1. Craig Roberts Physics Division
A Year in the Life of
Dyson-Schwinger Equations
Craig Roberts
Physics Division
60+30
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2. Published collaborations in 2010/2011
Adnan BASHIR (U Michoacan);
Stan BRODSKY (SLAC);
Lei CHANG (ANL & PKU);
Huan CHEN (BIHEP);
Ian CLOËT (UW);
Bruno EL-BENNICH (Sao Paulo);
Xiomara GUTIERREZ-GUERRERO (U Michoacan);
Roy HOLT (ANL);
Mikhail IVANOV (Dubna);
Yu-xin LIU (PKU);
Trang NGUYEN (KSU);
Si-xue QIN (PKU);
Hannes ROBERTS (ANL, FZJ, UBerkeley);
Robert SHROCK (Stony Brook);
Peter TANDY (KSU);
David WILSON (ANL)
Students
Early-career
scientists
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

3. Top Open Questions in Physics
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

4. Excerpt from the Top-10
Can we quantitatively understand quark and gluon confinement in quantum chromodynamics and the existence of a mass gap?
Quantum chromodynamics, or QCD, is the theory describing the strong nuclear force. Carried by gluons, it binds quarks into particles like protons and neutrons. Apparently, the tiny subparticles are permanently confined: one can't pull a quark or a gluon from a proton because the strong force gets stronger with distance and snaps them right back inside.
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

5. Nuclear Science Advisory Council Long Range Plan
A central goal of nuclear physics is to understand the structure and properties of protons and neutrons, and ultimately atomic nuclei, in terms of the quarks and gluons of QCD
So, what’s the problem?
They are legion …
Confinement
Dynamical chiral symmetry breaking
A fundamental theory of unprecedented complexity
QCD defines the difference between nuclear and particle physicists:
Nuclear physicists try to solve this theory
Particle physicists run away to a place where tree-level computations are all that’s necessary; perturbation theory, the last refuge of a scoundrel
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

6. Understanding NSAC’s Long Range Plan
What are the quarks and gluons of QCD?
Is there such a thing as a constituent quark, a constituent-gluon?
After all, these are the concepts for which Gell-Mann won the Nobel Prize.
Do they – can they – correspond to well-defined quasi-particle degrees-of-freedom?
If not, with what should they be replaced?
What is the meaning of the NSAC Challenge?
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

7. DSEs
Dyson (1949) & Schwinger (1951) One can derive a system of coupled integral equations relating all the Green functions for a theory, one to another.
Gap equation:
fermion self energy
gauge-boson propagator
fermion-gauge-boson vertex
These are nonperturbative equivalents in quantum field theory to the Lagrange equations of motion.
Essential in simplifying the general proof of renormalisability of gauge field theories.
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Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

8. Dyson-Schwinger Equations
Well suited to Relativistic Quantum Field Theory
Simplest level: Generating Tool for Perturbation Theory Materially Reduces Model-Dependence … Statement about long-range behaviour of quark-quark interaction
NonPerturbative, Continuum approach to QCD
Hadrons as Composites of Quarks and Gluons
Qualitative and Quantitative Importance of:
Dynamical Chiral Symmetry Breaking
– Generation of fermion mass from nothing
Quark & Gluon Confinement
– Coloured objects not detected,
Not detectable?
Approach yields
Schwinger functions; i.e.,
propagators and vertices
Cross-Sections built from
Schwinger Functions
Hence, method connects
observables with long-
range behaviour of the
running coupling
Experiment ↔ Theory
comparison leads to an
understanding of long-
range behaviour of
strong running-coupling
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

9. Confinement Forschungszentrum Julich, IKP: 20 July. 71/81
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

10. X Confinement Empirical fact. However Quark and Gluon Confinement
No matter how hard one strikes the proton, or any other hadron, one cannot liberate an individual quark or gluon
Empirical fact. However
There is no agreed, theoretical definition of light-quark confinement
Static-quark confinement is irrelevant to real-world QCD
There are no long-lived, very-massive quarks
Confinement entails quark-hadron duality; i.e., that all observable consequences of QCD can, in principle, be computed using an hadronic basis.
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

11. Confinement
Confinement is expressed through a violent change in the analytic structure of propagators for coloured particles & can almost be read from a plot of a states’ dressed-propagator
Gribov (1978); Munczek (1983); Stingl (1984); Cahill (1989);
Krein, Roberts & Williams (1992); Tandy (1994); …
Normal particle
Confined particle
complex-P2
complex-P2
timelike axis: P2<0
Real-axis mass-pole splits, moving into pair(s) of complex conjugate poles or branch points
Spectral density no longer positive semidefinite
& hence state cannot exist in observable spectrum
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

12. Dressed-gluon propagator
A.C. Aguilar et al., Phys.Rev. D80 (2009)
Gluon propagator satisfies
a Dyson-Schwinger Equation
Plausible possibilities
for the solution
DSE and lattice-QCD
agree on the result
Confined gluon
IR-massive but UV-massless
mG ≈ 2-4 ΛQCD
IR-massive but UV-massless, confined gluon
perturbative, massless gluon
massive , unconfined gluon
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

