Craig Roberts Physics Division

Explanation and Prediction of Observables using Continuum Strong QCD

Craig Roberts

Physics Division

Explaining Observables in Continuum Strong QCD (156p)

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Collaborators: 2011-Present1. Rocio BERMUDEZ (U Michoácan);2. Xiomara GUTIERREZ-GUERRERO (U Michoácan);3. S. HERNÁNDEZ (U Michoácan);4. Trang NGUYEN (KSU);5. Khépani RAYA (U Michoácan);6. Hannes ROBERTS (ANL, FZJ, UBerkeley);7. Chien-Yeah SENG (UW-Mad)8. Kun-lun WANG (PKU);9. Chen CHEN (USTC);10. J. Javier COBOS-MARTINEZ (U.Sonora);11. Mario PITSCHMANN (ANL & UW-Mad);12. Si-xue QIN (U. Frankfurt am Main);13. Jorge SEGOVIA (ANL);14. David WILSON (ODU);15. Lei CHANG (U.Adelaide); 16. Ian CLOËT (ANL);17. Bruno EL-BENNICH (São Paulo);

Craig Roberts: Nanjing University, October 2013

18. Adnan BASHIR (U Michoácan);19. Stan BRODSKY (SLAC);20. Gastão KREIN (São Paulo)21. Roy HOLT (ANL);22. Mikhail IVANOV (Dubna);23. Yu-xin LIU (PKU);24. Michael RAMSEY-MUSOLF (UW-Mad)25. Alfredo RAYA (U Michoácan);26. Sebastian SCHMIDT (IAS-FZJ & JARA);27. Robert SHROCK (Stony Brook);28. Peter TANDY (KSU);29. Tony THOMAS (U.Adelaide)30. Shaolong WAN (USTC)

StudentsPostdocsAsst. Profs.


3Craig Roberts: Nanjing University, October 2013

Enormous progress since

2010

arXiv:1310.2651 [nucl-th]

http://inspirehep.net/record/1257993?ln=en





Standard Model

Quantum Chromodynamics

QCD: The piece of the Standard Model that describes strong interactions.

Hadron Physics is a nonperturbative problem in QCD

Notwithstanding the 2013 Nobel Prize in Physics, the origin of 98% of the visible mass in the Universe is – somehow – found within QCD

Craig Roberts: Nanjing University, October 2013Explaining Observables in Continuum Strong QCD (156p)

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Hadrons

Hadron: Any of a class of subatomic particles that are composed of quarks and/or gluons and take part in the strong interaction. Examples: proton, neutron, & pion.International Scientific Vocabulary:

hadr- thick, heavy (from Greek hadros thick) + 2onFirst Known Use: 1962

Baryon: hadron with half-integer-spinMeson: hadron with integer-spin



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FacilitiesCraig Roberts: Nanjing University, October 2013


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FacilitiesQCD Machines

China– Beijing Electron-Positron Collider

Germany– COSY (Jülich Cooler Synchrotron)– ELSA (Bonn Electron Stretcher and Accelerator)– MAMI (Mainz Microtron)– Facility for Antiproton and Ion Research,

under construction near Darmstadt.New generation experiments in 2018 (perhaps)

Japan– J-PARC (Japan Proton Accelerator Research Complex),

under construction in Tokai-Mura, 150km NE of Tokyo.New generation experiments to begin soon

− KEK: Tsukuba, Belle Collaboration Switzerland (CERN)

– Large Hadron Collider: ALICE Detector and COMPASS Detector“Understanding deconfinement and chiral-symmetry restoration”



http://www.ihep.ac.cn/english/E-Bepc/index.htm

http://www2.fz-juelich.de/ikp/cosy/en/



http://www-elsa.physik.uni-bonn.de/Hadronenphysik/index_en.html

http://www.kph.uni-mainz.de/eng/index.php



http://www.fair-center.eu/Location.208.0.html

http://j-parc.jp/index-e.html

http://belle.kek.jp/

http://aliweb.cern.ch/

http://wwwcompass.cern.ch/

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USA– Thomas Jefferson National Accelerator Facility,

Newport News, VirginiaNature of cold hadronic matterUpgrade underway

Construction cost $310-million New generation experiments in 2014

– Relativistic Heavy Ion Collider, Brookhaven National Laboratory, Long Island, New YorkStrong phase transition, 10μs after Big Bang



A three dimensional view of the calculated particle paths resulting from collisions occurring within RHIC's STAR detector

http://jlab.org/index.html

http://www.bnl.gov/rhic/

http://www.bnl.gov/rhic/STAR.asp

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USA– Thomas Jefferson National Accelerator Facility,

Newport News, VirginiaNature of cold hadronic matterUpgrade underway

Construction cost $310-million New generation experiments in 2014



http://jlab.org/index.html


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Science Challenges for the coming decade: 2013-2022

Exploit opportunities provided by new data on hadron elastic and transition form factors– Chart infrared evolution of QCD’s coupling and

dressed-masses – Reveal correlations that are key to nucleon structure– Expose the facts and fallacies in modern descriptions

of hadron structure


All composite systems have “Form Factors”, which describe the distribution of an observable quantity amongst the constituents.


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Science Challenges for the coming decade: 2013-2022

Precision experimental study of (far) valence region (Bjorken-x > 0.5), and theoretical computation of distribution functions and distribution amplitudes– Computation is critical– Without it, no amount of data will reveal anything

about the theory underlying the phenomena of strong interaction physics


Parton distribution functions (PDFs) and distribution amplitudes (PDAs) are a quantum field theory analogue of wave functions. They have a probability interpretation and hence relate to concepts familiar from quantum mechanics.

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Overarching Science Challenges for the coming

decade: 2013-2022 Discover the meaning of confinement Determine its connection with DCSB

(dynamical chiral symmetry breaking) Elucidate their signals in observables

… so experiment and theory together can map the nonperturbative behaviour of the strong interactionIn my view, it is unlikely that two phenomena, so critical in the Standard Model and tied to the dynamical generation of a single mass-scale, can have different origins and fates.




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What is QCD?Craig Roberts: Nanjing University, October 2013

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QCD is a Theory Very likely a self-contained, nonperturbatively renormalisable

and hence well defined Quantum Field TheoryThis is not true of QED – cannot be defined nonperturbatively

No confirmed breakdown over an enormous energy domain: 0 GeV < E < 8000 GeV

Increasingly likely that any extension of the Standard Model will be based on the paradigm established by QCD – Extended Technicolour: electroweak symmetry breaks via a

fermion bilinear operator in a strongly-interacting non-Abelian theory. (Andersen et al. “Discovering Technicolor” Eur.Phys.J.Plus 126 (2011) 81)Higgs sector of the SM becomes an effective description of a more fundamental fermionic theory, similar to the Ginzburg-Landau theory of superconductivity



(not an effective theory)



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Contrast: so-called Effective Field Theories

EFTs applicable over a very restricted energy domain; e.g., ChPT known to breakdown for E > 2mπ

Can be used to help explore how features of QCD influence observables



QCD appears valid at all energy scales that have been tested so far: no breakdown below

E ≈ 60000 mπ

Cannot be used to test QCD Any mismatch between EF-Theory and experiment owes to an error in the formulation of one or conduct of the other

Can Cannot

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What is QCD?

Lagrangian of QCD– G = gluon fields– Ψ = quark fields

The key to complexity in QCD … gluon field strength tensor

Generates gluon self-interactions, whose consequences are quite extraordinary



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QED is the archetypal gauge field theory Perturbatively simple

but nonperturbatively undefined

Chracteristic feature: Light-by-light scattering; i.e., photon-photon interaction – leading-order contribution takesplace at order α4. Extremely small probability because α4 ≈10-9 !

cf.Quantum Electrodynamics



Relativistic Quantum Gauge Field Theory: Interactions mediated by vector boson exchange Vector bosons are perturbatively-massless

Similar interaction in QED Special feature of QCD – gluon self-interactions

What is QCD?


