Dyson-Schwinger Equations - PowerPoint Presentation

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Dyson-Schwinger Equations & Continuum QCD

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

StudentsEarly-career scientists

Published collaborations in 2010/2011

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Recommended readingR.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. 2991-3044C.D. Roberts , “Hadron Properties and Dyson-Schwinger Equations,”  arXiv:0712.0633 [nucl-th], Prog. Part. Nucl. Phys. 61

 (2008) pp. 50-65 C.D. Roberts, M.S. Bhagwat, A. Höll and S.V. Wright, “Aspects of Hadron Physics,” Eur. Phys. J. Special Topics 140 (2007) pp. 53-116A. Höll, C.D. Roberts and S.V. Wright, nucl-th/0601071, “Hadron Physics and Dyson-Schwinger Equations” (103 pages)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 (2002): “Primer for Quantum Field Theory in Hadron Physics” C. D. Roberts and S. M. Schmidt, “Dyson-Schwinger equations: Density, temperature and continuum strong QCD,” Prog. Part. Nucl. Phys. 45 (2000) S1C. D. Roberts and A. G. Williams,“Dyson-Schwinger equations and their application to hadronic physics,” Prog. Part. Nucl. Phys. 33 (1994) 477

DSFdB2011, IRMA France, 27 June - 1 July. 62pgs

Craig Roberts: Dyson-Schwinger Equations and Continuum QCD, I2

Introductory-level presentationsSlide3

DSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I3

Standard Model of Particle PhysicsSlide4

Standard Model- HistoryIn the early 20th Century, the only matter particles known to exist were the proton, neutron, and electron. DSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I4With 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. Slide5

Standard Model- The Pieces ElectromagnetismQuantum electrodynamics, 1946-1950Feynman, Schwinger, TomonagaNobel Prize (1965): "for their fundamental work in quantum electrodynamics, with deep-ploughing consequences for the physics of elementary particles".DSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I5

Weak interactionRadioactive decays, parity-violating decays, electron-neutrino scatteringGlashow, Salam, Weinberg - 1963-1973Nobel 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". Slide6

Standard Model- The Pieces Strong interactionExistence and composition of the vast bulk of visible matter in the Universe: proton, neutron the forces that form them and bind them to form nucleiresponsible for more than 98% of the visible matter in the UniverseDSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I6

Politzer, Gross and Wilczek – 1973-1974 Quantum Chromodynamics – QCDNobel Prize (2004): "for the discovery of asymptotic freedom in the theory of the strong interaction". NB. Perhaps worth noting that the nature of 95% of the matter in the Universe is completely unknownSlide7

Standard Model- FormulationDSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I7The 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 symmetry19 parameters, which must be determined through comparison with experimentPhysics is an experimental scienceSUL(2)xUY(1) represents the electroweak theory17 of the parameters are here, most of them tied to the Higgs boson, the model’s only fundamental scalar, which has never been seenThis sector is essentially perturbative, so the parameters are readily determinedSUc(3) represents the strong interaction component Just 2 of the parameters are intrinsic to SUc(3) – QCDHowever, 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 knownSlide8

Standard Model- FormulationKnown particle content of the Standard Model Higgs boson has not yet been foundDiscovery of the Higgs boson is one of the primary missions of the Large Hadron Collider Large Hadron ColliderLHCConstruction cost of $7 billionAccelerate particles to almost the speed of light, in two parallel beams in a 27km tunnel 175m underground, before colliding them at interaction pointsDuring a ten hour experiment , each beam will travel 10-billion km; i.e., almost 100-times the earth-sun distanceThe energy of each collision will reach 14 TeV (14 x 1012 eV)DSFdB2011, IRMA France, 27 June - 1 July. 62pgs

Craig Roberts: Dyson-Schwinger Equations and Continuum QCD, I8Slide9

Standard Model- FormulationVery compact expression of the fundamental interactions that govern the composition of the bulk of known matter in the UniverseThis is the most important part; viz., gauge-boson self-interaction in QCDResponsible for 98% of visible matter in the UniverseQCD will be my primary focusDSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I

