nucleon anddelta...

Nucleon and Delta Elastic and Transition Form Factors

Based on: - Phys. Rev. C88 (2013) 032201(R),- Few-Body Syst. 54 (2013) 1-33,

- Few-body Syst. 55 (2014) 1185-1222.

Jorge Segovia

Instituto Universitario de Fısica Fundamental y MatematicasUniversidad de Salamanca, Spain

Ian C. Cloet and Craig D. Roberts

Argonne National Laboratory, USA

Sebastian M. Schmidt

Institute for Advanced Simulation, Forschungszentrum Julich and JARA, Germany

Fachbereich Theoretische Physik / Institut fur Physik

Karl-Franzens-Universitat Graz

March 10th, 2015

Jorge Segovia ([email protected]) Nucleon and Delta Elastic and Transition Form Factors 1/37

The challenge of QCD

Quantum Chromodynamics is the only known example in nature of a nonperturvativefundamental quantum field theory

☞ QCD have profound implications for our understanding of the real-world:

Explain how quarks and gluons bind together to form hadrons.

Origin of the 98% of the mass in the visible universe.

☞ Given QCD’s complexity:

The best promise for progress is a strong interplay between experiment and theory.

Emergent phenomena

ւ ցQuark and gluon confinement Dynamical chiral symmetry breaking

↓ ↓Colored particles

have never been seenisolated

Hadrons do notfollow the chiralsymmetry pattern

Neither of these phenomena is apparent in QCD’s Lagrangianyet!

They play a dominant role determining characteristics of real-world QCD


Emergent phenomena: Confinement

Confinement is associated with dramatic, dynamically-driven changes in the analyticstructure of QCD’s propagators and vertices (QCD’s Schwinger functions)

☞ Dressed-propagator for a colored state:

An observable particle is associated with a poleat timelike-P2.

When the dressing interaction is confining:

Real-axis mass-pole splits, moving into a pairof complex conjugate singularities.

No mass-shell can be associated with a particlewhose propagator exhibits such singularity.

☞ Dressed-gluon propagator:

Confined gluon.

IR-massive but UV-massless.

mG ∼ 2− 4ΛQCD (ΛQCD ≃ 200MeV).

Any 2-point Schwinger function with an inflexionpoint at p2 > 0:→ Breaks the axiom of reflexion positivity→ No physical observable related with


Emergent phenomena: Dynamical chiral symmetry breaking

Spectrum of a theory invariant under chiral transformations should exhibitdegenerate parity doublets

π JP = 0− m = 140MeV cf. σ JP = 0+ m = 500MeV

ρ JP = 1− m = 775MeV cf. a1 JP = 1+ m = 1260MeV

N JP = 1/2+ m = 938MeV cf. N(1535) JP = 1/2− m = 1535MeV

Splittings between parity partners are greater than 100-times the light quark massscale: mu/md ∼ 0.5, md = 4MeV

☞ Dynamical chiral symmetry breaking

Mass generated from the interaction of quarks withthe gluon-medium.

Quarks acquire a HUGE constituent mass.

Responsible of the 98% of the mass of the proton.

☞ (Not) spontaneous chiral symmetry breaking

Higgs mechanism.

Quarks acquire a TINY current mass.

Responsible of the 2% of the mass of the proton. 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


Electromagnetic form factors of nucleon excited states

A central goal of Nuclear Physics: understand the structure and properties of protonsand neutrons, and ultimately atomic nuclei, in terms of the quarks and gluons of QCD.

Elastic and transition form factors

ւ ցUnique window into its

quark and gluon structureHigh-Q2 reach by

experiments

↓ ↓Distinctive information on theroles played by confinement

and DCSB in QCD

Probe the excited nucleonstructures at perturbative andnon-perturbative QCD scales

CEBAF Large Acceptance Spectrometer (CLAS)

☞ Most accurate results for the electroexcitation amplitudesof the four lowest excited states.

☞ They have been measured in a range of Q2 up to:

8.0GeV2 for ∆(1232)P33 and N(1535)S11 .

4.5GeV2 for N(1440)P11 and N(1520)D13 .

☞ The majority of new data was obtained at JLab.

Upgrade of CLAS up to 12GeV → CLAS12 (New generation experiments in 2015)


Theory tool: Dyson-Schwinger equations

Confinement and Dynamical Chiral Symmetry Breaking (DCSB) can be identified withproperties of dressed-quark and -gluon propagators and vertices (Schwinger functions)

Dyson-Schwinger equations (DSEs)

The quantum equations of motion of QCD whose solutionsare the Schwinger functions.

