Relativistic four-quark equations and cryptoexotic mesons spectroscopy.





Gerasyuta S.M.*, Kochkin V.I.

Department of Theoretical Physics, St. Petersburg State University, 198904, St. Petersburg, Russia.





Abstract

The four-quark equations are found in the framework of the dispersion relation technique. The

approximate solutions of these equations using the method based on the extraction of leading

singularities of the amplitudes are obtained. The four-quark amplitudes of cryptoexotic mesons

including the quarks of three flavours (u, d, s) are calculated. The mass values of low-lying cryptoexotic

mesons are calculated.





* Present address: Department of Physics, LTA, Institutski

Per. 5, St. Petersburg 194021, Russia



1


1. Introduction.


The history of qqqq states dates back to 1967 when Astier [1] suggested that KK bound states

could explain the low mass I = 1 enhancements seen in pp KK p . In 1968, Rosner [2] and Harari

[3] noticed that the dual t-channel of baryon-antibaryon scattering through meson exchanges represents
qqqq system, "baryonia". The first extensive calculation was by Jaffe [4] who applied a semi-classical

approximation of the MIT-bag model to qqqq spectroscopy. He noted that colour magnetic

interactions drive the multiplet splitting and he suggested that the lowest scalar nonet were qqqq

bound states, hence explaining many of their peculiar properties.
In 1977, Chan and Hogaasen [5] used bag and string models to investigate baryonia in their unmixed
colour states. They predicted a tower of (qq) (q q )
6 S-wave states held apart by a high angular
6


momentum barrier. In 1980 Wicklund [6] suggested that an isoscalar KK bound state explained
pp f ( ) KK
975 data. Weinstein and Isgur [7] have performed extensive calculations in the four-
0

quark system with the quark potential model. Their results suggest that the a ( 9 8 0 ) and f ( )
975 are
0 0

indeed KK bound states but that no other qqqq states exist (at least in the sectors they studied). These

authors noted the mounting evidence in favour of the molecular interpretation of the a and f . For
0 0

example, the qqqq bound state scenario provides an explanation for the narrow width of these

particles. Barnes [8] has argued that the gg -decay widths imply that the a and f are qqqq states
0 0

rather than qq states. Finally, the near degenerate of the a and f suggest that if they are qq states
0 0

then they are approximately ideally mixed and hence the ratio of decay width to pp should be roughly
4 rather than the observed value of 1/2 [9].
The bag model [10, 11] provides another important vehicle for investigating multiquark physics. It
has the benefit of utilizing an attractive physical picture of confinement, it incorporates asymptotic
freedom by describing the quarks inside of the bag with the free Dirac equation.
Jaffe and Low introduced a quantity they called the P-matrix whose poles are expected to correspond
to the energies calculated in the bag model [12]. Unfortunately their program is computationally
difficult.
It should be stressed that all predictions of multiquark bound states are entirely model-dependent,
thus the predictions of the quark potential model serve as an important test [13, 14].
In recent papers [15, 16] the relativistic Faddeev equations are represented in the form of the
dispersion over the two-body subenergy. The behavior of the low-energy three-particle amplitude is
determined by its leading singularities in the pair invariant masses. The suggested method of
approximate solution of the relativistic Faddeev equation was verified on the example of the S-wave
+ +
1 3
baryons. The calculated mass values of two baryonic multiplets J P = , are in good agreement
2 2
with the experimental ones. The dispersion relations technique allows us to consider relativistic effects
in the composite systems. Double dispersion relations over the masses of the composite particles were
used in [16] for the consideration of the form factors. The behavior of the nucleon electromagnetic form
factors at small and intermediate momentum transfer Q2 < , GeV 2
1 5 are determined.
In the present paper the relativistic four-quark equations are constructed in the form of the dispersion
relation over the two-body subenergy. We calculated the masses of the cryptoexotic multiplets using the
method based on the extraction of leading singularities of the amplitude.
In Section 2 the relativistic four-quark equations are constructed in the form of the dispersion
relation over the two-body subenergy. The approximate solutions of these equations using the method
2


