Nucleon and Delta Elastic and Transition Form Factors Based on: - - - PowerPoint PPT Presentation

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 F´ ısica Fundamental y Matem´ aticas Universidad de Salamanca, Spain Ian C. Clo¨ et and Craig D. Roberts Argonne National Laboratory, USA Sebastian M. Schmidt Institute for Advanced Simulation, Forschungszentrum J¨ ulich and JARA, Germany Fachbereich Theoretische Physik / Institut f¨ ur Physik Karl-Franzens-Universit¨ at Graz March 10th, 2015

Jorge Segovia (segonza@usal.es) 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 nonperturvative fundamental 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 ց Quark and gluon confinement Dynamical chiral symmetry breaking ↓ ↓ Colored particles have never been seen isolated Hadrons do not follow the chiral symmetry pattern Neither of these phenomena is apparent in QCD’s Lagrangian yet! They play a dominant role determining characteristics of real-world QCD

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 2/37

Emergent phenomena: Confinement

Confinement is associated with dramatic, dynamically-driven changes in the analytic structure of QCD’s propagators and vertices (QCD’s Schwinger functions) ☞ Dressed-propagator for a colored state: An observable particle is associated with a pole at timelike-P2. When the dressing interaction is confining: Real-axis mass-pole splits, moving into a pair

• f complex conjugate singularities.

No mass-shell can be associated with a particle whose propagator exhibits such singularity. ☞ Dressed-gluon propagator: Confined gluon. IR-massive but UV-massless. mG ∼ 2 − 4ΛQCD (ΛQCD ≃ 200 MeV). Any 2-point Schwinger function with an inflexion point at p2 > 0: → Breaks the axiom of reflexion positivity → No physical observable related with

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 3/37

Emergent phenomena: Dynamical chiral symmetry breaking

Spectrum of a theory invariant under chiral transformations should exhibit degenerate parity doublets π JP = 0− m = 140 MeV cf. σ JP = 0+ m = 500 MeV ρ JP = 1− m = 775 MeV cf. a1 JP = 1+ m = 1260 MeV N JP = 1/2+ m = 938 MeV cf. N(1535) JP = 1/2− m = 1535 MeV Splittings between parity partners are greater than 100-times the light quark mass scale: mu/md ∼ 0.5, md = 4 MeV ☞ Dynamical chiral symmetry breaking

Mass generated from the interaction of quarks with the 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.

1 2 3 p [GeV] 0.1 0.2 0.3 0.4 M(p) [GeV]

m = 0 (Chiral limit) m = 30 MeV m = 70 MeV

effect of gluon cloud Rapid acquisition of mass is

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 4/37

Electromagnetic form factors of nucleon excited states

A central goal of Nuclear Physics: understand the structure and properties of protons and neutrons, and ultimately atomic nuclei, in terms of the quarks and gluons of QCD. Elastic and transition form factors ւ ց Unique window into its quark and gluon structure High-Q2 reach by experiments ↓ ↓ Distinctive information on the roles played by confinement and DCSB in QCD Probe the excited nucleon structures at perturbative and non-perturbative QCD scales CEBAF Large Acceptance Spectrometer (CLAS) ☞ Most accurate results for the electroexcitation amplitudes

• f the four lowest excited states.

☞ They have been measured in a range of Q2 up to: 8.0 GeV2 for ∆(1232)P33 and N(1535)S11. 4.5 GeV2 for N(1440)P11 and N(1520)D13. ☞ The majority of new data was obtained at JLab. Upgrade of CLAS up to 12 GeV → CLAS12 (New generation experiments in 2015)

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 5/37

Theory tool: Dyson-Schwinger equations

Confinement and Dynamical Chiral Symmetry Breaking (DCSB) can be identified with properties of dressed-quark and -gluon propagators and vertices (Schwinger functions) Dyson-Schwinger equations (DSEs) The quantum equations of motion of QCD whose solutions are the Schwinger functions. → Propagators and vertices. Generating tool for perturbation theory. → No model-dependence. Nonperturbative tool for the study of continuum strong QCD. → Any model-dependence should be incorporated here. Allows the study of the interaction between light quarks in the whole range of momenta. → Analysis of the infrared behaviour is crucial to disentangle confinement and DCSB. Connect quark-quark interaction with experimental observables. → It is via the Q2 evolution of form factors that one gains access to the running

• f QCD’s coupling and masses from the infrared into the ultraviolet.

