Craig Roberts Physics Division 1 2 3 4

Craig Roberts Physics Division 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 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) Published Kun-lun WANG (PKU); collaborations: Lei CHANG (ANL, FZJ, PKU); Huan CHEN (BIHEP); 2010 -present Ian CLOËT (UAdelaide); Bruno EL-BENNICH (São Paulo); Mario PITSCHMANN (ANL & UW-Mad) David WILSON (ANL); Adnan BASHIR (U Michoácan); Students Stan BRODSKY (SLAC); Early-career Gastão KREIN (São Paulo) scientists 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)

Introductory-level presentations Recommended reading Ø C. D. Roberts, “Strong QCD and Dyson-Schwinger Equations, ” ar. Xiv: 1203. 5341 [nucl-th]. Notes based on 5 lectures to the conference on “Dyson-Schwinger Equations & Faà di Bruno Hopf Algebras in Physics and Combinatorics (DSFd. B 2011), ” Institut de Recherche Mathématique Avancée, 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) Ø 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 Ø A. Höll, C. D. Roberts and S. V. Wright, nucl-th/0601071, “Hadron Physics and Dyson-Schwinger Equations” (103 pages) Ø C. D. Roberts (2002): “Primer for Quantum Field Theory in Hadron Physics” (http: //www. phy. anl. gov/theory/ztfr/Lec. Notes. pdf) Ø C. D. Roberts and A. G. Williams, “Dyson-Schwinger equations and their application to hadronic physics, ” Prog. Part. Nucl. Phys. 33 (1994) 477 Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 2

Research-level presentations Recommended reading Ø 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, ” ar. Xiv: 1201. 3366 [nucl-th], Commun. Theor. Phys. 58 (2012) pp. 79 -134 Ø R. J. Holt and C. D. Roberts, “Distribution Functions of the Nucleon and Pion in the Valence Region, ” ar. Xiv: 1002. 4666 [nucl-th], Rev. Mod. Phys. 82 (2010) pp. 2991 -3044 Ø C. D. Roberts , “Hadron Properties and Dyson-Schwinger Equations, ” ar. Xiv: 0712. 0633 [nucl-th], Prog. Part. Nucl. Phys. 61 (2008) pp. 50 -65 Ø P. Maris and C. D. Roberts, “Dyson-Schwinger equations: A tool for hadron physics, ” Int. J. Mod. Phys. E 12, 297 (2003) Ø C. D. Roberts and S. M. Schmidt, “Dyson-Schwinger equations: Density, temperature and continuum strong QCD, ” Prog. Part. Nucl. Phys. 45 (2000) S 1 Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 3

Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 4

Ø In the early 20 th Century, the only matter particles known to exist were the proton, neutron, and electron. Standard Model - History Ø With the advent of cosmic ray science and particle accelerators, numerous additional particles were discovered: o muon (1937), pion (1947), kaon (1947), Roper resonance (1963), … Ø By the mid-1960 s, it was apparent that not all the particles could be fundamental. o A new paradigm was necessary. Ø Gell-Mann's and Zweig's constituent-quark theory (1964) was a critical step forward. o Gell-Mann, Nobel Prize 1969: "for his contributions and discoveries concerning the classification of elementary particles and their interactions". Ø Over the more than forty intervening years, theory now called the Standard Model of Particle Physics has passed almost all tests. Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 5

Standard Model - The Pieces Ø 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". Ø Weak interaction – Radioactive decays, parity-violating decays, electron-neutrino scattering – Glashow, Salam, Weinberg - 1963 -1973 • Nobel Prize (1979): "for their contributions to 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 (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 6

Standard Model - The Pieces Ø 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 – Politzer, Gross and Wilczek – 1973 -1974 Quantum Chromodynamics – QCD • Nobel Prize (2004): "for the discovery of asymptotic freedom in theory of the strong interaction". Ø NB. Worth noting that the nature of 95% of the matter in the Universe is completely unknown Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 7

Standard Model - Formulation Ø The Standard Model of Particle Physics is a local gauge field theory, which can be completely expressed in a very compact form Ø Lagrangian possesses SUc(3)x. SUL(2)x. UY(1) gauge symmetry – 19 parameters, which must be determined through comparison with experiment • Physics is an experimental science – SUL(2)x. UY(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 Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 8

Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 9

Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 10

Excerpt from the top-10, or top-24, or … Ø Can we quantitatively understand quark and gluon confinement in quantum chromodynamics and the existence of a mass gap? – Quantum chromodynamics, or QCD, is 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. Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 11

Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 12

What is QCD? Ø Lagrangian of QCD – G = gluon fields – Ψ = quark fields Ø The key to complexity in QCD … gluon field strength tensor Ø Generates gluon self-interactions, whose consequences are quite extraordinary Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 13

cf. Quantum Electrodynamics Ø QED is the archetypal gauge field theory Ø Perturbatively simple but nonperturbatively undefined Ø Chracteristic feature: Light-by-light scattering; i. e. , photon-photon interaction – leading-order contribution takes place at order α 4. Extremely small probability because α 4 ≈10 -9 ! Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 14

What is QCD? Relativistic Quantum Gauge Field Theory: Ø Interactions mediated by vector boson exchange Ø Vector bosons are perturbatively-massless 3 -gluon vertex Ø Similar interaction in QED Ø Special feature of QCD – gluon self-interactions 4 -gluon vertex Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 15

What is QCD? Ø Novel feature of QCD – Tree-level interactions between gauge-bosons – O(αs) cross-section cf. O(αem 4) in QED Ø One might guess that this is going to have a big impact Ø Elucidating part of that impact is the origin of the 2004 Nobel Prize to Politzer, and Gross & Wilczek 3 -gluon vertex 4 -gluon vertex Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 16

