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