dyson-schwinger equations and quantum chromodynamics · · 2008-08-25first contents back...
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First Contents Back Conclusion
Dyson-Schwinger Equationsand Quantum Chromodynamics
Craig D. Roberts
Physics Division
Argonne National Laboratory
http://www.phy.anl.gov/theory/staff/cdr.htmlCraig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 1/38
First Contents Back Conclusion
Room with a View
Room with a View
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 2/38
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Universal Truths
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 3/38
First Contents Back Conclusion
Universal Truths
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 3/38
First Contents Back Conclusion
Universal Truths
Form factors give information about distribution of hadron’s
characterising properties amongst its QCD constituents.
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 3/38
First Contents Back Conclusion
Universal Truths
Form factors give information about distribution of hadron’s
characterising properties amongst its QCD constituents.
Calculations at Q2 > 1GeV2 require a Poincaré-covariant
approach.
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 3/38
First Contents Back Conclusion
Universal Truths
Form factors give information about distribution of hadron’s
characterising properties amongst its QCD constituents.
Calculations at Q2 > 1GeV2 require a Poincaré-covariant
approach. Covariance requires existence of quark orbital
angular momentum in hadron’s rest-frame wave function.
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 3/38
First Contents Back Conclusion
Universal Truths
Form factors give information about distribution of hadron’s
characterising properties amongst its QCD constituents.
Calculations at Q2 > 1GeV2 require a Poincaré-covariant
approach. Covariance requires existence of quark orbital
angular momentum in hadron’s rest-frame wave function.
DCSB is most important mass generating mechanism for
matter in the Universe.
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 3/38
First Contents Back Conclusion
Universal Truths
Form factors give information about distribution of hadron’s
characterising properties amongst its QCD constituents.
Calculations at Q2 > 1GeV2 require a Poincaré-covariant
approach. Covariance requires existence of quark orbital
angular momentum in hadron’s rest-frame wave function.
DCSB is most important mass generating mechanism for
matter in the Universe. Higgs mechanism is irrelevant to
light-quarks.
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 3/38
First Contents Back Conclusion
Universal Truths
Form factors give information about distribution of hadron’s
characterising properties amongst its QCD constituents.
Calculations at Q2 > 1GeV2 require a Poincaré-covariant
approach. Covariance requires existence of quark orbital
angular momentum in hadron’s rest-frame wave function.
DCSB is most important mass generating mechanism for
matter in the Universe. Higgs mechanism is irrelevant to
light-quarks.
Challenge: understand relationship between parton properties
on the light-front and rest frame structure of hadrons.
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 3/38
First Contents Back Conclusion
Universal Truths
Form factors give information about distribution of hadron’s
characterising properties amongst its QCD constituents.
Calculations at Q2 > 1GeV2 require a Poincaré-covariant
approach. Covariance requires existence of quark orbital
angular momentum in hadron’s rest-frame wave function.
DCSB is most important mass generating mechanism for
matter in the Universe. Higgs mechanism is irrelevant to
light-quarks.
Challenge: understand relationship between parton properties
on the light-front and rest frame structure of hadrons. Problem
because, e.g., DCSB - an established keystone of low-energy
QCD and the origin of constituent-quark masses - has not
been realised in the light-front formulation.Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 3/38
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Form Factors: Why?
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 4/38
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Form Factors: Why?
The nucleon and pion hold special places in non-perturbativestudies of QCD.
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 4/38
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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: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 4/38
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: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 4/38
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: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 4/38
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: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 4/38
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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: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 5/38
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Some Questions
Where is the transition from non-pQCD to pQCD inthe pion and nucleon electromagnetic formfactors?
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 5/38
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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: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 5/38
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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: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 5/38
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: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 5/38
First Contents Back Conclusion
Some Questions
How accurately can the pion form factor beextracted from the ep → e′nπ+ reaction?
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 5/38
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Status
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 6/38
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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: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 6/38
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: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 6/38
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Answer for the pion
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 7/38
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Answer for the pion
Two → Infinitely many . . .
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 7/38
First Contents Back Conclusion
Answer for the pion
Two → Infinitely many . . .Handle thatproperly inquantumfield theory
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 7/38
First Contents Back Conclusion
Answer for the pion
Two → Infinitely many . . .Handle thatproperly inquantumfield theory. . .momentum-dependentdressing
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 7/38
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: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 7/38
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Pion Form Factor
Procedure Now Straightforward
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 8/38
First Contents Back Conclusion
Pion Form Factor
Solve Gap Equation⇒ Dressed-Quark Propagator, S(p)
Σ=
D
γΓS
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 8/38
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: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 8/38
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: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 8/38
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: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 8/38
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: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 8/38
First Contents Back Conclusion
Calculated Pion Form Factor
0 1 2 3 4
Q2 [GeV
2]
0
0.1
0.2
0.3
0.4
0.5
Q2 F
π(Q2 )
[GeV
2 ]
Amendolia et al.Ackermann et al.Brauel et al.Tadevosyan et al.
Horn et al.Maris and Tandy, 2005
Calculation first published in 1999; No Parameters VariedNumerical method improved in 2005
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 9/38
First Contents Back Conclusion
Calculated Pion Form Factor
0 1 2 3 4
Q2 [GeV
2]
0
0.1
0.2
0.3
0.4
0.5
Q2 F
π(Q2 )
[GeV
2 ]
Amendolia et al.Ackermann et al.Brauel et al.Tadevosyan et al.
Horn et al.Maris and Tandy, 2005
Calculation first published in 1999; No Parameters VariedNumerical method improved in 2005
Data publishedin 2001.Subsequentlyrevised
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 9/38
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Timelike Pion Form Factor
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 10/38
First Contents Back Conclusion
Timelike Pion Form FactorAb initio calculation into timelike regionDeeper than ground-state ρ-meson pole
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 10/38
First Contents Back Conclusion
Timelike Pion Form Factor
-1 -0.5 0 0.5 1 1.5 2 2.5 3
Q2 [GeV
2]
10-1
100
101
|Fπ(Q
2 )|Amendolia et al.Ackermann et al.Brauel et al.Tadevosyan et al.
Horn et al.Barkov et al.DSE calculation
Ab initio calculation into timelike regionDeeper than ground-state ρ-meson pole
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 10/38
First Contents Back Conclusion
Timelike Pion Form Factor
-1 -0.5 0 0.5 1 1.5 2 2.5 3
Q2 [GeV
2]
10-1
100
101
|Fπ(Q
2 )|Amendolia et al.Ackermann et al.Brauel et al.Tadevosyan et al.
Horn et al.Barkov et al.DSE calculation
Ab initio calculation into timelike regionDeeper than ground-state ρ-meson poleρ-meson not put in “by hand” – generated dynamically as a bound-state of dressed-quark and dressed-antiquark
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 10/38
First Contents Back Conclusion
Dimensionless product: rπ fπ
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 11/38
First Contents Back Conclusion
Dimensionless product: rπ fπ
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 11/38
First Contents Back Conclusion
Dimensionless product: rπ fπ
Improved rainbow-ladder interaction
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 11/38
First Contents Back Conclusion
Dimensionless product: rπ fπ
Improved rainbow-ladder interaction
Repeating Fπ(Q2) calculation
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 11/38
First Contents Back Conclusion
Dimensionless product: rπ fπ
Improved rainbow-ladder interaction
Repeating Fπ(Q2) calculation
Great strides towards placing nucleon studies on same
footing as mesons
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 11/38
First Contents Back Conclusion
Dimensionless product: rπ fπ
Improved rainbow-ladder interaction
Repeating Fπ(Q2) calculation
Experimentally: rπfπ = 0.315 ± 0.005
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 11/38
First Contents Back Conclusion
Dimensionless product: rπ fπ
Improved rainbow-ladder interaction
Repeating Fπ(Q2) calculation
Experimentally: rπfπ = 0.315 ± 0.005
DSE prediction
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 11/38
First Contents Back Conclusion
Dimensionless product: rπ fπ
Improved rainbow-ladder interaction
Repeating Fπ(Q2) calculation
Experimentally: rπfπ = 0.315 ± 0.005
DSE prediction
Lattice results
– James Zanotti [UK QCD]
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 11/38
First Contents Back Conclusion
Dimensionless product: rπ fπ
Improved rainbow-ladder interaction
Repeating Fπ(Q2) calculation
Experimentally: rπfπ = 0.315 ± 0.005
DSE prediction
Lattice results
– James Zanotti [UK QCD]
Fascinating result:
DSE and Lattice
– Experimental value
obtains independent of
current-quark mass.Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 11/38
First Contents Back Conclusion
Dimensionless product: rπ fπ
Improved rainbow-ladder interaction
Repeating Fπ(Q2) calculation
Experimentally: rπfπ = 0.315 ± 0.005
DSE prediction
Fascinating result:
DSE and Lattice
– Experimental value
obtains independent of
current-quark mass.
