First Contents Back Conclusion

DCSB & Confinementt

0.5 1. 1.5 2.p

0.1

0.2

0.3

0.4

0.5MHpL

Dressed-quark Mass Function

Impact of DCSB onMeson Structure and Interactions

Craig D. Robertscdroberts@anl.gov

Physics Division & School of Physics

Argonne National Laboratory Peking University

http://www.phy.anl.gov/theory/staff/cdr.htmlCraig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 1/30

First Contents Back Conclusion

Universal Truths

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 2/30

First Contents Back Conclusion

Universal Truths

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 2/30

First Contents Back Conclusion

Universal TruthsSpectrum of excited states, and elastic and transition form

factors provide unique information about long-range

interaction between light-quarks and distribution of hadron’s

characterising properties amongst its QCD constituents.

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 2/30

First Contents Back Conclusion

Universal TruthsSpectrum of excited states, and elastic and transition form

factors provide unique information about long-range

interaction between light-quarks and distribution of hadron’s

characterising properties amongst its QCD constituents.

Dynamical Chiral Symmetry Breaking (DCSB) is most

important mass generating mechanism for visible matter in the

Universe.

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 2/30

First Contents Back Conclusion

Universal TruthsSpectrum of excited states, and elastic and transition form

factors provide unique information about long-range

interaction between light-quarks and distribution of hadron’s

characterising properties amongst its QCD constituents.

Dynamical Chiral Symmetry Breaking (DCSB) is most

important mass generating mechanism for visible matter in the

Universe. Higgs mechanism is irrelevant to light-quarks.

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 2/30

First Contents Back Conclusion

Universal TruthsSpectrum of excited states, and elastic and transition form

factors provide unique information about long-range

interaction between light-quarks and distribution of hadron’s

characterising properties amongst its QCD constituents.

Dynamical Chiral Symmetry Breaking (DCSB) is most

important mass generating mechanism for visible matter in the

Universe. Higgs mechanism is irrelevant to light-quarks.

Running of quark mass entails that calculations at even

modest Q2 require a Poincaré-covariant approach.

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 2/30

First Contents Back Conclusion

Universal TruthsSpectrum of excited states, and elastic and transition form

factors provide unique information about long-range

interaction between light-quarks and distribution of hadron’s

characterising properties amongst its QCD constituents.

Dynamical Chiral Symmetry Breaking (DCSB) is most

important mass generating mechanism for visible matter in the

Universe. Higgs mechanism is irrelevant to light-quarks.

Running of quark mass entails that calculations at even

modest Q2 require a Poincaré-covariant approach. Covariance

requires existence of quark orbital angular momentum in

hadron’s rest-frame wave function.

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 2/30

First Contents Back Conclusion

Universal TruthsChallenge: understand relationship between parton

properties on the light-front and rest frame structure of

hadrons.

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 3/30

First Contents Back Conclusion

Universal TruthsChallenge: understand relationship between parton

properties on the light-front and rest frame structure of

hadrons.

E.g., one problem: DCSB - an established keystone oflow-energy QCD and the origin of constituent-quarkmasses - has not yet been realised in the light-frontformulation.

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 3/30

First Contents Back Conclusion

Universal TruthsChallenge: understand relationship between parton

properties on the light-front and rest frame structure of

hadrons.

E.g., one problem: DCSB - an established keystone oflow-energy QCD and the origin of constituent-quarkmasses - has not yet been realised in the light-frontformulation.

Resolution– So-called vacuum condensates can be understood as aproperty of hadrons themselves, which is expressed, forexample, in their Bethe-Salpeter or light-front wavefunctions.– DCSB obtained via coherent contribution from countableinfinity of higher Fock-state components in LF-wavefunction.

Brodsky, Roberts, Shrock, Tandy – arXiv:1005.4610 [nucl-th].

d

γ5π–

u

u–

u–

u–

+

+

––

–

–

–

d

γ5

δm

π–

u–

(a)

(b)

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 3/30

First Contents Back Conclusion

QCD’s Challenges

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 4/30

First Contents Back Conclusion

QCD’s Challenges

Quark and Gluon Confinement

No matter how hard one strikes the proton, one

cannot liberate an individual quark or gluon

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 4/30

First Contents Back Conclusion

QCD’s Challenges

Quark and Gluon Confinement

No matter how hard one strikes the proton, one

cannot liberate an individual quark or gluon

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=−

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 4/30

First Contents Back Conclusion

QCD’s Challenges

Quark and Gluon Confinement

No matter how hard one strikes the proton, one

cannot liberate an individual quark or gluon

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=−

Neither of these phenomena is apparent in QCD’s

Lagrangian yet they are the dominant determining

characteristics of real-world QCD.

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 4/30

First Contents Back Conclusion

QCD’s ChallengesUnderstand Emergent Phenomena

Quark and Gluon Confinement

No matter how hard one strikes the proton, one

cannot liberate an individual quark or gluon

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=−

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 – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 4/30

First Contents Back Conclusion

Charting the Interactionbetween light-quarks

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 5/30

First Contents Back Conclusion

Charting the Interactionbetween light-quarks

Confinement can be related to the analytic properties of

QCD’s Schwinger functions

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 5/30

First Contents Back Conclusion

Charting the Interactionbetween light-quarks

Confinement can be related to the analytic properties of

QCD’s Schwinger functions

Question of light-quark confinement can be translated into

the challenge of charting the infrared behavior of QCD’s

universal β-function

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 5/30

First Contents Back Conclusion

Charting the Interactionbetween light-quarks

Confinement can be related to the analytic properties of

QCD’s Schwinger functions

Question of light-quark confinement can be translated into

the challenge of charting the infrared behavior of QCD’s

universal β-function

This function may depend on the scheme chosen to

renormalise the quantum field theory but it is unique

within a given scheme.

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 5/30

First Contents Back Conclusion

Charting the Interactionbetween light-quarks

Confinement can be related to the analytic properties of

QCD’s Schwinger functions

Question of light-quark confinement can be translated into

the challenge of charting the infrared behavior of QCD’s

universal β-function

This function may depend on the scheme chosen to

renormalise the quantum field theory but it is unique

within a given scheme.

Of course, the behaviour of the β-function on the

perturbative domain is well known.

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 5/30

First Contents Back Conclusion

Charting the Interactionbetween light-quarks

Confinement can be related to the analytic properties of

QCD’s Schwinger functions

Question of light-quark confinement can be translated into

the challenge of charting the infrared behavior of QCD’s

universal β-function

This function may depend on the scheme chosen to

renormalise the quantum field theory but it is unique

within a given scheme.

Of course, the behaviour of the β-function on the

perturbative domain is well known.

This is a well-posed problem whose solution is an elemental

goal of modern hadron physics.

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 5/30

First Contents Back Conclusion

What is the light-quarkLong-Range Potential?

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 6/30

First Contents Back Conclusion

What is the light-quarkLong-Range Potential?

Potential between static (infinitely heavy) quarksmeasured in simulations of lattice-QCD is not relatedin any known way to the light-quark interaction.

