Dyson-Schwinger Equations – Theory and Phenomenology · Infinitely Many Coupled Equations...

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Dyson-Schwinger Equations– Theory and Phenomenology

Craig D. Roberts

[email protected]

Physics Division

Argonne National Laboratory

IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 1/52


Chicago from Space



Argonne National Laboratory



Dichotomy of the Pion




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





Not Allowed to do it by fine-tuning

Must exhibit m2

π ∝ mq

Current Algebra . . . 1968






Must exhibit m2

π ∝ mq


The correct understanding of pion observables;e.g. mass, decay constant and form factors,requires an approach to contain a well-defined andvalid chiral limit, and an accurate realisation ofdynamical chiral symmetry breaking.






Must exhibit m2

π ∝ mq



Requires detailed understanding of Connectionbetween Current-quark and Constituent-quarkmasses






Must exhibit m2

π ∝ mq



Requires detailed understanding of Connectionbetween Current-quark and Constituent-quarkmasses Using DSEs,

we’ve provided this.IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 4/52


Dyson-Schwinger Equations




A Modern Method for 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

· Quark & Gluon Confinement












⇒ Understanding InfraRed (long-range)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . behaviour of αs(Q2)













. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . behaviour of αs(Q2)

Method yields Schwinger Functions ≡ Propagators













. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . behaviour of αs(Q2)

Method yields Schwinger Functions ≡ Propagators

Cross-Sections built from Schwinger FunctionsIVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 5/52


Contemporary Reviews

Dyson-Schwinger Equations:Density, Temperature and Continuum Strong QCDC.D. Roberts and S.M. Schmidt, nu-th/0005064,Prog. Part. Nucl. Phys. 45 (2000) S1

The IR behavior of QCD Green’s functions:Confinement, DCSB, and hadrons . . .R. Alkofer and L. von Smekal, he-ph/0007355,Phys. Rept. 353 (2001) 281

Dyson-Schwinger equations:A Tool for Hadron PhysicsP. Maris and C.D. Roberts, nu-th/0301049,Int. J. Mod. Phys. E 12 (2003) pp. 297-365



Persistent Challenge




Infinitely Many Coupled Equations

Σ=

D

γΓS





Solutions are Schwinger Functions(Euclidean Green Functions)

Same VEVs measured in Lattice-QCD simulations







Coupling between equations necessitates truncation








Weak coupling expansion ⇒ Perturbation Theory








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







We introduced a systematic nonperturbative,symmetry-preserving truncation schemeGoldstone Theorem and Diquark Confinement Beyond RainbowLadder Approximation, A. Bender, C. D. Roberts and L. VonSmekal, Phys. Lett. B 380 (1996) 7


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







Has Enabled Proof of EXACT Results in QCD










And Formulation of Practical Phenomenological Tool to

Illustrate Exact Results










And Formulation of Practical Phenomenological Tool to

Make Predictions with Readily Quantifiable Errors




Perturbative Dressed-quarkPropagator




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!



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 (GeV

2)

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

Gap EquationGap Equation’s Kernel Enhanced on IR domain

⇒ IR Enhancement of M(p2)

10−2

10−1

100

101

102

p2 (GeV

2)

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

Euclidean Constituent–Quark

Mass: MEf : p2 = M(p2)2

flavour u/d s c b

ME

mζ∼ 102

∼ 10 ∼ 1.5 ∼ 1.1




Longstanding Prediction of Dyson-SchwingerEquation Studies


http://www.slac.stanford.edu/spires/find/hep/www?key=2909685





E.g., Dyson-Schwinger equations and theirapplication to hadronic physics,C. D. Roberts and A. G. Williams,Prog. Part. Nucl. Phys. 33 (1994) 477








Long used as basis for efficacious hadron physicsphenomenology








Long used as basis for efficacious hadron physicsphenomenology

Electromagnetic pion form-factor and neutralpion decay width,C. D. Roberts,Nucl. Phys. A 605 (1996) 475





Quenched-QCDDressed-Quark Propagator

0.0 1.0 2.0 3.0 4.0p (GeV)

0.0

0.1

0.2

0.3

0.4

0.5

0.0 1.0 2.0 3.0 4.0p (GeV)

0.0

0.2

0.4

0.6

0.8

1.0

M(p) Z(p)

M(p)





Quenched-QCDDressed-Quark Propagator2002

0.0 1.0 2.0 3.0 4.0p (GeV)

0.0

0.1

0.2

0.3

0.4

0.5

0.0 1.0 2.0 3.0 4.0p (GeV)

0.0

0.2

0.4

0.6

0.8

1.0

M(p) Z(p)

M(p)

“data:” Quenched Lattice Meas.– Bowman, Heller, Leinweber, Williams: he-lat/0209129






0.0 1.0 2.0 3.0 4.0p (GeV)

0.0

0.1

0.2

0.3

0.4

0.5

0.0 1.0 2.0 3.0 4.0p (GeV)

0.0

0.2

0.4

0.6

0.8

1.0

M(p) Z(p)

M(p)

“data:” Quenched Lattice Meas.– Bowman, Heller, Leinweber, Williams: he-lat/0209129current-quark masses: 30 MeV, 50 MeV, 100 MeV






0.0 1.0 2.0 3.0 4.0p (GeV)

0.0

0.1

0.2

0.3

0.4

0.5

0.0 1.0 2.0 3.0 4.0p (GeV)

0.0

0.2

0.4

0.6

0.8

1.0

M(p) Z(p)

M(p)

“data:” Quenched Lattice Meas.– Bowman, Heller, Leinweber, Williams: he-lat/0209129current-quark masses: 30 MeV, 50 MeV, 100 MeVCurves: Quenched DSE Cal.