→ Propagators and vertices.

Generating tool for perturbation theory.

→ No model-dependence.

Nonperturbative tool for the study of continuum strongQCD.

→ Any model-dependence should be incorporated here.

Allows the study of the interaction between light quarks in the whole range ofmomenta.

→ Analysis of the infrared behaviour is crucial to disentangle confinement andDCSB.

Connect quark-quark interaction with experimental observables.

→ It is via the Q2 evolution of form factors that one gains access to the runningof QCD’s coupling and masses from the infrared into the ultraviolet.


The simplest example of DSEs: The gap equation

The quark propagator is given by the gap equation:

S−1(p) = Z2(iγ · p +mbm) + Σ(p)

Σ(p) = Z1

∫ Λ

qg2Dµν(p − q)

λa

2γµS(q)

λa

2Γν(q, p)

General solution:

S(p) =Z(p2)

iγ · p +M(p2)

Kernel involves:Dµν (p − q) - dressed gluon propagatorΓν(q, p) - dressed-quark-gluon vertex

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

M(p2) exhibits dynamicalmass generation

Each of which satisfies its own Dyson-Schwinger equation

↓Infinitely many coupled equations

↓Coupling between equations necessitates truncation


Ward-Takahashi identities (WTIs)

Symmetries should be preserved by any truncation

↓Highly nontrivial constraint → Failure implies loss of any connection with QCD

↓Symmetries in QCD are implemented by WTIs → Relate different Schwinger functions

For instance, axial-vector Ward-Takahashi identity:

These observations show that symmetries relate the kernel of the gap equation – aone-body problem – with that of the Bethe-Salpeter equation – a two-body problem –


Bethe-Salpeter and Faddeev equations

Hadrons are studied via Poincare covariant bound-state equations

☞ Mesons

A 2-body bound state problem in quantumfield theory.

Properties emerge from solutions ofBethe-Salpeter equation:

Γ(k;P) =

∫

d4q

(2π)4K(q, k;P)S(q+P) Γ(q;P) S(q)

The kernel is that of the gap equation.

=

iΓ

iS

iΓ

K

iS

☞ Baryons

A 3-body bound state problem in quantumfield theory.

Structure comes from solving the Faddeevequation.

Faddeev equation: Sums all possible quantumfield theoretical exchanges and interactionsthat can take place between the threedressed-quarks.

=aΨ

P

pq

pd Γb

Γ−a

pd

pq

bΨP

q


Diquarks inside baryons

The attractive nature of quark-antiquark correlations in a color-singlet meson is alsoattractive for 3c quark-quark correlations within a color-singlet baryon

☞ Diquark correlations:

Empirical evidence in support of strong diquarkcorrelations inside the nucleon.

A dynamical prediction of Faddeev equationstudies.

In our approach: Non-pointlike color-antitripletand fully interacting. Thanks to G. Eichmann.

Diquark composition of the Nucleon (N) and Delta (∆)

Positive parity states

ւ ցpseudoscalar and vector diquarks scalar and axial-vector diquarks

↓ ↓Ignored

wrong paritylarger mass-scales

Dominantright parity

shorter mass-scales

→ N ⇒ 0+, 1+ diquarks∆ ⇒ only 1+ diquark


Baryon-photon vertex (in the quark-diquark picture)

One must specify how the photon couples to the constituents within the hadrons:

Nature of the coupling of the photon to the quark.

Nature of the coupling of the photon to the diquark.

Six contributions to the current

1 Coupling of the photon to thedressed quark.

2 Coupling of the photon to thedressed diquark:

➫ Elastic transition.

➫ Induced transition.

3 Exchange and seagull terms.

Ingredients in the contributions

1 Ψi,f ≡ Faddeev amplitudes.

2 Single line ≡ Quark prop.

3 Double line ≡ Diquark prop.

4 Γ ≡ Diquark BS amplitudes.

5 Xµ ≡ Seagull vertices.

One-loop diagrams

i

iΨ ΨPf

f

P

Q

i

iΨ ΨPf

f

P

Q

scalaraxial vector

i

iΨ ΨPf

f

P

Q

Two-loop diagrams

i

iΨ ΨPPf

f

Q

Γ−

Γ

µ

i

i

X

Ψ ΨPf

f

Q

P Γ−

µi

i

X−

Ψ ΨPf

f

P

Q

ΓJorge Segovia ([email protected]) Nucleon and Delta Elastic and Transition Form Factors 11/37

Contact-interaction (CI) framework

Symmetry preserving Dyson-Schwinger equation treatment of a vector ⊗ vectorcontact interaction

☞ Gluon propagator: Contact interaction.

g2Dµν(p − q) = δµν4παIR

m2G

☞ Truncation scheme: Rainbow-ladder.

