Tetraquarks with colour-blind forces in chiral quark models

Fl. Stancu; J.-M. Richard; M. Genovese; S.Pepin

REVIEW

Tetraquarks with colour-blind forces in chiral quark models

S.Pepin, Fl. Stancu, M. Genovese, J.-M. Richard

High Energy Physics - Phenomenology · Submitted 13 Sep 1996 · 19 references

Pith's verdict: A clean, honest model calculation: the Glozman meson-exchange force binds heavy tetraquarks where one-gluon-exchange does not, but the binding rides on a short-range term the authors themselves call ad hoc.

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arXiv IDhep-ph/9609348 (version 1)

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CitationFl. Stancu, J.-M. Richard, M. Genovese, S.Pepin. "Tetraquarks with colour-blind forces in chiral quark models." arXiv:hep-ph/9609348 (1996). https://pith.science/paper/CKVJMHGV

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Editor's note

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Worth your time if you follow exotics, but read it as a model-conditional statement, not a prediction. The one thing to take away: within Glozman et al.'s chiral quark model, cc̄qq̄ and bb̄qq̄ sit below the two-meson threshold by 0.19–0.50 GeV (Table 2), while the one-gluon-exchange model of Silvestre-Brac and Semay leaves cc̄qq̄ unbound by 19 MeV. That contrast is the paper's real content, and it is a genuine new model output — nobody had run the meson-exchange force on tetraquarks before this. The Hamiltonian, Eqs. (12)–(13), is written out explicitly, the colour-spin-flavour reduction is spelled out, and the four-body variational ansatz is stated. You can reconstruct the calculation from the paper, which is more than you can say for a lot of this literature. The baryon sector is checked against Glozman's Faddeev results and against Λc and Σc, and it comes out reasonable. That is real validation work and it earns credit. The soft spot is the one the authors flag themselves: the attraction is carried by the 9Vπ term in V34, and Vπ's short-range piece is the regularized delta of Eq. (7), which they describe as ad hoc. Fair enough — they say it. What they do not do is decompose the binding into Yukawa and Gaussian pieces, or vary r0 and ε. Given that the Gaussian coefficient 4ε³/√π is roughly sixty times the Yukawa tail at r0, the 'pion exchange' here is really a 0.43 fm finite-range attraction, and a sensitivity scan would cost them a page. I would ask for it. Second, the argument for dropping D and Ds exchange — that the two terms of Eq. (6) cancel for large µ — does not survive the regularized form, where the Gaussian width is set by ε, not µ. The Q-q̄ pairs in Eq. (13) currently carry only confinement, so this is a live gap, not a cosmetic one. Third, the abstract's '0.2–0.4 GeV' understates the bb̄ number in Table 2; minor, but fix it. Fourth, the q6 and QQq4 negative results rest on group-theory selections from an in-press book chapter and an unpublished preprint, with no numbers shown. Send it out. It is short, it is checkable, and the disagreement with the gluon-exchange camp is exactly the kind of thing referees should adjudicate.

Referee report

Significance

If the result holds, it is a useful and falsifiable statement: the replacement of chromomagnetic gluon exchange by meson exchange, which has been argued to describe the light baryon spectrum, would qualitatively change the stability pattern of heavy multiquarks, and the paper explicitly connects this to planned doubly charmed tetraquark searches at Fermilab and CERN. The calculation is transparent — the Hamiltonian is written out in Eqs. (12)-(13), the colour-spin-flavour reduction is displayed in Eq. (8), and the baryon sector is validated against an independent Faddeev treatment, which is a genuine consistency check rather than a plausibility argument. The paper is also commendably explicit about the model dependence of its inputs and about the fact that the (C1) potential was fitted to light baryons only. The main reservation is that the binding is carried by an ingredient the authors themselves describe as ad hoc, and no sensitivity study of that ingredient is presented.

