r l jaffe frascati may 2005 multiquark dynamics r. l. jaffe lnf frascati may 2005 r. l. jaffe lnf...

R L Jaffe Frascati May 2005


Multiquark DynamicsMultiquark Dynamics


R. L. Jaffe LNF Frascati May 2005

R. L. Jaffe LNF Frascati May 2005

• The problem of exotics in QCD• The aufbau principle for hadrons• The scalar mesons• Dynamical correlations ---

diquarks, a new field for study in QCD?

• Quark states and scattering

• The problem of exotics in QCD• The aufbau principle for hadrons• The scalar mesons• Dynamical correlations ---

diquarks, a new field for study in QCD?

• Quark states and scattering




I. General overview

A. Historical overview

a. Definition of exotic: flavor and CP

b. History of searches for exotics

c. State of affairs in 2003: absence of exotics & extra scalar nonet

B. The Theta (2003-2005) requiescat in pace?

a. Definition, quantum numbers, and significance

b. Very brief review of history and experiments

c. Summary of negative evidence

C. Models of hadrons and exotics --- once over quickly

a. Quark models and chiral soliton models

b. Implications of the death of the Theta for models

I. General overview

A. Historical overview

a. Definition of exotic: flavor and CP

b. History of searches for exotics

c. State of affairs in 2003: absence of exotics & extra scalar nonet

B. The Theta (2003-2005) requiescat in pace?

a. Definition, quantum numbers, and significance

b. Very brief review of history and experiments

c. Summary of negative evidence

C. Models of hadrons and exotics --- once over quickly

a. Quark models and chiral soliton models

b. Implications of the death of the Theta for models




II. Quarks and Diquarks

A. Introduction

a. Naive quark model

b. Spectroscopy versus dynamics

B. Correlations and spectroscopy: the case for diquarks

a. Correlations in QCD

b. Spectroscopy, Delta I=1/2, hadron spin splittings

c. Distribution and fragmentation function regularities, higher twist

d. Defining and extracting the properties of diquarks

E. A coherent qualitative picture of multiquark hadrons

a. Diquarks and the general absence of exotics

b. Where, if anywhere, to find multiquark hadrons

c. A program in color non-singlet spectroscopy?

II. Quarks and Diquarks

A. Introduction

a. Naive quark model

b. Spectroscopy versus dynamics

B. Correlations and spectroscopy: the case for diquarks

a. Correlations in QCD

b. Spectroscopy, Delta I=1/2, hadron spin splittings

c. Distribution and fragmentation function regularities, higher twist

d. Defining and extracting the properties of diquarks

E. A coherent qualitative picture of multiquark hadrons

a. Diquarks and the general absence of exotics

b. Where, if anywhere, to find multiquark hadrons

c. A program in color non-singlet spectroscopy?




III. Aspects of Multiquark Dynamics

A. General discussion.a. Qualitative expections for conventional hadrons

(Zero width decoupling)b. Expectations for multiquark hadrons (continuum subsidence)

B. Lessons from Large Nc

a. Mesons at large Nc --- a reminderb. Multiquark states at large Nc --- color substructure at leading non-trivial order in 1/N

C. Wigner’s boundary conditon methoda. The physical foundation for Wigner’s R-matrixb. Why hadrons are not K-matrix polesc. Insight from another boundary condition: the P-matirx

c . The P-matrix and hadron resonances

D. Feshbach-Fano resonancesa. Very low energy scatteringb. Potential scatteringc. Bound states in the continuum

III. Aspects of Multiquark Dynamics

A. General discussion.a. Qualitative expections for conventional hadrons

(Zero width decoupling)b. Expectations for multiquark hadrons (continuum subsidence)

B. Lessons from Large Nc

a. Mesons at large Nc --- a reminderb. Multiquark states at large Nc --- color substructure at leading non-trivial order in 1/N

C. Wigner’s boundary conditon methoda. The physical foundation for Wigner’s R-matrixb. Why hadrons are not K-matrix polesc. Insight from another boundary condition: the P-matirx

c . The P-matrix and hadron resonances

D. Feshbach-Fano resonancesa. Very low energy scatteringb. Potential scatteringc. Bound states in the continuum




Rigorous Approaches to QCDRigorous Approaches to QCD

Perturbative QCD at High Perturbative QCD at High

Lattice QCD for Simple Questions(quenched, away from chiral limit...)Lattice QCD for Simple Questions

(quenched, away from chiral limit...)VOODOO QCDVOODOO QCD**

* J D Bjorken 1986* J D Bjorken 1986

Chiral dynamics at very low energiesChiral dynamics at very low energies




I. General overviewI. General overview




Exotics at the beginning....Exotics at the beginning....




Non-ExoticsNon-Exotics

• Nuclei

• Exotic Mesons• Exotic Baryons

• Baryons

• MesonsAnd ExoticsAnd Exotics




(CP Exotics and other oddities)(CP Exotics and other oddities)• Certain C & P quantum numbers are excluded to

mesons in non-relativistic, two-body, Schroedinger quantum mechanics

• Certain C & P quantum numbers are excluded to mesons in non-relativistic, two-body, Schroedinger quantum mechanics

• Violation?• Violation?

Relativistic effects (transformation to center of mass doesn’t exist) or constituent gluonsRelativistic effects (transformation to center of mass doesn’t exist) or constituent gluons

This multiplet does not otherwise occur low in the spectrumThis multiplet does not otherwise occur low in the spectrum

• Baryon analogues: Center of mass excitations of nucleon octet and decuplet

• Baryon analogues: Center of mass excitations of nucleon octet and decuplet




Major efforts in ‘60’s and ‘70’sMajor efforts in ‘60’s and ‘70’s

• Meson-meson scattering phase shift analyses found resonances in all non-exotic channels and no structure in

• Meson-meson scattering phase shift analyses found resonances in all non-exotic channels and no structure in

• Meson-baryon partial wave analyses found resonances in all non-exotic channels and no

• Meson-baryon partial wave analyses found resonances in all non-exotic channels and no

• Hadron spectroscopy was a premier field of high energy physics: flagship experiments and very sophisticated analysis

• Hadron spectroscopy was a premier field of high energy physics: flagship experiments and very sophisticated analysis




Elastic resonance would be full circle in Argand diagram

Dots are spaced by 50 MeV lab energy.