Γaν(q, p) = (λa/2)γν

☞ Quark propagator: Gap equation.

S−1(p) = iγ · p +m+ Σ(p)

= iγ · p +M

M ∼ 0.4GeV. Implies momentumindependent constituent quark mass.

Implies momentum independent BSAs.

☞ Baryons: Faddeev equation.

mN = 1.14GeV m∆ = 1.39GeV

(masses reduced by meson-cloud effects)

☞ Form Factors: Two-loop diagrams notincorporated.

Exchange diagram

It is zero because our treatment of thecontact interaction model

i

iΨ ΨPPf

f

Q

Γ−

Γ

Seagull diagrams

They are zero

µ

i

i

X

Ψ ΨPf

f

Q

P Γ−

µi

i

X−

Ψ ΨPf

f

P

Q

Γ


Series of papers establishes its strengths and limitations

CI framework has judiciously been applied to a large body of hadron phenomena.Produces results in qualitative agreement with those obtained using

most-sophisticated interactions.

1 Features and flaws of a contact interaction treatment of the kaonC. Chen, L. Chang, C.D. Roberts, S.M. Schmidt S. Wan and D.J. WilsonPhys. Rev. C 87 045207 (2013). arXiv:1212.2212 [nucl-th]

2 Spectrum of hadrons with strangenessC. Chen, L. Chang, C.D. Roberts, S. Wan and D.J. WilsonFew Body Syst. 53 293-326 (2012). arXiv:1204.2553 [nucl-th]

3 Nucleon and Roper electromagnetic elastic and transition form factorsD.J. Wilson, I.C. Cloet, L. Chang and C.D. RobertsPhys. Rev. C 85, 025205 (2012). arXiv:1112.2212 [nucl-th]

4 π- and ρ-mesons, and their diquark partners, from a contact interactionH.L.L. Roberts, A. Bashir, L.X. Gutierrez-Guerrero, C.D. Roberts and D.J. WilsonPhys. Rev. C 83, 065206 (2011). arXiv:1102.4376 [nucl-th]

5 Masses of ground and excited-state hadronsH.L.L. Roberts, L. Chang, I.C. Cloet and C.D. RobertsFew Body Syst. 51, 1-25 (2011). arXiv:1101.4244 [nucl-th]

6 Abelian anomaly and neutral pion productionH.L.L. Roberts, C.D. Roberts, A. Bashir, L.X. Gutierrez-Guerrero and P.C. TandyPhys. Rev. C 82, 065202 (2010). arXiv:1009.0067 [nucl-th]


Weakness of the contact-interaction framework

A truncation which produces Faddeev amplitudes that are independent of relativemomentum:

Underestimates the quark orbital angular momentum content of the bound-state.

Eliminates two-loop diagram contributions in the EM currents.

Produces hard form factors.

Momentum dependence in the gluon propagator

↓QCD-based framework

↓Contrasting the results obtained for the same observablesone can expose those quantities which are most sensitiveto the momentum dependence of elementary quantities

in QCD.


QCD-kindred framework

☞ Gluon propagator: 1/k2-behaviour.

☞ Truncation scheme: Rainbow-ladder.

Γaν(q, p) = (λa/2)γν

☞ Quark propagator: Gap equation.

S−1(p) = Z2(iγ · p +mbm) + Σ(p)

=[

1/Z(p2)] [

iγ · p +M(p2)]

M(p2 = 0) ∼ 0.33GeV. Implies momentumdependent constituent quark mass.

Implies momentum dependent BSAs.

☞ Baryons: Faddeev equation.

mN = 1.18GeV m∆ = 1.33GeV

(masses reduced by meson-cloud effects)

☞ Form Factors: Two-loop diagramsincorporated.

