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Craig Roberts Physics Division http://www.phy.anl.gov/theory/staff/cdr.html

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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) Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 2 Published collaborations in 2010/2011 Students Early-career scientists

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Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 3

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Excerpt from the Top-10 Can we quantitatively understand quark and gluon confinement in quantum chromodynamics and the existence of a mass gap? Quantum chromodynamics, or QCD, is the theory describing the strong nuclear force. Carried by gluons, it binds quarks into particles like protons and neutrons. Apparently, the tiny subparticles are permanently confined: one can't pull a quark or a gluon from a proton because the strong force gets stronger with distance and snaps them right back inside. Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 4

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Nuclear Science Advisory Council Long Range Plan A central goal 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 So, what’s the problem? They are legion … –Confinement –Dynamical chiral symmetry breaking –A fundamental theory of unprecedented complexity QCD defines the difference between nuclear and particle physicists: –Nuclear physicists try to solve this theory –Particle physicists run away to a place where tree-level computations are all that’s necessary; perturbation theory, the last refuge of a scoundrel Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 5

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Understanding NSAC’s Long Range Plan Craig Roberts: A Year in the life of DSEs 6 Do they – can they – correspond to well-defined quasi-particle degrees-of-freedom? If not, with what should they be replaced? What is the meaning of the NSAC Challenge? What are the quarks and gluons of QCD? Is there such a thing as a constituent quark, a constituent-gluon? After all, these are the concepts for which Gell-Mann won the Nobel Prize. Forschungszentrum Julich, IKP: 20 July. 71/81

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DSEs Craig Roberts: A Year in the life of DSEs 7 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 These are nonperturbative equivalents in quantum field theory to the Lagrange equations of motion. Essential in simplifying the general proof of renormalisability of gauge field theories. Forschungszentrum Julich, IKP: 20 July. 71/81

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Dyson-Schwinger Equations Well suited to Relativistic Quantum Field Theory Simplest level: Generating Tool for Perturbation Theory... Materially Reduces Model- Dependence … Statement about long-range behaviour of quark-quark interaction NonPerturbative, Continuum approach to QCD Hadrons as Composites of Quarks and Gluons Qualitative and Quantitative Importance of: Dynamical Chiral Symmetry Breaking – Generation of fermion mass from nothing Quark & Gluon Confinement – Coloured objects not detected, Not detectable? Craig Roberts: A Year in the life of DSEs 8 Approach yields Schwinger functions; i.e., propagators and vertices Cross-Sections built from Schwinger Functions Hence, method connects observables with long- range behaviour of the running coupling Experiment ↔ Theory comparison leads to an understanding of long- range behaviour of strong running-coupling Forschungszentrum Julich, IKP: 20 July. 71/81

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Craig Roberts: A Year in the life of DSEs 9

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Confinement Quark and Gluon Confinement –No matter how hard one strikes the proton, or any other hadron, one cannot liberate an individual quark or gluon 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 Confinement entails quark-hadron duality; i.e., that all observable consequences of QCD can, in principle, be computed using an hadronic basis. Craig Roberts: A Year in the life of DSEs 10 X Forschungszentrum Julich, IKP: 20 July. 71/81

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Confinement 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); … Craig Roberts: A Year in the life of DSEs 11 complex-P 2 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 Normal particle Confined particle Forschungszentrum Julich, IKP: 20 July. 71/81 timelike axis: P 2 <0

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Dressed-gluon propagator Gluon propagator satisfies a Dyson-Schwinger Equation Plausible possibilities for the solution DSE and lattice-QCD agree on the result –Confined gluon –IR-massive but UV-massless –m G ≈ 2-4 Λ QCD Craig Roberts: A Year in the life of DSEs 12 perturbative, massless gluon massive, unconfined gluon IR-massive but UV-massless, confined gluon A.C. Aguilar et al., Phys.Rev. D80 (2009) 085018Phys.Rev. D80 (2009) 085018 Forschungszentrum Julich, IKP: 20 July. 71/81

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Charting the interaction between light-quarks Confinement can be related to the analytic properties of QCD's Schwinger functions. Question of light-quark confinement can be translated into the challenge of charting the infrared behavior of QCD's 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: A Year in the life of DSEs 13 This is a well-posed problem whose solution is an elemental goal of modern hadron physics. The answer provides QCD’s running coupling. Forschungszentrum Julich, IKP: 20 July. 71/81

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Charting the interaction between light-quarks Through QCD's Dyson-Schwinger equations (DSEs) the pointwise behaviour of the β-function determines the pattern of chiral symmetry breaking. DSEs connect β-function to experimental observables. Hence, comparison between computations and observations of o Hadron mass spectrum o Elastic and transition form factors can be used to chart β-function’s long-range behaviour. Extant studies show that the properties of hadron states with masses 1-2GeV are a great deal more sensitive to the long-range behaviour of the β- function than those of the π&ρ ground states. Craig Roberts: A Year in the life of DSEs 14 Forschungszentrum Julich, IKP: 20 July. 71/81

