First Contents Back Conclusion

Dynamical Chiral Symmetry Breakingand Hadron Structure

Craig D. Roberts

cdroberts@anl.gov

Physics Division

Argonne National LaboratoryDynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 1/40

First Contents Back Conclusion

Modern Miraclesin Hadron Physics

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 2/40

First Contents Back Conclusion

Modern Miraclesin Hadron Physics

proton = three constituent quarks

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 2/40

First Contents Back Conclusion

Modern Miraclesin Hadron Physics

proton = three constituent quarks

Mproton ≈ 1 GeV

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 2/40

First Contents Back Conclusion

Modern Miraclesin Hadron Physics

proton = three constituent quarks

Mproton ≈ 1 GeV

guess Mconstituent−quark ≈ 1 GeV

3≈ 350 MeV

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 2/40

First Contents Back Conclusion

Modern Miraclesin Hadron Physics

proton = three constituent quarks

Mproton ≈ 1 GeV

guess Mconstituent−quark ≈ 1 GeV

3≈ 350 MeV

pion =

constituent quark + constituent antiquark

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 2/40

First Contents Back Conclusion

Modern Miraclesin Hadron Physics

proton = three constituent quarks

Mproton ≈ 1 GeV

guess Mconstituent−quark ≈ 1 GeV

3≈ 350 MeV

pion =

constituent quark + constituent antiquark

guess Mpion ≈ 2 × Mproton

3≈ 700 MeV

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 2/40

First Contents Back Conclusion

Modern Miraclesin Hadron Physics

proton = three constituent quarks

Mproton ≈ 1 GeV

guess Mconstituent−quark ≈ 1 GeV

3≈ 350 MeV

pion =

constituent quark + constituent antiquark

guess Mpion ≈ 2 × Mproton

3≈ 700 MeV

WRONG . . . . . . . . . . . . . . . . . . . . . . Mpion = 140 MeV

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 2/40

First Contents Back Conclusion

Modern Miraclesin Hadron Physics

proton = three constituent quarks

Mproton ≈ 1 GeV

guess Mconstituent−quark ≈ 1 GeV

3≈ 350 MeV

pion =

constituent quark + constituent antiquark

guess Mpion ≈ 2 × Mproton

3≈ 700 MeV

WRONG . . . . . . . . . . . . . . . . . . . . . . Mpion = 140 MeV

Another meson:

. . . . . . . . . . . Mρ = 770 MeV . . . . . . . . . . . No Surprises Here

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 2/40

First Contents Back Conclusion

Modern Miraclesin Hadron Physics

proton = three constituent quarks

Mproton ≈ 1 GeV

guess Mconstituent−quark ≈ 1 GeV

3≈ 350 MeV

pion =

constituent quark + constituent antiquark

guess Mpion ≈ 2 × Mproton

3≈ 700 MeV

WRONG . . . . . . . . . . . . . . . . . . . . . . Mpion = 140 MeV

What is “wrong” with the pion?

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 2/40

First Contents Back Conclusion

Dichotomy of the Pion

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 3/40

First Contents Back Conclusion

Dichotomy of the Pion

How does one make an almost massless particle. . . . . . . . . . . from two massive constituent-quarks?

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 3/40

First Contents Back Conclusion

Dichotomy of the Pion

How does one make an almost massless particle. . . . . . . . . . . from two massive constituent-quarks?

Not Allowed to do it by fine-tuning a potential

Must exhibit m2

π ∝ mq

Current Algebra . . . 1968

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 3/40

First Contents Back Conclusion

Dichotomy of the Pion

How does one make an almost massless particle. . . . . . . . . . . from two massive constituent-quarks?

Not Allowed to do it by fine-tuning a potential

Must exhibit m2

π ∝ mq

Current Algebra . . . 1968

The correct understanding of pion observables;e.g. mass, decay constant and form factors,requires an approach to contain a well-defined andvalid chiral limit, and an accurate realisation ofdynamical chiral symmetry breaking.

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 3/40

First Contents Back Conclusion

Dichotomy of the Pion

How does one make an almost massless particle. . . . . . . . . . . from two massive constituent-quarks?

Not Allowed to do it by fine-tuning a potential

Must exhibit m2

π ∝ mq

Current Algebra . . . 1968

The correct understanding of pion observables;e.g. mass, decay constant and form factors,requires an approach to contain a well-defined andvalid chiral limit, and an accurate realisation ofdynamical chiral symmetry breaking.

Requires detailed understanding of Connectionbetween Current-quark and Constituent-quarkmasses

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 3/40

First Contents Back Conclusion

Dichotomy of the Pion

How does one make an almost massless particle. . . . . . . . . . . from two massive constituent-quarks?

Not Allowed to do it by fine-tuning a potential

Must exhibit m2

π ∝ mq

Current Algebra . . . 1968

The correct understanding of pion observables;e.g. mass, decay constant and form factors,requires an approach to contain a well-defined andvalid chiral limit, and an accurate realisation ofdynamical chiral symmetry breaking.

Requires detailed understanding of Connectionbetween Current-quark and Constituent-quarkmasses Using DSEs,

we’ve provided this.Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 3/40

First Contents Back Conclusion

QCD’s Emergent Phenomena

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 4/40

First Contents Back Conclusion

QCD’s Emergent Phenomena

Complex behaviour arises from apparently simple rules

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 4/40

First Contents Back Conclusion

QCD’s Emergent Phenomena

Complex behaviour arises from apparently simple rules

Quark and Gluon Confinement

No matter how hard one strikes the proton, one cannot

liberate an individual quark or gluon

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 4/40

First Contents Back Conclusion

QCD’s Emergent Phenomena

Complex behaviour arises from apparently simple rules

Quark and Gluon Confinement

No matter how hard one strikes the proton, one cannot

liberate an individual quark or gluon

Dynamical Chiral Symmetry Breaking

Very unnatural pattern of bound state masses

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 4/40

First Contents Back Conclusion

QCD’s Emergent Phenomena

Complex behaviour arises from apparently simple rules

Quark and Gluon Confinement

No matter how hard one strikes the proton, one cannot

liberate an individual quark or gluon

Dynamical Chiral Symmetry Breaking

Very unnatural pattern of bound state masses

Neither of these phenomena is apparent in QCD’s

Lagrangian yet they are the dominant determining

characteristics of real-world QCD.

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 4/40

First Contents Back Conclusion

QCD’s Emergent Phenomena

Complex behaviour arises from apparently simple rules

Quark and Gluon Confinement

No matter how hard one strikes the proton, one cannot

liberate an individual quark or gluon

Dynamical Chiral Symmetry Breaking

Very unnatural pattern of bound state masses

Neither of these phenomena is apparent in QCD’s

Lagrangian yet they are the dominant determining

characteristics of real-world QCD.

NSAC – Understanding these phenomena is one of the

greatest intellectual challenges in physicsDynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 4/40

First Contents Back Conclusion

What’s the Problem?

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 5/40

First Contents Back Conclusion

What’s the Problem?

Must calculate the hadron’s wave function

– Can’t be done using perturbation theory

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 5/40

First Contents Back Conclusion

What’s the Problem?

Must calculate the hadron’s wave function

– Can’t be done using perturbation theory

So what? Same is true of hydrogen atom

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 5/40

First Contents Back Conclusion

What’s the Problem?

Must calculate the hadron’s wave function

– Can’t be done using perturbation theory

So what? Same is true of hydrogen atom

Differences

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 5/40

First Contents Back Conclusion

What’s the Problem?

Must calculate the hadron’s wave function

– Can’t be done using perturbation theory

So what? Same is true of hydrogen atom

Differences

Here relativistic effects are crucial

– virtual particles

Quintessence of Relativistic Quantum Field Theory

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 5/40

First Contents Back Conclusion

What’s the Problem?

Must calculate the hadron’s wave function

– Can’t be done using perturbation theory

So what? Same is true of hydrogen atom

Differences

Here relativistic effects are crucial

– virtual particles

Quintessence of Relativistic Quantum Field Theory

Interaction between quarks – the Interquark Potential –

Unknown

throughout > 98% of the pion’s/proton’s volume

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 5/40

First Contents Back Conclusion

What’s the Problem?

Must calculate the hadron’s wave function

– Can’t be done using perturbation theory

So what? Same is true of hydrogen atom

Differences

Here relativistic effects are crucial

– virtual particles

Quintessence of Relativistic Quantum Field Theory

Interaction between quarks – the Interquark Potential –

Unknown

throughout > 98% of the pion’s/proton’s volume

Determination of hadrons’s wave function requires

ab initio nonperturbative solution

of fully-fledged relativistic quantum field theoryDynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 5/40

First Contents Back Conclusion

What’s the Problem?

