On the nature of light scalar mesons from their large $N_c$ behavior

J. R. Pelaez (U. Complutense; Madrid)

REVIEW

On the nature of light scalar mesons from their large $N_c$ behavior

J. R. Pelaez (U. Complutense, Madrid)

High Energy Physics - Phenomenology · Submitted 25 Sep 2003 · 23 references

Pith's verdict: A genuinely cleaner large-Nc test of the light scalars than the earlier cutoff-based attempts, with the vector calibration doing real work — but the numerical robustness of the scalar 'disappearance' is asserted rather than shown.

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arXiv IDhep-ph/0309292 (version 2)

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CitationJ. R. Pelaez (U. Complutense, Madrid). "On the nature of light scalar mesons from their large $N_c$ behavior." arXiv:hep-ph/0309292 (2003). https://pith.science/paper/27PXPTIS

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Editor's note

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Worth your time if you care about the scalar nonet, with one caveat about error propagation that the paper does not address. The real advance over [12,17] is technical but not cosmetic: by working at O(p4) with fully renormalized, scale-independent amplitudes, the Nc dependence sits entirely in the L_i and cannot hide in a cutoff or a regularization scale. That closes an obvious loophole in the earlier leading-order treatments, and it is the reason to take this version seriously rather than as a rerun. The rho and K* calibration is the strongest part of the paper — constant mass, width scaling as 1/Nc, matching the qqbar expectation within the mu band — and it earns the right to then judge the scalars by the same rule. The diagrammatic argument in Fig. 3 is also largely independent of the precise L_i values, which matters for what follows. The a0(980) is handled honestly: the peak survives at mu = 0.5 GeV and the paper says so instead of burying it. The note added discloses prior workshop presentation and independent confirmation by Uehara, which is the right way to handle precedence. My main reservation is the one the stress-test flags, and it holds up on reading. The shaded bands in Figs. 2, 4 and 5 cover only the variation of mu in 0.5-1 GeV; the fit errors on the input L_i from Table I are never propagated. That is uncomfortable where the errors are largest — L_8 = 0.78 ± 0.7 is roughly a 90% uncertainty, and L_5 and L_8 are exactly the parameters carrying the strange-quark dependence relevant to the kappa and f0(980). The paper says 'the results are similar for other sets' without showing them. I would not call this fatal, because the structural argument does not rest on the central fit values, but the abstract's 'disappear for large Nc' is a stronger statement than the figures establish. A ±1sigma scan over the L_i box, shown for at least the kappa and f0(980), would settle it. The mu-calibration loop the reader flags is mild and I would not weight it heavily. Recommendation: send it out. It is a careful, appropriately hedged methods paper that the scalar community should read and argue with.

Referee report

Significance

If the method holds, it is a genuinely useful bridge: it converts a fit to low-energy meson-meson data into a statement about the underlying quark-gluon composition, without introducing a model of the scalars. The design has real strengths that should be named. The vector channel is used as a falsification test rather than an afterthought, and it passes quantitatively, not merely qualitatively. The diagrammatic counting of Fig. 3 is largely independent of the precise numerical values of the L_i, so the structural part of the argument does not rest on the fit. The a0(980) is reported as ambiguous rather than forced into the preferred conclusion, and the scale uncertainty is displayed rather than hidden. The result is falsifiable by independent unitarization schemes and by lattice input, and the note added records an early cross-check by Uehara. The paper is short and the claim is appropriately hedged in the abstract (in support of), which is the right register for a result of this kind.

Major objections (3)

