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Why the heavy-flavored strange pentaquark does not exist
NUCLEAR PHYSICS A Nuclear Physics A 612 (1997) 449-456
Why the heavy-flavored strange pentaquark does not exist M. Shmatikov Russian Research Center “Kurchatov Institute”,
123182 Moscow, Russia
Received 29 July 1996; revised 23 September 1996
Abstract The possible existence of a molecular-type heavy-flavored strange pentaquark is considered. It is shown that no narrow state of this type exists. PACS:
14.4O.J~;14.403; 12.4O.Qq
1. Introduction The most impressive achievement of QCD in the nonperturbative domain is the classification of hadrons. The bulk of the observed hadrons can be successfully classified at present as ijq or qqq configurations. A few controversial candidates for a beyond-minimal quark configuration are under debate [ 11. The challenging question of the existence of exotics remains unanswered. One such exotic state is the pentaquark: a baryon containing five quarks G G gqqqq where Q (= b, c) is a heavy quark and q (= u, d, s) is a light one. Theoretical investigations of the pentaquark have yielded controversial results. Its existence has been claimed in various approaches. In the constituent quark model the existence was confirmed provided (i) one of the light quarks is a strange one and (ii) the binding color forces are assumed to be flavor-independent [ 2,3]. Stated differently in the limit of “flavor-blind” color forces the F& is expected to exist. However, with realistic breaking of SU( 3) ,G and, even more important, proper treatment of the kinetic energy, the pentaquark proved to be unstable with respect to strong decay [ 4,5 1. An alternative approach to the issue is to consider the pentaquark as the bound state of a Skyrme-type soliton (representing the nucleon) and a heavy meson. This model has been applied in the nonstrange sector only and the results obtained are far from decisive. The bound state was shown to exist in both b- and c-sectors (corresponding to the Pr, 037%9414/91/$17.00 Copyright @ PII SO375-9474(96)00410-l
1997 Elsevier Science B.V. All rights reserved.
ht. Shmatikov/Nuclear Physics A 612 (1997) 449-456
450
and Pz type pentaquarks)
in the limit of the infinite heavy-quark
due to the finite mass of the heavy quark, however, unbound [ 81. Finally, a molecular-type
pentaquark
mass [6]. Corrections
may make the pentaquark
P6 state has been predicted
states
[ 71. It is bound by
the long-range one-pion-exchange force, ensuring its genuine molecular structure. These forces, however, are not strong enough to bind a Pt pentaquark. In view of this situation, it is of interest
2. Quantum
to explore whether molecular-type
numbers
A molecular-type
of a molecular-type
exotic
Pb pentaquark
nonstrange extension
of this approach
strange meson pS( z (6s)) A viable alternative H* 1 and a hyperon
hadron
GS pentaquarks
strange heavy-flavored pentaquark
can be characterized
is a loosely
bound
the nucleon
fails because the pS-meson state consisting
Y. Two possibilities
by its particle
state of the B*N
to the system containing
is a molecular
exist.
The
A “naive”
and a bottom-flavored
does not couple to the r-meson.
of a (nonstrange)
are conceivable
system.
content.
depending
heavy quark meson on the total isospin
of the system T. Consider first the T = $ case. Then the pentaquark-to-be necessarily has the Hi.7 particle content. The “effective” strength constant of the driving one-pionexchange potential in the T = ; channel equals then v = fzznfHHry, where y is a spin
factor. In the limit of the W(3) symmetry fxzr = 2ap fNNT,where fNN,, is the constant of the NNn- coupling and (up is the F/( F + D) ratio for the pseudoscalar mesons. The n--meson is known to couple to the isovector particles only and, hence, it is uncoupled from the isoscalar
heavy meson. Introducing the constant g of the q-meson coupling to the constituent light quark we arrive at ~HH~ = g. The coupling constant g is related also to the NN?r coupling constant: g = i ft,rp,,,. Combining the factors above we get the following expression for the “effective” strength constant of the one-pion-exchange in the T = ; channel:
$wYfiww
vHP=
It is instructive to compare this strength constant to its counterpart system. The latter, taking into account the spin-isospin factors, is VNN
=
VNN in the NN
3f;N,y
The ratio of vH~ and VNN strength constants vHP/vNN=
is thus
+F'y.