– Bhagwat, Pichowsky, Roberts, Tandy nu-th/0304003






0.0 1.0 2.0 3.0 4.0p (GeV)

0.0

0.1

0.2

0.3

0.4

0.5

0.0 1.0 2.0 3.0 4.0p (GeV)

0.0

0.2

0.4

0.6

0.8

1.0

M(p) Z(p)

M(p)

“data:” Quenched Lattice Meas.– Bowman, Heller, Leinweber, Williams: he-lat/0209129current-quark masses: 30 MeV, 50 MeV, 100 MeVCurves: Quenched DSE Cal.

– Bhagwat, Pichowsky, Roberts, Tandy nu-th/0304003Linear extrapolation of lattice data to chiral limit is inaccurate





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



Hadrons

• Established understandingof two- and three-point functions



Hadrons

• Established understandingof two- and three-point functions

• What about bound states?



Hadrons

• Without bound states,Comparison with experiment isimpossible



Hadrons


• They appear as pole contributionsto n ≥ 3-point colour-singletSchwinger functions



Hadrons


• Bethe-Salpeter Equation

QFT Generalisation of Lippman-Schwinger Equation.



Hadrons




• What is the kernel, K?



Hadrons




• What is the kernel, K?

or What is the long-range potential in 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 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



Systematic Nonpert. Trunc.– Key Contributions




Existence . . .

Munczek, Phys. Rev. D 52(1995) 4736;

Bender, Roberts, von Smekal,Phys. Lett. B 380 (1996) 7.




Existence . . .



Convergence . . .

Bender, Detmold, Roberts, Thomas,Phys. Rev. C 65 (2002) 065203.




Existence . . .



Convergence . . .

Bender, Detmold, Roberts, Thomas,Phys. Rev. C 65 (2002) 065203.

Interactions . . .

P. Bicudo, Phys. Rev. C 67 (2003) 035201;

Höll, Krassnigg, Maris, Roberts, Wright,Phys. Rev. C 71 (2005) 065204



Andreas Krassnigg

FWF “ErwinSchrödinger Fellow,”ANL 2003-2005



Andreas Krassnigg

Almost BloodRelative of Arnold. . . Future

President?. . . Executioner?



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

=






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






nu-th/9707003 )

fH m2H = − ρH

ζ MH

Light-quarks; i.e., mq ∼ 0

fH → f0H & ρH

ζ →−〈qq〉0ζ

f0H

, Independent of mq

Hence m2H =

−〈qq〉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 →−〈qq〉0ζ

f0H

, Independent of mq

Hence m2H =

−〈qq〉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 resultIVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 19/52




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,

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

Dynamical Chiral Symmetry Breaking

– Goldstone’s Theorem –

impacts upon every pseudoscalar mesonIVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 19/52




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:SQQ = 1 ⊕ LF = 1 ⇒ J = 0

& LF = 1 ⇒ 3S1 ⊕ 3S1 (QQ) 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”IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 20/52


Radial Excitations& Chiral Symmetry




nu-th/0406030

Fundamental properties of QCD




nu-th/0406030


If chiral symmetry is dynamically broken,

then in the chiral limit every pseudoscalar meson is blind

to the weak interaction except π(140).

0 0.2 0.4 0.6mq [GeV]

0

0.1

0.2

f π [

GeV

]

fπ0

fπ1




nu-th/0406030


If chiral symmetry is dynamically broken,

then in the chiral limit every pseudoscalar meson is blind

to the weak interaction except π(140).

0 0.2 0.4 0.6mq [GeV]

0

0.1

0.2

f π [

GeV

]

fπ0

fπ1

If chiral symmetry is not broken,

then NO pseudoscalar meson experiences the weak

interaction.



Two-photon Couplings ofPseudoscalar MesonsHöll, Krassnigg, Maris, et al.,

“Electromagnetic properties of ground andexcited state pseudoscalar mesons,”nu-th/0503043

π0n(P ) y

y

y�

�

��

@@R

@@

6

��

��

γ(k1)

γ(k2)

T π0n

µν (k1, k2) =α

πiεµνρσk1ρk2σ Gπ0

n(k1, k2)

Define: Tπ0n

(P 2, Q2) = Gπ0n(k1, k2)

∣

∣

∣

k21=Q2=k2

2

This is a transition form factor.

Physical Processes described by couplings:gπ0

0γγ := Tπ00(−m2

π00, 0)

Width: Γπ0n

γγ = α2em

m3πn

16π3g2

πnγγ



Two-photon Couplings:Goldstone ModeHöll, Krassnigg, Maris, et al.,


π00(P ) y

y

y�

�

��

@@R

@@

6

��

��

γ(k1)

γ(k2)

T π00

µν (k1, k2) =α


0(k1, k2)

Chiral limit, model-independent and algebraic result

gπ00γγ := Tπ0

0(−m2

π00

= 0, 0)=1

2

1

fπ0

So long as truncation preserves chiral symmetry and thepattern of its dynamical breakdown

The most widely known consequence of the Abelian anomalyIVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 23/52


Two-photon Couplings:Transition Form FactorHöll, Krassnigg, Maris, et al.,


π0n(P ) y

y

y�

�

��

@@R

@@

6

��

��

γ(k1), k21 = Q2

γ(k2), k21 = Q2

T π0n

µν (k1, k2) =α


n(k1, k2)

So long as truncation preserves chiral symmetry and thepattern of its dynamical breakdown, and the one-looprenormalisation group properties of QCD: model-independentresult – ∀n:

Tπ0n

(P 2, Q2) = Gπ0n(k1, k2)

∣

∣

∣

k21=Q2=k2

2

Q2≫Λ2QCD

=4π2

3

fπn

Q2



Transition Form Factor:Chiral limitHöll, Krassnigg, Maris, et al.,


Chiral limit with DCSB: fπ06= 0






BUT, fπn≡ 0, ∀n!







