The a4 family is defined by the quantum number assignment IGJPC = 1−4++, meaning isospin one, positive G-parity, and a spin-four resonance with positive parity and charge conjugation. For such states the spin is S = 1 and the total angular momentum J = 4, which, through the constraints of parity and spin-orbit coupling, restricts the relative orbital angular momentum between the quark and antiquark to two possible values: L = 3, corresponding to F-wave states, and L = 5, corresponding to H-wave states. Each of these orbital families can then be stacked with radial excitations, producing a ladder of predicted particles: 1F, 2F, 3F, 1H, 2H, and beyond. Until recently, only the ground state of this ladder was firmly established.

That ground state, a4(1970), has a long experimental pedigree stretching back to 1977, when the Omega Group at CERN observed a spin-four enhancement near 2 GeV in partial wave analyses of the reaction π−p → 3πn. Over the following decades, the state was confirmed in a remarkable range of channels, including π±p → K0SK±p, π−p → ηπ0n, and π−p → ωπ−π0p, accumulating one of the densest experimental dossiers of any high-spin light meson. The most precise measurement comes from COMPASS itself, which reported a mass of 1952.2 ± 1.8 (+3/−3.5) MeV and a width of 327 ± 4 ± 6 MeV. A second candidate, a4(2255), surfaced in 2001 in the reactions p p̄ → π0η, 3π0, and π0η′, and was later corroborated by Fermilab E835 data in p p̄ → ηηπ0.

To test whether these observations fit into a coherent quark-model picture, the team employed two complementary phenomenological tools. The first is a modified version of the Godfrey-Isgur (GI) potential model, originally proposed in 1985 and long the workhorse of meson spectroscopy. The modified GI model introduces a screened confinement term, S(r) = b(1 − e^−μr)/μ + c, in which the screening parameter μ = 0.0779 GeV encodes the color-screening effect: at large quark separations, the linear confining potential softens, reflecting the fact that a flux tube spanning more than about a fermi can break by popping light quark pairs out of the vacuum. This single conceptual upgrade, combined with smearing functions for the one-gluon-exchange Coulomb term and momentum-dependent relativistic correction factors, allowed the model to achieve a global fit to 44 experimental light meson masses with a mean relative error of only 1.3 percent.

The second tool is the quark-pair creation model, also known as the ³P0 model, which describes how a meson decays strongly. In this framework, the decaying quark-antiquark pair produces a new q q̄ pair from the vacuum with vacuum quantum numbers JPC = 0++, and the three resulting quark lines rearrange into two outgoing mesons. The strength of this pair creation is controlled by a dimensionless parameter γ, taken here as 10.16, and the decay amplitude is computed from overlap integrals of the initial and final meson wave functions. Crucially, the spatial wave functions feeding the decay calculation come directly from the modified GI model, so the two frameworks form a closed predictive pipeline: masses and wave functions from the potential model, then partial and total widths from the ³P0 machinery, folded through the Jacob-Wick helicity formalism.

The pipeline first had to prove itself on the known states. For the ground-state a4(1970), the model predicts a mass of 1973 MeV and a total width of 312 MeV, both in excellent agreement with experiment. The dominant decay modes come out as ρω (106 MeV), πρ (68.5 MeV), πb1 (55.3 MeV), and πf2 (33.7 MeV), and the calculated branching ratios match the data strikingly well: Γ(ρπ)/Γ(f2π) is predicted to be 2.0 against an experimental value of 1.7 (+0.9/−0.8), while Γ(η′π)/Γ(ηπ) is 0.2 versus a measured 0.23 ± 0.07. For a4(2255), the model yields a mass of 2243 MeV and, when interpreted as the 2³F4 state, the first radial F-wave excitation, a total width of 222 MeV, consistent with the measured widths of roughly 280 to 440 MeV. The tiny ηπ branching fraction of 0.5 percent even explains how the state was originally discovered in that channel despite its rarity.

