Nucleon, Delta and Nucleon to Delta Electromagnetic Form Factors in DSEs

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1 Nucleon, Delta and Nucleon to Delta Electromagnetic Form Factors in DSEs Jorge Segovia, Ian C. Cloët and Craig D. Roberts Argonne National Laboratory Chen Chen and Shaolong Wan University of Science and Technology of China International Workshop on Nonperturbative Phenomena in Hadron and Particle Physics May 5-10, 2014, Ubatuba, São Paulo, Brazil Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 1/34

2 The challenge of CD uantum Chromodynamics is the only known example in nature of a nonperturvative fundamental quantum field theory CD 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 CD s complexity: The best promise for progress is a strong interplay between experiment and theory. Emergent phenomena ւ uark and gluon confinement Colored particles have never been seen isolated ց Dynamical chiral symmetry breaking Hadrons do not follow the chiral symmetry pattern Neither of these phenomena is apparent in CD s Lagrangian yet! They play a dominant role determining characteristics of real-world CD Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 2/34

3 Emergent phenomena: Confinement Confinement is associated with dramatic, dynamically-driven changes in the analytic structure of CD s propagators and vertices (CD s Schwinger functions) Dressed-propagator for a colored state: An observable particle is associated with a pole at timelike-p 2. When the dressing interaction is confining: Real-axis mass-pole splits, moving into a pair of 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. m G 2 4Λ CD (Λ CD 200MeV). Any 2-point Schwinger function with an inflexion point at p 2 > 0: Breaks the axiom of reflexion positivity No physical observable related with Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 3/34

4 Emergent phenomena: Dynamical chiral symmetry breaking Spectrum of a theory invariant under chiral transformations should exhibit degenerate parity doublets π J P = 0 m = 140MeV cf. σ J P = 0 + m = 500MeV ρ J P = 1 m = 775MeV cf. a 1 J P = 1 + m = 1260MeV N J P = 1/2 + m = 938MeV cf. N(1535) J P = 1/2 m = 1535MeV Splittings between parity partners are greater than 100-times the light quark mass scale: m u/m d 0.5, m d = 4MeV Dynamical chiral symmetry breaking Mass generated from the interaction of quarks with the gluon-medium. uarks acquire a HUGE constituent mass. Responsible of the 98% of the mass of the proton. (Not) spontaneous chiral symmetry breaking Higgs mechanism. uarks acquire a TINY current mass. Responsible of the 2% of the mass of the proton M(p) [GeV] Rapid acquisition of mass is effect of gluon cloud p [GeV] m = 0 (Chiral limit) m = 30 MeV m = 70 MeV Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 4/34

5 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 CD 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 CD. 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. Elastic and transition form factors can be used to illuminate CD (at infrared momenta). M(p) (GeV) µp GEp/GMp α = 2.0 α = 1.8 α = 1.4 α = p (GeV) α = 2.0 α = 1.8 α = 1.4 α = (GeV 2 ) Ian C. Cloet and Craig D. Roberts arxiv:nul-th/ Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 5/34

6 The simplest example of DSEs: The gap equation The quark propagator is given by the gap equation: S 1 (p) = Z 2 (iγ p +m bm )+Σ(p) Σ(p) = Z 1 Λ General solution: q g 2 D µν(p q) λa 2 γµs(q)λa 2 Γν(q,p) S(p) = Z(p 2 ) iγ p +M(p 2 ) Kernel involves: D µν(p q) - dressed gluon propagator Γ ν(q,p) - dressed-quark-gluon vertex M(p) [GeV] Each of which satisfies its own Dyson-Schwinger equation Infinitely many coupled equations Coupling between equations necessitates truncation Rapid acquisition of mass is effect of gluon cloud p [GeV] m = 0 (Chiral limit) m = 30 MeV m = 70 MeV M(p 2 ) exhibits dynamical mass generation Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 6/34

7 Vector and Axial-vector Ward-Green-Takahashi identities Symmetries should be preserved by any truncation Highly nontrivial constraint failure implies loss of any connection with CD Symmetries in FT are implemented by WTIs which relate different Schwinger functions For instance, axial-vector Ward-Takahashi identity: These observations show that symmetries relate the kernel of the gap equation a one-body problem with that of the Bethe-Salpeter equation a two-body problem Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 7/34

