Post on 16-Apr-2020
First Contents Back Conclusion
Hadron Physics andContinuum Strong QCD
Craig D. Robertscdroberts@anl.gov
Physics Division & School of Physics
Argonne National Laboratory Peking University
http://www.phy.a nl.gov/theory/staff/cdr.htmlCraig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 1/48
First Contents Back Conclusion
Form Factors: Why?
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 2/48
First Contents Back Conclusion
Form Factors: Why?
The nucleon and pion hold special places in non-perturbativestudies of QCD.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 2/48
First Contents Back Conclusion
Form Factors: Why?
The nucleon and pion hold special places in non-perturbativestudies of QCD.
An explanation of nucleon and pion structure and interactions iscentral to hadron physics – they are respectively the archetypesfor baryons and mesons.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 2/48
First Contents Back Conclusion
Form Factors: Why?
The nucleon and pion hold special places in non-perturbativestudies of QCD.
An explanation of nucleon and pion structure and interactions iscentral to hadron physics – they are respectively the archetypesfor baryons and mesons.
Form factors have long been recognized as a basic tool forelucidating bound state properties. They can be studied from verylow momentum transfer, the region of non-perturbative QCD, up toa region where perturbative QCD predictions can be tested.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 2/48
First Contents Back Conclusion
Form Factors: Why?
The nucleon and pion hold special places in non-perturbativestudies of QCD.
An explanation of nucleon and pion structure and interactions iscentral to hadron physics – they are respectively the archetypesfor baryons and mesons.
Form factors have long been recognized as a basic tool forelucidating bound state properties. They can be studied from verylow momentum transfer, the region of non-perturbative QCD, up toa region where perturbative QCD predictions can be tested.
Experimental and theoretical studies of nucleon electromagneticform factors have made rapid and significant progress during thelast several years, including new data in the time like region, andmaterial gains have been made in studying the pion form factor.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 2/48
First Contents Back Conclusion
Form Factors: Why?
The nucleon and pion hold special places in non-perturbativestudies of QCD.
An explanation of nucleon and pion structure and interactions iscentral to hadron physics – they are respectively the archetypesfor baryons and mesons.
Form factors have long been recognized as a basic tool forelucidating bound state properties. They can be studied from verylow momentum transfer, the region of non-perturbative QCD, up toa region where perturbative QCD predictions can be tested.
Experimental and theoretical studies of nucleon electromagneticform factors have made rapid and significant progress during thelast several years, including new data in the time like region, andmaterial gains have been made in studying the pion form factor.
Despite this, many urgent questions remain unanswered.Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 2/48
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JLab
Thomas Jefferson National Accelerator Facility
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 3/48
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JLab
Thomas Jefferson National Accelerator Facility
World’s Premier Hadron Physics Facility
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 3/48
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JLab
Thomas Jefferson National Accelerator Facility
World’s Premier Hadron Physics Facility
Design goal (4 GeV) experiments began in 1995
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 3/48
First Contents Back Conclusion
JLab
Thomas Jefferson National Accelerator Facility
World’s Premier Hadron Physics Facility
Design goal (4 GeV) experiments began in 1995
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 3/48
First Contents Back Conclusion
JLab
Thomas Jefferson National Accelerator Facility
World’s Premier Hadron Physics Facility
Design goal (4 GeV) experiments began in 1995
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 3/48
First Contents Back Conclusion
JLab
Thomas Jefferson National Accelerator Facility
World’s Premier Hadron Physics Facility
Design goal (4 GeV) experiments began in 1995
Electrons accelerated by
repeated journeys along linacs
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 3/48
First Contents Back Conclusion
JLab
Thomas Jefferson National Accelerator Facility
World’s Premier Hadron Physics Facility
Design goal (4 GeV) experiments began in 1995
Electrons accelerated by
repeated journeys along linacs
Once desired energy is
reached, Beam is directed into
Experimental Halls A, B and C
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 3/48
First Contents Back Conclusion
JLab
Thomas Jefferson National Accelerator Facility
World’s Premier Hadron Physics Facility
Design goal (4 GeV) experiments began in 1995
Electrons accelerated by
repeated journeys along linacs
Once desired energy is
reached, Beam is directed into
Experimental Halls A, B and C
Current Peak
Electron Beam Energy
Nearly 6 GeV
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 3/48
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JLab Hall-A
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 4/48
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JLab Hall-A
Measured Ratio of
Proton’s Electric and Magnetic Form Factors
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 4/48
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JLab Hall-A
0 1 2 3 4 5 6Q
2 [GeV2]
0
0.2
0.4
0.6
0.8
1
1.2
µ p GEp/ G
Mp
SLACJLab 1JLab 2
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 4/48
First Contents Back Conclusion
JLab Hall-A
0 1 2 3 4 5 6Q
2 [GeV2]
0
0.2
0.4
0.6
0.8
1
1.2
µ p GEp/ G
Mp
SLACJLab 1JLab 2
Walker et al., Phys.Rev. D 49, 5671(1994). (SLAC)
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 HallA Collaboration, Phys.Rev. Lett. 88 092301(2002)
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 4/48
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JLab Hall-A
0 1 2 3 4 5 6Q
2 [GeV2]
0
0.2
0.4
0.6
0.8
1
1.2
µ p GEp/ G
Mp
SLACJLab 1JLab 2
If JLab Correct, then
Completely
Unexpected Result:
In the Proton
– On Relativistic
Domain
– Distribution of
Quark-Charge
Not Equal
Distribution of
Quark-Current!
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 4/48
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Some Questions
What is the role of pion cloud in nucleonelectromagnetic structure?
Can we understand the pion cloud in a morequantitative and, perhaps, model-independentway?
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 5/48
First Contents Back Conclusion
Some Questions
Where is the transition from non-pQCD to pQCD inthe pion and nucleon electromagnetic formfactors?
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 5/48
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Some Questions
Do we understand the high Q2 behavior of theproton form factor ratio in the space-like region?
Can we make model-independent statementsabout the role of relativity or orbital angularmomentum in the nucleon?
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 5/48
First Contents Back Conclusion
Some Questions
Can we understand the rich structure of thetime-like proton form factors in terms ofresonances?
What do we expect for the proton form factor ratioin the time-like region?
What is the relation between proton and neutronform factor in the time-like region?
How do we understand the ratio between time-likeand space-like form factors?
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 5/48
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Some Questions
What is the role of two-photon exchangecontributions in understanding the discrepancybetween the polarization and Rosenbluthmeasurements of the proton form factor ratio?
What is the impact of these contributions on otherform factor measurements?
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 5/48
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Some Questions
How accurately can the pion form factor beextracted from the ep → e′nπ+ reaction?
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 5/48
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Status
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 6/48
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StatusCurrent status is described in
J. Arrington, C. D. Roberts and J. M. Zanotti“Nucleon electromagnetic form factors,”J. Phys. G 34, S23 (2007); [arXiv:nucl-th/0611050].
C. F. Perdrisat, V. Punjabi and M. Vanderhaeghen,“Nucleon electromagnetic form factors,”Prog. Part. Nucl. Phys. 59, 694 (2007);[arXiv:hep-ph/0612014].
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 6/48
First Contents Back Conclusion
StatusCurrent status is described in
J. Arrington, C. D. Roberts and J. M. Zanotti“Nucleon electromagnetic form factors,”J. Phys. G 34, S23 (2007); [arXiv:nucl-th/0611050].
C. F. Perdrisat, V. Punjabi and M. Vanderhaeghen,“Nucleon electromagnetic form factors,”Prog. Part. Nucl. Phys. 59, 694 (2007);[arXiv:hep-ph/0612014].
