Inha Nuclear Physics Group

Relativistic Relativistic

Approach Approach

to QQ Potentialto QQ Potential2007. 12. 14

Jin-Hee Yoon

Dept. of Physics, Inha University

Inha Nuclear Physics Group

Introduction

Relativistic Treatment of 2-Body Problem Composite systems with light quarks Systems with large coupling strengthDissociation and recombination of JJD + D+

Useful information on the suppression or the enhancement of in high-energy heavy-ion collisions

2-body Dirac equations of constraint dynamics- successfully tested on relativistic 2-body bound states in QE

D, QCD, & 2-body NN scattering. (Horace W. Crater et al., PRD74, 054028 (2006))

- Will be tested on heavy-quarkonium dissociation

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2-Body Constraint Dynamics

In a separable two-body basisParticles interacts pairwise through scalar and vector interactions.Spin-dependence is determined naturally.Yields simple Schrodinger-type equation.No cutoff parameter

Inha Nuclear Physics Group

2-Body Constraint Dynamics

P. van Alstine et al., J. Math. Phys.23, 1997 (1982) Two free spinless particles with the mass-shell constraint

Introducing the Poincare invariant world scalar S(x,p1,p2)

Dynamic mass-shell constraint

Schrodinger-like equation

0 ,0 2

1

2

1

0

1

2

1

2

1

0

1 mpHmpH

),,(),,(

),,(),,(

2122122

2112111

ppxMppxSmm

ppxMppxSmm

),,( 212222 ppxVmpMpH iiiiii

w

mmm

w

mmw

mwb

wbVpw

HHH

ww

ww

2122

21

2

222

222211

and 2

with

)( where

0)(

Inha Nuclear Physics Group

2-Body Constraint Dynamics

2-body Dirac Equation

where

)()( 22 wbVp w

TSSSSTSOSOSOSODw VVVVVVVVV 3210

)()12(cosh22sinh'2

'')12(cosh'2 222 rm

r

KF

r

KKFK

r

KFV D

22

0 22 AASSmV ww

r

KK

r

K

r

KF

r

FLV SO

2sinh')12(cosh)12(cosh'')(

2211

KqKlLV SO 2sinh'2cosh')( 212

KlKqLiV SO 2sinh'2cosh')( 213

r

KF

r

K

r

K

r

KKLrrV TSO

2sinh''2sinh)12(cosh')(ˆˆ

22121

r

K

rKF

r

K

rKFrkV SS

)12(cosh1''

2sinh1'')(21

KKF

r

K

rKF

r

K

rKFrnrrV TSS

2

21 ''2)12(cosh1

'3'2sinh1

''3)(ˆˆ

VSO2 & VSO3 =0

when m1=m2

Inha Nuclear Physics Group

2-Body Constraint Dynamics

HereF=(L-3G)/2 and K=(G+L)/2 G(A)=-(1/2) ln(1-2A/w) L=(1/2) ln[(1+2(S-A)/w+S2/m2)/(1-2A/w)] w : center of mass energy

l’(r)= -(1/2r)(E2M2-E1M1)/(E2M1+E1M2)(L-G)’

q’(r)=(1/2r) )(E1M2-E2M1)/(E2M1+E1M2)(L-G)’

m(r)=-(1/2)∇2G+(3/4)G’2+G’F’-K’2

k(r)=(1/2)∇2G-(1/2)G’2-G’F’-(1/2)(G’/r)+(K’/r)n(r)=∇2K-(1/2)∇2G-2K’F’+G’F’+(3G’/2r)-(3K’/r)

Inha Nuclear Physics Group

2-Body Constraint Dynamics

VSO2=0 and VSO3=0 when m1=m2

For singlet state, no spin-orbit contribution ←VD+VTSO+VSS+VTSS=0

H=p2+2mwS+S2+2wA-A2

0221 SLL

Inha Nuclear Physics Group

Relativistic Application(1)

Applied to the Binding Energy of chamoniumWithout spin-spin interaction M(exp)=3.067 GeV Compare the result with PRC 65, 034902 (2002)

T/Tc BE(non-rel.) BE(rel.) Rel. Error(%)0 -0.67 -0.61 9.00.6 -0.56 -0.52 7.10.7 -0.44 -0.41 6.80.8 -0.31 -0.30 3.20.9 -0.18 -0.17 4.01.0 -0.0076 -0.0076

0.0 At zero temperature, 10% difference at most!

Inha Nuclear Physics Group

Relativistic Application(2)

With spin-spin interaction M(S=0) = 3.09693 GeV

M(S=1) = 2.9788 GeV At T=0, relativistic treatment gives

BE(S=0) = -0.682 GeVBE(S=1) = -0.586 GeV

Spin-spin splitting ∼100 MeV

Inha Nuclear Physics Group

Overview of QQ Potential(1)

Pure Coulomb :

BE=-0.0148 GeV for color-singlet=-0.0129 GeV for color-triplet(no convergence)

+ Log factor :

BE=-0.0124 GeV for color-singlet=-0.0122 GeV for color-triplet

+ Screening :

BE=-0.0124 GeV for color-singletNo bound state for color-triplet

0 and 2.0

Sr

A

0 and 1

ln21

2.0

22

2

S

rer

A

0 and 1

ln21

2.0

22

2

S

rer

eA

r

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Overview of QQ Potential(2)

+ String tension(with no spin-spin interaction)

When b=0.17 BE=-0.3547 GeVWhen b=0.2 BE=-0.5257 GeVToo much sensitive to parameters!

brS

rer

eA

r

and 1

ln21

2.0

22

2

Inha Nuclear Physics Group

QQ PotentialModified Richardson Potential

27

12

32

11

4

f

s

n

22

2

1ln

213

4

rer

eA

r

s

rS 2

27

8

and

Parameters : And mass=m(T)

A : color-Coulomb interaction with the screeningS : linear interaction for confinement

Inha Nuclear Physics Group

V(J/)

Too Much Attractive!

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V(J/) *r2

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V(J/) *r3

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Vso(J/)

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V(J/)*r2.5

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V(J/) at small range

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V(J/) with mixing

For J/, S=1 and J=1.

Without mixing(L=0 only), splitting is reversed.

Therefore there has to be mixing between L=0 and L=2 states.

Inha Nuclear Physics Group

V(J/) with mixing

Inha Nuclear Physics Group

Work on process

To solve the S-eq. numerically,We introduce basis functionsn(r)=Nnrlexp(-n2r2/2) Ylm

n(r)=Nnrlexp(-r/n) Ylm

n(r)=Nnrlexp(-r/√n) Ylm

… None of the above is orthogonal. We can calculate <p2> analytically, but all the other terms has t

o be done numerically. The solution is used as an input again → need an iteration Basis ftns. depend on the choice of quite sensitively and there

fore on the choice of the range of r.

Inha Nuclear Physics Group

Future Work

Find the QQ potential which describes the mass spectrum of mesons and quarkonia well.Extends this potential to non-zero temperature.Find the dissociation temperature and cross section of a heavy quarkonium.And more…

Inha Nuclear Physics Group

Thank YouFor your attention.