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 6/30

First Contents Back Conclusion

Charting the Interactionbetween light-quarks

Through QCD’s Dyson-Schwinger equations (DSEs) the

pointwise behaviour of the β-function determines pattern of

chiral symmetry breaking

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 7/30

First Contents Back Conclusion

Charting the Interactionbetween light-quarks

Through QCD’s Dyson-Schwinger equations (DSEs) the

pointwise behaviour of the β-function determines pattern of

chiral symmetry breaking

DSEs connect β-function to experimental observables.

Hence, comparison between computations and

observations of, e.g.,

hadron mass spectrum;

elastic and transition form factors

can be used to chart β-function’s long-range behaviour

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 7/30

First Contents Back Conclusion

Charting the Interactionbetween light-quarks

Through QCD’s Dyson-Schwinger equations (DSEs) the

pointwise behaviour of the β-function determines pattern of

chiral symmetry breaking

DSEs connect β-function to experimental observables.

Hence, comparison between computations and

observations of, e.g.,

hadron mass spectrum;

elastic and transition form factors

can be used to chart β-function’s long-range behaviour

E.g.: Extant studies of mesons show that the properties of

hadron excited states are a great deal more sensitive to the

long-range behaviour of β-function than those of the ground

stateCraig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 7/30

First Contents Back Conclusion

Charting the Interactionbetween light-quarks

Through DSEs the pointwise behaviour of the β-function

determines pattern of chiral symmetry breaking

DSEs connect β-function to experimental observables.

Hence, comparison between computations and

observations can be used to chart β-function’s long-range

behaviour

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 8/30

First Contents Back Conclusion

Charting the Interactionbetween light-quarks

Through DSEs the pointwise behaviour of the β-function

determines pattern of chiral symmetry breaking

DSEs connect β-function to experimental observables.

Hence, comparison between computations and

observations can be used to chart β-function’s long-range

behaviour

To realise this goal, a nonperturbative symmetry-preserving

DSE truncation is necessary

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 8/30

First Contents Back Conclusion

Charting the Interactionbetween light-quarks

Through DSEs the pointwise behaviour of the β-function

determines pattern of chiral symmetry breaking

DSEs connect β-function to experimental observables.

Hence, comparison between computations and

observations can be used to chart β-function’s long-range

behaviour

To realise this goal, a nonperturbative symmetry-preserving

DSE truncation is necessary

Steady quantitative progress is being made with a

scheme that is systematically improvable

(See nucl-th/9602012 and references thereto)

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 8/30

First Contents Back Conclusion

Charting the Interactionbetween light-quarks

Through DSEs the pointwise behaviour of the β-function

determines pattern of chiral symmetry breaking

DSEs connect β-function to experimental observables.

Hence, comparison between computations and

observations can be used to chart β-function’s long-range

behaviour

To realise this goal, a nonperturbative symmetry-preserving

DSE truncation is necessary

Steady quantitative progress is being made with a

scheme that is systematically improvable

(See nucl-th/9602012 and references thereto)

Enabled proof or exact results in QCD:

e.g., BRST – arXiv:1005.4610 [nucl-th]; and . . .Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 8/30

First Contents Back Conclusion

Radial Excitations& Chiral Symmetry

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 9/30

First Contents Back Conclusion

Radial Excitations& Chiral Symmetry(Maris, Roberts, Tandy

nu-th/9707003 )

fH m2H = − ρH

ζ MH

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 9/30

First Contents Back Conclusion

Radial Excitations& Chiral Symmetry(Maris, Roberts, Tandy

nu-th/9707003 )

fH m2H = − ρH

ζ MH

• Mass2 of pseudoscalar hadron

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 9/30

First Contents Back Conclusion

Radial Excitations& Chiral Symmetry(Maris, Roberts, Tandy

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)

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 9/30

First Contents Back Conclusion

Radial Excitations& Chiral Symmetry(Maris, Roberts, Tandy

nu-th/9707003 )

fH m2H = − ρH

ζ MH

fH pµ = Z2

∫ Λ

q

12tr

{

(

TH)tγ5γµ 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

=

−i〈0|qγ5γµq|π〉

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 9/30

First Contents Back Conclusion

Radial Excitations& Chiral Symmetry(Maris, Roberts, Tandy

nu-th/9707003 )

fH m2H = − ρH

ζ MH

iρHζ = Z4

∫ Λ

q

12tr

{

(

TH)tγ5 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|qγ5q|π〉

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 9/30

First Contents Back Conclusion

Radial Excitations& Chiral Symmetry(Maris, Roberts, Tandy

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

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 9/30

First Contents Back Conclusion

Radial Excitations& Chiral Symmetry(Maris, Roberts, Tandy

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√

mHand ρH

ζ ∝ √mH

Hence, mH ∝ mq

. . . QCD Proof of Potential Model resultCraig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 9/30

First Contents Back Conclusion

Radial Excitations& Chiral SymmetryHöll, Krassnigg, Roberts

nu-th/0406030

fH m2H = − ρH

ζ MH

Valid for ALL Pseudoscalar mesons

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 10/30

First Contents Back Conclusion

Radial Excitations& Chiral SymmetryHöll, Krassnigg, Roberts

nu-th/0406030

fH m2H = − ρH

ζ MH

Valid for ALL Pseudoscalar mesons

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

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 10/30

First Contents Back Conclusion

Radial Excitations& Chiral SymmetryHöll, Krassnigg, Roberts

nu-th/0406030

fH m2H = − ρH

ζ MH

Valid for ALL Pseudoscalar mesons

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

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

m2πn 6=0

> m2πn=0

= 0, in chiral limit

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 10/30

First Contents Back Conclusion

Radial Excitations& Chiral SymmetryHöll, Krassnigg, Roberts

nu-th/0406030

fH m2H = − ρH

ζ MH

Valid for ALL Pseudoscalar mesons

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

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

m2πn 6=0

> m2πn=0

= 0, in chiral limit

⇒ fH = 0

ALL pseudoscalar mesons except π(140) in chiral limit

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 10/30

First Contents Back Conclusion

Radial Excitations& Chiral SymmetryHöll, Krassnigg, Roberts

nu-th/0406030

fH m2H = − ρH

ζ MH

Valid for ALL Pseudoscalar mesons

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

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

m2πn 6=0

> m2πn=0

= 0, in chiral limit

⇒ fH = 0

ALL pseudoscalar mesons except π(140) in chiral limit

Dynamical Chiral Symmetry Breaking

– Goldstone’s Theorem –

impacts upon every pseudoscalar mesonCraig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 10/30

First Contents Back Conclusion

Radial Excitations& Lattice-QCDMcNeile and Michael

he-la/0607032

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 11/30

First Contents Back Conclusion

Radial Excitations& Lattice-QCDMcNeile and Michael

he-la/0607032

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

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 11/30

First Contents Back Conclusion

Radial Excitations& Lattice-QCDMcNeile and Michael

he-la/0607032

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

CLEO: τ → π(1300) + ντ

⇒ fπ1< 8.4 MeV

Diehl & Hillerhe-ph/0105194

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 11/30

First Contents Back Conclusion

Radial Excitations& Lattice-QCDMcNeile and Michael

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

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

CLEO: τ → π(1300) + ντ

⇒ fπ1< 8.4 MeV

Diehl & Hillerhe-ph/0105194

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

⇒ fπ1

fπ

= 0.078 (93)

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 11/30

First Contents Back Conclusion

Radial Excitations& Lattice-QCDMcNeile and Michael

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

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

CLEO: τ → π(1300) + ντ

⇒ fπ1< 8.4 MeV

Diehl & Hillerhe-ph/0105194

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

⇒ 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 – Impact of DCSB on meson structure and interactions

11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 11/30

First Contents Back Conclusion

Radial Excitations& Lattice-QCDMcNeile and Michael

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

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

CLEO: τ → π(1300) + ντ

⇒ fπ1< 8.4 MeV

Diehl & Hillerhe-ph/0105194

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

⇒ 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 – Impact of DCSB on meson structure and interactions

11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 11/30

First Contents Back Conclusion

Charting the Interactionbetween light-quarks

Through DSEs the pointwise behaviour of the β-function

determines pattern of chiral symmetry breaking

DSEs connect β-function to experimental observables.