Most recently:“ECT∗ Workshop on Hadron Electromagnetic Form Factors”Organisers: Alexandrou, Arrington, Friedrich, Maas, RobertsPresentations, etc., available on-linehttp://ect08.phy.anl.gov/
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 6/48
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QCD’s Challenges
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 7/48
First Contents Back Conclusion
QCD’s Challenges
Quark and Gluon Confinement
No matter how hard one strikes the proton, one
cannot liberate an individual quark or gluon
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 7/48
First Contents Back Conclusion
QCD’s Challenges
Quark and Gluon Confinement
No matter how hard one strikes the proton, one
cannot liberate an individual quark or gluon
Dynamical Chiral Symmetry Breaking
Very unnatural pattern of bound state masses
e.g., Lagrangian (pQCD) quark mass is small but . . .
no degeneracy between JP=+ and JP=−
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 7/48
First Contents Back Conclusion
QCD’s Challenges
Quark and Gluon Confinement
No matter how hard one strikes the proton, one
cannot liberate an individual quark or gluon
Dynamical Chiral Symmetry Breaking
Very unnatural pattern of bound state masses
e.g., Lagrangian (pQCD) quark mass is small but . . .
no degeneracy between JP=+ and JP=−
Neither of these phenomena is apparent in QCD’s
Lagrangian yet they are the dominant determining
characteristics of real-world QCD.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 7/48
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QCD’s ChallengesUnderstand Emergent Phenomena
Quark and Gluon Confinement
No matter how hard one strikes the proton, one
cannot liberate an individual quark or gluon
Dynamical Chiral Symmetry Breaking
Very unnatural pattern of bound state masses
e.g., Lagrangian (pQCD) quark mass is small but . . .
no degeneracy between JP=+ and JP=−
Neither of these phenomena is apparent in QCD’s
Lagrangian yet they are the dominant determining
characteristics of real-world QCD.
QCD – Complex behaviour
arises from apparently simple rulesCraig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 7/48
First Contents Back Conclusion
Confinement
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 8/48
First Contents Back Conclusion
Confinement
Infinitely Heavy Quarks . . . Picture in Quantum Mechanics
integration of the force-3 loops
bosonic string
V (r) = σ r − π
12
1
r
√σ ∼ 470 MeV
Necco & Sommer
he-la/0108008
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 8/48
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Confinement
Illustrate this in terms of the action density . . . analogous to
plotting the Force = FQ̄Q(r) = σ +π
12
1
r2
Bali, et al.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 8/48
First Contents Back Conclusion
Confinement
What happens in the real world; namely, in the presence of
light-quarks?
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 8/48
First Contents Back Conclusion
Confinement
What happens in the real world; namely, in the presence of
light-quarks? No one knows . . . but Q̄Q + 2 × s̄s
Bali, et al.
he-la/0512018
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 8/48
First Contents Back Conclusion
Confinement
What happens in the real world; namely, in the presence of
light-quarks? No one knows . . . but Q̄Q + 2 × s̄s
Bali, et al.
he-la/0512018“The breaking of the string appears to be an instantaneous
process, with de-localized light quark pair creation.”
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 8/48
First Contents Back Conclusion
Confinement
What happens in the real world; namely, in the presence of
light-quarks? No one knows . . . but Q̄Q + 2 × s̄s
Bali, et al.
he-la/0512018“The breaking of the string appears to be an instantaneous
process, with de-localized light quark pair creation.”
Energy stored in string at instant before disappearance:
Ec ≃ 1.25 GeV
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 8/48
First Contents Back Conclusion
Confinement
What happens in the real world; namely, in the presence of
light-quarks? No one knows . . . but Q̄Q + 2 × s̄s
Bali, et al.
he-la/0512018“The breaking of the string appears to be an instantaneous
process, with de-localized light quark pair creation.”
Energy stored in string at instant before disappearance:
Ec ≃ MS+MS̄, where MS is s-quark constituent-mass
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 8/48
First Contents Back Conclusion
Confinement
What happens in the real world; namely, in the presence of
light-quarks? No one knows . . . but Q̄Q + 2 × s̄s
Bali, et al.
he-la/0512018“The breaking of the string appears to be an instantaneous
process, with de-localized light quark pair creation.”
Energy stored in string at instant before disappearance:
Ec ≃ MS+MS̄, where MS is s-quark constituent-mass
Flux tube collapses instantly and entirely when the energy it
contains exceeds that required to produce the lightest
constituent quark-antiquark pair.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 8/48
First Contents Back Conclusion
ConfinementTherefore . . . No information onpotential between light-quarks.
What happens in the real world; namely, in the presence of
light-quarks? No one knows . . . but Q̄Q + 2 × s̄s
Bali, et al.
he-la/0512018“The breaking of the string appears to be an instantaneous
process, with de-localized light quark pair creation.”
Energy stored in string at instant before disappearance:
Ec ≃ MS+MS̄, where MS is s-quark constituent-mass
Flux tube collapses instantly and entirely when the energy it
contains exceeds that required to produce the lightest
constituent quark-antiquark pair.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 8/48
First Contents Back Conclusion
Dyson-Schwinger EquationsEuler-Lagrange equations for quantum field theory
Well suited to Relativistic Quantum Field Theory
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 9/48
First Contents Back Conclusion
Dyson-Schwinger EquationsEuler-Lagrange equations for quantum field theory
Well suited to Relativistic Quantum Field Theory
Simplest level: Generating Tool for Perturbation Theory. . . . . . . . . . . . . . . . . . . . . . . . . Materially Reduces Model Dependence
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 9/48
First Contents Back Conclusion
Dyson-Schwinger EquationsEuler-Lagrange equations for quantum field theory
Well suited to Relativistic Quantum Field Theory
Simplest level: Generating Tool for Perturbation Theory. . . . . . . . . . . . . . . . . . . . . . . . . Materially Reduces Model Dependence
NonPerturbative, Continuum approach to QCD
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 9/48
First Contents Back Conclusion
Dyson-Schwinger EquationsEuler-Lagrange equations for quantum field theory
Well suited to Relativistic Quantum Field Theory
Simplest level: Generating Tool for Perturbation Theory. . . . . . . . . . . . . . . . . . . . . . . . . Materially Reduces Model Dependence
NonPerturbative, Continuum approach to QCD
Hadrons as Composites of Quarks and GluonsQualitative and Quantitative Importance of:· Dynamical Chiral Symmetry Breaking
– Generation of fermion mass from nothing· Quark & Gluon Confinement
– Coloured objects not detected, not detectable?
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 9/48
First Contents Back Conclusion
Dyson-Schwinger EquationsEuler-Lagrange equations for quantum field theory
Well suited to Relativistic Quantum Field Theory
Simplest level: Generating Tool for Perturbation Theory. . . . . . . . . . . . . . . . . . . . . . . . . Materially Reduces Model Dependence
NonPerturbative, Continuum approach to QCD
Hadrons as Composites of Quarks and GluonsQualitative and Quantitative Importance of:· Dynamical Chiral Symmetry Breaking
– Generation of fermion mass from nothing· Quark & Gluon Confinement
– Coloured objects not detected, not detectable?
Understanding ⇒ InfraRed behaviour of αs(Q2)
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 9/48
First Contents Back Conclusion
Dyson-Schwinger EquationsEuler-Lagrange equations for quantum field theory
Well suited to Relativistic Quantum Field Theory
Simplest level: Generating Tool for Perturbation Theory. . . . . . . . . . . . . . . . . . . . . . . . . Materially Reduces Model Dependence
NonPerturbative, Continuum approach to QCD
Hadrons as Composites of Quarks and GluonsQualitative and Quantitative Importance of:· Dynamical Chiral Symmetry Breaking
– Generation of fermion mass from nothing· Quark & Gluon Confinement
– Coloured objects not detected, not detectable?
Understanding ⇒ InfraRed behaviour of αs(Q2)
Method yields Schwinger Functions ≡ Propagators
Cross-Sections built from Schwinger FunctionsCraig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 9/48
First Contents Back Conclusion
Schwinger Functions
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 10/48
First Contents Back Conclusion
Schwinger Functions
Solutions are Schwinger Functions(Euclidean Green Functions)
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 10/48
First Contents Back Conclusion
Schwinger Functions
Solutions are Schwinger Functions(Euclidean Green Functions)
Not all are Schwinger functions are experimentallyobservable
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 10/48
First Contents Back Conclusion
Schwinger Functions
Solutions are Schwinger Functions(Euclidean Green Functions)
Not all are Schwinger functions are experimentallyobservable but . . .
all are same VEVs measured in numericalsimulations of lattice-regularised QCDopportunity for comparisons atpre-experimental level . . . cross-fertilisation
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 10/48
First Contents Back Conclusion
Schwinger Functions
Solutions are Schwinger Functions(Euclidean Green Functions)
Not all are Schwinger functions are experimentallyobservable but . . .
all are same VEVs measured in numericalsimulations of lattice-regularised QCDopportunity for comparisons atpre-experimental level . . . cross-fertilisation
Proving fruitful.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 10/48
First Contents Back Conclusion
World . . .