Exchange diagram

Play an important role

i

iΨ ΨPPf

f

Q

Γ−

Γ

Seagull diagrams

They are less important

µ

i

i

X

Ψ ΨPf

f

Q

P Γ−

µi

i

X−

Ψ ΨPf

f

P

Q

ΓJorge Segovia ([email protected]) Nucleon and Delta Elastic and Transition Form Factors 15/37

The Nucleon’s electromagnetic current

☞ The electromagnetic current can be generally written as:

Jµ(K ,Q) = ie Λ+(Pf ) Γµ(K ,Q) Λ+(Pi )

Incoming/outgoing nucleon momenta: P2i = P2

f = −m2N .

Photon momentum: Q = Pf − Pi , and total momentum: K = (Pi + Pf )/2.

The on-shell structure is ensured by the Nucleon projection operators.

☞ Vertex decomposes in terms of two form factors:

Γµ(K ,Q) = γµF1(Q2) +

1

2mNσµνQνF2(Q

2)

☞ The electric and magnetic (Sachs) form factors are a linear combination of theDirac and Pauli form factors:

GE (Q2) = F1(Q

2)− Q2

4m2N

F2(Q2)

GM(Q2) = F1(Q2) + F2(Q

2)

☞ They are obtained by any two sensible projection operators. Physical interpretation:

GE → Momentum space distribution of nucleon’s charge.

GM → Momentum space distribution of nucleon’s magnetization.


Electromagnetic form factors of the Nucleon (Revisited)

☞ Bibliography:Current quark mass dependence of nucleon magnetic moments and radii

I.C. Cloet et al., Few-Body Syst. 42 (2008) 91-113. arXiv:0804.3118 [nucl-th].

Survey of nucleon electromagnetic form factors

I.C. Cloet et al., Few-Body Syst. 46 (2012) 1-36. arXiv:0812.0416 [nucl-th].

Revealing dressed-quarks via the proton’s charge distribution

I.C. Cloet et al., Phys. Rev. Lett. 111 (2013) 101803. arXiv:1304.0855 [nucl-th].

☞ Modifications:

Photon-diquark vertices revisited in order to accomodate ∆γ∆ and Nγ∆:

Photon–axial-axial diquark transition.

Photon–axial-scalar diquark transition.

Anomalous magnetic moment of the dressed quark.

☞ Results:

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4

GE

Q2 (GeV2)

(a) CloetSegovia

Segovia-AMM

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3 3.5 4

GM

Q2 (GeV2)

(b) CloetSegovia

Segovia-AMM


Sachs electric and magnetic form factors (I)

☞ Q2 dependence of proton form factors:

0 1 2 3 4

0.0

0.5

1.0

x=Q2�mN

2

GEp

0 1 2 3 40.0

1.0

2.0

3.0

x=Q2�mN

2

GMp

☞ Proton static properties:

r2E r2EM2N r2M r2MM2

N µ

Theory (0.61 fm)2 (4.31)2 (0.53 fm)2 (3.16)2 2.50Experiment (0.88 fm)2 (4.19)2 (0.84 fm)2 (4.00)2 2.79

☞ Observations:

The QCD-kindred results are in fair agreement with experiment.

The contact-interaction results are typically too hard ⇔ Qualitative agreenent.


Sachs electric and magnetic form factors (II)

☞ Q2 dependence of neutron form factors:

0 1 2 3 40.00

0.04

0.08

x=Q2�mN

2

GEn

0 1 2 3 4

0.0

1.0

2.0

x=Q2�mN

2

GMn

☞ Neutron static properties:

r2E r2EM2N r2M r2MM2

N µ

Theory −(0.28 fm)2 −(1.99)2 (0.70 fm)2 (4.95)2 −1.83Experiment −(0.34 fm)2 −(1.62)2 (0.89 fm)2 (4.24)2 −1.91

☞ Observations:

The QCD-kindred results are in fair agreement with experiment ⇔ GnE is small

and hence is much affected by subdominant effects that we have neglected.

The defects of a contact-interaction are expressed with greatest force in theneutron electric form factor.


Unit-normalised ratio of Sachs electric and magnetic form factors

Both CI and QCD-kindred frameworks predict a zero crossing in µpGpE/G

pM

ææ

æ

æ

ææ æ

æ æà

à

à

à

òò

ò

ò òòòò

ò ò

÷

÷

÷

ì

ì

ì

0 2 4 6 8 100.0

0.5

1.0

Q2@GeV2

D

ΜpG

Ep�G

Mp

æ

æ

æ

à

à

à

0 2 4 6 8 10 120.0

0.2

0.4

0.6

Q2@GeV2

D

ΜnG

En�G

Mn

The possible existence and location of the zero in µpGpE/G

pM is a fairly direct measure

of the nature of the quark-quark interaction

0

0.1

0.2

0.3

0.4

M(p)

(GeV)

0 1 2 3 4

p (GeV)

α = 2.0

α = 1.8

α = 1.4

α = 1.0

0 2 4 6 8 10 120.0

0.2

0.4

0.6

0.8

1.0

Q2@GeV2

D

ΜN

GEN�G

MN


Flavour separation

The presence of strong diquark correlations within the nucleon is sufficient to explainthe empirically verified behaviour of the flavour-separated form factors.

