Relativistic quark model and pentaquark spectroscopy.



Gerasyuta S.M. , 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.

3. Forschungzentrum Julich, Institut fur Kernphysik (Theorie)

D-52425 Julich, Germany.









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 leading singularities of the amplitudes are obtained. The five-quark amplitudes for the low-

lying pentaquarks are calculated under the condition that flavor SU )
3
( symmetry holds. The

poles of five-quark amplitudes determine the masses of the lowest pentaquarks. The mass

spectra of pentaquarks which contain only light quarks are calculated. The calculation of

pentaquark amplitudes estimates the contributions of three subamplitudes: molecular

subamplitude BM , Mqqq subamplitude and q
Bq subamplitude. The main contributions to

the pentaquark amplitude are determined by the subamplitudes, which include the meson

states M.





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












1


I. Introduction


The existence of particles made of more than three quarks is an important issue of QCD

inspired models. The role of stable exotic against strong decays is crucial for understanding of

some aspects of the strong interactions. Their properties could be an important test for the

validity of various quark models. The five-quark systems (pentaquarks) was proposed

independently Gignoux, Silvestre-Brac, Richard [1] and Lipkin [2]. The more realistic

calculations [3] which take into account the flavor SU )
3
( - breaking, lead to stable systems

against strong decays. In the De Rujula-Georgi-Glashow model [4] the stable pentaquarks have

negative parity and require strangeness [3].

Accoding to recent experiments, no convincing evidence for the production of charmed-

strange pentaquarks has been observed either. However the existence of pentaquarks is not

ruled out. The analysis done so far can provide a good starting point for the future search in

high statistics charm experiments at CERN [5] or Fermilab [6]. The extension of the Skyrmion

approach to the heavy flavor sectors [7, 8] allowed to calculate the spectra of the low-lying

pentaquarks containing charm and bottom antiquarks. The conclusion of this model are similar

to those of chiral model with Goldstone boson exchange (GBE) [9]. In the GBE the candidates

of stability are not necessarely strange and have positive parity [10]. The Skyrmion approach

allowed to calculate the low-lying exotic baryon +
Z (spin , isospin 0, strangeness +1) with

mass 1530 MeV [11]. The binding of pentaquarks due to the long range one pion exchange has

been observed to show a molecular type structure [12]. The lattice gauge calculations became

also recently avaible [13] and they may shed additional light on the interaction of quarks.

In [14, 15] a relativistic generalization of the three-body Faddeev - Yakubovsky

equations was obtained in the form of dispersion relations in the pair energy of two interacting

particles. The mass spectrum of S-wave baryons including u, d and s quarks was calculated by

method based on isolating of leading singularities in the amplitude. We found the approximate

solution of integral three-quark equations by taking into account two-particle and triangle

singularities, all the weaker ones being neglected. If we considered the approximation, which

corresponds to taking into account two-body and triangle singularities, and defined all the

smooth functions of the subenergy variables (as compared with the singular part of the





2


amplitude) in the middle point of physical region of Dalitz-plot, then the problem reduces to

one of solving simple algebraic system equations.

In our paper [16] the relativistic generalization of the four-body Faddeev - Yakubovsky

type equations are represented in the form of dispersion relations of the two-body subenergy.

We investigated the relativistic scattering four-body amplitudes of the constituent quarks of

two flavors (u, d). The poles of these amplitudes determine the masses of the lowest hybrid

mesons. The constituent quark is color triplet and quark amplitudes obey the global color

symmetry. We used the results of the bootstrap quark model [17] and introduced the q
q  state

in the color octet channel with 3& --
- = and isospin I = 0. This bound state is identified as a
constituent gluon. In our model we take into account the color state with 3& --
- = and isospin

I = 1, which determines with the constituent gluon the hybrid state. In addition, TTTT states

are also predicted.

We derived the mixing of the hybrid and TTTT states. This state was called the hybrid

meson. The mass spectrum of lowest hybrid mesons with isospin I = 1 both with exotic

quantum numbers (non - q
q ) 3& -+
- = , --
and ordinary quantum numbers 3& ++
- = , ++
,

++
, -+
, --
was calculated. The important result of this model is the calculation of hybrid

meson amplitudes, which contain the contribution of two subamplitudes: four-quark amplitude

and hybrid amplitude. The main contribution corresponds to the four-quark amplitude. The

hybrid amplitude gives rise to only less 40 % of the hybrid meson contribution.