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DSE Studies – Phenomenology of gluon Wide-ranging study of π & ρ properties Effective coupling –Agrees with pQCD in ultraviolet –Saturates in infrared α(0)/π = 9 - 15 α(m G 2 )/π = 2 – 4 Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 15 Qin, Chang, Roberts, et al., in progress at IKP Running gluon mass –Gluon is massless in ultraviolet in agreement with pQCD –Massive in infrared m G (0) = 0.69 – 0.81 GeV m G (m G 2 ) = 0.30 GeV

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Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 16

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Dynamical Chiral Symmetry Breaking SStrong-interaction: Q CD CConfinement –E–Empirical feature –M–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 –S–Still circumstantial, no proof yet of confinement OOn the other hand, DCSB is a fact in QCD –I–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: A Year in the life of DSEs 17 Forschungszentrum Julich, IKP: 20 July. 71/81

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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: A Year in the life of DSEs 18 Forschungszentrum Julich, IKP: 20 July. 71/81 C.D. Roberts, Prog. Part. Nucl. Phys. 61 (2008) 50Prog. Part. Nucl. Phys. 61 (2008) 50 M. Bhagwat & P.C. Tandy, AIP Conf.Proc. 842 (2006) 225-227AIP Conf.Proc. 842 (2006) 225-227

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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: A Year in the life of DSEs 19 DSE prediction of DCSB confirmed Mass from nothing! Forschungszentrum Julich, IKP: 20 July. 71/81

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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: A Year in the life of DSEs 20 Hint of lattice-QCD support for DSE prediction of violation of reflection positivity Forschungszentrum Julich, IKP: 20 July. 71/81

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12GeV 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: A Year in the life of DSEs 21 Jlab 12GeV: Scanned by 2
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Craig Roberts: A Year in the life of DSEs 22

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Strong-interaction: QCD Gluons and quarks acquire momentum-dependent masses –characterised by an infrared mass-scale m ≈ 2-4 Λ QCD 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: γ μ. –Obvious, because with A(p 2 ) ≠ 1 and B(p 2 ) ≠ constant, the bare vertex cannot satisfy the Ward-Takahashi identity; viz., 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: A Year in the life of DSEs 23 Dressed-quark-gluon vertex Forschungszentrum Julich, IKP: 20 July. 71/81

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Dressed- quark-gluon vertex 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: A Year in the life of DSEs 24 Forschungszentrum Julich, IKP: 20 July. 71/81

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Gap Equation General Form D μν (k) – dressed-gluon propagator Γ ν (q,p) – dressed-quark-gluon vertex Until 2009, all studies of other hadron phenomena used the leading-order term in a symmetry-preserving truncation scheme; viz., –D μν (k) = dressed, as described previously –Γ ν (q,p) = γ μ … plainly, key nonperturbative effects are missed and cannot be recovered through any step-by-step improvement procedure Craig Roberts: A Year in the life of DSEs 25 Bender, Roberts & von Smekal Phys.Lett. B380 (1996) 7-12 Forschungszentrum Julich, IKP: 20 July. 71/81

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Gap Equation General Form D μν (k) – dressed-gluon propagator good deal of information available Γ ν (q,p) – dressed-quark-gluon vertex Information accumulating Suppose one has in hand – from anywhere – the exact form of the dressed-quark-gluon vertex What is the associated symmetry- preserving Bethe-Salpeter kernel?! Craig Roberts: A Year in the life of DSEs 26 If kernels of Bethe-Salpeter and gap equations don’t match, one won’t even get right charge for the pion. Forschungszentrum Julich, IKP: 20 July. 71/81

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Bethe-Salpeter Equation Bound-State DSE K(q,k;P) – fully amputated, two-particle irreducible, quark-antiquark scattering kernel Textbook material. Compact. Visually appealing. Correct Blocked progress for more than 60 years. Craig Roberts: A Year in the life of DSEs 27 Forschungszentrum Julich, IKP: 20 July. 71/81

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Bethe-Salpeter Equation General Form 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 Enables derivation of a Ward-Takahashi identity for Λ(q,k;P) Craig Roberts: A Year in the life of DSEs 28 Lei Chang and C.D. Roberts 0903.5461 [nucl-th] Phys. Rev. Lett. 103 (2009) 081601 Forschungszentrum Julich, IKP: 20 July. 71/81

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Ward-Takahashi Identity Bethe-Salpeter Kernel 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 This enables the identification and elucidation of a wide range of novel consequences of DCSB Craig Roberts: A Year in the life of DSEs 29 Lei Chang and C.D. Roberts 0903.5461 [nucl-th] Phys. Rev. Lett. 103 (2009) 081601 iγ5iγ5 iγ5iγ5 Forschungszentrum Julich, IKP: 20 July. 71/81

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Relativistic quantum mechanics Dirac equation (1928): Pointlike, massive fermion interacting with electromagnetic field Craig Roberts: A Year in the life of DSEs 30 Spin Operator Forschungszentrum Julich, IKP: 20 July. 71/81

