Craig Roberts Physics Division

Dyson-Schwinger Equations & Continuum QCD

Adnan BASHIR (U Michoacan);Stan BRODSKY (SLAC);Lei CHANG (ANL & PKU); Huan CHEN (BIHEP);Ian CLOËT (UW);Bruno EL-BENNICH (Sao Paulo);Xiomara GUTIERREZ-GUERRERO (U Michoacan);Roy HOLT (ANL);Mikhail IVANOV (Dubna);Yu-xin LIU (PKU);Trang NGUYEN (KSU);Si-xue QIN (PKU);Hannes ROBERTS (ANL, FZJ, UBerkeley);Robert SHROCK (Stony Brook);Peter TANDY (KSU);David WILSON (ANL)

Craig Roberts

Physics Division

StudentsEarly-career scientists

Published collaborations in 2010/2011

QCD’s Challenges

Dynamical Chiral Symmetry Breaking Very unnatural pattern of bound state masses;

e.g., Lagrangian (pQCD) quark mass is small but . . . no degeneracy between JP=+ and JP=− (parity partners)

Neither of these phenomena is apparent in QCD’s Lagrangian Yet they are the dominant determining characteristics

of real-world QCD.

QCD – Complex behaviour arises from apparently simple rules.Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.

2

Quark and Gluon ConfinementNo matter how hard one strikes the proton, one cannot liberate an individual quark or gluon

Understand emergent phenomena

LightCone 2011, SMU 23-27 May - 77pgs

Universal Truths

Hadron spectrum, and elastic and transition form factors provide unique information about long-range interaction between light-quarks and distribution of hadron's characterising properties amongst its QCD constituents.

Dynamical Chiral Symmetry Breaking (DCSB) is most important mass generating mechanism for visible matter in the Universe.

Higgs mechanism is (almost) irrelevant to light-quarks. Running of quark mass entails that calculations at even modest Q2 require a

Poincaré-covariant approach. Covariance + M(p2) require existence of quark orbital angular momentum in

hadron's rest-frame wave function. Confinement is expressed through a violent change of the propagators for

coloured particles & can almost be read from a plot of a states’ dressed-propagator.

It is intimately connected with DCSB.

Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.

3LightCone 2011, SMU 23-27 May - 77pgs


4

Strong-interaction: QCD

Asymptotically free– Perturbation theory is valid

and accurate tool at large-Q2

– Hence chiral limit is defined Essentially nonperturbative

for Q2 < 2 GeV2


Nature’s only example of truly nonperturbative, fundamental theory A-priori, no idea as to what such a theory can produce

5



Confinement


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Confinement

Quark and Gluon Confinement– No matter how hard one strikes the proton, or any other

hadron, one cannot liberate an individual quark or gluon Empirical fact. However

– There is no agreed, theoretical definition of light-quark confinement

– Static-quark confinement is irrelevant to real-world QCD• There are no long-lived, very-massive quarks

Confinement entails quark-hadron duality; i.e., that all observable consequences of QCD can, in principle, be computed using an hadronic basis.

X



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Confinement

Infinitely heavy-quarks plus 2 flavours with mass = ms – Lattice spacing = 0.083fm– String collapses

within one lattice time-stepR = 1.24 … 1.32 fm

– Energy stored in string at collapse Ec

sb = 2 ms – (mpg made via

linear interpolation) No flux tube between

light-quarks

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

Bs anti-Bs

“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.”


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



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Confinement Confinement is expressed through a violent change in

the analytic structure of propagators for coloured particles & can almost be read from a plot of a states’ dressed-propagator– Gribov (1978); Munczek (1983); Stingl (1984); Cahill (1989);

Krein, Roberts & Williams (1992); Tandy (1994); …

complex-P2 complex-P2

o Real-axis mass-pole splits, moving into pair(s) of complex conjugate poles or branch pointso Spectral density no longer positive semidefinite & hence state cannot exist in observable spectrum

Normal particle Confined particle


timelike axis: P2<0


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Dressed-gluon propagator

Gluon propagator satisfies a Dyson-Schwinger Equation

Plausible possibilities for the solution

DSE and lattice-QCDagree on the result– Confined gluon– IR-massive but UV-massless– mG ≈ 2-4 ΛQCD

perturbative, massless gluon

massive , unconfined gluon

IR-massive but UV-massless, confined gluon

A.C. Aguilar et al., Phys.Rev. D80 (2009) 085018




Charting the interaction between light-quarks

Confinement can be related to the analytic properties of QCD's Schwinger functions.

