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**Craig Roberts Physics Division**

A Year in the Life of Dyson-Schwinger Equations Craig Roberts Physics Division 60+30 Go to ”Insert (View) | Header and Footer" to add your organization, sponsor, meeting name here; then, click "Apply to All"

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**Published collaborations in 2010/2011**

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

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**Top Open Questions in Physics**

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

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

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

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**Understanding NSAC’s Long Range Plan**

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

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DSEs Dyson (1949) & Schwinger (1951) One can derive a system of coupled integral equations relating all the Green functions for a theory, one to another. Gap equation: fermion self energy gauge-boson propagator fermion-gauge-boson vertex These are nonperturbative equivalents in quantum field theory to the Lagrange equations of motion. Essential in simplifying the general proof of renormalisability of gauge field theories. (16) Craig Roberts: A Year in the life of DSEs 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? Approach yields Schwinger functions; i.e., propagators and vertices Cross-Sections built from Schwinger Functions Hence, method connects observables with long- range behaviour of the running coupling Experiment ↔ Theory comparison leads to an understanding of long- range behaviour of strong running-coupling Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

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**Confinement Forschungszentrum Julich, IKP: 20 July. 71/81**

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

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**X Confinement Empirical fact. However 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 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); … Normal particle Confined particle complex-P2 complex-P2 timelike axis: P2<0 Real-axis mass-pole splits, moving into pair(s) of complex conjugate poles or branch points Spectral density no longer positive semidefinite & hence state cannot exist in observable spectrum Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

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**Dressed-gluon propagator**

A.C. Aguilar et al., Phys.Rev. D80 (2009) 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 mG ≈ 2-4 ΛQCD IR-massive but UV-massless, confined gluon perturbative, massless gluon massive , unconfined gluon Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

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**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. 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 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 Hadron mass spectrum 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 Forschungszentrum Julich, IKP: 20 July. 71/81

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**DSE Studies – Phenomenology of gluon**

Qin, Chang, Roberts, et al., in progress at IKP 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 = α(mG2)/π = 2 – 4 Running gluon mass Gluon is massless in ultraviolet in agreement with pQCD Massive in infrared mG(0) = 0.69 – 0.81 GeV mG(mG2) = 0.30 GeV Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

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**Dynamical Chiral Symmetry Breaking Mass Gap**

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

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**Dynamical Chiral Symmetry Breaking**

Strong-interaction: QCD 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 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: A Year in the life of DSEs 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. (20) C.D. Roberts, Prog. Part. Nucl. Phys. 61 (2008) 50 M. Bhagwat & P.C. Tandy, AIP Conf.Proc. 842 (2006) Craig Roberts: A Year in the life of DSEs 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. Mass from nothing! DSE prediction of DCSB confirmed Craig Roberts: A Year in the life of DSEs 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. Hint of lattice-QCD support for DSE prediction of violation of reflection positivity Craig Roberts: A Year in the life of DSEs 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. Jlab 12GeV: Scanned by 2<Q2<9 GeV2 elastic & transition form factors. Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

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**Dynamical Chiral Symmetry Breaking Importance of being well-dressed for quarks & mesons**

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

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**Strong-interaction: QCD**

Dressed-quark-gluon vertex Gluons and quarks acquire momentum-dependent masses characterised by an infrared mass-scale m ≈ 2-4 ΛQCD 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(p2) ≠ 1 and B(p2) ≠ constant, the bare vertex cannot satisfy the Ward-Takahashi identity; viz., Number of contributors is too numerous to list completely (300 citations to 1st 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 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 Forschungszentrum Julich, IKP: 20 July. 71/81

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**Gap Equation General Form**

Dμν(k) – dressed-gluon propagator Γν(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., Dμν(k) = dressed, as described previously Γν(q,p) = γμ … plainly, key nonperturbative effects are missed and cannot be recovered through any step-by-step improvement procedure Bender, Roberts & von Smekal Phys.Lett. B380 (1996) 7-12 Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

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**Gap Equation General Form**

