Construction of $a_4$ family

TL;DR

This paper investigates the mass spectra and decay properties of the $a_4$ meson family using quark models, proposing $a_4(2610)$ as a potential $2H$ state and predicting properties of other states.

Contribution

It applies modified Godfrey-Isgur and quark-pair creation models to analyze the $a_4$ family, offering new assignments and predictions for unobserved states.

Findings

$a_4(2610)$ is a promising $2H$ state candidate.

Predicted masses and widths for $a_4(1H)$ and $a_4(3F)$ states.

Supported identification of $a_4(2610)$ with theoretical models.

Abstract

The COMPASS Collaboration recently reported the observation of a new resonance, $a_4(2610)$, which has sparked our interest in studying the $a_4$ family with {$I^GJ^{PC}=1^{-}4^{++}$}. In this work, we investigate the mass spectra and Okubo-Zweig-Iizuka-allowed two-body strong decays of the $a_4$ family using the modified Godfrey-Isgur quark model and the quark-pair creation model. We also explore the possibility of identifying $a_4(2610)$ as a $4F$ or $2H$ state, {and our numerical results suggest that it could be a promising candidate for the $a_4(2H)$ state. In addition, we predict the masses and the widths of the $a_4(1H)$ and $a_4(3F)$ states.}

Tables

Table 1. Table 1: Resonance parameters of the a 4 a_{4} states. The units of the mass and width are MeV.

State	Mass	Width
$a_4\left(1970\right)$	$1967\pm 16$	${324}_{-18}^{+15}$ Navas and others (2024)
$a_4\left(2255\right)$	$2237\pm 55$	$291\pm 12$ Uman et al. (2006)
	$2255\pm 40$	${330}_{-50}^{+110}$ Anisovich et al. (2001)
$a_4\left(2610\right)$	$2608\pm 9_{-38}^{+5}$	$609\pm {22}_{-311}^{+35}$ Beckers and Haas (2025)

Table 2. Table 2: Parameters of the MGI model  Wang et al. ( 2025 ) . m u ​ ( d ) m_{u(d)} and m s m_{s} are the masses of the u ​ ( d ) u(d) and s s quarks (or antiquarks), μ \mu is the screening parameter, b b is the confining parameter, and c c is the vacuum constant. σ 0 \sigma_{0} is the universal parameter in Eq. ( 2.6 ), while s s is a parameter related to heavy quarkonium masses. ϵ c \epsilon_{c} , ϵ t \epsilon_{t} , ϵ so ​ ( v ) \epsilon_{\rm so(v)} , and ϵ so ​ ( s ) \epsilon_{\rm so(s)} represent the relativistic corrections to the potential terms V ~ cont \tilde{V}^{\mathrm{cont}} , V ~ tens \tilde{V}^{\mathrm{tens}} , V ~ so ​ ( v ) \tilde{V}^{\mathrm{so(v)}} , and V ~ so ​ ( s ) \tilde{V}^{\mathrm{so(s)}} , respectively.

Parameter	value	Parameter	value
$m_{u\left(d\right)}$(GeV)	0.162	$s$	0.711
$m_s$ (GeV)	0.377	$\mu$ (GeV)	0.0779
$b$ (GeV2)	0.222	$c$ (GeV)	$-0.228$
${ϵ}_c$	-0.137	${ϵ}_{so\left(v\right)}$	0.0550
${ϵ}_{so\left(s\right)}$	0.366	${ϵ}_t$	0.493
${\sigma }_0$ (GeV)	1.791	…	…

Table 3. Table 3: The mass spectra of the fitted light meson states. “Expe.” refers to the experimental value, with the units in MeV. The experimental values and most of the errors are taken from the PDG Navas and others ( 2024 ) .

Table 4. Table 4: The mass spectra of a 4 a_{4} states. The unit is MeV.

Table 5. Table 5: Total and partial decay widths of the a 4 ​ ( 1 ​ F ) a_{4}(1F) and a 4 ​ ( 2 ​ F ) a_{4}(2F) states. “Br” denotes the branching ratio of the decay channel. The unit of width is MeV. Channels with widths smaller than 1 MeV are omitted.

Table 6. Table 6: Total and partial decay widths of the a 4 ​ ( 2 ​ H ) a_{4}(2H) and a 4 ​ ( 4 ​ F ) a_{4}(4F) states. The unit of width is MeV. Channels with widths smaller than 1 MeV are omitted.

Table 7. Table 7: Total and partial decay widths of the predicted a 4 ​ ( 1 ​ H ) a_{4}(1H) and a 4 ​ ( 3 ​ F ) a_{4}(3F) states. The unit of width is MeV. Channels with widths smaller than 1 MeV are omitted.

Equations

$$\tilde{H}=\sum _i{(m_i^2+{\mathbf{p}}_i^2)}^{1/2}+{\tilde{V}}^{\mathrm{eff}},$$

$${\tilde{V}}^{\mathrm{eff}}={\tilde{G}}_{12}+{\tilde{V}}^{\mathrm{cont}}+{\tilde{V}}^{\mathrm{tens}}+{\tilde{V}}^{\mathrm{so}(\mathrm{v})}+{\tilde{S}}_{12}(r)+{\tilde{V}}^{\mathrm{so}(\mathrm{s})},$$

$$\tilde{G}(r)=-\sum _k\frac{4{\alpha }_k}{3r}\left[\frac{2}{\sqrt{\pi }}{\int }_0^{{\gamma }_{k}r}{e}^{-x^2}dx\right],$$

$$S(r)=\frac{b(1-e^{-\mu r})}{\mu }+c,$$

$${\rho }_{ij}\left(\mathbf{r}-{\mathbf{r}}^{\prime }\right)=\frac{{\sigma }_{ij}^3}{{\pi }^{3/2}}{e}^{-{\sigma }_{ij}^2{\left(\mathbf{r}-{\mathbf{r}}^{\prime }\right)}^2},$$

$${\sigma }_{ij}^2={\sigma }_0^2[\frac{1}{2}+\frac{1}{2}{\left(\frac{4m_i{m}_j}{(m_i+m_j{)}^2}\right)}^4]+s^2{\left(\frac{2m_i{m}_j}{m_i+m_j}\right)}^2,$$

$$\begin{array}{rl}{\tilde{G}}_{ij}(r)=& \int {\mathrm{d}}^3{\mathbf{r}}^{\prime }{\rho }_{ij}(\mathbf{r}-{\mathbf{r}}^{\prime })G(r^{\prime })=\sum _k-\frac{4{\alpha }_k}{3r}\mathrm{erf}({\tau }_{kij}r),\end{array}$$