9Slide10

Standard Model- Complete?There are certainly phenomena Beyond the Standard ModelNeutrinos have mass, which is not true within the Standard ModelEmpirical evidence: νe ↔ νμ, ντ … neutrino flavour is not a constant of motionThe first experiment to detect the effects of neutrino oscillations was Ray Davis's Homestake Experiment in the late 1960s, which observed a deficit in the flux of solar neutrinos νeVerified and quantified in experiments at the Sudbury Neutrino ObservatoryDSFdB2011, IRMA France, 27 June - 1 July. 62pgs

Craig Roberts: Dyson-Schwinger Equations and Continuum QCD, I10A number of experimental hints and, almost literally, innumerably many theoretical speculations about other phenomenaSlide11

Top Open Questions in PhysicsDSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I11Slide12

Excerpts from the top-10, or top-24, or … What is dark matter? There seems to be a halo of mysterious invisible material engulfing galaxies, which is commonly referred to as dark matter. Existence of dark (=invisible) matter is inferred from the observation of its gravitational pull, which causes the stars in the outer regions of a galaxy to orbit faster than they would if there was only visible matter present. Another indication is that we see galaxies in our own local cluster moving toward each other.What is dark energy? The discovery of dark energy goes back to 1998. A group of scientists had recorded several dozen supernovae, including some so distant that their light had started to travel toward Earth when the universe was only a fraction of its present age. Contrary to their expectation, the scientists found that the expansion of the universe is not slowing, but accelerating.DSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I12Slide13

Excerpts from the top-10, or top-24, or … What is the lifetime of the proton and how do we understand it?  It used to be considered that protons, unlike, say, neutrons, live forever, never decaying into smaller pieces. Then in the 1970's, theorists realized that their candidates for a grand unified theory, merging all the forces except gravity, implied that protons must be unstable. Wait long enough and, very occasionally, one should break down. Must Grand Unification work this way?What physics explains the enormous disparity between the gravitational scale and the typical mass scale of the elementary particles?  In other words, why is gravity so much weaker than the other forces, like electromagnetism? A magnet can pick up a paper clip even though the gravity of the whole earth is pulling back on the other end.DSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I13Slide14

Excerpts from the top-10, or top-24, or … 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. According to the theory, the tiny subparticles are permanently confined. You 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.DSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I14Slide15

DSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I15

Quantum ChromodynamicsSlide16

What is QCD? Lagrangian of QCDG = gluon fieldsΨ = quark fieldsThe key to complexity in QCD … gluon field strength tensorGenerates gluon self-interactions, whose consequences are quite extraordinary

DSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I16Slide17

QED is the archetypal gauge field theoryPerturbatively simple but nonperturbatively undefinedChracteristic 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 !cf.Quantum ElectrodynamicsDSFdB2011, IRMA France, 27 June - 1 July. 62pgs

Craig Roberts: Dyson-Schwinger Equations and Continuum QCD, I17Slide18

Relativistic Quantum Gauge Field Theory:Interactions mediated by vector boson exchangeVector bosons are perturbatively-massless

Similar interaction in QEDSpecial feature of QCD – gluon self-interactionsWhat is QCD?

Craig Roberts: Dyson-Schwinger Equations and Continuum QCD, I

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3-gluon vertex

4-gluon vertex

DSFdB2011, IRMA France, 27 June - 1 July. 62pgsSlide19

What is QCD? Novel feature of QCDTree-level interactions between gauge-bosonsO(αs) cross-section cf. O(αem4) in QEDOne might guess that this is going to have a big impactElucidating that impact is the origin of the 2004 Nobel Prize to Politzer,

and Gross & WilczekDSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I193-gluon vertex4-gluon vertexSlide20

Running couplingsQuantum gauge-field theories are all typified by the feature that Nothing is ConstantDistribution 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 themparticle number is not conserved in quantum field theoryCouplings and masses are renormalised via processes involving virtual-particles. Such effects make these quantities depend on the energy scale at which one observes themDSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I20Slide21

