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Excited nucleon electromagnetic form factors from broken spin-flavor symmetry

Excited nucleon electromagnetic form factors from broken spin-flavor symmetry. Alfons Buchmann Universität Tübingen. Introduction Strong interaction symmetries SU(6) and 1/N expansion of QCD Electromagnetic form factor relations Group theoretical argument Summary.

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Excited nucleon electromagnetic form factors from broken spin-flavor symmetry

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  1. Excited nucleon electromagnetic form factors from broken spin-flavor symmetry Alfons Buchmann Universität Tübingen • Introduction • Strong interaction symmetries • SU(6) and 1/N expansion of QCD • Electromagnetic form factor relations • Group theoretical argument • Summary Nstar 2009, Beijing, 20 April 2009

  2. 1. Introduction

  3. Spatial extension of proton rp proton radial distribution Measurement of proton charge radius rp(exp) = 0.862(12) fm Simon et al., Z. Naturf. 35a (1980) 1

  4. N‘ e‘  Q  charge magnetic N e Elastic electron-nucleon scattering ...scattering angle Elastic form factors Q... four-momentum transfer Q²= -(²-q²) ...energy transfer q... three-momentum transfer ... photon N... nucleon (p,n) e... electron

  5. Geometric shape of proton charge distribution angular distribution Extraction of Ntransition quadrupole (C2) moment from data Q N(exp) = -0.0846(33) fm² Tiator et al., EPJ A17 (2003) 357

  6. (1232) N*(1440) N*(1520) orbital excitation E1, M2 radial excitation C0, M1 spin-isospin excitation M1, E2, C2 N(939) J=3/2- J=3/2+ J=1/2+ Proton excitation spectrum C2 multipole transition to (1232) is sensitive to angular shape of nucleon ground state ...

  7. Inelastic electron-nucleon scattering  N‘ e‘  Q  N e Additional information on nucleon ground state structure

  8. Properties of the nucleon • finite spatial extension (size) • nonspherical charge distribution (shape) • excited states (spectrum) What can we learn about these structural features using strong interaction symmetries as a guide?

  9. 2. Strong interaction symmetries

  10. Strong interaction symmetries • Strong interactions • are • approximately invariant • under • SU(2) isospin, • SU(3) flavor, • SU(6) spin-flavor • symmetry transformations.

  11. SU(3) flavor symmetry Gell-Mann, Ne‘eman1962 Flavor symmetry combines hadron isospin multiplets with different T and Y into larger multiplets, e.g., flavor octet and flavor decuplet.

  12. S n p D- D0 D+ D++ S*+ S*- S*0 S+ S0 S- L0 X*- X*0 X- X0 W SU(3) flavor symmetry 0 -1 -2 decuplet octet -3 J=3/2 J=1/2 T3 -1 -1/2 0 +1/2 +1 -3/2 -1/2 +1/2 +3/2

  13. Y hypercharge S strangeness Symmetry breaking along strangeness direction through hypercharge operator Y T3 isospin second order SU(3) symmetry breaking first order SU(3) symmetry breaking SU(3) invariant term SU(3) symmetry breaking mass operator M0, M1, M2 experimentally determined

  14. Group algebra relates symmetry breaking within a multiplet (Wigner-Eckart theorem) Relations between observables

  15. baryon octet Gell-Mann & Okubo mass formula baryon decuplet „equal spacing rule“ (M/M)exp ~ 1%

  16. SU(6) spin-flavor symmetry combines SU(3) multiplets with different spin and flavor to SU(6) spin-flavor supermultiplets. Gürsey, Radicati, Sakita, Beg, Lee, Pais, Singh,... (1964)

  17. S T3 SU(6) spin-flavor supermultiplet baryon supermultiplet

  18. e.g. Gürsey-Radicati SU(6) mass formula SU(6) symmetry breaking term Relations between octet and decuplet baryon masses

  19. proton/neutron magnetic moment ratio Successes of SU(6) • explains why Gell-Mann Okubo formula works for • octet and decuplet baryons with the same coefficients M0, M1, M2 • predicts fixed ratio between F and D type octet couplings • in agreement with experiment F/D=2/3 Higher predictive power than independent spin and flavor symmetries

  20. 3. Spin-flavor symmetry and 1/N expansion of QCD

  21. SU(6) spin-flavor as QCD symmetry SU(6) symmetry is exact in the limit NC  . NC ... number of colors For finite NC,spin-flavor symmetry is broken. Symmetry breaking operators can be classified according to the 1/NC expansion scheme. Gervais, Sakita, Dashen, Manohar,.... (1984)

  22. strong coupling two-body three-body 1/NC expansion of QCD processes NC ... number of colors

  23. SU(6) spin-flavor as QCD symmetry This results in the following hierarchy O[1] (1/NC0) > O[2] (1/NC1)> O[3] (1/NC2) one-quark operator two-quark operator three-quark operator i.e., higher order symmetry breaking operators are suppressed by higher powers of 1/NC.

