the resonance in the ϕη ′ channel can describe the data, with the former being more preferred.However, the assumption of J P = 0 − was excluded as it seemed to deviate significantly from the data in the analysis of Ref. [50].
It is important to note that the contribution of the PHSP process, in both Ref. [50] and our fit result, are much larger than that of the introduced resonance.However, the η polar angular distribution from the PHSP process is not settled based on the published data in Ref. [50], and it is not necessarily the same as that of the resonance.The authors in Ref. [50] did not consider the contribution of the PHSP process to the η polar angular distribution.Here we assume that the η polar angular distribution from the PHSP process is flat as a consequence of the S-wave nature of the background term in Eq. ( 50), the total η polar angular distribution can be predicted as follows: where α1 = 0.815 and α0 = 0.835 represent the fraction of the PHSP process obtained in our 1 −− and 0 −− fits, respectively.The comparison between the predictions in Eq. ( 51) and the data are shown in Fig. 4. From Fig. 4 it is evident that both 1 −− and 0 −− assumptions provide TABLE I.The parameters from the best fits together with the pole positions, E b , relative to the K * K1 threshold at 2145 MeV.
1 −− 1 (fixed) −1.6 ± 0.9 0 (fixed) 37.0 ± 2.9 1.44 ± 0.06 0.28 ± 0.02 −2.5 ± 4. a satisfactory description of the data.Consequently, we cannot definitely conclude whether the resonance signal is from the ϕ(2100), ϕ 0 (2100), or that both manifest in the J/ψ → ϕη ′ η decay.To address this, we propose to conduct an analysis of this process using the complete set of J/ψ events recorded by the BESIII detector [59], which is one order of magnitude larger than the sample size utilized in the previous study [50].The difference of the two curves in Fig. 4 may be disentangled with the full dataset.Furthermore, performing a partial wave analysis on the polar angular distribution of the η, as well as the helicity angular distribution of ϕη ′ , one may be able to ascertain the presence of the 1 −− and 0 −− K * K1 bound states.The information obtained from the ϕη channel is also of great value, as the K * K1 molecules can also decay into ϕη.
C. K * K1 in both ϕη ′ and ϕη channels In this subsection we try to include the contribution of K * K1 from the ϕη channel by letting P b free.As discussed before and confirmed by the fits in the previous subsection, the P a term is of higher order.To reduce the number of parameters, we fix P a = 0 in the following calculation.We still have 5 free parameters in total, P b , P c , α, β and C (1) or C (0) .The parameters from the best fit are listed in Table I with P a = 0 fixed.The fitting results are shown in Figs. 5 and 6.The resulting pole positions of the ϕ(2100) and the ϕ 0 (2100) hardly change.
The ϕη invariant mass distributions of J/ψ → ϕη ′ η decay are also predicted by the following expression, where only the contribution of the ϕ(2100) or the ϕ 0 (2100) is included since the lineshape of the PHSP contribution in the ϕη invariant mass distribution is not available from the published data in Ref. [50].The predicted ϕη invariant mass distribution is shown in Fig. 7, and one sees a peak near 2.1 GeV.In fact, from the Dalitz plot reported by the BESIII Collaboration [50], there seems a accumulation of events at m ϕη ≃ 2.1 GeV.

