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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
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