[Table I and Figs. 2, 4, 5] The shaded bands in Figs. 2, 4 and 5 cover only the variation of the renormalization scale mu in 0.5-1 GeV; the fit uncertainties on the input L_i of Table I are never propagated. This matters most where the errors are largest and where they are most relevant to the states under discussion: L8 = 0.78 +/- 0.7 (roughly ninety percent), L5 = 1.8 +/- 0.08, and the O(1) combination 2L1 - L2 = 0.0 +/- 0.1. L5 and L8 carry the quark-mass dependence and therefore enter precisely the strange channels (kappa, f0(980)) and the a0(980), which are the states where the conclusion is weakest or, for the a0, explicitly ambiguous. The headline statement of abstract:s2, that the sigma, kappa and f0(980) 'disappear for large Nc', is read off these figures, so the numerical robustness of the central claim is not established as the paper stands. A scan over the +/-1 sigma box of the L_i (at minimum varying L5 and L8, and the O(1) combination) showing that the pole trajectories retain their qualitative behavior would settle this; if the disappearance is driven by the central fit values, the claim needs to be qualified accordingly.
[Discussion of Fig. 4 (f0(980) and a0(980) region)] The rho, K*, sigma and kappa are diagnosed through pole positions on the appropriate sheet (Fig. 2), but the f0(980) and a0(980) are diagnosed instead from the modulus and imaginary part of the amplitude at a fixed energy, sqrt(s) = 960 MeV (Fig. 4). Near the KK threshold a dip in |t| can be a threshold artifact rather than the signature of a state, and the paper itself notes the 'distortions caused by the nearby KK threshold'. Since the f0(980) sits essentially on that threshold, the pole position is the meaningful object, and the two diagnostics are not obviously equivalent. The manuscript should either provide pole trajectories for the f0(980) and a0(980) analogous to Fig. 2, or justify explicitly why the fixed-energy amplitude diagnostic carries the same information in this region.
[Scale calibration and the a0(980) verdict (Figs. 2 and 5)] The band mu = 0.5-1 GeV is validated by requiring the vectors to reproduce qqbar behavior ('which confirms that the expected scale range where the large Nc scaling applies is correct'), and the same band is then used to certify the scalars. The vector check is a consistency check, not an independent determination, and the a0(980) verdict flips between mu = 0.55 GeV (peak disappears) and mu = 0.5 GeV (peak survives, Fig. 5). Because the a0 conclusion is decided at the extreme lower edge of the band, the manuscript should state what independent information fixes that lower edge, and should say plainly whether the a0 ambiguity is a one-sigma effect or an edge-of-band effect. This is a correctness-risk concern about the robustness of the stated uncertainty, not a claim that the logic is circular.

Minor objections (6)

[Text following Table I] The statement that 'the results are similar for other sets' is asserted without showing any alternative fit. Given the size of the fit systematics, at least one alternative IAM set should be displayed, or the statement should be softened to indicate that this was checked privately.
[Note added] The note added cites [23] as having confirmed the results 'without the scale uncertainties' using the approximated IAM of [9]. It would help to say explicitly whether [23] agrees on the a0(980) as well, since that is the one state where the present analysis is inconclusive.
[Fig. 2 caption] The dashed lines are described only as 'different Nc scaling laws'. The caption should identify which curve is 3/Nc, which is sqrt(Nc/3), and which is Nc/3, so the figure can be read without consulting the body text.
[Fig. 3 caption] The Nc-scaling labels on the three diagrams are garbled in the current version ('1/N 1/N 1/N_c^2'). Please check that the labels in the published figure are legible and correctly attached to panels a, b and c.
[Introduction and conclusion] The paper does not say how the picture fits the heavier scalar nonet, in particular the f0(1370) and f0(1500), which are the usual qqbar candidates. One or two sentences on whether the present method can be extended to them, and on what it would predict, would strengthen the spectroscopic argument considerably.
[Whole manuscript] There are several typographical issues to fix: 't should satisfy an elastic unitarity condition' in the paragraph introducing Eq. (2) is missing the initial 'I'; 'distorsions' appears in the discussion of Fig. 4; the qqbar-qqbar notation is written inconsistently as 'qqbar qqbar' and as 'qbar q qq' in different places; and the reference list has inconsistent punctuation, for example 'Phys. Rev. D15 267 (1977)'.

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The paper

The paper claims

By dialing up the number of colors in QCD, the paper argues, the light scalar mesons vanish while the vector mesons survive as quark–antiquark pairs.

The reading

The paper asks a spectroscopic question that ordinary chiral perturbation theory cannot answer on its own: are the lightest scalar resonances — the $f_0(500)$ or $sigma$, the $K_0^*(700)$ or $kappa$, the $f_0(980)$ and the $a_0(980)$ — ordinary quark–antiquark states, four-quark states, or something else? The strategy is to take the meson–meson scattering amplitudes of the QCD low-energy effective theory, unitarized so that they generate resonances, and then to vary the number of colors $N_c$ according to the known large-$N_c$ counting of the chiral parameters and of the pion decay constant. If a resonance is a quark–antiquark bound state, its mass should stay put while its width shrinks like $1/N_c$; if it is not, it should melt into the continuum. The vectors $rho(770)$ and $K^*(892)$ pass this test cleanly, which validates the method and locates the scale at which the counting applies. The scalars fail it: the $sigma$, $kappa$ and $f_0(980)$ lose their peaks as $N_c$ grows, which the author reads as evidence for a dominant four-quark (or two-meson) composition, with the $a_0(980)$ left genuinely undecided.

