Craig Roberts Physics Division

1. Truncations in DSE QCD 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); Chien-Yeah SENG (UW-Mad) Kun-lun WANG (PKU); Lei CHANG (ANL, FZJ, PKU); Huan CHEN (BIHEP); Ian CLOËT (UAdelaide); Bruno EL-BENNICH (São Paulo); Mario PITSCHMANN (ANL & UW-Mad) David WILSON (ANL); 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) Sebastian SCHMIDT (IAS-FZJ & JARA); Robert SHROCK (Stony Brook); Peter TANDY (KSU) Shaolong WAN (USTC) Published collaborations: 2010-present Craig Roberts Physics Division Students Early-career scientists

2. Introductory-level presentations Recommended reading Craig Roberts: Truncations in DSE-QCD (114p) C. D. Roberts, “Strong QCD and Dyson-Schwinger Equations,”arXiv:1203.5341 [nucl-th]. Notes based on 5 lectures to the conference on “Dyson-Schwinger Equations & Faàdi Bruno Hopf Algebras in Physics and Combinatorics (DSFdB2011),” Institut de RechercheMathématiqueAvancée, l'Universite de Strasbourg et CNRS, Strasbourg, France, 27.06-01.07/2011. To appear in “IRMA Lectures in Mathematics & Theoretical Physics,” published by the European Mathematical Society (EMS) C.D. Roberts, M.S. Bhagwat, A. Höll and S.V. Wright, “Aspects of Hadron Physics,” Eur. Phys. J. Special Topics 140 (2007) pp. 53-116 A. Höll, C.D. Roberts and S.V. Wright, nucl-th/0601071, “Hadron Physics and Dyson-Schwinger Equations” (103 pages) C.D. Roberts (2002): “Primer for Quantum Field Theory in Hadron Physics” (http://www.phy.anl.gov/theory/ztfr/LecNotes.pdf) C. D. Roberts and A. G. Williams,“Dyson-Schwinger equations and their application to hadronic physics,” Prog. Part. Nucl. Phys. 33 (1994) 477

3. Research-level presentations Recommended reading Craig Roberts: Truncations in DSE-QCD (114p) A. Bashir, Lei Chang, Ian C. Cloët, Bruno El-Bennich, Yu-xin Liu, Craig D. Roberts and Peter C. Tandy, “Collective perspective on advances in Dyson-Schwinger Equation QCD,” arXiv:1201.3366 [nucl-th], Commun. Theor. Phys. 58 (2012) pp. 79-134 R.J. Holt and C.D. Roberts, “Distribution Functions of the Nucleon and Pion in the Valence Region,” arXiv:1002.4666 [nucl-th], Rev. Mod. Phys. 82 (2010) pp. 2991-3044 C.D. Roberts , “Hadron Properties and Dyson-Schwinger Equations,”  arXiv:0712.0633 [nucl-th], Prog. Part. Nucl. Phys. 61 (2008) pp. 50-65  P. Maris and C. D. Roberts, “Dyson-Schwinger equations: A tool for hadron physics,” Int. J. Mod. Phys. E 12, 297 (2003) C. D. Roberts and S. M. Schmidt, “Dyson-Schwinger equations: Density, temperature and continuum strong QCD,” Prog. Part. Nucl. Phys. 45 (2000) S1

4. Standard Model of Particle Physics Craig Roberts: Truncations in DSE-QCD (114p)

5. Standard Model- History • With the advent of cosmic ray science and particle accelerators, numerous additional particles were discovered: • muon (1937), pion (1947), kaon (1947), Roper resonance (1963), … • By the mid-1960s, it was apparent that not all the particles could be fundamental. • A new paradigm was necessary. • Gell-Mann's and Zweig's constituent-quark theory (1964) was a critical step forward. • Gell-Mann, Nobel Prize 1969: "for his contributions and discoveries concerning the classification of elementary particles and their interactions". • Over the more than forty intervening years, the theory now called the Standard Model of Particle Physics has passed almost all tests. Craig Roberts: Truncations in DSE-QCD (114p) In the early 20th Century, the only matter particles known to exist were the proton, neutron, and electron.

