a r X i v :h e p -l a t /0304008v 1 15 A p r 2003
Behind the Success of the Quark Model
T.T.Takahashi ?,H.Suganuma ?,H.Ichie ?,H.Matsufuru §and Y.Nemoto ?
February 1,2008
Abstract
The ground-state three-quark (3Q)potential V g .s .
3Q and the excited-state 3Q poten-tial V e .s .3Q are studied using SU(3)lattice QCD at the quenched level.For more than
300patterns of the 3Q systems,the ground-state potential V g .s .
3Q is investigated in de-tail in lattice QCD with 123×24at β=5.7and with 163×32at β=5.8,6.0.As
a result,the ground-state potential V g .s .
3Q is found to be well described with Y-ansatz
within the 1%-level deviation.From the comparison with the Q-ˉQ
potential,we ?nd the universality of the string tension as σ3Q ?σQ ˉQ and the one-gluon-exchange result as A 3Q ?1
?
RCNP,Osaka University,Mihogaoka 10-1,Ibaraki,Osaka 567-0047,Japan ?
Faculty of Science,Tokyo Institute of Technology,Tokyo 152-8551,Japan ?
Humboldt Univ.zu Berlin,Inst.f¨u r Phys.,Invalidenstrasse,D-10115Berlin,Germany §
Yukawa Institute for Theoretical Physics,Kyoto University,Kyoto 606-8502,Japan ?
RIKEN-BNL Research Center,Brookhaven National Laboratory,Upton 11973,USA
3Q potential,although the3Q potential is responsible for the baryon properties and of great importance also for the quark-con?nement mechanism in baryons.On the ground-state3Q
potential V g.s.
3Q ,the detailed study has been recently performed,and Y-ansatz is now almost
conclusive[1,2].
In the latter half of this paper,we consider the connection between QCD and the quark model in terms of the excited-state inter-quark potential.The low-lying hadron properties, especially for baryons,can be successfully reproduced in the framework of the simple non-relativistic quark model[3],which has only quark degrees of freedom and has no gluonic modes.The non-relativistic treatment can be justi?ed by spontaneous chiral-symmetry breaking,which gives rise to a large constituent quark mass of about300MeV.On the other hand,we have no reason which supports the absence of the gluonic excitation modes in the low-lying hadron spectra.To give a solution to this mystery,we investigate the gluonic excitation modes.In spite of several lattice studies on the gluonic excitation mode in the Q-ˉQ system,the gluonic excited-state potential in the3Q system has not been investigated in
lattice QCD.We show the?rst result on the1st excited-state potential V e.s.
3Q in the spatially-
?xed3Q system in SU(3)lattice QCD[4].
2The ground-state3Q potential–Lattice QCD evi-dences of Y-ansatz
The Q-ˉQ potential is described by a simple form as V g.s.
QˉQ
(r)=?A QˉQ
|r i?r j|+σ3Q L min+C3Q(Y-ansatz)(1) with the minimal value L min of the total length of color-?ux-tubes linking three quarks.
We extract the static3Q potential V g.s.
3Q from the3Q Wilson loop[1,2]using SU(3)lattice
QCD calculations in the model-independent way.For the accurate measurement of V g.s.
3Q ,we
adopt the gauge-covariant smearing method,which enhances the ground-state overlap.For
more than300patterns of the3Q systems,we investigate V g.s.
3Q in lattice QCD with123×24
lattice atβ=5.7and with163×32lattices atβ=5.8,6.0.
Figure 1:The lattice QCD result for the ?ux-tube pro?le in the spatially-?xed 3Q system in the maximally-abelian projected QCD [5].The distance between the junction and each quark is about 0.5fm.
As a result,the 3Q potential can be accurately described by the Y-ansatz form in Eq.(1)
within the 1%-level deviation.From the comparison with the Q-ˉQ
potential,we ?nd the universality of the string tension as σ3Q ?σQ ˉQ and the one-gluon-exchange result as A 3Q ?
1
b 0+b 1r +b 2r 2+
c 0[7].However,the vibrational modes of the Y-type ?ux-tube system are
considered to be much more complicated and chaotic than those of the simple Q-ˉQ
?ux-tube,because of the possible interference among the vibrations of the three ?ux-tubes connected at the physical junction.
We investigate the 1st excited-state potential in the spatially-?xed 3Q system using lattice QCD with 163×32lattice at β=5.8for 24patterns of the 3Q systems.In Fig.2,we plot the
ground-state potential V g .s .3Q and the 1st excited-state potential V e .s .
3Q as the function of L min in the physical unit.
00.5
1 1.5
L min [fm]
1
2
3
4
3Q p o t e n t i a l [G e V ]
Figure 2:The lattice QCD results of the ground-state 3Q potential V g .s .
3Q (open circles)and
the 1st excited-state 3Q potential V e .s .
3Q (?lled circles)as the function of L min .The gluonic excitation energy is found to be more than 1GeV in the hadronic scale.
As a remarkable fact,the lowest gluonic excitation energy ?E =V e .s .
3Q ?V g .s .3Q
is found to be about 1GeV in the hadronic scale as L min ?0.5?1.5fm.This is rather large in comparison
with the low-lying excitation energy of the quark origin.(Also for the Q-ˉQ
system,a large gluonic excitation energy is reported in recent lattice studies [7].)Such a gluonic excitation contribution would be signi?cant in the highly-excited baryons with the excitation energy above 1GeV,and the lowest hybrid baryon [8],which is described as qqqG in the valence picture,is expected to have a large mass of about 2GeV.
The large gluonic excitation energy corresponding to the ?ux-tube vibrational energy is considered to originate from the “tight”?ux-tube with a large string tension,as a result of the strong con?nement e?ect.Due to the large gluonic excitation energy ?E ?1GeV,the gluonic excitation modes are invisible in the low-lying excitations of hadrons,which can be considered as a reason of the success of the simple quark model without gluonic modes [4].(See Fig.3.)
References
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Figure3:Connection from QCD to the success of the quark model for low-lying hadrons. The large gluonic excitation energy?E?1GeV leads to the absence of the gluonic mode in the low-lying hadrons and brings about the great success of the quark model.
[6]J.M.Cornwall,Phys.Rev.D54(1996)6527.
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