Hence, comparison between computations and

observations can be used to chart β-function’s long-range

behaviour

To realise this goal, a nonperturbative symmetry-preserving

DSE truncation is necessary

Steady quantitative progress is being made with a

scheme that is systematically improvable

(See nucl-th/9602012 and references thereto)

Enabled proof or exact results in QCD,

as we’ve just seenCraig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 12/30

First Contents Back Conclusion

Charting the Interactionbetween light-quarks

Through DSEs the pointwise behaviour of the β-function

determines pattern of chiral symmetry breaking

DSEs connect β-function to experimental observables.

Hence, comparison between computations and

observations can be used to chart β-function’s long-range

behaviour

To realise this goal, a nonperturbative symmetry-preserving

DSE truncation is necessary

On other hand, at present significant qualitative

advances possible with symmetry-preserving kernel

Ansätze that express important additional

nonperturbative effects – M(p2) – difficult/impossible to

capture in any finite sum of contributionsCraig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 12/30

First Contents Back Conclusion

Gap EquationGeneral Form

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 13/30

First Contents Back Conclusion

Gap EquationGeneral Form

Σ=

D

γΓS

Sf (p)−1 = Z2 (iγ · p+mbmf ) + Σf (p) ,

Σf (p) = Z1

∫ Λ

qg2Dµν(p− q)

λa

2γµSf (q)

λa

2Γf

ν (q, p),

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 13/30

First Contents Back Conclusion

Gap EquationGeneral Form

Σ=

D

γΓS

Sf (p)−1 = Z2 (iγ · p+mbmf ) + Σf (p) ,

Σf (p) = Z1

∫ Λ

qg2Dµν(p− q)

λa

2γµSf (q)

λa

2Γf

ν (q, p),

Z1,2(ζ2,Λ2) are respectively the vertex and quark wave

function renormalisation constants, with ζ the

renormalisation point

mbm(Λ) is the Lagrangian current-quark bare mass

Dµν(k) is the dressed-gluon propagator

Γfν (q, p) is the dressed-quark-gluon vertex

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 13/30

First Contents Back Conclusion

Gap EquationGeneral Form

Σ=

D

γΓS

Sf (p)−1 = Z2 (iγ · p+mbmf ) + Σf (p) ,

Σf (p) = Z1

∫ Λ

qg2Dµν(p− q)

λa

2γµSf (q)

λa

2Γf

ν (q, p),

Z1,2(ζ2,Λ2) are respectively the vertex and quark wave

function renormalisation constants, with ζ the

renormalisation point

mbm(Λ) is the Lagrangian current-quark bare mass

Dµν(k) is the dressed-gluon propagator

Γfν (q, p) is the dressed-quark-gluon vertex

Suppose one has in-hand the exact form of Γfν (q, p)

What is the associatedSymmetry-preserving Bethe-Salpeter Kernel?

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 13/30

First Contents Back Conclusion

Bound-state DSE

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 14/30

First Contents Back Conclusion

Bound-state DSEBethe-Salpeter Equation

Standard form, familiar from textbooks

[

Γjπ(k;P )

]

tu=

∫ Λ

q

[S(q + P/2)Γjπ(q;P )S(q − P/2)]sr K

rstu (q, k;P )

K(q, k;P ): Fully-amputated, 2-particle-irreducible,quark-antiquark scattering kernel

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 14/30

First Contents Back Conclusion

Bound-state DSEBethe-Salpeter Equation

Standard form, familiar from textbooks

[

Γjπ(k;P )

]

tu=

∫ Λ

q

[S(q + P/2)Γjπ(q;P )S(q − P/2)]sr K

rstu (q, k;P )

K(q, k;P ): Fully-amputated, 2-particle-irreducible,quark-antiquark scattering kernel

Compact. Visually appealing. Correct.

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 14/30

First Contents Back Conclusion

Bound-state DSEBethe-Salpeter Equation

Standard form, familiar from textbooks

[

Γjπ(k;P )

]

tu=

∫ Λ

q

[S(q + P/2)Γjπ(q;P )S(q − P/2)]sr K

rstu (q, k;P )

K(q, k;P ): Fully-amputated, 2-particle-irreducible,quark-antiquark scattering kernel

Compact. Visually appealing. Correct.

Blocked progress for more than 60 years.

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 14/30

First Contents Back Conclusion

Bethe-Salpeter EquationGeneral FormL. Chang and C. D. Roberts

0903.5461 [nucl-th], Phys. Rev. Lett. 103 (2009) 081601

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 15/30

First Contents Back Conclusion

Bethe-Salpeter EquationGeneral FormL. Chang and C. D. Roberts

0903.5461 [nucl-th], Phys. Rev. Lett. 103 (2009) 081601

Equivalent exact form:

Γfg5µ(k;P ) = Z2γ5γµ

−

∫

qg2Dαβ(k − q)

λa

2γαSf (q+)Γfg

5µ(q;P )Sg(q−)λa

2Γg

β(q−, k−)

+

∫

qg2Dαβ(k − q)

λa

2γαSf (q+)

λa

2Λfg

5µβ(k, q;P ),

(Poincaré covariance, hence q± = q ± P/2, etc., without loss of generality.)

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 15/30

First Contents Back Conclusion

Bethe-Salpeter EquationGeneral FormL. Chang and C. D. Roberts

0903.5461 [nucl-th], Phys. Rev. Lett. 103 (2009) 081601

Equivalent exact form:

Γfg5µ(k;P ) = Z2γ5γµ

−

∫

qg2Dαβ(k − q)

λa

2γαSf (q+)Γfg

5µ(q;P )Sg(q−)λa

2Γg

β(q−, k−)

+

∫

qg2Dαβ(k − q)

λa

2γαSf (q+)

λa

2Λfg

5µβ(k, q;P ),

(Poincaré covariance, hence q± = q ± P/2, etc., without loss of generality.)

