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REVIEW 5 major objections 4 minor 100 references

Spinless glueballs in generalized linear sigma model

T0 review · 5 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read A generalized linear sigma model with two pseudoscalar glueball fields reproduces all seven eta masses and finds that η(2225)/X(2370) is almost entirely glue.

desk verdict Strong effort and a new two-glueball construction, but the 'all seven eta masses' claim is an artifact of combining scenarios, and the glue-content conclusion depends on post-hoc filtering. read the letter →

arxiv 2508.09474 v1 pith:KYZBOY7M submitted 2025-08-13 hep-ph hep-exhep-latnucl-th

Amir H. Fariborz This is my paper · ORCID
classification hep-phhep-exhep-latnucl-th PACS 14.80.Bn11.30.Rd12.39.Fe
keywords glueballslinearsigmamodelaxialanomalyetamesonsscalarchiralnonetsglueballcondensateQCDphenomenology
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to show that a single effective Lagrangian — a generalized linear sigma model containing a quark-antiquark nonet, a four-quark nonet, a scalar glueball, and two pseudoscalar glueballs — can describe the whole isosinglet scalar and pseudoscalar spectrum below about 2.2 GeV. The central move is to satisfy the U(1)A axial anomaly with two pseudoscalar glueball fields: one physical and one unphysical that is integrated out to produce an effective instanton-type term. At leading order (terms with at most eight quark/antiquark lines), the model then reproduces all seven η masses and the f0 masses in agreement with experiment. The central conclusion is that η(2225) — or, practically indistinguishable by mass, X(2370) — is about 95% glue, while the scalar glueball content among f0(1370), f0(1500), and f0(1710) is left undetermined until the scalar glueball condensate h0 is pinned down.

What carries the argument

The central object is the generalized linear $\sigma$ model Lagrangian expanded to leading order in quark/antiquark lines (≤8), containing two chiral nonets $M$ (quark-antiquark) and $M'$ (four-quark), a scalar glueball $h$, and a pseudoscalar glueball $g$. The load-bearing identity is the axial-anomaly ansatz $G=(1-\xi)h^{3}g+\xi h^{3}g'$: integrating out the unphysical $g'$ turns the anomaly into Eq. (15), a linear coupling of $g$ to $\ln(\det M/\det M^{\dagger})$ plus a squared instanton-type term $-c_{3}\xi^{2}[\ln(\det M/\det M^{\dagger})]^{2}$. That squared term supplies the extra U(1)A splitting among the η masses, and the vacuum value $h_0=\langle h\rangle$ controls the glue content of

What would settle it

A decisive observation would be a precise measurement of J/ψ radiative decays into η(2225)/X(2370): a nearly pure pseudoscalar glueball should be produced with a distinctive rate and angular distribution relative to η′(958). If its J/ψ radiative branching ratio matches ordinary quark-antiquark isosinglet expectations, the claim that η(2225) is almost entirely glue is falsified.

Watch

Extended reading notes

Core claim

The paper's central claim is that agreement with the seven η masses forces the axial anomaly to be modeled with two pseudoscalar glueballs: a physical one g and an unphysical one g′ integrated out to generate an instanton-type term. With $G=(1-\xi)h^{3}g+\xi h^{3}g'$, integrating out $g'$ gives Eq. (15); at leading order (≤8 quark lines) a 20-parameter Monte Carlo scan then reproduces all seven experimental η masses, favoring η3, η4, η5 = η(1295), η(1760), η(2225). Filtering to the favored high range h0 = 0.65–0.99 GeV, η(2225)/X(2370) is 95% glue, while the scalar glueball content among f0(1370), f0(1500), f0(1710) stays h0-dependent, with the SU(3)-favored h0 ≈ 0.75–0.825 GeV selecting f0(

Load-bearing premise

The whole eta-mass result rests on the assumption that the QCD axial anomaly can be represented by the specific linear combination G=(1−ξ)$h^{3}$ g + ξ $h^{3}$ g′ of two pseudoscalar glueball fields, one unphysical and integrated out; if that combination is not how the anomaly enters the effective Lagrangian, the seven-eta agreement and the glue content of η(2225) do not follow.

