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Published collaborations in 2010/2011 Craig Roberts Physics Division Adnan BASHIR (U Michoacan); Students Stan BRODSKY (SLAC); Early-career scientists 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) QCD’s Challenges Understand emergent phenomena Quark and Gluon Confinement No matter how hard one strikes the proton, one cannot liberate an individual quark or gluon Dynamical Chiral Symmetry Breaking Very unnatural pattern of bound state masses; e.g., Lagrangian (pQCD) quark mass is small but . . . no degeneracy between JP=+ and JP=− (parity partners) Neither of these phenomena is apparent in QCD’s Lagrangian Yet they are the dominant determining characteristics of real-world QCD. QCD – Complex behaviour arises from apparently simple rules. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 2 Universal Truths Hadron spectrum, and elastic and transition form factors provide unique information about long-range interaction between lightquarks and distribution of hadron's characterising properties amongst its QCD constituents. Dynamical Chiral Symmetry Breaking (DCSB) is most important mass generating mechanism for visible matter in the Universe. Higgs mechanism is (almost) irrelevant to light-quarks. Running of quark mass entails that calculations at even modest Q2 require a Poincaré-covariant approach. Covariance + M(p2) require existence of quark orbital angular momentum in hadron's rest-frame wave function. Confinement is expressed through a violent change of the propagators for coloured particles & can almost be read from a plot of a states’ dressed-propagator. It is intimately connected with DCSB. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 3 Strong-interaction: QCD Asymptotically free – Perturbation theory is valid and accurate tool at large-Q2 – Hence chiral limit is defined Essentially nonperturbative for Q2 < 2 GeV2 Nature’s only example of truly nonperturbative, fundamental theory A-priori, no idea as to what such a theory can produce Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 4 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 5 X 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: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 6 G. Bali et al., PoS LAT2005 (2006) 308 “Note that the time is not a linear function of the distance but dilated within the string breaking region. On a linear time scale string breaking takes place rather rapidly. […] light pair creation seems to occur non-localized and instantaneously.” Confinement Infinitely heavy-quarks plus 2 flavours with mass = ms – Lattice spacing = 0.083fm – String collapses within one lattice time-step R = 1.24 … 1.32 fm – Energy stored in string at collapse Ecsb = 2 ms – (mpg made via linear interpolation) No flux tube between light-quarks Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. Bs anti-Bs LightCone 2011, SMU 23-27 May - 77pgs 7 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); … Confined particle Normal particle complex-P2 complex-P2 timelike axis: P2<0 o Real-axis mass-pole splits, moving into pair(s) of complex conjugate poles or branch points o Spectral density no longer positive semidefinite & hence state cannot exist in observable spectrum Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 8 Dressed-gluon propagator A.C. Aguilar et al., Phys.Rev. D80 (2009) 085018 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: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 9 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: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 10 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 excited states are a great deal more sensitive to the long-range behaviour of the β-function than those of the ground states. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 11 Maris & Tandy, Phys.Rev. C60 (1999) 055214 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 = 3.2 • α(1 GeV2)/π = 0.35 Running gluon mass – Gluon is massless in ultraviolet in agreement with pQCD – Massive in infrared • mG(0) = 0.76 GeV • mG(1 GeV2) = 0.46 GeV Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 12 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 13 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: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 14 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: Dyson-Schwinger Equations and Continuum QCD. C.D. Roberts, Prog. Part. Nucl. Phys. 61 (2008) 50 M. Bhagwat & P.C. Tandy, AIP Conf.Proc. 842 (2006) 225-227 LightCone 2011, SMU 23-27 May - 77pgs 15 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: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 16 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: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 17 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 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. elastic & transition form factors. LightCone 2011, SMU 23-27 May - 77pgs 18 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 19 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) m m 2 This is impossible in quantum mechanics, for which one always finds: mbound state mconstituent Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 20 Gell-Mann – Oakes – Renner Behavior of current divergences under SU(3) x SU(3). Relation Murray Gell-Mann, R.J. Oakes , B. Renner Phys.Rev. 175 (1968) 2195-2199 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 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 21 Gell-Mann – Oakes – Renner Behavior of current divergences under SU(3) x SU(3). Relation Murray Gell-Mann, R.J. Oakes , B. Renner Phys.Rev. 175 (1968) 2195-2199 PCAC hypothesis; viz., pion field dominates the divergence of Zhou Guangzhao 周光召 the axial-vector current Born 1929 Changsha, Hunan province Soft-pion theorem Commutator is chiral rotation Therefore, isolates explicit chiral-symmetry breaking term in the putative Hamiltonian In QCD, this is m qq and one therefore has Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 22 Gell-Mann – Oakes – Renner Relation Theoretical physics at its best. - (0.25GeV)3 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. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 23 Note of Warning Chiral Magnetism (or Magnetohadrochironics) A. Casher and L. Susskind, Phys. Rev. D9 (1974) 436 These authors argue that dynamical chiralsymmetry breaking can be realised as a property of hadrons, instead of via a nontrivial vacuum exterior to the measurable degrees of freedom Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. 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. LightCone 2011, SMU 23-27 May - 77pgs 24 QCD Sum Rules QCD and Resonance Physics. Sum Rules. M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov Nucl.Phys. B147 (1979) 385-447; citations: 3713 Introduction of the gluon vacuum condensate and development of “sum rules” relating properties of low-lying hadronic states to vacuum condensates Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 25 QCD Sum Rules QCD and Resonance Physics. Sum Rules. M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov Nucl.Phys. B147 (1979) 385-447; 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 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 26 “quark condensate” 1960-1980 Instantons in non-perturbative QCD vacuum, MA Shifman, AI Vainshtein… - Nuclear Physics B, 1980 Instanton density in a theory with massless quarks, MA Shifman, AI Vainshtein… - Nuclear Physics B, 1980 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 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 27 Universal Conventions Wikipedia: (http://en.wikipedia.org/wiki/QCD_vacuum) “The QCD vacuum is the vacuum state of quantum chromodynamics (QCD). It is an example of a nonperturbative vacuum state, characterized by many nonvanishing condensates such as the gluon condensate or the quark condensate. These condensates characterize the normal phase or the confined phase of quark matter.” Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 28 Universal Misapprehensions Since 1979, DCSB has commonly been associated literally with a spacetimeindependent 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 QCDcondensates 8 GN Experimentally, the answer is 4 QCD 2 0 3H 10 46 Ωcosm. const. = 0.76 This mismatch is a bit of a problem. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 29 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. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 30 Paradigm shift: In-Hadron Condensates Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201 Brodsky and Shrock, PNAS 108, 45 (2011) M. Burkardt, Phys.Rev. D58 (1998) 096015 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. QCD Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 31 Paradigm shift: In-Hadron Condensates Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201 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: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 32 Paradigm shift: In-Hadron Condensates Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201 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. Chiral limit – No qualitative difference (0; ) qq 0 between fπ and ρπ – And Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 33 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 QCD 46 picture of nothingness may not be needed. A calmer view of the vacuum would QCDcondensates 8 GN 10 2 also help resolve a nagging inconsistency with dark energy, the elusive force 3H 0 thought to be speeding up the expansion of the universe.” 4 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: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 34 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 35 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.Roberts, R. Delbourgo, A.G. Williams, H.J. Munczek, M.R. Pennington, A. Bashir, A. Kizilersu, P.Tandy, L. Chang, Y.-X. Liu … Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 36 Dressedquark-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: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 37 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., Bender, Roberts & von Smekal – Dμν(k) = dressed, as described previously – Γν(q,p) = γμ Phys.Lett. B380 (1996) 7-12 • … plainly, key nonperturbative effects are missed and cannot be recovered through any step-by-step improvement procedure Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 38 Gap Equation General Form Dμν(k) – dressed-gluon propagator good deal of information available Γν(q,p) – dressed-quark-gluon vertex Information accumulating If kernels of Bethe-Salpeter and gap equations don’t match, one won’t even get right charge for the pion. Suppose one has in hand – from anywhere – the exact form of the dressed-quark-gluon vertex What is the associated symmetrypreserving Bethe-Salpeter kernel?! Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 39 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: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 40 Bethe-Salpeter Equation Lei Chang and C.D. Roberts General Form 0903.5461 [nucl-th] Phys. Rev. Lett. 103 (2009) 081601 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: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 41 Ward-Takahashi Identity Lei Chang and C.D. Roberts Bethe-Salpeter Kernel 0903.5461 [nucl-th] Phys. Rev. Lett. 103 (2009) 081601 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: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 42 Relativistic quantum mechanics Dirac equation (1928): Pointlike, massive fermion interacting with electromagnetic field Spin Operator Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 43 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: 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), 0.001 16 – vertex correction The agreement with e experiment established e quantum electrodynamics as a valid tool. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 44 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: • F1(k2) = 1 • F2(k2=0) = α /[2 π] Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 45 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: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 46 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: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 47 Schwinger’s result? One-loop calculation: Plainly, one obtains Schwinger’s result for me2 ≠ 0 However, e e F2(k2) = 0 when me2 = 0 There is no Gordon identity: m=0 So, no mixing γμ ↔ σμν Results are unchanged at every order in perturbation theory … owing to symmetry … magnetic moment interaction is forbidden in a theory with manifest chiral symmetry Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 48 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 αs/2π ” ~ (– ⅙) QED-result But, in QED and QCD, the anomalous chromo- and electromagnetic moments vanish identically in the chiral limit! Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 49 L. Chang, Y. –X. Liu and C.D. Roberts arXiv:1009.3458 [nucl-th] Phys. Rev. Lett. 106 (2011) 072001 Dressed-quark anomalous magnetic moments DCSB Three strongly-dressed and essentiallynonperturbative contributions to dressed-quark-gluon vertex: Ball-Chiu term •Vanishes if no DCSB •Appearance driven by STI Anom. chrom. mag. mom. contribution to vertex •Similar properties to BC term •Strength commensurate with lattice-QCD Skullerud, Bowman, Kizilersu et al. hep-ph/0303176 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 50 Lattice-QCD Dressed-quark anomalous – m = 115 MeV chromomagnetic moment Nonperturbative result is two orders-of-magnitude larger than the perturbative computation – This level of Quenched Skullerud, Kizilersu et al. magnification is lattice-QCD JHEP 0304 (2003) 047 typical of DCSB – cf. Quark mass function: M(p2=0)= 400MeV M(p2=10GeV2)=4 MeV Prediction from perturbative QCD Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 51 L. Chang, Y. –X. Liu and C.D. Roberts arXiv:1009.3458 [nucl-th] Phys. Rev. Lett. 106 (2011) 072001 Dressed-quark anomalous magnetic moments DCSB Three strongly-dressed and essentiallynonperturbative contributions to dressed-quark-gluon vertex: Ball-Chiu term •Vanishes if no DCSB •Appearance driven by STI Anom. chrom. mag. mom. contribution to vertex •Similar properties to BC term •Strength commensurate with lattice-QCD Skullerud, Bowman, Kizilersu et al. hep-ph/0303176 Role and importance is novel discovery •Essential to recover pQCD •Constructive interference with Γ5 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 52 L. Chang, Y. –X. Liu and C.D. Roberts arXiv:1009.3458 [nucl-th] Phys. Rev. Lett. 106 (2011) 072001 Dressed-quark anomalous magnetic moments Formulated and solved general Bethe-Salpeter equation Obtained dressed electromagnetic vertex Confined quarks don’t have a mass-shell Factor of 10 magnification o Can’t unambiguously define magnetic moments o But can define magnetic moment distribution AEM is opposite in sign but of roughly equal magnitude as ACM Full vertex ME κACM κAEM 0.44 -0.22 0.45 0 0.048 Rainbow-ladder 0.35 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 53 L. Chang, Y. –X. Liu and C.D. Roberts arXiv:1009.3458 [nucl-th] Phys. Rev. Lett. 106 (2011) 072001 Dressed-quark anomalous magnetic moments Formulated and solved general Bethe-Salpeter equation Obtained dressed electromagnetic vertex Confined quarks don’t have a mass-shell Factor of 10 magnification 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. 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. (I.C. Cloët et al.) Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 54 Location of zero marks –m2meson Dressed Vertex & Meson Spectrum Experiment Rainbow- One-loop ladder corrected a1 1230 759 885 ρ 770 644 764 Mass splitting 455 115 121 Ball-Chiu Full vertex Splitting known experimentally for more than 35 years Hitherto, no explanation Systematic symmetry-preserving, Poincaré-covariant DSE truncation scheme of nucl-th/9602012. o Never better than ∼ ⅟₄ of splitting Constructing kernel skeleton-diagram-by-diagram, DCSB cannot be faithfully expressed: Full impact of M(p2) cannot be realised! Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 55 BC: zero moves deeper Full vertex: for both ρ &deeper a1 zero moves for a1 Bothshallower masses grow but for ρ Problem solved Dressed Vertex & Meson Spectrum Experiment Rainbow- One-loop ladder corrected Ball-Chiu Full vertex a1 1230 759 885 1066 1128 1270 ρ 770 644 764 924 919 790 Mass splitting 455 115 121 142 209 480 Fully consistent treatment of Ball-Chiu vertex o Retain λ3 – term but ignore Γ4 & Γ5 o Some effects of DCSB built into vertex & Bethe-Salpeter kernel Big impact on σ – π complex But, clearly, not the complete answer. Fully-consistent treatment of complete vertex Ansatz Lei Chang & C.D. Roberts, Promise of 1st reliable prediction arXiv:1104.4821 [nucl-th] Tracing massess of ground-state of light-quark hadron spectrum light-quark mesons Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 56 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 57 Unification of Meson & Baryon Properties Correlate the masses of meson and baryon ground- and excited-states within a single, symmetry-preserving framework Symmetry-preserving means: Poincaré-covariant & satisfy relevant Ward-Takahashi identities Constituent-quark model has hitherto been the most widely applied spectroscopic tool; whilst its weaknesses are emphasized by critics and acknowledged by proponents, it is of continuing value because there is nothing better that is yet providing a bigger picture. Nevertheless, no connection with quantum field theory & therefore not with QCD not symmetry-preserving & therefore cannot veraciously connect meson and baryon properties Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 58 R.T. Cahill et al., Austral. J. Phys. 42 (1989) 129-145 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 observation that an interaction which describes colour-singlet mesons also generates nonpointlike quark-quark (diquark) correlations in the colour-antitriplet channel Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. rqq ≈ rπ LightCone 2011, SMU 23-27 May - 77pgs 59 R.T. Cahill et al., Austral. J. Phys. 42 (1989) 129-145 Faddeev Equation quark exchange ensures Pauli statistics quark Linear, Homogeneous Matrix equation diquark composed of stronglydressed quarks bound by dressed-gluons 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: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 60 Diquarks in QCD 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 “Spectrum” of nonpointlike quark-quark correlations Observed in H.L.L. Roberts et al., 1102.4376 [nucl-th] Phys. Rev. C in press – DSE studies in QCD – 0+ & 1+ in Lattice-QCD Scalar diquark form factor – r0+ ≈ rπ Axial-vector diquarks – r1+ ≈ rρ BOTH essential CraigZero relation with old notion of pointlike constituent-like diquarks Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 61 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 Baryons & diquarks Provided numerous insights into baryon structure; e.g., There is a causal connection between mΔ - mN & m1+- m0+ mΔ mN Physical splitting grows rapidly with increasing diquark mass difference mΔ - mN Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 62 Provided numerous insights into baryon structure; e.g., mN ≈ 3 M & mΔ ≈ M+m1+ Baryons & diquarks Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 63 Vertex contains dressed-quark anomalous magnetic moment Photon-nucleon current Composite nucleon must interact with photon via nontrivial current constrained by Ward-Takahashi identities DSE, BSE, Faddeev equation, current → nucleon form factors Oettel, Pichowsky, Smekal Eur.Phys.J. A8 (2000) 251-281 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 64 Survey of nucleon electromagnetic form factors I.C. Cloët et al, arXiv:0812.0416 [nucl-th], Few Body Syst. 46 (2009) pp. 1-36 Nucleon Form Factors Unification of meson and nucleon form factors. Very good description. Quark’s momentumdependent anomalous magnetic moment has observable impact & materially improves agreement in all cases. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 65 I.C. Cloët, C.D. Roberts, et al. arXiv:0812.0416 [nucl-th] I.C. Cloët, C.D. Roberts, et al. In progress pG (Q ) p E p M 2 2 G (Q ) 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: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 66 I.C. Cloët, C.D. Roberts, et al. In progress Quark anomalous magnetic moment has big impact on proton ratio nG (Q ) n E n M 2 2 G (Q ) But little impact on the neutron ratio Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 67 I.C. Cloët, C.D. Roberts, et al. arXiv:0812.0416 [nucl-th] nG (Q ) n E n M 2 2 G (Q ) New JLab data: S. Riordan et al., arXiv:1008.1738 [nucl-ex] Phys.Rev.Lett. 105 (2010) 262302 DSE-prediction This evolution is very sensitive to momentum-dependence of dressed-quark propagator Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 68 Legend: Particle Data Group H.L.L. Roberts et al. EBAC Jülich Hadron Spectrum o Symmetry-preserving unification of the computation of meson & baryon masses o rms-rel.err./deg-of-freedom = 13% o PDG values (almost) uniformly overestimated in both cases - room for the pseudoscalar meson cloud?! Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. Masses of ground and excited-state hadrons H.L.L. Roberts et al., arXiv:1101.4244 [nucl-th] Few Body Systems (2011) pp. 1-25 69 LightCone 2011, SMU 23-27 May - 77pgs Baryon Spectrum In connection with EBAC's analysis, dressed-quark Faddeev-equation predictions for bare-masses agree within rms-relative-error of 14%. Notably, EBAC finds a dressed-quark-core for the Roper resonance, at a mass which agrees with Faddeev Eq. prediction. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 70 N. Suzuki et al., Phys.Rev.Lett. 104 (2010) 042302 EBAC examined the dynamical origins of the two poles associated with the Roper resonance are examined. Both of them, together with the next higher resonance in the P11 partial wave were found to have the same originating bare state Coupling to the mesonbaryon continuum induces multiple observed resonances from the same bare state. All PDG identified resonances consist of a core state and meson-baryon components. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. EBAC & the Roper resonance LightCone 2011, SMU 23-27 May - 77pgs 71 Legend: Particle Data Group H.L.L. Roberts et al. EBAC Now and Jülich Hadron Spectrum for the foreseeable future, QCD-based theory will provide only dressed-quark core masses; EBAC or EBAC-like tools necessary for mesons and baryons Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 72 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 Baryon Spectrum diquark content Fascinating suggestion: nucleon’s first excited state = almost entirely axial-vector diquark, despite the dominance of scalar diquark in ground state 0+ 77% 1+ 23% 100% 100% 0- 51% 43% 1- 49% 57% 100% 100% 100% Owes fundamentally to close relationship between scalar diquark and pseudoscalar meson And this follows from dynamical chiral symmetry breaking Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 73 Nucleon to Roper Transition Form Factors Extensive CLAS @ JLab Programme has produced the first measurements of nucleon-to-resonance transition form factors I. Aznauryan et al., Results of the N* Program at JLab arXiv:1102.0597 [nucl-ex] Theory challenge is to explain the measurements Notable result is zero in F2p→N*, explanation of which is a real challenge to theory. DSE study connects appearance of zero in F2p→N* with axial-vector-diquark dominance in Roper resonance and structure of form factors of J=1 state I.C. Cloët, C.D. Roberts, et al. Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. In progress LightCone 2011, SMU 23-27 May - 77pgs 74 Nucleon to Roper Transition Form Factors Tiator and Vanderhaeghen – in progress – Empirically-inferred light-front-transverse charge density – Positive core plus negative annulus Readily explained by dominance of axial-vector diquark – Considering isospin and charge – Negative d-quark twice as likely to be delocalised from always-positive core than positive u-quark d 2 {uu} u + 1 {ud} Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 75 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 76 Epilogue Confinement is almost Certainly the origin of DCSB Dynamical chiral symmetry breaking (DCSB) – mass from nothing for 98% of visible matter – is a reality o Expressed in M(p2), with observable signals in experiment Poincaré covariance Crucial in description of contemporary data e.g., BaBar Fully-self-consistent treatment of an interaction anomaly Comprehensive treatment of vast body of observables Essential if experimental data is truly to be understood. Dyson-Schwinger equations: o single framework, with IR model-input turned to advantage, “almost unique in providing unambiguous path from a defined interaction → Confinement & DCSB → Masses → radii → form McLerran & Pisarski factors → distribution functions → etc.” Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. arXiv:0706.2191 [hep-ph] LightCone 2011, SMU 23-27 May - 77pgs 77 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 78 I.C. Cloët, C.D. Roberts, et al. arXiv:0812.0416 [nucl-th] New JLab data: S. Riordan et al., arXiv:1008.1738 [nucl-ex] DSE-prediction p ,d 2 p ,u 2 F1 (Q ) F1 (Q ) Location of zero measures relative strength of scalar and axial-vector qq-correlations Brooks, Bodek, Budd, Arrington fit to data: hep-ex/0602017 Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. LightCone 2011, SMU 23-27 May - 77pgs 79 I.C. Cloët, C.D. Roberts, et al. arXiv:0812.0416 [nucl-th] Neutron Structure Function at high x SU(6) symmetry Deep inelastic scattering – the Nobel-prize winning quark-discovery experiments Reviews: S. Brodsky et al. NP B441 (1995) W. Melnitchouk & A.W.Thomas PL B377 (1996) 11 N. Isgur, PRD 59 (1999) R.J. Holt & C.D. Roberts RMP (2010) Craig Roberts: Dyson-Schwinger Equations and Continuum QCD. pQCD, uncorrelated Ψ DSE: 0+ & 1+ qq 0+ qq only Distribution of neutron’s momentum amongst quarks on the valence-quark domain LightCone 2011, SMU 23-27 May - 77pgs 80