The decisive test was a4(2610). The team considered two competing assignments: the 4F state, the second radial F-wave excitation with a predicted mass of 2640 MeV, or the 2H state, the first radial H-wave excitation with a predicted mass of 2589 MeV. On mass alone the 2H option wins, sitting closer to the measured 2608 MeV. But the decay patterns are far more discriminating. If a4(2610) is the 4F state, the model predicts a total width of only 174 MeV, dominated by ρa2(1700), πb1(2240), ρa2, and πb1(1960) — far too narrow for the observed 609 MeV. If instead it is the 2H state, the predicted width balloons to 665 MeV, with πρ4(2G), πb3(2244), and πη4(2G) as the main channels, landing squarely on the experimental value. A scan of the pair-creation strength γ from 6 to 14 reinforces the conclusion: the 2H interpretation matches the COMPASS width for γ between 6.8 and 10, while the 4F option only overlaps at γ > 13.3, where the theoretical width exceeds 500 MeV in tension with the Particle Data Group average.

The authors are careful to flag a conceptual subtlety that gives this kind of analysis its edge — and its limits. A state as broad as a4(2610) has a lifetime of order 1/Γ less than a fermi, comparable to or shorter than the size of the hadron itself, meaning the quark-antiquark pair can decay before completing even a single classical orbit. In that regime, the naive picture of a stable bound state breaks down, and alternative interpretations cannot be excluded. The screened potential of the modified GI model partially mitigates this problem, because color screening mimics the average effect of meson-loop coupled-channel corrections, so the calculated wave functions effectively describe dressed rather than bare quark-model states. Still, the team notes that the huge experimental uncertainty on the a4(2610) width — 609 (+35/−311) MeV — means future data will be essential for a definitive verdict.

Beyond settling the fate of a4(2610), the framework delivers concrete hunting licenses for the missing family members. The unobserved 1H state is predicted at a mass of 2405 MeV with a very large width of 685 MeV, decaying mainly through πb3 (20.4 percent), πρ4(2230) (15.6 percent), and πf3(1F) (10.3 percent). The 3F state should sit at 2466 MeV with a more modest width of about 250 MeV, dominated by πb1(1960), πf2(1950), ρω, ρa2, and πρ3. Across the scanned range of the pair-creation parameter, the total widths of these states span roughly 250 to 1300 MeV for the 1H and 100 to 470 MeV for the 3F, giving experimentalists a clear envelope within which to search. Meanwhile, the 4F state, if distinct from a4(2610), should appear as a comparatively narrow 170 MeV resonance.

The stakes here go beyond one quirky family of spin-four mesons. Light meson spectroscopy in the 2 to 3 GeV region is a dense forest where quark-model states, excited molecules, and threshold effects all overlap, and each newly classified resonance sharpens the map that any exotic candidate must be measured against. With three members — a4(1970), a4(2255), and now plausibly a4(2610) — the a4 family joins the ranks of systematically understood light meson multiplets, and two further members now carry published masses, widths, and dominant decay channels awaiting discovery. As high-statistics experiments such as COMPASS continue to mine diffractive dissociation data, the predicted 1H and 3F states offer a direct test of whether the screened quark model and the ³P0 decay framework can guide experimentalists to particles that have so far hidden in plain sight.

Subject of Research: Mass spectrum and strong decay analysis of the a4 meson family with JPC = 4++ in the quark model

Article Title: Construction of the (a_4) family

Article References: Wang, Y.-R., Pang, C.-Q., Chen, H., & Liu, X.-H. (2026). Construction of the $$a_4$$ family. The European Physical Journal C, 86(9), Article 1054. https://doi.org/10.1140/epjc/s10052-026-16301-y

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16301-y

Keywords: a4(2610), meson spectroscopy, COMPASS Collaboration, quark model, Godfrey-Isgur model, 3P0 model, strong decays, light mesons, color screening, quantum chromodynamics, spin-four resonances, particle physics