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Constituent quark model calculation for a possible JP=0−, T=0 dibaryon
Prog. Par?. Nucl. Phys.. Vol. 34, pp. 133-135,
1995 Copyright 0 1995 Elsevier Science Ltd Rioted io Great Britain. AU rights reserved 0146-6410195 $29.00
0146-6410(95)00010-0
Constituent Quark Model Calculation for a Possible P=O-, T=O Dibaryon G.
WAGNER, A. FAESSLER
L. YA. GLOZMAN,
A. BUCHMANN
and
institute of Theoretical Physics, University of Tiibingen, Auf der Morgenstelle D-72076 Tiibingen, Germany
14.
Abstract The assumption JP=O-,
of a narrow
(r .-., 5 MeV)
dibaryoii-resonance
T=O and a mass of 2065 MeV allowed recently
exchange
cross-sections
with these quantum
for various nuciei numbers
[I]. W e Investigate .
in the framework
with quantum
the mass of a B-quark
of the nonrelativistic
model.
Our shell-model
in the range of 2500 MeV. Since this result depends strongly
of the confining
potential,
we cannot
the usual assumption
including exclude
constituent.
a large basis of excited
the existence
of a quark-quark
numbers
all pionic double charge
masses example
diagonalisation
to describe
states
results
on the explicit
of the proposed
pair-confinement
state quark
dibaryon.
mechanism
might
in
form For not
be valid in the Gq-system.
It was recently exhibits supposed r -
a very
dibaryon
5 MeV
reported
narrow
explains
peak
resonance nicely
that near
the pionic
double
T, = 50 MeV,
with quantum all existing
data
charge nearly
numbers
Jp
exchange independent
reaction
(7rt, X-
of the nuclear
) on nuclei target.
A
=O- , T=O, mass 2065 MeV and width
[l].
The purpose of this work is to see, what mass is assigned to a Gq-system with the above quantum numbers J’=O-, T=O within the constituent quark model. The advantages of the present calculation are a proper treatment of the Pauii principle and the exact removal of the center-of-mass motion. The constituent-quark Hamiltonian (here for two flavours) describes in a nonrelativistic framework the dynamics of the effective degrees of freedom, “constituent quarks” of mass m,,, interacting via the following two-body potentials. i) Motivated by the linear u-model, in which the quarks acquire their constituent mass through the spontaneous breaking of chiral symmetry, and in which simultaneously the pion and its chiral partner, the o-meson, is introduced, we include in our Hamiltonian non-relativistic, Yukawa-like K- and a-exchange potentials. The structure of the constituent quarks and mesons is included by a form factor describing the extended quark-meson vertex, with a cut-off mass A in the range of the chiral symmetry breaking scale A N 750 MeV. ii) Furthermore, we have the effective “one-gluon exchange” potential [2] with effective strong coupling constant cy,q,responsible for the very short-range phenomena. iii) The confinement potential, which mainly determines the medium- and long-range (on the quark scale) phenomena, is taken to be linear, introducing two more parameters. the strength
a, (slope) and the constant
C (offset) of the confinement
potential.
An essential property of the quark-quark interaction is confinement. Therefore, the Translationally Invariant Shell-Model (TISM) basis is well suited for quark systems where possible
G. Wagner et al.
134
color-singlet - color-singlet clusterizations are forbidden, or at least suppressed. The range of the shell-model wave-functions is characterized by the Gaussian width parameter bN for the baryons (and b6 for the Gq-system). The full six-quark Hamiltonian is very different from the harmonic oscillator Hamiltonian. So, we diagonalize the CQM Hamiltonian in the basis of 11 states listed below, and interpret the lowest eigenvalue as the dibaryon mass. We use two-particle fractional parentage coefficients (f.p.c.) to evaluate the nratrix elements of the two-body potentials. The theory of how to construct the two-particle f.p.c. in the six-quark system is discussed in Ref.[3], where the necessary f.p.c. are also tabulated. In the above negative parity Gq-system, the possible number of oscillator quanta N in the shell nrodel basis are restricted to N = 1,3,5, . . . . The unambigious classification of states is provided by the number of internal excitation
quanta N, the Elliot synrbol (X/L)determining
the
SU(3) harnronic oscillator nrultiplet. the Young pattern [f]x defining the spatial permutational Se-symmetry, further the total orbital angular nronrentunr L, total spin S and total isospin T of all the quarks. The intermediate SU(6) cs synrmetry is needed, because in general, several states with different interniediate
N lflx _1
3
[511
synnnetries
(X/l) (10)
[421 (3ow)
exist.