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3-gluon vertex

4-gluon vertex



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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 (now known) example of a truly nonperturbative, fundamental theory A-priori, no idea as to what such a theory can produce


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What is Confinement?


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Light quarks & Confinement

A unit area placed midway between the quarks and perpendicular to the line connecting them intercepts a constant number of field lines, independent of the distance between the quarks. This leads to a constant force between the quarks – and a large force at that, equal to about 16 metric tons.”



Folklore … JLab Hall-D Conceptual Design Report(5) “The color field lines between a quark and an anti-quark form flux tubes.

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Problem: 16 tonnes of force makes a lot of pions.



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Problem: 16 tonnes of force makes a lot of pions.



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Light quarks & Confinement In the presence of

light quarks, pair creation seems to occur non-localized and instantaneously

No flux tube in a theory with light-quarks.

Flux-tube is not the correct paradigm for confinement in hadron physics



G. Bali et al., PoS LAT2005 (2006) 308

http://inspirebeta.net/record/694488?ln=en



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Confinement QFT Paradigm: – Confinement is expressed through a dramatic

change in the analytic structure of propagators for coloured states

– It can almost be read from a plot of the dressed-propagator for a coloured state


complex-P2 complex-P2

o Real-axis mass-pole splits, moving into pair(s) of complex conjugate singularities, (or other qualitatively analogous structures chracterised by a dynamically generated mass-scale)o State described by rapidly damped wave & hence state cannot exist in observable spectrum

Normal particle Confined particle

timelike axis: P2<0

s ≈ 1/Im(m) ≈ 1/2ΛQCD ≈ ½fm

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In the study of hadrons, attention should turn from equal-time potential models toward the continuum bound-state problem in quantum field theory

Such approaches offer the possibility of posing simultaneously the questions – What is confinement?– What is dynamical chiral symmetry breaking?– How are they related?– What are their empirical signals?



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Dynamical Chiral Symmetry Breaking



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Mass from NothingCraig Roberts: Nanjing University, October 2013


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DCSB is a fact in QCD– Dynamical, not spontaneous

• Add nothing to QCD , no Higgs field, nothing! • Effect achieved purely through the quark+gluon dynamics.

– It’s the most important mass generating mechanism for visible matter in the Universe. • Responsible for ≈98% of the proton’s mass.• Higgs mechanism is (almost) irrelevant to light-quarks.




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Gap EquationCraig Roberts: Nanjing University, October 2013


DCSB

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Mass from nothing!


C.D. Roberts, Prog. Part. Nucl. Phys. 61 (2008) 50M. Bhagwat & P.C. Tandy, AIP Conf.Proc. 842 (2006) 225-227 In QCD, all “constants” of

quantum mechanics are actually strongly momentum dependent: couplings, number density, mass, etc.

So, a quark’s mass depends on its momentum.

Mass function can be calculated and is depicted here.

Continuum- and Lattice-QCD are in agreement: the vast bulk of the light-quark mass comes from a cloud of gluons, dragged along by the quark as it propagates.






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Just one of the terms that are summed in a solution of the rainbow-ladder gap equation

Where does the mass come from?

Deceptively simply picture Corresponds to the sum of a countable infinity of diagrams.

NB. QED has 12,672 α5 diagrams Impossible to compute this in perturbation theory.

The standard algebraic manipulation tools are just inadequate



αS23

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Enigma of MassCraig Roberts: Nanjing University, October 2013


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Bound-states in Quantum Field Theory

Mass and “Wave Function” are obtained from a Bethe-Salpeter equation– Generalisation of the Lippmann-Schwinger equation

The pion … Nature’s strong-interaction messenger … is a critical example



Sketching the Bethe-Salpeter kernel, Lei Chang and Craig D. Roberts, arXiv:0903.5461 [nucl-th], Phys. Rev. Lett. 103 (2009) 081601 (4 pages)

http://www.slac.stanford.edu/spires/find/hep/www?eprint=arXiv:0903.5461



http://link.aps.org/doi/10.1103/PhysRevLett.103.081601





Pion’s Goldberger-Treiman relation


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Pion’s Bethe-Salpeter amplitudeSolution of the Bethe-Salpeter equation

Dressed-quark propagator

Axial-vector Ward-Takahashi identity entails

Pseudovector componentsnecessarily nonzero.

Cannot be ignored!

Exact inChiral QCD


Miracle: two body problem solved, almost completely, once solution of one body problem is known

Maris, Roberts and Tandynucl-th/9707003, Phys.Lett. B420 (1998) 267-273



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Dichotomy of the pionGoldstone mode and bound-state

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



fπ Eπ(p2) = B(p2)

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Enigma of mass

The quark level Goldberger-Treiman relation shows that DCSB has a very deep and far reaching impact on physics within the strong interaction sector of the Standard Model; viz.,

Goldstone's theorem is fundamentally an expression of equivalence between the one-body problem and the two-body problem in the pseudoscalar channel.

This emphasises that Goldstone's theorem has a pointwise expression in QCD

Hence, pion properties are an almost direct measure of the dressed-quark mass function.

Thus, enigmatically, the properties of the massless pion are the cleanest expression of the mechanism that is responsible for almost all the visible mass in the universe.



fπ Eπ(p2) = B(p2)

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In QCD, Gluons, too, become massive

Not just quarks … Gluons also have a

gap equation …1/k2 behaviour signals essential singularity in the running coupling:

Impossible to reach in perturbation theory



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422 )(

kkm

g

gg

)( 2kconst

e

Interaction model for the gap equationSi-xue Qin, Lei Chang, Y.-x.Liu, C.D. Roberts and D.J. WilsonarXiv:1108.0603 [nucl-th], Phys. Rev. C 84 (2011) 042202(R) [5 pages]




http://link.aps.org/doi/10.1103/PhysRevC.84.042202



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Vacuum Condensates?Craig Roberts: Nanjing University, October 2013


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“Orthodox Vacuum” Vacuum = “frothing sea” Hadrons = bubbles in that “sea”,

containing nothing but quarks & gluonsinteracting perturbatively, unless they’re near the bubble’s boundary, whereat they feel they’re trapped!



u

u

ud

u ud

du


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However, just like gluons and quarks, and for the same reasons:Condensates are confined within hadrons. There are no vacuum condensates.

Historically, DCSB came to be associated with a presumed existence of spacetime-independent condensates that permeated the universe.


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Background Worth noting that nonzero vacuum expectation values of local

operators in QCD—the so-called vacuum condensates—are phenomenological parameters, which were introduced at a time of limited computational resources in order to assist with the theoretical estimation of essentially nonperturbative strong-interaction matrix elements.

A universality of these condensates was assumed, namely, that the properties of all hadrons could be expanded in terms of the same condensates. While this helps to retard proliferation, there are nevertheless infinitely many of them.

As qualities associated with an unmeasurable state (the vacuum), such condensates do not admit direct measurement. Practitioners have attempted to assign values to them via an internally consistent treatment of many separate empirical observables.

However, only one, the so-called quark condensate, is attributed a value with any confidence.



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Dichotomy of the pionMass Formula for 0— Mesons

Consider the case of light quarks; namely, mq ≈ 0– If chiral symmetry is dynamically broken, then

• fH5 → fH50 ≠ 0

• ρH5 → – < q-bar q> / fH50 ≠ 0

both of which are independent of mq

Hence, one arrives at the corollary



Gell-Mann, Oakes, Renner relation1968mm 2

The so-called “vacuum quark condensate.” More later about this.


We now have sufficient information to address the question of just what is this so-called “vacuum quark condensate.”