6. Standard Model- The Pieces • Weak interaction • Radioactive decays, parity-violating decays, electron-neutrino scattering • Glashow, Salam, Weinberg - 1963-1973 • Nobel Prize (1979): • "for their contributions to the theory of the unified weak and electromagnetic interaction between elementary particles, including, inter alia, the prediction of the weak neutral current". Craig Roberts: Truncations in DSE-QCD (114p) • Electromagnetism • Quantum electrodynamics, 1946-1950 • Feynman, Schwinger, Tomonaga • Nobel Prize (1965): "for their fundamental work in quantum electrodynamics, with deep-ploughing consequences for the physics of elementary particles".

7. Standard Model- The Pieces • Politzer, Gross and Wilczek – 1973-1974 • Quantum Chromodynamics – QCD • Nobel Prize (2004): • "for the discovery of asymptotic freedom in the theory of the strong interaction". • NB. • Worth noting that the nature of 95% of the matter in the Universe is completely unknown Craig Roberts: Truncations in DSE-QCD (114p) • Strong interaction • Existence and composition of the vast bulk of visible matter in the Universe: • proton, neutron • the forces that form them and bind them to form nuclei • responsible for more than 98% of the visible matter in the Universe

8. Standard Model- Formulation Craig Roberts: Truncations in DSE-QCD (114p) • The Standard Model of Particle Physics is a local gauge field theory, which can be completely expressed in a very compact form • Lagrangian possesses SUc(3)xSUL(2)xUY(1) gauge symmetry • 19 parameters, which must be determined through comparison with experiment • Physics is an experimental science • SUL(2)xUY(1) represents the electroweak theory • 17 of the parameters are here, most of them tied to the Higgs boson, the model’s only fundamental scalar, which might now have been seen • This sector is essentially perturbative, so the parameters are readily determined • SUc(3) represents the strong interaction component • Just 2 of the parameters are intrinsic to SUc(3) – QCD • However, this is the really interesting sector because it is Nature’s only example of a truly and essentially nonperturbative fundamental theory • Impact of the 2 parameters is not fully known

9. 1900: 23 Questions “The Problems of Mathematics” Craig Roberts: Truncations in DSE-QCD (114p)

10. Top Open Questions in Physics Craig Roberts: Truncations in DSE-QCD (114p)

11. Excerpt from the top-10, or top-24, or … Craig Roberts: Truncations in DSE-QCD (114p) • Can we quantitatively understand quark and gluon confinement in quantum chromodynamics and the existence of a mass gap? • Quantum chromodynamics, or QCD, is the theory describing the strong nuclear force. • Carried by gluons, it binds quarks into particles like protons and neutrons. • Apparently, the tiny subparticles are permanently confined: one can't pull a quark or a gluon from a proton because the strong force gets stronger with distance and snaps them right back inside.

12. Quantum Chromodynamics Craig Roberts: Truncations in DSE-QCD (114p)

13. What is QCD? Craig Roberts: Truncations in DSE-QCD (114p) • 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

14. cf.Quantum Electrodynamics Craig Roberts: Truncations in DSE-QCD (114p) 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 !

15. 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 Craig Roberts: Truncations in DSE-QCD (114p)

16. What is QCD? 3-gluon vertex 4-gluon vertex Craig Roberts: Truncations in DSE-QCD (114p) • Novel feature of QCD • Tree-level interactions between gauge-bosons • O(αs) cross-section cf. O(αem4) in QED • One might guess that this is going to have a big impact • Elucidating part of that impact is the origin of the 2004 Nobel Prize to Politzer, and Gross & Wilczek

17. Running couplings Craig Roberts: Truncations in DSE-QCD (114p) • Quantum gauge-field theories are all typified by the feature that Nothing is Constant • Distribution of charge and mass, the number of particles, etc., indeed, all the things that quantum mechanics holds fixed, depend upon the wavelength of the tool used to measure them • particle number is not conserved in quantum field theory • Couplings and masses are renormalised via processes involving virtual-particles. Such effects make these quantities depend on the energy scale at which one observes them