Major objections (4)

[Eq. (7) and Table 2] The binding in Table 2 is dominated by the short-range piece of the regularized pion-exchange potential, Eq. (7), which enters V34 of Eq. (13) with a factor 9Vπ. The authors state in the text that 'it is somewhat of a surprise to see the delta-term of Eq. (6) taken seriously, and with an ad-hoc regularisation playing a crucial role in the quark dynamics', and they are right to flag it. What is missing is any test of how much of the 0.19-0.50 GeV binding survives if that regularized delta term is removed or its parameters are varied. A decomposition of ΔE into the Yukawa and Gaussian contributions of Eq. (7), and a variation of r0 and ε by, say, ±20% around the values imported from Ref. [13], would take a few lines of calculation and would tell the reader whether the central number is robust or is an artefact of one parameter choice. As it stands, the load-bearing ingredient of the central claim is neither decomposed nor varied, and the acknowledged ad hoc character of the regularization is left unquantified.
[Text following Eq. (8), and Eq. (13)] Heavy-meson exchange is dropped on the argument that 'when the meson mass µ reaches values of a few GeV as for D or ηc the two terms in Eq. (6) basically cancel each other'. That argument applies to the unregularized form of Eq. (6), in which both the Yukawa and the delta term scale with µ in a way that produces cancellation at large µ. It does not apply to the regularized form actually used, Eq. (7), where the short-range piece is a Gaussian whose width is set by ε = 0.573 GeV and whose coefficient 4ε³/√π is independent of µ. The cancellation invoked to justify dropping D and Ds exchange therefore does not survive the regularization that the calculation actually employs. This matters because the four Q-q̄ pairs in Eq. (13) currently carry only the confining potential; a D-exchange contribution to those pairs would act in the same channel as the pion exchange that produces the binding. The authors should either estimate the size of the omitted D-exchange term within their own regularized framework, or state explicitly that the cancellation argument is being applied outside its domain of validity and explain why the omission is nevertheless safe.
[Eq. (14) and Table 2] The binding energies in Table 2 are small differences, 0.19-0.50 GeV, between total energies of order 4-5 GeV computed with a single Gaussian trial function containing three width parameters. For the meson sector the authors state that the variational error 'never exceeds a few MeV with respect to the exact value', but no comparable convergence statement is given for the four-body system, where the trial space is far more restricted. Because the quantity of interest is a difference of two separately minimized variational energies, it is not protected by the usual variational bound, and a few tens of MeV of differential error would be a non-negligible fraction of the quoted result. The authors should report the optimal variational parameters, the separate values of E(QQ̄q̄q̄) and 2E(Qq̄), and a check that ΔE is stable when the trial function is enlarged, for example by admitting different widths for the Q-Q and q̄-q̄ pairs or a small admixture of an additional Gaussian.
[Paragraph beginning 'Due to arguments at the beginning of this letter'] The truncation to the 3̄3 colour configuration with mixing to 6̄6 neglected is justified by appeal to Ref. [3], where the argument was made for deeply bound heavy systems with a mass ratio m(Q)/m(q) larger than about 15. Here the binding is only 0.2-0.5 GeV and the mass ratios are m_c/m ≈ 4-5.5 and m_b/m ≈ 14-16, so the stated justification does not straightforwardly apply, particularly in the charm sector. Since the 6̄6 component is precisely the one that would be favoured by a spin-dependent short-range attraction of the kind that is driving the result, the neglect should be supported by an estimate rather than by citation. At minimum the authors should soften the claim that the mixing 'plays a negligible role' and indicate what size of 6̄6 admixture would be needed to alter the sign or magnitude of ΔE.

