NO ORDINARY MESONS: Light scalars Nc and quark mass dependence

Departamento de Física Teórica II. Universidad Complutense de Madrid

José R. Peláez

NO ORDINARY MESONS:Light scalars Nc and quark mass

dependence from Unitarized

Chiral Perturbation Theory:

INT.Seattle.

November

20th

2009

Outline

Scalar

Mesons:

A long story

in short

Unitarized

Chiral Perturbation

TheoryThe

Inverse

Amplitude

Method

(IAM)

Nc

behavior

of

scalars

Chiral Extrapolation: the

quark mass

dependence

One

loop

SU(3) ChPT: scalar

nonet

Two

loop

SU(2) ChPT: the

JRP, Phys.Rev.Lett. 92:102001,2004

G. Ríos and

JRP, Phys.Rev.Lett.97:242002,2006.

C. Hanhart, G. Ríos and


Scalar Mesons:

A long story in short

Motivation: The

f0

(600)

or

σ, half

a century

around

I=0, J=0

exchange

very

important

for

nucleon-nucleon

attraction!!

Crude

Sketch

of

NN potential:

From

C.N. Booth

Scalar-isoscalar

field

already

proposed

by Johnson

& Teller

in 1955

Soon

interpreted

within

“Linear sigma model”

(Gell-Mann) or

Nambu

Jona Lasinio

-

like

models, in the

60’s. Exchange in t-channel, not sensitive to

details in NN

Other

scalar

mesons

narrower

like

f0(980), a0(980) well

established

but not

well

understood

Is there a light scalar nonet as chiral partner

of the light pseudoscalar nonet?

The problem with the data. The S0 wave

From Ke (Kl4

)Pions on-shell. Very precise, but 00

-11. Recent data below mK.

Geneva-Saclay (77), E865

(01)

Example:CERN-Munich5 different analysis of same pn data !!

from Grayer et al. NPB (1974)

Systematic errors of 10o !!

NN

Initial state not well defined, model dependent off-shell

extrapolations (OPE, absorption, A2

exchange...) Phase shift ambiguities, etc...

From N scattering

Motivation: The

f0

(600) controversy

Very

controversial since

the

60’s.

The

reason: The

f0

(600) (and

kappa) is

a EXTREMELY WIDE.

Usually

refereed

to

its

pole:

Mostly

“observed”

in

scattering, but

no “resonance

peak”.

“not

well

established”

0+ state

in PDG until

1974

Removed from

1976 until

1994.

Back in PDG in 1996

2/ iMspole

1987

1979

1973

1972

The

light

scalar

controversy. The theory side...

Scalar

SU(3) multiplets

identification

controversial

f 0

a0

Some

people

claim/claimed

some

of

these

did/do not

exist(Pennington, Minkowski, Ochs, Narison…σ

as glueball

supportersin

general)

Possible

exotic

nature: tetraquarks

(Jaffe

’77) (issues

with

chiral symmerty

and

excess

of

states),

Too

many

resonances?Two

multiplets

+

glueball?..

Glueball

search: Characteristic

feature

of

non-abelian

QCD nature

f0

(980)

(800)

a0

(980)

Light scalar

nonet:

σ(500)Non-strange

heavier!!Inverted

hierarchy

Molecules (Achasov, Jülich-Bonn),…

Modified quark-antiquark models with meson interactionsVan Beveren, Rupp

Why the renewed interest in recent years?

New experimental data

Theoretical progress

“Recent“

progress: The

experiment

scattering

data from

the

70’s poor. Was

main

source

for

sigma discussions

Remarkable

improvement

from

Kl4 decays

in NA/48 (2008)

Different experiments:

(the

ones

that

most

convinced

PDG)Fermilab E791, Focus, Belle, KLOE, BES“Production” from J/Ψ, B-

and

D-

mesons, and

Φ

radiative

decays.

