Electromagnetic multipole moments

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Electromagnetic multipole moments of baryons
Alfons Buchmann University of Tübingen

• Introduction
• Method
• Observables
• Results
• Summary

NSTAR 2007, Bonn, 5-8 September 2007
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1. Introduction
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What can we learn from electromagnetic multipole
moments?
Electromagnetic multipole moments of baryons are
interesting observables.

• They are directly connected with spatial
  distributions of
• charges and currents inside baryons.
• They provide fundamental information on baryonic
• structure,
• size,
• shape .

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Example Proton magnetic moment
Experimental discovery by Frisch and Stern in
1933 ?p (exp) ? 2.5 ?N ? ?p is very
different from value predicted by Dirac equation
?p (Dirac) 1.0 ?N Conclusion proton has an
internal structure. Measurement of proton charge
radius by Hofstadter et al. in 1956 r²p(exp) ?
(0.81 fm)²
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Multipole expansion of charge distribution ? for
a system of point charges
J0 J1 J2
J3
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• Advantage of multipole expansion
• multipole operators MJ transfer definite
  angular momentum and parity
• angular momentum and parity selection rules
  apply,
• e.g. only even charge multipoles
• few multipoles suffice to describe charge and
  current density

Baryon B with total angular momentum Ji ? 2
Ji 1 electromagnetic multipoles
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Multipole expansion of baryon charge density
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Example N ?? quadrupole moment Recent
electron-proton scattering experiments provide
evidence for a nonzero p? ?(1232) transition
quadrupole moment
data
theory
neutron charge radius
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What can be learned from these results?

• both N and ? have nonspherical charge
  distributions
• to learn more about the geometric shape of both
  systems,
• one has to calculate their intrinsic quadrupole
  moments Q0

? concentrated along z-axis 3z²- term dominates
Q0 gt 0
? concentrated in equatorial plane r² -term
dominates Q0 lt 0
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Intrinsic (Q0) vs. spectroscopic (Q) quadrupole
moment
J1/2 ? Q0 even if Q0 ?0.
projection factor
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Buchmann and Henley (PRC 63 (2001) 015202) have
calculated Q0 in three nucleon models. All
models agree as to the sign of Q0. For example,
in the quark model they find
Q0 (N) -rn2 gt 0 Q0 (?) rn2 lt 0.
Neutron charge radius determines sign and size
of N and ? intrinsic quadrupole moments.
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Interpretation in pion cloud model
Q0 gt 0
Q0 lt 0
prolate
oblate
A. J. Buchmann and E. M. Henley, Phys. Rev. C63,
015202 (2001)
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Summary In the last decade interesting
experimental and theoretical results on the
charge quadrupole structure of the N and ?
system have been obtained. Based on model
calculations of the intrinsic quadrupole moments
of both systems, it has been proposed that
the charge distributions of N(939) and ?(1232)
possess considerable prolate and oblate
deformations.
Review V. Pascalutsa, M. Vanderhaeghen, S.N.
Yang, Phys. Rep. 437, 125 (2007)
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What about higher multipole moments?
Presently, practically nothing is known
about magnetic octupole moments of decuplet
baryons.
Information needed to reveal structural details
of spatial current distributions in baryons.
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2. Method
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General parametrization method

• Basic idea
• 1) Define for observable at hand
• a QCD operator Ô and QCD eigenstates ?B?
• 2) Rewrite QCD matrix element ltBÔB? in terms of
  
• spin-flavor space matrix elements
• including all spin-flavor operators allowed
  by
• Lorentz and inner QCD symmetries (G.
  Morpurgo, 1989)

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General spin-flavor operator O
constants A, B, and C ? parametrize orbital and
color
matrix elements. These are
determined from experiment.
Oi ? all allowed invariants in spin-flavor
space
Which spin-flavor operators are allowed? Operator
structures determined from symmetry principles.
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Strong interaction symmetries
Strong interactions are approximately invariant
under SU(3) flavor and SU(6) spin-flavor
symmetry transformations.
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S
SU(3) flavor multiplets
0
-1
-2
octet
decuplet
-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(6) spin-flavor symmetry
ties together SU(3) multiplets with different
spin and flavor into SU(6) spin-flavor
supermultiplets
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SU(6) spin-flavor supermultiplet
ground state baryon supermultiplet
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SU(6) spin-flavor is a symmetry of QCD
SU(6) symmetry is exact in the large NC limit of
QCD. For finite NC, the symmetry is broken. The
symmetry breaking operators can be classified
according to powers of 1/NC attached to
them. This leads to a hierarchy in importance of
one-, two-, and three-quark operators, i.e.,
higher order symmetry breaking operators are
suppressed by higher powers of 1/NC.
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1/NC expansion of QCD processes
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SU(6) spin-flavor symmetry breaking by
spin-flavor dependent two- and three-quark
operators
sk
m
k
These lift the degeneracy between octet and
decuplet baryons.
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Example quadrupole moment operator
no one-quark contribution
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SU(6) spin-flavor symmetry breaking by
spin-flavor dependent two- and three-quark
operators
e.g. electromagnetic current operator
ei ... quark charge si ... quark spin
mi ... quark mass
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Origin of these operator structures
1-quark operator
2-quark operators (exchange currents)
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Summary
Parametrization method