We have understood this
Implications far-reaching.Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 11/38
First Contents Back ConclusionCraig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 12/38
First Contents Back Conclusion
New Challenges
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 13/38
First Contents Back Conclusion
New Challenges
Next Steps . . . Applications to excited states and
axial-vector mesons, e.g., will improve understanding of
confinement interaction between light-quarks.
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 13/38
First Contents Back Conclusion
New Challenges
Next Steps . . . Applications to excited states and
axial-vector mesons, e.g., will improve understanding of
confinement interaction between light-quarks.
Move on to the problem of a symmetry preserving treatment
of hybrids and exotics.
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 13/38
First Contents Back Conclusion
New Challenges
Another Direction . . . Also want/need information about
three-quark systems
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 13/38
First Contents Back Conclusion
New Challenges
Another Direction . . . Also want/need information about
three-quark systems
With this problem . . . most wide-ranging studies employ
expertise familiar from meson applications circa ∼1995.
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 13/38
First Contents Back Conclusion
New Challenges
Another Direction . . . Also want/need information about
three-quark systems
With this problem . . . most wide-ranging studies employ
expertise familiar from meson applications circa ∼1995.
Namely . . . Model-building and Phenomenology,
constrained by the DSE results outlined already.Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 13/38
First Contents Back Conclusion
New Challenges
Another Direction . . . Also want/need information about
three-quark systems
With this problem . . . most wide-ranging studies employ
expertise familiar from meson applications circa ∼1995.
However, that is beginning to change . . .
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 13/38
First Contents Back Conclusion
New Challenges
Another Direction . . . Also want/need information about
three-quark systems
With this problem . . . most wide-ranging studies employ
expertise familiar from meson applications circa ∼1995.
However, that is beginning to change . . .
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 13/38
First Contents Back Conclusion
Nucleon . . .Three-body Problem?
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 14/38
First Contents Back Conclusion
Nucleon . . .Three-body Problem?
What is the picture in quantum field theory?
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 14/38
First Contents Back Conclusion
Nucleon . . .Three-body Problem?
What is the picture in quantum field theory?
Three →infinitelymany!
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 14/38
First Contents Back Conclusion
Faddeev equation
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 15/38
First Contents Back Conclusion
Faddeev equation
=aΨ
P
pq
pd Γb
Γ−a
pd
pq
bΨP
q
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 15/38
First Contents Back Conclusion
Faddeev equation
=aΨ
P
pq
pd Γb
Γ−a
pd
pq
bΨP
q
Linear, Homogeneous Matrix equation
Yields wave function (Poincaré Covariant Faddeev
Amplitude) that describes quark-diquark relative motion
within the nucleon
Scalar and Axial-Vector Diquarks . . . In Nucleon’s Rest
Frame Amplitude has . . . s−, p− & d−wave correlations
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 15/38
First Contents Back Conclusion
Diquark correlations
QUARK-QUARKCraig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 16/38
First Contents Back Conclusion
Diquark correlations
QUARK-QUARK
Same interaction that
describes mesons also
generates three coloured
quark-quark correlations:
blue–red, blue–green,
green–red
Confined . . . Does not
escape from within baryon.
Scalar is isosinglet,
Axial-vector is isotriplet
DSE and lattice-QCD
m[ud]0+
= 0.74 − 0.82
m(uu)1+
= m(ud)1+
= m(dd)1+
= 0.95 − 1.02
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 16/38
First Contents Back Conclusion
Nucleon EM Form Factors: A PrécisHoll, et al. : nu-th/0412046 & nu-th/0501033
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 17/38
First Contents Back Conclusion
Nucleon EM Form Factors: A Précis
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 17/38
First Contents Back Conclusion
Nucleon EM Form Factors: A Précis
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 18/38
First Contents Back Conclusion
Nucleon EM Form Factors: A PrécisCloet, et al. :arXiv:0710.2059, arXiv:0710.5746 & arXiv:0804.3118
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 18/38
First Contents Back Conclusion
Nucleon EM Form Factors: A PrécisCloet, et al. :arXiv:0710.2059, arXiv:0710.5746 & arXiv:0804.3118
• Interpreting expts. with GeV electromagnetic probes
requires Poincaré covariant treatment of baryons
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 18/38
First Contents Back Conclusion
Nucleon EM Form Factors: A PrécisCloet, et al. :arXiv:0710.2059, arXiv:0710.5746 & arXiv:0804.3118
• Interpreting expts. with GeV electromagnetic probes
requires Poincaré covariant treatment of baryons
⇒ Covariant dressed-quark Faddeev Equation
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 18/38
First Contents Back Conclusion
Nucleon EM Form Factors: A PrécisCloet, et al. :arXiv:0710.2059, arXiv:0710.5746 & arXiv:0804.3118
• Interpreting expts. with GeV electromagnetic probes
requires Poincaré covariant treatment of baryons
⇒ Covariant dressed-quark Faddeev Equation
• Excellent mass spectrum (octet and decuplet)
Easily obtained:(
1
NH
∑
H
[M expH − M calc
H ]2
[M expH ]2
)1/2
= 2%
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 18/38
First Contents Back Conclusion
Nucleon EM Form Factors: A PrécisCloet, et al. :arXiv:0710.2059, arXiv:0710.5746 & arXiv:0804.3118
• Interpreting expts. with GeV electromagnetic probes
requires Poincaré covariant treatment of baryons
⇒ Covariant dressed-quark Faddeev Equation
• Excellent mass spectrum (octet and decuplet)
Easily obtained:(
1
NH
∑
H
[M expH − M calc
H ]2
[M expH ]2
)1/2
= 2%
(Oettel, Hellstern, Alkofer, Reinhardt: nucl-th/9805054)
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 18/38
First Contents Back Conclusion
Nucleon EM Form Factors: A PrécisCloet, et al. :arXiv:0710.2059, arXiv:0710.5746 & arXiv:0804.3118
• Interpreting expts. with GeV electromagnetic probes
requires Poincaré covariant treatment of baryons
⇒ Covariant dressed-quark Faddeev Equation
• Excellent mass spectrum (octet and decuplet)
Easily obtained:(
1
NH
∑
H
[M expH − M calc
H ]2
[M expH ]2
)1/2
= 2%
• But is that good?
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 18/38
First Contents Back Conclusion
Nucleon EM Form Factors: A PrécisCloet, et al. :arXiv:0710.2059, arXiv:0710.5746 & arXiv:0804.3118
• Interpreting expts. with GeV electromagnetic probes
requires Poincaré covariant treatment of baryons
⇒ Covariant dressed-quark Faddeev Equation
• Excellent mass spectrum (octet and decuplet)
Easily obtained:(
1
NH
∑
H
[M expH − M calc
H ]2
[M expH ]2
)1/2
= 2%
• But is that good?
• Cloudy Bag: δMπ−loop+ = −300 to −400 MeV!
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 18/38
First Contents Back Conclusion
Nucleon EM Form Factors: A PrécisCloet, et al. :arXiv:0710.2059, arXiv:0710.5746 & arXiv:0804.3118
• Interpreting expts. with GeV electromagnetic probes
requires Poincaré covariant treatment of baryons
⇒ Covariant dressed-quark Faddeev Equation
• Excellent mass spectrum (octet and decuplet)
Easily obtained:(
1
NH
∑
H
[M expH − M calc
H ]2
[M expH ]2
)1/2
= 2%
• But is that good?
• Cloudy Bag: δMπ−loop+ = −300 to −400 MeV!
