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## Craig Roberts Physics Division

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**Tackling Continuum**Strong QCD Published collaborations: 2010-present Rocio BERMUDEZ (U Michoácan); Chen CHEN (ANL, IIT, USTC); Xiomara GUTIERREZ-GUERRERO (U Michoácan); Trang NGUYEN (KSU); Si-xue QIN (PKU); Hannes ROBERTS (ANL, FZJ, UBerkeley); Lei CHANG (ANL, FZJ, PKU); Huan CHEN (BIHEP); Ian CLOËT (UAdelaide); Bruno EL-BENNICH (São Paulo); David WILSON (ANL); Adnan BASHIR (U Michoácan); Stan BRODSKY (SLAC); Gastão KREIN (São Paulo) Roy HOLT (ANL); Mikhail IVANOV (Dubna); Yu-xin LIU (PKU); Robert SHROCK (Stony Brook); Peter TANDY (KSU) Craig Roberts Physics Division Students Early-career scientists**Confinement**Craig Roberts: Tackling Continuum Strong QCD**X**Confinement Coloursinglets Craig Roberts: Tackling Continuum Strong QCD • Gluon and Quark Confinement • No coloured states have yet been observed to reach a detector • 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 • But light-quarks are ubiquitous • Confinement entails quark-hadron duality; i.e., that all observable consequences of QCD can, in principle, be computed using an hadronic basis.**G. Bali et al., PoS LAT2005 (2006) 308**Confinement “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.” anti-Bs Bs Craig Roberts: Tackling Continuum Strong QCD • 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, nor in qQ systems**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, • or more complicated nonanalyticities … • Spectral density no longer positive semidefinite • & hence state cannot exist in observable spectrum Craig Roberts: Tackling Continuum Strong QCD • Confinement is expressed through a dramatic 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); Roberts, Williams & Krein (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: Tackling Continuum Strong QCD • 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**Qin et al., Phys. Rev. C 84 042202(Rapid Comm.) (2011)**Rainbow-ladder truncation DSE Studies – Phenomenology of gluon • Running gluon mass • Gluon is massless in ultraviolet in agreement with pQCD • Massive in infrared • mG(0) = 0.67-0.81 GeV • mG(mG2) = 0.53-0.64 GeV Craig Roberts: Tackling Continuum Strong QCD • Wide-ranging study of π & ρ properties • Effective coupling • Agrees with pQCDin ultraviolet • Saturates in infrared • α(0)/π = 8-15 • α(mG2)/π = 2-4**Dynamical ChiralSymmetry Breaking**Craig Roberts: Tackling Continuum Strong QCD**Dynamical ChiralSymmetry Breaking**Craig Roberts: Tackling Continuum Strong QCD • Whilst confinement is contentious … • 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 Mass from nothing! DSE prediction of DCSB confirmed Craig Roberts: Tackling Continuum Strong QCD 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**C.D. Roberts, Prog. Part. Nucl. Phys. 61 (2008) 50 M. Bhagwat & P.C. Tandy, AIP Conf.Proc. 842 (2006) 225-227 Hint of lattice-QCD support for DSE prediction of violation of reflection positivity Craig Roberts: Tackling Continuum Strong QCD 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**Craig Roberts: Tackling Continuum Strong QCD Jlab 12GeV: This region scanned by 2<Q2<9 GeV2 elastic & transition form factors.**π or K**The Future of Drell-Yan N Craig Roberts: Tackling Continuum Strong QCD Valence-quark PDFs and PDAs probe this critical and complementary region**Science Challenges for the coming decade: 2013-2022**Craig Roberts: Tackling Continuum Strong QCD Search for exotic hadrons Exploit opportunities provided by new data on nucleon elastic and transition form factors Precision experimental study of valence region, and theoretical computation of distribution functions and distribution amplitudes Develop QCD as a probe for physics beyond the Standard Model**Overarching Science Challenges for the coming decade:**2013-2022 Discover meaning of confinement, and its relationship to DCSB – the origin of visible mass • Search for exotic hadrons • Exploit opportunities provided by new data on nucleon elastic and transition form factors • Precision experimental study of valence region, and theoretical computation of distribution functions and distribution amplitudes • Develop QCD as a probe for physics beyond the Standard Model Craig Roberts: Tackling Continuum Strong QCD**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: Tackling Continuum Strong QCD • 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 • Through QCD's 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 spectrum, Elastic & transition form factors, Parton distribution fns can be used to chart β-function’s long-range behaviour.