$${\tau }_{kij}=\frac{1}{\sqrt{\frac{1}{{\sigma }_{ij}^2}+\frac{1}{{\gamma }_k^2}}}.$$

$${\tilde{S}}_{12}(r)$$

$${\tilde{G}}_{12}(r)\to {\tilde{G}}_{12}={\left(1+\frac{{\mathbf{p}}^2}{E_1{E}_2}\right)}^{1/2}{\tilde{G}}_{12}(r){\left(1+\frac{{\mathbf{p}}^2}{E_1{E}_2}\right)}^{1/2}.$$

$${\tilde{V}}_{\alpha \beta }^i(r)\to {\tilde{V}}_{\alpha \beta }^i={\left(\frac{m_{\alpha }{m}_{\beta }}{E_{\alpha }{E}_{\beta }}\right)}^{1/2+{ϵ}_i}{\tilde{V}}_{\alpha \beta }^i(r){\left(\frac{m_{\alpha }{m}_{\beta }}{E_{\alpha }{E}_{\beta }}\right)}^{1/2+{ϵ}_i},$$

$$\begin{array}{r}{\tilde{V}}^{\mathrm{cont}}=\frac{2{\mathbf{S}}_1\cdot {\mathbf{S}}_2}{3m_1{m}_2}{\nabla }^2{\tilde{G}}_{12}^c,\end{array}$$

$$\begin{array}{rl}{\tilde{V}}^{\mathrm{tens}}=& -\left(\frac{3{\mathbf{S}}_1\cdot \mathbf{r}{\mathbf{S}}_2\cdot \mathbf{r}/r^2-{\mathbf{S}}_1\cdot {\mathbf{S}}_2}{3m_1{m}_2}\right)\left(\frac{{\partial }^2}{\partial r^2}-\frac{1}{r}\frac{\partial }{\partial r}\right){\tilde{G}}_{12}^t,\end{array}$$

$$\begin{array}{rl}{\tilde{V}}^{\mathrm{so}(\mathrm{v})}=& \frac{{\mathbf{S}}_1\cdot \mathbf{L}}{2m_1^2}\frac{1}{r}\frac{\partial {\tilde{G}}_{11}^{\mathrm{so}(\mathrm{v})}}{\partial r}+\frac{{\mathbf{S}}_2\cdot \mathbf{L}}{2m_2^2}\frac{1}{r}\frac{\partial {\tilde{G}}_{22}^{\mathrm{so}(\mathrm{v})}}{\partial r}\\ & +\frac{({\mathbf{S}}_1+{\mathbf{S}}_2)\cdot \mathbf{L}}{m_1{m}_2}\frac{1}{r}\frac{\partial {\tilde{G}}_{12}^{\mathrm{so}(\mathrm{v})}}{\partial r},\end{array}$$

$$\begin{array}{rl}{\tilde{V}}^{\mathrm{so}(\mathrm{s})}=& -\frac{{\mathbf{S}}_1\cdot \mathbf{L}}{2m_1^2}\frac{1}{r}\frac{\partial {\tilde{S}}_{11}^{\mathrm{so}(\mathrm{s})}}{\partial r}-\frac{{\mathbf{S}}_2\cdot \mathbf{L}}{2m_2^2}\frac{1}{r}\frac{\partial {\tilde{S}}_{22}^{\mathrm{so}(\mathrm{s})}}{\partial r}.\end{array}$$

$$\begin{array}{r}{\psi }_{nL{M}_L}^{SHO}(\mathbf{r})=R_{nL}^{SHO}(r,\beta )Y_{L{M}_L}({\Omega }_r),\\ {\psi }_{nL{M}_L}^{SHO}(\mathbf{p})=R_{nL}^{SHO}(p,\beta )Y_{L{M}_L}({\Omega }_p),\end{array}$$

$$R_{nL}^{SHO}(r,\beta )=N_{nL}{\beta }^{3/2}(\beta r{)}^L{e}^{\frac{-r^2{\beta }^2}{2}}{L}_{n-1}^{L+1/2}({\beta }^2{r}^2),$$

$$R_{nL}^{SHO}(p,\beta )=\frac{(-1{)}^{(n-1)}(-i{)}^L}{{\beta }^{3/2}}{N}_{nL}{e}^{-\frac{p^2}{2{\beta }^2}}{(\frac{p}{\beta })}^L\times L_{n-1}^{L+1/2}(\frac{p^2}{{\beta }^2}),$$

$$N_{nL}=\sqrt{\frac{2(n-1)!}{\Gamma (n+L+1/2)}},$$

$$R_{nL}(r)=\sum _{n=1}^{n_{\max }}{C}_n{R}_{nL}^{\mathrm{SHO}}(r,\beta ),$$

$$R_{nL}(p)=\sum _{n=1}^{n_{\max }}{C}_n{R}_{nL}^{\mathrm{SHO}}(p,\beta )$$

$$\mathcal{T}=$$

$$\times {\mathcal{Y}}_{1m}\left(\frac{{\text{p}}_3-{\mathbf{p}}_4}{2}\right){\chi }_{1,-m}^{34}{\varphi }_0^{34}{\left({\omega }_0^{34}\right)}_{ij}{b}_{3i}^{†}({\mathbf{p}}_3)d_{4j}^{†}({\mathbf{p}}_4).$$

$$⟨\mathrm{BC}|\mathcal{T}|\mathrm{A}⟩={\delta }^3({\mathbf{P}}_{\mathrm{B}}+{\mathbf{P}}_{\mathrm{C}}){\mathcal{M}}^{M_{J_{\mathrm{A}}}{M}_{J_{\mathrm{B}}}{M}_{J_{\mathrm{C}}}},$$

$${\Gamma }_{\mathrm{A}\to \mathrm{BC}}$$

$${\mathcal{M}}^{JL}(\mathbf{P})=$$

$$\times ⟨J_{\mathrm{B}}{M}_{J_{\mathrm{B}}};J_{\mathrm{C}}{M}_{J_{\mathrm{C}}}|J_{\mathrm{A}}{M}_{J_{\mathrm{A}}}⟩{\mathcal{M}}^{M_{J_{\mathrm{A}}}{M}_{J_{\mathrm{B}}}{M}_{J_{\mathrm{C}}}},$$

