Craig Roberts Physics Division

A Year in the Life of Dyson-Schwinger Equations

Craig Roberts

Physics Division

http://www.phy.anl.gov/theory/staff/cdr.html

http://www.phy.anl.gov/theory/staff/cdr.html

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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)

Forschungszentrum Julich, IKP: 20 July. 71/81

Craig Roberts: A Year in the life of DSEs

Published collaborations

in 2010/2011StudentsEarly-career scientists

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Top Open Questions in

PhysicsForschungszentrum Julich, IKP: 20 July. 71/81


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Excerpt from the Top-10

Can we quantitatively understand quark and gluon confinement in quantum chromodynamics and the existence of a mass gap?

Quantum chromodynamics, or QCD, is the theory describing the strong nuclear force. Carried by gluons, it binds quarks into particles like protons and neutrons. Apparently, the tiny subparticles are permanently confined: one can't pull a quark or a gluon from a proton because the strong force gets stronger with distance and snaps them right back inside.



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Nuclear Science Advisory CouncilLong Range Plan

A central goal of nuclear physics is to understand the structure and properties of protons and neutrons, and ultimately atomic nuclei, in terms of the quarks and gluons of QCD

So, what’s the problem?They are legion … – Confinement– Dynamical chiral symmetry breaking– A fundamental theory of unprecedented complexity

QCD defines the difference between nuclear and particle physicists: – Nuclear physicists try to solve this theory– Particle physicists run away to a place where tree-level computations

are all that’s necessary; perturbation theory, the last refuge of a scoundrel



Understanding NSAC’sLong Range Plan


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Do they – can they – correspond to well-defined quasi-particle degrees-of-freedom?

If not, with what should they be replaced?

What is the meaning of the NSAC Challenge?

What are the quarks and gluons of QCD? Is there such a thing as a constituent quark, a

constituent-gluon? After all, these are the concepts for which Gell-Mann won the Nobel Prize.


DSEs


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Dyson (1949) & Schwinger (1951) . . . One can derive a system of coupled integral equations relating all the Green functions for a theory, one to another.Gap equation:

o fermion self energy o gauge-boson propagatoro fermion-gauge-boson vertex

These are nonperturbative equivalents in quantum field theory to the Lagrange equations of motion.

Essential in simplifying the general proof of renormalisability of gauge field theories.

)(1)(

ppipS



Dyson-SchwingerEquations

Well suited to Relativistic Quantum Field Theory Simplest level: Generating Tool for Perturbation

Theory . . . Materially Reduces Model-Dependence … Statement about long-range behaviour of quark-quark interaction

NonPerturbative, Continuum approach to QCD Hadrons as Composites of Quarks and Gluons Qualitative and Quantitative Importance of:

Dynamical Chiral Symmetry Breaking– Generation of fermion mass from nothing Quark & Gluon Confinement

– Coloured objects not detected, Not detectable?

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Approach yields Schwinger functions; i.e., propagators and vertices Cross-Sections built from Schwinger Functions Hence, method connects observables with long- range behaviour of the running coupling Experiment ↔ Theory comparison leads to an understanding of long- range behaviour of strong running-coupling


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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 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


http://inspirebeta.net/record/823162?ln=en


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: A Year in the life of DSEs

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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 states with masses 1-2GeV are a great deal more sensitive to the long-range behaviour of the β-function than those of the π&ρ ground states.


14Forschungszentrum Julich, IKP: 20 July. 71/81


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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 = 9 - 15 • α(mG

2)/π = 2 – 4


Qin, Chang, Roberts, et al., in progress at IKP

Running gluon mass– Gluon is massless in ultraviolet

in agreement with pQCD– Massive in infrared

• mG(0) = 0.69 – 0.81 GeV• mG(mG

2) = 0.30 GeV

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Dynamical Chiral Symmetry Breaking

Mass GapForschungszentrum Julich, IKP: 20 July. 71/81



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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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Importance of being well-dressed for quarks

& mesonsForschungszentrum Julich, IKP: 20 July. 71/81



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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.D. Roberts, R. Delbourgo, A.G. Williams, H.J. Munczek, M.R. Pennington, A. Bashir, A. Kizilersu, L. Chang, Y.-X. Liu …

Dressed-quark-gluon vertex



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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?



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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?



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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: A Year in the life of DSEs



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.


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.










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Looking deeperForschungszentrum Julich, IKP: 20 July. 71/81


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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



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Pion’s valence-quark distributions

Owing to absence of pion targets, the pion’s valence-quark distribution functions are measured via the Drell-Yan process: π p → μ+ μ− X

Three experiments: CERN (1983 & 1985)and FNAL (1989). No more recent experiments because theory couldn’t even explain these!