In this form . . . Λfg5µβ

is completely defined via the dressed-quark self-energy

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 15/30

First Contents Back Conclusion

Bethe-Salpeter KernelL. Chang and C. D. Roberts

0903.5461 [nucl-th], Phys. Rev. Lett. 103 (2009) 081601

Bethe-Salpeter equation introduced in 1951

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 16/30

First Contents Back Conclusion

Bethe-Salpeter Kernel60 year problemL. Chang and C. D. Roberts

0903.5461 [nucl-th], Phys. Rev. Lett. 103 (2009) 081601

Bethe-Salpeter equation introduced in 1951

Newly-derived Ward-Takahashi identity

PµΛfg5µβ(k, q;P ) = Γf

β(q+, k+) iγ5 + iγ5 Γgβ(q−, k−)

− i[mf (ζ) +mg(ζ)]Λfg5β(k, q;P ),

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 16/30

First Contents Back Conclusion

Bethe-Salpeter Kernel60 year problemL. Chang and C. D. Roberts

0903.5461 [nucl-th], Phys. Rev. Lett. 103 (2009) 081601

Bethe-Salpeter equation introduced in 1951

Newly-derived Ward-Takahashi identity

PµΛfg5µβ(k, q;P ) = Γf

β(q+, k+) iγ5 + iγ5 Γgβ(q−, k−)

− i[mf (ζ) +mg(ζ)]Λfg5β(k, q;P ),

For first time: can construct Ansatz for Bethe-Salpeter

kernel consistent with any reasonable quark-gluon vertex

Consistent means - all symmetries preserved!Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 16/30

First Contents Back Conclusion

Bethe-Salpeter Kernel60 year problemL. Chang and C. D. Roberts

0903.5461 [nucl-th], Phys. Rev. Lett. 103 (2009) 081601

Bethe-Salpeter equation introduced in 1951

Newly-derived Ward-Takahashi identity

PµΛfg5µβ(k, q;P ) = Γf

β(q+, k+) iγ5 + iγ5 Γgβ(q−, k−)

− i[mf (ζ) +mg(ζ)]Λfg5β(k, q;P ),

For first time: can construct Ansatz for Bethe-Salpeter

kernel consistent with any reasonable quark-gluon vertex

Procedure & results to expect . . .see arXiv:1003.5006 [nucl-th]

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 16/30

First Contents Back Conclusion

Mass Splittinga1 – ρ

exp.

mass a1 1230

mass ρ 775

mass-

splitting 455

Splitting known experimentally for more than 35 years.

Hitherto, no explanation.

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 17/30

First Contents Back Conclusion

Mass Splittinga1 – ρ

exp. rainbow- one-loop

ladder

mass a1 1230 759 885

mass ρ 775 644 764

mass-

splitting 455 115 121

Systematic, symmetry-preserving, Poincaré-covariant DSE

truncation scheme of nucl-th/9602012.

Never better than ∼ 14

of splitting.

Constructing kernel skeleton-diagram-by-diagram, DCSB

cannot be faithfully expressed: M(p2) is absent!

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 17/30

First Contents Back Conclusion

Mass Splittinga1 – ρ

exp. rainbow- one-loop Ball-Chiu

ladder consistent

mass a1 1230 759 885 1066

mass ρ 775 644 764 924

mass-

splitting 455 115 121 142

New nonperturbative, symmetry-preserving

Poincaré-covariant Bethe-Salpeter equation formulation of

arXiv:0903.5461 [nucl-th]

Ball-Chiu Ansatz for quark-gluon vertex

ΓBCµ (k, p) = . . . + (k + p)µ

B(k)−B(p)k2−p2

Some effects of DCSB built into vertex

Explains π – σ splitting but not this problemCraig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 17/30

First Contents Back Conclusion

Mass Splittinga1 – ρChang & Roberts arXiv:1003.5006 [nucl-th]

exp. rainbow- one-loop Ball-Chiu Ball-Chiu plus

ladder consistent anom. cm mom.

mass a1 1230 759 885 1066 1230

mass ρ 775 644 764 924 745

mass-

splitting 455 115 121 142 485

New nonperturbative, symmetry-preserving

Poincaré-covariant Bethe-Salpeter equation formulation of

arXiv:0903.5461 [nucl-th]

Ball-Chiu augmented by quark anomalous chromomagnetic

moment term: Γµ(k, p) = ΓBCµ + σµν(k − p)ν

B(k)−B(p)k2−p2

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 17/30

First Contents Back Conclusion

Mass Splittinga1 – ρChang & Roberts arXiv:1003.5006 [nucl-th]

exp. rainbow- one-loop Ball-Chiu Ball-Chiu plus

ladder consistent anom. cm mom.

mass a1 1230 759 885 1066 1230

mass ρ 775 644 764 924 745

mass-

splitting 455 115 121 142 485

New nonperturbative, symmetry-preserving

Poincaré-covariant Bethe-Salpeter equation formulation of

arXiv:0903.5461 [nucl-th]

DCSB is the answer. Subtle interplay between competing

effects, which can only now be explicated

Promise of first reliable prediction of light-quark meson

spectrum, including the so-called hybrid and exotic states.Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 17/30

First Contents Back Conclusion

Quark AnomalousMagnetic MomentsChang & Roberts, in progress

Massless fermion can’t possess an anomalous magnetic moment

Interaction term∫

d4x 12g ψ(x)σµνψ(x)Fµν(x)

explicitly breaks chiral symmetry

0 1 2 3 4 5 6-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

p/ME

aem

acm

ME=0.52 MeVaem(ME)=+0.228acm(ME)= -0.297

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 18/30

First Contents Back Conclusion

Quark AnomalousMagnetic MomentsChang & Roberts, in progress

Massless fermion can’t possess an anomalous magnetic moment

Interaction term∫

d4x 12g ψ(x)σµνψ(x)Fµν(x)

explicitly breaks chiral symmetry

However, DCSB can generate a large anomalouschromomagnetic moment even in chiral limit– This explains the a1-ρ mass-splitting

0 1 2 3 4 5 6-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

p/ME

aem

acm

ME=0.52 MeVaem(ME)=+0.228acm(ME)= -0.297

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 18/30

First Contents Back Conclusion

Quark AnomalousMagnetic MomentsChang & Roberts, in progress

Massless fermion can’t possess an anomalous magnetic moment

Interaction term∫

d4x 12g ψ(x)σµνψ(x)Fµν(x)

explicitly breaks chiral symmetry

New BSE formulation (arXiv:0903.5461 [nucl-th]) enablescomputation of dressed-quark electromagnetic moment givendressed-quark-gluon vertex with ACM-term

0 1 2 3 4 5 6-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

p/ME

aem

acm

ME=0.52 MeVaem(ME)=+0.228acm(ME)= -0.297

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 18/30

First Contents Back Conclusion

Quark AnomalousMagnetic MomentsChang & Roberts, in progress

Massless fermion can’t possess an anomalous magnetic moment

Interaction term∫

d4x 12g ψ(x)σµνψ(x)Fµν(x)

explicitly breaks chiral symmetry

New BSE formulation (arXiv:0903.5461 [nucl-th]) enablescomputation of dressed-quark electromagnetic moment givendressed-quark-gluon vertex with ACM-term

0 1 2 3 4 5 6-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

p/ME

aem

acm

ME=0.52 MeVaem(ME)=+0.228acm(ME)= -0.297

M(p2) ⇒ κ(p2)

Preliminary result forµ distributions

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 18/30

First Contents Back Conclusion

Quark AnomalousMagnetic MomentsChang & Roberts, in progress

Massless fermion can’t possess an anomalous magnetic moment

Interaction term∫

d4x 12g ψ(x)σµνψ(x)Fµν(x)

explicitly breaks chiral symmetry

New BSE formulation (arXiv:0903.5461 [nucl-th]) enablescomputation of dressed-quark electromagnetic moment givendressed-quark-gluon vertex with ACM-term

0 1 2 3 4 5 6-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

p/ME

aem

acm

ME=0.52 MeVaem(ME)=+0.228acm(ME)= -0.297

M(p2) ⇒ κ(p2)

Preliminary result forµ distributions

Cloët & RobertsEffect on hadron form factors?

