Craig Roberts Physics Division

1. Dyson-Schwinger Equations & Continuum QCD 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 Physics Division

2. QCD’s Challenges Understand emergent phenomena • Quark and Gluon Confinement • No matter how hard one strikes the proton, • one cannot liberate an individual quark or gluon Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • 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=− (parity partners) • Neither of these phenomena is apparent in QCD’s LagrangianYetthey are the dominant determiningcharacteristics of real-world QCD. • QCD – Complex behaviour arises from apparently simple rules.

3. Universal Truths Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • Hadron spectrum, and elastic and transition form factors provide unique information about long-range interaction between light-quarks and distribution of hadron'scharacterising 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 (almost) irrelevant to light-quarks. • Running of quark mass entails that calculations at even modest Q2 require a Poincaré-covariant approach. Covariance + M(p2) require existence of quark orbital angular momentum in hadron's rest-frame wave function. • Confinement is expressed through a violent change of the propagators for coloured particles & can almost be read from a plot of a states’ dressed-propagator. It is intimately connected with DCSB.

4. Strong-interaction: QCD • Nature’sonly example of truly nonperturbative, • fundamental theory • A-priori, no idea as to what such a theory • can produce Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • Asymptotically free • Perturbation theory is valid and accurate tool at large-Q2 • Hence chiral limit is defined • Essentiallynonperturbative for Q2 < 2 GeV2

5. Confinement Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.

6. X Confinement Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • 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.

7. G. Bali et al., PoS LAT2005 (2006) 308 Confinement “Note that the time is not a linear function of the distance but dilated within the string breaking region. On a linear time scale string breaking takes place rather rapidly. […] light pair creation seems to occur non-localized and instantaneously.” anti-Bs Bs Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • Infinitely heavy-quarks plus 2 flavours with mass = ms • Lattice spacing = 0.083fm • String collapses within one lattice time-step R = 1.24 … 1.32 fm • Energy stored in string at collapse Ecsb = 2 ms • (mpg made via linear interpolation) • No flux tube between light-quarks

8. Confinement Confined particle Normal 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: Dyson-Schwinger Equations and Continuum QCD. • 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); …

9. Dressed-gluon propagator A.C. Aguilar et al., Phys.Rev. D80 (2009) 085018 IR-massive but UV-massless, confined gluon perturbative, massless gluon massive , unconfined gluon Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • 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

10. 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. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • 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 universalβ-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.

11. Charting the interaction between light-quarks Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • Through QCD's Dyson-Schwinger equations (DSEs) the pointwisebehaviour 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 excited states are a great deal more sensitive to the long-range behaviour of the β-function than those of the ground states.

12. Maris & Tandy, Phys.Rev. C60 (1999) 055214 DSE Studies – Phenomenology of gluon • Running gluon mass • Gluon is massless in ultraviolet in agreement with pQCD • Massive in infrared • mG(0) = 0.76 GeV • mG(1 GeV2) = 0.46 GeV Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • Wide-ranging study of π & ρ properties • Effective coupling • Agrees with pQCDin ultraviolet • Saturates in infrared • α(0)/π = 3.2 • α(1 GeV2)/π = 0.35

13. Dynamical Chiral Symmetry Breaking Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.

14. Dynamical Chiral Symmetry Breaking Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • 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.

15. Frontiers of Nuclear Science:Theoretical Advances C.D. Roberts, Prog. Part. Nucl. Phys. 61 (2008) 50 M. Bhagwat & P.C. Tandy, AIP Conf.Proc. 842 (2006) 225-227 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 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.

16. Frontiers of Nuclear Science:Theoretical Advances Mass from nothing! DSE prediction of DCSB confirmed Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 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.

17. Frontiers of Nuclear Science:Theoretical Advances Hint of lattice-QCD support for DSE prediction of violation of reflection positivity Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 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.

18. 12GeVThe Future of JLab Jlab 12GeV: Scanned by 2<Q2<9 GeV2 elastic & transition form factors. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 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.

19. Dynamical Chiral Symmetry BreakingCondensates? Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.

20. Dichotomy of the pion Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • 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:

21. Gell-Mann – Oakes – RennerRelation Behavior of current divergences under SU(3) x SU(3). Murray Gell-Mann, R.J. Oakes , B. Renner Phys.Rev. 175 (1968) 2195-2199 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • This paper derives a relation between mπ2 and the expectation-value < π|u0|π>, where uois 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

22. Gell-Mann – Oakes – RennerRelation Behavior of current divergences under SU(3) x SU(3). Murray Gell-Mann, R.J. Oakes , B. Renner Phys.Rev. 175 (1968) 2195-2199 Zhou Guangzhao周光召 Born 1929 Changsha, Hunan province Commutator is chiral rotation Therefore, isolates explicit chiral-symmetry breaking term in the putative Hamiltonian Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • PCAC hypothesis; viz., pion field dominates the divergence of the axial-vector current • Soft-pion theorem • In QCD, this is and one therefore has

23. Gell-Mann – Oakes – RennerRelation - (0.25GeV)3 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • 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.