Running couplings Ø Quantum gauge-field theories are all typified by the feature that Nothing is Constant Ø 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 Ø 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 Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 17

QED cf. QCD? ü 2004 Nobel Prize in Physics : Gross, Politzer and Wilczek 5 x 10 -5 Add 3 -gluon self-interaction gluon antiscreening Craig Roberts: Truncations in DSE-QCD (114 p) fermion screening 8 -12/10/12: Math Aspects of Hadron Physics 18

What is QCD? Ø This momentum-dependent coupling translates into a coupling that depends strongly on separation. Ø Namely, the interaction between quarks, between gluons, and between quarks and gluons grows rapidly with separation Ø Coupling is huge at separations r = 0. 2 fm ≈ ⅟₄ rproton 0. 5 0. 4 0. 3 αs(r) ↔ 0. 2 0. 1 0. 002 fm 0. 02 fm 0. 2 fm Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 19

0. 5 Confinement in QCD 0. 4 0. 2 0. 1 0. 002 fm 0. 02 fm 0. 2 fm αs(r) 0. 3 Ø A peculiar circumstance; viz. , an interaction that becomes stronger as the participants try to separate Ø 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? Ø The Confinement Hypothesis: Colour-charged particles cannot be isolated and therefore cannot be directly observed. They clump together in colour-neutral boundstates Ø Confinement is an empirical fact. Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 20

Strong-interaction: QCD Ø Asymptotically free – Perturbation theory is valid and accurate tool at large-Q 2 – Hence chiral limit is defined Ø Essentially nonperturbative for Q 2 < 2 Ge. V 2 Ø Nature’s only example of truly nonperturbative, fundamental theory Ø A-priori, no idea as to what such a theory can produce Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 21

Millennium prize of $1, 000 for proving that SUc(3) gauge theory is mathematically welldefined, which will necessarily prove or disprove the confinement conjecture Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 22

The study of nonperturbative QCD is the puriew of … Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 23

Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 24

pion proton The structure of matter Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 25

Problem: Nature chooses to build things, us included, from matter fields Quarks instead of gauge fields. Ø Quarks are the problem with QCD Ø Pure-glue QCD is far simpler – Bosons are the only degrees of freedom & QCD In perturbation theory, quarks don’t seem to do much, just a little bit of very-normal charge screening. • 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 Ø Provides the Area Law & Linearly Rising Potential between static sources, so long identified with confinement Craig Roberts: Truncations in DSE-QCD (114 p) 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". 8 -12/10/12: Math Aspects of Hadron Physics 26

Contrast with Minkowksi metric: infinitely many four-vectors satisfy p 2 = p 0 p 0 – pipi = 0; e. g. , p= μ (1, 0, 0, 1), μ any number Formulating QCD Euclidean Metric Ø In order to translate QCD into a computational problem, Wilson had to employ a Euclidean Metric x 2 = 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. Ø 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. Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 27

Formulating Quantum Field Theory Euclidean Metric Ø 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, 3 rd 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). Ø For some theorists, interested in essentially nonperturbative QCD, this is always in the back of our minds Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 28

Formulating QCD Euclidean Metric Ø 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 spacelike momenta correspond to a Euclidean vector space Ø The continuation to Minkowski space rests on many assumptions about Schwinger functions that are demonstrably valid only in perturbation theory. Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 29

Euclidean Metric & Wick Rotation Ø It is assumed that a Wick rotation is valid; namely, that QCD dynamics don’t nonperturbatively generate Perturbative propagator singularity anything unnatural Ø This is a brave assumption, which turns out to be very, very false in the case of coloured states. Perturbative propagator Ø Hence, QCD MUST be defined in singularity Euclidean space. Ø The properties of the real-world are then determined only from a Aside: QED is only defined perturbatively. It possesses an infrared stable fixed point; and continuation of colour-singlet masses and couplings are regularised and renormalised in the vicinity of k 2=0. Wick quantities. Craig Roberts: Truncations in DSE-QCD (114 p) rotation is always valid in this context. 8 -12/10/12: Math Aspects of Hadron Physics 30

The Problem with QCD Ø This is a RED FLAG in QCD because nothing elementary is a colour singlet Ø Must somehow solve real-world problems – the spectrum and interactions of complex two- and three-body bound-states before returning to the real world Ø This is going to require a little bit of imagination and a very good toolbox: Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 31

Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 32

Nature’s strong messenger – Pion q 1947 – Pion discovered by Cecil Frank Powell q Studied tracks made by cosmic rays using photographic emulsion plates q Despite the fact that Cavendish Lab said method is incapable of “reliable and reproducible precision measurements. ” q Mass measured in scattering ≈ 250 -350 me Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 33

Nature’s strong messenger – Pion q The beginning of Particle Physics q Then came § Disentanglement of confusion between (1937) muon and pion – similar masses § Discovery of particles with “strangeness” (e. g. , kaon 1947 -1953) q Subsequently, a complete spectrum of mesons and baryons with mass below ≈1 Ge. V § 28 states π 140 Me. V ρ 780 Me. V q Became clear that P 940 Me. V pion is “too light” - hadrons supposed to be heavy, yet … Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 34

q 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) q Pion = Two quantum-mechanical constituent -quarks - particle+antiparticle - interacting via a potential Simple picture - Pion Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 35

Some of the Light Mesons IG(JPC) 140 Me. V 780 Me. V Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 36

Modern Miracles in Hadron Physics o proton = three constituent quarks • Mproton ≈ 1 Ge. V • Therefore guess Mconstituent−quark ≈ ⅓ × Ge. V ≈ 350 Me. V o pion = constituent quark + constituent antiquark • Guess Mpion ≈ ⅔ × Mproton ≈ 700 Me. V o WRONG. . . . . Mpion = 140 Me. V o Rho-meson • Also constituent quark + constituent antiquark – just pion with spin of one constituent flipped • Mrho ≈ 770 Me. V ≈ 2 × Mconstituent−quark What is “wrong” with the pion? Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 37

