Craig Roberts Physics Division

1. Explanation and Prediction of Observables using Continuum Strong QCD Craig Roberts Physics Division

2. Students Postdocs Asst. Profs. Collaborators: 2011-Present • Adnan BASHIR (U Michoácan); • Stan BRODSKY (SLAC); • Gastão KREIN (São Paulo) • Roy HOLT (ANL); • Mikhail IVANOV (Dubna); • Yu-xin LIU (PKU); • Michael RAMSEY-MUSOLF (UW-Mad) • Alfredo RAYA (U Michoácan); • Sebastian SCHMIDT (IAS-FZJ & JARA); • Robert SHROCK (Stony Brook); • Peter TANDY (KSU); • Tony THOMAS (U.Adelaide) • Shaolong WAN (USTC) Explaining Observables in Continuum Strong QCD (156p) Rocio BERMUDEZ (U Michoácan); Xiomara GUTIERREZ-GUERRERO (U Michoácan); S. HERNÁNDEZ(U Michoácan); Trang NGUYEN (KSU); Khépani RAYA (U Michoácan); Hannes ROBERTS (ANL, FZJ, UBerkeley); Chien-Yeah SENG (UW-Mad) Kun-lun WANG (PKU); Chen CHEN (USTC); J. JavierCOBOS-MARTINEZ (U.Sonora); Mario PITSCHMANN (ANL & UW-Mad); Si-xue QIN(U. Frankfurt am Main); Jorge SEGOVIA (ANL); David WILSON (ODU); Lei CHANG (U.Adelaide); Ian CLOËT (ANL); Bruno EL-BENNICH (São Paulo);

3. Enormous progress since 2010 arXiv:1310.2651 [nucl-th] Explaining Observables in Continuum Strong QCD (156p)

4. Standard Model Explaining Observables in Continuum Strong QCD (156p)

5. Quantum Chromodynamics • QCD: The piece of the Standard Model that describes strong interactions. • Hadron Physics is a nonperturbative problem in QCD • Notwithstanding the 2013 Nobel Prize in Physics, the origin of 98% of the visible mass in the Universe is – somehow – found within QCD Explaining Observables in Continuum Strong QCD (156p)

6. Explaining Observables in Continuum Strong QCD (156p)

7. Hadron: Any of a class of subatomic particles that are composed of quarks and/or gluons and take part in the strong interaction.  Examples: proton, neutron, & pion. International Scientific Vocabulary: hadr- thick, heavy (from Greek hadros thick) + 2on First Known Use: 1962 Baryon: hadron with half-integer-spin Meson: hadron with integer-spin Hadrons Explaining Observables in Continuum Strong QCD (156p)

8. Facilities Explaining Observables in Continuum Strong QCD (156p)

9. FacilitiesQCD Machines Explaining Observables in Continuum Strong QCD (156p) • China • Beijing Electron-Positron Collider • Germany • COSY (Jülich Cooler Synchrotron) • ELSA (Bonn Electron Stretcher and Accelerator) • MAMI (Mainz Microtron) • Facility for Antiproton and Ion Research, under construction near Darmstadt. New generation experiments in 2018 (perhaps) • Japan • J-PARC (Japan Proton Accelerator Research Complex), under construction in Tokai-Mura, 150km NE of Tokyo. New generation experiments to begin soon • KEK: Tsukuba, Belle Collaboration • Switzerland (CERN) • Large Hadron Collider: ALICE Detector and COMPASS Detector “Understanding deconfinement and chiral-symmetry restoration”

10. FacilitiesQCD Machines A three dimensional view of the calculated particle paths resulting from collisions occurring within RHIC's STAR detector Explaining Observables in Continuum Strong QCD (156p) • USA • Thomas Jefferson National Accelerator Facility, Newport News, Virginia Nature of cold hadronic matter Upgrade underway Construction cost $310-million New generation experiments in 2014 • Relativistic Heavy Ion Collider, Brookhaven National Laboratory, Long Island, New York Strong phase transition, 10μs after Big Bang

11. FacilitiesQCD Machines Explaining Observables in Continuum Strong QCD (156p) • USA • Thomas Jefferson National Accelerator Facility, Newport News, Virginia Nature of cold hadronic matter Upgrade underway Construction cost $310-million New generation experiments in 2014

12. Science Challenges for the coming decade: 2013-2022 All composite systems have “Form Factors”, which describe the distribution of an observable quantity amongst the constituents. Explaining Observables in Continuum Strong QCD (156p) • Exploit opportunities provided by new data on hadron elastic and transition form factors • Chart infrared evolution of QCD’s coupling and dressed-masses • Reveal correlations that are key to nucleon structure • Expose the facts and fallacies in modern descriptions of hadron structure

