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First Contents Back Conclusion
Dyson-Schwinger Equations– Theory and Phenomenology
Craig D. Roberts
cdroberts@anl.gov
Physics Division
Argonne National Laboratory
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 1/52
First Contents Back Conclusion
Chicago from Space
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 2/52
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Argonne National Laboratory
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 3/52
First Contents Back Conclusion
Argonne National Laboratory
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 3/52
First Contents Back Conclusion
Dichotomy of the Pion
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 4/52
First Contents Back Conclusion
Dichotomy of the Pion
How does one make an almost massless particle. . . . . . . . . . . from two massive constituent-quarks?
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 4/52
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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
Must exhibit m2
π ∝ mq
Current Algebra . . . 1968
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 4/52
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
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.
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 4/52
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
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 4/52
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
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.IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 4/52
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Dyson-Schwinger Equations
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 5/52
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Dyson-Schwinger Equations
A Modern Method for Relativistic Quantum Field Theory
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 5/52
First Contents Back Conclusion
Dyson-Schwinger Equations
A Modern Method for Relativistic Quantum Field Theory
Simplest level: Generating Tool for Perturbation Theory
. . . . . . . . . . . . . . . . . . . . Materially Reduces Model Dependence
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 5/52
First Contents Back Conclusion
Dyson-Schwinger Equations
A Modern Method for Relativistic Quantum Field Theory
Simplest level: Generating Tool for Perturbation Theory
. . . . . . . . . . . . . . . . . . . . Materially Reduces Model Dependence
NonPerturbative, Continuum approach to QCD
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 5/52
First Contents Back Conclusion
Dyson-Schwinger Equations
A Modern Method for 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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 5/52
First Contents Back Conclusion
Dyson-Schwinger Equations
A Modern Method for 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
· Quark & Gluon Confinement
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 5/52
First Contents Back Conclusion
Dyson-Schwinger Equations
A Modern Method for 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
· Quark & Gluon Confinement
⇒ Understanding InfraRed (long-range)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . behaviour of αs(Q2)
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 5/52
First Contents Back Conclusion
Dyson-Schwinger Equations
A Modern Method for 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
· Quark & Gluon Confinement
⇒ Understanding InfraRed (long-range)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . behaviour of αs(Q2)
Method yields Schwinger Functions ≡ Propagators
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 5/52
First Contents Back Conclusion
Dyson-Schwinger Equations
A Modern Method for 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
· Quark & Gluon Confinement
⇒ Understanding InfraRed (long-range)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . behaviour of αs(Q2)
Method yields Schwinger Functions ≡ Propagators
Cross-Sections built from Schwinger FunctionsIVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 5/52
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Contemporary Reviews
Dyson-Schwinger Equations:Density, Temperature and Continuum 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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 6/52
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Persistent Challenge
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 7/52
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Persistent Challenge
Infinitely Many Coupled Equations
Σ=
D
γΓS
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 7/52
First Contents Back Conclusion
Persistent Challenge
Infinitely Many Coupled Equations
Solutions are Schwinger Functions(Euclidean Green Functions)
Same VEVs measured in Lattice-QCD simulations
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 7/52
First Contents Back Conclusion
Persistent Challenge
Infinitely Many Coupled Equations
Solutions are Schwinger Functions(Euclidean Green Functions)
Same VEVs measured in Lattice-QCD simulations
Coupling between equations necessitates truncation
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 7/52
First Contents Back Conclusion
Persistent Challenge
Infinitely Many Coupled Equations
Solutions are Schwinger Functions(Euclidean Green Functions)
Same VEVs measured in Lattice-QCD simulations
Coupling between equations necessitates truncation
Weak coupling expansion ⇒ Perturbation Theory
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 7/52
First Contents Back Conclusion
Persistent Challenge
Infinitely Many Coupled Equations
Solutions are Schwinger Functions(Euclidean Green Functions)
Same VEVs measured in Lattice-QCD simulations
Coupling between equations necessitates truncation
Weak coupling expansion ⇒ Perturbation TheoryNot useful for the nonperturbative problemsin which we’re interested
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 7/52
First Contents Back Conclusion
Persistent Challenge
Infinitely Many Coupled Equations
Solutions are Schwinger Functions(Euclidean Green Functions)
Same VEVs measured in Lattice-QCD simulations
We introduced a systematic nonperturbative,symmetry-preserving truncation schemeGoldstone Theorem and Diquark Confinement Beyond RainbowLadder Approximation, A. Bender, C. D. Roberts and L. VonSmekal, Phys. Lett. B 380 (1996) 7
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 8/52
First Contents Back Conclusion
Persistent Challenge
Infinitely Many Coupled Equations
Solutions are Schwinger Functions(Euclidean Green Functions)
Same VEVs measured in Lattice-QCD simulations
We introduced a systematic nonperturbative,symmetry-preserving truncation schemeGoldstone Theorem and Diquark Confinement Beyond RainbowLadder Approximation, A. Bender, C. D. Roberts and L. VonSmekal, Phys. Lett. B 380 (1996) 7
Has Enabled Proof of EXACT Results in QCD
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 8/52
First Contents Back Conclusion
Persistent Challenge
Infinitely Many Coupled Equations
Solutions are Schwinger Functions(Euclidean Green Functions)
Same VEVs measured in Lattice-QCD simulations
We introduced a systematic nonperturbative,symmetry-preserving truncation schemeGoldstone Theorem and Diquark Confinement Beyond RainbowLadder Approximation, A. Bender, C. D. Roberts and L. VonSmekal, Phys. Lett. B 380 (1996) 7
Has Enabled Proof of EXACT Results in QCD
And Formulation of Practical Phenomenological Tool to
Illustrate Exact Results
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 8/52
First Contents Back Conclusion
Persistent Challenge
Infinitely Many Coupled Equations
Solutions are Schwinger Functions(Euclidean Green Functions)
Same VEVs measured in Lattice-QCD simulations
We introduced a systematic nonperturbative,symmetry-preserving truncation schemeGoldstone Theorem and Diquark Confinement Beyond RainbowLadder Approximation, A. Bender, C. D. Roberts and L. VonSmekal, Phys. Lett. B 380 (1996) 7
Has Enabled Proof of EXACT Results in QCD
And Formulation of Practical Phenomenological Tool to
Make Predictions with Readily Quantifiable Errors
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 8/52
First Contents Back Conclusion
Perturbative Dressed-quarkPropagator
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 9/52
First Contents Back Conclusion
Perturbative Dressed-quarkPropagator
S(p) =Z(p2)
iγ · p + M(p2)Σ
=D
γΓS
Gap Equation
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 9/52
First Contents Back Conclusion
Perturbative Dressed-quarkPropagator
S(p) =Z(p2)
iγ · p + M(p2)Σ
=D
γΓS
Gap Equationdressed-quark propagator
S(p) =1
iγ · pA(p2) + B(p2)
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 9/52
First Contents Back Conclusion
Perturbative Dressed-quarkPropagator
S(p) =Z(p2)
iγ · p + M(p2)Σ
=D
γΓS
Gap Equationdressed-quark propagator
S(p) =1
iγ · pA(p2) + B(p2)
Weak Coupling ExpansionReproduces Every Diagram in Perturbation Theory
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 9/52
First Contents Back Conclusion
Perturbative Dressed-quarkPropagator
S(p) =Z(p2)
iγ · p + M(p2)Σ
=D
γΓS
Gap Equationdressed-quark propagator
S(p) =1
iγ · pA(p2) + B(p2)
Weak Coupling ExpansionReproduces Every Diagram in Perturbation Theory
But in Perturbation Theory
B(p2) = m
(
1 − α
πln
[
p2
m2
]
+ . . .
)
m→0→ 0
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 9/52
First Contents Back Conclusion
Perturbative Dressed-quarkPropagator
S(p) =Z(p2)
iγ · p + M(p2)Σ
=D
γΓS
Gap Equationdressed-quark propagator
S(p) =1
iγ · pA(p2) + B(p2)
Weak Coupling ExpansionReproduces Every Diagram in Perturbation Theory
But in Perturbation Theory
B(p2) = m
(
1 − α
πln
[
p2
m2
]
+ . . .
)
m→0→ 0
No DCSBHere!