Must calculate the hadron’s wave function

– Can’t be done using perturbation theory

So what? Same is true of hydrogen atom

Determination of hadron’s wave function requires

ab initio nonperturbative solution

of fully-fledged relativistic quantum field theory

Modern Physics & Mathematics

– Still quite some way from being able to do that

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 5/40

First Contents Back Conclusion

Intranucleon Interaction

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 6/40

First Contents Back Conclusion

Intranucleon Interaction

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 6/40

First Contents Back Conclusion

Intranucleon Interaction

98% of the volumeDynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 6/40

First Contents Back Conclusion

Dyson-Schwinger Equations

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 7/40

First Contents Back Conclusion

Dyson-Schwinger Equations

Well suited to Relativistic Quantum Field Theory

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 7/40

First Contents Back Conclusion

Dyson-Schwinger Equations

Well suited to Relativistic Quantum Field Theory

Simplest level: Generating Tool for Perturbation Theory

. . . . . . . . . . . . . . . . . . . . Materially Reduces Model Dependence

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 7/40

First Contents Back Conclusion

Dyson-Schwinger Equations

Well suited to Relativistic Quantum Field Theory

Simplest level: Generating Tool for Perturbation Theory

. . . . . . . . . . . . . . . . . . . . Materially Reduces Model Dependence

NonPerturbative, Continuum approach to QCD

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 7/40

First Contents Back Conclusion

Dyson-Schwinger Equations

Well suited to Relativistic Quantum Field Theory

Simplest level: Generating Tool for Perturbation Theory

. . . . . . . . . . . . . . . . . . . . Materially Reduces Model Dependence

NonPerturbative, Continuum approach to QCD

Hadrons as Composites of Quarks and Gluons

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 7/40

First Contents Back Conclusion

Dyson-Schwinger Equations

Well suited to Relativistic Quantum Field Theory

Simplest level: Generating Tool for Perturbation Theory

. . . . . . . . . . . . . . . . . . . . Materially Reduces Model Dependence

NonPerturbative, Continuum approach to QCD

Hadrons as Composites of Quarks and Gluons

Qualitative and Quantitative Importance of:

· Dynamical Chiral Symmetry Breaking

– Generation of fermion mass from nothing

· Quark & Gluon Confinement

– Coloured objects not detected, not detectable?

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 7/40

First Contents Back Conclusion

Dyson-Schwinger Equations

Well suited to Relativistic Quantum Field Theory

Simplest level: Generating Tool for Perturbation Theory

. . . . . . . . . . . . . . . . . . . . Materially Reduces Model Dependence

NonPerturbative, Continuum approach to QCD

Hadrons as Composites of Quarks and Gluons

Qualitative and Quantitative Importance of:

· Dynamical Chiral Symmetry Breaking

– Generation of fermion mass from nothing

· Quark & Gluon Confinement

– Coloured objects not detected, not detectable?

⇒ Understanding InfraRed (long-range)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . behaviour of αs(Q2)

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 7/40

First Contents Back Conclusion

Dyson-Schwinger Equations

Well suited to Relativistic Quantum Field Theory

Simplest level: Generating Tool for Perturbation Theory

. . . . . . . . . . . . . . . . . . . . Materially Reduces Model Dependence

NonPerturbative, Continuum approach to QCD

Hadrons as Composites of Quarks and Gluons

Qualitative and Quantitative Importance of:

· Dynamical Chiral Symmetry Breaking

– Generation of fermion mass from nothing

· Quark & Gluon Confinement

– Coloured objects not detected, not detectable?

Method yields Schwinger Functions ≡ Propagators

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 7/40

First Contents Back Conclusion

Dyson-Schwinger Equations

Well suited to Relativistic Quantum Field Theory

Simplest level: Generating Tool for Perturbation Theory

. . . . . . . . . . . . . . . . . . . . Materially Reduces Model Dependence

NonPerturbative, Continuum approach to QCD

Hadrons as Composites of Quarks and Gluons

Qualitative and Quantitative Importance of:

· Dynamical Chiral Symmetry Breaking

– Generation of fermion mass from nothing

· Quark & Gluon Confinement

– Coloured objects not detected, not detectable?

Cross-Sections built from Schwinger Functions

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 7/40

First Contents Back Conclusion

Persistent Challenge

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 8/40

First Contents Back Conclusion

Persistent Challenge

Infinitely Many Coupled Equations

Σ=

D

γΓS

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 8/40

First Contents Back Conclusion

Persistent Challenge

Infinitely Many Coupled Equations

Solutions are Schwinger Functions(Euclidean Green Functions)

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 8/40

First Contents Back Conclusion

Persistent Challenge

Infinitely Many Coupled Equations

Solutions are Schwinger Functions(Euclidean Green Functions)

Not all are Schwinger functions are experimentallyobservable but all are same VEVs measured inLattice-QCD simulations . . . opportunity forcomparisons at pre-experimental level . . .cross-fertilisation

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 8/40

First Contents Back Conclusion

Persistent Challenge

Infinitely Many Coupled Equations

Solutions are Schwinger Functions(Euclidean Green Functions)

Coupling between equations necessitates truncation

Weak coupling expansion ⇒ Perturbation Theory

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 8/40

First Contents Back Conclusion

Persistent Challenge

Infinitely Many Coupled Equations

Solutions are Schwinger Functions(Euclidean Green Functions)

Coupling between equations necessitates truncation

Weak coupling expansion ⇒ Perturbation TheoryNot useful for the nonperturbative problemsin which we’re interested

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 8/40

First Contents Back Conclusion

Persistent Challenge

Infinitely Many Coupled Equations

Solutions are Schwinger Functions(Euclidean Green Functions)

There is at least one systematic nonperturbative,symmetry-preserving truncation schemeH.J. Munczek Phys. Rev. D 52 (1995) 4736Dynamical chiral symmetry breaking, Goldstone’stheorem and the consistency of the Schwinger-Dysonand Bethe-Salpeter EquationsA. Bender, C. D. Roberts and L. von Smekal, Phys.Lett. B 380 (1996) 7Goldstone Theorem and Diquark Confinement BeyondRainbow Ladder Approximation

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 9/40

First Contents Back Conclusion

Persistent Challenge

Infinitely Many Coupled Equations

Solutions are Schwinger Functions(Euclidean Green Functions)

There is at least one systematic nonperturbative,symmetry-preserving truncation scheme

Has Enabled Proof of EXACT Results in QCD

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 9/40

First Contents Back Conclusion

Persistent Challenge

Infinitely Many Coupled Equations

Solutions are Schwinger Functions(Euclidean Green Functions)

There is at least one systematic nonperturbative,symmetry-preserving truncation scheme

Has Enabled Proof of EXACT Results in QCD

And Formulation of Practical Phenomenological Tool to

Illustrate Exact Results

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 9/40

First Contents Back Conclusion

Persistent Challenge

Infinitely Many Coupled Equations

Solutions are Schwinger Functions(Euclidean Green Functions)

There is at least one systematic nonperturbative,symmetry-preserving truncation scheme

Has Enabled Proof of EXACT Results in QCD

And Formulation of Practical Phenomenological Tool to

Make Predictions with Readily Quantifiable Errors

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 9/40

First Contents Back Conclusion

Dressed-Quark Propagator

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 10/40

First Contents Back Conclusion

Dressed-Quark Propagator

S(p) =Z(p2)

iγ · p + M(p2)Σ

=D

γΓS

Gap Equation

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 10/40

First Contents Back Conclusion

Dressed-Quark Propagator

S(p) =Z(p2)

iγ · p + M(p2)Σ

=D

γΓS

Gap EquationGap Equation’s Kernel Enhanced on IR domain

⇒ IR Enhancement of M(p2)

10−2 10−1 100 101 102

p2 (GeV2)

10−3

10−2

10−1

100

101

M(p

2 ) (G

eV)

b−quarkc−quarks−quarku,d−quarkchiral limitM

2(p

2) = p

2

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 10/40

First Contents Back Conclusion

Dressed-Quark Propagator

S(p) =Z(p2)

iγ · p + M(p2)Σ

=D

γΓS

Gap EquationGap Equation’s Kernel Enhanced on IR domain

⇒ IR Enhancement of M(p2)

10−2 10−1 100 101 102

p2 (GeV2)

10−3

10−2

10−1

100

101

M(p

2 ) (G

eV)

b−quarkc−quarks−quarku,d−quarkchiral limitM

2(p

2) = p

2

Euclidean Constituent–Quark

Mass: MEf : p2 = M(p2)2

flavour u/d s c b

ME

mζ∼ 102 ∼ 10 ∼ 1.5 ∼ 1.1

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 10/40

First Contents Back Conclusion

Dressed-QuarkPropagator

Longstanding Prediction of

Dyson-Schwinger Equation

Studies

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 11/40

First Contents Back Conclusion

Dressed-QuarkPropagator

Longstanding Prediction of

Dyson-Schwinger Equation

Studies

E.g., Dyson-Schwinger

equations and their

application to hadronic

physics,

C. D. Roberts and

A. G. Williams,

Prog. Part. Nucl. Phys.

