Enhanced non-qqbar and non-glueball Nc behavior of light scalar mesons Guillermo Ríos Universidad de Murcia In collaboration with Jenifer Nebreda and José R. Peláez Phys. Rev. D84, 074003 (2011)

Introduction Light scalar mesons are of great interest in hadron and nuclear physics but their properties are the subject of an intense debate There are many resonances, some very wide and difficult to observe experimentally In the case of the kappa, even its existence is not well established according to PDG It is not clear how to fit them in SU(3) multiplets Debate on their spectroscopic classification: qqbars, glueballs, meson molecules, tetraquarks We study the spectroscopic nature of the σ and κ from the QCD 1/Nc expansion

Introduction The 1/Nc expansion gives clear definitions of different spectroscopic components qqbar states: Glueball states:

Introduction In [1] it was studied the spectroscopic nature resonances with unitarized ChPT (UChPT) by extrapolating to unphysical Nc values and checking the Nc scaling of the masses and widths of resonances The σ and κ resonances where shown to be non predominantly qqbar states Although a subdominant qqbar component in the σ may arise at larger Nc Here we study quantities which are highly suppressed (~1/Nc2, ~1/Nc3) in the 1/Nc expansion and check the 1/Nc predictions with real data at Nc=3 without the need to extrapolate to unphysical values These quantities are associated with elastic scattering phase shift evaluated at the "pole" mass of qqbar or gluebal resonances [1] J.R. Pelaez PRL92 (2004) 102001, J.R. Pelaez, G.Rios PRL97 (2006) 242002

Note that each order is suppressed by a (1/Nc)2 factor Highly 1/Nc suppressed observables Consider a resonance appearing in elastic two body scattering as a pole located at It was shown in [2] that if the resonance is a qqbar, The phase shift and its derivative, evaluated at mR, satisfy Note that each order is suppressed by a (1/Nc)2 factor [2]: Nieves, Ruiz Arriola, PLB679,449(2009)

Highly 1/Nc suppressed observables This 1/Nc counting, as shown in [2], comes from an expansion around mR2 of the “real” and “imaginary” parts of the pole equation A resonance appears as a pole on the partial wave in the second Riemann sheet analytic functions that coincide with the real and imaginary parts of t-1 on the real axis The amplitude on the second sheet is obtained crossing the cut in a continuous way So that and the pole position sR is given by [2]: Nieves, Ruiz Arriola, PLB679,449(2009)

Highly 1/Nc suppressed observables Since sR = mR2 + i mRΓR we expand the pole equation around mR2 in terms of imRΓR. For a qqbar state: Since the expansion parameter is purely imaginary, the different orders are real and imaginary alternatively Taking real and imaginary parts of the pole equation we obtain and remembering

Highly 1/Nc suppressed observables Then, Re t-1 when evaluated at mR2 scales as Nc-1 instead of Nc From , the phase shift δ satisfies We define from the above equations the following quantitites For a qqbar state, the coefficients a and b should be O(1) Large suppression at Nc=3, not extrapolation needed We only need experimental data

Highly 1/Nc suppressed observables We evaluate the observables Δ1 and Δ2 for the scalar and vector resonances appearing in elastic ππ and πK scattering: the scalars σ(600) and κ(800), and the vectors ρ(770) and K*(892) We use the output of the experimental data analyses based on dispersion techniques: For ππ scattering: R. García-Martín, R. Kaminski, J. Peláez, J. Ruiz de Elvira, and F. Ynduráin, PRD83 (2011) 074004 (yesterday's talk from Pelaez) For πK scattering in the scalar channel: S. Descotes-Genon and B. Moussallam, EPJC48 (2006) 553 For the πK scattering vector channel we use the elastic Inverse Amplitude Method (IAM), that gives a good description of scattering phase shift data

Calculation of coefficients assuming qqbar behavior If the resonance is predominantly qqbar the phase shift should satisfy the above equation with a natural value (O(1)) of the coefficient "a" Unnaturally small? Unnaturally large coefficients for the scalars Small values can be easily explained from cancellations with higher orders But also… σ and κ NOT predominantly qqbar