QED cf. QCD? Craig Roberts: Dyson-Schwinger Equations and Continuum QCD, I212004 Nobel Prize in Physics : Gross, Politzer and Wilczek

fermionscreeninggluonantiscreening

DSFdB2011, IRMA France, 27 June - 1 July. 62pgs

Add 3-gluon self-interaction

5 x10

-5Slide22

What is QCD?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 separationCoupling is huge at separations r = 0.2fm ≈ ⅟₄ rprotonDSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I

22↔0.002fm 0.02fm0.2fmαs(r)0.10.20.30.40.5Slide23

Confinement in QCD A peculiar circumstance; viz., an interaction that becomes stronger as the participants try to separateIf coupling grows so strongly with separation, thenperhaps 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?DSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I23

0.002fm 0.02fm0.2fmαs(r)0.10.20.30.40.5The Confinement Hypothesis: Colour

-charged particles cannot be isolated and therefore cannot be directly observed. They clump together in

colour-neutral bound-states

This is hitherto an empirical fact.Slide24

What is the interaction throughout more than 98% of the proton’s volume?

The Problem with QCD DSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I24Perhaps?!What we know unambiguously …

Is that we know too little!Slide25

Strong-interaction: QCDAsymptotically freePerturbation theory is valid and accurate tool at large-Q2 Hence chiral limit is definedEssentially nonperturbative for Q2 < 2 GeV

2DSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I25 Nature’s only example of truly nonperturbative, fundamental theory A-priori, no idea as to what such a theory can produceSlide26

Confinement?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

DSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I26Slide27

Hadron PhysicsThe study of

nonperturbative QCD is the puriew of …DSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I27Slide28

HadronsHadron: 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) +

2

on

First Known Use: 1962

Baryon:

hadron

with half-integer-spin

Meson:

hadron

with integer-spin

DSFdB2011, IRMA France, 27 June - 1 July. 62pgs

Craig Roberts:

Dyson-Schwinger Equations and Continuum QCD, I

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Hadron PhysicsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I29“Hadron physics is unique at the cutting edge of modern science because Nature has provided us with just one instance of a fundamental strongly-interacting theory; i.e., Quantum Chromodynamics (QCD). The community of science has never before confronted such a challenge as solving this theory.”

DSFdB2011, IRMA France, 27 June - 1 July. 62pgsSlide30

Nuclear Science Advisory Council 2007 – Long Range PlanCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I30 “A central goal of (the DOE Office of ) Nuclear Physics is to understand the structure and properties of protons and neutrons, and ultimately atomic nuclei, in terms of the quarks and gluons of QCD.”DSFdB2011, IRMA France, 27 June - 1 July. 62pgs

Internationally, this is an approximately $1-billion/year effort in experiment and theory, with roughly $375-million/year in the USA. Roughly 90% of these funds are spent on experiment$1-billion/year is the order of the operating budget of CERNSlide31

FacilitiesDSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I31Slide32

FacilitiesQCD MachinesChinaBeijing Electron-Positron ColliderGermanyCOSY (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 2015 (perhaps)JapanJ-PARC (Japan Proton Accelerator Research Complex), under construction in Tokai-Mura, 150km NE of Tokyo.

New generation experiments to begin toward end-2012KEK: Tsukuba, Belle CollaborationSwitzerland (CERN)Large Hadron Collider: ALICE Detector “Understanding deconfinement and chiral-symmetry restoration”DSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I32Slide33

FacilitiesQCD MachinesUSAThomas Jefferson National Accelerator Facility, Newport News, Virginia Nature of cold hadronic matter Upgrade underway Construction cost $310-million New generation experiments in 2016Relativistic Heavy Ion Collider, Brookhaven National Laboratory, Long Island, New York Strong phase transition, 10μs after Big BangDSFdB2011, IRMA France, 27 June - 1 July. 62pgs