The UP factor in the limit of the SU( 6) symmetry is 0.4, while in the Nijmegen potential ap = 0.485 [ 91. The value of the spin factor y depends on the spin and orbital momenta ’ Since it does not result in confusion we hereafter will denote for notational brevity the heavy vector meson as H.
M. Shmatikov/Nuclear Physics A 612 (1997) 449-456
of the considered
state. For the most favorable
2St,2-4D1,2
451
configuration
(see [ 71) it
is y = 2. Thus we arrive at
showing reduced
that VHz M ~VNN. One more factor which is important for binding is the mass of the system. The ratio of reduced masses for the HZ and NN systems
equals 1
mHZ‘ - 2% mN
mNN
1 +
This ratio equals reduced
MZ/mH
’
in the case of D- (B-)
1.55 (2.08)
mass of the HZ system as compared
the smallness
of the effective
strength constant
heavy
meson.
The increased
to the NN case still cannot overbalance VHp, allowing
us to safely conclude
that
the molecular-type H.Z pentaquark with T = 3 does not exist. An alternative possibility is the HY molecular state with total isospin T = l. This case comprises two coupled channels: HA and HZ. As stated above, the configuration most favorable from the point of view of binding by one-pion-exchange has the J” = k quantum numbers, corresponding to the 2St,2-4Dt,2 at the system of four coupled Schrodinger equations
coupled channels. Thus we arrive which can be written in the matrix
form
where 4 is the column
of wave functions
with the following
channels
labels:
(i) HA
(S-wave); (ii) HA (D-wave); (iii) HZ (S-wave); (iv) HZ (D-wave) and K* is the diagonal 4 x 4 matrix of the c.m.s. momenta squared (plus the matrix of centrifugal potentials)
: 2 KAH I&
+ 6/r*
(2) 4H ~~~
+
6/r* 1.
squared in Eq. (2) for the kinetic energy E in the HA channel is where Am is the mass difference K,,~ = 2rn,iHE and Shihdy ~~~ = 2rnzH( E + Am), of the Z- and A-hyperons (Am = rns - m,l M 76 MeV) and mm (Y = A, 2) is the reduced mass in the corresponding channel. In Eq. ( 1) m is the diagonal 4 x 4 matrix with the mAH and mzH elements. Finally, V is the matrix of driving forces reading: The c.m.s. momenta 2
V” = Vo [T 8 C,jb(m,r)
where VO is the potential
+ T 8 C,92(m,r)]
unit introduced
,
(3)
in [ 101 (4)
M. ShmatikodNuclear
452
Physics A 612 (1997) 449-456
where g is the r-meson-constituent-quark stant (jr functions
= 132 MeV).
coupling
In Eq. (3) 71 (I = 0,2)
~o:O(X) = YO(X)
-
YO(AX)
-
j2(x)
-
A3y2(Ax)
(A
-
and f,, is the n--meson
are the regularized
decay con-
(at small X) radial
1) exp(-Ax),
(5)
and (P= y2(x)
-
where y0,2 (x) are the modified YO(X)
=
exp(-xl . x ’
exp(-Ax)
spherical
l)(Ax+
1) ’
2x
Bessel functions
y2=exyx) (I+:+;)
and A = A/m,, with .4 being a cut-off parameter (see below). The Cs(,) matrix contains spin coefficients for the central (tensor) components of the one-pion-exchange forces:
Finally, T is the matrix of isospin coefficients
which in the limit of the SU( 3) symmetry
limit reads HA
I
H-2
I
(7)
where cyp is defined above.
3. Binding of a molecular-type The complexity methods
of the system
mandatory.
us consider
under consideration
Before proceeding
the conditions
we apply the criterion
to the numerical
makes application solution
of a zero-energy
of numerical
of the system
under which a loosely bound state is favorable.
for the appearance
central force is operative to be investigated
pentaquark
level in a system
[ 11 I. For the Yukawa forces under consideration
(1) let
To this end where a
the parameter
is
vo s = 0.5953 x 2my--, mZ where fi is the reduced mass of the system and y is the spin-isospin factor. The only component of the system under consideration where the Yukawa forces are operative is the S-wave HZ channel. To make our conclusions more transparent we compare the SIH value to the (reference) value of its NN counterpart SNN: SZH SNN
--
YZH YNN
1
x23 mN1
+mZ/mH'
(8)
M. Shmatikov/Nuclear Physics A 612 (1997) 449-456
The spin-isospin
yW factor equals the product of corresponding
453
elements
of the C, and
T matrices: YzH = 2 x %jffp. In the same notations
Substituting
the YNN equals
these expressions
SXH
into Eq. (8) we arrive at
4aP
-2
1 +
SNN
mP/mH
This ratio depends (B-)
’
upon the mass of the heavy meson mH and for the case of the D-
meson it is S~H M SNN X 1.0 (1.3).
(9)
The value SNN M 0.33 is known to be smaller than that required
for the emergence
of
a zero-energy
level (s 2 1) [ 111; nevertheless the existence of the deuteron and the virtual bound state in the ‘So channel suggests SNN as a benchmark for the appearance of a loosely bound state. Inspection of (9) shows that the conditions for the binding of the
HZ pair are at least not worse than in the NN system. It should be emphasized, however, that this conclusion should be considered only as guide. Indeed, the pentaquark-to-be spends part of the time in the HA channel where the interaction is absent. The resulting attraction
is weakened
and the existence
numerical solution of the system ( 1) . The equations to be solved involve
of a bound
two parameters
state can be explored controlling
the strength
only by of the
interaction. The first is the constant g, which determines the value of the potential unit (4). The value of g can be extracted from the well-known constant of the pseudovector NNT coupling: g =
(10)
!.f~N77,
where fiNlr/47r
= 0.08. However,
the coupling
of the n--meson
to a constituent
quark
bound in the heavy H-meson may be altered by the presence of the heavy quark. Another parameter affecting the strength of one-pion exchange is the form-factor cut-off A (see (5) and (6)). The larger A, the more the potentials are peaked at small X, and the stronger the resulting potential. The value of A is poorly known: in different models of the NN interaction it varies from 0.8 to 1.5 GeV. Note, however, that variations of g and A affect the strength of the interaction in the same direction: when either of these parameter increases the one-pion-exchange potential becomes stronger. The system (l), up to obvious modifications, describes as well the NN interaction in the 3St -3 D1 state. In this case the isotopic T matrix (7) becomes diagonal with all matrix elements equal to - y, and the system ( 1) splits into two identical 2 x 2 subsystems. The value of g which follows from (10) is g M 0.6, and the corresponding
454
hf. ShmatikodNuclearPhysicsA 612 (1997) 449-456
10
8
6 Eb [MeV]
2
4
2 ,
I
0 0.7
0.75
0.8 g
I! 0.85
0.9
Fig. 1. Dependence of the binding energy & in the AH-XH system with the J”(T) = l- (0) quantum numbers upon the form-factor cut-off A and the potential parameter g. Curve 1 (2) corresponds to A = 1500 ( 1200) MeV.
