Enhanced non-quark-antiquark and non-glueball Nc behavior of light scalar mesons
aa r X i v : . [ h e p - ph ] O c t Enhanced non-quark-antiquark and non-glueball N c behavior of light scalar mesons. J. Nebreda, J. R. Pel´aez, and G. R´ıos
Departamento de F´ısica Te´orica II, Universidad Complutense de Madrid, 28040 Madrid, Spain
We show that the latest and very precise dispersive data analyses require a large and very unnat-ural fine-tuning of the 1 /N c expansion at N c = 3 if the f (600) and K (800) light scalar mesons areto be considered predominantly ¯ qq states, which is not needed for light vector mesons. For this, weuse scattering observables whose 1 /N c corrections are suppressed further than one power of 1 /N c for ¯ qq or glueball states, thus enhancing contributions of other nature. This is achieved withoutusing unitarized ChPT, but if it is used we can also show that it is not just that the coefficients ofthe 1 /N c expansion are unnatural, but that the expansion itself does not even follow the expected1 /N c scaling of a glueball or a ¯ qq meson. PACS numbers: 12.39.Mk,12.39.Fe,11.15.Pg,13.75.Lb,14.40.Cs
Light scalar resonances play a relevant role for severalfields of Physics: For the nucleon-nucleon interaction, be-cause they are largely responsible for the attractive part[1] (with cosmological and anthropic implications). forthe QCD non-abelian nature, because some of these res-onances have the quantum numbers of the lightest glue-ball, also common to the vacuum and hence of relevancefor the spontaneous chiral symmetry breaking. Moreover,they are also of interest for the saturation [2] of the lowenergy constants of Chiral Perturbation Theory (ChPT)[3]. However, the precise properties of these mesons,their nature, spectroscopic classification, and even theirexistence—as for the K (800) or κ —are still the objectof an intense debate. In particular, different models [4]suggest that they may not be ordinary quark-antiquarkmesons, but tetraquarks, meson molecules, glueballs, or acomplicated mixture of all these. The problem, of course,is that we do not know how to solve QCD at low energies.However, since the QCD 1 /N c expansion is applica-ble at all energies, and the mass and width N c depen-dence of ¯ qq mesons and glueballs is well known [5], the N c scaling of resonances becomes a powerful tool to clas-sify them and understand their nature. In [6, 7] someof us studied the mass and width behavior of light res-onances using ChPT—which is the QCD low energy ef-fective Lagrangian—and unitarization with a dispersionrelation. It was found that the poles of the ρ (770) and K ∗ (982) vectors behave predominantly as expected for¯ qq states whereas those of the f (600), also called σ , and K (800) scalars do not [6]. Still, a possible subdominant¯ qq component for the f (600) may arise naturally at twoloops [7] within ChPT (less so at one loop), but with amass around 1 GeV or more.Of course, all these conclusions rely on unitarizedChPT and the assumption that corrections, suppressedjust by 1 /N c , are of natural size. Since N c =3 in real life,this may not seem as a large suppression, even more whenthe meaning of “natural size” may not be clear for dimen-sional parameters. For that reason, unitarized ChPT wasuseful to change N c , and reveal the 1 /N c scaling, no mat-ter how unnatural the coefficients may appear.Here we will provide adimensional observables withcorrections suppressed further than 1 /N c , that can also be applied directly to real data at N c = 3, without theneed to extrapolate to larger N c using unitarized ChPT.In particular, resonances appearing in elastic two-bodyscattering are commonly identified by three criteria. The N c behavior of one of these criteria—the associated polein the unphysical sheet—was already studied in [6, 7].A second possibility is to define the mass as the energywhere the phase shift reaches π/
2, which both for ππ or πK scattering occurs relatively far from the f (600) and K (800) pole positions. This criterion was studied in [8]for the f (600) with a relatively inconclusive result aboutits assumed ¯ qq behavior. A more reliable parametrizationand better data were called for and we will provide themhere together with more conclusive results. Third, thephase increases very fast in the resonance region and themass can be identified with the maximum of the phasederivative. All three criteria roughly coincide for nar-row resonances, but the most physical definition is thelatest, since it identifies the resonance as a metastablestate whose lifetime is the inverse of the width. Notethat this is the less evident feature both for the f (600)and K (800) and thus the phase derivative will becomeour preferred observable to test their N c dependence.Let us then recall that partial waves generically scaleas 1 /N c , except at the resonance mass m R . Actually,it has been found [8] that if a resonance pole at s R = m R − im R Γ R behaves as a ¯ qq [5], i.e. m R ∼ O (1) andΓ R ∼ O (1 /N c ), then the phase shift satisfies [22]: δ ( m R ) = π − Re t − σ (cid:12)(cid:12)(cid:12) m R | {z } O ( N − c ) + O ( N − c ) , (1) δ ′ ( m R ) = − (Re t − ) ′ σ (cid:12)(cid:12)(cid:12) m R | {z } O ( N c ) + O ( N − c ) , (2)where t ( s ) is the partial wave, σ = 2 k/ √ s and k is themeson center of mass momentum. Derivatives are takenwith respect to s . The 1 /N c counting of the differentterms in the equations above comes from the following ρ (770) K ∗ (892) f (600) K (800) a − . ± .