Hyslop, Ardnt, Roper, Workman

1992

Elastic resonance would be full circle in Argand diagram

Dots are spaced by 50 MeV lab energy.

Hyslop, Ardnt, Roper, Workman

1992

No KN I=0 resonance

resonance would be a state decaying to with same spin and parity as the nucleon:

resonance would be a state decaying to with same spin and parity as the nucleon:

No KN I=1 resonanceNo KN I=1 resonance




Meanwhile, as SU(3) multiplets of mesons and baryons filled up...

It became clear that there is a problem with the scalar mesons:

• Too many• Too light• Lightest have peculiar mass spectrum

Meanwhile, as SU(3) multiplets of mesons and baryons filled up...

It became clear that there is a problem with the scalar mesons:

• Too many• Too light• Lightest have peculiar mass spectrum




Summary through 2002 --Summary through 2002 --

•No exotics anywhere

•A nonet of supernumerary scalar mesons

•Aufbau principle differs from nuclei & atoms

•No exotics anywhere

•A nonet of supernumerary scalar mesons

•Aufbau principle differs from nuclei & atoms

2003 brought new discoveries and controversy!2003 brought new discoveries and controversy!




One manifestly exotic baryon seen by many experiments.One manifestly exotic baryon seen by many experiments.

Another exotic and two non-exotic partners seen by NA49Another exotic and two non-exotic partners seen by NA49

Experimental Discoveries Experimental Discoveries

Charm-analog seen by H1 but not by Zeus (3/04)

Charm-analog seen by H1 but not by Zeus (3/04)




• Positive sightings of baryons since 2003

• Positive sightings of baryons since 2003

Dzierba, Meyer, Szczepaniak hep-ex/0412077Dzierba, Meyer, Szczepaniak hep-ex/0412077




• Negative non-sightings of baryons since 2003

• Negative non-sightings of baryons since 2003

Dzierba, Meyer, Szczepaniak hep-ex/0412077Dzierba, Meyer, Szczepaniak hep-ex/0412077




Recent re-examinations in the KN systemRecent re-examinations in the KN system• Nussinov hep-ph/0307357

• Arndt, Strakovsky, Workmannucl-th/0311030

• Cahn & Trillinghep-ph/0311245

• Sibirtsev, Haldenbatter, Krewald, Meissnerhep-ph/0405099

• Nussinov hep-ph/0307357

• Arndt, Strakovsky, Workmannucl-th/0311030

• Cahn & Trillinghep-ph/0311245

• Sibirtsev, Haldenbatter, Krewald, Meissnerhep-ph/0405099

No sign of any resonance: interpreted as limits on width -- very stringent: < 1--4 MeVNo sign of any resonance: interpreted as limits on width -- very stringent: < 1--4 MeV




• Very recent results -- first of a second generation

• Very recent results -- first of a second generation• Reported by R. De Vita for CLAS at APS Tampa 4/17/05

• Reported by R. De Vita for CLAS at APS Tampa 4/17/05




• More detail• More detail




Same reaction, Phys. Lett. B572 (2003)




• Mass determinations are inconsistent

• Mass determinations are inconsistent

The is not officially dead yet, but ....The is not officially dead yet, but ....• More low energy

photoproduction experiments to report soon

• More low energy photoproduction experiments to report soon• No decisive experimental flaws, but see DMS

• No decisive experimental flaws, but see DMS

• Width limits are extreme and contradictory

• Width limits are extreme and contradictory

• Lattice calculations find no positive parity resonance

• Lattice calculations find no positive parity resonance

• New experiments contradict old, lower statistics sightings

• New experiments contradict old, lower statistics sightings

??




• Only significant, qualitative failure of naive quark assignments is the existence of an extra nonet of scalar mesons with masses below 1 GeV.

• Only significant, qualitative failure of naive quark assignments is the existence of an extra nonet of scalar mesons with masses below 1 GeV.

• Equivalently, the building of hadrons is radically different than the building of atoms and nuclei -- to make atoms add more electrons in the nuclear Coulomb field; to make nuclei, add protons and neutronsneutrons in the nuclear mean field; to make hadrons, stop at QQQ or QQ.

• Equivalently, the building of hadrons is radically different than the building of atoms and nuclei -- to make atoms add more electrons in the nuclear Coulomb field; to make nuclei, add protons and neutronsneutrons in the nuclear mean field; to make hadrons, stop at QQQ or QQ.

Situation post Theta

• No exotics




Quark ModelsQuark Models• Match naturally to DIS degrees of freedom• Mesons & Baryons -- spectroscopy & electroweak

interactions• Never fully consistent with relativity -- relativistic

single particle models but not field theoretic• Not the basis of a systematic expansion• Quark models have got a bad rap over the past

20 years. They are still the most powerful, broad, and heuristic tool for the hadron spectrum, and any hadron must have a quark interpretation...

• Match naturally to DIS degrees of freedom• Mesons & Baryons -- spectroscopy & electroweak

interactions• Never fully consistent with relativity -- relativistic

single particle models but not field theoretic• Not the basis of a systematic expansion• Quark models have got a bad rap over the past

20 years. They are still the most powerful, broad, and heuristic tool for the hadron spectrum, and any hadron must have a quark interpretation...

Models of hadrons and the Death of the ThetaModels of hadrons and the Death of the Theta




• Never could accommodate narrow width and

apparent positive parity

• Never could accommodate narrow width and

apparent positive parity

• No prediction because absolute mass scale of

could not be determined

• No prediction because absolute mass scale of

could not be determined

• Absence of Theta limits the strength of the

correlation (diquark) that has other important

spectroscopic and dynamical implications

• Absence of Theta limits the strength of the

correlation (diquark) that has other important

spectroscopic and dynamical implications

?? ?? ??