value
of the potential
2.2 MeV is reproduced
Vo M 1.3 MeV [lo]. The deuteron binding energy ED N with the g value increased by 15% (30%) for values of the
unit
cut-off parameter ,4 = 1500 ( 1200) MeV. Note that these values of the potential-strength parameters are somewhat higher than those obtained in [ 10 J . With the same set of parameters we have solved Eqs. (1) for the B/I-B2 system under consideration, in order to determine whether or not a loosely bound state can exist. According
to expectations,
the binding
energy increases
with both g and n (see
Fig. 1). Due to the small value of binding energy the mass of the state is close to the sum of masses of its components: m M mB + rnh x 6.4 GeV. At the same time, a bound state of the DA-D2 potential-strength parameters. In spite of this encouraging
system
was not found for any reasonable
result the BA-BZ
values of the
bound state seems not to be observable
as a resonance structure. Indeed, the hadronic composition of this object when translated into the quark content reads (f@-( sqq). This implies that the five-quark baryonic state can recombine into a tightly bound heavy baryon (bqq) (or its strange counterpart (bsq)) with the emission of mesons built of light quarks. The mass of the lightest bottom-flavored baryon & is about 5.75 GeV [ 121. Because of the large energy release (about 700 MeV), the width of such a decay will be large, and any resonance-like structure will be completely smeared out. A state with genuine pentaquark quantum numbers (i.e. the (6sqqq) configuration) would correspond to a (loosely bound) state in the i?A-BZ system. However, due to negative G-parity of the rr-meson, the onepion-exchange potential in such a system will have sign opposite to that in the BA-BX
M. Shmatikov/Nuclear
system.
Thus
the mechanism
bound i?A-gZ
Physics A 612 (1997) 449-456
considered
will generate
repulsion
455
and, as a result,
no
state should exist.
4. Conclusion We have strangeness,
considered
the possible
Gs E (gsqqq).
existence
of a heavy-flavored
As was stated in the introduction,
pentaquark
with
such a multiquark
state
definitely does not exist as a system of five quarks bound by the color QCD forces. In the approach where a pentaquark is considered as a bound state of a soliton (nucleon) and a heavy meson the problem has yet to be explored. Two points, however, should be taken into account. First, even in the nonstrange sector this model does not yield any decisive conclusions.
Second, the soliton-based
model is ill-suited
to the investigation
of
the %s pentaquark. Indeed, in this model the hyperon treated as a strange counterpart of the nucleon is already considered as a bound state of the Skyrme-type soliton (nucleon) and the K-meson [ 131. Then the consistent description of the heavy meson and the hyperon interaction would imply the investigation of a three-body system comprising the heavy meson, the K-meson and the nucleon (soliton). An alternative soliton-based approach,
where the pentaquark
is a bound state of the nucleon
and the (isoscalar)
B,
meson would imply that the long-range one-pion exchange vanishes and the interaction is controlled by forces of smaller range, with the corresponding ambiguities related to the short-distance In this paper
structure
of the soliton.
we have investigated
One-pion-exchange
proves
to be strong
the existence enough
of a molecular-type
to produce
a loosely
pentaquark. bound
state in
the BA-BZ system with the quantum numbers J”(T) = i-(i). However, because of of this bound state, it can decay rapidly into a bottomthe quark content (b@-(sqq) flavored baryon &, or its strange analogue, with the emission of light-quark mesons. Because of the large energy release the width of such a decay will be rather large precluding possible observation of a resonance-like structure. Thus we can conclude that investigations approaches
of the heavy-flavored
favor the conclusion
strange pentaquark
Gs
that such an exotic multiquark
in various
theoretical
state either does not
exist or is unobservable.
Acknowledgements The author is indebted
to N. Tomqvist
and K. Maltman
for discussions.
References I I 1 Proc. 6th lnt. Conf. on Hadron Spectroscopy, Manchester, UK, July 1995, eds. M.C. Birse, G.D. Lafferty and J.A. McGovern (World Scientific, Singapore, 1996). 121 C. Gignoux et al., Phys. Lett. B 193 (1987) 323. 131 H.J. Lipkin, Phys. Lett. B 195 (1987) 484.
456
M. Shmatikov/Nuclear Physics A 612 (1997) 449-456
[4] S. Fleck et al., Phys. L&t. B 220 (1989) 616.
