1. 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) Craig Roberts Physics Division Students Early-career scientists Published collaborations in 2010/2011

2. QCD’s Challenges  Dynamical Chiral Symmetry Breaking Very unnatural pattern of bound state masses; e.g., Lagrangian (pQCD) quark mass is small but... no degeneracy between J P =+ and J P =− (parity partners)  Neither of these phenomena is apparent in QCD’s Lagrangian Yet they are the dominant determining characteristics of real-world QCD. QCDQCD – Complex behaviour arises from apparently simple rules. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 2  Quark and Gluon Confinement No matter how hard one strikes the proton, one cannot liberate an individual quark or gluon Understand emergent phenomena LightCone 2011, SMU 23-27 May - 77pgs

3. Universal Truths HHadron spectrum, and elastic 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. D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. RRunning of quark mass entails that calculations at even modest Q 2 require a Poincaré-covariant approach. Covariance + M(p 2 ) require existence of quark orbital angular momentum in hadron's rest-frame wave function. C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. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 3 LightCone 2011, SMU 23-27 May - 77pgs

4. Strong-interaction: QCD  Asymptotically free –Perturbation theory is valid and accurate tool at large-Q 2 –Hence chiral limit is defined  Essentially nonperturbative for Q 2 < 2 GeV 2 LightCone 2011, SMU 23-27 May - 77pgs Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 4  Nature’s only example of truly nonperturbative, fundamental theory  A-priori, no idea as to what such a theory can produce

5. LightCone 2011, SMU 23-27 May - 77pgs Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 5

6. Confinement  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: Dyson-Schwinger Equations and Continuum QCD. 6 X LightCone 2011, SMU 23-27 May - 77pgs

7. Confinement  Infinitely heavy-quarks plus 2 flavours with mass = m s –Lattice spacing = 0.083fm –String collapses within one lattice time-step R = 1.24 … 1.32 fm –Energy stored in string at collapse E c sb = 2 m s –(mpg made via linear interpolation)  No flux tube between light-quarks Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 7 G. Bali et al., PoS LAT2005 (2006) 308PoS LAT2005 (2006) 308 BsBs anti -B s “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.” LightCone 2011, SMU 23-27 May - 77pgs

8. 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); … Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 8 complex-P 2 o Real-axis mass-pole splits, moving into pair(s) of complex conjugate poles or branch points o Spectral density no longer positive semidefinite & hence state cannot exist in observable spectrum Normal particle Confined particle LightCone 2011, SMU 23-27 May - 77pgs timelike axis: P 2 <0

9. Dressed-gluon propagator  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 –m G ≈ 2-4 Λ QCD Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 9 perturbative, massless gluon massive, unconfined gluon IR-massive but UV-massless, confined gluon A.C. Aguilar et al., Phys.Rev. D80 (2009) 085018Phys.Rev. D80 (2009) 085018 LightCone 2011, SMU 23-27 May - 77pgs

10. Charting the interaction between light-quarks  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: Dyson-Schwinger Equations and Continuum QCD. 10 This is a well-posed problem whose solution is an elemental goal of modern hadron physics. The answer provides QCD’s running coupling. LightCone 2011, SMU 23-27 May - 77pgs

11. 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 o Hadron mass spectrum o 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. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 11 LightCone 2011, SMU 23-27 May - 77pgs

12. 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 = 3.2 α(1 GeV 2 )/π = 0.35 LightCone 2011, SMU 23-27 May - 77pgs Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 12 Maris & Tandy, Phys.Rev. C60 (1999) 055214Phys.Rev. C60 (1999) 055214  Running gluon mass –Gluon is massless in ultraviolet in agreement with pQCD –Massive in infrared m G (0) = 0.76 GeV m G (1 GeV 2 ) = 0.46 GeV

13. LightCone 2011, SMU 23-27 May - 77pgs Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 13

14. Dynamical Chiral Symmetry Breaking SStrong-interaction: Q CD CConfinement –E–Empirical feature –M–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 –S–Still circumstantial, no proof yet of confinement OOn the other hand, DCSB is a fact in QCD –I–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: Dyson-Schwinger Equations and Continuum QCD. 14 LightCone 2011, SMU 23-27 May - 77pgs

