RECENT ADVANCES IN THE CALCULATION OF HADRON FORM FACTORS USING DYSON-SCHWINGER EQUATIONS OF QCD

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1 RECENT ADVANCES IN THE CALCULATION OF HADRON FORM FACTORS USING DYSON-SCHWINGER EQUATIONS OF QCD Jorge Segovia Argonne National Laboratory Thomas Jefferson National Accelerator Facility Newport News (Virginia) December 4th, 013 Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 1/45

2 Collaborators Argonne National Laboratory Craig D. Roberts (supervisor) Ian C. Cloët University of Science and Technology of China Chen Chen Shaolong Wan Forschungszentrum Jülich Sebastian M. Schmidt University of Adelaide Lei Chang Nanjing University Hong-Shi Zong Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs /45

3 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 Quark 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 QCD s Lagrangian yet! They play a dominant role determining characteristics of real-world QCD Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 3/45

4 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-p. 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 4Λ QCD (Λ QCD 00MeV). Any -point Schwinger function with an inflexion point at p > 0: Breaks the axiom of reflexion positivity No physical observable related with Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 4/45

5 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 = 160MeV N J P = 1/ + m = 938MeV cf. N(1535) J P = 1/ 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. 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 % 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) Recent advances in the calculation of hadron form factors using DSEs 5/45

6 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. Elastic and transition form factors can be used to illuminate QCD (at infrared momenta). M(p) (GeV) µp GEp/GMp α =.0 α = 1.8 α = 1.4 α = p (GeV) α =.0 α = 1.8 α = 1.4 α = Q (GeV ) Ian C. Cloet and Craig D. Roberts arxiv:nul-th/ Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 6/45

7 The simplest example of DSEs: The gap equation The quark propagator is given by the gap equation: S 1 (p) = Z (iγ p +m bm )+Σ(p) Σ(p) = Z 1 Λ General solution: q g D µν(p q) λa γµs(q)λa Γν(q,p) S(p) = Z(p ) iγ p +M(p ) 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 ) exhibits dynamical mass generation Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 7/45

8 Ward-Takahashi identities Symmetries should be preserved by any truncation Highly nontrivial constraint failure implies loss of any connection with QCD Symmetries in QFT 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) Recent advances in the calculation of hadron form factors using DSEs 8/45

9 Bethe-Salpeter and Faddeev equations Hadrons are studied via Poincaré covariant bound-state equations Mesons A -body bound state problem in quantum field theory. Properties emerge from solutions of Bethe-Salpeter equation: Γ(k;P) = d 4 q (π) 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) Recent advances in the calculation of hadron form factors using DSEs 9/45

10 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. 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) Recent advances in the calculation of hadron form factors using DSEs 10/45

11 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 QCD. Unique window into its quark and gluon structure Distinctive information on the roles played by confinement and DCSB in QCD 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 Q up to: 8.0GeV for (13)P 33 and N(1535)S GeV for N(1440)P 11 and N(150)D 13. The majority of new data was obtained at JLab. High-Q reach by experiments Probe the excited nucleon structures at perturbative and non-perturbative QCD scales Upgrade of CLAS up to 1GeV CLAS1 (New generation experiments in 015) Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 11/45

12 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 Q 8GeV. 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 (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 1/45

13 The electromagnetic current The electromagnetic current can be generally written as: J µλ (K,Q) = Λ +(P f )R λα (P f )iγ 5 Γ αµ(k,q)λ +(P i ) Incoming nucleon: P i = m N, and outgoing delta: P f = m. Photon momentum: Q = P f P i, and total momentum: K = (P i +P f )/. 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 Q (m m N ) 0 K Q 0 We take instead ˆK µ = TQ µν ˆK ν and ˆQ Vertex decomposes in terms of three (Jones-Scadron) form factors: [ λm Γ αµ = k (GM λ G E )γ 5ε αµγδ ˆK γ ˆQ δ GE TQ αγ TK γµ iς ] GC + λ ˆQ αˆk µ, m Magnetic dipole G M Electric quadrupole G E Coulomb quadrupole G C Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 13/45

