quantum chromodynamics: the origin of mass as we know it craig d. roberts physics division argonne...

Quantum Chromodynamics:The Origin of Mass as We Know itCraig D. Roberts

Physics DivisionArgonne National Laboratory

&

School of PhysicsPeking University

Transition Region

Argonne National Laboratory

Craig Roberts, Physics Division: QCD - Origin of Mass as We Know It

2IIT Physics Colloquium: 7 Oct 2010

Argonne National Laboratory

Physics DivisionATLAS Tandem Linac:

International User Facility for Low Energy Nuclear Physics

37 PhD Scientific StaffAnnual Budget:

$27million



Length-Scales of Physics



Physics Division


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Research sponsored primarily by Department of Energy: Office of Nuclear Physics Nuclear HadronTests of Standard Model

IIT Physics Colloquium: 7 Oct 2010

Physics Division


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Research sponsored primarily by Department of Energy: Office of Nuclear Physics Nuclear HADRONTests of Standard Model


Hadron Physics


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“Hadron physics is unique at the cutting edge of modern science because Nature has provided us with just one instance of a fundamental strongly-interacting theory; i.e., Quantum Chromodynamics (QCD). The community of science has never before confronted such a challenge as solving this theory.”


NSACLong Range Plan


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“A central goal of (DOE Office 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.”


Quarks and Nuclear Physics


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Standard Model of Particle Physics:Six quark flavours




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Real WorldNormal matter – only two light-quark flavours are active




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Real WorldNormal matter – only two light-quark flavours are activeOr, perhaps, three




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Real WorldNormal matter – only two light-quark flavours are activeOr, perhaps, three

For numerous good reasons, much research also focuses on accessible heavy-quarks Nevertheless, I will focus on the light-quarks; i.e., u & d.


What is QCD?



What is QCD?


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


What is QCD?


15


Similar interaction in QED


What is QCD?


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

completely change the character of the theory

3-gluon vertex

4-gluon vertex


QED cf. QCD? Running coupling


17

e

QED

mQ

Qln

32

1)(


QED cf. QCD? Running coupling


18

e

QED

mQ

Qln

32

1)(

Add 3-gluon self-interaction


QED cf. QCD?


19

e

QED

mQ

Qln

32

1)(

Q

NQ

f

QCD

ln)233(

6)(

fermionscreening

gluonantiscreening


QED cf. QCD?


20

2004 Nobel Prize in Physics : Gross, Politzer and Wilczek

e

QED

mQ

Qln

32

1)(

Q

NQ

f

QCD

ln)233(

6)(

fermionscreening

gluonantiscreening


Simple picture- Proton


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Three quantum-mechanical constituent-quarks interacting via a potential, derived from one constituent-gluon exchange


Simple picture- Pion


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Two quantum-mechanical constituent-quarks - particle+antiparticle -interacting via a potential, derived from one constituent-gluon exchange


Modern Miraclesin Hadron Physics


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o proton = three constituent quarks• Mproton ≈ 1GeV

• Therefore guess Mconstituent−quark ≈ ⅓ × GeV ≈ 350MeV




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o pion = constituent quark + constituent antiquark• Guess Mpion ≈ ⅔ × Mproton ≈ 700MeV




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o WRONG . . . . . . . . . . . . . . . . . . . . . . Mpion = 140MeV




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o WRONG . . . . . . . . . . . . . . . . . . . . . . Mpion = 140MeVo Rho-meson

• Also constituent quark + constituent antiquark – just pion with spin of one constituent flipped

• Mrho ≈ 770MeV ≈ 2 × Mconstituent−quark




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o WRONG . . . . . . . . . . . . . . . . . . . . . . Mpion = 140MeVo Rho-meson

• Also constituent quark + constituent antiquark – just pion with spin of one constituent flipped

• Mrho ≈ 770MeV ≈ 2 × Mconstituent−quark

What is “wrong” with the pion?IIT Physics Colloquium: 7 Oct 2010

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

However: current-algebra (1968) This is impossible in quantum mechanics, for which one

always finds:

mm 2

tconstituenstatebound mm


NSACLong Range Plan?