8 Bethe-Salpeter and Faddeev equations Hadrons are studied via Poincaré 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) = d 4 q (2π) 4 K(q,k;P)S(q+P)Γ(q;P)S(q) The kernel is that of the gap equation. iγ = iγ is is K 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 interactions that can take place between the three quarks that define its valence quark content. p q p d a Ψ P = p q p d Γ a q Γ b b Ψ P Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 8/34

9 Diquarks inside baryons The attractive nature of quark-antiquark correlations in a color-singlet meson is also attractive for 3 c 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. pseudoscalar and vector diquarks Ignored wrong parity larger mass-scales Diquark composition of the nucleon and Positive parity states ւ ց Dominant right parity shorter mass-scales scalar and axial-vector diquarks N 0 +, 1 + diquarks only 1 + diquark Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 9/34

10 Study of electromagnetic form factors - Motivation 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 CD. Unique window into its quark and gluon structure Distinctive information on the roles played by confinement and DCSB in CD Elastic and transition form factors ւ ց CEBAF Large Acceptance Spectrometer (CLAS) Most accurate results for the electroexcitation amplitudes of the four lowest excited states. They have been measured in a range of 2 up to: 8.0GeV 2 for (1232)P 33 and N(1535)S GeV 2 for N(1440)P 11 and N(1520)D 13. The majority of new data was obtained at JLab. High- 2 reach by experiments Probe the excited nucleon structures at perturbative and non-perturbative CD scales Upgrade of CLAS up to 12GeV CLAS12 (New generation experiments in 2015) Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 10/34

11 The γ N reaction Two ways in order to analyze the structure of the -resonances ւ π-mesons as a probe complex BUT: B( γn) 1% ց photons as a probe relatively simple 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 2 8GeV 2. 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 pcd to processes involving moderate momentum transfers. The role that experiments on resonance electroproduction can play in exposing non-perturbative phenomena in CD: The nature of confinement and Dynamical Chiral Symmetry Breaking. Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 11/34

12 The electromagnetic current The electromagnetic current can be generally written as: J µλ (K,) = Λ +(P f )R λα (P f )iγ 5 Γ αµ(k,)λ +(P i ) Incoming nucleon: P 2 i = m 2 N, and outgoing delta: P2 f = m 2. Photon momentum: = P f P i, and total momentum: K = (P i +P f )/2. The on-shell structure is ensured by the N- and -baryon projection operators. The composition of the 4-point function Γ αµ is determined by Poincaré covariance: Convenient to work with orthogonal momenta Simplify its structure considerably Not yet the case for K and (m m N ) 0 K 0 We take instead ˆK µ = T µν ˆK ν and ˆ Vertex decomposes in terms of three (Jones-Scadron) form factors: [ λm Γ αµ = k (GM 2λ G E )γ 5ε αµγδ ˆK γ ˆ δ GE T αγ TK γµ iς ] GC + λ ˆ αˆk µ, m Magnetic dipole G M Electric quadrupole G E Coulomb quadrupole G C Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 12/34

13 Extraction of the form factors The Jones-Scadron form factors are: G M = 3(s 2 +s 1 ), G E = s 2 s 1, G C = s 3. G M,Ash = G M,J S GM,Ash vs G M,J S ( 1+ 2 (m +m N ) 2 ) 1 2 The scalars are obtained from the following Dirac traces and momentum contractions: ς(1 + 2d) s 1 = n T K d ˆK µν λ ς Tr[γ5J µλγ ν], s 2 = n λ+ λ m T K µλ Tr[γ5J µλ], s 3 = 3n λ+ (1 + 2d) ˆK λ m d ˆK µ λ ς Tr[γ5J µλ]. We have used the following notation: 1 4d 2 n =, λ ± = (m ±m N ) ikλ m 2(m 2, ς = +m2 N ) 2(m 2 + m2 N ), d = m2 m2 N 2(m 2 + m2 N ), λm = ( 3 λ +λ, k = 1 + m ). 2 m N G. Eichmann et al., Phys. Rev. D 85, (2012). Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 13/34

14 Experimental results and theoretical expectations I.G. Aznauryan and V.D. Burkert Prog. Part. Nucl Phys. 67, 1-54 (2012) G* M,Ash /3G D R EM (%) The R EM ratio is measured to be minus a few percent R SM (%) The R SM ratio does not seem to settle to a constant at large (GeV 2 ) (GeV 2 ) pcd predictions CM predictions SU(6) predictions For 2 GM 1/4. R EM +100%. R SM constant. Without quark orbital angular momentum: R EM 0. R SM 0. p µ + = n µ 0 p µ + = 2 n µ n Data do not support these predictions Our aim: try to understand this longstanding puzzle Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 14/34