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 11/48
First Contents Back Conclusion
World . . .DSE Perspective
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 11/48
First Contents Back Conclusion
Persistent Challenge
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 12/48
First Contents Back Conclusion
Persistent Challenge
Infinitely Many Coupled Equations
Σ=
D
γΓS
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 12/48
First Contents Back Conclusion
Persistent Challenge
Infinitely Many Coupled Equations
Σ=
D
γΓS
Coupling between equations necessitates truncation
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 12/48
First Contents Back Conclusion
Persistent Challenge
Infinitely Many Coupled Equations
Σ=
D
γΓS
Coupling between equations necessitates truncation
Weak coupling expansion ⇒ Perturbation Theory
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 12/48
First Contents Back Conclusion
Persistent Challenge
Infinitely Many Coupled Equations
Σ=
D
γΓS
Coupling between equations necessitates truncation
Weak coupling expansion ⇒ Perturbation TheoryNot useful for the nonperturbative problemsin which we’re interested
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 12/48
First Contents Back Conclusion
Persistent Challenge
Infinitely Many Coupled Equations
There is at least one systematic nonperturbative,symmetry-preserving truncation schemeH.J. Munczek Phys. Rev. D 52 (1995) 4736Dynamical chiral symmetry breaking, Goldstone’stheorem and the consistency of the Schwinger-Dysonand Bethe-Salpeter EquationsA. Bender, C. D. Roberts and L. von Smekal, Phys.Lett. B 380 (1996) 7Goldstone Theorem and Diquark Confinement BeyondRainbow Ladder Approximation
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 13/48
First Contents Back Conclusion
Persistent Challenge
Infinitely Many Coupled Equations
There is at least one systematic nonperturbative,symmetry-preserving truncation scheme
Has Enabled Proof of EXACT Results in QCD
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 13/48
First Contents Back Conclusion
Persistent Challenge
Infinitely Many Coupled Equations
There is at least one systematic nonperturbative,symmetry-preserving truncation scheme
Has Enabled Proof of EXACT Results in QCD
And Formulation of Practical Phenomenological Tool to
Illustrate Exact Results
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 13/48
First Contents Back Conclusion
Persistent Challenge
Infinitely Many Coupled Equations
There is at least one systematic nonperturbative,symmetry-preserving truncation scheme
Has Enabled Proof of EXACT Results in QCD
And Formulation of Practical Phenomenological Tool to
Illustrate Exact Results
Make Predictions with Readily Quantifiable Errors
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 13/48
First Contents Back Conclusion
Persistent Challenge
Infinitely Many Coupled Equations
There is at least one systematic nonperturbative,symmetry-preserving truncation scheme
Has Enabled Proof of EXACT Results in QCD
And Formulation of Practical Phenomenological Tool to
Illustrate Exact Results
Make Predictions with Readily Quantifiable Errors
Examples:MIT – The Net Advance of PhysicsReview Articles and Tutorials in an Encyclopædic Formatweb.mit.edu/redingtn/www/netadv/Xdysonschw.html
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 13/48
First Contents Back Conclusion
PerturbativeDressed-quark Propagator
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 14/48
First Contents Back Conclusion
PerturbativeDressed-quark Propagator
S(p) =Z(p2)
iγ · p + M(p2)Σ
=D
γΓS
Gap Equation
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 14/48
First Contents Back Conclusion
PerturbativeDressed-quark Propagator
S(p) =Z(p2)
iγ · p + M(p2)Σ
=D
γΓS
Gap Equationdressed-quark propagator
S(p) =1
iγ · pA(p2) + B(p2)
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 14/48
First Contents Back Conclusion
PerturbativeDressed-quark Propagator
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
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 14/48
First Contents Back Conclusion
PerturbativeDressed-quark Propagator
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
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 14/48
First Contents Back Conclusion
PerturbativeDressed-quark Propagator
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!
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 14/48
First Contents Back Conclusion
Explanation?
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 15/48
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).
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 16/48
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
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 16/48
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
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 17/48
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
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 17/48
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
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 17/48
First Contents Back Conclusion
Dressed-Quark Propagator
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 18/48
First Contents Back Conclusion
Dressed-Quark Propagator
S(p) =Z(p2)
iγ · p + M(p2)Σ
=D
γΓS
Gap Equation
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 18/48
First Contents Back Conclusion
Dressed-Quark Propagator
S(p) =Z(p2)
iγ · p + M(p2)Σ
=D
γΓS
Gap EquationGap Equation’s Kernel Enhanced on IR domain
⇒ IR Enhancement of M(p2)
10−2 10−1 100 101 102
p2 (GeV2)
10−3
10−2
10−1
100
101
M(p
2 ) (G
eV)
b−quarkc−quarks−quarku,d−quarkchiral limitM
2(p
2) = p
2
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 18/48
First Contents Back Conclusion
Dressed-Quark Propagator
S(p) =Z(p2)
iγ · p + M(p2)Σ
=D
γΓS
Gap EquationGap Equation’s Kernel Enhanced on IR domain
⇒ IR Enhancement of M(p2)
10−2 10−1 100 101 102
p2 (GeV2)
10−3
10−2
10−1
100
101
M(p
2 ) (G
eV)
b−quarkc−quarks−quarku,d−quarkchiral limitM
2(p
2) = p
2
Euclidean Constituent–Quark
Mass: MEf : p2 = M(p2)2
flavour u/d s c b
ME
mζ∼ 102 ∼ 10 ∼ 1.5 ∼ 1.1
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 18/48
First Contents Back Conclusion
Dressed-Quark Propagator
S(p) =Z(p2)
iγ · p + M(p2)Σ
=D
γΓS
Gap EquationGap Equation’s Kernel Enhanced on IR domain
⇒ IR Enhancement of M(p2)
10−2 10−1 100 101 102
p2 (GeV2)
10−3
10−2
10−1
100
101
M(p
2 ) (G
eV)
b−quarkc−quarks−quarku,d−quarkchiral limitM
2(p
2) = p
2
Euclidean Constituent–Quark
Mass: MEf : p2 = M(p2)2
flavour u/d s c b
ME
mζ∼ 102 ∼ 10 ∼ 1.5 ∼ 1.1
Predictions confirmed innumerical simulations of lattice-QCD
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 18/48
First Contents Back Conclusion
Dressed-QuarkPropagator
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 19/48
First Contents Back Conclusion
Dressed-QuarkPropagator
Longstanding Prediction of
Dyson-Schwinger Equation
Studies
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 19/48
First Contents Back Conclusion
Dressed-QuarkPropagator
Longstanding Prediction of
Dyson-Schwinger Equation
Studies
E.g., Dyson-Schwinger
equations and their
application to hadronic
physics,
C. D. Roberts and
A. G. Williams,
Prog. Part. Nucl. Phys.
33 (1994) 477
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 19/48
First Contents Back Conclusion
Dressed-QuarkPropagator
Longstanding Prediction of
Dyson-Schwinger Equation
Studies
E.g., Dyson-Schwinger
equations and their
application to hadronic
physics,
C. D. Roberts and
A. G. Williams,
Prog. Part. Nucl. Phys.
33 (1994) 477
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 19/48
First Contents Back Conclusion
Dressed-QuarkPropagator
Longstanding Prediction of
Dyson-Schwinger Equation
Studies
E.g., Dyson-Schwinger
equations and their
application to hadronic
physics,
C. D. Roberts and
A. G. Williams,
Prog. Part. Nucl. Phys.