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 6/37

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

q

g2Dµν(p − q) λa 2 γµS(q)λa 2 Γν(q, p) General solution: S(p) = Z(p2) iγ · p + M(p2) Kernel involves:

Dµν(p − q) - dressed gluon propagator Γν(q, p)

• dressed-quark-gluon vertex

1 2 3 p [GeV] 0.1 0.2 0.3 0.4 M(p) [GeV]

m = 0 (Chiral limit) m = 30 MeV m = 70 MeV

effect of gluon cloud Rapid acquisition of mass is

M(p2) exhibits dynamical mass generation Each of which satisfies its own Dyson-Schwinger equation ↓ Infinitely many coupled equations ↓ Coupling between equations necessitates truncation

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 7/37

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 – a

• ne-body problem – with that of the Bethe-Salpeter equation – a two-body problem –

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 8/37

Bethe-Salpeter and Faddeev equations

Hadrons are studied via Poincar´ e covariant bound-state equations ☞ Mesons A 2-body bound state problem in quantum field theory. Properties emerge from solutions of Bethe-Salpeter equation: Γ(k; P) =

• d4q

(2π)4 K(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 quantum field theory. Structure comes from solving the Faddeev equation. Faddeev equation: Sums all possible quantum field theoretical exchanges and interactions that can take place between the three dressed-quarks.

=

a

Ψ P pq pd Γb Γ −a pd pq

b

Ψ P

q

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 9/37

Diquarks inside baryons

The attractive nature of quark-antiquark correlations in a color-singlet meson is also attractive for ¯ 3c quark-quark correlations within a color-singlet baryon ☞ Diquark correlations: Empirical evidence in support of strong diquark correlations inside the nucleon. A dynamical prediction of Faddeev equation studies. In our approach: Non-pointlike color-antitriplet and fully interacting.

Thanks to G. Eichmann.

Diquark composition of the Nucleon (N) and Delta (∆) Positive parity states ւ ց pseudoscalar and vector diquarks scalar and axial-vector diquarks ↓ ↓ Ignored wrong parity larger mass-scales Dominant right parity shorter mass-scales → N ⇒ 0+, 1+ diquarks ∆ ⇒ only 1+ diquark

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 10/37

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 the dressed quark.

2

Coupling of the photon to the dressed 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 i

Ψ Ψ P

f f

P Q

i i

Ψ Ψ P

f f

P Q

scalar axial vector i i

Ψ Ψ P

f f

P Q Two-loop diagrams

i i

Ψ Ψ P P

f f

Q

Γ

−

Γ

µ

i i

X

Ψ Ψ P

f f

Q P

Γ

−

µ

i i

X

− Ψ Ψ P

f f

P Q

Γ

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 11/37

Contact-interaction (CI) framework

Symmetry preserving Dyson-Schwinger equation treatment of a vector ⊗ vector contact interaction ☞ Gluon propagator: Contact interaction. g2Dµν(p − q) = δµν 4παIR m2

G

☞ Truncation scheme: Rainbow-ladder. Γa

ν(q, p) = (λa/2)γν

☞ Quark propagator: Gap equation. S−1(p) = iγ · p + m + Σ(p) = iγ · p + M M ∼ 0.4 GeV. Implies momentum independent constituent quark mass. Implies momentum independent BSAs. ☞ Baryons: Faddeev equation. mN = 1.14 GeV m∆ = 1.39 GeV (masses reduced by meson-cloud effects) ☞ Form Factors: Two-loop diagrams not incorporated. Exchange diagram It is zero because our treatment of the contact interaction model

i i

Ψ Ψ P P

f f

Q

Γ

−

Γ

Seagull diagrams They are zero

µ

i i

X

Ψ Ψ P

f f

Q P

Γ

−

µ

i i

X

− Ψ Ψ P

f f

P Q

Γ

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 12/37

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 kaon

• C. Chen, L. Chang, C.D. Roberts, S.M. Schmidt S. Wan and D.J. Wilson
• Phys. Rev. C 87 045207 (2013). arXiv:1212.2212 [nucl-th]

2 Spectrum of hadrons with strangeness

• C. Chen, L. Chang, C.D. Roberts, S. Wan and D.J. Wilson

Few Body Syst. 53 293-326 (2012). arXiv:1204.2553 [nucl-th]

3 Nucleon and Roper electromagnetic elastic and transition form factors

D.J. Wilson, I.C. Clo¨ et, L. Chang and C.D. Roberts

• Phys. Rev. C 85, 025205 (2012). arXiv:1112.2212 [nucl-th]