Minor objections (6)

[Abstract] The abstract quotes a binding of 0.2-0.4 GeV, whereas Table 2 reports −0.497 GeV for bb̄q̄q̄ with potential (C2). The abstract range should be brought into agreement with the table, or the table entry should be discussed.
[Abstract and §1] There are several typographical and grammatical slips that should be corrected: 'binds a heavy tetraquark systems' in the abstract, 'one could rise the question' in the paragraph before the q6 discussion, and 'T able 1' in the table caption.
[Eq. (8)] The notation V^{us}_η, V^{uc}_D and V^{sc}_{Ds} appears in the [2]_F, I = 1/2 entry of Eq. (8) without being defined in the text; only V^{uu}_η and V^{uu}_{ηc} are identified. A short definition of the superscript convention would make the flavour reduction checkable by the reader.
[Paragraph on QQqqqq and q6] The statement that the QQqqqq and q6 systems are 'largely insufficient to bind' and 'not sufficiently bound' is given without numbers. Quoting the computed ΔE for these systems, even as a bound, would let the reader judge how close they are to threshold and would strengthen the contrast with the tetraquark case.
[Table 1] The caption should state explicitly which masses are inputs and which are outputs of the fit, and the meaning of the superscripts a) and b) should be repeated in the caption rather than left to the footnotes.
[Baryon discussion, Eq. (10)] The reproduction of the Faddeev results of Ref. [13] with a Gaussian trial function is a valuable check and deserves one sentence of quantification: quoting the actual N, Δ, Λc and Σc energies obtained here alongside the Ref. [13] values would make the agreement concrete rather than asserted.

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The paper

The paper claims

Swapping gluon exchange for pion exchange turns heavy tetraquarks from unbound into bound, by 0.2 to 0.5 GeV.

The reading

The paper asks a narrow but consequential question: if the chromomagnetic (one-gluon-exchange) hyperfine force between quarks is replaced by pseudoscalar-meson exchange, as in the chiral quark model that reproduces the light baryon spectrum, what happens to multiquark states? The authors compute the energies of $QQbar q bar q$ systems ($Q=c,b$; $q=u,d$) variationally with Gaussian trial wave functions and find them bound relative to the lowest two-meson threshold by 0.185 and 0.226 GeV with the weak linear confinement of the chiral model, and by 0.332 and 0.497 GeV with a more conventional Coulomb-plus-linear confinement fitted to $J/psi$ and $Upsilon$. That is the opposite of what the standard one-gluon-exchange potential models give, where $ccbar q bar q$ sits slightly above threshold. The claim matters because it says the stability of exotic multiquarks is decided not by how strong the attraction is but by which colour-blind force supplies it: a force that is negligible between colour-octet-dominated heavy mesons can be strongly attractive inside a colour-$bar 3$ diquark paired with a colour-$3$ antidiquark.

What carries the argument

The load-bearing object is the meson-exchange term of the Hamiltonian, $ -sum_{i<j} tilde{lambda}^F_i cdot tilde{lambda}^F_j , vec{sigma}_i cdot vec{sigma}_j V_F(r_{ij})$, with the flavour operators extended from SU(3) to SU(4) so that pion, kaon, eta, eta-prime and (neglected) $D$-meson exchanges can act on a pair containing a charmed quark. For the chosen tetraquark configuration the colour and spin matrix elements collapse to a compact potential: the light-antiquark pair carries $9V_pi - V_eta - 2V_{eta'}$, while every other pair contributes only a quarter of the confining potential. The radial shape $V_mu$ is the regularized one of the chiral model: a Yukawa tail $mu^2 e^{-mu r}/r$ cut off below $r_0 = 2.18$ GeV$^{-1}$ and matched to a Gaussian representation of the delta function with width parameter $epsilon = 0.573$ GeV. A three-parameter Gaussian trial wave function in the Jacobi coordinates $x$, $y$, $z$ is minimized to obtain the four-body energies.

