Universidad de Murcia

Universidad de Murcia

Guillermo Ríos

Enhanced non-qqbar and non-glueball Nc behavior of light scalar mesons

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

Here we study quantities which are highly suppressed (~1/Nc2, ~1/Nc

3) 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

Although a subdominant qqbar component in the σ may arise at larger Nc

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

The amplitude on the second sheet is obtained crossing the cut in a continuous way

and the pole position sR is given by

analytic functions that coincide with the real and imaginary parts of t-1 on the real axis

So that

[2]: Nieves, Ruiz Arriola, PLB679,449(2009)


Since sR = mR2 + i mRΓR we expand the pole equation around mR

2 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


, the phase shift δ satisfies From

Then, Re t-1 when evaluated at mR2 scales as Nc

-1 instead of Nc

We define from the above equations the following quantitites

For a qqbar state, the coefficientsa and b should be O(1)

Large suppression at Nc=3, not extrapolation needed

We only need experimental data


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:

R. García-Martín, R. Kaminski, J. Peláez, J. Ruiz de Elvira, and F. Ynduráin, PRD83 (2011) 074004

For ππ scattering:

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

(yesterday's talk from Pelaez)

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 large coefficients for

the scalars

σ and κ NOT predominantly qqbar

Unnaturally small?

Small values can be easily explained from cancellations

with higher orders

But also…


We can interpret the O(Nc-3) corrections to Δ1 as the cube of a natural O(Nc

-1) term

Since Δ1 comes from the expansion

Now it is which should be of natural size

Still rather unnatural

Natural O(1) size


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


Now we calculate the coefficient "b" of Δ2

Unnaturally large

Natural O(1) size


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

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/Nc

2 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

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)

For the σ we obtainVery unnatural values

also for glueball interpretation

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)


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





Δ1-1 follows the qqbar 1/Nc3 scaling for the vectors


The same happens with Δ2-1


The same happens with Δ2-1


The scalars does not follow the qqbar (nor glueball) scaling


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)

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