Excited nucleon electromagnetic form factors from broken spin-flavor symmetry

Alfons Buchmann

Universität Tübingen

1. Introduction

2. Strong interaction symmetries

3. SU(6) and 1/N expansion of QCD

4. Electromagnetic form factor relations

5. Group theoretical argument

6. Summary

Nstar 2009, Beijing, 20 April 2009

1. Introduction

Spatial extension of proton

rpproton

Measurement of proton charge radius

rp(exp) = 0.862(12) fm

Simon et al., Z. Naturf. 35a (1980) 1

ρ(r)ρ

radial distribution

Elastic electron-nucleon scattering

N... nucleon (p,n) e... electron

Q... four-momentum transfer Q²= -(²- q²)

...energy transferq... three-momentum transfer

... photon

Elastic form factors

e

e‘

Q

N

N‘

...scattering angle

)(QG 2NMC,

magnetic )(QG 2NM

)(QG 2NC charge

Geometric shape of proton charge distribution

Extraction of N transition quadrupole (C2) moment from data

Q N (exp) = -0.0846(33) fm²

Tiator et al., EPJ A17 (2003) 357

φ)θ,ρ(r,)rρ(ρ

angular distribution

Proton excitation spectrum

N(939)

N*(1440)

radi

al e

xcita

tion

C0

, M

1

N*(1520)

orbi

tal e

xcita

tion

E1, M

2

(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 ofnucleon ground state

e

e‘

Q

N

N‘

Inelastic electron-nucleon scattering

)(QG 2ΔN

2C2,E1,M

Additional information on nucleon ground state structure

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?

2. Strong interaction symmetries

Strong interaction symmetries

Strong interactions are

approximately invariant under

•SU(2) isospin, •SU(3) flavor,•SU(6) spin-flavor

symmetry transformations.

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.

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

n p

SU(3) flavor symmetry

octet decuplet

Y hyperchargeS strangeness

T3 isospin BSY

Symmetry breaking alongstrangeness direction through

hypercharge operator Y

SU(3) symmetry breaking

M0, M1, M2 experimentally determined

4-)(MMM

2

210

Y1TTY1M

SU(3) invariant termfirst order SU(3) symmetry breaking

second order SU(3) symmetry breaking

mass operator

Group algebra relates symmetry breaking within a multiplet

(Wigner-Eckart theorem)

Relations between observables

Gell-Mann & Okubo mass formula

M3M4

1MM

2

1N

baryon octet

M-MM-MMM ****

baryon decuplet „equal spacing rule“

(M/M)exp ~ 1%

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)

SU(6) spin-flavor supermultiplet

spin flavorspin flavor

4,102,856

S

T3

baryon supermultiplet

)(M-)(MMM 3

2

210 1JJ

4

Y1TTY1M

Gürsey-Radicati SU(6) mass formula

Relations between octet and decuplet baryon masses

SU(6) symmetry breaking term

MMMM **e.g.

ji σσ~

Successes of SU(6)

2

3

μ

μ

n

p • proton/neutron magnetic moment ratio

• 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

Higher predictive power than independent spin and flavor symmetries

3. Spin-flavor symmetry and

1/N expansion of QCD

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)

1/NC expansion of QCD processes

CN

1~g

CN

1~g

CN1

O

two-body

2O

CN

1

three-body

CC

s N1

N

α

2

2

f

222

ΛQ

ln)N2(11

π124π

)(Qg)(Qstrong

coupling

NC ... number of colors

SU(6) spin-flavor as QCD symmetry

This results in the following hierarchy

O[1] (1/NC0) > O[2] (1/NC

1) > 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.

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, 096005 (2000); PRD66, 056002 (2002); PRD67, 016002 (2003)

4. Electromagnetic form factor relations

For NC=3 we may just as well usethe simpler spin-flavor parametrization method

developed by G. Morpurgo (1989).

Application to quadrupole and octupole moments

Buchmann and Henley, PRD 65, 073017 (2002); Eur. Phys. J. A 35, 267 (2008)

O[i] all allowed invariants in spin-flavor space for observable under investigation

]3[]2[]1[ ΟCΟBΟAΟ

one-quark two-quark three-quark

Spin-flavor operator O

constants A, B, C parametrize orbital- and color matrix elements; determined from experiment

Which spin-flavor operators are allowed?

tensorspin

jijziz

scalarspin

ji

3

jii[2] σσσσ3σσ2eBρ

Multipole expansion in spin-flavor space

• 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

• for neutron and quadrupole transition no contribution from one-body operator

SU(6) spin-flavor symmetry breaking

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

imiσ

jmjσ

imiσ

jmjσ

ei

ek

3-quark current 2-quark current

SU(6) symmetry breaking via spin and flavor dependent two- and three-quark currents

Neutron and N charge form factors

B456ρ56r n[2]n2n

B2256ρ56Q p[2]p

neutron charge radius

Ntransition quadrupole moment

2nr

2

1Q Δp

spin scalarspin tensor

neutron charge radiusN quadrupole moment

Buchmann,Hernandez,Faessler,PRC 55, 448

Extraction of p +(1232) transition quadrupole momentfrom electron-proton and photon-proton scattering data

2n(1232)p

fm )-0.0821(20rQ 2

2

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

experminent

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

Experimental N quadrupole moment

Including three-quark operators

tensorspin

jijziz

scalarspin

ji

3

ji

3

kjiki]3[[2]