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 18/30

First Contents Back Conclusion

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

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 19/30

First Contents Back Conclusion

Frontiers of Nuclear Science:Theoretical Advances

Σ=

D

γΓS

Gap Equation

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 19/30

First Contents Back Conclusion

Frontiers of Nuclear Science:Theoretical Advances

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

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 19/30

First Contents Back Conclusion

Frontiers of Nuclear Science:Theoretical Advances

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)30 MeV70 MeV

Lattice QCD confirms DSE prediction of DCSB

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.

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 19/30

First Contents Back Conclusion

Frontiers of Nuclear Science:Theoretical Advances

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)30 MeV70 MeV

for DSE-predictedHint of support in Lattice-QCD results

confinement signal

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.

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 19/30

First Contents Back Conclusion

Frontiers of Nuclear Science:Theoretical Advances

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)30 MeV70 MeV

for DSE-predictedHint of support in Lattice-QCD results

confinement signal

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 Elastic and Transition Form FactorsCraig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 19/30

First Contents Back Conclusion

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

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 20/30

First Contents Back Conclusion

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 )]

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 20/30

First Contents Back Conclusion

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 )]

• Dressed-quark Propagator: S(p) =1

iγ · pA(p2) +B(p2)

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 20/30

First Contents Back Conclusion

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 )]

• Dressed-quark Propagator: S(p) =1

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

⇒ fπEπ(k;P = 0) = B(p2)

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 20/30

First Contents Back Conclusion

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 )]

• Dressed-quark Propagator: S(p) =1

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

⇒ 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

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 20/30

First Contents Back Conclusion

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

Pseudovectorcomponentsnecessarilynonzero

• 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 )]

• Dressed-quark Propagator: S(p) =1

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

⇒ 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

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 20/30

First Contents Back Conclusion

GT for pion– QCD and F em

π (Q2)Maris, Robertsnucl-th/9804062

What does this mean for observables?

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 21/30

First Contents Back Conclusion

GT for pion– QCD and F em

π (Q2)Maris, Robertsnucl-th/9804062

What does this mean for observables?

0.0 5.0 10.0 15.0q

2 (GeV

2)

0.00

0.10

0.20

0.30

0.40

q2 Fπ(

q2 ) (G

eV2 )

Only Eπ −> 1/Q4

Including Fπ −> 1/Q2

0.19

0.12

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 21/30

First Contents Back Conclusion

GT for pion– QCD and F em

π (Q2)Maris, Robertsnucl-th/9804062

What does this mean for observables?

0.0 5.0 10.0 15.0q

2 (GeV

2)

0.00

0.10

0.20

0.30

0.40

q2 Fπ(

q2 ) (G

eV2 )

Only Eπ −> 1/Q4

Including Fπ −> 1/Q2

0.19

0.12

(

Q

2

)2

= 2GeV2

⇒ Q2 = 8GeV2

Pseudovector components dominate ultraviolet behaviour of

electromagnetic form factorCraig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 21/30

First Contents Back Conclusion

GT for pion– Contact Interaction

Guttierez, Bashir, Cloët, Roberts:arXiv:1002.1968 [nucl-th]

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 22/30

First Contents Back Conclusion

GT for pion– Contact Interaction

Guttierez, Bashir, Cloët, Roberts:arXiv:1002.1968 [nucl-th]

Bethe-Salpeter amplitude can’t depend on relative

momentum

⇒ General Form Γπ(P ) = γ5[iEπ(P ) +1

MQγ · PFπ(P )]

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 22/30

First Contents Back Conclusion

GT for pion– Contact Interaction

Guttierez, Bashir, Cloët, Roberts:arXiv:1002.1968 [nucl-th]

Bethe-Salpeter amplitude can’t depend on relative

momentum

⇒ General Form Γπ(P ) = γ5[iEπ(P ) +1

MQγ · PFπ(P )]

Solve chiral-limit gap and Bethe-Salpeter equations

P 2 = 0 : MQ = 0.40 , Eπ = 0.98 ,Fπ

MQ= 0.50

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 22/30

First Contents Back Conclusion

GT for pion– Contact Interaction

Guttierez, Bashir, Cloët, Roberts:arXiv:1002.1968 [nucl-th]

Bethe-Salpeter amplitude can’t depend on relative

momentum

⇒ General Form Γπ(P ) = γ5[iEπ(P ) +1

MQγ · PFπ(P )]

Solve chiral-limit gap and Bethe-Salpeter equations

P 2 = 0 : MQ = 0.40 , Eπ = 0.98 ,Fπ

MQ= 0.50

Origin of pseudovector component: Eπ drives Fπ

RHS Bethe-Salpeter equation:

γµS(k + P/2)iγ5EπS(k − P/2)γµ

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 22/30

First Contents Back Conclusion

GT for pion– Contact Interaction

Guttierez, Bashir, Cloët, Roberts:arXiv:1002.1968 [nucl-th]

Bethe-Salpeter amplitude can’t depend on relative

momentum

⇒ General Form Γπ(P ) = γ5[iEπ(P ) +1

MQγ · PFπ(P )]

Solve chiral-limit gap and Bethe-Salpeter equations

P 2 = 0 : MQ = 0.40 , Eπ = 0.98 ,Fπ

MQ= 0.50

Origin of pseudovector component: Eπ drives Fπ

RHS Bethe-Salpeter equation:

γµS(k + P/2)iγ5EπS(k − P/2)γµ

Has pseudovector component

∼ Eπ[σS(k+)σV (k−) + σS(k−)σV (k+)]

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 22/30

First Contents Back Conclusion

GT for pion– Contact Interaction

Guttierez, Bashir, Cloët, Roberts:arXiv:1002.1968 [nucl-th]

Bethe-Salpeter amplitude can’t depend on relative

momentum

⇒ General Form Γπ(P ) = γ5[iEπ(P ) +1

MQγ · PFπ(P )]

Solve chiral-limit gap and Bethe-Salpeter equations

P 2 = 0 : MQ = 0.40 , Eπ = 0.98 ,Fπ

MQ= 0.50

Origin of pseudovector component: Eπ drives Fπ

RHS Bethe-Salpeter equation:

γµS(k + P/2)iγ5EπS(k − P/2)γµ

Hence Fπ on LHS is forced to be nonzero

because Eπ on RHS is nonzero owing to DCSB

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 22/30

First Contents Back Conclusion

GT for pion– Contact Interaction

Guttierez, Bashir, Cloët, Roberts:arXiv:1002.1968 [nucl-th]

Bethe-Salpeter amplitude: General Form

Γπ(P ) = γ5[iEπ(P ) +1

MQγ · PFπ(P )]

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 23/30

First Contents Back Conclusion

GT for pion– Contact Interaction

Guttierez, Bashir, Cloët, Roberts:arXiv:1002.1968 [nucl-th]

