indico.nucleares.unam.mx · First Contents Back Conclusion Hadron Physics and Continuum Strong QCD...

First Contents Back Conclusion

Hadron Physics andContinuum Strong QCD

Craig D. [email protected]

Physics Division & School of Physics

Argonne National Laboratory Peking University

http://www.phy.a nl.gov/theory/staff/cdr.htmlCraig Roberts: Hadron Physics and Continuum Strong QCD

XII Mexican Workshop on Particles and Fields: Mini-courses, 4-8 Nov. 2009. . . 48 – p. 1/48


Form Factors: Why?

Craig Roberts: Hadron Physics and Continuum Strong QCD



Form Factors: Why?

The nucleon and pion hold special places in non-perturbativestudies of QCD.




Form Factors: Why?


An explanation of nucleon and pion structure and interactions iscentral to hadron physics – they are respectively the archetypesfor baryons and mesons.




Form Factors: Why?



Form factors have long been recognized as a basic tool forelucidating bound state properties. They can be studied from verylow momentum transfer, the region of non-perturbative QCD, up toa region where perturbative QCD predictions can be tested.




Form Factors: Why?




Experimental and theoretical studies of nucleon electromagneticform factors have made rapid and significant progress during thelast several years, including new data in the time like region, andmaterial gains have been made in studying the pion form factor.




Form Factors: Why?




Experimental and theoretical studies of nucleon electromagneticform factors have made rapid and significant progress during thelast several years, including new data in the time like region, andmaterial gains have been made in studying the pion form factor.

Despite this, many urgent questions remain unanswered.Craig Roberts: Hadron Physics and Continuum Strong QCD



JLab

Thomas Jefferson National Accelerator Facility




JLab


World’s Premier Hadron Physics Facility




JLab



Design goal (4 GeV) experiments began in 1995




JLab




Electrons accelerated by

repeated journeys along linacs




JLab






Once desired energy is

reached, Beam is directed into

Experimental Halls A, B and C




JLab






Once desired energy is

reached, Beam is directed into

Experimental Halls A, B and C

Current Peak

Electron Beam Energy

Nearly 6 GeV




JLab Hall-A




JLab Hall-A

Measured Ratio of

Proton’s Electric and Magnetic Form Factors




JLab Hall-A

0 1 2 3 4 5 6Q

2 [GeV2]

0

0.2

0.4

0.6

0.8

1

1.2

µ p GEp/ G

Mp

SLACJLab 1JLab 2




JLab Hall-A

0 1 2 3 4 5 6Q

2 [GeV2]

0

0.2

0.4

0.6

0.8

1

1.2

µ p GEp/ G

Mp

SLACJLab 1JLab 2

Walker et al., Phys.Rev. D 49, 5671(1994). (SLAC)

Jones et al., JLab HallA Collaboration, Phys.Rev. Lett. 84, 1398(2000)

Gayou, et al., Phys.Rev. C 64, 038202(2001)

Gayou, et al., JLab HallA Collaboration, Phys.Rev. Lett. 88 092301(2002)




JLab Hall-A

0 1 2 3 4 5 6Q

2 [GeV2]

0

0.2

0.4

0.6

0.8

1

1.2

µ p GEp/ G

Mp

SLACJLab 1JLab 2

If JLab Correct, then

Completely

Unexpected Result:

In the Proton

– On Relativistic

Domain

– Distribution of

Quark-Charge

Not Equal

Distribution of

Quark-Current!




Some Questions

What is the role of pion cloud in nucleonelectromagnetic structure?

Can we understand the pion cloud in a morequantitative and, perhaps, model-independentway?




Some Questions

Where is the transition from non-pQCD to pQCD inthe pion and nucleon electromagnetic formfactors?




Some Questions

Do we understand the high Q2 behavior of theproton form factor ratio in the space-like region?

Can we make model-independent statementsabout the role of relativity or orbital angularmomentum in the nucleon?




Some Questions

Can we understand the rich structure of thetime-like proton form factors in terms ofresonances?

What do we expect for the proton form factor ratioin the time-like region?

What is the relation between proton and neutronform factor in the time-like region?

How do we understand the ratio between time-likeand space-like form factors?




Some Questions

What is the role of two-photon exchangecontributions in understanding the discrepancybetween the polarization and Rosenbluthmeasurements of the proton form factor ratio?

What is the impact of these contributions on otherform factor measurements?




Some Questions

How accurately can the pion form factor beextracted from the ep → e′nπ+ reaction?




Status




StatusCurrent status is described in

J. Arrington, C. D. Roberts and J. M. Zanotti“Nucleon electromagnetic form factors,”J. Phys. G 34, S23 (2007); [arXiv:nucl-th/0611050].

C. F. Perdrisat, V. Punjabi and M. Vanderhaeghen,“Nucleon electromagnetic form factors,”Prog. Part. Nucl. Phys. 59, 694 (2007);[arXiv:hep-ph/0612014].




StatusCurrent status is described in

J. Arrington, C. D. Roberts and J. M. Zanotti“Nucleon electromagnetic form factors,”J. Phys. G 34, S23 (2007); [arXiv:nucl-th/0611050].

C. F. Perdrisat, V. Punjabi and M. Vanderhaeghen,“Nucleon electromagnetic form factors,”Prog. Part. Nucl. Phys. 59, 694 (2007);[arXiv:hep-ph/0612014].

Most recently:“ECT∗ Workshop on Hadron Electromagnetic Form Factors”Organisers: Alexandrou, Arrington, Friedrich, Maas, RobertsPresentations, etc., available on-linehttp://ect08.phy.anl.gov/




QCD’s Challenges




QCD’s Challenges

Quark and Gluon Confinement

No 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=−




QCD’s Challenges








Neither of these phenomena is apparent in QCD’s

Lagrangian yet they are the dominant determining

characteristics of real-world QCD.