15. 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. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 15 LightCone 2011, SMU 23-27 May - 77pgs C.D. Roberts, Prog. Part. Nucl. Phys. 61 (2008) 50Prog. Part. Nucl. Phys. 61 (2008) 50 M. Bhagwat & P.C. Tandy, AIP Conf.Proc. 842 (2006) 225-227AIP Conf.Proc. 842 (2006) 225-227

16. 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. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 16 DSE prediction of DCSB confirmed Mass from nothing! LightCone 2011, SMU 23-27 May - 77pgs

17. 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. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 17 Hint of lattice-QCD support for DSE prediction of violation of reflection positivity LightCone 2011, SMU 23-27 May - 77pgs

18. 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. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 18 Jlab 12GeV: Scanned by 2<Q 2 <9 GeV 2 elastic & transition form factors. LightCone 2011, SMU 23-27 May - 77pgs

19. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 19

20. Dichotomy of the pion Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 20  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: LightCone 2011, SMU 23-27 May - 77pgs

21. Gell-Mann – Oakes – Renner Relation Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 21  This paper derives a relation between m π 2 and the expectation-value, where u o 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 u 0 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=m π 2 → t=0 to take → … in-pion → in-vacuum LightCone 2011, SMU 23-27 May - 77pgs Behavior of current divergences under SU(3) x SU(3). Murray Gell-Mann, R.J. Oakes, B. Renner Phys.Rev. 175 (1968) 2195-2199

22. Gell-Mann – Oakes – Renner Relation Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 22  PCAC hypothesis; viz., pion field dominates the divergence of the axial-vector current  Soft-pion theorem  In QCD, this is and one therefore has LightCone 2011, SMU 23-27 May - 77pgs Behavior of current divergences under SU(3) x SU(3). Murray Gell-Mann, R.J. Oakes, B. Renner Phys.Rev. 175 (1968) 2195-2199 Commutator is chiral rotation Therefore, isolates explicit chiral-symmetry breaking term in the putative Hamiltonian Zhou Guangzhao 周光召 Born 1929 Changsha, Hunan province

23. Gell-Mann – Oakes – Renner Relation Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 23  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. LightCone 2011, SMU 23-27 May - 77pgs - (0.25GeV) 3

24. Note of Warning Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 24 LightCone 2011, SMU 23-27 May - 77pgs 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 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.

25. QCD Sum Rules  Introduction of the gluon vacuum condensate and development of “sum rules” relating properties of low-lying hadronic states to vacuum condensates Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 25 LightCone 2011, SMU 23-27 May - 77pgs QCD and Resonance Physics. Sum Rules. M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov Nucl.Phys. B147 (1979) 385-447; citations: 3713

26. QCD Sum Rules  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 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 26 LightCone 2011, SMU 23-27 May - 77pgs QCD and Resonance Physics. Sum Rules. M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov Nucl.Phys. B147 (1979) 385-447; citations: 3781

27. “quark condensate” 1960-1980  Instantons in non-perturbative QCD vacuum, Instantons in non-perturbative QCD vacuum MA Shifman, AI Vainshtein… - Nuclear Physics B, 1980  Instanton density in a theory with massless quarks, Instanton density in a theory with massless quarks MA Shifman, AI Vainshtein… - Nuclear Physics B, 1980  Exotic new quarks and dynamical symmetry breaking, Exotic new quarks and dynamical symmetry breaking WJ Marciano - Physical Review D, 1980  The pion in QCD The pion in QCD J Finger, JE Mandula… - Physics Letters B, 1980 No references to this phrase before 1980 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 27 LightCone 2011, SMU 23-27 May - 77pgs

28. Universal Conventions  Wikipedia: (http://en.wikipedia.org/wiki/QCD_vacuum)(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.” LightCone 2011, SMU 23-27 May - 77pgs Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 28

29. 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 Ω cosm. const. = 0.76  This mismatch is a bit of a problem. LightCone 2011, SMU 23-27 May - 77pgs Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 29

30. Resolution?  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. LightCone 2011, SMU 23-27 May - 77pgs Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 30

31. Paradigm shift: In-Hadron Condensates  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. LightCone 2011, SMU 23-27 May - 77pgs Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 31 QCDQCD Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201Phys. Rev. C82 (Rapid Comm.) (2010) 022201 Brodsky and Shrock, PNAS 108, 45 (2011)PNAS 108, 45 (2011) M. Burkardt, Phys.Rev. D58 (1998) 096015Phys.Rev. D58 (1998) 096015