In our paper [18] the relativistic generalization of five-quark equations (like Faddeev 

Yakubovsky approach) are constructed in the form of the dispersion relation. The five-quark

amplitudes for the low-lying hybrid baryons contain only light quarks and are calculated under

the condition that flavor SU(3) symmetry holds. In should be noted, that the calculated masses

of low-lying hybrid baryons agree with data [19] and with the results obtained in the flux-tube

model [20].

In our relativistic quark model with four-fermion interaction the octet color q
q bound

state was found, which corresponds to the constituent gluon G with mass M = 0.67 GeV
G


[17]. This approach is similar to the large N limit [21-23]. In diquark channel we have the
c

diquark level D with P +
J = 0 and the mass m = 0.72 GeV (in the color state 3 ). The
ud c





3


diquark state with P +
J = 1 in color state 3 also has an attractive interaction, but smaller than
c

that of the diquark with P +
J = 0 , therefore there is only the correlation of quarks, not a bound

state [17].

The calculated five-quark amplitude consists of four subamplitudes: qDG , qqqG , q
D D

and q
qq D , where D and G are the diquark state and exited constituent gluon state

respectively. The main contributions to the hybrid baryon amplitude are determined by the

subamplitudes, which include the exited gluon states.

The present paper is devoted to the construction of relativistic five-quark equations for

the pentaquarks. The five-quark amplitudes for the lowest pentaquarks contain only light

quarks and take into account the flavor SU(3) symmetry. The poles of these amplitudes

determine the masses of the low-lying pentaquarks. The constituent quark is the color triplet

and the quark amplitudes obey the global color symmetry. The interesting result of this model

is the calculation of pentaquark amplitudes which contain the contribution of three

subamplitudes: molecular subamplitude BM , Mqqq subamplitude and q
Bq subamplitude.

Here B corresponds to the lowest baryon (nucleon and isobar baryon). M are the low-

lying mesons with the quantum numbers: PC ++ ++ ++ -+ --
J = 0 1
, ,2 ,0 1
, and isospin I = 0. We call

P + P +
the pentaquark with J = 1 as the N pentaquark and the pentaquark with J = 3 as the
2 2

isobar pentaquark.

The mass values of the low-lying pentaquarks are calculated (Table 1, 2). The lowest

P +
mass of N pentaquark with J = 1 is equal M=1553 MeV. The pentaquark amplitudes take
2


into account the contribution of three subamplitudes. The main contributions to the pentaquark

amplitude are determined by subamplitudes which include the low-lying meson with

PC ++ ++ ++ -+ --
J = 0 1
, ,2 0
, 1
, .

The paper is organized as follows.

After this introduction, we discuss the five-quark amplitudes which contain only light

quarks (section 2).

In the section 3, we report our numerical results (Tables 1, 2) and the last section is

devoted to our discussion and conclusion.





4


In the Appendix A we give the relations, which allow to pass from the integration of the

cosines of the angles to the integration of the subenergies.

In the Appendix B we describe the integration contours of functions J , J
, J
, which
1 2 3


are determined by the interaction of the five quarks.

In the Appendix C we obtain the determinant of the algebraic equations, which allows

one to calculate the mass spectra of the pentaquarks.

In the Appendix D the quark-quark and quark-antiquark vertex functions and phase

spaces for the pentaquarks are given respectively (Tables 3, 4).



II. Pentaquark amplitudes.



We derived the relativistic five-quark equations in the framework of the dispersion

relation technique. For the sake of simplicity one considers the case of the SU ( )
3 - symmetry,
f


that the masses of all quarks are equal. We use only planar diagrams, the other diagrams due to

the rules of 1 / N expansion [21-23] are neglected. The correct equations for the amplitude are
c


obtained by taking into account the all possible subamplitudes. It corresponds to the division

complete system into subsystems from the smaller number of particles. Then one should

represent five-particle amplitude as a sum of ten subamplitudes:

A = A + A + A + A + A + A + A + A + A + A . In our case all particles are
12 13 14 15 23 24 25 34 35 45


identical, therefore we need to consider only one group of diagrams and the amplitude

corresponding to them, for example $ . The set of diagrams associated with the amplitude

$ can be further broken down into three groups corresponding to amplitudes

A (s, s , s , s ) , A (s, s , s , s ) , A (s, s , s , s ) (Fig. 1). The antiquark is shown
1 1234 12 34 2 1234 12 123 3 1234 25 125


by the arrow, the other lines correspond to the quarks. The coefficients are determined by the

permutation of quarks [24, 25].