ProblemConway et al. Phys. Rev. D 39, 92 (1989)Wijesooriya et al. Phys.Rev. C 72 (2005) 065203

Behaviour at large-x is inconsistent with pQCD; viz,

expt. (1-x)1+ε cf. QCD (1-x)2+γ



Pion








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First QCD-based calculation



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Computation of qvπ(x)

In the Bjorken limit; viz., q2 → ∞, P · q → − ∞but x := −q2/2P · q fixed,

Wμν(q;P) ~ qvπ(x)

Plainly, this can be computed given all the information we have at our disposal from the DSEs



T+

T–

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Computation of qvπ(x)

Before the first DSE computation, which used the running dressed-quark mass described previously, numerous authors applied versions of the Nambu–Jona-Lasinio model and were content to reproduce the data, arguing therefrom that the inferences from pQCD were wrong



Hecht, Roberts, Schmidt Phys.Rev. C 63 (2001) 025213

After the first DSE computation, experimentalists again became interested in the process because– DSEs agreed with pQCD

but disagreed with the data, and other models

Disagreement on the “valence domain,” which is uniquely sensitive to M(p2)

2.61/1.27= factor of 2 in the exponent





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Reanalysis of qvπ(x)

After the first DSE computation, the “Conway et al.” data were reanalysed, this time at next-to-leading-order (Wijesooriya et al. Phys.Rev. C 72 (2005) 065203)

The new analysis produced a much larger exponent than initially obtained; viz., β=1.87, but now it disagreed equally with NJL-model results and the DSE prediction NB. Within pQCD, one can readily understand why adding a higher-order correction

leads to a suppression of qvπ(x) at large-x.



Hecht, Roberts, Schmidt Phys.Rev. C 63 (2001) 025213

New experiments were proposed … for accelerators that do not yet exist but the situation remained otherwise unchanged

Until the publication of Distribution Functions of the Nucleon and Pion in the Valence Region, Roy J. Holt and Craig D. Roberts, arXiv:1002.4666 [nucl-th], Rev. Mod. Phys. 82 (2010) pp. 2991-3044















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Reanalysis of qvπ(x)

This article emphasised and explained the importance of the persistent discrepancy between the DSE result and experiment as a challenge to QCD

It prompted another reanalysis of the data, which accounted for a long-overlooked effect: viz., “soft-gluon resummation,” – Compared to previous analyses, we include next-to-leading-

logarithmic threshold resummation effects in the calculation of the Drell-Yan cross section. As a result of these, we find a considerably softer valence distribution at high momentum fractions x than obtained in previous next-to-leading-order analyses, in line with expectations based on perturbative-QCD counting rules or Dyson-Schwinger equations.



Distribution Functions of the Nucleon and Pion in the Valence Region, Roy J. Holt and Craig D. Roberts, arXiv:1002.4666 [nucl-th], Rev. Mod. Phys. 82 (2010) pp. 2991-3044

Aicher, Schäfer, Vogelsang, “Soft-Gluon Resummation and the Valence Parton Distribution Function of the Pion,” Phys. Rev. Lett. 105 (2010) 252003










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Current status of qv

π(x) DSE prediction and modern representation of the data are

indistinguishable on the valence-quark domain

Emphasises the value of using a single internally-consistent, well-constrained framework to correlate and unify the description of hadron observables

What about valence-quark distributions in the kaon?



Trang, Bashir, Roberts & Tandy, “Pion and kaon valence-quark parton distribution functions,” arXiv:1102.2448 [nucl-th], Phys. Rev. C 83, 062201(R) (2011) [5 pages]




http://link.aps.org/doi/10.1103/PhysRevC.83.062201





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qvπ(x) & qv

K(x) ms ≈ 24 mu & Ms ≈ 1.25 Mu

Expect the s-quark to carry more of the kaon’s momentum than the u-quark, so that xsK(x) peaks at larger value of x than xuK(x)

Expectation confirmed in computations, with s-quark distribution peaking at 15% larger value of x

Even though deep inelastic scattering is a high-Q2 process, constituent-like mass-scale explains the shift




xuπ(x)

xsK(x)xuK(x)









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uK(x)/uπ(x) Drell-Yan experiments at

CERN (1980 & 1983) provide the only extant measurement of this ratio

DSE result in complete accord with the measurement

New Drell-Yan experiment running now at FNAL is capable of validating this comparison

It should be done so that complete understanding can be claimed




Value of ratio at x=1 is a fixed point of the evolution equationsHence, it’s a very strong test of nonperturbative dynamics

Value of ratio at x=0 will approach “1” under evolution to higher resolving scales. This is a feature of perturbative dynamics

Using DSEs in QCD, one derives that the x=1 value is ≈ (fπ/fK)2 (Mu /Ms)4 = 0.3









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Grand UnificationForschungszentrum Julich, IKP: 20 July. 71/81


Unification of Meson & Baryon Spectra

Dyson-Schwinger Equations have been applied extensively to the spectrum and interactions of mesons with masses less than 1 GeV

On this domain the rainbow-ladder approximation, which is the leading-order in a systematic & symmetry-preserving truncation scheme – nucl-th/9602012, is a well-understood tool that is accurate for pseudoscalar and vector mesons: e.g., Prediction of elastic pion and kaon form factors: nucl-th/0005015 Anomalous neutral pion processes – γπγ & BaBar anomaly: 1009.0067 [nucl-th] Pion and kaon valence-quark distribution functions: 1102.2448 [nucl-th] Unification of these and other observables – ππ scattering: hep-ph/0112015

It can readily be extended to explain properties of the light neutral pseudoscalar mesons: 0708.1118 [nucl-th]



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http://arxiv.org/abs/1102.2448









DSEs & 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 the 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


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 correlations56

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



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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 plus current → nucleon form factors

Vertex must contain the dressed-quark anomalous magnetic moment: Lecture IV

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



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












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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

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





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DCSB & EDMsForschungszentrum Julich, IKP: 20 July. 71/81


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T-violation Invariance under C*P*T is a symmetry of any

local quantum field theory C*P alone is not a symmetry of the Standard

Model – Therefore Time-Reversal-Invariance is

violated.