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 9/52
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Dressed-Quark Propagator
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 10/52
First Contents Back Conclusion
Dressed-Quark Propagator
S(p) =Z(p2)
iγ · p + M(p2)Σ
=D
γΓS
Gap Equation
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 10/52
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 (GeV
2)
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 10/52
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 (GeV
2)
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 10/52
First Contents Back Conclusion
Dressed-Quark Propagator
Longstanding Prediction of Dyson-SchwingerEquation Studies
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 11/52
First Contents Back Conclusion
Dressed-Quark Propagator
Longstanding Prediction of Dyson-SchwingerEquation Studies
E.g., Dyson-Schwinger equations and theirapplication to hadronic physics,C. D. Roberts and A. G. Williams,Prog. Part. Nucl. Phys. 33 (1994) 477
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 11/52
First Contents Back Conclusion
Dressed-Quark Propagator
Longstanding Prediction of Dyson-SchwingerEquation Studies
E.g., Dyson-Schwinger equations and theirapplication to hadronic physics,C. D. Roberts and A. G. Williams,Prog. Part. Nucl. Phys. 33 (1994) 477
Long used as basis for efficacious hadron physicsphenomenology
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 11/52
First Contents Back Conclusion
Dressed-Quark Propagator
Longstanding Prediction of Dyson-SchwingerEquation Studies
E.g., Dyson-Schwinger equations and theirapplication to hadronic physics,C. D. Roberts and A. G. Williams,Prog. Part. Nucl. Phys. 33 (1994) 477
Long used as basis for efficacious hadron physicsphenomenology
Electromagnetic pion form-factor and neutralpion decay width,C. D. Roberts,Nucl. Phys. A 605 (1996) 475
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 11/52
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)
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 12/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 12/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 12/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 12/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 12/52
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).
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 13/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 13/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 14/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 14/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 14/52
First Contents Back Conclusion
Hadrons
• Established understandingof two- and three-point functions
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 15/52
First Contents Back Conclusion
Hadrons
• Established understandingof two- and three-point functions
• What about bound states?
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 15/52
First Contents Back Conclusion
Hadrons
• Without bound states,Comparison with experiment isimpossible
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 15/52
First Contents Back Conclusion
Hadrons
• Without bound states,Comparison with experiment isimpossible
• They appear as pole contributionsto n ≥ 3-point colour-singletSchwinger functions
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 15/52
First Contents Back Conclusion
Hadrons
• Without bound states,Comparison with experiment isimpossible
• Bethe-Salpeter Equation
QFT Generalisation of Lippman-Schwinger Equation.
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 15/52
First Contents Back Conclusion
Hadrons
• Without bound states,Comparison with experiment isimpossible
• Bethe-Salpeter Equation
QFT Generalisation of Lippman-Schwinger Equation.
• What is the kernel, K?
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 15/52
First Contents Back Conclusion
Hadrons
• Without bound states,Comparison with experiment isimpossible
• Bethe-Salpeter Equation
QFT Generalisation of Lippman-Schwinger Equation.
• What is the kernel, K?
or What is the long-range potential in QCD?
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 15/52
First Contents Back Conclusion
Bethe-Salpeter Kernel
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 16/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 16/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 16/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 16/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 16/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 16/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 16/52
First Contents Back Conclusion
Systematic Nonpert. Trunc.– Key Contributions
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 17/52
First Contents Back Conclusion
Systematic Nonpert. Trunc.– Key Contributions
Existence . . .
Munczek, Phys. Rev. D 52(1995) 4736;
Bender, Roberts, von Smekal,Phys. Lett. B 380 (1996) 7.
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 17/52
First Contents Back Conclusion
Systematic Nonpert. Trunc.– Key Contributions
Existence . . .
Munczek, Phys. Rev. D 52(1995) 4736;
Bender, Roberts, von Smekal,Phys. Lett. B 380 (1996) 7.
Convergence . . .
Bender, Detmold, Roberts, Thomas,Phys. Rev. C 65 (2002) 065203.
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 17/52
First Contents Back Conclusion
Systematic Nonpert. Trunc.– Key Contributions
Existence . . .
Munczek, Phys. Rev. D 52(1995) 4736;
Bender, Roberts, von Smekal,Phys. Lett. B 380 (1996) 7.
Convergence . . .
Bender, Detmold, Roberts, Thomas,Phys. Rev. C 65 (2002) 065203.
Interactions . . .
P. Bicudo, Phys. Rev. C 67 (2003) 035201;
Höll, Krassnigg, Maris, Roberts, Wright,Phys. Rev. C 71 (2005) 065204
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 17/52
First Contents Back Conclusion
Andreas Krassnigg
FWF “ErwinSchrödinger Fellow,”ANL 2003-2005
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 18/52
First Contents Back Conclusion
Andreas Krassnigg
Almost BloodRelative of Arnold. . . Future
President?. . . Executioner?
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 18/52
First Contents Back Conclusion
Radial Excitations& Chiral Symmetry
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 19/52
First Contents Back Conclusion
Radial Excitations& Chiral Symmetry(Maris, Roberts, Tandy
nu-th/9707003 )
fH m2H = − ρH
ζ MH
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 19/52
First Contents Back Conclusion
Radial Excitations& Chiral Symmetry(Maris, Roberts, Tandy
nu-th/9707003 )
fH m2H = − ρH
ζ MH
• Mass2 of pseudoscalar hadron
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 19/52
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)
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 19/52
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
=
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 19/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 19/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 19/52
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 resultIVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 19/52
First Contents Back Conclusion
Radial Excitations& Chiral SymmetryHöll, Krassnigg, Roberts
nu-th/0406030
fH m2H = − ρH
ζ MH
Valid for ALL Pseudoscalar mesons
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 19/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 19/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 19/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 19/52
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 mesonIVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 19/52
First Contents Back Conclusion
Radial Excitations
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 20/52
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)
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 20/52
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?
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 20/52
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?
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 20/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 20/52
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”IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 20/52
First Contents Back Conclusion
Radial Excitations& Chiral Symmetry
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 21/52
First Contents Back Conclusion
Radial Excitations& Chiral SymmetryHöll, Krassnigg, Roberts
nu-th/0406030
Fundamental properties of QCD
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 21/52
First Contents Back Conclusion
Radial Excitations& Chiral SymmetryHöll, Krassnigg, Roberts
nu-th/0406030
Fundamental properties of QCD
If chiral symmetry is dynamically broken,
then in the chiral limit every pseudoscalar meson is blind
to the weak interaction except π(140).
0 0.2 0.4 0.6mq [GeV]
0
0.1
0.2
f π [
GeV
]
fπ0
fπ1
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 21/52
First Contents Back Conclusion
Radial Excitations& Chiral SymmetryHöll, Krassnigg, Roberts
nu-th/0406030
Fundamental properties of QCD
If chiral symmetry is dynamically broken,
then in the chiral limit every pseudoscalar meson is blind
to the weak interaction except π(140).
0 0.2 0.4 0.6mq [GeV]
0
0.1
0.2
f π [
GeV
]
fπ0
fπ1
If chiral symmetry is not broken,
then NO pseudoscalar meson experiences the weak
interaction.
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 21/52
First Contents Back Conclusion
Two-photon Couplings ofPseudoscalar MesonsHöll, Krassnigg, Maris, et al.,
“Electromagnetic properties of ground andexcited state pseudoscalar mesons,”nu-th/0503043
π0n(P ) y
y
y�
�
��
@@R
@@
6
������������������������
������������������������
γ(k1)
γ(k2)
T π0n
µν (k1, k2) =α
πiεµνρσk1ρk2σ Gπ0
n(k1, k2)
Define: Tπ0n
(P 2, Q2) = Gπ0n(k1, k2)
∣
∣
∣
k21=Q2=k2
2
This is a transition form factor.