18. QED cf. QCD? 5 x10-5 Add 3-gluon self-interaction gluon antiscreening fermion screening Craig Roberts: Truncations in DSE-QCD (114p) • 2004 Nobel Prize in Physics : Gross, Politzer and Wilczek

19. What is QCD? 0.5 0.4 ↔ 0.3 αs(r) 0.2 0.1 0.002fm 0.02fm 0.2fm Craig Roberts: Truncations in DSE-QCD (114p) This momentum-dependent coupling translates into a coupling that depends strongly on separation. Namely, the interaction between quarks, between gluons, and between quarks and gluons grows rapidly with separation Coupling is hugeat separations r = 0.2fm ≈ ⅟₄ rproton

20. 0.5 Confinement in QCD 0.4 0.3 αs(r) 0.2 0.1 0.002fm 0.02fm 0.2fm • The Confinement Hypothesis: • Colour-charged particles cannot be isolated and therefore cannot be directly observed. They clump together in colour-neutral bound-states • Confinementis an empirical fact. Craig Roberts: Truncations in DSE-QCD (114p) • A peculiar circumstance; viz., an interaction that becomes stronger as the participants try to separate • If coupling grows so strongly with separation, then • perhaps it is unbounded? • perhaps it would require an infinite amount of energy in order to extract a quark or gluon from the interior of a hadron?

21. 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: Truncations in DSE-QCD (114p) • Asymptotically free • Perturbation theory is valid and accurate tool at large-Q2 • Hence chiral limit is defined • Essentiallynonperturbative for Q2 < 2 GeV2

22. Millennium prize of $1,000,000 for proving that SUc(3) gauge theory is mathematically well-defined, which will necessarily prove or disprove the confinement conjecture Confinement? Craig Roberts: Truncations in DSE-QCD (114p)

23. The study of nonperturbative QCD is the puriew of … Hadron Physics Craig Roberts: Truncations in DSE-QCD (114p)

24. 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 Craig Roberts: Truncations in DSE-QCD (114p)

25. proton pion The structure of matter Hadron Theory Craig Roberts: Truncations in DSE-QCD (114p)

26. Problem: Nature chooses to build things, us included, from matter fields instead of gauge fields. Quarks & QCD In perturbation theory, quarks don’t seem to do much, just a little bit of very-normal charge screening. K.G. Wilson, formulated lattice-QCD in 1974 paper: “Confinement of quarks”. Wilson Loop Nobel Prize (1982): "for his theory for critical phenomena in connection with phase transitions". Craig Roberts: Truncations in DSE-QCD (114p) • Quarks are the problem with QCD • Pure-glue QCD is far simpler • Bosons are the only degrees of freedom • Bosons have a classical analogue – see Maxwell’s formulation of electrodynamics • Generating functional can be formulated as a discrete probability measure that is amenable to direct numerical simulation using Monte-Carlo methods • No perniciously nonlocal fermion determinant • Provides the Area Law & Linearly Rising Potential between static sources, so long identified with confinement

27. Contrast with Minkowksi metric: infinitely many four-vectors satisfy p2 = p0p0 – pipi= 0; • e.g., p= μ (1,0,0,1), μ any number Formulating QCD Euclidean Metric Craig Roberts: Truncations in DSE-QCD (114p) • In order to translate QCD into a computational problem, Wilson had to employ a Euclidean Metric x2 = 0 possible if and only if x=(0,0,0,0) because Euclidean-QCD action defines a probability measure, for which many numerical simulation algorithms are available. • However, working in Euclidean space is more than simply pragmatic: • Euclidean lattice field theory is currently a primary candidate for the rigorous definition of an interacting quantum field theory. • This relies on it being possible to define the generating functional via a proper limiting procedure.