æææææææ

ææ

ææ

æ

æ

ààààààà àà

àà à à

0 1 2 3 4 5 6 7

0.0

1.0

2.0

x=Q2�MN

2

x2F

1p

d,x

2F

1p

u

æææææææææ

æ

æ

ææ

0 1 2 3 40.0

0.2

0.4

0.6

x=Q2�MN

2

x2F

2p

u�Κ

puà

àààààà

àà

àà à

à

0 1 2 3 40.0

0.2

0.4

0.6

x=Q2�MN

2

x2F

2p

d�Κ

pd

☞ Observations:

F d1p is suppressed respect F u

1p at large Q2.Valence content of the proton = uud → Imagine only scalar diquarks: u[ud]0+ .d-quark contribution comes from only photon-diquark diagram.u-quark contribution comes from photon-quark and photon-diquark diagrams.

The location of the zero in F d1p depends on the relative probability of finding 1+

and 0+ diquarks in the proton.

One would expect same behaviour in F2 → more contributions playing animportant role like anomalous magnetic moment → They are not the key toexplaining the data’s basic features.


The ∆’s electromagnetic current

The small-Q2 behaviour of the ∆ elastic form factors is a necessary element in

computing the γ∗N → ∆ transition form factors


Jµ,λω(K ,Q) = Λ+(Pf )Rλα(Pf ) Γµ,αβ(K ,Q) Λ+(Pi )Rβω(Pi )

☞ Vertex decomposes in terms of four form factors:

Γµ,αβ(K ,Q) =

[

(F∗

1 + F∗

2 )iγµ − F∗

2

m∆Kµ

]

δαβ −[

(F∗

3 + F∗

4 )iγµ − F∗

4

m∆Kµ

]

QαQβ

4m2∆

☞ The multipole form factors:

GE0(Q2), GM1(Q

2), GE2(Q2), GM3(Q

2),

are functions of F∗

1 , F∗

2 , F∗

3 and F∗

4 .

☞ They are obtained by any four sensible projection operators. Physical interpretation:

GE0 and GM1 → Momentum space distribution of ∆’s charge and magnetization.

GE2 and GM3 → Shape deformation of the ∆-baryon.


No experimental data

☞ Since one must deal with the very short ∆-lifetime (τ∆ ∼ 10−16τπ+ ):

Little is experimentally known about the elastic form factors.

Lattice-regularised QCD results are usually used as a guide.

☞ Lattice-regularised QCD produce ∆-resonance masses that are very large:

Approach mπ mρ m∆

Unquenched I 0.691 0.986 1.687Unquenched II 0.509 0.899 1.559Unquenched III 0.384 0.848 1.395Hybrid 0.353 0.959 1.533Quenched I 0.563 0.873 1.470Quenched II 0.490 0.835 1.425Quenched III 0.411 0.817 1.382

☞ We artificially inflate the mass of the ∆-baryon (right panels on the next...)

Black-solid line: DSEs + QCD-kindred interaction (with/without m∆ ∼ 1.7GeV)

Blue-dashed line: DSEs + contact interaction (with/without m∆ ∼ 1.7GeV)