32. Paradigm shift: In-Hadron Condensates  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 f π and ρ π LightCone 2011, SMU 23-27 May - 77pgs Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 32 Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201Phys. Rev. C82 (Rapid Comm.) (2010) 022201 Brodsky and Shrock, PNAS 108, 45 (2011)PNAS 108, 45 (2011)

33. Paradigm shift: In-Hadron Condensates  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 f π and ρ π –And LightCone 2011, SMU 23-27 May - 77pgs Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 33 Chiral limit Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201Phys. Rev. C82 (Rapid Comm.) (2010) 022201 Brodsky and Shrock, PNAS 108, 45 (2011)PNAS 108, 45 (2011)

34. “ 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.”dark energy LightCone 2011, SMU 23-27 May - 77pgs Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 34 “Void that is truly empty solves dark energy puzzle” Rachel Courtland, New Scientist 4 th Sept. 2010 Cosmological Constant: Putting QCD condensates back into hadrons reduces the mismatch between experiment and theory by a factor of 10 46 Possibly by far more, if technicolour-like theories are the correct paradigm for extending the Standard Model Paradigm shift: In-Hadron Condensates

35. LightCone 2011, SMU 23-27 May - 77pgs Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 35

36. Strong-interaction: QCD  Gluons and quarks acquire momentum-dependent masses –characterised by an infrared mass-scale m ≈ 2-4 Λ QCD  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: γ μ. –Obvious, because with A(p 2 ) ≠ 1 and B(p 2 ) ≠ constant, the bare vertex cannot satisfy the Ward-Takahashi identity; viz.,  Number of contributors is too numerous to list completely (300 citations to 1 st 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 … Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 36 Dressed-quark-gluon vertex LightCone 2011, SMU 23-27 May - 77pgs

37. 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: Dyson-Schwinger Equations and Continuum QCD. 37 LightCone 2011, SMU 23-27 May - 77pgs

38. Gap Equation General Form  D μν (k) – dressed-gluon propagator  Γ ν (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., –D μν (k) = dressed, as described previously –Γ ν (q,p) = γ μ … plainly, key nonperturbative effects are missed and cannot be recovered through any step-by-step improvement procedure Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 38 Bender, Roberts & von Smekal Phys.Lett. B380 (1996) 7-12 LightCone 2011, SMU 23-27 May - 77pgs

39. Gap Equation General Form  D μν (k) – dressed-gluon propagator  good deal of information available  Γ ν (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?! Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 39 If kernels of Bethe-Salpeter and gap equations don’t match, one won’t even get right charge for the pion. LightCone 2011, SMU 23-27 May - 77pgs

40. 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: Dyson-Schwinger Equations and Continuum QCD. 40 LightCone 2011, SMU 23-27 May - 77pgs

41. Bethe-Salpeter Equation General Form  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) Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 41 Lei Chang and C.D. Roberts 0903.5461 [nucl-th] Phys. Rev. Lett. 103 (2009) 081601 LightCone 2011, SMU 23-27 May - 77pgs

42. Ward-Takahashi Identity Bethe-Salpeter Kernel  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: Dyson-Schwinger Equations and Continuum QCD. 42 Lei Chang and C.D. Roberts 0903.5461 [nucl-th] Phys. Rev. Lett. 103 (2009) 081601 iγ5iγ5 iγ5iγ5 LightCone 2011, SMU 23-27 May - 77pgs

43. Relativistic quantum mechanics  Dirac equation (1928): Pointlike, massive fermion interacting with electromagnetic field Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 43 Spin Operator LightCone 2011, SMU 23-27 May - 77pgs

44. 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).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),Phys. Rev. 73, 416 (1948) –vertex correction  The agreement with experiment established quantum electrodynamics as a valid tool. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 44 e e 0.001 16 LightCone 2011, SMU 23-27 May - 77pgs

45. Fermion electromagnetic current – General structure with k = p f - p i  F 1 (k 2 ) – Dirac form factor; and F 2 (k 2 ) – Pauli form factor –Dirac equation: F 1 (k 2 ) = 1 F 2 (k 2 ) = 0 –Schwinger: F 1 (k 2 ) = 1 F 2 (k 2 =0) = α /[2 π] Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 45 LightCone 2011, SMU 23-27 May - 77pgs