Physical Processes described by couplings:gπ0
0γγ := Tπ00(−m2
π00, 0)
Width: Γπ0n
γγ = α2em
m3πn
16π3g2
πnγγ
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 22/52
First Contents Back Conclusion
Two-photon Couplings:Goldstone ModeHöll, Krassnigg, Maris, et al.,
“Electromagnetic properties of ground andexcited state pseudoscalar mesons,”nu-th/0503043
π00(P ) y
y
y�
�
��
@@R
@@
6
������������������������
������������������������
γ(k1)
γ(k2)
T π00
µν (k1, k2) =α
πiεµνρσk1ρk2σ Gπ0
0(k1, k2)
Chiral limit, model-independent and algebraic result
gπ00γγ := Tπ0
0(−m2
π00
= 0, 0)=1
2
1
fπ0
So long as truncation preserves chiral symmetry and thepattern of its dynamical breakdown
The most widely known consequence of the Abelian anomalyIVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 23/52
First Contents Back Conclusion
Two-photon Couplings:Transition Form FactorHöll, Krassnigg, Maris, et al.,
“Electromagnetic properties of ground andexcited state pseudoscalar mesons,”nu-th/0503043
π0n(P ) y
y
y�
�
��
@@R
@@
6
������������������������
������������������������
γ(k1), k21 = Q2
γ(k2), k21 = Q2
T π0n
µν (k1, k2) =α
πiεµνρσk1ρk2σ Gπ0
n(k1, k2)
So long as truncation preserves chiral symmetry and thepattern of its dynamical breakdown, and the one-looprenormalisation group properties of QCD: model-independentresult – ∀n:
Tπ0n
(P 2, Q2) = Gπ0n(k1, k2)
∣
∣
∣
k21=Q2=k2
2
Q2≫Λ2QCD
=4π2
3
fπn
Q2
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 24/52
First Contents Back Conclusion
Transition Form Factor:Chiral limitHöll, Krassnigg, Maris, et al.,
“Electromagnetic properties of ground andexcited state pseudoscalar mesons,”nu-th/0503043
Chiral limit with DCSB: fπ06= 0
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 25/52
First Contents Back Conclusion
Transition Form Factor:Chiral limitHöll, Krassnigg, Maris, et al.,
“Electromagnetic properties of ground andexcited state pseudoscalar mesons,”nu-th/0503043
Chiral limit with DCSB: fπ06= 0
BUT, fπn≡ 0, ∀n!
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 25/52
First Contents Back Conclusion
Transition Form Factor:Chiral limitHöll, Krassnigg, Maris, et al.,
“Electromagnetic properties of ground andexcited state pseudoscalar mesons,”nu-th/0503043
Chiral limit with DCSB: fπ06= 0
BUT, fπn≡ 0, ∀n!
Model-independent result, in chiral limit: ∀n ≥ 1
limm→0
Tπ0n
(−m2πn
, Q2)
Q2≫Λ2QCD
=4π2
3F (2)
n (−m2πn
)lnγ Q2/ω2
πn
Q4
∣
∣
∣
∣
∣
m=0where:
γ is an anomalous dimension
ωπnis a width mass-scale
both determined, in part, by properties of the meson’sBethe-Salpeter wave function.
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 25/52
First Contents Back Conclusion
Transition Form Factor:Chiral limitHöll, Krassnigg, Maris, et al.,
“Electromagnetic properties of ground andexcited state pseudoscalar mesons,”nu-th/0503043
Chiral limit with DCSB: fπ06= 0
BUT, fπn≡ 0, ∀n!
Model-independent result, in chiral limit: ∀n ≥ 1
limm→0
Tπ0n
(−m2πn
, Q2)
Q2≫Λ2QCD
=4π2
3F (2)
n (−m2πn
)lnγ Q2/ω2
πn
Q4
∣
∣
∣
∣
∣
m=0where:
γ is an anomalous dimension
ωπnis a width mass-scale
both determined, in part, by properties of the meson’sBethe-Salpeter wave function.
Importantly, F (2)n (−m2
πn
) 6∝ fπn. Instead, it is determined by
DCSB mass-scales for πn that do not vanish in the chiral limit.
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 25/52
First Contents Back Conclusion
Are we there yet?
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 26/52
First Contents Back Conclusion
Nucleon Properties
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 27/52
First Contents Back Conclusion
Nucleon Properties
Maris & Tandy . . . series of five papers . . . excellent
description of light pseudoscalar and vector mesons . . .
basket of 31 masses/couplings/radii with r.m.s. error of 15%
. . . moreover, prediction of Fπ(Q2) measured in Hall A.
Pieter Maris Peter Tandy
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 27/52
First Contents Back Conclusion
Nucleon Properties
Maris & Tandy . . . series of five papers . . . excellent
description of light pseudoscalar and vector mesons . . .
basket of 31 masses/couplings/radii with r.m.s. error of 15%
. . . moreover, prediction of Fπ(Q2) measured in Hall A.
One parameter model . . . parameter specifies long-range
interaction between light-quarks . . . model-independent
results in ultraviolet
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 27/52
First Contents Back Conclusion
Nucleon Properties
Maris & Tandy . . . series of five papers . . . excellent
description of light pseudoscalar and vector mesons . . .
basket of 31 masses/couplings/radii with r.m.s. error of 15%
. . . moreover, prediction of Fπ(Q2) measured in Hall A.
One parameter model . . . parameter specifies long-range
interaction between light-quarks . . . model-independent
results in ultraviolet
Next Steps . . . Applications to excited states and
axial-vector mesons, e.g., will improve understanding of
confinement interaction between light-quarks
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 27/52
First Contents Back Conclusion
Nucleon Properties
Another Direction . . . Also want/need information about
three-quark systems
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 27/52
First Contents Back Conclusion
Nucleon Properties
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.
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 27/52
First Contents Back Conclusion
Nucleon Properties
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.
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 27/52
First Contents Back Conclusion
Proton Form Factors:Modern Experiment
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 28/52
First Contents Back Conclusion
Proton Form Factors:Modern Experiment
Rosenbluth and Polarization-Transfer Extractions of
Ratio of Proton’s Electric and Magnetic Form Factors
0 2 4 6Q
2 [GeV2]
0
0.5
1
µ p GEp/ G
Mp
Rosenbluthprecision Rosenbluthpolarization transferpolarization transfer
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 28/52
First Contents Back Conclusion
Proton Form Factors:Modern Experiment
0 2 4 6Q
2 [GeV2]
0
0.5
1
µ p GEp/ G
Mp
Rosenbluthprecision Rosenbluthpolarization transferpolarization transfer
Walker et al., Phys. Rev. D 49, 5671 (1994)
Qattan et al., Phys.Rev. Lett. 94 142301(2005)
Jones et al., JLab HallA Collaboration, Phys.Rev. Lett. 84, 1398(2000)
Gayou, et al., Phys.Rev. C 64, 038202(2001)
Gayou, et al., JLab Hall A Collaboration,Phys. Rev. Lett. 88 092301 (2002)
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 28/52
First Contents Back Conclusion
Proton Form Factors:Modern Experiment
0 2 4 6Q
2 [GeV2]
0
0.5
1
µ p GEp/ G
Mp
Rosenbluthprecision Rosenbluthpolarization transferpolarization transfer
If Pol. Trans. Correct,
then Completely
Unexpected Result:
In the Proton
– On Relativistic
Domain
– Distribution of
Quark-Charge
Not Equal
Distribution of
Quark-Current!
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 28/52
First Contents Back Conclusion
Arne Höll
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 29/52
First Contents Back Conclusion
Arne Höll
Closing in onsomething
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 29/52
First Contents Back Conclusion
Nucleon EM Form Factors: A Précis
Holl, Kloker, et al. : nu-th/0412046 & nu-th/0501033
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 30/52
First Contents Back Conclusion
Nucleon EM Form Factors: A Précis
Holl, Kloker, et al. : nu-th/0412046 & nu-th/0501033
• Interpreting expts. with GeV electromagnetic probes
requires Poincaré covariant treatment of baryons
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 30/52
First Contents Back Conclusion
Nucleon EM Form Factors: A Précis
Holl, 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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 30/52
First Contents Back Conclusion
Nucleon EM Form Factors: A Précis
Holl, 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%
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 30/52
First Contents Back Conclusion
Nucleon EM Form Factors: A Précis
Holl, 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)
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 30/52
First Contents Back Conclusion
Nucleon EM Form Factors: A Précis
Holl, 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?
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 30/52
First Contents Back Conclusion
Nucleon EM Form Factors: A Précis
Holl, 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!
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 30/52
First Contents Back Conclusion
Nucleon EM Form Factors: A Précis
Holl, 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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 30/52
First Contents Back Conclusion
Harry LeePions and Form Factors
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 31/52
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)
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 31/52
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.