QCD’s ChallengesUnderstand Emergent Phenomena








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 rulesCraig Roberts: Hadron Physics and Continuum Strong QCD



Confinement




Confinement

Infinitely Heavy Quarks . . . Picture in Quantum Mechanics

integration of the force-3 loops

bosonic string

V (r) = σ r − π

12

1

r

√σ ∼ 470 MeV

Necco & Sommer

he-la/0108008




Confinement

Illustrate this in terms of the action density . . . analogous to

plotting the Force = FQ̄Q(r) = σ +π

12

1

r2

Bali, et al.




Confinement

What happens in the real world; namely, in the presence of

light-quarks?




Confinement


light-quarks? No one knows . . . but Q̄Q + 2 × s̄s

Bali, et al.

he-la/0512018




Confinement



Bali, et al.

he-la/0512018“The breaking of the string appears to be an instantaneous

process, with de-localized light quark pair creation.”




Confinement



Bali, et al.



Energy stored in string at instant before disappearance:

Ec ≃ 1.25 GeV




Confinement



Bali, et al.




Ec ≃ MS+MS̄, where MS is s-quark constituent-mass




Confinement



Bali, et al.





Flux tube collapses instantly and entirely when the energy it

contains exceeds that required to produce the lightest

constituent quark-antiquark pair.




ConfinementTherefore . . . No information onpotential between light-quarks.



Bali, et al.





Flux tube collapses instantly and entirely when the energy it

contains exceeds that required to produce the lightest

constituent quark-antiquark pair.




Dyson-Schwinger EquationsEuler-Lagrange equations for quantum field theory

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 GluonsQualitative and Quantitative Importance of:· Dynamical Chiral Symmetry Breaking

– Generation of fermion mass from nothing· Quark & Gluon Confinement

– Coloured objects not detected, not detectable?











Understanding ⇒ InfraRed behaviour of αs(Q2)











Understanding ⇒ InfraRed behaviour of αs(Q2)

Method yields Schwinger Functions ≡ Propagators

Cross-Sections built from Schwinger FunctionsCraig Roberts: Hadron Physics and Continuum Strong QCD



Schwinger Functions




Schwinger Functions

Solutions are Schwinger Functions(Euclidean Green Functions)




Schwinger Functions


Not all are Schwinger functions are experimentallyobservable




Schwinger Functions


Not all are Schwinger functions are experimentallyobservable but . . .

all are same VEVs measured in numericalsimulations of lattice-regularised QCDopportunity for comparisons atpre-experimental level . . . cross-fertilisation




Schwinger Functions


Not all are Schwinger functions are experimentallyobservable but . . .

all are same VEVs measured in numericalsimulations of lattice-regularised QCDopportunity for comparisons atpre-experimental level . . . cross-fertilisation

Proving fruitful.




World . . .




World . . .DSE Perspective




Persistent Challenge





Infinitely Many Coupled Equations

Σ=

D

γΓS






Σ=

D

γΓS

Coupling between equations necessitates truncation






Σ=

D

γΓS


Weak coupling expansion ⇒ Perturbation Theory






Σ=

D

γΓS


Weak coupling expansion ⇒ Perturbation TheoryNot useful for the nonperturbative problemsin which we’re interested






There is at least one systematic nonperturbative,symmetry-preserving truncation schemeH.J. Munczek Phys. Rev. D 52 (1995) 4736Dynamical chiral symmetry breaking, Goldstone’stheorem and the consistency of the Schwinger-Dysonand Bethe-Salpeter EquationsA. Bender, C. D. Roberts and L. von Smekal, Phys.Lett. B 380 (1996) 7Goldstone Theorem and Diquark Confinement BeyondRainbow Ladder Approximation



http://www.slac.stanford.edu/spires/find/hep/www?eprint=hep-ph&eprint=9411239

http://www.slac.stanford.edu/spires/find/hep/www?eprint=NUCL-TH&eprint=9602012

http://web.mit.edu/redingtn/www/netadv/Xdysonschw.html




There is at least one systematic nonperturbative,symmetry-preserving truncation scheme

Has Enabled Proof of EXACT Results in QCD









And Formulation of Practical Phenomenological Tool to

Illustrate Exact Results











Make Predictions with Readily Quantifiable Errors











Make Predictions with Readily Quantifiable Errors

Examples:MIT – The Net Advance of PhysicsReview Articles and Tutorials in an Encyclopædic Formatweb.mit.edu/redingtn/www/netadv/Xdysonschw.html





PerturbativeDressed-quark Propagator





S(p) =Z(p2)

iγ · p + M(p2)Σ

=D

γΓS

Gap Equation





S(p) =Z(p2)

iγ · p + M(p2)Σ

=D

γΓS

Gap Equationdressed-quark propagator

S(p) =1

iγ · pA(p2) + B(p2)





S(p) =Z(p2)

iγ · p + M(p2)Σ

=D

γΓS


S(p) =1

iγ · pA(p2) + B(p2)

Weak Coupling ExpansionReproduces Every Diagram in Perturbation Theory





S(p) =Z(p2)

iγ · p + M(p2)Σ

=D

γΓS


S(p) =1

iγ · pA(p2) + B(p2)


But in Perturbation Theory

B(p2) = m

(

1 − α

πln

[

p2

m2

]

+ . . .

)

m→0→ 0





S(p) =Z(p2)

iγ · p + M(p2)Σ

=D

γΓS


S(p) =1

iγ · pA(p2) + B(p2)


But in Perturbation Theory

B(p2) = m

(

1 − α

πln

[

p2

m2

]

+ . . .

)

m→0→ 0

No DCSBHere!




Explanation?




