First Contents Back Conclusion Charting the Interaction Between Light Quarks Craig D. Roberts [email protected]Physics Division Argonne National Laboratory http://www.phy.anl.gov/theory/staff/cdr.html Craig Roberts: Charting the interaction between light quarks CLAS12 European Workshop ... 23 transparencies – p. 1/40
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http://www.phy.anl.gov/theory/staff/cdr.htmlCraig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 1/40
First Contents Back Conclusion
Universal Truths
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 2/40
First Contents Back Conclusion
Universal Truths
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 2/40
First Contents Back Conclusion
Universal Truths
Spectrum of excited states and transition form factors
provide unique information about long-range interaction
between light-quarks and distribution of hadron’s
characterising properties amongst its QCD constituents.
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 2/40
First Contents Back Conclusion
Universal Truths
Spectrum of excited states and transition form factors
provide unique information about long-range interaction
between light-quarks and distribution of hadron’s
characterising properties amongst its QCD constituents.
Dynamical Chiral Symmetry Breaking (DCSB) is most
important mass generating mechanism for visible matter in the
Universe.
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 2/40
First Contents Back Conclusion
Universal Truths
Spectrum of excited states and transition form factors
provide unique information about long-range interaction
between light-quarks and distribution of hadron’s
characterising properties amongst its QCD constituents.
Dynamical Chiral Symmetry Breaking (DCSB) is most
important mass generating mechanism for visible matter in the
Universe. Higgs mechanism is irrelevant to light-quarks.
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 2/40
First Contents Back Conclusion
Universal Truths
Spectrum of excited states and transition form factors
provide unique information about long-range interaction
between light-quarks and distribution of hadron’s
characterising properties amongst its QCD constituents.
Dynamical Chiral Symmetry Breaking (DCSB) is most
important mass generating mechanism for visible matter in the
Universe. Higgs mechanism is irrelevant to light-quarks.
Running of quark mass entails that calculations at even
modest Q2 require a Poincaré-covariant approach.
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 2/40
First Contents Back Conclusion
Universal Truths
Spectrum of excited states and transition form factors
provide unique information about long-range interaction
between light-quarks and distribution of hadron’s
characterising properties amongst its QCD constituents.
Dynamical Chiral Symmetry Breaking (DCSB) is most
important mass generating mechanism for visible matter in the
Universe. Higgs mechanism is irrelevant to light-quarks.
Running of quark mass entails that calculations at even
modest Q2 require a Poincaré-covariant approach. Covariance
requires existence of quark orbital angular momentum in
hadron’s rest-frame wave function.
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 2/40
First Contents Back Conclusion
Universal Truths
Spectrum of excited states and transition form factors
provide unique information about long-range interaction
between light-quarks and distribution of hadron’s
characterising properties amongst its QCD constituents.
Dynamical Chiral Symmetry Breaking (DCSB) is most
important mass generating mechanism for visible 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: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 2/40
First Contents Back Conclusion
Universal Truths
Spectrum of excited states and transition form factors
provide unique information about long-range interaction
between light-quarks and distribution of hadron’s
characterising properties amongst its QCD constituents.
Dynamical Chiral Symmetry Breaking (DCSB) is most
important mass generating mechanism for visible 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: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 2/40
First Contents Back Conclusion
QCD’s Challenges
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 3/40
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: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 3/40
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: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 3/40
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: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 3/40
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: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 3/40
First Contents Back Conclusion
Dichotomy of Pion– Goldstone Mode and Bound state
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 4/40
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: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 4/40
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 m2
π ∝ mq
Current Algebra . . . 1968
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 4/40
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 m2
π ∝ mq
Current Algebra . . . 1968
The correct understanding of pion observables;e.g. mass, decay constant and form factors,requires an approach to contain a
well-defined and valid chiral limit;
and an accurate realisation ofdynamical chiral symmetry breaking.
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 4/40
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 m2
π ∝ mq
Current Algebra . . . 1968
The correct understanding of pion observables;e.g. mass, decay constant and form factors,requires an approach to contain a
well-defined and valid chiral limit;
and an accurate realisation ofdynamical chiral symmetry breaking.