13. Charting the interaction between light-quarks
This is a well-posed problem whose solution is an elemental goal of modern hadron physics.
The answer provides QCD’s running coupling.
Confinement can be related to the analytic properties of QCD's Schwinger functions.
Question of light-quark confinement can be translated into the challenge of charting the infrared behavior
of QCD's universal β-function
This function may depend on the scheme chosen to renormalise the quantum field theory but it is unique within a given scheme.
Of course, the behaviour of the β-function on the perturbative domain is well known.
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

14. Charting the interaction between light-quarks
Through QCD's Dyson-Schwinger equations (DSEs) the pointwise behaviour of the β-function determines the pattern of chiral symmetry breaking.
DSEs connect β-function to experimental observables. Hence, comparison between computations and observations of
Hadron mass spectrum
Elastic and transition form factors
can be used to chart β-function’s long-range behaviour.
Extant studies show that the properties of hadron states with masses 1-2GeV are a great deal more sensitive to the long-range behaviour of the β-function than those of the π&ρ ground states.
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

15. DSE Studies – Phenomenology of gluon
Qin, Chang, Roberts, et al., in progress at IKP
DSE Studies – Phenomenology of gluon
Wide-ranging study of π & ρ properties
Effective coupling
Agrees with pQCD in ultraviolet
Saturates in infrared
α(0)/π =
α(mG2)/π = 2 – 4
Running gluon mass
Gluon is massless in ultraviolet
in agreement with pQCD
Massive in infrared
mG(0) = 0.69 – 0.81 GeV
mG(mG2) = 0.30 GeV
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

16. Dynamical Chiral Symmetry Breaking Mass Gap
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

17. Dynamical Chiral Symmetry Breaking
Strong-interaction: QCD
Confinement
Empirical feature
Modern theory and lattice-QCD support conjecture
that light-quark confinement is a fact
associated with violation of reflection positivity; i.e., novel analytic structure for propagators and vertices
Still circumstantial, no proof yet of confinement
On the other hand, DCSB is a fact in QCD
It is the most important mass generating mechanism for visible matter in the Universe.
Responsible for approximately 98% of the proton’s mass.
Higgs mechanism is (almost) irrelevant to light-quarks.
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

18. Frontiers of Nuclear Science: Theoretical Advances
In QCD a quark's effective mass depends on its momentum. The function describing this can be calculated and is depicted here. Numerical simulations of lattice QCD (data, at two different bare masses) have confirmed model predictions (solid curves) that the vast bulk of the constituent mass of a light quark comes from a cloud of gluons that are dragged along by the quark as it propagates. In this way, a quark that appears to be absolutely massless at high energies (m =0, red curve) acquires a large constituent mass at low energies.
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C.D. Roberts, Prog. Part. Nucl. Phys. 61 (2008) 50
M. Bhagwat & P.C. Tandy, AIP Conf.Proc. 842 (2006)
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

19. Frontiers of Nuclear Science: Theoretical Advances
In QCD a quark's effective mass depends on its momentum. The function describing this can be calculated and is depicted here. Numerical simulations of lattice QCD (data, at two different bare masses) have confirmed model predictions (solid curves) that the vast bulk of the constituent mass of a light quark comes from a cloud of gluons that are dragged along by the quark as it propagates. In this way, a quark that appears to be absolutely massless at high energies (m =0, red curve) acquires a large constituent mass at low energies.
Mass from nothing!
DSE prediction of DCSB confirmed
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

20. Frontiers of Nuclear Science: Theoretical Advances
In QCD a quark's effective mass depends on its momentum. The function describing this can be calculated and is depicted here. Numerical simulations of lattice QCD (data, at two different bare masses) have confirmed model predictions (solid curves) that the vast bulk of the constituent mass of a light quark comes from a cloud of gluons that are dragged along by the quark as it propagates. In this way, a quark that appears to be absolutely massless at high energies (m =0, red curve) acquires a large constituent mass at low energies.
Hint of lattice-QCD support
for DSE prediction of violation of reflection positivity
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

21. 12GeV The Future of JLab
Numerical simulations of lattice QCD (data, at two different bare masses) have confirmed model predictions (solid curves) that the vast bulk of the constituent mass of a light quark comes from a cloud of gluons that are dragged along by the quark as it propagates. In this way, a quark that appears to be absolutely massless at high energies (m =0, red curve) acquires a large constituent mass at low energies.
Jlab 12GeV: Scanned by 2<Q2<9 GeV2
elastic & transition form factors.
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

22. Dynamical Chiral Symmetry Breaking Importance of being well-dressed for quarks & mesons
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

23. Strong-interaction: QCD
Dressed-quark-gluon vertex
Gluons and quarks acquire momentum-dependent masses
characterised by an infrared mass-scale m ≈ 2-4 ΛQCD
Significant body of work, stretching back to 1980, which shows that, in the presence of DCSB, the dressed-fermion-photon vertex is materially altered from the bare form: γμ.
Obvious, because with
A(p2) ≠ 1 and B(p2) ≠ constant,
the bare vertex cannot satisfy the Ward-Takahashi identity; viz.,
Number of contributors is too numerous to list completely (300 citations to 1st J.S. Ball paper), but prominent contributions by:
J.S. Ball, C.J. Burden, C.D. Roberts, R. Delbourgo, A.G. Williams,
H.J. Munczek, M.R. Pennington, A. Bashir, A. Kizilersu, L. Chang, Y.-X. Liu …
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

