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Keitaro Nagata, Chung-Yuan Christian University

Atsushi Hosaka, RCNP, Osaka Univ.

Structure of the nucleon and Roper Resonance with Diquark Correlations

Chiral 07 @ Osaka University, 13-16 November, 2007

N and R in QD Model : K.N, A.H, J.Phys. G32,777 (‘06).

EM structures : K.N, A.H, arXiv: 0708.3471.

1. QD- Description of the Roper with (i) Relativistic description of the nucleon (ii) Diquark correlations (iii) Chiral symmetry

2. Electric properties of the Roper

Roper Resonance: N(1440) I(J)P=1/2(1/2)+

The mechanism of the E.E. of Roper and its structures are longstanding problem.Various descriptions have been investigated; unharmonicity in QM, collective excitation, deformation, Goldstone boson exchange, two-pole, gluonic-hybrid…

Today, I want to talk about

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Wave-function of N ([S1+S2,S3]STotal)

Quarks with (0s)3 config.

2/12/1 2/1,1,2/1,0

2/12/1 ]2/1,1[]2/1,0[ N2/12/1 ]2/1,1[]2/1,0[ N

N and Roper in NRQM

In the non-relativistic description or the spin-flavor symmetry of N, the E.E of Roper is about 1GeV (>> 0.5 GeV).

Pauli principle

Relativistic description (local interpolator)

4

ckbjTaiijabc

Sk qqCqB )( 5

ckbjijTaiabc

Ak qqiCqB

52 ))((

a,b,c: color

i,j,k: isospin

(Ioffe, Z.Phys C18, 67 (83))

AS

AS

BBR

BBN

cossin

sincos

forbidden in NRQM

2/12/1,1

2/12/1,0

There are 5 possible operators for N, 2 among 5 are independent (Fierz transformation).

NRLimit

We choose the following operators (good NRLimit)

5

qDA

qDqqqB SS )~(

qDqqqB AA

555 )~(

Diquark correlation with

NMM SA MM spin-spin

interaction

attraction

repulsion

)0(0)( JI

)1(1)( JI

If there is an interaction, (e.g., spin-spin), the two nucleon states have the mass diff. ~ M-MN

Recent lattice calculations: MA-MS ~100-400 MeV

Babich et.al. PRD76,074021(‘07), Alexandrou,PRL97,222002(‘06),

Orginos,hep-lat/0510082.

good diquark

bad

Jaffe, Phys. Rept. 409, 1(05)

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Chiral quark-Diquark modelMesons ~ q q-bar in NJL model

Two diquarks: DS [I(J)=0(0)], DA [I(J)=1(1)]

qD interaction ~ chiral invariant four point int.

Two nucleons: BS=qDS , BA= qDA

Non-linear realization of chiral sym.

Auxiliary field method : qD model -> chiral MB Lagrangian

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DS

DS

DA

DA DA

DS

q

Chiral Q-D interaction (three types)

Scalar channel

Axial-vector

channel

SG AG v

Mixing betweentwo channels

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B1

q

DS B1 B2

q

DA B2

B1

BGv

vGB

GBBL

S

A

A

S

|ˆ|

1

0

0

B1,2 B1,2

G

Masses of two states

]GeV[6.0

]GeV[05.1

]GeV[65.0

]GeV[39.0

A

S

q

m

m

m

]GeV[23

]GeV[11

]GeV[103

1

1

1

v

G

G

A

S

18

]GeV[44.1

]GeV[94.0

R

N

M

M

[K.N, A.H, J. Phys. G32, 777 (2006)]

scalardominance of N

input

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)()()()2(

)(4

4

kpSkpSkkd

ZNiQq SScqSq

554

4

30

)(),()()2(

)2(3)(

kSppkpkd

ZiNq

AAA

AcAD

)()(),()()2(

)(4

4

kSkpppkpkd

ZNiQq SSSScSSD

554

4

30

)()()()2(

)(2

3)(

kpSkpSkkd

ZiNq

A

AcAq

BS BS

Scalar

BABA

Axial-vector

iso-doublet space

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Intrinsic diquark form factor (IDFF)

