Studying $X(2100)$ hadronic decays and predicting its pion and kaon induced productions

The newly observed $X(2100)$ by the BESIII Collaboration inspires our interest in studying the light meson system, especially axial-vector mesons. Since the $X(2100)$ has $J^P=1^+$ possibilities but cannot be distinguished only by mass, we make use of flux-tube model to study the strong decay behavior of $X(2100)$ under this assignment. The experimental width of the newly reported $X(2100)$ can be reproduced in our calculation, which favors an assignment of $X(2100)$ as the second radial excitation of $h_1(1380)$ with $I(J^P)=0(1^+)$. And the $\mathcal{B}(X(2100)\to \phi \eta^\prime)$ has a sizable contribution to the total width. Furthermore, we focus on the production of $X(2100)$ and its flavour partner $h_1(1965)$ induced by pion and kaon on a proton target with the Feynman model and the Regge model, which is an available platform to further identify their nature. The numerical results indicate that the total cross section are similar in the two models. When the range of momentum ${\mathrm{p_{Lab}}}$ is 10 to 30 GeV/$c$, the total cross sections for $\pi^-p\to X(2100)n$ and $K^-p\to X(2100)\Lambda$ are predicted to be at an order of magnitude of 0.1 $\mu$b. Whereas, the total cross section for $\pi^-p\to h_1(1965)n$ is near an order of magnitude of 10 $\mu$b when $p_{\mathrm{Lab}}$ is from 10 to 30 GeV/$c$, and much larger than that of reaction $K^-p\to h_1(1965)\Lambda$. These predictions can provide some valuable information to search for $X(2100)$ and $h_1(1965)$ in experiments at J-PARC, COMPASS, OKA@U-70 and SPS@CERN.

Very recently, BESIII continues to bring us a surprise with the discovery of a structure appearing in the φη ′ invariant mass spectrum of the J/ψ → φηη ′ decay [24]. With assumption of the spin-parity quantum number J P = 1 − , its measured resonance parameters are M = 2002.1 ± 27.5 ± 15.0 MeV and Γ = 129 ± 17 ± 7 MeV. If making a rough comparison, we find that its resonance parameter is close to the observed Y(2175). It is easy to speculate whether there exists connection between this resonance structure and Y(2175). However, if carefully checking their resonance parameters, it is obvious that we should be careful to make such conjecture. Just shown in Fig. 1, we list the resonance parameters of X(2100) and Y(2175) given by different experiments, there exists obvious difference since the mass and width of this structure are deviated from the corresponding parameters of Y(2175).
Considering this fact, we need to reveal the properties of the observed structure. Since this structure exists in the φη ′ invariant mass spectrum of the J/ψ → φηη ′ decay [24], there exist two possibilities of its J PC quantum number: 1) This structure has J PC = 1 −− , where J/ψ → Xη occurs via P-wave Thus, this structure is named as X(2100) tentatively, which is also applied to the following discussions when exploring its properties.
In this work, we can make scaling of the second radial excitation of h 1 meson family by adopting these reported h 1 states listed in Particle Data Group (PDG), which include h 1 (1170), h 1 (1380), h 1 (1595), h 1 (1965) and h 1 (2215), where the analysis of Regge trajectories [25,26] is employed. We may find that this analysis supports our scenario of X(2100) as an isoscalar axial-vector meson. Based on this study, we further investigate Okubo-Zweig-Iizuka (OZI)-allowed decay of X(2100) under this assignment, which may provide important information of total width and partial decay widths. By the comparison of theoretical and experimental results, we find that the conclusion of X(2100) as an isoscalar axial-vector meson is further enforced. According to the obtained decay behavior of X(2100), we also suggest that experimentalists should pay attention to J/ψ → φηη, where X(2100) as an axial-vector meson may exist in the corresponding φη invariant mass spectrum. In the next section, we illustrate the details of our study.
