Construction of the f₄ meson family

The paper does what it says on the tin: it applies the MGI mass model and QPC decay framework, already used by this group for a4, 3++, 5++, and kaon families, to the isoscalar f4 sector. The new content is the f4-specific spectrum, the MGI-QPC evaluation of f4(2300) as 2³F4, and decay tables for 3F, 4F, 1H, 2H states. The calculations are internally consistent and the paper is clearly written. The predictions for the strange 2F partner at 2453 MeV and the decay tables for higher excitations are falsifiable and useful for experimental searches. Credit where due: the paper provides detailed decay tables with branching fractions, and the authors acknowledge that the sbar-s 1F assignment cannot be excluded (sentence b194).

The soft spot is the load-bearing width comparison. The paper argues that 232 MeV (2F nbar-n) vs 375 MeV (1F sbar-s) favors the nonstrange assignment because 375 MeV exceeds the PDG upper limit of 330 MeV. But the calculation includes only OZI-allowed two-body channels; three-body contributions are omitted with no estimate. For a state near 2.3 GeV, three-body decays could easily add 50-100 MeV. Moreover, the pair-creation strength gamma=6.90 is calibrated on f4(2050) under the assumption that its two-body sum saturates the total width. If f4(2050) itself has non-negligible three-body decays, the true gamma would be lower, reducing both calculated widths and potentially bringing the 375 MeV value closer to the experimental range. The mixing angles are set to 6 degrees, but Fig. 1 shows strong variation with angle; the 375 MeV value is specific to that choice. The authors themselves note the sbar-s assignment cannot be excluded (b194), which undercuts the strength of the claim. The paper's conclusion is appropriately hedged, but the width discriminant is presented as more decisive than it actually is.

Who this is for: hadron spectroscopists working on light mesons, especially experimentalists looking for search channels. The paper deserves a serious referee because the calculations are well-documented and the predictions are falsifiable. The referee should ask for an estimate of three-body contributions and a discussion of how the width comparison changes under variation of mixing angles and gamma within reasonable ranges.

Recommendation: send to peer review. The paper is solid within its framework; the main concern is about the strength of the central claim, not about the validity of the work.

The central argument that f4(2300) is mainly nonstrange 2^3F4 rather than mainly strange 1^3F4 relies on the comparison of calculated total widths: 232 MeV (2F nbar-n) vs 375 MeV (1F sbar-s) at the common input mass 2320 MeV. The 375 MeV exceeds the PDG upper limit of 330 MeV (250±80), while 232 MeV falls inside. However, the calculation includes only OZI-allowed two-body channels; three-body decays (e.g., ρππ, ωππ, Kππ) are omitted. For a state near 2.3 GeV with many open channels, three-body contributions could easily add 50–100 MeV to the width. The authors provide no estimate of this omission. Moreover, the pair-creation strength γ=6.90 is calibrated on f4(2050) under the assumption that its two-body sum saturates the total width. If f4(2050) itself has non-negligible three-body decays (the PDG lists no three-body branching fractions but does not rule them out), the true γ would be lower, reducing both calculated widths and potentially bringing the 375 MeV value closer to the experimental range. The mixing angles θ_{1^3F4} and θ_{2^3F4} are both set to 6°, but the paper shows in Fig. 1 that the widths vary strongly with angle; the 375 MeV value is specific to 6° and could drop significantly at a different angle. The authors acknowledge that the sbar-s 1F assignment cannot be excluded (sentence b194), which undercuts the strength of the claim. The load-bearing concern is that the width comparison is presented as decisive but depends on unquantified model systematics (three-body channels, mixing-angle uncertainty, calibration assumption) that could reverse the preference.

The reader's weakest_assumption correctly identifies that the calibration of gamma assumes saturation of the f4(2050) width by two-body channels, and that no estimate of three-body contributions or model systematics is provided. My stress-test focuses on the same point: the width comparison that discriminates the two assignments for f4(2300) is fragile because it depends on this unvalidated saturation assumption and on the specific mixing angle. The paper itself acknowledges that the sbar-s assignment cannot be excluded (b194), which is consistent with a CONDITIONAL verdict. I agree with the reader that the paper is well-documented and the predictions are falsifiable, but the central claim is not yet robust enough for ACCEPT. The concrete test I propose would directly address whether the width discriminant survives when systematics are included. I do not change the verdict because the reader already assigned CONDITIONAL; my analysis reinforces that judgment.