Implications for quark models and

others...

Implications for quark models and

others...




• Three flavors• Three flavors

History: Duality ⇒ Strings ⇒ [Nc→ ∞ ] ⇒ Skyrme ⇒

CSM• There is nothing fundamentally “3-ish” about baryons in the CSM, therefore they are teeming with exotics

• Two flavors• Two flavors

Chiral Soliton Models ? ??

Three flavour exotics in the chiral soliton

model:

• Manohar, Chemtob (1984/85)• Praszalowicz uudds* at 1540 MeV

• Diakonov, Petrov, Polyakov: Narrow (1997)

• Weigel: excellent balanced summary (1998)




Implications of the death of the Theta

for CSM

Implications of the death of the Theta

for CSM• SU(3) Chiral Soliton Models (Diakonov, Petrov,

Polyakov)

• SU(3) Chiral Soliton Models (Diakonov, Petrov,

Polyakov)• Strong prediction asserted: M=1540 MeV,

= 15 MeV

• Strong prediction asserted: M=1540 MeV,

= 15 MeV• Apparently the model is unreliable, why?• Apparently the model is unreliable, why?

? Truncation of chiral effective lagrangian?? Truncation of chiral effective lagrangian?

? Adiabatic (rigid) excitation?? Adiabatic (rigid) excitation?

? Perturbative implimentation of SU(3)

violation?

? Perturbative implimentation of SU(3)

violation?

Dynamical

balance

Dynamical

balanceAssume that rotational excitations neither deform soliton nor mix with radial excitations. No separation of scales.

Assume that rotational excitations neither deform soliton nor mix with radial excitations. No separation of scales.

No justification to

ignore

No justification to

ignore

? Questionable relation to QCD in the first

place

? Questionable relation to QCD in the first

place




II. Quarks and DiquarksII. Quarks and Diquarks




Naive Quark Model• Assume you know the basics

• Uncorrelated quarks in a mean field:

NON-RELATIVISTIC POTENTIAL MODELS

RELATIVISTIC BAG MODELS

YOUR FAVORITE MODEL

• Uncorrelated quarks in a mean field:

NON-RELATIVISTIC POTENTIAL MODELS

RELATIVISTIC BAG MODELS

YOUR FAVORITE MODEL

• Good description of super-multiplets of light meson and baryon states

• Towers of multiquark states: vast number and no information about widths

• Widths in the quark model: finessed for QQ* and QQQ, it becomes essential for states that can fall apart into mesons and baryons

• Good description of super-multiplets of light meson and baryon states

• Towers of multiquark states: vast number and no information about widths

• Widths in the quark model: finessed for QQ* and QQQ, it becomes essential for states that can fall apart into mesons and baryonsDYNAMICSDYNAMICS




Correlations and

classification

Correlations and

classification• Confinement• Confinement

• Chiral symmetry breaking

• Chiral symmetry breaking

• correlations ?

• correlations ?

• Color, flavor, spin antisymmetry

• Color, flavor, spin antisymmetry




Classification

Fermi

statistics

Fermi

statistics

– Parity – Parity

+

Parity

+

Parity

ColorColor

Color Color

Leaves only two diquarks in the low energy spectrum

Leaves only two diquarks in the low energy spectrum




Diquarks

Flavor Color Spin

-8

8/3

“Good”“Good”“Bad”“Bad”

Good Good

BadBad




• Condensation in quark matter at high density

• Condensation in quark matter at high density• condenses in flavor antisymmetric channel

generating color-flavor locked superconductivity

• condenses in flavor antisymmetric channel generating color-flavor locked superconductivity

Long history in QCD, but never in the mainstream (D. Lichtenberg)Long history in QCD, but never in the mainstream (D. Lichtenberg)

Phenomenological evidence for diquarksPhenomenological evidence for diquarks

• Certain regularities in spectroscopy• Certain regularities in spectroscopy• Absence of• Absence of

• Systematic analysis of baryon and meson resonances.

More later -- A. Selem & F. Wilczek in preparation

• Systematic analysis of baryon and meson resonances.

More later -- A. Selem & F. Wilczek in preparation

• rule in nonleptonic weak decays

• rule in nonleptonic weak decays• dominance gives good description of non-

perturbative effects.• dominance gives good description of non-

perturbative effects.

• Systematic study by Neubert, Stech & collaborators in late 1980’s.

• Systematic study by Neubert, Stech & collaborators in late 1980’s.




Diquark regularities in DIS

• Baryon parton distribution function regularities follow from

• Baryon parton distribution function regularities follow from

• Regularities in fragmentation ratios in to hadrons

• Regularities in fragmentation ratios in to hadrons

Fragmentatio

n ratios

measured

at LEP

(Delphi)

Fragmentatio

n ratios

measured

at LEP

(Delphi)

Suggests

dominance of

-- favored

diquark.

Suggests

dominance of

-- favored

diquark.

Known since 1960’sKnown since 1960’s

Recent JLab results

nucl-ex/0308011

Recent JLab results

nucl-ex/0308011




Good diquark, strange

Good diquark, strange

• Formally define color antitriplet diquarks in the presence of an infinitely heavy spectator quark (or Polyakov line)

• Formally define color antitriplet diquarks in the presence of an infinitely heavy spectator quark (or Polyakov line)

• Awaiting lattice calculations, estimate from charm and strange systems:

• Awaiting lattice calculations, estimate from charm and strange systems:

Characterizing diquarksCharacterizing diquarks

Good diquark, non-strange

Good diquark, non-strange

Bad diquark, strange

Bad diquark, strange

Bad diquark, non-strange

Bad diquark, non-strange




• Other estimates from charm sector• Other estimates from charm sector

Good - bad mass difference decreases with quark mass.

Good - bad mass difference decreases with quark mass.