Model-independent result, in chiral limit: ∀n ≥ 1

limm→0

Tπ0n

(−m2πn

, Q2)

Q2≫Λ2QCD

=4π2

3F (2)

n (−m2πn

)lnγ Q2/ω2

πn

Q4

∣

∣

∣

∣

∣

m=0where:

γ is an anomalous dimension

ωπnis a width mass-scale

both determined, in part, by properties of the meson’sBethe-Salpeter wave function.







Model-independent result, in chiral limit: ∀n ≥ 1

limm→0

Tπ0n

(−m2πn

, Q2)

Q2≫Λ2QCD

=4π2

3F (2)

n (−m2πn

)lnγ Q2/ω2

πn

Q4

∣

∣

∣

∣

∣

m=0where:

γ is an anomalous dimension

ωπnis a width mass-scale

both determined, in part, by properties of the meson’sBethe-Salpeter wave function.

Importantly, F (2)n (−m2

πn

) 6∝ fπn. Instead, it is determined by

DCSB mass-scales for πn that do not vanish in the chiral limit.



Are we there yet?



Nucleon Properties



Nucleon Properties

Maris & Tandy . . . series of five papers . . . excellent

description of light pseudoscalar and vector mesons . . .

basket of 31 masses/couplings/radii with r.m.s. error of 15%

. . . moreover, prediction of Fπ(Q2) measured in Hall A.

Pieter Maris Peter Tandy



Nucleon Properties





One parameter model . . . parameter specifies long-range

interaction between light-quarks . . . model-independent

results in ultraviolet



Nucleon Properties





One parameter model . . . parameter specifies long-range

interaction between light-quarks . . . model-independent

results in ultraviolet

Next Steps . . . Applications to excited states and

axial-vector mesons, e.g., will improve understanding of

confinement interaction between light-quarks



Nucleon Properties

Another Direction . . . Also want/need information about

three-quark systems



Nucleon Properties


three-quark systems

With this problem . . . current expertise at approximately

same point as studies of mesons in 1995.



Nucleon Properties


three-quark systems

With this problem . . . current expertise at approximately

same point as studies of mesons in 1995.

Namely . . . Model-building and Phenomenology,

constrained by the DSE results outlined already.



Proton Form Factors:Modern Experiment




Rosenbluth and Polarization-Transfer Extractions of

Ratio of Proton’s Electric and Magnetic Form Factors

0 2 4 6Q

2 [GeV2]

0

0.5

1

µ p GEp/ G

Mp

Rosenbluthprecision Rosenbluthpolarization transferpolarization transfer




0 2 4 6Q

2 [GeV2]

0

0.5

1

µ p GEp/ G

Mp


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

Qattan et al., Phys.Rev. Lett. 94 142301(2005)

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 Hall A Collaboration,Phys. Rev. Lett. 88 092301 (2002)




0 2 4 6Q

2 [GeV2]

0

0.5

1

µ p GEp/ G

Mp


If Pol. Trans. Correct,

then Completely

Unexpected Result:

In the Proton

– On Relativistic

Domain

– Distribution of

Quark-Charge

Not Equal

Distribution of

Quark-Current!



Arne Höll



Arne Höll

Closing in onsomething



Nucleon EM Form Factors: A Précis

Holl, Kloker, et al. : nu-th/0412046 & nu-th/0501033







• Interpreting expts. with GeV electromagnetic probes

requires Poincaré covariant treatment of baryons









⇒ Covariant dressed-quark Faddeev Equation










• Excellent mass spectrum (octet and decuplet)

Easily obtained:(

1

NH

∑

H

[M expH − M calc

H ]2

[M expH ]2

)1/2

= 2%











Easily obtained:(

1

NH

∑

H

[M expH − M calc

H ]2

[M expH ]2

)1/2

= 2%

(Oettel, Hellstern, Alkofer, Reinhardt: nucl-th/9805054)











Easily obtained:(

1

NH

∑

H

[M expH − M calc

H ]2

[M expH ]2

)1/2

= 2%

• But is that good?











Easily obtained:(

1

NH

∑

H

[M expH − M calc

H ]2

[M expH ]2

)1/2

= 2%


• Cloudy Bag: δMπ−loop+ = −300 to −400 MeV!











Easily obtained:(

1

NH

∑

H

[M expH − M calc

H ]2

[M expH ]2

)1/2

= 2%


• Cloudy Bag: δMπ−loop+ = −300 to −400 MeV!

