Revisiting the assignment of $1^{3}D_{1}$ meson nonet

Quantum Chromodynamics (QCD) is widely accepted as a non-Abelian gauge field theory to describe strong interaction. QCD successfully explains many experimental data, in the high-energy region, the coupling constant is small, one can use perturbative method to handle the interaction. However, in the low energy region, the coupling constant is large and the perturbative method is not applicable. So far, no other analytical methods have been developed to solve this problem. In other words, the understanding of strong interaction is still incomplete. On the other hand, the properties of mesons are dominated by the non-perturbative effects of QCD, so mesons have become an ideal laboratory for the study of strong interactions in the strongly coupled non-perturbative regime Godfrey:1998pd ; Li:2004gu . The investigation of the meson spectrum is of great scientific significance for a better understanding of the non-perturbative effects of QCD.

In this work, we will focus on the assignment of $1^{3}D_{1}$ meson nonet. In the past few years, there have been a lot of analysis on the assignment of $1^{3}D_{1}$ meson nonet, but until now, there are still many confusing aspects about this problem Liang:2013jmx . In the recent edition of Particle Data Group, the $1^{3}D_{1}$ meson nonet assignments have been presented in the quark model ParticleDataGroup:2020ssz . Here we list the masses and decays of $1^{3}D_{1}$ meson nonet in Tab. I.

There is some controversy about $K^{*}(1680)$ as the kaon in the $1^{3}D_{1}$ meson nonet. To specify this question, we need to briefly review the kaon state $K^{*}(1410)$ in the $2^{3}S_{1}$ meson nonet. In PDG, the $K^{*}(1410)$ as candidate for $2^{3}S_{1}$ meson nonet, but there are still some problems about this assignment. On the one hand, the mass of $K^{*}(1410)$($1414\pm 15MeV$) is smaller compared with other states $\rho(1450)$ and $\omega(1420)$ of $2^{3}S_{1}$ meson nonet. In the past few years, people have analyzed the mass of $K^{*}(1410)$ with the different models, e.g. Godfrey-Isgur quark model ($\sim 1580MeV$) Godfrey:1985xj , constituent quark model ($\sim 1620MeV$) Vijande:2004he , semirelativistic potential model ($\sim 1600MeV$) Brau:2002zpy and Regge trajectories ($\sim 1608MeV$) Tornqvist:1990fv . The mass range does not support such assignment. On the other hand, the reported $\pi K$ branching fraction deviate significantly from the theoretical prediction Barnes:2002mu . Burakovsky also indicated the mass of $K^{\ast}(1410)$ seems too light to be the $2^{3}S_{1}$ meson nonet Burakovsky:1997ch . In Ref. Tornqvist:1990fv , the existence of state $K^{\ast}(1410)$ is doubted by $T\ddot{o}rnquist$. In our previous work Feng:2007zze , we also suggested that the assignment of $K^{\ast}(1410)$ should be tested in the future experiments. Many references above suggested that the assignment of state $K^{\ast}(1410)$ should be revisited in the future. The earlier edition of PDG recommended that the $K^{\ast}(1410)$ could be replaced by the $K^{\ast}(1680)$ as the $2^{3}S_{1}$ state ParticleDataGroup:2006fqo . If the replacement is testified reasonably in experiment, we would encounter another interesting puzzle, which state should be the candidate for kaon in the $1^{3}D_{1}$ meson nonet. The state $K^{\ast}(1680)$ with the mass $1717\pm 27MeV$ and full width $322\pm 110MeV$ is observed in the $\pi K$, $\rho K$ and $K^{\ast}(892)\pi$ decay modes at present ParticleDataGroup:2020ssz . However, Barnes indicated this state should have large branch fraction in $\pi K_{1}(1273)$ decay mode in the ${}^{3}P_{0}$ model Barnes:2002mu . Similarly, $K^{*}(1680)$ mass seems too light if we accept the $\rho(1700)$ and $\omega(1650)$ as $1^{3}D_{1}$ $n\bar{n}$ states. Moreover, the $K^{\ast}(1410)$ and $K^{\ast}(1680)$ could be mixtures of $2^{3}S_{1}$ and $1^{3}D_{1}$ states, this situation makes the problem more interesting Pang:2017dlw .