Highly NontrivialCraig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 4/40
First Contents Back Conclusion
What’s the Problem?
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 5/40
First Contents Back Conclusion
What’s the Problem?
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: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 5/40
First Contents Back Conclusion
What’s the Problem?
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.
Means . . . must calculate hadron wave functions
– Can’t be done using perturbation theory
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 5/40
First Contents Back Conclusion
What’s the Problem?
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.
Means . . . must calculate hadron wave functions
– Can’t be done using perturbation theory
Why problematic? Isn’t same true in quantum mechanics?
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 5/40
First Contents Back Conclusion
What’s the Problem?
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.
Means . . . must calculate hadron wave functions
– Can’t be done using perturbation theory
Why problematic? Isn’t same true in quantum mechanics?
Differences!
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 5/40
First Contents Back Conclusion
What’s the Problem?Relativistic QFT!
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.
Differences!
Here relativistic effects are crucial – virtual particles,
quintessence of Relativistic Quantum Field Theory –
must be included
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 5/40
First Contents Back Conclusion
What’s the Problem?Relativistic QFT!
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.
Differences!
Here relativistic effects are crucial – virtual particles,
quintessence of Relativistic Quantum Field Theory –
must be included
Interaction between quarks – the Interquark “Potential” –
unknown throughout > 98% of a hadron’s volume
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 5/40
First Contents Back Conclusion
Intranucleon Interaction
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 6/40
First Contents Back Conclusion
Intranucleon Interaction
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 6/40
First Contents Back Conclusion
Intranucleon Interaction
98% of the volume
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 6/40
First Contents Back Conclusion
Intranucleon Interaction?What is the
98% of the volume
The question must berigorously defined, and theanswer mapped out usingexperiment and theory.
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 6/40
First Contents Back Conclusion
Frontiers of Nuclear Science:A Long Range Plan (2007)
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 7/40
First Contents Back Conclusion
Frontiers of Nuclear Science:Theoretical Advances
Σ=
D
γΓS
Gap Equation
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 7/40
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: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 7/40
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: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 7/40
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: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 7/40
First Contents Back ConclusionCraig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 8/40
First Contents Back Conclusion
• Established understanding oftwo- and three-point functions
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 8/40
First Contents Back Conclusion
Hadrons
• Established understanding oftwo- and three-point functions
• What about bound states?
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 8/40
First Contents Back Conclusion
Hadrons
• Without bound states, Comparison withexperiment is impossible
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 8/40
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: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 8/40
First Contents Back Conclusion
Hadrons
• Without bound states, Comparison withexperiment is impossible
• Bethe-Salpeter Equation
QFT Generalisation of Lippmann-Schwinger Equation.
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 8/40
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: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 8/40
First Contents Back Conclusion
What is the light-quarkLong-Range Potential?