28. Formulating Quantum Field Theory Euclidean Metric Craig Roberts: Truncations in DSE-QCD (114p) • Constructive Field Theory Perspectives: • Symanzik, K. (1963) in Local Quantum Theory (Academic, New York) edited by R. Jost. • Streater, R.F. and Wightman, A.S. (1980), PCT, Spin and Statistics, and All That (Addison-Wesley, Reading, Mass, 3rd edition). • Glimm, J. and Jaffee, A. (1981), Quantum Physics. A Functional Point of View (Springer-Verlag, New York). • Seiler, E. (1982), Gauge Theories as a Problem of Constructive Quantum Theory and Statistical Mechanics (Springer-Verlag, New York). • For some theorists, interested in essentially nonperturbativeQCD, this is always in the back of our minds

29. Formulating QCD Euclidean Metric Craig Roberts: Truncations in DSE-QCD (114p) • However, there is another very important reason to work in Euclidean space; viz., Owing to asymptotic freedom, all results of perturbation theory are strictly valid only at spacelike-momenta. • The set of spacelikemomenta correspond to a Euclidean vector space • The continuation to Minkowski space rests on many assumptions about Schwinger functions that are demonstrably valid only in perturbation theory.

30. Euclidean Metric& Wick Rotation Perturbative propagator singularity Perturbative propagator singularity Aside: QED is only defined perturbatively. It possesses an infrared stable fixed point; and masses and couplings are regularised and renormalised in the vicinity of k2=0. Wick rotation is always valid in this context. Craig Roberts: Truncations in DSE-QCD (114p) It is assumed that a Wick rotation is valid; namely, that QCD dynamics don’t nonperturbatively generate anything unnatural This is a brave assumption, which turns out to be very, very false in the case of coloured states. Hence, QCDMUST be defined in Euclidean space. The properties of the real-world are then determined only from a continuation of colour-singlet quantities.

31. The Problem with QCD Craig Roberts: Truncations in DSE-QCD (114p) • This is a RED FLAG in QCD because nothingelementaryis a colour singlet • Must somehow solve real-world problems • the spectrum and interactions of complex two- and three-body bound-states before returning to the real world • This is going to require a little bit of imagination and a very good toolbox: Dyson-Schwinger equations

32. Never before seen by the human eye Craig Roberts: Truncations in DSE-QCD (114p)

33. Nature’s strong messenger – Pion Craig Roberts: Truncations in DSE-QCD (114p) • 1947 – Pion discovered by Cecil Frank Powell • Studied tracks made by cosmic rays using photographic emulsion plates • Despite the fact that Cavendish Lab said method is incapable of “reliable and reproducible precision measurements.” • Mass measured in scattering ≈ 250-350 me

34. Nature’s strong messenger – Pion Craig Roberts: Truncations in DSE-QCD (114p) • The beginning of Particle Physics • Then came • Disentanglement of confusion between (1937) muon and pion – similar masses • Discovery of particles with “strangeness” (e.g., kaon1947-1953) • Subsequently, a complete spectrum of mesons and baryons with mass below ≈1 GeV • 28 states • Became clear that pion is “too light” - hadrons supposed to be heavy, yet …

35. Simple picture- Pion • Gell-Mann and Ne’eman: • Eightfold way(1961) – a picture based • on group theory: SU(3) • Subsequently, quark model – • where the u-, d-, s-quarks • became the basis vectors in the • fundamental representation • of SU(3) • Pion = • Two quantum-mechanical constituent-quarks - particle+antiparticle - • interacting via a potential Craig Roberts: Truncations in DSE-QCD (114p)

36. Some of the Light Mesons IG(JPC) 140 MeV 780 MeV Craig Roberts: Truncations in DSE-QCD (114p)

37. Modern Miraclesin Hadron Physics Craig Roberts: Truncations in DSE-QCD (114p) • proton = three constituent quarks • Mproton ≈ 1GeV • Therefore guess Mconstituent−quark ≈ ⅓ × GeV ≈ 350MeV • pion = constituent quark + constituent antiquark • Guess Mpion ≈ ⅔ × Mproton≈ 700MeV • WRONG . . . . . . . . . . . . . . . . . . . . . . Mpion = 140MeV • Rho-meson • Also constituent quark + constituent antiquark – just pion with spin of one constituent flipped • Mrho ≈ 770MeV ≈ 2 × Mconstituent−quark What is “wrong” with the pion?

