Diapositiva 1

1
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 arXiv1009.6204 hep-ph
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Outline
?Introduction
? UChPT and the 1/Nc expansion.
? FESR and local duality.
? Results
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Introduction and motivation
Light scalars, and particularly the sigma are of
interest for nuleon-nucleon attraction,
glueballs, chiral symmetry breaking, Chiral
Perturbation Theory etc
Actually, NLO ChPT dispersion relations finds
different Nc behaviours JRP, Phys.Rev.Lett.
92102001,2004,
The s becomes broader and its contribution to
the amplitude decreases
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Introduction and motivation
PROBLEM
Local duality requires cancellation between the s
and ? . IF SIGMA DISAPPEARS AT LARGER
Nc Possible contradiction with local duality?
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Outline
?Introduction
? UChPT and the 1/Nc expansion.
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Chiral Perturbation Theory
Weinberg, Gasser Leutwyler
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
pp scattering
leading 1/Nc behavior known from QCD !!!
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Elastic two-body Unitarity Constraints One
channel
EXACT unitarity not satisfied by ChPT series (or
any other series)
Badly violated if ChPT series extrapolated to
high energies or resonance region How to fix that?
We can use ChPT for Re 1/t But it is better to
use this info inside a dispersion relation
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The Inverse Amplitude Method Dispersive
Derivation THE REAL THING
We have just seen that, for physical s
and
PC is O(p6) and we neglect it or use ChPT
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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 ?,
?(770), ?(800), K(892),
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The 1/Nc expansion
ChPT parameters Leading 1/Nc behavior known and
model Independent
UChPT predicts 1/Nc Behavior of resonances
The IAM reliable for Nc lt 15 30 at most beyond
that, just a qualitative model (since QCD weakly
interacting for large Nc)
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LIGHT VECTOR MESONS
qqbar states
Nc
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What about scalars ?
JRP, Phys.Rev.Lett. 92102001,2004
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
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Results O(p6) the sigma
G. Ríos and JRPelaez, Phys.Rev.Lett.97242002,2006
For Nc 10 tor 12
Mixing?
The O(p6) calculation suggests a subdominant
qqbar component for the s with a LARGER MASS
2.5 Ms 1 to 1.2GeV
This subdominant qqbar component can fix the
duality problem of a non-qqbar interpretattion for
the sigma
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Outline
?Introduction
? UChPT and the 1/Nc expansion.
? FESR and local duality.
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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 pp
scattering s-channel
I 2 t-channel exchange should be suppressed
respect to other isospin
Crossing relates t-channel I2 amplitude to
s-channel amplitudes
s
?
T
Very small
The I2 suppression requires strong s-?
cancellation
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Local duality FESR
On the average-cancellation" properly defined
via Finite Energy Sum Rules.
Regge theory interpretation is
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Local duality vs. non-qqbar sigma
The I2 pp scattering s-channel remains non
resonant with Nc. In t-channel suppressed
respect to other isospins
The Regge parameters dont depend on Nc. (at LO)
The I2 FESR should be still suppressed for any
Nc.
s - ? cancellation needed for all Nc
But if s - ? behave differently with Nc, this
cancellation does not occur!!
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Outline
?Introduction
? UChPT and the 1/Nc expansion.
? FESR and local duality.
? Results
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FESR for Nc 3. Check with real data
First point Check the FESR suppression for Nc3
Using real data parametrizations, we have
checked
Kaminski, JRP and Yndurain, PRD77054015,2008
for t th
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FESR and IAM
For n 2, 3, this cancellation occurs below 1-1.5
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
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FESR and Nc. Case with vanishing s
Local duality implies a s - ? cancellation with
Nc.
However, the s and ? mesons show a different Nc
behaviour.
If we take a case where the s amplitude vanishes
(typically the NLO IAM) the ? dominates the
FESR.
T
T
Vanish with Nc
SMALL
Local duality spoilt at larger Nc!!
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FESR and Nc. Case with vanishing s
At higher Nc
The s amplitude vanishes there is no s-?
cancellation.
Local duality fails
CONFLICT WITH LOCAL DUALITY IF THE SIGMA
DISAPPEARS COMPLETELY This is the expected problem
FESR suppression, checked using a real
parametrization.
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FESR and Nc. Case with subdominant
quark-antiquark mixture
There is still a cancellation between the s and
? amplitudes.
The FESR are still suppressed with Nc
Local duality is still satisfied
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FESR and Nc. Case with subdominant
quark-antiquark mixture
FESR remain small with Nc.
The subleading qqbar s 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
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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
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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.
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FESR and Nc With and without subdominant
quark-antiqurk admixture
No subdminant component (typical _at_NLO) No FESR
suppression
With subdominant component (natural _at_NNLO) FESR
suppression
Local duality is satisfied.
Local duality fails
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We have even extrapolated to (too) large Nc. The
cancellation continues
The suppresion continues. It is an stable efffect
But IAM reliable for Nc lt 15 30 at most beyond
that, just a qualitative model (since QCD weakly
interacting for large Nc)
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Summary
Light scalars and particularly the s seem likely
non ordinary quark-antiquark mesons
All non-qqbar scenarios where the s completely
disapperars from the spectrum (typical
_at_NLO-UChPT, pure tetraquark, pure molecule,
etc), suffer CONFLICT WITH LOCAL DUALITY