Spontaneous(Dynamical)Chiral Symmetry Breaking

The 2008 Nobel Prize in Physics was divided, one half awarded to Yoichiro Nambu

"for the discovery of the mechanism of spontaneous broken symmetry in subatomic physics"



Nambu – Jona-LasinioModel


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Treats a chirally-invariant four-fermion Lagrangian & solves the gap equation in Hartree-Fock approximation (analogous to rainbow truncation)

Possibility of dynamical generation of nucleon mass is elucidated Essentially inequivalent vacuum states are identified (Wigner and

Nambu states) & demonstration thatthere are infinitely many, degenerate but distinct Nambu vacua, related by a chiral rotation

Nontrivial Vacuum is “Born”Craig Roberts: Nanjing University, October 2013

Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. I

Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345–358 Dynamical Model Of Elementary Particles

Based On An Analogy With Superconductivity. IIY. Nambu, G. Jona-Lasinio, Phys.Rev. 124 (1961) 246-254

Gell-Mann – Oakes – RennerRelation


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This paper derives a relation between mπ

2 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


Behavior of current divergences under SU(3) x SU(3).Murray Gell-Mann, R.J. Oakes , B. Renner Phys.Rev. 175 (1968) 2195-2199



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PCAC hypothesis; viz., pion field dominates the divergence of the axial-vector current

Soft-pion theorem

In QCD, this is and one therefore has


Behavior of current divergences under SU(3) x SU(3).Murray Gell-Mann, R.J. Oakes , B. Renner Phys.Rev. 175 (1968) 2195-2199

Commutator is chiral rotationTherefore, isolates explicit chiral-symmetry breaking term in the putative Hamiltonian

qqm

Zhou Guangzhao 周光召Born 1929 Changsha, Hunan province



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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 matrix element, there is no problem.

Problem arises if one over-interprets this number, which textbooks have been doing for a VERY LONG TIME.


- (0.25GeV)3

Original Note of Warning



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 aproperty of hadrons, instead of via a nontrivial vacuum exterior to the measurable degrees of freedom

The essential ingredient required for a spontaneous symmetry breakdown in a composite system is the existence of a divergent number of constituents – DIS provided evidence for divergent sea of low-momentum partons – parton model.

QCD Sum Rules

Introduction of the gluon vacuum condensate

and development of “sum rules” relating properties of low-lying hadronic states to vacuum condensates



QCD and Resonance Physics. Sum Rules.M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov Nucl.Phys. B147 (1979) 385-447; citations: 3713

QCD Sum Rules

Introduction of the gluon vacuum condensate

and development of “sum rules” relating properties of low-lying hadronic states to vacuum condensates

At this point (1979), the cat was out of the bag: a physical reality was seriously attributed to a plethora of vacuum condensates



QCD and Resonance Physics. Sum Rules.M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov Nucl.Phys. B147 (1979) 385-447; citations: 3781

“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 QCDJ Finger, JE Mandula… - Physics Letters B, 1980

No references to this phrase before 1980Explaining Observables in Continuum Strong QCD (156p)


8300+ REFERENCES TO THIS PHRASE SINCE 1980

http://linkinghub.elsevier.com/retrieve/pii/0550321380903041








http://link.aps.org/doi/10.1103/PhysRevD.21.2425




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 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.”



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http://en.wikipedia.org/wiki/QCD_vacuum

QCD

How should one approach this problem, understand it, within Quantum ChromoDynamics?

1) Are the quark and gluon “condensates” theoretically well-defined?

2) Is there a physical meaning to this quantity or is it merely just a mass-dimensioned parameter in a theoretical computation procedure?



0||0 qq 1973-1974

QCD

Why does it matter?



0||0 qq 1973-1974

“Dark Energy”

Two pieces of evidence for an accelerating universe1) Observations of type Ia supernovae

→ the rate of expansion of the Universe is growing2) Measurements of the composition of the Universe point to a

missing energy component with negative pressure: CMB anisotropy measurements indicate that the Universe is at

Ω0 = 1 ⁺⁄₋ 0.04. In a flat Universe, the matter density and energy density must sum to the critical density. However, matter only contributes about ⅓ of the critical density,

ΩM = 0.33 ⁺⁄₋ 0.04. Thus, ⅔ of the critical density is missing.



“Dark Energy”

In order not to interfere with the formation of structure (by inhibiting the growth of density perturbations) the energy density in this component must change more slowly than matter (so that it was subdominant in the past).

Accelerated expansion can be accommodated in General Relativity through the Cosmological Constant, Λ. Einstein introduced the repulsive effect of the cosmological

constant in order to balance the attractive gravity of matter so that a static universe was possible. He promptly discarded it after the discovery of the expansion of the Universe.



In order to have escaped detection, the missing energy must be smoothly distributed.

412 )10(8

GeVG

obs

Contemporary cosmological observations mean:

“Dark Energy”

The only possible covariant form for the energy of the (quantum) vacuum; viz.,

is mathematically equivalent to the cosmological constant.

“It is a perfect fluid and precisely spatially uniform”“Vacuum energy is almost the perfect candidate for

dark energy.”



“The advent of quantum field theory made consideration of the cosmological constant obligatory not optional.”Michael Turner, “Dark Energy and the New Cosmology”

obsQCD 4610

“Dark Energy”

QCD vacuum contributionIf chiral symmetry breaking is expressed in a nonzero

expectation value of the quark bilinear, then the energy difference between the symmetric and broken phases is of order

MQCD≈0.3 GeVOne obtains therefrom:



“The biggest embarrassment in theoretical physics.”

Mass-scale generated by spacetime-independent condensate

Enormous and even greater contribution from Higgs VEV!

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GMOR Relation




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GMOR Relation Valuable to highlight the precise form of the Gell-Mann–Oakes–

Renner (GMOR) relation: Eq. (3.4) in Phys.Rev. 175 (1968) 2195

o mπ is the pion’s mass o Hχsb is that part of the hadronic Hamiltonian density which

explicitly breaks chiral symmetry. Crucial to observe that the operator expectation value in this

equation is evaluated between pion states. Moreover, the virtual low-energy limit expressed in the equation is

purely formal. It does not describe an achievable empirical situation.


Expanding the concept of in-hadron condensatesLei Chang, Craig D. Roberts and Peter C. TandyarXiv:1109.2903 [nucl-th], Phys. Rev. C85 (2012) 012201(R)






http://prc.aps.org/abstract/PRC/v85/i1/e012201




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GMOR Relation In terms of QCD quantities, GMOR relation entails

o mudζ = mu

ζ + mdζ … the current-quark masses

o S π

ζ(0) is the pion’s scalar form factor at zero momentum transfer, Q2=0

RHS is proportional to the pion σ-term Consequently, using the connection between the σ-term and the

Feynman-Hellmann theorem, GMOR relation is actually the statement










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GMOR Relation Using

it follows that

This equation is valid for any values of mu,d, including the neighbourhood of the chiral limit, wherein













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GMOR Relation Consequently, in the neighbourhood of the chiral limit

This is a QCD derivation of the commonly recognised form of the GMOR relation.

Neither PCAC nor soft-pion theorems were employed in the analysis.

Nature of each factor in the expression is abundantly clear; viz., chiral limit values of matrix elements that explicitly involve the hadron.









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Expanding the Concept




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In-Hadron Condensates

Plainly, the in-pseudoscalar-meson condensate can be represented through the pseudoscalar meson’s scalar form factor at zero momentum transfer Q2 = 0.

Using an exact mass formula for scalar mesons, one proves the in-scalar-meson condensate can be represented in precisely the same way.

By analogy, and with appeal to demonstrable results of heavy-quark symmetry, the Q2 = 0 values of vector- and pseudovector-meson scalar form factors also determine the in-hadron condensates in these cases.

This expression for the concept of in-hadron quark condensates is readily extended to the case of baryons.

Via the Q2 = 0 value of any hadron’s scalar form factor, one can extract the value for a quark condensate in that hadron which is a reasonable and realistic measure of dynamical chiral symmetry breaking.









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Hadron Charges

Hadron Form factor matrix elements Scalar charge of a hadron is an intrinsic property of

that hadron … no more a property of the vacuum than the hadron’s electric charge, axial charge, tensor charge, etc. …



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Confinement contains

condensatesCraig Roberts: Nanjing University, October 2013



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Confinement Confinement is essential to the validity of the notion of in-hadron

condensates. Confinement makes it impossible to construct gluon or quark

quasiparticle operators that are nonperturbatively valid. So, although one can define a perturbative (bare) vacuum for QCD,

it is impossible to rigorously define a ground state for QCD upon a foundation of gluon and quark quasiparticle operators.