based on the extraction of leading singularities of the amplitude are obtained. The four-quark
amplitudes of cryptoexotic mesons are calculated.
Section 3 is devoted to the calculation results for the cryptoexotic meson mass spectrum (Table 1).
Table 2 shows the contributions of different subamplitudes to the four-quark amplitude.
In the Conclusion the status of the considered model is discussed.
In the Appendix A the quark-antiquark vertex functions and the phase spaces for the cryptoexotic
mesons are given (Tables 3, 4).
In the Appendix B we search the integration contours of functions J , , , which are determined
1 J2 J3

by the interaction of the four quarks.



2. Four-quark amplitudes of the cryptoexotic mesons.


In the recent papers [17, 18] the relativistic four-quark equations are constructed. We consider only
the planar diagrams, the other diagrams due to the rules of 1/Nc - expansion [19, 20] are neglected. For
the sake of simplicity one considered the case of the SU (3) f -symmetry that the masses of all particles

are equal.
In the present paper we investigate scattering amplitudes of the constituent quarks of three flavours
(u, d, s). The poles of these amplitudes determine the masses of the cryptoexotic mesons. The masses of
the constituent quarks u and d are of the order of 300-400 MeV, the strange quark is 100-150 MeV
heavier. The strange quark violates the flavour SU (3) f -symmetry. The constituent quark is color

triplet and quark amplitudes obey the global color symmetry.
We derived the relativistic four-quark equations in the framework of the dispersion relation
technique. Let some current produce two pairs of quark-antiquark (Fig.1,2). The consideration of
diagrams in Fig.1,2 allows to present graphically the equations for the four-quark amplitudes. However,
the correct equations for the amplitude are obtained at the account of all possible subamplitudes. It
corresponds to the division complete system on the subsystems from the smaller number of particles.
Then one should present four-particle amplitude as a sum of six subamplitudes:
A = A + A + A + A + A + A
12 13 14 23 24 34 . It defines the division of the diagrams in the groups according to

the last interaction of particles. In this case we need to consider only one group of diagrams and the
amplitude corresponding to them, for example A12 . We must take into account each sequence of the

inclusion of interaction. For instance, the process beginning with interaction of the particles 1 and 2 can
proceed by the three ways: particle 3 and 4 consistently join a chosen pair, or begin to interact among
themselves, and each of the three ways of the connection there should correspond to their own
amplitudes [21, 22]. Therefore the diagrams corresponding to amplitude A12 are divided in three group
A ( s, s , s ) ( , , ) ( , , ) ( , , )
1 12 123 , A s s s
1 12 124 and A s s s
2 12 34 (moreover the subamplitudes A s s s
1 12 123 and
A ( s, s , s )
1 12 124 are analogous).

In the present paper one supposed that the transitions q q q q
a a b b (a, b = u, d, s) for the colour

singlet states are absent [4]. The similar considerations are not valid for the colour octet states.
The equations for the four-quark amplitudes in the graphic form are presented (Fig.1,2). The
coefficients are determined by the permutation of quarks [21]. The four-quark amplitude A is defined
by contribution of equal subamplitudes A = A
12 34 . The other subamplitudes A , A (describing the
13 24

interactions of diquarks) and A , A (satisfying the Okubo - Zweig - Iisuka rule) do not contribute to
14 23

the amplitude A . Then we can consider only one group of diagrams corresponding to the amplitude
A12 .