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 31/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 31/52
First Contents Back Conclusion
Faddeev equation
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 32/52
First Contents Back Conclusion
Faddeev equation
=aΨ
P
pq
pd Γb
Γ−a
pd
pq
bΨP
q
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 32/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 32/52
First Contents Back Conclusion
Parametrising diquark properties
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 33/52
First Contents Back Conclusion
Parametrising diquark properties
Dressed-quark . . . fixed by DSE and Meson Studies
. . . Burden, Roberts, Thomson, Phys. Lett. B 371, 163 (1996)
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 33/52
First Contents Back Conclusion
Parametrising diquark properties
• Bethe-Salpeter-Like Amplitudes
Γ0+
(k;K) =1
N 0+Ha Ciγ5 iτ2 F(k2/ ω2
0+ ) ,
tiΓ1+
µ (k;K) =1
N 1+Ha iγµC t
i F(k2/ ω21+ )
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 33/52
First Contents Back Conclusion
Parametrising diquark properties
• Bethe-Salpeter-Like Amplitudes
Γ0+
(k;K) =1
N 0+Ha Ciγ5 iτ2 F(k2/ ω2
0+ ) ,
tiΓ1+
µ (k;K) =1
N 1+Ha iγµC t
i F(k2/ ω21+ )
• Colour matrices:
{H1 = iλ7,H2 = −iλ5,H3 = iλ2} , ǫc1c2c3 = (Hc3)c1c2
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 33/52
First Contents Back Conclusion
Parametrising diquark properties
• Bethe-Salpeter-Like Amplitudes
Γ0+
(k;K) =1
N 0+Ha Ciγ5 iτ2 F(k2/ ω2
0+ ) ,
tiΓ1+
µ (k;K) =1
N 1+Ha iγµC t
i F(k2/ ω21+ )
• Two parameters: ω0+ , ω1+
∼ Inverse of diquarks’ configuration-space size
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 33/52
First Contents Back Conclusion
Parametrising diquark properties
• Pseudoparticle Propagators
∆0+
(K) =1
m20+
F(K2/ω20+) ,
∆1+
µν (K) =
(
δµν +KµKν
m21+
)
1
m21+
F(K2/ω21+)
F(x) =1 − exp(−x)
xAbsence of a Spectral Representation
Realisation of Confinement
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 33/52
First Contents Back Conclusion
Parametrising diquark properties
• Pseudoparticle Propagators
∆0+
(K) =1
m20+
F(K2/ω20+) ,
∆1+
µν (K) =
(
δµν +KµKν
m21+
)
1
m21+
F(K2/ω21+)
• Two parameters: m0+ , m1+
∼ Inverse of diquarks’ configuration-space correlation length
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 33/52
First Contents Back Conclusion
Parametrising diquark properties
• Total of four parameters
. . . reduce that via Normalisation Condition
d
dK2
(
1
m2JP
F(K2/ω2JP )
)
−1∣
∣
∣
∣
∣
∣
K2=0
= 1 ⇒ ω2JP =
1
2m2
JP ,
Accentuates free-particle-like propagation characteristics of the
diquarks within hadron.
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 33/52
First Contents Back Conclusion
Parametrising diquark properties
• Total of four parameters
. . . reduce that via Normalisation Condition
d
dK2
(
1
m2JP
F(K2/ω2JP )
)
−1∣
∣
∣
∣
∣
∣
K2=0
= 1 ⇒ ω2JP =
1
2m2
JP ,
Accentuates free-particle-like propagation characteristics of the
diquarks within hadron.
Two Parameter Faddeev Equation Model of Nucleon
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 33/52
First Contents Back Conclusion
Parametrising diquark properties
• Total of four parameters
. . . reduce that via Normalisation Condition
d
dK2
(
1
m2JP
F(K2/ω2JP )
)
−1∣
∣
∣
∣
∣
∣
K2=0
= 1 ⇒ ω2JP =
1
2m2
JP ,
Accentuates free-particle-like propagation characteristics of the
diquarks within hadron.
Two Parameter Faddeev Equation Model of Nucleon
Solve Faddeev Equation
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 33/52
First Contents Back Conclusion
Parametrising diquark properties
• Total of four parameters
. . . reduce that via Normalisation Condition
d
dK2
(
1
m2JP
F(K2/ω2JP )
)
−1∣
∣
∣
∣
∣
∣
K2=0
= 1 ⇒ ω2JP =
1
2m2
JP ,
Accentuates free-particle-like propagation characteristics of the
diquarks within hadron.
Two Parameter Faddeev Equation Model of Nucleon
Solve Faddeev Equation
Vary m0+ and m1+ to obtain desired masses for N and ∆
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 33/52
First Contents Back Conclusion
Results: Nucleonand ∆ Masses
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 34/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 34/52
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 attractionIVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 34/52
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 + ∂µπIVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 34/52
First Contents Back Conclusion
Nucleon-Photon Vertex
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 35/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 35/52
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
ΓIVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 35/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 36/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 36/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 36/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 36/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 36/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 36/52
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.
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 36/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 36/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 36/52
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
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 36/52
First Contents Back Conclusion
Epilogue
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 37/52
First Contents Back Conclusion
Epilogue
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 37/52
First Contents Back Conclusion
Epilogue
Dyson-Schwinger Equations
Provide Understanding of
Dynamical Chiral Symmetry Breaking:
⇒ π is quark-antiquark Bound State
AND QCD’s Goldstone Mode
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 37/52
First Contents Back Conclusion
Epilogue
Dyson-Schwinger Equations
Provide Understanding of
Dynamical Chiral Symmetry Breaking:
⇒ π is quark-antiquark Bound State
AND QCD’s Goldstone ModeFoundation for Proof of
Exact Results in QCD
e.g., Quark Goldberger-Treiman
Properties of Pseudoscalar Mesons
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 37/52
First Contents Back Conclusion
Epilogue
Dyson-Schwinger Equations
Provide Understanding of
Dynamical Chiral Symmetry Breaking:
⇒ π is quark-antiquark Bound State
AND QCD’s Goldstone ModeFoundation for Proof of
Exact Results in QCD
e.g., Quark Goldberger-Treiman
Properties of Pseudoscalar Mesons
Renormalisation-Group-Improved Rainbow-Ladder
⇒ Practical Phenomenological Tool
Corrections QuantifiableIVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 37/52
First Contents Back Conclusion
Epilogue
Poincaré Covariant
Faddeev Equation
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 37/52
First Contents Back Conclusion
Epilogue
Poincaré Covariant
Faddeev Equation
Nonpointlike scalar and axial-vector diquark correlations
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 37/52
First Contents Back Conclusion
Epilogue
Poincaré Covariant
Faddeev Equation
Nonpointlike scalar and axial-vector diquark correlations
s−, p−, d−wave quark angular momentum
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 37/52
First Contents Back Conclusion
Epilogue
Poincaré Covariant
Faddeev Equation
Nonpointlike scalar and axial-vector diquark correlations
s−, p−, d−wave quark angular momentum
Quark core, relaxed to allow for pion cloud
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 37/52
First Contents Back Conclusion
Epilogue
Poincaré Covariant
Faddeev Equation
Nonpointlike scalar and axial-vector diquark correlations
s−, p−, d−wave quark angular momentum
Quark core, relaxed to allow for pion cloud
Predicts zero in GPE(Q2) at Q2 ≈ 6.5 GeV2
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 37/52
First Contents Back Conclusion
Epilogue
Appearance of a zero in GE(Q2) – Completely Unexpected
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 37/52
First Contents Back Conclusion
Epilogue
Appearance of a zero in GE(Q2) – Completely Unexpected
0 0.2 0.4 0.6 0.8 1r (fm)
0
0.002
0.004
0.006
0.008
0.01
ρ(r)
Peak Position and Height -- as zero moves in toward q*q=0
peak moves out and falls in magnitude-- redistribution of charge ... moves to larger r
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 37/52
First Contents Back Conclusion
Epilogue
Appearance of a zero in GE(Q2) – Completely Unexpected
0 0.2 0.4 0.6 0.8 1r (fm)
0
0.002
0.004
0.006
0.008
0.01
ρ(r)
Peak Position and Height -- as zero moves in toward q*q=0
peak moves out and falls in magnitude-- redistribution of charge ... moves to larger r
However, Current Density remains peaked at r = 0!