Extended reading notes

Core claim

Within the chiral quark model of Glozman and collaborators, where the short-range spin-spin force comes from pseudoscalar-meson exchange rather than gluon exchange, the heavy tetraquarks $ccbar q bar q$ and $bbbar q bar q$ are bound. Using the colour-$bar 3$ diquark / colour-$3$ antidiquark configuration with total spin $S=1$ and isospin $I=0$, and neglecting mixing with the $6-bar 6$ channel, the authors obtain binding energies $Delta E = E(QQbar q bar q) - 2E(Qbar q)$ of $-0.185$ and $-0.226$ GeV for the weak-confinement parameter set and $-0.332$ and $-0.497$ GeV for the standard Coulomb-plus-linear set. The comparison column is decisive for the paper's argument: with the same conventional confinement but one-gluon-exchange hyperfine instead of meson exchange, $ccbar q bar q$ comes out at $+0.019$ GeV, i.e. unbound, and $bbbar q bar q$ only weakly bound at $-0.135$ GeV. The authors also check the six-quark channels: neither $QQqqqq$ nor $q^6$ is bound below the corresponding two-baryon threshold, so the effect does not immediately proliferate into a zoo of multiquarks. As a calibration, they reproduce the nucleon, $Delta$, $Lambda_c$ and $Sigma_c$ masses with the same variational Ansatz before applying it to four-body systems.

Load-bearing premise

The binding rests almost entirely on the short-distance delta-function piece of the pion-exchange potential, regularized with a cutoff and width chosen by hand; the authors themselves call it 'somewhat of a surprise' that this term is taken seriously in quark dynamics, so if the regularization is not physically justified the $9V_pi$ attraction loses its basis.

Editorial extensions

If this is right

Doubly-charmed and doubly-bottom tetraquarks should appear as resonances below the $D D$ and $B B$ thresholds, at masses roughly 0.2 to 0.5 GeV under twice the average heavy-meson mass, and should decay into a heavy-meson pair only via finite-width or mixing effects.
The stability of an exotic multiquark is a diagnostic of the underlying quark-quark force: measuring whether $ccbar q bar q$ is bound discriminates between gluon-exchange and meson-exchange pictures of the hyperfine interaction.
Because the six-quark systems $QQqqqq$ and $q^6$ remain unbound in the same model, the chiral force does not predict a dense spectrum of multiquark states — the binding is specific to the tetraquark colour topology.
The large difference between the two confinement choices ($-0.19$ vs $-0.33$ GeV for charm) shows that a chiral potential fitted only to light baryons is not quantitatively reliable for heavy-flavour predictions; a chiral model fitted across light and heavy sectors is needed before the numbers can be trusted.

Reading between the lines

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

If the effect is real, the same mechanism should bind tetraquarks with one heavy quark rather than two, e.g. $c bar q bar q bar q$ or $c bar c q bar q$ in the appropriate colour channel; the paper does not compute these, and they would be a cheap consistency test.
The binding scales with the number of light-antiquark pairs that can exchange pions, so a tetraquark with strange antiquarks should be less bound than the $u,d$ case — a trend that lattice studies or future experiments could check independently of the model details.
Because the attraction lives in the short-range regularized delta piece, an alternative regularization that preserves the same long-range Yukawa tail but suppresses the contact term would likely erase most of the binding; that makes the prediction sensitive to a piece of the potential that is currently fixed by fitting baryon masses rather than derived.
The comparison with the one-gluon-exchange result suggests the two pictures differ mainly in the colour-$bar 3$ diquark channel, which is exactly where lattice QCD can compute the static quark-quark potential; a lattice determination of the spin-dependent potential in that channel would arbitrate between them without needing a tetraquark to be observed.

Original abstract

We discuss the stability of multiquark systems within the recent model of Glozman et al. where the chromomagnetic hyperfine interaction is replaced by pseudoscalar-meson exchange contributions. We find that such an interaction binds a heavy tetraquark systems $QQ\bar q\bar q$ ($Q=c,b and q=u,d)$ by 0.2-0.4 GeV. This is at variance with results of previous models where $cc\bar q\bar q$ is unstable.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,985 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.11 · human participants 0.01 · controlled comparison 0.20 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.98 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We discuss the stability of multiquark systems within the recent model of Glozman et al. where the chromomagnetic hyperfine interaction is replaced by pseudoscalar-meson exchange contributions. We find that such an interaction binds a heavy tetraquark systems $QQ\bar q\bar q$ ($Q=c,b and q=u,d)$ by 0.2-0.4 GeV. This is at variance with results of previous models where $cc\bar q\bar q$ is unstable.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
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s0	We discuss the stability of multiquark systems within the recent model of Glozman et al. where the chromomagnetic hyperfine interaction is replaced by pseudoscalar-meson exchange contributions.	contribution 0.48	0.26	0.06	asserted 0.61
s1	We find that such an interaction binds a heavy tetraquark systems $QQ\bar q\bar q$ ($Q=c,b and q=u,d)$ by 0.2-0.4 GeV.	result 1.00	0.95	0.79	asserted 1.00
s2	This is at variance with results of previous models where $cc\bar q\bar q$ is unstable.	interpretation 0.63	0.18	0.18	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.95 · evidence from simulation or computation 0.70.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (19)

