1. The nature of the lightest scalar meson, its Nc behavior
Departamento de Física Teórica II Universidad Complutense de Madrid
The nature of the lightest scalar meson, its Nc behavior
and semi-local duality
J.R. Peláez
In collaboration with:
J. Ruiz de Elvira, M. Pennigton
and D. Wilson
arXiv: [hep-ph]

2. ● UChPT and the 1/Nc expansion.
Outline
●Introduction
● UChPT and the 1/Nc expansion.
● FESR and local duality.
● Results

3. The scalar nonet may appear above 1 GeV
Introduction and motivation
Light scalars, and particularly the sigma are of interest for nuleon-nucleon
attraction, glueballs, chiral symmetry breaking, Chiral Perturbation Theory etc…
In general they are hard to accommodate as ordinary mesons
Actually, there is mounting evidence that these states may not be ordinary quark-antiquark states
Jaffe, van Beveren,, Rupp, Tornqvist, Roos, Close, Schecter, Sannino, Fariborz, Black, Oset, Oller, JRP, Hanhart, Achasov, Kalashnikova, Maiani Polosa, Riquer and many others…
The scalar nonet may appear above 1 GeV
Actually, NLO ChPT+ dispersion relations finds different Nc behaviours
JRP, Phys.Rev.Lett. 92:102001,2004,
The ρ becomes narrower with Nc, as expected for a meson.
The σ becomes broader and its contribution to the amplitude decreases

4. Local duality requires cancellation between the σ and ρ .
Introduction and motivation
In general, non states have DIFFERENT Nc dependence than the ρ
PROBLEM:
Local duality requires cancellation between the σ and ρ .
IF SIGMA “DISAPPEARS AT LARGER Nc
Possible contradiction with local duality?
At NNLO a subdominant component suggested for the σ around >1 GeV.
(probably related to the ordinary nonet around that region )
G. Ríos and JRP Phys.Rev.Lett.97:242002,2006,
Here we show that this >1 GeV subdominant component
ensures that local duality is still satisfied.

5. ● UChPT and the 1/Nc expansion.
Outline
●Introduction
● UChPT and the 1/Nc expansion.

6. Chiral Perturbation Theory
Weinberg, Gasser & Leutwyler
’s Goldstone Bosons
of the spontaneous
chiral symmetry breaking
SU(2)V SU(2)A  SU(2)V
QCD degrees of freedom
at low energies << 4f~1 GeV
ChPT is the low energy EFFECTIVE THEORY OF QCD
most general low-energy expansion
of a pion lagrangian with the QCD symmetries
Leading order parameters: ,
At higher orders, QCD
dynamics encoded in
Low Energy Constants
determined from experiment
ππ scattering : :
leading 1/Nc behavior known from QCD !!!
ChPT limited to low energies

7. Elastic two-body Unitarity Constraints: One channel
Partial wave UNITARITY
(On the real axis above threshold)
KNOWN EXACTLY (kinematics)
EXACT unitarity not satisfied by ChPT series
(or any other series)
Unitarity bound
Badly violated if ChPT series
extrapolated to high energies
or resonance region
How to fix that?
exactly unitary !!
We only need the
Real part of 1/t
(dynamics)
We can use ChPT for Re 1/t
But it is better to use this info inside a dispersion relation

8. IAM Define We have just seen that, for physical s and PHYSICAL cut
The Inverse Amplitude Method: Dispersive Derivation: THE REAL THING
Define
We have just seen that, for physical s
and
PHYSICAL cut
EXACTLY Opposite
to each other
Write dispersion relations for G and t4
Subtraction Constants
from ChPT expansion
OK since s=0
G(0)=t2(0)-t4(0)
Up to NLO ChPT
Opposite to each other
IAM
All together…we find AGAIN
PC is O(p6) and
we neglect it
or use ChPT

9. Mass Width/2 Very simple. Systematic extension to higher orders
The Inverse Amplitude Method: Results for one channel
Truong ‘89, Truong,Dobado,Herrero,’90, Dobado JRP,‘93,‘96
Very simple. Systematic extension to higher orders
Simultaneously: Unitarity + Chiral expansion
ChPT used ONLY at low energies: subtraction constants and left cut, NOT in resonance region
Dispersion relation allows us to go to complex plane.
Generates Poles of Resonances:
f0(600) or “”, ρ(770), (800), K*(892),
=f0(600)
(770)
K*(890)
Width/2
Mass
f0(600) pole: 440-i245 MeV
Dobado, JRP ‘96

10. The 1/Nc expansion provides a clear definition of states
ChPT parameters:
Leading 1/Nc behavior known and model Independent
UChPT predicts 1/Nc
Behavior of resonances
The IAM reliable for Nc < 15 – 30 at most
beyond that, just a qualitative model
(since QCD weakly interacting for large Nc)

11. The IAM generates the expected Nc scaling of established qq states
LIGHT VECTOR MESONS
The IAM generates the expected
Nc scaling of established qq states
JRP, Phys.Rev.Lett. 92:102001,2004
qqbar states:
The (770)
The K*(892)
MN/M3
N/3
MN/M3
N/3 Nc
MN/M3
N/3 Nc

12. What about scalars ? The  (=770MeV) The  (=500MeV) MN/M3 N/3
JRP, Phys.Rev.Lett. 92:102001,2004
The  (=770MeV)
The  (=500MeV) Nc
MN/M3
N/3
MN/M3
N/3 Nc
Similar conclusions for the f0(980) and a0(980)
Complicated by the presence of THRESHOLDS and except in a corner of parameter space for the a0(980)
Requires coupled channel formalism

13. results to those at O(p4): Robust Non qqbar dominant component
Results O(p6): the sigma
G. Ríos and JRPelaez, Phys.Rev.Lett.97:242002,2006
Near Nc = 3 similar
results to those at O(p4):
Robust Non qqbar dominant
component
M becomes constant ~ 1GeV
For Nc ~ 10 tor 12
Γ starts decreasing
Mixing?
The O(p6) calculation suggests
a subdominant qqbar
component for the σ with a
LARGER MASS
~ 2.5 Mσ ~ 1 to 1.2GeV
This subdominant qqbar component
can fix the duality problem
of a non-qqbar interpretattion
for the sigma

14. ● UChPT and the 1/Nc expansion.
Outline
●Introduction
● UChPT and the 1/Nc expansion.
● FESR and local duality.