QCD & Interaction BetweenLight-Quarks

Kernel of Gap Equation: Dµν(p − q) Γν(q)

Dressed-gluon propagator and dressed-quark-gluon vertex

Reliable DSE studies of Dressed-gluon propagator:

R. Alkofer and L. von Smekal, The infrared behavior of QCDGreen’s functions . . . , Phys. Rept. 353, 281 (2001).




QCD & Interaction BetweenLight-Quarks

Kernel of Gap Equation: Dµν(p − q) Γν(q)

Dressed-gluon propagator and dressed-quark-gluon vertex

Reliable DSE studies of Dressed-gluon propagator:

R. Alkofer and L. von Smekal, The infrared behavior of QCDGreen’s functions . . . , Phys. Rept. 353, 281 (2001).

Dressed-gluon propagator – lattice-QCD simulations confirm thatbehaviour:

D. B. Leinweber, J. I. Skullerud, A. G. Williams and C.Parrinello [UKQCD Collaboration], Asymptotic scaling andinfrared behavior of the gluon propagator, Phys. Rev. D 60,094507 (1999) [Erratum-ibid. D 61, 079901 (2000)].

Exploratory DSE and lattice-QCD studiesof dressed-quark-gluon vertex




Dressed-gluon PropagatorAlkofer, Detmold, Fischer,Maris: he-ph/0309078

Dµν(k) =

(

δµν − kµkν

k2

)

Z(k2)

k2

10-2

10-1

100

101

102

103

p2 [GeV

2]

10-1

100

Z(p

2 )

lattice, Nf=0

DSE, Nf=0

DSE, Nf=3

Fit to DSE, Nf=3

Suppressionmeans ∃ IR gluonmass-scale≈1 GeV

Naturally, thisscale has thesame origin asΛQCD




Dressed-Quark Propagator





S(p) =Z(p2)

iγ · p + M(p2)Σ

=D

γΓS

Gap Equation





S(p) =Z(p2)

iγ · p + M(p2)Σ

=D

γΓS

Gap EquationGap Equation’s Kernel Enhanced on IR domain

⇒ IR Enhancement of M(p2)

10−2 10−1 100 101 102

p2 (GeV2)

10−3

10−2

10−1

100

101

M(p

2 ) (G

eV)

b−quarkc−quarks−quarku,d−quarkchiral limitM

2(p

2) = p

2





S(p) =Z(p2)

iγ · p + M(p2)Σ

=D

γΓS



10−2 10−1 100 101 102

p2 (GeV2)

10−3

10−2

10−1

100

101

M(p

2 ) (G

eV)


2(p

2) = p

2

Euclidean Constituent–Quark

Mass: MEf : p2 = M(p2)2

flavour u/d s c b

ME

mζ∼ 102 ∼ 10 ∼ 1.5 ∼ 1.1





S(p) =Z(p2)

iγ · p + M(p2)Σ

=D

γΓS



10−2 10−1 100 101 102

p2 (GeV2)

10−3

10−2

10−1

100

101

M(p

2 ) (G

eV)


2(p

2) = p

2

Euclidean Constituent–Quark

Mass: MEf : p2 = M(p2)2

flavour u/d s c b

ME

mζ∼ 102 ∼ 10 ∼ 1.5 ∼ 1.1

Predictions confirmed innumerical simulations of lattice-QCD




Dressed-QuarkPropagator



http://www.slac.stanford.edu/spires/find/hep/www?key=2979683



Longstanding Prediction of

Dyson-Schwinger Equation

Studies








Studies

E.g., Dyson-Schwinger

equations and their

application to hadronic

physics,

C. D. Roberts and

A. G. Williams,

Prog. Part. Nucl. Phys.

33 (1994) 477









Studies


equations and their


physics,

C. D. Roberts and

A. G. Williams,


33 (1994) 477

Long used as basis for

efficacious hadron physics

phenomenology









Studies


equations and their


physics,

C. D. Roberts and

A. G. Williams,


33 (1994) 477

Long used as basis for

efficacious hadron physics

phenomenology

Electromagnetic pion

form-factor and neutral

pion decay width,

C. D. Roberts,

Nucl. Phys. A 605(1996) 475






Frontiers of Nuclear Science:A Long Range Plan (2007)




Frontiers of Nuclear Science:Theoretical Advances

Σ=

D

γΓS

Gap Equation





Σ=

D

γΓS

Gap Equation

S(p) =Z(p2)

iγ · p + M(p2)

0 1 2 3

p [GeV]

0

0.1

0.2

0.3

0.4

M(p

) [G

eV] m = 0 (Chiral limit)

m = 30 MeVm = 70 MeV

effect of gluon cloudRapid acquisition of mass is





S(p) =Z(p2)

iγ · p + M(p2)

0 1 2 3

p [GeV]

0

0.1

0.2

0.3

0.4

M(p

) [G




Mass from nothing .

In QCD a quark’s effective massdepends on its momentum. Thefunction describing this can becalculated and is depicted here.Numerical simulations of latticeQCD (data, at two different baremasses) have confirmed modelpredictions (solid curves) that thevast bulk of the constituent massof a light quark comes from acloud of gluons that are draggedalong by the quark as itpropagates. In this way, a quarkthat appears to be absolutelymassless at high energies(m = 0, red curve) acquires alarge constituent mass at lowenergies.





S(p) =Z(p2)

iγ · p + M(p2)

0 1 2 3

p [GeV]

0

0.1

0.2

0.3

0.4

M(p

) [G




Mass from nothing .

In QCD a quark’s effective massdepends on its momentum. Thefunction describing this can becalculated and is depicted here.Numerical simulations of latticeQCD (data, at two different baremasses) have confirmed modelpredictions (solid curves) that thevast bulk of the constituent massof a light quark comes from acloud of gluons that are draggedalong by the quark as itpropagates. In this way, a quarkthat appears to be absolutelymassless at high energies(m = 0, red curve) acquires alarge constituent mass at lowenergies.

↑ ↑

Scanned by Q2 ∈ [2,9] GeV2 Baryon Form FactorsCraig Roberts: Hadron Physics and Continuum Strong QCD




In QCD

a quark’s mass must depend on

its momentum



First Contents Back ConclusionCraig Roberts: Hadron Physics and Continuum Strong QCD



• Established understanding oftwo- and three-point functions




Hadrons

• Established understanding oftwo- and three-point functions

• What about bound states?