33 (1994) 477

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 11/40

First Contents Back Conclusion

Dressed-QuarkPropagator

Longstanding Prediction of

Dyson-Schwinger Equation

Studies

E.g., Dyson-Schwinger

equations and their

application to hadronic

physics,

C. D. Roberts and

A. G. Williams,

Prog. Part. Nucl. Phys.

33 (1994) 477

Long used as basis for

efficacious hadron physics

phenomenology

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 11/40

First Contents Back Conclusion

Dressed-QuarkPropagator

Longstanding Prediction of

Dyson-Schwinger Equation

Studies

E.g., Dyson-Schwinger

equations and their

application to hadronic

physics,

C. D. Roberts and

A. G. Williams,

Prog. Part. Nucl. Phys.

33 (1994) 477

Long used as basis for

efficacious hadron physics

phenomenology

Electromagnetic pion

form-factor and neutral

pion decay width,

C. D. Roberts,

Nucl. Phys. A 605(1996) 475

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 11/40

First Contents Back Conclusion

Mandar Bhagwat

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 12/40

First Contents Back Conclusion

Mandar Bhagwat

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 12/40

First Contents Back Conclusion

Mandar BhagwatEmulatingsupervisor’sapproach tophysics

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 12/40

First Contents Back Conclusion

Quenched-QCDDressed-Quark Propagator

0.0 1.0 2.0 3.0 4.0p (GeV)

0.0

0.1

0.2

0.3

0.4

0.5

0.0 1.0 2.0 3.0 4.0p (GeV)

0.0

0.2

0.4

0.6

0.8

1.0

M(p) Z(p)

M(p)

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 13/40

First Contents Back Conclusion

Quenched-QCDDressed-Quark Propagator2002

0.0 1.0 2.0 3.0 4.0p (GeV)

0.0

0.1

0.2

0.3

0.4

0.5

0.0 1.0 2.0 3.0 4.0p (GeV)

0.0

0.2

0.4

0.6

0.8

1.0

M(p) Z(p)

M(p)

“data:” Quenched Lattice Meas.– Bowman, Heller, Leinweber, Williams: he-lat/0209129

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 13/40

First Contents Back Conclusion

Quenched-QCDDressed-Quark Propagator2002

0.0 1.0 2.0 3.0 4.0p (GeV)

0.0

0.1

0.2

0.3

0.4

0.5

0.0 1.0 2.0 3.0 4.0p (GeV)

0.0

0.2

0.4

0.6

0.8

1.0

M(p) Z(p)

M(p)

“data:” Quenched Lattice Meas.– Bowman, Heller, Leinweber, Williams: he-lat/0209129current-quark masses: 30 MeV, 50 MeV, 100 MeV

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 13/40

First Contents Back Conclusion

Quenched-QCDDressed-Quark Propagator2002

0.0 1.0 2.0 3.0 4.0p (GeV)

0.0

0.1

0.2

0.3

0.4

0.5

0.0 1.0 2.0 3.0 4.0p (GeV)

0.0

0.2

0.4

0.6

0.8

1.0

M(p) Z(p)

M(p)

“data:” Quenched Lattice Meas.– Bowman, Heller, Leinweber, Williams: he-lat/0209129current-quark masses: 30 MeV, 50 MeV, 100 MeVCurves: Quenched DSE Cal.

– Bhagwat, Pichowsky, Roberts, Tandy nu-th/0304003

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 13/40

First Contents Back Conclusion

Quenched-QCDDressed-Quark Propagator2002

0.0 1.0 2.0 3.0 4.0p (GeV)

0.0

0.1

0.2

0.3

0.4

0.5

0.0 1.0 2.0 3.0 4.0p (GeV)

0.0

0.2

0.4

0.6

0.8

1.0

M(p) Z(p)

M(p)

“data:” Quenched Lattice Meas.– Bowman, Heller, Leinweber, Williams: he-lat/0209129current-quark masses: 30 MeV, 50 MeV, 100 MeVCurves: Quenched DSE Cal.

– Bhagwat, Pichowsky, Roberts, Tandy nu-th/0304003Linear extrapolation of lattice data to chiral limit is inaccurate

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 13/40

First Contents Back Conclusion

QCD & Interaction BetweenLight-Quarks

Kernel of Gap Equation: Dµν(p − q) Γν(q)

Dressed-gluon propagator and dressed-quark-gluon vertex

Reliable DSE studies of Dressed-gluon propagator:

R. Alkofer and L. von Smekal, The infrared behavior of QCDGreen’s functions . . . , Phys. Rept. 353, 281 (2001).

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 14/40

First Contents Back Conclusion

QCD & Interaction BetweenLight-Quarks

Kernel of Gap Equation: Dµν(p − q) Γν(q)

Dressed-gluon propagator and dressed-quark-gluon vertex

Reliable DSE studies of Dressed-gluon propagator:

R. Alkofer and L. von Smekal, The infrared behavior of QCDGreen’s functions . . . , Phys. Rept. 353, 281 (2001).

Dressed-gluon propagator – lattice-QCD simulations confirm thatbehaviour:

D. B. Leinweber, J. I. Skullerud, A. G. Williams and C.Parrinello [UKQCD Collaboration], Asymptotic scaling andinfrared behavior of the gluon propagator, Phys. Rev. D 60,094507 (1999) [Erratum-ibid. D 61, 079901 (2000)].

Exploratory DSE and lattice-QCD studiesof dressed-quark-gluon vertex

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 14/40

First Contents Back Conclusion

Dressed-gluon PropagatorAlkofer, Detmold, Fischer,Maris: he-ph/0309078

Dµν(k) =

(

δµν −kµkν

k2

)

Z(k2)

k2

10-2

10-1

100

101

102

103

p2 [GeV

2]

10-1

100

Z(p

2 )

lattice, Nf=0

DSE, Nf=0

DSE, Nf=3

Fit to DSE, Nf=3

Suppressionmeans ∃ IR gluonmass-scale≈1 GeV

Naturally, thisscale has thesame origin asΛQCD

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 15/40

First Contents Back Conclusion

Dressed-gluon PropagatorAlkofer, Detmold, Fischer,Maris: he-ph/0309078

Dµν(k) =

(

δµν −kµkν

k2

)

Z(k2)

k2

10-2

10-1

100

101

102

103

p2 [GeV

2]

10-1

100

Z(p

2 )

lattice, Nf=0

DSE, Nf=0

DSE, Nf=3

Fit to DSE, Nf=3

Suppressionmeans ∃ IR gluonmass-scale≈1 GeV

Naturally, thisscale has thesame origin asΛQCD

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 15/40

First Contents Back Conclusion

Dressed-gluon PropagatorAlkofer, Detmold, Fischer,Maris: he-ph/0309078

Dµν(k) =

(

δµν −kµkν

k2

)

Z(k2)

k2

10-2

10-1

100

101

102

103

p2 [GeV

2]

10-1

100

Z(p

2 )

lattice, Nf=0

DSE, Nf=0

DSE, Nf=3

Fit to DSE, Nf=3

Suppressionmeans ∃ IR gluonmass-scale≈1 GeV

Naturally, thisscale has thesame origin asΛQCD

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 15/40

First Contents Back Conclusion

Consituent-quark σ-term

Impact of Dynamical chiral symmetry breaking . . . exhibited

via constituent-quark σ-term

σf := mf (ζ)∂ME

f

∂mf (ζ), (ME)2 := s | s = M(s)2 .

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 16/40

First Contents Back Conclusion

Consituent-quark σ-term

Impact of Dynamical chiral symmetry breaking . . . exhibited

via constituent-quark σ-term

σf := mf (ζ)∂ME

f

∂mf (ζ), (ME)2 := s | s = M(s)2 .

Renormalisation-group-invariant and determined from

solutions of the gap equation

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 16/40

First Contents Back Conclusion

Consituent-quark σ-term

Impact of Dynamical chiral symmetry breaking . . . exhibited

via constituent-quark σ-term

σf := mf (ζ)∂ME

f

∂mf (ζ), (ME)2 := s | s = M(s)2 .