Diquark -- heavy quark spin interaction decreases with heavy quark mass and with light quark mass

Diquark -- heavy quark spin interaction decreases with heavy quark mass and with light quark mass

• Conclude: charm baryon masses allow estimate of diquark masses in a heavy quark background. Correlation is ~ 200 MeV. Not huge, but important

• Conclude: charm baryon masses allow estimate of diquark masses in a heavy quark background. Correlation is ~ 200 MeV. Not huge, but important




• QCD explains the absence of exotics in general, • Makes explicit predictions for non-exotic two quark, two antiquark states,• Which have been verified by experiment,• (Though this is not uncontroversial),• And the same ideas suggest the channels in which exotic baryons are most likely...• And the key to the dynamics is the concept of

DIQUARKS

• QCD explains the absence of exotics in general, • Makes explicit predictions for non-exotic two quark, two antiquark states,• Which have been verified by experiment,• (Though this is not uncontroversial),• And the same ideas suggest the channels in which exotic baryons are most likely...• And the key to the dynamics is the concept of

DIQUARKS✺ RLJ (1977), RLJ and F. Low (1979)✺ RLJ (1977), RLJ and F. Low (1979)

✺✺Remarkably: it has been known since 1977 thatRemarkably: it has been known since 1977 that

Spectroscopic consequences of diquark correlationsSpectroscopic consequences of diquark correlations

PreviewPreview




Spectra Spectra

• Baryons• BaryonsFermi statistics kills the singlet and allows only one octet which is a mixture of good and bad diquark except for and

Fermi statistics kills the singlet and allows only one octet which is a mixture of good and bad diquark except for and

• Mesons• Mesons

No Exotics !No Exotics !Lightest multiplet is 0++ nonetLightest multiplet is 0++ nonet




Tetraquark Scalar NonetTetraquark Scalar Nonet




Scalar mesons: a supernumerary nonetScalar mesons: a supernumerary nonet




• Pentaquarks• Pentaquarks

Only possible exotic

Only possible exotic

First combine two diquarksFirst combine two diquarks

Where subscripts denote symmetry in flavor exchangeWhere subscripts denote symmetry in flavor exchange

Then combine with antiquark:Then combine with antiquark:

Diquarks antisymmetric in flavorDiquarks antisymmetric in flavor

Diquarks symmetric in flavorDiquarks symmetric in flavor




(Good diquark is a boson)(Good diquark is a boson)

Non-exoticNegative parityNonet

Non-exoticNegative parityNonet

Case I: Case I:

Symmetric in spin. Antisymmetric in color and flavorSymmetric in space

Symmetric in spin. Antisymmetric in color and flavorSymmetric in space

Examine symmetry of diquark-diquark state:Examine symmetry of diquark-diquark state:

has odd parity

has odd parity

• Lowest mass• Even parity

• Lowest mass• Even parity




Note difference in charm

sector

Note difference in charm

sectorSU(3)-flavor triplet of exotic (positive baryon number, negative charm) charmed baryons.

Lesson: What is not exotic in one sector may show up as exotic in another.

SU(3)-flavor triplet of exotic (positive baryon number, negative charm) charmed baryons.

Lesson: What is not exotic in one sector may show up as exotic in another.

For c and b quarks the flavor antisymmetric [qq][qq] states are also exotic. Perhaps very lightFor c and b quarks the flavor antisymmetric [qq][qq] states are also exotic. Perhaps very light

Lipkin Stewart, Wessling, Wise

hep-ph/0402076

Lipkin Stewart, Wessling, Wise

hep-ph/0402076




Case

II:

Case

II:

(Good diquark is a boson)(Good diquark is a boson)

Antiymmetric in space:

Antiymmetric in space:

Symmetric in spin. Antisymmetric in color Symmetric in flavor

Symmetric in spin. Antisymmetric in color Symmetric in flavor

Examine symmetry of diquark-diquark state:Examine symmetry of diquark-diquark state:

has even parity

has even parity

• Heavier(!)• Odd parity

• Heavier(!)• Odd parity




SummarySummary PentaquarksPentaquarks

• Lightest• Lightest

Negative parity, non-exotic, s-waveNegative parity, non-exotic, s-wave

Positive parity, exotic, p-wavePositive parity, exotic, p-wave

• Heavier• Heavier

• Charmed• Charmed

ExoticExotic s-wave s-waveIt’s existence is It’s existence is independent of independent of evidence for Thetaevidence for Theta

ExoticExotic s-wave s-waveIt’s existence is It’s existence is independent of independent of evidence for Thetaevidence for Theta

Obscure for dynamical reasons --- see later

Obscure for dynamical reasons --- see later

Would be prominent if light enough, but diquark correlation fights with angular momentum.

Would be prominent if light enough, but diquark correlation fights with angular momentum.

Not precluded by anti-Theta evidenceNot precluded by anti-Theta evidence




In Deep Inelastic Processes In Deep Inelastic Processes

• Structure function regularities, especially as • Structure function regularities, especially as

• Fragmentation functions• Fragmentation functions

• If diquarks are strongly correlated, baryon fragmentation functions should not be dramatically smaller than meson f.f.

• If diquarks are strongly correlated, baryon fragmentation functions should not be dramatically smaller than meson f.f.

• Data on two particular baryons...• Data on two particular baryons...

Good [u,d] diquark in s-wave

Good [u,d] diquark in s-waveGood [u,d] diquark in p-wave

Good [u,d] diquark in p-wave




Limits on diquarks from higher twist...Limits on diquarks from higher twist... A. Vainshteyn & RLJ

(unpublished)

“But aren’t strong correlations in QCD ruled out by the absence of large twist-four corrections to DIS?”

“How pointlike can diquarks be?”

“But aren’t strong correlations in QCD ruled out by the absence of large twist-four corrections to DIS?”

“How pointlike can diquarks be?” corrections to DIS are known to be small, and limit non-perturbative scales in QCD beyond

corrections to DIS are known to be small, and limit non-perturbative scales in QCD beyond These limits constrain diquarks because twist-four operators include ones sensitive to diquark correlations...