3


In order to present the amplitudes A ( s, s , s ) ( , , )
l 12 123 ( l =1-4) and A s s s
p 12 34 (p=5-7) in form of the

dispersion relation it is necessary to define the amplitude of two-quark interaction a ( s )
j ik . One uses the

results of the bootstrap quark model [23,24] and writes down the pair quarks amplitude in the form:
G 2 ( s )
j ik
(1) a ( s ) = ,
j ik 1 - B ( s )
j ik

r 2
ik ' '
ds' (s G
) (s )
j ik j ik
(2) B (s = ik
) ,
j ik p
2 s' -
+ s
( m m ik ik
i k )

Here G ( s ) ( ) ( s )
j ik is the quark-antiquark vertex function. B s
j ik , rj ik are the Chew-Mandelstam

function [25] and the phase space respectively. We introduced the cut-off parameter ik . There j=1

corresponds to pair of quarks qq with J PC = 0++ , 1++ , 2++ , 0-+ , 1-- , 1+- (colour singlet SU (3)c ) and

j=2 defines the quark pair with J PC = 1-- in colour channel 8c (constituent gluon). The vertex

functions are shown in the Table 3, the functions r ( s )
j ik are given in the Appendix A (see Table 4). In

the case in question the interacting quarks do not produce bound state, then the integration in (3)-(9) is
2
carried out from the threshold (m + m )
i k to the cut-off ik . The integral equation systems,

corresponding to Fig 1,2, have the following form:
l B uu ( s ) G ( s )
1 1 12 1 12 ' $ ' '
(3) A ( s, s , s ) = + 2 J
[ $ A ( s, s , s ) + J A ( s, s , s )]
1 12 123 1 1 13 123 3 6 13 24
1 - B uu ( s ) 1 - B uu ( s )
1 12 1 12

l B ss ( s ) G ( s )
ss ' $ ss
2 1 12 1 12 ' '
(4) A ( s, s , s ) = + 2 J
[ $ A ( s, s , s ) + J A ( s, s , s )]
2 12 123 1 2 13 123 3 5 13 24
1 - B ss ( s ) 1 - B ss ( s )
1 12 1 12

l B us ( s ) G ( s ) $ s
3 1 12 1 12 ' '
(5) A ( s, s , s ) = + 2 J A ( s, s , s )
3 12 123 3 7 13 24
1 - B us ( s ) 1 - B us ( s )
1 12 1 12

l B ud ( s ) G ( s )
4 1 12 1 12 $ ' '
(6) A ( s, s , s ) = + 2 J A ( s, s , s )
4 12 123 3 6 13 24
1 - B ud ( s ) 1 - B ud ( s )
1 12 1 12

l B ss ( s B ss
) ( s ) G ( s G
) ( s )
A ( s, s , s 5 2 12 2 34 2 12 2 34
) = + 4 
5 12 34 ss ss ss ss
(7) 1
[ - B ( s 1
)][ - B ( s )] 1
[ - B ( s 1
)][ - B ( s )]
2 12 2 34 2 12 2 34


 J
[ $ A ( s, s ' , s ' ) + J$ A ( s, s ' , s ' ) + J$ s A ( s, s ' , s ' ) + J$ ss A ( s, s ' , s ' )]
2 1 13 134 2 4 13 134 2 3 13 134 2 2 13 134

l B uu ( s B uu
) ( s ) G ( s G
) ( s )
A ( s, s , s 6 2 12 2 34 2 12 2 34
) = + 4 
6 12 34 uu uu uu uu
(8) 1
[ - B ( s 1
)][ - B ( s )] 1
[ - B ( s 1
)][ - B ( s )]
2 12 2 34 2 12 2 34


 J
[ $ A ( s, s ' , s ' ) + J$ A ( s, s ' , s ' ) + J$ s A ( s, s ' , s ' ) + J$ ss A ( s, s ' , s ' )]
2 1 13 134 2 4 13 134 2 3 13 134 2 2 13 134

l B ss ( s B uu
) ( s ) G ( s G
) ( s )
A ( s, s , s 7 2 12 2 34 2 12 2 34
) = + 4 
7 12 34 ss uu ss uu
- - - -
(9) 1
[ B ( s 1
)][ B ( s )] 1
[ B ( s 1
)][ B ( s )]
2 12 2 34 2 12 2 34


 J
[ $ A ( s, s ' , s ' ) + J$ A ( s, s ' , s ' ) + J$ s A ( s, s ' , s ' ) + J$ ss A ( s, s ' , s ' )] ,