• Critical to anticipate pion cloud effects
Roberts, Tandy, Thomas, et al., nu-th/02010084
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 18/38
First Contents Back Conclusion
Nucleon’s self-energy - pion loop
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 19/38
First Contents Back Conclusion
Nucleon’s self-energy - pion loop
Σ(P ) = 3
∫
d4k
(2π)4g2PV (P, k)∆π((P − k)2)
× γ · (P − k)γ5 G(k) γ · (P − k)γ5
= iγ · k [A(k2) − 1] + B(k2)
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 19/38
First Contents Back Conclusion
Nucleon’s self-energy - pion loop
Σ(P ) = 3
∫
d4k
(2π)4g2PV (P, k)∆π((P − k)2)
× γ · (P − k)γ5 G(k) γ · (P − k)γ5
= iγ · k [A(k2) − 1] + B(k2)
• Pseudovector coupling
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 19/38
First Contents Back Conclusion
Nucleon’s self-energy - pion loop
Σ(P ) = 3
∫
d4k
(2π)4g2PV (P, k)∆π((P − k)2)
× γ · (P − k)γ5 G(k) γ · (P − k)γ5
= iγ · k [A(k2) − 1] + B(k2)
• Pseudovector coupling
• Completely equivalent to pseudoscalar coupling
IF that is treated completely
• Tadpole contribution can’t be neglected
(Hecht, Oettel, Roberts, Schmidt, Tandy, Thomas: nucl-th/0201084)
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 19/38
First Contents Back Conclusion
Nucleon’s self-energy - pion loop
Σ(P ) = 3
∫
d4k
(2π)4g2PV (P, k) ∆π((P − k)2)
× γ · (P − k)γ5 G(k) γ · (P − k)γ5
= iγ · k [A(k2) − 1] + B(k2)
gPV (P, k), πN vertex function
Calculated using Γπ and ΨN
Always soft: Monopole λ ∼ 0.6 GeV
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 19/38
First Contents Back Conclusion
Nucleon’s self-energy - pion loop
Σ(P ) = 3
∫
d4k
(2π)4g2PV (P, k) ∆π((P − k)2)
× γ · (P − k)γ5 G(k) γ · (P − k)γ5
= iγ · k [A(k2) − 1] + B(k2)
gPV (P, k), πN vertex function
Calculated using Γπ and ΨN
Always soft: Monopole λ ∼ 0.6 GeV
Corresponds to range rλ ∼ 0.8 fm
. . . pion cloud does not
penetrate deeply within nucleon. 0.0 0.5 1.0 1.5k (GeV)
0.0
0.2
0.4
0.6
0.8
1.0
u2 (k
)Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 19/38
First Contents Back Conclusion
Nucleon’s self-energy - pion loop
Σ(P ) = 3
∫
d4k
(2π)4g2PV (P, k)∆π((P − k)2)
× γ · (P − k)γ5 G(k) γ · (P − k)γ5
= iγ · k [A(k2) − 1] + B(k2)
G(k) = 1/[iγ · k + M + Σ(P )] Pole Position Not
= −iγ · k σV(k2) + σS(k2) Known a priori
Mass shift calculated via self-consistent solution
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 19/38
First Contents Back Conclusion
Nucleon’s self-energy - pion loop
Σ(P ) = 3
∫
d4k
(2π)4g2PV (const.)∆π((P − k)2)
× γ · (P − k)γ5 G(k) γ · (P − k)γ5
= iγ · k [A(k2) − 1] + B(k2)
Obtain Integral Equation Kernels∫
dΩk f((P − k)2) =2
π
∫ 1
−1dz√
1 − z2 f(P 2 + k2 − 2Pkz)
E.g.ωB(P 2, k2) =
∫
dΩk(P − k)2
(P − k)2 + m2π
= 1 − 2m2π
a +√
a2 − b2,
a = P 2 + k2 + m2π, b = 2Pk
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 19/38
First Contents Back Conclusion
Nucleon’s self-energy - pion loop
Σ(P ) = 3
∫
d4k
(2π)4g2PV (P, k)∆π((P − k)2)
× γ · (P − k)γ5 G(k) γ · (P − k)γ5
= iγ · k [A(k2) − 1] + B(k2)
But gPV = gPV (P 2, k2, P · k)
Therefore, In General, Kernel only known Numerically
Complicates analysis . . .
locating, incorporating poles in integrand
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 19/38
First Contents Back Conclusion
Nucleon Self Energy: Chiral Limit
Hecht, et al., nu-th/0201084
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 20/38
First Contents Back Conclusion
Nucleon Self Energy: Chiral Limit
Hecht, et al., nu-th/0201084
Let’s look what happens when mπ → 0
Minkowski Space
Pseudovector Coupling
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 20/38
First Contents Back Conclusion
Nucleon Self Energy: Chiral Limit
Hecht, et al., nu-th/0201084
Let’s look what happens when mπ → 0
Minkowski Space
Pseudovector Coupling
One-loop nucleon self energy
Σ(P) = 3ig2
4M2
∫
d4k
(2π)4∆(k2, m2
π) 6kγ5 G0(P − k) 6kγ5 .
This integral is divergent. Assume a Poincaré covariant regularisation,characterised by a mass-scale λ
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 20/38
First Contents Back Conclusion
Nucleon Self Energy: Chiral Limit
Hecht, et al., nu-th/0201084
Let’s look what happens when mπ → 0
Minkowski Space
Pseudovector Coupling
One-loop nucleon self energy
Σ(P) = 3ig2
4M2
∫
d4k
(2π)4∆(k2, m2
π) 6kγ5 G0(P − k) 6kγ5 .
This integral is divergent. Assume a Poincaré covariant regularisation,characterised by a mass-scale λ
Decompose nucleon propagator into positive and negative energy components
G0(P ) =1
6P − M0
= G+
0(P ) + G−
0(P )
=M
ωN ( ~P )
[
Λ+( ~P )1
P0 − ωN ( ~P ) + iε+ Λ−( ~P )
1
P0 + ωN ( ~P ) − iε
]
(4)
ω2N (~P) = ~P2 + M2, and Λ±(~P) = ( 6 P ± M)/(2M), P = (ω(~P), ~P)
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 20/38
First Contents Back Conclusion
Nucleon Self Energy: Chiral Limit
Hecht, et al., nu-th/0201084
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 21/38
First Contents Back Conclusion
Nucleon Self Energy: Chiral Limit
Hecht, et al., nu-th/0201084
One-loop nucleon self energy
Σ(P) = 3ig2
4M2
∫
d4k
(2π)4∆(k2, m2
π) 6kγ5 G0(P − k) 6kγ5 .
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 21/38
First Contents Back Conclusion
Nucleon Self Energy: Chiral Limit
Hecht, et al., nu-th/0201084
One-loop nucleon self energy
Σ(P) = 3ig2
4M2
∫
d4k
(2π)4∆(k2, m2
π) 6kγ5 G0(P − k) 6kγ5 .
Shift in the mass of a positive energy nucleon nucleon:
δM+ = 12trD
[
Λ+(~P = 0) Σ(P0 = M, ~P = 0)]
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 21/38
First Contents Back Conclusion
Nucleon Self Energy: Chiral Limit
Hecht, et al., nu-th/0201084
One-loop nucleon self energy
Σ(P) = 3ig2
4M2
∫
d4k
(2π)4∆(k2, m2
π) 6kγ5 G0(P − k) 6kγ5 .
Shift in the mass of a positive energy nucleon nucleon:
δM+ = 12trD
[
Λ+(~P = 0) Σ(P0 = M, ~P = 0)]
Focus on positive energy nucleon’s contribution to the loop integral; i.e.,∆(k) G+(P − k), which we denote: δF M
++
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 21/38
First Contents Back Conclusion
Nucleon Self Energy: Chiral Limit
Hecht, et al., nu-th/0201084
One-loop nucleon self energy
Σ(P) = 3ig2
4M2
∫
d4k
(2π)4∆(k2, m2
π) 6kγ5 G0(P − k) 6kγ5 .
Shift in the mass of a positive energy nucleon nucleon:
δM+ = 12trD
[
Λ+(~P = 0) Σ(P0 = M, ~P = 0)]
Focus on positive energy nucleon’s contribution to the loop integral; i.e.,∆(k) G+(P − k), which we denote: δF M
++
To evaluate k0 integral, close contour in lower half-plane, thereby encirclingonly the positive-energy pion pole.