15 Electromagnetic current description in the quark-diquark picture To compute the electromagnetic properties of the γ N reaction in a given framework, one must specify how the photon couples to its constituents. Six contributions to the current One-loop diagrams Two-loop diagrams 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. P f Ψf Ψi P i P f Ψf Ψi P i P f P f Γ i Ψ Ψ P f i Γ Γ i Ψ Ψ P i f X µ Ingredients in the contributions 1 Ψ i,f Faddeev amplitudes. 2 Single line uark prop. 3 Double line Diquark prop. 4 Γ Diquark BS amplitudes. P i Ψ Ψ P f f i axial vector scalar µ P i Ψ Ψ P f f i Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 15/34 X Γ

16 Elastic form factor of the proton in the quark-diquark picture (I) Each diagram can be expressed in a similar way: Γ µ = Λ +(P f )J µ(k,)λ +(P i ) Photon coupling directly to a dressed-quark with the diquark acting as a bystander Initial state and final state: Proton Two axial-vector diquark isospin states: (I,I z) = (1,1) flavor content: {uu} P f Ψ f Ψ i Pi (I,I z) = (1,0) flavor content: {ud} In the isospin limit, they appear with relative weighting: ( 2/3) : ( 1/3) scalar Therefore 1 1 J scalar µ = eu I{ud} µ J axial µ = e d I {uu} µ eu I{ud} µ = 0 P f Ψ f axial vector Ψ i P i Hard contributions appear in the microscopic description of the elastic form factor of the proton Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 16/34

17 Elastic form factor of the proton in the quark-diquark picture (II) Remaining diagrams: Photon interacting with diquarks H.L.L. Roberts et al. Phys. Rev. C 83, (2011) I.C. Cloet et al., Few Body Syst. 46, 1-36 (2012) P f P f Ψ f Ψ f Ψ i Pi 1 G 1 E,M,, G 1 M G Ψ i Pi 0.2 axial scalar GeV 2 Composite object Electromagnetic radius is nonzero (r qq r π) Softer contribution to the form factors G 0 Γ 1, G ΠΓΡ GeV 2 Soft contributions appear in the microscopic description of the elastic form factor of the proton Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 17/34

18 Transition form factor of γ N in the quark-diquark picture Diagrams in which the photon interact with diquarks appear Photon coupling directly to a dressed-quark with the diquark acting as a bystander Initial state: Proton Two axial-vector diquark isospin states: (I,I z) = (1,1) flavor content: {uu} P f Ψ f Ψ i Pi (I,I z) = (1,0) flavor content: {ud} In the isospin limit, they appear with relative weighting: ( 2/3) : ( 1/3) scalar Final state: + Same isospin states of axial-vector diquark. Different weighting due to I = 3/2: ( 1/3) : ( 2/3) P f Ψ f axial vector Ψ i P i Therefore J 1,axial µα = e d I 1{uu} µα + eu I1{ud} µα = I1{qq} µα (K,) Soft and still hard contributions appear in the microscopic description of the γ N electromagnetic reaction Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 18/34

19 General observation Similar contributions in both cases: G p M vs G p M G p M should fall asymptotically at the same rate as Gp M. By isospin considerations: G n M should fall asymptotically at the same rate as G p M. Hold SU(6): p µ + n µ 0 p µ p. Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 19/34

20 Simple framework Symmetry preserving Dyson-Schwinger equation treatment of a vector vector contact interaction Gluon propagator: Contact interaction. g 2 D µν(p q) = δ µν 4πα IR m 2 G Truncation scheme: Rainbow-ladder. Γ a ν (q,p) = (λa /2)γ ν uark propagator: Gap equation. Form Factors: Two-loop diagrams not incorporated. Exchange diagram It is zero because our treatment of the contact interaction model Γ Pf i Ψ Ψ P f i Γ S 1 (p) = iγ p +m+σ(p) = iγ p +M M 0.4GeV. Implies momentum independent constituent quark mass. Implies momentum independent BSAs. Baryons: Faddeev equation. m N = 1.14GeV m = 1.39GeV (masses reduced by meson-cloud effects) P f Seagull diagrams They are zero Γ i Ψ Ψ P i f Xµ µ P i Ψ Ψ P f f i Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 20/34 X Γ