In order to represent the subamplitudes A (s, s , s , s ) , A (s, s , s , s ) ,
1 1234 12 34 2 1234 12 123


and A (s, s , s , s ) in form of the dispersion relation it is necessary to define the
3 1234 25 125


amplitudes of quark-quark and quark-antiquark interaction b (s ) . The quark amplitudes
n ik





5


T
T TT and qq qq are calculated in the framework of the dispersion N/D method with

the input four-fermion interaction with quantum numbers of the gluon [17]. We use the results

of our relativistic quark model [17] and write down the pair quarks amplitude in the form:



2
G (s )
b (s ) n ik
= , (1)
n ik 1 - B (s )
n ik

ds' (s' G2
) (s' )
ik n ik n ik
B (s ) =
n ik .
(2)
2 s' - s
ik ik
4m




Here VLN is the two-particle subenergy squared, VLMN corresponds to the energy squared of

particles i, j, k
, s is the four-particle subenergy squared and V is the system total energy
ijkl


squared. )
G (s are the quark-quark and quark-antiquark vertex functions (Table 3). B (s ) ,
n ik n ik

(s ) are the Chew-Mandelstam functions with the cut  off [26] and the phase spaces
n ik

respectively (Appendix D, Table 4). There n=1 corresponds to qq -pair with P +
J = 0 in the 3
c

color state, n=2 describes qq -pair with P +
J = 1 in the 3 color state and n=3 defines the q
q -
c

pairs, with correspond to the mesons with quantum numbers: PC ++ ++ ++ -+ --
J = 0 1
, 2

, ,0 1
, and

isospin I = 0.

In the case in question the interacting quarks do not produce a bound state, therefore the

integration in (3) - (5) is carried out from the threshold 4 2
m to the cut-off . The integral

equation systems, corresponding to Fig. 1 (the meson state with PC ++
J = 0 and diquark with
P +
J = 0 ), can be described as:



B (s )B (s )
A (s, s , s , s ) 1 3 12 1 34
= + 6J )
1
,
3
( A (s, s , '
s , '
s ) +
1 1234 12 34 1
[ - B (s 1
)][ - B (s )] 2 3 1234 23 234
3 12 1 34 , (3)

+ 2J )
1
,
3
( A (s, s , '
s , '
s ) + 6J )
3
( A (s, s , '
s , s ) + 2J )
3
( A (s, s , '
s , s )
2 2 1234 13 134 1 2 1234 15 125 1 3 1234 25 125

B (s )
A (s, s , s , s ) 2 3 12
= + 4J )
3
( A (s, s , '
s , '
s ) , (4)
2 1234 12 123 1 - B (s ) 3 1 1234 13 24
3 12





6


B (s )
A (s, s , s , s ) 3 1 25
= + 2J )
1
( A (s, s , '
s , '
s ) , (5)
3 1234 25 125 1 - B (s ) 3 1 1234 35 21
1 25




were are the current constants. We introduced the integral operators:
i







+
*
O V GV *O V O V G]
- O =
, (6)



> - %
O
V @ V - V
P -

*O V *S V
- O S = 

> - %O V
@> -
%S V @ , (7)






+ +
GV *
O V O V GV *S V S V
 G]
G]





V - V V - V
P P - -

*O V a
- O = 
- O
% V a

a + ,
(8)


a
+ + ]
GV *
O V
 O V
G]
G] G]



V - V -
P - - ] - ] - ] - ] + ]] ]


were l, p are equal 1 or 3. If we use the diquark state with P +
J = 1 and the meson with

PC ++ ++ ++ -+ --
J = 0 1
, 2

, ,0 1
, , l, p are equal 2 or 3. There m is a quark mass.

Hereafter we suggest that some unknown (not large) contribution from small distances

which might be taken into account by the cut-off procedure. In the (6)  (8) we choose the

"hard" cutting, but we can use also the "soft" cutting, for instance

G (s ) . It does not change essentially the calculated mass
n ik = G exp
n ( 2 2 2
- (sik - 4m ) / )

spectrum.

In the equations (6) and (8) z is the cosine of the angle between the relative momentum
1


of the particles 1 and 2 in the intermediate state and the momentum of the particle 3 in the final

state, is taken in the c.m. of particles 1 and 2. In the equation (8) z is the cosine of the angle

between the momenta of the particles 3 and 4 in the final state, is taken in the c.m. of particles

1 and 2. z is the cosine of the angle between the relative momentum of particles 1 and 2 in the
2


intermediate state and the momentum of the particle 4 in the final state, is taken in the c.m. of

particles 1 and 2. In the equation (7): z is the cosine of the angle between relative momentum
3





7


of particles 1 and 2 in the intermediate state and the relative momentum of particles 3 and 4 in

the intermediate state, is taken in the c.m. of particles 1 and 2. z is the cosine of the angle
4


between the relative momentum of the particles 3 and 4 in the intermediate state and

momentum of the particle 1 in the intermediate state, is taken in the c.m. of particles 3, 4.

Using the relation of Appendix A we can pass from the integration of the cosines of the

angles to the integration of the subenergies.

Let us extract two-particle singularities in the amplitudes A (s, s , s , s ) ,
1 1234 12 34


A (s, s , s , s ) and A (s, s , s , s ) :
2 1234 12 123 3 1234 25 125


(s, s , s , s )B (s )B (s )
A (s, s , s , s ) 1 1234 12 34 3 12 1 34
= . (9)
1 1234 12 34 1
[ - B (s 1
)][ - B (s )]
3 12 1 34

(s, s , s , s )B (s )
A (s, s , s , s ) 2 1234 12 123 3 12
= , (10)
2 1234 12 123 1 - B (s )
3 12

(s, s , s , s )B (s )
A (s, s , s , s ) 3 1234 25 125 1 25
= , (11)
3 1234 25 125 1 - B (s )
1 25


We do not extract three- and four-particle singularities, because they are weaker than

two-particle singularities.

We used the classification of singularities, which was proposed in paper [27] for the two

and three particle singularities. The construction of approximate solution of the (3) - (5) is

based on the extraction of the leading singularities of the amplitudes. The main singularities in

s 4m2 are from pair rescattering of the particles i and k. First of all there are threshold
ik


square-root singularities. Also possible singularities are pole singularities which correspond to

the bound states. The diagrams of Fig.1 apart from two-particle singularities have the

triangular singularities, the singularities define the interaction of four and five particles. Such

classification allowed us to find the corresponding solution of (3) - (5) by taking into account

some definite number of leading singularities and neglecting all the weaker ones. We

considered the approximation which defines two-particle, triangle, four- and five-particle

singularities. The functions (s, s , s , s ) , (s, s , s , s ) and (s, s , s , s ) are
1 1234 12 34 2 1234 12 123 3 1234 25 125


the smooth functions of s , V s , s as compared with the singular part of the
ik LMN , ijkl


amplitudes, hence they can be expanded in a series in the singularity point and only the first




8


term of this series should be employed further. Using this classification one define the reduced

amplitudes , , as well as the B-functions in the middle point of the physical region of
1 2 3


Dalitz-plot at the point s :
0


s + 15 2
m
sik = s = (12)
0 0 10

2
s = 3s - 3m 2
, s = 6s - 8m
123 0 1234 0


Such a choice of point s allows to replace the integral equations (3) - (5) (Fig. 1) by
0


the algebraic equations (13) - (15) respectively:

= + 6 J )
1
,
1
,
3
( + 2 J )
3
,
1
,
3
( + 6 J )
1
,
3
,
3
( + 2 J )
1
,
1
,
3
( , (13)
1 1 3 2 2 2 2 1 3 1
= + 4 J )
1
,
3
,
3
( , (14)
2 2 1 3
= + 2 J )
3
,
1
,
1
( . (15)
3 3 1 3


We use the functions J (l, p) , J (l, p, r) , J (l, p, r) ( l, p, r = 1, 2, 3):
1 2 3



* +
O
V %S
V

GV O V

G]
- =
O S
% , (16)
- -
O
V

V V - %
P S V



2
G ( 12
s ) 2
G ( 34
s )B ( 13
s )
J (l, p, r) l 0 p 0 r 0
= 
2 B ( 12
s )B ( 34
s )
l 0 p 0 (17)
'
' + +
ds ( 's ) '
ds (s ) 1 1
p 34 dz dz 1
12 l 12 34 3 4

' 12 ' 34
s - s s - s - B s
2 2 2 2 1 ( ' )
4m 12 0 4 34 0 1
- 1
- r
m 13


2 12 ~
G (s , )B ( 13
s )B ( 24
s ) -
l 0 p 0 r 0 1 B ( 12
s )
J (l, p, r) l 0
= 
3 12 ~
1 - B (s , ) B ( 12
s )
l 0 l 0
~ + .(18)
+ + z
1 '
ds ( 's ) 1 1 2
dz 1 1
12 l 12 1
 dz dz