4 π- and ρ-mesons, and their diquark partners, from a contact interaction

H.L.L. Roberts, A. Bashir, L.X. Gutierrez-Guerrero, C.D. Roberts and D.J. Wilson

• Phys. Rev. C 83, 065206 (2011). arXiv:1102.4376 [nucl-th]

5 Masses of ground and excited-state hadrons

H.L.L. Roberts, L. Chang, I.C. Clo¨ et and C.D. Roberts Few Body Syst. 51, 1-25 (2011). arXiv:1101.4244 [nucl-th]

6 Abelian anomaly and neutral pion production

H.L.L. Roberts, C.D. Roberts, A. Bashir, L.X. Gutierrez-Guerrero and P.C. Tandy

• Phys. Rev. C 82, 065202 (2010). arXiv:1009.0067 [nucl-th]

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 13/37

Weakness of the contact-interaction framework

A truncation which produces Faddeev amplitudes that are independent of relative momentum: 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 observables

• ne can expose those quantities which are most sensitive

to the momentum dependence of elementary quantities in QCD.

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 14/37

QCD-kindred framework

☞ Gluon propagator: 1/k2-behaviour. ☞ Truncation scheme: Rainbow-ladder. Γa

ν(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) ∼ 0.33 GeV. Implies momentum

dependent constituent quark mass. Implies momentum dependent BSAs. ☞ Baryons: Faddeev equation. mN = 1.18 GeV m∆ = 1.33 GeV (masses reduced by meson-cloud effects) ☞ Form Factors: Two-loop diagrams incorporated. Exchange diagram Play an important role

i i

Ψ Ψ P P

f f

Q

Γ

−

Γ

Seagull diagrams They are less important

µ

i i

X

Ψ Ψ P

f f

Q P

Γ

−

µ

i i

X

− Ψ Ψ P

f f

P Q

Γ

Jorge Segovia (segonza@usal.es) 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: P2

i = P2 f = −m2 N.

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 2mN σµνQνF2(Q2) ☞ The electric and magnetic (Sachs) form factors are a linear combination of the Dirac and Pauli form factors: GE (Q2) = F1(Q2) − Q2 4m2

N

F2(Q2) GM(Q2) = F1(Q2) + F2(Q2) ☞ 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.

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 16/37

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:

0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5 3 3.5 4 GE Q2 (GeV2)

(a)

Cloet Segovia Segovia-AMM 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 3.5 4 GM Q2 (GeV2)

(b)

Cloet Segovia Segovia-AMM

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 17/37

Sachs electric and magnetic form factors (I)

☞ Q2 dependence of proton form factors: 1 2 3 4 0.0 0.5 1.0 xQ2mN

2

GE

p

1 2 3 4 0.0 1.0 2.0 3.0 xQ2mN

2

GM

p

☞ Proton static properties: r2

E

r2

E M2 N

r2

M

r2

MM2 N

µ Theory (0.61 fm)2 (4.31)2 (0.53 fm)2 (3.16)2 2.50 Experiment (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.

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 18/37

Sachs electric and magnetic form factors (II)

☞ Q2 dependence of neutron form factors: 1 2 3 4 0.00 0.04 0.08 xQ2mN

2

GE

n

1 2 3 4 0.0 1.0 2.0 xQ2mN

2

GM

n

☞ Neutron static properties: r2

E

r2

E M2 N

r2

M

r2

MM2 N

µ Theory −(0.28 fm)2 −(1.99)2 (0.70 fm)2 (4.95)2 −1.83 Experiment −(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 ⇔ G n

E 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 the neutron electric form factor.

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 19/37

Unit-normalised ratio of Sachs electric and magnetic form factors

Both CI and QCD-kindred frameworks predict a zero crossing in µpG p

E/G p M

• 2

4 6 8 10 0.0 0.5 1.0 Q2 GeV2 ΜpGE

pGM p

• 2

4 6 8 10 12 0.0 0.2 0.4 0.6 Q2 GeV2 ΜnGE

nGM n

The possible existence and location of the zero in µpG p

E /G p M is a fairly direct measure

• f the nature of the quark-quark interaction

0.1 0.2 0.3 0.4

M(p) (GeV)