GE0 and GM1 with(out) an inflated quark core mass

ò

ò

ò

òòò

òòò ò

òò ò ò

æ

æ

ææ

æ æ

æ

æ æ æææ æ æ

à

à

à

à

àà à

à à à à à àà

ì

ì

ì

ì

ìì

ììììì

ìì

ìììììììì

0 1 2

0.0

0.5

1.0

x=Q2�mD

2

GE

0

ò

ò

ò

òòò

òòò ò

òò ò ò

æ

æ

ææ

æ æ

æ

æ æ æææ æ æ

à

à

à

à

àà à

à à à à à àà

ì

ì

ì

ì

ìì

ììììì

ìì

ìììììììì

0 1 2

0.0

0.5

1.0

x=Q2�mD

2

GE

0ò

ò

ò

òòò

òòò ò ò ò ò

ò

æ

æ

æ

ææ æ

ææ æ æ æ æ æ æ

à

àà

à

à à

à à à àà à à à

ì

ì

ìì

ìì

ìì

ììì

ìì

ìììììììì

0 1 20.0

1.0

2.0

3.0

4.0

x=Q2�mD

2

GM

1

ò

ò

ò

òòò

òòò ò ò ò ò

ò

æ

æ

æ

ææ æ

ææ æ æ æ æ æ æ

à

àà

à

à à

à à à àà à à à

ì

ì

ìì

ìì

ìì

ììì

ìì

ìììììììì

0 1 20.0

1.0

2.0

3.0

4.0

x=Q2�mD

2

GM

1


GE2 and GM3 with(out) an inflated quark core mass

òò ò ò ò ò ò ò ò

ò ò

æ ææ æ

ææ æ æ æ æ

æ

à à à àà

à

àà

àà à

ì

ì ìì

ìì

ì

ìì

ì

ììì

ì

ìì

ì

ì

0 1 2

-6

-4

-2

0

x=Q2�mD

2

GE

2òò ò ò ò ò ò ò ò

ò ò

æ ææ æ

ææ æ æ æ æ

æ

à à à àà

à

àà

àà à

ì

ì ìì

ìì

ì

ìì

ì

ììì

ì

ìì

ì

ì

0 1 2

-6

-4

-2

0

x=Q2�mD

2

GE

2ò

ò ò

òòòòòòòò

æ

ææ

æ ææ

æææ

ææà

à

à

à àà

ààà

à

à

0 1 2

-3

-2

-1

0

1

x=Q2�mD

2

GM

3

ò

ò ò

òòòòòòòò

æ

ææ

æ ææ

æææ

ææà

à

à

à àà

ààà

à

à

0 1 2

-3

-2

-1

0

1

x=Q2�mD

2

GM

3


Static electromagnetic properties of the ∆(1232)+

Approach GM1(0) GE2(0) GM3(0)DSE-QCD-kindred +2.86 −6.67 −3.00DSE-contact +3.83 −2.82 +0.80

Exp +3.6+1.3−1.7 ± 2.0± 4 - -

The. Average 3.5± 0.7 −2.5± 1.9 No consensus

Lattice-QCD (hybrid) +3.0± 0.2 −2.06+1.27−2.35 0.00

1/Nc + N → ∆ - −1.87± 0.08 -Faddeev equation +2.38 −0.67 > 0Covariant χPT+qLQCD +3.74± 0.03 −0.9± 0.6 −0.9± 2.1QCD-SR +4.2± 1.1 −0.6± 0.2 −0.7± 0.2χQSM +3.1 −2.0 -General Param. Method - −4.4 −2.6QM+exchange-currents +4.6 −4.6 -1/Nc +ms -expansion +3.8± 0.3 - -RQM +3.1 - -HBχPT +2.8± 0.3 −1.2± 0.8 -nrCQM +3.6 −1.8 -


The γ∗N → ∆ reaction

Two ways in order to analyze the structure of the ∆-resonances

ւ ցπ-mesons as a probe photons as a probe

↓ ↓complex relatively simple

BUT: B(∆ → γN) . 1%

This became possible with the advent of intense, energetic electron-beam facilities

Reliable data on the γ∗p → ∆+ transition:

☞ Available on the entire domain 0 ≤ Q2 . 8GeV2.

Isospin symmetry implies γ∗n → ∆0 is simply related with γ∗p → ∆+.

γ∗p → ∆+ data has stimulated a great deal of theoretical analysis:

Deformation of hadrons.

The relevance of pQCD to processes involving moderate momentum transfers.

The role that experiments on resonance electroproduction can play in exposingnon-perturbative phenomena in QCD:

☞ The nature of confinement and Dynamical Chiral Symmetry Breaking.


The γ∗N → ∆ transtion current (I)


Jµλ(K ,Q) = Λ+(Pf )Rλα(Pf ) iγ5 Γαµ(K ,Q) Λ+(Pi )

Incoming nucleon: P2i = −m2

N , and outgoing delta: P2f = −m2

∆.

Photon momentum: Q = Pf − Pi , and total momentum: K = (Pi + Pf )/2.

The on-shell structure is ensured by the N- and ∆-baryon projection operators.