Question of light-quark confinement can be translated into the challenge of charting the infrared behavior of QCD's universal β-function– This function may depend on the scheme chosen to renormalise

the quantum field theory but it is unique within a given scheme.– Of course, the behaviour of the β-function on the

perturbative domain is well known.Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.

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This is a well-posed problem whose solution is an elemental goal of modern hadron physics.The answer provides QCD’s running coupling.


Charting the interaction between light-quarks

Through QCD's Dyson-Schwinger equations (DSEs) the pointwise behaviour of the β-function determines the pattern of chiral symmetry breaking.

DSEs connect β-function to experimental observables. Hence, comparison between computations and observations ofo Hadron mass spectrumo Elastic and transition form factorscan be used to chart β-function’s long-range behaviour.

Extant studies show that the properties of hadron excited states are a great deal more sensitive to the long-range behaviour of the β-function than those of the ground states.




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DSE Studies – Phenomenology of gluon

Wide-ranging study of π & ρ properties Effective coupling

– Agrees with pQCD in ultraviolet – Saturates in infrared

• α(0)/π = 3.2 • α(1 GeV2)/π = 0.35


Maris & Tandy, Phys.Rev. C60 (1999) 055214

Running gluon mass– Gluon is massless in ultraviolet

in agreement with pQCD– Massive in infrared

• mG(0) = 0.76 GeV• mG(1 GeV2) = 0.46 GeV



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Dynamical Chiral Symmetry Breaking




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Strong-interaction: QCD Confinement

– Empirical feature– Modern theory and lattice-QCD support conjecture

• that light-quark confinement is a fact• associated with violation of reflection positivity; i.e., novel analytic

structure for propagators and vertices– Still circumstantial, no proof yet of confinement

On the other hand, DCSB is a fact in QCD– It is the most important mass generating mechanism for visible

matter in the Universe. Responsible for approximately 98% of the proton’s

mass.Higgs mechanism is (almost) irrelevant to light-quarks.



Frontiers of Nuclear Science:Theoretical Advances

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.


C.D. Roberts, Prog. Part. Nucl. Phys. 61 (2008) 50M. Bhagwat & P.C. Tandy, AIP Conf.Proc. 842 (2006) 225-227










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DSE prediction of DCSB confirmed

Mass from nothing!





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Hint of lattice-QCD support for DSE prediction of violation of reflection positivity



12GeVThe Future of JLab

Numerical simulations of lattice QCD (data, at two different bare masses) have confirmed model predictions (solid curves) that the vast bulk of the constituent mass of a light quark comes from a cloud of gluons that are dragged along by the quark as it propagates. In this way, a quark that appears to be absolutely massless at high energies (m =0, red curve) acquires a large constituent mass at low energies.

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Jlab 12GeV: Scanned by 2<Q2<9 GeV2 elastic & transition form factors.


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Condensates?LightCone 2011, SMU 23-27 May - 77pgs


Dichotomy of the pion


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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:

mm 2

tconstituenstatebound mm


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 transition matrix element, there is no problem.

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


- (0.25GeV)3

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 1980Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.


6550+ 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

Universal Misapprehensions

Since 1979, DCSB has commonly been associated literally with a spacetime-independent mass-dimension-three “vacuum condensate.”

Under this assumption, “condensates” couple directly to gravity in general relativity and make an enormous contribution to the cosmological constant

Experimentally, the answer is

Ωcosm. const. = 0.76 This mismatch is a bit of a problem.



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4620

4

103

8

HG QCDNscondensateQCD

Resolution?

Quantum Healing Central: “KSU physics professor [Peter Tandy] publishes groundbreaking

research on inconsistency in Einstein theory.” Paranormal Psychic Forums:

“Now Stanley Brodsky of the SLAC National Accelerator Laboratory in Menlo Park, California, and colleagues have found a wayto get rid of the discrepancy. “People have just been taking it on faith that this quark condensate is present throughout the vacuum,” says Brodsky.