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What is a constituent quark, a constituent-gluon?




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




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If not, with what should they be replaced?






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If not, with what should they be replaced?What is the meaning of the NSAC Challenge?




What is themeaning of all this?


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If mπ=mρ , then repulsive and attractive forces in the Nucleon-Nucleon potential have the SAME range and there is NO intermediate range attraction.




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Under these circumstances: Can 12C be stable? Is the deuteron stable; can Big-Bang Nucleosynthesis occur? Many more existential questions …





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Under these circumstances: Can 12C be stable? Is the deuteron stable; can Big-Bang Nucleosynthesis occur?

(Many more existential questions …)

Probably not … but it wouldn’t matter because we wouldn’t be around to worry about it.



QCD’s Challenges


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


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)


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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.


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QCD’s ChallengesUnderstand emergent phenomena

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.


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Why don’t we juststop talking & solve the

problem? Emergent phenomena can’t be studied using perturbation theory




problem? Emergent phenomena can’t be studied using perturbation theory So what? Same is true of bound-state problems in quantum

mechanics!





mechanics! Differences:

Here relativistic effects are crucial – virtual particlesQuintessence of Relativistic Quantum Field Theory

Interaction between quarks – the Interquark Potential – Unknown throughout > 98% of the pion’s/proton’s volume!





mechanics! Differences:

Here relativistic effects are crucial – virtual particlesQuintessence of Relativistic Quantum Field Theory

Interaction between quarks – the Interquark Potential – Unknown throughout > 98% of the pion’s/proton’s volume!

Understanding requires ab initio nonperturbative solution of fully-fledged interacting relativistic quantum field theory



Universal Truths

Spectrum of hadrons (ground, excited and exotic states), and hadron 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.



Universal Truths


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.



Universal Truths



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 requires existence of quark orbital angular momentum in hadron's rest-frame wave function.



Universal Truths



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 requires 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, Physics Division: QCD - Origin of Mass as We Know It


How can we tackle the SM’sStrongly-interacting piece?




The Traditional Approach – Modelling




The Traditional Approach – Modelling

– has its problems.




Lattice-QCD


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– Spacetime becomes an hypercubic lattice– Computational challenge, many millions of degrees of freedom



Lattice-QCD –


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– Spacetime becomes an hypercubic lattice– Computational challenge, many millions of degrees of freedom– Approximately 500 people worldwide & 20-30 people per collaboration.


A Compromise?Dyson-Schwinger Equations




1994 . . . “As computer technology continues to improve, lattice gauge theory [LGT] will become an increasingly useful means of studying hadronic physics through investigations of discretised quantum chromodynamics [QCD]. . . . .”




1994 . . . “However, it is equally important to develop other complementary nonperturbative methods based on continuum descriptions. In particular, with the advent of new accelerators such as CEBAF (VA) and RHIC (NY), there is a need for the development of approximation techniques and models which bridge the gap between short-distance, perturbative QCD and the extensive amount of low- and intermediate-energy phenomenology in a single covariant framework. . . . ”




1994 . . . “Cross-fertilisation between LGT studies and continuum techniques provides a particularly useful means of developing a detailed understanding of nonperturbative QCD.”





C. D. Roberts and A. G. Williams, “Dyson-Schwinger equations and their application to hadronic physics,” Prog. Part. Nucl. Phys. 33, 477 (1994) [arXiv:hep-ph/9403224].(473 citations)



http://inspirebeta.net/record/37474?ln=en





C. D. Roberts and A. G. Williams, “Dyson-Schwinger equations and their application to hadronic physics,” Prog. Part. Nucl. Phys. 33, 477 (1994) [arXiv:hep-ph/9403224].(473 citations)






A Compromise?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.


A Compromise?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

)(

1)(

ppipS


A Compromise?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

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?