14 Extraction of the form factors The Jones-Scadron form factors are: G M = 3(s +s 1 ), G E = s s 1, G C = s 3. G M,Ash = G M,J S GM,Ash vs G M,J S ( 1+ Q (m +m N ) ) 1 The scalars are obtained from the following Dirac traces and momentum contractions: ς(1 + d) s 1 = n T K d ˆK µν λ ς Tr[γ5J µλγ ν], s = n λ+ λ m T K µλ Tr[γ5J µλ], s 3 = 3n λ+ (1 + d) ˆK λ m d ˆK µ λ ς Tr[γ5J µλ]. We have used the following notation: 1 4d n =, λ ± = (m ±m N ) +Q Q 4ikλ m (m, ς = +m N ) (m + m N ), d = m m N (m + m N ), λm = ( 3 λ +λ, k = 1 + m ). m N G. Eichman et al., Phys. Rev. D 85, (01). Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 14/45

15 Experimental results and theoretical expectations I.G. Aznauryan and V.D. Burkert Prog. Part. Nucl Phys. 67, 1-54 (01) 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 Q Q (GeV ) Q (GeV ) pqcd predictions CQM predictions SU(6) predictions For Q GM 1/Q4. R EM +100%. R SM constant. Without quark orbital angular momentum: R EM 0. R SM 0. p µ + = n µ 0 p µ + = n µ n Data do not support these predictions Our aim: try to understand this longstanding puzzle Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 15/45

16 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. There are six contributions to the current. The picture shows the one-loop diagrams 1 Coupling of the photon to the dressed quark. Coupling of the photon to the dressed diquark: Elastic transition. Induced transition. Q P f Ψf Ψi P i P f Ψf Ψi P i scalar diquark correlations are absent from the -resonance Only axial-vector diquark correlations contribute in the top and middle diagrams Each diagram can be expressed like the electromagnetic current: Γ µλ = Λ +(P f )R λα (P f )J µα(k,q)λ +(P i ) Q P i Ψ Ψ P f f i axial vector scalar Q Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 16/45

17 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,q)λ +(P i ) Photon coupling directly to a dressed-quark with the diquark acting as a bystander Initial state and final state: Proton Q 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: ( /3) : ( 1/3) scalar Q Therefore 1 1 J scalar µ = eu I{ud} µ J axial µ = 3 3 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) Recent advances in the calculation of hadron form factors using DSEs 17/45

18 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, (011) P f P f Ψ f Ψ f Q Q Ψ i Pi 1 G 1 E,M,Q, G 1 M G Q Ψ i Pi 0. axial scalar Q GeV G 0 Γ 1, G ΠΓΡ Q GeV Composite object Electromagnetic radius is nonzero (r qq r π) Softer contribution to the form factors Soft contributions appear in the microscopic description of the elastic form factor of the proton Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 18/45

19 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 Q 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: ( /3) : ( 1/3) scalar Q Final state: + Same isospin states of axial-vector diquark. Different weighting due to I = 3/: ( 1/3) : ( /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,Q) Soft and still hard contributions appear in the microscopic description of the γ N electromagnetic reaction Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 19/45

20 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) Recent advances in the calculation of hadron form factors using DSEs 0/45

21 Simple framework Symmetry preserving Dyson-Schwinger equation treatment of a vector vector contact interaction Gluon propagator: Contact interaction. g D µν(p q) = δ µν 4πα IR m G Σ Gap equation = D Truncation scheme: Rainbow-ladder. Γ a ν (q,p) = (λa /)γ ν Quark propagator: Gap equation. S 1 (p) = iγ p +m+σ(p) = iγ p +M M 0.4GeV = constant. Implies momentum independent BSAs. Baryons: Faddeev equation. m N = 1.14GeV m = 1.39GeV (masses reduced by meson-cloud effects) p q p d γ Bethe-Salpeter equation is iγ a Ψ = iγ S is Faddeev equation p q P = p d Γ a q Γ b K Γ b Ψ P Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 1/45

22 Series of papers establishes strengths and limitations Used judiciously, produces results indistinguishable from most-sophisticated interactions employed in the rainbow ladder truncation of QCDs 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 (013). arxiv:11.1 [nucl-th] Spectrum of hadrons with strangeness C. Chen, L. Chang, C.D. Roberts, S. Wan and D.J. Wilson Few Body Syst (01). 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, 0505 (01). arxiv:111.1 [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, (011). 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, 1-5 (011). 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 8, 0650 (010). 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, 0650 (010). arxiv: [nucl-th] Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs /45