The paper addresses a genuine open question in light-meson spectroscopy: the identity of f4(2300) and the mass and decay pattern of the strange partner of f4(2050). Its central conclusion disagrees with the Regge-based ssbar assignment of Refs. [5,16], so it is a falsifiable contribution rather than a restatement of consensus. The strengths are concrete: the full channel lists in Tables V-VIII are transparent; branching fractions and the ratio of the two competing total widths are independent of the common gamma at fixed inputs; the prediction for f'_4(2F) at 2453 MeV with K* anti-K*(1410) and K* anti-K* as leading channels is sharp and directly searchable; and the comparison with the earlier Barnes et al. 3P0 results for f4(2050) is a useful cross-check. The assignment logic is not circular, because f4(2300) data do not enter the calibration, and the paper itself concedes in Sec. III.B.2 that the strange assignment cannot be excluded outright. The principal weakness is that the absolute width scale, on which the quantitative exclusion of the ssbar assignment rests, is calibrated on a single fitted gamma with no estimate of three-body or other model systematics.

The paper aims to pin down the spectroscopic identity of the f4(2300) meson and to map out the whole quark-antiquark family with $I^G(J^{PC}) = 0^+(4^{++})$, the isoscalar spin-4 counterpart of the established $a_4$ family. It combines masses from the modified Godfrey-Isgur (MGI) quark model, whose confining potential is screened rather than linear, with two-body strong-decay widths from the quark-pair-creation (3P0) model, calibrating the single pair-creation strength $\gamma$ against the measured total width of the well-established f4(2050). The mass spectrum and the decay pattern both favor f4(2300) as the mainly nonstrange radial excitation $2\,{}^3F_4$: at the common input mass of 2320 MeV this assignment gives 232 MeV total width, inside the measured $250\pm 80$ MeV, whereas the mainly strange $1\,{}^3F_4$ alternative gives 375 MeV and decays dominantly into kaon channels. If the assignment is right, the paper delivers a searchable map of the family — a strange partner f'_4(2F) at 2453 MeV with width 216 MeV and leading modes $K^*\bar K^*(1410)$ and $K^*\bar K^*$, plus full width predictions for the 3F, 4F, 1H, and 2H flavor pairs.

What carries the argument. The carrying mechanism is the pairing of two quark models. The modified Godfrey-Isgur (MGI) model replaces the linear confining potential with a screened form $S(r) = \frac{b}{\mu}(1-e^{-\mu r}) + c$ and diagonalizes the relativized Hamiltonian in an oscillator basis, producing masses and numerical momentum-space wave functions; for the decay calculations the parent masses are taken from experiment while the wave functions come from the MGI diagonalization. The quark-pair-creation (QPC, or 3P0) model generates OZI-allowed two-body amplitudes by creating a $0^{++}$ quark-antiquark pair from the vacuum and recombining it with the parent quarks, with all amplitudes controlled by one dimensionless strength $\gamma$. The load-bearing step is the calibration: matching the sum of the computed f4(2050) two-body widths to its measured total width $237\pm 18$ MeV fixes $\gamma = 6.90^{+0.26}_{-0.27}$, and that single number converts every partial width of every other state into an absolute width. The discriminating observables are then the total width (232 versus 375 MeV for the two f4(2300) candidates) and kaon-versus-nonstrange partial-width ratios such as $\Gamma_{K^*\bar K^*}/\Gamma_{\rho\rho}$, which differ by roughly three orders of magnitude between the two flavor assignments.