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 9/40
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: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 9/40
First Contents Back Conclusion
Bethe-Salpeter Kernel
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 10/40
First Contents Back Conclusion
Bethe-Salpeter Kernel
Axial-vector Ward-Takahashi identity
Pµ Γl5µ(k;P ) = S−1(k+)
1
2λl
f iγ5 +1
2λl
f iγ5 S−1(k−)
−Mζ iΓl5(k;P ) − iΓl
5(k;P ) Mζ
QFT Statement of Chiral Symmetry
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 10/40
First Contents Back Conclusion
Bethe-Salpeter Kernel
Axial-vector Ward-Takahashi identity
Pµ Γl5µ(k;P ) = S−1(k+)
1
2λl
f iγ5 +1
2λl
f iγ5 S−1(k−)
−Mζ iΓl5(k;P ) − iΓl
5(k;P ) Mζ
Satisfies BSE Satisfies DSE
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 10/40
First Contents Back Conclusion
Bethe-Salpeter Kernel
Axial-vector Ward-Takahashi identity
Pµ Γl5µ(k;P ) = S−1(k+)
1
2λl
f iγ5 +1
2λl
f iγ5 S−1(k−)
−Mζ iΓl5(k;P ) − iΓl
5(k;P ) Mζ
Satisfies BSE Satisfies DSEKernels very differentbut must be intimately related
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 10/40
First Contents Back Conclusion
Bethe-Salpeter Kernel
Axial-vector Ward-Takahashi identity
Pµ Γl5µ(k;P ) = S−1(k+)
1
2λl
f iγ5 +1
2λl
f iγ5 S−1(k−)
−Mζ iΓl5(k;P ) − iΓl
5(k;P ) Mζ
Satisfies BSE Satisfies DSEKernels very differentbut must be intimately related
• Relation must be preserved by truncation
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 10/40
First Contents Back Conclusion
Bethe-Salpeter Kernel
Axial-vector Ward-Takahashi identity
Pµ Γl5µ(k;P ) = S−1(k+)
1
2λl
f iγ5 +1
2λl
f iγ5 S−1(k−)
−Mζ iΓl5(k;P ) − iΓl
5(k;P ) Mζ
Satisfies BSE Satisfies DSEKernels very differentbut must be intimately related
• Relation must be preserved by truncation• Nontrivial constraint
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 10/40
First Contents Back Conclusion
Bethe-Salpeter Kernel
Axial-vector Ward-Takahashi identity
Pµ Γl5µ(k;P ) = S−1(k+)
1
2λl
f iγ5 +1
2λl
f iγ5 S−1(k−)
−Mζ iΓl5(k;P ) − iΓl
5(k;P ) Mζ
Satisfies BSE Satisfies DSEKernels very differentbut must be intimately related
• Relation must be preserved by truncation• Failure ⇒ Explicit Violation of QCD’s Chiral Symmetry
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 10/40
First Contents Back Conclusion
Persistent Challenge
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 11/40
First Contents Back Conclusion
Persistent Challenge
Infinitely Many Coupled Equations
Σ=
D
γΓS
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 11/40
First Contents Back Conclusion
Persistent Challenge
Infinitely Many Coupled Equations
Σ=
D
γΓS
Coupling between equations necessitates truncation
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 11/40
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: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 11/40
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: Charting the interaction between light quarks
CLAS12 European Workshop . . . 23 transparencies – p. 11/40
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: Charting the interaction between light quarks
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Full ALPHA formulation is required to see suppression, becausePCAC relation is at the heart of the conditions imposed forimprovement (determining coefficients of irrelevant operators)Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 30/40
First Contents Back Conclusion
Radial Excitations& Lattice-QCDMcNeile and Michael
he-la/0607032
0 0.5 1 1.5 2 2.5 3 3.5 4
( r0 mπ )
2
0
0.2
0.4
0.6
0.8
f π’/f
πnot improvedNP improvedExpt. bound
When we first heard about [this result] our first reaction was acombination of “that is remarkable” and “unbelievable”.
The suppression of fπ1is a useful benchmark that can be used to
tune and validate lattice QCD techniques that try to determine theproperties of excited states mesons.Craig Roberts: Charting the interaction between light quarks
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First Contents Back Conclusion
Pion Form Factor
Procedure Now Straightforward
Craig Roberts: Charting the interaction between light quarks
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First Contents Back Conclusion
Pion Form Factor
Solve Gap Equation⇒ Dressed-Quark Propagator, S(p)
Σ=
D
γΓS
Craig Roberts: Charting the interaction between light quarks
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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: Charting the interaction between light quarks
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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: Charting the interaction between light quarks
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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: Charting the interaction between light quarks
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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: Charting the interaction between light quarks
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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: Charting the interaction between light quarks
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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: Charting the interaction between light quarks
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First Contents Back Conclusion
Timelike Pion Form Factor
Craig Roberts: Charting the interaction between light quarks
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First Contents Back Conclusion
Timelike Pion Form Factor
Ab initio calculation intotimelike region. Deeper thanground-state ρ-meson pole
Craig Roberts: Charting the interaction between light quarks
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First Contents Back Conclusion
Timelike Pion Form Factor
-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 calculationQCDSF/UKQCD, monopole fit + error band
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
Amendolia et al.Ackermann et al.Brauel et al.Tadevosyan et al.