$${\mathcal{M}}^{M_{J_{\mathrm{A}}}{M}_{J_{\mathrm{B}}}{M}_{J_{\mathrm{C}}}}$$

$$=\gamma \sum _{\begin{array}{c}{M}_{L_{\mathrm{A}}},M_{S_{\mathrm{A}}},M_{L_{\mathrm{B}}},\\ M_{S_{\mathrm{B}}}{M}_{L_{\mathrm{C}},M_S},m\end{array}}⟨L_{\mathrm{A}}{M}_{L_{\mathrm{A}}}{S}_{\mathrm{A}}{M}_{S_{\mathrm{A}}}\mid J_{\mathrm{A}}{M}_{J_{\mathrm{A}}}⟩$$

$$\times ⟨L_{\mathrm{B}}{M}_{L_{\mathrm{B}}}{S}_{\mathrm{B}}{M}_{S_{\mathrm{B}}}\mid J_{\mathrm{B}}{M}_{J_{\mathrm{B}}}⟩⟨L_{\mathrm{C}}{M}_{L_{\mathrm{C}}}{S}_{\mathrm{C}}{M}_{S_{\mathrm{C}}}\mid J_{\mathrm{C}}{M}_{J_{\mathrm{C}}}⟩$$

$$\times ⟨1m1-m\mid 0⟩⟨{\chi }_{S_{\mathrm{B}}{M}_{S_{\mathrm{B}}}}^{14}{\chi }_{S_{\mathrm{C}}{M}_{S_{\mathrm{C}}}}^{32}\mid {\chi }_{S_{\mathrm{A}}{M}_{S_{\mathrm{A}}}}^{12}{\chi }_{1-m}^{34}⟩$$

$$\times [⟨{\varphi }_{\mathrm{B}}^{14}{\varphi }_{\mathrm{C}}^{32}\mid {\varphi }_{\mathrm{A}}^{12}{\varphi }_0^{34}⟩I\left(\mathbf{P},m_1,m_2,m_3\right)+$$

$$(-1{)}^{1+S_{\mathrm{A}}+S_{\mathrm{B}}+S_{\mathrm{C}}+L_{\mathrm{C}}}⟨{\varphi }_{\mathrm{B}}^{32}{\varphi }_{\mathrm{C}}^{14}\mid {\varphi }_{\mathrm{A}}^{12}{\varphi }_0^{34}⟩I\left(-\mathbf{P},m_2,m_1,m_3\right)],$$

$$I\left(\mathbf{P},m_1,m_2,m_3\right)$$

$$=\int d^3\mathbf{k}{\psi }_{\mathrm{B}}^{\ast }(\mathbf{k}+U\mathbf{P}){\psi }_{\mathrm{C}}^{\ast }(\mathbf{k}+V\mathbf{P}){\psi }_{\mathrm{A}}(\mathbf{k}-\mathbf{P}){\mathcal{Y}}_1^m(\mathbf{k}),$$

Full text

I introduction

Very recently, the COMPASS Collaboration announced a new resonance [we denote it as $a_{4}(2610)$], with a mass around 2.6 GeV in the $KK$ final state Beckers and Haas (2025). The corresponding mass and width were determined to be $2608\pm 9^{+5}_{-38}$ MeV and $609\pm 22^{+35}_{-311}$ MeV, respectively. Meanwhile, we note that the PDG “Further States” also lists an $a_{4}(2255)$ meson Navas and others (2024), which has sparked our interest in investigating the internal structure of both $a_{4}(2255)$ and $a_{4}(2610)$. Together with the ground state, the $a_{4}(1970)$ meson, establishing the $a_{4}$ meson family would be highly significant for completing the light meson family.

The $a_{4}$ meson family is characterized by $I^{G}J^{PC}=1^{-}4^{++}$, which corresponds to spin $S=1$ and total angular momentum $J=4$. Due to the constraints from parity and spin-orbit coupling, the relative orbital angular momentum can take two possible values, $L=3$ and $L=5$, corresponding to the $F$-wave and $H$-wave, respectively.

In 1977, the Omega Group at CERN observed an $a_{4}$ state with a mass of $2030$ MeV and a width of $510\pm 200$ MeV in a partial wave analysis (PWA) of $3\pi$ system production in the reaction $\pi^{-}p\to 3\pi n$ at 12 and 15 GeV/$c$ Corden and others (1978), which is the $a_{4}(1970)$ listed in PDG Navas and others (2024). $a_{4}(1970)$ was also found in the reactions $\pi^{\pm}p\to K_{s}^{0}K^{\pm}p$ Cleland and others (1982), $\pi^{-}p\to\eta\pi^{0}n$ Alde and others (1996), $\pi^{-}A\to\omega\pi^{-}\pi^{0}A^{*}$ Amelin and others (1999), $\pi^{-}p\to\eta^{\prime}\pi^{-}p$ Ivanov and others (2001), $\pi^{-}p\to\omega\pi^{-}\pi^{0}p$ Lu and others (2005), and $\pi^{-}P_{b}\to\omega\pi^{-}\pi^{-}\pi^{+}P_{b}^{\prime}$ Lu and others (2005). Further evidence for this state was also found in many other experiments Baldi et al. (1978); Anisovich et al. (2001); Chung and others (2002); Uman et al. (2006); Alekseev and others (2010); Adolph and others (2015); Aghasyan and others (2018). The latest observation of $a_{4}(1970)$ was reported by the COMPASS Collaboration, which provides the most precise measurement of the resonance parameters $M=1952.2\pm 1.8^{+3}_{-3.5}$ MeV and $\Gamma=324\pm 4\pm 6$ MeV Beckers and Haas (2025).

In 2001, Anisovich et al. reported evidence for $a_{4}(2255)$ in the reactions $p\bar{p}\to\pi^{0}\eta,3\pi^{0},\text{and}~\pi^{0}\eta^{\prime}$ Anisovich et al. (2001). Uman et al. confirmed the state $a_{4}(2255)$ use the data from the Fermilab E835 experiment in the reaction $p\bar{p}\to\eta\eta\pi^{0}$ Uman et al. (2006).

$a_{4}(1970)$ is well established as the ground state of the $a_{4}$ family Bramon and Masso (1981); Godfrey and Isgur (1985); Ebert et al. (2009); Navas and others (2024). In our previous work, $a_{4}(2255)$ was suggested to be the $2^{3}F_{4}$ state, the first radial excitation of $a_{4}(1970)$ Pang et al. (2015), which is consistent with the conclusions of Refs. Afonin (2007); Anisovich et al. (2000); Anisovich (2004a, b); Masjuan et al. (2012).

Ebert et al. obtained a $1^{3}H_{4}$ state with a mass of $M=2234$ MeV, close to that of $a_{4}(2255)$ Ebert et al. (2009). However, the PWA in Ref. Anisovich et al. (2001) showed that $a_{4}(2255)$ corresponds to a ${}^{3}F_{4}$ state.