**Mass function exhibits inflexion**• point at QIR ≈ mG ≈ 0.6GeV • So … pQCD is definitely invalid • for momentaQ<QIR • E.g., use of DGLAP equations • cannot be justified in QCD at • Q<QIR=0.6GeV, irrespective of order. • 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 Necessary Precondition Essentially nonperturbative Craig Roberts: Tackling Continuum Strong QCD • Experiment ↔ Theory comparison leads to an understanding of long-range behaviour of strong running-coupling • However, if one wants to draw reliable conclusions about Q2-dependence of QCD’s running coupling, • Then, approach must veraciously express Q2-dependence of QCD’s running masseS • True for ALL observables • From spectrum … • through elastic & transition form factors … • to PDFs and GPDs … etc.**Nature’s Strong Messenger**Craig Roberts: Tackling Continuum Strong QCD**Nature’s strong messenger – Pion**Craig Roberts: Tackling Continuum Strong QCD • 1947 – Pion discovered by Cecil Frank Powell • Studied tracks made by cosmic rays using photographic emulsion plates • Despite the fact that Cavendish Lab said method is incapable of “reliable and reproducible precision measurements.” • Mass measured in scattering ≈ 250-350 me**Nature’s strong messenger – Pion**Craig Roberts: Tackling Continuum Strong QCD • The beginning of Particle Physics • Then came • Disentanglement of confusion between (1937) muon and pion – similar masses • Discovery of particles with “strangeness” (e.g., kaon1947-1953) • Subsequently, a complete spectrum of mesons and baryons with mass below ≈1 GeV • 28 states • Became clear that pion is “too light” - hadrons supposed to be heavy, yet …**Dichotomy of the pion**Craig Roberts: Tackling Continuum Strong QCD • 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:**Dichotomy of the pionGoldstone mode and bound-state**HIGHLY NONTRIVIAL Impossible in quantum mechanics Only possible in asymptotically-free gauge theories Craig Roberts: Tackling Continuum Strong QCD • The correct understanding of pion observables; e.g. mass, decay constant and form factors, requires an approach to contain a • well-defined and validchiral limit; • and an accurate realisation of dynamical chiral symmetry breaking.**Persistent Challenge**Truncation Craig Roberts: Tackling Continuum Strong QCD**Persistent challenge in application of DSEs**Invaluable check on practical truncation schemes Craig Roberts: Tackling Continuum Strong QCD • Infinitely many coupled equations: Kernel of the equation for the quark self-energy involves: • Dμν(k) – dressed-gluon propagator • Γν(q,p) – dressed-quark-gluon vertex each of which satisfies its own DSE, etc… • Coupling between equations necessitates a truncation • Weak coupling expansion ⇒ produces every diagram in perturbation theory • Otherwise useless for the nonperturbative problems in which we’re interested**Relationship must be preserved by any truncation**Highly nontrivial constraint FAILURE has an extremely high cost – loss of any connection with QCD Persistent challenge- truncation scheme Quark propagator satisfies a gap equation Axial-Vector vertex Satisfies an inhomogeneous Bethe-Salpeter equation Kernels of these equations are completely different But they must be intimately related Craig Roberts: Tackling Continuum Strong QCD • Symmetries associated with conservation of vector and axial-vector currents are critical in arriving at a veracious understanding of hadron structure and interactions • Example: axial-vector Ward-Takahashi identity • Statement of chiral symmetry and the pattern by which it’s broken in quantum field theory**Persistent challenge- truncation scheme**quark-antiquark scattering kernel Craig Roberts: Tackling Continuum Strong QCD These observations show that symmetries relate the kernel of the gap equation – nominally a one-body problem, with that of the Bethe-Salpeter equation – considered to be a two-body problem Until 1995/1996 people had no idea what to do Equations were truncated, sometimes with good phenomenological results, sometimes with poor results Neither good nor bad could be explained**Persistent challenge- truncation scheme**Craig Roberts: Tackling Continuum Strong QCD • There are now two nonperturbative & symmetry preserving truncation schemes • 1995 – H.J. Munczek, Phys. Rev. D 52 (1995) 4736,Dynamical chiral symmetry breaking, Goldstone’s theorem and the consistency of the Schwinger-Dyson and Bethe-Salpeter Equations 1996 – A. Bender, C.D. Roberts and L. von Smekal, Phys.Lett. B 380 (1996) 7, Goldstone Theorem and Diquark Confinement Beyond Rainbow Ladder Approximation • 2009 – Lei Chang and C.D. Roberts, Phys. Rev. Lett. 103 (2009) 081601, 0903.5461 [nucl-th], Sketching the Bethe-Salpeter kernel • Enables proof of numerous exact results**Some of