Likewise, it is impossible to construct an interacting vacuum – a BCS-like trial state – and hence DCSB in QCD cannot rigorously be expressed via a spacetime-independent coherent state built upon the ground state of perturbative QCD.

Whilst this does not prevent one from following this path to build practical models for use in hadron physics phenomenology, it does invalidate any claim that theoretical artifices in such models are empirical.


Confinement Contains CondensatesS.J. Brodsky, C.D. Roberts, R. Shrock and P.C. TandyarXiv:1202.2376 [nucl-th]

http://arxiv.org/abs/1202.2376



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“Orthodox Vacuum” Vacuum = “frothing sea” Hadrons = bubbles in that “sea”,

containing nothing but quarks & gluonsinteracting perturbatively, unless they’re near the bubble’s boundary, whereat they feel they’re trapped!



u

u

ud

u ud

du

73

New Paradigm Vacuum = hadronic fluctuations

but no condensates Hadrons = complex, interacting systems

within which perturbative behaviour is restricted to just 2% of the interior



u

u

ud

u ud

du

“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.”



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“Void that is truly empty solves dark energy puzzle”Rachel Courtland, New Scientist 4th Sept. 2010

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

4620

4

103

8

HG QCDNscondensateQCD

Paradigm shift:In-Hadron Condensates

http://www.newscientist.com/article/mg19325911.700-dark-energy-seeking-the-heart-of-darkness.html


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Relevant References arXiv:1310.2651 [nucl-th]

Explanation and Prediction of Observables using Continuum Strong QCDIan C. Cloët and Craig D. Roberts

arXiv:1202.2376, Phys. Rev. C 85 (2012) 065202 Confinement contains condensatesStanley 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 Henrik Zinkernagel.







http://dx.doi.org/10.1103/PhysRevC.85.065202



























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Parton structure of

hadronsCraig Roberts: Nanjing University, October 2013


Valence quarks

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Deep inelastic scattering

Quark discovery experiment at SLAC (1966-1978, Nobel Prize in 1990)

Completely different to elastic scattering– Blow the target to pieces instead of keeping only

those events where it remains intact. Cross-section is interpreted as a measurement of

the momentum-fraction probability distribution for quarks and gluons within the target hadron: q(x), g(x)



Probability that a quark/gluon within the target will carry a fraction x of the bound-state’s light-front momentumDistribution 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




http://rmp.aps.org/abstract/RMP/v82/i4/p2991_1



78

Parton Structure of Hadrons

Valence-quark structure of hadrons– Definitive of a hadron.

After all, it’s how we distinguish a proton from a neutron– Expresses charge; flavour; baryon number; and other

Poincaré-invariant macroscopic quantum numbers– Via evolution, determines background at LHC

Sea-quark distributions– Flavour content, asymmetry, intrinsic: yes or no?

Answers are essentially nonperturbative features of QCD



79

Parton Structure of Hadrons Light front provides a link with quantum mechanics

– If a probability interpretation is ever valid, then it’s in the infinite-momentum frame

Enormous amount of intuitively expressive information about hadrons & processes involving them is encoded in – Parton distribution functions – Generalised parton distribution functions – Transverse-momentum-dependent parton distribution

functions Information will be revealed by the measurement of

these functions – so long as they can be calculatedSuccess of programme demands very close collaboration between experiment and theory



80

Parton Structure of Hadrons Need for calculation is emphasised by Saga of pion’s

valence-quark distribution:o 1989: uv

π ~ (1-x)1 – inferred from LO-Drell-Yan & disagrees with QCD;

o 2001: DSE- QCD predictsuv

π ~ (1-x)2 argues that distribution inferred from data can’t be correct;



Valence quark distributions in the pion, M.B. Hecht, Craig D. Roberts, S.M. Schmidt, nucl-th/0008049, Phys.Rev. C63 (2001) 025213 .





81

Parton Structure of Hadrons Need for calculation is emphasised by Saga of pion’s

valence-quark distribution:o 1989: uv

π ~ (1-x)1 – inferred from LO-Drell-Yan & disagrees with QCD;

o 2001: DSE- QCD predicts uv

π ~ (1-x)2 argues that distribution inferred from data can’t be correct;

o 2010: NLO reanalysis including soft-gluon resummation, inferred distribution agrees with DSE and QCD



Soft-gluon resummation and the valence parton distribution function of the pion, M. Aicher, A. Schafer, W. Vogelsang, Phys.Rev.Lett. 105 (2010) 252003, arXiv:1009.2481 [hep-ph]

Valence quark distributions in the pion, M.B. Hecht, Craig D. Roberts, S.M. Schmidt, nucl-th/0008049, Phys.Rev. C63 (2001) 025213 .

http://dx.doi.org/10.1103/PhysRevLett.105.252003










82

Exact expression in QCD for the pion’s valence-quark parton distribution amplitude

Expression is Poincaré invariant but a probability interpretation is only valid on the light-front because only thereupon does one have particle-number conservation.

Probability that a valence-quark or antiquark carries a fraction x=k+ / P+

of the pion’s light-front momentum { n2=0, n.P = -mπ}

Pion’s valence-quark Distribution Amplitude


Pion’s Bethe-Salpeter wave function

Whenever a nonrelativistic limit is realistic, this would correspond to the Schroedinger wave function.

83


Methods have recently been developed to compute φπ(x); viz., the projection of the pion’s Poincaré-covariant wave-function onto the light-front

Results have been obtained with rainbow-ladder DSE kernel, simplest symmetry preserving form; and the best DCSB-improved kernel that is currently available.

xα (1-x)α, with α=0.3



Imaging dynamical chiral symmetry breaking: pion wave function on the light front, Lei Chang, et al., arXiv:1301.0324 [nucl-th], Phys. Rev. Lett. 110 (2013) 132001 (2013) [5 pages].




http://prl.aps.org/abstract/PRL/v110/i13/e132001







84


Both kernels agree: marked broadening of φπ(x), which owes to DCSB



Asymptotic

RL

DB

This may be claimed because PDA is computed at a low renormalisation scale in the chiral limit, whereat the quark mass function owes entirely to DCSB.

Difference between RL and DB results is readily understood: B(p2) is more slowly varying with DB kernel and hence a more balanced result












85


Both kernels agree: marked broadening of φπ(x), which owes to DCSB



Asymptotic

RL

DB

This may be claimed because PDA is computed at a low renormalisation scale in the chiral limit, whereat the quark mass function owes entirely to DCSB.

Difference between RL and DB results is readily understood: B(p2) is more slowly varying with DB kernel and hence a more balanced result

These computations are the first to directly expose DCSB – pointwise – on the light-front; i.e., in the infinite momentum frame.












86




Established a one-to-one connection between DCSB and the pointwise form of the pion’s wave function.

Dilation measures the rate at which dressed-quark approaches the asymptotic bare-parton limit

Experiments at JLab12 can empirically verify the behaviour of M(p), and hence chart the IR limit of QCD

C.D. Roberts, Prog. Part. Nucl. Phys. 61 (2008) 50

Dilation of pion’s wave function is measurable in

pion’s electromagnetic form factor at JLab12

A-rated: E12-06-10






http://www.jlab.org/exp_prog/proposals/06/PR12-06-101.pd











87

Lattice comparisonPion’s valence-quark PDA

Employ the generalised-Gegenbauer method described in Phys. Rev. Lett. 110 (2013) 132001 (2013) [5 pages] to analyse lattice-QCD output



Lattice-QCD => one nontrivial moment:

<(2x-1)2> = 0.27 ± 0.04 Legend

• Solid = DB (Best) DSE• Dashed = RL DSE• Dotted (black) = 6 x (1-x)• Dot-dashed = midpoint

lattice; and the yellow shading exhibits band allowed by lattice errors

φπ~ xα (1-x)α

α=0.35+0.32 = 0.67- 0.24 = 0.11

DB α=0.31 but 10% a2<0RL α=0.29 and 0% a2

V. Braun et al., PRD 74 (2006) 074501

Pion distribution amplitude from lattice-QCD, I. C. Cloët, L. Chang, C. D. Roberts, S. M. Schmidt and P. C. Tandy, arXiv:1306.2645 [nucl-th], Phys. Rev. Lett. 111 (2013) 092001 [5 pages]








http://prd.aps.org/abstract/PRD/v74/i7/e074501











88

Lattice comparisonPion’s valence-quark PDA

Establishes that contemporary DSE- and lattice-QCD computations, at the same scale, agree on the pointwise form of the pion's PDA, φπ(x).