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Massive point-fermion Anomalous magnetic moment Dirac’s prediction held true for the electron until improvements in experimental techniques enabled the discovery of a small deviation: H. M. Foley and P. Kusch, Phys. Rev. 73, 412 (1948).Phys. Rev. 73, 412 (1948) –Moment increased by a multiplicative factor: 1.001 19 ± 0.000 05. This correction was explained by the ﬁrst systematic computation using renormalized quantum electrodynamics (QED): J.S. Schwinger, Phys. Rev. 73, 416 (1948),Phys. Rev. 73, 416 (1948) –vertex correction The agreement with experiment established quantum electrodynamics as a valid tool. Craig Roberts: A Year in the life of DSEs 31 e e 0.001 16 Forschungszentrum Julich, IKP: 20 July. 71/81

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Fermion electromagnetic current – General structure with k = p f - p i F 1 (k 2 ) – Dirac form factor; and F 2 (k 2 ) – Pauli form factor –Dirac equation: F 1 (k 2 ) = 1 F 2 (k 2 ) = 0 –Schwinger: F 1 (k 2 ) = 1 F 2 (k 2 =0) = α /[2 π] Craig Roberts: A Year in the life of DSEs 32 Forschungszentrum Julich, IKP: 20 July. 71/81

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Plainly, can’t simply take the limit m → 0. Standard QED interaction, generated by minimal substitution: Magnetic moment is described by interaction term: –Invariant under local U(1) gauge transformations –but is not generated by minimal substitution in the action for a free Dirac ﬁeld. Transformation properties under chiral rotations? –Ψ(x) → exp(iθγ 5 ) Ψ(x) Magnetic moment of a massless fermion? Craig Roberts: A Year in the life of DSEs 33 Forschungszentrum Julich, IKP: 20 July. 71/81

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Standard QED interaction, generated by minimal substitution: –Unchanged under chiral rotation –Follows that QED without a fermion mass term is helicity conserving Magnetic moment interaction is described by interaction term: –NOT invariant –picks up a phase-factor exp(2iθγ 5 ) Magnetic moment interaction is forbidden in a theory with manifest chiral symmetry Magnetic moment of a massless fermion? Craig Roberts: A Year in the life of DSEs 34 Forschungszentrum Julich, IKP: 20 July. 71/81

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OOne-loop calculation: PPlainly, one obtains Schwinger’s result for m e 2 ≠ 0 HHowever, F 2 (k 2 ) = 0 when m e 2 = 0 TThere is no Gordon identity: RResults are unchanged at every order in perturbation theory … owing to symmetry … magnetic moment interaction is forbidden in a theory with manifest chiral symmetry Schwinger’s result? Craig Roberts: A Year in the life of DSEs 35 e e m=0 So, no mixing γ μ ↔ σ μν Forschungszentrum Julich, IKP: 20 July. 71/81

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QCD and dressed-quark anomalous magnetic moments Schwinger’s result for QED: pQCD: two diagrams o (a) is QED-like o (b) is only possible in QCD – involves 3-gluon vertex Analyse (a) and (b) o (b) vanishes identically: the 3-gluon vertex does not contribute to a quark’s anomalous chromomag. moment at leading-order o (a) Produces a finite result: “ – ⅙ α s /2π ” ~ (– ⅙) QED-result But, in QED and QCD, the anomalous chromo- and electro- magnetic moments vanish identically in the chiral limit! Craig Roberts: A Year in the life of DSEs 36 Forschungszentrum Julich, IKP: 20 July. 71/81

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Dressed-quark anomalous magnetic moments Three strongly-dressed and essentially- nonperturbative contributions to dressed-quark-gluon vertex: Craig Roberts: A Year in the life of DSEs 37 DCSB Ball-Chiu term Vanishes if no DCSB Appearance driven by STI Anom. chrom. mag. mom. contribution to vertex Similar properties to BC term Strength commensurate with lattice-QCD Skullerud, Bowman, Kizilersu et al. hep-ph/0303176 L. Chang, Y. –X. Liu and C.D. Roberts arXiv:1009.3458 [nucl-th] arXiv:1009.3458 [nucl-th] Phys. Rev. Lett. 106 (2011) 072001 Forschungszentrum Julich, IKP: 20 July. 71/81

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Dressed-quark anomalous chromomagnetic moment Lattice-QCD –m = 115 MeV Nonperturbative result is two orders-of-magnitude larger than the perturbative computation –This level of magnification is typical of DCSB –cf. Craig Roberts: A Year in the life of DSEs 38 Skullerud, Kizilersu et al. JHEP 0304 (2003) 047 Prediction from perturbative QCD Quenched lattice-QCD Quark mass function: M(p 2 =0)= 400MeV M(p 2 =10GeV 2 )=4 MeV Forschungszentrum Julich, IKP: 20 July. 71/81