24. Dressed- quark-gluon vertex
Single most important feature
Perturbative vertex is helicity-conserving:
Cannot cause spin-flip transitions
However, DCSB introduces nonperturbatively generated structures that very strongly break helicity conservation
These contributions
Are large when the dressed-quark mass-function is large
Therefore vanish in the ultraviolet; i.e., on the perturbative domain
Exact form of the contributions is still the subject of debate but their existence is model-independent - a fact.
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

25. Gap Equation General Form
Dμν(k) – dressed-gluon propagator
Γν(q,p) – dressed-quark-gluon vertex
Until 2009, all studies of other hadron phenomena used the leading-order term in a symmetry-preserving truncation scheme; viz.,
Dμν(k) = dressed, as described previously
Γν(q,p) = γμ
… plainly, key nonperturbative effects are missed and cannot be recovered through any step-by-step improvement procedure
Bender, Roberts & von Smekal
Phys.Lett. B380 (1996) 7-12
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

26. Gap Equation General Form
Dμν(k) – dressed-gluon propagator
good deal of information available
Γν(q,p) – dressed-quark-gluon vertex
Information accumulating
Suppose one has in hand – from anywhere – the exact form of the dressed-quark-gluon vertex
What is the associated symmetry-
preserving Bethe-Salpeter kernel?!
If kernels of Bethe-Salpeter
and gap equations don’t match,
one won’t even get
right charge for the pion.
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

27. Bethe-Salpeter Equation Bound-State DSE
K(q,k;P) – fully amputated, two-particle irreducible,
quark-antiquark scattering kernel
Textbook material.
Compact. Visually appealing. Correct
Blocked progress for more than 60 years.
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

28. Bethe-Salpeter Equation General Form
Lei Chang and C.D. Roberts
[nucl-th]
Phys. Rev. Lett. 103 (2009)
Equivalent exact bound-state equation but in this form
K(q,k;P) → Λ(q,k;P)
which is completely determined by dressed-quark self-energy
Enables derivation of a Ward-Takahashi identity for Λ(q,k;P)
(30)
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

29. Ward-Takahashi Identity Bethe-Salpeter Kernel
Lei Chang and C.D. Roberts
[nucl-th]
Phys. Rev. Lett. 103 (2009)
iγ5
iγ5
Now, for first time, it’s possible to formulate an Ansatz for
Bethe-Salpeter kernel given any form for the dressed-quark-gluon vertex by using this identity
This enables the identification and elucidation of a wide range of novel consequences of DCSB
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

30. Relativistic quantum mechanics
Dirac equation (1928):
Pointlike, massive fermion interacting with electromagnetic field
Spin Operator
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

31. Massive point-fermion Anomalous magnetic moment
Dirac’s prediction held true for the electron until improvements in experimental techniques enabled the discovery of a small deviation: H. M. Foley and P. Kusch, Phys. Rev. 73, 412 (1948).
Moment increased by a multiplicative factor: ±
This correction was explained by the first systematic computation using renormalized quantum electrodynamics (QED):
J.S. Schwinger, Phys. Rev. 73, 416 (1948),
vertex correction
The agreement with
experiment established
quantum electrodynamics as a valid tool. e e
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

32. Fermion electromagnetic current – General structure
with k = pf - pi
F1(k2) – Dirac form factor; and F2(k2) – Pauli form factor
Dirac equation:
F1(k2) = 1
F2(k2) = 0
Schwinger:
F2(k2=0) = α /[2 π]
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

33. Magnetic moment of a massless fermion?
Plainly, can’t simply take the limit m → 0.
Standard QED interaction, generated by minimal substitution:
Magnetic moment is described by interaction term:
Invariant under local U(1) gauge transformations
but is not generated by minimal substitution in the action for a free Dirac field.
Transformation properties under chiral rotations?
Ψ(x) → exp(iθγ5) Ψ(x)
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

34. Magnetic moment of a massless fermion?
Standard QED interaction, generated by minimal substitution:
Unchanged under chiral rotation
Follows that QED without a fermion mass term is helicity conserving
Magnetic moment interaction is described by interaction term:
NOT invariant
picks up a phase-factor exp(2iθγ5)
Magnetic moment interaction is forbidden in a theory with manifest chiral symmetry
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

35. Schwinger’s result? F2(k2) = 0 when me2 = 0 One-loop calculation:
Plainly, one obtains Schwinger’s result for me2 ≠ 0
However,
F2(k2) = 0 when me2 = 0
There is no Gordon identity:
Results are unchanged at every order in perturbation theory … owing to symmetry … magnetic moment interaction is forbidden in a theory with manifest chiral symmetry e e
m=0
So, no mixing
γμ ↔ σμν
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

36. QCD and dressed-quark anomalous magnetic moments
Schwinger’s result for QED:
pQCD: two diagrams
(a) is QED-like
(b) is only possible in QCD – involves 3-gluon vertex
Analyse (a) and (b)
(b) vanishes identically: the 3-gluon vertex does not contribute to a quark’s anomalous chromomag. moment at leading-order
(a) Produces a finite result: “ – ⅙ αs/2π ”
~ (– ⅙) QED-result
But, in QED and QCD, the anomalous chromo- and electro-magnetic moments vanish identically in the chiral limit!
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

37. Dressed-quark anomalous magnetic moments
L. Chang, Y. –X. Liu and C.D. Roberts arXiv: [nucl-th]
Phys. Rev. Lett. 106 (2011)
Dressed-quark anomalous magnetic moments
DCSB
Three strongly-dressed and essentially-
nonperturbative contributions to dressed-quark-gluon vertex:
Ball-Chiu term
Vanishes if no DCSB
Appearance driven by STI
Anom. chrom. mag. mom.
contribution to vertex
Similar properties to BC term
Strength commensurate with lattice-QCD
Skullerud, Bowman, Kizilersu et al.
hep-ph/
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

38. Dressed-quark anomalous chromomagnetic moment
Lattice-QCD
m = 115 MeV
Nonperturbative result is two orders-of-magnitude larger than the perturbative computation
This level of
magnification is
typical of DCSB
cf.
Quenched
lattice-QCD
Skullerud, Kizilersu et al.
JHEP 0304 (2003) 047
Quark mass function:
M(p2=0)= 400MeV
M(p2=10GeV2)=4 MeV
Prediction from
perturbative QCD
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

39. Dressed-quark anomalous magnetic moments
L. Chang, Y. –X. Liu and C.D. Roberts arXiv: [nucl-th]
Phys. Rev. Lett. 106 (2011)
Dressed-quark anomalous magnetic moments
DCSB
Three strongly-dressed and essentially-
nonperturbative contributions to dressed-quark-gluon vertex:
Ball-Chiu term
Vanishes if no DCSB
Appearance driven by STI
Anom. chrom. mag. mom.
contribution to vertex
Similar properties to BC term
Strength commensurate with lattice-QCD
Skullerud, Bowman, Kizilersu et al.
hep-ph/
Role and importance is
novel discovery
Essential to recover pQCD
Constructive interference with Γ5
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

40. Dressed-quark anomalous magnetic moments
L. Chang, Y. –X. Liu and C.D. Roberts arXiv: [nucl-th]
Phys. Rev. Lett. 106 (2011)
Dressed-quark anomalous magnetic moments
Formulated and solved general
Bethe-Salpeter equation
Obtained dressed
electromagnetic vertex
Confined quarks
don’t have a mass-shell
Can’t unambiguously define
magnetic moments
But can define
magnetic moment distribution
Factor of 10
magnification
AEM is opposite in sign but of
roughly equal magnitude
as ACM ME
κACM
κAEM
Full vertex
0.44
-0.22
0.45
Rainbow-ladder
0.35
0.048
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

41. Dressed-quark anomalous magnetic moments
L. Chang, Y. –X. Liu and C.D. Roberts arXiv: [nucl-th]
Phys. Rev. Lett. 106 (2011)
Dressed-quark anomalous magnetic moments
Formulated and solved general
Bethe-Salpeter equation
Obtained dressed
electromagnetic vertex
Confined quarks
don’t have a mass-shell
Can’t unambiguously define
magnetic moments
But can define
magnetic moment distribution
Factor of 10
magnification
Contemporary theoretical estimates:
1 – 10 x 10-10
Largest value reduces discrepancy expt.↔theory from 3.3σ to below 2σ.
Potentially important for elastic and transition form factors, etc.
Significantly, also quite possibly for muon g-2 – via Box diagram,
which is not constrained by extant data.
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

42. Looking deeper Forschungszentrum Julich, IKP: 20 July. 71/81
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

43. Deep inelastic scattering
Quark discovery experiment at SLAC
( , Nobel Prize in 1990)
Completely different to elastic scattering
Blow the target to pieces instead of keeping only those events where it remains intact.
Cross-section is interpreted as a measurement of the momentum-fraction probability distribution for quarks and gluons within the target hadron: q(x), g(x)
Probability that a quark/gluon within the target will carry a fraction x of the bound-state’s light-front momentum
Distribution Functions of the Nucleon and Pion in the Valence Region, Roy J. Holt and Craig D. Roberts, arXiv: [nucl-th], Rev. Mod. Phys. 82 (2010) pp
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

44. Pion’s valence-quark distributions
Owing to absence of pion targets, the pion’s valence-quark distribution functions are measured via the Drell-Yan process:
π p → μ+ μ− X
Three experiments: CERN (1983 & 1985)
and FNAL (1989). No more recent
experiments because theory couldn’t
even explain these!
Problem
Conway et al. Phys. Rev. D 39, 92 (1989)
Wijesooriya et al. Phys.Rev. C 72 (2005)
Behaviour at large-x is inconsistent with
pQCD; viz,
expt. (1-x)1+ε
cf. QCD (1-x)2+γ
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

45. First QCD-based calculation
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

46. Computation of qvπ(x) T+
In the Bjorken limit; viz.,
q2 → ∞, P · q → − ∞
but x := −q2/2P · q fixed,
Wμν(q;P) ~ qvπ(x)
Plainly, this can be computed given all the information we have at our disposal from the DSEs T+ T–
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

47. Hecht, Roberts, Schmidt
Phys.Rev. C 63 (2001)
Computation of qvπ(x)
Before the first DSE computation, which used the running dressed-quark mass described previously, numerous authors applied versions of the Nambu–Jona-Lasinio model and were content to reproduce the data, arguing therefrom that the inferences from pQCD were wrong
After the first DSE computation, experimentalists again became interested in the process because
DSEs agreed with pQCD but disagreed with the data, and other models
Disagreement on the “valence domain,” which is uniquely sensitive to M(p2)
2.61/1.27= factor of 2 in the exponent
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

48. Hecht, Roberts, Schmidt
Phys.Rev. C 63 (2001)
Reanalysis of qvπ(x)
After the first DSE computation, the “Conway et al.” data were reanalysed, this time at next-to-leading-order (Wijesooriya et al. Phys.Rev. C 72 (2005) )
The new analysis produced a much larger exponent than initially obtained; viz., β=1.87, but now it disagreed equally with NJL-model results and the DSE prediction
NB. Within pQCD, one can readily understand why adding a higher-order correction leads to a suppression of qvπ(x) at large-x.
New experiments were proposed … for accelerators that do not yet exist but the situation remained otherwise unchanged
Until the publication of Distribution Functions of the Nucleon and Pion in the Valence Region,
Roy J. Holt and Craig D. Roberts, arXiv: [nucl-th], Rev. Mod. Phys. 82 (2010) pp
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

49. Distribution Functions of the Nucleon and Pion in the Valence Region, Roy J. Holt and Craig D. Roberts, arXiv: [nucl-th], Rev. Mod. Phys. 82 (2010) pp
Reanalysis of qvπ(x)
This article emphasised and explained the importance of the persistent discrepancy between the DSE result and experiment as a challenge to QCD
It prompted another reanalysis of the data, which accounted for a long-overlooked effect: viz., “soft-gluon resummation,”
Compared to previous analyses, we include next-to-leading-logarithmic threshold resummation effects in the calculation of the Drell-Yan cross section. As a result of these, we find a considerably softer valence distribution at high momentum fractions x than obtained in previous next-to-leading-order analyses, in line with expectations based on perturbative-QCD counting rules or Dyson-Schwinger equations.
Aicher, Schäfer, Vogelsang, “Soft-Gluon Resummation and the Valence Parton Distribution Function of the Pion,” Phys. Rev. Lett. 105 (2010)
Craig Roberts: A Year in the life of DSEs
Forschungszentrum Julich, IKP: 20 July. 71/81

50. Current status of qvπ(x)
Trang, Bashir, Roberts & Tandy, “Pion and kaon valence-quark parton distribution functions,” arXiv: [nucl-th], Phys. Rev. C 83, (R) (2011) [5 pages]
Current status of qvπ(x)
DSE prediction and modern representation of the data are
indistinguishable
on the valence-quark domain
Emphasises the value of using a single internally-consistent, well-constrained framework to correlate and unify the description of hadron observables
What about valence- quark distributions in the kaon?
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51. qvπ(x) & qvK(x) ms ≈ 24 mu & Ms ≈ 1.25 Mu
Trang, Bashir, Roberts & Tandy, “Pion and kaon valence-quark parton distribution functions,” arXiv: [nucl-th], Phys. Rev. C 83, (R) (2011) [5 pages]
qvπ(x) & qvK(x)
ms ≈ 24 mu & Ms ≈ 1.25 Mu
Expect the s-quark to carry more of the kaon’s momentum than the u-quark, so that xsK(x) peaks at larger value of x than xuK(x)
Expectation confirmed in computations, with s-quark distribution peaking at 15% larger value of x
Even though deep inelastic scattering is a high-Q2 process, constituent-like mass-scale explains the shift
xuK(x)
xsK(x)
xuπ(x)
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52. Trang, Bashir, Roberts & Tandy, “Pion and kaon valence-quark parton distribution functions,” arXiv: [nucl-th], Phys. Rev. C 83, (R) (2011) [5 pages]
uK(x)/uπ(x)
Drell-Yan experiments at CERN (1980 & 1983) provide the only extant measurement of this ratio
DSE result in complete accord with the measurement
New Drell-Yan experiment running now at FNAL is capable of validating this comparison
It should be done so that complete understanding can be claimed
Value of ratio at x=0 will approach “1” under evolution to higher resolving scales. This is a feature of perturbative dynamics
Using DSEs in QCD, one derives that the x=1 value is
≈ (fπ/fK)2 (Mu /Ms)4 = 0.3
Value of ratio at x=1 is a fixed point of the evolution equations
Hence, it’s a very strong test of nonperturbative dynamics
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53. Grand Unification Forschungszentrum Julich, IKP: 20 July. 71/81
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54. Unification of Meson & Baryon Spectra
Dyson-Schwinger Equations have been applied extensively to the spectrum and interactions of mesons with masses less than 1 GeV
On this domain the rainbow-ladder approximation, which is the leading-order in a systematic & symmetry-preserving truncation scheme – nucl-th/ , is a well-understood tool that is accurate for pseudoscalar and vector mesons:
e.g.,
Prediction of elastic pion and kaon form factors: nucl-th/
Anomalous neutral pion processes – γπγ & BaBar anomaly: [nucl-th]
Pion and kaon valence-quark distribution functions: [nucl-th]
Unification of these and other observables – ππ scattering: hep-ph/
It can readily be extended to explain properties of the light neutral pseudoscalar mesons: [nucl-th]
Craig Roberts: A Year in the life of DSEs
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55. DSEs & Baryons Dynamical chiral symmetry breaking (DCSB)
R.T. Cahill et al.,
Austral. J. Phys. 42 (1989)
DSEs & Baryons
Dynamical chiral symmetry breaking (DCSB)
– has enormous impact on meson properties.
Must be included in description
and prediction of baryon properties.
DCSB is essentially a quantum field theoretical effect.
In quantum field theory
Meson appears as pole in four-point quark-antiquark Green function
→ Bethe-Salpeter Equation
Nucleon appears as a pole in a six-point quark Green function
→ Faddeev Equation.
Poincaré covariant Faddeev equation sums all possible exchanges and interactions that can take place between three dressed-quarks
Tractable equation is based on the observation that an interaction which describes colour-singlet mesons also generates nonpointlike quark-quark (diquark) correlations in the colour-antitriplet channel
SUc(3):
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56. Faddeev Equation Linear, Homogeneous Matrix equation
R.T. Cahill et al.,
Austral. J. Phys. 42 (1989)
Faddeev Equation
quark exchange
ensures Pauli statistics
quark
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 . . .
Both have “correct” parity and “right” masses
In Nucleon’s Rest Frame Amplitude has
s−, p− & d−wave correlations
diquark
composed of strongly-dressed quarks bound by dressed-gluons
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57. Photon-nucleon current
L. Chang, Y. –X. Liu and C.D. Roberts arXiv: [nucl-th]
Phys. Rev. Lett. 106 (2011)
Photon-nucleon current
Vertex must contain the dressed-quark anomalous magnetic moment: Lecture IV
Composite nucleon must interact with photon via nontrivial current constrained by Ward-Takahashi identities
DSE → BSE → Faddeev equation plus current → nucleon form factors
In a realistic calculation, the last three diagrams represent 8-dimensional integrals, which can be evaluated using Monte-Carlo techniques
Oettel, Pichowsky, Smekal
Eur.Phys.J. A8 (2000)
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58. critical importance of DCSB in explanation of real-world observables.
I.C. Cloët, C.D. Roberts, et al.
arXiv: [nucl-th]
I.C. Cloët, C.D. Roberts, et al.
In progress
DSE result Dec 08
DSE result
– including the
anomalous
magnetic
moment distribution
Highlights again the
critical importance of
DCSB in explanation of
real-world observables.
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59. DCSB & EDMs Forschungszentrum Julich, IKP: 20 July. 71/81
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60. Hecht, Roberts, Schmidt
Phys.Rev. C64 (2001)
T-violation
Invariance under C*P*T is a symmetry of any local quantum field theory
C*P alone is not a symmetry of the Standard Model
Therefore Time-Reversal-Invariance is violated.
Possession of an EDM by J=½ particle is a violation of T-symmetry
Standard Model: |dn,p |< ecm
Current experiment:
|dp |<7.9 x ecm & |dn |<1.6 x ecm
Suppose nonzero dp or dn is measured
What is the relation to current-quark EDMs: du,d,s,… ?
SU(6) Quark Model: dp=⅓(4du-dd) & dn=⅓(4dd-du)
But these are constituent-quark EDMs,
which aren’t readily defined in SM!
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61. Nucleon EDM Nucleon EDM described by
Hecht, Roberts, Schmidt
Phys.Rev. C64 (2001)
Nucleon EDM
Nucleon EDM described by
