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 protonrp
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 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)

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

6. 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
...

7. Inelastic electron-nucleon scattering e‘ N‘  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
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 T3 SU(3) flavor symmetry octet decuplet J=1/2 J=3/2 W S+ S- S0 L0 X0n 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

13. 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

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

15. Gell-Mann & Okubo mass formula
baryon octet
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. SU(6) spin-flavor supermultiplet
baryon
supermultiplet

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

19. 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

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

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, (2000);
PRD66, (2002);
PRD67, (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, (2002);
Eur. Phys. J. A 35, 267 (2008)

27. 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?

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. 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

30. 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

31. 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)
Tiator et al., EPJ A17 (2003) 357
theory
Buchmann et al., PRC 55 (1997) 448
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
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)

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

37. 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, (2004).

38. 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

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

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

41. 5. Group theoretical argument

42. 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

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 =
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. 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)

48. Construction of 56 tensor
decuplet
octet
examples:

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

50. 6. Summary

51. 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.

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