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Dressed-quark anomalous magnetic moments Three strongly-dressed and essentially- nonperturbative contributions to dressed-quark-gluon vertex: Craig Roberts: A Year in the life of DSEs 39 DCSB Ball-Chiu term Vanishes if no DCSB Appearance driven by STI Anom. chrom. mag. mom. contribution to vertex Similar properties to BC term Strength commensurate with lattice-QCD Skullerud, Bowman, Kizilersu et al. hep-ph/0303176 Role and importance is novel discovery Essential to recover pQCD Constructive interference with Γ 5 L. Chang, Y. –X. Liu and C.D. Roberts arXiv:1009.3458 [nucl-th] arXiv:1009.3458 [nucl-th] Phys. Rev. Lett. 106 (2011) 072001 Forschungszentrum Julich, IKP: 20 July. 71/81

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Dressed-quark anomalous magnetic moments Craig Roberts: A Year in the life of DSEs 40 Formulated and solved general Bethe-Salpeter equation Obtained dressed electromagnetic vertex Confined quarks don’t have a mass-shell o Can’t unambiguously define magnetic moments o But can define magnetic moment distribution MEME κ ACM κ AEM Full vertex0.44-0.220.45 Rainbow-ladder0.3500.048 AEM is opposite in sign but of roughly equal magnitude as ACM L. Chang, Y. –X. Liu and C.D. Roberts arXiv:1009.3458 [nucl-th] arXiv:1009.3458 [nucl-th] Phys. Rev. Lett. 106 (2011) 072001 Factor of 10 magnification Forschungszentrum Julich, IKP: 20 July. 71/81

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Dressed-quark anomalous magnetic moments Craig Roberts: A Year in the life of DSEs 41 Potentially important for elastic and transition form factors, etc. Significantly, also quite possibly for muon g-2 – via Box diagram, which is not constrained by extant data. L. Chang, Y. –X. Liu and C.D. Roberts arXiv:1009.3458 [nucl-th] arXiv:1009.3458 [nucl-th] Phys. Rev. Lett. 106 (2011) 072001 Factor of 10 magnification Formulated and solved general Bethe-Salpeter equation Obtained dressed electromagnetic vertex Confined quarks don’t have a mass-shell o Can’t unambiguously define magnetic moments o But can define magnetic moment distribution Contemporary theoretical estimates: 1 – 10 x 10 -10 Largest value reduces discrepancy expt.↔theory from 3.3σ to below 2σ. Forschungszentrum Julich, IKP: 20 July. 71/81

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Craig Roberts: A Year in the life of DSEs 42

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Deep inelastic scattering Quark discovery experiment at SLAC (1966-1978, Nobel Prize in 1990) Completely different to elastic scattering –Blow the target to pieces instead of keeping only those events where it remains intact. Cross-section is interpreted as a measurement of the momentum-fraction probability distribution for quarks and gluons within the target hadron: q(x), g(x) Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 43 Probability that a quark/gluon within the target will carry a fraction x of the bound-state’s light-front momentum Distribution Functions of the Nucleon and Pion in the Valence Region, Roy J. Holt and Craig D. Roberts, arXiv:1002.4666 [nucl-th], Rev. Mod. Phys. 82 (2010) pp. 2991-3044 arXiv:1002.4666 [nucl-th]Rev. Mod. Phys. 82 (2010) pp. 2991-3044

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Pion’s valence-quark distributions Owing to absence of pion targets, the pion’s valence-quark distribution functions are measured via the Drell-Yan process: π p → μ + μ − X Three experiments: CERN (1983 & 1985) and FNAL (1989). No more recent experiments because theory couldn’t even explain these! Problem Conway et al. Phys. Rev. D 39, 92 (1989)Phys. Rev. D 39, 92 (1989) Wijesooriya et al. Phys.Rev. C 72 (2005) 065203Phys.Rev. C 72 (2005) 065203 Behaviour at large-x is inconsistent with pQCD; viz, expt. (1-x) 1+ε cf. QCD (1-x) 2+γ Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 44 Pion

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Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 45

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Computation of q v π (x) In the Bjorken limit; viz., q 2 → ∞, P · q → − ∞ but x := −q 2 /2P · q fixed, W μν (q;P) ~ q v π (x) Plainly, this can be computed given all the information we have at our disposal from the DSEs Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 46 T+T+ T–T–

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Computation of q v π (x) Before the first DSE computation, which used the running dressed-quark mass described previously, numerous authors applied versions of the Nambu–Jona-Lasinio model and were content to reproduce the data, arguing therefrom that the inferences from pQCD were wrong Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 47 Hecht, Roberts, Schmidt Phys.Rev. C 63 (2001) 025213 After the first DSE computation, experimentalists again became interested in the process because –DSEs agreed with p QCD but disagreed with the data, and other models Disagreement on the “valence domain,” which is uniquely sensitive to M(p 2 ) 2.61/1.27= factor of 2 in the exponent