a third form factor in the photon-nucleon current:
Current-quark EDMs, from any source, will produce nucleon EDM, which can be computed if-and-only-if one has a framework for connecting current-quarks to hadron observables
Craig Roberts: A Year in the life of DSEs
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62. Quark EDMs Faddeev equation approach
Hecht, Roberts, Schmidt
Phys.Rev. C64 (2001)
Quark EDMs
Faddeev equation approach
EDM appears in dressed-quark–photon vertex: two possible additions because confined-quarks don’t have a mass-shell –
Q= electric- charge matrix
m = current-quark mass
H± = diag[hu±,hd±] – current-quark gyroelectric ratios
May now compute relation between hq & hN
Craig Roberts: A Year in the life of DSEs
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63. Quark EDMs Nucleon structure factors
Hecht, Roberts, Schmidt
Phys.Rev. C64 (2001)
Quark EDMs Nucleon structure factors
Structure factors Quark charges Gyroelectric ratios
Matrix of gyroelectric ratios
Nonzero because dressed-quarks are not Dirac particles
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64. Hecht, Roberts, Schmidt
Phys.Rev. C64 (2001)
Gyroelectric ratios
Neutron and proton (hN=e/2MN) related to current-quarks
NB. h+ = 0 in constituent-quark models
coefficient-ratio(hd–/hu–)= 3.4
cf. SU(6) CQMs coefficient-ratio(hd–/hu–)= 2.0
SU(6) is not a good approximation
Example: Charged-Higgs-boson-exchange models yield hd>>hu
hn = – 40.6 ( 0.85 hd hd- )
hp = 5.1 ( hd hd- )
Suppose hu>>hd
hp = 81.2 ( 0.85 hu hu- )
Craig Roberts: A Year in the life of DSEs
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65. Hecht, Roberts, Schmidt
Phys.Rev. C64 (2001)
Quark EDMs
The scale of DCSB in QCD amplifies the contribution from a current-quark’s EDM to that of the bound state containing it
minimal enhancement factor is roughly the ratio of the constituent-quark and current-quark masses; two-orders of magnitude
it is understandable and calculable through the necessary momentum dependence of the dressed-quark mass function.
Confinement & compositeness entail that the dressed-quarks comprising bound state are not on shell & hence the two C P- & T-violating operator structures that are indistinguishable in the free-quark limit yield materially different contributions to the EDM of the bound state.
These operator structures must be analysed and their strengths determined independently in any model that provides for C P and T violation.
Both these effects should be accounted for in using dN as a means of constraining extensions of the Standard Model.
Craig Roberts: A Year in the life of DSEs
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66. Dynamical Chiral Symmetry Breaking Condensates?
Craig Roberts: A Year in the life of DSEs
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67. Connecting QCD with coupled-channels calculations
Craig Roberts: A Year in the life of DSEs
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68. Man kann über Alles sprechen Aber nicht über eine Stunde
Craig Roberts: A Year in the life of DSEs
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69. Epilogue Forschungszentrum Julich, IKP: 20 July. 71/81
Craig Roberts: A Year in the life of DSEs
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70. Masses of ground and excited-state hadrons Hannes L. L
Masses of ground and excited-state hadrons Hannes L.L. Roberts, Lei Chang, Ian C. Cloët
and Craig D. Roberts,
arXiv: [nucl-th]
Few Body Systems  (2011) pp. 1-25
Epilogue
New perspectives on the quark condensate
Brodsky, Roberts, Shrock, Tandy,
Phys. Rev. C82 (Rapid Comm.) (2010)
QCD is the most interesting part of the Standard Model and Nature’s only example of an essentially nonperturbative fundamental theory
Dyson-Schwinger equations:
Described some highlights from 2010/2011:
Confinement is a dynamical phenomenon
It cannot in principle be expressed via a potential
Gluons are nonperturbatively massive & quarks are not Dirac particles
2001 Prediction of pion valence-quark distribution function confirmed in 2010,
via modern reanalysis of extant Drell-Yan data
Dynamical chiral symmetry breaking is a fact.
It’s responsible for 98% of the mass of visible matter in the Universe
Provides an enhancement factor between hadron EDMs and current-quark EDMs
Omitted more highlights from 2010/2011; e.g.,
Condensates are contained within hadrons
studies relating dressed-quark hadron-spectrum to EBAC and Jülich coupled-channels models
Craig Roberts: A Year in the life of DSEs
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71. Return to nucleon EDM with Mario P., Michael R-M & student
Spectrum of exotics
Nucleon -> Roper transition form factor
Spectrum of baryons with strangeness
Continuing work on PDFs with extension to nucleon
Computation of meson-nucleon form factors for possible use in coupled channels computations
Review articles Form.Factors=Rev.Mod.Phys., DSEs=Prog.Part.Nucl.Phys. & Eur.Math.Soc. & Chin.J.Phys.
Where do we go from here?
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72. Dynamical Chiral Symmetry Breaking Condensates?
Craig Roberts: A Year in the life of DSEs
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73. Dichotomy of the pion
How does one make an almost massless particle from two massive constituent-quarks?
Naturally, one could always tune a potential in quantum mechanics so that the ground-state is massless
– but some are still making this mistake
However:
current-algebra (1968)
This is impossible in quantum mechanics, for which one always finds:
(8)
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74. Gell-Mann – Oakes – Renner Relation
Behavior of current divergences under SU(3) x SU(3).
Murray Gell-Mann, R.J. Oakes , B. Renner
Phys.Rev. 175 (1968)
This paper derives a relation between
mπ2 and the expectation-value < π|u0|π>,
where uo is an operator that is linear in the putative Hamiltonian’s explicit chiral-symmetry breaking term
NB. QCD’s current-quarks were not yet invented, so u0 was not expressed in terms of current-quark fields
PCAC-hypothesis (partial conservation of axial current) is used in the derivation
Subsequently, the concepts of soft-pion theory
Operator expectation values do not change as t=mπ2 → t=0
to take < π|u0|π> → < 0|u0|0> … in-pion → in-vacuum
(8)
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75. Gell-Mann – Oakes – Renner Relation