δF M++ = −3g2
∫
d3k
(2π)3ωN (~k2) − M0
4 ωN (~k2)
1
ωπ(~k2) [ωπ(~k2) + ωN (~k2) − M0](9)
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 21/38
First Contents Back Conclusion
Nucleon Self Energy: Chiral Limit
Hecht, et al., nu-th/0201084
δF M++ = −3g2
∫
d3k
(2π)3ωN (~k2) − M0
4 ωN (~k2)
1
ωπ(~k2) [ωπ(~k2) + ωN (~k2) − M0](10)
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 22/38
First Contents Back Conclusion
Nucleon Self Energy: Chiral Limit
Hecht, et al., nu-th/0201084
δF M++ = −3g2
∫
d3k
(2π)3ωN (~k2) − M0
4 ωN (~k2)
1
ωπ(~k2) [ωπ(~k2) + ωN (~k2) − M0](14)
On the domain for which the regularised integral has significant support, assumethat M0 is very much greater than all other mass scales.
ωN (~k2) − M ≈~k2
2 M(15)
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 22/38
First Contents Back Conclusion
Nucleon Self Energy: Chiral Limit
Hecht, et al., nu-th/0201084
δF M++ = −3g2
∫
d3k
(2π)3ωN (~k2) − M0
4 ωN (~k2)
1
ωπ(~k2) [ωπ(~k2) + ωN (~k2) − M0](18)
On the domain for which the regularised integral has significant support, assumethat M0 is very much greater than all other mass scales.
ωN (~k2) − M ≈~k2
2 M(19)
Then δF M++ ≈ −3g2
∫
d3k
(2π)3
~k2
8 M2
1
ω2λi
(~k2)(20)
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 22/38
First Contents Back Conclusion
Nucleon Self Energy: Chiral Limit
Hecht, et al., nu-th/0201084
δF M++ = −3g2
∫
d3k
(2π)3ωN (~k2) − M0
4 ωN (~k2)
1
ωπ(~k2) [ωπ(~k2) + ωN (~k2) − M0](22)
On the domain for which the regularised integral has significant support, assumethat M0 is very much greater than all other mass scales.
ωN (~k2) − M ≈~k2
2 M(23)
Then δF M++ ≈ −3g2
∫
d3k
(2π)3
~k2
8 M2
1
ω2λi
(~k2)(24)
So thatd2 δF M
++
(dm2π)2
≈ −3g2
4M2
∫
d3k
(2π)3
~k2
ω6π(~k2)
= −9
128π
g2
M2
1
mπ. (25)
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 22/38
First Contents Back Conclusion
Nucleon Self Energy: Chiral Limit
Hecht, et al., nu-th/0201084
δF M++ = −3g2
∫
d3k
(2π)3ωN (~k2) − M0
4 ωN (~k2)
1
ωπ(~k2) [ωπ(~k2) + ωN (~k2) − M0](26)
On the domain for which the regularised integral has significant support, assumethat M0 is very much greater than all other mass scales.
ωN (~k2) − M ≈~k2
2 M(27)
Then δF M++ ≈ −3g2
∫
d3k
(2π)3
~k2
8 M2
1
ω2λi
(~k2)(28)
So thatd2 δF M
++
(dm2π)2
≈ −3g2
4M2
∫
d3k
(2π)3
~k2
ω6π(~k2)
= −9
128π
g2
M2
1
mπ. (29)
Namely δF M++ = −
3
32π
g2
M2m3
π + f+(1)
(λ1, λ2) m2π + f
+(0)
(λ1, λ2)
where the last two terms express the necessary contribution from the regulator.
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 22/38
First Contents Back Conclusion
Nucleon Self Energy: Chiral Limit
Hecht, et al., nu-th/0201084
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 23/38
First Contents Back Conclusion
Nucleon Self Energy: Chiral Limit
Hecht, et al., nu-th/0201084
Nucleon’s self energy
δF M++ = −
3
32π
g2
M2m3
π + f+(1)
(λ1, λ2) m2π + f
+(0)
(λ1, λ2)
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 23/38
First Contents Back Conclusion
Nucleon Self Energy: Chiral Limit
Hecht, et al., nu-th/0201084
Nucleon’s self energy
δF M++ = −
3
32π
g2
M2m3
π + f+(1)
(λ1, λ2) m2π + f
+(0)
(λ1, λ2)
Given that m2π ∝ m in the neighbourhood of the chiral limit, the m3
π isnonanalytic in the current-quark mass on this domain.
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 23/38
First Contents Back Conclusion
Nucleon Self Energy: Chiral Limit
Hecht, et al., nu-th/0201084
Nucleon’s self energy
δF M++ = −
3
32π
g2
M2m3
π + f+(1)
(λ1, λ2) m2π + f
+(0)
(λ1, λ2)
Given that m2π ∝ m in the neighbourhood of the chiral limit, the m3
π isnonanalytic in the current-quark mass on this domain.
This is the Leading Nonanalytic Contribution much touted in effective fieldtheory.
Its form is completely fixed by chiral symmetry and the pattern of itsdynamical breaking.
NB. Contribution from negative energy nucleon is ∝1
M3.
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 23/38
First Contents Back Conclusion
Nucleon Self Energy: Chiral Limit
Hecht, et al., nu-th/0201084
Nucleon’s self energy
δF M++ = −
3
32π
g2
M2m3
π + f+(1)
(λ1, λ2) m2π + f
+(0)
(λ1, λ2)
Given that m2π ∝ m in the neighbourhood of the chiral limit, the m3
π isnonanalytic in the current-quark mass on this domain.
This is the Leading Nonanalytic Contribution much touted in effective fieldtheory.
Its form is completely fixed by chiral symmetry and the pattern of itsdynamical breaking.
NB. Contribution from negative energy nucleon is ∝1
M3.
The remaining terms are regular in the current-quark mass. Their exact naturedepends on the explicit form of regularisation procedure.
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 23/38
First Contents Back Conclusion
Nucleon Self Energy: Chiral Limit
Hecht, et al., nu-th/0201084
Nucleon’s self energy
δF M++ = −
3
32π
g2
M2m3
π + f+(1)
(λ1, λ2) m2π + f
+(0)
(λ1, λ2)
Given that m2π ∝ m in the neighbourhood of the chiral limit, the m3
π isnonanalytic in the current-quark mass on this domain.
The Leading Nonanalytic Contribution is a model-independent result.
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 23/38
First Contents Back Conclusion
Nucleon Self Energy: Chiral Limit
Hecht, et al., nu-th/0201084
Nucleon’s self energy
δF M++ = −
3
32π
g2
M2m3
π + f+(1)
(λ1, λ2) m2π + f
+(0)
(λ1, λ2)
Given that m2π ∝ m in the neighbourhood of the chiral limit, the m3
π isnonanalytic in the current-quark mass on this domain.
The Leading Nonanalytic Contribution is a model-independent result.
Unfortunately, it is of limited relevance. In a calculation of the nucleon’s mass, theactual value of the pion loop’s contribution is almost completely determined by theregularisation dependent terms.
It is essential for a framework to veraciously express the leading nonanalyticcontribution . . . it serves as a check that DCSB is truly described.
However, beyond that, one must accept that the world is complex.The pion has a finite size. So does the nucleon.These sizes set the mass-scale which determines the nucleon’s massshift.
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 23/38
First Contents Back Conclusion
Model pion-nucleon coupling
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 24/38
First Contents Back Conclusion
Model pion-nucleon coupling
gPV (P, k) =g
2Mexp(−(P − k)2/Λ2)
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 24/38
First Contents Back Conclusion
Model pion-nucleon coupling
gPV (P, k) =g
2Mexp(−(P − k)2/Λ2)
• B-Kernel∫
dΩk g2PV ((P − k)2)
[
1 − 2m2π
(P − k)2 + m2π
]
Clearly the sum of two independent terms.