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Paradigm shift:In-Hadron Condensates

Resolution– Whereas it might sometimes be convenient in computational

truncation schemes to imagine otherwise, “condensates” do not exist as spacetime-independent mass-scales that fill all spacetime.

– So-called vacuum condensates can be understood as a property of hadrons themselves, which is expressed, for example, in their Bethe-Salpeter or light-front wavefunctions.

– GMOR cf.



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QCD

Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201Brodsky and Shrock, PNAS 108, 45 (2011)M. Burkardt, Phys.Rev. D58 (1998) 096015













– No qualitative difference between fπ and ρπ



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Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201Brodsky and Shrock, PNAS 108, 45 (2011)











– No qualitative difference between fπ and ρπ

– And



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0);0( qq

Chiral limit

Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201Brodsky and Shrock, PNAS 108, 45 (2011)







“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


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

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Importance of being well-dressed for quarks

& mesonsLightCone 2011, SMU 23-27 May - 77pgs



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Strong-interaction: 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



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Dressed-quark-gluon vertex

Single most important feature– Perturbative vertex is helicity-conserving:

• Cannot cause spin-flip transitions– However, DCSB introduces nonperturbatively generated

structures that very strongly break helicity conservation– These contributions

• Are large when the dressed-quark mass-function is large– Therefore vanish in the ultraviolet;

i.e., on the perturbative domain– Exact form of the contributions is still the subject of debate

but their existence is model-independent - a fact.


Gap EquationGeneral Form

Dμν(k) – dressed-gluon propagator Γν(q,p) – dressed-quark-gluon vertex Until 2009, all studies of other hadron phenomena used the

leading-order term in a symmetry-preserving truncation scheme; viz., – Dμν(k) = dressed, as described previously– Γν(q,p) = γμ

• … plainly, key nonperturbative effects are missed and cannot be recovered through any step-by-step improvement procedure


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Bender, Roberts & von SmekalPhys.Lett. B380 (1996) 7-12




Gap EquationGeneral Form

Dμν(k) – dressed-gluon propagator good deal of information available

Γν(q,p) – dressed-quark-gluon vertex Information accumulating

Suppose one has in hand – from anywhere – the exact form of the dressed-quark-gluon vertex

What is the associated symmetry-preserving Bethe-Salpeter kernel?!


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If kernels of Bethe-Salpeter and gap equations don’t match,one won’t even get right charge for the pion.


Bethe-Salpeter EquationBound-State DSE

K(q,k;P) – fully amputated, two-particle irreducible, quark-antiquark scattering kernel

Textbook material. Compact. Visually appealing. Correct

Blocked progress for more than 60 years.




Bethe-Salpeter EquationGeneral Form

Equivalent exact bound-state equation but in this form K(q,k;P) → Λ(q,k;P)

which is completely determined by dressed-quark self-energy Enables derivation of a Ward-Takahashi identity for Λ(q,k;P)

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Lei Chang and C.D. Roberts0903.5461 [nucl-th]Phys. Rev. Lett. 103 (2009) 081601






Ward-Takahashi IdentityBethe-Salpeter Kernel

Now, for first time, it’s possible to formulate an Ansatz for Bethe-Salpeter kernel given any form for the dressed-quark-gluon vertex by using this identity

This enables the identification and elucidation of a wide range of novel consequences of DCSB

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Lei Chang and C.D. Roberts0903.5461 [nucl-th]Phys. Rev. Lett. 103 (2009) 081601

iγ5 iγ5






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Relativistic quantum mechanics

Dirac equation (1928):Pointlike, massive fermion interacting with electromagnetic field

Spin Operator



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Massive point-fermion Anomalous magnetic moment

Dirac’s prediction held true for the electron until improvements in experimental techniques enabled the discovery of a small deviation: H. M. Foley and P. Kusch, Phys. Rev. 73, 412 (1948).– Moment increased by a multiplicative factor: 1.001 19 ± 0.000 05.