• Critical to anticipate pion cloud effects

Roberts, Tandy, Thomas, et al., nu-th/02010084






Harry LeePions and Form Factors








Dynamical coupled-channels model . . . Analyzed extensive JLabdata . . . Completed a study of the ∆(1236)

Meson Exchange Model for πN Scattering and γN → πN Reaction, T. Satoand T.-S. H. Lee, Phys. Rev. C 54, 2660 (1996)

Dynamical Study of the ∆ Excitation in N(e, e′π) Reactions, T. Sato andT.-S. H. Lee, Phys. Rev. C 63, 055201/1-13 (2001)











Pion cloud effects are large in the low Q2 region.

0

1

2

3

0 1 2 3 4

Q2(GeV/c)2

DressedBare

Ratio of the M1 form factor in γN → ∆

transition and proton dipole form factor GD .Solid curve is G∗

M(Q2)/GD(Q2) including

pions; Dotted curve is GM (Q2)/GD(Q2)

without pions.











Pion cloud effects are large in the low Q2 region.

0

1

2

3

0 1 2 3 4

Q2(GeV/c)2

DressedBare

Ratio of the M1 form factor in γN → ∆

transition and proton dipole form factor GD .Solid curve is G∗

M(Q2)/GD(Q2) including

pions; Dotted curve is GM (Q2)/GD(Q2)

without pions.

Quark Core

Responsible for only 2/3 ofresult at small Q2

Dominant for Q2 >2 – 3 GeV2







Faddeev equation



Faddeev equation

=aΨ

P

pq

pd Γb

Γ−a

pd

pq

bΨP

q



Faddeev equation

=aΨ

P

pq

pd Γb

Γ−a

pd

pq

bΨP

q

Linear, Homogeneous Matrix equation

Yields wave function (Poincaré Covariant Faddeev

Amplitude) that describes quark-diquark relative motion

within the nucleon

Scalar and Axial-Vector Diquarks . . . In Nucleon’s Rest

Frame Amplitude has . . . s−, p− & d−wave correlations



Parametrising diquark properties




Dressed-quark . . . fixed by DSE and Meson Studies

. . . Burden, Roberts, Thomson, Phys. Lett. B 371, 163 (1996)




• Bethe-Salpeter-Like Amplitudes

Γ0+

(k;K) =1

N 0+Ha Ciγ5 iτ2 F(k2/ ω2

0+ ) ,

tiΓ1+

µ (k;K) =1

N 1+Ha iγµC t

i F(k2/ ω21+ )





Γ0+

(k;K) =1


0+ ) ,

tiΓ1+

µ (k;K) =1

N 1+Ha iγµC t

i F(k2/ ω21+ )

• Colour matrices:

{H1 = iλ7,H2 = −iλ5,H3 = iλ2} , ǫc1c2c3 = (Hc3)c1c2





Γ0+

(k;K) =1


0+ ) ,

tiΓ1+

µ (k;K) =1

N 1+Ha iγµC t

i F(k2/ ω21+ )

• Two parameters: ω0+ , ω1+

∼ Inverse of diquarks’ configuration-space size




• Pseudoparticle Propagators

∆0+

(K) =1

m20+

F(K2/ω20+) ,

∆1+

µν (K) =

(

δµν +KµKν

m21+

)

1

m21+

F(K2/ω21+)

F(x) =1 − exp(−x)

xAbsence of a Spectral Representation

Realisation of Confinement




• Pseudoparticle Propagators

∆0+

(K) =1

m20+

F(K2/ω20+) ,

∆1+

µν (K) =

(

δµν +KµKν

m21+

)

1

m21+

F(K2/ω21+)

• Two parameters: m0+ , m1+

∼ Inverse of diquarks’ configuration-space correlation length




• Total of four parameters

. . . reduce that via Normalisation Condition

d

dK2

(

1

m2JP

F(K2/ω2JP )

)

−1∣

∣

∣

∣

∣

∣

K2=0

= 1 ⇒ ω2JP =

1

2m2

JP ,

Accentuates free-particle-like propagation characteristics of the

diquarks within hadron.






d

dK2

(

1

m2JP

F(K2/ω2JP )

)

−1∣

∣

∣

∣

∣

∣

K2=0

= 1 ⇒ ω2JP =

1

2m2

JP ,



Two Parameter Faddeev Equation Model of Nucleon






d

dK2

(

1

m2JP

F(K2/ω2JP )

)

−1∣

∣

∣

∣

∣

∣

K2=0

= 1 ⇒ ω2JP =

1

2m2

JP ,




Solve Faddeev Equation






d

dK2

(

1

m2JP

F(K2/ω2JP )

)

−1∣

∣

∣

∣

∣

∣

K2=0

= 1 ⇒ ω2JP =

1

2m2

JP ,




Solve Faddeev Equation

Vary m0+ and m1+ to obtain desired masses for N and ∆



Results: Nucleonand ∆ Masses




Mass-scale parameters (in GeV) for the scalar and axial-vector

diquark correlations, fixed by fitting nucleon and ∆ masses

Set A – fit to the actual masses was required; whereas for

Set B – fitted mass was offset to allow for “π-cloud” contributions

set MN M∆ m0+ m1+ ω0+ ω1+

A 0.94 1.23 0.63 0.84 0.44=1/(0.45 fm) 0.59=1/(0.33 fm)

B 1.18 1.33 0.79 0.89 0.56=1/(0.35 fm) 0.63=1/(0.31 fm)

m1+ → ∞: MAN = 1.15 GeV; MB

N = 1.46 GeV









A 0.94 1.23 0.63 0.84 0.44=1/(0.45 fm) 0.59=1/(0.33 fm)