(51 [6] [7] [ 81 [9]
S. Zouzou and J.-M. Richard, Few-Body Syst. 16 ( 1994) 1. D.O. Riska and N.N. Scoccola, Phys. Lett. B 299 (1993) 338. M. Shmatikov, Phys. Lett. B 349 (1995) 411. Y. Oh, B.-Y. Park and D.-P Min. Phys. L&t. B 331 ( 1994) 362. M.M. Nagels, T.A. Rijken and J.J. de Swart, Phys. Rev. D 12 (1975) 744. [lo] N.A. Tiimqvist, Phys. Rev. L.ett. 67 (1992) 556. [ 111 J.M. Blatt and V.F. Weisskopf, Theoretical nuclear physics (Wiley, New York, 1952). [ 121 Particle Data Group, Phys. Rev. D 50 (1994) 1173. [ 131 C.D. Callan, K. Hombostel and I. Klebanov, Phys. Len. B 202 ( 1988) 269.
Why the heavy-flavored strange pentaquark does not exist M. Shmatikov Russian Research Center “Kurchatov Institute”,
123182 Moscow, Russia
Received 29 July 1996; revised 23 September 1996
Abstract The possible existence of a molecular-type heavy-flavored strange pentaquark is considered. It is shown that no narrow state of this type exists. PACS:
14.4O.J~;14.403; 12.4O.Qq
1. Introduction The most impressive achievement of QCD in the nonperturbative domain is the classification of hadrons. The bulk of the observed hadrons can be successfully classified at present as ijq or qqq configurations. A few controversial candidates for a beyond-minimal quark configuration are under debate [ 11. The challenging question of the existence of exotics remains unanswered. One such exotic state is the pentaquark: a baryon containing five quarks G G gqqqq where Q (= b, c) is a heavy quark and q (= u, d, s) is a light one. Theoretical investigations of the pentaquark have yielded controversial results. Its existence has been claimed in various approaches. In the constituent quark model the existence was confirmed provided (i) one of the light quarks is a strange one and (ii) the binding color forces are assumed to be flavor-independent [ 2,3]. Stated differently in the limit of “flavor-blind” color forces the F& is expected to exist. However, with realistic breaking of SU( 3) ,G and, even more important, proper treatment of the kinetic energy, the pentaquark proved to be unstable with respect to strong decay [ 4,5 1. An alternative approach to the issue is to consider the pentaquark as the bound state of a Skyrme-type soliton (representing the nucleon) and a heavy meson. This model has been applied in the nonstrange sector only and the results obtained are far from decisive. The bound state was shown to exist in both b- and c-sectors (corresponding to the Pr, 037%9414/91/$17.00 Copyright @ PII SO375-9474(96)00410-l
1997 Elsevier Science B.V. All rights reserved.
ht. Shmatikov/Nuclear Physics A 612 (1997) 449-456
450
and Pz type pentaquarks)
in the limit of the infinite heavy-quark
due to the finite mass of the heavy quark, however, unbound [ 81. Finally, a molecular-type
pentaquark
mass [6]. Corrections
may make the pentaquark
P6 state has been predicted
states
[ 71. It is bound by
the long-range one-pion-exchange force, ensuring its genuine molecular structure. These forces, however, are not strong enough to bind a Pt pentaquark. In view of this situation, it is of interest
2. Quantum
to explore whether molecular-type
numbers
A molecular-type
of a molecular-type
exotic
Pb pentaquark
nonstrange extension
of this approach
strange meson pS( z (6s)) A viable alternative H* 1 and a hyperon
hadron
GS pentaquarks
strange heavy-flavored pentaquark
can be characterized
is a loosely
bound
the nucleon
fails because the pS-meson state consisting
Y. Two possibilities
by its particle
state of the B*N
to the system containing
is a molecular
exist.
