Electromagnetic multipole momentsof baryons

Alfons BuchmannUniversity of Tübingen

1. Introduction

2. Method

3. Observables

4. Results

5. Summary

NSTAR 2007, Bonn, 5-8 September 2007

1. Introduction

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 .

Experimental discovery by Frisch and Stern in 1933p (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 1956r²p(exp) (0.81 fm)²

Example: Proton magnetic moment

Multipole expansion of charge distribution for a system of point charges

q0 + d1 + Q2 + 3 + ~

octupolequadrupoledipolemonopole

+q

–q

+q

+q

+q –q

–q –q

–q

–q

–q

+q

+q

+q

+q

+++

J=0 J=1 J=2 J=3

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

Example: N(939) Ji=1/2charge monopole C0magnetic dipole M1

Baryon B with total angular momentum Ji

2 Ji+ 1 electromagnetic multipoles

fifiiJ

f JJJJJif0JBMJB

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

Example: N Example: N quadrupole moment quadrupole moment

Recent electron-proton scattering experiments provide evidence Recent electron-proton scattering experiments provide evidence for a nonzero pfor a nonzero p +(1232) transition quadrupole moment(1232) transition quadrupole moment

2n(1232)p

fm )-0.0821(20rQ 2

2

1Buchmann et al., PRC 55 (1997) 448

data

2)33(0846.0 fmQ (exp)(1232)p

Tiator et al., EPJ A17 (2003) 357

2)9(108.0 fmQ (exp)(1232)p

Blanpied et al., PRC 64 (2001) 025203

theory

neutron charge radius

What can be learned from these results?

Definition of intrinsic quadrupole moment

)3()( 2230 rzrdrQ

• 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 in equatorial plane r² -term dominates Q0 < 0

concentrated along z-axis 3z²- term dominates Q0 > 0 prolate

oblate

Intrinsic (Q0) vs. spectroscopic (Q)quadrupole moment

0

2

02

02 12

131cos3

2

1cos Q

)J(J

)J(JJQΘQΘ) (PQ z

2cos

J

JΘ z

body-fixed frame

lab frame

projection factor

J=1/2 Q=0

even if Q0 0.

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

Neutron charge radius determines sign and size

of N and intrinsic quadrupole moments.

Q0 (N) = -rn2 > 0

Q0 () = rn2 < 0.

Interpretation in pion cloud model

A. J. Buchmann and E. M. Henley, Phys. Rev. C63, 015202 (2001)

prolate oblate

Q0 > 0 Q0 < 0

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 thecharge 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)

What about higher multipole moments?

Presently, practically nothing is known about

magnetic octupole momentsof decuplet baryons.

Information needed to reveal structural details of spatial current distributions in baryons.

2. Method

General parametrization method

Basic idea1) Define for observable at hand a QCD operator Ô and QCD eigenstates

2) Rewrite QCD matrix element B|Ô|B in terms of spin-flavor space matrix elements including all spin-flavor operators allowed by Lorentz and inner QCD symmetries (G. Morpurgo, 1989)

B | ... exact QCD state containing q, (qq), (qq)2, g, ...

| B ... auxilliary 3 quark QCD state

V ... unitary operator dresses auxilliary state with quark-antiquark pairs and gluons

W | B ... spin-flavor state

O ... spin-flavor operator

W||W |V O V | B | O | B B B B B Oˆˆ

QCD matrix element spin-flavor matrix elementV ...unitary operator

B ...auxilliary 3q states

O[i] all allowed invariants in spin-flavor space

]3[]2[]1[ ΟCΟBΟAΟ

one-body two-body three-body

General spin-flavor operator O

constants A, B, and C parametrize orbital and color matrix elements. These are determined from experiment.

Which spin-flavor operators are allowed?Operator structures determined from symmetry principles.

Strong interaction symmetries

Strong interactions are

approximately invariant under

SU(3) flavor and SU(6) spin-flavor symmetry transformations.

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

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

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 operatorsare suppressed by higher powers of 1/NC.

1/NC expansion of QCD processes

CN

1~g

CN

1~g

CN1

O

two-body

2O

CN

1

three-body

Cs N

1α

2

2

fC

222

ΛQ

ln)N2N(11

π124π

)(Qg)(Qstrong 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

km

σk

Example: quadrupole moment operator

jijziz

3

kjik

jijziz

3

jii

σσσσ3eC

σσσσ3eBQ

no one-quark contribution

two-body

CN1

O

three-body

2O

CN

1

SU(6) spin-flavor symmetry breakingby spin-flavor dependent

two- and three-quark operators

e.g. electromagnetic current operator ei ... quark charge i ... quark spin mi ... quark mass

imiσ

jmjσ

imiσ

jmjσ

ei

ek

3-quark current 2-quark current

1-quark operator 2-quark operators(exchange currents)

Origin of these operator structures

• based on symmetries of QCD

• quark-gluon dynamics reflected in 1/Nc hierarchy

• employs complete operator basis in SU(6) spin-flavor space

Summary

Parametrization method

3. Observables

Magnetic octupole moments ofdecuplet baryons

Decuplet baryons have Ji=3/2 4 electromagnetic multipoles

Example: +(1232)

charge monopole moment q+=1

charge quadrupole moment Q+

rn2

magnetic dipole moment + p

magnetic octupole moment + = ?