Dyson-SchwingerEquations

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

Theory . . . Materially Reduces Model-Dependence

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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In doing this, arrive at understanding of long- range behaviour of strong running-coupling

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


Mass from Nothing?!Perturbation Theory

QCD is asymptotically-free (2004 Nobel Prize) Chiral-limit is well-defined;

i.e., one can truly speak of a massless quark. NB. This is nonperturbatively impossible in QED.






Dressed-quark propagator: Weak coupling expansion of

gap equation yields every diagram in perturbation theory In perturbation theory:


65

...ln1)(2

22 p

mpM





Dressed-quark propagator: Weak coupling expansion of

gap equation yields every diagram in perturbation theory In perturbation theory: If m=0, then M(p2)=0

Start with no mass,Always have no mass.


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...ln1)(2

22 p

mpM


Dynamical (Spontaneous)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"



Mass from Nothing?!Nonperturbative DSEs


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Gap equation is a nonlinear integral equationModern computers enable it to be solved, self-consistently,

with ease In the last ten years, we have learnt a great deal

about the nature of its kernelWhat do the self-consistent,

nonperturbative solutions tell us?


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.






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DSE prediction of DCSB confirmed

Mass from nothing!





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Hint of lattice-QCD supportfor DSE prediction of violation of reflection positivity IIT Physics Colloquium: 7 Oct 2010

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.



Building on the concepts and theory that produces the features that have been described, one can derive numerous exact results in QCD.





One of them explains the peculiar nature of the pion’s mass; i.e., it’s relationship to the Lagrangian current-quark mass m(ς):


74

P. Maris, C.D. Roberts & P.C. Tandynucl-th/9707003






One of them explains the peculiar nature of the pion’s mass; i.e., it’s relationship to the Lagrangian current-quark mass m(ς):


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This is the promised, peculiar,non-quantum-mechanical relationship.

What are the constants of proportionality, physically?

P. Maris, C.D. Roberts & P.C. Tandynucl-th/9707003




Gell-Mann – Oakes – RennerRelation (1968)




Pion’s leptonic decay constant, mass-dimensioned observable which describes rate of process π+→μ+ν





Vacuum quark condensate





Vacuum quark condensate With the GMOR, without the authors’ intention, began the story of

vacuum condensates Through the intervening years it became commonplace to believe

that condensates are “REAL”; Namely, spacetime-independent mass-dimensioned vacuum expectation values, which have measurable consequences.



Universal “Truths”

Suppose, as is widely held, that vacuum condensates are spacetime-independent, mass-dimensioned physical quantities



Universal “Truths”


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.”



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

Universal Misapprehensions


Then they couple to gravity in general relativity and make an enormous contribution to the cosmological constant


82

4520

4

103

8

HG QCDNscondensateQCD




Then they couple to gravity in general relativity and make anenormous contribution to the cosmological constant

Experimentally, the answer is

Ωcosm. const. = 0.76


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4520

4

103

8





Then they couple to gravity in general relativity and make anenormous contribution to the cosmological constant

Experimentally, the answer is

Ωcosm. const. = 0.76

This appalling mismatch is a bit of a problem.Craig Roberts, Physics Division: QCD - Origin of Mass as We Know It

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4520

4

103

8



Paradigm shift:In-Hadron Condensates

B 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.


85

Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201


QCD


B Resolution



– GMOR cf.


86



QCD


B Resolution



– GMOR cf.


87




B Resolution



– No qualitative difference between fπ and ρπ


88




B Resolution




– And


89




B Resolution




– And


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0);0( qq

Chiral limit



“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.”

Cosmological Constant: – Putting QCD condensates back into hadrons reduces the mismatch

between experiment and theory by a factor of 1045

– Possibly by far more, if technicolour-like theories are the correct paradigm for extending the Standard Model


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


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

Nature of the Pion:QCD’s Goldstone Mode



Nature of the Pion:QCD’s Goldstone Mode


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2 → many or infinitely many

Nature and number of constituents depends on the wavelengthof the probe

Constituent-quarks are replaced by thedressed-quarksand –gluons of QCD


Charting the interaction between light-quarks

We’ve covered Dynamical Chiral Symmetry Breaking in detail. It’s the origin of 98% of all the visible matter in the Universe

What about confinement, the other and probably most fundamental of the emergent phenomena?