IV. SUMMARY
The interaction between K * and K1 has been investigated in the OBE model, where t-channel vector meson and pseudoscalar meson exchanges are taken into account.There are two parameters in the potential to FIG. 3. The best fit of the J/ψ → ϕη ′ η data [50] with 0 −− K * K1 rescattering only in the ϕη ′ channel (P b = 0).See the caption of Fig. 2. FIG. 4. The η polar angular distribution in J/ψ → ϕη ′ η.The data are taken from Ref. [50] and the lines are the predictions from the best fits shown in Figs. 2 and 3.
absorb the cutoff dependence of physical observables and they can be determined by experimental data.Additionally, the u-channel π exchange, which plays a crucial role in the decay width of the K * K1 molecule, is also considered, where the three body effects of K * K * π are carefully examined.
In order to investigate the cause of the observed enhancement near 2.1 GeV in the ϕη ′ final states in the J/ψ → ϕη ′ η process [50], we conduct a fit analysis of the invariant mass distribution of ϕη ′ .The inclusion of the K * K1 channel in the analysis yields satisfactory results, as both the ϕ(2100) and the ϕ 0 (2100) are able to adequately describe the data.However, it is difficult to determine whether one or both of these states contribute to the J/ψ → ϕη ′ η decay, even when considering the η polar angular distribution.It is worth noting that the K * K1 bound states are also capable of decaying into ϕη.There-fore, valuable insights into the K * K1 bound states can be obtained by analyzing the invariant mass distribution of ϕη, which is predicted in Fig. 7.The Dalitz plots in Ref. [50] reveal an accumulation of data within the range of [4, 4.5] GeV 2 in the ϕη final states.Consequently, we propose conducting a study on the J/ψ → ϕη ′ η decay using the entire dataset of J/ψ events collected by BE-SIII [59], which is approximately eight times larger than the dataset used in Ref. [50].By performing a partial wave analysis of the polar and helicity angular distributions, one may be able to disentangle the contribution of ϕ(2100) and ϕ 0 (2100) to the J/ψ → ϕη ′ η decay.Furthermore, other decays of J/ψ into ηK K, ηK * K * , ηK K * and ϕηη can also be explored to study the resonance(s) around 2.1 GeV.While the ϕ(2100) should contribute to all these processes, the ϕ 0 (2100) can only couple to the last two.
served in experiments.Such amplitude is obtained by taking the integral path of z in Eq. (A1) to be [−1, 1] along the real z axis.In practice, when the pole of the integrand, say z 0 , is close to the integral path [−1, 1], we deform the integral path away from the pole for better numerical performance.On the other hand, when searching for poles on the complex energy plane, E should move from the physical axis to the pole position so that this pole has direct influences on the amplitude on the physical axis.In this process, z 0 may also move and possibly cross the integral path [−1, 1], which will result in a discontinuity, namely a cut, of the amplitude.To cross this cut continuously, the integral path should be deformed accordingly to avoid the cross with the trajectory of z 0 .This is the main logic to obtain the amplitude that is connected to the physical axis directly.
We will show how to choose the integral path for the S-wave projection of the potential from t-channel and uchannel meson exchanges in the following.Note that when calculating the amplitude on the physical axis, all the off-shell, half-on-shell and on-shell potentials are needed while when searching for poles on the physical Riemann sheet,3 only the off-shell potentials are relevant.Therefore, we can pay attention just to the trajectory of z 0 when varying E for the off-shell potential.

t-channel meson exchange
The potential from the t-channel meson exchange takes the form of and the pole position of the integrand reads For the off-shell potential where k, k ′ ≥ 0, z 0 is independent of E and lies beyond the integral path [−1, 1].For the on-shell or half-on-shell potential, z 0 has an imaginary part due to the finite width of K * and K 1 , and hence z 0 is also far from the integral path [−1, 1].Therefore, the integral path of the t channel meson exchange needs no deformation.

u-channel π exchange
The analytical structure is much more complicated for the u-channel π exchange.In the LSE, the propagator of the exchanged π in Eq. ( 32) should be rewritten as where with √ s the total energy, m 1 = m K * − iΓ K * /2, m 2 = m K1 − iΓ K1 /2 and W (m, q) = q 2 + m 2 .
When performing the S-wave projection for the onshell and half-on-shell potentials, the poles of the integrand may come from the roots of d 1 and W (m π , q), denoted by z 0d and z 0π , respectively.We should first determine the pole positions and then deform the integral path accordingly to make the poles not so close to the integral path, as shown in Fig. 8. Precisely, we define a region M where −1 <Re[z]< 1 and −0.3 <Im[z]< 0.3, and then choose the integral path properly by determining if z 0d and z 0π belong to M .Specific situations can be divided into several cases: