Relativistic five-quark equations and negative parity pentaquarks

Abstract

The relativistic five-quark equations are found in the framework of the dispersion relation technique. The solutions of these equations using the method based on the extraction of the leading singularities of the amplitudes are obtained. The five-quark amplitudes for the low-lying pentaquarks including the u, d, s- quarks are calculated. The poles of these amplitudes determine the masses of the negative parity pentaquarks with I = 0, 1 and spin 3/2-, 5/2-. The mass of the lowest pentaquark with I = 0 and spin 3/2- is equal to 1514 MeV. Comment: 18 pages, pdf, published version

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Relativistic five-quark equations and negative parity pentaquarks.

Gerasyuta S.M.1,2 , Kochkin V.I.1

1. Department of Theoretical Physics, St. Petersburg State University, 198904,
St. Petersburg, Russia.
2. Department of Physics, LTA, 194021, St. Petersburg, Russia,
E-mail: gerasyuta@sg6488.spb.edu




Abstract

The relativistic five-quark equations are found in the framework of the dispersion relation
technique. The solutions of these equations using the method based on the extraction of the
leading singularities of the amplitudes are obtained. The five-quark amplitudes for the low-
lying pentaquarks including the u, d, s- quarks are calculated. The poles of these amplitudes
determine the masses of the negative parity pentaquarks with I = 0, 1 and
−−
=
2
5
2
3,
P
J
. The
mass of the lowest pentaquark with I = 0 and
−
=2
3
P
J
is equal to 1514 MeV.





PACS number: 11.55Fv, 12.39 Ki, 12.39Mk, 12.40 Yx

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I. Introduction.

Recent observation of an exotic baryon state with positive strangeness,
+
θ (1540), by
LEPS collaboration in Spring – 8 [1] and subsequent experiments [2 - 10] has raised great
interest in hadron physics. This state cannot be an ordinary three-quark baryon since it has
positive strangeness, and therefore the minimal quark content is
sudud . The most striking
feature of
+
θ (1540) is that the width is unusually small: Γ< 25 MeV despite the fact that it lies
about 100 MeV above the NK threshold.
The discovery of
+
θ has triggered intensive theoretical studies to understand the structure
of the
+
θ [11 - 20]. One of the main issues is to clarify the quantum numbers, especially, the
spin and the parity, which are key properties to understand the abnormally small width. Recent
analysis of the
+
K scattering from the xenon or deuteron implies even smaller value Γ< 1
MeV [21 - 23]. The chiral soliton model has predicted the masses and widths of the pentaquark
baryons with less theoretical ambiguity based on the
f
SU
) 3 (
flavor algebra [24]. The model
indicates the width of
+
θ around a few ten MeV [25]. In the non-relativistic quark model (A.
Hosaka et al) [26] found that the negative parity
+
θ width becomes very large which is of order
of several hundreds MeV, while it is about a several tens MeV for the positive parity. By
assuming additionally diquark correlations, the width is reduced to be of order 10 MeV.
However the possibility of
−
=2
3
P
J
is not excluded. If
+
θ carries
−
=2
3
P
J
, it decays into
nucleons and kaons in the D-wave. The phase space suppression factor is ~
5)/(
θ
mpk
~
5
10−,
where
k p is the decay momentum of kaon. The decay width could be well below ten MeV
even if the coupling constant
KN
gθ
is big due to θ s strong overlap with KN .
There have been a few theoretical papers on the possible
−
=2
3
P
J
pentaquarks using
different models. Page et al. suggested I = 2,
−
−
=
2
3
2
1,
P
J
for
+
θ to resolve the narrow width
puzzle [17]. Jaffe and Wilczek discussed the
−
=2
3
P
J
assignment for Ξ pentaquark [27]. The
mass spectrum of
−
=2
3
P
J
pentaquarks were studied with the perturbative chiral quark model
[28]. Takeuchi and Shimizu suggested the observed θ resonance as I = 0,
−
=2
3
P
J

*
NK bound

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state using the quark model [29]. With the flux tube model, Kanada-Enyo et al. studied the
mass and decay width of the I = 1,
−
=2
3
P
J
pentaquark [30]. Huang et al. proposed
+
θ as
molecular state of
π
NK with I = 1,
−
=2
3
P
J
using the chiral ) 3 (
SU
quark model [31]. The
phenomenology of
2
3
=
P
J
pentaquarks such as the mixing scheme and mass pattern was
discussed in [32]. Pentaquark states with
2
3
=
P
J
and I = 0, 1 were studied using currents
composed of one scalar diquark and one vector diquark [33]. Shi-Lin Zhu investigated the
possible existence of the spin 3/2 pentaquark states using the finite energy sum rule [34].
In our previous paper the relativistic five-quark equations for the family of the θ
pentaquarks are constructed [35]. The five-quark amplitudes for the low-lying θ pentaquarks
are calculated. The poles of these amplitudes determine the masses of the θ pentaquarks. The
masses of the constituent u, d, s -quarks coincide with the quark masses of the ordinary baryons
in our quark model [36]:
du
m,= 410 MeV,
s
m = 557 MeV. The model has only three
parameters. The cut-off parameters
+
Λ0=16.5 and
+
Λ1=20.12 are similar to those [37]. The
gluon coupling constant g =0.456 is fitted by fixing
+
θ pentaquark mass m (1540). We
received the isotensor θ pentaquarks with
±
±
=
2
3
2
1,
P
J
and predict the masses of
++
θ
(
0
θ ) and
+++
θ
(
−
θ ) pentaquarks [35].
The present paper is devoted to the construction of relativistic five-quark equations for
the low-lying negative parity pentaquarks, which decay into nucleons and kaons in the D-wave.
The pentaquarks built with s quark in D-wave. The five-quark amplitudes for the negative
parity pentaquarks are constructed. The poles of these amplitudes determine the masses of the
negative parity pentaquarks with I = 0, 1 and
−−
=
2
5
2
3,
P
J
. The mass of the lowest pentaquark
with I = 0 and
−
=2
3
P
J
is equal to 1514 MeV. We use the parameters of the previous paper.
The paper is organized as follows. After this introduction, we discuss the five-quark
amplitudes, which contain s - antiquark in the D-wave and four nonstrange quarks (Section 2).
In section 3, we report our numerical results (Tables I, II) and the last section is devoted to the
discussion and conclusion.

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II. Five-quark amplitudes for the negative parity pentaquarks.

We derive the relativistic five-quark equations in the framework of the dispersion relation
technique. We use only planar diagrams; the other diagrams due to the rules of
c
N
/ 1 expansion
[38 - 40] are neglected.
The current generates a five-quark system. Their successive pair interactions lead to the
diagrams shown in Figs. 1, 2. The correct equations for the amplitude are obtained by taking
into account all possible subamplitudes. It corresponds to the division of complete system into
subsystems smaller number of particles. Then one should represent a five-particle amplitude as
a sum of ten subamplitudes:
45 353425 2423 15 141312
AAAAAAAAAAA
+++++++++=
. (1)
This defines the division of the diagrams into group according to the certain pair interaction of
particles. The total amplitude can be represented graphically as a sum of diagrams.
We need to consider only one group of diagrams and the amplitude corresponding to
them, for example
12
A . We shall consider the derivation of the relativistic generalization of the
Faddeev-Yakubovsky approach for
+
θ pentaquark. We shall construct the five-quark
amplitude of four up quarks and one strange antiquark, in which the pair interactions with the
quantum numbers of
+
0 and
+
1 diquarks are included. The set of diagrams associated with the
amplitude
12
A can further be broken down into groups corresponding to amplitudes:
),,,(
34121234
s
1
sssA
, ),,,(
35 121234
s
2
sssA
, ),,,(
34 251234
s
3
sssA
, ),,,(
35 241234
s
4
sssA
,
),,,(
134
s
131234
s
5
ssA
, ),,,(
234231234
s
6
sssA
, ),,,(
234241234
s
7
sssA
(Fig. 1). The s is shown by the
arrow and other lines correspond to the four nonstrange quarks. In the case of the
++
θ
(
0
θ )
pentaquarks we need to use six (Fig. 2) subamplitudes.
Here
ik s is the two-particle subenergy squared,
ijk
s corresponds to the energy squared of
particles i j k
, , ,
ijkl
s
is the four-particle subenergy squared and s is the system total energy
squared.
The system of graphical equations are determined by the subamplitudes using the self-
consistent method. If we consider the
+
θ state, we use seven subamplitudes (Fig. 1). In the

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case of the
++
θ
(
0
θ ) pentaquarks we derive six (Fig. 2) subamplitudes. The coefficients are
determined by the permutation of quarks [41, 42]. We do not extract the initial four- and three-
particle singularities in the subamplitude ),,,(
34 121234
s
1
sssA
because they are weaker than the
two-particle singularities.
In order to represent the subamplitudes ),,,(
34121234
s
1
sssA
, ),,,(
35 121234
s
2
sssA
,
),,,(
34 251234
s
3
sssA
,
),,,(
35 241234
s
4
sssA
,
),,,(
134
s
131234
s
5
ssA
,
),,,(
234 23 1234
s
6
sssA
,
),,,(
23424 1234
s
7
sssA
in the form of a dispersion relation it is necessary to define the amplitudes
of quark-quark and quark-antiquark interaction )(
iknsb
. The pair quarks amplitudes
qqqq →

and
qqqq →
are calculated in the framework of the dispersion N/D method with the input
four-fermion interaction [43- 45] with quantum numbers of the gluon [46]. The regularization
of the dispersion integral for the D-function is carried out with the cut-off parameters
n
Λ .
The four-quark interaction are considered as an input [46]:
2)()(2
)() )((2)(
ssgssqIqgqIqg
ss
f
s
f
µνµµνµν
λγ λγ λγγλ++
r
(2)
Here
fI is the unity matrix in the flavor space (u, d), λ are the color Gell-Mann matrices.
Dimentional constants of the four-fermion interaction
ν g ,
)(s
gν and
)(ss
gν
are parameters of the
model. At
ν g =
)(s
gν =
)(ss
gν
the flavor
f
SU
) 3 (
symmetry occurs. The strange quark violates the
flavor
f
SU
) 3 (
symmetry. In order to avoid an additional violation parameters we introduce the
scale shift of the dimentional parameters [46]:
)(
2
2
s
)(
2
2
2
2
4
)(
sss
s
g
m
π
g
mm
g
m
π
ννν
π
γ=
+
==
(3)
2)(
)(
m
4
m
ki
n
n
ik
+
Λ
=
λ
, n= 1, 2, 3. (4)
Here
i
m and
k
m are the quark masses in the intermediate state of the quark loop.
Dimensionless parameters γ and
n
λ are supposed to be constants which are independent of the
quark interaction type. The applicability of (Eq. 2) is verified by the success of De Rujula-
Georgi-Glashow quark model [47], where only the short-range part of Breit potential connected
with the gluon exchange is responsible for the mass splitting in hadron multiplets.