Horn et al.Barkov et al.DSE calculationQCDSF/UKQCD, monopole fit + error band
QCDSF/UKQCD, simulation result
Ab initio calculation intotimelike region. Deeper thanground-state ρ-meson pole
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 33/40
First Contents Back Conclusion
Timelike Pion Form Factor
-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 calculationQCDSF/UKQCD, monopole fit + error band
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
Amendolia et al.Ackermann et al.Brauel et al.Tadevosyan et al.
Horn et al.Barkov et al.DSE calculationQCDSF/UKQCD, monopole fit + error band
QCDSF/UKQCD, simulation result
Ab initio calculation intotimelike region. Deeper thanground-state ρ-meson poleρ-meson not put in “by hand” – generated dynamically as a bound-state of dressed-quark and dressed-antiquark
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 33/40
First Contents Back Conclusion
Dimensionless product: rπ fπ
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 34/40
First Contents Back Conclusion
Dimensionless product: rπ fπ
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 34/40
First Contents Back Conclusion
Dimensionless product: rπ fπ
Improved rainbow-ladder interaction
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 34/40
First Contents Back Conclusion
Dimensionless product: rπ fπ
Improved rainbow-ladder interaction
Repeating Fπ(Q2) calculation
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 34/40
First Contents Back Conclusion
Dimensionless product: rπ fπ
Improved rainbow-ladder interaction
Repeating Fπ(Q2) calculation
Experimentally: rπfπ = 0.315 ± 0.005
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 34/40
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: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 34/40
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: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 34/40
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: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 34/40
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: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 34/40
First Contents Back Conclusion
Pion Form Factors
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 35/40
First Contents Back Conclusion
Pion Form Factors
There is a sense in which it is easy to fabricate amodel that can reproduce the elasticelectromagnetic pion form factor
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 35/40
First Contents Back Conclusion
Pion Form Factors
There is a sense in which it is easy to fabricate amodel that can reproduce the elasticelectromagnetic pion form factor
However, a veracious description of the pion willsimultaneously predict the elastic electromagneticform factor, Fπ(Q2) AND the γ∗π → γ transitionform factor
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 35/40
First Contents Back Conclusion
Pion Form FactorsInfidelity without simultaneity
There is a sense in which it is easy to fabricate amodel that can reproduce the elasticelectromagnetic pion form factor
However, a veracious description of the pion willsimultaneously predict the elastic electromagneticform factor, Fπ(Q2) AND the γ∗π → γ transitionform factor
The latter is connected with the Abelian anomaly –therefore fundamentally connected with chiralsymmetry and its dynamical breaking – no meremodel can successfully describe this without finetuning
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 35/40
First Contents Back Conclusion
Pion Form FactorsInfidelity without simultaneity
There is a sense in which it is easy to fabricate amodel that can reproduce the elasticelectromagnetic pion form factor
However, a veracious description of the pion willsimultaneously predict the elastic electromagneticform factor, Fπ(Q2) AND the γ∗π → γ transitionform factor
The latter is connected with the Abelian anomaly –therefore fundamentally connected with chiralsymmetry and its dynamical breaking – no meremodel can successfully describe this without finetuning
Must similarly require prediction of γ∗π → ππ andall other anomalous processes
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 35/40
First Contents Back Conclusion
Answer for the pion
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 36/40
First Contents Back Conclusion
Answer for the pion
Two → Infinitely many . . .