46.  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(iθγ 5 ) Ψ(x) Magnetic moment of a massless fermion? Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 46 LightCone 2011, SMU 23-27 May - 77pgs

47.  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(2iθγ 5 )  Magnetic moment interaction is forbidden in a theory with manifest chiral symmetry Magnetic moment of a massless fermion? Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 47 LightCone 2011, SMU 23-27 May - 77pgs

48. OOne-loop calculation: PPlainly, one obtains Schwinger’s result for m e 2 ≠ 0 HHowever, F 2 (k 2 ) = 0 when m e 2 = 0 TThere is no Gordon identity: RResults are unchanged at every order in perturbation theory … owing to symmetry … magnetic moment interaction is forbidden in a theory with manifest chiral symmetry Schwinger’s result? Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 48 e e m=0 So, no mixing γ μ ↔ σ μν LightCone 2011, SMU 23-27 May - 77pgs

49. QCD and dressed-quark anomalous magnetic moments  Schwinger’s result for QED:  pQCD: two diagrams o (a) is QED-like o (b) is only possible in QCD – involves 3-gluon vertex  Analyse (a) and (b) o (b) vanishes identically: the 3-gluon vertex does not contribute to a quark’s anomalous chromomag. moment at leading-order o (a) Produces a finite result: “ – ⅙ α s /2π ” ~ (– ⅙) QED-result  But, in QED and QCD, the anomalous chromo- and electro- magnetic moments vanish identically in the chiral limit! Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 49 LightCone 2011, SMU 23-27 May - 77pgs

50. Dressed-quark anomalous magnetic moments  Three strongly-dressed and essentially- nonperturbative contributions to dressed-quark-gluon vertex: Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 50 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, Kizilersu et al. hep-ph/0303176 L. Chang, Y. –X. Liu and C.D. Roberts arXiv:1009.3458 [nucl-th] arXiv:1009.3458 [nucl-th] Phys. Rev. Lett. 106 (2011) 072001 LightCone 2011, SMU 23-27 May - 77pgs

51. 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. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 51 Skullerud, Kizilersu et al. JHEP 0304 (2003) 047 Prediction from perturbative QCD Quenched lattice-QCD Quark mass function: M(p 2 =0)= 400MeV M(p 2 =10GeV 2 )=4 MeV LightCone 2011, SMU 23-27 May - 77pgs

52. Dressed-quark anomalous magnetic moments  Three strongly-dressed and essentially- nonperturbative contributions to dressed-quark-gluon vertex: Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 52 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, Kizilersu et al. hep-ph/0303176 Role and importance is novel discovery Essential to recover pQCD Constructive interference with Γ 5 L. Chang, Y. –X. Liu and C.D. Roberts arXiv:1009.3458 [nucl-th] arXiv:1009.3458 [nucl-th] Phys. Rev. Lett. 106 (2011) 072001 LightCone 2011, SMU 23-27 May - 77pgs

53. Dressed-quark anomalous magnetic moments Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 53  Formulated and solved general Bethe-Salpeter equation  Obtained dressed electromagnetic vertex  Confined quarks don’t have a mass-shell o Can’t unambiguously define magnetic moments o But can define magnetic moment distribution MEME κ ACM κ AEM Full vertex0.44-0.220.45 Rainbow-ladder0.3500.048  AEM is opposite in sign but of roughly equal magnitude as ACM L. Chang, Y. –X. Liu and C.D. Roberts arXiv:1009.3458 [nucl-th] arXiv:1009.3458 [nucl-th] Phys. Rev. Lett. 106 (2011) 072001 Factor of 10 magnification LightCone 2011, SMU 23-27 May - 77pgs

54. Dressed-quark anomalous magnetic moments Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 54  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. (I.C. Cloët et al.) L. Chang, Y. –X. Liu and C.D. Roberts arXiv:1009.3458 [nucl-th] arXiv:1009.3458 [nucl-th] Phys. Rev. Lett. 106 (2011) 072001 Factor of 10 magnification  Formulated and solved general Bethe-Salpeter equation  Obtained dressed electromagnetic vertex  Confined quarks don’t have a mass-shell o Can’t unambiguously define magnetic moments o But can define magnetic moment distribution Contemporary theoretical estimates: 1 – 10 x 10 -10 Largest value reduces discrepancy expt.↔theory from 3.3σ to below 2σ. LightCone 2011, SMU 23-27 May - 77pgs