21 Series of papers establishes strengths and limitations Used judiciously, produces results indistinguishable from most-sophisticated interactions employed in the rainbow ladder truncation of CDs DSEs 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 (2013). arxiv: [nucl-th] 2 Spectrum of hadrons with strangeness C. Chen, L. Chang, C.D. Roberts, S. Wan and D.J. Wilson Few Body Syst (2012). arxiv: [nucl-th] 3 Nucleon and Roper electromagnetic elastic and transition form factors D.J. Wilson, I.C. Cloët, L. Chang and C.D. Roberts Phys. Rev. C 85, (2012). arxiv: [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, (2011). arxiv: [nucl-th] 5 Masses of ground and excited-state hadrons H.L.L. Roberts, L. Chang, I.C. Cloët and C.D. Roberts Few Body Syst. 51, 1-25 (2011). arxiv: [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, (2010). arxiv: [nucl-th] 7 Pion form factor from a contact interaction L.X. Gutierrez-Guerrero, A. Bashir, I.C. Cloët and C.D. Roberts Phys. Rev. C 81, (2010). arxiv: [nucl-th] Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 21/34

22 Weakness of contact-interaction 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. GM (2 = 0) ind.-p DSE kernels dep.-p DSE kernels axial-diquark( ) axial-diquark(p) axial-diquark( ) scalar-diquark(p) Improved version Rescale the axial-diquark( ) scalar-diquark(p) diagram using: 1+ g as/aa 1+ 2 /m 2 ρ axial( )-scalar(p) = axial( )-axial(p) for G M (2 = 0) Incorporate dressed quark-anomalous magnetic moment Consequence of the DCSB. Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 22/34

23 More sophisticated framework Gluon propagator: 1/k 2 -behaviour. Truncation scheme: Rainbow-ladder. Γ a ν (q,p) = (λa /2)γ ν uark propagator: Gap equation. S 1 (p) = Z 2 (iγ p +m bm )+Σ(p) = [ 1/Z(p 2 ) ][ iγ p +M(p 2 ) ] M 0.33 GeV. Implies momentum dependent constituent quark mass. Implies momentum dependent BSAs. Baryons: Faddeev equation. m N = 1.18GeV m = 1.33GeV (masses reduced by meson-cloud effects) Form Factors: Two-loop diagrams incorporated. Exchange diagram Play an important role Γ Pf i Ψ Ψ P f i Γ Seagull diagrams They are less important P Γ f i Ψ Ψ P f i X µ µ P i Ψ Ψ P f f i X Γ Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 23/34

24 Electromagnetic form factors of the Nucleon Bibliography: Current quark mass dependence of nucleon magnetic moments and radii I.C. Cloet et al., Few Body Syst. 42, (2008). arxiv: [nucl-th]. Survey of nucleon electromagnetic form factors I.C. Cloet et al., Few Body Syst. 46, 1-36 (2012). arxiv: [nucl-th]. Revealing dressed-quarks via the proton s charge distribution I.C. Cloet et al., Phys. Rev. Lett. 111, (2013). arxiv: [nucl-th]. Observations: Model parameters revisited in order to accomodate and N. Axial-vector diquark properties. Axial-scalar transition. Anomalous magnetic moment of the dressed quark. Results: G E (a) Cloet Segovia Segovia-AMM G M (b) Cloet Segovia Segovia-AMM µ G E /G M (c) Gayou et al Punjabi et al Puckett et al Puckett et al cloet-raa040 cloet-raa080 Segovia Segovia-AMM (GeV 2 ) 2 (GeV 2 ) 2 (GeV 2 ) Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 24/34

25 2 -behaviour of G M,Jones Scadron (I) GM,J S cf. Experimental data and dynamical models G M Observations: x 2 m 2 All curves are in marked disagreement at infrared momenta. Similarity between Solid-black and Dot-Dashed-green. Solid-black: Sophisticated interaction. Dashed-blue: Contact interaction. Dot-Dashed-green: Dynamical + no meson-cloud The discrepancy at infrared comes from omission of meson-cloud effects. Both curves are consistent with data for m GeV2. Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 25/34