Unambiguous probe of impact of explicit chiral symmetry

breaking on the mass function

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 16/40

First Contents Back Conclusion

Consituent-quark σ-term

Impact of Dynamical chiral symmetry breaking . . . exhibited

via constituent-quark σ-term

σf := mf (ζ)∂ME

f

∂mf (ζ), (ME)2 := s | s = M(s)2 .

Ratioσf

MEf

=EXPLICIT

EXPLICIT + DYNAMICAL

measures effect of EXPLICIT chiral symmetry breaking on

dressed-quark mass-function

cf. SUM of effects of EXPLICIT AND DYNAMICAL CHIRAL

SYMMETRY BREAKING

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 16/40

First Contents Back Conclusion

Consituent-quark σ-term

Impact of Dynamical chiral symmetry breaking . . . exhibited

via constituent-quark σ-term

σf := mf (ζ)∂ME

f

∂mf (ζ), (ME)2 := s | s = M(s)2 .

10-2

10-1

100

mζ [GeV]

0

0.2

0.4

0.6

0.8

1

σ Q /

ME

mu,d

ms

mc

mb

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 16/40

First Contents Back Conclusion

Consituent-quark σ-term

Impact of Dynamical chiral symmetry breaking . . . exhibited

via constituent-quark σ-term

σf := mf (ζ)∂ME

f

∂mf (ζ), (ME)2 := s | s = M(s)2 .

10-2

10-1

100

mζ [GeV]

0

0.2

0.4

0.6

0.8

1

σ Q /

ME

mu,d

ms

mc

mb

Obvious: ratio vanishes for

light-quarks because

magnitude of their

constituent-mass owes

primarily to DCSB. On the

other hand, for heavy-quarks

it approaches one.

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 16/40

First Contents Back Conclusion

Consituent-quark σ-term

Impact of Dynamical chiral symmetry breaking . . . exhibited

via constituent-quark σ-term

σf := mf (ζ)∂ME

f

∂mf (ζ), (ME)2 := s | s = M(s)2 .

10-2

10-1

100

mζ [GeV]

0

0.2

0.4

0.6

0.8

1

σ Q /

ME

mu,d

ms

mc

mb

Essentially dynamical

component of chiral

symmetry breaking, and

manifestation in all its

order parameters,

vanishes with increasing

current-quark mass

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 16/40

First Contents Back Conclusion

Hadrons

• Established understandingof two- and three-point functions

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 17/40

First Contents Back Conclusion

Hadrons

• Established understandingof two- and three-point functions

• What about bound states?

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 17/40

First Contents Back Conclusion

Hadrons

• Without bound states,Comparison with experiment isimpossible

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 17/40

First Contents Back Conclusion

Hadrons

• Without bound states,Comparison with experiment isimpossible

• They appear as pole contributionsto n ≥ 3-point colour-singletSchwinger functions

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 17/40

First Contents Back Conclusion

Hadrons

• Without bound states,Comparison with experiment isimpossible

• Bethe-Salpeter Equation

QFT Generalisation of Lippmann-Schwinger Equation.

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 17/40

First Contents Back Conclusion

Hadrons

• Without bound states,Comparison with experiment isimpossible

• Bethe-Salpeter Equation

QFT Generalisation of Lippmann-Schwinger Equation.

• What is the kernel, K?

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 17/40

First Contents Back Conclusion

Hadrons

• Without bound states,Comparison with experiment isimpossible

• Bethe-Salpeter Equation

QFT Generalisation of Lippmann-Schwinger Equation.

• What is the kernel, K?

orDynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 17/40

First Contents Back Conclusion

What is the Long-Range Potential?

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 18/40

First Contents Back Conclusion

What is the Long-Range Potential?

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 18/40

First Contents Back Conclusion

Bethe-Salpeter Kernel

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 19/40

First Contents Back Conclusion

Bethe-Salpeter Kernel

Axial-vector Ward-Takahashi identity

Pµ Γl5µ(k;P ) = S−1(k+)

1

2λl

f iγ5 +1

2λl

f iγ5 S−1(k−)

−Mζ iΓl5(k;P ) − iΓl

5(k;P ) Mζ

QFT Statement of Chiral Symmetry

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 19/40

First Contents Back Conclusion

Bethe-Salpeter Kernel

Axial-vector Ward-Takahashi identity

Pµ Γl5µ(k;P ) = S−1(k+)

1

2λl

f iγ5 +1

2λl

f iγ5 S−1(k−)

−Mζ iΓl5(k;P ) − iΓl

5(k;P ) Mζ

Satisfies BSE Satisfies DSE

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 19/40

First Contents Back Conclusion

Bethe-Salpeter Kernel

Axial-vector Ward-Takahashi identity

Pµ Γl5µ(k;P ) = S−1(k+)

1

2λl

f iγ5 +1

2λl

f iγ5 S−1(k−)

−Mζ iΓl5(k;P ) − iΓl

5(k;P ) Mζ

Satisfies BSE Satisfies DSEKernels must be intimately related

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 19/40

First Contents Back Conclusion

Bethe-Salpeter Kernel

Axial-vector Ward-Takahashi identity

Pµ Γl5µ(k;P ) = S−1(k+)

1

2λl

f iγ5 +1

2λl

f iγ5 S−1(k−)

−Mζ iΓl5(k;P ) − iΓl

5(k;P ) Mζ

Satisfies BSE Satisfies DSEKernels must be intimately related

• Relation must be preserved by truncation

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 19/40

First Contents Back Conclusion

Bethe-Salpeter Kernel

Axial-vector Ward-Takahashi identity

Pµ Γl5µ(k;P ) = S−1(k+)

1

2λl

f iγ5 +1

2λl

f iγ5 S−1(k−)

−Mζ iΓl5(k;P ) − iΓl

5(k;P ) Mζ

Satisfies BSE Satisfies DSEKernels must be intimately related

• Relation must be preserved by truncation• Nontrivial constraint

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 19/40

First Contents Back Conclusion

Bethe-Salpeter Kernel

Axial-vector Ward-Takahashi identity

Pµ Γl5µ(k;P ) = S−1(k+)

1

2λl

f iγ5 +1

2λl

f iγ5 S−1(k−)

−Mζ iΓl5(k;P ) − iΓl

5(k;P ) Mζ

Satisfies BSE Satisfies DSEKernels must be intimately related

• Relation must be preserved by truncation• Failure ⇒ Explicit Violation of QCD’s Chiral Symmetry

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 19/40

First Contents Back Conclusion

Pion and . . .Pseudoscalar Mesons?

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 20/40

First Contents Back Conclusion

Pion and . . .Pseudoscalar Mesons?

Can a bound-state of massive constituents truly bemassless . . . without fine-tuning?

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 20/40

First Contents Back Conclusion

Radial Excitations& Chiral Symmetry

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 21/40

First Contents Back Conclusion

Radial Excitations& Chiral Symmetry(Maris, Roberts, Tandy

nu-th/9707003 )

fH m2H = − ρH

ζ MH

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 21/40

First Contents Back Conclusion

Radial Excitations& Chiral Symmetry(Maris, Roberts, Tandy

nu-th/9707003 )

fH m2H = − ρH

ζ MH

• Mass2 of pseudoscalar hadron

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 21/40

First Contents Back Conclusion

Radial Excitations& Chiral Symmetry(Maris, Roberts, Tandy

nu-th/9707003 )

fH m2H = − ρH

ζ MH

MH := trflavour

[

M (µ)

{

TH ,(

TH)t}]

= mq1+mq2

• Sum of constituents’ current-quark masses

• e.g., TK+

= 12

(

λ4 + iλ5)

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 21/40

First Contents Back Conclusion

Radial Excitations& Chiral Symmetry(Maris, Roberts, Tandy

nu-th/9707003 )

fH m2H = − ρH

ζ MH

fH pµ = Z2

∫ Λ

q

12tr{

(

TH)t

γ5γµ S(q+)ΓH(q;P )S(q−)}

• Pseudovector projection of BS wave function at x = 0

• Pseudoscalar meson’s leptonic decay constant

i

i

i

iAµπ kµ

πf

k

Γ

S

(τ/2)γµ γ

S

555

=

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 21/40

First Contents Back Conclusion

Radial Excitations& Chiral Symmetry(Maris, Roberts, Tandy

nu-th/9707003 )

fH m2H = − ρH

ζ MH

iρHζ = Z4

∫ Λ

q

12tr{

(

TH)t

γ5 S(q+)ΓH(q;P )S(q−)}

• Pseudoscalar projection of BS wave function at x = 0

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 21/40

First Contents Back Conclusion

Radial Excitations& Chiral Symmetry(Maris, Roberts, Tandy

nu-th/9707003 )

fH m2H = − ρH

ζ MH

Light-quarks; i.e., mq ∼ 0

fH → f0H & ρH

ζ →−〈qq〉0ζ

f0H

, Independent of mq

Hence m2H =

−〈qq〉0ζ(f0

H)2mq . . . GMOR relation, a corollary

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 21/40

First Contents Back Conclusion

Radial Excitations& Chiral Symmetry(Maris, Roberts, Tandy

nu-th/9707003 )

fH m2H = − ρH

ζ MH

Light-quarks; i.e., mq ∼ 0

fH → f0H & ρH

ζ →−〈qq〉0ζ

f0H

, Independent of mq

Hence m2H =

−〈qq〉0ζ(f0

H)2mq . . . GMOR relation, a corollary

Heavy-quark + light-quark

⇒ fH ∝ 1√

mH

and ρHζ ∝ √

mH

Hence, mH ∝ mq

. . . QCD Proof of Potential Model resultDynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 21/40

First Contents Back Conclusion

Andreas Krassnigg

FWF “ErwinSchrödinger Fellow,”ANL 2003-2005

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 22/40

First Contents Back Conclusion

Andreas Krassnigg

Almost BloodRelative of Arnold. . . Future

President?. . . Executioner?

