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## Craig Roberts Physics Division

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**A Year in the Life of**Dyson-Schwinger Equations Craig Roberts Physics Division http://www.phy.anl.gov/theory/staff/cdr.html**Published collaborations in 2010/2011**Students Early-career scientists Craig Roberts: A Year in the life of DSEs 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)**Top Open Questions in Physics**Craig Roberts: A Year in the life of DSEs**Excerpt from the Top-10**Craig Roberts: A Year in the life of DSEs • 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.**Nuclear Science Advisory CouncilLong Range Plan**Craig Roberts: A Year in the life of DSEs • 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**Understanding NSAC’sLong 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. Craig Roberts: A Year in the life of DSEs • 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?**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. Craig Roberts: A Year in the life of DSEs**Dyson-SchwingerEquations**• 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 • 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?**Confinement**Craig Roberts: A Year in the life of DSEs**X**Confinement Craig Roberts: A Year in the life of DSEs • 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.**Confinement**Confined particle Normal 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 • 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); …**Dressed-gluon propagator**A.C. Aguilar et al., Phys.Rev. D80 (2009) 085018 IR-massive but UV-massless, confined gluon perturbative, massless gluon massive , unconfined gluon Craig Roberts: A Year in the life of DSEs • 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**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. Craig Roberts: A Year in the life of DSEs • 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 universalβ-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.**Charting the interaction between light-quarks**Craig Roberts: A Year in the life of DSEs • Through QCD's Dyson-Schwinger equations (DSEs) the pointwisebehaviour 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.**Qin, Chang, Roberts, et al., in progress at IKP**DSE Studies – Phenomenology of gluon • 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 • Wide-ranging study of π & ρ properties • Effective coupling • Agrees with pQCDin ultraviolet • Saturates in infrared • α(0)/π = 9 - 15 • α(mG2)/π = 2 – 4**Dynamical Chiral Symmetry BreakingMass Gap**Craig Roberts: A Year in the life of DSEs**Dynamical Chiral Symmetry Breaking**Craig Roberts: A Year in the life of DSEs • 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.**Frontiers of Nuclear Science:Theoretical Advances**C.D. Roberts, Prog. Part. Nucl. Phys. 61 (2008) 50 M. Bhagwat & P.C. Tandy, AIP Conf.Proc. 842 (2006) 225-227 Craig Roberts: A Year in the life of DSEs 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.**Frontiers of Nuclear Science:Theoretical Advances**Mass from nothing! DSE prediction of DCSB confirmed Craig Roberts: A Year in the life of DSEs 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.**Frontiers of Nuclear Science:Theoretical Advances**Hint of lattice-QCD support for DSE prediction of violation of reflection positivity Craig Roberts: A Year in the life of DSEs 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.**12GeVThe Future of JLab**Jlab 12GeV: Scanned by 2<Q2<9 GeV2 elastic & transition form factors. Craig Roberts: A Year in the life of DSEs 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.**Dynamical Chiral Symmetry BreakingImportance of being**well-dressed for quarks & mesons Craig Roberts: A Year in the life of DSEs**Strong-interaction: QCD**Dressed-quark-gluon vertex Craig Roberts: A Year in the life of DSEs • 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 …**Dressed-quark-gluon vertex**Craig Roberts: A Year in the life of DSEs • 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.**Gap EquationGeneral Form**Bender, Roberts & von Smekal Phys.Lett. B380 (1996) 7-12 Craig Roberts: A Year in the life of DSEs • 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**Gap EquationGeneral Form**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 • 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?!**Bethe-Salpeter EquationBound-State DSE**Craig Roberts: A Year in the life of DSEs • 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.**Bethe-Salpeter EquationGeneral Form**Lei Chang and C.D. Roberts 0903.5461 [nucl-th] Phys. Rev. Lett. 103 (2009) 081601 Craig Roberts: A Year in the life of DSEs • Equivalent exact bound-state equation but in thisform 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)**Ward-Takahashi IdentityBethe-Salpeter Kernel**Lei Chang and C.D. Roberts 0903.5461 [nucl-th] Phys. Rev. Lett. 103 (2009) 081601 iγ5 iγ5 Craig Roberts: A Year in the life of DSEs • Now, for first time, it’s possible to formulate an Ansatz for Bethe-Salpeter kernel given anyform 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**Relativistic quantum mechanics**Spin Operator Craig Roberts: A Year in the life of DSEs • Dirac equation (1928): Pointlike, massive fermion interacting with electromagnetic field**Massive point-fermionAnomalous magnetic moment**0.001 16 e e Craig Roberts: A Year in the life of DSEs • 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), • vertex correction • The agreement with experiment established quantum electrodynamics as a valid tool.