Editorial extensions

If this is right

  • If the paper is right, η(2225)/X(2370) is the leading experimental candidate for the long-sought pseudoscalar glueball.
  • The same Lagrangian gives sizeable glue admixtures to f0(1370), f0(1500), and f0(1710), so pinning down the scalar glueball condensate h0 becomes the decisive next measurement.
  • The simulations split into a low range h0 = 0.40–0.65 GeV and a high range h0 = 0.65–0.99 GeV with contradictory predictions; the decay widths and the SU(3) limit favor the high range, effectively ruling out the low range.
  • Because η(2225) and X(2370) differ in mass by only about 3%, either state can serve as the model's fifth pseudoscalar, and the nearly-pure-glue conclusion applies to whatever state occupies that slot.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the two-glueball anomaly ansatz generalizes, other effective Lagrangians that keep a physical pseudoscalar glueball will also need a second, integrated-out field to preserve the instanton term that drives η–η′ splitting.
  • A future lattice QCD value of the scalar glueball condensate at the percent level could decide which f0 is the scalar glueball, since the model's scalar compositions change sharply with h0 even within 0.65–0.99 GeV.
  • The near-mass degeneracy of η(2225) and X(2370) means the model cannot assign the glueball label to one uniquely; radiative-decay data could break the degeneracy in a way the mass spectrum alone cannot.
  • Because the leading-order truncation yields degenerate bare four-quark nonets, measuring mass splittings among the heavy scalar and pseudoscalar states would test whether subleading terms are needed.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 4 minor

Summary. The manuscript develops a generalized linear sigma model with two chiral nonets (quark-antiquark and two-quark-two-antiquark), a scalar glueball h, and a physical pseudoscalar glueball g. To implement the U(1)A anomaly, it additionally introduces an unphysical pseudoscalar glueball g' which is integrated out, yielding an effective instanton-type term. Working at leading order (at most eight quark/antiquark lines) and with a simplified trace anomaly, the author derives 5x5 mass matrices for isosinglet scalars and pseudoscalars, performs a Monte Carlo scan over about 20 parameters, and accepts parameter sets satisfying the chi condition (37) against the experimental eta masses. The paper claims agreement with all seven eta masses, identifies eta(2225) as approximately 95% glue, and analyzes scalar glueball contents and decay widths, ultimately filtering to a high scalar-glueball-condensate range.

Significance. If the claimed agreement were genuine, the framework would be a useful phenomenological tool for glueball-meson mixing. The paper is substantial: it gives explicit mass matrices and three-point couplings, and the numerical scan is extensive. However, the central claim as stated is not supported. The model has only five eta eigenstates; the 'seven eta masses' are reproduced only by stitching together different scenarios. The eta-mass agreement is in-sample because the acceptance condition uses those masses. After filtering, the predicted scalar and kaon masses in Table III deviate by 100-300 MeV from PDG values. The headline glueball assignment depends on discarding the low-h0 region. These issues are load-bearing, so the paper's main conclusions do not follow.