LS,T 1,170
w
[flcs 13211 [421, [3211, L3131, [231, w41
There is only one state with N=l which is compatible with Jp=O-, T=O, and its quantum numbers are given in the above Table. The full classification of the orbital parts with N=3 can be found in [4].In the actual calculation, we restrict ourselves to the ten states (out of 31 possible states with N=3) listed above. They give an essentially new contribution in orbital space, and are sufficient to see on a qualitative level the effect of the configuration mixing. The constituent quarks, as well as the r, 0, gluon and confinement potentials of the present CQM are effective degrees of freedom, supposed to simulate the dominant features of nonperturbative QCD at low energy. The above introduced parameters m,,, A, a,$, a,, C, bN are usually fitted to describe the baryon spectrum, in our case to describe simultaneously the N(939), A( 1232) and N*( 1535) baryons with shell-model ground state wave-functions, assuring mininral adniixtures of excited states through the so-called nucleon-stability condition % = 0 as an additional constraint. With fixed parameters of the quark-quark interaction one should minimize the lowest eigenvalue with respect to the harm-osc. parameter bG in the six-quark trial wave function. Of course, the harm-osc. parameters of the baryon (bN) and the Gq-system (b,j) need not be the sanie. Our results with various possible paranreter sets (with and without chiral interactions) show, that the values for the dibaryon mass fl%e systematically higher than the experimental number Mpyl, = 2065 MeV by at least 400 MeV. A unique description of the baryons and this supposed dibaryon is not possible within our model. However, we are not able to rule out the dibaryon interpretation of the narrow peak in (r+, r-) reactions, since taking an “unusually” small confinement strength a,., we are able to get 2065 MeV, but at the expense of a very large dibaryon (b6 -1.25 fni), and the fact, that we are not able to describe the negative parity baryons N* anymore. Since our results depend strongly on the explicit form of the confinement potential, the confirmation of such a dibaryon-resonance could probably improve our understanding of confinement. The concept of confinement is barely understood, and might not be extended straightforwardly from the 3q- to more complicated multi-quark systems.
A Possible Jp=O‘, T=O Dibaryon References (1993), 42; (l] RBilger, H.A.Cl enlent and M.G.Schepkin Phys.Rev.Lett.71 R.Bilger et al, Z.Phys.A343 (1992), 491. (1975), 147. [2] A.DeRi?jula, H.G eorgi, S.L.Glashow, Phys.Rev.Dl2 (31 I.T.Obukhovsky Z.Phys.A308 (1982), 253. (41 L.Ya.Glonnan, A.Buchmann and Amand Faessler, Jour.Phys.G20( 1994), L49; L.Ya.Glozman. Georg Wagner, A.Buclmann, Amand Faessler, to be published.
135
1995 Copyright 0 1995 Elsevier Science Ltd Rioted io Great Britain. AU rights reserved 0146-6410195 $29.00
0146-6410(95)00010-0
Constituent Quark Model Calculation for a Possible P=O-, T=O Dibaryon G.
WAGNER, A. FAESSLER
L. YA. GLOZMAN,
A. BUCHMANN
and
institute of Theoretical Physics, University of Tiibingen, Auf der Morgenstelle D-72076 Tiibingen, Germany
14.
Abstract The assumption JP=O-,
of a narrow
(r .-., 5 MeV)
dibaryoii-resonance
T=O and a mass of 2065 MeV allowed recently
exchange
cross-sections
with these quantum
for various nuciei numbers
[I]. W e Investigate .
in the framework
with quantum
the mass of a B-quark
of the nonrelativistic
model.
Our shell-model
in the range of 2500 MeV. Since this result depends strongly
of the confining
potential,
we cannot
the usual assumption
including exclude
constituent.
a large basis of excited
the existence
of a quark-quark
numbers
all pionic double charge
masses example
diagonalisation
to describe
states
results
on the explicit
of the proposed
pair-confinement
state quark
dibaryon.
mechanism
might
in
form For not
be valid in the Gq-system.
It was recently exhibits supposed r -
a very
dibaryon
5 MeV
reported
narrow
explains
peak
resonance nicely
that near
the pionic
double
T, = 50 MeV,
with quantum all existing
data
charge nearly
numbers
Jp
exchange independent
reaction
(7rt, X-
of the nuclear
) on nuclei target.
A
=O- , T=O, mass 2065 MeV and width
[l].
The purpose of this work is to see, what mass is assigned to a Gq-system with the above quantum numbers J’=O-, T=O within the constituent quark model. The advantages of the present calculation are a proper treatment of the Pauii principle and the exact removal of the center-of-mass motion. The constituent-quark Hamiltonian (here for two flavours) describes in a nonrelativistic framework the dynamics of the effective degrees of freedom, “constituent quarks” of mass m,,, interacting via the following two-body potentials. i) Motivated by the linear u-model, in which the quarks acquire their constituent mass through the spontaneous breaking of chiral symmetry, and in which simultaneously the pion and its chiral partner, the o-meson, is introduced, we include in our Hamiltonian non-relativistic, Yukawa-like K- and a-exchange potentials. The structure of the constituent quarks and mesons is included by a form factor describing the extended quark-meson vertex, with a cut-off mass A in the range of the chiral symmetry breaking scale A N 750 MeV. ii) Furthermore, we have the effective “one-gluon exchange” potential [2] with effective strong coupling constant cy,q,responsible for the very short-range phenomena. iii) The confinement potential, which mainly determines the medium- and long-range (on the quark scale) phenomena, is taken to be linear, introducing two more parameters. the strength
a, (slope) and the constant
C (offset) of the confinement
potential.