55. Dressed Vertex & Meson Spectrum  Splitting known experimentally for more than 35 years  Hitherto, no explanation  Systematic symmetry-preserving, Poincaré-covariant DSE truncation scheme of nucl-th/9602012.nucl-th/9602012 o Never better than ∼ ⅟₄ of splitting  Constructing kernel skeleton-diagram-by-diagram, DCSB cannot be faithfully expressed: Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 55 ExperimentRainbow- ladder One-loop corrected Ball-ChiuFull vertex a11230 ρ 770 Mass splitting 455 Full impact of M(p 2 ) cannot be realised! ExperimentRainbow- ladder One-loop corrected Ball-ChiuFull vertex a11230 759 885 ρ 770 644 764 Mass splitting 455 115 121 Location of zero marks –m 2 meson LightCone 2011, SMU 23-27 May - 77pgs

56. Dressed Vertex & Meson Spectrum  Fully consistent treatment of Ball-Chiu vertex o Retain λ 3 – term but ignore Γ 4 & Γ 5 o Some effects of DCSB built into vertex & Bethe-Salpeter kernel  Big impact on σ – π complex  But, clearly, not the complete answer.  Fully-consistent treatment of complete vertex Ansatz  Promise of 1 st reliable prediction of light-quark hadron spectrum Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 56 ExperimentRainbow- ladder One-loop corrected Ball-ChiuFull vertex a11230 759 8851066 ρ 770 644 764 924 Mass splitting 455 115 121 142 ExperimentRainbow- ladder One-loop corrected Ball-ChiuFull vertex a11230 759 88511281270 ρ 770 644 764 919 790 Mass splitting 455 115 121 209 480 BC: zero moves deeper for both ρ & a 1 Both masses grow Full vertex: zero moves deeper for a 1 but shallower for ρ Problem solved LightCone 2011, SMU 23-27 May - 77pgs Lei Chang & C.D. Roberts, arXiv:1104.4821 [nucl-th] arXiv:1104.4821 [nucl-th] Tracing massess of ground-state light-quark mesons

57. LightCone 2011, SMU 23-27 May - 77pgs Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 57

58. Unification of Meson & Baryon Properties  Correlate the masses of meson and baryon ground- and excited-states within a single, symmetry-preserving framework  Symmetry-preserving means: Poincaré-covariant & satisfy relevant Ward-Takahashi identities  Constituent-quark model has hitherto been the most widely applied spectroscopic tool; whilst its weaknesses are emphasized by critics and acknowledged by proponents, it is of continuing value because there is nothing better that is yet providing a bigger picture.  Nevertheless,  no connection with quantum field theory & therefore not with QCD  not symmetry-preserving & therefore cannot veraciously connect meson and baryon properties Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 58 LightCone 2011, SMU 23-27 May - 77pgs

59. 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 observation that an interaction which describes colour-singlet mesons also generates nonpointlike quark-quark (diquark) correlations in the colour-antitriplet channel Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 59 R.T. Cahill et al., Austral. J. Phys. 42 (1989) 129-145 r qq ≈ r π LightCone 2011, SMU 23-27 May - 77pgs

60. Faddeev Equation  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 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 60 R.T. Cahill et al., Austral. J. Phys. 42 (1989) 129-145 diquark quark quark exchange ensures Pauli statistics composed of strongly- dressed quarks bound by dressed-gluons LightCone 2011, SMU 23-27 May - 77pgs

61.  “Spectrum” of nonpointlike quark-quark correlations  Observed in –DSE studies in QCD –0 + & 1 + in Lattice-QCD  Scalar diquark form factor –r 0+ ≈ r π  Axial-vector diquarks –r 1+ ≈ r ρ Diquarks in QCD LightCone 2011, SMU 23-27 May - 77pgs Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 61  Zero relation with old notion of pointlike constituent-like diquarks BOTH essential H.L.L. Roberts et al., 1102.4376 [nucl-th]1102.4376 [nucl-th] Phys. Rev. C in press Masses of ground and excited-state hadrons Hannes L.L. Roberts, Lei Chang, Ian C. Cloët and Craig D. Roberts, arXiv:1101.4244 [nucl-th]arXiv:1101.4244 [nucl-th] Few Body Systems (2011) pp. 1-25