26 2 -behaviour of G M,Jones Scadron (II) 1.1 Transition cf. elastic magnetic form factors ΜΜ G M GM x 2 m 2 Solid-black: Sophisticated interaction. Dashed-blue: Contact interaction. Fall-off rate of G M,J S (2 ) in the γ p + must much that of G M ( 2 ). 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, 2 1GeV 2 Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 26/34

27 2 -behaviour of G M,Ash Presentations of experimental data typically use the Ash convention G M,Ash (2 ) falls faster than a dipole 1.0 3G D G M,Ash x 2 m 2 No sound reason to expect: GM,Ash /G M constant Jones-Scadron should exhibit: GM,J S /G M constant Meson-cloud effects More than 30% for m 2. Very soft disappear rapidly. G M,Ash vs G M,J S A factor 1/ of difference. Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 27/34

28 Electric and coulomb quadrupoles R EM = R SM = 0 in SU(6)-symmetric CM. Deformation of the hadrons involved. Modification of the structure of the transition current. R SM R SM : Good description of the rapid fall at large momentum transfer x 2 m 2 R EM R EM : A particularly sensitive measure of orbital angular momentum correlations x 2 m 2 Zero Crossing in the transition electric form factor Contact interaction at m GeV2 Sophisticated interaction at m GeV2 Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 28/34

29 Large 2 -behaviour of the quadrupole ratios Helicity conservation arguments in pcd should apply equally to an internally-consistent symmetry-preserving treatment of a contact interaction 1.0 R EM 2 = 1, R SM 2 = constant R SM, R EM x 2 m Ρ 2 Observations: Truly asymptotic 2 is required before predictions are realized. R EM = 0 at an empirical accessible momentum and then R EM 1. R SM constant. Curve contains the logarithmic corrections expected in CD. Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 29/34

30 The elastic form factors The small- 2 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: J µ,λω (K,) = Λ +(P f )R λα (P f )Γ µ,αβ (K,)Λ +(P i )R βω (P i ) Vertex decomposes in terms of four form factors: [ ] [ ] Γ µ,αβ (K,) = (F1 +F 2 )iγµ F 2 K µ δ αβ (F3 m +F 4 )iγµ F 4 α β K µ m 4m 2 The multipole form factors: G E0 ( 2 ), G M1 ( 2 ), G E2 ( 2 ), G M3 ( 2 ), are functions of F1, F 2, F 3 and F 4. They are obtained by any four sensible projection operators. Physical interpretation: G E0 and G M1 Momentum space distribution of s charge and magnetization. G E2 and G M3 Shape deformation of the -baryon. Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 30/34

31 No experimental data Since one must deal with the very short -lifetime (τ τ π +): Little is experimentally known about the elastic form factors. Lattice-regularised CD results are usually used as a guide. Lattice-regularised CD produce -resonance masses that are very large: Approach m π m ρ m Unquenched I Unquenched II Unquenched III Hybrid uenched I uenched II uenched III We artificially inflate the mass of the -baryon (right panels on the next...) Black solid line: DSEs + sophisticated interaction (with/without m = 1.8GeV) Blue dashed line: DSEs + contact interaction (with/without m = 1.8GeV) Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 31/34

32 G E0 and G M1 with(out) an inflated quark core mass G E G E x 2 2 m x 2 2 m G M G M x 2 2 m x 2 2 m Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 32/34

33 G E2 and G M3 with(out) an inflated quark core mass G E G E x 2 2 m x 2 2 m 1 G M G M x 2 2 m x 2 2 m Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 33/34

34 Conclusions The γ N transition form factors Jones-Scadron G p M : G p M falls asymptotically at the same rate as Gp M. Compatible with isospin symmetry and pcd predictions. Data do not fall unexpectedly rapid once the kinematic relation between Jones-Scadron and Ash conventions is properly account for. R EM and R SM : Limits of pcd, R EM 1 and R SM constant, are apparent in our calculation but truly asymptotic 2 is required before the predictions are realized. Strong diquark correlations within baryons produce a zero in the transition electric quadrupole at 2 2GeV 2. The elastic form factors The elastic form factors are very sensitive to m. Lattice-regularised CD produce -resonance masses that are very large, the form factors obtained therewith should be interpreted carefully. Jorge Segovia (Argonne National Laboratory) Nucleon, Delta and Nucleon to Delta EM Form Factors in DSEs 34/34