1. 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, (2011)

2. 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

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

4. 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) , J.R. Pelaez, G.Rios PRL97 (2006)

5. 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)

6. 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)

7. 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

8. 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

9. 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)
(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

10. 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

11. 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

12. 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

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

14. 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

15. 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

16. 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

17. 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)

18. 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

19. Nc evolution of corrections

20. Nc evolution of corrections

21. Nc evolution of corrections

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

23. Nc evolution of corrections
The same happens with Δ2-1

24. Nc evolution of corrections
The same happens with Δ2-1

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

26. 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)

27. 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)