These limits constrain diquarks because twist-four operators include ones sensitive to diquark correlations...




Leading twistLeading twist

However: “Good” diquark is spinless and does not contribute at twist four !However: “Good” diquark is spinless and does not contribute at twist four !

Dimension-6, spin-2 ⇒ twist-4, but only if quarks are coupled to maximum spin.Dimension-6, spin-2 ⇒ twist-4, but only if quarks are coupled to maximum spin.

Twist four -- diquark operator

Twist four -- diquark operator




QCD spectroscopy in color non-singlet sectors

• Neutralize with spectator Wilson line (= infinitely heavy quark)

• (Or with uniform color background charge (how?))• Compare with bottom hadron spectroscopy• And with phenomenological models

• Neutralize with spectator Wilson line (= infinitely heavy quark)

• (Or with uniform color background charge (how?))• Compare with bottom hadron spectroscopy• And with phenomenological models

Mesons

Mesons

Baryons

Baryons

Tetraquark mesonsTetraquark mesonsPentaquark baryonsPentaquark baryons

Even color sextet light quark statesEven color sextet light quark states

• Study correlations in QCD by studying color non-singlet spectroscopy!

• Study correlations in QCD by studying color non-singlet spectroscopy!

• The “ affair” suggests that correlations in QCD can be studied by studying color non-singlet light quark systems neutralized by heavy quark spectator.




• Study correlations in QCD by studying color non-singlet spectroscopy!• Study correlations in QCD by studying color non-singlet spectroscopy!




Example: Classification of ground state Example: Classification of ground state

“Exotics” with unique SU(3) assignments.

“Exotics” with unique SU(3) assignments.“Exotics” mixed by SU(3) violating interactions.

“Exotics” mixed by SU(3) violating interactions.

Total J=3/2 states with Total J=3/2 states with (Ignore for simplicity -- dynamics?)(Ignore for simplicity -- dynamics?)




“Exotics” with unique SU(3) assignments.

“Exotics” with unique SU(3) assignments.“Exotics” mixed by SU(3) violating interactions.

“Exotics” mixed by SU(3) violating interactions.

Non-exotics, mix with single quark states.

Non-exotics, mix with single quark states.

Total J=1/2 states with Total J=1/2 states with (Ignore for simplicity -- dynamics?)(Ignore for simplicity -- dynamics?)




Color triplet spectroscopy Color triplet spectroscopy Spin Isospin Strangeness SU(3)

3/2 1 +1 [15]

3/2 3/2 0 [15]

3/2 1/2 0 [15] + [3] + [6]*

3/2 1 -1 [15] + [6]*

3/2 0 -1 [15] + [3]

3/2 1/2 -2 [15]

3/2 0 +1 [6]*

1/2 1 +1 [15]

1/2 3/2 0 [15]

1/2 1 -1 [15] + [6]

1/2 1/2 -2 [15]

1/2 0 +1 [6]

* This representation only occurs when qq are coupled to 6* This representation only occurs when qq are coupled to 6




III. Aspects of multiquark dynamicsIII. Aspects of multiquark dynamics




General pictureGeneral picture

• Ordinary mesons (qq*) and baryons (qqq) --- zero width resonances when quark pair creation is suppressed. Properties modified with finite widths.

• Ordinary mesons (qq*) and baryons (qqq) --- zero width resonances when quark pair creation is suppressed. Properties modified with finite widths.

• Multiquark mesons (qqq*q*) and baryons (qqqqq*) --- part of the meson-meson or meson-baryon continuum, modified by QCD interactions. “Multiquark states fall apart if they are above threshold”. Should be treated in a dynamical picture where they revert to the non-interacting continuum as QCD interactions are turned off.

• Multiquark mesons (qqq*q*) and baryons (qqqqq*) --- part of the meson-meson or meson-baryon continuum, modified by QCD interactions. “Multiquark states fall apart if they are above threshold”. Should be treated in a dynamical picture where they revert to the non-interacting continuum as QCD interactions are turned off.




Large Nc counting (for mesons and qqq*q*)Large Nc counting (for mesons and qqq*q*)

• Ordinary mesons• Ordinary mesons

• Widths go to zero as 1/Nc• Widths go to zero as 1/Nc

• Meson-meson scattering goes to zero as N goes to infinity except at the narrow resonances

• Meson-meson scattering goes to zero as N goes to infinity except at the narrow resonances

• qqq*q* • qqq*q* RLJ SLAC PUB 951 (1981)

RLJ SLAC PUB 951 (1981)

by dualityby duality




• So “binding” of exotic qqq*q* must vanish as N goes to infinity

• Study...

• So “binding” of exotic qqq*q* must vanish as N goes to infinity

• Study...

where color, flavor, and spin couplings are suppressed, normed so

where color, flavor, and spin couplings are suppressed, normed so

Any D(x) can be decomposed in terms of color singlet bilinears

Any D(x) can be decomposed in terms of color singlet bilinears

However, in a fixed basis, eg. (12)(34) an arbitrary D(x) will include “hidden color”, ie qq*-octet, statesHowever, in a fixed basis, eg. (12)(34) an arbitrary D(x) will include “hidden color”, ie qq*-octet, states




Large N counting

Large N counting

• Color adjoint mixing is O(1/N)

• Color adjoint mixing is O(1/N)

Conclude

• In the limit, qqq*q* is unbound. It is merely the meson-meson continuum

Conclude

• In the limit, qqq*q* is unbound. It is merely the meson-meson continuum

• Disconnected diagrams are O(1)

• Color singlet exchange is O(1/N2)

• In order color correlations mix into the (qq*)(qq*) system

• In order color correlations mix into the (qq*)(qq*) system




Immediate consequences:• qqq*q* states should generically be very broad• Exception would be if they are below natural

decay thresholds• If the N-dependence of meson-meson scattering

can be studied --- eg. on the lattice or using chiral dynamics --- then the distinction between qqq*q* and qq* states should be clear:• qq* states decouple by vanishing width at large N• qqq*q* states subside into the continuum at large N