2 1 13 134 2 4 13 134 2 3 13 134 2 2 13 134

l are the current constants. Here we introduce the integral operators:
i
+
12 ' r ' ' 1
$ ds (s ) G (s ) dz
(10) J ( ;
s m ; m ; m ; m ) = 12 1 12 1 12 1

1 1 2 3 4 p ' -
+ - 2
2 s s
( m m ) 12 12 1
1 2
+ +
12 ' r ' ' 34 ' r ' ' 1 1
$ ds (s ) G (s ) ds (s ) G (s ) dz dz
(11) J ( ;
s m ; m ; m ; m ) = 12 2 12 2 12 34 2 34 2 34 3 4

2 1 2 3 4 p ' - p ' -
+ + - 2 - 2
2 s s 2 s s
( m m ) 12 12 ( m m ) 34 34 1 1
1 2 3 4





4


1
$
J = 
3 ( ;
s m1; m2 ; m3 ; m4 ) 4 p

(12) + + +
1 2 ' 1 1 z
ds 2
r 1
1 2 1 ( '
s12 ) G1 ( '
s12 ) dz1
 dz dz2 ,
p ' -
- - - - +
2
+ s 2 2 2
2 1 2
1 2 s12
m - -
1 m 1 1
2 z2 z z1 z2 zz1 z
( ) 2

One used the following indications:
$ $
J ( ;
s ;
m ;
m ;
m m) = J
i i
$ $
J ( ;
s m ; m ; m ; m ) = J ss , here i=1,2,3
i s s s s i
$ $ $ $
J ( ;
s ;
m ;
m m ; m ) = J s , J = ,
3 ( ;
s ;
m m ; ;
m m ) J s
2 s s 2 s s 3

there m and m are the masses of nonstrange and strange quarks respectively. The B qq -functions are
s

defined by the composition of qq -quark pair for the final state. In the equations (10) and (12) z is the
1

cosine of the angle between the relative momentum of the particles 1 and 2 in the intermediate state and
that of the particle 3 in the final state, which is taken in the c.m. of particles 1 and 2. In the equation
(12) z is the cosine of the angle between the momentum of the particles 3 and 4 in the final state,
which is taken in the c.m. of particles 1 and 2. z is the cosine of the angle between the relative
2

momentum of particles 1 and 2 in the intermediate state and the momentum of the particle 4 in the final
state, which is taken in the c.m. of particles 1 and 2. In the equation (11) we have defined: z is the
3

cosine of the angle between relative momentum of particles 1 and 2 in the intermediate state and that of
the relative momentum of particles 3 and 4 in the intermediate state, which is taken in the c.m. of
particles 1 and 2; z is the cosine of the angle between the relative momentum of the particles 3 and 4
4

in the intermediate state and that of the momentum of the particle 1 in the intermediate state which is
taken in the c.m. of particles 3, 4. Using (13)-(17) we can pass from the integration over the cosines of
the angles to the integration over the subenergies. The choice of integration contours of functions
J , , do not differ from the papers [17, 18] (see Appendix B).
1 J2 J3
( s - s' - m 2 )( s' - m 2 + m 2 )
s' = m 2 + m 2 123 12 3 12 2 1
+ +
13 1 3 2s'12
(13) z1
+ [( s - s' - m 2 2
) - 4s' m 2 ][( s' - m 2 + m 2 2
) - 4s' m 2 ]
2s' 123 12 3 12 3 12 2 1 12 1
12

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

2s' ( s + s' - s - s' ) - ( s - s' - m 2 )( s' - s' - m 2 )
12 12 123 124 123 12 3 124 12 4
(15) z = [( s - s' - m 2 2
) - 4m 2 s' ][( s' - s' - m 2 2
) - 4m 2 s ' ]
123 12 3 3 12 124 12 4 4 12





2
' ' '
- - - 4
2 s s12 s34 z3 s12 m
(16) 1 2
s' ' ' '
= + + + - - - 4 ' '
[( ) ]
1 3 4 m1 s34 s s12 s34 s12 s34
2 2 s'12


s' - s' - m 2 z s' - 4m 2
s' = m 2 + m 2 134 34 1 4 34 3
+ + [ s' - s' - m2 2 - 4m2s'
( ) ]
(17) 13 1 2 s' 134 34 1 1 34
2 2
34