Dμν(k) – dressed-gluon propagator good deal of information available Γν(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?! If kernels of Bethe-Salpeter and gap equations don’t match, one won’t even get right charge for the pion. Craig Roberts: A Year in the life of DSEs 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 Forschungszentrum Julich, IKP: 20 July. 71/81

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**Bethe-Salpeter Equation General Form**

Lei Chang and C.D. Roberts [nucl-th] Phys. Rev. Lett. 103 (2009) 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) (30) Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

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**Ward-Takahashi Identity Bethe-Salpeter Kernel**

Lei Chang and C.D. Roberts [nucl-th] Phys. Rev. Lett. 103 (2009) iγ5 iγ5 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 Forschungszentrum Julich, IKP: 20 July. 71/81

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**Relativistic quantum mechanics**

Dirac equation (1928): Pointlike, massive fermion interacting with electromagnetic field Spin Operator Craig Roberts: A Year in the life of DSEs 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). Moment increased by a multiplicative factor: ± This correction was explained by the ﬁrst systematic computation using renormalized quantum electrodynamics (QED): J.S. Schwinger, Phys. Rev. 73, 416 (1948), vertex correction The agreement with experiment established quantum electrodynamics as a valid tool. e e Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

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**Fermion electromagnetic current – General structure**

with k = pf - pi F1(k2) – Dirac form factor; and F2(k2) – Pauli form factor Dirac equation: F1(k2) = 1 F2(k2) = 0 Schwinger: F2(k2=0) = α /[2 π] Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

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**Magnetic moment of a massless fermion?**

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(iθγ5) Ψ(x) Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

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**Magnetic moment of a massless fermion?**

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(2iθγ5) Magnetic moment interaction is forbidden in a theory with manifest chiral symmetry Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

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**Schwinger’s result? F2(k2) = 0 when me2 = 0 One-loop calculation:**

Plainly, one obtains Schwinger’s result for me2 ≠ 0 However, F2(k2) = 0 when me2 = 0 There is no Gordon identity: Results are unchanged at every order in perturbation theory … owing to symmetry … magnetic moment interaction is forbidden in a theory with manifest chiral symmetry e e m=0 So, no mixing γμ ↔ σμν Craig Roberts: A Year in the life of DSEs 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 (a) is QED-like (b) is only possible in QCD – involves 3-gluon vertex Analyse (a) and (b) (b) vanishes identically: the 3-gluon vertex does not contribute to a quark’s anomalous chromomag. moment at leading-order (a) Produces a finite result: “ – ⅙ αs/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 Forschungszentrum Julich, IKP: 20 July. 71/81

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**Dressed-quark anomalous magnetic moments**

L. Chang, Y. –X. Liu and C.D. Roberts arXiv: [nucl-th] Phys. Rev. Lett. 106 (2011) Dressed-quark anomalous magnetic moments DCSB Three strongly-dressed and essentially- nonperturbative 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/ Craig Roberts: A Year in the life of DSEs 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. Quenched lattice-QCD Skullerud, Kizilersu et al. JHEP 0304 (2003) 047 Quark mass function: M(p2=0)= 400MeV M(p2=10GeV2)=4 MeV Prediction from perturbative QCD Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

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**Dressed-quark anomalous magnetic moments**

L. Chang, Y. –X. Liu and C.D. Roberts arXiv: [nucl-th] Phys. Rev. Lett. 106 (2011) Dressed-quark anomalous magnetic moments DCSB Three strongly-dressed and essentially- nonperturbative 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/ Role and importance is novel discovery Essential to recover pQCD Constructive interference with Γ5 Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

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**Dressed-quark anomalous magnetic moments**

L. Chang, Y. –X. Liu and C.D. Roberts arXiv: [nucl-th] Phys. Rev. Lett. 106 (2011) Dressed-quark anomalous magnetic moments Formulated and solved general Bethe-Salpeter equation Obtained dressed electromagnetic vertex Confined quarks don’t have a mass-shell Can’t unambiguously define magnetic moments But can define magnetic moment distribution Factor of 10 magnification AEM is opposite in sign but of roughly equal magnitude as ACM ME κACM κAEM Full vertex 0.44 -0.22 0.45 Rainbow-ladder 0.35 0.048 Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