Dichotomy of the pion Ø How does one make an almost massless particle from two massive constituent-quarks? Ø Naturally, one could always tune a potential in quantum mechanics so that the ground-state is massless – but some are still making this mistake Ø However: current-algebra (1968) Ø This is impossible in quantum mechanics, for which one always finds: Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 38

Dichotomy of the pion Goldstone mode and bound-state Ø The correct understanding of pion observables; e. g. mass, decay constant and form factors, requires an approach to contain a – well-defined and valid chiral limit; – and an accurate realisation of dynamical chiral symmetry breaking. HIGHLY NONTRIVIAL Impossible in quantum mechanics Only possible in asymptotically-free gauge theories Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 39

Chiral QCD mt = 40, 000 mu Why? Ø Current-quark masses – External paramaters in QCD – Generated by the Higgs boson, within the Standard Model – Raises more questions than it answers Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 40

Chiral Symmetry Ø Interacting gauge theories, in which it makes sense to speak of massless fermions, have a nonperturbative chiral symmetry Ø A related concept is Helicity, which is the projection of a particle’s spin, J, onto it’s direction of motion: Ø For a massless particle, helicity is a Lorentz-invariant spinobservable 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 Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 41

Chiral Symmetry Ø Chirality operator is γ 5 – Chiral transformation: Ψ(x) → exp(i γ 5 θ) Ψ(x) – Chiral rotation through θ = ⅟₄ π • Composite particles: JP=+ ↔ JP= • Equivalent to the operation of parity conjugation Ø Therefore, a prediction of chiral symmetry is the existence of degenerate parity partners in theory’s spectrum Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 42

Chiral Symmetry Ø Perturbative QCD: u- & d- quarks are very light mu /md ≈ 0. 5 & md ≈ 4 Me. V (a generation of high-energy experiments) H. Leutwyler, 0911. 1416 [hep-ph] Ø However, splitting between parity partners is greater -than 100 -times this mass-scale; e. g. , JP Mass ⅟₂+ (p) 940 Me. V ⅟₂1535 Me. V Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 43

Dynamical Chiral Symmetry Breaking Ø Something is happening in QCD – some inherent dynamical effect is dramatically changing the pattern by which the Lagrangian’s chiral symmetry is expressed Ø Qualitatively different from spontaneous symmetry breaking aka the Higgs mechanism – Nothing is added to theory – Have only fermions & gauge-bosons Yet, the mass-operator generated by theory produces a spectrum with no sign of chiral symmetry Craig D Roberts John D Roberts Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 44

QCD’s Challenges Understand emergent phenomena Ø Quark and Gluon Confinement No matter how hard one strikes the proton, one cannot liberate an individual quark or gluon Ø Dynamical Chiral Symmetry Breaking Very unnatural pattern of bound state masses; e. g. , Lagrangian (p. QCD) quark mass is small but . . . no degeneracy between JP=+ and JP=− (parity partners) Ø Neither of these phenomena is apparent in QCD’s Lagrangian Yet they are the dominant determining characteristics of real-world QCD. Ø QCD – Complex behaviour arises from apparently simple rules. Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 45

Dyson-Schwinger Equations Ø 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: o fermion self energy o gauge-boson propagator o fermion-gauge-boson vertex Ø These are nonperturbative equivalents in quantum field theory to the Lagrange equations of motion. Ø Essential in simplifying the general proof of renormalisability of gauge field theories. Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 46

Dyson-Schwinger Equations Ø Well suited to Relativistic Quantum Field Theory Ø Simplest level: Generating Tool for Perturbation Theory . . . Materially Reduces Model. Dependence … Statement about long-range behaviour of quark-quark interaction Ø Non. Perturbative, Continuum approach to QCD Ø Hadrons as Composites of Quarks and Gluons Ø Qualitative and Quantitative Importance of: v Dynamical Chiral Symmetry Breaking – Generation of fermion mass from nothing v Quark & Gluon Confinement – Coloured objects not detected, Not detectable? ØApproach yields Schwinger functions; i. e. , propagators and vertices ØCross-Sections built from Schwinger Functions ØHence, method connects observables with long range behaviour of the running coupling ØExperiment ↔ Theory comparison leads to an understanding of long range behaviour of strong running-coupling Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 47

Mass from Nothing? ! Perturbation Theory Ø QCD is asymptotically-free (2004 Nobel Prize) v Chiral-limit is well-defined; i. e. , one can truly speak of a massless quark. v NB. This is nonperturbatively impossible in QED. Ø Dressed-quark propagator: Ø Weak coupling expansion of gap equation yields every diagram in perturbation theory Ø In perturbation theory: If m=0, then M(p 2)=0 Start with no mass, Always have no mass. Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 48

Craig D Roberts John D Roberts Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 49

Nambu—Jona-Lasinio Model Ø Recall the gap equation Ø NJL gap equation Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 50

Nambu—Jona-Lasinio Model Ø Multiply the NJL gap equation by (-iγ∙p); trace over Dirac indices: – Angular integral vanishes, therefore A(p 2) = 1. – This owes to the fact that the NJL model is defined by a four-fermion contact-interaction in configuration space, which entails a momentum -independent interaction in momentum space. Ø Simply take Dirac trace of NJL gap equation: – Integrand is p 2 -independent, therefore the only solution is B(p 2) = constant = M. Ø General form of the propagator for a fermion dressed by the NJL interaction: S(p) = 1/[ i γ∙p + M ] Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 51