38. Dichotomy of the pion Craig Roberts: Truncations in DSE-QCD (114p) • 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:

39. Dichotomy of the pionGoldstone mode and bound-state HIGHLY NONTRIVIAL Impossible in quantum mechanics Only possible in asymptotically-free gauge theories Craig Roberts: Truncations in DSE-QCD (114p) • 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.

40. Chiral QCD mt = 40,000 mu Why? Craig Roberts: Truncations in DSE-QCD (114p) • Current-quark masses • External paramaters in QCD • Generated by the Higgs boson, within the Standard Model • Raises more questions than it answers

41. Chiral Symmetry Craig Roberts: Truncations in DSE-QCD (114p) • Interacting gauge theories, in which it makes sense to speak of massless fermions, have a nonperturbativechiral symmetry • A related concept is Helicity, which is the projection of a particle’s spin, J, onto it’s direction of motion: • For a massless particle, helicity is a Lorentz-invariant spin-observable λ = ±; i.e., it’s parallel or antiparallel to the direction of motion • Obvious: • massless particles travel at speed of light • hence no observer can overtake the particle and thereby view its momentum as having changed sign

42. Chiral Symmetry Craig Roberts: Truncations in DSE-QCD (114p) • Chirality operator is γ5 • Chiral transformation: Ψ(x) → exp(i γ5θ) Ψ(x) • Chiral rotation through θ = ⅟₄ π • Composite particles: JP=+ ↔ JP=- • Equivalent to the operation of parity conjugation • Therefore, a prediction of chiral symmetry is the existence of degenerate parity partners in the theory’s spectrum

43. Chiral Symmetry Craig Roberts: Truncations in DSE-QCD (114p) • Perturbative QCD: u- & d- quarks are very light mu /md≈ 0.5 & md≈ 4MeV (a generation of high-energy experiments) H. Leutwyler, 0911.1416 [hep-ph] • However, splitting between parity partners is greater-than 100-times this mass-scale; e.g.,

44. Dynamical Chiral Symmetry Breaking Craig D Roberts John D Roberts Craig Roberts: Truncations in DSE-QCD (114p) • Something is happening in QCD • some inherent dynamical effect is dramatically changing the pattern by which the Lagrangian’schiral symmetry is expressed • Qualitatively different from spontaneous symmetry breaking aka the Higgs mechanism • Nothing is added to the theory • Have only fermions & gauge-bosons Yet, the mass-operator generated by the theory produces a spectrum with no sign of chiral symmetry

45. 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: Truncations in DSE-QCD (114p) • 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.

46. Dyson-SchwingerEquations • Dyson (1949) & Schwinger (1951) . . . One can derive a system of coupled integral equations relating all the Green functions for a theory, one to another. • Gap equation: • fermion self energy • gauge-boson propagator • fermion-gauge-boson vertex • These are nonperturbative equivalents in quantum field theory to the Lagrange equations of motion. • Essential in simplifying the general proof of renormalisability of gauge field theories. Craig Roberts: Truncations in DSE-QCD (114p)

47. 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: Truncations in DSE-QCD (114p) • 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?

48. Mass from Nothing?!Perturbation Theory Craig Roberts: Truncations in DSE-QCD (114p) • QCD is asymptotically-free (2004 Nobel Prize) • Chiral-limit is well-defined; i.e., one can truly speak of a massless quark. • NB. This is nonperturbativelyimpossible in QED. • Dressed-quark propagator: • Weak coupling expansion of gap equation yields every diagram in perturbation theory • In perturbation theory: If m=0, then M(p2)=0 Start with no mass, Always have no mass.

49. Craig D Roberts John D Roberts Dynamical Chiral Symmetry Breaking Craig Roberts: Truncations in DSE-QCD (114p)

50. Nambu—Jona-Lasinio Model Craig Roberts: Truncations in DSE-QCD (114p) Recall the gap equation NJL gap equation