Hadrons

• Without bound states, Comparison withexperiment is impossible




Hadrons


• They appear as pole contributions to n ≥ 3-pointcolour-singlet Schwinger functions




Hadrons


• Bethe-Salpeter Equation

QFT Generalisation of Lippmann-Schwinger Equation.




Hadrons


• Bethe-Salpeter Equation

QFT Generalisation of Lippmann-Schwinger Equation.

• What is the kernel, K?

or What is the long-range potential in QCD?Craig Roberts: Hadron Physics and Continuum Strong QCD



What is the light-quarkLong-Range Potential?




What is the light-quarkLong-Range Potential?

Potential between static (infinitely heavy) quarksmeasured in simulations of lattice-QCD is not relatedin any simple way to the light-quark interaction.Craig Roberts: Hadron Physics and Continuum Strong QCD



Bethe-Salpeter Kernel





Axial-vector Ward-Takahashi identity

Pµ Γl5µ(k;P ) = S−1(k+)

1

2λl

f iγ5 +1

2λl

f iγ5 S−1(k−)

−Mζ iΓl5(k;P ) − iΓl

5(k;P ) Mζ

QFT Statement of Chiral Symmetry






Pµ Γl5µ(k;P ) = S−1(k+)

1

2λl

f iγ5 +1

2λl

f iγ5 S−1(k−)


5(k;P ) Mζ

Satisfies BSE Satisfies DSE






Pµ Γl5µ(k;P ) = S−1(k+)

1

2λl

f iγ5 +1

2λl

f iγ5 S−1(k−)


5(k;P ) Mζ

Satisfies BSE Satisfies DSEKernels very differentbut must be intimately related






Pµ Γl5µ(k;P ) = S−1(k+)

1

2λl

f iγ5 +1

2λl

f iγ5 S−1(k−)


5(k;P ) Mζ


• Relation must be preserved by truncation






Pµ Γl5µ(k;P ) = S−1(k+)

1

2λl

f iγ5 +1

2λl

f iγ5 S−1(k−)


5(k;P ) Mζ


• Relation must be preserved by truncation• Nontrivial constraint






Pµ Γl5µ(k;P ) = S−1(k+)

1

2λl

f iγ5 +1

2λl

f iγ5 S−1(k−)


5(k;P ) Mζ


• Relation must be preserved by truncation• Failure ⇒ Explicit Violation of QCD’s Chiral Symmetry




Goldstone’s Theorem

In the chiral limit the QCD Action possesses chiral symmetry

The chiral limit is a good approximation in QCDfor u- and d-quarks

If this SU(Nf = 2) chiral symmetry is dynamically broken, thenthere is a massless composite particle associated with eachgenerator of chiral transformations; i.e., three Goldstone Bosons

These three Goldstone Bosons have long been identified with thepions: π+ , π0 , π−









E.g., V (x, y) = (σ2 + π2 − 1)2

– Hamiltonian: T + V , is Rotationally Invariant

-1

-0.5

0

0.5

1-1

-0.5

0

0.5

1

0

0.5

1

1.5

-1

-0.5

0

0.5

1

•

Ground State

Ball at any (σ, π)

for which σ2 + π2 = 1

All Positions have Same (Minimum) EnergyBut not invariant under rotationsCraig Roberts: Hadron Physics and Continuum Strong QCD








If one assumes the s-quark is also light; namely, assumes thatSU(Nf = 3) chiral symmetry is a good approximation, then thekaons are four more Goldstone Bosons




Pion and . . .Pseudoscalar Mesons?




Pion and . . .Pseudoscalar Mesons?

Can a bound-state of massive constituents truly bemassless . . . without fine-tuning?




Dichotomy of Pion– Goldstone Mode and Bound state





How does one make an almost massless particle. . . . . . . . . . . from two massive constituent-quarks?






Not Allowed to do it by fine-tuning a potential

Must exhibit m2π ∝ mq

Current Algebra . . . 1968









The correct understanding of pion observables;e.g. mass, decay constant and form factors,requires an approach to contain a

well-defined and valid chiral limit;

and an accurate realisation ofdynamical chiral symmetry breaking.









The correct understanding of pion observables;e.g. mass, decay constant and form factors,requires an approach to contain a

well-defined and valid chiral limit;

and an accurate realisation ofdynamical chiral symmetry breaking.

Highly NontrivialCraig Roberts: Hadron Physics and Continuum Strong QCD



Resolving the Dichotomy

Minimal requirements

detailed understanding of connection between

Current-quark and Constituent-quark masses;

and systematic, symmetry preserving means of realising

this connection in bound-states.




Resolving the Dichotomy

Minimal requirements

detailed understanding of connection between

Current-quark and Constituent-quark masses;

and systematic, symmetry preserving means of realising

this connection in bound-states.

Satisfying these requirements enables

Proof of numerous exact results for pseudoscalar

mesons

Formulation of reliable models

To illustrate those results

Make predictions of observables with quantifiable

errors




Goldberger-Treiman for pionMaris, Roberts, Tandynucl-th/9707003





• Pseudoscalar Bethe-Salpeter amplitude

Γπj (k;P ) = τπj

γ5

[

iEπ(k;P ) + γ · PF π(k;P )

+ γ · k k · P Gπ(k;P ) + σµν kµPν Hπ(k;P )]







γ5

[



• Dressed-quark Propagator: S(p) =1

iγ · pA(p2) + B(p2)







γ5

[




iγ · pA(p2) + B(p2)• Axial-vector Ward-Takahashi identity

⇒ fπEπ(k;P = 0) = B(p2)







γ5

[





⇒ fπEπ(k;P = 0) = B(p2)

FR(k; 0) + 2 fπFπ(k; 0) = A(k2)

GR(k; 0) + 2 fπGπ(k; 0) = 2A′(k2)

HR(k; 0) + 2 fπHπ(k; 0) = 0





Pseudovectorcomponentsnecessarilynonzero



γ5

[





⇒ fπEπ(k;P = 0) = B(p2)