Critical coupling for dynamical mass generation? NJL model & a mass gap? Ø Evaluate the integrals Ø Λ defines the model’s mass-scale. Henceforth set Λ = 1, then all other dimensioned quantities are given in units of this scale, in which case the gap equation can be written Ø Chiral limit, m=0 – Solutions? Ø Chiral limit, m=0 – Suppose, on the other hand that M≠ 0, and thus may be • One is obvious; viz. , M=0 cancelled This is the perturbative result • This nontrivial solution … start with no mass, end up with no mass can exist if-and-only-if one may satisfy Craig Roberts: Truncations in DSE-QCD (114 p) 3π2 m. G 2 = C(M 2, 1) 52 8 -12/10/12: Math Aspects of Hadron Physics

Critical coupling for dynamical mass generation! NJL model & a mass gap? Ø Can one satisfy 3π2 m. G 2 = C(M 2, 1) ? – C(M 2, 1) = 1 − M 2 ln [ 1 + 1/M 2 ] • Monotonically decreasing function of M • Maximum value at M = 0; viz. , C(M 2=0, 1) = 1 Ø Consequently, there is a solution iff 3π2 m. G 2 < 1 – Typical scale for hadron physics: Λ = 1 Ge. V • There is a M≠ 0 solution iff m. G 2 < (Λ/(3 π2)) = (0. 2 Ge. V)2 Ø Interaction strength is proportional to 1/m. G 2 – Hence, if interaction is strong enough, then one can start with no mass but end up with a massive, perhaps very massive fermion Dynamical Chiral Symmetry Breaking Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 53

Solution of gap equation Ø Weak coupling corresponds to m. G large, in which case M≈m Ø On the other hand, strong coupling; i. e. , m. G small, M>>m This is the defining characteristic of dynamical chiral symmetry breaking NJL Model Dynamical Mass Critical m. G=0. 186 Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 54

NJL Model and Confinement? Ø Confinement: no free-particle-like quarks Ø Fully-dressed NJL propagator Ø This is merely a free-particle-like propagator with a shifted mass p 2 + M 2 = 0 → Minkowski-space mass = M Ø Hence, whilst NJL model exhibits dynamical chiral symmetry breaking it does not confine. NJL-fermion still propagates as a plane wave Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 55

Munczek-Nemirovsky Model Ø Munczek, H. J. and Nemirovsky, A. M. (1983), “The Ground State q-q. bar Mass Spectrum In QCD, ” Phys. Rev. D 28, 181. Antithesis of NJL model; viz. , Ø Delta-function in momentum space NOT in configuration space. In this case, G sets the mass scale Ø MN Gap equation Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 56

MN Model’s Gap Equation Ø The gap equation yields the following pair of coupled, algebraic equations (set G = 1 Ge. V 2) Ø Consider the chiral limit form of the equation for B(p 2) – Obviously, one has the trivial solution B(p 2) = 0 – However, is there another? Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 57

MN model Ø The existence of a B(p 2) ≠ 0 solution; i. e. , a solution and DCSB that dynamically breaks chiral symmetry, requires (in units of G) p 2 A 2(p 2) + B 2(p 2) = 4 Ø Substituting this result into the equation for A(p 2) one finds A(p 2) – 1 = ½ A(p 2) → A(p 2) = 2, which in turn entails B(p 2) = 2 ( 1 – p 2 )½ Ø Physical requirement: quark self-energy is real on the domain of spacelike momenta → complete chiral limit solution Craig Roberts: Truncations in DSE-QCD (114 p) NB. Self energies are momentum-dependent because the interaction is momentum-dependent. Should expect the same in QCD. 8 -12/10/12: Math Aspects of Hadron Physics 58

MN Model and Confinement? Ø Solution we’ve found is continuous and defined for all p 2, even p 2 < 0; namely, timelike momenta Ø Examine the propagator’s denominator p 2 A 2(p 2) + B 2(p 2) = 4 This is greater-than zero for all p 2 … – There are no zeros – So, the propagator has no pole Ø This is nothing like a free-particle propagator. It can be interpreted as describing a confined degree-of-freedom Ø Note that, in addition there is no critical coupling: The nontrivial solution exists so long as G > 0. Ø Conjecture: All confining theories exhibit DCSB – NJL model demonstrates that converse is not true. Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 59

Massive solution in MN Model Ø In the chirally asymmetric case the gap equation yields Ø Second line is a quartic equation for B(p 2). Can be solved algebraically with four solutions, available in a closed form. Ø Only one solution has the correct p 2 → ∞ limit; viz. , B(p 2) → m. This is the unique physical solution. Ø NB. The equations and their solutions always have a smooth m → 0 limit, a result owing to the persistence of the DCSB solution. Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 60

Ø Large-s: M(s) ∼ m Ø Small-s: M(s) ≫ m This is the essential characteristic of DCSB Ø We will see that p 2 -dependent massfunctions are a quintessential feature of QCD. Ø No solution of s +M(s)2 = 0 → No plane-wave propagation Confinement? ! Munczek-Nemirovsky Dynamical Mass These two curves never cross: Confinement Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 61

What happens in the real world? Ø Strong-interaction: QCD – Asymptotically free – • Perturbation theory is valid and accurate tool at large-Q 2 & hence chiral limit is defined Essentially nonperturbative for Q 2 < 2 Ge. V 2 • Nature’s only example of truly nonperturbative, fundamental theory • A-priori, no idea as to what such a theory can produce Ø Possibilities? Essentially nonperturbative – G(0) < 1: M(s) ≡ 0 is only solution for m = 0. – G(0) ≥ 1: M(s) ≠ 0 is possible and energetically favoured: DCSB. – M(0) ≠ 0 is a new, dynamically generated mass-scale. If it’s large enough, can explain how a theory that is apparently massless (in the Lagrangian) possesses the spectrum of a massive theory. Perturbative domain Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 62

Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 63

Overview Ø Confinement and Dynamical Chiral Symmetry Breaking are Key Emergent Phenomena in QCD Ø Understanding requires Nonperturbative Solution of Fully-Fledged Relativistic Quantum Field Theory – Mathematics and Physics still far from being able to accomplish that Ø Confinement and DCSB are expressed in QCD’s propagators and vertices – Nonperturbative modifications should have observable consequences Ø Dyson-Schwinger Equations are a useful analytical and numerical tool for nonperturbative study of relativistic quantum field theory Ø Simple models (NJL) can exhibit DCSB – DCSB ⇒ Confinement Ø Simple models (MN) can exhibit Confinement – Confinement ⇒ DCSB Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 64

Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 65

Wilson Loop & the Area Law τ z Ø C is a closed curve in space, P is the path order operator Ø Now, place static (infinitely heavy) fermionic sources of colour charge at positions z 0=0 & z=½L Ø Then, evaluate <WC(z, τ)> as a functional integral over gauge-field configurations Ø In the strong-coupling limit, the result can be Linear potential obtained algebraically; viz. , <WC(z, τ)> = exp(-V(z) τ ) Craig Roberts: Truncations in DSE-QCD (114 p) σ = String tension where V(z) is the potential between the static sources, which behaves as V(z) = σ z 8 -12/10/12: Math Aspects of Hadron Physics 66

Wilson Loop & Area Law Ø Typical result from a numerical simulation of pure-glue QCD (hep-lat/0108008) Ø r 0 is the Sommer-parameter, which relates to the force between static quarks at intermediate distances. Ø The requirement r 02 F(r 0) = 1. 65 provides a connection between pure-glue QCD and potential models for mesons, and produces r 0 ≈ 0. 5 fm Dotted line: Bosonic-string model V(r) = σ r – π/(12 r) √σ = 1/(0. 85 r 0)=1/(0. 42 fm) = 470 Me. V Solid line: 3 -loop result in perturbation theory Breakdown at r = 0. 3 r 0 = 0. 15 fm Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 67

Ø Illustration in terms of Action – density, which is analogous to plotting the force: F(r) = σ – (π/12)(1/r 2) Ø It is pretty hard to overlook the flux tube between the static source and sink Ø Phenomenologists embedded in quantum mechanics and string theorists have been nourished by this result for many, many years. Flux Tube Models of Hadron Structure BUT … the Real World has light quarks … what then? ! Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 68

X Confinement Ø Quark and Gluon Confinement – No matter how hard one strikes the proton, or any other hadron, one cannot liberate an individual quark or gluon Ø 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 Ø Confinement entails quark-hadron duality; i. e. , that all observable consequences of QCD can, in principle, be computed using an hadronic basis. Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 69

G. Bali et al. , Po. S LAT 2005 (2006) 308 “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. ” Confinement Ø Infinitely heavy-quarks plus 2 flavours with mass = ms – Lattice spacing = 0. 083 fm – 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) Ø No flux tube between light-quarks Craig Roberts: Truncations in DSE-QCD (114 p) Bs anti-Bs 8 -12/10/12: Math Aspects of Hadron Physics 70

Confinement Ø Static-quark confinement is irrelevant to real-world QCD – There are no long-lived, very-massive quarks Ø Indeed, potential models are irrelevant to light-quark physics, something which should have been plain from the start: copious production of light particleantiparticle pairs ensures that a potential model description is meaningless for light-quarks in QCD Bs anti-Bs Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 71

Confinement Ø Confinement is expressed through a violent change in the analytic structure of propagators for coloured particles & can almost be read from a plot of a states’ dressed-propagator – Gribov (1978); Munczek (1983); Stingl (1984); Cahill (1989); Krein, Roberts & Williams (1992); Tandy (1994); … Confined particle Normal particle complex-P 2 timelike axis: P 2<0 o Real-axis mass-pole splits, moving into pair(s) of complex conjugate poles or branch points o Spectral density no longer positive semidefinite & hence state cannot exist in observable spectrum Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 72

Dressed-gluon propagator A. C. Aguilar et al. , Phys. Rev. D 80 (2009) 085018 Ø Gluon propagator satisfies a Dyson-Schwinger Equation Ø Plausible possibilities for the solution Ø DSE and lattice-QCD agree on the result – Confined gluon – IR-massive but UV-massless – m. G ≈ 2 -4 ΛQCD IR-massive but UV-massless, confined gluon perturbative, massless gluon massive , unconfined gluon Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 73

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. Ø Confinement can be related to the analytic properties of QCD's Schwinger functions. Ø Question of light-quark confinement can be translated into the challenge of charting the infrared behavior of QCD's universal β-function – This function may depend on the scheme chosen to renormalise the quantum field theory but it is unique within a given scheme. – Of course, the behaviour of the β-function on the perturbative domain is well known. Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 74

Charting the interaction between light-quarks Ø Through QCD's Dyson-Schwinger equations (DSEs) the pointwise behaviour of the β-function determines the pattern of chiral symmetry breaking. Ø DSEs connect β-function to experimental observables. Hence, comparison between computations and observations of o Hadron mass spectrum o Elastic and transition form factors o Parton distribution functions can be used to chart β-function’s long-range behaviour. Ø Extant studies show that the properties of hadron excited states are a great deal more sensitive to the long-range behaviour of the βfunction than those of the ground states. Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 75

Qin et al. , Phys. Rev. C 84 042202(Rapid Comm. ) (2011) Rainbow-ladder truncation DSE Studies – Phenomenology of gluon Ø Wide-ranging study of π & ρ properties Ø Effective coupling – Agrees with p. QCD in ultraviolet – Saturates in infrared • α(0)/π = 8 -15 • α(m. G 2)/π = 2 -4 Ø Running gluon mass – Gluon is massless in ultraviolet in agreement with p. QCD – Massive in infrared • m. G(0) = 0. 67 -0. 81 Ge. V • m. G(m. G 2) = 0. 53 -0. 64 Ge. V Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 76

Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 77

Dynamical Chiral Symmetry Breaking Ø Strong-interaction: QCD Ø 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 Ø 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. Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 78

Frontiers of Nuclear Science: Theoretical Advances In QCD a quark's effective mass depends on its momentum. The function describing this can be calculated and is depicted here. Numerical simulations of lattice QCD (data, at two different bare masses) have confirmed model predictions (solid curves) that the vast bulk of the constituent mass of a light quark comes from a cloud of gluons that are dragged along by the quark as it propagates. In this way, a quark that appears to be absolutely massless at high energies (m =0, red curve) acquires a large constituent mass at low energies. Craig Roberts: Truncations in DSE-QCD (114 p) C. D. Roberts, Prog. Part. Nucl. Phys. 61 (2008) 50 M. Bhagwat & P. C. Tandy, AIP Conf. Proc. 842 (2006) 225 -227 8 -12/10/12: Math Aspects of Hadron Physics 79

Frontiers of Nuclear Science: Theoretical Advances In QCD a quark's effective mass depends on its momentum. The function describing this can be calculated and is depicted here. Numerical simulations of lattice QCD (data, at two different bare masses) have confirmed model predictions (solid curves) that the vast bulk of the constituent mass of a light quark comes from a cloud of gluons that are dragged along by the quark as it propagates. In this way, a quark that appears to be absolutely massless at high energies (m =0, red curve) acquires a large constituent mass at low energies. Craig Roberts: Truncations in DSE-QCD (114 p) Mass from nothing! DSE prediction of DCSB confirmed 8 -12/10/12: Math Aspects of Hadron Physics 80

Frontiers of Nuclear Science: Theoretical Advances In QCD a quark's effective mass depends on its momentum. The function describing this can be calculated and is depicted here. Numerical simulations of lattice QCD (data, at two different bare masses) have confirmed model predictions (solid curves) that the vast bulk of the constituent mass of a light quark comes from a cloud of gluons that are dragged along by the quark as it propagates. In this way, a quark that appears to be absolutely massless at high energies (m =0, red curve) acquires a large constituent mass at low energies. Hint of lattice-QCD support for DSE prediction of violation of reflection positivity Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 81

12 Ge. V The Future of JLab Numerical simulations of lattice QCD (data, at two different bare masses) have confirmed model predictions (solid curves) that the vast bulk of the constituent mass of a light quark comes from a cloud of gluons that are dragged along by the quark as it propagates. In this way, a quark that appears to be absolutely massless at high energies (m =0, red curve) acquires a large constituent mass at low energies. Craig Roberts: Truncations in DSE-QCD (114 p) Jlab 12 Ge. V: Scanned by 2<Q 2<9 Ge. V 2 elastic & transition form factors. 8 -12/10/12: Math Aspects of Hadron Physics 82

K or π The Future of Drell-Yan N Ø Valence-quark PDFs and PDAs probe this critical and complementary region Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 83

Ø Search for exotic hadrons Ø Exploit opportunities provided by new data on nucleon elastic and transition form factors Ø Precision experimental study of valence region, and theoretical computation of distribution functions and distribution amplitudes Ø Develop QCD as a probe for physics beyond the Standard Model Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 84

Discover meaning of confinement, and its relationship to DCSB – the origin of visible mass Ø Search for exotic hadrons Ø Exploit opportunities provided by new data on nucleon elastic and transition form factors Ø Precision experimental study of valence region, and theoretical computation of distribution functions and distribution amplitudes Ø Develop QCD as a probe for physics beyond the Standard Model Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 85

Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 86

Persistent challenge in application of DSEs Ø Infinitely many coupled equations: Kernel of the equation for the quark self-energy involves: – Dμν(k) – dressed-gluon propagator – Γν(q, p) – dressed-quark-gluon vertex each of which satisfies its own DSE, etc… Ø Coupling between equations necessitates a truncation Invaluable check on – Weak coupling expansion practical truncation ⇒ produces every diagram in perturbation theory schemes – Otherwise useless for the nonperturbative problems in which we’re interested Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 87

Relationship must be preserved by any truncation Highly nontrivial constraint FAILURE has an extremely high cost – loss of any connection with QCD Persistent challenge - truncation scheme Ø Symmetries associated with conservation of vector and axial-vector currents are critical in arriving at a veracious understanding of hadron structure and interactions Ø Example: axial-vector Ward-Takahashi identity – Statement of chiral symmetry and the pattern by which it’s broken in quantum field theory Quark propagator satisfies a gap equation Axial-Vector vertex Satisfies an inhomogeneous Bethe-Salpeter equation Kernels of these equations are completely different But they must be intimately related Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 88

Persistent challenge - truncation scheme Ø These observations show that symmetries relate the kernel of the gap equation – nominally a one-body problem, with that of the Bethe-Salpeter equation – considered to be a two-body problem Ø Until 1995/1996 people had quark-antiquark no idea what to do scattering kernel Ø Equations were truncated, sometimes with good phenomenological results, sometimes with poor results Ø Neither good nor bad could be explained Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 89

Persistent challenge - truncation scheme Ø Happily, that changed, and there is now at least one systematic, nonperturbative and symmetry preserving truncation scheme – 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 – A. Bender, C. D. Roberts and L. von Smekal, Phys. Lett. B 380 (1996) 7, Goldstone Theorem and Diquark Confinement Beyond Rainbow Ladder Approximation Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 90