Additionally, we notice that the first and the third radial excitations of h 1 (1380) is still missing in experiment. Thus, in this work we also predict their properties, which provide valuable information to further experimental exploration to them. By this theoretical and experimental effort, we hope to establish the h 1 meson family step by step, since it is a crucial block of whole light hadron spectrum. This paper is organized as follows. After introduction, we discuss X(2100) as an axial-vector meson by the mass spec- Y(2175) PDG [27] Y(2175) BESIII [6] J/ψ → ηφπ + π − Y(2175) BABAR [28] e + e − → φπ + π − Y(2175) BES [5] J/ψ → ηφ f 0 (980) Y(2175) BABAR [29] e + e − → φη Y(2175) BELLE [30]   Here, the hollow circle and solid circle with blue denote theoretical and experimental values, respectively. The newly observed X(2100) under the J P = 1 + assignment is marked by the red star. trum analysis and the two-body OZI-allowed decay channel calculation in Sec. II, and the properties of five reported h 1 mesons listed in PDG and two predicted h 1 states are also given. The paper ends with a short summary in Sec. III. When performing the analysis of mass spectrum of light meson, the approach of Regge trajectories [25,26] was extensively adopted and tested in different meson systems (see Refs. [18,19,[33][34][35][36] for more details). Thus, in this work, we still apply Regge trajectories to analyze isoscalar axial-vector light mesons.
In general, a relation between mass and the corresponding radial quantum number can be found, i.e., where M 0 is the ground state mass and M denotes the mass of radial excitation with radial quantum number n. µ 2 is the slope parameter of the trajectory [33]. If taking Eq. (1) to study isoscalar axial-vector light mesons, we need the experimental information of these particles. Just shown in PDG [27], there are h 1 (1170), h 1 (1380), h 1 (1595), h 1 (1965) and h 1 (2215). Usually, h 1 (1170) and its strangeonium partner h 1 (1380) are ground states of isoscalar axial-vector mesons. And then, h 1 (1170), h 1 (1595), h 1 (1965) and h 1 (2215) are composed of a typical Regge trajectory, which can be described by Eq. (1) with µ 2 = 1.19 GeV 2 [34]. If fixing slope parameter µ 2 = 1.19 GeV 2 and taking the mass of h 1 (1380) as input, we can construct another Regge trajectory, and find that the mass of the second radial excitation of h 1 (1380) is 2087 MeV, which overlaps with the measured mass of X(2100) by BESIII with the J P = 1 + assumption [24]. It supports to assign X(2100) into the isoscalar axial-vector meson family.
As by product, we predict the masses of the first and the third radial excitations of h 1 (1380), which are 1780 MeV and 2355 MeV, respectively. Thus, we name these two missing h 1 states as h 1 (1780) and h 1 (2355) for the convenience when illustrating their decay behaviors. The analysis of Regge trajectories for these discussed h 1 states is shown in Fig. 2. In this following discussion, we focus on OZI-allowed strong decay behavior of the newly observed X(2100) as an axial-vector meson, by which the reliability of this assignment can be tested further. In the present study, we adopt the fluxtube model [19,[37][38][39][40], which is applied to quantitatively give the decay information of hadron. Thus, we briefly introduce it in this section.
In the flux-tube model, a quark and antiquark connected by a tube of chromoelectric flux construct a meson, where this tube can be treated as a vibrating string. For the meson, the string is in the vibrational ground state (vibrational excitation corresponds to hybrid). When a meson decay occurs, the vibrational string breaks at a point, and simultaneously quarkantiquark pair is created from vacuum to connect to the free end of string and further form two outgoing mesons.
In this phenomenological model, a dimensionless parameter γ should be introduced by expression [40] where γ 0 as a phenomenological parameter is fixed as 14.8 by experimental data [19]. b is string tension with the typical value 0.18 GeV 2 [40]. And, w min is shortest distance from line connecting the quark-antiquark pair of initial meson to the location of creating quark-antiquark pair from vacuum. Under the framework of the flux-tube model, the expression of partial wave amplitude depicting a decay A → B + C is expressed as [40] Here, P is the momentum of meson B. S and L denote the total spin and relative orbital angular momentum between mesons B and C, respectively. E B is the total energy of meson B. L i and J i (i = A, B, C) are the orbital angular momentum and total spin for a meson, respectively, while M L i and M J i denote the corresponding magnetic quantum numbers. Additionally, m 1 and m 2 are quark masses within meson A. m 3 represents the mass of the quark and antiquark created from the vacuum. χ and φ with subscripts and superscripts denote the spin and flavor wave functions of the corresponding mesons or quark pair created from the vacuum, respectively. I ft is momentum space integral. Its calculation is related to the overlap of spatial wave functions of involved mesons. We use the simple harmonic oscillator (SHO) wave function to describe the spatial wave function of the mesons, where the β value in SHO is fixed through the calculated root-mean-square momentum [41]. The total decay width of the meson is The details of the flux-tube model can be refereed to Refs. [37,40].
