Excited nucleon electromagnetic form factors from broken spin-flavor symmetry

1
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
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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
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
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?
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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,
Neeman1962 Flavor symmetry combines hadron
isospin multiplets with different T and Y into
larger multiplets, e.g., flavor octet and
flavor decuplet.
12
S
SU(3) flavor symmetry
0
-1
-2
decuplet
octet
-3
J3/2
J1/2
T3
-1/2
1/2
-1
0
1
-3/2
-1/2
3/2
1/2
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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
Gell-Mann Okubo mass formula
baryon decuplet equal spacing rule
(?M/M)exp 1
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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)
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SU(6) spin-flavor supermultiplet
baryon supermultiplet
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Gürsey-Radicati SU(6) mass formula
SU(6) symmetry breaking term
Relations between octet and decuplet baryon
masses
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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/D2/3

Higher predictive power than independent spin and
flavor symmetries
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3. Spin-flavor symmetry and 1/N expansion of
QCD
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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)
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1/NC expansion of QCD processes
NC ... number of colors
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SU(6) spin-flavor as QCD symmetry
This results in the following hierarchy
O1 (1/NC0) gt O2 (1/NC1) gt
O3 (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.
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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)
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4. Electromagnetic form factor relations
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For NC3 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)
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Spin-flavor operator O
Oi ? 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?
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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
SU(6) spin-flavor symmetry breaking
SU(6) symmetry breaking via spin and flavor
dependent two- and three-quark currents
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Neutron and N?? charge form factors
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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
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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
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Use two-parameter Galster formula for GCn
Grabmayr and Buchmann, Phys. Rev. Lett. 86 (2001)
2237
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data electro-pionproduction curves elastic
neutron form factors
Maid 2007 reanalysis
from A.J. Buchmann, Phys. Rev. Lett. 93, 212301
(2004).
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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
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Limiting values
best fit of data (MAID 2007) with d1.75
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5. Group theoretical argument
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Spin-flavor selection rules
M ? 0 only if ?R transforms according to one of
the representations R on the right hand side
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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.

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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.
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Decomposition of SU(6) tensor 405 into SU(3) and
SU(2) tensors
scalar J0
vector J1
tensor J2
First entry dimension of SU(3) flavor operator
Second entry dimension of SU(2) spin operator
2J1
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.
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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
2J1
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.
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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)
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Construction of 56 tensor
examples
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Explicit construction of 35 tensor
alltogether 35 generators
405 tensor
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6. Summary
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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.
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END Thank you for your attention.