)()()()( intrinsic, qFqqq DTotalADSD

point

Scalar )96.0/1/(1)( 2qqF S

22 )7.0/1/(1)( qqF A

(Monopole and dipole shape from Kroll et al. PLB316,546('93))

0.5 fm

0.6-0.9 -> 0.8 fm

Weiss, et al,PLB312,6,(93)

Axial

)51.0/1/(1)( 2qqF S in SD calculationMaris, nucl-th/0412059

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EM form factors of p, n, p*, n*

BS BSBS BS BA

BA BA BA

Nucleon Breit frame

),,,(

00

ADAqSDSq

qiF

Fi

kjijkM

E

22

22

sin3

1

3

1cos

3

1

3

1

sin0cos3

1

3

2

EAD

EAq

ESD

ESq

En

EAD

EAq

ESD

ESq

Ep

FFFFG

FFFFG

22

22

sin3

1

3

1cos

3

1

3

12/

sin0cos3

1

3

22/

MAD

MAq

MSD

MSqN

Mn

MAD

MAq

MSD

MSqN

Mp

FFFFMG

FFFFMG

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Electric form factors(with IDFF)

IDFF of scalar improve both GE of proton and neutron

axial improve GE of proton but not of neutron.

q2[GeV]

Proton Neutron

q2[GeV]

IDFF

Scalar

Axial

Both

Neither

fm8.0A

r

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Electric form factor of Roper with IDFF

Charge radii of Roper resonance

for proton : p* is slightly larger than p

for neutron : n* is slightly smaller than n (~0)

Q2[GeV] q2[GeV] q2[GeV]

n*

n

p

p*

Proton(p*) Neutron(n*)

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Summary and Conclusion

QD picture for the nucleon and Roper resonance.In a relativistic framework, two kinds of the wave-functions are available for the the nucleon.

With diquark correlations, the mass difference between the two states are about 500 MeV.

The charge radii of the Roper are almost comparable to that of the nucleon.

Future work : helicity amplitude (off-diagonal terms)

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Discussion proton component

(R.M.S) size of Bs and BA are almost the same.

ud

u BE =50MeV

E

SDE

SqEp FFG

3

1

3

2~ E

ADE

AqEp FFG 0~*

0.6fm 0.8fm

Charge radii of N and R

uu, ud

d, u

0.8fm

p p*

0.8 0.6+0.1 0.8 0.8+0.1

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Discussion neutron component

(R.M.S) size of Bs and BA are almost the same.

Charge radii of N and R

E

ADE

AqER FFG

3

1

3

1~

E

SDE

SqEn FFG

3

1

3

1~

ud

d

0.6fm 0.8fm

ud, dd

d, u

0.8fm

n n*

0.8 0.6+0.1 0.8 0.8+0.1

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•Roper-like excitation mode for octet,

•Not confirmed for decuplet.

Roper in SU(3)

N

G.S.

Excitation Energy[GeV]

J=1/2 J=3/2

1

N(940)

N(1440)

N(1535)

N(1650)N(1710)

(1115)

(1405)

(1600)

(1670)

(1800)(1810)

(1190)

(1660)

(1750)

(1315) (1232)

(1600)

(1700)

(1920)

(1385)

(1670)

(1940)

(1530)

(1820)

(1670)

(1690)

(1950)

(2030)

(2250)

(2250)

Assuming the non-relativistic description or the spin-flavor symmetry of thenucleon, the E.E of Roper is about 1GeV. Relativistic description of N Two types of wave-functions (and)is available. Each w.f. independently satisfies Pauli priciple. (spin flavor symmetry is not a good symmetry there)

The second nucleon states (orthogonal to N(940)) ? (i) Relativistic descriptions of the nucleon (ii) Chiral symmetry (iii) Diquark correlations

What is the structure of the Roper ?

-> Nucleon and Roper resonance