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 22/40

First Contents Back Conclusion

Radial Excitations& Chiral SymmetryHöll, Krassnigg, Roberts

nu-th/0406030

fH m2H = − ρH

ζ MH

Valid for ALL Pseudoscalar mesons

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 23/40

First Contents Back Conclusion

Radial Excitations& Chiral SymmetryHöll, Krassnigg, Roberts

nu-th/0406030

fH m2H = − ρH

ζ MH

Valid for ALL Pseudoscalar mesons

ρH ⇒ finite, nonzero value in chiral limit, MH → 0

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 23/40

First Contents Back Conclusion

Radial Excitations& Chiral SymmetryHöll, Krassnigg, Roberts

nu-th/0406030

fH m2H = − ρH

ζ MH

Valid for ALL Pseudoscalar mesons

ρH ⇒ finite, nonzero value in chiral limit, MH → 0

“radial” excitation of π-meson,

m2πn 6=0

> m2πn=0

= 0, in chiral limit

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 23/40

First Contents Back Conclusion

Radial Excitations& Chiral SymmetryHöll, Krassnigg, Roberts

nu-th/0406030

fH m2H = − ρH

ζ MH

Valid for ALL Pseudoscalar mesons

ρH ⇒ finite, nonzero value in chiral limit, MH → 0

“radial” excitation of π-meson,

m2πn 6=0

> m2πn=0

= 0, in chiral limit

⇒ fH = 0

ALL pseudoscalar mesons except π(140) in chiral limit

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 23/40

First Contents Back Conclusion

Radial Excitations& Chiral SymmetryHöll, Krassnigg, Roberts

nu-th/0406030

fH m2H = − ρH

ζ MH

Valid for ALL Pseudoscalar mesons

ρH ⇒ finite, nonzero value in chiral limit, MH → 0

“radial” excitation of π-meson,

m2πn 6=0

> m2πn=0

= 0, in chiral limit

⇒ fH = 0

ALL pseudoscalar mesons except π(140) in chiral limit

Dynamical Chiral Symmetry Breaking

– Goldstone’s Theorem –

impacts upon every pseudoscalar mesonDynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 23/40

First Contents Back Conclusion

Radial Excitations& Lattice-QCDMcNeile and Michael

he-la/0607032

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 24/40

First Contents Back Conclusion

Radial Excitations& Lattice-QCDMcNeile and Michael

he-la/0607032

When we first heard about [this result] our first reaction was acombination of “that is remarkable” and “unbelievable”.

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 24/40

First Contents Back Conclusion

Radial Excitations& Lattice-QCDMcNeile and Michael

he-la/0607032

When we first heard about [this result] our first reaction was acombination of “that is remarkable” and “unbelievable”.

CLEO: τ → π(1300) + ντ

⇒ fπ1< 8.4 MeV

Diehl & Hillerhe-ph/0105194

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 24/40

First Contents Back Conclusion

Radial Excitations& Lattice-QCDMcNeile and Michael

he-la/0607032

0 0.5 1 1.5 2 2.5 3 3.5 4

( r0 mπ )

2

0

0.2

0.4

0.6

0.8

f π’/f

πnot improvedNP improvedExpt. bound

When we first heard about [this result] our first reaction was acombination of “that is remarkable” and “unbelievable”.

CLEO: τ → π(1300) + ντ

⇒ fπ1< 8.4 MeV

Diehl & Hillerhe-ph/0105194

Lattice-QCD check:163 × 32,a ∼ 0.1 fm,two-flavour, unquenched

⇒ fπ1

fπ

= 0.078 (93)

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 24/40

First Contents Back Conclusion

Radial Excitations& Lattice-QCDMcNeile and Michael

he-la/0607032

0 0.5 1 1.5 2 2.5 3 3.5 4

( r0 mπ )

2

0

0.2

0.4

0.6

0.8

f π’/f

πnot improvedNP improvedExpt. bound

When we first heard about [this result] our first reaction was acombination of “that is remarkable” and “unbelievable”.

CLEO: τ → π(1300) + ντ

⇒ fπ1< 8.4 MeV

Diehl & Hillerhe-ph/0105194

Lattice-QCD check:163 × 32,a ∼ 0.1 fm,two-flavour, unquenched

⇒ fπ1

fπ

= 0.078 (93)

Full ALPHA formulation is required to see suppression, becausePCAC relation is at the heart of the conditions imposed forimprovement (determining coefficients of irrelevant operators)Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 24/40

First Contents Back Conclusion

Radial Excitations& Lattice-QCDMcNeile and Michael

he-la/0607032

0 0.5 1 1.5 2 2.5 3 3.5 4

( r0 mπ )

2

0

0.2

0.4

0.6

0.8

f π’/f

πnot improvedNP improvedExpt. bound

When we first heard about [this result] our first reaction was acombination of “that is remarkable” and “unbelievable”.

CLEO: τ → π(1300) + ντ

⇒ fπ1< 8.4 MeV

Diehl & Hillerhe-ph/0105194

Lattice-QCD check:163 × 32,a ∼ 0.1 fm,two-flavour, unquenched

⇒ fπ1

fπ

= 0.078 (93)

The suppression of fπ1is a useful benchmark that can be used to

tune and validate lattice QCD techniques that try to determine theproperties of excited states mesons.Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 24/40

First Contents Back Conclusion

Radial Excitations

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 25/40

First Contents Back Conclusion

Radial Excitations

Spectrum contains 3 pseudoscalars [IG(JP )L = 1−(0−)S]

masses below 2 GeV: π(140) ; π(1300); and π(1800)

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 25/40

First Contents Back Conclusion

Radial Excitations

Spectrum contains 3 pseudoscalars [IG(JP )L = 1−(0−)S]

masses below 2 GeV: π(140) ; π(1300); and π(1800)

The Pion

Consituent-Q Model: 1st three members ofn 1S0 trajectory; i.e., ground state plus radial excitations?

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 25/40

First Contents Back Conclusion

Radial Excitations

Spectrum contains 3 pseudoscalars [IG(JP )L = 1−(0−)S]

masses below 2 GeV: π(140) ; π(1300); and π(1800)

The Pion

Consituent-Q Model: 1st three members ofn 1S0 trajectory; i.e., ground state plus radial excitations?

But π(1800) is narrow (Γ = 207 ± 13) & decay pattern mightindicate some “flux tube angular momentum” content:SQQ = 1 ⊕ LF = 1 ⇒ J = 0

& LF = 1 ⇒ 3S1 ⊕ 3S1 (QQ) decays suppressed?

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 25/40

First Contents Back Conclusion

Radial Excitations

Spectrum contains 3 pseudoscalars [IG(JP )L = 1−(0−)S]

masses below 2 GeV: π(140) ; π(1300); and π(1800)

The Pion

Consituent-Q Model: 1st three members ofn 1S0 trajectory; i.e., ground state plus radial excitations?

But π(1800) is narrow (Γ = 207 ± 13) & decay pattern mightindicate some “flux tube angular momentum” content:

Radial excitations & Hybrids & Exotics ⇒ Long-range radial wavefunctions ⇒ sensitive to confinement

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 25/40

First Contents Back Conclusion

Radial Excitations

Spectrum contains 3 pseudoscalars [IG(JP )L = 1−(0−)S]

masses below 2 GeV: π(140) ; π(1300); and π(1800)

The Pion

Consituent-Q Model: 1st three members ofn 1S0 trajectory; i.e., ground state plus radial excitations?