33 (1994) 477
Long used as basis for
efficacious hadron physics
phenomenology
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 19/48
First Contents Back Conclusion
Dressed-QuarkPropagator
Longstanding Prediction of
Dyson-Schwinger Equation
Studies
E.g., Dyson-Schwinger
equations and their
application to hadronic
physics,
C. D. Roberts and
A. G. Williams,
Prog. Part. Nucl. Phys.
33 (1994) 477
Long used as basis for
efficacious hadron physics
phenomenology
Electromagnetic pion
form-factor and neutral
pion decay width,
C. D. Roberts,
Nucl. Phys. A 605(1996) 475
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 19/48
First Contents Back Conclusion
Frontiers of Nuclear Science:A Long Range Plan (2007)
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 20/48
First Contents Back Conclusion
Frontiers of Nuclear Science:Theoretical Advances
Σ=
D
γΓS
Gap Equation
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 20/48
First Contents Back Conclusion
Frontiers of Nuclear Science:Theoretical Advances
Σ=
D
γΓS
Gap Equation
S(p) =Z(p2)
iγ · p + M(p2)
0 1 2 3
p [GeV]
0
0.1
0.2
0.3
0.4
M(p
) [G
eV] m = 0 (Chiral limit)
m = 30 MeVm = 70 MeV
effect of gluon cloudRapid acquisition of mass is
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 20/48
First Contents Back Conclusion
Frontiers of Nuclear Science:Theoretical Advances
S(p) =Z(p2)
iγ · p + M(p2)
0 1 2 3
p [GeV]
0
0.1
0.2
0.3
0.4
M(p
) [G
eV] m = 0 (Chiral limit)
m = 30 MeVm = 70 MeV
effect of gluon cloudRapid acquisition of mass is
Mass from nothing .
In QCD a quark’s effective massdepends on its momentum. Thefunction describing this can becalculated and is depicted here.Numerical simulations of latticeQCD (data, at two different baremasses) have confirmed modelpredictions (solid curves) that thevast bulk of the constituent massof a light quark comes from acloud of gluons that are draggedalong by the quark as itpropagates. In this way, a quarkthat appears to be absolutelymassless at high energies(m = 0, red curve) acquires alarge constituent mass at lowenergies.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 20/48
First Contents Back Conclusion
Frontiers of Nuclear Science:Theoretical Advances
S(p) =Z(p2)
iγ · p + M(p2)
0 1 2 3
p [GeV]
0
0.1
0.2
0.3
0.4
M(p
) [G
eV] m = 0 (Chiral limit)
m = 30 MeVm = 70 MeV
effect of gluon cloudRapid acquisition of mass is
Mass from nothing .
In QCD a quark’s effective massdepends on its momentum. Thefunction describing this can becalculated and is depicted here.Numerical simulations of latticeQCD (data, at two different baremasses) have confirmed modelpredictions (solid curves) that thevast bulk of the constituent massof a light quark comes from acloud of gluons that are draggedalong by the quark as itpropagates. In this way, a quarkthat appears to be absolutelymassless at high energies(m = 0, red curve) acquires alarge constituent mass at lowenergies.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 20/48
First Contents Back Conclusion
Frontiers of Nuclear Science:Theoretical Advances
S(p) =Z(p2)
iγ · p + M(p2)
0 1 2 3
p [GeV]
0
0.1
0.2
0.3
0.4
M(p
) [G
eV] m = 0 (Chiral limit)
m = 30 MeVm = 70 MeV
effect of gluon cloudRapid acquisition of mass is
Mass from nothing .
In QCD a quark’s effective massdepends on its momentum. Thefunction describing this can becalculated and is depicted here.Numerical simulations of latticeQCD (data, at two different baremasses) have confirmed modelpredictions (solid curves) that thevast bulk of the constituent massof a light quark comes from acloud of gluons that are draggedalong by the quark as itpropagates. In this way, a quarkthat appears to be absolutelymassless at high energies(m = 0, red curve) acquires alarge constituent mass at lowenergies.
↑ ↑
Scanned by Q2 ∈ [2,9] GeV2 Baryon Form FactorsCraig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 20/48
First Contents Back Conclusion
Frontiers of Nuclear Science:Theoretical Advances
In QCD
a quark’s mass must depend on
its momentum
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 21/48
First Contents Back ConclusionCraig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 22/48
First Contents Back Conclusion
• Established understanding oftwo- and three-point functions
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 22/48
First Contents Back Conclusion
Hadrons
• Established understanding oftwo- and three-point functions
• What about bound states?
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 22/48
First Contents Back Conclusion
Hadrons
• Without bound states, Comparison withexperiment is impossible
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 22/48
First Contents Back Conclusion
Hadrons
• Without bound states, Comparison withexperiment is impossible
• They appear as pole contributions to n ≥ 3-pointcolour-singlet Schwinger functions
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 22/48
First Contents Back Conclusion
Hadrons
• Without bound states, Comparison withexperiment is impossible
• Bethe-Salpeter Equation
QFT Generalisation of Lippmann-Schwinger Equation.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 22/48
First Contents Back Conclusion
Hadrons
• Without bound states, Comparison withexperiment is impossible
• Bethe-Salpeter Equation
QFT Generalisation of Lippmann-Schwinger Equation.
• What is the kernel, K?
or What is the long-range potential in QCD?Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 22/48
First Contents Back Conclusion
What is the light-quarkLong-Range Potential?
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 23/48
First Contents Back Conclusion
What is the light-quarkLong-Range Potential?
Potential between static (infinitely heavy) quarksmeasured in simulations of lattice-QCD is not relatedin any simple way to the light-quark interaction.Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 23/48
First Contents Back Conclusion
Bethe-Salpeter Kernel
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 24/48
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
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 24/48
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
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 24/48
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 very differentbut must be intimately related
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 24/48
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 very differentbut must be intimately related
• Relation must be preserved by truncation
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 24/48
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 very differentbut must be intimately related
• Relation must be preserved by truncation• Nontrivial constraint
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 24/48
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 very differentbut must be intimately related
• Relation must be preserved by truncation• Failure ⇒ Explicit Violation of QCD’s Chiral Symmetry
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 24/48
First Contents Back Conclusion
Goldstone’s Theorem
In the chiral limit the QCD Action possesses chiral symmetry
The chiral limit is a good approximation in QCDfor u- and d-quarks
If this SU(Nf = 2) chiral symmetry is dynamically broken, thenthere is a massless composite particle associated with eachgenerator of chiral transformations; i.e., three Goldstone Bosons
These three Goldstone Bosons have long been identified with thepions: π+ , π0 , π−
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 25/48
First Contents Back Conclusion
Goldstone’s Theorem
In the chiral limit the QCD Action possesses chiral symmetry
The chiral limit is a good approximation in QCDfor u- and d-quarks
If this SU(Nf = 2) chiral symmetry is dynamically broken, thenthere is a massless composite particle associated with eachgenerator of chiral transformations; i.e., three Goldstone Bosons
These three Goldstone Bosons have long been identified with thepions: π+ , π0 , π−
E.g., V (x, y) = (σ2 + π2 − 1)2
– Hamiltonian: T + V , is Rotationally Invariant
-1
-0.5
0
0.5
1-1
-0.5
0
0.5
1
0
0.5
1
1.5
-1
-0.5
0
0.5
1
•
Ground State
Ball at any (σ, π)
for which σ2 + π2 = 1
All Positions have Same (Minimum) EnergyBut not invariant under rotationsCraig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 25/48
First Contents Back Conclusion
Goldstone’s Theorem
In the chiral limit the QCD Action possesses chiral symmetry
The chiral limit is a good approximation in QCDfor u- and d-quarks
If this SU(Nf = 2) chiral symmetry is dynamically broken, thenthere is a massless composite particle associated with eachgenerator of chiral transformations; i.e., three Goldstone Bosons
These three Goldstone Bosons have long been identified with thepions: π+ , π0 , π−
If one assumes the s-quark is also light; namely, assumes thatSU(Nf = 3) chiral symmetry is a good approximation, then thekaons are four more Goldstone Bosons
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 25/48
First Contents Back Conclusion
Pion and . . .Pseudoscalar Mesons?
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 26/48
First Contents Back Conclusion
Pion and . . .Pseudoscalar Mesons?