23 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. Suppresses the two-loop diagrams. GM (Q = 0) ind.-p DSE kernels dep.-p DSE kernels axial-diquark( ) axial-diquark(p) axial-diquark( ) scalar-diquark(p) Two sets of results 1 Original result. Improved version: Rescale the axial-diquark( ) scalar-diquark(p) diagram using: 1 + g as/aa 1 + Q /m ρ axial( )-scalar(p) = axial( )-axial(p) for G M (Q = 0) Incorporate dressed quark-anomalous magnetic moment Consequence of the DCSB. Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 3/45

24 Q -behaviour of G M,Jones Scadron (I) GM,J S cf. Experimental data and dynamical models Solid-black: Original result. Dashed-blue: Improved version. Dot-Dashed-green: Dynamical model without meson-cloud effects. Dotted-brown: Estimation with a sophisticated interaction. G M x Q m Ρ Both computed curves are consistent with data for Q mρ. They are in marked disagreement at infrared momenta. Similarity between Dashed-blue and Dot-Dashed-green. The discrepancy results from the omission of meson-cloud effects. Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 4/45

25 Q -behaviour of G M,Jones Scadron (II) 1.1 Transition cf. elastic magnetic form factors Μ n Μ N GM N GM n x Q m Ρ Solid-black: Proton. Red-points: Neutron. Fall-off rate of G M,J S (Q ) in the γ p + must much that of G M (Q ). 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, Q GeV Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 5/45

26 Q -behaviour of G M,Ash Presentations of experimental data typically use the Ash convention G M,Ash (Q ) falls faster than a dipole. There is no sound reason to expect: G M,Ash /Gn M constant Jones-Scadron should exhibit: G M,J S /Gn M constant G D G M,Ash 3 1 Two main reasons x Q m Ρ Meson-cloud effects Provide more than 30% for Q m ρ These contributions are very soft They disappear rapidly GM,Ash vs G p ( M,J S GM,Ash = Q G M,J S 1+ (m +m N ) A factor 1/Q of difference ) 1 Provides material damping for Q 4m ρ Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 6/45

27 Electric and coulomb quadrupoles R EM = R SM = 0 in SU(6)-symmetric CQM. Deformation of the hadrons involved. Modification of the structure of the transition current. R SM : Good description of the rapid fall at large momentum transfer. R SM x Q m Ρ R EM : A particularly sensitive measure of orbital angular momentum correlations. R EM x Q m Ρ Solid-black: Original Dashed-blue: Improved Dotted-brown: Sophisticated R EM = 0 at x 1.50 (contact) at x 3.5 (sophisticated). Even a contact interaction produces correlations that are comparable with those observed empirically. Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 7/45

28 Large Q -behaviour of the quadrupole ratios Helicity conservation arguments in pqcd should apply equally to an internally-consistent symmetry-preserving treatment of a contact interaction R EM Q = 1, R SM Q = constant 1.0 R SM, R EM Observations: x Q m Ρ Truly asymptotic Q 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 QCD. Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 8/45

29 The elastic form factors The small-q behavior 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,Q) = Λ +(P f )R λα (P f )Γ µ,αβ (K,Q)Λ +(P i )R βω (P i ) Vertex decomposes in terms of four form factors: [ ] [ ] Γ µ,αβ (K,Q) = (F1 +F )iγµ F K µ δ αβ (F3 m +F 4 )iγµ F 4 QαQ β K µ m 4m The multipole form factors: G E0 (Q ), G M1 (Q ), G E (Q ), G M3 (Q ), are functions of F1, F, 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 E and G M3 Shape deformation of the -baryon. Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 9/45

30 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 QCD results are usually used as a guide. Lattice-regularised QCD produce -resonance masses that are very large: Approach m π m ρ m Unquenched I Unquenched II Unquenched III Hybrid Quenched I Quenched II Quenched III We artificially inflate the mass of the -baryon Black solid line: DSEs + contact interaction + m = 1.8GeV Blue dashed line: DSEs + sophisticated interaction (+ m = 1.8GeV) Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 30/45

31 G E0 and G M1 with(out) an inflated quark core mass G E G E x Q m Ρ x Q m Ρ G M G M x Q m Ρ x Q m Ρ Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 31/45

32 G E and G M3 with(out) an inflated quark core mass G E G M x Q m Ρ x Q m Ρ G E G M x Q m Ρ x Q m Ρ Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 3/45

33 Observation The elastic form factors are very sensitive to m Lattice regularised QCD produce -resonance masses that are very large. Form factors obtained therewith are a poor guide to the s properties. Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 33/45