Immediate consequences:• qqq*q* states should generically be very broad• Exception would be if they are below natural

decay thresholds• If the N-dependence of meson-meson scattering

can be studied --- eg. on the lattice or using chiral dynamics --- then the distinction between qqq*q* and qq* states should be clear:• qq* states decouple by vanishing width at large N• qqq*q* states subside into the continuum at large NNote --- exactly this analysis has been performed

for the scalar mesons in chiral effective theory by J. Peláez & collaborators

Note --- exactly this analysis has been performed for the scalar mesons in chiral effective theory by J. Peláez & collaborators

Phys. Rev. Lett. 92 (2004) 102001, hep-ph/0307018, 0306063, 0411107

Phys. Rev. Lett. 92 (2004) 102001, hep-ph/0307018, 0306063, 0411107




But in general, more phenomenological methods are needed...

But in general, more phenomenological methods are needed...How should quark model eigenstates be

represented in the hadron-hadron S-matrix?How should quark model eigenstates be

represented in the hadron-hadron S-matrix?

“Unitarization”“Unitarization”

When a state is studied in QCD models, its decays are usually ignored...

quark modelsQCD sum ruleschiral soliton models

So they are “zero width approximations”

When a state is studied in QCD models, its decays are usually ignored...

quark modelsQCD sum ruleschiral soliton models

So they are “zero width approximations”Two classes of unitarization approaches interest me --- Two classes of unitarization approaches interest me --- both are qualitative, neither has been developed in a fully both are qualitative, neither has been developed in a fully relativistic, many channel world. Nevertheless they offer relativistic, many channel world. Nevertheless they offer insight into the nature of the problem...insight into the nature of the problem...

•• Boundary condition approachesBoundary condition approaches

•• Feshbach resonancesFeshbach resonances




Boundary condition approachesBoundary condition approachesWigner’s R-matrixWigner’s R-matrix

• Invented for neutron scattering from complex nuclei. At very low energies, (eg. thermal neutrons!)

• Invented for neutron scattering from complex nuclei. At very low energies, (eg. thermal neutrons!)

• Because the wavelength outside the region of interaction is so long, the amplitude outside is >> inside, and to a good approximation the wavefunction vanishes at r = 0, whence the phase shift is zero, unless ...

• Because the wavelength outside the region of interaction is so long, the amplitude outside is >> inside, and to a good approximation the wavefunction vanishes at r = 0, whence the phase shift is zero, unless ...

• The slope of the wavefunction vanishes at the edge of the nucleus. In that case the amplitudes inside and outside are equal and the phase shift is π/2.

• The slope of the wavefunction vanishes at the edge of the nucleus. In that case the amplitudes inside and outside are equal and the phase shift is π/2.




Typical energy, small scale

Typical energy, small scale

interaction regioninteraction region

Typical energy, large scaleTypical energy, large scale

Wavefunction outside is free and vanishes at origin, so

Wavefunction outside is free and vanishes at origin, so

Special energy where Special energy where

At this energy, the amplitudes of the wavefunctions outside and inside are the same. The phase shift at r = b is π/2:

At this energy, the amplitudes of the wavefunctions outside and inside are the same. The phase shift at r = b is π/2:




View from outsideView from outside

At the energy where the slope of the wavefunction vanishes at the boundary of the interaction region

At the energy where the slope of the wavefunction vanishes at the boundary of the interaction region

• Interior Hamiltonian has an eigenstate at Ej obeying

• Interior Hamiltonian has an eigenstate at Ej obeying

View from inside

View from inside

• Phase shift is π/2 as measured from r=b at Ej

• Phase shift is π/2 as measured from r=b at Ej




R-Matrix formalismR-Matrix formalism

The boundary condition method connects the inside to the outside avoiding detailed statements about how the internal state couples to the scattering channel

The boundary condition method connects the inside to the outside avoiding detailed statements about how the internal state couples to the scattering channel

Inside: Solve the Hamiltonian eigenvalue equation subject to the funny boundary condition

Inside: Solve the Hamiltonian eigenvalue equation subject to the funny boundary conditionThis provide the momenta at which the R-matrix has poles. [The residue of the poles in the R-matrix is related to dE/db.]

This provide the momenta at which the R-matrix has poles. [The residue of the poles in the R-matrix is related to dE/db.]

Outside: the logarithmic derivative of the wavefunction at the endge of the interaction region suffices to construct the scattering state:

Outside: the logarithmic derivative of the wavefunction at the endge of the interaction region suffices to construct the scattering state:




• S(E) is unitary• At very low energies, S(E) is approximately unity except

near poles in R(b,E)• S(E) is independent of b as long as interaction vanishes

for r > b. This gives a constraint on the b-dependence of R, such that dS/db=0 (Potential scattering can be included for r > b)

• Poles in S(E) are narrow if poles in R(b,E) have small residue

• S(E) is unitary• At very low energies, S(E) is approximately unity except

near poles in R(b,E)• S(E) is independent of b as long as interaction vanishes

for r > b. This gives a constraint on the b-dependence of R, such that dS/db=0 (Potential scattering can be included for r > b)

• Poles in S(E) are narrow if poles in R(b,E) have small residueUNITARIZATION: assume a pole and

residue...and get a finite width resonance in S(E)

UNITARIZATION: assume a pole and residue...

and get a finite width resonance in S(E)

A beautiful formalism, but what is it’s relation to QCD, multiquark hadrons, meson-meson scattering, etc.?

A beautiful formalism, but what is it’s relation to QCD, multiquark hadrons, meson-meson scattering, etc.?




The K-matrix... The standard approach to unitarization. The K-matrix... The standard approach to unitarization. Suppose that the interaction region is very small compared to the de Broglie wavelength of the scattering particles: Then,

Suppose that the interaction region is very small compared to the de Broglie wavelength of the scattering particles: Then,• Approximate b = 0, • Approximate b = 0,

• Physics? On resonance the scattering wave appears already phase shifted by π/2 at the origin!