FR(k; 0) + 2 fπFπ(k; 0) = A(k2)

GR(k; 0) + 2 fπGπ(k; 0) = 2A′(k2)

HR(k; 0) + 2 fπHπ(k; 0) = 0

Exact in

Chiral QCD




Radial Excitations& Chiral Symmetry



http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+key+3584445



Radial Excitations& Chiral Symmetry(Maris, Roberts, Tandy

nu-th/9707003 )

fH m2H = − ρH

ζ MH







nu-th/9707003 )

fH m2H = − ρH

ζ MH

• Mass2 of pseudoscalar hadron







nu-th/9707003 )

fH m2H = − ρH

ζ MH

MH := trflavour

[

M (µ)

{

TH ,(

TH)t

}]

= mq1+mq2

• Sum of constituents’ current-quark masses

• e.g., TK+

= 12

(

λ4 + iλ5)







nu-th/9707003 )

fH m2H = − ρH

ζ MH

fH pµ = Z2

∫ Λ

q

12tr

{

(

TH)t

γ5γµ S(q+)ΓH(q;P )S(q−)}

• Pseudovector projection of BS wave function at x = 0

• Pseudoscalar meson’s leptonic decay constant

i

i

i

iAµπ kµ

πf

k

Γ

S

(τ/2)γµ γ

S

555

=

〈0|q̄γ5γµq|π〉







nu-th/9707003 )

fH m2H = − ρH

ζ MH

iρHζ = Z4

∫ Λ

q

12tr

{

(

TH)t

γ5 S(q+)ΓH(q;P )S(q−)}

• Pseudoscalar projection of BS wave function at x = 0

i

i

i

iP π

πρ

k

Γ

S

(τ/2) γ

S

555

=

〈0|q̄γ5q|π〉







nu-th/9707003 )

fH m2H = − ρH

ζ MH

Light-quarks; i.e., mq ∼ 0

fH → f0H & ρH

ζ →−〈q̄q〉0ζ

f0H

, Independent of mq

Hence m2H =

−〈q̄q〉0ζ(f0

H)2mq . . . GMOR relation, a corollary







nu-th/9707003 )

fH m2H = − ρH

ζ MH

Light-quarks; i.e., mq ∼ 0

fH → f0H & ρH

ζ →−〈q̄q〉0ζ

f0H

, Independent of mq

Hence m2H =

−〈q̄q〉0ζ(f0

H)2mq . . . GMOR relation, a corollary

Heavy-quark + light-quark

⇒ fH ∝ 1√

mH

and ρHζ ∝ √

mH

Hence, mH ∝ mq

. . . QCD Proof of Potential Model resultCraig Roberts: Hadron Physics and Continuum Strong QCD





Radial Excitations




Radial Excitations

Spectrum contains 3 pseudoscalars [IG(JP )L = 1−(0−)S]

masses below 2 GeV: π(140) ; π(1300); and π(1800)




Radial Excitations



The Pion

Consituent-Q Model: 1st three members ofn 1S0 trajectory; i.e., ground state plus radial excitations?




Radial Excitations



The Pion


But π(1800) is narrow (Γ = 207 ± 13) & decay pattern mightindicate some “flux tube angular momentum” content:SQ̄Q = 1 ⊕ LF = 1 ⇒ J = 0

& LF = 1 ⇒ 3S1 ⊕ 3S1 (Q̄Q) decays suppressed?




Radial Excitations



The Pion


But π(1800) is narrow (Γ = 207 ± 13) & decay pattern mightindicate some “flux tube angular momentum” content:

Radial excitations & Hybrids & Exotics ⇒ Long-range radial wavefunctions ⇒ sensitive to confinement




Radial Excitations



The Pion


But π(1800) is narrow (Γ = 207 ± 13) & decay pattern mightindicate some “flux tube angular momentum” content:

Radial excitations & Hybrids & Exotics ⇒ Long-range radial wavefunctions ⇒ sensitive to confinement

NSAC Long-Range Plan, 2002: . . . an understanding ofconfinement “remains one of the

greatest intellectual challenges in physics”Craig Roberts: Hadron Physics and Continuum Strong QCD



Radial Excitations& Chiral SymmetryHöll, Krassnigg, Roberts

nu-th/0406030

fH m2H = − ρH

ζ MH

Valid for ALL Pseudoscalar mesons






nu-th/0406030

fH m2H = − ρH

ζ MH


ρH ⇒ finite, nonzero value in chiral limit, MH → 0






nu-th/0406030

fH m2H = − ρH

ζ MH



“radial” excitation of π-meson, not the ground state, so

m2πn 6=0

> m2πn=0

= 0, in chiral limit






nu-th/0406030

fH m2H = − ρH

ζ MH




m2πn 6=0

> m2πn=0


⇒ fH = 0

ALL pseudoscalar mesons except π(140) in chiral limit






nu-th/0406030

fH m2H = − ρH

ζ MH




m2πn 6=0

> m2πn=0


⇒ fH = 0

ALL pseudoscalar mesons except π(140) in chiral limit


– Goldstone’s Theorem –

impacts upon every pseudoscalar mesonCraig Roberts: Hadron Physics and Continuum Strong QCD




Charge NeutralPseudoscalar Mesons





non-Abelian Anomaly and η-η′ mixing






Mesons containing s̄-s are special: η & η′







Problem: η′ is a pseudoscalar meson but it’s much more

massive than the other eight constituted from light-quarks.







Problem: η′ is a pseudoscalar meson but it’s much more

massive than the other eight constituted from light-quarks.

Origin: While the classical action associated with QCD is

invariant under UA(1) (Abelian axial transformations

generated by λ0γ5), the quantum field theory is not!







Flavour mixing takes place in singlet channel: λ0 ⇔ λ8







This is a perturbative diagram.

It has almost nothing to do with η ⇔ η′ mixing.