This unification of DSE- and lattice-QCD results expresses a deeper equivalence between them, expressed, in particular, via the common behaviour they predict for the dressed-quark mass-function, which is both – a definitive signature of dynamical chiral symmetry

breaking – and the origin of the distribution amplitude's dilation.



Pion distribution amplitude from lattice-QCD, I. C. Cloët, L. Chang, C. D. Roberts, S. M. Schmidt and P. C. Tandy, arXiv:1306.2645 [nucl-th], Phys. Rev. Lett. 111 (2013) 092001 [5 pages]









89

When is asymptotic PDA valid?

PDA is a wave function not directly observable

but PDF is. φπ

asy(x) can only be a good approximation to the pion's PDA when it is accurate to write

uvπ (x) ≈ δ(x)

for the pion's valence-quark distribution function.

This is far from valid at currently accessible scales



Q2=27 GeV2

This is not δ(x)!

Explanation and Prediction of Observables using Continuum Strong QCD, I.C. Cloët & C.D. Roberts 1310.2651 [nucl-th]





90

When is asymptotic PDA valid? When is asymptopia reached?

If uvπ(x) ≈ δ(x), then <x> = ∫0

1 dx x uvπ(x) = 0;

i.e., the light-front momentum fraction carried by valence-quarks is ZERO Asymptopia is reached when <x> is “small”

As usual, the computed valence-quark distribution produces (π = u+dbar)

2<x>2GeV = 44% When is <x> small?


NLO evolution of PDF, computation of <x>. Even at LHC energies, light-front fraction of

the π momentum:<x>dressed valence-quarks = 25% <x>glue = 54%, <x>sea-quarks = 21%

LHC: 16TeV

Evolution in QCD is LOGARITHMIC

JLab 2GeV






91

When is asymptotic PDA valid? When is asymptopia reached?

If uvπ(x) ≈ δ(x), then <x> = ∫0

1 dx x uvπ(x) = 0;

i.e., the light-front momentum fraction carried by valence-quarks is ZERO Asymptopia is reached when <x> is “small”

As usual, the computed valence-quark distribution produces (π = u+dbar)

2<x>2GeV = 44% When is <x> small?


NLO evolution of PDF, computation of <x>. Even at LHC energies, light-front fraction of

the π momentum:<x>dressed valence-quarks = 25% <x>glue = 54%, <x>sea-quarks = 21%

LHC: 16TeV


JLab 2GeV

Even at LHC energy scales, nonperturbative effects, such as DCSB, are playing a crucial role in setting the scales in PDAs and PDFs.






92

At the “Planck scale”


Explanation and Prediction of Observables using Continuum Strong QCD, I.C. Cloët & C.D. Roberts


In the truly asymptotic domain, way, way beyond LHC energy scales, gluons and sea-quarks share the momentum of a hadron, each with roughly 50% of the momentum


93

Elastic Scattering


94

Form FactorsElastic Scattering



Elastic Form factors have long been recognised as a basic tool for elucidating bound-state properties.

They are of particular value in hadron physics because they provide information on structure as a function of Q2, the squared momentum-transfer:– Small-Q2 is the nonperturbative domain– Large-Q2 is the perturbative domain– Nonperturbative methods in hadron physics must explain the

behaviour from Q2=0 through the transition domain, whereupon the behaviour is currently being measured

Experimental and theoretical studies of hadron electromagnetic form factors have made rapid and significant progress during the last several years, including new data in the time like region, and material gains have been made in studying the pion form factor.



Charged pion elastic form factor

P. Maris & P.C. Tandy, Phys.Rev. C62 (2000) 055204: numerical procedure is practically useless for Q2>4GeV2, so prediction ends!

Pion electromagnetic form factor at spacelike momentaL. Chang, I. C. Cloët, C. D. Roberts, S. M. Schmidt and P. C. Tandy, arXiv:1307.0026 [nucl-th], Phys. Rev. Lett. 111, 141802 (2013)










96

New AlgorithmCraig Roberts: Nanjing University, October 2013

97

Perturbation Theory Integral Representation

Noboru Nakanishi – a generalised type of spectral representation Use fits to dressed-quark propagator and Bethe-Salpeter

amplitudes in order to define continuation into the complex plane Expressions involved enable use of all the “tricks” familiar from

perturbation theory but just add another integral over a spectral density, ρ(z)



Δ(x) = 1/(x+Λ2)




P. Maris & P.C. Tandy, Phys.Rev. C62 (2000) 055204: numerical procedure is practically useless for Q2>4GeV2, so prediction ends!

Algorithm developed for pion PDA overcomes this obstacle

Solves the practical problem of continuing from Euclidean metric formulation to Minkowski space













Improved DSE interaction kernel, based on DSE and lattice-QCD studies of gluon sectorS.-x. Qin, L. Chang et al. Phys.Rev. C84 (2011) 042202(R)

New prediction in better agreement with available data than old DSE result

Prediction extends from Q2=0 to arbitrarily large Q2, without interruption, unifying both domains

DSE 2000 … Breakdown here













Unlimited domain of validity emphasised in this figure

Depict prediction on domain 0<Q2<20GeV2 but have computed the result to Q2=100GeV2.

If it were necessary, reliable results could readily be obtained at even higher values.

DSE 2013











Predict a maximum at 6-GeV2, which lies within domain that is accessible to JLab12

Difficult, however, to distinguish DSE prediction from Amendolia-1986 monopole

What about comparison with perturbative QCD?

Amendolia et al.

DSE 2013

ρ-meson pole VMD

maximum

A-rated: E12-06-10


http://www.jlab.org/exp_prog/proposals/06/PR12-06-101.pd









Charged pion elastic form factor Prediction of pQCD

obtained when the pion valence-quark PDA has the form appropriate to the scale accessible in modern experiments is markedly different from the result obtained using the asymptotic PDA

Near agreement between the pertinent perturbative QCD prediction and DSE-2013 prediction is striking.

DSE 2013

pQCD obtained with φπasy(x)

pQCD obtained with φπ(x;2GeV), i.e., the PDA appropriate to the scale of the experiment

15%

Single DSE interaction kernel, determined fully by just one parameter and preserving the one-loop renormalisation group behaviour of QCD, has unified the Fπ(Q2) and φπ(x) (and numerous other quantities)










Charged pion elastic form factor Leading-order, leading-twist

QCD prediction, obtained with φπ(x) evaluated at a scale appropriate to the experiment underestimates DSE-2013 prediction by merely an approximately uniform 15%.

Small mismatch is explained by a combination of higher-order, higher-twist corrections & and shortcomings in the rainbow-ladder truncation.

DSE 2013

pQCD obtained with φπasy(x)

pQCD obtained φπ(x;2GeV), i.e., the PDA appropriate to the scale of the experiment

15%

Hence, one should expect dominance of hard contributions to the pion form factor for Q2>8GeV2.

Nevertheless, the normalisation of the form factor is fixed by a pion wave-function whose dilation with respect to φπ

asy(x) is a definitive signature of DCSB









104

Achieved a longstanding goal

We now have a comprehensive understanding of the nature and structure of QCD’s dichotomous Goldstone Mode!