For h 1 (1170), ρπ contributes its nearly total decay width. By Fig. 3, we can find that the experimental width of h 1 (1170) can be reproduced well with theoretical calculation. For h 1 (1380), three decay channels (KK * , πρ, and ηω) are OZIallowed. Our calculation shows that h 1 (1380) is a good candidate of ground state of the h 1 family. Its ss main component can be reflected by the dominant KK * decay channel just shown in Fig. 3. The h 1 (1380) decays into πρ and ηω are subordinate decays.
Since h 1 (1170) and h 1 (1380) are well established ground states in the h 1 meson family, the above study is also good test to the flux tube model and these adopted parameters, which will be applied to the following calculation.
For h 1 (1595), its partner is still missing. In Sec. II, we predict its mass to be around 1780 MeV by the Regge trajectory analysis. Here, we tentatively name it as h 1 (1780). There still exists a mixing scheme for h 1 (1595) and h 1 (1780),  (5). Here, the mass labeled by superscript † is calculated by mass relation in Eq. (8), which can be as input to estimate the masses of other K( 1 P 1 ) states with higher radial quantum number through the Regge trajectories formula (1) with µ 2 = 1.19 GeV 2 . These masses of higher K(n 1 P 1 ) states is marked by superscript ‡. By adopting the similar way, we also estimate the masses of b 1 with radial quantum number n = 2 and h ′ 1 with radial quantum numbers n = 2, 4, which are also marked by superscript ‡. Additionally, the mass of the newly observed X(2100) under the J P = 1 + assignment is labeled by superscript ♮ . Other experimental masss are taken from PDG [27]. Radial quantum number m b 1 m K(n 1 P 1 ) m h ′ . The R dependence of the total and partial decay widths of h 1 (1170) [48][49][50] and h 1 (1380) [45,[51][52][53], where listed the corresponding experimental data (dashed lines with gray band) for comparison with our theoretical calculation. The selected R range already contains the suggested R value for ground state of the h 1 family in Ref. [54].
In Fig. 4, we present the dependence of the total and partial decay widths of h 1 (1595) and the predicted h 1 (1780) on the R values, where we select the R range can contain the suggested R values for the first radial excitations in Ref. [54].
For h 1 (1595), its dominant decay channel is ρπ while the ηω and KK * decay channels have subordinate contribution to the total width. We need to indicate that the calculated total width is smaller than the experimental width as shown in Fig.  4. At present, there only is one experimental measurement for h 1 (1595) [55], i.e., experimentalist announced h 1 (1595) via analyzing the π − p → ωηn reaction. Due to this reason, h 1 (1595) is omitted from the summary table of PDG. Considering this situation, more precise measurement of the resonance parameter of h 1 (1595) is needed, which will be next task for experimentalist. It will be helpful to clarify this difference between experimental and our theoretical results. In this work, we also predict the decay property of h 1 (1780). h 1 (1780) dominantly decays into KK * since h 1 (1780) has dominant ss component. Thus, the contribution of h 1 (1780) → ηφ, πρ, πρ(1450) are not obvious to its total decay width. Experimental search for h 1 (1780) is still an interesting research topic, where h 1 (1780) → KK * is a suggested ideal channel for hunting h 1 (1780). Additionally, we may conclude that h 1 (1780) is a broad state with order of magnitude of several tens MeV according to our calculation. The detailed information of the decay behavior of h 1 (1780) can be found by Fig. 4. 3. X(2100) and its partner h 1 (1965) The key point of the present work is to test the possibility of the newly observed X(2100) as the second radial excitation of the h 1 family, which becomes a partner of h 1 (1965). X(2100) and h 1 (1965) satisfy the below relation |h 1 (1965) |X(2100) = sin θ 3 cos θ 3 cos θ 3 − sin θ 3 |nn |ss .
When taking R = 4.8 GeV −1 , the experimental width of X(2100) can be reproduced well (see Fig. 7), which shows that X(2100) can be a good candidate of the second radial excitation of the h 1 meson family.
And then, the corresponding branching ratios dependent on the R value are given in Fig. 7, by which πρ, KK * , KK * (1410) as main decay channels and πρ(1450), ηφ, K * K * , KK * 0 (1430), η ′ φ, and KK * 2 (1430) as subordinate decays can be identified. In fact, the main decay channels to X(2100) is KK * and KK * (1410). But, there still exist some challenges to identify kaon since weak interaction dominates the decay behavior of kaon, which makes the whole reconstruction efficiency of X(2100) via these kaon final states is low. It is the reason why X(2100) cannot be identified firstly from the kaon final states.   Since X(2100) → φη ′ has a sizable contribution to the total width, X(2100) firstly observed in the φη ′ final state can be understood. Our calculation gives that the branching ratio of X(2100) → φη ′ can reach up to 2.69% as seen in Fig.  5. The BESIII Collaboration measured the product branching fraction B(J/ψ → ηX(2100)) × B(X(2100) → φη ′ ) = (9.6 ± 1.4 ± 1.6) × 10 −5 . Combining our theoretical result and this experimental data, we may extract B(J/ψ → ηX(2100)) = 3.24 × 10 −3 .
