Craig Roberts Physics Division

1. Continuum Strong QCD - Opportunities & Challenges Published collaborations: 2010-present Rocio BERMUDEZ (U Michoácan); Chen CHEN (ANL, IIT, USTC); Xiomara GUTIERREZ-GUERRERO (U Michoácan); Trang NGUYEN (KSU); Si-xue QIN (PKU); Hannes ROBERTS (ANL, FZJ, UBerkeley); Lei CHANG (ANL, FZJ, PKU); Huan CHEN (BIHEP); Ian CLOËT (UAdelaide); Bruno EL-BENNICH (São Paulo); David WILSON (ANL); Adnan BASHIR (U Michoácan); Stan BRODSKY (SLAC); Gastão KREIN (São Paulo) Roy HOLT (ANL); Mikhail IVANOV (Dubna); Yu-xin LIU (PKU); Robert SHROCK (Stony Brook); Peter TANDY (KSU) Craig Roberts Physics Division Students Early-career scientists

2. Excited Baryons D. Wilson (Mon.), T.-S.H. Lee (Tues.) • R. Gothe (Wed.) QCD’s Challenges Understand emergent phenomena • Quark and Gluon Confinement • No matter how hard one strikes the proton, • one cannot liberate an individual gluon or quark Craig Roberts: O & C in Strong 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 from apparently simple rules.

3. Universal Truths Craig Roberts: O & C in Strong 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. Dyson-SchwingerEquations • 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: O & C in Strong QCD • 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?

5. Mass function exhibits inflexion • point at QIR≈0.5GeV • So … pQCD is definitely invalid • for momentaQ<QIR • E.g., use of DGLAP equations • cannot be justified in QCD at • Q<QIR=0.5GeV, irrespective of order. • Distribution Functions of the Nucleon and Pion in the Valence Region, Roy J. Holt and Craig D. RobertsarXiv:1002.4666 [nucl-th], • Rev. Mod. Phys. 82 (2010) pp. 2991-3044 • & presentation by Peter Tandy on Tuesday Necessary Precondition Essentially nonperturbative Craig Roberts: O & C in Strong QCD • Experiment ↔ Theory comparison leads to an understanding of long-range behaviour of strong running-coupling • However, if one wants to draw reliable conclusions about Q2-dependence of QCD’s running coupling, • Then, approach must veraciously express Q2-dependence of QCD’s running masses • True for ALL observables • From spectrum … • through elastic & transition form factors … • to PDFs and GPDs … etc.

6. Confinement Craig Roberts: O & C in Strong QCD

7. X Confinement Coloursinglets Craig Roberts: O & C in Strong QCD • Gluon and Quark Confinement • No coloured states have yet been observed to reach a detector • 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 • But light-quarks are ubiquitous • Confinement entails quark-hadron duality; i.e., that all observable consequences of QCD can, in principle, be computed using an hadronic basis.

8. 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: O & C in Strong 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, nor in qQ systems

9. 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, • or more complicated nonanalyticities … • Spectral density no longer positive semidefinite • & hence state cannot exist in observable spectrum Craig Roberts: O & C in Strong QCD • Confinement is expressed through a dramatic 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); Roberts, Williams & Krein (1992); Tandy (1994); …

10. 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: O & C in Strong 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

11. 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: O & C in Strong 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.

12. Charting the interaction between light-quarks Craig Roberts: O & C in Strong 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.

13. Qin et al., Phys. Rev. C 84 042202(Rapid Comm.) (2011) Rainbow-ladder truncation DSE Studies – Phenomenology of gluon • Running gluon mass • Gluon is massless in ultraviolet in agreement with pQCD • Massive in infrared • mG(0) = 0.67-0.81 GeV • mG(mG2) = 0.53-0.64 GeV Craig Roberts: O & C in Strong QCD • Wide-ranging study of π & ρ properties • Effective coupling • Agrees with pQCDin ultraviolet • Saturates in infrared • α(0)/π = 8-15 • α(mG2)/π = 2-4

14. Dynamical ChiralSymmetry Breaking Craig Roberts: O & C in Strong QCD

15. Dynamical Chiral Symmetry Breaking Craig Roberts: O & C in Strong 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.

16. 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: O & C in Strong 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 Mass from nothing! DSE prediction of DCSB confirmed Craig Roberts: O & C in Strong 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. Frontiers of Nuclear Science:Theoretical Advances Hint of lattice-QCD support for DSE prediction of violation of reflection positivity Craig Roberts: O & C in Strong 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.