What carries the argument

The engine is the Inverse Amplitude Method (IAM): elastic unitarity requires $mathrm{Im}, t = sigma |t|^2$, hence $t = 1/(mathrm{Re}, t^{-1} - i sigma)$, so a chiral perturbative expansion can be used for $mathrm{Re}, t^{-1}$ while unitarity is imposed exactly. This resummation generates the $rho$, $K^*$, $sigma$ and $kappa$ poles from amplitudes that contain no resonances order by order, while still respecting the low-energy chiral expansion, and it generalizes to a coupled-channel matrix form $T simeq (mathrm{Re}, T^{-1} - i Sigma)^{-1}$ that covers $pi pi$, $pi K$ and $K ar{K}$. The second ingredient is the large-$N_c$ counting of the chiral parameters, which converts the question 'what is this state made of?' into a question about how a pole moves when $N_c$ is varied. Because the amplitudes are fully renormalized and scale independent, the author argues that all $N_c$ dependence enters through the $L_i$ and cannot hide in an arbitrary cutoff.

Extended reading notes

Core claim

The central claim is that the large-$N_c$ behavior of unitarized chiral amplitudes discriminates between spectroscopic hypotheses for the light scalars. Using the one-loop $O(p^4)$ amplitudes of chiral perturbation theory resummed with the Inverse Amplitude Method, the author scales $f_0 to f_0 sqrt{N_c/3}$ and $L_i to L_i (N_c/3)$ for $i = 2, 3, 5, 8$, holding masses and the combinations $2L_1 - L_2$, $L_4$, $L_6$, $L_7$ fixed, and tracks the pole positions as $N_c$ runs from 3 to 30. The $rho$ and $K^*$ poles keep their masses and see their widths fall roughly as $3/N_c$, matching the quark–antiquark expectation within the band of scales $mu = 0.5$–$1$ GeV where the counting is taken to apply. In the $sigma$ and $kappa$ regions the amplitudes instead shrink with growing $N_c$, with $mathrm{Im}, t sim O(1/N_c^2)$ and $mathrm{Re}, t sim O(1/N_c)$, and the widths grow rather than shrink — behavior the author argues is incompatible with a quark–antiquark assignment and natural for a $ar{q}ar{q}qq$ or two-meson state, which is expected to become unbound as $N_c to infty$. The $f_0(980)$ shows a dip that disappears at large $N_c$, following a $1/N_c^2$ pattern. The $a_0(980)$ is the exception: its peak vanishes when the scaling is applied at $mu = 0.55$–$1$ GeV but survives at $mu = 0.5$ GeV, so the author explicitly declines to rule out a quark–antiquark nature or sizable mixing for it.

Load-bearing premise

The argument assumes that the leading large-$N_c$ counting of the chiral parameters and of the pion decay constant can be applied at a single scale somewhere between 0.5 and 1 GeV, with the subleading $1/N_c$ corrections small enough at $N_c = 3$ not to move the poles — even though the paper itself notes that the leading and subleading parts of the measured parameters cannot be separated and that the logarithmic terms can be rather large at $N_c = 3$.

Editorial extensions

If this is right

The $sigma$, $kappa$ and $f_0(980)$ should not appear as narrow states in any setting where the number of colors is effectively large, and their widths should grow rather than shrink with $N_c$.
The $rho(770)$ and $K^*(892)$ serve as a calibration: their observed $1/N_c$ width scaling fixes the window $mu simeq 0.5$–$1$ GeV in which the large-$N_c$ counting of chiral parameters is meaningful.
The $a_0(980)$ remains a live candidate for a quark–antiquark assignment or for substantial mixing, and its fate depends on resolving the renormalization-scale ambiguity rather than on new dynamics.
The same procedure — scale the effective couplings by their fundamental-theory counting and watch the poles — can be exported to other unitarized effective theories, such as heavy-baryon chiral perturbation theory or the strongly interacting symmetry-breaking sector.
A four-quark or molecular assignment for the light scalars implies that their couplings to two mesons are leading in $N_c$, so their phenomenology should be organized around meson–meson channels rather than around quark-model spectroscopy.