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.




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.


96

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



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





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 of mesons 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.





DSEs connect β-function to experimental observables. Hence, comparison between computations and observations can be used to chart β-function’s long-range behaviour.

To realise this goal, a nonperturbative symmetry-preserving DSE truncation is necessary:o Steady quantitative progress is being made with a scheme that is

systematically improvable (Bender, Roberts, von Smekal – nucl-th/9602012)o Leading-order is called the rainbow-ladder truncation.







DSEs connect β-function to experimental observables. Hence, comparison between computations and observations can be used to chart β-function’s long-range behaviour.

To realise this goal, a nonperturbative symmetry-preserving DSE truncation is necessary:o On the other hand, at significant qualitative advances are possible with

symmetry-preserving kernel Ansätze that express important additional nonperturbative effects – M(p2) – difficult/impossible to capture in any finite sum of contributions.


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Can’t walk beyond the rainbow, but must leap!


Gap EquationGeneral Form




Dμν(k) – dressed-gluon propagator Γν(q,p) – dressed-quark-gluon vertex




Dμν(k) – dressed-gluon propagator Γν(q,p) – dressed-quark-gluon vertex Suppose one has in hand – from anywhere – the exact

form of the dressed-quark-gluon vertex

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



Bethe-Salpeter EquationBound-State DSE

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

Textbook material. Compact. Visually appealing. Correct



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)


107

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


108


iγ5 iγ5





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


109


iγ5 iγ5





Dressed-quark anomalousmagnetic moments

Schwinger’s result for QED:




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






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, Physics Division: QCD - Origin of Mass as We Know It



Interaction term that describes magnetic-moment coupling to gauge fieldo Straightforward to show that it mixes left ↔ righto Thus, explicitly violates chiral symmetry





Follows that in fermion’s e.m. current γμF1 does cannot mix with σμνqνF2

No Gordon Identityo Hence massless fermions cannot not possess a measurable

chromo- or electro-magnetic moment





Follows that in fermion’s e.m. current γμF1 does cannot mix with σμνqνF2

No Gordon Identityo Hence massless fermions cannot not possess a measurable

chromo- or electro-magnetic moment But what if the chiral symmetry is dynamically

broken, strongly, as it is in QCD?Craig Roberts, Physics Division: QCD - Origin of Mass as We Know It



Three strongly-dressed and essentially-

nonperturbative contributions to dressed-quark-gluon vertex:


117

DCSB

Lei Chang, Yu-Xin Liu and Craig D. RobertsarXiv:1009.3458 [nucl-th]


http://inspirebeta.net/record/868745







118

DCSB

Ball-Chiu term•Vanishes if no DCSB•Appearance driven by STI










119

DCSB


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










120

DCSB


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








121

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








122

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 ACMo Potentially important for transition form factors, etc.o Muon g-2 ?





Dressed Vertex & Meson Spectrum

Splitting known experimentally for more than 35 years Hitherto, no explanation


123

Experiment Rainbow-ladder

One-loop corrected

Ball-Chiu Full vertex

a1 1230

ρ 770

Mass splitting 455



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:


124


One-loop corrected


a1 1230 759 885

ρ 770 644 764

Mass splitting 455 115 121

Full impact of M(p2) cannot be realised!





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.


125


One-loop corrected


a1 1230 759 885 1066

ρ 770 644 764 924

Mass splitting 455 115 121 142



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


126


One-loop corrected


a1 1230 759 885 1066 1230

ρ 770 644 764 924 745

Mass splitting 455 115 121 142 485



Fully-consistent treatment of complete vertex Ansatz Subtle interplay between competing effects, which can only

now be explicated Promise of first reliable prediction of light-quark hadron

spectrum, including the so-called hybrid and exotic states.