Let us extract two-particle singularities in the amplitudes A ( s, s , s ) ( , , )
l 12 123 and A s s s
p 12 34 :



5


a ( s, s , s B
) ( s )
l 12 123 1 12
(18) A ( s, s , s ) =
l 12 123 1 - B ( s )
1 12

a (s, s12 , s34 )B2 (s12 )B2 (s34 )
p
(19) A (s, s =
1 2 , s3 4 ) ,
p [1 - ][1 - ]
B2 (s12 ) B2 (s34 )
l =1-4, p=5-7.
In the amplitude A ( s, s , s )
l 12 123 we do not extract three-particle singularity, because it is weaker than
two-particle and taking into account in the function a (s, s , s )
l 12 123 .

We used the classification of singularities, which was proposed in papers [17, 18]. The construction
of approximate solution of the (18) and (19) is based on the extraction of the leading singularities of the
amplitudes. The main singularities in s are from pair rescattering of the particles i and k. First of all
ik

there are threshold square root singularities. Also possible are pole singularities which correspond to
the bound states. They are situated on the first sheet of complex s plane in case of real bound state
ik

and on the second sheet in case of virtual bound state. The diagrams Fig.1, 2 apart from two-particle
singularities have their specific triangular singularities and the singularities correspond to the
interaction of four particles. Such classification allows us to search the approximate solution of (18)
and (19) by taking into account some definite number of leading singularities and neglecting all the
weaker ones. We consider the approximation, which corresponds to the single interaction of all four
particles (two-particle, triangle and four-particle singularities). The functions a (s, s , s )
l 12 123 and
a (s, s , s ) are the smooth functions of s
p 12 34 ik , sijk as compared with the singular part of the

amplitudes, hence they can be expanded in a series in the singularity point and only the first term of this
series should be employed further. Using this classification one define the functions a (s, s , s )
l 12 123 and
a (s, s , s ) as well as the B-functions in the middle point of the physical region of Dalitz-plot at the
p 12 34

point s0 :
2
+ 8
0 0 s m
s = =
1 s
i j 4 6

0 2 2 2 2
4 - 2 + 2 + + +
[ ( )]
0 s s1 m1 m2 m3 m4
(20) s = i
2 2 2 2
(m + + + + + + +
) ( ) ( ) ( )
1 m m1 m4 m2 m3 m m
j i 4

0 1 2 0 1 2 0 2 2 2
s = + + + + - - -
( ) ( )
1 2 3 s1 4 m1 m s 4 m2 m3 s m1 m2 m
i j 3

$ $ $
Here one suggested i=2, j=3 for J and J ; i=3, j=2 for J . Moreover, the other choice of the middle
1 2 3


point does not change essentially the obtained results. Such a choice of points s0 , s0 allows as to
12 34

replace the integral equations (3)-(9) by the algebraic equations (21)-(27) respectively:
uu 0
(21) a = l + 2 a
[ J + a J ] / B ( s )
1 1 1 1 6 3 1 12

ss ss ss 0
(22) a = l + 2 a
[ J + a J ] / B ( s )
2 2 2 1 5 3 1 12

s us 0
(23) a = l + a
2 J / B ( s )
3 3 7 3 1 12

ud 0
(24) a = l + a
2 J / B ( s )
4 4 6 3 1 12

s ss ss 0 ss 0
(25) a = l + 4 a
[( + a )J + a J + a J ] / [B ( s )B ( s )]
5 5 1 4 2 3 2 2 2 2 12 2 34

s ss uu 0 uu 0
(26) a = l + 4 a
[( + a )J + a J + a J ] / [B ( s )B ( s )]
6 6 1 4 2 3 2 2 2 2 12 2 34

s ss us 0 us 0
(27) a = l + 4 a
[( + a )J + a J + a J ] / [B ( s )B ( s )]
7 7 1 4 2 3 2 2 2 2 12 2 34