19. 12GeVThe Future of JLab Jlab 12GeV: Scanned by 2<Q2<9 GeV2 elastic & transition form factors. Craig Roberts: O & C in Strong 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.

20. Persistent Challenge Truncation Craig Roberts: O & C in Strong QCD

21. Dichotomy of the pionGoldstone mode and bound-state HIGHLY NONTRIVIAL Impossible in quantum mechanics Only possible in asymptotically-free gauge theories Craig Roberts: O & C in Strong QCD • The correct understanding of pion observables; e.g. mass, decay constant and form factors, requires an approach to contain a • well-defined and validchiral limit; • and an accurate realisation of dynamical chiral symmetry breaking.

22. Persistent challenge- truncation scheme Craig Roberts: O & C in Strong QCD • There are now two nonperturbative & symmetry preserving truncation schemes • 1995 – H.J. Munczek, Phys. Rev. D 52 (1995) 4736,Dynamical chiral symmetry breaking, Goldstone’s theorem and the consistency of the Schwinger-Dyson and Bethe-Salpeter Equations 1996 – A. Bender, C.D. Roberts and L. von Smekal, Phys.Lett. B 380 (1996) 7, Goldstone Theorem and Diquark Confinement Beyond Rainbow Ladder Approximation • 2009 – Lei Chang and C.D. Roberts, Phys. Rev. Lett. 103 (2009) 081601, 0903.5461 [nucl-th], Sketching the Bethe-Salpeter kernel • Enables proof of numerous exact results

23. Some of many Exact Results Craig Roberts: O & C in Strong QCD

24. Maris, Roberts and Tandy nucl-th/9707003, Phys.Lett. B420 (1998) 267-273  Pion’s Goldberger-Treiman relation Pseudovector components necessarily nonzero. Cannot be ignored! Miracle: two body problem solved, almost completely, once solution of one body problem is known Exact in Chiral QCD Craig Roberts: O & C in Strong QCD • Pion’s Bethe-Salpeter amplitude Solution of the Bethe-Salpeter equation • Dressed-quark propagator • Axial-vector Ward-Takahashi identity entails

25. Dichotomy of the pionGoldstone mode and bound-state fπ Eπ(p2) = B(p2) Craig Roberts: O & C in Strong QCD • Goldstone’s theorem has a pointwise expression in QCD; Namely, in the chiral limit the wave-function for the two-body bound-state Goldstone mode is intimately connected with, and almost completely specified by, the fully-dressed one-body propagator of its characteristic constituent • The one-body momentum is equated with the relative momentum of the two-body system

26. Dynamical Chiral Symmetry BreakingImportance of being well-dressed for quarks & mesons Craig Roberts: O & C in Strong QCD

27. Strong-interaction: QCD Dressed-quark-gluon vertex Craig Roberts: O & C in Strong 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 …

28. Dressed-quark-gluon vertex Craig Roberts: O & C in Strong 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.

29. Gap EquationGeneral Form Bender, Roberts & von Smekal Phys.Lett. B380 (1996) 7-12 Craig Roberts: O & C in Strong 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 (or LO+NLO) in a symmetry-preserving truncation scheme; viz., • Dμν(k) = fully-dressed • Γν(q,p) = γμ • … plainly, key nonperturbative effects are missed and cannot be recovered through any step-by-step improvement procedure

30. 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: O & C in Strong 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?!

31. Bethe-Salpeter EquationBound-State DSE Craig Roberts: O & C in Strong 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.

32. Bethe-Salpeter EquationGeneral Form Lei Chang and C.D. Roberts 0903.5461 [nucl-th] Phys. Rev. Lett. 103 (2009) 081601 Craig Roberts: O & C in Strong 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)

33. 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: O & C in Strong 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

34. QCD and dressed-quark anomalous magnetic moments Craig Roberts: O & C in Strong 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! • In QCD, chiral symmetry is dynamically broken, strongly • What then?

35. 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: O & C in Strong QCD • Three strongly-dressed and essentially- nonperturbative contributions to dressed-quark-gluon vertex:

36. Dressed-quark anomalous chromomagnetic moment Quenched lattice-QCD Skullerud, Kizilersuet al. JHEP 0304 (2003) 047 Quark mass function: M(p2=0)= 400MeV M(p2=10GeV2)=4 MeV Prediction from perturbative QCD Craig Roberts: O & C in Strong QCD • 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.