An essential property of the quark-quark interaction is confinement. Therefore, the Translationally Invariant Shell-Model (TISM) basis is well suited for quark systems where possible
G. Wagner et al.
134
color-singlet - color-singlet clusterizations are forbidden, or at least suppressed. The range of the shell-model wave-functions is characterized by the Gaussian width parameter bN for the baryons (and b6 for the Gq-system). The full six-quark Hamiltonian is very different from the harmonic oscillator Hamiltonian. So, we diagonalize the CQM Hamiltonian in the basis of 11 states listed below, and interpret the lowest eigenvalue as the dibaryon mass. We use two-particle fractional parentage coefficients (f.p.c.) to evaluate the nratrix elements of the two-body potentials. The theory of how to construct the two-particle f.p.c. in the six-quark system is discussed in Ref.[3], where the necessary f.p.c. are also tabulated. In the above negative parity Gq-system, the possible number of oscillator quanta N in the shell nrodel basis are restricted to N = 1,3,5, . . . . The unambigious classification of states is provided by the number of internal excitation
quanta N, the Elliot synrbol (X/L)determining
the
SU(3) harnronic oscillator nrultiplet. the Young pattern [f]x defining the spatial permutational Se-symmetry, further the total orbital angular nronrentunr L, total spin S and total isospin T of all the quarks. The intermediate SU(6) cs synrmetry is needed, because in general, several states with different interniediate
N lflx _1
3
[511
synnnetries
(X/l) (10)
[421 (3ow)
exist.
LS,T 1,170
w
[flcs 13211 [421, [3211, L3131, [231, w41
There is only one state with N=l which is compatible with Jp=O-, T=O, and its quantum numbers are given in the above Table. The full classification of the orbital parts with N=3 can be found in [4].In the actual calculation, we restrict ourselves to the ten states (out of 31 possible states with N=3) listed above. They give an essentially new contribution in orbital space, and are sufficient to see on a qualitative level the effect of the configuration mixing. The constituent quarks, as well as the r, 0, gluon and confinement potentials of the present CQM are effective degrees of freedom, supposed to simulate the dominant features of nonperturbative QCD at low energy. The above introduced parameters m,,, A, a,$, a,, C, bN are usually fitted to describe the baryon spectrum, in our case to describe simultaneously the N(939), A( 1232) and N*( 1535) baryons with shell-model ground state wave-functions, assuring mininral adniixtures of excited states through the so-called nucleon-stability condition % = 0 as an additional constraint. With fixed parameters of the quark-quark interaction one should minimize the lowest eigenvalue with respect to the harm-osc. parameter bG in the six-quark trial wave function. Of course, the harm-osc. parameters of the baryon (bN) and the Gq-system (b,j) need not be the sanie. Our results with various possible paranreter sets (with and without chiral interactions) show, that the values for the dibaryon mass fl%e systematically higher than the experimental number Mpyl, = 2065 MeV by at least 400 MeV. A unique description of the baryons and this supposed dibaryon is not possible within our model. However, we are not able to rule out the dibaryon interpretation of the narrow peak in (r+, r-) reactions, since taking an “unusually” small confinement strength a,., we are able to get 2065 MeV, but at the expense of a very large dibaryon (b6 -1.25 fni), and the fact, that we are not able to describe the negative parity baryons N* anymore. Since our results depend strongly on the explicit form of the confinement potential, the confirmation of such a dibaryon-resonance could probably improve our understanding of confinement. The concept of confinement is barely understood, and might not be extended straightforwardly from the 3q- to more complicated multi-quark systems.
A Possible Jp=O‘, T=O Dibaryon References (1993), 42; (l] RBilger, H.A.Cl enlent and M.G.Schepkin Phys.Rev.Lett.71 R.Bilger et al, Z.Phys.A343 (1992), 491. (1975), 147. [2] A.DeRi?jula, H.G eorgi, S.L.Glashow, Phys.Rev.Dl2 (31 I.T.Obukhovsky Z.Phys.A308 (1982), 253. (41 L.Ya.Glonnan, A.Buchmann and Amand Faessler, Jour.Phys.G20( 1994), L49; L.Ya.Glozman. Georg Wagner, A.Buclmann, Amand Faessler, to be published.
135