Here we introduce following functions:
+1
1 2 ' r ' 1
2 0 ds ( )
1 2 1 s12 dz1
(28) J ( ; ; ; ; ) = ( )
1 s m1 m2 m3 m4 G1 B1 s13 0
p ' - 2 1 -
+ - ( ' )
2 s12 s12 1 B1 s13
( m1 m2 )



6


J (s; m ; m ; m ; m ) = G 4 B (s0 ) 
2 1 2 3 4 2 1 13
+ +
(29) 12 ds' r (s' 34
) ds' r (s' 1
) dz 1 dz 1
 12 2 12 34 2 34 3 4

p ' - p ' - 2 - 2 1 -
+ + - ( ' )
2 s s0 2 s s0 B s
( m m 12 12
) ( m m 34 34
) 1 1 1 13
1 2 3 4




0
1 - 1
2 0 0 B ( , )
1 s12 1 2
J ( ; ; ; ; ) = ( ) ( ) 
3 s m1 m2 m3 m4 G1 B2 s13 B2 s24 0 ~
1 - B 4
( , ) p
1 s12 1 2
(30) ~ +
1
+ 1
+
1 2 z
ds 2
' r ' 1 1
( )
1 2 1 s12 dz1

0 dz dz2
p ' - '
- - - - + (s24 )]
+ - - [ - ( ' )][ -
2 s 2 2 2
2 1 2 1 1
1 2 s12 1 1
( 1 2 ) 2 z z 2 1 3 2
1 z2 zz1 z B s B
m m z 2

As the integration region the physical region of the reaction should be chosen, therefore -
1 z 1 (
i
' ' ' '
i=1,2,3,4 ). From these conditions we can define the regions of the integration over s , s
, s
, s

13 24 134 124 .

Let us consider the integration region over s' . For this purpose we use equation (15). This condition
124

corresponds to 0 2
z 1. By consideration of this inequality one can obtain:
( s - s - m 2 )( s + s' - m 2 )
s' = s' + m 2 123 4 123 12 3
+ 
124 12 4 2s123
(31) 1
 [(s - s' - m 2 2
) - 4m 2 s' ][( s - s - m 2 2
) - 4m 2 s ]
2s 123 12 3 3 12 123 4 4 123
123


We must take into account the upper restriction of the integration region over s' in J :
12 3

2

~ , if ( s + m )
12 12 123 3
(32) =
12 ( s + m )2 , if
> ( s + m )2


123 3 12 123 3


The integration contours of the functions J , J , J are given in the Appendix B. The function J takes
1 2 3 3

into account the singularity, which corresponds to the simultaneous vanishing of all propagators in the
four-particle diagram like those in Fig.1. In the case in question the functions a (s) are determined as:
i

(33) a ( s ) = F ( s, l ) / (s )
i i i

There ( s ) is the determinant:
s s ss ss ss ss
(34) (s ) = (1 - J
8 J - J
8 J )(1 - J
2 )(1 - J
2 ) - J
8 J (1 - J
2 ) - J
8 J (1 - J
2 )
2 3 2 3 1 1 2 3 1 2 3 1

The right-hand sides of (33) might have a pole in s which corresponds to the bound state of the four
quarks. The poles of rescattering amplitudes for the cryptoexotic mesons J PC = 0++ , 1++ , 2++ , 0-+ ,
1-- , 1+- correspond to the bound state and determine the masses of the cryptoexotic mesons.