01 0.02 − +119 − -2527 b . +0 . − . +28 − /N c expansion fordifferent resonances. For ¯ qq resonances, all them are expectedto be of order one or less. expansions at s = m R [23]:Re t − = m R Γ R h m R Γ R t − ) ′′ − σ ′ i + O ( N − c ) , (3) m R Γ R = σ (Re t − ) ′ + O ( N − c ) . (4)In brief, the corrections in Eqs. (1) to (4) are suppressedby a further 1 /N c power due to an expansion on theimaginary part of the pole, which scales like Γ ∼ /N c .As nicely shown in [8], by expanding separately the realand imaginary parts of t − , only the 1 /N n +1 c powers arekept on each expansion, leading to Eqs. (3) and (4).Since we are interested in adimensional observableswhose corrections are suppressed further than just 1 /N c ,we can recast Eqs.(1) and (2) as follows: π − Re t − /σδ (cid:12)(cid:12)(cid:12) m R ≡ ∆ = 1 + aN c , (5) − [Re t − ] ′ δ ′ σ (cid:12)(cid:12)(cid:12) m R ≡ ∆ = 1 + bN c . (6)Note that we have normalized each equation and ex-tracted the leading 1 /N c dependence so that the coef-ficients a and b should naturally be O (1) or less. It isrelatively simple to make a and b much smaller than onewith cancellations with natural higher order 1 /N c contri-butions, but very unnatural to make them much larger.Now, in Table I we show the resulting a and b for thelightest resonances found in ππ and πK elastic scattering.Before describing in detail the calculations, let us observethat for the ρ (770) and K ∗ (892) vector resonances allparameters are of order one or less, as expected for ¯ qq states . In contrast, for the f (600) and K (800) scalarresonances we find that all parameters are larger, by twoorders of magnitude, than expected for ¯ qq states . Thisis one of the main results of this work and make the ¯ qq interpretation of both scalars extremely unnatural.Let us now describe in detail our calculations and theirdifferent degree of precision and reliability. As com-mented above, the f (600) “Breit-Wigner” mass was al-ready studied [8] using Eq. (1), but no conclusion wasreached on whether the deviations were consistent withthe 1 /N c suppression or not. This was partly attributedto the limited reliability of the conformal parametrizationor unitarized ChPT—whose phase never reaches π/ output of the dataanalysis in [9] constrained to satisfy once subtracted cou-pled dispersion relations—or GKPY equations—as well as Roy equations, which is therefore model independentand specially suited to obtain the f (600) pole [10]. Notethat this analysis incorporates the very recent and re-liable data on K l decays from NA48/2 [10, 11], whichis a key factor in attaining high levels of precision. Theanalysis in [9] is also in good agreement with previous dis-persive results based on standard Roy equations [12]. Wehave followed the same rigorous approach for the ρ (770),although, being so narrow, the conformal unconstraineddata analysis and the IAM yield very similar results. Theuncertainties we quote for both the f (600) and ρ (770)cover the uncertainties in the output of the dispersiverepresentation.In this work we also deal with strange resonances in πK scattering. For the scalar K (800) we have also used a rig-orous dispersive calculation, namely, that in [13], whichuses Roy-Steiner equations to determine the isospin 1/2scalar channel of πK scattering, although this time wecan only provide a central value. Note, however, thatthe value of m R obtained in that analysis is located be-low threshold, so that the phase shift is ill defined at m R . Nevertheless, we have been using the m R mass def-inition to allow for an easier comparison with [8], butthe definition √ s R = m − i Γ / N c scaling of Eqs. (1) and (2)does not change if we evaluate the quantities at s = m ,instead of m R , since m differs from m R in Γ /