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 24/38
First Contents Back Conclusion
Model pion-nucleon coupling
gPV (P, k) =g
2Mexp(−(P − k)2/Λ2)
• B-Kernel∫
dΩk g2PV ((P − k)2)
[
1 − 2m2π
(P − k)2 + m2π
]
• First term can be evaluated exactly
g2PV (P 2, k2) =
∫
dΩk g2PV ((P − k)2)
=g2
4M2e−2(P 2+k2)/Λ2 Λ2
2PkI1(4Pk/Λ2) ,
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 24/38
First Contents Back Conclusion
Model pion-nucleon coupling
gPV (P, k) =g
2Mexp(−(P − k)2/Λ2)
• B-Kernel∫
dΩk g2PV ((P − k)2)
[
1 − 2m2π
(P − k)2 + m2π
]
• Second term can be approximated
ωg2(P 2, k2) = 2m2π
∫
dΩkg2PV ((P − k)2)
(P − k)2 + m2π
≈ g2PV (|P − k|2) 2m2
π
a +√
a2 − b2
• Reliable when analytic
structure of gPV is not key to that of solutionCraig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 24/38
First Contents Back Conclusion
Model pion-nucleon coupling
gPV (P, k) =g
2Mexp(−(P − k)2/Λ2)
• B-Kernel∫
dΩk g2PV ((P − k)2)
[
1 − 2m2π
(P − k)2 + m2π
]
• Total Kernel:
≈ g2PV (P 2, k2) − g2
PV (|P 2 − k2|) 2m2π
a +√
a2 − b2,
=: g2PV (P 2, k2) − g2
PV (P 2, k2))2m2
π
a +√
a2 − b2,
• Analytic structure is transparent
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 24/38
First Contents Back Conclusion
Nucleon’s self energy and mass shift
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 25/38
First Contents Back Conclusion
Nucleon’s self energy and mass shift
• Solve DSE Nonperturbatively
M2DA2(−M2
D) = [M + B(−M2D)]2
δM = MD − M
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 25/38
First Contents Back Conclusion
Nucleon’s self energy and mass shift
• Solve DSE Nonperturbatively
M2DA2(−M2
D) = [M + B(−M2D)]2
δM = MD − M
• Vector self energy
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 25/38
First Contents Back Conclusion
Nucleon’s self energy and mass shift
• Solve DSE Nonperturbatively
M2DA2(−M2
D) = [M + B(−M2D)]2
δM = MD − M
• Scalar self energy
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 25/38
First Contents Back Conclusion
Nucleon’s self energy and mass shift
• Solve DSE Nonperturbatively
M2DA2(−M2
D) = [M + B(−M2D)]2
δM = MD − M
(Λ,ΛN ) (Λ,ΛN ) (Λ,ΛN )
(0.9,∞) (0.9, 1.5) (0.9, 2.0)
− δM (MeV) 222 61 99
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 25/38
First Contents Back Conclusion
Nucleon’s self energy and mass shift
• Solve DSE Nonperturbatively
M2DA2(−M2
D) = [M + B(−M2D)]2
δM = MD − M
(Λ,ΛN ) (Λ,ΛN ) (Λ,ΛN )
(0.9,∞) (0.9, 1.5) (0.9, 2.0)
− δM (MeV) 222 61 99
No suppression for nucleon off-shell in self-energy loop;
i.e, gPV ((P − k2), P 2, k2)
Neglected this dependence
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 25/38
First Contents Back Conclusion
Nucleon’s self energy and mass shift
• Solve DSE Nonperturbatively
M2DA2(−M2
D) = [M + B(−M2D)]2
δM = MD − M
(Λ,ΛN ) (Λ,ΛN ) (Λ,ΛN )
(0.9,∞) (0.9, 1.5) (0.9, 2.0)
− δM (MeV) 222 61 99
gPV (P 2, k2, P · k) =g
2Me−(P−k)2/Λ2
e−(P 2+M2+k2+M2)/Λ2N
Correct on-shell limit:
gPV (P 2 = −M2, k2 = −M2, (P − k)2 = 0) =g
2MCraig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 25/38
First Contents Back Conclusion
Nucleon’s self energy and mass shift
• Solve DSE Nonperturbatively
M2DA2(−M2
D) = [M + B(−M2D)]2
δM = MD − M Range from mesonexchange model phen.
(Λ,ΛN ) (Λ,ΛN ) (Λ,ΛN )
(0.9,∞) (0.9, 1.5) (0.9, 2.0)
− δM (MeV) 222 61 99
gPV (P 2, k2, P · k) =g
2Me−(P−k)2/Λ2
e−(P 2+M2+k2+M2)/Λ2N
ΛN → ∞ ⇒ pointlike nucleon
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 25/38
First Contents Back Conclusion
Pion loop’s effect
Nonpointlike πN -loop
. . . reduces nucleon’s mass by ∼ 100 MeV
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 26/38
First Contents Back Conclusion
Pion loop’s effect
Nonpointlike πN -loop
. . . reduces nucleon’s mass by ∼ 100 MeV
There’s also a π∆-loop
. . . reduces nucleon’s mass by not more than 100 MeV
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 26/38
First Contents Back Conclusion
Pion loop’s effect
Nonpointlike πN -loop
. . . reduces nucleon’s mass by ∼ 100 MeV
There’s also a π∆-loop
. . . reduces nucleon’s mass by not more than 100 MeV
−δMN ∼ 200 MeV
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 26/38
First Contents Back Conclusion
Pion loop’s effect
Nonpointlike πN -loop
. . . reduces nucleon’s mass by ∼ 100 MeV
There’s also a π∆-loop
. . . reduces nucleon’s mass by not more than 100 MeV
−δMN ∼ 200 MeV
Qualitative effect of this?
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 26/38
First Contents Back Conclusion
Too much of a good thing
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 27/38
First Contents Back Conclusion
Too much of a good thing
• Refit Faddeev model parameters,
allowing for heavier “quark-core” mass
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 27/38
First Contents Back Conclusion
Too much of a good thing
ω0+ ω1+ MN M∆ ωf1ωf2
R
0+ 0.45 - 1.44 - 0.36 0.35 2.32
0+ & 1+ 0.45 1.36 1.14 1.33 0.44 0.36 0.54
0+ 0.64 - 1.59 - 0.39 0.41 1.28
0+ & 1+ 0.64 1.19 0.94 1.23 0.49 0.44 0.25
50% reduction in role of axial-vector diquark
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 27/38
First Contents Back Conclusion
Too much of a good thing
ω0+ ω1+ MN M∆ ωf1ωf2
R
0+ 0.45 - 1.44 - 0.36 0.35 2.32
0+ & 1+ 0.45 1.36 1.14 1.33 0.44 0.36 0.54
0+ 0.64 - 1.59 - 0.39 0.41 1.28
0+ & 1+ 0.64 1.19 0.94 1.23 0.49 0.44 0.25
50% reduction in role of axial-vector diquark
10% increase in role of scalar diquark
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 27/38
First Contents Back Conclusion
Too much of a good thing
ω0+ ω1+ MN M∆ ωf1ωf2
R
0+ 0.45 - 1.44 - 0.36 0.35 2.32
0+ & 1+ 0.45 1.36 1.14 1.33 0.44 0.36 0.54
0+ 0.64 - 1.59 - 0.39 0.41 1.28
0+ & 1+ 0.64 1.19 0.94 1.23 0.49 0.44 0.25
Unsurprisingly:
Requiring Exact Fit to N , ∆ masses
with only q, (qq)JP Degrees of Freedom
⇒ Forces 1+ to mimic, in part, effect of π
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 27/38
First Contents Back Conclusion
Pseudoscalar mesonsand Form Factors
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 28/38
First Contents Back Conclusion
Pseudoscalar mesonsand Form Factors
Light mass of pseudoscalar mesons means they play a very
important role in many aspects of hadron physics.
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 28/38
First Contents Back Conclusion
Pseudoscalar mesonsand Form Factors
Light mass of pseudoscalar mesons means they play a very
important role in many aspects of hadron physics.
Indeed, no approach to low-energy hadron physics that
does not explicitly account for pseudoscalar meson degrees
of freedom can be valid.
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 28/38
First Contents Back Conclusion
Pseudoscalar mesonsand Form Factors
Light mass of pseudoscalar mesons means they play a very
important role in many aspects of hadron physics.
Indeed, no approach to low-energy hadron physics that
does not explicitly account for pseudoscalar meson degrees
of freedom can be valid.
Another example . . . pseudoscalar mesons also contribute
materially to form factors.
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 28/38
First Contents Back Conclusion
Pseudoscalar mesonsand Form Factors
Light mass of pseudoscalar mesons means they play a very
important role in many aspects of hadron physics.
Indeed, no approach to low-energy hadron physics that
does not explicitly account for pseudoscalar meson degrees
of freedom can be valid.
Another example . . . pseudoscalar mesons also contribute
materially to form factors.
Illustrate with γN → ∆ transition form factor. Focus on the
M1 (spin-flip) form factor, G∗M(Q2).