Craig Roberts: Dyson-Schwinger Equations and Continuum QCD, I33A three dimensional view of the calculated particle paths resulting from collisions occurring within RHIC's STAR detectorSlide34

Relativistic Quantum Field TheoryA theoretical understanding of the phenomena of Hadron Physics requires the use of the full machinery of relativistic quantum field theory. Relativistic quantum field theory is the ONLY way to reconcile quantum mechanics with special relativity.Relativistic quantum field theory is based on the relativistic quantum mechanics of Dirac.Unification of special relativity (Poincaré covariance) and quantum mechanics took some time. Questions still remain as to a practical implementation of an Hamiltonian formulation of the relativistic quantum mechanics of interacting systems.Poincaré group has ten generators: six associated with the Lorentz transformations (rotations and boosts) four associated with translationsQuantum mechanics describes the time evolution of a system with interactions. That evolution is generated by the Hamiltonian.DSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I

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Relativistic Quantum Field TheoryRelativistic quantum mechanics predicts the existence of antiparticles; i.e., the equations of relativistic quantum mechanics admit negative energy solutions. However, once one allows for particles with negative energy, then particle number conservation is lost: Esystem = Esystem + (Ep1 + Eanti-p1 ) + . . . ad infinitumThis is a fundamental problem for relativistic quantum mechanics – Few particle systems can be studied in relativistic quantum mechanics but the study of (infinitely) many bodies is difficult. No general theory currently exists.This feature entails that, if a theory is formulated with an interacting Hamiltonian, then boosts will fail to commute with the Hamiltonian. Hence, the state vector calculated in one momentum frame will not be kinematically related to the state in another frame. That makes a new calculation necessary in every frame.DSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts:

Dyson-Schwinger Equations and Continuum QCD, I35Slide36

Relativistic Quantum Field TheoryHence the discussion of scattering, which takes a state of momentum p to another state with momentum p′, is problematic. (See, e.g., B.D. Keister and W.N. Polyzou (1991), “Relativistic Hamiltonian dynamics in nuclear and particle physics,” Adv. Nucl. Phys. 20, 225.)Relativistic quantum field theory is an answer. The fundamental entities are fields, which can simultaneously represent an uncountable infinity of particles; Thus, the nonconservation of particle number is not a problem. This is crucial because key observable phenomena in hadron physics are essentially connected with the existence of virtual particles.Relativistic quantum field theory has its own problems, however. The question of whether a given relativistic quantum field theory is rigorously well defined is unsolved.

DSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I36Slide37

Relativistic Quantum Field TheoryAll relativistic quantum field theories admit analysis in perturbation theory. Perturbative renormalisation is a well-defined procedure and has long been used in Quantum Electrodynamics (QED) and Quantum Chromodynamics (QCD).A rigorous definition of a theory, however, means proving that the theory makes sense nonperturbatively. This is equivalent to proving that all the theory’s renormalisation constants are nonperturbatively well-behaved.Hadron Physics involves QCD. While it makes excellent sense perturbatively, it is not known to be a rigorously well-defined theory. Hence it cannot truly be said to be THE theory of the strong interaction (hadron physics).Nevertheless, physics does not wait on mathematics. Physicists make assumptions and explore their consequences. Practitioners assume that QCD is (somehow) well-defined and follow where that leads us.DSFdB2011, IRMA France, 27 June - 1 July. 62pgs

Craig Roberts: Dyson-Schwinger Equations and Continuum QCD, I37Slide38

Relativistic Quantum Field TheoryExperiment’s task: explore and map the hadron physics landscape with well-understood probes, such as the electron at JLab and Mainz.Theory’s task: employ established mathematical tools, and refine and invent others in order to use the Lagangian of QCD to predict what should be observable real-world phenomena.A key aim of the worlds’ hadron physics programmes in experiment & theory: determine whether there are any contradictions with what we can truly prove in QCD. Hitherto, there are none. But that doesn’t mean there are no puzzles!Interplay between Experiment and Theory is the engine of discovery and progress. The Discovery Potential of both is high. Much learnt in the last five years.These lectures will provide a perspective on the meaning of these discoveriesFurthermore, I expect that many surprises remain in Hadron Physics.

DSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I38Slide39

Quarks & QCDQuarks are the problem with QCDPure-glue QCD is far simplerBosons are the only degrees of freedomBosons have a classical analogue – see Maxwell’s formulation of electrodynamicsGenerating functional can be formulated as a discrete probability measure that is amenable to direct numerical simulation using Monte-Carlo methodsNo perniciously nonlocal fermion determinantProvides the Area Law & Linearly Rising Potential

between static sources, so long identified with confinementDSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I39K.G. Wilson, formulated lattice-QCD in 1974 paper: “Confinement of quarks”. Wilson LoopNobel Prize (1982): "for his theory for critical phenomena in connection with phase transitions".Problem: Nature chooses to build things, us included, from matter fields instead of gauge fields. In perturbation theory, quarks don’t seem to do much, just a little bit of very-normal charge screening.Slide40

Formulating QCD Euclidean MetricIn order to translate QCD into a computational problem, Wilson had to employ a Euclidean Metric because the 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.DSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I40Slide41

Formulating QCD Euclidean MetricThe moments of the measure; i.e., “vacuum expectation values” of the fields, are the n-point Schwinger functions; and the quantum field theory is completely determined once all its Schwinger functions are known. The time-ordered Green functions of the associated Minkowski space theory can be obtained in a formally well-defined fashion from the Schwinger functions. This is all

formally true.DSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I41Slide42

Formulating Quantum Field Theory Euclidean MetricConstructive 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 nonperturbative Q

CD, this is always in the back of our mindsDSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I42Slide43

Formulating QCD Euclidean MetricHowever, 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 spacelike momenta correspond to a Euclidean vector spaceThe continuation to Minkowski space rests on many assumptions about Schwinger functions that are demonstrably valid only in perturbation theory. DSFdB2011, IRMA France, 27 June - 1 July. 62pgs

Craig Roberts: Dyson-Schwinger Equations and Continuum QCD, I43Slide44

Euclidean Metric& Wick RotationIt is assumed that a Wick rotation is valid; namely, that QCD dynamics don’t nonperturbatively generate anything unnaturalThis is a brave assumption, which turns out to be very, very false in the case of coloured states.Hence, QCD MUST be defined in Euclidean space.The properties of the real-world are then determined only from a continuation of colour

-singlet quantities.DSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I44Perturbative propagatorsingularityPerturbative propagatorsingularityAside: 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.Slide45

The Problem with QCD This is a RED FLAG in QCD because nothing elementary is a colour singletMust somehow solve real-world problemsthe spectrum and interactions of complex two- and three-body bound-states before returning to the real worldThis is going to require a little bit of imagination and a very good toolbox:

Dyson-Schwinger equationsDSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I45Slide46

Euclidean Metric ConventionsDSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I46Slide47

Euclidean Transcription FormulaeDSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I47It is possible to derive every equation of Euclidean QCD by assuming certain analytic properties of the integrands. However, the derivations can be sidestepped using the following transcription rules

:Slide48

Hadron TheoryThe structure of matter

DSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I48pionprotonSlide49

Quarks and Nuclear PhysicsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I49Standard Model of Particle Physics: Six quark flavoursReal World Normal matter – only two light-quark

flavours are active Or, perhaps, threeFor numerous good reasons, much research also focuses on accessible heavy-quarks Nevertheless, I will mainly focus on the light-quarks; i.e., u & d.