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Reanalysis of q v π (x) After the first DSE computation, the “Conway et al.” data were reanalysed, this time at next-to-leading-order ( Wijesooriya et al. Phys.Rev. C 72 (2005) 065203 )Phys.Rev. C 72 (2005) 065203 The new analysis produced a much larger exponent than initially obtained; viz., β=1.87, but now it disagreed equally with NJL-model results and the DSE prediction NB. Within pQCD, one can readily understand why adding a higher-order correction leads to a suppression of q v π (x) at large-x. Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 48 Hecht, Roberts, Schmidt Phys.Rev. C 63 (2001) 025213 New experiments were proposed … for accelerators that do not yet exist but the situation remained otherwise unchanged Until the publication of Distribution Functions of the Nucleon and Pion in the Valence Region, Roy J. Holt and Craig D. Roberts, arXiv:1002.4666 [nucl-th], Rev. Mod. Phys. 82 (2010) pp. 2991-3044 arXiv:1002.4666 [nucl-th]Rev. Mod. Phys. 82 (2010) pp. 2991-3044

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Reanalysis of q v π (x) This article emphasised and explained the importance of the persistent discrepancy between the DSE result and experiment as a challenge to QCD It prompted another reanalysis of the data, which accounted for a long-overlooked effect: viz., “soft-gluon resummation,” –Compared to previous analyses, we include next-to-leading- logarithmic threshold resummation effects in the calculation of the Drell-Yan cross section. As a result of these, we find a considerably softer valence distribution at high momentum fractions x than obtained in previous next-to-leading-order analyses, in line with expectations based on perturbative-QCD counting rules or Dyson- Schwinger equations. Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 49 Distribution Functions of the Nucleon and Pion in the Valence Region, Roy J. Holt and Craig D. Roberts, arXiv:1002.4666 [nucl-th], Rev. Mod. Phys. 82 (2010) pp. 2991-3044 arXiv:1002.4666 [nucl-th]Rev. Mod. Phys. 82 (2010) pp. 2991-3044 Aicher, Schäfer, Vogelsang, “Soft-Gluon Resummation and the Valence Parton Distribution Function of the Pion,” Phys. Rev. Lett. 105 (2010) 252003 Phys. Rev. Lett. 105 (2010) 252003

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Current status of q v π (x) DSE prediction and modern representation of the data are indistinguishable on the valence-quark domain Emphasises the value of using a single internally- consistent, well- constrained framework to correlate and unify the description of hadron observables What about valence- quark distributions in the kaon? Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 50 Trang, Bashir, Roberts & Tandy, “Pion and kaon valence- quark parton distribution functions,” arXiv:1102.2448 [nucl-th], Phys. Rev. C 83, 062201(R) (2011) [5 pages]arXiv:1102.2448 [nucl-th],Phys. Rev. C 83, 062201(R) (2011) [5 pages]

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q v π (x) & q v K (x) m s ≈ 24 m u & M s ≈ 1.25 M u Expect the s-quark to carry more of the kaon’s momentum than the u- quark, so that xs K (x) peaks at larger value of x than xu K (x) Expectation confirmed in computations, with s-quark distribution peaking at 15% larger value of x Even though deep inelastic scattering is a high-Q 2 process, constituent-like mass-scale explains the shift Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 51 Trang, Bashir, Roberts & Tandy, “Pion and kaon valence- quark parton distribution functions,” arXiv:1102.2448 [nucl-th], Phys. Rev. C 83, 062201(R) (2011) [5 pages]arXiv:1102.2448 [nucl-th],Phys. Rev. C 83, 062201(R) (2011) [5 pages] xu π (x) xs K (x) xu K (x)

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u K (x)/u π (x) Drell-Yan experiments at CERN (1980 & 1983) provide the only extant measurement of this ratio DSE result in complete accord with the measurement New Drell-Yan experiment running now at FNAL is capable of validating this comparison It should be done so that complete understanding can be claimed Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 52 Trang, Bashir, Roberts & Tandy, “Pion and kaon valence- quark parton distribution functions,” arXiv:1102.2448 [nucl-th], Phys. Rev. C 83, 062201(R) (2011) [5 pages]arXiv:1102.2448 [nucl-th],Phys. Rev. C 83, 062201(R) (2011) [5 pages] Value of ratio at x=1 is a fixed point of the evolution equations Hence, it’s a very strong test of nonperturbative dynamics Value of ratio at x=0 will approach “1” under evolution to higher resolving scales. This is a feature of perturbative dynamics Using DSEs in QCD, one derives that the x=1 value is ≈ (f π /f K ) 2 (M u /M s ) 4 = 0.3

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Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 53

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Unification of Meson & Baryon Spectra Dyson-Schwinger Equations have been applied extensively to the spectrum and interactions of mesons with masses less than 1 GeV On this domain the rainbow-ladder approximation, which is the leading-order in a systematic & symmetry-preserving truncation scheme – nucl-th/9602012, is a well-understood tool that is accurate for pseudoscalar and vector mesons:nucl-th/9602012 e.g., Prediction of elastic pion and kaon form factors: nucl-th/0005015nucl-th/0005015 Anomalous neutral pion processes – γπγ & BaBar anomaly: 1009.0067 [nucl-th]1009.0067 [nucl-th] Pion and kaon valence-quark distribution functions: 1102.2448 [nucl-th]1102.2448 [nucl-th] Unification of these and other observables – ππ scattering: hep-ph/0112015hep-ph/0112015 It can readily be extended to explain properties of the light neutral pseudoscalar mesons: 0708.1118 [nucl-th]0708.1118 [nucl-th] Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 54