Driver is the non-Abelian anomaly








KA ∼

f1 f2

IS

IS

e.g. IS =

f1 f2








KA ∼

f1 f2

IS

IS

e.g. IS =

f1 f2

Contribution to the

Bethe-Salpeter kernel

associated with the

non-Abelian anomaly.

All terms have the

“hairpin” structure.

No finite sum of such intermediate states is sufficient to

veraciously represent the anomaly.





PµΓa5µ(k;P ) = S−1(k+)iγ5Fa + iγ5FaS−1(k−)

−2iMabΓb5(k;P ) −Aa(k;P )







{Fa| a = 0, . . . , N2f − 1} are the generators of U(Nf )

S = diag[Su, Sd, Ss, Sc, Sb, . . .]

Mab = trF

[

{Fa,M}F b]

,

M = diag[mu,md,ms,mc,mb, . . .] = matrix of current-quark

bare masses







{Fa| a = 0, . . . , N2f − 1} are the generators of U(Nf )

S = diag[Su, Sd, Ss, Sc, Sb, . . .]

Mab = trF

[

{Fa,M}F b]

,

M = diag[mu,md,ms,mc,mb, . . .] = matrix of current-quark

bare masses

The final term in the second line expresses the non-Abelian

axial anomaly.







Aa(k;P ) = S−1(k+) δa0 AU (k;P )S−1(k−)

AU (k;P ) =

∫

d4xd4y ei(k+·x−k−·y)Nf

⟨

F0q(x)Q(0) q̄(y)⟩







Aa(k;P ) = S−1(k+) δa0 AU (k;P )S−1(k−)

AU (k;P ) =

∫


⟨

F0q(x)Q(0) q̄(y)⟩

Q(x) = iαs

4πtrC [ǫµνρσFµνFρσ(x)] = ∂µKµ(x)

. . . The topological charge density operator.







Aa(k;P ) = S−1(k+) δa0 AU (k;P )S−1(k−)

AU (k;P ) =

∫


⟨

F0q(x)Q(0) q̄(y)⟩

Q(x) = iαs



(Trace is over colour indices & Fµν = 12λaF a

µν .)







Aa(k;P ) = S−1(k+) δa0 AU (k;P )S−1(k−)

AU (k;P ) =

∫


⟨

F0q(x)Q(0) q̄(y)⟩

Q(x) = iαs



Important that only Aa=0 is nonzero.







Aa(k;P ) = S−1(k+) δa0 AU (k;P )S−1(k−)

AU (k;P ) =

∫


⟨

F0q(x)Q(0) q̄(y)⟩

Q(x) = iαs



NB. While Q(x) is gauge invariant, the associated

Chern-Simons current, Kµ, is not ⇒ in QCD no physical

boson can couple to Kµ and hence no physical states can

contribute to resolution of UA(1) problem.Craig Roberts: Hadron Physics and Continuum Strong QCD



Charge NeutralPseudoscalar MesonsBhagwat, Chang, Liu, Roberts, Tandy

nucl-th/arXiv:0708.1118





Bhagwat, Chang, Liu, Roberts, Tandynucl-th/arXiv:0708.1118

Only A0 6≡ 0 is interesting






Only A0 6≡ 0 is interesting . . . otherwise all pseudoscalar

mesons are Goldstone Modes!






Anomaly term has structureA0(k;P ) = F0γ5 [iEA(k;P ) + γ · PFA(k;P )

+γ · kk · PGA(k;P ) + σµνkµPνHA(k;P )]






AVWTI gives generalised Goldberger-Treiman relations

2f0η′EBS(k; 0) = 2B0(k

2) − EA(k; 0),

F 0R(k; 0) + 2f0

η′FBS(k; 0) = A0(k2) − FA(k; 0),

G0R(k; 0) + 2f0

η′GBS(k; 0) = 2A′0(k

2) − GA(k; 0),

H0R(k; 0) + 2f0

η′HBS(k; 0) = −HA(k; 0),

A0, B0 characterise gap equation’s chiral limit solution.






AVWTI gives generalised Goldberger-Treiman relations

2f0η′EBS(k; 0) = 2B0(k

2) − EA(k; 0),

F 0R(k; 0) + 2f0

η′FBS(k; 0) = A0(k2) − FA(k; 0),

G0R(k; 0) + 2f0

η′GBS(k; 0) = 2A′0(k

2) − GA(k; 0),

H0R(k; 0) + 2f0

η′HBS(k; 0) = −HA(k; 0),

A0, B0 characterise gap equation’s chiral limit solution.

Follows that EA(k; 0) = 2B0(k2) is necessary and

sufficient condition for absence of massless η′ bound-state.






EA(k; 0) = 2B0(k2)

Discussing the chiral limit

B0(k2) 6= 0 if, and only if, chiral symmetry is dynamically

broken.

Hence, absence of massless η′ bound-state is only

assured through existence of intimate connection

between DCSB and an expectation value of the

topological charge density.






EA(k; 0) = 2B0(k2)

Discussing the chiral limit

B0(k2) 6= 0 if, and only if, chiral symmetry is dynamically

broken.

Hence, absence of massless η′ bound-state is only

assured through existence of intimate connection

between DCSB and an expectation value of the

topological charge density.

Further highlighted . . . proved〈q̄q〉0ζ = − lim

Λ→∞Z4(ζ

2,Λ2) trCD

∫ Λ

qS0(q, ζ)

= Nf

∫

d4x 〈q̄(x)iγ5q(x)Q(0)〉0 .






AVWTI ⇒ QCD mass formulae for neutral pseudoscalar mesons







Implications of mass formulae illustrated using elementarydynamical model, which includes Ansatz for that part of theBethe-Salpeter kernel related to the non-Abelian anomaly








Employed in an analysis of pseudoscalar- and vector-mesonbound-states








Despite its simplicity, model is elucidative and phenomenologicallyefficacious; e.g., it predicts

η–η′ mixing angles of ∼ −15◦ (Expt.: −13.3◦ ± 1.0◦)

π0–η angles of ∼ 1.2◦ (Expt. p d → 3He π0: 0.6◦ ± 0.3◦)

Strong neutron-proton mass difference . . .

