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