Reading between the lines

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

The scale ambiguity that keeps the $a_0(980)$ undecided is not a nuisance but a measurable quantity: a lattice or dispersive determination of the $L_i$ at a fixed scale would convert the whole argument from a band of outcomes into a single prediction.
If the $sigma$ is predominantly a two-meson object, its pole position should shift measurably with the light quark masses in a way that tracks the $pi pi$ threshold rather than the quark-mass dependence of a confined pair, which is in principle checkable with lattice QCD at varying pion mass.
The same $N_c$ diagnostic applied to the tensor mesons and to the heavier scalar nonet would test whether the four-quark assignment is a property of the lightest multiplet only or a pattern that repeats at higher mass.
Because the method relies on the leading term of a $1/N_c$ expansion evaluated at $N_c = 3$, a natural cross-check is to repeat the analysis with the known subleading chiral parameters included, which would show whether the scalar pole trajectories are stable or merely an artifact of truncation.

Original abstract

We show how to obtain information about the states of an effective field theory in terms of the underlying fundamental theory. In particular we analyze the spectroscopic nature of meson resonances from the meson-meson scattering amplitudes of the QCD low energy effective theory, combined with the expansion in the large number of colors. The vectors follow a qqbar behavior, whereas the sigma, kappa and f_0(980) scalars disappear for large N_c, in support of a qqqbarqbar-like nature. The a_0 shows a similar pattern, but the uncertainties are large enough to accommodate both interpretations.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.90 · human participants 0.01 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.17 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We show how to obtain information about the states of an effective field theory in terms of the underlying fundamental theory. In particular we analyze the spectroscopic nature of meson resonances from the meson-meson scattering amplitudes of the QCD low energy effective theory, combined with the expansion in the large number of colors. The vectors follow a qqbar behavior, whereas the sigma, kappa and f_0(980) scalars disappear for large N_c, in support of a qqqbarqbar-like nature. The a_0 shows a similar pattern, but the uncertainties are large enough to accommodate both interpretations.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We show how to obtain information about the states of an effective field theory in terms of the underlying fundamental theory.	contribution 0.86	0.86	0.06	asserted 1.00
s1	In particular we analyze the spectroscopic nature of meson resonances from the meson-meson scattering amplitudes of the QCD low energy effective theory, combined with the expansion in the large number of colors.	method 0.52	0.90	0.08	asserted 0.94
s2	The vectors follow a qqbar behavior, whereas the sigma, kappa and f_0(980) scalars disappear for large N_c, in support of a qqqbarqbar-like nature.	result 0.85	0.87	0.82	asserted 0.94
s3	The a_0 shows a similar pattern, but the uncertainties are large enough to accommodate both interpretations.	result 0.59	0.88	0.07	hedged possibility 0.62

What kind of paper the classifier reports this is: theoretical result 0.98 · evidence from proof or derivation 0.90.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
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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

Figures

**Figure 3.** FIG. 3: Representative diagrams contributing to meson

Showing 3 of the paper's 5 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-ph/0309292 by the authors.

References

Works cited (23)

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How later papers used it

51 later papers cite it

Evidence: substantial. 47 of the citing papers share no author with it, and 8 of those carry a confident label from Pith's citation classifier.

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2 later papers by other authors are listed below: 2 cited it as support.

Cites it as support 2 of 5 papers with this label

Said that the sigma field in its model is a qqbar state, citing this paper.

Vacuum Properties of Mesons in a Linear Sigma Model with Vector Mesons and Global Chiral Invariance (2010) · 1.00 · paraphrased: the licence does not allow quoting
Said that, in agreement with this view, the large-N_C behavior of the corresponding pole confirms this feature, citing this paper.

How the axial anomaly controls flavor mixing among mesons (2017) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

47 arXiv papers cite this in 75 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 5 papers

We obtain the result that vector mesons such as rho and K* survive as narrow width resonances, but all of the scalar meson nonet below 1 GeV fade out as Nc exceeds a rather small number about 6.