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 37/52
First Contents Back Conclusion
Epilogue
Appearance of a zero in GE(Q2) – Completely Unexpected
Wave Function is complex and correlated mix of virtual
particles and antiparticles: s−, p− and d−waves
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 37/52
First Contents Back Conclusion
Epilogue
Appearance of a zero in GE(Q2) – Completely Unexpected
Wave Function is complex and correlated mix of virtual
particles and antiparticles: s−, p− and d−waves
Simple three-quark bag-model picture that is still being
taught at many universities is profoundly incorrect
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 37/52
First Contents Back Conclusion
Epilogue
Appearance of a zero in GE(Q2) – Completely Unexpected
Wave Function is complex and correlated mix of virtual
particles and antiparticles: s−, p− and d−waves
Simple three-quark bag-model picture that is still being
taught at many universities is profoundly incorrect
Can Expect
Contemporary Nuclear Physicsto Yield Many More Surprises
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 37/52
First Contents Back Conclusion
Contents1. Pion Dichotomy
2. DSEs
3. Persistent Challenge
4. Perturbative Quark Propagator
5. Dressed-Quark Propagator
6. Lattice cf. DSE
7. Light-Quark Interaction
8. Dressed-gluon Propagator
9. Hadrons
10. Bethe-Salpeter Kernel
11. Nonpert. Trunc. – Contributions
12. Excitations & Chiral Symmetry
13. Radial Excitations
14. Radial Excitations II
15. Two-photon Couplings
16. Proton FF
17. Nucleon EM Form Factors
18. Pions and Form Factors
19. Faddeev equation
20. Parametrising diquark properties
21. Results: Nucleon & ∆ Masses
22. Form Factor Ratio: GE/GM
23. Dressed-Vertex
24. Dressed-Vertex II
25. Goldberger-Treiman for pion
26. Colour-singlet BSE
27. Two-photon: small Q2
28. E.M. Charge Radii
29. DIS
30. Valence Distribution
31. Handbag Diagrams
32. General Calc. cf. Data
33. Form Factor Ratio: Q ∗F2/F1
34. Form Factor Ratio: alternative F2/F1IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 38/52
First Contents Back Conclusion
Quenched-QCDDressed-quark-gluon Vertex
0 1 2 3p (GeV)
0
0.5
1
1.5
2
2.5λ
1(p) Quenched Lattice
4p2λ
2(p) Quenched Lattice
-2pλ3(p) Quenched Lattice
λ1(p) DSE--Lat (quenched)
4p2λ
2(p) DSE--Lat
-2pλ3(p) DSE--Lat
λ1(p) Abelian Ansatz (WI)
-2pλ3(p) Abelian Ansatz (WI)
Bhagwat, et al.:
nu-th/0304003
nu-th/0403012
he-ph/0407163
share 65 citations
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 39/52
First Contents Back Conclusion
Quenched-QCDDressed-quark-gluon Vertex
0 1 2 3p (GeV)
0
0.5
1
1.5
2
2.5λ
1(p) Quenched Lattice
4p2λ
2(p) Quenched Lattice
-2pλ3(p) Quenched Lattice
λ1(p) DSE--Lat (quenched)
4p2λ
2(p) DSE--Lat
-2pλ3(p) DSE--Lat
λ1(p) Abelian Ansatz (WI)
-2pλ3(p) Abelian Ansatz (WI)
Bhagwat, et al.:
nu-th/0304003
nu-th/0403012
he-ph/0407163
share 65 citations
Lattice – Skullerud,et al.: he-ph/0303176
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 39/52
First Contents Back Conclusion
Quenched-QCDDressed-quark-gluon Vertex
0 1 2 3p (GeV)
0
0.5
1
1.5
2
2.5λ
1(p) Quenched Lattice
4p2λ
2(p) Quenched Lattice
-2pλ3(p) Quenched Lattice
λ1(p) DSE--Lat (quenched)
4p2λ
2(p) DSE--Lat
-2pλ3(p) DSE--Lat
λ1(p) Abelian Ansatz (WI)
-2pλ3(p) Abelian Ansatz (WI)
Bhagwat, et al.:
nu-th/0304003
nu-th/0403012
he-ph/0407163
share 65 citations
Lattice – Skullerud,et al.: he-ph/0303176
Parameter Free DSEPrediction confirmslattice simulation ofλ1, λ3
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 39/52
First Contents Back Conclusion
Quenched-QCDDressed-quark-gluon Vertex
0 1 2 3p (GeV)
0
0.5
1
1.5
2
2.5λ
1(p) Quenched Lattice
4p2λ
2(p) Quenched Lattice
-2pλ3(p) Quenched Lattice
λ1(p) DSE--Lat (quenched)
4p2λ
2(p) DSE--Lat
-2pλ3(p) DSE--Lat
λ1(p) Abelian Ansatz (WI)
-2pλ3(p) Abelian Ansatz (WI)
Bhagwat, et al.:
nu-th/0304003
nu-th/0403012
he-ph/0407163
share 65 citations
Lattice – Skullerud,et al.: he-ph/0303176
Parameter Free DSEPrediction confirmslattice simulation ofλ1, λ3
Parameter FreeDSE Prediction suggests lattice result for λ2 erroneous– owing to systematic errors
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 39/52
First Contents Back Conclusion
Goldberger-Treiman for pion
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 40/52
First Contents Back Conclusion
Goldberger-Treiman for pion
• Pseudoscalar Bethe-Salpeter amplitude
Γπj (k;P ) = τπj
γ5
[
iEπ(k;P ) + γ · PF π(k;P )
+ γ · k k · P Gπ(k;P ) + σµν kµPν Hπ(k;P )]
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 40/52
First Contents Back Conclusion
Goldberger-Treiman for pion
• Pseudoscalar Bethe-Salpeter amplitude
Γπj (k;P ) = τπj
γ5
[
iEπ(k;P ) + γ · PF π(k;P )
+ γ · k k · P Gπ(k;P ) + σµν kµPν Hπ(k;P )]
• Dressed-quark Propagator: S(p) =1
iγ · pA(p2) + B(p2)
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 40/52
First Contents Back Conclusion
Goldberger-Treiman for pion
• Pseudoscalar Bethe-Salpeter amplitude
Γπj (k;P ) = τπj
γ5
[
iEπ(k;P ) + γ · PF π(k;P )
+ γ · k k · P Gπ(k;P ) + σµν kµPν Hπ(k;P )]
• Dressed-quark Propagator: S(p) =1
iγ · pA(p2) + B(p2)• Axial-vector Ward-Takahashi identity
⇒ fπEπ(k;P = 0) = B(p2)
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 40/52
First Contents Back Conclusion
Goldberger-Treiman for pion
• Pseudoscalar Bethe-Salpeter amplitude
Γπj (k;P ) = τπj
γ5
[
iEπ(k;P ) + γ · PF π(k;P )
+ γ · k k · P Gπ(k;P ) + σµν kµPν Hπ(k;P )]
• Dressed-quark Propagator: S(p) =1
iγ · pA(p2) + B(p2)• Axial-vector Ward-Takahashi identity
⇒ fπEπ(k;P = 0) = B(p2)
FR(k; 0) + 2 fπFπ(k; 0) = A(k2)
GR(k; 0) + 2 fπGπ(k; 0) = 2A′(k2)
HR(k; 0) + 2 fπHπ(k; 0) = 0
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 40/52
First Contents Back Conclusion
Goldberger-Treiman for pion
Pseudovectorcomponentsnecessarilynonzero
• Pseudoscalar Bethe-Salpeter amplitude
Γπj (k;P ) = τπj
γ5
[
iEπ(k;P ) + γ · PF π(k;P )
+ γ · k k · P Gπ(k;P ) + σµν kµPν Hπ(k;P )]
• Dressed-quark Propagator: S(p) =1
iγ · pA(p2) + B(p2)• Axial-vector Ward-Takahashi identity
⇒ fπEπ(k;P = 0) = B(p2)
FR(k; 0) + 2 fπFπ(k; 0) = A(k2)
GR(k; 0) + 2 fπGπ(k; 0) = 2A′(k2)
HR(k; 0) + 2 fπHπ(k; 0) = 0
Exact in
Chiral QCD
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 40/52
First Contents Back Conclusion
Colour-singletBethe-Salpeter equation
Detmold et al., nu-th/0202082Bhagwat, et al., nu-th/0403012
�a�(k; p) = + + + : : :
�M =�n �n� �M + �a;n�
�a;n� = �M�n�1� + �M�n�1� + �a;n�1�
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 41/52
First Contents Back Conclusion
Colour-singletBethe-Salpeter equation