Core claim. The central claim is that f4(2300) is mainly the nonstrange $2\,{}^3F_4$ state of the quark model. In the MGI spectrum the nonstrange $2\,{}^3F_4$ level sits at 2243 MeV and the mainly strange $1\,{}^3F_4$ level at 2170 MeV, so the radial nonstrange assignment is closer to the PDG central estimate of 2320 MeV; at the adopted mixing angles the two assignments' total two-body widths at the common input mass of 2320 MeV are 232 and 375 MeV, and only the former falls inside the measured $250\pm 80$ MeV interval. The two options also predict opposite decay chemistries: the nonstrange state feeds $\rho\rho$, $\omega\omega$, $\pi\pi$, and $a_2(1700)\pi$, matching the final states seen in experiment, while the strange state is dominated by $K^*\bar K^*$ and $K\bar K^*$, with $\Gamma_{K^*\bar K^*}/\Gamma_{\rho\rho}\simeq 64$ for the strange assignment but only $\simeq 0.025$ for the nonstrange one. The paper itself notes that it cannot exclude a mainly strange $1\,{}^3F_4$ contribution to the $K\bar K$ observation of f4(2300), since that state's mass is also close to 2320 MeV. Its predictions for the rest of the family put the mainly strange partner f'_4(2F) at 2453 MeV with total width 216 MeV and leading channels $K^*\bar K^*(1410)$ and $K^*\bar K^*$, and give complete width tables and leading modes for the 3F, 4F, 1H, and 2H flavor pairs.

The premise it rests on. The whole absolute-width comparison rests on the assumption that the OZI-allowed two-body channels listed for f4(2050) saturate its measured total width, so that the single fitted pair-creation strength $\gamma = 6.90$ applies unchanged to every other state and the 232-versus-375 MeV discrimination is trustworthy.

Search for the strange partner f'_4(2F) near 2.45 GeV in $K^*\bar K^*(1410)$ and $K^*\bar K^*$; the predicted partial-width ratio of these two channels is 1.50 and, at fixed masses, wave functions, and mixing angles, is independent of $\gamma$.

If f4(2300) is the nonstrange $2\,{}^3F_4$ state, it should be seen mainly in $\rho\rho$, $a_2(1700)\pi$, $\pi\pi(1300)$, $a_1(1640)\pi$, $\pi_2(1880)\pi$, and $\rho\rho(1450)$, with kaon final states strongly suppressed; measuring $\Gamma_{K^*\bar K^*}/\Gamma_{\rho\rho}$ tests the flavor assignment directly.

The H-wave states f4(1H) at 2406 MeV and f4(2H) at 2589 MeV are predicted to be more than twice as broad as the nearby F-wave states, with characteristic $a_3(1875)\pi$ and $\pi_4$ modes, giving an angular-distribution test for separating the nearly degenerate orbital excitations.

The strange H-wave states f'_4(1H) at 2607 MeV and f'_4(2H) at 2802 MeV should be very broad (549 and 391 MeV) with leading channels $K\bar K'_2(1D)$, $K\bar K'_3(1F)$, and $K_1(1270)\bar K^*$.

The $a_2(4P)\pi$ branching fraction rises from 1.57% in f4(3F) to 14.6% in f4(4F), a signature that distinguishes the two higher radial excitations.

If the radial nonstrange assignment holds, f4(2050) and f4(2300) become successive members of one $n\,{}^3F_4$ sequence with a spacing of roughly 270 MeV in the nonstrange sector; comparing that spacing with the isovector $a_4$ family would show whether the same screened potential and decay machinery reproduce the observed pattern across isospin partners.

The calibration scheme implicitly assumes that three-body and non-OZI two-body decays of f4(2050) are negligible; a future measurement of its $\rho\rho$ branching fraction (predicted here at 44%) would test whether the single-$\gamma$ procedure is capturing the decay dynamics or merely absorbing missing channels.

The strong dependence of the primed-state widths on the flavor-mixing angle suggests that, once the strange partner f'_4(2F) is found, its measured width could be inverted to determine the $2\,{}^3F_4$ mixing angle rather than taking it as an input.

Because the F- and H-wave states at 2400-2600 MeV are nearly degenerate in mass but differ by more than a factor of two in width, the predicted width contrast itself could serve partial-wave analysts as a prior for disentangling overlapping $4^{++}$ signals in that region.

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