From quarks and gluons to baryon form factors...Form factors directly related to properties of pseudoscalar, vector, axial-vector quark-antiquark vertices: Baryon form factors inherit

From quarks and gluonsto baryon form factors

Gernot EichmannUniversity of Giessen, Germany

Erice, September 20, 2011

Gernot Eichmann (Giessen U.) 1 / 15

Hadron phenomenology:

QCD:

Mass spectrum

Dynamical chiral symmetry breakingConfinement

UA(1) anomalyInfrared structure of Green functions

Hadron deformationCharge and magnetization structureQuark and gluon distribution in hadrons,spin and OAM structure

Chiral properties from pion cloudTransition between perturbativeand non-perturbative regions Experiment

(JLab, MAMI, MIT-Bates, RHIC, CERN-SPS, FAIR, ...)

Lattice QCD & ChPT

Quark models

Bridges between perturbativeand non-perturbative QCD(GPDs/TMDs, ...)


Hadron phenomenology:

QCD:

Mass spectrum

Dynamical chiral symmetry breakingConfinement

UA(1) anomalyInfrared structure of Green functions

Hadron deformationCharge and magnetization structureQuark and gluon distribution in hadrons,spin and OAM structure

ab-initionon-perturbativecovariantcontinuumlight & heavy quarks

Chiral properties from pion cloudTransition between perturbativeand non-perturbative regions

Truncation of DSEsDyson-Schwinger equationsfor QCD’s Green functions

Covariant bound-state equationsfor hadron wave functions / amplitudes


Nucleon em. FFsGE, PRD 84 (2011)

Delta em. FFsD. Nicmorus, GE, R. Alkofer, PRD 82 (2010)

N∆γ (em. transition)GE & D. Nicmorus, in preparation

N∆π (ps. transition)V. Mader, GE, M. Blank, A. Krassnigg, PRD 84 (2011)

Nucleon axial & ps. FFsGE & C. S. Fischer, in preparation

Nucleon and Delta form factors:


Building blocks

quark propagator

hadron amplitudesand masses

offshellT-matrix

qqq, qq kernels

BSE & Faddeev equation

Scatteringequation

DSE

Truncation of DSEs

currents, form factors,hadron structure

“Gauging”of T-matrix

Ansatz

Haberzettl, PRC 56 (1997)Kvinikhidze, Blankleider, PRC 60 (1999)


++

Quark-quark correlationsassumed as dominant structure in baryons. Hints: lattice QCD, BSE, hadron spectrum, ...

=

Irreducible 3-bodydiagrams3-gluon coupling to each quark, ...

+

Three-body equation


++

Quark-quark correlationsassumed as dominant structure in baryons. Hints: lattice QCD, BSE, hadron spectrum, ...

Poincaré covariance

Dynamical chiral symmetry breaking

GE, Alkofer, Krassnigg, Nicmorus, PRL 104 (2010); GE, PRD 84 (2011)

Sanchis-Alepuz, GE, Villalba-Chávez, Alkofer, 1109.0199 [hep-ph]

Same setup for mesons and baryons: we need quark propagator & qq / qq kernel

Delta: 128 basis elements (s, p, d, f waves)

Faddeev equation

Nucleon: 64 basis elements (s, p, d waves)

orbital angular momentum in the bound-state amplitudes:

mass generation at quark & hadron level


Beyond rainbow-ladder:

( )2

( )2

Effective coupling is (the only!) model input

Rainbow-ladder truncation

in quark DSE

Satisfies Vector WTI ( e.m. current conservation) and Axial WTI ( Goldberger-Treiman, GMOR)

Pion cloud: chiral region, low- structure in FFs. Not included “Quark core”

Decay channels ( , ): so far only bound states“Non-resonant corrections”: cancel pion cloud in some channels ( , ?, ?), dominant in others (scalar, axialvector mesons)Fischer, Williams, PRL 103 (2009), Chang, Liu, Roberts, PRL 103 (2009)

in meson BSE &Faddeev equation

( )2

Infrared scale adjusted to , width kept as parameter

( ) = , + ( ²) 2 ²²

Maris, Tandy: PRC 60 (1999)

12

15

9

6

3

00 0.5 1 1.5 2

[ ]

( )

1.61.71.81.92.0


Mass results

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

0.10.0 0.2 0.3 0.4 0.5

[ ]

[ ]