34 The Ω elastic form factors - General remarks The Ω -baryon s electromagnetic properties must be more amenable to measurement Magnetic dipole moment has measured with some precision: µ Ω = (.0±0.05)µ N The Ω consists of three valence s-quarks: Only decays via the weak interaction Significantly more stable. Ω s Lifetime: τ Ω τ 10 3 τ π +. kaon-cloud effects are smaller than pion-cloud effects: OZI rule: Meson-loops dominated by kaons in Ω. Kaons are heavier than pions Meson-loop effects are smaller than in the baryon. Dressed-quark-core should be a good approximation for the Ω m the = 1.76GeV 1.67GeV = m exp m the, = 1.39GeV 1.3GeV = m exp, The value m π should affect less to masses of higher excited states in Lattice-regularised QCD: Therefore one expects results in better agreement. Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 34/45

35 The Ω elastic form factors - Results G E0 G E x Q m Ρ x Q m Ρ G M1 G M x Q m Ρ x Q m Ρ Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 35/45

36 Bound state amplitudes (BSAs) Exclusive reactions can be described in terms of Poincaré-covariant hadron bound-state amplitudes (BSAs) Bethe-Salpeter amplitudes Mesons Faddeev amplitudes Baryons This approach has been used widely: Elastic and transition electromagnetic form factors. Strong decays of hadrons. Semileptonic and nonleptonic weak decays of heavy mesons. The BSAs are predictions of the framework and the associated computational scheme is applicable on the entire domain of accessible momentum transfers. Truncations must be employed in formulating the problem. Issues related to the construction of veracious truncation schemes. In equal-time quantization, the wave function for a hadron is a frame dependent concept: As it is defined by observations of different space points at a fixed time. Boost operators are interaction dependent, i.e. are dynamical. Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 36/45

37 Parton distribution amplitudes (PDAs) In high energy scattering experiments particles move at near speed of light. Natural to quantize a theory at equal light-front time: τ = (t +z)/. Light-front wave functions, ψ(x i, k i ), have many remarkable properties: Provide a frame-independent representation of hadrons. Have a probability interpretation - as close as QFT gets to QM. Do not depend on the hadrons 4-momentum; only internal variables: x i and k i. Boosts are kinematical - not dynamical!! PDAs are (almost) observables and are related to light-front wave functions: ϕ(x i ) = d k i ψ(x i, k i ) x i L-F fraction of the bound-state total-momentum carried by the quark. Hard exclusive hadronic reactions can be expressed in terms of the PDAs. Example: Electromagnetic form factor of light pseudoscalar meson: Q 0 >Λ QCD Q F π(q ) Q >Q0 16πα s(q )fπw ϕ, w ϕ = 1 1 dx x ϕπ(x), The scale Q 0 and PDAs are not determined by the analysis framework. Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 37/45

38 Connection between BSAs and PDAs PDAs may be obtained as light-front projections of the hadron BSAs. Lei Chang et al., Phys. Rev. Lett. 110, (013); 111, (013). Ian C. Cloet et al., Phys. Rev. Lett. 111, (013). Example: The pion s PDA: f π ϕ π(x) = Z d 4 k (π) δ(k+ xp + )Tr [ γ + γ 5 S(k)Γ π(k,p)s(k p) ] S(k)Γ π(k,p)s(k p) is the pion s Bethe-Salpeter amplitude. ϕ π(x) is the axial-vector projection of the pion s BSA onto the light-front. Two features emerged in developing the connection between BSAs and PDAs: The so-called asymptotic PDA: ϕ asy (x) = 6x(1 x) provides an unacceptably poor description of meson internal structure at all scales which are either currently accessible or foreseeable in experiments. Expected!! evolution with energy scale in QCD is logarithmic. The leading twist PDAs for light-quark mesons are concave functions at all energy scales: Eliminates the possibility of humped distributions. Enables one to obtain the meson s PDA from a very limited number of moments. Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 38/45

39 Computing PDAs from moments (I) Energy scale dependence of PDA (c.f. DGLAP equations for PDFs): τ d 1 dτ ϕ(x,τ) = dy V(x,y)ϕ(y,τ) 0 This evolution equation has a solution of the form: [ ϕ(x;τ) = ϕ asy (x) 1+ a 3/ j (τ)c (3/) j=1,,... ] j (x 1) QCD is invariant under the collinear conformal group SL(;R) in Q limit. Gegenbauer-α = 3/ polynomials are the irreducible representations of this group. Nonperturbative methods in QCD typically provide access to moments of the PDA: 1 (x x) m τ ϕ = dx (x x) m ϕ(x;τ), x = 1 x Accurate approx. to ϕ(x; τ) is obtained by using just first few terms of expansion. 0.5 Leads to ϕ(x) whose behaviour is not concave humped distributions. Slow convergence and spurious oscillations x Φ Π x Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 39/45

