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Excited nucleon electromagnetic form factors from broken spin-flavor symmetry. Alfons Buchmann Universität Tübingen. Introduction Strong interaction symmetries SU(6) and 1/N expansion of QCD Electromagnetic form factor relations Group theoretical argument Summary. - PowerPoint PPT Presentation
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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.