Very

good

statisticsClear

initial

states

and

different

systematic

uncertainties.

Strong experimental claims for wide and light

around 500 MeV

Personal Caveat: not really model independent analysis (Isobar-

Kmatrix models, etc…)

“Strong”

experimental claims for wide and light

around 800 MeV

Lots of data on decays, couplings, etc... for all light scalars

Motivation: The

f0

(600) and recent data

Very

controversial since

the

60’s.

The

reason: The

f0

(600) (and

kappa) is

a EXTREMELY WIDE.

Usually

refereed

to

its

pole:

Mostly

“observed”

in

scattering, but

no “resonance

peak”.

“not

well

established”

0+ state

in PDG until

1974

Removed from

1976 until

1994.

Back in PDG in 1996

PDG2002: “σ

well

established”

However, since

1996 still

quoted

as

Mass= 400-1200 MeV

Width= 600-1000 MeV

2/ iMspole

After 2000 also observed in production process

’95 model

by one

of

the

authors

of

PDG review

1987

1979

1973

1972

“Recent”

progress: The

theory

Many

old an new theoretical

studies

based

on

crude/simple models,

Issues

with

chiral symmetry, Adler

zeros

Often

ignoring

mixing: arguing

about

decay

modes

too simple

Often

neglecting

widths!!

Spurious

parameters

At

best…. “QCD inspired”.

Lattice

still

far, but

maybe

in the

not

too distant

future…

Hadronization missing

Suspicion: What you put in is what you get out??Improvements: With the new data this is no longer possibleModels are more respectful with chiral symmetry and include some

mixing

Dominant quark-antiquark component very disfavouredGlueball very unlikely

Tetraquark-molecule preferred

The real improvement:

Analyticity and Effective Lagrangians

The 60’s and early 70’s: Strong constraints on amplitudes from ANALYTICITY

in the form of

dispersion relations

But poor input on some parts of the integrals and poor knowledge/understanding of subtraction constants = amplitudes at

low

energy values

The 80’s and early 90’s: Development of Chiral Perturbation Theory (ChPT).(Weinberg, Gasser, Leutwyler)

It is the effective low energy theory of QCD. Provides information/understanding on low energy amplitudes

The 90’s and early 2000’s: Combination of Analyticity and ChPT(Truong, Dobado, Herrero, Donoghe, JRP, Gasser, Leutwyler, Bijnens, Colangelo, Caprini, Zheng, Zhou, Pennington...)

QCD: quarks, gluons, dynamics at high energy. Symmetries

Analyticity and Effective Lagrangians: two approaches

1st approach.

(Roy-like equations)

Use data on all waves + high energy + ChPT numerical predictions to determine amplitudes. The most precise

and model independent way of

getting poles. (Colangelo Gasser, Letwyler group, Moussallam, Paris group, Lesniak Krakow group,

UCM-UAM-Madrid Group)

2nd approach.

(Unitarized ChPT)

Use ChPT amplitudes inside dispersive integrals.

Not so precise, but all input controlled with the effective Lagrangian and QCD.

Reliable for understanding and connecting with QCD

sigma and kappa existence , mass and

width

firmly established

Data after

2000 both

scattering

and

productionDispersive-

model

independent

approaches

Chiral symmetry

correct

Outline

Scalar

Mesons:

A long story

in short

Unitarized

Chiral Perturbation

TheoryThe

Inverse

Amplitude

Method

(IAM)

Nc

behavior

of

scalars


quark mass

dependence

One

loop

SU(3) ChPT: scalar

nonet

Two

loop

SU(2) ChPT: the


G. Ríos and




Unitarized Chiral Perturbation TheoryThe Inverse Amplitude Method (IAM)

Chiral Perturbation Theory in the meson sector Weinberg, Gasser & Leutwyler

ChPT

is the most general expansion

in energies of a lagrangian made only of pions, kaons and etas

compatible with the QCD symmetry breaking

Leading order parameters: breaking scale f0

and masses

At 1-loop, QCD dynamics encoded inchiral parameters:

L1

...L8Determined from

EXPERIMENT

leading 1/Nc

behavior known from QCD

ChPT is the QCD Effective Theorybut is limited to low energies

, K ,

Goldstone Bosons of the spontaneous

chiral symmetry breakingSU(Nf

)V

SU(Nf

)A

SU(Nf

)V

QCD degrees of freedomat low energies << 4f~1.2 GeV

Partial wave unitarity(On the real axis above threshold)

t1Im

itt

1Re

1

2Im tt

exactly unitary !!

ChPT = series in p2

perturbative unitarity 2

24Im tt

...42 ttt

...)Re(Re 422

21 tttt

provides ....

2242

22

Re titttt

42

22

tttt

IAM

Unitarity and the Inverse Amplitude Method: One channel –one loop NAIVE DERIVATION

Truong

‘89, Truong,Dobado,Herrero,’90, Dobado, JRP,‘93,‘96

The

Inverse

Amplitude

Method: Results

for

one

channel

Fit

and K ELASTIC scattering

data

Preliminary Update: J. Nebrera and JRP ‘09

Truong ‘89, Truong,Dobado,Herrero,’90, Dobado JRP,‘93,‘96

EXTREMELY SIMPLE

Originally obtained from dispersion relation This allows us to go to the complex plane.

Dynamically Generates Poles: Resonances: f0

(600) or “sigma”, K*, ρ

The Inverse Amplitude Method: Results for one channel

f

(770) K*(890)

Width/2

Mass

f

pole: 440-i245 MeVDobado, Pelaez ‘96

Systematic extension to higher orders

Unitarity + Chiral Low energy expansion

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

it

or use ChPT

Partial wave unitarity(On the real axis above all thresholds)

1ImT

11 )(Re iTT

*Im TTT

ChPT = series in p2

perturbative unitarity

224Im TTT

...42 TTT

1242

12

1 ...)Re(Re TTTTTprovides

....

exactly unitary !!

21

22422 )Re( TTTiTTTT

21

422 )( TTTTT Coupled

channel

IAM

Oller,

Oset,

JRP, PRL80(1998)3452, PRD59,(1999)074001A. Gomez Nicola and J.R.P.

PRD65:054009, (2002)

Unitarity and the Inverse Amplitude Method: Multiple channels–one loop

To the DATA !!

One-loop ChPT IAM fit to meson-meson scattering (+3% syst.)

f

f0

(980)

a0

κ(900)K* (892)ρ

low-energy 00

-1100

d

/dElab

KK

3/2 0

KK

1/2 1

KK

1/2 0

KK

00

Gómez-Nicola, JRP, Phys. Rev. D65:054009, (2002)

20

11

Incompatible sets of Data. Customarily add systematic error:

MINUIT fit :

3%,

Final error: MINUIT error + Systematic error

Identical curvesbut variation in

parameters1%, 5%

ChPT(=M) IAM fit (+3%) IAM fits L1 0.4 0.3 0.561 0.008 0.6 0.1 L2 1.35 0.3 1.211 0.001 1.2 0.1 L3 -3.51.1 -2.79 0.02 -2.79 0.14L4 -0.3 0.5 -0.36 0.02 -0.36 0.17L5 1.4 0.5 1.39 0.02 1.4 0.5 L6 -0.2 0.3 0.07 0.03 0.07 0.08 L7 -0.4 0.2 -0.444 0.03 -0.44 0.15L8 0.9 0.3 0.78 0.02 0.8 0.2

Complete Meson-meson Scattering in Unitarized Chiral Perturbation Theory

Gómez-Nicola, JRP, Phys. Rev. D65:054009, (2002)

Simultaneous description of low energy and resonances

Fully renormalized and with parameters compatible with ChPT.