major comments (5)
  1. [Abstract; Sec. IV.C; Eq. (27); Table I; Fig. 9] The abstract claims the model 'accurately generate[s] all seven eta masses' and is in 'complete agreement with experiment.' Yet the eta mass matrix is 5x5 (Eq. 27), so each parameter set has five eigenstates. The paper itself states 'the model contains five etas' and uses ten assignment scenarios (Table I) to match these to five of the seven PDG states. Fig. 9 shows that eta(1405)/eta(1475) appear only in the low-h0 range (0.40-0.65 GeV) while eta(2225) appears only in the high-h0 range (0.65-0.99 GeV). After filtering to high h0, Table III lists only eta(547), eta'(958), eta(1295), eta(1760), eta(2225). Thus no single parameter set reproduces all seven masses; the apparent complete agreement is an artifact of combining incompatible scenarios.
  2. [Sec. IV.B-C; Eqs. (35)-(37)] The acceptance condition (37) is built from Eq. (35), which compares the model's eta masses with the experimental eta masses in Eq. (34). Parameter sets are retained only if this chi is below the experimental chi. Consequently, the subsequent 'remarkable agreement' with the eta spectrum is in-sample, not a prediction; the statement in Sec. IV.C that 'this is not a fit' is contradicted by the selection procedure. The model is not independently predicting the eta masses; it is being filtered by them.
  3. [Table III; Sec. IV.C.1] Table III reports predictions with substantial deviations from PDG values: K0*(700) at 1.116 GeV vs 0.838 GeV, K0*(1430) at 1.576 GeV vs 1.425 GeV, f0(980) at 1.145 GeV vs 0.990 GeV, and f0(1370) around 1.46-1.50 GeV. These are 100-300 MeV off. The text argues that unitarity corrections will lower some of these masses, but those corrections are not computed here. At the level of the Lagrangian masses presented, 'complete agreement with experiment' is not accurate.
  4. [Sec. VI; Figs. 9, 11; Table IV] The conclusion that eta(2225) is ~95% glue (Table IV) is obtained only after filtering out the low-h0 range (Sec. VI). In the unfiltered scan, Fig. 11 shows that in the low-h0 range eta(1760) carries the largest glue content. The high-h0 filter is justified by decay-width comparisons and other external evidence, but not by a model-internal criterion. The headline assignment therefore depends on choosing one of two contradictory regimes. This is a selection effect, not a robust prediction.
  5. [Sec. III.A; Eqs. (13)-(15)] The axial anomaly is implemented through the ad hoc combination G=(1-xi)h^3 g + xi h^3 g', with g' an unphysical field having an arbitrary mass and xi a free parameter. Integrating out g' yields the instanton term that drives the eta masses. The paper does not derive this construction from QCD or show it is a controlled approximation to the U(1)A anomaly. Since this term is the main source of the eta masses, the agreement with eta data tests this ansatz, not directly the QCD anomaly. This is a load-bearing assumption that needs independent support.
minor comments (4)
  1. [Throughout] Typos and spelling errors: 'pseusoscalar', 'tohether', 'antiquak', 'nonlienar sigme model', 'P ARAMETER DETERMINA TION'. A careful proofreading pass is needed.
  2. [Sec. IV.B] In the sentence 'from the values of u1h2 0, u3h2 0 and u4h2 0', the last combination should presumably be u4 h0 (not u4 h0^2), since u4 was combined with h0 in the previous steps; please clarify notation.
  3. [Eq. (18)] The displayed expression for fSB appears to have a missing closing parenthesis and ambiguous superscript; please correct the typesetting.
  4. [Table III] The experimental PDG values in Table III are given with varying precision; for consistency, include the uncertainty of eta(2225) in the table column headers.

Circularity Check

1 steps flagged · score 6.0 of 10

The seven-eta-mass 'agreement' is an in-sample fit: condition (37) selects parameter sets by comparing the five model eta eigenstates to five chosen PDG eta masses, so Table III's eta entries are not predictions; the all-seven claim is also an aggregate over ten scenarios and two h0 ranges that no single parameter set realizes.

  1. fitted input called prediction [Sec. IV B-C, Eqs. (34)-(37), Table I, Fig. 3, Table III]
    "To determine each acceptable q, we target experimental values for the eta masses [3]: ... (34). ... χk(p,q) = Σ_i |m theo.[η_i] − m exp.[η_i]| / m exp.[η_i] (35). ... a scenario is considered acceptable if the resulting chi value is less than the corresponding experimental chi value for that scenario. Later: 'Note again that this is not a fit to the experimental data, but rather obtained by imposing the condition (37) which is just a collective constraint on the combination of the eta masses.'"