1 2 3 4

p (GeV) α = 2.0 α = 1.8 α = 1.4 α = 1.0

2 4 6 8 10 12 0.0 0.2 0.4 0.6 0.8 1.0 Q2 GeV2 ΜNGE

NGM N

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 20/37

Flavour separation

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

• 1

2 3 4 5 6 7 0.0 1.0 2.0 xQ

2MN 2

x2F1 p

d ,x2F1 p u

• 1

2 3 4 0.0 0.2 0.4 0.6 xQ

2MN 2

x2F2 p

u Κp u

• 1

2 3 4 0.0 0.2 0.4 0.6 xQ

2MN 2

x2F2 p

d Κp d

☞ Observations: F d

1p 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 d

1p depends on the relative probability of finding 1+

and 0+ diquarks in the proton. One would expect same behaviour in F2 → more contributions playing an important role like anomalous magnetic moment → They are not the key to explaining the data’s basic features.

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 21/37

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 ☞ The electromagnetic current can be generally written as: 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(Q2), GE2(Q2), GM3(Q2), 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.

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 22/37

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 1.687 Unquenched II 0.509 0.899 1.559 Unquenched III 0.384 0.848 1.395 Hybrid 0.353 0.959 1.533 Quenched I 0.563 0.873 1.470 Quenched II 0.490 0.835 1.425 Quenched 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∆ ∼ 1.7 GeV) Blue-dashed line: DSEs + contact interaction (with/without m∆ ∼ 1.7 GeV)

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 23/37

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

• 1

2 0.0 0.5 1.0 xQ2m

2

GE0

• 1

2 0.0 0.5 1.0 xQ2m

2

GE0

• 1

2 0.0 1.0 2.0 3.0 4.0 xQ2m

2

GM1

• 1

2 0.0 1.0 2.0 3.0 4.0 xQ2m

2

GM1

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 24/37

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

• 1

2 6 4 2 xQ2m

2

GE2

• 1

2 6 4 2 xQ2m

2

GE2

• 1

2 3 2 1 1 xQ2m

2

GM3

• 1

2 3 2 1 1 xQ2m

2

GM3

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 25/37

Static electromagnetic properties of the ∆(1232)+

Approach GM1(0) GE2(0) GM3(0) DSE-QCD-kindred +2.86 −6.67 −3.00 DSE-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 > 0 Covariant χPT+qLQCD +3.74 ± 0.03 −0.9 ± 0.6 −0.9 ± 2.1 QCD-SR +4.2 ± 1.1 −0.6 ± 0.2 −0.7 ± 0.2 χQSM +3.1 −2.0

• General Param. Method
• −4.4

−2.6 QM+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

• Jorge Segovia (segonza@usal.es)

Nucleon and Delta Elastic and Transition Form Factors 26/37

The γ∗N → ∆ reaction

Two ways in order to analyze the structure of the ∆-resonances ւ ց π-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 8 GeV2. 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 exposing non-perturbative phenomena in QCD: ☞ The nature of confinement and Dynamical Chiral Symmetry Breaking.

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 27/37

The γ∗N → ∆ transtion current (I)

☞ The electromagnetic current can be generally written as: Jµλ(K, Q) = Λ+(Pf ) Rλα(Pf ) iγ5 Γαµ(K, Q) Λ+(Pi) Incoming nucleon: P2

i = −m2 N, and outgoing delta: P2 f = −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 Poincar´ e covariance: Convenient to work with orthogonal momenta ↔ Simplify its structure considerably ↓ Not yet the case for K and Q ↔ ∆(m∆ − mN) = 0 ⇒ K · Q = 0 ↓ We take instead ˆ K ⊥

µ = TQ µν ˆ

Kν and ˆ Q ☞ Vertex decomposes in terms of three (Jones-Scadron) form factors: Γαµ = k λm 2λ+ (G ∗

M − G ∗ E )γ5εαµγδ ˆ

K ⊥

γ ˆ

Qδ − G ∗

E TQ αγTK γµ − iς

λm G ∗

C ˆ

Qα ˆ K ⊥

µ

• ,

Magnetic dipole ⇒ G ∗

M

Electric quadrupole ⇒ G ∗

E

Coulomb quadrupole ⇒ G ∗

C

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 28/37

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 − 1

2

The scalars are obtained from the following Dirac traces and momentum contractions: s1 = n