This correction was explained by the first systematic computation using renormalized quantum electrodynamics (QED): J.S. Schwinger, Phys. Rev. 73, 416 (1948), – vertex correction

The agreement with experiment established quantum electrodynamics as a valid tool.

e e

0.001 16


http://prola.aps.org/abstract/PR/v73/i4/p412_1

http://prola.aps.org/abstract/PR/v73/i4/p416_1


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Fermion electromagnetic current – General structure

with k = pf - pi F1(k2) – Dirac form factor; and F2(k2) – Pauli form factor

– Dirac equation: • F1(k2) = 1 • F2(k2) = 0

– Schwinger: • F1(k2) = 1• F2(k2=0) = α /[2 π]



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Plainly, can’t simply take the limit m → 0. Standard QED interaction, generated by minimal substitution:

Magnetic moment is described by interaction term:

– Invariant under local U(1) gauge transformations – but is not generated by minimal substitution in the action for a free

Dirac field. Transformation properties under chiral rotations?

– Ψ(x) → exp(iθγ5) Ψ(x)

Magnetic moment of a massless fermion?



47

Standard QED interaction, generated by minimal substitution:

– Unchanged under chiral rotation– Follows that QED without a fermion mass term is helicity conserving

Magnetic moment interaction is described by interaction term:

– NOT invariant– picks up a phase-factor exp(2iθγ5)

Magnetic moment interaction is forbidden in a theory with manifest chiral symmetry

Magnetic moment of a massless fermion?



48

One-loop calculation:

Plainly, one obtains Schwinger’s result for me2 ≠ 0

However,

F2(k2) = 0 when me2 = 0

There is no Gordon identity:

Results are unchanged at every order in perturbation theory … owing to symmetry … magnetic moment interaction is forbidden in a theory with manifest chiral symmetry

Schwinger’s result?

e e

m=0 So, no mixingγμ ↔ σμν


QCD and dressed-quark anomalous magnetic moments

Schwinger’s result for QED: pQCD: two diagrams

o (a) is QED-likeo (b) is only possible in QCD – involves 3-gluon vertex

Analyse (a) and (b)o (b) vanishes identically: the 3-gluon vertex does not contribute to

a quark’s anomalous chromomag. moment at leading-ordero (a) Produces a finite result: “ – ⅙ αs/2π ”

~ (– ⅙) QED-result But, in QED and QCD, the anomalous chromo- and electro-

magnetic moments vanish identically in the chiral limit!Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.



Dressed-quark anomalousmagnetic moments

Three strongly-dressed and essentially-

nonperturbative contributions to dressed-quark-gluon vertex:

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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, Kizilersu et al.hep-ph/0303176

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


http://inspirebeta.net/record/868745



http://link.aps.org/doi/10.1103/PhysRevLett.106.072001






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Dressed-quark anomalous chromomagnetic moment

Lattice-QCD– m = 115 MeV

Nonperturbative result is two orders-of-magnitude larger than the perturbative computation– This level of

magnification istypical of DCSB

– cf.

Skullerud, Kizilersu et al.JHEP 0304 (2003) 047

Prediction from perturbative QCD

Quenched lattice-QCD

Quark mass function:M(p2=0)= 400MeVM(p2=10GeV2)=4 MeV





Three strongly-dressed and essentially-

nonperturbative contributions to dressed-quark-gluon vertex:

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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, Kizilersu et al.hep-ph/0303176

Role and importance isnovel discovery•Essential to recover pQCD•Constructive interference with Γ5













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Formulated and solved general Bethe-Salpeter equation Obtained dressed electromagnetic vertex Confined quarks don’t have a mass-shello Can’t unambiguously define

magnetic momentso But can define

magnetic moment distribution

ME κACM κAEM

Full vertex 0.44 -0.22 0.45

Rainbow-ladder 0.35 0 0.048

AEM is opposite in sign but of roughly equal magnitude as ACM


Factor of 10 magnification












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Potentially important for elastic and transition form factors, etc. Significantly, also quite possibly for muon g-2 – via Box diagram,

which is not constrained by extant data. (I.C. Cloët et al.)


Factor of 10 magnification

Formulated and solved general Bethe-Salpeter equation Obtained dressed electromagnetic vertex Confined quarks don’t have a mass-shello Can’t unambiguously define

magnetic momentso But can define

magnetic moment distribution

Contemporary theoretical estimates:1 – 10 x 10-10

Largest value reduces discrepancy expt.↔theory from 3.3σ to below 2σ.