    The acceptance condition (37) is defined by comparing the model's five eta eigenvalues with five of the seven PDG eta masses, for each of the ten assignment scenarios in Table I. Any parameter set retained in S_II is therefore selected precisely because it reproduces a chosen subset of the eta masses. Table III then lists those eta masses as 'predictions', and the abstract reports 'complete agreement' for all seven. This is a fitted input renamed as a prediction: the agreement is in-sample by construction. Additionally, the eta mass matrix is 5×5 (Eq. 27), so no single parameter set has seven eta eigenstates; Fig. 9 shows η(1405)/η(1475) only in the low-h0 range and η(2225) only in the high-h0 range, so the 'all seven' claim is obtained by stitching together incompatible parameter regions

full rationale

The paper's headline claim of reproducing all seven eta masses is the clearest circular step. Equations (34)-(37) make the experimental eta masses the targets of the acceptance filter, and the text explicitly insists this is 'not a fit' even though the goodness function χ_k is a sum of relative mass deviations against those same masses. The ten scenarios in Table I and the h0-dependent results in Fig. 9 further show that the model, having only five eta eigenstates, cannot match all seven PDG states at a single parameter set; the 'complete agreement' is an aggregate over scenarios and h0 ranges. This reduces the central 'all seven eta masses' claim to a fit plus a categorization procedure. The glue-content conclusion for η(2225) is less directly circular: it is an output of the diagonalized mass matrix, but it is obtained only after filtering to the high-h0 range, a filter supported partly by the author's own SU(3)-limit studies [29,30] and partly by decay-width comparisons. Because the decay-width data provide independent filtering support and because the paper does produce genuinely external outputs (kaon masses, f0 masses, decay widths), the circularity is partial rather than total. Score 6 reflects that the central seven-eta prediction reduces by construction, while some independent model content remains.

Assumptions & free parameters 10 free parameters · 5 assumptions · 2 invented entities

The paper contributes a specific Lagrangian ansatz and a numerical scan, but the physics output is controlled by about 20 parameters and several modeling assumptions. The key free parameters are the glueball condensate h0, the two-glueball mixing xi, and the anomaly coefficients c3 and gamma1, all fitted to meson data or fixed by hand.