∼< 75 % current-quark mass-difference




Ab-Initio Calculations





Pieter Maris Peter Tandy





Maris & Tandy, Series of Five Articles: 1999 – Present

Perfected a Renormalisation-Group ImprovedRainbow-Ladder Model of Quark-Quark Interaction







• Rainbow-Ladder = First Orderin Truncation Described Above

• Anticipate Accurate for 0− & 1− Mesons







• One Parameter = Interaction Energy:E ≈ 700 MeV

• Dressed-Glue Mass scale:Characterises DCSB and light-quark Confinement

• Both Phenomena Disappear for E . 200 MeV







• One Parameter = Interaction Energy:E ≈ 700 MeV

• Dressed-Glue Mass scale:Characterises DCSB and light-quark Confinement

• Both Phenomena Disappear for E . 200 MeV

• Dyson-Schwinger equations:A Tool for Hadron Physics

P. Maris and C.D. Roberts, nu-th/0301049Craig Roberts: Hadron Physics and Continuum Strong QCD



Interaction

0 1 2 3 4 5

q2 (GeV

2)

0

1

2

3

4

αeff (q

2 )Perturbative evolution

Nonperturbative IR Enhancement

ε=720 MeV

Kernel ofBethe-SalpeterEquation

K(p, k;P ) ≈αeff((p − k)2)

(p − k)2




Interaction

0 1 2 3 4 5

q2 (GeV

2)

0

1

2

3

4

αeff (q



ε=720 MeV


K(p, k;P ) ≈αeff((p − k)2)

(p − k)2

Prescribes GapEquation’s Kernel




Interaction

0 1 2 3 4 5

q2 (GeV

2)

0

1

2

3

4

αeff (q



ε=720 MeV


K(p, k;P ) ≈αeff((p − k)2)

(p − k)2


Connects Ansatz for long-range part of QCD’s interactionwith Observables.




Interaction

0 1 2 3 4 5

q2 (GeV

2)

0

1

2

3

4

αeff (q



ε=720 MeV


K(p, k;P ) ≈αeff((p − k)2)

(p − k)2


IR-Enhancement at long-range agrees semi-quantitativelywith Bhagwat, et al .




Pion Form Factor

Procedure Now Straightforward




Pion Form Factor

Solve Gap Equation⇒ Dressed-Quark Propagator, S(p)

Σ=

D

γΓS




Pion Form Factor

Use that to Complete Bethe Salpeter Kernel, K

Solve Homogeneous Bethe-Salpeter Equation for PionBethe-Salpeter Amplitude, Γπ




Pion Form Factor

Use that to Complete Bethe Salpeter Kernel, K

Solve Homogeneous Bethe-Salpeter Equation for PionBethe-Salpeter Amplitude, Γπ

Solve Inhomogeneous Bethe-Salpeter Equation forDressed-Quark-Photon Vertex, Γµ




Pion Form Factor

Now have all elements for Impulse Approximation toElectromagnetic Pion Form factor

Γπ(k;P )

Γµ(k;P )

S(p)




Pion Form Factor

Now have all elements for Impulse Approximation toElectromagnetic Pion Form factor

Γπ(k;P )

Γµ(k;P )

S(p)

Evaluate this final,three-dimensional integral




Calculated Pion Form Factor

Calculation published in 1999; No Parameters Varied

0 1 2 3 4

Q2 [GeV

2]

0

0.1

0.2

0.3

0.4

0.5

Q2 F

π(Q2 )

[G

eV2 ]

AmendoliaBrauel, re-analyzedVolmerDSE calculationVMD ρ monopole, mρ=770 MeV






0 1 2 3 4

Q2 [GeV

2]

0

0.1

0.2

0.3

0.4

0.5

Q2 F

π(Q2 )

[G

eV2 ]


Data published in 2001






0 1 2 3 4

Q2 [GeV

2]

0

0.1

0.2

0.3

0.4

0.5

Q2 F

π(Q2 )

[G

eV2 ]


Data published in 2001

Many subsequentsuccessful applications. . Again, parameters Fixed

Notably π π Scattering

Maris, et al., Phys. Rev. D 65, 076008

Bicudo, Phys. Rev. C 67, 035201Craig Roberts: Hadron Physics and Continuum Strong QCD



Pion . . . J = 0

but . . .

Pseudoscalar meson Bethe-Salpeter amplitude

χπ(k;P ) = γ5 [iEπn(k;P ) + γ · PFπn(k;P )

γ · k k · P Gπn(k;P ) + σµν kµPν Hπn(k;P )]




Pion . . . J = 0

but . . .




Orbital angular momentum is not a Poincaré invariant.

However, if absent in a particular frame, it will appear in

another frame related via a Poincaré transformation.




Pion . . . J = 0

but . . .




Nonzero quark orbital angular momentum is thus a

necessary outcome of a Poincaré covariant description.




Pion . . . J = 0

but . . .




In QCD, a Poincaré invariant theory with interactions, E 6= 0

forces nonzero results for F , G and H




Pion . . . J = 0

but . . .




In QCD, a Poincaré invariant theory with interactions, E 6= 0

forces nonzero results for F , G and H

J = 0 . . . but while E and F are purely L = 0 in the rest

frame, the G and H terms are associated with L = 1. Thus a

pseudoscalar meson Bethe-Salpeter wave function always

contains both S- and P -wave components.