Can a bound-state of massive constituents truly bemassless . . . without fine-tuning?
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 26/48
First Contents Back Conclusion
Dichotomy of Pion– Goldstone Mode and Bound state
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 27/48
First Contents Back Conclusion
Dichotomy of Pion– Goldstone Mode and Bound state
How does one make an almost massless particle. . . . . . . . . . . from two massive constituent-quarks?
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 27/48
First Contents Back Conclusion
Dichotomy of Pion– Goldstone Mode and Bound state
How does one make an almost massless particle. . . . . . . . . . . from two massive constituent-quarks?
Not Allowed to do it by fine-tuning a potential
Must exhibit m2π ∝ mq
Current Algebra . . . 1968
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 27/48
First Contents Back Conclusion
Dichotomy of Pion– Goldstone Mode and Bound state
How does one make an almost massless particle. . . . . . . . . . . from two massive constituent-quarks?
Not Allowed to do it by fine-tuning a potential
Must exhibit m2π ∝ 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 and valid chiral limit;
and an accurate realisation ofdynamical chiral symmetry breaking.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 27/48
First Contents Back Conclusion
Dichotomy of Pion– Goldstone Mode and Bound state
How does one make an almost massless particle. . . . . . . . . . . from two massive constituent-quarks?
Not Allowed to do it by fine-tuning a potential
Must exhibit m2π ∝ 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 and valid chiral limit;
and an accurate realisation ofdynamical chiral symmetry breaking.
Highly NontrivialCraig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 27/48
First Contents Back Conclusion
Resolving the Dichotomy
Minimal requirements
detailed understanding of connection between
Current-quark and Constituent-quark masses;
and systematic, symmetry preserving means of realising
this connection in bound-states.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 28/48
First Contents Back Conclusion
Resolving the Dichotomy
Minimal requirements
detailed understanding of connection between
Current-quark and Constituent-quark masses;
and systematic, symmetry preserving means of realising
this connection in bound-states.
Satisfying these requirements enables
Proof of numerous exact results for pseudoscalar
mesons
Formulation of reliable models
To illustrate those results
Make predictions of observables with quantifiable
errors
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 28/48
First Contents Back Conclusion
Goldberger-Treiman for pionMaris, Roberts, Tandynucl-th/9707003
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 29/48
First Contents Back Conclusion
Goldberger-Treiman for pionMaris, Roberts, Tandynucl-th/9707003
• 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 )]
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 29/48
First Contents Back Conclusion
Goldberger-Treiman for pionMaris, Roberts, Tandynucl-th/9707003
• 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)
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 29/48
First Contents Back Conclusion
Goldberger-Treiman for pionMaris, Roberts, Tandynucl-th/9707003
• 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)
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 29/48
First Contents Back Conclusion
Goldberger-Treiman for pionMaris, Roberts, Tandynucl-th/9707003
• 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
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 29/48
First Contents Back Conclusion
Goldberger-Treiman for pionMaris, Roberts, Tandynucl-th/9707003
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
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 29/48
First Contents Back Conclusion
Radial Excitations& Chiral Symmetry
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 30/48
First Contents Back Conclusion
Radial Excitations& Chiral Symmetry(Maris, Roberts, Tandy
nu-th/9707003 )
fH m2H = − ρH
ζ MH
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 30/48
First Contents Back Conclusion
Radial Excitations& Chiral Symmetry(Maris, Roberts, Tandy
nu-th/9707003 )
fH m2H = − ρH
ζ MH
• Mass2 of pseudoscalar hadron
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 30/48
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)
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 30/48
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
=
〈0|q̄γ5γµq|π〉
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 30/48
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
i
i
i
iP π
πρ
k
Γ
S
(τ/2) γ
S
555
=
〈0|q̄γ5q|π〉
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 30/48
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 →−〈q̄q〉0ζ
f0H
, Independent of mq
Hence m2H =
−〈q̄q〉0ζ(f0
H)2mq . . . GMOR relation, a corollary
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 30/48
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 →−〈q̄q〉0ζ
f0H
, Independent of mq
Hence m2H =
−〈q̄q〉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 resultCraig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 30/48
First Contents Back Conclusion
Radial Excitations
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 31/48
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)
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 31/48
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?
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 31/48
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:SQ̄Q = 1 ⊕ LF = 1 ⇒ J = 0
& LF = 1 ⇒ 3S1 ⊕ 3S1 (Q̄Q) decays suppressed?
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 31/48
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
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 31/48
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”Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 31/48
First Contents Back Conclusion
Radial Excitations& Chiral SymmetryHöll, Krassnigg, Roberts
nu-th/0406030
fH m2H = − ρH
ζ MH
Valid for ALL Pseudoscalar mesons
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 32/48
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
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 32/48
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, not the ground state, so
m2πn 6=0
> m2πn=0
= 0, in chiral limit
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 32/48
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, not the ground state, so
m2πn 6=0
> m2πn=0
= 0, in chiral limit
⇒ fH = 0
ALL pseudoscalar mesons except π(140) in chiral limit
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 32/48
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, not the ground state, so
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 mesonCraig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 32/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 33/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
non-Abelian Anomaly and η-η′ mixing
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 33/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
non-Abelian Anomaly and η-η′ mixing
Mesons containing s̄-s are special: η & η′
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 33/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
non-Abelian Anomaly and η-η′ mixing
Mesons containing s̄-s are special: η & η′
Problem: η′ is a pseudoscalar meson but it’s much more
massive than the other eight constituted from light-quarks.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 33/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
non-Abelian Anomaly and η-η′ mixing
Mesons containing s̄-s are special: η & η′
Problem: η′ is a pseudoscalar meson but it’s much more
massive than the other eight constituted from light-quarks.
Origin: While the classical action associated with QCD is
invariant under UA(1) (Abelian axial transformations
generated by λ0γ5), the quantum field theory is not!
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 33/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
non-Abelian Anomaly and η-η′ mixing
Mesons containing s̄-s are special: η & η′
Flavour mixing takes place in singlet channel: λ0 ⇔ λ8
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 33/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
non-Abelian Anomaly and η-η′ mixing
Mesons containing s̄-s are special: η & η′
Flavour mixing takes place in singlet channel: λ0 ⇔ λ8
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 33/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
non-Abelian Anomaly and η-η′ mixing
Mesons containing s̄-s are special: η & η′
This is a perturbative diagram.
It has almost nothing to do with η ⇔ η′ mixing.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 33/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
non-Abelian Anomaly and η-η′ mixing
Mesons containing s̄-s are special: η & η′
Driver is the non-Abelian anomaly
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 33/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
non-Abelian Anomaly and η-η′ mixing
Mesons containing s̄-s are special: η & η′
Driver is the non-Abelian anomaly
KA ∼
f1 f2
IS
IS
e.g. IS =
f1 f2
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 33/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
non-Abelian Anomaly and η-η′ mixing
Mesons containing s̄-s are special: η & η′
Driver is the non-Abelian anomaly
KA ∼
f1 f2
IS
IS
e.g. IS =
f1 f2
Contribution to the
Bethe-Salpeter kernel
associated with the
non-Abelian anomaly.
All terms have the
“hairpin” structure.
No finite sum of such intermediate states is sufficient to
veraciously represent the anomaly.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 33/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
PµΓa5µ(k;P ) = S−1(k+)iγ5Fa + iγ5FaS−1(k−)
−2iMabΓb5(k;P ) −Aa(k;P )
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 34/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
PµΓa5µ(k;P ) = S−1(k+)iγ5Fa + iγ5FaS−1(k−)
−2iMabΓb5(k;P ) −Aa(k;P )
{Fa| a = 0, . . . , N2f − 1} are the generators of U(Nf )
S = diag[Su, Sd, Ss, Sc, Sb, . . .]
Mab = trF
[
{Fa,M}F b]
,
M = diag[mu,md,ms,mc,mb, . . .] = matrix of current-quark
bare masses
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 34/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
PµΓa5µ(k;P ) = S−1(k+)iγ5Fa + iγ5FaS−1(k−)
−2iMabΓb5(k;P ) −Aa(k;P )
{Fa| a = 0, . . . , N2f − 1} are the generators of U(Nf )
S = diag[Su, Sd, Ss, Sc, Sb, . . .]