But π(1800) is narrow (Γ = 207 ± 13) & decay pattern mightindicate some “flux tube angular momentum” content:

Radial excitations & Hybrids & Exotics ⇒ Long-range radial wavefunctions ⇒ sensitive to confinement

NSAC Long-Range Plan, 2002: . . . an understanding ofconfinement “remains one of the

greatest intellectual challenges in physics”Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 25/40

First Contents Back Conclusion

Pion . . . J = 0

but . . .Orbital angular momentum is not a Poincaré invariant.

However, if absent in a particular frame, it will appear in

another frame related via a Poincaré transformation.

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 26/40

First Contents Back Conclusion

Pion . . . J = 0

but . . .Nonzero quark orbital angular momentum is thus a

necessary outcome of a Poincaré covariant description.

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 26/40

First Contents Back Conclusion

Pion . . . J = 0

but . . .Pseudoscalar meson Bethe-Salpeter amplitude

χπ(k;P ) = γ5 [iEπn(k;P ) + γ · PFπn

(k;P )

γ · k k · P Gπn(k;P ) + σµν kµPν Hπn

(k;P )]

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 26/40

First Contents Back Conclusion

Pion . . . J = 0

but . . .Pseudoscalar meson Bethe-Salpeter amplitude

χπ(k;P ) = γ5 [iEπn(k;P ) + γ · PFπn

(k;P )

γ · k k · P Gπn(k;P ) + σµν kµPν Hπn

(k;P )]

J = 0 . . . but while E and F are purely L = 0 in the rest

frame, the G and H terms are associated with L = 1. Thus a

pseudoscalar meson Bethe-Salpeter wave function always

contains both S- and P -wave components.

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 26/40

First Contents Back Conclusion

Pion . . . J = 0

but . . .J = 0 . . . but while E and F are purely L = 0 in the rest

frame, the G and H terms are associated with L = 1. Thus a

pseudoscalar meson Bethe-Salpeter wave function always

contains both S- and P -wave components.Introduce mixing

angle θπ such that

χπ ∼ cos θπ|L = 0〉+ sin θπ|L = 1〉

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 26/40

First Contents Back Conclusion

Pion . . . J = 0

but . . .J = 0 . . . but while E and F are purely L = 0 in the rest

frame, the G and H terms are associated with L = 1. Thus a

pseudoscalar meson Bethe-Salpeter wave function always

contains both S- and P -wave components.Introduce mixing

angle θπ such that

χπ ∼ cos θπ|L = 0〉+ sin θπ|L = 1〉

0 2 4 6 8 100

-+n=0 meson mass (GeV)

0

10

20

30

θ π n=0

(d

egre

es)

1.2 1.6 2 2.40

-+n meson mass (GeV)

8

12

16

θ π n (

deg

rees

)

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 26/40

First Contents Back Conclusion

Pion . . . J = 0

but . . .J = 0 . . . but while E and F are purely L = 0 in the rest

frame, the G and H terms are associated with L = 1. Thus a

pseudoscalar meson Bethe-Salpeter wave function always

contains both S- and P -wave components.Introduce mixing

angle θπ such that

χπ ∼ cos θπ|L = 0〉+ sin θπ|L = 1〉

0 2 4 6 8 100

-+n=0 meson mass (GeV)

0

10

20

30

θ π n=0

(d

egre

es)

1.2 1.6 2 2.40

-+n meson mass (GeV)

8

12

16

θ π n (

deg

rees

)

L is significant in

the neighbourhood

of the chiral limit,

and decreases with

increasing

current-quark mass.Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 26/40

First Contents Back Conclusion

New Challenges

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 27/40

First Contents Back Conclusion

New Challenges

Next Steps . . . Applications to excited states and

axial-vector mesons, e.g., will improve understanding of

confinement interaction between light-quarks.

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 27/40

First Contents Back Conclusion

New Challenges

Next Steps . . . Applications to excited states and

axial-vector mesons, e.g., will improve understanding of

confinement interaction between light-quarks.

Move on to the problem of a symmetry preserving treatment

of hybrids and exotics.

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 27/40

First Contents Back Conclusion

New Challenges

Another Direction . . . Also want/need information about

three-quark systems

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 27/40

First Contents Back Conclusion

New Challenges

Another Direction . . . Also want/need information about

three-quark systems

With this problem . . . current expertise at approximately

same point as studies of mesons in 1995.

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 27/40

First Contents Back Conclusion

New Challenges

Another Direction . . . Also want/need information about

three-quark systems

With this problem . . . current expertise at approximately

same point as studies of mesons in 1995.

Namely . . . Model-building and Phenomenology,

constrained by the DSE results outlined already.Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 27/40

First Contents Back Conclusion

Nucleon EM Form Factors: A PrécisHoll, Kloker, et al. : nu-th/0412046 & nu-th/0501033

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 28/40

First Contents Back Conclusion

Nucleon EM Form Factors: A PrécisHoll, Kloker, et al. : nu-th/0412046 & nu-th/0501033

• Interpreting expts. with GeV electromagnetic probes

requires Poincaré covariant treatment of baryons

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 28/40

First Contents Back Conclusion

Nucleon EM Form Factors: A PrécisHoll, Kloker, et al. : nu-th/0412046 & nu-th/0501033

• Interpreting expts. with GeV electromagnetic probes

requires Poincaré covariant treatment of baryons

⇒ Covariant dressed-quark Faddeev Equation

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 28/40

First Contents Back Conclusion

Nucleon EM Form Factors: A PrécisHoll, Kloker, et al. : nu-th/0412046 & nu-th/0501033

• Interpreting expts. with GeV electromagnetic probes

requires Poincaré covariant treatment of baryons

⇒ Covariant dressed-quark Faddeev Equation

• Excellent mass spectrum (octet and decuplet)

Easily obtained:(

1

NH

∑

H

[M expH − M calc

H ]2

[M expH ]2

)1/2

= 2%

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 28/40

First Contents Back Conclusion

Nucleon EM Form Factors: A PrécisHoll, Kloker, et al. : nu-th/0412046 & nu-th/0501033

• Interpreting expts. with GeV electromagnetic probes

requires Poincaré covariant treatment of baryons

⇒ Covariant dressed-quark Faddeev Equation

• Excellent mass spectrum (octet and decuplet)

Easily obtained:(

1

NH

∑

H

[M expH − M calc

H ]2

[M expH ]2

)1/2

= 2%

(Oettel, Hellstern, Alkofer, Reinhardt: nucl-th/9805054)

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 28/40

First Contents Back Conclusion

Nucleon EM Form Factors: A PrécisHoll, Kloker, et al. : nu-th/0412046 & nu-th/0501033

• Interpreting expts. with GeV electromagnetic probes

requires Poincaré covariant treatment of baryons

⇒ Covariant dressed-quark Faddeev Equation

• Excellent mass spectrum (octet and decuplet)

Easily obtained:(

1

NH

∑

H

[M expH − M calc

H ]2

[M expH ]2

)1/2

= 2%

• But is that good?

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 28/40

First Contents Back Conclusion

Nucleon EM Form Factors: A PrécisHoll, Kloker, et al. : nu-th/0412046 & nu-th/0501033

• Interpreting expts. with GeV electromagnetic probes

requires Poincaré covariant treatment of baryons

⇒ Covariant dressed-quark Faddeev Equation

• Excellent mass spectrum (octet and decuplet)

Easily obtained:(

1

NH

∑

H

[M expH − M calc

H ]2

[M expH ]2

)1/2

= 2%

• But is that good?

• Cloudy Bag: δMπ−loop+ = −300 to −400 MeV!

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 28/40

First Contents Back Conclusion

Nucleon EM Form Factors: A PrécisHoll, Kloker, et al. : nu-th/0412046 & nu-th/0501033

• Interpreting expts. with GeV electromagnetic probes

requires Poincaré covariant treatment of baryons

⇒ Covariant dressed-quark Faddeev Equation

• Excellent mass spectrum (octet and decuplet)

Easily obtained:(

1

NH

∑

H

[M expH − M calc

H ]2

[M expH ]2

)1/2

= 2%

• But is that good?

• Cloudy Bag: δMπ−loop+ = −300 to −400 MeV!

• Critical to anticipate pion cloud effects

Roberts, Tandy, Thomas, et al., nu-th/02010084

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 28/40

First Contents Back Conclusion

Faddeev equation

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 29/40

First Contents Back Conclusion

Faddeev equation

=aΨ

P

pq

pd Γb

Γ−a

pd

pq

bΨP

q

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 29/40

First Contents Back Conclusion

Faddeev equation

=aΨ

P

pq

pd Γb

Γ−a

pd

pq

bΨP

q

Linear, Homogeneous Matrix equation

Yields wave function (Poincaré Covariant Faddeev

Amplitude) that describes quark-diquark relative motion

within the nucleon

Scalar and Axial-Vector Diquarks . . . In Nucleon’s Rest

Frame Amplitude has . . . s−, p− & d−wave correlations

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 29/40

First Contents Back Conclusion

Diquark correlations

QUARK-QUARKDynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 30/40

First Contents Back Conclusion

Diquark correlations

QUARK-QUARK

Same interaction that

describes mesons also

generates three coloured

quark-quark correlations:

blue–red, blue–green,

green–red

Confined . . . Does not

escape from within baryon.