Pion . . . J = 0

but . . .J = 0 . . . but while E and F are purely L = 0 in the rest



contains both S- and P -wave components.Introduce mixing

angle θπ such that

χπ ∼ cos θπ|L = 0〉+sin θπ|L = 1〉




Pion . . . J = 0







0 2 4 6 8 100

-+n=0 meson mass (GeV)

0

10

20

30

θ π n=0

(d

egre

es)

1.2 1.6 2 2.40

-+n meson mass (GeV)

8

12

16

θ π n (

deg

rees

)




Pion . . . J = 0







0 2 4 6 8 100

-+n=0 meson mass (GeV)

0

10

20

30

θ π n=0

(d

egre

es)

1.2 1.6 2 2.40

-+n meson mass (GeV)

8

12

16

θ π n (

deg

rees

)

L is significant in

the neighbourhood

of the chiral limit,

and decreases with

increasing

current-quark mass.Craig Roberts: Hadron Physics and Continuum Strong QCD



Deep-inelastic scattering





Looking for Quarks





Looking for Quarks

Signature Experiment for QCD:

Discovery of Quarks at SLAC





Looking for Quarks

Signature Experiment for QCD:

Discovery of Quarks at SLAC

Cross-section: Interpreted as Measurement ofMomentum-Fraction Prob. Distribution: q(x), g(x)




Pion’s valence quark distn





π is Two-Body System: “Easiest” Bound State in QCD

However, NO π Targets!







Existing Measurement Inferred from Drell-Yan:

πN → µ+µ−X







Existing Measurement Inferred from Drell-Yan:

πN → µ+µ−X

Proposal (Holt & Reimer, ANL, nu-ex/0010004)

e−5GeV – p25 GeV Collider → Accurate “Measurement”

p n

πγ




Handbag diagrams

��

��

��

��

��

��

��

��

P P

k

q q

k-q-P

k-qk-q

k+q

��

��

��

��

��

��

��

��

P P

k

qq

k+q+P

k+q




Handbag diagrams

��

��

��

��

��

��

��

��

P P

k

q q

k-q-P

k-qk-q

k+q

��

��

��

��

��

��

��

��

P P

k

qq

k+q+P

k+q

Wµν(q;P ) =1

2πIm

[

T+µν(q;P ) + T−

µν(q;P )]

T+µν(q, P ) = tr

∫

d4k

(2π)4τ−Γ̄π(k− 1

2

;−P )S(k−0) ieQΓν(k−0, k)

×S(k) ieQΓµ(k, k−0)S(k−0)τ+Γπ(k− 1

2

;P )S(k−−)




Handbag diagrams

Bjorken Limit: q2 → ∞ , P · q → −∞

but x := − q2

2P · q fixed.

Numerous algebraic simplifications

��

��

��

��

��

��

��

��

P P

k

q q

k-q-P

k-qk-q

k+q

��

��

��

��

��

��

��

��

P P

k

qq

k+q+P

k+q

Wµν(q;P ) =1

2πIm

[

T+µν(q;P ) + T−

µν(q;P )]

T+µν(q, P ) = tr

∫

d4k

(2π)4τ−Γ̄π(k− 1

2

;−P )S(k−0) ieQΓν(k−0, k)

×S(k) ieQΓµ(k, k−0)S(k−0)τ+Γπ(k− 1

2

;P )S(k−−)




Calc. uV (x) cf. Drell-Yan dataHecht, Roberts, Schmidtnucl-th/0008049





0.0 0.2 0.4 0.6 0.8 1.0x

0.0

0.2

0.4

0.6

xuv(

x)

x uv(x;q0=0.54GeV)x uv(x;q=2GeV)E615 πN Drell−Yan 4GeVSMRS 92 Fitx uv(x;q=4GeV)Fit x

α(1−x)

β

Hec

ht, R

ober

ts, S

chm

idt,

nucl

−th

/000

8049





0.0 0.2 0.4 0.6 0.8 1.0x

0.0

0.2

0.4

0.6

xuv(

x)


α(1−x)

β

Hec

ht, R

ober

ts, S

chm

idt,

nucl

−th

/000

8049

Resolving Scale: q0 = 0.54GeV = 1/(0.37 fm)




Calc. uV (x) cf. Drell-Yan data

0.0 0.2 0.4 0.6 0.8 1.0x

0.0

0.2

0.4

0.6

xuv(

x)


α(1−x)

β

Hec

ht, R

ober

ts, S

chm

idt,

nucl

−th

/000

8049

q =

2GeV Calc. Fit, 92 Latt., 97

〈x〉q 0.24 0.24 ± 0.01 0.27 ± 0.01

〈x2〉q 0.10 0.10 ± 0.01 0.11 ± 0.3

〈x3〉q 0.050 0.058 ± 0.004 0.048 ± 0.020Craig Roberts: Hadron Physics and Continuum Strong QCD



Extant theory vs. experiment

K. Wijersooriya, P. Reimer and R. Holt,

nu-ex/0509012 ... Phys. Rev. C (Rapid)

0.0 0.2 0.4 0.6 0.8 1.0x

0.0

0.1

0.2

0.3

0.4

x u

v(x)

E615 πN Drell−Yan 4GeVNLO Analysis of E615 ... β=1.87DSE ... β= 2.61NJL ... β= 1.27




Answer for the pion




Answer for the pion

Two → Infinitely many . . .




Answer for the pion

Two → Infinitely many . . .Handle thatproperly inquantumfield theory




Answer for the pion

Two → Infinitely many . . .Handle thatproperly inquantumfield theory. . .momentum-dependentdressing




Answer for the pion

Two → Infinitely many . . .Handle thatproperly inquantumfield theory. . .momentum-dependentdressing. . .perceiveddistribution ofmass dependson the resolving scale




Exegesis




Exegesis

Hadron Physics is ∼ $300-million/year effort in USA alone




Exegesis


Subject is QCD . . . in the nonperturbative domain




Exegesis



Keystones are the Emergent Phenomena

Confinement– quarks and gluons never alone reach a detector

Dynamical Chiral Symmetry Breaking– counter-intuitive pattern of bound state masses andinteractions




Exegesis



Keystones are the Emergent Phenomena

Confinement– quarks and gluons never alone reach a detector

Dynamical Chiral Symmetry Breaking– counter-intuitive pattern of bound state masses andinteractions

Review presentation in Mazatlan:

Elastic electromagnetic pion form factor

Nature of Baryons



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