Bethe-Salpeter amplitude: General Form

Γπ(P ) = γ5[iEπ(P ) +1

MQγ · PFπ(P )]

Asymptotic form of electromagnetic pion form factor

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 23/30

First Contents Back Conclusion

GT for pion– Contact Interaction

Guttierez, Bashir, Cloët, Roberts:arXiv:1002.1968 [nucl-th]

Bethe-Salpeter amplitude: General Form

Γπ(P ) = γ5[iEπ(P ) +1

MQγ · PFπ(P )]

Asymptotic form of electromagnetic pion form factor

E2π-term ⇒ F em

π E(Q2) ∼ M2

Q2, photon(Q)

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 23/30

First Contents Back Conclusion

GT for pion– Contact Interaction

Guttierez, Bashir, Cloët, Roberts:arXiv:1002.1968 [nucl-th]

Bethe-Salpeter amplitude: General Form

Γπ(P ) = γ5[iEπ(P ) +1

MQγ · PFπ(P )]

Asymptotic form of electromagnetic pion form factor

E2π-term ⇒ F em

π E(Q2) ∼ M2

Q2, photon(Q)

EπFπ-term.

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 23/30

First Contents Back Conclusion

GT for pion– Contact Interaction

Guttierez, Bashir, Cloët, Roberts:arXiv:1002.1968 [nucl-th]

Bethe-Salpeter amplitude: General Form

Γπ(P ) = γ5[iEπ(P ) +1

MQγ · PFπ(P )]

Asymptotic form of electromagnetic pion form factor

E2π-term ⇒ F em

π E(Q2) ∼ M2

Q2, photon(Q)

EπFπ-term. Breit Frame:

pion(P = (0, 0, −Q/2, iQ/2))

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 23/30

First Contents Back Conclusion

GT for pion– Contact Interaction

Guttierez, Bashir, Cloët, Roberts:arXiv:1002.1968 [nucl-th]

Bethe-Salpeter amplitude: General Form

Γπ(P ) = γ5[iEπ(P ) +1

MQγ · PFπ(P )]

Asymptotic form of electromagnetic pion form factor

E2π-term ⇒ F em

π E(Q2) ∼ M2

Q2, photon(Q)

EπFπ-term. Breit Frame:

pion(P = (0, 0, −Q/2, iQ/2))

F emπ EF (Q2) ∼ 2 Sγ · (P + Q)FπSγ4SEπ

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 23/30

First Contents Back Conclusion

GT for pion– Contact Interaction

Guttierez, Bashir, Cloët, Roberts:arXiv:1002.1968 [nucl-th]

Bethe-Salpeter amplitude: General Form

Γπ(P ) = γ5[iEπ(P ) +1

MQγ · PFπ(P )]

Asymptotic form of electromagnetic pion form factor

E2π-term ⇒ F em

π E(Q2) ∼ M2

Q2, photon(Q)

EπFπ-term. Breit Frame:

pion(P = (0, 0, −Q/2, iQ/2))

F emπ EF (Q2) ∼ 2 Sγ · (P + Q)FπSγ4SEπ

⇒ F emπ EF (Q2) ∝ Q2

M2Q

Fπ

Eπ× E2

π−term = constant!

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 23/30

First Contents Back Conclusion

GT for pion– Contact Interaction

Guttierez, Bashir, Cloët, Roberts:arXiv:1002.1968 [nucl-th]

Bethe-Salpeter amplitude: General Form

Γπ(P ) = γ5[iEπ(P ) +1

MQγ · PFπ(P )]

Asymptotic form of electromagnetic pion form factor

E2π-term ⇒ F em

π E(Q2) ∼ M2

Q2, photon(Q)

EπFπ-term. Breit Frame:

pion(P = (0, 0, −Q/2, iQ/2))

F emπ EF (Q2) ∼ 2 Sγ · (P + Q)FπSγ4SEπ

⇒ F emπ EF (Q2) ∝ Q2

M2Q

Fπ

Eπ× E2

π−term = constant!

This behaviour dominates for Q2 & M2Q

Eπ

Fπ> 0.8 GeV2

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 23/30

First Contents Back Conclusion

Computation : ElasticPion Form Factor

Guttierez, Bashir, Cloët, Roberts:arXiv:1002.1968 [nucl-th]

DSE prediction: M(p2); i.e., interaction1

|x − y|2

cf. M(p2) = Constant; i.e., interaction δ4(x − y)

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 24/30

First Contents Back Conclusion

Computation : ElasticPion Form Factor

Guttierez, Bashir, Cloët, Roberts:arXiv:1002.1968 [nucl-th]

DSE prediction: M(p2); i.e., interaction1

|x − y|2

cf. M(p2) = Constant; i.e., interaction δ4(x − y)

Single mass-scale parameterin both studies

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 24/30

First Contents Back Conclusion

Computation : ElasticPion Form Factor

Guttierez, Bashir, Cloët, Roberts:arXiv:1002.1968 [nucl-th]

DSE prediction: M(p2); i.e., interaction1

|x − y|2

cf. M(p2) = Constant; i.e., interaction δ4(x − y)

Single mass-scale parameterin both studies

Same predictions forQ2 = 0 properties

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 24/30

First Contents Back Conclusion

Computation : ElasticPion Form Factor

Guttierez, Bashir, Cloët, Roberts:arXiv:1002.1968 [nucl-th]

DSE prediction: M(p2); i.e., interaction1

|x − y|2

cf. M(p2) = Constant; i.e., interaction δ4(x − y)

0 2 4 6Q

2 [GeV

2]

0

0.1

0.2

0.3

0.4

0.5

Q2 F

π(Q2 )

[GeV

2 ]

Maris-Tandy-2000

VMD ρ pole

CERN ’80s

JLab, 2001

JLab at 12 GeV

pert. QCD

JLab, 2006b

JLab, 2006a

NJL Model

Single mass-scale parameterin both studies

Same predictions forQ2 = 0 properties

Disagreement > 20%for Q2 > M2

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 24/30

First Contents Back Conclusion

Pion’svalence distributionHolt & Roberts: arXiv:1002.4666 [nucl-th]

0.2 0.4 0.6 0.8 1.0x

0.1

0.2

0.3

0.4

0.5

0.6

x uVΠ

HxL

point

0.5 1. 1.5 2.

H L

Function

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 25/30

First Contents Back Conclusion

Pion’svalence distributionHolt & Roberts: arXiv:1002.4666 [nucl-th]

0.2 0.4 0.6 0.8 1.0x

0.1

0.2

0.3

0.4

0.5

0.6

x uVΠ

HxL

α(q2)

q2

q2

∼>M2D

∝

(

1

q2

)1+κ

point

0.5 1. 1.5 2.

H L

Function

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 25/30

First Contents Back Conclusion

Pion’svalence distributionHolt & Roberts: arXiv:1002.4666 [nucl-th]

0.2 0.4 0.6 0.8 1.0x

0.1

0.2

0.3

0.4

0.5

0.6

x uVΠ

HxL

α(q2)

q2

q2

∼>M2D

∝

(

1

q2

)1+κ

⇒ B(k2)k2

∼>M2D

∝

(

1

k2

)1+κ

point

0.5 1. 1.5 2.