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DSEs & Baryons Dynamical chiral symmetry breaking (DCSB) – has enormous impact on meson properties. Must be included in description and prediction of baryon properties. DCSB is essentially a quantum field theoretical effect. In quantum field theory Meson appears as pole in four-point quark-antiquark Green function → Bethe-Salpeter Equation Nucleon appears as a pole in a six-point quark Green function → Faddeev Equation. Poincaré covariant Faddeev equation sums all possible exchanges and interactions that can take place between three dressed-quarks Tractable equation is based on the observation that an interaction which describes colour-singlet mesons also generates nonpointlike quark-quark (diquark) correlations in the colour-antitriplet channel Craig Roberts: A Year in the life of DSEs 55 R.T. Cahill et al., Austral. J. Phys. 42 (1989) 129-145 Forschungszentrum Julich, IKP: 20 July. 71/81 SU c (3):

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Faddeev Equation Linear, Homogeneous Matrix equation Yields wave function (Poincaré Covariant Faddeev Amplitude) that describes quark-diquark relative motion within the nucleon Scalar and Axial-Vector Diquarks... Both have “correct” parity and “right” masses In Nucleon’s Rest Frame Amplitude has s−, p− & d−wave correlations Craig Roberts: A Year in the life of DSEs 56 diquark quark quark exchange ensures Pauli statistics composed of strongly- dressed quarks bound by dressed-gluons Forschungszentrum Julich, IKP: 20 July. 71/81 R.T. Cahill et al., Austral. J. Phys. 42 (1989) 129-145

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Photon-nucleon current Composite nucleon must interact with photon via nontrivial current constrained by Ward-Takahashi identities DSE → BSE → Faddeev equation plus current → nucleon form factors Craig Roberts: A Year in the life of DSEs 57 Vertex must contain the dressed-quark anomalous magnetic moment: Lecture IV Oettel, Pichowsky, Smekal Eur.Phys.J. A8 (2000) 251-281 Forschungszentrum Julich, IKP: 20 July. 71/81 L. Chang, Y. –X. Liu and C.D. Roberts arXiv:1009.3458 [nucl-th] arXiv:1009.3458 [nucl-th] Phys. Rev. Lett. 106 (2011) 072001 In a realistic calculation, the last three diagrams represent 8-dimensional integrals, which can be evaluated using Monte-Carlo techniques

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Craig Roberts: A Year in the life of DSEs 58 I.C. Cloët, C.D. Roberts, et al. arXiv:0812.0416 [nucl-th] Highlights again the critical importance of DCSB in explanation of real-world observables. DSE result Dec 08 DDSE result – including the anomalous magnetic moment distribution I.C. Cloët, C.D. Roberts, et al. In progress Forschungszentrum Julich, IKP: 20 July. 71/81

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Craig Roberts: A Year in the life of DSEs 59

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T-violation Invariance under C*P*T is a symmetry of any local quantum field theory C*P alone is not a symmetry of the Standard Model –Therefore Time-Reversal-Invariance is violated. Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 60 Hecht, Roberts, Schmidt Phys.Rev. C64 (2001) 025204 Possession of an EDM by J=½ particle is a violation of T-symmetry Standard Model: |d n,p |< 10 -32 ecm –Current experiment: |d p |<7.9 x 10 -25 ecm & |d n |<1.6 x 10 -26 ecm Suppose nonzero d p or d n is measured –What is the relation to current-quark EDMs: d u,d,s,… ? –SU(6) Quark Model: d p =⅓(4d u -d d ) & d n =⅓(4d d -d u ) But these are constituent-quark EDMs, which aren’t readily defined in SM!

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Nucleon EDM Nucleon EDM described by a third form factor in the photon-nucleon current: Current-quark EDMs, from any source, will produce nucleon EDM, which can be computed if-and-only-if one has a framework for connecting current-quarks to hadron observables Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 61 Hecht, Roberts, Schmidt Phys.Rev. C64 (2001) 025204

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Quark EDMs Faddeev equation approach EDM appears in dressed-quark–photon vertex: two possible additions because confined-quarks don’t have a mass-shell – Q= electric- charge matrix m = current-quark mass H ± = diag[h u ±,h d ± ] – current-quark gyroelectric ratios May now compute relation between h q & h N Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 62 Hecht, Roberts, Schmidt Phys.Rev. C64 (2001) 025204

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Quark EDMs Nucleon structure factors Matrix of gyroelectric ratios Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 63 Hecht, Roberts, Schmidt Phys.Rev. C64 (2001) 025204 Structure factors Quark charges Gyroelectric ratios Nonzero because dressed-quarks are not Dirac particles