Coupled channel IAM Oller, Oset, JRP, PRL80(1998)3452, PRD59,(1999)074001A. Gomez Nicola and J.R.P.

PRD65:054009, (2002)

400 600 800 1000 1200-300

-200

-100

0

100

400 600 800 1000 1200-300

-200

-100

0

100

900 950 1000 1050 1100 1150 1200-300

-200

-100

0

100

1000 1020 1040 1060 1080 1100-100

-80

-60

-40

-20

0

700 800 900 1000 1100 1200-300

-200

-100

0

100

700 800 900 1000 1100-300

-200

-100

0

100

f0(600) and f0

(980)

κ a0

K* (octet)

With the full one-loop SU(3) unitarized ChPT, we GENERATE,the following resonances, not present in the ChPT Lagrangian,as poles in the second Riemann sheet

J.R.P, hep-ph/0301049. AIP Conf.Proc.660:102-115,2003

Brief review: Mod.Phys.Lett.A19:2879-2894,2004

without a priori assumptions on on their existence or nature

a0 (980)

f0

(600)

f0 (980)

754 18

440

8

753

52

74

10

212

15

235

33

973 +39-127 11+189

-11

12 +43-12

Resonance POLE POSITIONS...

CompleteIAM

CompleteIAM

1117+24-320

cusp?

K* 889

13 24

4

Light scalar nonet(Only statistical errors)

(MeV)MRe poles (MeV)2/Im poles

Intro -

Summary

Unitarized Chiral Perturbation Theory

The Inverse Amplitude Method (IAM)Description of meson-meson scattering compatible with ChPTNo model dependence in one-channel (not in coupled channels)Both LIGHT VECTORS and SCALARS as polesNLO (1 loop) and NNLO(2 loops) IAM availableNo supurious parameter dependence in cutoffs or other parameters

Outline

Scalar

Mesons:

A long story

in short

Unitarized

Chiral Perturbation

TheoryThe

Inverse

Amplitude

Method

(IAM)

Nc

behavior

of

scalars


quark mass

dependence

One

loop

SU(3) ChPT: scalar

nonet

Two

loop

SU(2) ChPT: the


G. Ríos and




Nc

behavior of scalars

(x10-3) ChPT (=M)

IAM fits Large Nc SCALING

2L1- L2 -0.6 0.6 0.0 0.2 O(1) L2 1.4 0.3 1.2 0.1 O(Nc) L3 -3.5 1.1 -2.79 0.14 O(Nc) L4 -0.3 0.5 -0.36 0.17 O(1) L5 1.4 0.5 1.4 0.5 O(Nc) L6 -0.2 0.3 0.07 0.08 O(1) L7 -0.4 0.2 -0.44 0.15 O(1) L8 0.9 0.3 0.8 0.2 O(Nc)

Leading 1/Nc

expansion

We cannot obtain the Li from QCD, BUT their 1/Nc expansion, is known and Model Independent

Pions, kaons and etas states: )/1(),1( cNOOM The qqbar meson masses M=O(1)

and their decay constants f=O(Nc

)

Our IAM ChPT amplitudes do not

have

any other parameter hiding Nc dependencelike cutoffs, subtractions, etc...

We can thus study

the Nc scaling of the resonances

The IAM generates the expected1/Nc

scaling of established qq statesJRP, Phys.Rev.Lett. 92:102001,2004

0 5 10 15 20

0.20.40.60.8

11.21.4

MN

/M3

N

/3

Nc 0 5 10 15 20

0.20.40.60.8

11.21.4

MN

/M3

N

/3

Nc

LIGHT VECTOR MESONSqqbar states:

)/1(),1( cNOOM

0 5 10 15 20

0.20.40.60.8

11.21.4

MN

/M3

N

/3

The (770)

400 600 800 1000 1200M

- 70

- 60

- 50

- 40

- 30

- 20

- 10

0

-iG

2

Nc=3

Nc=5

Nc=10

Nc=20

-i

/2

The K*(892)

700 800 900 1000 1100 1200M

- 20

- 15

- 10

- 5

0

-iG

2

Nc=3

Nc=5

Nc=10

Nc=20

-i

/2

0 5 10 15 20

0.250.5

0.751

1.251.5

1.752

What about scalars

?

