Excited nucleon electromagnetic form factors from broken spinflavor

Excited nucleon electromagnetic form factors from broken spin-flavor symmetry Alfons Buchmann Universität Tübingen 1. Introduction 2. Strong interaction symmetries 3. SU(6) and 1/N expansion of QCD 4. Electromagnetic form factor relations 5. Group theoretical argument 6. Summary Nstar 2009, Beijing, 20 April 2009

1. Introduction

Spatial extension of proton rp radial distribution Measurement of proton charge radius rp(exp) = 0. 862(12) fm Simon et al. , Z. Naturf. 35 a (1980) 1

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

Geometric shape of proton charge distribution angular distribution Extraction of N transition quadrupole (C 2) moment from data Q N (exp) = -0. 0846(33) fm² Tiator et al. , EPJ A 17 (2003) 357

Proton excitation spectrum N*(1520) at cit (1232) E 1 , M 2 ex J=1/2+ bit al N(939) or C 0, M 1 radial excitation N*(1440) ne i p s tio a t i c n x so i 2 n i C p , s 2 E , M 1 C 2 multipole transition to (1232) is sensitive to angular shape of nucleon ground state J=3/2 - J=3/2+ . . .

Inelastic electron-nucleon scattering N‘ e‘ Q e N 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 Strong interactions are approximately invariant under • SU(2) isospin, • SU(3) flavor, • SU(6) spin-flavor symmetry transformations.

SU(3) flavor symmetry Gell-Mann, Ne‘eman 1962 Flavor symmetry combines hadron isospin multiplets with different T and Y into larger multiplets, e. g. , flavor octet and flavor decuplet.

S SU(3) flavor symmetry n 0 -1 S - p S 0 - S + 0 S *- + S ++ S*+ *0 L 0 X -2 - -3 -1 -1/2 X X*- 0 octet decuplet J=1/2 J=3/2 0 +1/2 +1 -3/2 -1/2 X*0 T 3 +1/2 +3/2

SU(3) symmetry breaking Symmetry breaking along strangeness direction through hypercharge operator Y Y hypercharge S strangeness T 3 isospin mass operator SU(3) invariant term first order SU(3) symmetry breaking second order SU(3) symmetry breaking M 0, M 1, M 2 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 S baryon supermultiplet T 3

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

Successes of SU(6) • explains why Gell-Mann Okubo formula works for octet and decuplet baryons with the same coefficients M 0, M 1, M 2 • 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/NC 0) > O[2] (1/NC 1) > O[3] (1/NC 2) 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, PRD 62, 096005 (2000); PRD 66, 056002 (2002); PRD 67, 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)

Spin-flavor operator O one-quark two-quark three-quark O[i] 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 ek ei 2 -quark current 3 -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 Buchmann, Hernandez, Faessler, PRC 55, 448 N quadrupole moment neutron charge radius spin tensor spin scalar

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 A 17 (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 C 2/M 1 ratio Insert form factor relations C 2/M 1 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 4 th moment of n(r) Grabmayr and Buchmann, Phys. Rev. Lett. 86 (2001) 2237

data: electro-pionproduction curves: elastic neutron form factors d=2. 80 d=1. 75 d=0. 80 JLab 2006 from: A. J. Buchmann, Phys. Rev. Lett. 93, 212301 (2004). Maid 2007 reanalysis

New MAID 2007 analysis C 2/M 1(Q²)=S 1+/M 1+(Q²) MAID 2003 . . Buchmann 2004 MAID 2007 from: Drechsel, Kamalov, Tiator, EPJ A 34 (2007) 69

New MAID 2007 analysis MAID 2003 . . Buchmann 2004 MAID 2007 JLab data analysis MAID 2007 reanalysis of same JLab data

Limiting values d=2. 8 best fit of data (MAID 2007) with d=1. 75 d=0. 8

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 ( 0 -body 1 -body second order 2 -body third order 3 -body )

SU(6) symmetry breaking operators 1. 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 2. 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 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 2 J+1 Charge operator transforms as flavor octet. Coulomb multipoles have even rank (odd dimension) in 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 2 J+1 Charge operator transforms as flavor octet. Coulomb multipoles have even rank (odd dimension) in 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 i. . . f. . . components of initial 56 components of final 56 components of operator provides relations between matrix elements of different components of 405 tensor and states 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 C 2/M 1 ratio in N transition predicted from empirical elastic neutron form factor ratio GCn/GMn agrees in sign and magnitude with C 2/M 1 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.

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