First Contents Back Conclusion Hadron Physics and Continuum Strong QCD Craig D. Roberts [email protected]Physics Division & School of Physics Argonne National Laboratory Peking University http://www.phy.anl.gov/theory/staff/cdr.html Craig Roberts: Hadron Physics and Continuum Strong QCD XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009... 48 – p. 1/48
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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
First Contents Back Conclusion
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
First Contents Back Conclusion
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
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
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
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First Contents Back Conclusion
JLab Hall-A
Craig Roberts: Hadron Physics and Continuum Strong QCD
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First Contents Back Conclusion
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
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
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
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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
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
First Contents Back Conclusion
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
First Contents Back Conclusion
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
First Contents Back Conclusion
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
First Contents Back Conclusion
Some Questions
How accurately can the pion form factor beextracted from the ep → e′nπ+ reaction?
Craig Roberts: Hadron Physics and Continuum Strong QCD
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First Contents Back Conclusion
Status
Craig Roberts: Hadron Physics and Continuum Strong QCD
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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].
Craig Roberts: Hadron Physics and Continuum Strong QCD
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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/
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First Contents Back Conclusion
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
First Contents Back Conclusion
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
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First Contents Back Conclusion
Confinement
Craig Roberts: Hadron Physics and Continuum Strong QCD
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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
First Contents Back Conclusion
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
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First Contents Back Conclusion
Dyson-Schwinger EquationsEuler-Lagrange equations for quantum field theory
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
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First Contents Back Conclusion
Schwinger Functions
Craig Roberts: Hadron Physics and Continuum Strong QCD
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First Contents Back Conclusion
Schwinger Functions
Solutions are Schwinger Functions(Euclidean Green Functions)
Craig Roberts: Hadron Physics and Continuum Strong QCD
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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
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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
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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
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First Contents Back Conclusion
World . . .
Craig Roberts: Hadron Physics and Continuum Strong QCD
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First Contents Back Conclusion
World . . .DSE Perspective
Craig Roberts: Hadron Physics and Continuum Strong QCD
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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
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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
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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
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Frontiers of Nuclear Science:A Long Range Plan (2007)
Craig Roberts: Hadron Physics and Continuum Strong QCD
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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
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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
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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
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
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
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