departamento de física teórica ii. universidad complutense de madrid
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Departamento de Física Teórica II. Universidad Complutense de Madrid. The nature of the lightest scalar meson, its N c behavior and semi-local duality. J.R. Peláez. In collaboration with: J. Ruiz de Elvira, M. Pennigton and D. Wilson arXiv:1009.6204 [hep-ph]. Outline. - PowerPoint PPT PresentationTRANSCRIPT
Departamento de Física Teórica II. Universidad Complutense de Madrid
J.R. Peláez
The nature of the lightest scalar meson, its Nc behavior and semi-local duality
In collaboration with:J. Ruiz de Elvira, M. Pennigton
and D. Wilson
arXiv:1009.6204 [hep-ph]
Outline
●Introduction
● UChPT and the 1/Nc expansion.
● FESR and local duality.
● Results
Introduction and motivation
Light scalars, and particularly the sigma are of interest for nuleon-nucleonattraction, glueballs, chiral symmetry breaking, Chiral Perturbation Theory etc…
Actually, NLO ChPT+ dispersion relations finds different Nc behaviours JRP, Phys.Rev.Lett. 92:102001,2004,
The σ becomes broader and its contribution to the amplitude decreases
The ρ becomes narrower with Nc, as expected for a meson.
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
Local duality requires cancellation between the σ and ρ .IF SIGMA “DISAPPEARS AT LARGER NcPossible 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.
In general, non- states have DIFFERENT Nc dependence than the ρ
Introduction and motivation
PROBLEM:
Outline
●Introduction
● UChPT and the 1/Nc expansion.
Chiral Perturbation Theory Weinberg, Gasser & Leutwyler
ChPTis the low energy EFFECTIVE THEORY OF QCDmost general low-energy expansion
of a pion lagrangian with the QCD symmetries
Leading order parameters:
At higher orders, QCD dynamics encoded in
Low Energy Constantsdetermined from experiment
’s Goldstone Bosons of the spontaneous
chiral symmetry breakingSU(2)V SU(2)A SU(2)V
QCD degrees of freedomat low energies << 4f~1 GeV
ππ scattering
,
::
ChPT limited to low energies
leading 1/Nc behavior known from QCD !!!
t1Im
Partial wave UNITARITY(On the real axis above threshold)
2Im tt
itt
1Re
1
exactly unitary !!
Elastic two-body Unitarity Constraints: One channel
sps CM /2)( KNOWN EXACTLY (kinematics)
We only need theReal part of 1/t
(dynamics)
We can use ChPT for Re 1/tBut it is better to use this info inside a dispersion relation
EXACT unitarity not satisfied by ChPT series(or any other series)
11
t
Unitarity bound
Badly violated if ChPT seriesextrapolated to high energies
or resonance regionHow to fix that?
The Inverse Amplitude Method: Dispersive Derivation: THE REAL THING
Write dispersion relations for G and t4
t1Im 2
24Im tt
We have just seen that, for physical s
and
,2
2
ttG Define
Gtt ImIm 224
PHYSICAL cutEXACTLY Opposite
to each other
Subtraction Constantsfrom ChPT expansion
OK since s=0G(0)=t2(0)-t4(0)
Up to NLO ChPTOpposite to each other
42
22
tttt
IAM
All together…we find AGAIN
PC is O(p6) andwe neglect itor use ChPT
Truong ‘89, Truong,Dobado,Herrero,’90, Dobado JRP,‘93,‘96
Very simple. Systematic extension to higher orders
Dispersion relation allows us to go to complex plane.
Generates Poles of Resonances: f0(600) or “”, ρ(770), (800), K*(892),
The Inverse Amplitude Method: Results for one channel
f (770) K*(890)
Width/2
Mass
fpole: 440-i245 MeV
Dobado, JRP ‘96
Simultaneously: Unitarity + Chiral expansion
ChPT used ONLY at low energies: subtraction constants and left cut, NOT in resonance region
The 1/Nc expansion
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 mostbeyond that, just a qualitative model
(since QCD weakly interacting for large Nc)
0 5 10 15 20
0.20.40.60.811.21.4
MN/M3
N/3
Nc 0 5 10 15 20
0.20.40.60.811.21.4
MN/M3
N/3
Nc
LIGHT VECTOR MESONSqqbar states:
)/1(),1( cNOOM
The (770)
400 600 800 1000 1200M
70
60
50
40
30
20
10
0
i2
Nc3
Nc5
Nc10
Nc20
The K*(892)
700 800 900 1000 1100 1200M
20
15
10
5
0
i2
Nc3
Nc5
Nc10
Nc20
0 5 10 15 20
0.20.40.60.811.21.4
MN/M3
N/3
The IAM generates the expectedNc scaling of established qq states
JRP, Phys.Rev.Lett. 92:102001,2004
0 5 10 15 20
0.250.50.751
1.251.51.752
What about scalars ?