H L

Function

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 25/30

First Contents Back Conclusion

Pion’svalence distributionHolt & Roberts: arXiv:1002.4666 [nucl-th]

0.2 0.4 0.6 0.8 1.0x

0.1

0.2

0.3

0.4

0.5

0.6

x uVΠ

HxL

α(q2)

q2

q2

∼>M2D

∝

(

1

q2

)1+κ

⇒ B(k2)k2

∼>M2D

∝

(

1

k2

)1+κ

⇒ qπv (x;Q0)

x∼1∝ (1 − x)2(1+κ) at Q0 ∼> MD .

point

0.5 1. 1.5 2.

H L

Function

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 25/30

First Contents Back Conclusion

Pion’svalence distributionHolt & Roberts: arXiv:1002.4666 [nucl-th]

0.2 0.4 0.6 0.8 1.0x

0.1

0.2

0.3

0.4

0.5

0.6

x uVΠ

HxL

α(q2)

q2

q2

∼>M2D

∝

(

1

q2

)1+κ

⇒ B(k2)k2

∼>M2D

∝

(

1

k2

)1+κ

⇒ qπv (x;Q0)

x∼1∝ (1 − x)2(1+κ) at Q0 ∼> MD .

MD ≈ 0.4 GeV

– p2-location of DCSB inflexion point in M(p2)DCSB inflexion point

0.5 1. 1.5 2.p

0.1

0.2

0.3

0.4

0.5MHpL

Dressed-quark Mass Function

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 25/30

First Contents Back Conclusion

Pion’svalence distributionHolt & Roberts: arXiv:1002.4666 [nucl-th]

0.2 0.4 0.6 0.8 1.0x

0.1

0.2

0.3

0.4

0.5

0.6

x uVΠ

HxL

α(q2)

q2

q2

∼>M2D

∝

(

1

q2

)1+κ

⇒ B(k2)k2

∼>M2D

∝

(

1

k2

)1+κ

⇒ qπv (x;Q0)

x∼1∝ (1 − x)2(1+κ) at Q0 ∼> MD .

MD ≈ 0.4 GeV

– p2-location of DCSB inflexion point in M(p2)DCSB inflexion point

0.5 1. 1.5 2.p

0.1

0.2

0.3

0.4

0.5MHpL

Dressed-quark Mass Function

κQCD = 0 ⇒ qπv (x; 1GeV2) ∝ (1 − x)2

DSE calculation shows this valid for Lx = {x|x > 0.86}.Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 25/30

First Contents Back Conclusion

Ratio – Kaon/Pionu-valence distribution

T. Nguyen: PhD Thesis (Kent State U.)Nguyen, Tandy, Bashir, Roberts, in progressHolt & Roberts: arXiv:1002.4666 [nucl-th]

0 0.2 0.4 0.6 0.8 1x

0

0.2

0.4

0.6

0.8

1

1.2

uk / u

π at q = 5 GeV (Full BSE)

uK

/ uπ at q

0 = 0.57 GeV (Full BSE)

uK

/ uπ at q

0 = 0.57 GeV (E

0, F

0)

uK

/ uπ at q = 5 GeV (E

0, F

0)

Evolved uk, u

π Pade fits at q

0 = 0.57 GeV to q = 5 GeVdata: Badier, et al. , Phys. Lett. B 93 (1980) 354

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 26/30

First Contents Back Conclusion

Ratio – Kaon/Pionu-valence distribution

T. Nguyen: PhD Thesis (Kent State U.)Nguyen, Tandy, Bashir, Roberts, in progressHolt & Roberts: arXiv:1002.4666 [nucl-th]

0 0.2 0.4 0.6 0.8 1x

0

0.2

0.4

0.6

0.8

1

1.2

uk / u

π at q = 5 GeV (Full BSE)

uK

/ uπ at q

0 = 0.57 GeV (Full BSE)

uK

/ uπ at q

0 = 0.57 GeV (E

0, F

0)

uK

/ uπ at q = 5 GeV (E

0, F

0)

Evolved uk, u

π Pade fits at q

0 = 0.57 GeV to q = 5 GeVdata: Badier, et al. , Phys. Lett. B 93 (1980) 354DSE–result obtained

using interaction that

predicted Fπ(Q2)

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 26/30

First Contents Back Conclusion

Ratio – Kaon/Pionu-valence distribution

T. Nguyen: PhD Thesis (Kent State U.)Nguyen, Tandy, Bashir, Roberts, in progressHolt & Roberts: arXiv:1002.4666 [nucl-th]

0 0.2 0.4 0.6 0.8 1x

0

0.2

0.4

0.6

0.8

1

1.2

uk / u

π at q = 5 GeV (Full BSE)

uK

/ uπ at q

0 = 0.57 GeV (Full BSE)

uK

/ uπ at q

0 = 0.57 GeV (E

0, F

0)

uK

/ uπ at q = 5 GeV (E

0, F

0)

Evolved uk, u

π Pade fits at q

0 = 0.57 GeV to q = 5 GeVdata: Badier, et al. , Phys. Lett. B 93 (1980) 354DSE–result obtained

using interaction that

predicted Fπ(Q2)

Influence of M(p2)

felt strongly for x > 0.5

QCD-M(p2) ⇒prediction:

uπ,K(x) ∝ (1 − x)2

at resolving-scale

Q0 = 0.6 GeV

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 26/30

First Contents Back Conclusion

Ratio – Kaon/Pionu-valence distribution

T. Nguyen: PhD Thesis (Kent State U.)Nguyen, Tandy, Bashir, Roberts, in progressHolt & Roberts: arXiv:1002.4666 [nucl-th]

0 0.2 0.4 0.6 0.8 1x

0

0.2

0.4

0.6

0.8

1

1.2

uk / u

π at q = 5 GeV (Full BSE)

uK

/ uπ at q

0 = 0.57 GeV (Full BSE)

uK

/ uπ at q

0 = 0.57 GeV (E

0, F

0)

uK

/ uπ at q = 5 GeV (E

0, F

0)

Evolved uk, u

π Pade fits at q

0 = 0.57 GeV to q = 5 GeVdata: Badier, et al. , Phys. Lett. B 93 (1980) 354DSE–result obtained

using interaction that

predicted Fπ(Q2)

Influence of M(p2)

felt strongly for x > 0.5

QCD-M(p2) ⇒prediction:

uπ,K(x) ∝ (1 − x)2

at resolving-scale

Q0 = 0.6 GeV

uπ,K(x = 1) invariant under DGLAP-evolution

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 26/30

First Contents Back Conclusion

Ratio – Kaon/Pionu-valence distribution

T. Nguyen: PhD Thesis (Kent State U.)Nguyen, Tandy, Bashir, Roberts, in progressHolt & Roberts: arXiv:1002.4666 [nucl-th]

0 0.2 0.4 0.6 0.8 1x

0

0.2

0.4

0.6

0.8

1

1.2

uk / u

π at q = 5 GeV (Full BSE)

uK

/ uπ at q

0 = 0.57 GeV (Full BSE)

uK

/ uπ at q

0 = 0.57 GeV (E

0, F

0)

uK

/ uπ at q = 5 GeV (E

0, F

0)