Dressed Vertex & Meson Spectrum

Splitting known experimentally for more than 35 years Hitherto, no explanation Systematic symmetry-preserving, Poincaré-covariant DSE

truncation scheme of nucl-th/9602012.o Never better than ⅟₄ of splitting∼

Constructing kernel skeleton-diagram-by-diagram, DCSB cannot be faithfully expressed:

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Experiment Rainbow-ladder

One-loop corrected

Ball-Chiu Full vertex

a1 1230

ρ 770

Mass splitting 455

Full impact of M(p2) cannot be realised!


One-loop corrected


a1 1230 759 885

ρ 770 644 764

Mass splitting 455 115 121

Location of zeromarks –m2

meson




Dressed Vertex & Meson Spectrum

Fully consistent treatment of Ball-Chiu vertexo Retain λ3 – term but ignore Γ4 & Γ5

o Some effects of DCSB built into vertex & Bethe-Salpeter kernel Big impact on σ – π complex But, clearly, not the complete answer.

Fully-consistent treatment of complete vertex Ansatz Promise of 1st reliable prediction

of light-quark hadron spectrumCraig Roberts: Dyson-Schwinger Equations and Continuum QCD.

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One-loop corrected


a1 1230 759 885 1066

ρ 770 644 764 924

Mass splitting 455 115 121 142


One-loop corrected


a1 1230 759 885 1128 1270

ρ 770 644 764 919 790

Mass splitting 455 115 121 209 480

BC: zero moves deeper for both ρ & a1 Both masses grow

Full vertex: zero moves deeper for a1 but shallower for ρProblem solved


Lei Chang & C.D. Roberts, arXiv:1104.4821 [nucl-th]Tracing massess of ground-state light-quark mesons




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Grand Unification



Unification of Meson & Baryon Properties

Correlate the masses of meson and baryon ground- and excited-states within a single, symmetry-preserving framework Symmetry-preserving means:

Poincaré-covariant & satisfy relevant Ward-Takahashi identities Constituent-quark model has hitherto been the most widely applied

spectroscopic tool; whilst its weaknesses are emphasized by critics and acknowledged by proponents, it is of continuing value because there is nothing better that is yet providing a bigger picture.

Nevertheless, no connection with quantum field theory & therefore not with QCD not symmetry-preserving & therefore cannot veraciously connect

meson and baryon properties




Baryons Dynamical chiral symmetry breaking (DCSB)

– has enormous impact on meson properties.Must be included in description and prediction of baryon properties.

DCSB is essentially a quantum field theoretical effect. In quantum field theory Meson appears as pole in four-point quark-antiquark Green function →

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

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

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

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

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

rqq ≈ rπ




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 correlations60

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

diquark

quark

quark exchangeensures Pauli statistics

composed of strongly-dressed quarks bound by dressed-gluons




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“Spectrum” of nonpointlike quark-quark correlations

Observed in– DSE studies in QCD– 0+ & 1+ in Lattice-QCD

Scalar diquark form factor– r0+ ≈ rπ

Axial-vector diquarks– r1+ ≈ rρ

Diquarks in QCD


Zero relation with old notion of pointlike constituent-like diquarks

BOTH essential

H.L.L. Roberts et al., 1102.4376 [nucl-th]Phys. Rev. C in press

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







Baryons & diquarks


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Provided numerous insights into baryon structure; e.g., There is a causal connection between mΔ - mN & m1+- m0+

mΔ - mN

mN

mΔ

Physical splitting grows rapidly with increasing diquark mass difference






Baryons & diquarks


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Provided numerous insights into baryon structure; e.g., mN ≈ 3 M & mΔ ≈ M+m1+



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Photon-nucleon current

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

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

Vertex contains dressed-quark anomalous magnetic moment

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




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Nucleon Form Factors



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

Unification of meson and nucleon form factors.

Very good description.

Quark’s momentum-dependent anomalous magnetic moment has observable impact & materially improves agreement in all cases.

http://www.slac.stanford.edu/spires/find/hep/www?eprint=arXiv:0812.0416




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

)(

)(

2

2

QG

QG

pM

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

I.C. Cloët, C.D. Roberts, et al.In progress





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Quark anomalous magnetic moment hasbig impact on proton ratio

But little impact on theneutron ratio



)(

)(2

2

QG

QGnM

nEn



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New JLab data: S. Riordan et al., arXiv:1008.1738 [nucl-ex]

DSE-prediction


)(

)(2

2

QG

QGnM

nEn

This evolution is very sensitive to momentum-dependence of dressed-quark propagator


Phys.Rev.Lett. 105 (2010) 262302









Hadron Spectrum


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Legend: Particle Data Group H.L.L. Roberts et al. EBAC Jülich

o Symmetry-preserving unification of the computation of meson & baryon masseso rms-rel.err./deg-of-freedom = 13%o PDG values (almost) uniformly overestimated in both cases - room for the pseudoscalar meson cloud?!