Scalar is isosinglet,

Axial-vector is isotriplet

DSE and lattice-QCD

m[ud]0+

= 0.74 − 0.82

m(uu)1+

= m(ud)1+

= m(dd)1+

= 0.95 − 1.02

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 30/40

First Contents Back Conclusion

Harry LeePions and Form Factors

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 31/40

First Contents Back Conclusion

Harry LeePions and Form Factors

Dynamical coupled-channels model . . . Analyzed extensive JLabdata . . . Completed a study of the ∆(1236)

Meson Exchange Model for πN Scattering and γN → πN Reaction, T. Satoand T.-S. H. Lee, Phys. Rev. C 54, 2660 (1996)

Dynamical Study of the ∆ Excitation in N(e, e′π) Reactions, T. Sato andT.-S. H. Lee, Phys. Rev. C 63, 055201/1-13 (2001)

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 31/40

First Contents Back Conclusion

Harry LeePions and Form Factors

Dynamical coupled-channels model . . . Analyzed extensive JLabdata . . . Completed a study of the ∆(1236)

Meson Exchange Model for πN Scattering and γN → πN Reaction, T. Satoand T.-S. H. Lee, Phys. Rev. C 54, 2660 (1996)

Dynamical Study of the ∆ Excitation in N(e, e′π) Reactions, T. Sato andT.-S. H. Lee, Phys. Rev. C 63, 055201/1-13 (2001)

Pion cloud effects are large in the low Q2 region.

0

1

2

3

0 1 2 3 4

Q2(GeV/c)2

DressedBare

Ratio of the M1 form factor in γN → ∆

transition and proton dipole form factor GD .Solid curve is G∗

M(Q2)/GD(Q2) including

pions; Dotted curve is GM (Q2)/GD(Q2)

without pions.

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 31/40

First Contents Back Conclusion

Harry LeePions and Form Factors

Dynamical coupled-channels model . . . Analyzed extensive JLabdata . . . Completed a study of the ∆(1236)

Meson Exchange Model for πN Scattering and γN → πN Reaction, T. Satoand T.-S. H. Lee, Phys. Rev. C 54, 2660 (1996)

Dynamical Study of the ∆ Excitation in N(e, e′π) Reactions, T. Sato andT.-S. H. Lee, Phys. Rev. C 63, 055201/1-13 (2001)

Pion cloud effects are large in the low Q2 region.

0

1

2

3

0 1 2 3 4

Q2(GeV/c)2

DressedBare

Ratio of the M1 form factor in γN → ∆

transition and proton dipole form factor GD .Solid curve is G∗

M(Q2)/GD(Q2) including

pions; Dotted curve is GM (Q2)/GD(Q2)

without pions.

Quark Core

Responsible for only 2/3 ofresult at small Q2

Dominant for Q2 >2 – 3 GeV2

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 31/40

First Contents Back Conclusion

Results: Nucleonand ∆ Masses

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 32/40

First Contents Back Conclusion

Results: Nucleonand ∆ Masses

Mass-scale parameters (in GeV)

for the scalar and axial-vector

diquark correlations, fixed by

fitting nucleon and ∆ masses

Set A – fit to the actual masses was required; whereas for

Set B – fitted mass was offset to allow for “π-cloud” contributions

set MN M∆ m0+ m1+ ω0+ ω1+

A 0.94 1.23 0.63 0.84 0.44=1/(0.45 fm) 0.59=1/(0.33 fm)

B 1.18 1.33 0.79 0.89 0.56=1/(0.35 fm) 0.63=1/(0.31 fm)

m1+ → ∞: MAN = 1.15 GeV; MB

N = 1.46 GeV

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 32/40

First Contents Back Conclusion

Results: Nucleonand ∆ Masses

Mass-scale parameters (in GeV)

for the scalar and axial-vector

diquark correlations, fixed by

fitting nucleon and ∆ masses

Set A – fit to the actual masses was required; whereas for

Set B – fitted mass was offset to allow for “π-cloud” contributions

set MN M∆ m0+ m1+ ω0+ ω1+

A 0.94 1.23 0.63 0.84 0.44=1/(0.45 fm) 0.59=1/(0.33 fm)

B 1.18 1.33 0.79 0.89 0.56=1/(0.35 fm) 0.63=1/(0.31 fm)

m1+ → ∞: MAN = 1.15 GeV; MB

N = 1.46 GeV

Axial-vector diquark provides significant attractionDynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 32/40

First Contents Back Conclusion

Results: Nucleonand ∆ Masses

Mass-scale parameters (in GeV)

for the scalar and axial-vector

diquark correlations, fixed by

fitting nucleon and ∆ masses

Set A – fit to the actual masses was required; whereas for

Set B – fitted mass was offset to allow for “π-cloud” contributions

set MN M∆ m0+ m1+ ω0+ ω1+

A 0.94 1.23 0.63 0.84 0.44=1/(0.45 fm) 0.59=1/(0.33 fm)

B 1.18 1.33 0.79 0.89 0.56=1/(0.35 fm) 0.63=1/(0.31 fm)

m1+ → ∞: MAN = 1.15 GeV; MB

N = 1.46 GeV

Constructive Interference: 1++-diquark + ∂µπDynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 32/40

First Contents Back Conclusion

Angular MomentumRest Frame

M. Oettel, et al.nucl-th/9805054Crude estimate based on

magnitudes ⇒ probability for a

u-quark to carry the proton’s

spin is Pu↑ ∼ 80 %, with

Pu↓ ∼ 5 %, Pd↑ ∼ 5 %,

Pd↓ ∼ 10 %.

Hence, by this reckoning ∼ 30%

of proton’s rest-frame spin is

located in dressed-quark

angular momentum.

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 33/40

First Contents Back Conclusion

Nucleon-Photon Vertex

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 34/40

First Contents Back Conclusion

Nucleon-Photon Vertex

M. Oettel, M. Pichowskyand L. von Smekal, nu-th/9909082

6 terms . . .constructed systematically . . . current conserved automatically

for on-shell nucleons described by Faddeev Amplitude

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 34/40

First Contents Back Conclusion

Nucleon-Photon Vertex

M. Oettel, M. Pichowskyand L. von Smekal, nu-th/9909082

6 terms . . .constructed systematically . . . current conserved automatically

for on-shell nucleons described by Faddeev Amplitude

i

iΨ ΨPf

f

P

Q i

iΨ ΨPf

f

P

Q

i

iΨ ΨPPf

f

Q

Γ−

Γ

scalaraxial vector

i

iΨ ΨPf

f

P

Q

µ

i

i

X

Ψ ΨPf

f

Q

P Γ−

µi

i

X−

Ψ ΨPf

f

P

Q

ΓDynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 34/40

First Contents Back Conclusion

Form Factor Ratio:GE/GM

0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp

Rosenbluthprecision Rosenbluthpolarization transferpolarization transfer

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 35/40

First Contents Back Conclusion

Form Factor Ratio:GE/GMCombine these elements . . .

0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp

Rosenbluthprecision Rosenbluthpolarization transferpolarization transfer

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 35/40

First Contents Back Conclusion

Form Factor Ratio:GE/GMCombine these elements . . .

Dressed-Quark Core

0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp

Rosenbluthprecision Rosenbluthpolarization transferpolarization transfer

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 35/40

First Contents Back Conclusion

Form Factor Ratio:GE/GMCombine these elements . . .

Dressed-Quark Core

Ward-Takahashi

Identity preserving

current

0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp

Rosenbluthprecision Rosenbluthpolarization transferpolarization transfer

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 35/40

First Contents Back Conclusion

Form Factor Ratio:GE/GMCombine these elements . . .

Dressed-Quark Core

Ward-Takahashi

Identity preserving

current

Anticipate and

Estimate Pion

Cloud’s Contribution

0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp

Rosenbluthprecision Rosenbluthpolarization transferpolarization transfer

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 35/40

First Contents Back Conclusion

Form Factor Ratio:GE/GMCombine these elements . . .

Dressed-Quark Core

Ward-Takahashi

Identity preserving

current

Anticipate and

Estimate Pion

Cloud’s Contribution

0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp

covariant Fadeev resultRosenbluthprecision Rosenbluthpolarization transferpolarization transfer

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 35/40

First Contents Back Conclusion

Form Factor Ratio:GE/GMCombine these elements . . .