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Gyroelectric ratios Neutron and proton (h N =e/2M N ) related to current-quarks NB. h + = 0 in constituent-quark models coefficient-ratio(h d – /h u – )= 3.4 cf. SU(6) CQMs coefficient-ratio(h d – /h u – )= 2.0 SU(6) is not a good approximation Example: Charged-Higgs-boson-exchange models yield h d >>h u h n = – 40.6 ( 0.85 h d + + 1.15 h d - ) h p = 5.1 ( -0.66 h d + + 1.34 h d - ) Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 64 Hecht, Roberts, Schmidt Phys.Rev. C64 (2001) 025204 Suppose h u >>h d h p = 81.2 ( 0.85 h u + + 1.15 h u - )

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Quark EDMs The scale of DCSB in QCD amplifies the contribution from a current- quark’s EDM to that of the bound state containing it –minimal enhancement factor is roughly the ratio of the constituent- quark and current-quark masses; two-orders of magnitude –it is understandable and calculable through the necessary momentum dependence of the dressed-quark mass function. Conﬁnement & compositeness entail that the dressed-quarks comprising bound state are not on shell & hence the two C P- & T- violating operator structures that are indistinguishable in the free- quark limit yield materially different contributions to the EDM of the bound state. –These operator structures must be analysed and their strengths determined independently in any model that provides for C P and T violation. Both these effects should be accounted for in using d N as a means of constraining extensions of the Standard Model. Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 65 Hecht, Roberts, Schmidt Phys.Rev. C64 (2001) 025204

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QCD is the most interesting part of the Standard Model and Nature’s only example of an essentially nonperturbative fundamental theory Dyson-Schwinger equations: o Described some highlights from 2010/2011: Confinement is a dynamical phenomenon It cannot in principle be expressed via a potential Gluons are nonperturbatively massive & quarks are not Dirac particles 2001 Prediction of pion valence-quark distribution function confirmed in 2010, via modern reanalysis of extant Drell-Yan data Dynamical chiral symmetry breaking is a fact. It’s responsible for 98% of the mass of visible matter in the Universe Provides an enhancement factor between hadron EDMs and current-quark EDMs o Omitted more highlights from 2010/2011; e.g., Condensates are contained within hadrons studies relating dressed-quark hadron-spectrum to EBAC and Jülich coupled- channels models Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 70 Masses of ground and excited-state hadrons Hannes L.L. Roberts, Lei Chang, Ian C. Cloët and Craig D. Roberts, arXiv:1101.4244 [nucl-th] Few Body Systems (2011) pp. 1-25 New perspectives on the quark condensate Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201

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Dichotomy of the pion Craig Roberts: A Year in the life of DSEs 73 How does one make an almost massless particle from two massive constituent-quarks? Naturally, one could always tune a potential in quantum mechanics so that the ground-state is massless – but some are still making this mistake However: current-algebra (1968) This is impossible in quantum mechanics, for which one always finds: Forschungszentrum Julich, IKP: 20 July. 71/81

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Gell-Mann – Oakes – Renner Relation Craig Roberts: A Year in the life of DSEs 74 This paper derives a relation between m π 2 and the expectation-value, where u o is an operator that is linear in the putative Hamiltonian’s explicit chiral-symmetry breaking term NB. QCD’s current-quarks were not yet invented, so u 0 was not expressed in terms of current-quark fields PCAC-hypothesis (partial conservation of axial current) is used in the derivation Subsequently, the concepts of soft-pion theory Operator expectation values do not change as t=m π 2 → t=0 to take → … in-pion → in-vacuum Forschungszentrum Julich, IKP: 20 July. 71/81 Behavior of current divergences under SU(3) x SU(3). Murray Gell-Mann, R.J. Oakes, B. Renner Phys.Rev. 175 (1968) 2195-2199

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Gell-Mann – Oakes – Renner Relation Craig Roberts: A Year in the life of DSEs 75 Theoretical physics at its best. But no one is thinking about how properly to consider or define what will come to be called the vacuum quark condensate So long as the condensate is just a mass-dimensioned constant, which approximates another well-defined transition matrix element, there is no problem. Problem arises if one over-interprets this number, which textbooks have been doing for a VERY LONG TIME. Forschungszentrum Julich, IKP: 20 July. 71/81 - (0.25GeV) 3

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Note of Warning Craig Roberts: A Year in the life of DSEs 76 Forschungszentrum Julich, IKP: 20 July. 71/81 Chiral Magnetism (or Magnetohadrochironics) A. Casher and L. Susskind, Phys. Rev. D9 (1974) 436 These authors argue that dynamical chiral- symmetry breaking can be realised as a property of hadrons, instead of via a nontrivial vacuum exterior to the measurable degrees of freedom The essential ingredient required for a spontaneous symmetry breakdown in a composite system is the existence of a divergent number of constituents – DIS provided evidence for divergent sea of low-momentum partons – parton model.