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B. Silvestre Brac and C. Semay, Z. Phys. C 59, 457 (1993); 61, 271 (1994)

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L.Ya. Glozman, Z. Papp and W. Plessas, Phys. Lett. B 381, 311 (1996)

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R.L. Jaﬀe, Phys. Rev. D 15, 267 (1977); 17, 1444 (1978)

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D.M. Brink and Fl. Stancu, Phys. Rev. D 49, 4665 (1994)

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G. Karl, Int. J. of Mod. Phys. E1, 491 (1992); Nucl. Phys. A 558, 113c (1993); N.A. T¨ ornqvist, Proc. of “Int. Europh. Conf. on High Energy Phys.”, Brussels (Belgium), edit. J. Lemonne et al., 84 (1995); L. G. Landsberg, Phys. of A tom. Nucl. 57, 42 (1994)

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[8]

Particle Data Group, Phys. Rev. D 54, 1 (1996)

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[9]

M.A. Moinester, Tel Aviv University preprint TAUP 2255- 95 (hep-ph/9506405), Z. Phys. A(in press)

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[10]

D.M. Kaplan, Proc. Int. Workshop “Production and Decay of Hyperons, Charm and Beauty Hadrons”, Strasbourg (France) September 5–8, 19 95

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[11]

COMPASS Collaboration (G. Baum et al.), CERN–SPSLC–96 -14, March 1996

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L.Ya. Glozman and D.O. Riska, Phys. Rep. 268, 263 (1996). 9

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L.Ya. Glozman and D.O. Riska, Nucl. Phys. A 603, 326 (1996)

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Fl. Stancu, Group Theory in Subnuclear Physics, Oxford University Press, 1996, (in press), chapter 4; S. Pepin and Fl. Stancu, Preprint ULG- PNT-96-1-J. 10

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How the literature uses it

33 arXiv papers cite this in 39 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

We find that such an interaction binds a heavy tetraquark system $QQ\bar q\bar q$ ($Q=c, b$ and $q=u, d)$ by $0.2-0.4$ GeV.

Multiquark Systems in a Constituent Quark Model with Chiral Dynamics · 1.00

Papers citing it · 17 sentences without a confident label

Cites it as support 1 paper

Calculations based on the flavor-spin interaction [ref] or the instanton induced interactions [ref] also show that the mass of $T_cc$ is below that of two charmed mesons.

Charmed Exotics in Heavy Ion Collisions · Introduction · 0.99

Cites it as background 20 papers

Their masses have been calculated in many different procedures, such as QCD sum rule [ref] , lattice [ref] , Bethe-Salpeter equation approach [ref] , heavy quark symmetry [ref] and several QCD inspired models [ref] .

Discovery potentials of double-charm tetraquarks · Introduction · 0.86
In one approach, $T_QQ$ is considered as a tetraquark state, in which the effective degrees of freedom are constituent quarks [ref] .

Exotic mesons with double charm and bottom flavor · Introduction · 0.91
After that, numerous theoretical studies had concentrated on double-heavy tetraquark systems using various approaches, including the color–magnetic interaction model [ref] , non-relativistic quark model [ref] , relativized quark model [ref] , relativistic quark model [ref] , hadronic molecular picture [ref] , QCD sum rule [ref] , lattice QCD [ref] , heavy-quark symmetry [ref] and others [ref] .

Doubly heavy tetraquark states in the constituent quark model using diffusion Monte Carlo method · Introduction · 1.00

Discussion

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