Detmold et al., nu-th/0202082Bhagwat, et al., nu-th/0403012
• Coupling-modified dressed-ladder vertex�a�(k; p) = + + + : : :C C2
�M =�n �n� �M + �a;n�
�a;n� = �M�n�1� + �M�n�1� + �a;n�1�
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 41/52
First Contents Back Conclusion
Colour-singletBethe-Salpeter equation
Detmold et al., nu-th/0202082Bhagwat, et al., nu-th/0403012
• Coupling-modified dressed-ladder vertex�a�(k; p) = + + + : : :C C2
• BSE consistent with vertex�M =�n �n� �M + �a;n�
�a;n� = �M�n�1� + �M�n�1� + �a;n�1�
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 41/52
First Contents Back Conclusion
Colour-singletBethe-Salpeter equation
Detmold et al., nu-th/0202082Bhagwat, et al., nu-th/0403012
• Coupling-modified dressed-ladder vertex�a�(k; p) = + + + : : :C C2
• BSE consistent with vertex�M =�n �n� �M + �a;n�• Bethe-Salpeter kernel . . . recursion relation
−1
8C �a;n� = �M�n�1� + �M�n�1� + �a;n�1�
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 41/52
First Contents Back Conclusion
Colour-singletBethe-Salpeter equation
Detmold et al., nu-th/0202082Bhagwat, et al., nu-th/0403012
• Coupling-modified dressed-ladder vertex�a�(k; p) = + + + : : :C C2
• BSE consistent with vertex�M =�n �n� �M + �a;n�• Bethe-Salpeter kernel . . . recursion relation
−1
8C �a;n� = �M�n�1� + �M�n�1� + �a;n�1�
• Kernel necessarily non-planar,even with planar vertex
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 41/52
First Contents Back Conclusion
π and ρ mesons
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 42/52
First Contents Back Conclusion
π and ρ mesons
Mn=0H Mn=1
H Mn=2H Mn=∞
H
π, m = 0 0 0 0 0
π, m = 0.011 0.147 0.135 0.139 0.138
ρ, m = 0 0.920 0.648 0.782 0.754
ρ, m = 0.011 0.936 0.667 0.798 0.770
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 42/52
First Contents Back Conclusion
π and ρ mesons
Mn=0H Mn=1
H Mn=2H Mn=∞
H
π, m = 0 0 0 0 0
π, m = 0.011 0.147 0.135 0.139 0.138
ρ, m = 0 0.920 0.648 0.782 0.754
ρ, m = 0.011 0.936 0.667 0.798 0.770
• π massless in chiral limit . . . NO Fine Tuning
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 42/52
First Contents Back Conclusion
π and ρ mesons
Mn=0H Mn=1
H Mn=2H Mn=∞
H
π, m = 0 0 0 0 0
π, m = 0.011 0.147 0.135 0.139 0.138
ρ, m = 0 0.920 0.648 0.782 0.754
ρ, m = 0.011 0.936 0.667 0.798 0.770
• π massless in chiral limit . . . NO Fine Tuning
• ALL π-ρ mass splitting present in chiral limit
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 42/52
First Contents Back Conclusion
π and ρ mesons
Mn=0H Mn=1
H Mn=2H Mn=∞
H
π, m = 0 0 0 0 0
π, m = 0.011 0.147 0.135 0.139 0.138
ρ, m = 0 0.920 0.648 0.782 0.754
ρ, m = 0.011 0.936 0.667 0.798 0.770
• π massless in chiral limit . . . NO Fine Tuning
• ALL π-ρ mass splitting present in chiral limitand with the Simplest kernel
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 42/52
First Contents Back Conclusion
π and ρ mesons
Mn=0H Mn=1
H Mn=2H Mn=∞
H
π, m = 0 0 0 0 0
π, m = 0.011 0.147 0.135 0.139 0.138
ρ, m = 0 0.920 0.648 0.782 0.754
ρ, m = 0.011 0.936 0.667 0.798 0.770
• π massless in chiral limit . . . NO Fine Tuning
• π-ρ mass splitting driven by DχSB mechanismNot constituent-quark-model-like hyperfine splitting
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 42/52
First Contents Back Conclusion
π and ρ mesons
Mn=0H Mn=1
H Mn=2H Mn=∞
H
π, m = 0 0 0 0 0
π, m = 0.011 0.147 0.135 0.139 0.138
ρ, m = 0 0.920 0.648 0.782 0.754
ρ, m = 0.011 0.936 0.667 0.798 0.770
• π massless in chiral limit . . . NO Fine Tuning
• π-ρ mass splitting driven by DχSB mechanismNot constituent-quark-model-like hyperfine splitting
• Extending kernel
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 42/52
First Contents Back Conclusion
π and ρ mesons
Mn=0H Mn=1
H Mn=2H Mn=∞
H
π, m = 0 0 0 0 0
π, m = 0.011 0.147 0.135 0.139 0.138
ρ, m = 0 0.920 0.648 0.782 0.754
ρ, m = 0.011 0.936 0.667 0.798 0.770
• π massless in chiral limit . . . NO Fine Tuning
• π-ρ mass splitting driven by DχSB mechanismNot constituent-quark-model-like hyperfine splitting
• Extending kernel: NO effect on mπ
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 42/52
First Contents Back Conclusion
π and ρ mesons
Mn=0H Mn=1
H Mn=2H Mn=∞
H
π, m = 0 0 0 0 0
π, m = 0.011 0.147 0.135 0.139 0.138
ρ, m = 0 0.920 0.648 0.782 0.754
ρ, m = 0.011 0.936 0.667 0.798 0.770
• π massless in chiral limit . . . NO Fine Tuning
• π-ρ mass splitting driven by DχSB mechanismNot constituent-quark-model-like hyperfine splitting
• Extending kernel: NO effect on mπ
For mρ – zeroth order, accurate to 20%
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 42/52
First Contents Back Conclusion
π and ρ mesons
Mn=0H Mn=1
H Mn=2H Mn=∞
H
π, m = 0 0 0 0 0
π, m = 0.011 0.147 0.135 0.139 0.138
ρ, m = 0 0.920 0.648 0.782 0.754
ρ, m = 0.011 0.936 0.667 0.798 0.770
• π massless in chiral limit . . . NO Fine Tuning
• π-ρ mass splitting driven by DχSB mechanismNot constituent-quark-model-like hyperfine splitting
• Extending kernel: NO effect on mπ
For mρ – zeroth order, accurate to 20%– one loop, accurate to 13%
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 42/52
First Contents Back Conclusion
π and ρ mesons
Mn=0H Mn=1
H Mn=2H Mn=∞
H
π, m = 0 0 0 0 0
π, m = 0.011 0.147 0.135 0.139 0.138
ρ, m = 0 0.920 0.648 0.782 0.754
ρ, m = 0.011 0.936 0.667 0.798 0.770
• π massless in chiral limit . . . NO Fine Tuning
• π-ρ mass splitting driven by DχSB mechanismNot constituent-quark-model-like hyperfine splitting
• Extending kernel: NO effect on mπ
For mρ – zeroth order, accurate to 20%– one loop, accurate to 13%– two loop, accurate to 4%
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 42/52
First Contents Back Conclusion
Calculated Transition Form Factor:RGI Rainbow-LadderHöll, Krassnigg, Maris, et al.,
“Electromagnetic properties of ground andexcited state pseudoscalar mesons,”nu-th/0503043
-0.2 0 0.2 0.4 0.6 0.8
Q2 [GeV
2]
-1
0
1
2
3
4
5
Tπ n(Q
2 ) [
GeV
-1]
n = 0 (ground state)n = 1 (radial excitation)
1 / ( 2 fπ0)
mu(1 GeV)
= md(1 GeV) = 5.5 MeV
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 43/52
First Contents Back Conclusion
Calculated Transition Form Factor:RGI Rainbow-LadderHöll, Krassnigg, Maris, et al.,
“Electromagnetic properties of ground andexcited state pseudoscalar mesons,”nu-th/0503043
-0.2 0 0.2 0.4 0.6 0.8
Q2 [GeV
2]
-1
0
1
2
3
4
5
Tπ n(Q
2 ) [
GeV
-1]
n = 0 (ground state)n = 1 (radial excitation)
1 / ( 2 fπ0)
mu(1 GeV)
= md(1 GeV) = 5.5 MeV
Tπ01(−m2
π1, Q2) < 0 , Q2 ≥ −m2
π1/4;
viz., it is negative on the entire kinematically accessible domain.