The
A “naive”
and a bottom-flavored
does not couple to the r-meson.
of a (nonstrange)
are conceivable
system.
content.
depending
heavy quark meson on the total isospin
of the system T. Consider first the T = $ case. Then the pentaquark-to-be necessarily has the Hi.7 particle content. The “effective” strength constant of the driving one-pionexchange potential in the T = ; channel equals then v = fzznfHHry, where y is a spin
factor. In the limit of the W(3) symmetry fxzr = 2ap fNNT,where fNN,, is the constant of the NNn- coupling and (up is the F/( F + D) ratio for the pseudoscalar mesons. The n--meson is known to couple to the isovector particles only and, hence, it is uncoupled from the isoscalar
heavy meson. Introducing the constant g of the q-meson coupling to the constituent light quark we arrive at ~HH~ = g. The coupling constant g is related also to the NN?r coupling constant: g = i ft,rp,,,. Combining the factors above we get the following expression for the “effective” strength constant of the one-pion-exchange in the T = ; channel:
$wYfiww
vHP=
It is instructive to compare this strength constant to its counterpart system. The latter, taking into account the spin-isospin factors, is VNN
=
VNN in the NN
3f;N,y
The ratio of vH~ and VNN strength constants vHP/vNN=
is thus
+F'y.
The UP factor in the limit of the SU( 6) symmetry is 0.4, while in the Nijmegen potential ap = 0.485 [ 91. The value of the spin factor y depends on the spin and orbital momenta ’ Since it does not result in confusion we hereafter will denote for notational brevity the heavy vector meson as H.
M. Shmatikov/Nuclear Physics A 612 (1997) 449-456
of the considered
state. For the most favorable
2St,2-4D1,2
451
configuration
(see [ 71) it
is y = 2. Thus we arrive at
showing reduced
that VHz M ~VNN. One more factor which is important for binding is the mass of the system. The ratio of reduced masses for the HZ and NN systems
equals 1
mHZ‘ - 2% mN
mNN
1 +
This ratio equals reduced
MZ/mH
’
in the case of D- (B-)
1.55 (2.08)
mass of the HZ system as compared
the smallness
of the effective
strength constant
heavy
meson.
The increased
to the NN case still cannot overbalance VHp, allowing
us to safely conclude
that
the molecular-type H.Z pentaquark with T = 3 does not exist. An alternative possibility is the HY molecular state with total isospin T = l. This case comprises two coupled channels: HA and HZ. As stated above, the configuration most favorable from the point of view of binding by one-pion-exchange has the J” = k quantum numbers, corresponding to the 2St,2-4Dt,2 at the system of four coupled Schrodinger equations
coupled channels. Thus we arrive which can be written in the matrix
form
where 4 is the column
of wave functions
with the following
channels
labels:
(i) HA
(S-wave); (ii) HA (D-wave); (iii) HZ (S-wave); (iv) HZ (D-wave) and K* is the diagonal 4 x 4 matrix of the c.m.s. momenta squared (plus the matrix of centrifugal potentials)
: 2 KAH I&
+ 6/r*
(2) 4H ~~~
+
6/r* 1.
squared in Eq. (2) for the kinetic energy E in the HA channel is where Am is the mass difference K,,~ = 2rn,iHE and Shihdy ~~~ = 2rnzH( E + Am), of the Z- and A-hyperons (Am = rns - m,l M 76 MeV) and mm (Y = A, 2) is the reduced mass in the corresponding channel. In Eq. ( 1) m is the diagonal 4 x 4 matrix with the mAH and mzH elements. Finally, V is the matrix of driving forces reading: The c.m.s. momenta 2
V” = Vo [T 8 C,jb(m,r)
where VO is the potential
+ T 8 C,92(m,r)]
unit introduced
,
(3)
in [ 101 (4)
M. ShmatikodNuclear
452
Physics A 612 (1997) 449-456
where g is the r-meson-constituent-quark stant (jr functions
= 132 MeV).
coupling
In Eq. (3) 71 (I = 0,2)
~o:O(X) = YO(X)
-
YO(AX)
-
j2(x)
-
A3y2(Ax)
(A
-
and f,, is the n--meson
are the regularized
decay con-
(at small X) radial
1) exp(-Ax),
(5)
and (P= y2(x)
-
where y0,2 (x) are the modified YO(X)
=
exp(-xl . x ’
exp(-Ax)
spherical
l)(Ax+
1) ’
2x
Bessel functions
y2=exyx) (I+:+;)
and A = A/m,, with .4 being a cut-off parameter (see below). The Cs(,) matrix contains spin coefficients for the central (tensor) components of the one-pion-exchange forces:
Finally, T is the matrix of isospin coefficients
which in the limit of the SU( 3) symmetry
limit reads HA
I
H-2
I
(7)
where cyp is defined above.