- (0.25GeV)3
Theoretical physics at its best.
But no one is thinking about how properly to consider or define what will come to be called the
vacuum quark condensate
So long as the condensate is just a
mass-dimensioned constant,
which approximates another
well-defined transition matrix
element, there is no problem.
Problem arises if one
over-interprets this number,
which textbooks have been doing
for a VERY LONG TIME.
(8)
Craig Roberts: A Year in the life of DSEs
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76. Note of Warning These authors argue that dynamical chiral-
Chiral Magnetism (or Magnetohadrochironics)
A. Casher and L. Susskind, Phys. Rev. D9 (1974) 436
These authors argue
that dynamical chiral-
symmetry breaking
can be realised as a
property of hadrons,
instead of via a
nontrivial vacuum exterior to the measurable degrees of freedom
(10)
The essential ingredient required for a spontaneous symmetry breakdown in a composite system is the existence of a divergent number of constituents
– DIS provided evidence for divergent sea of low-momentum partons – parton model.
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77. QCD Sum Rules Introduction of the gluon vacuum condensate
QCD and Resonance Physics. Sum Rules.
M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov
Nucl.Phys. B147 (1979) ; citations: 3713
Introduction of the gluon vacuum condensate
and development of “sum rules” relating properties of low-lying hadronic states to vacuum condensates
(10)
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78. QCD Sum Rules
QCD and Resonance Physics. Sum Rules.
M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov
Nucl.Phys. B147 (1979) ; citations: 3781
Introduction of the gluon vacuum condensate
and development of “sum rules” relating properties of low-lying hadronic states to vacuum condensates
At this point (1979), the cat was out of the bag: a physical reality was seriously attributed to a plethora of vacuum condensates
(10)
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79. No references to this phrase before 1980
“quark condensate”
Instantons in non-perturbative QCD vacuum,
MA Shifman, AI Vainshtein… - Nuclear Physics B, 1980
Instanton density in a theory with massless quarks,
Exotic new quarks and dynamical symmetry breaking,
WJ Marciano - Physical Review D, 1980
The pion in QCD
J Finger, JE Mandula… - Physics Letters B, 1980
No references to this phrase before 1980
(10)
6550+ references to this phrase since 1980
Craig Roberts: A Year in the life of DSEs
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80. Universal Misapprehensions
Since 1979, DCSB has commonly been associated literally with a spacetime-independent mass-dimension-three “vacuum condensate.”
Under this assumption, “condensates” couple directly to gravity in general relativity and make an enormous contribution to the cosmological constant
Experimentally, the answer is
Ωcosm. const. = 0.76
This mismatch is a bit of a problem.
Craig Roberts: A Year in the life of DSEs
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81. Resolution? Quantum Healing Central: Paranormal Psychic Forums:
“KSU physics professor [Peter Tandy] publishes groundbreaking research on inconsistency in Einstein theory.”
Paranormal Psychic Forums:
“Now Stanley Brodsky of the SLAC National
Accelerator Laboratory in Menlo Park,
California, and colleagues have found a way
to get rid of the discrepancy. “People have
just been taking it on faith that this quark
condensate is present throughout the
vacuum,” says Brodsky. 
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82. Paradigm shift: In-Hadron Condensates
Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010)
Brodsky and Shrock, PNAS 108, 45 (2011)
Resolution
Whereas it might sometimes be convenient in computational truncation schemes to imagine otherwise, “condensates” do not exist as spacetime-independent mass-scales that fill all spacetime.
So-called vacuum condensates can be understood as a property of hadrons themselves, which is expressed, for example,
in their Bethe-Salpeter or
light-front wavefunctions.
GMOR
cf.
M. Burkardt, Phys.Rev. D58 (1998)
QCD
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83. Paradigm shift: In-Hadron Condensates
Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010)
Brodsky and Shrock, PNAS 108, 45 (2011)
Resolution
Whereas it might sometimes be convenient in computational truncation schemes to imagine otherwise, “condensates” do not exist as spacetime-independent mass-scales that fill all spacetime.
So-called vacuum condensates can be understood as a property of hadrons themselves, which is expressed, for example,
in their Bethe-Salpeter or
light-front wavefunctions.
No qualitative difference
between fπ and ρπ
Craig Roberts: A Year in the life of DSEs
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84. Paradigm shift: In-Hadron Condensates
Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010)
Brodsky and Shrock, PNAS 108, 45 (2011)
Resolution
Whereas it might sometimes be convenient in computational truncation schemes to imagine otherwise, “condensates” do not exist as spacetime-independent mass-scales that fill all spacetime.
So-called vacuum condensates can be understood as a property of hadrons themselves, which is expressed, for example,
in their Bethe-Salpeter or
light-front wavefunctions.
No qualitative difference
between fπ and ρπ
And
Chiral limit
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85. Paradigm shift: In-Hadron Condensates
“Void that is truly empty solves dark energy puzzle”
Rachel Courtland, New Scientist 4th Sept. 2010
“EMPTY space may really be empty. Though quantum theory suggests that a vacuum should be fizzing with particle activity, it turns out that this paradoxical picture of nothingness may not be needed. A calmer view of the vacuum would also help resolve a nagging inconsistency with dark energy, the elusive force thought to be speeding up the expansion of the universe.”
Cosmological Constant:
Putting QCD condensates back into hadrons reduces the
mismatch between experiment and theory by a factor of 1046
Possibly by far more, if technicolour-like theories are the correct
paradigm for extending the Standard Model
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