B 1.18 1.33 0.79 0.89 0.56=1/(0.35 fm) 0.63=1/(0.31 fm)

m1+ → ∞: MAN = 1.15 GeV; MB

N = 1.46 GeV

Axial-vector diquark provides significant attractionIVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 34/52








A 0.94 1.23 0.63 0.84 0.44=1/(0.45 fm) 0.59=1/(0.33 fm)

B 1.18 1.33 0.79 0.89 0.56=1/(0.35 fm) 0.63=1/(0.31 fm)

m1+ → ∞: MAN = 1.15 GeV; MB

N = 1.46 GeV

Constructive Interference: 1++-diquark + ∂µπIVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 34/52


Nucleon-Photon Vertex




M. Oettel, M. Pichowskyand L. von Smekal, nu-th/9909082

6 terms . . .constructed systematically . . . current conserved automatically

for on-shell nucleons described by Faddeev Amplitude




M. Oettel, M. Pichowskyand L. von Smekal, nu-th/9909082

6 terms . . .constructed systematically . . . current conserved automatically

for on-shell nucleons described by Faddeev Amplitude

i

iΨ ΨPf

f

P

Q i

iΨ ΨPf

f

P

Q

i

iΨ ΨPPf

f

Q

Γ−

Γ

scalaraxial vector

i

iΨ ΨPf

f

P

Q

µ

i

i

X

Ψ ΨPf

f

Q

P Γ−

µi

i

X−

Ψ ΨPf

f

P

Q

ΓIVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 35/52


Form Factor Ratio:GE/GM

0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp




Form Factor Ratio:GE/GMCombine these elements . . .

0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp





Dressed-Quark Core

0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp





Dressed-Quark Core

Ward-Takahashi

Identity preserving

current

0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp





Dressed-Quark Core

Ward-Takahashi

Identity preserving

current

Anticipate and

Estimate Pion

Cloud’s Contribution

0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp





Dressed-Quark Core

Ward-Takahashi

Identity preserving

current

Anticipate and

Estimate Pion


0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp

covariant Fadeev resultRosenbluthprecision Rosenbluthpolarization transferpolarization transfer




Dressed-Quark Core

Ward-Takahashi

Identity preserving

current

Anticipate and

Estimate Pion


0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp


All parameters fixed in

other applications . . . Not varied.




Dressed-Quark Core

Ward-Takahashi

Identity preserving

current

Anticipate and

Estimate Pion


0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp




Agreement with Pol. Trans. data at Q2 ∼> 2 GeV2




Dressed-Quark Core

Ward-Takahashi

Identity preserving

current

Anticipate and

Estimate Pion


0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp





Correlations in Faddeev amplitude – quark orbital

angular momentum – essential to that agreement




Dressed-Quark Core

Ward-Takahashi

Identity preserving

current

Anticipate and

Estimate Pion


0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp





Correlations in Faddeev amplitude – quark orbital

angular momentum – essential to that agreement

Predict Zero at Q2 ≈ 6.5GeV2



Epilogue



Epilogue


Provide Understanding of

Dynamical Chiral Symmetry Breaking:

⇒ π is quark-antiquark Bound State

AND QCD’s Goldstone Mode



Epilogue





AND QCD’s Goldstone ModeFoundation for Proof of

Exact Results in QCD

e.g., Quark Goldberger-Treiman

Properties of Pseudoscalar Mesons



Epilogue





AND QCD’s Goldstone ModeFoundation for Proof of

Exact Results in QCD

e.g., Quark Goldberger-Treiman

Properties of Pseudoscalar Mesons

Renormalisation-Group-Improved Rainbow-Ladder

⇒ Practical Phenomenological Tool

Corrections QuantifiableIVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 37/52


Epilogue

Poincaré Covariant

Faddeev Equation



Epilogue

Poincaré Covariant

Faddeev Equation

Nonpointlike scalar and axial-vector diquark correlations



Epilogue

Poincaré Covariant

Faddeev Equation


s−, p−, d−wave quark angular momentum



Epilogue

Poincaré Covariant

Faddeev Equation



Quark core, relaxed to allow for pion cloud



Epilogue

Poincaré Covariant

Faddeev Equation



Quark core, relaxed to allow for pion cloud

Predicts zero in GPE(Q2) at Q2 ≈ 6.5 GeV2



Epilogue

Appearance of a zero in GE(Q2) – Completely Unexpected



Epilogue


0 0.2 0.4 0.6 0.8 1r (fm)

0

0.002

0.004

0.006

0.008

0.01

ρ(r)

Peak Position and Height -- as zero moves in toward q*q=0

peak moves out and falls in magnitude-- redistribution of charge ... moves to larger r



Epilogue


0 0.2 0.4 0.6 0.8 1r (fm)

0

0.002

0.004

0.006

0.008

0.01

ρ(r)

Peak Position and Height -- as zero moves in toward q*q=0

peak moves out and falls in magnitude-- redistribution of charge ... moves to larger r

However, Current Density remains peaked at r = 0!