Now, with the discovery of $a_{4}(2610)$, the $a_{4}$ family already contains three members: $a_{4}(1970)$, $a_{4}(2255)$, and $a_{4}(2610)$, as listed in Table 1. The study of internal structure of $a_{4}(2610)$ and the construction of the $a_{4}$ family has become an interesting and useful issue.

Mass spectral analysis and two-body strong decay are always used to probe the internal structure of mesons. Strong interactions in light meson systems cannot be calculated directly from first principles (except through Lattice Quantum Chromodynamics(QCD)), owing to the non-perturbative nature of QCD Prelovsek (2025). Consequently, phenomenological frameworks such as QCD sum rules, potential models Godfrey and Isgur (1985); Ebert et al. (2009), and the ${}^{3}P_{0}$ model Micu (1969) are routinely employed to describe their spectra and decays. In fact, both lattice calculations and QCD sum rules remain challenging, especially for higher excitations. Phenomenological approaches are commonly adopted to explore their properties. For example, the Godfrey-Isgur (GI) potential model is often used to investigate meson spectra Godfrey and Isgur (1985); Godfrey et al. (2016), While the GI model adequately reproduces the spectra of low-lying mesons, its extension to highly excited states requires the inclusion of color screening effects Laermann et al. (1986); Born et al. (1989); Knechtli and Sommer (2000); Chao et al. (1992); Ding et al. (1993). Song et al. modified the GI model (named as the MGI model) by taking into account the color screening effects, which has proven successful in describing charmed mesons Song et al. (2015). Subsequently, this framework has been successfully applied to doubly heavy flavor mesons, light mesons and baryons Wang et al. (2018, 2019); Weng et al. (2024). For describing the nature of higher excitations of the $a_{4}$ family, the MGI model is adopted to study their property. The quark pair creation (QPC) model, also known as the ${}^{3}P_{0}$ model, is an effective model to study the two-body strong decays of mesons Micu (1969); Ackleh et al. (1996), which is well suited for this work.

In our previous work, we adopted the MGI model and the QPC model to predict the spectrum and two-body decay properties of the $5^{++}$ meson family, respectively Pang (2019). In this work, these two models are suited to study the spectrum and two-body decays of the $a_{4}$ family. We hope that our effort will be helpful in revealing the internal structure of $a_{4}(2255)$ and $a_{4}(2610)$, and in establishing the $a_{4}$ meson family.

The paper is organized as follows: In Sec. II, we briefly review the MGI model and the ${}^{3}P_{0}$ model. In Sec. III, our numerical results for the $a_{4}$ family are presented. We first verify the assignment of the $a_{4}(2040)$ $a_{4}(1970)$ and the $a_{4}(2255)$. Then, we study the newly observed $a_{4}(2610)$ and provide a prediction of the $a_{4}(1H)$ and $a_{4}(3F)$. Finally, a conclusion is given in Sec. IV.

II Models employed in this work

In this section, we introduce the MGI model and the QPC model employed in this work.

II.1 The modified GI model

Based on the GI model, which was proposed by Godfrey and Isgur in 1985 to describe relativistic meson spectra with great success, especially for low-lying mesons Godfrey and Isgur (1985), Song et al. proposed the MGI model Song et al. (2015), in which a screened potential term was introduced to describe the charmed and charmed-strange meson’s excited states better. Since then, the MGI model has been applied to study the light meson spectroscopy Pang et al. (2019); Wang et al. (2022b, a); Feng et al. (2022); Wang et al. (2024, 2025), double heavy quarkonium Wang et al. (2018, 2019), and baryons Weng et al. (2024). The Hamiltonian of the MGI model reads:

[TABLE]

where $m_{i}$ denotes the mass of the quark (or antiquark). The mass of the strange quark (or antiquark) $m_{s}=0.377$ GeV, the up and down quark (or antiquark) $m_{u(d)}=0.162$ GeV are adopted. The effective potential $\tilde{V}^{\mathrm{eff}}$ includes the following items:

[TABLE]

where the individual terms are interpreted as follows:

• •

$\tilde{G}_{12}$ is referred to as the Coulomb term, which is understood to arise from one-gluon exchange.

• •

$\tilde{V}^{\mathrm{cont}}$ is referred to as the contact term, which is introduced to account for the short-range interaction between quarks.

• •

$\tilde{V}^{\mathrm{tens}}$ is denoted as the tensor term, which is generated by the spin–spin interaction from one-gluon exchange.

• •

$\tilde{V}^{\mathrm{so(v)}}$ is referred to as the vector spin–orbit term.

• •

$\tilde{S}_{12}(r)$ is denoted as the screened confinement term, which is introduced to reflect the color-screening effect.

• •

$\tilde{V}^{\mathrm{so(s)}}$ is referred to as the scalar spin–orbit term.

The explicit forms and detailed discussions of these potential terms are presented below.

The spin-independent terms of the nonrelativistic potential are given by

[TABLE]

with $\alpha_{k}=(0.25,0.15,0.2)$ and $\gamma_{k}=(1/2{\,\text{GeV}},\sqrt{10}/2{\,\text{GeV}},\sqrt{1000}/2{\,\text{GeV}})$ for $k=1,2,3$ Godfrey and Isgur (1985), and

[TABLE]

where $\mu=0.0779$ GeV is adopted as the screening parameter from our previous work Wang et al. (2025). This parameter characterizes the strength of the color screening effect, which is absent in the GI model, and thus represents an improvement of the MGI model. The confining parameter is taken as $b=0.222$ GeV2, while the vacuum constant is $c=-0.228$ GeV Wang et al. (2025).