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QCD Sum Rules Introduction of the gluon vacuum condensate and development of “sum rules” relating properties of low-lying hadronic states to vacuum condensates Craig Roberts: A Year in the life of DSEs 77 Forschungszentrum Julich, IKP: 20 July. 71/81 QCD and Resonance Physics. Sum Rules. M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov Nucl.Phys. B147 (1979) 385-447; citations: 3713

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QCD Sum Rules Introduction of the gluon vacuum condensate and development of “sum rules” relating properties of low-lying hadronic states to vacuum condensates At this point (1979), the cat was out of the bag: a physical reality was seriously attributed to a plethora of vacuum condensates Craig Roberts: A Year in the life of DSEs 78 Forschungszentrum Julich, IKP: 20 July. 71/81 QCD and Resonance Physics. Sum Rules. M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov Nucl.Phys. B147 (1979) 385-447; citations: 3781

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“quark condensate” 1960-1980 Instantons in non-perturbative QCD vacuum, Instantons in non-perturbative QCD vacuum MA Shifman, AI Vainshtein… - Nuclear Physics B, 1980 Instanton density in a theory with massless quarks, Instanton density in a theory with massless quarks MA Shifman, AI Vainshtein… - Nuclear Physics B, 1980 Exotic new quarks and dynamical symmetry breaking, Exotic new quarks and dynamical symmetry breaking WJ Marciano - Physical Review D, 1980 The pion in QCD The pion in QCD J Finger, JE Mandula… - Physics Letters B, 1980 No references to this phrase before 1980 Craig Roberts: A Year in the life of DSEs 79 Forschungszentrum Julich, IKP: 20 July. 71/81

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Universal Misapprehensions Since 1979, DCSB has commonly been associated literally with a spacetime- independent mass-dimension-three “vacuum condensate.” Under this assumption, “condensates” couple directly to gravity in general relativity and make an enormous contribution to the cosmological constant Experimentally, the answer is Ω cosm. const. = 0.76 This mismatch is a bit of a problem. Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 80

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Resolution? Quantum Healing Central: “KSU physics professor [Peter Tandy] publishes groundbreaking research on inconsistency in Einstein theory.” Paranormal Psychic Forums: “Now Stanley Brodsky of the SLAC National Accelerator Laboratory in Menlo Park, California, and colleagues have found a way to get rid of the discrepancy. “People have just been taking it on faith that this quark condensate is present throughout the vacuum,” says Brodsky. Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 81

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Paradigm shift: In-Hadron Condensates Resolution –Whereas it might sometimes be convenient in computational truncation schemes to imagine otherwise, “condensates” do not exist as spacetime-independent mass-scales that fill all spacetime. –So-called vacuum condensates can be understood as a property of hadrons themselves, which is expressed, for example, in their Bethe-Salpeter or light-front wavefunctions. –GMOR cf. Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 82 QCDQCD Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201Phys. Rev. C82 (Rapid Comm.) (2010) 022201 Brodsky and Shrock, PNAS 108, 45 (2011)PNAS 108, 45 (2011) M. Burkardt, Phys.Rev. D58 (1998) 096015Phys.Rev. D58 (1998) 096015

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Paradigm shift: In-Hadron Condensates Resolution –Whereas it might sometimes be convenient in computational truncation schemes to imagine otherwise, “condensates” do not exist as spacetime-independent mass-scales that fill all spacetime. –So-called vacuum condensates can be understood as a property of hadrons themselves, which is expressed, for example, in their Bethe-Salpeter or light-front wavefunctions. –No qualitative difference between f π and ρ π Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 83 Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201Phys. Rev. C82 (Rapid Comm.) (2010) 022201 Brodsky and Shrock, PNAS 108, 45 (2011)PNAS 108, 45 (2011)

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Paradigm shift: In-Hadron Condensates Resolution –Whereas it might sometimes be convenient in computational truncation schemes to imagine otherwise, “condensates” do not exist as spacetime-independent mass-scales that fill all spacetime. –So-called vacuum condensates can be understood as a property of hadrons themselves, which is expressed, for example, in their Bethe-Salpeter or light-front wavefunctions. –No qualitative difference between f π and ρ π –And Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 84 Chiral limit Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201Phys. Rev. C82 (Rapid Comm.) (2010) 022201 Brodsky and Shrock, PNAS 108, 45 (2011)PNAS 108, 45 (2011)

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“ EMPTY space may really be empty. Though quantum theory suggests that a vacuum should be fizzing with particle activity, it turns out that this paradoxical picture of nothingness may not be needed. A calmer view of the vacuum would also help resolve a nagging inconsistency with dark energy, the elusive force thought to be speeding up the expansion of the universe.”dark energy Forschungszentrum Julich, IKP: 20 July. 71/81 Craig Roberts: A Year in the life of DSEs 85 “Void that is truly empty solves dark energy puzzle” Rachel Courtland, New Scientist 4 th Sept. 2010 Cosmological Constant: Putting QCD condensates back into hadrons reduces the mismatch between experiment and theory by a factor of 10 46 Possibly by far more, if technicolour-like theories are the correct paradigm for extending the Standard Model Paradigm shift: In-Hadron Condensates

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