free parameters (10)
  • h0 (scalar glueball condensate) = 0.786 GeV average in high range; full acceptable range 0.40-0.99 GeV
    Key parameter; determines glue contents; filtered to high range based on decay widths and the SU(3) limit.
  • xi (mixing between two pseudoscalar glueballs) = 0.817 +/- 0.100 (Table II)
    Controls relative weight of physical versus unphysical glueball in the axial anomaly, Eq. (13).
  • gamma1 (log-anomaly mixing parameter) = 0.00655 +/- 0.00188 (Table II)
    Weight of det M versus Tr(M M'+) log terms in the U(1)A anomaly, Eq. (14).
  • c3 (axial anomaly coefficient) = -4.76e-4 +/- 1.13e-4 (Table II)
    Strength of induced instanton-type term after integrating out g'.
  • u6 (mass coefficient of physical pseudoscalar glueball) = 4.049 +/- 1.706 (Table II)
    Sets the pseudoscalar glueball mass term.
  • u1, u3, u4 (glueball-nonet couplings) = Derived from u1 h0^2, u3 h0^2, u4 h0 determined in Appendix A
    Chiral nonet-glueball interaction strengths entering all mass matrices.
  • u2 (quartic coupling) = 0.388 (Table II)
    Set by isotriplet mass inputs.
  • alpha1, alpha3, beta1, beta3 (scalar VEVs) = approx 0.0607, 0.0783, 0.0247, 0.0192 GeV (Table II)
    VEVs determined from input masses and minimum equations (28).
  • A1, A3 (quark mass parameters) = A1 approx 0.000666, A3 approx 0.01955 GeV; A3/A1 constrained to 28.3-30
    Set explicit chiral symmetry breaking.
  • gamma_m, lambda1, lambda2, lambda3 (trace anomaly parameters) = gamma_m=2, lambda1=1/4, lambda2=lambda3=0 (Eq. 21)
    Simplifying limit saturating the trace anomaly with the glueball only; not fitted, but central to the mass matrices.
assumptions (5)
  • domain assumption The two chiral nonets M and M' span the low-energy meson degrees of freedom below about 2 GeV.
    Sec. II, based on Refs. [68,69]; no independent derivation is given.
  • ad hoc to paper Effective terms with at most eight quark and antiquark lines dominate (leading order).
    Sec. I and Sec. III A, approximation inherited from Ref. [69].
  • ad hoc to paper The U(1)A anomaly is exactly modeled by log terms with one physical and one unphysical pseudoscalar glueball, and integrating out the unphysical one yields the instanton term.
    Sec. III A, Eqs. (13)-(15); this is the paper's central construction.
  • ad hoc to paper The trace anomaly is saturated by the scalar glueball term alone, with lambda1=1/4, lambda2=lambda3=0, gamma_m=2.
    Eq. (21) and Sec. IV A; this limit is used for all numerical results.
  • domain assumption Tree-level potential and minimum equations define the vacuum and mass matrices.
    Sec. III B; no loop corrections or unitarization are included in the mass spectrum.
invented entities (2)
  • Unphysical pseudoscalar glueball g'
    purpose: Integrated out to generate the instanton-type term needed for eta masses; has no asymptotic state.
    Introduced ad hoc in Eq. (13) and eliminated by its equation of motion; only the combination c3 xi^2 enters, so no direct observable signature is predicted.
  • Physical pseudoscalar glueball g independent evidence
    purpose: Mixes with eta mesons; predicted to dominate eta(2225)/X(2370).
    A standard QCD glueball degree of freedom, supported by lattice estimates of a pseudoscalar glueball around 2.3 to 2.6 GeV; the paper predicts a dominant-glue state above 2 GeV.

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Cite this review

Pith. "Pith review of Spinless glueballs in generalized linear sigma model." pith.science (2026). https://pith.science/paper/KYZBOY7M

@misc{pith2026250809474,
  author       = {Pith},
  title        = {Pith review of: Spinless glueballs in generalized linear sigma model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KYZBOY7M}},
  note         = {Machine review of arXiv:2508.09474}
}
read the original abstract

Within the framework of the generalized linear sigma model, a comprehensive analysis of scalar and pseudoscalar glueballs and their mixing with mesons is presented. The Lagrangian of the model contains two chiral nonets, a quark-antiquark type and a two-quark two-antiquark type. The pseudoscalar and scalar glueballs are introduced through their connections with the axial and the trace anomalies of QCD, respectively. It is found that in order to satisfy the axial anomaly and at the same time accurately generate all seven eta masses, it is necessary to include at least two pseudoscalar glueballs, a physical one and an unphysical one that gets integrated out and yields an effective instanton-type term which is needed in generation of the eta masses. At the leading order, which corresponds to keeping effective terms in the Lagrangian that contain no more than eight underlying quarks and antiquarks, the mass spectrum of the model is worked out and shown to be in complete agreement with experiment. The quark and glue contents of the isosinglet scalars below 2 GeV, and of the isosinglet pseudoscalars up to around 2.2 GeV, are analyzed in detail and their correlations with the scalar glueball condensate are examined. Decay widths of isosinglet scalars as well as different self consistencies within this framework are used to probe the glueball condensate and thereby estimate the quark and glue contents of these states. In the pseudoscalar sector, the state that is dominantly made of glue is clearly a state with mass above 2 GeV. In the scalar sector, the identification of glue contents is less certain and in principle the three isosinglets in the 1.5-2.0 GeV can contain substantial glue. These glue contents are determined as functions of the scalar glueball condensate which is the key quantity in this analysis.