Krassnigg, PRD 80 (2009)Alkofer, Fischer, Williams, EPJ A38 (2008)Fischer, Williams, PRL 103 (2009)Chang, Roberts, PRL 103 (2009)

Consistent descriptionof , , , ground states(dominated by s waves)

Only one input parameter:scale , adjusted to

scalar, axial-vector mesonsexcited mesons

– ´

What doesn’t work so well in rainbow-ladder:

GE, PRD 84 (2011)Nucleon mass:

Maris, Tandy, Nucl. Phys.Proc. Suppl. 161 (2006)

–meson mass:

Sanchis-Alepuz et al., 1109.0199 [hep-ph]

Delta mass:


Hadron current

2-quarkkernel

++

Impulse approximation

Coupling to 2-quark kernel


General expression for a baryon’s non-perturbative current: GE, PRD 84 (2011)

+

+ + +


Hadron current

=

Form factors directly related to properties of pseudoscalar, vector, axial-vector quark-antiquark vertices:

Baryon form factors inherit meson bound-state poles:“vector-meson dominance” for em. FFs

Vector WTI em. current conservation; AXWTI Goldberger-Treiman relation

gluonexchange kernel

++

Impulse approximation



General expression for a baryon’s non-perturbative current: GE, PRD 84 (2011)

+

+ + +(rainbow-ladder) (Faddeev truncation)



match dataat large , missing pion cloudbelow ~2

same shape up tostrange-quark:~30% p-wavesin amplitude

agree with latticeat large quark massand ChPT quark corein chiral region

Sachs FFsvs. momentumtransfer

, for ,

Passchier, Herberg, Zhu, Bermuth, Warren

Geis

Plaster/Madey

Riordan

Glazier

0

1

2

0 1 2 3

/

/

| ( )|Anderson

Lachniet

Anklin

Lung

Rock

Kubon

00

1

2

3

1 2 3

/

( )

( )

( )Crawford

Paolone

Zhan

Punjabi

Ron

0.0

1.0

0.4

0.2

0.6

0.8

0.00

0.10

0.04

0.02

0.06

0.08

0 1 2 3 0 1 2 3 64 5/



Electric proton form factor at large momenta:

, for ,

Difference likely due totwo-photon corrections

Rosenbluth method suggested / = const., in agreement

with perturbative scaling

Polarization experiments at JLAB showed falloff in / , with possible zero crossing

Faddeev result consistent with data, suggests zero crossing at larger photon momentum

OAM in nucleon amplitude!

Guichon, Vanderhaeghen, PRL 91 (2003)

Warren

Plaster/Madey

Riordan

Glazier

0.0

1.0

1.2

0.4

0.2

0.0

0.4

0.2

0.3

0.1

0.0

0.4

0.5

0.3

0.1

0.2

0.6

0.0

2.0

1.0

1.5

0.5

-0.2

0.6

0.8

0 2 4 6 8

0 2 4 6 8 0 2 4 6 8

0 2 4 6 8

/

/

/

/

Crawford

Paolone

Zhan

Gayou/Puckett

Puckett

Punjabi

Ron

/ /( )



Pauli-to-Dirac ratios at large momenta:

, for ,

Updated scaling prediction for Pauli-to-Dirac ratios:Belitsky, Ji, Yuan, PRL 91 (2003)

Faddeev result consistent with data. Perturbative behaviorbuilt in by gluon exchange, produces ~30% OAM in rest frame.

~ OAM

const. ~ ²( ²/ ²)

²

Logarithmic scaling implies zero crossing in / .

Warren

Plaster/Madey

Riordan

Glazier

0.0

0.4

0.2

0.3

0.1

0.0

2.0

1.0

1.5

0.5

0 2 4 6 8 0 2 4 6 8

/

/

/

/


Nucleon axial & ps. FFsGE & C. S. Fischer, in preparation , ,

~ axial-longitudinal vertex, pseudoscalar poles

~ pseudoscalar vertex, pseudoscalar poles

related by AXWTI

at =0:related by analyticity

~ axial-transverse vertex, axial-vector poles

(0) = (0)

Goldberger-Treimanaccurately reproducedfor all quark masses:

0.0 0.1 0.2 0.3 0.4 0.5

[ ]0.0 0.1 0.2 0.3 0.4 0.5

[ ]

0.8

1.0

1.2

1.4

0.0

1.0

0.2

0.4

0.6

0.8 [ ]