37. 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 • Role and importance is • novel discovery • Essential to recover pQCD • Constructive interference with Γ5 AdnanBashir, Wed.: Dynamical chiral symmetry breaking and the fermion--gauge-boson vertex, Bashir, Bermudez, Chang & Roberts, arXiv:1112.4847 [nucl-th], Phys. Rev. C in press Craig Roberts: O & C in Strong QCD • Three strongly-dressed and essentially- nonperturbative contributions to dressed-quark-gluon vertex:

38. L. Chang, Y. –X. Liu and C.D. RobertsarXiv:1009.3458 [nucl-th] Phys. Rev. Lett. 106 (2011) 072001 Dressed-quark anomalousmagnetic 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 ACM • AEM is opposite in sign but of • roughly equal magnitude • as ACM Craig Roberts: O & C in Strong QCD

39. L. Chang, Y. –X. Liu and C.D. RobertsarXiv:1009.3458 [nucl-th] Phys. Rev. Lett. 106 (2011) 072001 Dressed-quark anomalousmagnetic 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 • Potentially important for elastic and transition form factors, etc. • Indeed, for any process involving photons • coupling to a dressed-quark Craig Roberts: O & C in Strong QCD

40. L. Chang, Y. –X. Liu and C.D. RobertsarXiv:1009.3458 [nucl-th] Phys. Rev. Lett. 106 (2011) 072001 Dressed-quark anomalousmagnetic 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 muong-2 – via Box diagram, • which is not constrained by extant data. Craig Roberts: O & C in Strong QCD

41. Location of zero marks –m2meson a1 – ρ mass splitting Full impact of M(p2) cannot be realised! Craig Roberts: O & C in Strong QCD • Known experimentally for more than 35 years • Hitherto, no explanation • Systematic symmetry-preserving, Poincaré-covariant DSE truncation scheme of nucl-th/9602012. • Never better than ∼ ⅟₄ of splitting • Constructing kernel skeleton-diagram-by-diagram, DCSB cannot be faithfully expressed:

42. Solves problem of a1 – ρ mass splitting Lei Chang & C.D. Roberts, Tracing masses of ground-state light-quark mesons, arXiv:1104.4821 [nucl-th], Phys. Rev. C (Rapid Comm.) in press M(p2) magnifies spin orbit splitting here, precisely as in σ-π comparison Craig Roberts: O & C in Strong QCD Fully nonperturbative BSE kernel that incorporates and expresses DCSB: establishes unambiguously that a1 & ρ are parity-partner bound-states of dressed light valence-quarks.

43. Dynamical Chiral Symmetry BreakingVacuum Condensates? Craig Roberts: O & C in Strong QCD

44. Universal Conventions Craig Roberts: O & C in Strong 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.”

45. Universal Misapprehensions Craig Roberts: O & C in Strong 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.

46. Resolution? Craig Roberts: O & C in Strong 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. 

47. New Paradigm“in-hadron condensates” Craig Roberts: O & C in Strong QCD

48. Won’t say more here but reserve comments for a workshop in São Paolo next week. Relevant References Craig Roberts: O & C in Strong QCD arXiv:1202.2376 Confinement contains condensates Stanley J. Brodsky, Craig D. Roberts, Robert Shrock, Peter C. Tandy arXiv:1109.2903 [nucl-th], Phys. Rev. C85 (2012) 012201(RapCom), Expanding the concept of in-hadron condensatesLei Chang, Craig D. Roberts and Peter C. Tandy arXiv:1005.4610 [nucl-th], Phys. Rev. C82 (2010) 022201(RapCom.) New perspectives on the quark condensate, Brodsky, Roberts, Shrock, Tandy arXiv:0905.1151 [hep-th], PNAS 108, 45 (2011) Condensates in Quantum Chromodynamics and the Cosmological Constant, Brodsky and Shrock, hep-th/0012253 The Quantum vacuum and the cosmological constant problem, Svend Erik Rugh and HenrikZinkernagel.

49. Grand Unification Craig Roberts: O & C in Strong QCD

50. Modern Challenges exotic mesons: quantum numbers not possible for quantum mechanical quark-antiquark systems hybrid mesons: normal quantum numbers but non-quark-model decay pattern BOTH suspected of having “constituent gluon” content ↔ Craig Roberts: O & C in Strong QCD Computation of spectrum of hybrid and exotic mesons Equally pressing, some might say more so, is the three-body problem; viz., baryons in QCD