inha nuclear physics group relativistic approach to qq potential 2007. 12. 14 jin-hee yoon dept. of...
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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
Inha Nuclear Physics Group
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
Inha Nuclear Physics Group
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!
Inha Nuclear Physics Group
V(J/) *r2
Inha Nuclear Physics Group
V(J/) *r3
Inha Nuclear Physics Group
Vso(J/)
Inha Nuclear Physics Group
V(J/)*r2.5
Inha Nuclear Physics Group
V(J/) at small range
Inha Nuclear Physics Group
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.
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