4 ' 12
s - s -
- - - - - + - B s - B s
2 2 2 2 2 2 1
[ ( ' 1
)][ ( ' )]
4m 12 0 1 1 z 1 z z z 2zz z p 13 r 24
2 1 2 1 2


The other choices of point s do not change essentially the contributions of , , and ,
0 1 2 3


therefore we omit the indexes ik
s . Since the vertex functions depend only slightly on energy it
0


is possible to treat them as constants in our approximation and determine them in a way similar

to that used in [28, 29].

The integration contours of functions J , J
, J
are given in the Appendix B (Figs. 3, 4,
1 2 3


5). The equations, which are similar to (13)  (15), correspond to other low-lying mesons with





9


isospin I = 0, PC ++ ++ ++ -+ --
J = 0 1
, 2

, ,0 1
, and diquarks with P + +
J = 0 1
, (graphic equations

Fig.1, 2) are considered in the Appendix C.

The solutions of the system of equations (Appendix C) are considered as:

(s) = F (s, ) / D(s) , (19)
i i i


where zeros of D(s) determinants define the masses of bound states of pentaquarks. F (s, )
i i


are the functions of s and . The functions F (s, ) determine the contributions of
i i i


subamplitudes to the pentaquark amplitude.





III. Calculation results.



The poles of the reduced amplitudes , , , correspond to the bound states and
1 2 3


determine the masses of N and  isobar pentaquarks. In the considered calculation the

quark mass are not fixed. In order to fix anyhow m , we assume


m = MeV
m m + . The model has only one new parameter as compared to our
5
2

model of hybrid baryons [18]. The gluon coupling constant J = 0.357 is determined by the

fixing of N pentaquark mass m + )
1990
( . The cut-off parameters are similar to the paper [18]:
5
2


the cut-off parameters
+ = 22 and + =32.4 for nucleon and  isobar pentaquarks
0 1


respectively. The calculated mass values of low-lying nucleon and  isobar pentaquarks are

P -
shown in Tables 1, 2. We found the lowest masses of N pentaquarks with J = 1 M=1378
2

P + P -
MeV, J = 1 M=1553 MeV and  isobar pentaquarks with J = 3 M=1150 MeV,
2 2

P +
J = 3 M=1290 MeV. If we increase the quark mass, the masses of the lowest  isobar
2


pentaquarks can be increased, but the masses of the pentaquarks will be most of the calculated

masses (Tables 1, 2). The low-lying  isobar pentaquark masses are smaller than the N

pentaquark masses. It is depended on the different interactions in the diquark channels

P + +
J = 0 1
, . The calculated values of the pentaquark masses are compared to the experimental

data [19]. We predict the degeneracy of some states. The calculation of pentaquark amplitude



10


estimates the contributions of three subamplitudes. The main contributions to the pentaquark

amplitude are determined by the subamplitudes, which include the low-lying meson states. The

Tables 1, 2 show the contributions of the following subamplitudes: A (BM ) , A (Mqqq) , A
1 2 3


( q
Mq ) . We sound that the contributions of A and A subamplitudes are about 40-50 % of the
1 2


pentaquark contribution. The contribution of the subamplitude A is less than 15 % of the
3

P -
pentaquark amplitude. The mass values of  isobar pentaquarks with J = 3 M=1150 MeV
2

P +
end J = 3 M=1290 MeV are depended on the large molecular contributions 71 % and 75 %
2


of the  isobar pentaquark amplitudes (Table 2). The lightest nucleon pentaquark with

P +
J = 1 M=1553 MeV might possibly be identified with Roper resonance [30-34]. The
2


structure of the Roper resonance can be described as the mixing of three-particle system and

the nucleon pentaquark.







IV Conclusion.



In strongly bound system of light quarks such as the baryons consideration, where

S P the approximation of nonrelativistic kinematics and dynamics not justified.

In our relativistic five-quark model (Faddeev  Yakubovsky type approach) we

calculated the masses of low-lying pentaquarks. We used SU ( )
3 symmetry. The quark
f


amplitudes obey the global color symmetry. The masses of the constituent quarks are equal to

405 MeV. We considered the scattering amplitudes of the constituent quarks. The poles of

these amplitudes determine the masses of low-lying pentaquarks. The derived five-quark

amplitude consists of three subamplitudes: BM , Mqqq , Bqq , where B and M are the

baryon and the meson respectively.

Unlike mesons, all half-integral spin and parity quantum numbers are allowed in the

baryon sector, so that experiments search for such pentaquark are not simple. Furthermore, no





11


decay channels are a priori forbidden. These two facts make identification of a pentaquark

difficult.

We manage with the quarks as with real particles. However, in the soft region the quark

diagrams should be treated as spectral integrals of the quark mass with the spectral density

( 2
m ) : the integration of the quark mass in the amplitudes eliminates the quark singularities

and introduces the hadron ones. One can believe that the approximation:

( 2
m ) ( 2 2
m - m ) (20)
q


could be possible for the low-lying hadrons (here m is the "mass" of the constituent quark).
q


We hope that the approach given by (20) is sufficiently good for the calculation of the low-

lying pentaquarks being carried out here.





Acknowledgments.

One of authors S.M. Gerasyuta is indebted to Institut fur Kernphysik Forschungzentrum

Julich for the hospitality where this work has been beginning. The authors would like to thank

T. Barnes, D.I. Diakonov, S. Krewald, N.N. Nikolaev, P.R. Page for useful discussions. This

research was supported in part by Russian Ministry of Education, Program "Universities of

RussiaXQGHU&RQWUD.W



APPENDIX A



We can go over from integration with respect of the cosines of angles to integration with

respect to the energy variables by using the relations:

' 2 '
s - s - m z s - m
' 2 4 2
s = 2 123 12 1 12
m + + [( ' 2
s - s - m )2 - 4 ' 2
s m ] (A1)
13 2 2 ' 123 12 12
s12

' 2 '
s - s - m z s - m
' 2 4 2
s = 2 124 12 1 12
m + + [( ' 2
s - s - m )2 - 4 ' 2
s m ] (A2)
24 2 2 ' 124 12 12
s12

2 '
s ( ' '
s + s - s - s ) - ( ' 2
s - s - m )( ' ' 2
s - s - m )
12 1234 12 123 124 123 12 124 12
z = (A3)
[( ' 2
s - s - m )2 - 4 2 '
m s ][( ' ' 2
s - s - m )2 - 4 2 '
m s ]
123 12 12 124 12 12






12


' ' '
s - s - s z s - m
' 2 ' 4 2
1234 12 34 3 12
s = m + s + + [( ' '
s - s - s )2 - 4 ' '
s s ] (A4)
134 34 2 2 ' 1234 12 34 12 34
s12

' ' 2 ' 2
s - s - m z s - 4m
' 2 134 34 4 34
s = 2m + + (s - s - m ) - 4m s (A5)
13 ' [ ' ' 2 2 2 '
134 34 34 ]
2 2 s34

The integration in consideration take on the physical region, where -1 z ( i = 1, 2,
i 1


3, 4). Then one can define the integration region on the invariant variables. Therefore for s'
124


we have condition 0 2
z 1,

s
 - s - m s + s - m

' ' 2 ( 2 )( ' 2 )
1234 123 123 12
s = s + m + 
124 12 2s123 (A6)
1
 [( ' 2
s - s - m )2 - 4 2 '
m s ][( 2
s - s - m )2 - 4 2
m s ]
2 123 12 12 1234 123 123
s123

and the region of integration on s' in J :
12 3


~


, if
( s m)
123 + 2
= (A7)
( s + 2
m)
, if ( s m)
123 > + 2
123










APPENDIX B



The integration contour 1 (Fig. 3) corresponds to the connection s ( s m 2
< - ) , the
123 12


contour 2 is defined by the connection ( s m)2 s ( s m)2
- < < + . The point
12 123 12


s ( s m 2
= - ) is not singular, that the round of this point at V + -
L and V L gives
123 12


identical result. s ( s m 2
= + ) is the singular point, but in our case the integration contour
123 12


can not pass through this point that the region in consideration is situated below the production

threshold of the four particles 2
s < 16m . The similar situation for the integration over
1234
V in

the function J is occurred. But the difference consists of the given integration region that is
3


conducted between the complex conjugate points (contour 2 Fig. 3). In Fig. 3, 4b, 5 the dotted



13


lines define the square-root cut of the Chew-Mandelstam functions. They correspond to two-

particles threshold and also three-particles threshold in Fig. 4a. The integration contour 1 (Fig.

4a) is determined by 2
s < ( s - s ) , the contour 2 corresponds to the case
1234 12 34


2 2
( s - s ) < s < ( s + s ) . 2
s = ( s - s ) is not singular point, that the
12 34 1234 12 34 1234 12 34


round of this point at s and s gives identical results. The integration contour 1
1234 -
i
1234 +
i

(Fig. 4b) is determined by region 2
s < ( s - s ) and s ( s m 2
< - ) , the integration
1234 12 34 134 34


contour 2 corresponds to 2
s < ( s - s ) and ( s m)2 s ( s m)2
- < + . The
1234 12 34 34 134 34


contour 3 is defined by 2 2
( s - s ) < s < ( s + s ) . Here the singular point would
12 34 1234 12 34


be s ( s m 2
= + ) . But in our case this point is not achievable, if one has the condition
134 34


2
s < 16m . We have to consider the integration over V .
1234 in the function J3

While s < s + m2
5 the integration is conducted along the complex axis (the contour 1, Fig.
124 12

5). If we come to the point s = s + m2
5 , that the output into the square-root cut of Chew-
124 12


Mandelstam function (contour 2, Fig. 5) is occurred. In this case the part of the integration

contour in nonphysical region is situated and the integration contour along the real axis is

conducted. The other part of integration contour corresponds to physical regions. This part of

integration contour along the complex axis is conducted. The suggested calculation shows that

the contribution of the integration over the nonphysical region is small [28, 29].





APPENDIX C



We considered the algebraic equations and determinants, which allow one to calculate

the poles of reduced amplitudes , , for the low-lying pentaquark. If we use the
1 2 3

diquark with P +
J = 0 (l, p, r are equal 1 or 3), we can calculate the spectrum of N

pentaquarks. If we use the diquark with P +
J = 1 (l, p, r are equal 2 or 3), we can calculate the

spectrum of - isobar pentaquarks.






14


Figure 1

= + 6 J ( )
1
,
1
,
3 + 2 J ( )
3
,
1
,
3 + 6 J ( )
1
,
3
,
3 + 2 J ( )
1
,
1
,
3
1 1 3 2 2 2 2 1 3 1
= + 4 J ( )
1
,
3
,
3 (C1)
2 2 1 3
= + 2 J )
3
,
1
,
1
(
3 3 1 3
D(s) = 1 - 8J ( J + J - J J + J
3 {)
1
,
3
,
3 ( )
3
,
1
,
3 3 (
2 1 })
1
,
3
,
3 4 3 {)
3
,
1
,
1
( 3 ( )
1
,
1
,
3 (
2 1 })
1
,
1
,
3



Figure 2

= + 2 J ( )
3
,
1
,
3 + 6 J ( )
1
,
3
,
3
1 1 2 2 2 1
= + 4 J ( )
1
,
3
,
3 (C2)
2 2 1 3
D(s) = 1 - 8J ( J + J
3 {)
1
,
3
,
3 ( )
3
,
1
,
3 3 (
2 1 }
)
1
,
3
,
3


Functions J (l, p, r) , J (l, p, r) and )
J (l, p, r correspond to (16)  (18), l, p, r = 1, 2, 3.
1 2 3





APPENDIX D



The vertex functions are shown in Table 3. The two-particle phase space for the equal quark

masses is defined as:

s s - 4m2
PC PC ik PC ik
(s , J ) = (J ,n) + (J , n) ,
n ik 4m2 sik

The coefficients (J PC
, n) and (J PC
, n) are given in Table 4.























15


Table I. Low-lying nucleon pentaquark masses and contributions of subamplitudes

BM , Mqqq and q
Bq to pentaquark amplitude in % (diquark with P +
J = 0 ).

)LJ Meson - 3& P
J Mass, MeV BM Mqqq q
Bq
1 ++
0 +
1 1553 P (1440) 49.35 38.54 12.11
2 11
1 ++ + +
1 3
, 1650 P (1710) 48.36 38.75 12.89
2 2 11
P (1720)
13
1 ++ +
3 1875 P (1900) 45.17 39.06 15.77
2 13
2 ++ +
5 1990 F (1990) 49.08 50.92 -
2 15
1 -+
0 -
1 1378 S (1535) 55.53 35.84 8.63
2 11
1 -- - -
1 3
, 1814 S (1650) 44.87 39.52 15.61
2 2 11
D (1700)
13

Parameters of model: quark mass m = 405 MeV, cut-off parameter =22; gluon coupling
constant J =0.357. Experimental mass values of nucleon pentaquark are given in parentheses
[19].




Table II. Low-lying - isobar pentaquark masses and contributions of subamplitudes

BM , Mqqq and Bqq to pentaquark amplitude in % (diquark with P +
J = 1 ).

Fig. Meson - 3& P
J Mass, MeV BM Mqqq q
Bq
1 ++
0 +
3 1290 ( - ) 75.05 16.62 8.33
2
1 ++ + + +
1 3 5
, , 1580 P (1750) 58.78 31.24 9.98
2 2 2 31
P (1600)
33

( - )
1 ++ + + +
1 3 5
, , 1845 P (1910) 50.70 36.12 13.18
2 2 2 31
P (1920)
33

F (1905)
35
2 ++ +
7 1970 F (1950) 52.26 47.74 -
37
2
1 -+
0 -
3 1150 ( - ) 71.19 22.92 5.89
2
1 -- - - -
1 3 5
, , 1782 S (1620) 51.51 35.76 12.73
2 2 2 31
D (1700)
33

D (1930)
35

Parameters of model: quark mass m = 405 MeV, cut-off parameter =32.4; gluon constant
J =0.357. Experimental mass values of - isobar pentaquarks are given in parentheses [19].



16


Table III. Vertex functions

- 3& 2
G
n
+
(n=1) 4g / 3 - 8 2
gm /(3s )
ik
+
(n=2) 2g/3
-+
0 (n=3) 8g / 3 -16 2
gm 3
/( s )
ik

-- (n=3) 4g / 3
++
0 (n=3) 8g / 3
++ (n=3) 4g / 3
++ (n=3) 4g / 3




Table IV. Coefficient of Chew-Mandelstam functions for n = 3 (meson states)

and diquarks n = 1 ( P +
J = 0 ), n = 2 ( P +
J = 1 ).

- 3& n (J PC
, n) (J PC
, n)
++
0 3 1/2 -1/2
++ 3 0
++ 3 3/10 1/5
-+
0 3 1/2 0
-- 3 1/3 1/6
+
1 1/2 0
+
2 1/3 1/6




Figure captions.



Fig.1. Graphic representation of the equations for the five-quark subamplitudes

A (s, s , s , s ) (BM ) , A (s, s , s , s ) (Mqqq) , and A (s, s , s , s ) ( q
Bq ) using
1 1234 12 34 2 1234 12 123 3 1234 25 125

the low-lying mesons with PC ++ ++ ++ -+ --
J = 0 1
, ,2 ,0 1
, and diquarks with P + +
J = 0 1
, .

Fig.2. Graphic representation of the equations for the five-quark subamplitudes

A (s, s , s , s ) (BM ) , )
A (s, s , s , s (Mqqq) .
1 1234 12 34 2 1234 12 123


Fig. 3. Contours of integration 1, 2 in the complex plane V for the functions -, - .





17


Fig. 4. Contours of integration 1, 2, 3 in the complex plane V (a) and V (b) for the function

- .

Fig. 5. Contours of integration 1, 2 in the complex plane V for the function - .





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18


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19


1
2



+ 0 + 0


+
1 2

33
55 44
+ 0

6

1 3
2
+ 0

5



22 Fig.1

2 3 1
1 44 2
+ + 0
+ 0 3 4 3 1
0 5 4
+ +
5 0 0

6 4 2
++ + +
1 2 55 22
2 3
1 5 4 +
5 4
+ 0
3 1
4 3
0
=+ = = 2
1 1 2 5
2 3 4 +
5 + 0
5
3 1 4
0
4 3


1 2

3
5 4
+ 0

6 Fig.2
+
1 3
2 4
+ 0 3 4
5 +
5 0

2 4

1 2
2 3
1 5 4 5
+ 3 4
0
=+ =+
1 11 22
2 3 4
5 + 5
3
0
4


Im(s 13 )

s +13

1 2 Re( s 13)
s _13 s +13
s _13


Fig. 3


Im(s 134 ) Im(s 13 )
s + + +
134 s 13 s 13
1 2 Re( s 134 ) 1 2 3 Re(s 13)
s _ _ +
134 s +134 s 13 s 13
_ _
s _ s s
134 13 13





Fig. 4a Fig. 4b

s
Im( 24 )
s +
s + 24
24 2
1 Re( s 24)

s _24
s _24


Fig. 5