**Fermion electromagnetic current – General structure**Craig Roberts: A Year in the life of DSEs 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]**Magnetic moment of a masslessfermion?**Craig Roberts: A Year in the life of DSEs • 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)**Magnetic moment of a masslessfermion?**Craig Roberts: A Year in the life of DSEs • 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**Schwinger’s result?**e e m=0 So, no mixing γμ↔ σμν Craig Roberts: A Year in the life of DSEs 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**QCD and dressed-quark anomalous magnetic moments**Craig Roberts: A Year in the life of DSEs • 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!**L. Chang, Y. –X. Liu and C.D. RobertsarXiv:1009.3458**[nucl-th] Phys. Rev. Lett. 106 (2011) 072001 Dressed-quark anomalousmagnetic moments DCSB • Ball-Chiu term • Vanishes if no DCSB • Appearance driven by STI • Anom. chrom. mag. mom. • contribution to vertex • Similar properties to BC term • Strength commensurate with lattice-QCD • Skullerud, Bowman, Kizilersuet al. • hep-ph/0303176 Craig Roberts: A Year in the life of DSEs • Three strongly-dressed and essentially- nonperturbative contributions to dressed-quark-gluon vertex:**Dressed-quark anomalous chromomagnetic moment**Quenched lattice-QCD Skullerud, Kizilersuet 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 • 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.**L. Chang, Y. –X. Liu and C.D. RobertsarXiv:1009.3458**[nucl-th] Phys. Rev. Lett. 106 (2011) 072001 Dressed-quark anomalousmagnetic moments DCSB • Ball-Chiu term • Vanishes if no DCSB • Appearance driven by STI • Anom. chrom. mag. mom. • contribution to vertex • Similar properties to BC term • Strength commensurate with lattice-QCD • Skullerud, Bowman, Kizilersuet al. • hep-ph/0303176 • Role and importance is • novel discovery • Essential to recover pQCD • Constructive interference with Γ5 Craig Roberts: A Year in the life of DSEs • Three strongly-dressed and essentially- nonperturbative contributions to dressed-quark-gluon vertex:**L. Chang, Y. –X. Liu and C.D. RobertsarXiv:1009.3458**[nucl-th] Phys. Rev. Lett. 106 (2011) 072001 Dressed-quark anomalousmagnetic 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 Craig Roberts: A Year in the life of DSEs**L. Chang, Y. –X. Liu and C.D. RobertsarXiv:1009.3458**[nucl-th] Phys. Rev. Lett. 106 (2011) 072001 Dressed-quark anomalousmagnetic 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 muong-2 – via Box diagram, • which is not constrained by extant data. Craig Roberts: A Year in the life of DSEs**Looking deeper**Craig Roberts: A Year in the life of DSEs**Deep inelastic scattering**Probability that a quark/gluon within the target will carry a fraction x of the bound-state’s light-front momentum Distribution Functions of the Nucleon and Pion in the Valence Region, Roy J. Holt and Craig D. Roberts, arXiv:1002.4666 [nucl-th], Rev. Mod. Phys. 82 (2010) pp. 2991-3044 Craig Roberts: A Year in the life of DSEs • Quark discovery experiment at SLAC (1966-1978, Nobel Prize in 1990) • Completely different to elastic scattering • Blow the target to pieces instead of keeping only those events where it remains intact. • Cross-section is interpreted as a measurement of the momentum-fraction probability distribution for quarks and gluons within the target hadron: q(x), g(x)**Pion’svalence-quark distributions**Pion Craig Roberts: A Year in the life of DSEs • 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) Wijesooriyaet al. Phys.Rev. C 72 (2005) 065203 Behaviour at large-x is inconsistent with pQCD; viz, expt. (1-x)1+ε cf. QCD (1-x)2+γ**First QCD-based calculation**Craig Roberts: A Year in the life of DSEs**Computation of qvπ(x)**T+ T– Craig Roberts: A Year in the life of DSEs 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**Hecht, Roberts, Schmidt**Phys.Rev. C 63 (2001) 025213 Computation of qvπ(x) • 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 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**Hecht, Roberts, Schmidt**Phys.Rev. C 63 (2001) 025213 Reanalysis of qvπ(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:1002.4666 [nucl-th], Rev. Mod. Phys. 82 (2010) pp. 2991-3044 Craig Roberts: A Year in the life of DSEs • After the first DSE computation, the “Conway et al.” data were reanalysed, this time at next-to-leading-order (Wijesooriyaet al. Phys.Rev. C 72 (2005) 065203) • The new analysis produced a much larger exponent than initially obtained; viz., β=1.87, but now it disagreed equally with NJL-model results and the DSE prediction • NB. Within pQCD, one can readily understand why adding a higher-order correction leads to a suppression of qvπ(x) at large-x.**Distribution Functions of the Nucleon and Pion in the**Valence Region, Roy J. Holt and Craig D. Roberts, arXiv:1002.4666 [nucl-th], Rev. Mod. Phys. 82 (2010) pp. 2991-3044 Reanalysis of qvπ(x) Aicher, Schäfer, Vogelsang, “Soft-Gluon Resummation and the Valence Parton Distribution Function of the Pion,”Phys. Rev. Lett. 105 (2010) 252003 Craig Roberts: A Year in the life of DSEs • 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.**Trang, Bashir, Roberts & Tandy,“Pion and kaon**valence-quark parton distribution functions,” arXiv:1102.2448 [nucl-th],Phys. Rev. C 83, 062201(R) (2011) [5 pages] Current status of qvπ(x) Craig Roberts: A Year in the life of DSEs 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?