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**Dressed-quark anomalous magnetic moments**

L. Chang, Y. –X. Liu and C.D. Roberts arXiv: [nucl-th] Phys. Rev. Lett. 106 (2011) Dressed-quark anomalous magnetic moments Formulated and solved general Bethe-Salpeter equation Obtained dressed electromagnetic vertex Confined quarks don’t have a mass-shell Can’t unambiguously define magnetic moments But can define magnetic moment distribution Factor of 10 magnification Contemporary theoretical estimates: 1 – 10 x 10-10 Largest value reduces discrepancy expt.↔theory from 3.3σ to below 2σ. 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. Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

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**Looking deeper Forschungszentrum Julich, IKP: 20 July. 71/81**

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

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**Deep inelastic scattering**

Quark discovery experiment at SLAC ( , 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) 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: [nucl-th], Rev. Mod. Phys. 82 (2010) pp Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

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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) Wijesooriya et al. Phys.Rev. C 72 (2005) Behaviour at large-x is inconsistent with pQCD; viz, expt. (1-x)1+ε cf. QCD (1-x)2+γ Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

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**First QCD-based calculation**

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

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**Computation of qvπ(x) T+**

In the Bjorken limit; viz., q2 → ∞, P · q → − ∞ but x := −q2/2P · q fixed, Wμν(q;P) ~ qvπ(x) Plainly, this can be computed given all the information we have at our disposal from the DSEs T+ T– Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

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Hecht, Roberts, Schmidt Phys.Rev. C 63 (2001) Computation of qvπ(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 After the first DSE computation, experimentalists again became interested in the process because DSEs agreed with pQCD but disagreed with the data, and other models Disagreement on the “valence domain,” which is uniquely sensitive to M(p2) 2.61/1.27= factor of 2 in the exponent Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

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Hecht, Roberts, Schmidt Phys.Rev. C 63 (2001) Reanalysis of qvπ(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) ) 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 qvπ(x) at large-x. 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: [nucl-th], Rev. Mod. Phys. 82 (2010) pp Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

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Distribution Functions of the Nucleon and Pion in the Valence Region, Roy J. Holt and Craig D. Roberts, arXiv: [nucl-th], Rev. Mod. Phys. 82 (2010) pp Reanalysis of qvπ(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. Aicher, Schäfer, Vogelsang, “Soft-Gluon Resummation and the Valence Parton Distribution Function of the Pion,” Phys. Rev. Lett. 105 (2010) Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

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**Current status of qvπ(x)**

Trang, Bashir, Roberts & Tandy, “Pion and kaon valence-quark parton distribution functions,” arXiv: [nucl-th], Phys. Rev. C 83, (R) (2011) [5 pages] Current status of qvπ(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? Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

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**qvπ(x) & qvK(x) ms ≈ 24 mu & Ms ≈ 1.25 Mu**

Trang, Bashir, Roberts & Tandy, “Pion and kaon valence-quark parton distribution functions,” arXiv: [nucl-th], Phys. Rev. C 83, (R) (2011) [5 pages] qvπ(x) & qvK(x) ms ≈ 24 mu & Ms ≈ 1.25 Mu Expect the s-quark to carry more of the kaon’s momentum than the u-quark, so that xsK(x) peaks at larger value of x than xuK(x) Expectation confirmed in computations, with s-quark distribution peaking at 15% larger value of x Even though deep inelastic scattering is a high-Q2 process, constituent-like mass-scale explains the shift xuK(x) xsK(x) xuπ(x) Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

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Trang, Bashir, Roberts & Tandy, “Pion and kaon valence-quark parton distribution functions,” arXiv: [nucl-th], Phys. Rev. C 83, (R) (2011) [5 pages] uK(x)/uπ(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 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 ≈ (fπ/fK)2 (Mu /Ms)4 = 0.3 Value of ratio at x=1 is a fixed point of the evolution equations Hence, it’s a very strong test of nonperturbative dynamics Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