many**Exact Results Craig Roberts: Tackling Continuum Strong QCD**Maris, Roberts and Tandy**nucl-th/9707003, Phys.Lett. B420 (1998) 267-273 Pion’s Goldberger-Treiman relation Pseudovector components necessarily nonzero. Cannot be ignored! Miracle: two body problem solved, almost completely, once solution of one body problem is known Exact in Chiral QCD Craig Roberts: Tackling Continuum Strong QCD • Pion’s Bethe-Salpeter amplitude Solution of the Bethe-Salpeter equation • Dressed-quark propagator • Axial-vector Ward-Takahashi identity entails**Dichotomy of the pionGoldstone mode and bound-state**fπ Eπ(p2) = B(p2) Craig Roberts: Tackling Continuum Strong QCD • Goldstone’s theorem has a pointwise expression in QCD; Namely, in the chiral limit the wave-function for the two-body bound-state Goldstone mode is intimately connected with, and almost completely specified by, the fully-dressed one-body propagator of its characteristic constituent • The one-body momentum is equated with the relative momentum of the two-body system**Dynamical Chiral Symmetry BreakingVacuum Condensates?**Craig Roberts: Tackling Continuum Strong QCD**Universal Conventions**Craig Roberts: Tackling Continuum Strong QCD • 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 non-perturbative vacuum state, characterized by many non-vanishing condensates such as the gluon condensate or the quark condensate. These condensates characterize the normal phase or the confined phase of quark matter.”**Universal Misapprehensions**Craig Roberts: Tackling Continuum Strong QCD • 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.**New Paradigm“in-hadron condensates”**Craig Roberts: Tackling Continuum Strong QCD**Won’t say more here but reserve comments for a workshop in**São Paolo next week. Relevant References Craig Roberts: Tackling Continuum Strong QCD arXiv:1202.2376 Confinement contains condensates Stanley J. Brodsky, Craig D. Roberts, Robert Shrock, Peter C. Tandy arXiv:1109.2903 [nucl-th], Phys. Rev. C85 (2012) 012201(RapCom), Expanding the concept of in-hadron condensatesLei Chang, Craig D. Roberts and Peter C. Tandy arXiv:1005.4610 [nucl-th], Phys. Rev. C82 (2010) 022201(RapCom.) New perspectives on the quark condensate, Brodsky, Roberts, Shrock, Tandy arXiv:0905.1151 [hep-th], PNAS 108, 45 (2011) Condensates in Quantum Chromodynamics and the Cosmological Constant, Brodsky and Shrock, hep-th/0012253 The Quantum vacuum and the cosmological constant problem, Svend Erik Rugh and HenrikZinkernagel.**Dynamical Chiral Symmetry BreakingImportance of being**well-dressed for quarks & mesons Craig Roberts: Tackling Continuum Strong QCD**Strong-interaction: QCD**Dressed-quark-gluon vertex Craig Roberts: Tackling Continuum Strong QCD • 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 …**Dressed-quark-gluon vertex**Craig Roberts: Tackling Continuum Strong QCD • 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: Tackling Continuum Strong QCD • 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 (or LO+NLO) in a symmetry-preserving truncation scheme; viz., • Dμν(k) = fully-dressed • 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: Tackling Continuum Strong QCD • 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: Tackling Continuum Strong QCD • 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: Tackling Continuum Strong QCD • 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: Tackling Continuum Strong QCD • 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**QCD and dressed-quark anomalous magnetic moments**Craig Roberts: Tackling Continuum Strong QCD • 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! • In QCD, chiral symmetry is dynamically broken, strongly • What then?**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: Tackling Continuum Strong QCD • 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: Tackling Continuum Strong QCD • 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 AdnanBashir, Wed.: Dynamical chiral symmetry breaking and the fermion--gauge-boson vertex, Bashir, Bermudez, Chang & Roberts, arXiv:1112.4847 [nucl-th], Phys. Rev. C in press Craig Roberts: Tackling Continuum Strong QCD • 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 ACM • AEM is opposite in sign but of • roughly equal magnitude • as ACM Craig Roberts: Tackling Continuum Strong QCD**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 • Potentially important for elastic and transition form factors, etc. • Indeed, for any process involving photons • coupling to a dressed-quark Craig Roberts: Tackling Continuum Strong QCD**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: Tackling Continuum Strong QCD

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