Epilogue


Wave Function is complex and correlated mix of virtual

particles and antiparticles: s−, p− and d−waves



Epilogue




Simple three-quark bag-model picture that is still being

taught at many universities is profoundly incorrect



Epilogue




Simple three-quark bag-model picture that is still being

taught at many universities is profoundly incorrect

Can Expect

Contemporary Nuclear Physicsto Yield Many More Surprises



Contents1. Pion Dichotomy

2. DSEs

3. Persistent Challenge

4. Perturbative Quark Propagator

5. Dressed-Quark Propagator

6. Lattice cf. DSE

7. Light-Quark Interaction

8. Dressed-gluon Propagator

9. Hadrons

10. Bethe-Salpeter Kernel

11. Nonpert. Trunc. – Contributions

12. Excitations & Chiral Symmetry

13. Radial Excitations

14. Radial Excitations II

15. Two-photon Couplings

16. Proton FF

17. Nucleon EM Form Factors

18. Pions and Form Factors

19. Faddeev equation

20. Parametrising diquark properties

21. Results: Nucleon & ∆ Masses

22. Form Factor Ratio: GE/GM

23. Dressed-Vertex

24. Dressed-Vertex II

25. Goldberger-Treiman for pion

26. Colour-singlet BSE

27. Two-photon: small Q2

28. E.M. Charge Radii

29. DIS

30. Valence Distribution

31. Handbag Diagrams

32. General Calc. cf. Data

33. Form Factor Ratio: Q ∗F2/F1

34. Form Factor Ratio: alternative F2/F1IVth International Conference on Quarks and Nuclear Physics, Madrid 5-10 June, 2006 – p. 38/52


Quenched-QCDDressed-quark-gluon Vertex

0 1 2 3p (GeV)

0

0.5

1

1.5

2

2.5λ

1(p) Quenched Lattice

4p2λ


-2pλ3(p) Quenched Lattice

λ1(p) DSE--Lat (quenched)

4p2λ

2(p) DSE--Lat

-2pλ3(p) DSE--Lat

λ1(p) Abelian Ansatz (WI)

-2pλ3(p) Abelian Ansatz (WI)

Bhagwat, et al.:

nu-th/0304003

nu-th/0403012

he-ph/0407163

share 65 citations




0 1 2 3p (GeV)

0

0.5

1

1.5

2

2.5λ


4p2λ




4p2λ

2(p) DSE--Lat

-2pλ3(p) DSE--Lat



Bhagwat, et al.:

nu-th/0304003

nu-th/0403012

he-ph/0407163

share 65 citations

Lattice – Skullerud,et al.: he-ph/0303176




0 1 2 3p (GeV)

0

0.5

1

1.5

2

2.5λ


4p2λ




4p2λ

2(p) DSE--Lat

-2pλ3(p) DSE--Lat



Bhagwat, et al.:

nu-th/0304003

nu-th/0403012

he-ph/0407163

share 65 citations


Parameter Free DSEPrediction confirmslattice simulation ofλ1, λ3




0 1 2 3p (GeV)

0

0.5

1

1.5

2

2.5λ


4p2λ




4p2λ

2(p) DSE--Lat

-2pλ3(p) DSE--Lat



Bhagwat, et al.:

nu-th/0304003

nu-th/0403012

he-ph/0407163

share 65 citations


Parameter Free DSEPrediction confirmslattice simulation ofλ1, λ3

Parameter FreeDSE Prediction suggests lattice result for λ2 erroneous– owing to systematic errors



Goldberger-Treiman for pion




• 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



Colour-singletBethe-Salpeter equation

Detmold et al., nu-th/0202082Bhagwat, et al., nu-th/0403012

�a�(k; p) = + + + : : :

�M =�n �n� �M + �a;n�

�a;n� = �M�n�1� + �M�n�1� + �a;n�1�





• Coupling-modified dressed-ladder vertex�a�(k; p) = + + + : : :C C2

�M =�n �n� �M + �a;n�

�a;n� = �M�n�1� + �M�n�1� + �a;n�1�






• BSE consistent with vertex�M =�n �n� �M + �a;n�

�a;n� = �M�n�1� + �M�n�1� + �a;n�1�






• BSE consistent with vertex�M =�n �n� �M + �a;n�• Bethe-Salpeter kernel . . . recursion relation

−1

8C �a;n� = �M�n�1� + �M�n�1� + �a;n�1�






• BSE consistent with vertex�M =�n �n� �M + �a;n�• Bethe-Salpeter kernel . . . recursion relation

−1

8C �a;n� = �M�n�1� + �M�n�1� + �a;n�1�

• Kernel necessarily non-planar,even with planar vertex



π and ρ mesons



π and ρ mesons

Mn=0H Mn=1

H Mn=2H Mn=∞

H

π, m = 0 0 0 0 0

π, m = 0.011 0.147 0.135 0.139 0.138

ρ, m = 0 0.920 0.648 0.782 0.754

ρ, m = 0.011 0.936 0.667 0.798 0.770



π and ρ mesons

Mn=0H Mn=1

H Mn=2H Mn=∞

H

π, m = 0 0 0 0 0

π, m = 0.011 0.147 0.135 0.139 0.138

ρ, m = 0 0.920 0.648 0.782 0.754

ρ, m = 0.011 0.936 0.667 0.798 0.770

• π massless in chiral limit . . . NO Fine Tuning



π and ρ mesons

Mn=0H Mn=1

H Mn=2H Mn=∞

H

π, m = 0 0 0 0 0

π, m = 0.011 0.147 0.135 0.139 0.138

ρ, m = 0 0.920 0.648 0.782 0.754

ρ, m = 0.011 0.936 0.667 0.798 0.770


• ALL π-ρ mass splitting present in chiral limit



π and ρ mesons

Mn=0H Mn=1

H Mn=2H Mn=∞

H

π, m = 0 0 0 0 0

π, m = 0.011 0.147 0.135 0.139 0.138

ρ, m = 0 0.920 0.648 0.782 0.754

ρ, m = 0.011 0.936 0.667 0.798 0.770


• ALL π-ρ mass splitting present in chiral limitand with the Simplest kernel



π and ρ mesons

Mn=0H Mn=1

H Mn=2H Mn=∞

H

π, m = 0 0 0 0 0

π, m = 0.011 0.147 0.135 0.139 0.138

ρ, m = 0 0.920 0.648 0.782 0.754

ρ, m = 0.011 0.936 0.667 0.798 0.770


• π-ρ mass splitting driven by DχSB mechanismNot constituent-quark-model-like hyperfine splitting