127


One-loop corrected


a1 1230 759 885 1066 1230

ρ 770 644 764 924 745

Mass splitting 455 115 121 142 485


Pion’s Goldberger-Treiman relation


128

Maris, Roberts and Tandynucl-th/9707003

Pion’s Bethe-Salpeter amplitude

Dressed-quark propagator






129




Axial-vector Ward-Takahashi identity entails

Exact inChiral QCD






130




Axial-vector Ward-Takahashi identity entails

Pseudovector componentsnecessarily nonzero.

Cannot be ignored!

Exact inChiral QCD




Pion’s GT relationImplications for observables?


131

Maris and Robertsnucl-th/9804062






132


Pseudovector componentsdominate in ultraviolet:(Q/2)2 = 2 GeV2

pQCD point for M(p2)→ pQCD at Q2 = 8GeV2






133


Pseudovector componentsdominate in ultraviolet:(Q/2)2 = 2 GeV2

pQCD point for M(p2)→ pQCD at Q2 = 8GeV2




Pion’s GT relation


134

Guttierez, Bashir, Cloët, RobertsarXiv:1002.1968 [nucl-th]







Pion’s GT relationContact interaction


135




Bethe-Salpeter amplitude can’t depend on relative momentum; propagator can’t be momentum-dependent

1 MQ







136





Solved gap and Bethe-Salpeter equationsP2=0: MQ=0.4GeV, Eπ=0.098, Fπ=0.5MQ

1 MQ







137





Solved gap and Bethe-Salpeter equationsP2=0: MQ=0.4GeV, Eπ=0.098, Fπ=0.5MQ

1 MQ

Nonzero and significant







138




Asymptotic form of Fπ(Q2)Eπ

2(P)→ Fπem(Q2) = MQ

2/Q2

1 MQ

For 20+ years it was imagined that contact-interaction produced a result that’s indistinguishable From pQCD counting rule







139




Asymptotic form of Fπ(Q2)Eπ

2(P)→ Fπem(Q2) = MQ

2/Q2

Eπ(P) Fπ(P) – cross-term

→ Fπem(Q2) = (Q2/MQ

2) * [Eπ(P)/Fπ(P)] * Eπ2(P)-term = CONSTANT!

1 MQ

For 20+ years it was imagined that contact-interaction produced a result that’s indistinguishable From pQCD counting rule





Pion’s ElectromagneticForm Factor


140


QCD-based DSE prediction: D(x-y) = produces M(p2)~1/p2

cf. contact-interaction: produces M(p2)=constant

)(2

1

yx

)(~)(4

yxyxD







141




)(2

1

yx

)(~)(4

yxyxD







142




)(2

1

yx

)(~)(4

yxyxD

Single mass parameter in both studies Same predictions for Q2=0 observables Disagreement >20% for Q2>MQ

2





BaBar Anomalyγ* γ → π0


143

H.L.L. Roberts, C.D. Roberts, Bashir, Guttierez, TandyarXiv:1009.0067 [nucl-th]



)(2

1

yx

)(~)(4

yxyxD







144




)(2

1

yx

)(~)(4

yxyxD

pQCD







145




)(2

1

yx

)(~)(4

yxyxD

No fully-self-consistent treatment of the pion can reproduce the BaBar data. All produce monotonically- increasing concave functions. BaBar data not a true measure of γ* γ → π0

Likely source of error is misidentification of π0 π0

events where 2nd π0 isn’t seen.

pQCD





Unifying Baryonsand Mesons


146

M(p2) – effects have enormous impact on meson properties.Must be included in description and prediction of baryon

properties.




147


properties. M(p2) 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.