Large Nc Behavior of Light Scalar Meson Nonet Revisited · 0.99
We study, within an approximate Inverse Amplitude Method to unitarize Chiral Perturbation Theory, whether low mass scalar mesons can survive in large N_c regime, and show that vector mesons such as \rho and K^* survive as narrow width resonances, but all of the scalar meson nonet below 1GeV fade out as N_c becomes large.

Can Low Mass Scalar Meson Nonet Survive in Large N_c Limit ? · 0.98
The vectors thus generated follow closely a qqbar behavior, whereas the light scalar poles follow the large Nc behavior expected for a dominant tetraquark or two-meson structure.

Light scalars as tetraquarks or two-meson states from large Nc and unitarized Chiral Perturbation Theory · 0.95
The scalars sigma, kappa f_0(980) and, possibly, the a_0(980) do not seem to behave as q-qbar states, in contrast to the vectors rho(770) and K*(892).

Large N_c behavior of light resonances in meson-meson scattering · 0.90
In contrast with the rho or K*, the scalar behavior is at odds with a qqbar "dominant component".

Comments on the large Nc behavior of light scalars · 0.89

Papers citing it · 54 sentences without a confident label

Uses its method or tool 2 papers

Alternatively, the Inverse Amplitude Method of Refs. [ref] can be used.

Vacuum Properties of Mesons in a Linear Sigma Model with Vector Mesons and Global Chiral Invariance · The Lagrangian · 0.92
Similar method is also used in [ref] .

Two-pole structures in a relativistic Friedrichs-Lee-QPC scheme · A brief review of relativistic Friedrichs-Lee-QPC scheme · 0.91

Uses its data or values 1 paper

[ref] se obtiene $M_S /M_V = 0.58,0.84,0.96, 0.98$, respectivamente, con $M_S$ y $M_V$ las partes reales de los polos en la segunda hoja de Riemann.] Por tanto, el intentar compatibilizar el límite de $N_c$ grande en la SRA con el modelo quark espectral produce una degeneración de los mesones escalar y vector.

Efectos de Temperatura Finita y Curvatura en QCD y Modelos de Quarks Quirales · Límite de $N_c$ grande y Dualidad · 0.89

Compares with it 1 paper

In a later study [ref] , similar $σ$ trajectories are obtained as those in Refs. [ref] in the region of $N_C≲30$.

$N_C$ evolution of the light meson resonances · Resonance trajectories from the IAM with $SU(3)$ $χPT$ at one loop · 0.91

Cites it as support 3 papers

The $N_C$ behaviors of the amplitudes relevant for the $f_0(980)$ and $a_0(980)$ show that most likely the $q̅q$ components play small roles in these two scalar resonances [ref] .

$N_C$ evolution of the light meson resonances · Resonance trajectories from the IAM with $SU(3)$ $χPT$ at one loop · 0.96
Thus, the sigma field in our model is a $q̅q$ state [ref] .

Vacuum Properties of Mesons in a Linear Sigma Model with Vector Mesons and Global Chiral Invariance · The $σ→ππ$ decay width · 1.00
As mentioned above, it has been successfully used in the past to accurately describe the scattering in the $ππ/πη$ and $KK̅$ systems (at physical values of the quark masses), see Refs. [ref] .

Unitarized Chiral Perturbation Theory in a finite volume: scalar meson sector · The framework based on UCHPT · 0.98

Cites it as background 11 papers

The first attempt to the investigate the $N_C$ evolution of the resonance poles in the IAM approach is carried out in Refs. [ref] , which is based on the one-loop two-meson scattering amplitudes of the $SU(3)$ $χ$PT [ref] .

$N_C$ evolution of the light meson resonances · Resonance trajectories from the IAM with $SU(3)$ $χPT$ at one loop · 0.84
The chiral unitary model has been extensively used for the study of hadron resonances as dynamically generated states [ref] .

Hadron resonances with coexistence of different natures · Hadronic composites in the chiral unitary model · 0.82
* Two resonances are candidates for the ground-state $q̅q$ resonance in the scalar-kaon channel (with alternative interpretations – just as in the case of the $a_0$ resonances – in terms of $q̅q̅qq$ structures or meson-meson molecules): $K_0^⋆(800)$/$κ$ and $K_0^⋆(1430)$; controversy still surrounds the first of these states [ref] .

Excited Scalar and Pseudoscalar Mesons in the Extended Linear Sigma Model · Lagrangian · 0.83

Discussion

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