Mab = trF
[
{Fa,M}F b]
,
M = diag[mu,md,ms,mc,mb, . . .] = matrix of current-quark
bare masses
The final term in the second line expresses the non-Abelian
axial anomaly.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 34/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
PµΓa5µ(k;P ) = S−1(k+)iγ5Fa + iγ5FaS−1(k−)
−2iMabΓb5(k;P ) −Aa(k;P )
Aa(k;P ) = S−1(k+) δa0 AU (k;P )S−1(k−)
AU (k;P ) =
∫
d4xd4y ei(k+·x−k−·y)Nf
⟨
F0q(x)Q(0) q̄(y)⟩
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 34/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
PµΓa5µ(k;P ) = S−1(k+)iγ5Fa + iγ5FaS−1(k−)
−2iMabΓb5(k;P ) −Aa(k;P )
Aa(k;P ) = S−1(k+) δa0 AU (k;P )S−1(k−)
AU (k;P ) =
∫
d4xd4y ei(k+·x−k−·y)Nf
⟨
F0q(x)Q(0) q̄(y)⟩
Q(x) = iαs
4πtrC [ǫµνρσFµνFρσ(x)] = ∂µKµ(x)
. . . The topological charge density operator.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 34/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
PµΓa5µ(k;P ) = S−1(k+)iγ5Fa + iγ5FaS−1(k−)
−2iMabΓb5(k;P ) −Aa(k;P )
Aa(k;P ) = S−1(k+) δa0 AU (k;P )S−1(k−)
AU (k;P ) =
∫
d4xd4y ei(k+·x−k−·y)Nf
⟨
F0q(x)Q(0) q̄(y)⟩
Q(x) = iαs
4πtrC [ǫµνρσFµνFρσ(x)] = ∂µKµ(x)
. . . The topological charge density operator.
(Trace is over colour indices & Fµν = 12λaF a
µν .)
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 34/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
PµΓa5µ(k;P ) = S−1(k+)iγ5Fa + iγ5FaS−1(k−)
−2iMabΓb5(k;P ) −Aa(k;P )
Aa(k;P ) = S−1(k+) δa0 AU (k;P )S−1(k−)
AU (k;P ) =
∫
d4xd4y ei(k+·x−k−·y)Nf
⟨
F0q(x)Q(0) q̄(y)⟩
Q(x) = iαs
4πtrC [ǫµνρσFµνFρσ(x)] = ∂µKµ(x)
. . . The topological charge density operator.
Important that only Aa=0 is nonzero.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 34/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
PµΓa5µ(k;P ) = S−1(k+)iγ5Fa + iγ5FaS−1(k−)
−2iMabΓb5(k;P ) −Aa(k;P )
Aa(k;P ) = S−1(k+) δa0 AU (k;P )S−1(k−)
AU (k;P ) =
∫
d4xd4y ei(k+·x−k−·y)Nf
⟨
F0q(x)Q(0) q̄(y)⟩
Q(x) = iαs
4πtrC [ǫµνρσFµνFρσ(x)] = ∂µKµ(x)
. . . The topological charge density operator.
NB. While Q(x) is gauge invariant, the associated
Chern-Simons current, Kµ, is not ⇒ in QCD no physical
boson can couple to Kµ and hence no physical states can
contribute to resolution of UA(1) problem.Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 34/48
First Contents Back Conclusion
Charge NeutralPseudoscalar MesonsBhagwat, Chang, Liu, Roberts, Tandy
nucl-th/arXiv:0708.1118
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 35/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
Bhagwat, Chang, Liu, Roberts, Tandynucl-th/arXiv:0708.1118
Only A0 6≡ 0 is interesting
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 35/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
Bhagwat, Chang, Liu, Roberts, Tandynucl-th/arXiv:0708.1118
Only A0 6≡ 0 is interesting . . . otherwise all pseudoscalar
mesons are Goldstone Modes!
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 35/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
Bhagwat, Chang, Liu, Roberts, Tandynucl-th/arXiv:0708.1118
Anomaly term has structureA0(k;P ) = F0γ5 [iEA(k;P ) + γ · PFA(k;P )
+γ · kk · PGA(k;P ) + σµνkµPνHA(k;P )]
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 35/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
Bhagwat, Chang, Liu, Roberts, Tandynucl-th/arXiv:0708.1118
AVWTI gives generalised Goldberger-Treiman relations
2f0η′EBS(k; 0) = 2B0(k
2) − EA(k; 0),
F 0R(k; 0) + 2f0
η′FBS(k; 0) = A0(k2) − FA(k; 0),
G0R(k; 0) + 2f0
η′GBS(k; 0) = 2A′0(k
2) − GA(k; 0),
H0R(k; 0) + 2f0
η′HBS(k; 0) = −HA(k; 0),
A0, B0 characterise gap equation’s chiral limit solution.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 35/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
Bhagwat, Chang, Liu, Roberts, Tandynucl-th/arXiv:0708.1118
AVWTI gives generalised Goldberger-Treiman relations
2f0η′EBS(k; 0) = 2B0(k
2) − EA(k; 0),
F 0R(k; 0) + 2f0
η′FBS(k; 0) = A0(k2) − FA(k; 0),
G0R(k; 0) + 2f0
η′GBS(k; 0) = 2A′0(k
2) − GA(k; 0),
H0R(k; 0) + 2f0
η′HBS(k; 0) = −HA(k; 0),
A0, B0 characterise gap equation’s chiral limit solution.
Follows that EA(k; 0) = 2B0(k2) is necessary and
sufficient condition for absence of massless η′ bound-state.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 35/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
Bhagwat, Chang, Liu, Roberts, Tandynucl-th/arXiv:0708.1118
EA(k; 0) = 2B0(k2)
Discussing the chiral limit
B0(k2) 6= 0 if, and only if, chiral symmetry is dynamically
broken.
Hence, absence of massless η′ bound-state is only
assured through existence of intimate connection
between DCSB and an expectation value of the
topological charge density.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 35/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
Bhagwat, Chang, Liu, Roberts, Tandynucl-th/arXiv:0708.1118
EA(k; 0) = 2B0(k2)
Discussing the chiral limit
B0(k2) 6= 0 if, and only if, chiral symmetry is dynamically
broken.
Hence, absence of massless η′ bound-state is only
assured through existence of intimate connection
between DCSB and an expectation value of the
topological charge density.
Further highlighted . . . proved〈q̄q〉0ζ = − lim
Λ→∞Z4(ζ
2,Λ2) trCD
∫ Λ
qS0(q, ζ)
= Nf
∫
d4x 〈q̄(x)iγ5q(x)Q(0)〉0 .
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 35/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
Bhagwat, Chang, Liu, Roberts, Tandynucl-th/arXiv:0708.1118
AVWTI ⇒ QCD mass formulae for neutral pseudoscalar mesons
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 36/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
Bhagwat, Chang, Liu, Roberts, Tandynucl-th/arXiv:0708.1118
AVWTI ⇒ QCD mass formulae for neutral pseudoscalar mesons
Implications of mass formulae illustrated using elementarydynamical model, which includes Ansatz for that part of theBethe-Salpeter kernel related to the non-Abelian anomaly
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 36/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
Bhagwat, Chang, Liu, Roberts, Tandynucl-th/arXiv:0708.1118
AVWTI ⇒ QCD mass formulae for neutral pseudoscalar mesons
Implications of mass formulae illustrated using elementarydynamical model, which includes Ansatz for that part of theBethe-Salpeter kernel related to the non-Abelian anomaly
Employed in an analysis of pseudoscalar- and vector-mesonbound-states
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 36/48
First Contents Back Conclusion
Charge NeutralPseudoscalar Mesons
Bhagwat, Chang, Liu, Roberts, Tandynucl-th/arXiv:0708.1118
AVWTI ⇒ QCD mass formulae for neutral pseudoscalar mesons
Implications of mass formulae illustrated using elementarydynamical model, which includes Ansatz for that part of theBethe-Salpeter kernel related to the non-Abelian anomaly
Despite its simplicity, model is elucidative and phenomenologicallyefficacious; e.g., it predicts
η–η′ mixing angles of ∼ −15◦ (Expt.: −13.3◦ ± 1.0◦)
π0–η angles of ∼ 1.2◦ (Expt. p d → 3He π0: 0.6◦ ± 0.3◦)
Strong neutron-proton mass difference . . .
