Structure of strange baryons
Alfons BuchmannUniversity of Tuebingen
1. Introduction
2. SU(6) spin-flavor symmetry
3. Observables
4. Results
5. Summary
Hyperon 2006, Mainz, 9-13 October 2006
1. Introduction
Hadrons with nonzero strangeness
• add a new dimension to matter
• provide evidence for larger symmetries
• are a testing ground of quantum field theories
• have important astrophysical implications
• improve our understanding of ordinary matter
yet
little is known about their spatial structure,such as their
size and shape
2. Strong interaction symmetries
Strong interactions are
approximately invariant under
SU(3) flavor and SU(6) spin-flavor symmetry transformations.
These symmetries lead to:
• conservation laws • degenerate hadron multiplets • relations between observables
n p
S
T3
0
-3
-2
-1
-1/2 +1/2-1 0 +1 -3/2 -1/2 +3/2+1/2
J=1/2 J=3/2
SU(3) flavor multiplets
octet decuplet
Group algebra relates symmetry breaking within a multiplet
(Wigner-Eckart theorem)
Y hyperchargeS strangeness
T3 isospin
4-)(MMM
2
210
Y1TTY1M
BSY
symmetry breaking alongstrangeness direction byhypercharge operator Y
Relations between observables
M0, M1, M2 from experiment
M3M4
1MM
2
1N
Gell-Mann & Okubo mass formula
MMM-Mor
M-MM-MMM
**
****
3
Equal spacing rule
SU(6) spin-flavor symmetry
ties together SU(3) multiplets
with different spin and flavor
into
SU(6) spin-flavor supermultiplets
SU(6) spin-flavor supermultiplet
spinflavor spinflavor
4,102,856
S
T3
ground state baryon supermultiplet
)(M-)(MMM 3
2
210 1JJ
4
Y1TTY1M
Gürsey-Radicati mass formula
Relations between octet and decuplet masses
npΔΔMMMM 0
SU(6) symmetry breaking part
e.g.
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 hierachy in importance of one-, two-, and three-quark operators, i.e., higher order symmetry breaking operatorsare suppressed by higher powers of 1/NC.
Cs N
1α
2
2
fC
222
ΛQ
ln)N2N(11
π124π
)(Qg)(Q
1/NC expansion of QCD processes
CN
1~g
CN
1~g
CN1
O
two-body
2O
CN
1
three-body
strong coupling
SU(6) spin-flavor symmetry breakingby spin-flavor dependent
two- and three-quark operators
These lift the degeneracy between octet and decuplet baryons.
imiσ
jmjσ
imiσ
jmjσ
ji
u
mm
m 2
jiji σσσσ
SU(3) symmetry breaking
s
u
mm
r
in the following r=0.6
SU(3) symmetry breaking parameter
O[i] all invariants in spin-flavor space that are allowed by Lorentz invariance and internal symmetries of QCD
]3[]2[]1[ CBA
one-body two-body three-body
General spin-flavor operator O
Constants A, B, and C
parametrize
orbital and color matrix elements.
They are determined from experiment.
3. Observables
Baryon structure information encoded e.g. in charge form factor:
• size (charge radii)• shape (quadrupole moments)
shapesizecharge
Qq61
rq61
1)(qρ B22
B22
B
Multipole expansion of baryon charge density
C0Bρ C2
Bρ
)q(Y)qρ(dΩ~ρ e.g. 20q
C2B ˆ
Charge radius operator
body3body2body1
σσeCσσeBeA
dqqdρ
6r
3
kjijik
3
jijii
3
1ii
0q2
2B2
B2
CN1
O
2O
CN
1
0O
CN
1
ei...quark chargei...quark spin
1-quark operator 2-quark operators(exchange currents)
Origin of these operator structures
SU(6) spin-flavor symmetry breakingby spin-flavor dependent
two- and three-quark operators
imiσ
jmjσ
imiσ
jmjσ
ei
ek
e.g. electromagnetic current operator ei ... quark charge i ... quark spin mi ... quark mass
3-quark current 2-quark current
What is the shape of octet and decuplet baryons?