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 43/52
First Contents Back Conclusion
Calculated Transition Form Factor:RGI Rainbow-LadderHöll, Krassnigg, Maris, et al.,
“Electromagnetic properties of ground andexcited state pseudoscalar mesons,”nu-th/0503043
-0.2 0 0.2 0.4 0.6 0.8
Q2 [GeV
2]
-1
0
1
2
3
4
5
Tπ n(Q
2 ) [
GeV
-1]
n = 0 (ground state)n = 1 (radial excitation)
1 / ( 2 fπ0)
mu(1 GeV)
= md(1 GeV) = 5.5 MeV
Tπ01(−m2
π1, Q2) < 0 , Q2 ≥ −m2
π1/4;
viz., it is negative on the entire kinematically accessible domain.
Γπ00γγ = 7.9 eV, Γπ0
1γγ = 240 eV
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 43/52
First Contents Back Conclusion
Calculated Transition Form Factor:RGI Rainbow-LadderHöll, Krassnigg, Maris, et al.,
“Electromagnetic properties of ground andexcited state pseudoscalar mesons,”nu-th/0503043
1 10 100 1000
Q2 [GeV
2]
10-5
10-4
10-3
10-2
10-1
100
|Tπ n(Q
2 )| [
GeV
-1]
n = 0n = 14 π2
fπn/ (3Q
2)
mu(1 GeV)
= md(1 GeV) = 5.5 MeV
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 44/52
First Contents Back Conclusion
Calculated Transition Form Factor:RGI Rainbow-LadderHöll, Krassnigg, Maris, et al.,
“Electromagnetic properties of ground andexcited state pseudoscalar mesons,”nu-th/0503043
1 10 100 1000
Q2 [GeV
2]
10-5
10-4
10-3
10-2
10-1
100
|Tπ n(Q
2 )| [
GeV
-1]
n = 0n = 14 π2
fπn/ (3Q
2)
mu(1 GeV)
= md(1 GeV) = 5.5 MeV
Predicted UV-behaviour is abundantly clear
precise for Q2 > 120 GeV2
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 44/52
First Contents Back Conclusion
Transition Form Factor ( Chiral ):RGI Rainbow-LadderHöll, Krassnigg, Maris, et al.,
“Electromagnetic properties of ground andexcited state pseudoscalar mesons,”nu-th/0503043
1 10 100 1000
Q2 [GeV
2]10
-7
10-6
10-5
10-4
10-3
10-2
10-1
|Tπ n(Q
2 )| [
GeV
-1]
up/down quarks
4 π2fπ1
/ (3Q2)
chiral limit (4 π2
/ 3) (0.22 GeV)3 / Q4
mu(1 GeV)
= md(1 GeV) = 0
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 45/52
First Contents Back Conclusion
Transition Form Factor ( Chiral ):RGI Rainbow-LadderHöll, Krassnigg, Maris, et al.,
“Electromagnetic properties of ground andexcited state pseudoscalar mesons,”nu-th/0503043
1 10 100 1000
Q2 [GeV
2]10
-7
10-6
10-5
10-4
10-3
10-2
10-1
|Tπ n(Q
2 )| [
GeV
-1]
up/down quarks
4 π2fπ1
/ (3Q2)
chiral limit (4 π2
/ 3) (0.22 GeV)3 / Q4
mu(1 GeV)
= md(1 GeV) = 0
Again, Predicted UV-behaviouris abundantly clear
precise for Q2 > 120 GeV2
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 45/52
First Contents Back Conclusion
Transition Form Factor ( Chiral ):RGI Rainbow-LadderHöll, Krassnigg, Maris, et al.,
“Electromagnetic properties of ground andexcited state pseudoscalar mesons,”nu-th/0503043
1 10 100 1000
Q2 [GeV
2]10
-7
10-6
10-5
10-4
10-3
10-2
10-1
|Tπ n(Q
2 )| [
GeV
-1]
up/down quarks
4 π2fπ1
/ (3Q2)
chiral limit (4 π2
/ 3) (0.22 GeV)3 / Q4
mu(1 GeV)
= md(1 GeV) = 0
Again, Predicted UV-behaviouris abundantly clear
precise for Q2 > 120 GeV2
F(2)1 (−m2
π1) lnγ Q2/ω2
π1
∣
∣
∣
m=0≈ (0.22 GeV)3 ≃ −〈qq〉0 (3)
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 45/52
First Contents Back Conclusion
Electromagnetic Charge Radii – RGIRainbow-LadderHöll, Krassnigg, Maris, et al.,
nu-th/0503043
0.3 0.32 0.34 0.36 0.38 0.4ω [GeV]
0.60
0.65
0.70
0.75
0.80
r π [fm
]
n = 0 (ground state)n = 1 (radial excitation)linear fit: 0.61 + 0.11 ωlinear fit: 0.09 + 1.76 ω
mu,d(1 GeV) = 5.5 MeV
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 46/52
First Contents Back Conclusion
Electromagnetic Charge Radii – RGIRainbow-LadderHöll, Krassnigg, Maris, et al.,
nu-th/0503043
0.3 0.32 0.34 0.36 0.38 0.4ω [GeV]
0.60
0.65
0.70
0.75
0.80
r π [fm
]
n = 0 (ground state)n = 1 (radial excitation)linear fit: 0.61 + 0.11 ωlinear fit: 0.09 + 1.76 ω
mu,d(1 GeV) = 5.5 MeV
Reminder:MT-model has oneIR-mass-scale – ω
ra := 1/ω
gauges the rangeof strong attraction
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 46/52
First Contents Back Conclusion
Electromagnetic Charge Radii – RGIRainbow-LadderHöll, Krassnigg, Maris, et al.,
nu-th/0503043
0.3 0.32 0.34 0.36 0.38 0.4ω [GeV]
0.60
0.65
0.70
0.75
0.80
r π [fm
]
n = 0 (ground state)n = 1 (radial excitation)linear fit: 0.61 + 0.11 ωlinear fit: 0.09 + 1.76 ω
mu,d(1 GeV) = 5.5 MeV
Reminder:MT-model has oneIR-mass-scale – ω
ra := 1/ω
gauges the rangeof strong attraction
Goldstone Mode’sproperties are insensitive to ra
Expected cf. T 6= 0, Goldstone mode’s properties do notchange until very near chiral symmetry restorationtemperature.
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 46/52
First Contents Back Conclusion
Electromagnetic Charge Radii – RGIRainbow-LadderHöll, Krassnigg, Maris, et al.,
nu-th/0503043
0.3 0.32 0.34 0.36 0.38 0.4ω [GeV]
0.60
0.65
0.70
0.75
0.80
r π [fm
]
n = 0 (ground state)n = 1 (radial excitation)linear fit: 0.61 + 0.11 ωlinear fit: 0.09 + 1.76 ω
mu,d(1 GeV) = 5.5 MeV
Reminder:MT-model has oneIR-mass-scale – ω
ra := 1/ω
gauges the rangeof strong attraction
1st excited state:orthogonal to Goldstone mode
Not protected . . . properties very sensitive to ra
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 46/52
First Contents Back Conclusion
Electromagnetic Charge Radii – RGIRainbow-LadderHöll, Krassnigg, Maris, et al.,
nu-th/0503043
0.3 0.32 0.34 0.36 0.38 0.4ω [GeV]
0.60
0.65
0.70
0.75
0.80
r π [fm
]
n = 0 (ground state)n = 1 (radial excitation)linear fit: 0.61 + 0.11 ωlinear fit: 0.09 + 1.76 ω
mu,d(1 GeV) = 5.5 MeV
Reminder:MT-model has oneIR-mass-scale – ω
ra := 1/ω
gauges the rangeof strong attraction
Best estimate rπ1= 1.4 rπ0
But rπ1< rπ0
is possible if confinement force is very strong
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 46/52
First Contents Back Conclusion
Electromagnetic Charge Radii – RGIRainbow-LadderHöll, Krassnigg, Maris, et al.,
nu-th/0503043
0.3 0.32 0.34 0.36 0.38 0.4ω [GeV]
0.60
0.65
0.70
0.75
0.80
r π [fm
]
n = 0 (ground state)n = 1 (radial excitation)linear fit: 0.61 + 0.11 ωlinear fit: 0.09 + 1.76 ω
mu,d(1 GeV) = 5.5 MeV
Reminder:MT-model has oneIR-mass-scale – ω
ra := 1/ω
gauges the rangeof strong attraction
Radial excitations areplainly useful to map outthe long-range part of interaction between light-quarks.