DSFdB2011, IRMA France, 27 June - 1 July. 62pgsSlide50

Nature’s strong messenger – PionCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I50DSFdB2011, IRMA France, 27 June - 1 July. 62pgs1947 – 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 meSlide51

Nature’s strong messenger – PionCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I51DSFdB2011, IRMA France, 27 June - 1 July. 62pgsThe beginning of Particle PhysicsThen came Disentanglement of confusion between (1937) muon

and pion – similar massesDiscovery of particles with “strangeness” (e.g., kaon1947-1953)Subsequently, a complete spectrum of mesons and baryons with mass below ≈1 GeV28 statesBecame clear that pion is “too light” - hadrons supposed to be heavy, yet …π140 MeVρ

780

MeV

P

940

MeVSlide52

Simple picture- PionCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I52 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 potentialDSFdB2011, IRMA France, 27 June - 1 July. 62pgsSlide53

Some of the Light MesonsDSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I53140 MeV780 MeVI

G(JPC)Slide54

Modern Miraclesin Hadron PhysicsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I54proton = three constituent quarksMproton ≈ 1GeVTherefore guess Mconstituent−quark ≈ ⅓ ×

GeV ≈ 350MeVpion = constituent quark + constituent antiquarkGuess Mpion ≈ ⅔ × Mproton ≈ 700MeVWRONG . . . . . . . . . . . . . . . . . . . . . . Mpion = 140MeVRho-mesonAlso constituent quark + constituent antiquark – just pion with spin of one constituent flippedMrho ≈ 770MeV ≈ 2 × Mconstituent−quark What is “wrong” with the pion?

DSFdB2011, IRMA France, 27 June - 1 July. 62pgsSlide55

Dichotomy of the pionCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I55How 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 mistakeHowever:

current-algebra (1968) This is impossible in quantum mechanics, for which one always finds: DSFdB2011, IRMA France, 27 June - 1 July. 62pgsSlide56

Dichotomy of the pionGoldstone mode and bound-stateThe correct understanding of pion observables; e.g. mass, decay constant and form factors, requires an approach to contain awell-defined and valid chiral limit;and an accurate realisation of dynamical chiral symmetry breaking.DSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I

56HIGHLY NONTRIVIALImpossible in quantum mechanicsOnly possible in asymptotically-free gauge theoriesSlide57

Chiral QCDCurrent-quark masses External paramaters in QCDGenerated by the Higgs boson, within the Standard ModelRaises more questions than it answersDSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I57

mt = 40,000 muWhy?Slide58

Chiral SymmetryInteracting gauge theories, in which it makes sense to speak of massless fermions, have a nonperturbative chiral symmetryA 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 motionObvious: massless

particles travel at speed of lighthence no observer can overtake the particle and thereby view its momentum as having changed signDSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I58Slide59

Chiral SymmetryChirality operator is γ5 Chiral transformation: Ψ(x) → exp(i γ5 θ) Ψ(x)Chiral rotation through θ = ⅟₄ πComposite 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 spectrumDSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I59Slide60

Chiral SymmetryCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I60Perturbative QCD: u- & d- quarks are very light mu /md ≈ 0.5 & md

≈ 4 MeV (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.,DSFdB2011, IRMA France, 27 June - 1 July. 62pgsJP⅟₂+ (p)⅟₂-Mass940 MeV1535 MeVSlide61

Dynamical Chiral Symmetry BreakingSomething is happening in QCDsome inherent dynamical effect is dramatically changing the pattern by which the Lagrangian’s chiral symmetry is expressedQualitatively different from spontaneous symmetry breaking aka the Higgs mechanismNothing is added to the theoryHave only fermions & gauge-bosons

Yet, the mass-operatorgenerated by the theory produces a spectrumwith no sign of chiral symmetryDSFdB2011, IRMA France, 27 June - 1 July. 62pgsCraig Roberts: Dyson-Schwinger Equations and Continuum QCD, I61Craig D RobertsJohn D RobertsSlide62

QCD’s ChallengesDynamical 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 Lagrangian Yet they are the dominant determining characteristics of real-world QCD.QCD – Complex behaviour arises from apparently simple rules.Craig Roberts: Dyson-Schwinger Equations and Continuum QCD, I62 Quark and Gluon ConfinementNo matter how hard one strikes the proton, one cannot liberate an individual quark or gluon

Understand emergent phenomena

DSFdB2011, IRMA France, 27 June - 1 July. 62pgs