☞ The composition of the 4-point function Γαµ is determined by Poincare covariance:

Convenient to work with orthogonal momenta ↔ Simplify its structure considerably

↓Not yet the case for K and Q ↔ ∆(m∆ −mN) 6= 0 ⇒ K · Q 6= 0

↓We take instead K⊥

µ = TQµν Kν and Q

☞ Vertex decomposes in terms of three (Jones-Scadron) form factors:

Γαµ = k

[

λm

2λ+(G∗

M − G∗

E )γ5εαµγδK⊥γ Qδ − G∗

ETQαγT

Kγµ − iς

λmG∗

C QαK⊥µ

]

,

Magnetic dipole ⇒ G∗

M Electric quadrupole ⇒ G∗

E Coulomb quadrupole ⇒ G∗

C


The γ∗N → ∆ transtion current (II)

The Jones-Scadron form factors are:

G ∗M = 3(s2 + s1),

G ∗E = s2 − s1,

G ∗C = s3.

G∗M,Ash

vs G∗M,J−S

G∗M,Ash = G∗

M,J−S

(

1 +Q2

(m∆ +mN)2

)− 12

The scalars are obtained from the following Dirac traces and momentum contractions:

s1 = n

√

ς(1 + 2d )

d − ςTKµν K

⊥

λ Tr[γ5Jµλγν ],

s2 = nλ+

λm

TKµλTr[γ5Jµλ],

s3 = 3nλ+

λm

(1 + 2d )

d − ςK⊥

µ K⊥

λ Tr[γ5Jµλ].

We have used the following notation:

n =

√

1 − 4d 2

4ik λm

, λ± =(m∆±mN )2+Q2

2(m2∆+m2

N)

, ς =Q2

2(m2∆ + m2

N),

d =m2

∆ − m2N

2(m2∆ + m2

N ), λm =

√

λ+λ−, k =

√

3

2

(

1 +m∆

mN

)

.

G. Eichmann et al., Phys. Rev. D85 (2012) 093004


Experimental results and theoretical expectations

I.G. Aznauryan and V.D. Burkert Prog. Part. Nucl Phys. 67 (2012) 1-54

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

10-1

1Q2 (GeV2)

G* M

,Ash

/3G

D

-7-6-5-4-3-2-1012

RE

M (

%)

-35

-30

-25

-20

-15

-10

-5

0

10-1

1Q2 (GeV2)

RS

M (

%)

☞ The REM ratio is measured to beminus a few percent.

☞ The RSM ratio does not seem to

settle to a constant at large Q2.

SU(6) predictions

〈p|µ|∆+〉 = 〈n|µ|∆0〉〈p|µ|∆+〉 = −

√2 〈n|µ|n〉

CQM predictions

(Without quark orbitalangular momentum)

REM → 0.

RSM → 0.

pQCD predictions

(For Q2 → ∞)

G∗

M → 1/Q4.

REM → +100%.

RSM → constant.

Experimental data do not support theoretical predictions


Q2-behaviour of G ∗M,Jones−Scadron

(I)

Transition cf. elastic magnetic form factors

0 2 40.8

0.9

1.0

1.1

x=Q2�mD

2

HΜ�Μ*LH

GM*�G

ML

0 2 40.5

0.7

0.9

1.1

x=Q2�mD

2

HΜ�Μ*LH

GM*�G

ML

Fall-off rate of G∗

M,J−S (Q2) in the γ∗p → ∆+ must follow that of GM (Q2).

With isospin symmetry:〈p|µ|∆+〉 = −〈n|µ|∆0〉

so same is true of the γ∗n → ∆0 magnetic form factor.

These are statements about the dressed quark core contributions

→ Outside the domain of meson-cloud effects, Q2 & 1.5GeV2


Q2-behaviour of G ∗M,Jones−Scadron

(II)

G∗

M,J−S cf. Experimental data and dynamical models

æ

æ

æ

æ

æææ

ææ

æææææææææææææææææææ

ææææ ææ æ æ

æ

0 0.5 1 1.50

1

2

3

x=Q2�mD

2

GM

,J-

S*

Solid-black:QCD-kindred interaction.

Dashed-blue:Contact interaction.

Dot-Dashed-green:Dynamical + no meson-cloud

☞ Observations:

All curves are in marked disagreement at infrared momenta.

Similarity between Solid-black and Dot-Dashed-green.

The discrepancy at infrared comes from omission of meson-cloud effects.

Both curves are consistent with data for Q2 & 0.75m2∆ ∼ 1.14GeV

2.