3. Binding of a molecular-type The complexity methods
of the system
mandatory.
us consider
under consideration
Before proceeding
the conditions
we apply the criterion
to the numerical
makes application solution
of a zero-energy
of numerical
of the system
under which a loosely bound state is favorable.
for the appearance
central force is operative to be investigated
pentaquark
level in a system
[ 11 I. For the Yukawa forces under consideration
(1) let
To this end where a
the parameter
is
vo s = 0.5953 x 2my--, mZ where fi is the reduced mass of the system and y is the spin-isospin factor. The only component of the system under consideration where the Yukawa forces are operative is the S-wave HZ channel. To make our conclusions more transparent we compare the SIH value to the (reference) value of its NN counterpart SNN: SZH SNN
--
YZH YNN
1
x23 mN1
+mZ/mH'
(8)
M. Shmatikov/Nuclear Physics A 612 (1997) 449-456
The spin-isospin
yW factor equals the product of corresponding
453
elements
of the C, and
T matrices: YzH = 2 x %jffp. In the same notations
Substituting
the YNN equals
these expressions
SXH
into Eq. (8) we arrive at
4aP
-2
1 +
SNN
mP/mH
This ratio depends (B-)
’
upon the mass of the heavy meson mH and for the case of the D-
meson it is S~H M SNN X 1.0 (1.3).
(9)
The value SNN M 0.33 is known to be smaller than that required
for the emergence
of
a zero-energy
level (s 2 1) [ 111; nevertheless the existence of the deuteron and the virtual bound state in the ‘So channel suggests SNN as a benchmark for the appearance of a loosely bound state. Inspection of (9) shows that the conditions for the binding of the
HZ pair are at least not worse than in the NN system. It should be emphasized, however, that this conclusion should be considered only as guide. Indeed, the pentaquark-to-be spends part of the time in the HA channel where the interaction is absent. The resulting attraction
is weakened
and the existence
numerical solution of the system ( 1) . The equations to be solved involve
of a bound
two parameters
state can be explored controlling
the strength
only by of the
interaction. The first is the constant g, which determines the value of the potential unit (4). The value of g can be extracted from the well-known constant of the pseudovector NNT coupling: g =
(10)
!.f~N77,
where fiNlr/47r
= 0.08. However,
the coupling
of the n--meson
to a constituent
quark
bound in the heavy H-meson may be altered by the presence of the heavy quark. Another parameter affecting the strength of one-pion exchange is the form-factor cut-off A (see (5) and (6)). The larger A, the more the potentials are peaked at small X, and the stronger the resulting potential. The value of A is poorly known: in different models of the NN interaction it varies from 0.8 to 1.5 GeV. Note, however, that variations of g and A affect the strength of the interaction in the same direction: when either of these parameter increases the one-pion-exchange potential becomes stronger. The system (l), up to obvious modifications, describes as well the NN interaction in the 3St -3 D1 state. In this case the isotopic T matrix (7) becomes diagonal with all matrix elements equal to - y, and the system ( 1) splits into two identical 2 x 2 subsystems. The value of g which follows from (10) is g M 0.6, and the corresponding
454
hf. ShmatikodNuclearPhysicsA 612 (1997) 449-456
10
8
6 Eb [MeV]
2
4
2 ,
I
0 0.7
0.75
0.8 g
I! 0.85
0.9
Fig. 1. Dependence of the binding energy & in the AH-XH system with the J”(T) = l- (0) quantum numbers upon the form-factor cut-off A and the potential parameter g. Curve 1 (2) corresponds to A = 1500 ( 1200) MeV.