• Physics? On resonance the scattering wave appears already phase shifted by π/2 at the origin!

• A historical accident: Dalitz initiated the study of hadron scattering when hadrons were thought to be pointlike

• When a resonance is very narrow it doesn’t matter what boundary condition you impose on H-interior, the physics is insensitive

• A historical accident: Dalitz initiated the study of hadron scattering when hadrons were thought to be pointlike

• When a resonance is very narrow it doesn’t matter what boundary condition you impose on H-interior, the physics is insensitive

That doesn’t mean it has the physics right!That doesn’t mean it has the physics right!

• Hadrons are not pointlike, so the phase shift extrapolated in to the origin has no significance

• Quark model calculations do not approximate the boundary condition that the derivative of the wavefunction vanishes at the hadron’s surface.

• Hadrons are not pointlike, so the phase shift extrapolated in to the origin has no significance

• Quark model calculations do not approximate the boundary condition that the derivative of the wavefunction vanishes at the hadron’s surface.




Single channel S-wave example

Pole in K-matrix occurs when phase shift is π/2. Certainly a reasonable description of a narrow resonance

For example,

Pole in K-matrix occurs when phase shift is π/2. Certainly a reasonable description of a narrow resonance

For example,

Choose any other boundary condition,and associated “matrix” will have a pole at an energy very close to K-matrix pole. So exact treatment is not necessary.

Choose any other boundary condition,and associated “matrix” will have a pole at an energy very close to K-matrix pole. So exact treatment is not necessary.




• But the energies at which the K-matrix has poles has no physical relation to the solutions to a QCD-motivated Hamiltonian eigenvalue equation!

• Particularly poorly motivated for channels where multiquark dynamics may be important

• But the energies at which the K-matrix has poles has no physical relation to the solutions to a QCD-motivated Hamiltonian eigenvalue equation!

• Particularly poorly motivated for channels where multiquark dynamics may be important

What would be a better physical construction in QCD, where quarks are confined in hadrons of radius ~ 1 fermi??

What would be a better physical construction in QCD, where quarks are confined in hadrons of radius ~ 1 fermi??

Standard approach to finding resonances in hadron scattering data:1. Assume the K-matrix is given by a sum of poles (plus a smooth background, ...)II. Fit the resulting S-matrix to the dataIII. Send the resulting energies and residues of K-matrix poles to the PDG!

Standard approach to finding resonances in hadron scattering data:1. Assume the K-matrix is given by a sum of poles (plus a smooth background, ...)II. Fit the resulting S-matrix to the dataIII. Send the resulting energies and residues of K-matrix poles to the PDG!




Quark model states resemble zeros in Wigner’s R-matrix (or poles in P=1/R) because they are confined at distance scales of order

Quark model states resemble zeros in Wigner’s R-matrix (or poles in P=1/R) because they are confined at distance scales of order

At a pole in P(b,E)At a pole in P(b,E)

Imagine this is the π π relative wave functionImagine this is the π π relative wave function

RLJ and F. E. Low 1979

RLJ and F. E. Low 1979

At a pole in P(b,E)At a pole in P(b,E)• On the inside• On the inside

• Define P as the inverse of R-matrix• Define P as the inverse of R-matrix

• Knowing P allows one to construct the scattering state

• Knowing P allows one to construct the scattering state

This is for 1-channel, s-wave. Generalization is straightforward

This is for 1-channel, s-wave. Generalization is straightforward




Interpretation of P-matrix polesInterpretation of P-matrix poles

• They occur at the energies at which the scattering wave function (eg. ππ) has a node at r = b, and therefore would match smoothly to internal confined quark state.

• They occur at the energies at which the scattering wave function (eg. ππ) has a node at r = b, and therefore would match smoothly to internal confined quark state.

• However: ππ wavefunction has nodes at r = b even when there is no interaction! (Here is where things get interesting).

• However: ππ wavefunction has nodes at r = b even when there is no interaction! (Here is where things get interesting).

• Poles occur at k = nπ/b (in the s-wave). These locations of P-matrix poles correspond to no interaction.

• Poles occur at k = nπ/b (in the s-wave). These locations of P-matrix poles correspond to no interaction.

• Interpretation:• Four quark interactions shift P-matrix poles and change their residues.• Two quark states (narrow) are added to the P-matrix

• Interpretation:• Four quark interactions shift P-matrix poles and change their residues.• Two quark states (narrow) are added to the P-matrix




In the case of no interaction --- the “reference” or “compensation P-matrix, P0

In the case of no interaction --- the “reference” or “compensation P-matrix, P0

Look at effect of shifting first pole in P up or down relative to its “compensation” value, k = π/b, (about 700 Mev for ππ scattering

Look at effect of shifting first pole in P up or down relative to its “compensation” value, k = π/b, (about 700 Mev for ππ scattering

Upward shift in P-pole location corresponds to repulsion! and shows up as negative phase shift. ππ phase carries information about repulsive quark interactions

Upward shift in P-pole location corresponds to repulsion! and shows up as negative phase shift. ππ phase carries information about repulsive quark interactionsDownward shift in P-pole location corresponds to attraction! and shows up as positive phase shift. ππ phase carries information about attractive quark interactions

Downward shift in P-pole location corresponds to attraction! and shows up as positive phase shift. ππ phase carries information about attractive quark interactions




P-matrix dynamics...P-matrix dynamics...

• Multiquark states are interpreted as information about the shifts in the poles in the compensating P-matrix.

• When a multiquark state is shifted down in mass by QCD interactions, the P-matrix pole energy is moved down, and the result is that the phase shift becomes positive in that region.

• The QCD attraction that shifted the multiquark mass down is observed as an attractive interaction in hadron-hadron scattering.