Figures

Figures reproduced from arXiv: 2508.09474 by the authors.

Figure 1
Figure 1. FIG. 1: Histograms for the masses of isotriplet systems of [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Histograms for the components of isotriplets and isodoublets found in numerical simulation of this model [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Simulation results for the eta masses predicted in this work using set [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (19 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Simulation results for the [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Histograms for the components of eta states found in numerical simulation of this model using set [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Histograms for the components of [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Histogram for unfiltered glueball condensate [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Goodness of simulations measured by function [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Simulation results for eta masses (dots/colors) over set [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Simulation results for [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: The glue content of [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: The glue content of the [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]
Figure 13
Figure 13. Figure 13: gives the decay width of f0(500) versus h0 together with the experimental bounds which are clearly in complete agreement over the entire range of h0 = 0.5 − 0.99 GeV [PITH_FULL_IMAGE:figures/full_fig_p021_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14: Decay widths (in GeV) of the [PITH_FULL_IMAGE:figures/full_fig_p022_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15: Decay widths (in GeV) of the [PITH_FULL_IMAGE:figures/full_fig_p023_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16: Decay widths (in GeV) of the [PITH_FULL_IMAGE:figures/full_fig_p024_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17: Decay widths (in GeV) of the [PITH_FULL_IMAGE:figures/full_fig_p025_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18: Decay ratios of the [PITH_FULL_IMAGE:figures/full_fig_p025_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19: Decay ratios of the [PITH_FULL_IMAGE:figures/full_fig_p026_19.png]
Figure 20
Figure 20. Figure 20: FIG. 20: Decay ratios of the [PITH_FULL_IMAGE:figures/full_fig_p026_20.png]
Figure 21
Figure 21. Figure 21: FIG. 21: Histogram representing the number of decay widths and decay ratios predicted by the model computed [PITH_FULL_IMAGE:figures/full_fig_p027_21.png]
Figure 22
Figure 22. Figure 22: FIG. 22: Masses of bare (unmixed) states (in GeV) in the scalar and pseudoscalar nonets together with the masses of [PITH_FULL_IMAGE:figures/full_fig_p035_22.png]

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Works this paper leans on

100 extracted references · 79 canonical work pages

  1. [1]

    [69] given in (6)

    (27) 9 These mass matrices are subject to the following minimum equations which are obtained from imposing limit (21) on minimum equations (20): ∂V0 ∂fa 0 = 2 √ 2 2u2α3 1 + 2α1β3h0u4 +u 1h2 0α1 + 2α3β1h0u4 −A 1 = 0 ∂V0 ∂fb 0 = 8α 1β1h0u4 + 4u2α3 3 + 2u1h2 0α3 −2A 3 = 0 ∂V0 ∂fc 0 = 2 √ 2h0 (2α1α3u4 +β 1h0u3) = 0 ∂V0 ∂fd 0 = 4α 2 1h0u4 + 2u3h2 0β3 = 0 ∂V0 ∂...

  2. [2]

    Isodoublet and isotriplet states Although these states have been studied in the framework of the generalized linear sigma model without glueballs in [69], in this section we revisit their case within the present extended framework that has been augmented with glueballs. While glueballs do not directly contribute to these states, as the framework is extend...

  3. [3]

    Isosinglet states As pointed out before, the properties of the isosinglet states involve both parameterspandqin setS II . Particularly, parameterh 0 (the glueball condensate) plays a key role in this analysis and a range of 0.75-0.825 was favored for this parameter in the analysis of this model in the SU(3) flavor limit in [30]. While we use this SU(3) re...