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**Grand Unification Forschungszentrum Julich, IKP: 20 July. 71/81**

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

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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/ , is a well-understood tool that is accurate for pseudoscalar and vector mesons: e.g., Prediction of elastic pion and kaon form factors: nucl-th/ Anomalous neutral pion processes – γπγ & BaBar anomaly: [nucl-th] Pion and kaon valence-quark distribution functions: [nucl-th] Unification of these and other observables – ππ scattering: hep-ph/ It can readily be extended to explain properties of the light neutral pseudoscalar mesons: [nucl-th] Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

55
**DSEs & Baryons Dynamical chiral symmetry breaking (DCSB)**

R.T. Cahill et al., Austral. J. Phys. 42 (1989) 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 SUc(3): Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

56
**Faddeev Equation Linear, Homogeneous Matrix equation**

R.T. Cahill et al., Austral. J. Phys. 42 (1989) Faddeev Equation quark exchange ensures Pauli statistics quark 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 diquark composed of strongly-dressed quarks bound by dressed-gluons Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

57
**Photon-nucleon current**

L. Chang, Y. –X. Liu and C.D. Roberts arXiv: [nucl-th] Phys. Rev. Lett. 106 (2011) Photon-nucleon current Vertex must contain the dressed-quark anomalous magnetic moment: Lecture IV Composite nucleon must interact with photon via nontrivial current constrained by Ward-Takahashi identities DSE → BSE → Faddeev equation plus current → nucleon form factors In a realistic calculation, the last three diagrams represent 8-dimensional integrals, which can be evaluated using Monte-Carlo techniques Oettel, Pichowsky, Smekal Eur.Phys.J. A8 (2000) Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81 Go to ”Insert (View) | Header and Footer" to add your organization, sponsor, meeting name here; then, click "Apply to All"

58
**critical importance of DCSB in explanation of real-world observables.**

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

59
**DCSB & EDMs Forschungszentrum Julich, IKP: 20 July. 71/81**

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

60
Hecht, Roberts, Schmidt Phys.Rev. C64 (2001) 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. Possession of an EDM by J=½ particle is a violation of T-symmetry Standard Model: |dn,p |< ecm Current experiment: |dp |<7.9 x ecm & |dn |<1.6 x ecm Suppose nonzero dp or dn is measured What is the relation to current-quark EDMs: du,d,s,… ? SU(6) Quark Model: dp=⅓(4du-dd) & dn=⅓(4dd-du) But these are constituent-quark EDMs, which aren’t readily defined in SM! Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

61
**Nucleon EDM Nucleon EDM described by**

Hecht, Roberts, Schmidt Phys.Rev. C64 (2001) 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 Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

62
**Quark EDMs Faddeev equation approach**

Hecht, Roberts, Schmidt Phys.Rev. C64 (2001) 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[hu±,hd±] – current-quark gyroelectric ratios May now compute relation between hq & hN Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

63
**Quark EDMs Nucleon structure factors**

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

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

65
Hecht, Roberts, Schmidt Phys.Rev. C64 (2001) 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 dN as a means of constraining extensions of the Standard Model. Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

66
**Dynamical Chiral Symmetry Breaking Condensates?**

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

67
**Connecting QCD with coupled-channels calculations**

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

68
**Man kann über Alles sprechen Aber nicht über eine Stunde**

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

69
**Epilogue Forschungszentrum Julich, IKP: 20 July. 71/81**

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

70
**Masses of ground and excited-state hadrons Hannes L. L**

Masses of ground and excited-state hadrons Hannes L.L. Roberts, Lei Chang, Ian C. Cloët and Craig D. Roberts, arXiv: [nucl-th] Few Body Systems (2011) pp. 1-25 Epilogue New perspectives on the quark condensate Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) QCD is the most interesting part of the Standard Model and Nature’s only example of an essentially nonperturbative fundamental theory Dyson-Schwinger equations: 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 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 Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