There are two methods to characterize the effective potential $\tilde{V}^{\mathrm{eff}}$ when relativistic effects in meson systems are taken into account. The first method introduces a smearing function, defined as follows:

[TABLE]

with

[TABLE]

where $\sigma_{0}=1.791$ GeV is a universal parameter and $s=0.711$. The values of these two parameters, listed in Table 2, are taken from Ref. Wang et al. (2025). The Coulomb term $\tilde{G}_{12}(r)$ is then written as

[TABLE]

where

[TABLE]

The confinement potential $\tilde{S}_{12}(r)$ can be expressed as

[TABLE]

The second method introduces the momentum-dependent factors

[TABLE]

The semirelativistic corrections of the spin-dependent terms are written as

[TABLE]

where $\tilde{V}^{i}_{\alpha\beta}(r)$ denote the contact term, the tensor term, the vector, and the scalar spin-orbit terms. The parameters $\epsilon_{i}=\epsilon_{c}$, $\epsilon_{t}$, $\epsilon_{\rm so(v)}$, and $\epsilon_{\rm so(s)}$ represent the relativistic corrections to $\tilde{V}^{\mathrm{cont}}$, $\tilde{V}^{\mathrm{tens}}$, $\tilde{V}^{\mathrm{so(v)}}$, and $\tilde{V}^{\mathrm{so(s)}}$, respectively Wang et al. (2022a). Then the explicit forms of the spin-dependent potentials are

[TABLE]

[TABLE]

[TABLE]

[TABLE]

We use the simple harmonic oscillator (SHO) basis to solve the spectrum of the light mesons. The SHO wave functions are given by

[TABLE]

with

[TABLE]

and normalization factor

[TABLE]

where $Y_{LM_{L}}(\Omega)$ denotes the spherical harmonic function, $L_{n-1}^{L+1/2}(x)$ the associated Laguerre polynomial and $\Gamma(n+L+1/2)$ the gamma function. And

[TABLE]

are the spatial wave functions of the mesons, where $C_{n}$ are the expansion coefficients, which can be derived through the process of diagonalizing the Hamiltonian (2.1). Then the SHO wave function depends only on a single parameter $\beta$, which is determined by minimizing the eigenvalue $E_{nL}$, i.e., $\partial E_{nL}/\partial\beta_{i}=0$ and $\partial^{2}E_{nL}/\partial\beta_{i}^{2}>0$, where $i$ labels different light mesons, and $n_{max}=21$ is adopted in the present calculation. The spatial wave functions of the mesons obtained with the MGI model are then applied to the calculation of strong decay processes.

Finally, it is important to note that the MGI model provides a global fit to the light meson spectrum with a $\chi^{2}$ value of about 40, as shown in Table 3. The 11 parameters used in the model are not set arbitrarily, but are determined by fitting to the experimental data. Although this $\chi^{2}$ value is not particularly small in the context of data fitting, it remains reasonable within the framework of the potential models for meson spectroscopy. If we use the mean relative error of 1.3% for the 44 experimental mass values, according to $\chi^{2}=\left(\frac{M_{\text{Expe.}}-M_{\text{theory}}}{\text{error}}\right)^{2}$, one can obtain that the theoretical mass $M_{\text{theory}}\approx M_{\text{Expe.}}\times(1\pm 0.1)$, indicating a relative error of about 10% in the fitting results. Therefore, the internal structure of $a_{4}(2610)$ cannot be determined solely from its mass spectrum. Further investigation into its strong decay properties is required, which will be facilitated by the ${}^{3}P_{0}$ model introduced in the following section.

II.2 The QPC model

The QPC model, also known as ${}^{3}P_{0}$ model, was first proposed by Micu Micu (1969) and further developed by the Orsay group Le Yaouanc et al. (1973, 1974, 1975, 1977b, 1977a). This model has been widely applied to the calculation of OZI-allowed two-body strong decays of mesons van Beveren et al. (1983); Titov et al. (1996); Ackleh et al. (1996); Blundell (1996); Bonnaz et al. (2002); Zhou et al. (2005); Lu et al. (2006); Zhang et al. (2007); Luo et al. (2009); Sun and Liu (2009); Liu et al. (2010); Sun et al. (2010); Rijken et al. (2010); Ye et al. (2012); Wang et al. (2012); He et al. (2013); Sun et al. (2013); Pang et al. (2019); Wang et al. (2023, 2022b); Li et al. (2023b, a); Wang et al. (2020); Pang et al. (2017); Wang et al. (2024, 2022a); Feng et al. (2021, 2022). Recent studies Bruschini et al. (2025); Alkofer et al. (2024) further provide theoretical support for the rationality of this model. In this model, the transition operator $\mathcal{T}$ describes the creation of a quark-antiquark pair (denoted by indices 3 and 4) from the vacuum with quantum numbers ${}^{2S+1}L_{J}=^{3}P_{0}$, and can be written as

[TABLE]

Here, the parameter $\gamma$ in QPC model represents the strength of $q\bar{q}$ pair creation from the vacuum, and in this work the value $\gamma=10.16$ is adopted Wang et al. (2025). $\mathcal{Y}_{l}^{m}(\bf{p})\equiv$ $p^{l}Y_{l}^{m}(\theta_{p},\phi_{p})$ denotes a solid harmonic. The symbols $\chi$, $\phi$, and $\omega$ denote the spin, flavor, and color wave functions, respectively. $\mathbf{p}_{3}$ and $\mathbf{p}_{4}$ are the three-momenta of the quark–antiquark pair created from the vacuum, while $i$ and $j$ are their color indices. With this transition operator, the decay amplitudes of mesons can be systematically calculated within the QPC framework. The amplitude $\mathcal{M}^{{M}_{J_{\mathrm{A}}}M_{J_{\mathrm{B}}}M_{J_{\mathrm{C}}}}$ is defined as

[TABLE]

where $\mathbf{P}_{\mathrm{B}}$ and $\mathbf{P}_{\mathrm{C}}$ are the three-momenta of mesons $\mathrm{B}$ and $\mathrm{C}$ in the rest frame of the meson $\mathrm{A}$, and ${M}_{J_{X}}$ (with $X=\mathrm{A},\mathrm{B},\mathrm{C}$) denotes the magnetic quantum number of the corresponding meson. Finally, the general form of the decay width can be expressed as

[TABLE]

where $m_{\mathrm{A}}$ is the mass of the initial meson $\mathrm{A}$, $\mathbf{J}=\mathbf{J}_{\mathrm{B}}+\mathbf{J}_{\mathrm{C}}$, $L$ is the relative orbital angular momentum between mssons $\mathrm{B}$ and $\mathrm{C}$, and $\mathbf{P}=\mathbf{P}_{\mathrm{B}}$. The partial-wave amplitude $\mathcal{M}^{JL}(\mathbf{P})$ is related to the amplitude $\mathcal{M}^{{M}_{J_{\mathrm{A}}}M_{J_{\mathrm{B}}}M_{J_{\mathrm{C}}}}$ via the Jacob–Wick formula Jacob and Wick (1959)

[TABLE]

in which

[TABLE]

with the overlap integral

[TABLE]

where

[TABLE]

Here, $m_{1}$ and $m_{2}$ are the masses of the quark and the antiquark in meson $\mathrm{A}$, respectively. In this work, $m_{1}=0.377$ GeV, $m_{2}=0.162$ GeV for $s\bar{n}$ (or $n\bar{s}$), $m_{1}=m_{2}=0.162$ GeV for $n\bar{n}$, and $m_{1}=m_{2}=0.377$ GeV for $s\bar{s}$. The mass of the created quark (or antiquark) from the vacuum is denoted as $m_{3}$, and is taken as 0.162 GeV for $n\bar{n}$ and 0.377 GeV for $s\bar{s}$.