4, whichis O ( N − c ). Thus, the values for the K (800) in Table Icorrespond to this choice. For the vector K ∗ (892) thereare no very precise purely dispersive descriptions of theexisting data and we therefore rely on a single partialwave dispersion relation and SU(3) ChPT to one-loopto determine its subtraction constants (this is known asChPT unitarized with the single channel Inverse Ampli-tude Method (IAM) [14]), which we will briefly explainin the next section. We have applied the same method tothe ρ (770) and the results lie within 50% of their centralvalue when using the GKPY dispersive representation.Since the K ∗ (892) is narrower than the ρ (770), the IAMis likely to provide a better approximation than in the ρ (770) case, but even with that 50% uncertainty, it isenough to check that the a and b parameters are smallerthan one.There is, of course, another way of interpreting ourresults, which is that due to the large 1 /N c coefficientsof the f (600) the series simply does not converge. Inparticular, Eq.(1), which was thoroughly considered in[8], is obtained as an expansion of arctan( x ) = x − x / .. .In this way we could explain why the a = − . ± . ρ (770): it is simply the effectof calculating a = ˜ a / a = 0 . +0 . − . , which is nownaturally of O (1). We could try the same procedure forthe f (600), assuming its series expansion is that of a ¯ qq ,to find ˜ a = 9 .
1, still rather unnatural, but of course, thisvalue makes no sense since the whole series would not beconverging and terms higher than 1 /N c would becomedominant.This is one of the reasons why despite being only sup-pressed by 1 /N c instead of 1 /N c , we also provide theexpansion in Eq.(6) obtained from the derivative of theamplitude. In this case the b/N c term is not the squareof a natural 1 /N c quantity, i.e., bN c = Re t − σ h σ ′ (Re t − ) ′ − Re t − σ i + O ( N − c ) . (7)Despite containing a cancellation between two 1 /N c terms, its value for the ρ (770) is rather natural. How-ever, once again, the value for the scalars is almost twoorders of magnitude larger than expected.In the previous analysis it is very relevant that thewidth of the resonance is suppressed with additional 1 /N c powers with respect to the mass. Actually, it is ratherstraightforward to extend the formalism to study the as-sumption that the f (600) could be predominantly a glue-ball, since then m R ∼ O (1) and Γ R ∼ O (1 /N c ) [5, 15].As a consequence, for the glueball case, the scaling ofEqs. (3) and (4) changes and so does that of δ ( m R ) and δ ′ ( m R ): δ ( m R ) = π − Re t − σ (cid:12)(cid:12)(cid:12) m R | {z } O ( N − c ) + O ( N − c ) , (8) δ ′ ( m R ) = (Re t − ) ′ σ (cid:12)(cid:12)(cid:12) m R | {z } O ( N c ) + O ( N − c ) . (9)Much as it was done in Eqs. (5) and (6), in order to makeexplicit this further N c suppression we can define somenew parameters a ′ and b ′ that should be of O (1) if theresonance was a glueball:∆ = 1 + a ′ N c , ∆ = 1 + b ′ N c . (10)Following the same procedure as before we obtain for the f (600), a ′ = − +3200 − and b ′ = 2080 +760 − . In otherwords, a very dominant or pure glueball nature for the f (600) is very disfavored by the 1 /N c expansion, evenmore than the ¯ qq interpretation. This is because it wouldrequire even more unnatural coefficients, this time toolarge by three to four orders of magnitude.Of course, as we did for the ¯ qq case, we could worryabout the fact that, due to the arctan( x ) = x − x / ... expansion, the a ′ should have been interpreted as a ′ =˜ a ′ /
