The potential higher radial excitations in the light pseudoscalar meson family

Inspired by the event accumulation around 2.6 GeV in the $\eta^\prime\pi^+\pi^-$ invariant mass spectrum of $J/\psi\to \gamma \eta^\prime\pi^+\pi^-$, which was reported by the BESIII Collaboration, we carry out the study of the mass spectrum and decay behavior of four radial excitations in the pseudoscalar meson family, which include $\eta^{(\prime)}(6S)$ and $\eta^{(\prime)}(7S)$. Combining with these analysis, we present the calculation of the reactions induced by pion or kaon on proton target which are relevant to these four discussed states. According to this information, we give concrete experimental suggestion of searching for them, which will become new task to future experiments.


I. INTRODUCTION
Studying hadron spectroscopy can provide valuable hints to deep our understanding of nonperturbative quantum chromodynamics (QCD). Among the hadron family, light mesons construct a special group since there exists extra abundant measured information of light mesons. In 1935, Yukawa predicted the existence of pion meson for quantitatively depicting nuclear force [1], which was discoveryed by Lattes, Occhialini and Powell [2]. By joint effort from experimentalist and theorist, the number of light mesons is increasing in the past decades. In recent years, the running experiment like BE-SIII is still playing crucial role to explore and discovery light meson. As emphasized in the "White paper on the Future Physics Program of BESIII" recently released by the BESIII Collaboration [3], investigating the light hadron spectrum will be an important issue in future ten years at BESIII. Here, besides establishing conventional light meson, what is more important is the search for exotic hadronic state like glueball [4].
Since 2011, Lanzhou group performed systematic studies on mass spectrum and strong decay behavior of light mesons, which include pseudoscalar states [5,6], vector state [7], ρ and ρ 3 states [8], tensor states [9], pseudotensor states [10], axial vector states [11], J PC = 2 − unflavored states [12], kaons [13], and high-spin states [14]. Besides the above work, other theoretical groups focused on this issue [15]. By these investigations, an overview of the properties of light meson family was presented, since the above studies almost contain these light mesons with allowed different spin-parity quantum numbers.
After finishing these theoretical work, we should consider how to continue the following studies of light meson. There should exist three possible directions: 1) Combining with concrete experimental observation of light meson, we can decode the properties of these observed states by giving mass spectrum analysis, decay behavior and production. A typical ex-ample is Ref. [16], where X(2100) observed in J/ψ → φηη can be categorized into the h 1 meson group which can be supported by mass and decay calculation. In addition, its production induced by pion and kaon was discussed. 2) Exploring high spin states. If checking the data from Particle Data Group (PDG) [17], we may find that the experimental information of higher spin states is not abundant. When establishing these high spin states in experiment, theoretical guidance is important. In Ref. [14], the authors presented a systematic studies on these high-spin states below 3 GeV. Additionally, Pang et al. predicted the mass and decay behavior of 5 ++ mesons [18]. 3) Exploring the potential higher radial excitations of light meson accessible at experiment is also very interesting, which will be main task in this work.
As frontier of precision in particle physics, hadron physics has entered a new era with the accumulation of high precision experimental data since 2003. We may find two typical examples to reflect the importance of experimental precision. One is hidden charm P c observation. In 2015, LHCb observed two P c states (P c (4380) and P c (4450)) in Λ b → J/ψpK [19]. After four years, LHCb reanalyzed this channel based on data from Run I and Run II and found that P c (4450) contains two substructures (P c (4440) and P c (4457)) and there also exists another P c (4312) [20], which give strong evidence of the existence of hidden-charm molecular pentaquark [21].
We noticed that BESIII already collected a sample of 10 billion J/ψ events in 2019. With more precision data, we have reason to believe that more η/η -like states with higher mass  can be found in future experiment. Thus, it is a suitable time to carry out the theoretical study of the properties of these potential higher radial excitations in the light isoscalar pseudoscalar meson family. Another topic of this work is to discuss the possible search for them by combing with the present BESIII experimental data [22][23][24]. However, we have to face a fact that the masses of η(7S ) and η (7S ) are close to that of η c (1S ), which results in the difficulty of detect the signals of η(7S ) and η (7S ). Thus, we propose that the production processes induced by pion or kaon can be as good platform to explore higher states in the η/η meson family. In this work, we take the production processes relevant to η ( ) (6S ) and η ( ) (7S ) as example to give a quantitative illustration.