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 28/38
First Contents Back Conclusion
Harry LeePions and Form Factors
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 29/38
First Contents Back Conclusion
Harry LeePions and Form Factors
Dynamical coupled-channels model . . . Analyzed extensive JLabdata . . . Completed a study of the ∆(1236)
Meson Exchange Model for πN Scattering and γN → πN Reaction, T. Satoand T.-S. H. Lee, Phys. Rev. C 54, 2660 (1996)
Dynamical Study of the ∆ Excitation in N(e, e′π) Reactions, T. Sato andT.-S. H. Lee, Phys. Rev. C 63, 055201/1-13 (2001)
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 29/38
First Contents Back Conclusion
Harry LeePions and Form Factors
Dynamical coupled-channels model . . . Analyzed extensive JLabdata . . . Completed a study of the ∆(1236)
Meson Exchange Model for πN Scattering and γN → πN Reaction, T. Satoand T.-S. H. Lee, Phys. Rev. C 54, 2660 (1996)
Dynamical Study of the ∆ Excitation in N(e, e′π) Reactions, T. Sato andT.-S. H. Lee, Phys. Rev. C 63, 055201/1-13 (2001)
Pion cloud effects are large in the low Q2 region.
0
1
2
3
0 1 2 3 4
Q2(GeV/c)2
DressedBare
Ratio of the M1 form factor in γN → ∆
transition and proton dipole form factor GD .Solid curve is G∗
M(Q2)/GD(Q2) including
pions; Dotted curve is GM (Q2)/GD(Q2)
without pions.
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 29/38
First Contents Back Conclusion
Harry LeePions and Form Factors
Dynamical coupled-channels model . . . Analyzed extensive JLabdata . . . Completed a study of the ∆(1236)
Meson Exchange Model for πN Scattering and γN → πN Reaction, T. Satoand T.-S. H. Lee, Phys. Rev. C 54, 2660 (1996)
Dynamical Study of the ∆ Excitation in N(e, e′π) Reactions, T. Sato andT.-S. H. Lee, Phys. Rev. C 63, 055201/1-13 (2001)
Pion cloud effects are large in the low Q2 region.
0
1
2
3
0 1 2 3 4
Q2(GeV/c)2
DressedBare
Ratio of the M1 form factor in γN → ∆
transition and proton dipole form factor GD .Solid curve is G∗
M(Q2)/GD(Q2) including
pions; Dotted curve is GM (Q2)/GD(Q2)
without pions.
Quark Core
Responsible for only 2/3 ofresult at small Q2
Dominant for Q2 >2 – 3 GeV2
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 29/38
First Contents Back Conclusion
Results: Nucleonand ∆ Masses
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 30/38
First Contents Back Conclusion
Results: Nucleonand ∆ Masses
Mass-scale parameters (in GeV)
for the scalar and axial-vector
diquark correlations, fixed by
fitting nucleon and ∆ masses
Set A – fit to the actual masses was required; whereas for
Set B – fitted mass was offset to allow for “π-cloud” contributions
set MN M∆ m0+ m1+ ω0+ ω1+
A 0.94 1.23 0.63 0.84 0.44=1/(0.45 fm) 0.59=1/(0.33 fm)
B 1.18 1.33 0.80 0.89 0.56=1/(0.35 fm) 0.63=1/(0.31 fm)
m1+ → ∞: MAN = 1.15 GeV; MB
N = 1.46 GeV
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 30/38
First Contents Back Conclusion
Results: Nucleonand ∆ Masses
Mass-scale parameters (in GeV)
for the scalar and axial-vector
diquark correlations, fixed by
fitting nucleon and ∆ masses
Set A – fit to the actual masses was required; whereas for
Set B – fitted mass was offset to allow for “π-cloud” contributions
set MN M∆ m0+ m1+ ω0+ ω1+
A 0.94 1.23 0.63 0.84 0.44=1/(0.45 fm) 0.59=1/(0.33 fm)
B 1.18 1.33 0.80 0.89 0.56=1/(0.35 fm) 0.63=1/(0.31 fm)
m1+ → ∞: MAN = 1.15 GeV; MB
N = 1.46 GeV
Axial-vector diquark provides significant attractionCraig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 30/38
First Contents Back Conclusion
Results: Nucleonand ∆ Masses
Mass-scale parameters (in GeV)
for the scalar and axial-vector
diquark correlations, fixed by
fitting nucleon and ∆ masses
Set A – fit to the actual masses was required; whereas for
Set B – fitted mass was offset to allow for “π-cloud” contributions
set MN M∆ m0+ m1+ ω0+ ω1+
A 0.94 1.23 0.63 0.84 0.44=1/(0.45 fm) 0.59=1/(0.33 fm)
B 1.18 1.33 0.80 0.89 0.56=1/(0.35 fm) 0.63=1/(0.31 fm)
m1+ → ∞: MAN = 1.15 GeV; MB
N = 1.46 GeV
Constructive Interference: 1++-diquark + ∂µπCraig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 30/38
First Contents Back Conclusion
Nucleon-Photon Vertex
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 31/38
First Contents Back Conclusion
Nucleon-Photon Vertex
M. Oettel, M. Pichowskyand L. von Smekal, nu-th/9909082
6 terms . . .constructed systematically . . . current conserved automatically
for on-shell nucleons described by Faddeev Amplitude
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 31/38
First Contents Back Conclusion
Nucleon-Photon Vertex
M. Oettel, M. Pichowskyand L. von Smekal, nu-th/9909082
6 terms . . .constructed systematically . . . current conserved automatically
for on-shell nucleons described by Faddeev Amplitude
i
iΨ ΨPf
f
P
Q i
iΨ ΨPf
f
P
Q
i
iΨ ΨPPf
f
Q
Γ−
Γ
scalaraxial vector
i
iΨ ΨPf
f
P
Q
µ
i
i
X
Ψ ΨPf
f
Q
P Γ−
µi
i
X−
Ψ ΨPf
f
P
Q
ΓCraig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 31/38
First Contents Back Conclusion
Form Factor Ratio:GE/GM
0 2 4 6 8 10Q
2 [GeV2]
-1
-0.5
0
0.5
1
1.5
2
µ p GEp/ G
Mp
Rosenbluthprecision Rosenbluthpolarization transferpolarization transfer
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 32/38
First Contents Back Conclusion
Form Factor Ratio:GE/GMCombine these elements . . .
0 2 4 6 8 10Q
2 [GeV2]
-1
-0.5
0
0.5
1
1.5
2
µ p GEp/ G
Mp
Rosenbluthprecision Rosenbluthpolarization transferpolarization transfer
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 32/38
First Contents Back Conclusion
Form Factor Ratio:GE/GMCombine these elements . . .
Dressed-Quark Core
0 2 4 6 8 10Q
2 [GeV2]
-1
-0.5
0
0.5
1
1.5
2
µ p GEp/ G
Mp
Rosenbluthprecision Rosenbluthpolarization transferpolarization transfer
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 32/38
First Contents Back Conclusion
Form Factor Ratio:GE/GMCombine these elements . . .
Dressed-Quark Core
Ward-Takahashi
Identity preserving
current
0 2 4 6 8 10Q
2 [GeV2]
-1
-0.5
0
0.5
1
1.5
2
µ p GEp/ G
Mp
Rosenbluthprecision Rosenbluthpolarization transferpolarization transfer
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 32/38
First Contents Back Conclusion
Form Factor Ratio:GE/GMCombine these elements . . .
Dressed-Quark Core
Ward-Takahashi
Identity preserving
current
Anticipate and
Estimate Pion
Cloud’s Contribution
0 2 4 6 8 10Q
2 [GeV2]
-1
-0.5
0
0.5
1
1.5
2
µ p GEp/ G
Mp
Rosenbluthprecision Rosenbluthpolarization transferpolarization transfer
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 32/38
First Contents Back Conclusion
Form Factor Ratio:GE/GMCombine these elements . . .
Dressed-Quark Core
Ward-Takahashi
Identity preserving
current
Anticipate and
Estimate Pion
Cloud’s Contribution
0 2 4 6 8 10Q
2 [GeV2]
-1
-0.5
0
0.5
1
1.5
2
µ p GEp/ G
Mp
covariant Fadeev resultRosenbluthprecision Rosenbluthpolarization transferpolarization transfer
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 32/38
First Contents Back Conclusion
Form Factor Ratio:GE/GMCombine these elements . . .