Evolved uk, u

π Pade fits at q

0 = 0.57 GeV to q = 5 GeVdata: Badier, et al. , Phys. Lett. B 93 (1980) 354DSE–result obtained

using interaction that

predicted Fπ(Q2)

Influence of M(p2)

felt strongly for x > 0.5

QCD-M(p2) ⇒prediction:

uπ,K(x) ∝ (1 − x)2

at resolving-scale

Q0 = 0.6 GeV

uπ,K(x = 1) invariant under DGLAP-evolution

Accessible at Upgraded JLab & Electron-Ion ColliderCraig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 26/30

First Contents Back Conclusion

Some projectscurrently underway

Elucidate effects of confinement & DCSB in

light-quark meson spectrum, including so-called hybrids and exotics, usingPoincaré-covariant symmetry-preserving Bethe-Salpeter equation(L. Chang, arXiv:0903.5461 [nucl-th])

hadron valence-quark distribution functions (A. Bashir, P.C. Tandy )

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 27/30

First Contents Back Conclusion

Some projectscurrently underway

Elucidate effects of confinement & DCSB in

light-quark meson spectrum, including so-called hybrids and exotics, usingPoincaré-covariant symmetry-preserving Bethe-Salpeter equation(L. Chang, arXiv:0903.5461 [nucl-th])

hadron valence-quark distribution functions (A. Bashir, P.C. Tandy )

Following extensive study of nucleon elastic electromagnetic form factors,Survey of nucleon electromagnetic form factorsI.C. Cloët et al., arXiv:0812.0416 [nucl-th], Few Body Syst. 46 (2009) pp. 1-36

with numerous predictions, either verified by experiment or serving to motivatenew experiments, studies are underway to elucidate signals of M(p2) inQ2-evolution of nucleon elastic and transition form factors; viz.,

N → ∆

N → P11(1440) e.g., Fd1(Q2) = 0 at Q2/M2 ≈ 5

κ(p2)

(M. Bhagwat, L. Chang, I. Cloët, H. Roberts)

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 27/30

First Contents Back Conclusion

Some projectscurrently underway

Elucidate effects of confinement & DCSB in

light-quark meson spectrum, including so-called hybrids and exotics, usingPoincaré-covariant symmetry-preserving Bethe-Salpeter equation(L. Chang, arXiv:0903.5461 [nucl-th])

hadron valence-quark distribution functions (A. Bashir, P.C. Tandy )

Following extensive study of nucleon elastic electromagnetic form factors,Survey of nucleon electromagnetic form factorsI.C. Cloët et al., arXiv:0812.0416 [nucl-th], Few Body Syst. 46 (2009) pp. 1-36

with numerous predictions, either verified by experiment or serving to motivatenew experiments, studies are underway to elucidate signals of M(p2) inQ2-evolution of nucleon elastic and transition form factors; viz.,

N → ∆

N → P11(1440) e.g., Fd1(Q2) = 0 at Q2/M2 ≈ 5

κ(p2)

(M. Bhagwat, L. Chang, I. Cloët, H. Roberts)

Incorporate “resonant contributions” (pion cloud) in kernels of bound-stateequations (e.g., Eichmann, Roberts et al. – 0802.1948 [nucl-th] & Cloët, Roberts –0811.2018 [nucl-th]; and Fischer, Williams – 0808.3372 [hep-ph])

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 27/30

First Contents Back Conclusion

Epilogue

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 28/30

First Contents Back Conclusion

EpilogueDCSB exists in QCD.

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 28/30

First Contents Back Conclusion

EpilogueDCSB exists in QCD.

It is manifest in dressed propagators and

vertices

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 28/30

First Contents Back Conclusion

EpilogueDCSB exists in QCD.

It is manifest in dressed propagators and

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: gπqq ≈ 4.3

pseudscalar mesons couple unnaturally

strongly to the lightest baryons

gπNN ≈ 12.8 ≈ 3gπqq

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 28/30

First Contents Back Conclusion

Epilogue

DCSB impacts dramatically upon observables

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 28/30

First Contents Back Conclusion

Epilogue

DCSB impacts dramatically upon observables

Spectrum; e.g., splittings: σ–π & a1–ρ

Elastic and Transition Form Factors

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 28/30

First Contents Back Conclusion

Epilogue

DCSB impacts dramatically upon observables

Spectrum; e.g., splittings: σ–π & a1–ρ

Elastic and Transition Form FactorsBut M(p2) is an essentially quantum field theoretical effect

Exposing & elucidating its effect in hadron physics requires

nonperturbative, symmetry preserving framework; i.e.,

Poincaré covariance, chiral and e.m. current conservation, etc.

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 28/30

First Contents Back Conclusion

Epilogue

DCSB impacts dramatically upon observables

Spectrum; e.g., splittings: σ–π & a1–ρ

Elastic and Transition Form FactorsBut M(p2) is an essentially quantum field theoretical effect

Exposing & elucidating its effect in hadron physics requires

nonperturbative, symmetry preserving framework; i.e.,

Poincaré covariance, chiral and e.m. current conservation, etc.

DSEs provide such a framework.

Studies underway will identify observable signals of M(p2),

the most important mass-generating mechanism for visible

matter in the Universe

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 28/30

First Contents Back Conclusion

Epilogue

DCSB impacts dramatically upon observables

Spectrum; e.g., splittings: σ–π & a1–ρ

Elastic and Transition Form FactorsBut M(p2) is an essentially quantum field theoretical effect

Exposing & elucidating its effect in hadron physics requires

nonperturbative, symmetry preserving framework; i.e.,

Poincaré covariance, chiral and e.m. current conservation, etc.

DSEs provide such a framework.

Studies underway will identify observable signals of M(p2),

the most important mass-generating mechanism for visible

matter in the Universe

DSEs: Tool enabling insight to be drawn from experiment

into long-range piece of interaction between light-quarksCraig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 28/30

First Contents Back Conclusion

Epilogue

Now is an exciting time . . .

Positioned to unify phenomena as apparently disparate as

Hadron spectrum

Elastic and transition form factors, from small- to large-Q2

Parton distribution functions

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 29/30

First Contents Back Conclusion

Epilogue

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

Now is an exciting time . . .

Positioned to unify phenomena as apparently disparate as

Hadron spectrum

Elastic and transition form factors, from small- to large-Q2

Parton distribution functions

Key: an understanding of both the fundamental origin of nuclear

mass and the far-reaching consequences of the mechanism

responsible; namely, Dynamical Chiral Symmetry BreakingCraig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 29/30

First Contents Back Conclusion

Contents

1. Universal Truths

2. QCD’s Challenges

3. Charting the Interaction

4. Radial Excitations & χ-Symmetry

5. Radial Excitations & χ-Symmetry II

6. Radial Excitations& Lattice-QCD

7. Bound-state DSE

8. BSE – General Form

9. a1 – ρ

10. Quark Anom. Mag. Moms.

11. Frontiers of Nuclear Science

12. Goldberger-Treiman for pion

13. GT – Contact Interaction

14. Computation: Fπ(Q2)

15. Pion’s valence distribution

16. Kaon/Pion u-valence distribution

17. Current Projects

Craig Roberts – Impact of DCSB on meson structure and interactions11th International Workshop on Mesons – Krakow, Poland, 10-15 June 2010 . . . 29 – p. 30/30