15. Introduction. Local Duality
Local duality implies that a large number of s-channel resonances are,
“on the average“, dual to t-channel Regge exchanges.
No resonances exchanged in repulsive I = 2 ππ scattering s-channel
I = 2 t-channel exchange should be suppressed respect to other isospin
Crossing relates t-channel I=2 amplitude to s-channel amplitudes: σ ρ T
Very small
The I=2 suppression requires strong σ-ρ cancellation

16. Local duality & FESR
“On the average-cancellation" properly defined via Finite Energy Sum Rules.
Regge theory interpretation is:

17. But if σ - ρ behave differently with Nc,
Local duality vs. non-qqbar sigma
The I=2 ππ scattering s-channel remains non resonant with Nc.
In t-channel suppressed respect to other isospins
The Regge parameters don’t depend on Nc. (at LO)
The I=2 FESR should be still suppressed for any Nc.
σ - ρ cancellation needed for all Nc
But if σ - ρ behave differently with Nc,
this cancellation does not occur!!

18. ● UChPT and the 1/Nc expansion.
Outline
●Introduction
● UChPT and the 1/Nc expansion.
● FESR and local duality.
● Results

19. For Nc =3, local duality is satisfied.
FESR for Nc = 3. Check with real data
First point: Check the FESR suppression for Nc=3
Using real data parametrizations, we have checked:
Kaminski, JRP and Yndurain, PRD77:054015,2008
for t = th
For Nc =3, local duality is satisfied.

20. For n= 2, 3, this cancellation occurs below 1-1.5 GeV.
FESR and IAM
For n= 2, 3, this cancellation occurs below GeV.
We can use the IAM to study local duality, but only applies for S0, P and S2 waves
We calculate the FESR using the IAM and check the influence of those waves.
The influence of higher waves is around 10%.
The IAM predicts correctly the FESR suppression.
We can use the IAM to study the FERS dependence on Nc

21. T T FESR and Nc. Case with vanishing σ
Local duality implies a σ - ρ cancellation with Nc.
However, the σ and ρ mesons show a different Nc behaviour.
If we take a case where the σ amplitude vanishes (typically the NLO IAM)
the ρ dominates the FESR. T T
Vanish with Nc
SMALL
Local duality spoilt at larger Nc!!

22. FESR and Nc. Case with vanishing σ
At higher Nc
The σ amplitude
vanishes: there is
no σ-ρ cancellation.
Local duality fails
FESR suppression, checked using a real parametrization.
CONFLICT WITH LOCAL DUALITY
IF THE SIGMA DISAPPEARS COMPLETELY
This is the expected problem

23. FESR and Nc. Case with subdominant quark-antiquark mixture
But if a subleading component for the σ emerges around 1 GeV,
As it happens naturally within two-loop ChPT.
There is still a cancellation between
the σ and ρ amplitudes.
The FESR are still suppressed with Nc
Local duality is still satisfied

24. FESR and Nc. Case with subdominant quark-antiquark mixture
FESR remain small with Nc.
The subleading qqbar σ component at 1 GeV , ensures local duality.
LOCAL DUALITY IS SATISFIED with Nc
Two loop UChPT solves the problem
naturally
FESR suppression, checked using a real parametrization

25. Case with subdominant quark-antiquark mixture. Other states
Cancellation occurs only if the subdominant state has a mass below 1.5 GeV
Important: a LARGE width when reaching back the real axis around1.2 GeV (FESR are 1/sn suppressed), otherwise no cancellation
Most likely this is an ordinary meson component common to other mesons in that region (J.Ruiz de Elvira, F. J. llanes Estrada, JRP in preparation)
OTHER mesons or qqbar components in that region are not enough for the cancellation at large Nc. They have a too narrow width for larger Nc

26. Case with subdominant quark-antiquark mixture. Other states
In particular f0(980) effect too small
We have also added a crude model of the f2(1275).
It contributes a littke to the cancellation, but not enough. The effect of the
Subdominant component is larger.

27. Local duality is satisfied.
FESR and Nc: With and without subdominant quark-antiqurk admixture
No subdminant
component:
No FESR
suppression
With subdominant
component
FESR suppression
Local duality is satisfied.
Local duality fails

28. The suppresion continues. It is an stable efffect
We have even extrapolated to (too) large Nc. The cancellation continues
The suppresion continues.
It is an stable efffect
But IAM reliable for Nc < 15 – 30 at most
beyond that, just a qualitative model
(since QCD weakly interacting for large Nc)

29. Summary
Light scalars and particularly the σ seem likely non ordinary quark-antiquark mesons
Actually, the 1/Nc expansion within UChPT shows that the σ meson is not predominantly a state, while genrating the correct ρ dependence.
All non-qqbar scenarios where the σ completely disapperars
from the spectrum pure tetraquark, pure molecule, etc…),
suffer
CONFLICT WITH LOCAL DUALITY
The σ 1/Nc behavior is predominantly that of a non ordinary meson, but
a subdominant component with the 1/Nc behavior naturally suggested by two-loop unitarized ChPT ensures that
local duality is still satisfied.