Hecht, Roberts, Schmidt Phys.Rev. C64 (2001) 025204

Possession of an EDM by J=½ particle is a violation of T-symmetry Standard Model: |dn,p |< 10-32 ecm

– Current experiment: |dp |<7.9 x 10-25 ecm & |dn |<1.6 x 10-26 ecm

Suppose nonzero dp or dn is measured– What is the relation to current-quark EDMs: du,d,s,… ?– SU(6) Quark Model: dp=⅓(4du-dd) & dn=⅓(4dd-du)

• But these are constituent-quark EDMs, which aren’t readily defined in SM!



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Nucleon EDM

Nucleon EDM described by a third form factor in the photon-nucleon current:

Current-quark EDMs, from any source, will produce nucleon EDM, which can be computed if-and-only-if one has a framework for connecting current-quarks to hadron observables






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Quark EDMs

Faddeev equation approach EDM appears in dressed-quark–photon vertex: two possible

additions because confined-quarks don’t have a mass-shell –

Q= electric- charge matrixm = current-quark mass H± = diag[hu

±,hd±] – current-quark gyroelectric ratios

May now compute relation between hq & hN






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Quark EDMsNucleon structure factors

Matrix of gyroelectric ratios




Structure factors Quark charges Gyroelectric ratios

Nonzero because dressed-quarks are not Dirac particles



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Gyroelectric ratios

Neutron and proton (hN=e/2MN) related to current-quarks

NB. h+ = 0 in constituent-quark modelscoefficient-ratio(hd

–/hu–)= 3.4

cf. SU(6) CQMs coefficient-ratio(hd–/hu

–)= 2.0 SU(6) is not a good approximation

Example: Charged-Higgs-boson-exchange models yield hd>>hu hn = – 40.6 ( 0.85 hd

+ + 1.15 hd- )

hp = 5.1 ( -0.66 hd+ + 1.34 hd

- )Forschungszentrum Julich, IKP: 20 July. 71/81



Suppose hu>>hd

hp = 81.2 ( 0.85 hu+ + 1.15 hu

- )



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Quark EDMs The scale of DCSB in QCD amplifies the contribution from a current-

quark’s EDM to that of the bound state containing it– minimal enhancement factor is roughly the ratio of the constituent-

quark and current-quark masses; two-orders of magnitude– it is understandable and calculable through the necessary momentum

dependence of the dressed-quark mass function. Confinement & compositeness entail that the dressed-quarks

comprising bound state are not on shell & hence the two C P- & T-violating operator structures that are indistinguishable in the free-quark limit yield materially different contributions to the EDM of the bound state. – These operator structures must be analysed and their strengths

determined independently in any model that provides for C P and T violation.

Both these effects should be accounted for in using dN as a means of constraining extensions of the Standard Model.






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Condensates?Forschungszentrum Julich, IKP: 20 July. 71/81


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Connecting QCD with coupled-channels

calculationsForschungszentrum Julich, IKP: 20 July. 71/81




MAN KANN ÜBER ALLES SPRECHENABER NICHT ÜBER EINE STUNDE



Epilogue

70

Epilogue QCD is the most interesting part of the Standard Model and Nature’s

only example of an essentially nonperturbative fundamental theory Dyson-Schwinger equations:

o Described some highlights from 2010/2011: Confinement is a dynamical phenomenon

It cannot in principle be expressed via a potential Gluons are nonperturbatively massive & quarks are not Dirac particles 2001 Prediction of pion valence-quark distribution function confirmed in 2010,

via modern reanalysis of extant Drell-Yan data Dynamical chiral symmetry breaking is a fact.

It’s responsible for 98% of the mass of visible matter in the UniverseProvides an enhancement factor between hadron EDMs and current-quark EDMs

o Omitted more highlights from 2010/2011; e.g., Condensates are contained within hadrons studies relating dressed-quark hadron-spectrum to EBAC and Jülich coupled-

channels models



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

New perspectives on the quark condensateBrodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201







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Where do we go from here?Forschungszentrum Julich, IKP: 20 July. 71/81


72


Condensates?Forschungszentrum Julich, IKP: 20 July. 71/81


Dichotomy of the pion


73

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


74

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

Gell-Mann – Oakes – RennerRelation


75

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: A Year in the life of DSEs


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 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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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( qqChiral 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

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

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