Masses of ground and excited-state hadronsH.L.L. Roberts et al., arXiv:1101.4244 [nucl-th] Few Body Systems (2011) pp. 1-25




Baryon Spectrum


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In connection with EBAC's analysis, dressed-quark Faddeev-equation predictions for bare-masses agree within rms-relative-error of 14%. Notably, EBAC finds a dressed-quark-core for the Roper resonance,

at a mass which agrees with Faddeev Eq. prediction.



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EBAC & the Roper resonance

EBAC examined the dynamical origins of the two poles associated with the Roper resonance are examined.

Both of them, together with the next higher resonance in the P11 partial wave were found to have the same originating bare state

Coupling to the meson-baryon continuum induces multiple observed resonances from the same bare state.

All PDG identified resonances consist of a core state and meson-baryon components.

N. Suzuki et al., Phys.Rev.Lett. 104 (2010) 042302




Hadron Spectrum


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Legend: Particle Data Group H.L.L. Roberts et al. EBAC Jülich

Now and for the foreseeable future, QCD-based theory will provide only dressed-quark core masses;EBAC or EBAC-like tools necessary for mesons and baryons


Baryon Spectrum



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0+ 77%

1+ 23% 100% 100% 100%

0- 51% 43%

1- 49% 57% 100% 100%

diquark contentFascinating suggestion: nucleon’s first excited state = almost entirely axial-vector diquark, despite the dominance of scalar diquark in ground state

Owes fundamentally to close relationship between scalar diquark and pseudoscalar mesonAnd this follows from dynamical chiral symmetry breaking




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Nucleon to Roper Transition Form Factors Extensive CLAS @ JLab Programme has produced the first

measurements of nucleon-to-resonance transition form factors

Theory challenge is toexplain the measurements

Notable result is zero in F2p→N*,

explanation of which is a realchallenge to theory.

DSE study connects appearanceof zero in F2

p→N* with axial-vector-diquark dominance in Roper resonance and structure of form factors of J=1 state



I. Aznauryan et al., Results of the N* Program at JLabarXiv:1102.0597 [nucl-ex]

Preliminary





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Nucleon to Roper Transition Form Factors

Tiator and Vanderhaeghen – in progress– Empirically-inferred light-front-transverse

charge density– Positive core plus negative annulus

Readily explained by dominance of axial-vector diquark – Considering isospin and charge– Negative d-quark twice as likely to be

delocalised from always-positive core than positive u-quark



{uu}

d

2 {ud}

u

1+

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Epilogue



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Epilogue

Dynamical chiral symmetry breaking (DCSB) – mass from nothing for 98% of visible matter – is a realityo Expressed in M(p2), with observable signals in experiment

Poincaré covarianceCrucial in description of contemporary data

Fully-self-consistent treatment of an interaction Comprehensive treatment of vast body of observablesEssential if experimental data is truly to be understood.

Dyson-Schwinger equations: o single framework, with IR model-input turned to advantage,

“almost unique in providing unambiguous path from a defined interaction → Confinement & DCSB → Masses → radii → form factors → distribution functions → etc.”


McLerran & PisarskiarXiv:0706.2191 [hep-ph]

Confinement is almost Certainly the origin of DCSB

e.g., BaBar anomaly





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Just in case …




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New JLab data: S. Riordan et al., arXiv:1008.1738 [nucl-ex]

DSE-prediction


)(

)(2,

1

2,1

QF

QFup

dp

Location of zero measures relative strength of scalar and axial-vector qq-correlations

Brooks, Bodek, Budd, Arrington fit to data: hep-ex/0602017









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Neutron Structure Function at high x

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: 0+ & 1+ qq


Distribution of neutron’s momentum amongst quarks on the valence-quark domainLightCone 2011, SMU 23-27 May - 77pgs




Craig Roberts Physics Division

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