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

1. Introduction

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

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

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

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

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

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?

2. Strong interaction symmetries

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

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.

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

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

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

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

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)

SU(6) spin-flavor supermultiplet baryon supermultiplet

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

Higher predictive power than independent spin and flavor symmetries 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 proton/neutron magnetic moment ratio Higher predictive power than independent spin and flavor symmetries

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

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)

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

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.

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)

4. Electromagnetic form factor relations

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)

Which spin-flavor operators are allowed? Spin-flavor operator O one-quark two-quark three-quark 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?

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

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

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

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

Including three-quark operators

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

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

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

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

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

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

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

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

5. Group theoretical argument

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

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.

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.

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].

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].

Wigner-Eckart theorem reduced matrix element same value for the entire multiplet 56 provides relations between matrix elements of different components of 405 tensor and states i... components of initial 56 f... components of final 56 ... components of operator 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)

Construction of 56 tensor decuplet octet examples:

Explicit construction of 35 tensor alltogether 35 generators 405 tensor:

6. Summary

Summary Broken SU(6) spin-flavor symmetry leads to a relation between the N  quadrupole and the neutron charge form factors. The C2/M1 ratio in N transition predicted from empirical elastic neutron form factor ratio GCn/GMn agrees in sign and magnitude with C2/M1 data over a wide range of momentum transfers (see MAID 2007 analysis). General group theoretical arguments based on the transformation properties of the states and operators and the Wigner-Eckart theorem support previous derivations of connection between N  transition and nucleon ground state form factors.

Thank you for your attention. END Thank you for your attention.