148


properties. M(p2) 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 founded on observation that an interaction

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

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



Faddeev Equation


149

Linear, Homogeneous Matrix equation


diquark

quark



Faddeev Equation


150

Linear, Homogeneous Matrix equationYields wave function (Poincaré Covariant Faddeev Amplitude)

that describes quark-diquark relative motion within the nucleon


diquark

quark

quark exchangeensures Pauli statistics



Faddeev Equation


151

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 correlations


diquark

quark

quark exchangeensures Pauli statistics



Spectrum of some known u- & d-quark baryons


152

Mesons & Diquarks

H.L.L. Roberts, L. Chang and C.D. RobertsarXiv:1007.4318 [nucl-th]H.L.L. Roberts, L. Chang, I.C. Cloët and C.D. Roberts arXiv:1007.3566 [nucl-th]

m0+ m1

+ m0- m1

- mπ mρ mσ ma1

0.72 1.01 1.17 1.31 0.14 0.80 1.06 1.23










153

Mesons & DiquarksCahill, Roberts, Praschifka: Phys.Rev. D36 (1987) 2804

Proof of mass ordering: diquark-mJ+ > meson-mJ-


m0+ m1

+ m0- m1

- mπ mρ mσ ma1

0.72 1.01 1.17 1.31 0.14 0.80 1.06 1.23












154




m0+ m1

+ m0- m1

- mπ mρ mσ ma1

0.72 1.01 1.17 1.31 0.14 0.80 1.06 1.23

Baryons: ground-states and 1st radial exciationsmN mN* mN(⅟₂) mN*(⅟₂-) mΔ mΔ* mΔ(3⁄₂-) mΔ*(3⁄₂-)

DSE 1.05 1.73 1.86 2.09 1.33 1.85 1.98 2.16

EBAC 1.76 1.80 1.39 1.98












155




m0+ m1

+ m0- m1

- mπ mρ mσ ma1

0.72 1.01 1.17 1.31 0.14 0.80 1.06 1.23


DSE 1.05 1.73 1.86 2.09 1.33 1.85 1.98 2.16

EBAC 1.76 1.80 1.39 1.98 mean-|relative-error| = 2%-Agreement

DSE dressed-quark-core masses cf. Excited Baryon Analysis Center (JLab) bare masses is significant ’cause no attempt was made to ensure this.












156




m0+ m1

+ m0- m1

- mπ mρ mσ ma1

0.72 1.01 1.17 1.31 0.14 0.80 1.06 1.23


DSE 1.05 1.73 1.86 2.09 1.33 1.85 1.98 2.16

EBAC 1.76 1.80 1.39 1.98 mean-|relative-error| = 2%-Agreement

DSE dressed-quark-core masses cf. Excited Baryon Analysis Center (JLab) bare masses is significant ’cause no attempt was made to ensure this.

1st radialExcitation ofN(1535)?










Nucleon ElasticForm Factors


157

Photon-baryon vertexOettel, Pichowsky and von Smekal, nucl-th/9909082

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

Form factors reveal how the observable properties of the nucleon – charge and magnetisation – are shared amongst its constituents





Nucleon ElasticForm Factors


158

Photon-baryon vertexOettel, Pichowsky and von Smekal, nucl-th/9909082


“Survey of nucleon electromagnetic form factors” – unification of meson and baryon observables; and prediction of nucleon elastic form factors to 15 GeV2






159

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


)(

)(2

2

QG

QGnM

nEn









160


DSE-prediction


)(

)(2

2

QG

QGnM

nEn

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









161



)(

)(2,

1

2,1

QF

QFup

dp









162



)(

)(2,

1

2,1

QF

QFup

dp

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









163


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









164

Neutron Structure Function at high x

SU(6) symmetry

pQCD

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) 11N. 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 domainIIT Physics Colloquium:

7 Oct 2010





165

Neutron Structure Function at high x

SU(6) symmetry

pQCD

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) 11N. 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 domainIIT Physics Colloquium:

7 Oct 2010





166

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

Confinement is almost Certainly the origin of DCSB



167

Epilogue


Poincaré covarianceCrucial in description of contemporary data




168

Epilogue



Fully-self-consistent treatment of an interaction Essential if experimental data is truly to be understood.




169

Epilogue



Fully-self-consistent treatment of an interaction Essential 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]






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