value
of the potential
2.2 MeV is reproduced
Vo M 1.3 MeV [lo]. The deuteron binding energy ED N with the g value increased by 15% (30%) for values of the
unit
cut-off parameter ,4 = 1500 ( 1200) MeV. Note that these values of the potential-strength parameters are somewhat higher than those obtained in [ 10 J . With the same set of parameters we have solved Eqs. (1) for the B/I-B2 system under consideration, in order to determine whether or not a loosely bound state can exist. According
to expectations,
the binding
energy increases
with both g and n (see
Fig. 1). Due to the small value of binding energy the mass of the state is close to the sum of masses of its components: m M mB + rnh x 6.4 GeV. At the same time, a bound state of the DA-D2 potential-strength parameters. In spite of this encouraging
system
was not found for any reasonable
result the BA-BZ
values of the
bound state seems not to be observable
as a resonance structure. Indeed, the hadronic composition of this object when translated into the quark content reads (f@-( sqq). This implies that the five-quark baryonic state can recombine into a tightly bound heavy baryon (bqq) (or its strange counterpart (bsq)) with the emission of mesons built of light quarks. The mass of the lightest bottom-flavored baryon & is about 5.75 GeV [ 121. Because of the large energy release (about 700 MeV), the width of such a decay will be large, and any resonance-like structure will be completely smeared out. A state with genuine pentaquark quantum numbers (i.e. the (6sqqq) configuration) would correspond to a (loosely bound) state in the i?A-BZ system. However, due to negative G-parity of the rr-meson, the onepion-exchange potential in such a system will have sign opposite to that in the BA-BX
M. Shmatikov/Nuclear
system.
Thus
the mechanism
bound i?A-gZ
Physics A 612 (1997) 449-456
considered
will generate
repulsion
455
and, as a result,
no
state should exist.
4. Conclusion We have strangeness,
considered
the possible
Gs E (gsqqq).
existence
of a heavy-flavored
As was stated in the introduction,
pentaquark
with
such a multiquark
state
definitely does not exist as a system of five quarks bound by the color QCD forces. In the approach where a pentaquark is considered as a bound state of a soliton (nucleon) and a heavy meson the problem has yet to be explored. Two points, however, should be taken into account. First, even in the nonstrange sector this model does not yield any decisive conclusions.
Second, the soliton-based
model is ill-suited
to the investigation
of
the %s pentaquark. Indeed, in this model the hyperon treated as a strange counterpart of the nucleon is already considered as a bound state of the Skyrme-type soliton (nucleon) and the K-meson [ 131. Then the consistent description of the heavy meson and the hyperon interaction would imply the investigation of a three-body system comprising the heavy meson, the K-meson and the nucleon (soliton). An alternative soliton-based approach,
where the pentaquark
is a bound state of the nucleon
and the (isoscalar)
B,
meson would imply that the long-range one-pion exchange vanishes and the interaction is controlled by forces of smaller range, with the corresponding ambiguities related to the short-distance In this paper
structure
of the soliton.
we have investigated
One-pion-exchange
proves
to be strong
the existence enough
of a molecular-type
to produce
a loosely
pentaquark. bound
state in
the BA-BZ system with the quantum numbers J”(T) = i-(i). However, because of of this bound state, it can decay rapidly into a bottomthe quark content (b@-(sqq) flavored baryon &, or its strange analogue, with the emission of light-quark mesons. Because of the large energy release the width of such a decay will be rather large precluding possible observation of a resonance-like structure. Thus we can conclude that investigations approaches
of the heavy-flavored
favor the conclusion
strange pentaquark
Gs
that such an exotic multiquark
in various
theoretical
state either does not
exist or is unobservable.
Acknowledgements The author is indebted
to N. Tomqvist
and K. Maltman
for discussions.
References I I 1 Proc. 6th lnt. Conf. on Hadron Spectroscopy, Manchester, UK, July 1995, eds. M.C. Birse, G.D. Lafferty and J.A. McGovern (World Scientific, Singapore, 1996). 121 C. Gignoux et al., Phys. Lett. B 193 (1987) 323. 131 H.J. Lipkin, Phys. Lett. B 195 (1987) 484.
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