13. Science Challenges for the coming decade: 2013-2022 Parton distribution functions (PDFs) and distribution amplitudes (PDAs) are a quantum field theory analogue of wave functions. They have a probability interpretation and hence relate to concepts familiar from quantum mechanics. Explaining Observables in Continuum Strong QCD (156p) • Precision experimental study of (far) valence region (Bjorken-x> 0.5), and theoretical computation of distribution functions and distribution amplitudes • Computation is critical • Without it, no amount of data will reveal anything about the theory underlying the phenomena of strong interaction physics

14. Overarching Science Challenges for the coming decade: 2013-2022 Explaining Observables in Continuum Strong QCD (156p) Discover the meaning of confinement Determine its connection with DCSB (dynamical chiral symmetry breaking) Elucidate their signals in observables … so experiment and theory together can map the nonperturbativebehaviour of the strong interaction In my view, it is unlikely that two phenomena, so critical in the Standard Model and tied to the dynamical generation of a single mass-scale, can have different origins and fates.

15. What is QCD? Explaining Observables in Continuum Strong QCD (156p)

16. QCD is a Theory (not an effective theory) Explaining Observables in Continuum Strong QCD (156p) • Very likely a self-contained, nonperturbativelyrenormalisable and hence well defined Quantum Field Theory This is not true of QED – cannot be defined nonperturbatively • No confirmed breakdown over an enormous energy domain: 0 GeV < E < 8000 GeV • Increasingly likely that any extension of the Standard Model will be based on the paradigm established by QCD • Extended Technicolour: electroweak symmetry breaks via a fermion bilinear operator in a strongly-interacting non-Abelian theory. (Andersen et al. “Discovering Technicolor” Eur.Phys.J.Plus 126 (2011) 81) Higgs sector of the SM becomes an effective description of a more fundamental fermionic theory, similar to the Ginzburg-Landau theory of superconductivity

17. Contrast: so-called Effective Field Theories Can Cannot • QCD appears valid at all energy scales that have been tested so far: no breakdown below • E ≈ 60000 mπ • Cannot be used to test QCD • Any mismatch between • EF-Theory and experiment owes to an error in the formulation of one or conduct of the other Explaining Observables in Continuum Strong QCD (156p) EFTs applicable over a very restricted energy domain; e.g., ChPT known to breakdown for E > 2mπ Can be used to help explore how features of QCD influence observables

18. What is QCD? Explaining Observables in Continuum Strong QCD (156p) • Lagrangian of QCD • G = gluon fields • Ψ = quark fields • The key to complexity in QCD … gluon field strength tensor • Generates gluon self-interactions, whose consequences are quite extraordinary

19. cf.Quantum Electrodynamics Explaining Observables in Continuum Strong QCD (156p) QED is the archetypal gauge field theory Perturbatively simple but nonperturbatively undefined Chracteristic feature: Light-by-light scattering; i.e., photon-photon interaction – leading-order contribution takes place at order α4. Extremely small probability because α4 ≈10-9 !

20. What is QCD? • Relativistic Quantum Gauge Field Theory: • Interactions mediated by vector boson exchange • Vector bosons are perturbatively-massless • Similar interaction in QED • Special feature of QCD – gluon self-interactions 3-gluon vertex 4-gluon vertex Explaining Observables in Continuum Strong QCD (156p)

21. Strong-interaction: QCD • Nature’sonly (now known) example of a truly nonperturbative, fundamental theory • A-priori, no idea as to what such a theory • can produce Explaining Observables in Continuum Strong QCD (156p) • Asymptotically free • Perturbation theory is valid and accurate tool at large-Q2 • Hence chiral limit is defined • Essentiallynonperturbative for Q2 < 2 GeV2

22. What is Confinement? Explaining Observables in Continuum Strong QCD (156p)

23. Light quarks & Confinement • Folklore … JLab Hall-DConceptual Design Report(5) “The color field lines between a quark and an anti-quark form flux tubes. Explaining Observables in Continuum Strong QCD (156p) A unit area placed midway between the quarks and perpendicular to the line connecting them intercepts a constant number of field lines, independent of the distance between the quarks. This leads to a constant force between the quarks – and a large force at that, equal to about 16 metric tons.”

24. Light quarks & Confinement Explaining Observables in Continuum Strong QCD (156p) • Problem: 16 tonnes of force makes a lot of pions.