40 Computing PDAs from moments (II) ALTERNATIVE Accept that, at all accessible scales, PDAs determined by nonperturbative dynamics PDAs should be reconstructed from moments by using Gegenbauer polynomials of order α, with this order determined by the moments themselves, not fixed beforehand. j s ] ϕ(x;τ) = N α[x(1 x)] [1+ α 1/ aj α (τ)c (α) j (x 1) j=1,,... Quick convergence only j s = needed for the pion (there is no j s = 1) One may project ϕ(x;τ) onto the asymptotic form a 3/ j (τ) = 3 j +3 (j +)(j +1) 1 0 dx C (3/) j (x 1)ϕ(x;τ), Therewith obtaining all coefficients necessary to represent any computed distribution in the conformal form without ambiguity or difficulty. One may determine the distribution at any τ τ using the ERBL evolution equations for the coefficients {a 3/ j (τ),i = 1,,...}. Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 40/45

41 Meson PDA moments from Lattice At most two nontrivial moments of ϕ(x) can be computed using numerical simulations of lattice-regularised QCD. Proposed procedure: Reconstruct PDAs from lattice-qcd moments using: ϕ(x) = x α (1 x) β /B(α,β). Valid also for mesons comprised from quarks with nondegenerated masses. Evolving the distribution obtained to another momentum scale: Project the formula onto the asymptotic form. Employ the ERBL evolution equations. Lattice-regularised QCD data: Indistinguishable: ϕ V ϕv ϕp The appearance of precision is misleading: 3% for pion!! 1/ = (x x) ϕ asy (x x) ϕ (x x) ϕ (x x) ϕ point = 1/3 meson (x x) n π n = 0.5(1)() 0.8(1)() ρ n = 0.5()() 0.7(1)() K n = ()() 0.036(1)() K n = (1)() 0.043()(3) K n = 0.5(1)() 0.6(1)() K n = 0.5(1)() 0.5()() Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 41/45

42 Light pseudoscalar and vector mesons with equal-mass valence-quarks A B A Φ du x Φ du x C x x Curve A: DSE prediction for the chiral pion Region B: , α ud = β ud = Region C: , α ud = β ud = Observations: PDAs obtained from the two different lattices have overlapping error bands. However, the differences are material. It appears that the lattice configurations produce a form of φ ud (x) that is too broad. Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 4/45

43 Light pseudoscalar and vector mesons with no equal-mass valence-quarks Φ su x D A Φ su x F E D x x Curve A: DSE prediction for the chiral pion Region D: , α us = βus = Region E: , α us = βus = Observations: Shift in position relative to the peak in the pion s PDA. SU(3)-flavor symmetry is broken nonperturbatively at the level of 10%. PDA shift toward φ asy when increasing energy scale. Region E: GeV 10GeV. PDA evolution is extremely slow. Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 43/45

44 ERBL evolution Evolution with energy scale in QCD is LOGARITHMIC Example: first moment: Vertical dashed line: Marks τ = 100 GeV. Horizontal dotted line: Marks 0.5 x x not reach until τ = 7.3TeV xx su LnΖΖ Positive value: s-quark carries more momentum than the ū-quark. It is responsible for the shift in position. Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 44/45

45 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 pqcd 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 pqcd, R EM 1 and R SM constant, are apparent in our calculation but truly asymptotic Q is required before the predictions are realized. Strong diquark correlations within baryons produce a zero in the transition electric quadrupole at Q GeV. The elastic form factors The elastic form factors are very sensitive to m. Lattice-regularised QCD produce -resonance masses that are very large, the form factors obtained therewith should be interpreted carefully. Parton distribution amplitudes (PDAs) The method introduced enables one to obtain a pointwise accurate approximation to meson PDAs from limited moments (information). PDAs of pseudoscalar and vector mesons with (non)-equal-mass valence-quarks are presented. At all energy scales, the PDAs are concave functions whose dilation and asymmetry express the strength of dynamical chiral symmetry breaking. Jorge Segovia (Argonne National Laboratory) Recent advances in the calculation of hadron form factors using DSEs 45/45