62. Baryons & diquarks Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 62  Provided numerous insights into baryon structure; e.g.,  There is a causal connection between m Δ - m N & m 1+ - m 0+ m Δ - m N mNmN mΔmΔ Physical splitting grows rapidly with increasing diquark mass difference LightCone 2011, SMU 23-27 May - 77pgs Masses of ground and excited-state hadrons Hannes L.L. Roberts, Lei Chang, Ian C. Cloët and Craig D. Roberts, arXiv:1101.4244 [nucl-th]arXiv:1101.4244 [nucl-th] Few Body Systems (2011) pp. 1-25

63. Baryons & diquarks Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 63  Provided numerous insights into baryon structure; e.g.,  m N ≈ 3 M & m Δ ≈ M+m 1+ LightCone 2011, SMU 23-27 May - 77pgs

64. Photon-nucleon current  Composite nucleon must interact with photon via nontrivial current constrained by Ward-Takahashi identities  DSE, BSE, Faddeev equation, current → nucleon form factors Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 64 Vertex contains dressed-quark anomalous magnetic moment Oettel, Pichowsky, Smekal Eur.Phys.J. A8 (2000) 251-281 LightCone 2011, SMU 23-27 May - 77pgs

65. Nucleon Form Factors LightCone 2011, SMU 23-27 May - 77pgs Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 65 Survey of nucleon electromagnetic form factors I.C. Cloët et al, arXiv:0812.0416 [nucl-th],arXiv:0812.0416 [nucl-th] Few Body Syst. 46 (2009) pp. 1-36 Unification of meson and nucleon form factors. Very good description. Quark’s momentum- dependent anomalous magnetic moment has observable impact & materially improves agreement in all cases.

66. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 66 I.C. Cloët, C.D. Roberts, et al. arXiv:0812.0416 [nucl-th]  Highlights again the critical importance of DCSB in explanation of real-world observables.  DSE result Dec 08 DDSE result – including the anomalous magnetic moment distribution I.C. Cloët, C.D. Roberts, et al. In progress LightCone 2011, SMU 23-27 May - 77pgs

67. Quark anomalous magnetic moment has big impact on proton ratio But little impact on the neutron ratio LightCone 2011, SMU 23-27 May - 77pgs Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 67 I.C. Cloët, C.D. Roberts, et al. In progress

68. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 68  New JLab data: S. Riordan et al., arXiv:1008.1738 [nucl-ex] arXiv:1008.1738 [nucl-ex]  DSE-prediction I.C. Cloët, C.D. Roberts, et al. arXiv:0812.0416 [nucl-th]  This evolution is very sensitive to momentum-dependence of dressed-quark propagator LightCone 2011, SMU 23-27 May - 77pgs Phys.Rev.Lett. 105 (2010) 262302

69. Hadron Spectrum Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 69 Legend: Particle Data Group H.L.L. Roberts et al. EBAC Jülich o Symmetry-preserving unification of the computation of meson & baryon masses o rms-rel.err./deg-of-freedom = 13% o PDG values (almost) uniformly overestimated in both cases - room for the pseudoscalar meson cloud?! LightCone 2011, SMU 23-27 May - 77pgs Masses of ground and excited-state hadrons H.L.L. Roberts et al., arXiv:1101.4244 [nucl-th]arXiv:1101.4244 [nucl-th] Few Body Systems (2011) pp. 1-25

70. Baryon Spectrum Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 70  In connection with EBAC's analysis, dressed-quark Faddeev-equation predictions for bare-masses agree within rms-relative-error of 14%.  Notably, EBAC finds a dressed-quark-core for the Roper resonance, at a mass which agrees with Faddeev Eq. prediction. LightCone 2011, SMU 23-27 May - 77pgs

71. EBAC & the Roper resonance  EBAC examined the dynamical origins of the two poles associated with the Roper resonance are examined.  Both of them, together with the next higher resonance in the P 11 partial wave were found to have the same originating bare state  Coupling to the meson- baryon continuum induces multiple observed resonances from the same bare state.  All PDG identified resonances consist of a core state and meson-baryon components. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 71 N. Suzuki et al., Phys.Rev.Lett. 104 (2010) 042302Phys.Rev.Lett. 104 (2010) 042302 LightCone 2011, SMU 23-27 May - 77pgs