24. Note of Warning Chiral Magnetism (or Magnetohadrochironics) A. Casher and L. Susskind, Phys. Rev. D9 (1974) 436 • These authors argue • thatdynamical chiral- • symmetry breaking • can be realised as a • property of hadrons, • instead of via a • nontrivial vacuum exterior to the measurable degrees of freedom 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. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.

25. QCD Sum Rules QCD and Resonance Physics. Sum Rules. M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov Nucl.Phys. B147 (1979) 385-447; citations: 3713 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • Introduction of the gluon vacuum condensate and development of “sum rules” relating properties of low-lying hadronic states to vacuum condensates

26. QCD Sum Rules QCD and Resonance Physics. Sum Rules. M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov Nucl.Phys. B147 (1979) 385-447; citations: 3781 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • 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

27. “quark condensate”1960-1980 6550+ references to this phrase since 1980 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • Instantons in non-perturbative QCD vacuum, MA Shifman, AI Vainshtein… - Nuclear Physics B, 1980 • Instanton density in a theory with massless quarks, MA Shifman, AI Vainshtein… - Nuclear Physics B, 1980 • 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

28. Universal Conventions Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • Wikipedia: (http://en.wikipedia.org/wiki/QCD_vacuum) “The QCD vacuum is the vacuum state of quantum chromodynamics (QCD). It is an example of a non-perturbative vacuum state, characterized by many non-vanishing condensates such as the gluon condensate or the quark condensate. These condensates characterize the normal phase or the confined phase of quark matter.”

29. Universal Misapprehensions Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • 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.

30. Resolution? Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • Quantum Healing Central: • “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. 

31. Paradigm shift:In-Hadron Condensates Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201 Brodsky and Shrock, PNAS 108, 45 (2011) M. Burkardt, Phys.Rev. D58 (1998) 096015 QCD Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • 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.

32. Paradigm shift:In-Hadron Condensates Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201 Brodsky and Shrock, PNAS 108, 45 (2011) Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • 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 fπand ρπ

33. Paradigm shift:In-Hadron Condensates Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201 Brodsky and Shrock, PNAS 108, 45 (2011) Chiral limit Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • 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 fπand ρπ • And

34. Paradigm shift:In-Hadron Condensates “Void that is truly empty solves dark energy puzzle” Rachel Courtland, New Scientist 4th Sept. 2010 • 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 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. “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.”

35. Dynamical Chiral Symmetry BreakingImportance of being well-dressed for quarks & mesons Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.

36. Strong-interaction: QCD Dressed-quark-gluon vertex Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • 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.Roberts, R. Delbourgo, A.G. Williams, H.J. Munczek, M.R. Pennington, A. Bashir, A. Kizilersu, P.Tandy, L. Chang, Y.-X. Liu …

37. Dressed-quark-gluon vertex Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • 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.

38. Gap EquationGeneral Form Bender, Roberts & von Smekal Phys.Lett. B380 (1996) 7-12 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • 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

39. Gap EquationGeneral Form If kernels of Bethe-Salpeter and gap equations don’t match, one won’t even get right charge for the pion. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • 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?!

40. Bethe-Salpeter EquationBound-State DSE Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • 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.

41. Bethe-Salpeter EquationGeneral Form Lei Chang and C.D. Roberts 0903.5461 [nucl-th] Phys. Rev. Lett. 103 (2009) 081601 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • Equivalent exact bound-state equation but in thisform 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)

42. Ward-Takahashi IdentityBethe-Salpeter Kernel Lei Chang and C.D. Roberts 0903.5461 [nucl-th] Phys. Rev. Lett. 103 (2009) 081601 iγ5 iγ5 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • Now, for first time, it’s possible to formulate an Ansatz for Bethe-Salpeter kernel given anyform 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

43. Relativistic quantum mechanics Spin Operator Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • Dirac equation (1928): Pointlike, massive fermion interacting with electromagnetic field

44. Massive point-fermionAnomalous magnetic moment 0.001 16 e e Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • 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: 1.001 19 ± 0.000 05. • 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.

45. Fermion electromagnetic current – General structure Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. with k = pf- pi • F1(k2) – Dirac form factor; and F2(k2) – Pauli form factor • Dirac equation: • F1(k2) = 1 • F2(k2) = 0 • Schwinger: • F1(k2) = 1 • F2(k2=0) = α /[2 π]

46. Magnetic moment of a masslessfermion? Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • 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)

47. Magnetic moment of a masslessfermion? Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • 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

48. Schwinger’s result? e e m=0 So, no mixing γμ↔ σμν Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 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

49. QCD and dressed-quark anomalous magnetic moments Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • 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!

50. L. Chang, Y. –X. Liu and C.D. RobertsarXiv:1009.3458 [nucl-th] Phys. Rev. Lett. 106 (2011) 072001 Dressed-quark anomalousmagnetic moments DCSB • 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, Kizilersuet al. • hep-ph/0303176 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. • Three strongly-dressed and essentially- nonperturbative contributions to dressed-quark-gluon vertex: