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First Contents Back Conclusion

Charting the InteractionBetween Light Quarks

Craig D. Roberts

[email protected]

Physics Division

Argonne National Laboratory

http://www.phy.anl.gov/theory/staff/cdr.htmlCraig Roberts: Charting the interaction between light quarks

CLAS12 European Workshop . . . 23 transparencies – p. 1/40


Universal Truths

Craig Roberts: Charting the interaction between light quarks



Universal Truths

Spectrum of excited states 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.




Universal Truths







Universe. Higgs mechanism is irrelevant to light-quarks.




Universal Truths








Running of quark mass entails that calculations at even

modest Q2 require a Poincaré-covariant approach.




Universal Truths








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








Challenge: understand relationship between parton properties

on the light-front and rest frame structure of hadrons.




Universal Truths








Challenge: understand relationship between parton properties

on the light-front and rest frame structure of hadrons. Problem

because, e.g., DCSB - an established keystone of low-energy

QCD and the origin of constituent-quark masses - has not

been realised in the light-front formulation.




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: Charting the interaction between light quarks



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 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 and valid chiral limit;

and an accurate realisation ofdynamical chiral symmetry breaking.







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 and valid chiral limit;

and an accurate realisation ofdynamical chiral symmetry breaking.

Highly NontrivialCraig Roberts: Charting the interaction between light quarks



What’s the Problem?





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.










Means . . . must calculate hadron wave functions

– Can’t be done using perturbation theory












Why problematic? Isn’t same true in quantum mechanics?












Why problematic? Isn’t same true in quantum mechanics?

Differences!




What’s the Problem?Relativistic QFT!






Differences!

Here relativistic effects are crucial – virtual particles,

quintessence of Relativistic Quantum Field Theory –

must be included




What’s the Problem?Relativistic QFT!






Differences!

Here relativistic effects are crucial – virtual particles,

quintessence of Relativistic Quantum Field Theory –

must be included

Interaction between quarks – the Interquark “Potential” –

unknown throughout > 98% of a hadron’s volume




Intranucleon Interaction




Intranucleon Interaction

98% of the volume




Intranucleon Interaction?What is the

98% of the volume

The question must berigorously defined, and theanswer mapped out usingexperiment and theory.




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

eV] m = 0 (Chiral limit)

m = 30 MeVm = 70 MeV

effect of gluon cloudRapid acquisition of mass is

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.



First Contents Back ConclusionCraig Roberts: Charting the interaction between light quarks



• 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: Charting the interaction between light quarks



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: Charting the interaction between light quarks



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




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




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








And Formulation of Practical Phenomenological Tool to

Illustrate Exact Results

Make Predictions with Readily Quantifiable Errors




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

i

i

i

iP π

πρ

k

Γ

S

(τ/2) γ

S

555

=







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: Charting the interaction between light quarks





New Challenges




New Challenges

Next Steps . . . Applications to excited states and

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

confinement interaction between light-quarks.




New Challenges

Next Steps . . . Applications to excited states and

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

confinement interaction between light-quarks.

Move on to the problem of a symmetry preserving treatment

of hybrids and exotics.




New Challenges

Another Direction . . . Also want/need information about

three-quark systems




New Challenges


three-quark systems

With this problem . . . most wide-ranging studies employ

expertise familiar from meson applications circa ∼1995.




New Challenges


three-quark systems



Namely . . . Model-building and Phenomenology,

constrained by the DSE results outlined already.Craig Roberts: Charting the interaction between light quarks



New Challenges


three-quark systems



However, that is beginning to change . . .




Unifying Studyof Mesons and Baryons





How does one incorporate dressed-quark mass function,

M(p2), in study of baryons?






M(p2), in study of baryons? Behaviour of M(p2) is es-

sentially a quantum field theoretical effect.








In quantum field theory a nucleon appears as a pole in a six-

point quark Green function.










Residue is proportional to nucleon’s Faddeev amplitude











Poincaré covariant Faddeev equation sums all possible

exchanges and interactions that can take place between

three dressed-quarks











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 quark-quark (diquark) correlations in the

colour-3̄ (antitriplet) channelCraig Roberts: Charting the interaction between light quarks



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




Diquark correlations

QUARK-QUARKCraig Roberts: Charting the interaction between light quarks



Diquark correlations

QUARK-QUARK

Same interaction that

describes mesons also

generates three coloured

quark-quark correlations:

blue–red, blue–green,

green–red

Confined . . . Does not

escape from within baryon.