• And vica versa for a repulsive QCD interaction shifting a state up

• As the interaction goes away, the effects subside into the continuum

• QQ* or QQQ states are added to the P-matrix since they do not appear in the continuum when the interactions are turned off

• States like the rho meson are added to the compensating P-matrix and appear as narrow resonances (if their residues are small)

• For all it’s limitations, the P-matrix method encodes the correct large N behavior and physically links QCD interactions to hadron-hadron phase shifts.

• Multiquark states are interpreted as information about the shifts in the poles in the compensating P-matrix.

• When a multiquark state is shifted down in mass by QCD interactions, the P-matrix pole energy is moved down, and the result is that the phase shift becomes positive in that region.

• The QCD attraction that shifted the multiquark mass down is observed as an attractive interaction in hadron-hadron scattering.

• And vica versa for a repulsive QCD interaction shifting a state up

• As the interaction goes away, the effects subside into the continuum

• QQ* or QQQ states are added to the P-matrix since they do not appear in the continuum when the interactions are turned off

• States like the rho meson are added to the compensating P-matrix and appear as narrow resonances (if their residues are small)

• For all it’s limitations, the P-matrix method encodes the correct large N behavior and physically links QCD interactions to hadron-hadron phase shifts.




One pole shifted up since I=2 π π corresponds to bad-bad

diquark

One pole shifted up since I=2 π π corresponds to bad-bad

diquark

P-matrix fits to π π scatteringP-matrix fits to π π scattering

One pole shifted down since I=0 π π

corresponds to good-good

diquark

One pole shifted down since I=0 π π

corresponds to good-good

diquark

One pole added

since I=1 π π

corresponds to a

confined qq* state

One pole added

since I=1 π π

corresponds to a

confined qq* state

P-matrix fits to π π scatteringP-matrix fits to π π scattering




Feshbach resonances and confinementFeshbach resonances and confinement

The provides an unique opportunity to consider how a hadron should be represented in the S-matrix at very low energies

The provides an unique opportunity to consider how a hadron should be represented in the S-matrix at very low energies

1. Quasi-non-relativistic1. Quasi-non-relativistic

2. Only one channel open2. Only one channel open

• A problem in Schrödinger quantum mechanics• A problem in Schrödinger quantum mechanics• How do you get striking effects at low

energy in non-relativistic quantum mechanics?

• How do you get striking effects at low energy in non-relativistic quantum mechanics?• Ordinary non-relativistic potential scattering• Ordinary non-relativistic potential scattering

• Bound state in a closed (= confined) channel in the continuum... A phenomenon first discovered by U. Fano (1935), now known as a Feshbach resonance

• Bound state in a closed (= confined) channel in the continuum... A phenomenon first discovered by U. Fano (1935), now known as a Feshbach resonance




What sort of effects are created by potential scattering in low partial waves at low energies?What sort of effects are created by potential scattering in low partial waves at low energies?

• Resonance results from interplay of attractive interation and long range angular momentum barrier.

• No potential resonances in the s-wave• Potential scattering resonances are typically narrow

only when 〈 kb 〉≪ 1• For example, to make a resonance at the mass of the Θ

• Resonance results from interplay of attractive interation and long range angular momentum barrier.

• No potential resonances in the s-wave• Potential scattering resonances are typically narrow

only when 〈 kb 〉≪ 1• For example, to make a resonance at the mass of the Θ




How, then, can low energy, narrow hadronic resonances appear in low partial waves?How, then, can low energy, narrow hadronic resonances appear in low partial waves?

• Bound state in the continuum (Feshbach-Fano)• Bound state in the continuum (Feshbach-Fano)

• As the confined state decouples (zero width)

• As the confined state decouples (zero width)

• Interaction “off resonance” can be arbitrarily weak without removing resonance. Confined channel is absent from continuum.

• Interaction “off resonance” can be arbitrarily weak without removing resonance. Confined channel is absent from continuum.

• For a discussion applied to Theta, see RLJ and A. Jain, hep-ph/0408046

• For a discussion applied to Theta, see RLJ and A. Jain, hep-ph/0408046




• Conjecture:

i. Ordinary (qq* and qqq) hadrons are bound states in confined channels that couple to scattering channel through quark pair creation. In low energy scattering they would manifest as Feshbach resonances.

ii. Multiquark hadrons are (generically) modulations of the continuum in open channels. In non-relativistic limit they would influence the open-channel potential.

iii. Consistent with P-matrix description.

• Conjecture:

i. Ordinary (qq* and qqq) hadrons are bound states in confined channels that couple to scattering channel through quark pair creation. In low energy scattering they would manifest as Feshbach resonances.

ii. Multiquark hadrons are (generically) modulations of the continuum in open channels. In non-relativistic limit they would influence the open-channel potential.

iii. Consistent with P-matrix description.




Conclusions• The absence of exotics in the hadron spectrum

is now qualitatively understood.• The Theta may have died, but it refocused

interest in the problem of correlations in QCD.• Diquarks are the obvious candidates for

spectroscopic correlations

• The absence of exotics in the hadron spectrum is now qualitatively understood.

• The Theta may have died, but it refocused interest in the problem of correlations in QCD.

• Diquarks are the obvious candidates for spectroscopic correlations• Lots of phenomenological evidence for

diquarks• Absence of exotics and scalar mesons• Role in DIS --- qualitative --- fragmentation

functions and higher twist. More to be done

• Lots of phenomenological evidence for diquarks

• Absence of exotics and scalar mesons• Role in DIS --- qualitative --- fragmentation

functions and higher twist. More to be done• A systematic study of correlations in QCD:

light quark spectroscopy in the color non-singlet sectors

• What is the correct dynamical framework for analyzing low energy data to make contact with QCD?

• A systematic study of correlations in QCD: light quark spectroscopy in the color non-singlet sectors

• What is the correct dynamical framework for analyzing low energy data to make contact with QCD?

r l jaffe frascati may 2005 multiquark dynamics r. l. jaffe lnf frascati may 2005 r. l. jaffe lnf...

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