This branching ratio provides important information of further theoretical study on the production of X(2100) associated with a η meson via the J/ψ decay.
In near future, BESIII will play an important role to explore light hadrons. Our results also show that h 1 (1965) may decay into ηω with comparable branching fraction with that of η ′ ω (see Fig. 6), i.e., which is stable for these discussed R range. We suggest BE-SIII to carry out the study of J/ψ → ηh 1 (1965) → ηη ′ ω and ηηω, especially checking the corresponding ωη and ωη ′ invariant mass spectra.
Since h 1 (1965) is a broad structure, how to identify this broad resonance will be a challenge for experimentalist.
C. Higher states in the h 1 meson family When discussing these higher states of the h 1 meson family, we mainly take h 1 (2215) and the predicted h 1 (2355) into account, where h 1 (2215) is an axial-vector listed in PDG as further state [27], which was reported by SPEC in the pp → ωη, ωπ 0 π 0 reactions [56].
For h 1 (2215), its main decay channels include ρπ and πρ(1450). In addition, sizable decays like KK * , KK * (1410), ηω, ρa 1 (1260) have obvious contribution to its total decay width. Thus, we also understand why h 1 (2215) exists in the reported pp → ωη reaction. Other decay behaviors of h 1 (2215) can be found in Fig. 8. Similar to the situation of the discussed h 1 (1965), we suggest BESIII to perform the analysis of J/ψ → η (′) ηω, by which h 1 (2215) should be confirmed in the η (′) ω invariant mass spectrum. In this work, we also predict the decay behaviors of h 1 (2355) (see Fig. 9 for more details), which is assigned as the partner of h 1 (2215). The calculated OZI-allowed decay results indicate that h 1 (2355) mainly decays into KK * , KK * (1410), πρ, ηφ, K * K * , ηω(1650) and K * K 1 (1400), which is presented in Fig. 9. In addition, the decay channel η ′ φ also has a nonnegligible contribution to total width, which shows that there should exist the evidence of this predicted h 1 state in the reported φη ′ invariant mass spectrum. When checking the BESIII data [24], we notice that there is an enhancement around 2.35 GeV in the φη ′ invariant mass spectrum of J/ψ → η ′ ηφ. This evidence should be confirmed by future experiment.

III. SUMMARY
As an important part of studying light hadron spectrum, the axial-vector light meson family is not well established. The observation of X(2100) in J/ψ → η ′ ηφ provides us a good chance to further perform the investigation of isoscalar axialvector light mesons. In this work, X(2100) is assigned as the second radial excitation of h 1 (1380), which can be supported by the analysis of the Regge trajectory analysis and the calculation of the corresponding OZI-allowed decays. We suggest a possible channel J/ψ → φηη to identify X(2100) in the corresponding φη invariant mass spectrum. In the present work, we also predict the mass positions and the decay properties of two missing h 1 mesons (h 1 (1780) and h 1 (2355)). Hunting for these missing h 1 states will be interesting research topics.
In the following several years, the BESIII experiment will still be the main force of exploring the light hadrons. These theoretical predictions presented in this work may provide valuable reference to future experimental studies on this issue. We are waiting for the progress on this filed, especially from BESIII.
For theorist, the physics around 2.1 GeV light hadrons should be paid more attentions, which has close relation to these higher states of the ρ, φ, ω and h 1 meson families, and exotic states. The former observed Y(2175) has inspired ex- . The R dependence of the total and partial decay widths of h 1 (2215). Here, we also list the SPEC result (dashed line with gray band) for the width of h 1 (2215) [56]. We need to specify that some tiny channels are not drawn. tensive discussion on this issue. When facing the different experimental observations of the states around 2.1 GeV, we should be very careful to directly treat them to be the same state as the observed Y(2175) only according to a simple comparison of their resonance parameters. Furthermore, the study of their decay and production will provide valuable information to identify their inner structure. The present work provide a typical example. We expect more theoretical groups to focus on the physics around 2.1 GeV light hadrons.