The paper is organized as follows. After the Introduction, we present the mass spectrum analysis in Sec. II. In Sec. III, we calculate the Okubo-Zweig-Iizuka (OZI) allowed decays of these discussed η/η states. And then, we explore the production processes of four discussed η/η states which are in-duced by pion and kaon (see Sec. IV). This paper ends with a summary.
The Regge trajectory theory was firstly proposed by Regge in 1959, and was later widely used to study the light hadron spectrum [5,[7][8][9][26][27][28][29]. In general, for light meson, the relationship between the square of the energy and the total angular momentum (J) is linear, which is the Regge-Chew-Frautschi relationship M 2 J = M 2 J + α 2 (J − J ). Here, J and J denote the total angular momentum of the discussed states. M J and M J are the masses of these states with quantum numbers J and J , repectively.
For discussing the fifth and the sixth radial excited states of pseudoscalar mesons, we adopt another version of Regge trajectory analysis, which was adopted in the study of different light meson systems [5, 7-9, 11, 16, 27, 28]. The relation of the mass and the radial quantum number n satisfies M 2 = M 2 0 + (n − 1)µ 2 , where M 0 is the mass of ground state and M is the mass of excited state with radial quantum number n. µ 2 represents the slope of the trajectory, and its value range is µ 2 = 1.25 ± 0.15 GeV 2 .
In Ref. [24], BESIII indicated that the evidence of a η/ηlike state around 2.6 GeV may exist in the η π + π − invariant mass spectrum. This possible enhancement structure may correspond to the η(2627) as the fifth radial excitation of η(548) predicted in this work. In the following section, we will further discuss the decay behavior of η(2627), which can provide valuable information to future experimental investigation. Additionally, we will illustrate the decay properties of the remaining three predicted η/η mesonic states

III. DECAY PROPERTIES OF THE DISCUSSED FOUR η/η EXCITATIONS
For calculating the OZI-allowed decay behavior of these four predicted η/η higher radiation excitations, we utilize the quark pair creation (QPC) model [30][31][32]. The QPC model was first proposed by Micu [33] and then further developed by Orsay group [34][35][36][37][38]. It was widely used to study the OZI-allowed strong decay of hadrons [5, 7-9, 11, 12, 39-43]. In the QPC model, when a meson decay occurs, a quarkantiquark pair is created from vacuum with the quantum number J PC = 0 ++ , which can be combined with the corresponding antiquark and quark in the initial meson to form two final mesons.
A transition operator T is introduced to describe a qq pair creation from the vacuum Here, parameter γ represents the probability that a quarkuntiquark pair is creation from the vacuum. k 3 and k 4 denote the three-momenta of quark and untiquark created from the vacuum, respectively. φ 34 0 = (uū + dd + ss)/ √ 3 describes the flavor singlet and ω 34 0 = δ α 3 α 4 / √ 3(α = 1, 2, 3) denotes the color singlet. χ 34 1,−m is a spin triplet state. i and j are the S U(3) color indices of the created quark pairs from the vacuum. Y m (k) ≡ |k| Y m (θ K , φ K ) represents the -th solid harmonic polynomial.
By the Jacob-Wick formula [30,44,45], the decay amplitude can be expressed as (2) and the general decay width reads In the concrete calculation, the harmonic oscillator wave the parameter R can be determined by reproducing the realistic root mean square radius of the corresponding meson state.
In Refs. [5,6], Lanzhou group performed a systematic studies on pseudoscalar mesonic states by combing with these observed pseudoscalar X states shown in the η π + π − invariant mass spectrum [22][23][24]. These investigations enforce the possibility of categorizing these reported pseudoscalar X states into pseudoscalar meson family. Along this line, we further present the decay properties of four η/η excitations (η(2627), η(2742), η(2867) and η(2973)) listed in body strong decay, which is dependent on the R range 1 . Additionally, we need to emphasize that a mixing scheme should be introduced when discussing four η ( ) mesons. Here, |η q (nS ) = 1 √ 2 (|uū + |dd ) and |η s (nS ) = |ss are the flavor wave functions. For the fifth and the sixth radial excitations, the information of mixing angle is still absent. Thus, we roughly take 4.18 • [6] in our concrete calculation under the assumption that this mixing angles for the fifth and the sixth radial excitation are same as that of the fourth radial excitation.
• In the following, we discuss the partner of η(2627), which corresponds to η(2742) as η (6S ). Our result shows that η(2742) is a broad state with width 396.3 − 590.1 MeV, which may result in the difficulty to identify it in experiment. Its dominant decay mode includes KK * , while K 1 (1270)K * , KK 0 (1430), KK * 2 (1430) and K * K * have main contribution to the total decay width of η(2742). In Fig. 4, more information of its partial decay widths can be found.
• From the analysis of Regge trajectory (see Fig.2), η(7S ) has mass 2867, which is referred to as η(2867) in this work. In Fig. 5, we present its total and partial decay widths. Similar to η(2625) mentioned above, η(2867) mainly decays into πa 2 (1320) and πa 0 (1450). But, the width of η(2867) is lightly broader than that of X(2625). Here, the suggested ideal channel of searching for η(2867) is still η π + π − which can be from J/ψ → γη π + π − . It is obvious that BESIII should try to hunt it with more experimental data.
• η(2973) as a η (7S ) state should have the decay behav-ior shown in Fig. 6, where its dominant decay channel is KK * . The total decay width of η(2973) is very broad. Thus, it is difficult to discovery such broad structure in experiment. In addition, we have to face the fact that η(2973) almost overlaps with η c (1S ).

IV. THE PRODUCTION RELEVANT TO THE DISCUSSED η/η (nS ) INDUCED BY PION OR KAON
Until now, the η/η mesons mostly were observed through the J/ψ radiative decay process. Searching for them in different reaction platforms is an interesting issue, which may provide more abundant information to decode these states. It is well known that the pion-proton and kaon-proton scattering processes are effective experimental tools in exploring light hadrons. A typical example is η(1295), which was first observed in the pion-proton scattering process π − p → nηπ + π − [46]. Therefore, pion-proton and kaon-proton scattering could be a peculiar way to investigate the η/η mesons. Based on this motivation, in this work we will explore the productions of η(6S ) ( ) and η ( ) (7S ) via the pion-proton and kaon-proton scattering processes, where effective Lagrangian approach is adopted. These calculated results are valuable for the further relevant experimental exploration, where several concrete experiments include J-PARC [47,48], COMPASS [49], and SPS@CERN [50].
The Feynman diagram of these discussed higher radial excitation of the pseudoscalar meson family produced via pion and kaon induced reactions on a proton target is illustrated in Fig. 7, where we only consider the t-channel diagram. Besides, the contributions from s-channel and u-channel are not considered in this work, since the s-channel is usually negligibly small and the u-channel always concentrates at backward angles. For the π − p → η * n reaction, we take the relevant effective Lagrangians [51,52] L a 0 πη * = g a 0 πη(nS ) a 0 · π η(nS ), where a 0 , η * , π, and N donate the a 0 (980), η(nS )/η (nS ), pion, and nucleon fields, respectively. The coupling constants g a 0 πη(nS ) (g a 0 πη(6S ) = 1.28 GeV, g a 0 πη(7S ) = 0.93 GeV, g a 0 πη (6S ) = 0.09 GeV, g a 0 πη (7S ) = 0.06 GeV) can be determined by the decay width of η * decaying into πa 0 (980) which is calculated by the QPC model in Sec. III. Besides, for the coupling constant g a 0 NN , we adopt g 2 a 0 NN /4π = 1.075 which is implied by the Bonn one-boson exchange model for the nucleon-nucleon interaction [53].
With the above preparation, the amplitudes of these discussed reactions shown in Fig. 7 can be written as where t = p 2 t = (p 3 − p 1 ) 2 ,g µα = −g µα + p µ t p α t /m 2 K * , and F x (t) is the form factor of t-channel exchange for each interaction vertex, which is taken as the monopole form [53,58] in this work. The cutoff Λ x in form factor can be parameterized as Λ x = m x + αΛ QCD with Λ QCD = 220 MeV. In general, the value of parameter α is taken around 1 [16,59]. Thus, we also take α = 1 in our calculation.
Here, we also introduce the Reggeized treatment to the tchannel in order to better describe the behavior of the hadron production at high momentum [16,[60][61][62][63]. To the Reggeized treatment for the t-channel meson exchange, we only need to replace the form factor in the Feynman amplitudes in Eqs. (9)-(10) as The scale factor s scale is fixed at 1 GeV, and we set the signature ξ = 1 for the a 0 exchange and ξ = −1 for the K * exchange. The Regge trajectories of α a 0 (t) and α K * (t) read as respectively. Besides, the Gamma function can suppress poles of the sin[πα(t)] when the α(t) ≤ 0, and the poles of α(t) > 0 can be avoided automatically since t < 0 for the t-channel leads to α(t) < 1. Now, all parameters have been determined. Then, we can calculate the cross sections of the productions of these four pion and kaon induced reactions on a proton target. For the 2 → 2 reaction process, the differential scattering cross section can be expressed as where s = (p 1 + p 2 ) 2 is the square of center of mass energy, p 1cm denotes the momentum of incident pion or kaon in the center of mass frame, and the overline on |M|  In Fig. 8, we show the numerical result of the cross sections of π − p → η(6S )n and K − p → η(6S )Λ as a function of the pion and kaon momenta in the laboratory system (P Lab ), respectively. For the π − p → η(6S )n reaction, both of the line shapes of the total cross sections in the Feynman model (red dotted line) and the Regge model (red solid line) sharply increase near the threshold, and then they begin to slowly decrease with increasing P Lab . In the Feynman model, the total cross section reaches up to a maximum of 6.3 nb at a momentum P Lab = 9.5 GeV/c. But in the Regge model, the maximum of the total cross section is 11.0 nb at P Lab = 11.1 GeV/c. Different from the Feynman case, the line shape of the total cross section in the Regge model decreases more rapidly with increasing P Lab when the total cross section has reached a maximum.
For the K − p → η(6S )Λ reaction, the obtained line shapes of the total cross sections in the Feynman model (blue dotted line) and the Regge model (blue solid line) also sharply increase near the threshold, but then slowly trend to a stable value. The line shape in the Feynman model (blue dotted line) is increasing slowly with increasing P Lab , while the line shape in the Regge model (blue solid line) is decreasing slowly with increasing p Lab when the total cross section reaches up to a maximum 0.05 nb at P Lab = 25.5 GeV/c. Compared to the Feytnman model, the Regge model gives a smaller cross section for the K − p → η(6S )Λ reaction. Hence, the line shape difference between these two models can be applied to distinguish the role of the Regge model in further experiment. Beside, we find that the total cross section of η(6S ) given by the π − p → η(6S )n reaction is significantly larger than the result obtained by the K − p → η(6S )Λ reaction, since η(6S ) coupling with πa 0 is stronger than η(6S ) interacting with KK * . According to our results, the pion-proton scattering may be more better platform than the kaon-proton scattering to explore the η(6S ) state. We also suggest that the P Lab range with 9.0 ∼ 12 GeV/c is a good momentum window for future experiment to hunt η(6S ) via the pion-proton scattering platform. If experimentalist want to find η(6S ) in the kaon-proton scattering process, the P Lab around 25 GeV/c is a suitable momentum window. Additionally, in Fig. 9, we also show the numerical results of the cross sections of η(7S ) produced through the pionproton scattering and the kaon-proton scattering, where the behavior of the lines shape of the total cross sections as a function of P Lab is similar to the case of η(6S ). The total cross section of η(7S ) produced through the π − p → η(7S )n reaction has a maximum 2.3 nb at P Lab = 11.3 GeV/c in the Feynman model, 3.3 nb at P Lab = 13.2 GeV/c in the Regge model. And, in the π − p → η(7S )n reaction, the total cross section tends to a stable value 1.5 nb for the Feynman model and 0.04 nb for the Regge model at P Lab = 25 GeV/c. Hence, the P Lab range of 11.3 ∼ 13.2 GeV/c and P Lab = 25 GeV/c may be a good momentum window to search for η(7S ) on the pion-proton and kaon-proton scattering, respectively.
In Fig. 10 and Fig. 11, we present the lines shape of total production cross sections of η (6S ) and η (7S ) with the pion-  proton scattering and the kaon-proton scattering, respectively. In fact, we notice that the behavior of these lines shape is similar to that of the reactions π − p → η(6S )n and K − p → η(7S )Λ, which is due to the similarity of the corresponding reaction amplitudes. However, there also exists difference, i.e., the η (6S ) and η (7S ) production cross sections obtained by the pion-proton scattering are smaller than the results obtained by the kaon-proton scattering. The main reason is that the coupling of η (6S )/η (7S ) to KK * is stronger than that of them to πa 0 . For the η (6S ) (η (7S )) production through the pionproton scattering, the total cross section has maximum 0.03 nb (0.01 nb) at P Lab = 10.5 GeV/c (P Lab = 12.2 GeV/c) in the Feynman model, and in the Regge model, respectively, where the maximum is 0.04 nb (0.01 nb) at P Lab = 13.6 GeV/c (P Lab = 13.6 GeV/c). The total cross sections of η (6S ) and η (7S ) in the kaon-proton scattering are much larger than the results of pion-proton scattering. The line shape of the total cross section of the K − p → η (6S ) (K − p → η (7S ) ) reaction in the Feynman model sharply increases near the threshold, then slowly trends to a stable value 40 nb (47 nb) at P Lab = 25 GeV/c (P Lab = 24 GeV/c). But in the Regge model, the total cross section of η (6S ) (η (7S )) in the kaon-proton scattering has a maximum of 1.4 nb (1.3 nb) at P Lab = 9.8 GeV/c (P Lab = 11.2 GeV/c). Hence, the P Lab range of 9.8 ∼ 25 GeV/c (11.2 ∼ 25 GeV/c) may be a good momentum window for future experiments to research η (6S ) (η (7S )) on the kaon-proton scattering platform. And, in the pion-proton scattering process, a suitable window is around P Lab = 13.6 GeV/c. Another notable behavior for these calculated lines shape is that there exists a small cusp near the production threshold for all of the Regge model results. They are relevant to the factor 1 + ξe −iπα(t) , which may result in dips at α a 0 (t) = −1, −3, −5, · · · for the pion-proton scattering reaction. And, for the kaon-proton scattering reaction, dips appear at α K * (t) = 0, −2, −4, · · · . These cusps also exist the X(2100) and h 1 (1965) productions by the pion-proton scattering and the kaon-proton scattering [16,64,65]. Such cusps is physical or only unphysical should be clarified by the future precise experimental data.