105

What’s left?Craig Roberts: Nanjing University, October 2013


106

New Challenges

Computation of spectrum of hybrid and exotic mesons

Equally pressing, some might say more so, is the three-body problem; viz., baryons in QCD



exotic mesons: quantum numbers not possible for quantum mechanical quark-antiquark systemshybrid mesons: normal quantum numbers but non-quark-model decay patternBOTH suspected of having “constituent gluon” content

↔

107

Grand UnificationCraig Roberts: Nanjing University, October 2013


Unification of Meson & Baryon Properties

Correlate the properties 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



109

Fully-covariant computationof nucleon form factors

First such calculations:– G. Hellstern et al., Nucl.Phys. A627 (1997) 679-709 , restricted to

Q2<2GeV2 – J.C.R. Bloch et al., Phys.Rev. C60 (1999) 062201(R) , restricted to

Q2<3GeV2

Exploratory:– Included some correlations within the nucleon, but far from the most

generally allowed– Used very simple photon-nucleon interaction current

Did not isolate and study GEp(Q2)/GM

p(Q2) How does one study baryons in QCD?








Baryon Structure Dynamical chiral symmetry breaking (DCSB)

– has enormous impact on meson properties. Must be included in description

and prediction of baryon properties. DCSB is essentially a quantum field theoretical effect.

In quantum field theory Meson appears as pole in four-point quark-antiquark Green function

→ Bethe-Salpeter Equation Nucleon appears as a pole in a six-point quark Green function

→ Faddeev Equation. Poincaré covariant Faddeev equation sums all possible exchanges and

interactions that can take place between three dressed-quarks Tractable equation is based on the observation that an interaction which

describes colour-singlet mesons also generates nonpointlike quark-quark (diquark) correlations in the colour-antitriplet channel

Craig Roberts: Nanjing University, October 2013 110

R.T. Cahill et al.,Austral. J. Phys. 42 (1989) 129-145

6333 SUc(3):



Faddeev Equation

Linear, Homogeneous Matrix equationYields 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 correlations111

diquark

quark

quark exchangeensures Pauli statistics

composed of strongly-dressed quarks bound by dressed-gluons


R.T. Cahill et al.,Austral. J. Phys. 42 (1989) 129-145


Why should a pole approximation produce reliable results?

Faddeev Equation



112

quark-quark scattering matrix - a pole approximation is used to arrive at the Faddeev-equation

Consider the rainbow-gap and ladder-Bethe-Salpeter equations

In this symmetry-preserving truncation, colour-antitriplet quark-quark correlations (diquarks) are described by a very similar homogeneous Bethe-Salpeter equation

Only difference is factor of ½ Hence, an interaction that describes mesons also generates

diquark correlations in the colour-antitriplet channel

Diquarks



113

Calculation of diquark masses in QCDR.T. Cahill, C.D. Roberts and J. PraschifkaPhys.Rev. D36 (1987) 2804






Faddeev Equation


scalar diquark component

axial-vector diquark component

Survey of nucleon electromagnetic form factorsI.C. Cloët et al, arXiv:0812.0416 [nucl-th], Few Body Syst. 46 (2009) pp. 1-36





Faddeev Equation







Faddeev Equation

Every one of these entries has a simple matrix structure Similar form for the kernel entries that involve axial-vector

diquark correlations Combining everything, one arrives at a linear homogeneous

matrix equation for the amplitudes S(k;P)u(P), A(k;P)u(P)






117

Baryon Structure

Remarks Diquark correlations are not inserted by hand

Such correlations are a dynamical consequence of strong-coupling in QCD

The same mechanism that produces an almost massless pion from two dynamically-massive quarks; i.e., DCSB, forces a strong correlation between two quarks in colour-antitriplet channels within a baryon – an indirect consequence of Pauli-Gürsey symmetry

Diquark correlations are not pointlike– Typically, r0+ ~ rπ & r1+ ~ rρ

(actually 10% larger)– They have soft form factors



SU(2) isospin symmetry of hadrons might emerge from mixing half-integer spin particles with their antiparticles.

Faddeev Equation

118

Voyage of DiscoveryCraig Roberts: Nanjing University, October 2013


119

Recapitulation One method by which to validate QCD is computation of its hadron spectrum and subsequent comparison with modern experiment. Indeed, this is an integral part of the international effort in nuclear physics.

For example, the N∗ programme and the search for hybrid and exotic mesons together address the questions:– which hadron states and resonances are produced by QCD? – how are they constituted?

This intense effort in hadron spectroscopy is a motivation to extend the research just described and treat ground- and excited-state hadrons with s-quark content. (New experiments planned in Japan)

Key elements in a successful spectrum computation are: – symmetries and the pattern by which they are broken; – the mass-scale associated with confinement and DCSB; – and full knowledge of the physical content of bound-state kernels.

All this is provided by the DSE approach.



Vector-vector contact interaction

mG = 800MeV is a gluon mass-scale – dynamically generated in QCD

Gap equation:

DCSB: M ≠ 0 is possible so long as αIR>αIRcritical =0.4π

Observables require αIR = 0.93π

Contact-Interaction Kernel



120


121

Contact Interaction arXiv:1212.2212 [nucl-th], Phys. Rev. C 87 92013) 045207 [15 pages]

Features and flaws of a contact interaction treatment of the kaonChen Chen, L. Chang, C. D. Roberts, S. M. Schmidt, Shaolong Wan and D. J. Wilson,

arXiv:1209.4352 [nucl-th], Phys. Rev. C87 (2013) 015205 [12 pages]Electric dipole moment of the rho-mesonM. Pitschmann, C.-Y. Seng, M. J. Ramsey-Musolf, C. D. Roberts, S. M. Schmidt and D. J. Wilson

arXiv:1204.2553 [nucl-th], Few Body Syst. (2012) DOI: 10.1007/s00601-012-0466-3 Spectrum of Hadrons with Strangeness, Chen Chen, L. Chang, C.D. Roberts, Shaolong Wan and D.J. Wilson

arXiv:1112.2212 [nucl-th], Phys. Rev. C85 (2012) 025205 [21 pages] Nucleon and Roper electromagnetic elastic and transition form factors, D. J. Wilson, I. C. Cloët, L. Chang and C. D. Roberts

arXiv:1102.4376 [nucl-th], Phys. Rev. C 83, 065206 (2011) [12 pages] , π- and ρ-mesons, and their diquark partners, from a contact interaction, H.L.L. Roberts, A. Bashir, L.X. Gutiérrez-Guerrero, C.D. Roberts and David J. Wilson

arXiv:1101.4244 [nucl-th], Few Body Syst. 51 (2011) pp. 1-25Masses of ground and excited-state hadronsH.L.L. Roberts, Lei Chang, Ian C. Cloët and Craig D. Roberts

arXiv:1009.0067 [nucl-th], Phys. Rev. C82 (2010) 065202 [10 pages]Abelian anomaly and neutral pion productionHannes L.L. Roberts, C.D. Roberts, A. Bashir, L. X. Gutiérrez-Guerrero & P. C. Tandy

arXiv:1002.1968 [nucl-th], Phys. Rev. C 81 (2010) 065202 (5 pages)Pion form factor from a contact interaction, L. Xiomara Gutiérrez-Guerrero, A. Bashir, I. C. Cloët & C. D. Roberts


Symmetry-preserving treatment of vector-vector contact-interaction: series of papers establishes strengths & limitations.















http://www.springerlink.com/content/023t3713p2743k76/
















http://www.springerlink.com/content/l277662502362386/















122

ContactInteraction



Symmetry-preserving treatment of vector×vector contact interaction is useful tool for the study of phenomena characterised by probe momenta less-than the dressed-quark mass.

Whilst this interaction produces form factors which are too hard, interpreted carefully, even they can be used to draw valuable insights; e.g., concerning relationships between different hadrons.

Studies employing a symmetry-preserving regularisation of the contact interaction serve as a useful surrogate, exploring domains which analyses using interactions that more closely resemble those of QCD are as yet unable to enter.

They’re critical at present in attempts to use data as tool for charting nature of the quark-quark interaction at long-range; i.e., identifying signals of the running of couplings and masses in QCD.