π and ρ mesons

Mn=0H Mn=1

H Mn=2H Mn=∞

H

π, m = 0 0 0 0 0

π, m = 0.011 0.147 0.135 0.139 0.138

ρ, m = 0 0.920 0.648 0.782 0.754

ρ, m = 0.011 0.936 0.667 0.798 0.770



• Extending kernel



π and ρ mesons

Mn=0H Mn=1

H Mn=2H Mn=∞

H

π, m = 0 0 0 0 0

π, m = 0.011 0.147 0.135 0.139 0.138

ρ, m = 0 0.920 0.648 0.782 0.754

ρ, m = 0.011 0.936 0.667 0.798 0.770



• Extending kernel: NO effect on mπ



π and ρ mesons

Mn=0H Mn=1

H Mn=2H Mn=∞

H

π, m = 0 0 0 0 0

π, m = 0.011 0.147 0.135 0.139 0.138

ρ, m = 0 0.920 0.648 0.782 0.754

ρ, m = 0.011 0.936 0.667 0.798 0.770




For mρ – zeroth order, accurate to 20%



π and ρ mesons

Mn=0H Mn=1

H Mn=2H Mn=∞

H

π, m = 0 0 0 0 0

π, m = 0.011 0.147 0.135 0.139 0.138

ρ, m = 0 0.920 0.648 0.782 0.754

ρ, m = 0.011 0.936 0.667 0.798 0.770




For mρ – zeroth order, accurate to 20%– one loop, accurate to 13%



π and ρ mesons

Mn=0H Mn=1

H Mn=2H Mn=∞

H

π, m = 0 0 0 0 0

π, m = 0.011 0.147 0.135 0.139 0.138

ρ, m = 0 0.920 0.648 0.782 0.754

ρ, m = 0.011 0.936 0.667 0.798 0.770




For mρ – zeroth order, accurate to 20%– one loop, accurate to 13%– two loop, accurate to 4%



Calculated Transition Form Factor:RGI Rainbow-LadderHöll, Krassnigg, Maris, et al.,


-0.2 0 0.2 0.4 0.6 0.8

Q2 [GeV

2]

-1

0

1

2

3

4

5

Tπ n(Q

2 ) [

GeV

-1]

n = 0 (ground state)n = 1 (radial excitation)

1 / ( 2 fπ0)

mu(1 GeV)

= md(1 GeV) = 5.5 MeV





-0.2 0 0.2 0.4 0.6 0.8

Q2 [GeV

2]

-1

0

1

2

3

4

5

Tπ n(Q

2 ) [

GeV

-1]


1 / ( 2 fπ0)

mu(1 GeV)

= md(1 GeV) = 5.5 MeV

Tπ01(−m2

π1, Q2) < 0 , Q2 ≥ −m2

π1/4;

viz., it is negative on the entire kinematically accessible domain.





-0.2 0 0.2 0.4 0.6 0.8

Q2 [GeV

2]

-1

0

1

2

3

4

5

Tπ n(Q

2 ) [

GeV

-1]


1 / ( 2 fπ0)

mu(1 GeV)

= md(1 GeV) = 5.5 MeV

Tπ01(−m2

π1, Q2) < 0 , Q2 ≥ −m2

π1/4;

viz., it is negative on the entire kinematically accessible domain.

Γπ00γγ = 7.9 eV, Γπ0

1γγ = 240 eV





1 10 100 1000

Q2 [GeV

2]

10-5

10-4

10-3

10-2

10-1

100

|Tπ n(Q

2 )| [

GeV

-1]

n = 0n = 14 π2

fπn/ (3Q

2)

mu(1 GeV)

= md(1 GeV) = 5.5 MeV





1 10 100 1000

Q2 [GeV

2]

10-5

10-4

10-3

10-2

10-1

100

|Tπ n(Q

2 )| [

GeV

-1]

n = 0n = 14 π2

fπn/ (3Q

2)

mu(1 GeV)

= md(1 GeV) = 5.5 MeV

Predicted UV-behaviour is abundantly clear

precise for Q2 > 120 GeV2



Transition Form Factor ( Chiral ):RGI Rainbow-LadderHöll, Krassnigg, Maris, et al.,


1 10 100 1000

Q2 [GeV

2]10

-7

10-6

10-5

10-4

10-3

10-2

10-1

|Tπ n(Q

2 )| [

GeV

-1]

up/down quarks

4 π2fπ1

/ (3Q2)

chiral limit (4 π2

/ 3) (0.22 GeV)3 / Q4

mu(1 GeV)