  4. [4]

    7 (beyond this range, the model has no acceptable solutions)

    Analysis of the glueball condensate over the range ofh0 = 0.40−0.99GeV The histogram forh 0 over its entire range of 0.40−0.99 Gev is given in Fig. 7 (beyond this range, the model has no acceptable solutions). The distribution is consistent with what was found in [29, 30] in the SU(3) limit of this model. We see roughly two ranges,h 0 = 0.40−0.65 GeV (low...

    2020

  5. [5]

    Tr X 2 a − 1 (2u3h2 0)2 det X 2 a −1 = 4u2 4α2 3h2 0 (u3h2 0)2 (A4) which would then allow determination of the combined parametersα 3u4h0 in terms of the combined parametersu 3h2 0 in (A3) and the experimental masses: (α3u4h0)2 = 1 64 (m2 a −m 2 a′)2 − 4 u3h2 0 −(m 2 a +m 2 a′) 2 ! .(A5) Then, the two combined quantities (A3) and (A5), together with the ...

  6. [6]

    (A6) Next, theπ−π ′ mixing angle is found from the diagonalization ofM 2 π: cos 2θπ = 4 u3h2 0 −m 2 π −m 2 π′ q 64 (α3u4h0)2 + 16 (u3h2 0)2 −8 (u 3h2

  7. [7]

    37 Next, we compute the combined quantityα 1u4h0 for an input ofA 3/A1 (the ratio of strange to non-strange quark mass)

    (m2π +m 2 π′) + (m2π +m 2 π′)2 .(A7) The axial current relates the pion decay constant to the mixing angle: Fπ = 2α1 cosθ π −2β 1 sinθ π (A8) which can then be used to computeα 1 in terms of the known quantities determined above: α1 = 1 2 Fπ cosθ π − β1 α1 sinθ π ,(A9) which in turn determinesβ 1 using the relationship (A6) above. 37 Next, we compute the ...

  8. [8]

    However, the first three quantities can be in turn expressed in terms of the target quantityα 1u4h0. Usingα 3u4h0 from (A5) andα 1 from (A9), we can expressα 3 in terms of α1u4h0: α3 = (α3u4h0) (α1u4h0) α1 (A11) Using now known quantityu 3h2 0 from (A3),α 1 from (A9), and the fourth minimum equation in (28), we can express β3 in terms ofα 1u4h0: β3 =− 2 (...

Show all 100 references
  1. [9]

    α1.(A12) Substitutingβ 3 from (A12) into the second equation in (A2), and solving foru 2 we find u2 = 1 8α 2 1 " m2 a +m 2 a′ −m 2 π −m 2 π′ −16 (α1u4h0)2 (u3h2 0) # .(A13) The quantityu 1h2 0 can be expressed in terms ofu 2 in (A13),α 1 in (A9), andu 3h2 0 in (A3). Starting f...

  2. [10]

    We can also substitute the necessary inputs into (A10) and findA 1 and A3. Furthermore, we can use these combined values to also compute: u4h0 = (u4h0α1) α1 .(A16) In summary, in this appendix we determined parametersα 1, β1, α3, β3, A1, A3, u2 and the combined parameters u1h2...

  3. [11]

    The interaction Lagrangian and the fields In this appendix we give the notation and formulas for the two-body decay widths of isosinglet scalars into two pseudoscalars. The relevant part of the Lagrangian can be written in the form −L= 1√ 2 X R,A,B γ FR ΠA ΠB FR ΠA ·Π B + 1√ 2...

  4. [12]

    Decay width formulas The decay widths of isosinglet scalars to two-pseudoscalars are: Γ [FR →Π A ΠB ] = 3 (2−δ AB )γ 2 FR ΠA ΠB q ΠA ΠB 8π m2 FR , Γ [FR →K A KB ] = (2−δ AB )γ 2 FR KA KB q KA KB 8π m2 FR , Γ [FR →η S ηT ] = (1 +δ ST )γ 2 FR ηS ηT qηS ηT 8π m2 FR ,(B15) whereqi...

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