Spectrum of Baryons

Static “approximation” Implements analogue of contact interaction in Faddeev-equation

In combination with contact-interaction diquark-correlations, generates Faddeev equation kernels which themselves are momentum-independent

The merit of this truncation is the dramatic simplifications which it produces

Used widely in hadron physics phenomenology; e.g., Bentz, Cloët, Thomas et al.



123

Mg

MpiB21

Variant of:A. Buck, R. Alkofer & H. Reinhardt, Phys. Lett. B286 (1992) 29.

H.L.L. Roberts, L. Chang, I.C. Cloët and C.D. Roberts, arXiv:1101.4244 [nucl-th], Few Body Syst. 51 (2011) pp. 1-25




124

Spectrum of Hadronswith Strangeness

Solve gap equation for u & s-quarks

Input ratio ms /mu = 24 is consistent with modern estimates Output ratio Ms /Mu = 1.43 shows dramatic impact of DCSB, even on

the s-quark κ = in-hadron condensate rises slowly with mass of hadron



arXiv:1204.2553 [nucl-th], Spectrum of hadrons with strangeness, Chen Chen, L. Chang, C.D. Roberts, Shaolong Wan and D.J. Wilson, Few Body Syst. (2012) DOI: 10.1007/s00601-012-0466-3






125

Spectrum of Mesonswith Strangeness

Solve Bethe-Salpeter equations for mesons and diquarks



arXiv:1204.2553 [nucl-th], Spectrum of hadrons with strangeness, Chen Chen, L. Chang, C.D. Roberts, Shaolong Wan and D.J. Wilson




126

Spectrum of Mesonswith Strangeness




Computed values for ground-states are greater than the empirical masses, where they are known.

Typical of DCSB-corrected kernels that omit resonant contributions; i.e., do not contain effects that mayphenomenologically be associated with a meson cloud.

Perhaps underestimate radial-ground splitting by 0.2GeV





127

Spectrum of Diquarkswith Strangeness








128

Spectrum of Diquarkswith Strangeness





Level ordering of diquark correlations is same as that for mesons. In all diquark channels, except scalar, mass of diquark’s partner meson is a fair guide to the diquark’s mass: o Meson mass bounds the diquark’s mass from below;o Splitting always less than 0.13GeV and decreases with

increasing meson massScalar channel “special” owing to DCSB




129

Bethe-Salpeter amplitudes

Bethe-Salpeter amplitudes are couplings in Faddeev Equation

Magnitudes for diquarks follow precisely the meson pattern




Owing to DCSB, FE couplings in ½- channels are 25-times weaker than in ½+ !




130

Spectrum of Baryonswith Strangeness

Solved all Faddeev equations, obtained masses and eigenvectors of the octet and decuplet baryons.







131

Spectrum of Baryonswith Strangeness





As with mesons, computed baryon masses lie uniformly above the empirical values.

Success because our results are those for the baryons’ dressed-quark cores, whereas empirical values include effects associated with meson-cloud, which typically produce sizable reductions.

JülichEBAC




132

Structure of Baryonswith Strangeness Baryon structure is flavour-blind




Diquark content




133

Structure of Baryonswith Strangeness Baryon structure is flavour-blind



arXiv:1204.2553 [nucl-th], Spectrum of hadrons with strangeness, Chen, Chang, Roberts, Wan and Wilson & Nucleon and Roper em elastic and transition form factors, D. J. Wilson, I. C. Cloët, L. Chang and C. D. Roberts, arXiv:1112.2212 [nucl-th], Phys. Rev. C85 (2012) 025205 [21 pages]

Diquark content

Jqq=0 content of J=½ baryons is almost independent of their flavour structure

Radial excitation of ground-state octet possess zero scalar diquark content!

This is a consequence of DCSB Ground-state (1/2)+ possess unnaturally

large scalar diquark content Orthogonality forces radial excitations to

possess (almost) none at all!

80%

50%

50%

0%










134






(1/2)+

(1/2)+

(1/2)-




135






(1/2)+

(1/2)+

(1/2)-

This level ordering has long been a problem in CQMs with linear or HO confinement potentials

Correct ordering owes to DCSB Positive parity diquarks have

Faddeev equation couplings 25-times greater than negative parity diquarks

Explains why approaches within which DCSB cannot be realised (CQMs) or simulations whose parameters suppress DCSB will both have difficulty reproducing experimental ordering




136

Getting realCraig Roberts: Nanjing University, October 2013


137

Charting the Interaction Interaction in QCD is not momentum-independent

– Behaviour for Q2>2GeV2 is well know; namely, renormalisation-group-improved one-gluon exchange

– Computable in perturbation theory



Known = there is a “freezing” of the interaction below a scale of roughly 0.4GeV, which is why momentum-independent interaction works

Unknown Infrared behavior of the interaction,

which is responsible for Confinement DCSB

How is the transition to pQCD made and is it possible to define a transition boundary?


138

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 = 8-15 • α(mG

2)/π = 2-4


Qin et al., Phys. Rev. C 84 042202(Rapid Comm.) (2011)Rainbow-ladder truncation

Running gluon mass– Gluon is massless in ultraviolet

in agreement with pQCD– Massive in infrared

• mG(0) = 0.67-0.81 GeV• mG(mG

2) = 0.53-0.64 GeV


139

Structure of Hadrons Elastic form factors

– Provide vital information about the structure and composition of the most basic elements of nuclear physics.

– They are a measurable and physical manifestation of the nature of the hadrons' constituents and the dynamics that binds them together.

Accurate form factor data are driving paradigmatic shifts in our pictures of hadrons and their structure; e.g., – role of orbital angular momentum and nonpointlike diquark

correlations– scale at which p-QCD effects become evident– strangeness content– meson-cloud effects– etc.



140

Jlab Highlights



)(

)(2

2

QG

QGpM

pEp

http://www.jlab.org/highlights/phys.html

Distribution of charge is different from distribution of magnetisation

http://www.jlab.org/highlights/phys.html


141

Photon-nucleon current

Composite nucleon must interact with photon via nontrivial current constrained by Ward-Green-Takahashi identities

DSE → BSE → Faddeev equation plus current → nucleon form factors

Vertex must contain the dressed-quark anomalous magnetic moment

Oettel, Pichowsky, SmekalEur.Phys.J. A8 (2000) 251-281


L. Chang, Y. –X. Liu and C.D. RobertsarXiv:1009.3458 [nucl-th]Phys. Rev. Lett. 106 (2011) 072001

In a realistic calculation, the last three diagrams represent 8-dimensional integrals, which can be evaluated using Monte-Carlo techniques



http://inspirebeta.net/record/868745









142

I.C. Cloët, C.D. Roberts, et al.arXiv:0812.0416 [nucl-th]

)(

)(2

2

QG

QGpM

pEp

Highlights again the critical importance of DCSB in explanation of real-world observables.

DSE result Dec 08

DSE result – including the anomalous magnetic moment distribution


L. Chang, Y. –X. Liu and C.D. RobertsarXiv:1009.3458 [nucl-th]Phys. Rev. Lett. 106 (2011) 072001













Visible Impacts of DCSB


Apparently small changes in M(p) within the domain 1<p(GeV)<3have striking effect on the proton’s electric form factor

The possible existence and location of the zero is determined by behaviour of Q2F2

p(Q2) Like the pion’s PDA, Q2F2

p(Q2) measures the rate at which dress-ed-quarks become parton-like: F2

p=0 for bare quark-partons Therefore, GE

p can’t be zero on the bare-parton domain

I.C. Cloët, C.D. Roberts, A.W. Thomas: Revealing dressed-quarks via the proton's charge distribution, arXiv: 1304.0855 [nucl-th]






Visible Impacts of DCSB


Follows that the possible existence and location

of a zero in the ratio of proton elastic form factors

[μpGEp(Q2)/GMp(Q2)] are a direct measure of the nature of the quark-quark interaction in the Standard Model.