The spatial wave functions of mesons obtained using the MGI model are employed in the calculation of the strong decays of the $a_{4}$ family within the QPC model as we mentioned previously.

For the final states, the mixing scheme of strange mesons with natural parity ($L=J$) can be expressed as

[TABLE]

where $\theta_{nL}$ denotes as the mixing angle between the $K(n^{1}L_{L})$ and $K(n^{3}L_{L})$ states. In this work, the masses of ${K(nL)}$ and ${K^{\prime}(nL)}$ are calculated using the MGI model Pang et al. (2025). The mixing angle is taken as $\theta_{1P}=-34^{\circ}$ Cheng (2013). For other cases, the mixing angle is given by $\theta_{nL}=-\text{arctan}(\sqrt{\frac{L}{L+1}})$ Asghar et al. (2019).

The flavor wave functions of isoscalar mesons can be expressed in the mixing form

[TABLE]

where $X$ and $X^{\prime}$ denote two isoscalar mesons (such as $\eta$ and $\eta^{\prime}$), $\phi_{x}$ is the mixing angle in the quark-flavor scheme, and the light nonstrange component is defined as $n\bar{n}=(u\bar{u}+d\bar{d})/\sqrt{2}$. The flavor mixing information of the isoscalar mesons used in this work is adopted from Ref. Pang et al. (2025).

III Numerical results and phenomenological analysis

The spectrum of the $a_{4}$ family is calculated using the MGI model and listed in Table 4. The OZI-allowed two-body strong decay properties of the $a_{4}$ family are presented in Tables 5-7. We now turn to a phenomenological analysis of the spectrum and the decay information of the $a_{4}$ family.

III.1 Verification of the assignment of the $a_{4}(2040)$ $a_{4}(1970)$ and the $a_{4}(2255)$

As the ground state of the $a_{4}$ meson family, $a_{4}(1970)$ has been well established both theoretically and experimentally Corden and others (1978); Baldi et al. (1978); Cleland and others (1982); Amelin and others (1999); Anisovich et al. (2001); Chung and others (2002); Bramon and Masso (1981); Godfrey and Isgur (1985); Ebert et al. (2009); Navas and others (2024).

In 1978, M. J. Corden et al. collected data on the charge-exchange reaction $\pi^{-}p\to 3\pi n$ at beam momenta of 12 and 15 GeV/$c$ using the CERN Omega Multiparticle Spectrometer. A natural spin-parity enhancement was observed at a mass of about 2 GeV/$c^{2}$ with $J^{P}=4^{+}$ preferred, which was denoted $A_{2}^{*}(2030)$ ($a_{4}(1970)$ in this work) Corden and others (1978). In the same year, R. Baldi et al. reported the observation of an $I^{G}J^{P}=1^{-}4^{+}$ ($a_{4}(1970)$ in this work) state in the reaction $\pi^{-}p\to K_{S}^{0}K^{-}p{)}$ at 10 GeV/$c$, measured with a nonmagnetic spectrometer at the CERN proton synchrotron (PS) Baldi et al. (1978). Evidence for the $a_{4}(1970)$ meson was obtained at a mass of about $1900$ MeV with a width of about 200 MeV Baldi et al. (1978). In 1982, W. E. Cleland et al. analysed the reaction $\pi^{\pm}p\to K_{S}^{0}K^{\pm}p$ at 30 and 50 GeV/$c$ Cleland and others (1982). This analysis confirmed the spin-4 $A_{2}(2040)$ ($a_{4}(1970)$ in this work) state Cleland and others (1982). A signal with nine standard deviations in both beam polarities at 50 GeV/$c$ yielded a $K\bar{K}$ cross section of $0.50\pm 0.09$ $\mu$b for this resonance, with mass and width determined as $M=2040\pm 30$ MeV and $\Gamma=380\pm 150$ MeV Cleland and others (1982). By 1999, $a_{4}(1970)$ was once again experimentally observed in the reaction $\pi^{-}A\to\omega\pi^{-}\pi^{0}A^{*}$ Amelin and others (1999). Since then, a considerable amount of experimental data on $a_{4}(1970)$ has emerged. In 2001, A. V. Anisovich et al. reported a combined analysis of $3\pi^{0}$, $\pi^{0}\eta$ and $\pi^{0}\eta^{\prime}$ data in the mass range 1960–2410 MeV and found $a_{4}(1970)$ again with mass and width of $2005^{+25}_{-45}$ MeV and $180\pm 30$ MeV Anisovich et al. (2001). Subsequently, $a_{4}(1970)$ was verified in reactions $\pi^{-}p\to\eta^{\prime}\pi^{-}p$ Ivanov and others (2001), $\pi^{-}p\to 3\pi p$ Chung and others (2002), $\pi^{-}p\to\omega\pi^{-}\pi^{0}p$ Lu and others (2005), and $\bar{p}p\to\eta\eta\pi^{0}$ Uman et al. (2006). In 2010, the COMPASS experiment at the CERN SPS studied the diffractive dissociation of negative pions into the $\pi^{-}\pi^{-}\pi^{+}$ final state using a 190 GeV/$c$ pion beam on a lead target, and clearly confirmed the $a_{4}(1970)$ with a mass of $1885\pm 13^{+50}_{-2}$ MeV and a width of $294\pm 25^{+46}_{-19}$ MeV Alekseev and others (2010). Later in 2015, the COMPASS Collaboration observed $a_{4}$ in the $\pi^{-}p\to\eta^{(\prime)}\pi^{-}p$ process with a mass and width of $1900^{+80}_{-20}$ MeV/$c^{2}$ and $300^{+80}_{-100}$ MeV/$c^{2}$, respectively Adolph and others (2015). In 2018, COMPASS performed a most comprehensive resonance-model fit of $\pi^{-}\pi^{-}\pi^{+}$ states based on their PWA of a large data set of diffractive-dissociation events from the reaction $\pi^{-}+p\to\pi^{-}\pi^{-}\pi^{+}+p_{\text{recoil}}$ with a 190 GeV/$c$ pion beam. The mass and width of $a_{4}(1970)$ were reported as $1935^{+11}_{-13}$ MeV and $333^{+16}_{-21}$ MeV, respectively Aghasyan and others (2018). The most recent measurement of $a_{4}(1970)$ by COMPASS indicates that the mass and width of $a_{4}(1970)$ are $1952.2\pm 1.8^{+3}_{-3.5}$ MeV and $327\pm 4\pm 6$ MeV, respectively Beckers and Haas (2025).