A. J. Buchmann and E. M. Henley, Phys. Rev. C63, 015202 (2001)
prolate oblate
Quadrupole moment operator
jijziz
3
kjik
jijziz
3
jii
σσσσ3eC
σσσσ3eBQ
two-body
three-body
no one-body contribution
CN1
O
2O
CN
1
4. Results
Some relations between charge radii
0)(r)(r2)(r
0)(r)(r2)(r
(*) r)O(1(n)r)(Σr(p)r
(n)r2
1)(Λr
(n)r)(Δr(p)r)(Δr
*20*2*2
2022
222
202
20222
from (*) r²(-)=0.676 (66) fm² (A. Buchmann, R. F. Lebed, Phys. Rev. D 67, 016002 (2003)) theoretical range due to size of SU(3) flavor symmetry breaking
r²(-)=0.61(12)(9) fm² (Selex experiment, I. Eschrich et al. PLB522, 233(2001))
equal spacing rule
A. J. B., R. F. Lebed, Phys. Rev. D 62, 096005 (2000)
2
2
2
4C)r4B4C4B
)/3rC)(r4B0
)/3r4C)(2r4B4C4B
4r)]/3-4C(1-r)[4B(24C4B
r)/32C)(1B0
2r)/34C)(14B4C4B
8C8B8C8B
4C4B4C4B
00
4C4B4C4B
baryon
(
2(
(
(2
(
0*
*
*
0*
*
0
1)Q(r1)Q(r
Decuplet quadrupole moments
Similar tablefor
octet-decuplet transition quadrupole moments
Relations between observables
There are 18 quadrupole moments, 10 diagonal and 8 tansitional.
These are expressed in terms of two constants B and C.
There must be 16 relations between them.
12 relations out of 16 hold irrespective of how badly SU(3) flavor symmetry is broken.
A. J. Buchmann and E. M. Henley, Phys. Rev. D65, 07317 (2002)
rulespacingequal0)Q(Q)Q(Q3
symmetryisospin
0QQ2Q0Q2Q0Q0QQ
ΔΩΣΞ
ΣΣΣ
ΔΔ
Δ
ΔΔ
**
**0*
-
0
-
Diagonal quadrupole moments
These and the following 7 relations hold irrespective of how badly SU(3) is broken.
0
3
2
2
1
2
1
2
1
6
1
2
1
022
2
00000*
*
0000
0
0
**
***
**0*
**
**
**0*
Σ
Σ
ΣΣ
ΣΣ
ΣΣ
ΣΣΣΣ
np
QQQ
0QQQQ
0QQQ
QQQQ
0QQQ
0QQ2Q
0QQ
p
Tra
nsiti
on q
uadr
upol
e m
omen
ts
4 r-dependent relations
0QQr
0QQ2Qr2
0QQ1r261
0QQ12r31
ΩΔ2
ΞΞΞΔp2
ΣΔp
ΣΔ
*0*00
*0
*
Numerical results
Determination of constant B from relation between
N transition quadrupole moment and
neutron charge radius rn2
A. Buchmann, E. Hernandez, A. Faessler, Phys. Rev. C 55, 448 (1997)
4r
BB22
Q2
1r
2
1Q
2n
2nΔN
comparison with experiment
22nΔN fm0.082(2)r
2
1Q
experiment2
ΔN fm0.0846(33)Q
experiment2
ΔN fm0.1080(90)Q
theory
Tiator et al.(2003)
Blanpied et al.(2001)
Buchmann et al.(1996)
data: electro-pionproductioncurves: elastic neutron form factors
A.J. Buchmann, Phys. Rev. Lett. 93, 212301 (2004).
0.0350.080ΞΞ
0.0130ΞΞ
0.0900.080ΣΣ
0.0420.069ΣΛ
0.0350.040ΣΣ
0.0210ΣΣ
0.0800.080Δn
0.0800.080Δp
baryon
0*0
*
*
0*0
0*0
*
0
1)Q(r1)Q(r
transition quadrupole moments
0.041 0.113
0.009- 0
0.059 0.113
0.105- 0.113-
0.008- 0
0.083 0.113-
0.226- 0.226-
0.113- 0.113-
0 0
0.113 0.113
baryon
0*
*
*
0*
*
0
1)Q(r 1)Q(r
diagonal quadrupole moments
Intrinsic quadrupole moment of nucleon
0 2n0 rQ
bab)-2(a
δ
2ba
r
δr54
ba52
Q 2220
a/b=1.1large!
Use r= 1 fm, Q0= 0.11 fm², then solve for a and b
A. J. Buchmann and E. M. Henley, Phys. Rev. C63, 015202 (2001)
5. Summary
• SU(6) spin-flavor analysis relations between baryon quadrupole moments
• decuplet baryons have negative quadrupole moments of the order of the neutron charge radius large oblate intrinsic deformation
• Experimental determination of Q is perhaps possible with Panda detector at GSI