Nc

MN

/M3

N

/3

0 5 10 15 20

0.250.5

0.751

1.251.5

1.752

Nc

MN

/M3

N

/3

Similar results follow 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)


The

(=770MeV)

400 600 800 1000 1200M

- 1000

- 800

- 600

- 400

- 200

0

-iG

2

Nc=3Nc=5

Nc=10

Nc=20

-i

/2

The

(=500MeV)

400 600 800 1000 1200M

- 1000

- 800

- 600

- 400

- 200

0

-iG

2

Nc=3Nc=5

Nc=10

-i

/2

Since for quark antiquark states )/1(),1( cNOOM

qqNOTisqqis 1,1 22

G. Ríos and JRP, Phys.Rev.Lett.97:242002,2006.

2-like function to measure how close a resonance is to a quark antiquark

One loop

Two loops

This

2-like can be used to:

1) check

if a resonance behaves as quark-antiquark

2) Try to force

a resonance to behaves as quark-antiquark by tuning parameters

The rho always comes out naturally as a quark-antiquark

The sigma cannot be made to behave as a quark-antiquarkeven by forcing it

G. Ríos and JRP, Phys.Rev.Lett.97:242002,2006.

Large Nc behavior of UNITARIZED

TWO LOOP

ChPT

The f0

(600)

still does NOT

behave DOMINANTLY

as quark-antiquark

BUT, from Nc>8 or 10, the f0

(600) we might

be seeing a

quark-antiquark subdominant

component whose large Nc mass is

1 GeV

quark-antiquark mixingmay emerge at larger Ncat M1 GeV

The sigma:

MN

/M3

N

/ 3

These results suggest what LIGHT SCALARS ARE NOT predominantly made of...

The light scalar nonet DOMINANT component

is NOT a quark-antiquark

state

Results consistent at higher orders.The

cannot

be forced to behave as a

quark-antiquark.

At two loops, a subdominant quark-antiquark component emerges

above

1 GeV!!

(consistent with two nonet picture, the lightestnon-quark antiquark)

In agreement with similar conclusions of quark models by Rupp-Van Beveren et al.or unitary chiral approach by Oset-Oller. Or report of Tornqvist & Close J.PhysG28:R249(2002)

Outline

Scalar

Mesons:

A long story

in short

Unitarized

Chiral Perturbation

TheoryThe

Inverse

Amplitude

Method

(IAM)

Nc

behavior

of

scalars


quark mass

dependence

One

loop

SU(3) ChPT: scalar

nonet

Two

loop

SU(2) ChPT: the


G. Ríos and




Chiral Extrapolation: the quark mass dependence

ChPT provides the correct QCD dependence of quark masses as an expansion...

The LATTICE provides rigorous and systematic QCD results in termsof quarks and gluons with growing interest in scattering and the

scalar

sector.

Caveat:

small, realistic, quark masses are hard to implement.

We can study the scalars in Unitarized ChPT for larger quark masses(chiral extrapolation)

and provide a reference for lattice studies

Motivation for Chiral extrapolation

Anthropic considerations...

Truong

‘89, Truong,Dobado,Herrero,’90, Dobado, JRP,‘93,‘96

The

Inverse

Amplitude

Method: Results

for

one

channel

Fit

and K ELASTIC scattering

data + LATTICE

Preliminary J. Nebreda, JRP 2009

The (770) The f0

(600) or

Becomes narrower,and gets closer to the

new thresholdAt some points enters the first

sheet (bound state)

Becomes narrower.When in real axis

“splits” in two real poles

Pole movements with

increasing m

UChPT

SU(2) single channel

calculation

(770) versus f0(600)

Pole movements with

increasing m

Pole movements with

increasing m

To follow the position relative to threshold: normalize to m

units

-

ρ

pole-

σ

pole

Conjugate poles reach the real axis AT THRESHOLD: The rho:

- one pole in the 1st

sheet (bound state).