25. Light quarks & Confinement Explaining Observables in Continuum Strong QCD (156p) Problem: 16 tonnes of force makes a lot of pions.

26. G. Bali et al., PoS LAT2005 (2006) 308 Light quarks & Confinement Explaining Observables in Continuum Strong QCD (156p) In the presence of light quarks, pair creation seems to occur non-localized and instantaneously No flux tube in a theory with light-quarks. Flux-tube is not the correct paradigm for confinement in hadron physics

27. Confinement Confined particle Normal particle complex-P2 complex-P2 timelike axis: P2<0 s ≈ 1/Im(m) ≈ 1/2ΛQCD≈ ½fm • Real-axis mass-pole splits, moving into pair(s) of complex conjugate singularities, • (or other qualitatively analogous structures chracterised by a dynamically generated mass-scale) • State described by rapidly damped wave & hence state cannot exist in observable spectrum Explaining Observables in Continuum Strong QCD (156p) • QFT Paradigm: • Confinement is expressed through a dramatic change in the analytic structure of propagators for coloured states • It can almost be read from a plot of the dressed-propagator for a coloured state

28. Light quarks & Confinement Explaining Observables in Continuum Strong QCD (156p) • In the study of hadrons, attention should turn from equal-time potential models toward the continuum bound-state problem in quantum field theory • Such approaches offer the possibility of posing simultaneously the questions • What is confinement? • What is dynamical chiral symmetry breaking? • How are they related? • What are their empirical signals?

29. Dynamical ChiralSymmetry Breaking Explaining Observables in Continuum Strong QCD (156p)

30. Mass from Nothing Explaining Observables in Continuum Strong QCD (156p)

31. Dynamical Chiral Symmetry Breaking Explaining Observables in Continuum Strong QCD (156p) • DCSB is a fact in QCD • Dynamical, not spontaneous • Add nothing to QCD , no Higgs field, nothing! • Effect achieved purely through the quark+gluon dynamics. • It’s the most important mass generating mechanism for visible matter in the Universe. • Responsible for ≈98% of the proton’s mass. • Higgs mechanism is (almost) irrelevant to light-quarks.

32. Gap Equation Explaining Observables in Continuum Strong QCD (156p)

33. DCSB C.D. Roberts, Prog. Part. Nucl. Phys. 61 (2008) 50 M. Bhagwat & P.C. Tandy, AIP Conf.Proc. 842 (2006) 225-227 • In QCD, all “constants” of quantum mechanics are actually strongly momentum dependent: couplings, number density, mass, etc. • So, a quark’s mass depends on its momentum. • Mass function can be calculated and is depicted here. • Continuum- and Lattice-QCD Mass from nothing! • are in agreement: the vast bulk of the light-quark mass comes from a cloud of gluons, dragged along by the quark as it propagates. Explaining Observables in Continuum Strong QCD (156p)

34. Where does the mass come from? αS23 Just one of the terms that are summed in a solution of the rainbow-ladder gap equation Explaining Observables in Continuum Strong QCD (156p) Deceptively simply picture Corresponds to the sum of a countable infinity of diagrams. NB. QED has 12,672 α5 diagrams Impossible to compute this in perturbation theory. The standard algebraic manipulation tools are just inadequate

35. Enigma of Mass Explaining Observables in Continuum Strong QCD (156p)

36. Bound-states in Quantum Field Theory Sketching the Bethe-Salpeter kernel, Lei Chang and Craig D. Roberts, arXiv:0903.5461 [nucl-th], Phys. Rev. Lett. 103 (2009) 081601 (4 pages) Explaining Observables in Continuum Strong QCD (156p) • Mass and “Wave Function” are obtained from a Bethe-Salpeter equation • Generalisation of the Lippmann-Schwinger equation • The pion … Nature’s strong-interaction messenger … is a critical example

37. 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 Explaining Observables in Continuum Strong QCD (156p) • Pion’s Bethe-Salpeter amplitude Solution of the Bethe-Salpeter equation • Dressed-quark propagator • Axial-vector Ward-Takahashi identity entails

38. Dichotomy of the pionGoldstone mode and bound-state fπ Eπ(p2) = B(p2) Explaining Observables in Continuum Strong QCD (156p) • 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

39. Enigma of mass fπ Eπ(p2) = B(p2) Explaining Observables in Continuum Strong QCD (156p) • The quark level Goldberger-Treiman relation shows that DCSB has a very deep and far reaching impact on physics within the strong interaction sector of the Standard Model; viz., Goldstone's theorem is fundamentally an expression of equivalence between the one-body problem and the two-body problem in the pseudoscalar channel.  • This emphasises that Goldstone's theorem has a pointwise expression in QCD • Hence, pion properties are an almost direct measure of the dressed-quark mass function.  • Thus, enigmatically, the properties of the masslesspion are the cleanest expression of the mechanism that is responsible for almost all the visible mass in the universe.