71
**Return to nucleon EDM with Mario P., Michael R-M & student**

Spectrum of exotics Nucleon -> Roper transition form factor Spectrum of baryons with strangeness Continuing work on PDFs with extension to nucleon Computation of meson-nucleon form factors for possible use in coupled channels computations Review articles Form.Factors=Rev.Mod.Phys., DSEs=Prog.Part.Nucl.Phys. & Eur.Math.Soc. & Chin.J.Phys. Where do we go from here? Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81 Go to ”Insert (View) | Header and Footer" to add your organization, sponsor, meeting name here; then, click "Apply to All"

72
**Dynamical Chiral Symmetry Breaking Condensates?**

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

73
Dichotomy of the pion 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: (8) Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

74
**Gell-Mann – Oakes – Renner Relation**

Behavior of current divergences under SU(3) x SU(3). Murray Gell-Mann, R.J. Oakes , B. Renner Phys.Rev. 175 (1968) This paper derives a relation between mπ2 and the expectation-value < π|u0|π>, where uo 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 u0 was not expressed in terms of current-quark fields PCAC-hypothesis (partial conservation of axial current) is used in the derivation Subsequently, the concepts of soft-pion theory Operator expectation values do not change as t=mπ2 → t=0 to take < π|u0|π> → < 0|u0|0> … in-pion → in-vacuum (8) Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

75
**Gell-Mann – Oakes – Renner Relation**

- (0.25GeV)3 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. (8) Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

76
**Note of Warning These authors argue that dynamical chiral-**

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

77
**QCD Sum Rules Introduction of the gluon vacuum condensate**

QCD and Resonance Physics. Sum Rules. M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov Nucl.Phys. B147 (1979) ; citations: 3713 Introduction of the gluon vacuum condensate and development of “sum rules” relating properties of low-lying hadronic states to vacuum condensates (10) Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

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QCD Sum Rules QCD and Resonance Physics. Sum Rules. M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov Nucl.Phys. B147 (1979) ; citations: 3781 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 (10) Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

79
**No references to this phrase before 1980**

“quark condensate” Instantons in non-perturbative QCD vacuum, MA Shifman, AI Vainshtein… - Nuclear Physics B, 1980 Instanton density in a theory with massless quarks, Exotic new quarks and dynamical symmetry breaking, WJ Marciano - Physical Review D, 1980 The pion in QCD J Finger, JE Mandula… - Physics Letters B, 1980 No references to this phrase before 1980 (10) 6550+ references to this phrase since 1980 Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

80
**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 Ωcosm. const. = 0.76 This mismatch is a bit of a problem. Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

81
**Resolution? Quantum Healing Central: Paranormal Psychic Forums:**

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

82
**Paradigm shift: In-Hadron Condensates**

Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) Brodsky and Shrock, PNAS 108, 45 (2011) 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. M. Burkardt, Phys.Rev. D58 (1998) QCD Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

83
**Paradigm shift: In-Hadron Condensates**

Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) Brodsky and Shrock, PNAS 108, 45 (2011) 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 fπ and ρπ Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

84
**Paradigm shift: In-Hadron Condensates**

Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) Brodsky and Shrock, PNAS 108, 45 (2011) 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 fπ and ρπ And Chiral limit Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

85
**Paradigm shift: In-Hadron Condensates**

“Void that is truly empty solves dark energy puzzle” Rachel Courtland, New Scientist 4th Sept. 2010 “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.” Cosmological Constant: Putting QCD condensates back into hadrons reduces the mismatch between experiment and theory by a factor of 1046 Possibly by far more, if technicolour-like theories are the correct paradigm for extending the Standard Model Craig Roberts: A Year in the life of DSEs Forschungszentrum Julich, IKP: 20 July. 71/81

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Baryon Properties from Continuum-QCD Craig D. Roberts Physics Division Argonne National Laboratory & School of Physics Peking University.

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