Nc
MN/M3
N/3
The (=770MeV)
400 600 800 1000 1200M
1000
800
600
400
200
0
i2
Nc3Nc5
Nc10
Nc20
The (=500MeV)
400 600 800 1000 1200M
1000
800
600
400
200
0
i2
Nc3Nc5
Nc10
0 5 10 15 20
0.250.50.751
1.251.51.752
Nc
MN/M3
N/3
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
JRP, Phys.Rev.Lett. 92:102001,2004
Results O(p6): the sigma
Near Nc = 3 similarresults to those at O(p4):Robust Non qqbar dominant component
M becomes constant ~ 1GeV 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
G. Ríos and JRPelaez, Phys.Rev.Lett.97:242002,2006
For Nc ~ 10 tor 12
This subdominant qqbar component can fix the duality problem
of a non-qqbar interpretattionfor the sigma
Outline
●Introduction
● UChPT and the 1/Nc expansion.
● FESR and local duality.
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
Crossing relates t-channel I=2 amplitude to s-channel amplitudes:
The I=2 suppression requires strong σ-ρ cancellation
I = 2 t-channel exchange should be suppressed respect to other isospin
Very small
σ ρ
T
“On the average-cancellation" properly defined via Finite Energy Sum Rules.
Regge theory interpretation is:
Local duality & FESR
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.
But if σ - ρ behave differently with Nc, this cancellation does not occur!!
σ - ρ cancellation needed for all Nc
Outline
●Introduction
● UChPT and the 1/Nc expansion.
● FESR and local duality.
● Results
FESR for Nc = 3. Check with real data
Using real data parametrizations, we have checked:Kaminski, JRP and Yndurain, PRD77:054015,2008
First point: Check the FESR suppression for Nc=3
for t = th
For Nc =3, local duality is satisfied.
FESR and IAM
We calculate the FESR using the IAM and check the influence of those waves.
The IAM predicts correctly the FESR suppression.
The influence of higher waves is around 10%.
We can use the IAM to study the FERS dependence on Nc
We can use the IAM to study local duality, but only applies for S0, P and S2 waves
For n= 2, 3, this cancellation occurs below 1-1.5 GeV.
Vanish with Nc
FESR and Nc. Case with vanishing σ
If we take a case where the σ amplitude vanishes (typically the NLO IAM)the ρ dominates the FESR.
However, the σ and ρ mesons show a different Nc behaviour.
Local duality implies a σ - ρ cancellation with Nc.
Local duality spoilt at larger Nc!!
SMALL
T
T
FESR suppression, checked using a real parametrization.
At higher Nc
Local duality fails
CONFLICT WITH LOCAL DUALITY IF THE SIGMA DISAPPEARS COMPLETELY
This is the expected problem
The σ amplitude vanishes: there is no σ-ρ cancellation.
FESR and Nc. Case with vanishing σ
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.
Local duality is still satisfied
The FESR are still suppressed with Nc
FESR suppression, checked using a real parametrization
FESR remain small with Nc.
The subleading qqbar σ component at 1 GeV , ensures local duality.
LOCAL DUALITY IS SATISFIED with NcTwo loop UChPT solves the problem
naturally
FESR and Nc. Case with subdominant quark-antiquark mixture
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
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
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)
Case with subdominant quark-antiquark mixture. Other states
In particular f0(980) effect too small
Case with subdominant quark-antiquark mixture. Other states
We have also added a crude model of the f2(1275).It contributes a littke to the cancellation, but not enough. The effect of theSubdominant component is larger.
FESR and Nc: With and without subdominant quark-antiqurk admixture
No subdminant component:(typical @NLO)No FESR
suppression
Local duality fails
With subdominantcomponent
(natural @NNLO)FESR
suppression
Local duality is satisfied.
We have even extrapolated to (too) large Nc. The cancellation continues
But IAM reliable for Nc < 15 – 30 at mostbeyond that, just a qualitative model
(since QCD weakly interacting for large Nc)
The suppresion continues.It is an stable efffect
Summary
Actually, the 1/Nc expansion within UChPT shows that the σ meson is not predominantly a state, while genrating the correct ρ dependence.
The σ 1/Nc behavior is predominantly that of a non ordinary meson, buta subdominant component with the 1/Nc behavior naturally suggested
by two-loop unitarized ChPT ensures that local duality is still satisfied.
Light scalars and particularly the σ seem likely non ordinary quark-antiquark mesons
All non-qqbar scenarios where the σ completely disapperars from the spectrum (typical @NLO-UChPT, pure tetraquark, pure molecule, etc…),
sufferCONFLICT WITH LOCAL DUALITY