∼< 75 % current-quark mass-difference
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 36/48
First Contents Back Conclusion
Ab-Initio Calculations
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 37/48
First Contents Back Conclusion
Ab-Initio Calculations
Pieter Maris Peter Tandy
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 37/48
First Contents Back Conclusion
Ab-Initio Calculations
Maris & Tandy, Series of Five Articles: 1999 – Present
Perfected a Renormalisation-Group ImprovedRainbow-Ladder Model of Quark-Quark Interaction
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 37/48
First Contents Back Conclusion
Ab-Initio Calculations
Maris & Tandy, Series of Five Articles: 1999 – Present
Perfected a Renormalisation-Group ImprovedRainbow-Ladder Model of Quark-Quark Interaction
• Rainbow-Ladder = First Orderin Truncation Described Above
• Anticipate Accurate for 0− & 1− Mesons
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 37/48
First Contents Back Conclusion
Ab-Initio Calculations
Maris & Tandy, Series of Five Articles: 1999 – Present
Perfected a Renormalisation-Group ImprovedRainbow-Ladder Model of Quark-Quark Interaction
• One Parameter = Interaction Energy:E ≈ 700 MeV
• Dressed-Glue Mass scale:Characterises DCSB and light-quark Confinement
• Both Phenomena Disappear for E . 200 MeV
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 37/48
First Contents Back Conclusion
Ab-Initio Calculations
Maris & Tandy, Series of Five Articles: 1999 – Present
Perfected a Renormalisation-Group ImprovedRainbow-Ladder Model of Quark-Quark Interaction
• One Parameter = Interaction Energy:E ≈ 700 MeV
• Dressed-Glue Mass scale:Characterises DCSB and light-quark Confinement
• Both Phenomena Disappear for E . 200 MeV
• Dyson-Schwinger equations:A Tool for Hadron Physics
P. Maris and C.D. Roberts, nu-th/0301049Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 37/48
First Contents Back Conclusion
Interaction
0 1 2 3 4 5
q2 (GeV
2)
0
1
2
3
4
αeff (q
2 )Perturbative evolution
Nonperturbative IR Enhancement
ε=720 MeV
Kernel ofBethe-SalpeterEquation
K(p, k;P ) ≈αeff((p − k)2)
(p − k)2
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 38/48
First Contents Back Conclusion
Interaction
0 1 2 3 4 5
q2 (GeV
2)
0
1
2
3
4
αeff (q
2 )Perturbative evolution
Nonperturbative IR Enhancement
ε=720 MeV
Kernel ofBethe-SalpeterEquation
K(p, k;P ) ≈αeff((p − k)2)
(p − k)2
Prescribes GapEquation’s Kernel
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 38/48
First Contents Back Conclusion
Interaction
0 1 2 3 4 5
q2 (GeV
2)
0
1
2
3
4
αeff (q
2 )Perturbative evolution
Nonperturbative IR Enhancement
ε=720 MeV
Kernel ofBethe-SalpeterEquation
K(p, k;P ) ≈αeff((p − k)2)
(p − k)2
Prescribes GapEquation’s Kernel
Connects Ansatz for long-range part of QCD’s interactionwith Observables.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 38/48
First Contents Back Conclusion
Interaction
0 1 2 3 4 5
q2 (GeV
2)
0
1
2
3
4
αeff (q
2 )Perturbative evolution
Nonperturbative IR Enhancement
ε=720 MeV
Kernel ofBethe-SalpeterEquation
K(p, k;P ) ≈αeff((p − k)2)
(p − k)2
Prescribes GapEquation’s Kernel
IR-Enhancement at long-range agrees semi-quantitativelywith Bhagwat, et al .
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 38/48
First Contents Back Conclusion
Pion Form Factor
Procedure Now Straightforward
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 39/48
First Contents Back Conclusion
Pion Form Factor
Solve Gap Equation⇒ Dressed-Quark Propagator, S(p)
Σ=
D
γΓS
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 39/48
First Contents Back Conclusion
Pion Form Factor
Use that to Complete Bethe Salpeter Kernel, K
Solve Homogeneous Bethe-Salpeter Equation for PionBethe-Salpeter Amplitude, Γπ
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 39/48
First Contents Back Conclusion
Pion Form Factor
Use that to Complete Bethe Salpeter Kernel, K
Solve Homogeneous Bethe-Salpeter Equation for PionBethe-Salpeter Amplitude, Γπ
Solve Inhomogeneous Bethe-Salpeter Equation forDressed-Quark-Photon Vertex, Γµ
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 39/48
First Contents Back Conclusion
Pion Form Factor
Now have all elements for Impulse Approximation toElectromagnetic Pion Form factor
Γπ(k;P )
Γµ(k;P )
S(p)
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 39/48
First Contents Back Conclusion
Pion Form Factor
Now have all elements for Impulse Approximation toElectromagnetic Pion Form factor
Γπ(k;P )
Γµ(k;P )
S(p)
Evaluate this final,three-dimensional integral
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 39/48
First Contents Back Conclusion
Calculated Pion Form Factor
Calculation published in 1999; No Parameters Varied
0 1 2 3 4
Q2 [GeV
2]
0
0.1
0.2
0.3
0.4
0.5
Q2 F
π(Q2 )
[G
eV2 ]
AmendoliaBrauel, re-analyzedVolmerDSE calculationVMD ρ monopole, mρ=770 MeV
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 40/48
First Contents Back Conclusion
Calculated Pion Form Factor
Calculation published in 1999; No Parameters Varied
0 1 2 3 4
Q2 [GeV
2]
0
0.1
0.2
0.3
0.4
0.5
Q2 F
π(Q2 )
[G
eV2 ]
AmendoliaBrauel, re-analyzedVolmerDSE calculationVMD ρ monopole, mρ=770 MeV
Data published in 2001
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 40/48
First Contents Back Conclusion
Calculated Pion Form Factor
Calculation published in 1999; No Parameters Varied
0 1 2 3 4
Q2 [GeV
2]
0
0.1
0.2
0.3
0.4
0.5
Q2 F
π(Q2 )
[G
eV2 ]
AmendoliaBrauel, re-analyzedVolmerDSE calculationVMD ρ monopole, mρ=770 MeV
Data published in 2001
Many subsequentsuccessful applications. . Again, parameters Fixed
Notably π π Scattering
Maris, et al., Phys. Rev. D 65, 076008
Bicudo, Phys. Rev. C 67, 035201Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 40/48
First Contents Back Conclusion
Pion . . . J = 0
but . . .
Pseudoscalar meson Bethe-Salpeter amplitude
χπ(k;P ) = γ5 [iEπn(k;P ) + γ · PFπn(k;P )
γ · k k · P Gπn(k;P ) + σµν kµPν Hπn(k;P )]
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 41/48
First Contents Back Conclusion
Pion . . . J = 0
but . . .
Pseudoscalar meson Bethe-Salpeter amplitude
χπ(k;P ) = γ5 [iEπn(k;P ) + γ · PFπn(k;P )
γ · k k · P Gπn(k;P ) + σµν kµPν Hπn(k;P )]
Orbital angular momentum is not a Poincaré invariant.
However, if absent in a particular frame, it will appear in
another frame related via a Poincaré transformation.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 41/48
First Contents Back Conclusion
Pion . . . J = 0
but . . .
Pseudoscalar meson Bethe-Salpeter amplitude
χπ(k;P ) = γ5 [iEπn(k;P ) + γ · PFπn(k;P )
γ · k k · P Gπn(k;P ) + σµν kµPν Hπn(k;P )]
Nonzero quark orbital angular momentum is thus a
necessary outcome of a Poincaré covariant description.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 41/48
First Contents Back Conclusion
Pion . . . J = 0
but . . .