Scalar is isosinglet,

Axial-vector is isotriplet

DSE and lattice-QCD

m[ud]0+

= 0.74 − 0.82

m(uu)1+

= m(ud)1+

= m(dd)1+

= 0.95 − 1.02




Ab-initio studyof mesons & nucleons

Eichmann et al.– arXiv:0802.1948 [nucl-th]– arXiv:0810.1222 [nucl-th]

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

0.10.0 0.2 0.3 0.4 0.5 0.6

Bethe-Salpeter & Faddeev

equations built from same

RG-improved rainbow-ladder

interaction






0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

0.10.0 0.2 0.3 0.4 0.5 0.6




interaction

Simultaneous calculation of

baryon & meson properties,

& prediction of their correlation






0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

0.10.0 0.2 0.3 0.4 0.5 0.6




interaction




Continuum prediction for

evolution of mρ & MN with

quantity that can methodically

be connected with the

current-quark mass in QCD






0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

0.10.0 0.2 0.3 0.4 0.5 0.6




interaction




Continuum prediction for

evolution of mρ & MN with

quantity that can methodically

be connected with the

current-quark mass in QCD

Systematically improvableCraig Roberts: Charting the interaction between light quarks



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

ΓCraig Roberts: Charting the interaction between light quarks



DSE-basedFaddeev Equation





Cloët et al.– arXiv:0710.2059 [nucl-th]– arXiv:0710.5746 [nucl-th]– arXiv:0804.3118 [nucl-th]






Faddeev equation input –algebraic parametrisations ofDSE results, constrained by π

and K observables







and K observables

Two parameters– M0+ = 0.8 GeV,M1+ = 0.9 GeV– chosen to giveMN = 1.18, M∆ = 1.33

– allow for pseudoscalar mesoncontributions






0 2 4 6 8 10Q

2 [GeV2]

0

0.5

1

µ p GEp/ G

Mp

r1+ = 0.4 fm

Punjabi (2005)Gayou (2002)


and K observables



Sensitivity to details of thecurrent – expressed throughdiquark radius






0 2 4 6 8 10Q

2 [GeV2]

0

0.5

1

µ p GEp/ G

Mp

r1+ = 0.4 fm



and K observables




On Q2

∼

< 4 GeV2 result lies below experiment. This can be attributed to omissionof pseudoscalar-meson-cloud contributions






0 2 4 6 8 10Q

2 [GeV2]

0

0.5

1

µ p GEp/ G

Mp

r1+ = 0.4 fm



and K observables




On Q2

∼

< 4 GeV2 result lies below experiment. This can be attributed to omissionof pseudoscalar-meson-cloud contributions

Always a zero but position depends on details of current




ab initioFaddeev Equation






0 1 2 3 4 5 6

0.2

0.0

0.4

0.6

0.8

1.0

1.2






0 1 2 3 4 5 6

0.2

0.0

0.4

0.6

0.8

1.0

1.2

Parameter-free rainbow-ladderFaddeev equation – resultqualitatively identical and insemiquantitative agreement






0 1 2 3 4 5 6

0.2

0.0

0.4

0.6

0.8

1.0

1.2


Improved numerical algorithmneeded to extend calculation tolarger Q2






0 1 2 3 4 5 6

0.2

0.0

0.4

0.6

0.8

1.0

1.2


Improved numerical algorithmneeded to extend calculation tolarger Q2

Calculation unifies π, ρ and nucleon properties – keystone

is behaviour of dressed-quark mass function and hence

veracious description of QCD’s Goldstone mode




Ratio of NeutronPauli & Dirac Form Factors

Q̂2

(ln Q̂2/Λ̂)2

F n2

(Q̂2)

F n1

(Q̂2)

Λ̂ = Λ/MN = 0.44

Ensures proton ratioconstant for Q̂2 ≥ 4





2 4 6 8

y=Q2/M

2

0.1

1

10

- [y

/Ln2 (y

M2 /Λ

2 )] F

n 2(y)/

κ nFn 1(y

)

DSE result- DSE -Hall A E02-013 Preliminary

Madey et al. nucl-ex/0308007

Q̂2

(ln Q̂2/Λ̂)2

F n2

(Q̂2)

F n1

(Q̂2)

Λ̂ = Λ/MN = 0.44






Q̂2

(ln Q̂2/Λ̂)2

F n2

(Q̂2)

F n1

(Q̂2)

Λ̂ = Λ/MN = 0.44


Brown band– ab initio RL result




Pion Cloud




Pion CloudF2 – neutron





0 1 2 3 4 5 6

Q2/M

2

0

0.05

0.1

0.15

F2n : D

SE

- K

elly

Pseudoscalar contribution20% of peak value

Comparisonbetween Faddeevequation resultand Kelly’sparametrisation

Faddeevequation set-upto describedressed-quarkcore





0 1 2 3 4 5 6

Q2/M

2

0

0.05

0.1

0.15

F2n : D

SE

- K

elly

Pseudoscalar contribution20% of peak value

Comparisonbetween Faddeevequation resultand Kelly’sparametrisation

Faddeevequation set-upto describedressed-quarkcore

Pseudoscalar meson cloud (and relatedeffects) significant for Q2

∼< 3 − 4 M2

N




Epilogue




EpilogueDCSB exists in QCD.





It is manifest in dressed propagators and

vertices






vertices

It predicts, amongst other things, that

light current-quarks become heavy

constituent-quarks: 4 → 400 MeV

pseudoscalar mesons are unnaturally

light: mρ = 770 cf. mπ = 140 MeV

pseudoscalar mesons couple unnaturally

strongly to light-quarks: gπq̄q ≈ 4.3

pseudscalar mesons couple unnaturally

strongly to the lightest baryons

gπN̄N ≈ 12.8 ≈ 3gπq̄q






vertices

It predicts, amongst other things, that

light current-quarks become heavy

constituent-quarks: 4 → 400 MeV

pseudoscalar mesons are unnaturally

light: mρ = 770 cf. mπ = 140 MeV

pseudoscalar mesons couple unnaturally

strongly to light-quarks: gπq̄q ≈ 4.3

pseudscalar mesons couple unnaturally

strongly to the lightest baryons

gπN̄N ≈ 12.8 ≈ 3gπq̄q

It impacts dramatically upon observables.Craig Roberts: Charting the interaction between light quarks



Epilogue

Dyson-Schwinger Equations

Poincaré covariant unification of meson and baryon

observables




Epilogue



observables

All global and pointwise corollaries of DCSB are

manifested naturally without fine-tuning




Epilogue



observables



Excited states:

Mesons already being studied

Baryons are within practical reach




Epilogue



observables



Excited states:



Ab-initio study of N → ∆ transition underway




Epiloguenothing!



observables



Excited states:



Ab-initio study of N → ∆ transition underway

Tool enabling insight to be drawn from experiment into

long-range piece of interaction between light-quarksCraig Roberts: Charting the interaction between light quarks



Contents

1. Universal Truths

2. QCD’s Challenges

3. Dichotomy of the Pion

4. Dressed-Quark Propagator

5. Frontiers of Nuclear Science

6. Hadrons

7. Confinement

8. Bethe-Salpeter Kernel

9. Persistent Challenge

10. Radial Excitations

11. Radial Excitations & Lattice-QCD

12. Pion FF

13. Calculated Pion FF

14. All Pion Form Factors

15. Nucleon Challenge

16. Unifying Meson & Nucleon

17. Faddeev equation

18. Diquark correlations

19. Ab-initio study of mesons &nucleons

20. rπ fπ

21. Nucleon-Photon Vertex

22. DSE-based Faddeev Equation

23. Ratio of Neutron FFs

24. Pion Cloud




Dyson-Schwinger EquationsDressed-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




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.

he-la/0512018




Confinement

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

light-quarks?




Confinement


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

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.

he-la/0512018

“The breaking of the string appears to be an instantaneous

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

Therefore . . . No

information on potential

between light-quarks.





Höll, Krassnigg, Robertsnu-th/0406030

fH m2H = − ρH

ζ MH

Valid for ALL Pseudoscalar mesons







fH m2H = − ρH

ζ MH


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







fH m2H = − ρH

ζ MH



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

m2πn 6=0

> m2πn=0

= 0, in chiral limit







fH m2H = − ρH

ζ MH




m2πn 6=0

> m2πn=0


⇒ fH = 0

ALL pseudoscalar mesons except π(140) in chiral limit







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: Charting the interaction between light quarks




Radial Excitations& Lattice-QCDMcNeile and Michael

he-la/0607032





he-la/0607032

When we first heard about [this result] our first reaction was acombination of “that is remarkable” and “unbelievable”.





he-la/0607032


CLEO: τ → π(1300) + ντ

⇒ fπ1< 8.4 MeV

Diehl & Hillerhe-ph/0105194





he-la/0607032

0 0.5 1 1.5 2 2.5 3 3.5 4

( r0 mπ )

2

0

0.2

0.4

0.6

0.8

f π’/f

πnot improvedNP improvedExpt. bound


CLEO: τ → π(1300) + ντ

⇒ fπ1< 8.4 MeV


Lattice-QCD check:163 × 32,a ∼ 0.1 fm,two-flavour, unquenched

⇒ fπ1

fπ

= 0.078 (93)





he-la/0607032

0 0.5 1 1.5 2 2.5 3 3.5 4

( r0 mπ )

2

0

0.2

0.4

0.6

0.8

f π’/f



CLEO: τ → π(1300) + ντ

⇒ fπ1< 8.4 MeV



⇒ fπ1

fπ

= 0.078 (93)

Full ALPHA formulation is required to see suppression, becausePCAC relation is at the heart of the conditions imposed forimprovement (determining coefficients of irrelevant operators)Craig Roberts: Charting the interaction between light quarks




he-la/0607032

0 0.5 1 1.5 2 2.5 3 3.5 4

( r0 mπ )

2

0

0.2

0.4

0.6

0.8

f π’/f



CLEO: τ → π(1300) + ντ

⇒ fπ1< 8.4 MeV



⇒ fπ1

fπ

= 0.078 (93)

The suppression of fπ1is a useful benchmark that can be used to

tune and validate lattice QCD techniques that try to determine theproperties of excited states mesons.Craig Roberts: Charting the interaction between light quarks



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

0 1 2 3 4

Q2 [GeV

2]

0

0.1

0.2

0.3

0.4

0.5

Q2 F

π(Q2 )

[GeV

2 ]

Amendolia et al.Ackermann et al.Brauel et al.Tadevosyan et al.

Horn et al.Maris and Tandy, 2005

Calculation first published in 1999; No Parameters VariedNumerical method improved in 2005




Calculated Pion Form Factor

0 1 2 3 4

Q2 [GeV

2]

0

0.1

0.2

0.3

0.4

0.5

Q2 F

π(Q2 )

[GeV

2 ]


Horn et al.Maris and Tandy, 2005

Calculation first published in 1999; No Parameters VariedNumerical method improved in 2005

Data publishedin 2001.Subsequentlyrevised




Timelike Pion Form Factor





Ab initio calculation intotimelike region. Deeper thanground-state ρ-meson pole





-0.5 0 0.5 1 1.5 2 2.5 3

Q2 [GeV

2]

10-1

100

101

|Fπ(Q

2 )|


Horn et al.Barkov et al.DSE calculationQCDSF/UKQCD, monopole fit + error band

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1



QCDSF/UKQCD, simulation result

Ab initio calculation intotimelike region. Deeper thanground-state ρ-meson pole





-0.5 0 0.5 1 1.5 2 2.5 3

Q2 [GeV

2]

10-1

100

101

|Fπ(Q

2 )|



0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1



QCDSF/UKQCD, simulation result

Ab initio calculation intotimelike region. Deeper thanground-state ρ-meson poleρ-meson not put in “by hand” – generated dynamically as a bound-state of dressed-quark and dressed-antiquark




Dimensionless product: rπ fπ





Improved rainbow-ladder interaction






Repeating Fπ(Q2) calculation







Experimentally: rπfπ = 0.315 ± 0.005








DSE prediction








DSE prediction

Lattice results

– James Zanotti [UK QCD]








DSE prediction

Lattice results

– James Zanotti [UK QCD]

Fascinating result:

DSE and Lattice

– Experimental value

obtains independent of

current-quark mass.Craig Roberts: Charting the interaction between light quarks







DSE prediction

Fascinating result:

DSE and Lattice

– Experimental value

obtains independent of

current-quark mass.

We have understood this

Implications far-reaching.Craig Roberts: Charting the interaction between light quarks



Pion Form Factors




Pion Form Factors

There is a sense in which it is easy to fabricate amodel that can reproduce the elasticelectromagnetic pion form factor




Pion Form Factors


However, a veracious description of the pion willsimultaneously predict the elastic electromagneticform factor, Fπ(Q2) AND the γ∗π → γ transitionform factor




Pion Form FactorsInfidelity without simultaneity



The latter is connected with the Abelian anomaly –therefore fundamentally connected with chiralsymmetry and its dynamical breaking – no meremodel can successfully describe this without finetuning




Pion Form FactorsInfidelity without simultaneity



The latter is connected with the Abelian anomaly –therefore fundamentally connected with chiralsymmetry and its dynamical breaking – no meremodel can successfully describe this without finetuning

Must similarly require prediction of γ∗π → ππ andall other anomalous processes




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






Leading-order truncation of DSEs – rainbow-ladder







Corrections vanish with increasing current-quark mass

⇒ rainbow-ladder exact in heavy-quark limit









However, at physical light-quark mass, corrections to

observables not protected by symmetries: uniformly ≈ 35%

Roughly 50/50-split between nonresonant and resonant

(pseudoscalar meson loop) contributions













Symmetry preserving and systematic approach can

elucidate and account for these effects













Symmetry preserving and systematic approach can

elucidate and account for these effects

Use this knowledge to constrain interaction in infrared

Interaction in ultraviolet predicted by perturbative

expansion of DSEsCraig Roberts: Charting the interaction between light quarks





ETMC

RBC/UKQCD

CP-PACS + Adelaide

Experiment

MILC

1.2

1.1

1.0

0.9

0.0 0.2 0.4 0.6 0.8

0.8

0.7






ETMC

RBC/UKQCD

CP-PACS + Adelaide

Experiment

MILC

1.2

1.1

1.0

0.9

0.0 0.2 0.4 0.6 0.8

0.8

0.7

Rainbow-Ladder DSE result

one parameter for IR – “confinement radius”

Results insensitive to value on material domain






ETMC

RBC/UKQCD

CP-PACS + Adelaide

Experiment

MILC

1.2

1.1

1.0

0.9

0.0 0.2 0.4 0.6 0.8

0.8

0.7




Numerical simulations of lattice-QCD






ETMC

RBC/UKQCD

CP-PACS + Adelaide

Experiment

MILC

1.2

1.1

1.0

0.9

0.0 0.2 0.4 0.6 0.8

0.8

0.7




Numerical simulations of lattice-QCD

FRR extrapolation of lattice CP-PACS resultCraig Roberts: Charting the interaction between light quarks





Precisely the same

interaction






0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

0.10.0 0.2 0.3 0.4 0.5 0.6

Precisely the same

interaction

Same ρ-meson curve






0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

0.10.0 0.2 0.3 0.4 0.5 0.6

Precisely the same

interaction

Same ρ-meson curve

m2π-dependence of 0+ and

1+ diquark masses

“unobservable” – show

marked sensitivity to

single model parameter;

viz., confinement radius






0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

0.10.0 0.2 0.3 0.4 0.5 0.6

Precisely the same

interaction

Same ρ-meson curve

m2π-dependence of 0+ and

1+ diquark masses

“unobservable” – show

marked sensitivity to

single model parameter;

viz., confinement radius

But . . . [mav − msc], mρ

& MN . . . are independent

of that parameter






0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

0.10.0 0.2 0.3 0.4 0.5 0.6

Parameter-independent

RL-DSE predictions, with

veracious description of

Goldstone mode






0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

0.10.0 0.2 0.3 0.4 0.5 0.6




Goldstone mode

DSE and lattice agree on

heavy-quark domain






0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

0.10.0 0.2 0.3 0.4 0.5 0.6




Goldstone mode

DSE and lattice agree on

heavy-quark domain

Prediction: at physical m2π,

Mquark−coreN = 1.26(2) GeV

cf. FRR+lattice-QCD,

Mquark−coreN = 1.27(2) GeV

⇒ subleading corrections,

including 0−-meson loops,

δMN = −320 MeV,

δmρ = −220 MeV Craig Roberts: Charting the interaction between light quarks


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