72. Hadron Spectrum Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 72 Legend: Particle Data Group H.L.L. Roberts et al. EBAC Jülich Now and for the foreseeable future, QCD-based theory will provide only dressed-quark core masses; EBAC or EBAC-like tools necessary for mesons and baryons LightCone 2011, SMU 23-27 May - 77pgs

73. Baryon Spectrum LightCone 2011, SMU 23-27 May - 77pgs Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 73 Masses of ground and excited-state hadrons Hannes L.L. Roberts, Lei Chang, Ian C. Cloët and Craig D. Roberts, arXiv:1101.4244 [nucl-th]arXiv:1101.4244 [nucl-th] Few Body Systems (2011) pp. 1-25 0+0+ 77% 1+1+ 23%100% 0-0- 51%43% 1-1- 49%57%100% diquark content Fascinating suggestion: nucleon’s first excited state = almost entirely axial-vector diquark, despite the dominance of scalar diquark in ground state Owes fundamentally to close relationship between scalar diquark and pseudoscalar meson And this follows from dynamical chiral symmetry breaking

74. Nucleon to Roper Transition Form Factors  Extensive CLAS @ JLab Programme has produced the first measurements of nucleon-to-resonance transition form factors  Theory challenge is to explain the measurements  Notable result is zero in F 2 p→N *, explanation of which is a real challenge to theory.  DSE study connects appearance of zero in F 2 p→N * with axial-vector-diquark dominance in Roper resonance and structure of form factors of J=1 state LightCone 2011, SMU 23-27 May - 77pgs Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 74 I. Aznauryan et al., Results of the N* Program at JLab arXiv:1102.0597 [nucl-ex] I.C. Cloët, C.D. Roberts, et al. In progress

75. Nucleon to Roper Transition Form Factors  Tiator and Vanderhaeghen – in progress –Empirically-inferred light-front-transverse charge density –Positive core plus negative annulus  Readily explained by dominance of axial-vector diquark –Considering isospin and charge –Negative d-quark twice as likely to be delocalised from always-positive core than positive u-quark LightCone 2011, SMU 23-27 May - 77pgs Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 75 {uu} d 2 {ud} u 1+

76. LightCone 2011, SMU 23-27 May - 77pgs Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 76

77. Epilogue  Dynamical chiral symmetry breaking (DCSB) – mass from nothing for 98% of visible matter – is a reality o Expressed in M(p 2 ), with observable signals in experiment  Poincaré covariance Crucial in description of contemporary data  Fully-self-consistent treatment of an interaction Comprehensive treatment of vast body of observables Essential if experimental data is truly to be understood.  Dyson-Schwinger equations: o single framework, with IR model-input turned to advantage, “almost unique in providing unambiguous path from a defined interaction → Confinement & DCSB → Masses → radii → form factors → distribution functions → etc.” Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 77 McLerran & Pisarski arXiv:0706.2191 [hep-ph] Confinement is almost Certainly the origin of DCSB e.g., BaBar anomaly LightCone 2011, SMU 23-27 May - 77pgs

78. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 78

79. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 79  New JLab data: S. Riordan et al., arXiv:1008.1738 [nucl-ex] arXiv:1008.1738 [nucl-ex]  DSE-prediction I.C. Cloët, C.D. Roberts, et al. arXiv:0812.0416 [nucl-th]  Location of zero measures relative strength of scalar and axial-vector qq-correlations Brooks, Bodek, Budd, Arrington fit to data: hep-ex/0602017 LightCone 2011, SMU 23-27 May - 77pgs

80. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 80 Neutron Structure Function at high x SU(6) symmetry pQCD, uncorrelated Ψ 0 + qq only Deep inelastic scattering – the Nobel-prize winning quark-discovery experiments Reviews:  S. Brodsky et al. NP B441 (1995)  W. Melnitchouk & A.W.Thomas PL B377 (1996) 11  N. Isgur, PRD 59 (1999)  R.J. Holt & C.D. Roberts RMP (2010) DSE: 0 + & 1 + qq I.C. Cloët, C.D. Roberts, et al. arXiv:0812.0416 [nucl-th] Distribution of neutron’s momentum amongst quarks on the valence-quark domain LightCone 2011, SMU 23-27 May - 77pgs