Apart from the kaon, the $s\bar{s}$ member of $1^{3}D_{1}$ meson nonet is even more disturbing, with the $\rho(1700)$ and $\omega(1650)$ as the well established $1^{3}D_{1}$ $n\bar{n}$ states, the $s\bar{s}$ member mass is investigated in different models, e.g. Godfrey-Isgur quark model Godfrey:1985xj , the flux-tube model Barnes:1995hc . Surprisingly, the $\phi(2170)$ is assigned as the the $s\bar{s}$ member of $1^{3}D_{1}$ meson nonet in PDG ParticleDataGroup:2020ssz . Because the mass of $\phi(2170)$ deviates greatly from the conventional predictions, we have doubts about such assignment. It is also pointed out that $\phi(2170)$ proposed as a tetraquark Agaev:2019coa . A more detailed discussion of this state can be found in Refs. Agaev:2019coa ; BESIII:2020gnc .

In this work, we will investigate the mass spectrum of $1^{3}D_{1}$ meson nonet in the framework of Regge phenomenology. In addition, the decays of $1^{3}D_{1}$ meson nonet are given. The article is organized as follows: In Sec. II, a brief review Regge phenomenology and the mass relations of $1^{3}D_{1}$ state are obtained. In Sec. III, decays of $\phi(???)$ are presented in the ${}^{3}P_{0}$ model, and a summary is given in Sec. IV.

Regge theory originated from the analysis of scattering amplitudes in complex angular momentum space in 1959 Regge:1959mz , and it is applied to the study of high energy particles physics. Regge theory involves almost all aspects of strong interactions, including hadron spectrum, the forces of particles, and the high energy behavior of scattering amplitudes Collins:1971ff ; Chen:2016spr ; Guo:2008he ; Anisovich:2000kxa ; Masjuan:2012gc ; Pang:2019ovr ; Chen:2018nnr ; Chen:2021kfw ; Chen:2022flh ; Li:2007px .In Regge theory, the pole of split wave amplitude in the plane of the complex angular momentum is called Regge pole, and the curve depended by it moving in the plane when energy changes is called Regge trajectory. The plots of Regge trajectories of hadrons in the $(J,M^{2})$ plane are usually called Chew-Frautschi plots (where $J$ and $M$ are respectively the total spins and the masses of the hadrons). In the past two decades, the quasilinear Regge trajectory was used for studying hadron spectra and result in reasonable description for the hadron spectroscopy Li:2004gu ; Guo:2008he ; Anisovich:2000kxa ; Li:2007px .

In the present work, we investigate the $1^{3}D_{1}$ meson nonet in the framework of quasilinear Regge trajectories. The quasi-linear Regge trajectory for a meson state is usually parameterized as Li:2004gu ; Guo:2008he ; Li:2007px

where $n$ ($n=u$ or $d$ quark), $s$ and $c$ refer to the quark constituents, $J$ and $M$ are the spin and mass of the meson state, respectively. $N$ is the radial quantum number, $L$ is orbital angular momentum. The $\alpha^{\prime}$ and $\alpha$ are the slope and intercept of the Regge trajectory. For a given meson state, the intercept and slope can be expressed by the following relations.

The intercept correlations (7), (8) and (9) were derived from in the dual-resonance model Berezinsky:1969erk , and are satisfied in two-dimensional QCD Brower:1977as , the dual-analytic model Kobylinsky:1978db , and the quark bremsstrahlung model Dixit:1979mz . The slope relations (10), (11) and (12) are obtained in the framework of topological expansion and the $q\bar{q}$-string picture of hadrons Kaidalov:1980bq .

Combining the relations (1)-(6) and (7)-(9) , one obtains

Apply the relations (13), (14) and (15) to the $1^{3}S_{1}$, $1^{3}P_{2}$ and $1^{3}D_{1}$ meson state, the following relations are obtained by eliminating the slopes. Our analysis is based on these assumptions that the slopes of parity partners trajectories coincide and the slopes of ground and radial excited states are the same, which is widely used in Refs. Anisovich:2000kxa ; Li:2007px ; Anisovich:2001ig ; Anisovich:2002us , that is to say, $\alpha^{\prime}_{n\bar{n}({N^{3}S_{1}})}=\alpha^{\prime}_{n\bar{n}({N^{3}P_{2}})}=\alpha^{\prime}_{n\bar{n}({N^{3}D_{1}})}$, $\alpha^{\prime}_{n\bar{s}({N^{3}S_{1}})}=\alpha^{\prime}_{n\bar{s}({N^{3}P_{2}})}=\alpha^{\prime}_{n\bar{s}({N^{3}D_{1}})}$, $\alpha^{\prime}_{s\bar{s}({N^{3}S_{1}})}=\alpha^{\prime}_{s\bar{s}({N^{3}P_{2}})}=\alpha^{\prime}_{s\bar{s}({N^{3}D_{1}})}$, $\alpha^{\prime}_{c\bar{c}({N^{3}S_{1}})}=\alpha^{\prime}_{c\bar{c}({N^{3}P_{2}})}$, $\alpha^{\prime}_{c\bar{n}({N^{3}S_{1}})}=\alpha^{\prime}_{c\bar{n}({N^{3}P_{2}})}$, $\alpha^{\prime}_{c\bar{s}({N^{3}S_{1}})}=\alpha^{\prime}_{c\bar{s}({N^{3}P_{2}})}$.