In fact, the above study gives the prediction of searching for higher radial excitations of pseudoscalar meson via the pionproton scattering and the kaon-pion scattering, which can be as new task for future experiment. Although the present experimental information is still absent, we try to test the validity of our framework adopted in this work. We notice that there exists measurement of the differential cross section of π − p → η(1295)n at p Lab = 8.95 GeV/c [66]. Thus, we calculate the differential cross section of π − p → η(1295)n at p Lab = 8.95 GeV/c and make a comparison of our theoretical result with the experimental data (see Fig. 12 for the details). Here, the blue doted line and red solid line are calculated in the Feynman model and the Regge model, respectively, and the experimental data from left-hand side to right-hand side are average values of differential cross section in the range of |t − t min | with 0 ∼ 0.05 (GeV/c) 2 , 0.05 ∼ 0.2 (GeV/c) 2 and 0.2 ∼ 0.6 (GeV/c) 2 . Generally, our result is comparable with the experimental data. In the small |t − t min | range, the Regge model result is more close to the experimental result than the Feynman model result. By this study combined with concrete experiment, we have reason to believe that the adopted theoretical framework in the present work can be used to study the production of higher radial excitation in the pseudoscalar meson family, and give reasonable theoretical prediction accessible at experiment.

V. SUMMARY
The study of hadron spectroscopy may provide valuable hint to understand the non-perturbative behavior of QCD. As an important group in whole hadron family, light hadron has attracted extensive attention of theorist and experimentalist. In this work, we still pay attention to the pseudoscalar meson family. Since 2003, more and more pseudoscalar states including X(1835), X(2120), X(2370) and X(2500) have been reported in experiments [22][23][24][25], which provide good chance to construct the pseudoscalar meson family [5,6]. It is obvious that it is not the ends of whole aspect. Especially, the BESIII measurement of the η π + π − invariant mass spectrum of J/ψ → γη π + π − [24] shows that there exists a possible event accumulation around 2.6 GeV. This experimental information also stimulates our interest in exploring higher radial excitations of the pseudoscalar meson family.
In the present work, we focus on the fifth and the sixth radial excitations of the pseudoscalar meson family. By performing the mass spectrum analysis and the two-body OZIallowed strong decay calculation, we may obtain the information of their resonance parameters and partial decay widths, which is crucial to hunt and identify these higher radial excitations of the pseudoscalar meson family in future experiment.
Of course, it should not limit us to the above issues since the investigation around these pseudoscalar mesons contains their production and so on. We also notice that the established η(1295) was firstly observed in the pion-proton scattering process [46]. Thus, in this work, we exam the possibility of searching for the discussed four pseudoscalar mesons via the production induced by the pion or kaon. By the effective Lagrangian approach, we estimate the corresponding production cross sections, and find that these physical quantities are sizable. Combining with these information, we further give theoretical suggestion of finding them.
As indicated in the "White paper on the Future Physics Program of BESIII" [3], BESIII still plays important role to explore light hadron. Especially, BESIII has 10 billion J/ψ data, which makes the discovery of these higher radial excited pseudoscalar mesons become possible. Although BESIII has special status of studying on light hadron, we still believe that it is not unique way to detect these states. The pion-proton and kaon-proton scattering experiments can be as a supplement as illustrated in the present work. exchange reaction at 8.95 GeV/c, Phys. Lett. B 267, 293 (1991).