3. But even with that interpretation we would stillfind ˜ a ′ = 27 +5 − , again rather unnatural. Once more, andas it happened in the ¯ qq case, the b ′ parameter does notcorrespond to the fourth power of any natural quantity,so that its value is genuinely unnatural, disfavoring theglueball interpretation.Let us remark that in the case of tetraquarks ormolecules, the width is not expected to be suppressedwith additional 1 /N c powers with respect to the mass ofthe resonance [15, 16]. Thus, our previous formalism does not apply. Furthermore, it is most likely that scalars area mixture of different components. Therefore our results,while showing that neither the ¯ qq or a glueball are favoredas dominant components of light scalars, do not excludethat these structures could be mixed with other compo-nents that would dominate the 1 /N c expansion with adifferent N c behavior [24].In summary, we have just shown that if, for the lightscalar mesons, we study ¯ qq or glueball 1 /N c expansionsas those in Eqs. (5), (6) and (10), their coefficients comeout very unnatural, suggesting that these resonances can-not be described as predominantly made of a quark andan antiquark or a glueball. Note that, contrary to ourprevious works [6, 7], this conclusion has been reached from dispersive analyses of data , without extrapolatingto N c = 3 using unitarized ChPT.However, unitarized ChPT will be used next to calcu-late the ∆ i − /N c expansion of ¯ qq or glueball states given inEqs. (5), (6) and (10), thus explaining the need for un-natural coefficients if a ¯ qq or glueball-like expansion isassumed. The Inverse Amplitude Method:
The elastic IAM[14] uses ChPT to evaluate the subtraction constants andthe left cut of a dispersion relation for the inverse of thepartial wave. The elastic right cut is exact, since the elas-tic unitarity condition Im t = σ | t | , fixes Im t − = − σ .Note that the IAM is derived only from elastic unitar-ity, analyticity in the form of a dispersion relation, andChPT, which is only used at low energies. It satisfiesexact elastic unitarity and reproduces meson-meson scat-tering data up to energies ∼ ρ (770) and f (600), as well as the K ∗ (892) and the K (800) resonances as poles in ππ and πK scattering am-plitudes, respectively.The dependence on the QCD number of colors, N c ,is implemented [6, 7] through the leading N c scaling ofthe ChPT low energy constants (LECs), and is modelindependent [3, 7, 17]. Fortunately, for Eqs.(5) and (6) tohold, only the leading 1 /N c behavior is needed. Note alsothat the IAM does not have any other parameters whereuncontrolled N c dependence could hide—as it happensin other unitarization methods—so that the IAM allowsus to check the scaling of the ∆ i − The SU (2) IAM:
Only the non-strange f (600) and ρ (770) resonances can be checked, but we can do it uni-tarizing with the IAM the corresponding partial waveseither to one or two loops. We simply scale f N c → f p N c /
3, the one loop constants as l ri, N c → l ri N c / r i, N c → r i ( N c / .Thus, in the two first columns of Fig. 1 we show, for the ρ (770) and f (600) resonances, the scaling of the ∆ i − N c = 3value, in order to cancel the leading part of the a and b ( ∆ -1) N c /( ∆ -1) SU(2) O(p ) ρ (770)f (600)(3/N c ) c ( ∆ -1) N c /( ∆ -1) SU(2) O(p ) ρ (770)f (600)(3/N c ) ( ∆ -1) N c /( ∆ -1) SU(2) O(p ) ρ (770)f (600)(3/N c ) c ( ∆ -1) N c /( ∆ -1) SU(2) O(p ) ρ (770)f (600)(3/N c ) ( ∆ -1) N c /( ∆ -1) SU(3) O(p ) ρ (770)f (600)(3/N c ) c ( ∆ -1) N c /( ∆ -1) SU(3) O(p )K*(892)K (800)(3/N c ) ( ∆ -1) N c /( ∆ -1) SU(3) O(p ) ρ (770)f (600)(3/N c ) c ( ∆ -1) N c /( ∆ -1) SU(3) O(p )K*(892)K (800)(3/N c ) FIG. 1: 1 /N c scaling of the ∆ i − N c = 3 value for light scalar and vector mesons, usingunitarized ChPT within SU (2) or SU (3) and to one or two loops— O ( p ) and O ( p ), respectively. coefficients and thus extract the leading 1 /N kc behaviorof Eqs. (5) and (6). For the one-loop calculations we usethe set of LECs in [18], whereas for the two-loop calcu-lation we use the fit D from [18, 19]. We have checkedthat similar results are obtained when using other sets ofLECs in these references or the estimates from resonancesaturation [2].We can observe that the scaling for the ρ (770) observ-ables overlaps with the expectation for the leading behav-ior of ¯ qq states. However, in the case of the f (600) thescaling is completely different. To one loop the f (600)observables grow instead of decreasing. Let us note how-ever, that, for N c larger than ∼