In the PDG, the $1^{3}S_{1}$ meson multiplet $\rho(770)$, $K^{*}(892)$, $J/\psi(1S)$, $D^{*}$ and $D_{s}^{*\pm}$, $1^{3}P_{2}$ meson multiplet $a_{2}(1320)$, $K_{2}^{*}(1430)$, $\chi_{c2}(1P)$, $D_{2}^{*}(2420)$ and $1^{3}D_{1}$ meson state $\rho(1700)$ are well established. Inserting the masses of these states to the relations (16)-(21), we obtain the following meson masses $M_{n\bar{s}({1^{3}D_{1}})}=1835.1\pm 6.3MeV$, $M_{s\bar{s}({1^{3}D_{1}})}=1944.1\pm 6.6MeV$, $M_{n\bar{s}({2^{3}S_{1}})}=1562.9\pm 9.1MeV$, $M_{s\bar{s}({2^{3}S_{1}})}=1655.9\pm 9.7MeV$.

Next, we discuss the masses of radial excited state of $1^{3}D_{1}$ meson nonet. In the present work, considering the fact that the ground and the radial excitations have the same slopes, we can have the following relations from (1)-(3).

$N$ and $N^{\prime}$ are the radial quantum numbers, $N^{\prime}=1$ is ground state. Based on the assumption that the dispersion of $\alpha_{q\bar{q^{\prime}}N}(0)-\alpha_{q\bar{q^{\prime}}1}(0)$ is flavor independent, the expression $\alpha_{n\bar{n}({N^{\prime 2S+1}L_{J}})}(0)-\alpha_{n\bar{n}({N^{2S+1}L_{J}})}(0)$ can be simplified to $N^{\prime}-N$, which is used in Refs. Anisovich:2000kxa ; Anisovich:2002us ; Anisovich:2003tm . However, by introducing the latest existing experimental data, one can find this assumption will lead to large deviation. After a comprehensive phenomenological analysis of some well-established mesons, Filipponi et al. pointed out that the values of $\alpha_{n\bar{n}({N^{\prime 2S+1}L_{J}})}(0)-\alpha_{n\bar{n}({N^{2S+1}L_{J}})}(0)$, $\alpha_{s\bar{s}({N^{\prime 2S+1}L_{J}})}(0)-\alpha_{s\bar{s}({N^{2S+1}L_{J}})}(0)$ and $\alpha_{n\bar{s}({N^{\prime 2S+1}L_{J}})}(0)-\alpha_{n\bar{s}({N^{2S+1}L_{J}})}(0)$ depend on the constituent quark masses through the combination $m_{i}+m_{j}$ ($m_{i}$ and $m_{j}$ are the constituent masses of quark and antiquark) in Refs. Filipponi:1997hb ; Filipponi:1997vf . In this work, we introduce parameters $f_{i\overline{j}}(m_{i}+m_{j})$ into relations (22)-(24) and result in the following relations (25)-(27)Li:2007px ; Liu:2010zzd .

In relations (25)-(27), we take $m_{n}=0.29GeV$, $m_{s}=0.45GeV$, $\alpha^{\prime}_{n\bar{n}}=0.8830GeV^{-2}$, $\alpha^{\prime}_{n\bar{s}}=0.8493GeV^{-2}$, $\alpha^{\prime}_{s\bar{s}}=0.8181GeV^{-2}$ as input Li:2007px ; Liu:2010zzd . With the aid of the masses $\rho(770)$, $\rho(1450)$, $M_{n\bar{s}}(1^{3}S_{1})$, $M_{n\bar{s}}(2^{3}S_{1})$, $M_{s\bar{s}}(2^{3}S_{1})$, the parameters $f_{n\bar{n}}$, $f_{n\bar{n}}$, $f_{n\bar{n}}$ are determined to be

Based on the relations (25)-(27), the radial excitation masses of the $N^{3}D_{1}$ multiplet can be estimated. Our predictions and those given by other references are listed in Tab. II and Fig. 1.