Dressed-Quark Core
Ward-Takahashi
Identity preserving
current
Anticipate and
Estimate Pion
Cloud’s Contribution
0 2 4 6 8 10Q
2 [GeV2]
-1
-0.5
0
0.5
1
1.5
2
µ p GEp/ G
Mp
covariant Fadeev resultRosenbluthprecision Rosenbluthpolarization transferpolarization transfer
All parameters fixed in
other applications . . . Not varied.
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 32/38
First Contents Back Conclusion
Form Factor Ratio:GE/GMCombine these elements . . .
Dressed-Quark Core
Ward-Takahashi
Identity preserving
current
Anticipate and
Estimate Pion
Cloud’s Contribution
0 2 4 6 8 10Q
2 [GeV2]
-1
-0.5
0
0.5
1
1.5
2
µ p GEp/ G
Mp
covariant Fadeev resultRosenbluthprecision Rosenbluthpolarization transferpolarization transfer
All parameters fixed in
other applications . . . Not varied.
Agreement with Pol. Trans. data at Q2∼> 2 GeV2
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 32/38
First Contents Back Conclusion
Form Factor Ratio:GE/GMCombine these elements . . .
Dressed-Quark Core
Ward-Takahashi
Identity preserving
current
Anticipate and
Estimate Pion
Cloud’s Contribution
0 2 4 6 8 10Q
2 [GeV2]
-1
-0.5
0
0.5
1
1.5
2
µ p GEp/ G
Mp
covariant Fadeev resultRosenbluthprecision Rosenbluthpolarization transferpolarization transfer
All parameters fixed in
other applications . . . Not varied.
Agreement with Pol. Trans. data at Q2∼> 2 GeV2
Correlations in Faddeev amplitude – quark orbital
angular momentum – essential to that agreement
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 32/38
First Contents Back Conclusion
Form Factor Ratio:GE/GMCombine these elements . . .
Dressed-Quark Core
Ward-Takahashi
Identity preserving
current
Anticipate and
Estimate Pion
Cloud’s Contribution
0 2 4 6 8 10Q
2 [GeV2]
-1
-0.5
0
0.5
1
1.5
2
µ p GEp/ G
Mp
covariant Fadeev resultRosenbluthprecision Rosenbluthpolarization transferpolarization transfer
All parameters fixed in
other applications . . . Not varied.
Agreement with Pol. Trans. data at Q2∼> 2 GeV2
Correlations in Faddeev amplitude – quark orbital
angular momentum – essential to that agreement
Predict Zero at Q2 ≈ 6.5GeV2
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 32/38
First Contents Back Conclusion
Density profileof charge and magnetisation
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 33/38
First Contents Back Conclusion
Density profileof charge and magnetisation
Proton’s Electromagnetic Form Factor
Appearance of a zero in GE(Q2) – Completely
Unexpected
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 33/38
First Contents Back Conclusion
Density profileof charge and magnetisation
Proton’s Electromagnetic Form Factor
Appearance of a zero in GE(Q2) – Completely
Unexpected
0 0.2 0.4 0.6 0.8 1r (fm)
0
0.002
0.004
0.006
0.008
0.01
ρ(r)
Peak Position and Height -- as zero moves in toward q*q=0
peak moves out and falls in magnitude-- redistribution of charge ... moves to larger r
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 33/38
First Contents Back Conclusion
Density profileof charge and magnetisation
Proton’s Electromagnetic Form Factor
Appearance of a zero in GE(Q2) – Completely
Unexpected
0 0.2 0.4 0.6 0.8 1r (fm)
0
0.002
0.004
0.006
0.008
0.01
ρ(r)
Peak Position and Height -- as zero moves in toward q*q=0
peak moves out and falls in magnitude-- redistribution of charge ... moves to larger r
However, Current Density remains peaked at r = 0!Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 33/38
First Contents Back Conclusion
Density profileof charge and magnetisation
Proton’s Electromagnetic Form Factor
Appearance of a zero in GE(Q2) – Completely
Unexpected
Wave Function is complex and correlated mix of virtual
particles and antiparticles: s−, p− and d−waves
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 33/38
First Contents Back Conclusion
Density profileof charge and magnetisation
Proton’s Electromagnetic Form Factor
Appearance of a zero in GE(Q2) – Completely
Unexpected
Wave Function is complex and correlated mix of virtual
particles and antiparticles: s−, p− and d−waves
Simple independent-particle three-quark bag-model picture
is profoundly incorrect
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 33/38
First Contents Back Conclusion
Improved current
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 34/38
First Contents Back Conclusion
Improved current
Composite axial-vector diquark correlation
Electromagnetic current can be complicated
Limited constraints on large-Q2 behaviour
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 34/38
First Contents Back Conclusion
Improved current
Composite axial-vector diquark correlation
Electromagnetic current can be complicated
Limited constraints on large-Q2 behaviour
Improved performance of code
Implemented corrections so that large-Q2 behaviour of
form factors could be reliably calculated
Exposed two weaknesses in rudimentary Ansatz
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 34/38
First Contents Back Conclusion
Improved current
Composite axial-vector diquark correlation
Improved performance of code
Implemented corrections so that large-Q2 behaviour of
form factors could be reliably calculated
Exposed two weaknesses in rudimentary Ansatz
Diquark effectively pointlike to hard probe
Didn’t account for diquark being off-shell in recoil
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 34/38
First Contents Back Conclusion
Improved current
Composite axial-vector diquark correlation
Minor but material improvements to current
Introduce form factor: radius 0.8 fm
Increase recoil mass by 10%
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 34/38
First Contents Back Conclusion
Improved current
Composite axial-vector diquark correlation
Minor but material improvements to current
Introduce form factor: radius 0.8 fm
Increase recoil mass by 10%
−2.0
−1.5
−1.0
−0.5
0
0.5
1.0
GE/G
MR
atio
s
0 2 4 6 8 10 12
Q2 (GeV2)
µp × Gp
E
Gp
M
µn × GnE
GnM
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 34/38
First Contents Back Conclusion
Improved current
Composite axial-vector diquark correlation
Minor but material improvements to current
Introduce form factor: radius 0.8 fm
Increase recoil mass by 10%
−2.0
−1.5
−1.0
−0.5
0
0.5
1.0
GE/G
MR
atio
s
0 2 4 6 8 10 12
Q2 (GeV2)
µp × Gp
E
Gp
M
µn × GnE
GnM
Proton – zero shifted:
6.5 → 8.0 GeV2
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 34/38
First Contents Back Conclusion
Improved current
Composite axial-vector diquark correlation
Minor but material improvements to current
Introduce form factor: radius 0.8 fm
Increase recoil mass by 10%
−2.0
−1.5
−1.0
−0.5
0
0.5
1.0
GE/G
MR
atio
s
0 2 4 6 8 10 12
Q2 (GeV2)
µp × Gp
E
Gp
M
µn × GnE
GnM
Proton – zero shifted:
6.5 → 8.0 GeV2
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 34/38
First Contents Back Conclusion
Improved current
Composite axial-vector diquark correlation
Minor but material improvements to current
Introduce form factor: radius 0.8 fm
Increase recoil mass by 10%
−2.0
−1.5
−1.0
−0.5
0
0.5
1.0
GE/G
MR
atio
s
0 2 4 6 8 10 12
Q2 (GeV2)
µp × Gp
E
Gp
M
µn × GnE
GnM
Proton – zero shifted:
6.5 → 8.0 GeV2
Neutron – peak shifted:
7.5 → 5.0 GeV2
& now predict zero
a little above 11 GeV2
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 34/38
First Contents Back Conclusion
Improved current
Composite axial-vector diquark correlation
Minor but material improvements to current
Introduce form factor: radius 0.8 fm
Increase recoil mass by 10%
−2.0
−1.5
−1.0
−0.5
0
0.5
1.0
GE/G
MR
atio
s
0 2 4 6 8 10 12
Q2 (GeV2)
µp × Gp
E
Gp
M
µn × GnE
GnM
Proton – zero shifted:
6.5 → 8.0 GeV2
Neutron – peak shifted:
7.5 → 5.0 GeV2
& now predict zero
a little above 11 GeV2
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 34/38
First Contents Back Conclusion
Pion Cloud
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 35/38
First Contents Back Conclusion
Pion CloudF1 – neutron
0 2 4 6 8 10
Q2/MN
2
-0.1
-0.05
0
F1n
Kelly ParametrisationDSE
Comparison
between Faddeev
equation result
and Kelly’s
parametrisation
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 35/38
First Contents Back Conclusion
Pion CloudF1 – neutron
0 2 4 6 8 10
Q2/MN
2
-0.1
-0.05
0
F1n
Kelly ParametrisationDSE
Comparison
between Faddeev
equation result
and Kelly’s
parametrisation
Faddeev equation
set-up to describe
dressed-quark
core
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 35/38
First Contents Back Conclusion
Pion CloudF1 – neutron
0 2 4 6 8 10
Q2/MN
2
-0.1
-0.05
0
F1n
Kelly ParametrisationDSE
Pion cloud contribution
Comparison
between Faddeev
equation result
and Kelly’s
parametrisation
Faddeev equation
set-up to describe
dressed-quark
core
Pseudoscalar
meson cloud (and
related effects)
significant for
Q2∼< 3 − 4 M2
N
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 35/38
First Contents Back Conclusion
Epilogue
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 36/38
First Contents Back Conclusion
Epilogue
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 36/38
First Contents Back Conclusion
Epilogue
DCSB exists in QCD.