Q2-behaviour of G ∗M,Ash

Presentations of experimental data typically use the Ash convention– G∗

M,Ash(Q2) falls faster than a dipole –

ææ

ææææææææææææææ

æ

æ ææ

æ

æ

0 1 2 3 4 50.0

0.5

1.0

x=Q2�mD

2

GM

,Ash

*�N

Mn

No sound reason to expect:

G∗

M,Ash/GM ∼ constant

Jones-Scadron should exhibit:

G∗

M,J−S/GM ∼ constant

Meson-cloud effects

Up-to 35% for Q2 . 2.0m2∆.

Very soft → disappear rapidly.

G∗

M,Ashvs G∗

M,J−S

A factor 1/√Q2 of difference.


Electric and coulomb quadrupoles

☞ REM = RSM = 0 in SU(6)-symmetric CQM.

Deformation of the hadrons involved.

Modification of the structure of the transitioncurrent. ⇔

☞ RSM : Good description of the rapid fallat large momentum transfer.

££ææ¢¢òòòòòòòòò

òò

ò ò

ò

òò

ò

ò ò

ò

ò

òò

0.0 1.0 2.0 3.0 4.0

0

-5

-10

-15

-20

-25

-30

x=Q2�mD

2

RS

MH%L

☞ REM : A particularly sensitive measure oforbital angular momentum correlations.

àà

ôô

ææ¢¢òò

ò

òòòò

ò

ò

ò

òò òò

ò

ò

ò ò ò

ò

ò

òò

0.0 1.0 2.0 3.0 4.0

0

-2

-4

-6

x=Q2�mD

2

RE

MH%L

Zero Crossing in the transition electric form factor

Contact interaction → at Q2 ∼ 0.75m2∆ ∼ 1.14GeV

2

QCD-kindred interaction → at Q2 ∼ 3.25m2∆ ∼ 4.93GeV

2


Large Q2-behaviour of the quadrupole ratios

Helicity conservation arguments in pQCD should apply equally to both the resultsobtained within our QCD-kindred framework and those produced by an

internally-consistent symmetry-preserving treatment of a contact interaction

REMQ2

→∞= 1, RSM

Q2→∞= constant

0 20 40 60 80 100-0.5

0.0

0.5

1.0

x=Q 2�m Ρ

2

RS

M,R

EM

Observations:

Truly asymptotic Q2 is required before predictions are realized.

REM = 0 at an empirical accessible momentum and then REM → 1.

RSM → constant. Curve contains the logarithmic corrections expected in QCD.


Summary

Unified study of nucleon and ∆ elastic and transition form factors that comparespredictions made by:

Contact-interaction,

QCD-kindred interaction,

within the framework of Dyson-Schwinger equations.

The comparison clearly establishes

☞ Experiments are sensitive to the momentum dependence of the running couplingsand masses in the strong interaction sector of the Standard Model.

☞ Experiment-theory collaboration can effectively constrain the evolution to infraredmomenta of the quark-quark interaction in QCD.

☞ New experiments using upgraded facilities will leave behind meson-cloud effects andwill gain access to the region of transition between nonperturbative and perturbativebehaviour of QCD’s running coupling and masses.


Conclusions

☞ The NγN elastic form factors:

GpE (Q

2)/GpM (Q2) possesses a zero at Q2 ∼ 10GeV

2.

Any change in the interaction which shifts a zero in the proton ratio to larger Q2

relocates a zero in GnE (Q

2)/GnM (Q2) to smaller Q2.

The presence of strong diquark correlations within the nucleon is sufficient tounderstand empirical extractions of the flavour-separated form factors.

☞ The ∆γ∆ elastic form factors:

The ∆ elastic form factors are very sensitive to m∆, particularly GE2 and GM3

form factors.

Lattice-regularised QCD produce ∆-resonance masses that are very large, theform factors obtained therewith should be interpreted carefully.

☞ The Nγ∆ transition form factors:

G∗pM,J−S falls asymptotically at the same rate as Gp

M . This is compatible with

isospin symmetry and pQCD predictions.

Data do not fall unexpectedly rapid once the kinematic relation betweenJones-Scadron and Ash conventions is properly account for.

Strong diquark correlations within baryons produce a zero in the transitionelectric quadrupole at Q2 ∼ 5GeV

2.

Limits of pQCD, REM → 1 and RSM → constant, are apparent in our calculationbut truly asymptotic Q2 is required before the predictions are realized.


nucleon anddelta...

Documents