10, the f (600) pole lieson the third quadrant of the complex plane. Before thathappens, the value of m R becomes less than 4 m π and thephase shift has no physical meaning so that Eqs. (5) and(6) do not hold. This behavior does not occur to twoloops. Actually we find again the f (600) behavior al-ready observed in [7], where, for N c close to 3, the widthgrows as in the one-loop case (and so do the observableshere), but for larger N c the f (600) starts behaving moreas a ¯ qq . Note that this ¯ qq behavior appears at a masssomewhat bigger than 1 GeV. This was a hint of the f (600) being a mixture of a predominantly non-¯ qq com-ponent and, at least, a subdominant ¯ qq component witha mass much heavier that the physical one, which is theone that survives at large N c . In terms of the ∆ i − N c = 3 and a decrease at larger N c . Therefore, it isnot only that the a and b coefficients of the f (600) aretoo large as shown in the previous section, but that thescaling itself does not correspond to a ¯ qq state (and evenless so to a glueball). To two-loops, the ρ (770) does not follow exactly the leading behavior of ¯ qq states but de-creases slightly faster, which can be naturally explaineddue to subleading effects or to a possible small pion cloudcontribution. The SU (3) IAM:
Now we can study the scaling of ∆ i − ρ (770) and f (600), but also for the K ∗ (892) and K (800) resonances, although in this casethe elastic unitarized amplitudes are available only toone loop [20, 21]. We have now eight LECs, called L i ( µ ),that scale [3, 17] as L i, N c → L i ( N c /
3) for i = 2 , , , L − L , L , L and L do not change with N c .In the third and fourth columns of Fig. 1 we show theresults found using the set of LECs called Fit II in [20].Similar results are obtained with Fit I or the estimatesfrom resonance saturation in [2]. In the upper panels wesimply reobtain within the SU (3) formalism the sameresults we obtained for the ρ (770) and f (600) withinthe SU (2) formalism to one loop. In the lower panels weshow the results for the light vector K ∗ (892), followingnicely the ¯ qq expectations, as well as the results for thescalar K (800), which has a very similar behavior to the f (600), at odds with a dominant ¯ qq or glueball nature. Summary:
In this work we have studied the 1 /N c expansion of the meson-meson scattering phase-shiftsaround the pole mass of a ¯ qq or glueball resonance. Inparticular we have defined observables whose correctionsare suppressed further than just one power of N c , pay-ing particular attention to the derivative of the phase,which provides a physical and intuitive definition of aresonance. Using recent and very precise dispersive dataanalyses we have shown that if we assume a ¯ qq or glueballbehavior for the f (600) and K (800), the coefficients ofthe expansion of such observables turn out unnaturallylarge. This is shown without using ChPT or extrapo-lating beyond N c = 3. Moreover, when using unitarizedChPT, we have shown that it is the very 1 /N c scaling ofthe observables which does not follow the pattern of the1 /N c expansion expected for ¯ qq or glueball states. Acknoledgments.
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