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Construction of the $a_4$ family

This paper investigates the mass spectra and strong decays of the $a_4$ family using the modified Godfrey-Isgur quark model and the quark-pair creation model, suggesting that the recently observed $a_4(2610)$ resonance is a promising candidate for the $a_4(2H)$ state while also predicting the properties of the $a_4(1H)$ and $a_4(3F)$ states.

Original authors: Ya-Rong Wang, Cheng-Qun Pang, Hao Chen, Xiao-Hai Liu

Published 2026-08-21

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CC BY 4.0

Original authors: Ya-Rong Wang, Cheng-Qun Pang, Hao Chen, Xiao-Hai Liu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The universe is built from a handful of fundamental particles, but when these particles bind together, they form a vast and complex zoo of composite objects called hadrons. Among these, mesons are pairs of a quark and an antiquark held together by the strong nuclear force, the most powerful interaction in nature. Physicists have spent decades cataloging these mesons, organizing them into families based on their mass, spin, and other quantum properties, much like biologists classify living species. However, the catalog is not yet complete. While the lighter, more stable members of these families are well understood, the heavier, more excited versions often appear as broad, fuzzy signals in experimental data rather than sharp, distinct points. These broad signals are difficult to interpret because they decay almost instantly, making it hard to tell if they are simple combinations of quarks or something more exotic. Understanding these fleeting structures is crucial for completing the map of how matter is built from the ground up.

Recently, the COMPASS Collaboration at CERN announced the discovery of a new, broad structure in the family of mesons known as the $a_4$ family. This new particle, tentatively named $a_4(2610)$ , has a mass of about 2,608 MeV and a very wide width of roughly 609 MeV, indicating it decays extremely rapidly. Alongside this new discovery, physicists have long known of two other members in this family: the ground state $a_4(1970)$ and a slightly heavier state called $a_4(2255)$ . The challenge has been to figure out exactly what these particles are made of and how they fit into the theoretical framework that predicts the existence of such families. In a new study, researchers have used advanced computer models to simulate the behavior of these particles, aiming to identify the nature of the new $a_4(2610)$ and predict the properties of other members of the family that have not yet been seen.

The researchers approached this problem using two complementary theoretical tools. The first is a model that calculates the mass of a meson by treating the quark and antiquark as particles moving within a specific potential energy field. This model has been refined to include a "screening" effect, which accounts for how the force between quarks changes as they move apart, a detail that is essential for accurately predicting the masses of heavier, excited states. The second tool is a model that simulates how these mesons break apart into two other particles. By combining these two approaches, the team could calculate not only what the mass of a hypothetical particle should be, but also how fast it would decay and into which specific particles it would split. This dual approach allows them to compare their theoretical predictions directly with the messy, real-world data collected by experiments.

The team first tested their models against the well-known members of the $a_4$ family to ensure their methods were reliable. They successfully reproduced the mass and decay properties of the ground state $a_4(1970)$ and the excited state $a_4(2255)$ . Their calculations confirmed that $a_4(2255)$ is indeed the first excited version of the ground state, a finding that aligns with previous theories and gives them confidence in their results. With the models validated, they turned their attention to the newly discovered $a_4(2610)$ . The central question was whether this new particle represented a specific type of excited state known as a "4F" state or a different type called a "2H" state. These labels refer to the specific way the quarks orbit each other and their total energy levels.

The results of the simulation strongly favored one possibility over the other. When the researchers assumed the new particle was a "4F" state, their model predicted a mass that was slightly too high and, more importantly, a decay width that was far too narrow. The model suggested a width of only about 174 MeV, which is much smaller than the 609 MeV observed by the COMPASS experiment. However, when they assumed the particle was a "2H" state, the predictions changed dramatically. The calculated mass of 2,589 MeV was very close to the observed 2,608 MeV, and the predicted decay width of 665 MeV matched the experimental measurement with remarkable precision. The researchers concluded that the new $a_4(2610)$ is most likely the "2H" state, a higher-energy configuration of the quark pair. This identification suggests that the "4F" state might still exist as a separate, narrower particle that has not yet been clearly observed.

Beyond solving the mystery of the new particle, the study also looked ahead to the rest of the family. The researchers predicted the existence of two other members of the $a_4$ family that have not yet been discovered: the $a_4(1H)$ and the $a_4(3F)$ states. They calculated that the $a_4(1H)$ should have a mass of about 2,405 MeV and a very broad decay width of roughly 685 MeV, while the $a_4(3F)$ should be slightly heavier at 2,466 MeV with a narrower width of about 250 MeV. The study also detailed exactly which particles these new states would likely decay into, providing a specific "fingerprint" for experimentalists to look for. By offering these concrete predictions, the work provides a clear roadmap for future experiments, helping physicists know where to look and what to expect as they continue to map out the full spectrum of these fundamental particles.