• ς(1 + 2d )

d − ς

TK

µν ˆ

K ⊥

λ Tr[γ5Jµλγν],

s2 = n λ+ λm TK

µλTr[γ5Jµλ],

s3 = 3n λ+ λm (1 + 2d )

d − ς

ˆ K ⊥

µ ˆ

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

∆ − m2 N

2(m2

∆ + m2 N) ,

λm =

• λ+λ−,

k =

• 3

2

• 1 + m∆

mN

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

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 29/37

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

1

Q2 (GeV2) G*M,Ash/3GD

• 7
• 6
• 5
• 4
• 3
• 2
• 1

1 2

REM (%)

• 35
• 30
• 25
• 20
• 15
• 10
• 5

10

• 1

1

Q2 (GeV2) RSM (%)

☞ The REM ratio is measured to be minus 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 orbital angular momentum) REM → 0. RSM → 0. pQCD predictions (For Q2 → ∞) G ∗

M → 1/Q4.

REM → +100%. RSM → constant. Experimental data do not support theoretical predictions

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 30/37

Q2-behaviour of G ∗

M,Jones−Scadron (I) Transition cf. elastic magnetic form factors 2 4 0.8 0.9 1.0 1.1 xQ2m

2

ΜΜGM

GM

2 4 0.5 0.7 0.9 1.1 xQ2m

2

ΜΜGM

GM

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 1.5 GeV2

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 31/37

Q2-behaviour of G ∗

M,Jones−Scadron (II) G ∗

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

• 0.5

1 1.5 1 2 3 xQ2m

2

GM,JS

• 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

∆ ∼ 1.14 GeV2.

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 32/37

Q2-behaviour of G ∗

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

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

• 1

2 3 4 5 0.0 0.5 1.0 xQ2m

2

GM,Ash

• NM

n

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,Ash vs G ∗ M,J−S

A factor 1/ √ Q2 of difference.

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 33/37

Electric and coulomb quadrupoles

☞ REM = RSM = 0 in SU(6)-symmetric CQM. Deformation of the hadrons involved. Modification of the structure of the transition current. ⇔ ☞ RSM: Good description of the rapid fall at large momentum transfer.

• 0.0

1.0 2.0 3.0 4.0 5 10 15 20 25 30 xQ2m

2

RSM ☞ REM: A particularly sensitive measure of

• rbital angular momentum correlations.
• 0.0

1.0 2.0 3.0 4.0 2 4 6 xQ2m

2

REM Zero Crossing in the transition electric form factor Contact interaction → at Q2 ∼ 0.75m2

∆ ∼ 1.14 GeV2

QCD-kindred interaction → at Q2 ∼ 3.25m2

∆ ∼ 4.93 GeV2

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 34/37

Large Q2-behaviour of the quadrupole ratios

Helicity conservation arguments in pQCD should apply equally to both the results

• btained within our QCD-kindred framework and those produced by an

internally-consistent symmetry-preserving treatment of a contact interaction REM

Q2→∞

= 1, RSM

Q2→∞

= constant

20 40 60 80 100 0.5 0.0 0.5 1.0 xQ 2m Ρ

2

RSM,REM

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.

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 35/37

Summary

Unified study of nucleon and ∆ elastic and transition form factors that compares predictions 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 couplings and masses in the strong interaction sector of the Standard Model. ☞ Experiment-theory collaboration can effectively constrain the evolution to infrared momenta of the quark-quark interaction in QCD. ☞ New experiments using upgraded facilities will leave behind meson-cloud effects and will gain access to the region of transition between nonperturbative and perturbative behaviour of QCD’s running coupling and masses.

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 36/37

Conclusions

☞ The NγN elastic form factors: G p

E (Q2)/G p M(Q2) possesses a zero at Q2 ∼ 10 GeV2.

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

E (Q2)/G n M(Q2) to smaller Q2.

The presence of strong diquark correlations within the nucleon is sufficient to understand 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, the form factors obtained therewith should be interpreted carefully. ☞ The Nγ∆ transition form factors: G ∗p

M,J−S falls asymptotically at the same rate as G p

• M. This is compatible with

isospin symmetry and pQCD predictions. Data do not fall unexpectedly rapid once the kinematic relation between Jones-Scadron and Ash conventions is properly account for. Strong diquark correlations within baryons produce a zero in the transition electric quadrupole at Q2 ∼ 5 GeV2. Limits of pQCD, REM → 1 and RSM → constant, are apparent in our calculation but truly asymptotic Q2 is required before the predictions are realized.

Jorge Segovia (segonza@usal.es) Nucleon and Delta Elastic and Transition Form Factors 37/37