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 36/38
First Contents Back Conclusion
Epilogue
DCSB exists in QCD.
It is manifest in dressed propagators and
vertices
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 36/38
First Contents Back Conclusion
Epilogue
DCSB exists in QCD.
It is manifest in dressed propagators and
vertices
It predicts, amongst other things, that
light current-quarks become heavy
constituent-quarks
pseudoscalar mesons are unnaturally
light
pseudoscalar mesons couple unnaturally
strongly to light-quarks
pseudscalar mesons couple unnaturally
strongly to the lightest baryons
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 36/38
First Contents Back Conclusion
Epilogue
DCSB exists in QCD.
It is manifest in dressed propagators and
vertices
It predicts, amongst other things, that
light current-quarks become heavy
constituent-quarks
pseudoscalar mesons are unnaturally
light
pseudoscalar mesons couple unnaturally
strongly to light-quarks
pseudscalar mesons couple unnaturally
strongly to the lightest baryons
It impacts dramatically upon observables.Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 36/38
First Contents Back Conclusion
Epilogue
Form Factors - progress anticipated in near- to medium-term
Quantifying pseudoscalar meson “cloud” effects
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 36/38
First Contents Back Conclusion
Epilogue
Form Factors - progress anticipated in near- to medium-term
Quantifying pseudoscalar meson “cloud” effects
Locating and explaining the transition from nonp-QCD to
p-QCD in the pion and nucleon electromagnetic form
factors
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 36/38
First Contents Back Conclusion
Epilogue
Form Factors - progress anticipated in near- to medium-term
Quantifying pseudoscalar meson “cloud” effects
Locating and explaining the transition from nonp-QCD to
p-QCD in the pion and nucleon electromagnetic form
factors
Explaining the high Q2 behavior of the proton form factor
ratio in the space-like region
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 36/38
First Contents Back Conclusion
Epilogue
Form Factors - progress anticipated in near- to medium-term
Quantifying pseudoscalar meson “cloud” effects
Locating and explaining the transition from nonp-QCD to
p-QCD in the pion and nucleon electromagnetic form
factors
Explaining the high Q2 behavior of the proton form factor
ratio in the space-like region
Detailing broadly the role of two-photon exchange
contributions
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 36/38
First Contents Back Conclusion
Epiloguenothing!
Form Factors - progress anticipated in near- to medium-term
Quantifying pseudoscalar meson “cloud” effects
Locating and explaining the transition from nonp-QCD to
p-QCD in the pion and nucleon electromagnetic form
factors
Explaining the high Q2 behavior of the proton form factor
ratio in the space-like region
Detailing broadly the role of two-photon exchange
contributions
Explaining relationship between parton properties on the
light-front and rest frame structure of hadronsCraig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 36/38
First Contents Back Conclusion
Vector piece of nucleon’s self energy
−1.0 0.0 1.0 2.0sign(t
2) |t| (GeV)
1.00
1.04
1.08
A(t
2 )
M = 0.94, mπ = 0.14
Λ = 0.9 GeV, gA = 1
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 37/38
First Contents Back Conclusion
Vector piece of nucleon’s self energy
−1.0 0.0 1.0 2.0sign(t
2) |t| (GeV)
1.00
1.04
1.08
A(t
2 )
M = 0.94, mπ = 0.14
Λ = 0.9 GeV, gA = 1
Solid line: One-loop;
numerically evaluated,
exact (numerical) kernel
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 37/38
First Contents Back Conclusion
Vector piece of nucleon’s self energy
−1.0 0.0 1.0 2.0sign(t
2) |t| (GeV)
1.00
1.04
1.08
A(t
2 )
M = 0.94, mπ = 0.14
Λ = 0.9 GeV, gA = 1
Solid line: One-loop;
numerically evaluated,
exact (numerical) kernel
Dotted line: One-loop;
algebraic; code test
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 37/38
First Contents Back Conclusion
Vector piece of nucleon’s self energy
−1.0 0.0 1.0 2.0sign(t
2) |t| (GeV)
1.00
1.04
1.08
A(t
2 )
M = 0.94, mπ = 0.14
Λ = 0.9 GeV, gA = 1
Solid line: One-loop;
numerically evaluated,
exact (numerical) kernel
Dotted line: One-loop;
algebraic; code test
Dashed line: Fully
self-consistent numerical
solution; extra pion loops
do little
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 37/38
First Contents Back Conclusion
Vector piece of nucleon’s self energy
−1.0 0.0 1.0 2.0sign(t
2) |t| (GeV)
1.00
1.04
1.08
A(t
2 )
M = 0.94, mπ = 0.14
Λ = 0.9 GeV, gA = 1
Solid line: One-loop;
numerically evaluated,
exact (numerical) kernel
Dotted line: One-loop;
algebraic; code test
Dot-dashed line: Fully
self-consistent numerical
solution; continuation to
timelike region return
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 37/38
First Contents Back Conclusion
Scalar piece of nucleon’s self energy
−1.0 0.0 1.0 2.0sign(t
2) |t| (GeV)
0.84
0.88
0.92
M+
B(t
2 ) (G
eV)
M = 0.94, mπ = 0.14
Λ = 0.9 GeV, gA = 1
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 38/38
First Contents Back Conclusion
Scalar piece of nucleon’s self energy
−1.0 0.0 1.0 2.0sign(t
2) |t| (GeV)
0.84
0.88
0.92
M+
B(t
2 ) (G
eV)
M = 0.94, mπ = 0.14
Λ = 0.9 GeV, gA = 1
Solid line: One-loop;
numerically evaluated,
exact (numerical) kernel
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 38/38
First Contents Back Conclusion
Scalar piece of nucleon’s self energy
−1.0 0.0 1.0 2.0sign(t
2) |t| (GeV)
0.84
0.88
0.92
M+
B(t
2 ) (G
eV)
M = 0.94, mπ = 0.14
Λ = 0.9 GeV, gA = 1
Solid line: One-loop;
numerically evaluated,
exact (numerical) kernel
Dotted line: One-loop;
algebraic; code test
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 38/38
First Contents Back Conclusion
Scalar piece of nucleon’s self energy
−1.0 0.0 1.0 2.0sign(t
2) |t| (GeV)
0.84
0.88
0.92
M+
B(t
2 ) (G
eV)
M = 0.94, mπ = 0.14
Λ = 0.9 GeV, gA = 1
Solid line: One-loop;
numerically evaluated,
exact (numerical) kernel
Dotted line: One-loop;
algebraic; code test
Dashed line: Fully
self-consistent numerical
solution; extra pion loops
do little
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 38/38
First Contents Back Conclusion
Scalar piece of nucleon’s self energy
−1.0 0.0 1.0 2.0sign(t
2) |t| (GeV)
0.84
0.88
0.92
M+
B(t
2 ) (G
eV)
M = 0.94, mπ = 0.14
Λ = 0.9 GeV, gA = 1
Solid line: One-loop;
numerically evaluated,
exact (numerical) kernel
Dotted line: One-loop;
algebraic; code test
Dot-dashed line: Fully
self-consistent numerical
solution; continuation to
timelike region return
Craig Roberts: Dyson Schwinger Equations and QCD
25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 38/38