Dressed-Quark Core

Ward-Takahashi

Identity preserving

current

Anticipate and

Estimate Pion

Cloud’s Contribution

0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp

covariant Fadeev resultRosenbluthprecision Rosenbluthpolarization transferpolarization transfer

All parameters fixed in

other applications . . . Not varied.

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 35/40

First Contents Back Conclusion

Form Factor Ratio:GE/GMCombine these elements . . .

Dressed-Quark Core

Ward-Takahashi

Identity preserving

current

Anticipate and

Estimate Pion

Cloud’s Contribution

0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp

covariant Fadeev resultRosenbluthprecision Rosenbluthpolarization transferpolarization transfer

All parameters fixed in

other applications . . . Not varied.

Agreement with Pol. Trans. data at Q2 ∼> 2 GeV2

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 35/40

First Contents Back Conclusion

Form Factor Ratio:GE/GMCombine these elements . . .

Dressed-Quark Core

Ward-Takahashi

Identity preserving

current

Anticipate and

Estimate Pion

Cloud’s Contribution

0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp

covariant Fadeev resultRosenbluthprecision Rosenbluthpolarization transferpolarization transfer

All parameters fixed in

other applications . . . Not varied.

Agreement with Pol. Trans. data at Q2 ∼> 2 GeV2

Correlations in Faddeev amplitude – quark orbital

angular momentum – essential to that agreement

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 35/40

First Contents Back Conclusion

Form Factor Ratio:GE/GMCombine these elements . . .

Dressed-Quark Core

Ward-Takahashi

Identity preserving

current

Anticipate and

Estimate Pion

Cloud’s Contribution

0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp

covariant Fadeev resultRosenbluthprecision Rosenbluthpolarization transferpolarization transfer

All parameters fixed in

other applications . . . Not varied.

Agreement with Pol. Trans. data at Q2 ∼> 2 GeV2

Correlations in Faddeev amplitude – quark orbital

angular momentum – essential to that agreement

Predict Zero at Q2 ≈ 6.5GeV2

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 35/40

First Contents Back Conclusion

Neutron Form Factors

0 2 4 6 8Q

2 [GeV2]

0

0.1

0.2

0.3

0.4

0.5

µ n GEn/ G

Mn

nucl-ex 0308007

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 36/40

First Contents Back Conclusion

Neutron Form Factors

0 2 4 6 8Q

2 [GeV2]

0

0.1

0.2

0.3

0.4

0.5

µ n GEn/ G

Mn

nucl-ex 0308007

Expt. Madey, et

al. nu-ex/0308007

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 36/40

First Contents Back Conclusion

Neutron Form Factors

0 2 4 6 8Q

2 [GeV2]

0

0.1

0.2

0.3

0.4

0.5

µ n GEn/ G

Mn

nucl-ex 0308007

Expt. Madey, et

al. nu-ex/0308007

Calc. Bhagwat, et

al. nu-th/0610080

µp

GnE(Q2)

GnM(Q2)

= −r2n

6Q2

Valid for r2nQ2 ∼< 1

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 36/40

First Contents Back Conclusion

Neutron Form Factors

0 2 4 6 8Q

2 [GeV2]

0

0.1

0.2

0.3

0.4

0.5

µ n GEn/ G

Mn

nucl-ex 0308007

Expt. Madey, et

al. nu-ex/0308007

Calc. Bhagwat, et

al. nu-th/0610080

µp

GnE(Q2)

GnM(Q2)

= −r2n

6Q2

Valid for r2nQ2 ∼< 1

No sign yet of a zero in GnE(Q2), even though calculation

predicts GpE(Q2 ≈ 6.5 GeV2) = 0

Data to Q2 = 3.4 GeV2 is being analysed (JLab E02-013)Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 36/40

First Contents Back Conclusion

Epilogue

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 37/40

First Contents Back Conclusion

Epilogue

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 37/40

First Contents Back Conclusion

Epilogue

DCSB exists in QCD.

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 37/40

First Contents Back Conclusion

Epilogue

DCSB exists in QCD.

It is manifest in the dressed light-quark propagator.

It impacts dramatically upon observables.

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 37/40

First Contents Back Conclusion

Epilogue

DCSB exists in QCD.

It is manifest in the dressed light-quark propagator.

It impacts dramatically upon observables.

Confinement

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 37/40

First Contents Back Conclusion

Epilogue

DCSB exists in QCD.

It is manifest in the dressed light-quark propagator.

It impacts dramatically upon observables.

Confinement

Can be realised in dressed propagators of elementary

excitations

Observables can be used to explore model realisations

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 37/40

First Contents Back Conclusion

Epilogue

DCSB exists in QCD.

It is manifest in the dressed light-quark propagator.

It impacts dramatically upon observables.

Confinement

Can be realised in dressed propagators of elementary

excitations

Observables can be used to explore model realisations

An excellent way to test conjectures and constrain the

possibilities

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 37/40

First Contents Back Conclusion

Epilogue

DCSB exists in QCD.

It is manifest in the dressed light-quark propagator.

It impacts dramatically upon observables.

Confinement

Can be realised in dressed propagators of elementary

excitations

Observables can be used to explore model realisations

An excellent way to test conjectures and constrain the

possibilities

Physics is an Experimental science

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 37/40

First Contents Back Conclusion

Contents

1. Dichotomy of the Pion

2. Dyson-Schwinger Equations

3. Persistent Challenge

4. Dressed-Quark Propagator

5. Quenched-QCD Dressed-Quark

6. Hadrons

7. Bethe-Salpeter Kernel

8. Excitations& Chiral Symmetry

9. Radial Excitations

10. Radial Excitations& Lattice-QCD

11. Pion J = 0

12. Nucleon EM Form Factors

13. Faddeev equation

14. Diquark correlations

15. Pions and Form Factors

16. Results: Nucleon & ∆ Masses

17. Angular Momentum

18. Nucleon-Photon Vertex

19. Form Factor Ratio: GE/GM

20. Neutron Form Factors

21. Parametrising diquark properties

22. Contemporary Reviews

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 38/40

First Contents Back Conclusion

Parametrisingdiquark properties

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 39/40

First Contents Back Conclusion

Parametrisingdiquark properties

Dressed-quark . . . fixed by DSE and Meson Studies

. . . Burden, Roberts, Thomson, Phys. Lett. B 371, 163 (1996)

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 39/40

First Contents Back Conclusion

Parametrisingdiquark properties

Dressed-quark . . . fixed by DSE and Meson Studies

. . . Burden, Roberts, Thomson, Phys. Lett. B 371, 163 (1996)

Non-pointlike scalar and pseudovector colour-antitriplet

diquark correlations – described by

Bethe-Salpeter amplitudes . . . width for each – ωJP

Confining propagators . . . mass for each – mJP

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 39/40

First Contents Back Conclusion

Parametrisingdiquark properties

Dressed-quark . . . fixed by DSE and Meson Studies

. . . Burden, Roberts, Thomson, Phys. Lett. B 371, 163 (1996)

Non-pointlike scalar and pseudovector colour-antitriplet

diquark correlations – described by

Bethe-Salpeter amplitudes . . . width for each – ωJP

Confining propagators . . . mass for each – mJP

Widths fixed by “asymptotic freedom” condition –

d

dK2

(

1

m2JP

F(K2/ω2JP )

)−1∣

∣

∣

∣

∣

∣

K2=0

= 1 ⇒ ω2JP =

1

2m2

JP ,

Only two parameters; viz., diquark “masses”: mJP

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 39/40

First Contents Back Conclusion

Contemporary Reviews

Dyson-Schwinger Equations: Density, Temperature andContinuum Strong QCDC.D. Roberts and S.M. Schmidt, nu-th/0005064,Prog. Part. Nucl. Phys. 45 (2000) S1

The IR behavior of QCD Green’s functions: Confinement, DCSB,and hadrons . . .R. Alkofer and L. von Smekal, he-ph/0007355,Phys. Rept. 353 (2001) 281

Dyson-Schwinger equations: A Tool for Hadron PhysicsP. Maris and C.D. Roberts, nu-th/0301049,Int. J. Mod. Phys. E 12 (2003) pp. 297-365

Infrared properties of QCD from Dyson-Schwinger equations.C. S. Fischer, he-ph/0605173,J. Phys. G 32 (2006) pp. R253-R291

Dynamical Chiral Symmetry Breaking and Hadron Structure

Strong Dynamics and Dynamical Chiral Symmetry Breaking, 4-5 June/07, ANL– p. 40/40