Technical Summary: Construction of the $a_4$ Family

Problem Statement
The paper addresses the need to establish the complete spectroscopy of the $a_4$ meson family, characterized by quantum numbers $I^G J^{PC} = 1^- 4^{++}$ . While the ground state $a_4(1970)$ is well-established, and the $a_4(2255)$ has been previously identified, the recent observation by the COMPASS Collaboration of a broad structure, $a_4(2610)$ , with a mass around 2.6 GeV, necessitates a systematic investigation. The primary challenge is to determine the internal structure (radial and orbital excitations) of the $a_4(2255)$ and the newly observed $a_4(2610)$ , and to predict the properties of higher excitations ( $a_4(1H)$ and $a_4(3F)$ ) that have not yet been observed. Due to the non-perturbative nature of QCD in light meson systems, direct first-principles calculations are difficult, requiring phenomenological approaches.

Methodology
The authors employ a two-model framework to investigate the mass spectra and strong decay properties of the $a_4$ family:

Modified Godfrey-Isgur (MGI) Model: To calculate mass spectra, the authors utilize the MGI model. This is an extension of the original Godfrey-Isgur (GI) potential model, incorporating a screened confinement potential to account for color-screening effects, which are crucial for describing highly excited states. The Hamiltonian includes kinetic energy terms, a screened Coulomb term (one-gluon exchange), a screened confinement term, and various spin-dependent interactions (contact, tensor, and spin-orbit terms). The model parameters are fitted to experimental light meson data, yielding a global fit with a $\chi^2$ value of approximately 40. The spatial wave functions are solved using a simple harmonic oscillator (SHO) basis.
Quark-Pair Creation (QPC) Model ( $^3P_0$ Model): To analyze the two-body strong decays, the authors use the QPC model. This effective model describes the decay of a meson into two mesons via the creation of a light quark-antiquark pair from the vacuum with quantum numbers $^{2S+1}L_J = ^3P_0$ . The decay amplitudes are calculated using the spatial wave functions derived from the MGI model. The strength of the pair creation is governed by the parameter $\gamma$ , set to 10.16 based on previous fits.

Key Contributions and Results

Verification of Established States:
- $a_4(1970)$ : Identified as the ground state ( $1^3F_4$ ). The MGI model predicts a mass of 1973 MeV, closely matching the experimental value ( $1967 \pm 16$ MeV). The calculated total width is 312 MeV, consistent with the experimental width of $324^{+15}_{-18}$ MeV. The dominant decay channels are $\rho\omega$ , $\pi\rho$ , $\pi b_1$ , and $\pi f_2$ .
- $a_4(2255)$ : Identified as the first radial excitation ( $2^3F_4$ ). The predicted mass is 2243 MeV, aligning well with experimental values ( $2237 \pm 55$ MeV and $2255 \pm 40$ MeV). The calculated width is 222 MeV, comparable to experimental data. The study confirms that $a_4(2255)$ is a viable candidate for the $2^3F_4$ state.
Analysis of the New State $a_4(2610)$ :
- The authors test two assignments for the COMPASS-observed $a_4(2610)$ : the $4^3F_4$ state or the $2^3H_4$ state.
- Mass Spectrum: The MGI model predicts the mass of the $a_4(2H)$ state at 2589 MeV and the $a_4(4F)$ state at 2640 MeV. The experimental mass ( $2608 \pm 9^{+5}_{-38}$ MeV) is closer to the $a_4(2H)$ prediction.
- Decay Widths: The calculated total width for the $a_4(2H)$ assignment is 665 MeV, which is in much better agreement with the experimental width ( $609 \pm 22^{+35}_{-311}$ MeV) than the narrow width of 174 MeV predicted for the $a_4(4F)$ assignment.
- Conclusion on $a_4(2610)$ : Based on both mass and width comparisons, the authors suggest that $a_4(2610)$ is more likely the $a_4(2H)$ state. They propose that the $a_4(4F)$ state may exist as a separate, narrower resonance with a width of approximately 170 MeV.
Predictions for Unobserved States:
- $a_4(1H)$ : Predicted mass is 2405 MeV with a total width of 685 MeV. Dominant decay modes include $\pi b_3$ , $\pi \rho_4(2230)$ , and $\pi f_3(1F)$ .
- $a_4(3F)$ : Predicted mass is 2466 MeV with a total width of 250 MeV. Dominant decay modes include $\pi b_1(1960)$ , $\pi f_2(1950)$ , $\rho\omega$ , and $\rho a_2$ .

Significance and Claims
The paper claims that the combined MGI and QPC framework provides a consistent description of the $a_4$ meson family. By successfully reproducing the properties of the established ground state and the first radial excitation, the model gains credibility for predicting higher excitations. The primary significance of this work lies in:

Providing a theoretical identification for the newly observed $a_4(2610)$ as the $a_4(2H)$ state.
Offering specific predictions for the masses and decay widths of the yet-unobserved $a_4(1H)$ and $a_4(3F)$ states, which can guide future experimental searches.
Highlighting the importance of color-screening effects in the MGI model for accurately describing the spectra of highly excited light mesons.

The authors maintain a modest stance regarding the broad nature of the $a_4(2610)$ , acknowledging that large uncertainties in the experimental width and the potential for coupled-channel effects (meson-loop effects) mean that the interpretation as a pure $q\bar{q}$ resonance requires further experimental verification. The work serves as a phenomenological diagnostic tool to assist in establishing the light meson family structure.

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