RBC/UKQCD (Yamazaki ‘09)

ETMC (Alexandrou ‘11)

QCDSF (Ali Khan ‘06)

LHPC (Bratt ‘10)


ETMC (Alexandrou ‘11)LHPC (Bratt ‘10)

Pion-cloud corrections?lattice: large box sizes


Nucleon axial & ps. FFsGE & C. S. Fischer, in preparation

ChPT: Procura et al., PRD 75 (2007)

, ,

~ axial-longitudinal vertex, pseudoscalar poles

~ pseudoscalar vertex, pseudoscalar poles

related by AXWTI

at =0:related by analyticity

~ axial-transverse vertex, axial-vector poles

(0) = (0)

Goldberger-Treimanaccurately reproducedfor all quark masses:

0.0 0.1 0.2 0.3 0.4 0.5

[ ]0.0 0.1 0.2 0.3 0.4 0.5

[ ]

0.8

1.0

1.2

1.4

0.0

1.0

0.2

0.4

0.6

0.8 [ ]


ETMC (Alexandrou ‘11)

QCDSF (Ali Khan ‘06)

LHPC (Bratt ‘10)


ETMC (Alexandrou ‘11)LHPC (Bratt ‘10)

Pion-cloud corrections?lattice: large box sizes



[ ]

[ ]

[%]

[%]

0.0

0

-2

-3

-4

-5

-1

0

-2

-4

-6

-8

-10

-12

-14

0.2 0.4 0.6 0.8 1.0 1.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2

MAMI (Beck ‘99)LEGS (Planpied ‘01)OOPS (Sparveris ‘05)MAMI (Stave ‘08)CLAS (Aznauryan ‘09)

OOPS (Sparveris ‘05)MAMI (Stave ‘08)

MAMI (Pospischil ‘00)

CLAS (Aznauryan ‘09)

, , *

*

& (electric & Coulomb quadrupole transitions) small & negative. Encode deformation. Quark model: need d-waves in or amplitude, or pion cloud.

pQCD scaling predictions: 1, const.Carlson, PRD 34 (1986)

(magnetic dipole transition) dominant: quark spin flip (s wave).

ChPT: strong chiral non-analyticities due to open decay channelPascalutsa, Vanderhaeghen, Phys. Rept. 437 (2007)

?

?Faddeev result (here: quark-diquark model)reproduces & even without pion cloud, and d-waves are typically small.



[%]

[ ]0.0 0.5 2.0 2.5

25

20

15

10

5

-5

0

1.51.0

s wavess + p wavesFull

[ ]0.0 0.5 2.0 2.51.51.0

0.08

0.06

0.04

0.02

0.00

-0.02

-0.04

( )

s waves

s + p waves

Full

*

, , *

dominated by p waves! Without OAM: small, becomes positive and grows ( pQCD!)

Poincaré covariance rich structure in and amplitudes, already in the quark-diquark model.

Non-zero OAM appears naturally (p, d, f waves). p waves much more important than d waves.

= 0

= 1

= 2

= 0

= 1

= 2

= 3

SC AX

= ¹ = ³


Nucleon and Delta ‘quark core’ dominated by quark-quark correlations: described by Faddeev equation; can be simplified to quark-diquark picture

Groundwork for systematic description of hadron properties in continuum QCD.Meson and baryon physics described by the same interaction.

Even in s-wave dominated ground-state baryons, Poincaré covarianceimplies orbital angular momentum (~ p waves) in their wave functions.

hadron deformation, perturbative behavior in form factors

Dynamical chiral symmetry breaking generates (quark & hadron) masses.

Missing structure at low momentum & in chiral region due to pion cloud.

Excited baryons, Tetraquarks

Summary

Outlook

Hadronic four-point functions: Compton scattering;pion electroproduction; GPDs & nucleon structure

Truncation beyondrainbow-ladder!!


Thanks for your attention.

Cheers to my collaborators:

R. Alkofer, M. Blank, C. S. Fischer, W. Heupel, A. Krassnigg, V. Mader, D. Nicmorus, H. Sanchis-Alepuz, S. Villalba-Chávez


IntroductionFaddeev equationForm FactorsSummary Outlook

Documents

From quarks and gluons to baryon form factors...Form factors directly related to properties of pseudoscalar, vector, axial-vector quark-antiquark vertices: Baryon form factors inherit