40. Interaction model for the gap equationSi-xue Qin, Lei Chang, Y.-x.Liu, C.D. Roberts and D.J. WilsonarXiv:1108.0603 [nucl-th],  Phys. Rev. C 84 (2011) 042202(R) [5 pages] In QCD, Gluons, too, become massive Explaining Observables in Continuum Strong QCD (156p) Not just quarks … Gluons also have a gap equation … 1/k2behaviour signals essential singularity in the running coupling: Impossible to reach in perturbation theory

41. Dynamical Chiral Symmetry BreakingVacuum Condensates? Explaining Observables in Continuum Strong QCD (156p)

42. “Orthodox Vacuum” u d u u d u u u d Explaining Observables in Continuum Strong QCD (156p) Vacuum = “frothing sea” Hadrons = bubbles in that “sea”, containing nothing but quarks & gluons interacting perturbatively, unless they’re near the bubble’s boundary, whereat they feel they’re trapped!

43. Historically, DCSB came to be associated with a presumed existence of spacetime-independent condensates that permeated the universe. However, just like gluons and quarks, and for the same reasons:Condensates are confined within hadrons. There are no vacuum condensates. Explaining Observables in Continuum Strong QCD (156p)

44. Background Explaining Observables in Continuum Strong QCD (156p) Worth noting that nonzero vacuum expectation values of local operators in QCD—the so-called vacuum condensates—are phenomenological parameters, which were introduced at a time of limited computational resources in order to assist with the theoretical estimation of essentially nonperturbative strong-interaction matrix elements. A universality of these condensates was assumed, namely, that the properties of all hadrons could be expanded in terms of the same condensates. While this helps to retard proliferation, there are nevertheless infinitely many of them. As qualities associated with an unmeasurable state (the vacuum), such condensates do not admit direct measurement. Practitioners have attempted to assign values to them via an internally consistent treatment of many separate empirical observables. However, only one, the so-called quark condensate, is attributed a value with any confidence.

45. Maris, Roberts and Tandy nucl-th/9707003, Phys.Lett. B420 (1998) 267-273  Dichotomy of the pionMass Formula for 0— Mesons We now have sufficient information to address the question of just what is this so-called “vacuum quark condensate.” The so-called “vacuum quark condensate.” More later about this. Gell-Mann, Oakes, Renner relation 1968 Explaining Observables in Continuum Strong QCD (156p) • Consider the case of light quarks; namely, mq ≈ 0 • If chiral symmetry is dynamically broken, then • fH5 → fH50 ≠ 0 • ρH5 → – < q-bar q> / fH50 ≠ 0 both of which are independent of mq • Hence, one arrives at the corollary

46. Spontaneous(Dynamical)Chiral Symmetry Breaking Explaining Observables in Continuum Strong QCD (156p) The 2008Nobel Prize in Physics was divided, one half awarded to YoichiroNambu "for the discovery of the mechanism of spontaneous broken symmetry in subatomic physics"

47. Nambu – Jona-LasinioModel Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. I Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345–358 Dynamical Model Of Elementary Particles Based On An Analogy With Superconductivity. II Y. Nambu, G. Jona-Lasinio, Phys.Rev. 124 (1961) 246-254 Explaining Observables in Continuum Strong QCD (156p) • Treats a chirally-invariant four-fermionLagrangian & solves the gap equation in Hartree-Fock approximation (analogous to rainbow truncation) • Possibility of dynamical generation of nucleon mass is elucidated • Essentially inequivalent vacuum states are identified (Wigner and Nambu states) & demonstration that there are infinitely many, degenerate but distinct Nambuvacua, related by a chiral rotation • Nontrivial Vacuum is “Born”

48. 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 Explaining Observables in Continuum Strong QCD (156p) • 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

49. 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 Explaining Observables in Continuum Strong QCD (156p) • PCAC hypothesis; viz., pion field dominates the divergence of the axial-vector current • Soft-pion theorem • In QCD, this is and one therefore has

50. Gell-Mann – Oakes – RennerRelation - (0.25GeV)3 Explaining Observables in Continuum Strong QCD (156p) • 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 matrix element, there is no problem. • Problem arises if one over-interprets this number, which textbooks have been doing for a VERY LONG TIME.