I.C. Cloët, C.D. Roberts, A.W. Thomas: Revealing dressed-quarks via the proton's charge distribution, arXiv: 1304.0855 [nucl-th]






145

Flavor separation of proton form factors

Very different behavior for u & d quarks Means apparent scaling in proton F2/F1 is purely accidental


Cates, de Jager, Riordan, Wojtsekhowski, PRL 106 (2011) 252003

Q4F2q/k

Q4 F1q


146

Diquark correlations!

Poincaré covariant Faddeev equation – Predicts scalar and axial-vector

diquarks Proton's singly-represented d-quark

more likely to be struck in association with 1+ diquark than with 0+

– form factor contributions involving 1+ diquark are softer


Cloët, Eichmann, El-Bennich, Klähn, Roberts, Few Body Syst. 46 (2009) pp.1-36Wilson, Cloët, Chang, Roberts, PRC 85 (2012) 045205

Doubly-represented u-quark is predominantly linked with harder 0+ diquark contributions

Interference produces zero in Dirac form factor of d-quark in proton– Location of the zero depends on the relative probability of finding

1+ & 0+ diquarks in proton– Correlated, e.g., with valence d/u ratio at x=1

d

u

=Q2/M2


147

Far valence domain x≃1



148

Far valence domain x≃1

Endpoint of the far valence domain: x 1, is especially significant≃– All familiar PDFs vanish at x=1; but ratios of any two need not– Under DGLAP evolution, the value of such a ratio is invariant.

Thus, e.g., – limx→1 dv(x)/uv(x)

is unambiguous, scale invariant, nonperturbative feature of QCD. keen discriminator between frameworks that claim to explain nucleon structure.

Furthermore, Bjorken-x=1 corresponds strictly to the situation in which the invariant mass of the hadronic final state is precisely that of the target; viz., elastic scattering. Structure functions inferred experimentally on x 1 ≃

are determined theoretically by target's elastic form factors.


Nucleon spin structure at very high-xCraig D. Roberts, Roy J. Holt and Sebastian M. SchmidtarXiv:1308.1236 [nucl-th], Phys. Lett. B in press




149

Neutron Structure Function at high-x

Valence-quark distributions at x=1– Fixed point under DGLAP evolution– Strong discriminator between theories

Algebraic formula

– P1p,s = contribution to the proton's charge arising from diagrams

with a scalar diquark component in both the initial and final state

– P1p,a = kindred axial-vector diquark contribution

– P1p,m = contribution to the proton's charge arising from diagrams

with a different diquark component in the initial and final state.



I.C. Cloët, C.D. Roberts, et al.arXiv:0812.0416 [nucl-th], Few Body Syst. 46 (2009) 1-36D. J. Wilson, I. C. Cloët, L. Chang and C. D. RobertsarXiv:1112.2212 [nucl-th], Phys. Rev. C85 (2012) 025205 [21 pages]

Measures relative strength of axial-vector/scalar diquarks in proton




http://www.springerlink.com/content/d0r372483080w110/








150

Neutron StructureFunction at high-x


d/u=1/2SU(6) symmetry

pQCD, uncorrelated Ψ

0+ qq only, d/u=0

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)

d/u=0.28DSE: “realistic”

Distribution of neutron’s momentum amongst quarks on the valence-quark domain

DSE: “contact”d/u=0.18

Melnitchouk, Accardi et al. Phys.Rev. D84 (2011) 117501

x>0.9

Melnitchouk, Arrington et al. Phys.Rev.Lett. 108 (2012) 252001

















151


SU(6) symmetry

pQCD, uncorrelated Ψ

0+ qq only

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)

DSE: “realistic”

Distribution of neutron’s momentum amongst quarks on the valence-quark domainCraig Roberts: Nanjing University, October 2013

DSE: “contact”

Melnitchouk, Accardi et al. Phys.Rev. D84 (2011) 117501

x>0.9

Melnitchouk, Arrington et al. Phys.Rev.Lett. 108 (2012) 252001


NB. d/u|x=1= 0 means there are

no valence d-quarks in the proton!

JLab12 can resolve this conundrum
















152


“While it is quite hazardous to extrapolate from our limited xB range all the way to xB = 1, these results appear to disfavor models of the proton with d/u=0 at xB = 1”


Short Range Correlations and the EMC Effect, L.B. Weinstein et al., Phys.Rev.Lett. 106 (2011) 052301, arXiv:1009.5666 [hep-ph]

Figure courtesy of D.W. Higinbotham

Observation: EMC effect measured in electron DIS at 0.35 < xB < 0.7, is linearly related to the Short Range Correlation (SRC) scale factor obtained from electron inclusive scattering at xB > 1.







153

Nucleon spin structure at very high x


Similar formulae for nucleon longitudinal structure functions.

Plainly, existing data cannot distinguish between modern pictures of nucleon structure

Empirical results for nucleon longitudinal spin asymmetries on x ≃ 1 promise to add greatly to our capacity for discriminating between contemporary pictures of nucleon structure.

NB. pQCD is actually model-dependent: assumes SU(6) spin-flavour wave function for the proton's valence-quarks and the corollary that a hard photon may interact only with a quark that possesses the same helicity as the target.

Nucleon spin structure at very high-xCraig D. Roberts, Roy J. Holt and Sebastian M. SchmidtarXiv:1308.1236 [nucl-th], Phys. Lett. B in press




154

Theory Lattice-QCD

– Significant progress in the last five years

– This must continue Bound-state problem in

continuum quantum field theory– Significant progress, too– This must continue

First Sino-Americas School & Workshop on the Continuum Bound-State Problem, Hefei, China. 22-26/Oct./2013



http://hepg-work.ustc.edu.cn/dse2013/

http://hepg-work.ustc.edu.cn/dse2013/



Epilogue

156

Epilogue The Physics of Hadrons is Unique:

– Confronting a fundamental theory in which the elementary degrees-of-freedom are intangible and only composites reach detectors

Confinement in real-world is NOT understood DCSB is crucial to any understanding of hadron

phenomena

They must have a common origin Experimental and theoretical study of the Bound-

state problem in continuum QCD promises to provide many more insights and answers.



157

This is not the end



158

Regge Trajectories? Martinus Veltmann, “Facts and Mysteries in Elementary Particle Physics” (World Scientific,

Singapore, 2003): In time the Regge trajectories thus became the cradle of string theory. Nowadays the Regge trajectories have largely disappeared, not in the least because these higher spin bound states are hard to find experimentally. At the peak of the Regge fashion (around 1970) theoretical physics produced many papers containing families of Regge trajectories, with the various (hypothetically straight) lines based on one or two points only!



Phys.Rev. D 62 (2000) 016006 [9 pages]

1993: "for elucidating the quantum structure of electroweak interactions in physics"

Systematics of radial and angular-momentum Regge trajectories of light non-strange qqbar-states“ P. Masjuan, E. Ruiz Arriola, W. Broniowski. arXiv:1305.3493 [hep-ph]





http://inspirehep.net/record/1233541/export/hx




159

Hybrid Hadrons & Lattice QCD – Robert Edwards, Baryons13

Heavy pions … so, naturally, constituent-quark like spectra To which potential does it correspond?


arXiv:1104.5152, 1201.2349


160

Hybrid meson models – Robert Edwards, Baryons13

With minimal quark content, , gluonic field can in a color singlet or octet

`constituent’ gluonin S-wave

`constituent’ gluonin P-wave

bag model

flux-tube model

arXiv:1104.5152, 1201.2349



161

Hybrid baryon models – Robert Edwards, Baryons13

Minimal quark content, , gluonic field can be in color singlet, octet or decuplet

bag model

flux-tube model

Now must take into account permutation symmetry of quarks and gluonic field

arXiv:1104.5152, 1201.2349


162

Table of Contents

I. IntroductionII. What is QCD?III. Confinement?IV. DCSBV. Enigma of MassVI. Condensates?VII. Pion valence-quark distributionVIII. Pion valence-quark parton

distribution amplitudeIX. Where is asymptopia?X. Charged pion elastic form factor



A. Baryon StructureB. Spectrum of BaryonsC. Nucleon form factorsD. Nucleon structure functions at l

arge-x

E. EpilogueF. Regge supplement

Mesons Baryons

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