The mass of $a_{4}(1970)$ obtained in our calculation is 1928 MeV, as shown in Table 4, which is slightly lower than that in Refs. Godfrey (1985); Ebert et al. (2009), and is closest to the experimental value Navas and others (2024). The OZI-allowed two-body strong decay behavior of $a_{4}(1970)$ is presented in Table 5. The total width of $a_{4}(1970)$ is 312 MeV according to our calculation, which agrees well with the experimental width of $324^{+15}_{-18}$ MeV Navas and others (2024). The partial decay widths of $\rho\omega$, $\pi\rho$, $\pi b_{1}$, and $\pi f_{2}$ are found to be 106 MeV, 68.5 MeV, 55.3 MeV and 33.7 MeV, respectively. It is worth noting that the branching ratio $\Gamma(\rho\pi)/\Gamma(f_{2}\pi)$ is calculated to be $2.0$, which is consistent with the experimental value of 1.7${}^{+0.9}_{-0.8}$ Chung and others (2002); Aghasyan and others (2018). In addition, the ratio $\Gamma(\eta^{\prime}\pi)/\Gamma(\eta\pi)$ is abtained obtained as $0.2$, in agreement with the experimental value of 0.23$\pm 0.07$ Alde and others (1996); Adolph and others (2015). We also consider the $\gamma$ dependence of the total decay width of $a_{4}(1970)$ in the range of 6$\sim$14, as shown in Fig. 1. The corresponding experimental data for comparison with our theoretical calculation are also presented. We find that when $\gamma$ is between 10 and 10.75, our theoretical calculation matches the experimental width of $a_{4}(1970)$, as depicted in Fig. 1. As mentioned in Ref. Pang et al. (2025), the ratios of the two-body strong decay channels are independent of the value of $\gamma$.

The $a_{4}(2255)$ is listed in the “Further States” in PDG. It was first observed in 2001 with a mass and width of $2255\pm 40$ MeV and $330^{+110}_{-50}$ MeV, respectively Anisovich et al. (2001). Uman et al. reported a strong resonance, $a_{4}(2237)$, decaying into $\eta\pi$, with $M/\Gamma=(2237\pm 5)/(291\pm 15)$ MeV Uman et al. (2006). The $a_{4}(2237)$ resonance is the $a_{4}(2255)$. The mass of $a_{4}(2255)$ we calculated by the MGI model is 2243 MeV, which is in better agreement with the experimental values of 2237 MeV Uman et al. (2006) and 2255 MeV Anisovich et al. (2001). The $a_{4}(2255)$ is considered to be a strong candidate for the $2F$ state and the OZI-allowed two-body strong decay width is calculated to be 222 MeV, which is comparable to the experimental values Anisovich et al. (2001); Uman et al. (2006). The decay channels $\pi f_{2}(1640)$, $\rho\omega$, $\pi b_{1}(2P)$, and $\pi\rho(1450)$ make significant contributions to the total width of the $a_{4}(2255)$. The channels $\rho a_{2}$, $\pi\rho_{3}$, $\pi\eta(1295)$ and $\rho h_{1}$ have branching ratios of $7.4\%$, $5.6\%$, $4.7\%$ and $3.4\%$, respectively. It is worth noting that the $\eta\pi$ channel, with a branching fraction of $0.5\%$, is one of the decay modes through which the $a_{4}(2255)$ resonance was discovered experimentally Anisovich et al. (2001); Uman et al. (2006). More details can been seen in Table 5. In Fig. 2, the $\gamma$ dependence of the total decay width of $a_{4}(2255)$ within the range of $\gamma=6\sim 14$ is presented. Our calculation is consistent with the experimental widths from SPEC Anisovich et al. (2001) when $\gamma$ is between 10 and 14, and with E835 Uman et al. (2006) when $\gamma$ is between 11.5 and 12.

Our results strongly favor interpreting $a_{4}(2255)$ as the $2^{3}F_{4}$ state, i.e., the first radial excitation of $a_{4}(1970)$, thereby reinforcing the reliability of the MGI+QPC framework in describing the $a_{4}$ meson family and providing a basis for the discussion of even higher excitations such as $a_{4}(2610)$.

III.2 Study of the new state $a_{4}(2610)$

Recently, COMPASS observed a broad shoulder at a mass above that of the $a_{4}(1970)$ in the $KK$ final state. Their fit indicated that, on top of the non-resonant background, an additional $a_{4}$ state was needed to accurately describe the spectrum of the $J^{PC}=4^{++}$ wave Beckers and Haas (2025). The corresponding mass and width were determined to be $2608\pm 9^{+5}_{-38}$ MeV and $609\pm 22^{+35}_{-311}$ Mev, respectively Beckers and Haas (2025).

For this newly observed $a_{4}(2610)$, we test the possibility of $a_{4}(2610)$ being the candidate for $a_{4}(4F)$ or $a_{4}(2H)$ by analyzing its mass spectrum and two-body strong decay properties.

According to the analysis of the mass spectrum (Table 4) by the MGI model, the theoretical mass of the $a_{4}(2H)$ state, $2589~\text{MeV}$, is in closer agreement with the experimental value of $2608\pm 9^{+5}_{-38}~\text{MeV}$ than the predicted mass of the $a_{4}(4F)$ state, $2640~\text{MeV}$.

Subsequently, we investigate the two-body strong decay behaviors based on that the $a_{4}(2610)$ is the candidate for $a_{4}(4F)$ or $a_{4}(2H)$, as shown in Table 6. When we assign $a_{4}(2610)$ as a candidate for the $a_{4}(4F)$ state, we obtain the total decay width of 174 MeV and the most important decay channels $\rho a_{2}(1700)$, $\pi b_{1}(2240)$, $\rho a_{2}$, $\pi b_{1}(1960)$ and $\pi\rho_{3}(1990)$. In addition, the decay modes $\pi f_{2}(1950)$, $\pi b_{1}$, $\rho\omega_{3}$, $\pi\rho_{3}$ and $\pi\rho$ also make contribution to the total decay width.