σσσσ3σσ2

eCeBρρ

2nr

2

1Q Δp

C)2-B(456ρρ56r n[3][2]n2n

C)2-B(2256ρρ56Q p[3][2]p

Relation remains intact after including three-quark currentsBuchmann and Lebed, PRD 67 (2003)

Relations between octet and decuplet electromagnetic form factors

nΔp μ2μ

)(QG2)(QG 2nM

2Δp1M

2nr

2

1Q Δp

)(QGQ

23)(QG 2n

C22Δp

2C

magnetic form factorsBeg, Lee, Pais, 1964

charge form factorsBuchmann, Hernandez, Faessler, 1997

Buchmann, 2000

)(QG

)(QG

6

Mq)(Q

1M

2C2Δp

1M

2Δp2CN2

)(QG

)(QG

Q2

M

Q

q)(Q

1M

2C2n

M

2nCN2

Definition of C2/M1 ratio

C2/M1 expressed via neutron elastic form factors

Insert form factor relations

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

Use two-parameter Galster formula for GCn

)(QGτd1

τa)(QG 2n

M2n

C )(QGμ)(QG 2

Dn2n

C

τd1

τa

Q2

M

Q

q)(Q

1M

2C N2

2N

2

M4

Qτ

2nr~a4nr~d

neutron charge radius

4th moment of n(r)Grabmayr and Buchmann, Phys. Rev. Lett. 86 (2001) 2237

data: electro-pionproductioncurves: elastic neutron form factors

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

d=0.80

d=1.75

d=2.80

JLab 2006

Maid 2007 reanalysis

New MAID 2007 analysis

C2/M1(Q²)=S1+/M1+(Q²)

MAID 2003 . . Buchmann 2004

MAID 2007

from: Drechsel, Kamalov, Tiator, EPJ A34 (2007) 69

JLab data analysis MAID 2007 reanalysis of same JLab data

MAID 2003 . . Buchmann 2004

MAID 2007

New MAID 2007 analysis

Limiting values

031.0μ

r

12

M

M2

MM0)(Q

1M

2C

n

2nN

Δ

2N

2Δ2

21.006.0d

a

M

M

4

1)(Q

1M

2C

Δ

N2

d=2.8 d=0.8

best fit of data (MAID 2007) with d=1.75 10.0)(Q1M

2C 2

5. Group theoretical argument

Spin-flavor selection rules

56Ω56M [R]

2695405351 5656

M 0 only if [R] transforms according to one of the

representations R on the right hand side

( 0-body 3-body ) 2-body 1-body first order second order third order

SU(6) symmetry breaking operators

1. First order SU(6) symmetry breaking operators transforming according to the 35 dimensional representation generated by a antiquark-quark bilinear 6* x 6 = 35 + 1

• do not split the octet and decuplet mass degeneracy• give a zero neutron charge radius • give a zero N quadrupole moment

2. 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.

SU(6) symmetry breaking

Second order spin-flavor symmetry breaking operators can be constructed from direct products of two first order operators.

4052802801893535135 35

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.

Decomposition of SU(6) tensor 405 into SU(3) and SU(2) tensors

)5,27()5,8()5,1()3,27()3,10()3,10()3,8(2

)1,27()1,8()1,1(405

First entry: dimension of SU(3) flavor operator Second entry: dimension of SU(2) spin operator 2J+1

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].

Charge operator transforms as flavor octet.Coulomb multipoles have even rank (odd dimension) in spin space.

scalar J=0

vector J=1

tensor J=2

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

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].

Charge operator transforms as flavor octet.Coulomb multipoles have even rank (odd dimension) in spin space.

....)7,8()5,8(2)3,8(2)1,8(2695

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)

t)coefficien(CG565656Ω56M ]405[]405[

if

reduced matrix element same value for the entire multiplet 56

provides relationsbetween matrix elementsof different componentsof 405 tensor and states

i... components of initial 56 f... components of final 56... components of operator

Wigner-Eckart theorem

Construction of 56 tensor

BD

CADjkiAD

BCDijkCD

ABDkij

ABCijkαβγ

bεχεbεχεbεχε23

1

dχB

decuplet

octet

examples: 124115 BB2

1

2

1p, zS

indexspin 1,2kj,i,

indexflavor 31,2,CB,A,

k)(C,γ j),(B,ßi),(A,α

functionwave1/2spin

functionwave3/2spinχ

tensoroctetflavorb

tensordecupletflavord

i

ijk

AB

ABC

124115 B2B2

1,

zS

Explicit construction of 35 tensor

81,P1,2,3;a;FS,F11,S:X PaPa[35]n

generatorspinflavorFS

generator spinS

generatorflavorF

Pa

a

P

j)(B,ßi),(A,α ,,,:X[35]n

B

APi

jaB

APji

BA

i

ja FSFS

[35]m

[35]n

[405]mn, XXX

alltogether 35 generators

405 tensor:

6. Summary

The C2/M1 ratio in N transition predicted from empirical

elastic neutron form factor ratio GCn/GM

n agrees in sign and magnitude with C2/M1 data over a wide range of momentum transfers (see MAID 2007 analysis).

Summary

General group theoretical arguments based on the transformationproperties of the states and operators and the Wigner-Eckart theorem support previous derivations of connection betweenN transition and nucleon ground state form factors.

Broken SU(6) spin-flavor symmetry leads to a relation between the N quadrupole and the neutron charge form factors.

ENDThank you for your attention.