Calculation of coefficients assuming qqbar behavior Since Δ1 comes from the expansion We can interpret the O(Nc-3) corrections to Δ1 as the cube of a natural O(Nc-1) term Now it is which should be of natural size Still rather unnatural Natural O(1) size

Calculation of coefficients assuming qqbar behavior Now we calculate the coefficient "b" of Δ2 This time it cannot be interpreted as the square of a natural O(Nc-1) quantity Evaluating it explicitly

Calculation of coefficients assuming qqbar behavior Now we calculate the coefficient "b" of Δ2 Natural O(1) size Unnaturally large

Calculation of coefficients assuming qqbar behavior If the σ(600) and κ(800) resonances are to be interpreted as predominantly as qqbar states, we need coefficients unnaturally large (by two orders of magnitude) to accommodate the 1/Nc expansion predictions at Nc=3 The qqbar interpretation of the σ and κ resonances is very unnatural from the 1/Nc expansion This is obtained from data at Nc=3, without extrapolating to unphysical Nc values

Now we have four 1/Nc powers between different orders Calculation of coefficients assuming glueball behavior The case of the glueball interpretation of the σ meson is even more unnatural since the width of a glueball goes as 1/Nc2 instead of only 1/Nc Because of this extra 1/Nc suppression in the glueball width we get even more suppressed observables. Now we have So that Now we have four 1/Nc powers between different orders

Very unnatural values also for glueball interpretation Calculation of coefficients assuming glueball behavior Nc scaling of the Δ1 and Δ2 observables in the glueball case a' and b' should be O(1) Very unnatural values also for glueball interpretation For the σ we obtain As before, we can interpret the corrections to Δ1 as the cube of a pure O(Nc-2) quantity Still large The glueball interpretation for the σ is very disfavoured from the 1/Nc expansion

Nc evolution of corrections One could still think that, even if the coefficients are unnatural, the evolution with Nc of the corrections is that of a qqbar (or glueball) We sudy the Nc behavior of the observables with Unitarized Chiral Perturbation Theory (Inverse Amplitude Method)

Nc evolution of corrections The elastic Inverse Amplitude Method (IAM) is a unitarization technique to obtain (elastic) unitary amplitudes matching ChPT at low energies It consits on evaluate a dispersion relation for t-1 The imaginay part along the elastic Right Cut is exactly known from unitarity, Im t-1 = -σ ) The Left Cut and substraction constants are evaluated using ChPT In the end we arrive to the simple formula Satisfies exact elastic unitarity and describes well data up to energies where inelasticities are important Matches the chiral expansion at low energies. Can be generalized at higher orders Has the correct analytic structure and we find poles on the 2nd sheet associated to resonances The correct leading Nc dependence of amplitudes is implemented through the chiral parameters. No spurious parameters where uncontrolled Nc dependence could hide

Nc evolution of corrections

Nc evolution of corrections

Nc evolution of corrections

Nc evolution of corrections Δ1-1 follows the qqbar 1/Nc3 scaling for the vectors

Nc evolution of corrections The same happens with Δ2-1

Nc evolution of corrections The same happens with Δ2-1

Nc evolution of corrections The scalars does not follow the qqbar (nor glueball) scaling

Nc evolution of corrections In the case of the σ we can also use the IAM up to O(p6) within SU(2) ChPT Near Nc=3 the observables grow, as in O(p4) At larger Nc they decrease quickly. As pointed out in [3]: possible mix with a subdominant qqbar component [3] J.R. Pelaez, G.Rios PRL97 (2006) 242002

Summary We study, from data at Nc=3, observables whose value is fixed by the 1/Nc expansion up to highly suppressed corrections for qqbar and glueball states If the σ and the κ are to be interpreted as qqbar or glueball states the subleading corrections need unnaturally large coefficients, by two orders of magnitude A dominant qqbar or glueball nature for the σ and κ is then very disfavoured by the 1/Nc expansion We have checked with UChPT that the suppressed corrections do not follow the qqbar scaling for the scalars (and they do for the vectors)