-

another in the 2nd

sheet in almost the same position

Pole movements with

increasing m

The sigma: 1) Conjugate poles reach the real axis

BELOW threshold:

2)

TWO real POLES on the 2nd sheet: “Splitting”

typical of scalars.

-

ρ

pole-

σ

pole

3) One moves towards threshold until it jumps to the 1st

sheet.The other remains on the 2nd

sheet in ASYMMETRIC position

If very asymmetric: sizable “molecular” componentMorgan, Pennington. PRD48 (1993) 1185

The rho mass grows

slower

than sigma

There is a “non-analyticity”

in the sigma mπ

dependence.

Resonane mass

m

dependence

Resonance width

m

dependence

Resonance width

m

dependence vs. phase space

For

a narrow

vector particle

(like

the

rho) the

decay

width

is

given

by

Phase

space

Coupling

topions

We

can calculate

the

width

variation

due

to

phase

space

reductionand

compare with

our

results. The

difference

gives

the

dependence

of

the

coupling

constant

on

the

pion

mass

Rho width

m


couplingalmost

independent

of

mπ(assumption in somelattice calculations)

Width behaviorexplained by phase space

Γρ

/ Γρphys

phase spaceΓρ

/ Γρphys

IAM

It

does not follow

the phase

space decrease of a Breit-Wigner:

The

dynamics

of

the

sigma decay

depends

strongly

on

the

pion(quark)

mass(Recall

that

some

pion-pion

vertices

in ChPT depend

on

the

pion

mass).

Very badapproximation

for

a wide

resonanceas the

sigma

g dependence

on

mπ

Γσ

/ Γσphys

phase spaceΓσ

/ Γσphys

IAM

Rho width

m


CAUTION!!! We

give

POLE MASS

in complex plane

Lattice caveats:Improved actions,Lattice spacing...Finite volume...WIDTHLESS rho

The best would be to use ChPT on the lattice....future work

Comparison with

lattice

results

for

the

rho

Comparison with

lattice

results

for

the

sigma

AGAIN CAUTION!!! We

give

POLE MASS

in complex plane + usuallattice caveats

IMPORTANT REMARKExtrapolations should take careof known scalar mass “splitting”

non-analyticity

Summary:Scalars in Unitarized Chiral Perturbation Theory

Simultaneously

resonances and low energy meson-meson scattering

with parameters compatible with ChPT

Generates Light Scalar

Nonet and vector octet

SUBDOMINANT quark-antiquark component around 1.1 GeV.(Suggests mixing with heavier ordinary scalar nonet)

SCALARS predominantly NOT

quark-antiquark states

quark-antiquark remarkably good for vectors

Nc

behavior of light resonances

M

behavior

qualitatively similar to lattice,

Caution: lattice non-zero width

and that we use pole masses

We

predict

the

parameters

giving

the

M

dependence

on M.

Consistent with chiral fits.

The

sigma mass

dependence is stronger than for the rho

In progress...

SU(3), strange resonances.... DONE !!. Surprises in KSFR.

Γ

just obeys the

BW phase

space reduction whereas Γ

does

not.Dynamical M

effects through 2 pion couplings!

Summary: 2nd part

Pion mass dependence of (770)

and f0(600) mass and width

IDEA FROM THIS WORKSHOP @ INT !!!Calculate also phase shift chiral extrapolations. Within ChPT and IAM. May be feasible in lattice in the not too distant future

Documents

NO ORDINARY MESONS: Light scalars Nc and quark mass dependence