Modified skeleton expansion in which the propagators are fully -dressed but the vertices are constructed term-by-term Cutting scheme Ø The procedure generates a Bethe-Salpeter kernel from the kernel of any gap equation whose diagrammatic content is known – That this is possible and achievable systematically is necessary and sufficient to prove some exact results in QCD dressed propagators gap eq. Leading-order: rainbow- ladder truncation Ø The procedure also enables the formulation of practical phenomenological models that can be used to illustrate the exact results and provide predictions for experiment with readily quantifiable errors. Craig Roberts: Truncations in DSE-QCD (114 p) BS kernel bare vertices In gap eq. , add 1 -loop vertex correction Then BS kernel has 3 new terms at this order 8 -12/10/12: Math Aspects of Hadron Physics 91

Now able to explain the dichotomy of the pion Ø How does one make an almost massless particle from two massive constituent-quarks? Ø Naturally, one could always tune a potential in quantum mechanics so that the ground-state is massless – but some are still making this mistake Ø However: current-algebra (1968) Ø This is impossible in quantum mechanics, for which one always finds: Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 92

Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 93

Maris, Roberts and Tandy nucl-th/9707003, Phys. Lett. B 420 (1998) 267 -273 Pion’s Goldberger -Treiman relation Ø Pion’s Bethe-Salpeter amplitude Solution of the Bethe-Salpeter equation Pseudovector components necessarily nonzero. Cannot be ignored! Ø Dressed-quark propagator Ø Axial-vector Ward-Takahashi identity entails Exact in Chiral QCD Miracle: two body problem solved, almost completely, once solution of one body problem is known Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 94

Dichotomy of the pion Goldstone mode and bound-state fπ Eπ(p 2) = B(p 2) Ø Goldstone’s theorem has a pointwise expression in QCD; Namely, in the chiral limit the wave-function for the twobody 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 Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 95

Maris, Roberts and Tandy nucl-th/9707003, Phys. Lett. B 420 (1998) 267 -273 Dichotomy of the pion Mass Formula for 0— Mesons Ø Mass-squared of the pseudscalar hadron Ø Sum of the current-quark masses of the constituents; e. g. , pion = muς + mdς , where “ς” is the renormalisation point Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 96

Maris, Roberts and Tandy nucl-th/9707003, Phys. Lett. B 420 (1998) 267 -273 Dichotomy of the pion Mass Formula for 0— Mesons Ø Pseudovector projection of the Bethe-Salpeter wave function onto the origin in configuration space – Namely, the pseudoscalar meson’s leptonic decay constant, which is the strong interaction contribution to the strength of the meson’s weak interaction Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 97

Maris, Roberts and Tandy nucl-th/9707003, Phys. Lett. B 420 (1998) 267 -273 Dichotomy of the pion Mass Formula for 0— Mesons Ø Pseudoscalar projection of the Bethe-Salpeter wave function onto the origin in configuration space – Namely, a pseudoscalar analogue of the meson’s leptonic decay constant Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 98

Maris, Roberts and Tandy nucl-th/9707003, Phys. Lett. B 420 (1998) 267 -273 Dichotomy of the pion Mass Formula for 0— Mesons Ø Consider the case of light quarks; namely, mq ≈ 0 – If chiral symmetry is dynamically broken, then • f. H 5 → f. H 50 ≠ 0 • ρH 5 → – < q-bar q> / f. H 50 ≠ 0 The so-called “vacuum quark condensate. ” More later about this. both of which are independent of mq Ø Hence, one arrives at the corollary Gell-Mann, Oakes, Renner relation 1968 Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 99

Maris, Roberts and Tandy nucl-th/9707003, Phys. Lett. B 420 (1998) 267 -273 Dichotomy of the pion Mass Formula for 0— Mesons Ø Consider a different case; namely, one quark mass fixed and the other becoming very large, so that mq /m. Q << 1 Ø Then Provides – f. H 5 ∝ 1/√m. H 5 QCD proof of – ρH 5 ∝ √m. H 5 potential model result and one arrives at m. H 5 ∝ m. Q Ivanov, Kalinovsky, Roberts Phys. Rev. D 60, 034018 (1999) [17 pages] Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 100

Strong-interaction: QCD Dressed-quark-gluon vertex Ø Gluons and quarks acquire momentum-dependent masses – characterised by an infrared mass-scale m ≈ 2 -4 ΛQCD Ø 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(p 2) ≠ 1 and B(p 2) ≠ constant, the bare vertex cannot satisfy the Ward-Takahashi identity; viz. , Ø Number of contributors is too numerous to list completely (300 citations to 1 st J. S. Ball paper), but prominent contributions by: J. S. Ball, C. J. Burden, C. D. Roberts, R. Delbourgo, A. G. Williams, H. J. Munczek, M. R. Pennington, A. Bashir, A. Kizilersu, L. Chang, Y. -X. Liu … Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 101

Dressedquark-gluon vertex Ø Single most important feature – Perturbative vertex is helicity-conserving: • Cannot cause spin-flip transitions – However, DCSB introduces nonperturbatively generated structures that very strongly break helicity conservation – These contributions • Are large when the dressed-quark mass-function is large – Therefore vanish in the ultraviolet; i. e. , on the perturbative domain – Exact form of the contributions is still the subject of debate but their existence is model-independent - a fact. Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 102

Gap Equation General Form Ø Dμν(k) – dressed-gluon propagator Ø Γν(q, p) – dressed-quark-gluon vertex Ø Until 2009, all studies of other hadron phenomena used the leading-order term in a symmetry-preserving truncation scheme; viz. , Bender, Roberts & von Smekal – Dμν(k) = dressed, as described previously – Γν(q, p) = γμ Phys. Lett. B 380 (1996) 7 -12 • … plainly, key nonperturbative effects are missed and cannot be recovered through any step-by-step improvement procedure Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 103