When we assign $a_{4}(2610)$ as a candidate for the $a_{4}(2H)$ state, the two-body strong decay width is found to be 665 MeV. Besides, $\pi\rho_{4}(2G)$, $\pi b_{3}(2244)$ and $\pi\eta_{4}(2G)$ are the mainly decay channels, while $\pi b_{3}$, $\pi\rho_{2}(1D)$, $\pi\eta_{2}$ and $\pi f_{3}$ serve as important decay modes. In addition, $a_{4}(2610)$ can also decay into $\pi f_{3}(1F)$, $\pi\rho_{4}$, $\pi b_{1}$ and other channels exhibited in Table 6. The predicted width of the $a_{4}(2H)$, $\Gamma^{\text{th}}_{a_{4}(2H)}=665~\text{MeV}$, is in much better agreement with the experimental value of $609\pm 22^{+35}_{-311}~\text{MeV}$, compared with the narrower theoretical width of the $a_{4}(4F)$, $\Gamma^{\text{th}}_{a_{4}(4F)}=174~\text{MeV}$.

On the other hand, the branching ratio of the $KK$ decay channel may provide useful information, since the $a_{4}(2610)$ was discovered in the $KK$ final state. In our calculation, $\text{Br}(KK)=0.005\%$ for $a_{4}(2H)$ and $\text{Br}(KK)=0.001\%$ for $a_{4}(4F)$, which are not listed in Table 6. This result implies that the $a_{4}(2610)$ is more likely to be assigned as the $a_{4}(2H)$ state.

We show the $\gamma$ dependence of the total decay width for $a_{4}(2610)$ in the range of $\gamma=6\sim 14$ in Fig. 3. From the figure, we find that for $\gamma$ between 7 and 12, if $a_{4}(2610)$ is interpreted as $a_{4}(2H)$, our theoretical calculation aligns with the experimental width of $a_{4}(2610)$ from COMPASS Beckers and Haas (2025). For $\gamma$ greater than 13.5 and with $a_{4}(2610)$ interpreted as $a_{4}(4F)$, the theoretical width overlaps with the experimental data. However, for $\gamma$ values above 13.5, the theoretical width of $a_{4}(1970)$ is larger than 500 MeV, which contradicts the experimental value Navas and others (2024).

Based on a comprehensive analysis of the mass spectrum and two-body strong decays under the $a_{4}(4F)$ and $a_{4}(2H)$ assignments, we suggest that the newly observed $a_{4}(2610)$ is more likely to be $a_{4}(2H)$ state and $a_{4}(4F)$ may be a narrow state with a width of 170 MeV. However, due to the large uncertainties ($609\pm 22^{+35}_{-311}$) on width, more data will be required in the future for further verification.

III.3 Prediction of the $a_{4}(1H)$ and $a_{4}(3F)$

The $a_{4}(1H)$ and $a_{4}(3F)$ states have not yet been observed in experiments. Here, we provide predictions for their mass and OZI-allowed two-body strong decay behaviors. According to the MGI model, the predicted masses are 2405 MeV for $a_{4}(1H)$ and 2466 MeV for $a_{4}(3F)$. Employing the spatial wave functions of $a_{4}(1H)$ and $a_{4}(3F)$ obtained from the MGI model as input, we calculated the OZI-allowed two-body strong decay widths using the ${}^{3}P_{0}$ model, with results summarized in Table 7.

As for $a_{4}(1H)$, we predict that its width is 685 MeV. The $\pi b_{3}$, $\pi\rho_{4}(2230)$, and $\pi f_{3}(1F)$ with branching ratios of $20.4\%$, $15.6\%$, and $10.3\%$ are the mainly decay modes. $\pi\rho_{2}(1D)$, $\pi\eta_{2}$, $\rho a_{1}$, and $\rho\omega$ also make significant contributions to the total decay width. More detailed decay channels and their corresponding branching ratios can be seen in Table 7.

As for $a_{4}(3F)$, the OZI-allowed two-body strong decay width is predicted to be about 250 MeV. $\pi b_{1}(1960)$, $\pi f_{2}(1950)$, $\rho\omega$, $\rho a_{2}$, and $\pi\rho_{3}$ are the most important decay final state of $a_{4}(1H)$. In addition, $a_{4}(1H)$ can also decay into $\pi\rho(1900)$, $\pi\rho_{3}(1990)$, $\pi f_{2}(1640)$, and $\pi\eta(1295)$. More detailed decay information is listed in Table 7.

We also present the $\gamma$ dependence of the total decay width of $a_{4}(1H)$ and $a_{4}(3F)$ in the range of $\gamma=6\sim 14$ in Fig. 4. We find that the predicted total width of $a_{4}(1H)$ and $a_{4}(3F)$ are in the ranges of 250$\sim$1300 MeV and 100$\sim$470 MeV, respectively.

IV CONCLUSION

In this work, we have systematically studied the mass spectra and OZI-allowed two-body strong decay behaviors of the $a_{4}$ family, especially the newly observed $a_{4}(2610)$ state.

Our study indicates that for the already established ground state of the $a_{4}$ meson family, namely the $a_{4}(1970)$, the mass and decay properties predicted by the model we used are in very good agreement with experimental data. Likewise, the $a_{4}(2255)$ can be well interpreted as the first radial excitation of the $a_{4}(1970)$, with the theoretical predictions for both its mass and width being consistent with measurements.

For the newly observed $a_{4}(2610)$, we have examined its possible assignments as the $a_{4}(4F)$ and $a_{4}(2H)$ states. Both the mass spectrum and the total width support that $a_{4}(2610)$ is more likely to be the $a_{4}(2H)$ state, while $a_{4}(4F)$ may correspond to a relatively narrow state with a width of 170 MeV.

In addition, we predict the masses and widths of the $a_{4}(3F)$ and $a_{4}(1H)$ states to be

[TABLE]

We also provide their dominant decay modes. These results may serve as valuable guidance for the experimental identification and future searches of the $a_{4}(3F)$, $a_{4}(4F)$, and $a_{4}(1H)$ states.

We look forward to upcoming experimental studies, which will be crucial for clarifying the nature of the newly observed light meson family members with $I^{G}J^{PC}=1^{-}4^{++}$, as well as in validating or exploring these theoretical predictions presented here.

Acknowledgements.

C.-Q. P. and Y.-R. W. contributed equally to this work and should be considered co-first authors. This work is supported by the National Natural Science Foundation of China under Grants No. 12235018, No. 11975165, No. 11965016, and No. 12247101, and by the Natural Science Foundation of Qinghai Province under Grant No. 2022-ZJ-939Q, the Fundamental Research Funds for the Central Universities (Grant No. lzujbky-2024-jdzx06).