= md(1 GeV) = 0





1 10 100 1000

Q2 [GeV

2]10

-7

10-6

10-5

10-4

10-3

10-2

10-1

|Tπ n(Q

2 )| [

GeV

-1]

up/down quarks

4 π2fπ1

/ (3Q2)

chiral limit (4 π2

/ 3) (0.22 GeV)3 / Q4

mu(1 GeV)

= md(1 GeV) = 0

Again, Predicted UV-behaviouris abundantly clear






1 10 100 1000

Q2 [GeV

2]10

-7

10-6

10-5

10-4

10-3

10-2

10-1

|Tπ n(Q

2 )| [

GeV

-1]

up/down quarks

4 π2fπ1

/ (3Q2)

chiral limit (4 π2

/ 3) (0.22 GeV)3 / Q4

mu(1 GeV)

= md(1 GeV) = 0

Again, Predicted UV-behaviouris abundantly clear


F(2)1 (−m2

π1) lnγ Q2/ω2

π1

∣

∣

∣

m=0≈ (0.22 GeV)3 ≃ −〈qq〉0 (3)



Electromagnetic Charge Radii – RGIRainbow-LadderHöll, Krassnigg, Maris, et al.,

nu-th/0503043

0.3 0.32 0.34 0.36 0.38 0.4ω [GeV]

0.60

0.65

0.70

0.75

0.80

r π [fm

]

n = 0 (ground state)n = 1 (radial excitation)linear fit: 0.61 + 0.11 ωlinear fit: 0.09 + 1.76 ω

mu,d(1 GeV) = 5.5 MeV





nu-th/0503043

0.3 0.32 0.34 0.36 0.38 0.4ω [GeV]

0.60

0.65

0.70

0.75

0.80

r π [fm

]



Reminder:MT-model has oneIR-mass-scale – ω

ra := 1/ω

gauges the rangeof strong attraction





nu-th/0503043

0.3 0.32 0.34 0.36 0.38 0.4ω [GeV]

0.60

0.65

0.70

0.75

0.80

r π [fm

]




ra := 1/ω


Goldstone Mode’sproperties are insensitive to ra

Expected cf. T 6= 0, Goldstone mode’s properties do notchange until very near chiral symmetry restorationtemperature.





nu-th/0503043

0.3 0.32 0.34 0.36 0.38 0.4ω [GeV]

0.60

0.65

0.70

0.75

0.80

r π [fm

]




ra := 1/ω


1st excited state:orthogonal to Goldstone mode

Not protected . . . properties very sensitive to ra





nu-th/0503043

0.3 0.32 0.34 0.36 0.38 0.4ω [GeV]

0.60

0.65

0.70

0.75

0.80

r π [fm

]




ra := 1/ω


Best estimate rπ1= 1.4 rπ0

But rπ1< rπ0

is possible if confinement force is very strong





nu-th/0503043

0.3 0.32 0.34 0.36 0.38 0.4ω [GeV]

0.60

0.65

0.70

0.75

0.80

r π [fm

]




ra := 1/ω


Radial excitations areplainly useful to map outthe long-range part of interaction between light-quarks.





nu-th/0503043

0.3 0.32 0.34 0.36 0.38 0.4ω [GeV]

0.60

0.65

0.70

0.75

0.80

r π [fm

]




ra := 1/ω



Same is true of orbital excitations; e.g., axial-vector mesons.





nu-th/0503043

0.3 0.32 0.34 0.36 0.38 0.4ω [GeV]

0.60

0.65

0.70

0.75

0.80

r π [fm

]




ra := 1/ω



Same is true of orbital excitations; e.g., axial-vector mesons.

Hall-D at JLab




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!






Proved on

22/July/2002, ANL






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




Proposal at JLab

(Holt, Reimer, Wijesooriya, et al.,

JLab at 12 GeV)

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−Γπ(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 · qfixed.

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−Γπ(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−−)



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



Form Factor Ratio: Q ∗F2/F1




0.0

1.0

2.0

3.0

4.0

Q2 F

2p/(

κ pF1p

) set BSLACJLab1JLab2

0 2 4 6 8 10Q

2 [GeV2]

0

0.5

1

1.5

Q F

2p/(

κ pF1p

)




0.0

1.0

2.0

3.0

4.0

Q2 F

2p/(

κ pF1p

) set BSLACJLab1JLab2

0 2 4 6 8 10Q

2 [GeV2]

0

0.5

1

1.5

Q F

2p/(

κ pF1p

)

Perhaps ≈ constant for 2 ∼< Q2 ∼< 6GeV2



Formln Factor Ratio: alternativeF2/F1




2 4 6 8 10Q

2 [GeV2]

0

0.5

1

1.5

(Q/ln

(Q2 /Λ

2 ))2 F

2p/(κ

pF1p

)

set BSLACJLab 1JLab 2




2 4 6 8 10Q

2 [GeV2]

0

0.5

1

1.5

(Q/ln

(Q2 /Λ

2 ))2 F

2p/(κ

pF1p

)

set BSLACJLab 1JLab 2

Q2

[ln Q2/Λ2]2F2(Q

2)

F1(Q2)= constant, Q2 ≫ Λ2 ≈ M2

N

Suggestive

NB. Framework

constructed to give

quark-counting

i.e., “pQCD” but with

wrong anomalous

dimensions but they’re

ignored in ln-power “2” of

this ratio


Dyson-Schwinger Equations – Theory and Phenomenology · Infinitely Many Coupled Equations...

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Transcript of Dyson-Schwinger Equations – Theory and Phenomenology · Infinitely Many Coupled Equations...