Definition of magnetic multipole operator

)()ˆ(1

2

12

4)0( 3 rJrrYrrd

J

M

JM J

MJNJ

M

zN

rJrrdM

M)(

2

1

2

)0( 310

special cases:

J=1

J=3 22330 3)(

8

3

2

)0(rzrJrrd

M

Mz

N

Magnetic octupole moment analogous to

charge quadrupole moment Q

223 3)(8

3rzrJrrd z

223 3)( rzrrdQ

replace magnetic moment density )(rJr

by charge density )(r

Magnetic octupole moment measures the deviation of the magnetic moment distribution from spherical symmetry

> 0 magnetic moment density is prolate

< 0 magnetic moment density is oblate

Construction ofoctupole moment operator

in spin-flavor space

~ MJ=3 ... tensor of rank 3

rank 3 tensor built from quark spin operators 1

~ i1 j

1 k1

involves Pauli spin matrices of three different quarks

is a three-quark operator

If two Pauli spin operators in had the same particle index

e.g. i=j=1

reduction to a single spin matrixdue to the SU(2) spin commutation relations

mklmlk σεi 111 2],[

is neccessarily a three-quark operator

Allowed spin-flavor operators

kjijziz

3

kjii σσσσσ3eC

]3[

kjijziz

3

kjik σσσσσ3eC'

]3[

ei...quark chargei...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?

Spin-flavor selection rules

56Ω56M R

2695405351 5656

M 0 only if R transforms according to

representations found in the product

0-body 3-body 2-body 1-body

There is a unique three-quark operatortransforming 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.

4. Results

Matrix elements

BBBB qCWW 4]3[

qB baryon charge

In the flavor symmetry limit, magnetic octupole moments of decuplet baryons are

proportional to the baryon charge

ji

u

mm

m2

3

jiji σσσσ

Introduce SU(3) symmetry breaking

s

u

mm

r SU(3) symmetry breaking parameter

flavor symmetry limit r=1

3

32

32

2

2

2

4Cr4C

)/3r-rC(2r0

)/3rr4C(r4C

)/3rr4C(2 4C

)/32r-r(1 2C0

)/3rr4C(14C

8C8C

4C4C

00

4C4C

baryon

4

2

0*

*

*

0*

*

0

1)(r1)(r

Baryon magnetic octupole moments

Efficient parametrization

of

baryon octupole moments

in terms of

just one constant C.

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.

0**0

ΔΞ

ΔΩΣΞ

ΣΣΣ

ΔΔ

Δ

ΔΔ

)()(

rulespacingequal0)()(3

symmetryisospin

020200

**

**

**0*

-

0

-

Diagonal octupole moment relations

These 6 relations hold irrespective of how badly SU(3) is broken.

3 r-dependent relations

0ΩΩr

0ΩΩr

0ΩΩrr131

ΩΔ3

ΞΣ

ΣΔ2

**

*

Typical size of SU(3) symmetry breaking parameterr=mu/md=0.6

Information on the geometric shape of current distribution in baryons

from sign and magnitude of .

Numerical results

Determine sign and magnitude of using various approaches

1) pion cloud model2) experimental N N*(1680) transition octupole moment3) current algebra approach

Estimate of in pion cloud model

spherical coresurrounded by

P-wave cloud

1

11

0

11

010

0'

''2

'2'23

1'

YnYn

YpYp

´

0 Y01

bare core cloud

)r(J)r(J)r(J)r(J γπNγπγN

N

N

Electromagnetic current of pion cloud model

)( TTrQ NN

)()( 3

rrrm

f)r(J NγπN

Insert spatial current density J(r)

only N interaction current contributes to

zrJrrzrd )(38

3 223

Matrix element

πNΔπNΔrQΔΩΔΩ

2fm11602 .rQnΔ

fm41

1.

mr

ππ

quark model relation pion Compton wave length

3Δ

fm0.16Ω

Comparison with experimental N N*(1680) transition octupole moment

A p½ = -15 6 10-3 GeV–1/2

A p3/2 = 133 12 10-3 GeV–1/2

Rewrite helicity amplitudes in terms of charge quadrupole and magnetic octupole form factors

Ji=1/2 Jf=5/2 transition between positive parity states only charge quadrupole and magnetic octupole can contribute

GM3 (0) = = 0.16 fm³

GC2 (0) = Q = 0.20 fm²

Order of magnitude agrees with estimate in pion cloud model

(+) = 4C = -0.16 fm³

C=-0.04 fm³

We can now determine the constant Cand predict the sign and size of

for all decuplet baryons

0.03 0.16

0.03 0

0.06 0.16

0.15- 0.16-

0.02- 0

0.10 0.16-

0.32- 0.32-

0.16- 0.16-

0 0

0.16 0.16

baryon

0*

*

*

0*

*

0

1)Q(r 1)Q(r

diagonal octupole moments [fm³]

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

ENDThank you for your attention.

n u

d

d

d d

u

0

Interpretation in quark model

2n0 r(n)Q

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.

Two-quark spin-spin operators are repulsive for quark pairs with spin 1.

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.

2n

00 r)(Q A. J. Buchmann, Can. J. Phys. 83 (2005)

Estimate of degree of nonsphericity

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)

Another configuration

+q

–2q

+q

+q

–q

+q +++

+q

–2q

–2q

+q

+

q + d1 + Q2 + 3 + (r) =

octupolequadrupoledipolemonopole

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

)(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.

Example: 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

Decomposition of 2695 into irred. reps. of SU(3)F and SU(2)S subgroups

2695 = (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.

flavor spin

)(23

)(

)(2)(

22

22

221

QGQ

QG

QGQG

nC

pC

nM

pM

Relations between N transition form factorsand elastic neutron form factors

Buchmann, Phys. Rev. Lett. 93 (2004) 212301

)(

)(

6)(

1

22

1

222

QG

QGMqQ

M

Cp

M

pCN

)(

)(

2)(

1

22

22

QG

QG

Q

M

Q

qQ

M

CnM

nCN

Definition of C2/M1 ratio

neutron elastic form factor ratio

data: electro-pionproductioncurves: elastic neutron form factors

A.J. Buchmann, Phys. Rev. Lett. 93, 212301 (2004).