Dynamical chiral symmetry breaking and the fermion--gauge-boson vertex, A. Bashir, R. Bermudez, L. Chang and C. D. Roberts, ar. Xiv: 1112. 4847 [nucl-th], Phys. Rev. C 85 (2012) 045205 [7 pages] Ø Dμν(k) – dressed-gluon propagator Ø good deal of information available Ø Γν(q, p) – dressed-quark-gluon vertex Ø Information accumulating Gap Equation General Form If kernels of Bethe-Salpeter and gap equations don’t match, one won’t even get right charge for the pion. Ø Suppose one has in hand – from anywhere – the exact form of the dressed-quark-gluon vertex What is the associated symmetrypreserving Bethe-Salpeter kernel? ! Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 104

Bethe-Salpeter Equation Bound-State DSE Ø K(q, k; P) – fully amputated, two-particle irreducible, quark-antiquark scattering kernel Ø Textbook material. Ø Compact. Visually appealing. Correct Blocked progress for more than 60 years. Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 105

Bethe-Salpeter Equation Lei Chang and C. D. Roberts General Form 0903. 5461 [nucl-th] Phys. Rev. Lett. 103 (2009) 081601 Ø Equivalent exact bound-state equation but in this form K(q, k; P) → Λ(q, k; P) which is completely determined by dressed-quark self-energy Ø Enables derivation of a Ward-Takahashi identity for Λ(q, k; P) Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 106

Ward-Takahashi Identity Lei Chang and C. D. Roberts Bethe-Salpeter Kernel 0903. 5461 [nucl-th] Phys. Rev. Lett. 103 (2009) 081601 iγ 5 Ø Now, for first time, it’s possible to formulate an Ansatz for Bethe-Salpeter kernel given any form for the dressed-quark-gluon vertex by using this identity Ø This enables the identification and elucidation of a wide range of novel consequences of DCSB Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 107

L. Chang, Y. –X. Liu and C. D. Roberts ar. Xiv: 1009. 3458 [nucl-th] Phys. Rev. Lett. 106 (2011) 072001 Dressed-quark anomalous magnetic moments Ø DCSB Three strongly-dressed and essentiallynonperturbative contributions to dressed-quark-gluon vertex: Ball-Chiu term • Vanishes if no DCSB • Appearance driven by STI Anom. chrom. mag. mom. contribution to vertex • Similar properties to BC term • Strength commensurate with lattice-QCD Skullerud, Bowman, Kizilersu et al. hep-ph/0303176 Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 108

Ø Lattice-QCD Dressed-quark anomalous – m = 115 Me. V chromomagnetic moment Ø Nonperturbative result is two orders-of-magnitude larger than the perturbative computation – This level of Quenched Skullerud, Kizilersu et al. magnification is lattice-QCD JHEP 0304 (2003) 047 typical of DCSB – cf. Quark mass function: M(p 2=0)= 400 Me. V M(p 2=10 Ge. V 2)=4 Me. V Prediction from perturbative QCD Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 109

L. Chang, Y. –X. Liu and C. D. Roberts ar. Xiv: 1009. 3458 [nucl-th] Phys. Rev. Lett. 106 (2011) 072001 Dressed-quark anomalous magnetic moments Ø DCSB Three strongly-dressed and essentiallynonperturbative contributions to dressed-quark-gluon vertex: Ball-Chiu term • Vanishes if no DCSB • Appearance driven by STI Anom. chrom. mag. mom. contribution to vertex • Similar properties to BC term • Strength commensurate with lattice-QCD Skullerud, Bowman, Kizilersu et al. hep-ph/0303176 Role and importance is novel discovery • Essential to recover p. QCD • Constructive interference with Γ 5 Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 110

L. Chang, Y. –X. Liu and C. D. Roberts ar. Xiv: 1009. 3458 [nucl-th] Phys. Rev. Lett. 106 (2011) 072001 Dressed-quark anomalous magnetic moments ØFormulated and solved general Bethe-Salpeter equation ØObtained dressed Factor of 10 electromagnetic vertex magnification ØConfined quarks don’t have a mass-shell o Can’t unambiguously define magnetic moments o But can define magnetic moment distribution Ø AEM is opposite in sign but of roughly equal magnitude as ACM Full vertex ME κACM κAEM 0. 44 -0. 22 0. 45 0 0. 048 Rainbow-ladder 0. 35 Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 111

Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 112

QCD is the most interesting part of the standard model - Nature’s only example of an essentially nonperturbative fundamental theory, Ø Confinement with light-quarks is not connected in any known way with a linear potential; not with a potential of any kind. Ø Confinement with light-quarks is associated with a dramatic change in the infrared structure of the parton propagators. Ø Dynamical chiral symmetry breaking, the origin of 98% of visible matter in universe, is manifested unambiguously and fundamentally in an equivalence between the one- and twobody problem in QCD Ø Appears that Experiment and Theory must work together to answer the questions posed by confinement and dynamical chiral symmetry breaking because Theory alone can’t. Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 113

QCD is the most interesting part of the standard model - Nature’s only example of an essentially nonperturbative fundamental theory, Ø An enormous body of hadron physics phenomenology is built upon the framework outlined above. Ø Questions: – Can the problem of symmetry-preserving DSE truncation be framed within the Hopf algebra approach to DSEs and renormalisation? – Can you pose and answer the question of DCSB? – If so, can tractable equations be produced that physicists can use to solve the meson bound-state problem? Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 114

Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 115

Contents Ø Ø Ø Standard Model – History Top Open Questions in Physics Quantum Chromodynamics Running couplings Formulating QCD Euclidean Metric Modern Miracles in Hadron Physics Dyson-Schwinger Equations Nambu—Jona-Lasinio Model Munczek-Nemirovsky Model Confinement Dynamical chiral symmetry breaking Epilogue Craig Roberts: Truncations in DSE-QCD (114 p) 8 -12/10/12: Math Aspects of Hadron Physics 116