Pseudoscalar meson Bethe-Salpeter amplitude
χπ(k;P ) = γ5 [iEπn(k;P ) + γ · PFπn(k;P )
γ · k k · P Gπn(k;P ) + σµν kµPν Hπn(k;P )]
In QCD, a Poincaré invariant theory with interactions, E 6= 0
forces nonzero results for F , G and H
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 41/48
First Contents Back Conclusion
Pion . . . J = 0
but . . .
Pseudoscalar meson Bethe-Salpeter amplitude
χπ(k;P ) = γ5 [iEπn(k;P ) + γ · PFπn(k;P )
γ · k k · P Gπn(k;P ) + σµν kµPν Hπn(k;P )]
In QCD, a Poincaré invariant theory with interactions, E 6= 0
forces nonzero results for F , G and H
J = 0 . . . but while E and F are purely L = 0 in the rest
frame, the G and H terms are associated with L = 1. Thus a
pseudoscalar meson Bethe-Salpeter wave function always
contains both S- and P -wave components.
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 41/48
First Contents Back Conclusion
Pion . . . J = 0
but . . .J = 0 . . . but while E and F are purely L = 0 in the rest
frame, the G and H terms are associated with L = 1. Thus a
pseudoscalar meson Bethe-Salpeter wave function always
contains both S- and P -wave components.Introduce mixing
angle θπ such that
χπ ∼ cos θπ|L = 0〉+sin θπ|L = 1〉
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 41/48
First Contents Back Conclusion
Pion . . . J = 0
but . . .J = 0 . . . but while E and F are purely L = 0 in the rest
frame, the G and H terms are associated with L = 1. Thus a
pseudoscalar meson Bethe-Salpeter wave function always
contains both S- and P -wave components.Introduce mixing
angle θπ such that
χπ ∼ cos θπ|L = 0〉+sin θπ|L = 1〉
0 2 4 6 8 100
-+n=0 meson mass (GeV)
0
10
20
30
θ π n=0
(d
egre
es)
1.2 1.6 2 2.40
-+n meson mass (GeV)
8
12
16
θ π n (
deg
rees
)
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 41/48
First Contents Back Conclusion
Pion . . . J = 0
but . . .J = 0 . . . but while E and F are purely L = 0 in the rest
frame, the G and H terms are associated with L = 1. Thus a
pseudoscalar meson Bethe-Salpeter wave function always
contains both S- and P -wave components.Introduce mixing
angle θπ such that
χπ ∼ cos θπ|L = 0〉+sin θπ|L = 1〉
0 2 4 6 8 100
-+n=0 meson mass (GeV)
0
10
20
30
θ π n=0
(d
egre
es)
1.2 1.6 2 2.40
-+n meson mass (GeV)
8
12
16
θ π n (
deg
rees
)
L is significant in
the neighbourhood
of the chiral limit,
and decreases with
increasing
current-quark mass.Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 41/48
First Contents Back Conclusion
Deep-inelastic scattering
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 42/48
First Contents Back Conclusion
Deep-inelastic scattering
Looking for Quarks
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 42/48
First Contents Back Conclusion
Deep-inelastic scattering
Looking for Quarks
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 42/48
First Contents Back Conclusion
Deep-inelastic scattering
Looking for Quarks
Signature Experiment for QCD:
Discovery of Quarks at SLAC
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 42/48
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)
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 42/48
First Contents Back Conclusion
Pion’s valence quark distn
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 43/48
First Contents Back Conclusion
Pion’s valence quark distn
π is Two-Body System: “Easiest” Bound State in QCD
However, NO π Targets!
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 43/48
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
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 43/48
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
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 43/48
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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������
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���������
P P
k
k+q+P
k+q
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 44/48
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−Γ̄π(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−−)
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 44/48
First Contents Back Conclusion
Handbag diagrams
Bjorken Limit: q2 → ∞ , P · q → −∞
but x := − q2
2P · q fixed.
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−Γ̄π(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−−)
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 44/48
First Contents Back Conclusion
Calc. uV (x) cf. Drell-Yan dataHecht, Roberts, Schmidtnucl-th/0008049
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 45/48
First Contents Back Conclusion
Calc. uV (x) cf. Drell-Yan dataHecht, Roberts, Schmidtnucl-th/0008049
0.0 0.2 0.4 0.6 0.8 1.0x
0.0
0.2
0.4
0.6
xuv(
x)
x uv(x;q0=0.54GeV)x uv(x;q=2GeV)E615 πN Drell−Yan 4GeVSMRS 92 Fitx uv(x;q=4GeV)Fit x
α(1−x)
β
Hec
ht, R
ober
ts, S
chm
idt,
nucl
−th
/000
8049
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 45/48
First Contents Back Conclusion
Calc. uV (x) cf. Drell-Yan dataHecht, Roberts, Schmidtnucl-th/0008049
0.0 0.2 0.4 0.6 0.8 1.0x
0.0
0.2
0.4
0.6
xuv(
x)
x uv(x;q0=0.54GeV)x uv(x;q=2GeV)E615 πN Drell−Yan 4GeVSMRS 92 Fitx uv(x;q=4GeV)Fit x
α(1−x)
β
Hec
ht, R
ober
ts, S
chm
idt,
nucl
−th
/000
8049
Resolving Scale: q0 = 0.54GeV = 1/(0.37 fm)
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 45/48
First Contents Back Conclusion
Calc. uV (x) cf. Drell-Yan data
0.0 0.2 0.4 0.6 0.8 1.0x
0.0
0.2
0.4
0.6
xuv(
x)
x uv(x;q0=0.54GeV)x uv(x;q=2GeV)E615 πN Drell−Yan 4GeVSMRS 92 Fitx uv(x;q=4GeV)Fit x
α(1−x)
β
Hec
ht, R
ober
ts, S
chm
idt,
nucl
−th
/000
8049
q =
2GeV Calc. Fit, 92 Latt., 97
〈x〉q 0.24 0.24 ± 0.01 0.27 ± 0.01
〈x2〉q 0.10 0.10 ± 0.01 0.11 ± 0.3
〈x3〉q 0.050 0.058 ± 0.004 0.048 ± 0.020Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 45/48
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
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 46/48
First Contents Back Conclusion
Answer for the pion
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 47/48
First Contents Back Conclusion
Answer for the pion
Two → Infinitely many . . .
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 47/48
First Contents Back Conclusion
Answer for the pion
Two → Infinitely many . . .Handle thatproperly inquantumfield theory
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 47/48
First Contents Back Conclusion
Answer for the pion
Two → Infinitely many . . .Handle thatproperly inquantumfield theory. . .momentum-dependentdressing
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 47/48
First Contents Back Conclusion
Answer for the pion
Two → Infinitely many . . .Handle thatproperly inquantumfield theory. . .momentum-dependentdressing. . .perceiveddistribution ofmass dependson the resolving scale
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 47/48
First Contents Back Conclusion
Exegesis
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 48/48
First Contents Back Conclusion
Exegesis
Hadron Physics is ∼ $300-million/year effort in USA alone
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 48/48
First Contents Back Conclusion
Exegesis
Hadron Physics is ∼ $300-million/year effort in USA alone
Subject is QCD . . . in the nonperturbative domain
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 48/48
First Contents Back Conclusion
Exegesis
Hadron Physics is ∼ $300-million/year effort in USA alone
Subject is QCD . . . in the nonperturbative domain
Keystones are the Emergent Phenomena
Confinement– quarks and gluons never alone reach a detector
Dynamical Chiral Symmetry Breaking– counter-intuitive pattern of bound state masses andinteractions
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 48/48
First Contents Back Conclusion
Exegesis
Hadron Physics is ∼ $300-million/year effort in USA alone
Subject is QCD . . . in the nonperturbative domain
Keystones are the Emergent Phenomena
Confinement– quarks and gluons never alone reach a detector
Dynamical Chiral Symmetry Breaking– counter-intuitive pattern of bound state masses andinteractions
Review presentation in Mazatlan:
Elastic electromagnetic pion form factor
Nature of Baryons
Craig Roberts: Hadron Physics and Continuum Strong QCD
XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 48/48