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 36/40
First Contents Back Conclusion
Answer for the pion
Two → Infinitely many . . .Handle thatproperly inquantumfield theory
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 36/40
First Contents Back Conclusion
Answer for the pion
Two → Infinitely many . . .Handle thatproperly inquantumfield theory. . .momentum-dependentdressing
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 36/40
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: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 36/40
First Contents Back Conclusion
Ab-initio studyof mesons & nucleons
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 37/40
First Contents Back Conclusion
Ab-initio studyof mesons & nucleons
Eichmann et al.– arXiv:0802.1948 [nucl-th]– arXiv:0810.1222 [nucl-th]
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 37/40
First Contents Back Conclusion
Ab-initio studyof mesons & nucleons
Eichmann et al.– arXiv:0802.1948 [nucl-th]– arXiv:0810.1222 [nucl-th]
Leading-order truncation of DSEs – rainbow-ladder
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 37/40
First Contents Back Conclusion
Ab-initio studyof mesons & nucleons
Eichmann et al.– arXiv:0802.1948 [nucl-th]– arXiv:0810.1222 [nucl-th]
Leading-order truncation of DSEs – rainbow-ladder
Corrections vanish with increasing current-quark mass
⇒ rainbow-ladder exact in heavy-quark limit
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 37/40
First Contents Back Conclusion
Ab-initio studyof mesons & nucleons
Eichmann et al.– arXiv:0802.1948 [nucl-th]– arXiv:0810.1222 [nucl-th]
Leading-order truncation of DSEs – rainbow-ladder
Corrections vanish with increasing current-quark mass
⇒ rainbow-ladder exact in heavy-quark limit
However, at physical light-quark mass, corrections to
observables not protected by symmetries: uniformly ≈ 35%
Roughly 50/50-split between nonresonant and resonant
(pseudoscalar meson loop) contributions
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 37/40
First Contents Back Conclusion
Ab-initio studyof mesons & nucleons
Eichmann et al.– arXiv:0802.1948 [nucl-th]– arXiv:0810.1222 [nucl-th]
Leading-order truncation of DSEs – rainbow-ladder
Corrections vanish with increasing current-quark mass
⇒ rainbow-ladder exact in heavy-quark limit
However, at physical light-quark mass, corrections to
observables not protected by symmetries: uniformly ≈ 35%
Roughly 50/50-split between nonresonant and resonant
(pseudoscalar meson loop) contributions
Symmetry preserving and systematic approach can
elucidate and account for these effects
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 37/40
First Contents Back Conclusion
Ab-initio studyof mesons & nucleons
Eichmann et al.– arXiv:0802.1948 [nucl-th]– arXiv:0810.1222 [nucl-th]
Leading-order truncation of DSEs – rainbow-ladder
Corrections vanish with increasing current-quark mass
⇒ rainbow-ladder exact in heavy-quark limit
However, at physical light-quark mass, corrections to
observables not protected by symmetries: uniformly ≈ 35%
Roughly 50/50-split between nonresonant and resonant
(pseudoscalar meson loop) contributions
Symmetry preserving and systematic approach can
elucidate and account for these effects
Use this knowledge to constrain interaction in infrared
Interaction in ultraviolet predicted by perturbative
expansion of DSEsCraig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 37/40
First Contents Back Conclusion
Ab-initio studyof mesons & nucleons
Eichmann et al.– arXiv:0802.1948 [nucl-th]– arXiv:0810.1222 [nucl-th]
ETMC
RBC/UKQCD
CP-PACS + Adelaide
Experiment
MILC
1.2
1.1
1.0
0.9
0.0 0.2 0.4 0.6 0.8
0.8
0.7
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 38/40
First Contents Back Conclusion
Ab-initio studyof mesons & nucleons
Eichmann et al.– arXiv:0802.1948 [nucl-th]– arXiv:0810.1222 [nucl-th]
ETMC
RBC/UKQCD
CP-PACS + Adelaide
Experiment
MILC
1.2
1.1
1.0
0.9
0.0 0.2 0.4 0.6 0.8
0.8
0.7
Rainbow-Ladder DSE result
one parameter for IR – “confinement radius”