  24. Large NC QCD provides a perturbative expansion scheme for QCD processes that works at all energy scales Application of 1/NC expansion to charge radii and quadrupole moments Buchmann, Hester, Lebed, PRD62, 096005 (2000); PRD66, 056002 (2002); PRD67, 016002 (2003)

  25. 4. Electromagnetic form factor relations

  26. For NC=3 we may just as well use the simpler spin-flavor parametrization method developed by G. Morpurgo (1989). Application to quadrupole and octupole moments Buchmann and Henley, PRD 65, 073017 (2002); Eur. Phys. J. A 35, 267 (2008)

  27. one-quark two-quark three-quark Spin-flavor operator O O[i] all allowed invariants in spin-flavor space for observable under investigation constants A, B, C  parametrize orbital- and color matrix elements; determined from experiment Which spin-flavor operators are allowed?

  28. Multipole expansion in spin-flavor space • for neutron and quadrupole transition no contribution from one-body operator • most general structure of two-body charge operator [2] in spin-flavor space • fixed ratio of factors multiplying spin scalar (+2) and spin tensor (-1) • sandwich between SU(6) wave functions

  29. ei ... charge si ... spin mi ... mass e.g. electromagnetic current operator ek ei g g 2-quark current 3-quark current SU(6) spin-flavor symmetry breaking SU(6) symmetry breaking via spin and flavor dependent two- and three-quark currents

  30. neutron charge radius N transition quadrupole moment Buchmann, Hernandez, Faessler, PRC 55, 448 N quadrupole moment neutron charge radius spin scalar spin tensor Neutron and N charge form factors

  31. Blanpied et al., PRC 64 (2001) 025203 Tiator et al., EPJ A17 (2003) 357 Buchmann et al., PRC 55 (1997) 448 Experimental N quadrupole moment Extraction of p +(1232) transition quadrupole moment from electron-proton and photon-proton scattering data experminent theory neutron charge radius

  32. Including three-quark operators

  33. Relation remains intact after including three-quark currents Buchmann and Lebed, PRD 67 (2003)

  34. Relations between octet and decuplet electromagnetic form factors magnetic form factors Beg, Lee, Pais, 1964 charge form factors Buchmann, Hernandez, Faessler, 1997 Buchmann, 2000

  35. Definition of C2/M1 ratio Insert form factor relations C2/M1 expressed via neutron elastic form factors A. J. Buchmann, Phys. Rev. Lett. 93 (2004) 212301

  36. neutron charge radius 4th moment of n(r) Use two-parameter Galster formula for GCn Grabmayr and Buchmann, Phys. Rev. Lett. 86 (2001) 2237

  37. d=2.80 d=1.75 d=0.80 JLab 2006 data: electro-pionproduction curves: elastic neutron form factors Maid 2007 reanalysis from: A.J. Buchmann, Phys. Rev. Lett. 93, 212301 (2004).

  38. New MAID 2007 analysis C2/M1(Q²)=S1+/M1+(Q²)   MAID 2003  .  .  Buchmann 2004 MAID 2007 from: Drechsel, Kamalov, Tiator, EPJ A34 (2007) 69

  39. New MAID 2007 analysis   MAID 2003  .  .  Buchmann 2004 MAID 2007 JLab data analysis  MAID 2007 reanalysis of same JLab data

  40. d=2.8 d=0.8 Limiting values best fit of data (MAID 2007) with d=1.75

  41. 5. Group theoretical argument

  42. ( 0-body 1-body 2-body 3-body ) first order second order third order Spin-flavor selection rules M 0 only if[R]transforms according to one of the representations R on the right hand side

  43. SU(6) symmetry breaking operators • First order SU(6) symmetry breaking operators transforming according to the 35 dimensional representation generated • by a antiquark-quark bilinear 6* x 6 = 35 + 1 • do not split the octet and decuplet mass degeneracy • give a zero neutron charge radius • give a zero N   quadrupole moment • We need second and third order SU(6) symmetry breaking • operators transforming according to the higher dimensional • 405 and 2695 reps in order to describe the above phenomena.

  44. SU(6) symmetry breaking Second order spin-flavor symmetry breaking operators can be constructed from direct products of two first order operators. However, only the 405 dimensional representation appears in the the direct product 56* x 56. Therefore, an allowed second order operator must transform according to the 405.

  45. Decomposition of SU(6) tensor 405 into SU(3) and SU(2) tensors scalar J=0 vector J=1 tensor J=2 First entry: dimension of SU(3) flavor operator Second entry: dimension of SU(2) spin operator 2J+1 Charge operator transforms as flavor octet. Coulomb multipoles have even rank (odd dimension) in spin space. Spin scalar (8,1) and spin tensor (8,5) are the only components of the SU(6) tensor 405 that can then contribute to [2].

  46. Decomposition of SU(6) tensor 2695 into SU(3) and SU(2) tensors First entry: dimension of SU(3) flavor operator Second entry: dimension of SU(2) spin operator 2J+1 Charge operator transforms as flavor octet. Coulomb multipoles have even rank (odd dimension) in spin space. Spin scalar (8,1) and spin tensor (8,5) are the only components of the SU(6) tensor 2695 that can then contribute to [3].

  47. reduced matrix element same value for the entire multiplet 56 i... components of initial 56 f... components of final 56 ... components of operator provides relations between matrix elements of different components of 405 tensor and states Wigner-Eckart theorem This explains why spin scalar (charge monopole) and spin tensor (charge quadrupole) operators and their matrix elements are related. A. Buchmann, AIP conference proceedings 904 (2007)

  48. decuplet octet Construction of 56 tensor examples:

  49. Explicit construction of 35 tensor alltogether 35 generators 405 tensor:

  50. 6. Summary

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