• based on symmetries of QCD
• quark-gluon dynamics reflected in 1/Nc hierarchy
  
• employs complete operator basis in SU(6)
  spin-flavor space

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3. Observables
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Magnetic octupole moments of decuplet baryons
Decuplet baryons have Ji3/2 ? 4
electromagnetic multipoles
charge monopole moment q?1 charge
quadrupole moment Q? ? rn2 magnetic dipole
moment ?? ? ?p magnetic octupole moment
?? ?
Example ?(1232)
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Definition of magnetic multipole operator
special cases
J1
J3
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Magnetic octupole moment ? analogous to charge
quadrupole moment Q
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Magnetic octupole moment measures the deviation
of the magnetic moment distribution from
spherical symmetry
? gt 0 magnetic moment density is prolate
? lt 0 magnetic moment density is oblate
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Construction of octupole moment operator ? in
spin-flavor space

• ? MJ3 ... tensor of rank 3
• rank 3 tensor built from quark spin operators ?1
  
• ? ?i1 ? ?j1 ? ?k1
• ?
• involves Pauli spin matrices of three different
  quarks

? is a three-quark operator
?
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If two Pauli spin operators in ? had the same
particle index e.g. ij1 ? reduction to a single
spin matrix due to the SU(2) spin commutation
relations
? is neccessarily a three-quark operator
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Allowed spin-flavor operators
ei...quark charge ?i...quark spin
two-quark quadrupole operator multiplied by spin
of third quark
three-quark quadrupole operator multiplied by
spin of third quark
Do we need both?
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Spin-flavor selection rules
M ? 0 only if ?R transforms according
to representations found in the
product
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There is a unique three-quark operator transformin
g according to 2695 dim. rep. of SU(6). ? We
can take either one of the two spin-flavor
structures. Explicit calculation of both
operators give the same results for decuplet
baryons.
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4. Results
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Matrix elements
qB baryon charge
In the flavor symmetry limit, magnetic octupole
moments of decuplet baryons are proportional to
the baryon charge
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Introduce SU(3) symmetry breaking
SU(3) symmetry breaking parameter
flavor symmetry limit r1
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Baryon magnetic octupole moments ?
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Efficient parametrization of baryon octupole
moments in terms of just one constant C.
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Relations among octupole moments
There are 10 diagonal octupole moments. These
are expressed in terms of 1 constant C. ? There
must be 9 relations between them.
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Diagonal octupole moment relations
These 6 relations hold irrespective of how
badly SU(3) is broken.
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3 r-dependent relations
Typical size of SU(3) symmetry breaking
parameter rmu/md0.6
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Information on the geometric shape of current
distribution in baryons from sign and magnitude
of ?.
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Numerical results
Determine sign and magnitude of ? using various
approaches 1) pion cloud model 2) experimental N
? N(1680) transition octupole moment 3) current
algebra approach
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Estimate of ? in pion cloud model
spherical ? core surrounded by P-wave ? cloud
?
? cloud
bare ? core
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Electromagnetic current of pion cloud model
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Insert spatial current density J(r) ? only ??N
interaction current contributes to ?
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Matrix element
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Comparison with experimental N ? N(1680)
transition octupole moment
Ji1/2 ? Jf5/2 transition between positive
parity states only charge quadrupole and
magnetic octupole can contribute
Rewrite helicity amplitudes in terms of charge
quadrupole and magnetic octupole form factors
GM3 (0) ? 0.16 fm³ GC2 (0) Q 0.20 fm²
Order of magnitude agrees with estimate in pion
cloud model
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We can now determine the constant C and predict
the sign and size of ? for all decuplet baryons

• (?) 4C -0.16 fm³
• C-0.04 fm³

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diagonal octupole moments fm³
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5. Summary

• magnetic octupole moments provide unique
  opportunity to learn more about three-quark
  currents
• SU(6) spin-flavor analysis ?
  relations between baryon octupole moments
• estimated sign and magnitude of ? in pion cloud
  model ? Q ?
• decuplet baryons have negative octupole moments
• oblate deformation of magnetic moment
  distribution
• experimental determination of ? ? is a challenge

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END Thank you for your attention.
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Interpretation in quark model
Two-quark spin-spin operators are repulsive for
quark pairs with spin 1.
In the ?0, all quark pairs have spin 1. Equal
distance between down-down and up-down pairs. ?
planar (oblate) charge distribution ? zero
charge radius.
In the neutron, the two down quarks are in a
spin 1 state, and are pushed further apart than
an up-down pair. ? elongated (prolate) charge
distribution ? negative neutron charge radius.
A. J. Buchmann, Can. J. Phys. 83 (2005)
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Estimate of degree of nonsphericity
Use r 1 fm, Q0 0.11 fm², then solve for a and b
a/b1.1 large!
A. J. Buchmann and E. M. Henley, Phys. Rev. C63,
015202 (2001)
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Another configuration
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Relations between observables
Group algebra relates symmetry breaking within a
multiplet (Wigner-Eckart theorem)
Y hypercharge S strangeness
T3 isospin
M0, M1, M2 from experiment
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Gell-Mann Okubo mass formula
Equal spacing rule
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Gürsey-Radicati mass formula
SU(6) symmetry breaking part
Relations between octet and decuplet masses
e.g.
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Example charge radius operator
ei...quark charge ?i...quark spin
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Decomposition of 2695 into irred. reps. of
SU(3)F and SU(2)S subgroups

• (1,7)
  (1,3)
• (8,7) 2 (8,5)
  (8,3) (8,1) ?

• There is only one (8,7) in the decomposition of
  2695
• unique three-quark magnetic octupole operator
• transforming as flavor octet.

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Relations between N? ? transition form
factors and elastic neutron form factors
Buchmann, Phys. Rev. Lett. 93 (2004) 212301
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Definition of C2/M1 ratio
neutron elastic form factor ratio
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data electro-pionproduction curves elastic
neutron form factors
A.J. Buchmann, Phys. Rev. Lett. 93, 212301
(2004).