Results insensitive to value on material domain
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 38/40
First Contents Back Conclusion
Ab-initio studyof mesons & nucleons
Eichmann et al.– arXiv:0802.1948 [nucl-th]– arXiv:0810.1222 [nucl-th]
ETMC
RBC/UKQCD
CP-PACS + Adelaide
Experiment
MILC
1.2
1.1
1.0
0.9
0.0 0.2 0.4 0.6 0.8
0.8
0.7
Rainbow-Ladder DSE result
one parameter for IR – “confinement radius”
Results insensitive to value on material domain
Numerical simulations of lattice-QCD
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 38/40
First Contents Back Conclusion
Ab-initio studyof mesons & nucleons
Eichmann et al.– arXiv:0802.1948 [nucl-th]– arXiv:0810.1222 [nucl-th]
ETMC
RBC/UKQCD
CP-PACS + Adelaide
Experiment
MILC
1.2
1.1
1.0
0.9
0.0 0.2 0.4 0.6 0.8
0.8
0.7
Rainbow-Ladder DSE result
one parameter for IR – “confinement radius”
Results insensitive to value on material domain
Numerical simulations of lattice-QCD
FRR extrapolation of lattice CP-PACS resultCraig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 38/40
First Contents Back Conclusion
Ab-initio studyof mesons & nucleons
Eichmann et al.– arXiv:0802.1948 [nucl-th]– arXiv:0810.1222 [nucl-th]
Precisely the same
interaction
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 39/40
First Contents Back Conclusion
Ab-initio studyof mesons & nucleons
Eichmann et al.– arXiv:0802.1948 [nucl-th]– arXiv:0810.1222 [nucl-th]
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
0.10.0 0.2 0.3 0.4 0.5 0.6
Precisely the same
interaction
Same ρ-meson curve
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 39/40
First Contents Back Conclusion
Ab-initio studyof mesons & nucleons
Eichmann et al.– arXiv:0802.1948 [nucl-th]– arXiv:0810.1222 [nucl-th]
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
0.10.0 0.2 0.3 0.4 0.5 0.6
Precisely the same
interaction
Same ρ-meson curve
m2π-dependence of 0+ and
1+ diquark masses
“unobservable” – show
marked sensitivity to
single model parameter;
viz., confinement radius
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 39/40
First Contents Back Conclusion
Ab-initio studyof mesons & nucleons
Eichmann et al.– arXiv:0802.1948 [nucl-th]– arXiv:0810.1222 [nucl-th]
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
0.10.0 0.2 0.3 0.4 0.5 0.6
Precisely the same
interaction
Same ρ-meson curve
m2π-dependence of 0+ and
1+ diquark masses
“unobservable” – show
marked sensitivity to
single model parameter;
viz., confinement radius
But . . . [mav − msc], mρ
& MN . . . are independent
of that parameter
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 39/40
First Contents Back Conclusion
Ab-initio studyof mesons & nucleons
Eichmann et al.– arXiv:0802.1948 [nucl-th]– arXiv:0810.1222 [nucl-th]
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
0.10.0 0.2 0.3 0.4 0.5 0.6
Parameter-independent
RL-DSE predictions, with
veracious description of
Goldstone mode
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 40/40
First Contents Back Conclusion
Ab-initio studyof mesons & nucleons
Eichmann et al.– arXiv:0802.1948 [nucl-th]– arXiv:0810.1222 [nucl-th]
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
0.10.0 0.2 0.3 0.4 0.5 0.6
Parameter-independent
RL-DSE predictions, with
veracious description of
Goldstone mode
DSE and lattice agree on
heavy-quark domain
Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 40/40
First Contents Back Conclusion
Ab-initio studyof mesons & nucleons
Eichmann et al.– arXiv:0802.1948 [nucl-th]– arXiv:0810.1222 [nucl-th]
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
0.10.0 0.2 0.3 0.4 0.5 0.6
Parameter-independent
RL-DSE predictions, with
veracious description of
Goldstone mode
DSE and lattice agree on
heavy-quark domain
Prediction: at physical m2π,
Mquark−coreN = 1.26(2) GeV
cf. FRR+lattice-QCD,
Mquark−coreN = 1.27(2) GeV
⇒ subleading corrections,
including 0−-meson loops,
δMN = −320 MeV,
δmρ = −220 MeV Craig Roberts: Charting the interaction between light quarks
CLAS12 European Workshop . . . 13 transparencies – p. 40/40