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 46/52
First Contents Back Conclusion
Electromagnetic Charge Radii – RGIRainbow-LadderHöll, Krassnigg, Maris, et al.,
nu-th/0503043
0.3 0.32 0.34 0.36 0.38 0.4ω [GeV]
0.60
0.65
0.70
0.75
0.80
r π [fm
]
n = 0 (ground state)n = 1 (radial excitation)linear fit: 0.61 + 0.11 ωlinear fit: 0.09 + 1.76 ω
mu,d(1 GeV) = 5.5 MeV
Reminder:MT-model has oneIR-mass-scale – ω
ra := 1/ω
gauges the rangeof strong attraction
Radial excitations areplainly useful to map outthe long-range part of interaction between light-quarks.
Same is true of orbital excitations; e.g., axial-vector mesons.
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 46/52
First Contents Back Conclusion
Electromagnetic Charge Radii – RGIRainbow-LadderHöll, Krassnigg, Maris, et al.,
nu-th/0503043
0.3 0.32 0.34 0.36 0.38 0.4ω [GeV]
0.60
0.65
0.70
0.75
0.80
r π [fm
]
n = 0 (ground state)n = 1 (radial excitation)linear fit: 0.61 + 0.11 ωlinear fit: 0.09 + 1.76 ω
mu,d(1 GeV) = 5.5 MeV
Reminder:MT-model has oneIR-mass-scale – ω
ra := 1/ω
gauges the rangeof strong attraction
Radial excitations areplainly useful to map outthe long-range part of interaction between light-quarks.
Same is true of orbital excitations; e.g., axial-vector mesons.
Hall-D at JLab
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 46/52
First Contents Back Conclusion
Deep-inelastic scattering
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 47/52
First Contents Back Conclusion
Deep-inelastic scattering
Looking for Quarks
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 47/52
First Contents Back Conclusion
Deep-inelastic scattering
Looking for Quarks
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 47/52
First Contents Back Conclusion
Deep-inelastic scattering
Looking for Quarks
Signature Experiment for QCD:
Discovery of Quarks at SLAC
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 47/52
First Contents Back Conclusion
Deep-inelastic scattering
Looking for Quarks
Signature Experiment for QCD:
Discovery of Quarks at SLAC
Cross-section: Interpreted as Measurement ofMomentum-Fraction Prob. Distribution: q(x), g(x)
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 47/52
First Contents Back Conclusion
Pion’s valence quark distn
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 48/52
First Contents Back Conclusion
Pion’s valence quark distn
π is Two-Body System: “Easiest” Bound State in QCD
However, NO π Targets!
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 48/52
First Contents Back Conclusion
Pion’s valence quark distn
π is Two-Body System: “Easiest” Bound State in QCD
However, NO π Targets!
Proved on
22/July/2002, ANL
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 48/52
First Contents Back Conclusion
Pion’s valence quark distn
π is Two-Body System: “Easiest” Bound State in QCD
However, NO π Targets!
Existing Measurement Inferred from Drell-Yan:
πN → µ+µ−X
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 48/52
First Contents Back Conclusion
Pion’s valence quark distn
π is Two-Body System: “Easiest” Bound State in QCD
However, NO π Targets!
Existing Measurement Inferred from Drell-Yan:
πN → µ+µ−X
Proposal (Holt & Reimer, ANL, nu-ex/0010004)
e−5GeV – p25 GeV Collider → Accurate “Measurement”
p n
πγ
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 48/52
First Contents Back Conclusion
Pion’s valence quark distn
Proposal at JLab
(Holt, Reimer, Wijesooriya, et al.,
JLab at 12 GeV)
p n
πγ
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 48/52
First Contents Back Conclusion
Handbag diagrams
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P P
k
q q
k-q-P
k-qk-q
k+q
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P P
k
k+q+P
k+q
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 49/52
First Contents Back Conclusion
Handbag diagrams
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P P
k
q q
k-q-P
k-qk-q
k+q
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P P
k
k+q+P
k+q
Wµν(q;P ) =1
2πIm[
T+µν(q;P ) + T−
µν(q;P )]
T+µν(q, P ) = tr
∫
d4k
(2π)4τ−Γπ(k
−1
2
;−P )S(k−0) ieQΓν(k−0, k)
×S(k) ieQΓµ(k, k−0)S(k−0)τ+Γπ(k−
1
2
;P )S(k−−)
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 49/52
First Contents Back Conclusion
Handbag diagrams
Bjorken Limit: q2 → ∞ , P · q → −∞
but x := −q2
2P · qfixed.
Numerous algebraic simplifications
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P P
k
q q
k-q-P
k-qk-q
k+q
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P P
k
k+q+P
k+q
Wµν(q;P ) =1
2πIm[
T+µν(q;P ) + T−
µν(q;P )]
T+µν(q, P ) = tr
∫
d4k
(2π)4τ−Γπ(k
−1
2
;−P )S(k−0) ieQΓν(k−0, k)
×S(k) ieQΓµ(k, k−0)S(k−0)τ+Γπ(k−
1
2
;P )S(k−−)
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 49/52
First Contents Back Conclusion
Extant theory vs. experiment
K. Wijersooriya, P. Reimer and R. Holt,
nu-ex/0509012 ... Phys. Rev. C (Rapid)
0.0 0.2 0.4 0.6 0.8 1.0x
0.0
0.1
0.2
0.3
0.4
x u
v(x)
E615 πN Drell−Yan 4GeVNLO Analysis of E615 ... β=1.87DSE ... β= 2.61NJL ... β= 1.27
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 50/52
First Contents Back Conclusion
Form Factor Ratio: Q ∗F2/F1
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 51/52
First Contents Back Conclusion
Form Factor Ratio: Q ∗F2/F1
0.0
1.0
2.0
3.0
4.0
Q2 F
2p/(
κ pF1p
) set BSLACJLab1JLab2
0 2 4 6 8 10Q
2 [GeV2]
0
0.5
1
1.5
Q F
2p/(
κ pF1p
)
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 51/52
First Contents Back Conclusion
Form Factor Ratio: Q ∗F2/F1
0.0
1.0
2.0
3.0
4.0
Q2 F
2p/(
κ pF1p
) set BSLACJLab1JLab2
0 2 4 6 8 10Q
2 [GeV2]
0
0.5
1
1.5
Q F
2p/(
κ pF1p
)
Perhaps ≈ constant for 2 ∼< Q2 ∼< 6GeV2
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 51/52
First Contents Back Conclusion
Formln Factor Ratio: alternativeF2/F1
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 52/52
First Contents Back Conclusion
Formln Factor Ratio: alternativeF2/F1
2 4 6 8 10Q
2 [GeV2]
0
0.5
1
1.5
(Q/ln
(Q2 /Λ
2 ))2 F
2p/(κ
pF1p
)
set BSLACJLab 1JLab 2
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 52/52
First Contents Back Conclusion
Formln Factor Ratio: alternativeF2/F1
2 4 6 8 10Q
2 [GeV2]
0
0.5
1
1.5
(Q/ln
(Q2 /Λ
2 ))2 F
2p/(κ
pF1p
)
set BSLACJLab 1JLab 2
Q2
[ln Q2/Λ2]2F2(Q
2)
F1(Q2)= constant, Q2 ≫ Λ2 ≈ M2
N
Suggestive
NB. Framework
constructed to give
quark-counting
i.e., “pQCD” but with
wrong anomalous
dimensions but they’re
ignored in ln-power “2” of
this ratio
IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 52/52
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