3. Calculation results.


In the bootstrap quark of model [23, 24] there is a bound state in the gluon channel with mass of the
order of 0.7 GeV. This bound state should be identified as a constituent gluon. It should be pointed out
that such a value of the mass agrees with the hard-process phenomenology [26]. An analogous
estimation of the gluon mass is obtained in the bag models [27, 28].
The strange quark in our model gives rise to the violation of the flavour SU (3) f -symmetry. In order

to avoid an additional violation parameter we introduce the scale shift of the dimensional parameter:
= + 2
(m m ) / 4 .
ik i k





7


In the considered calculation the quark masses ( m and ms ) are not fixed. In order to fix anyhow m
1 1
and m = 368 ( (1430)) = 513 ( (2010))
s we assume m M eV m m f and m M eV m m .
4 s s f
2 4 2



The model under consideration proceeds from the assumption that the quark interaction forces are
the two-component ones. The long-range component is due to the confinement. When the low-lying
mesons are considered, the long-range component of the forces is neglected. The creation of ordinary
mesons is mainly due to the constituent gluon exchange (Fig. 3a). But for the excited mesons the long-
range forces are important. Namely, the confinement of the qq pair with comparatively large energy is

actually realized as the production of the new qq pairs. This means that in the transition qq qq the

forces appear which are connected with the process of the Fig. 3b type. These box-diagrams can be
important in the formation of hadron spectra [29]. We do not see any difficult in taking into account the
box-diagrams with the help of the dispersion technique. For the sake of simplicity we restrict ourselves
to the introduction of quark mass shift , which are defined by the contributions of the nearest
production thresholds of pair mesons pp , ph , KK , Kh and so on. We suggest that the parameter
eff eff
takes into account the confinement potential effectively: m = m + , m = m +
s s and changes the

behavior of pair quarks amplitude (1). It allows us to construct the excited cryptoexotic mesons
amplitudes and calculate the mass spectrum by analogy with the P-wave meson spectrum in the
bootstrap quark model [30]. The model in consideration have another two parameters: cut-off
parameter and gluon constant g for each group of mesons. The subenergy cut-off and the vertex
function g can be determined by mean of fixing of cryptoexotic mesons mass values ( J PC = -+ ++
0 2
, ).

The vertex functions of various types of the interactions are given in Table 3. The calculated values of
mass cryptoexotic mesons (groups I-III) are shown in the Table 1. The results are in good agreement
with the experimental data [31] and the other model results [32-37]. However, for the lowest J PC = ++
0
multiplet the discrepancy between calculated and observed values of masses M = 1010 (985
) MeV is
f0

more than others. It is possible that this is due to the admixture of the qq scalar state, moreover the

mass value of two-quark state with J PC = ++
0 in the bootstrap quark model M = 870 (985
) MeV is
f0

obtained.



4. Conclusion.


In the present paper in the framework of approximate method of solution four-particle relativistic
problem the mass spectrum of cryptoexotic mesons, including u, d, s - quarks, are calculated. The mass
values of already detected candidates of cryptoexotic mesons are calculated. The interesting result of
this model is the calculation of cryptoexotic meson amplitudes, which contain the contributions of
seven subamplitudes: four four-quark amplitudes and three glueball amplitudes. The contributions of
glueball subamplitudes (groups I-III) are given in Table 2. The decay width of cryptoexotic mesons can
be calculated in the framework this model. The suggested approximate method allows to construct the
four-quark amplitudes, including heavy quarks Q=c, b and calculate the mass spectrum of heavy
cryptoexotic mesons.





8


APPENDIX A


The two-particle phase space for the unequal quark masses is defined as:
- 2
r PC = a s
PC ik + b PC + d (m m )
(s , J ) ( J ) ( J ) ( J PC ) i k 
1 ik + 2
(m m ) s
i k ik

- + 2 - - 2
[s (m m ) ][s (m m ) ]
ik i k ik i k

sik
r = r --
(s ) (s 1
, )
2 ik 1 ik
1
B ( s ) = [ B
2 ( s , m, m ) + B ( s , m , m )]
2 ik 2 ik 2 ik s s
3

The coefficients a(J PC ) , b(J PC ) and d (J PC ) are given in Table 4.



APPENDIX B


2
The integration contour 1 (Fig.4) corresponds to the connection s < -
( ) , the contour 2 is
1 2 3 s12 m3

2 2 2
defined by the connection ( s - < < + . The point s = -
( ) is not
1 2 3 s12 m
1 2 m3 ) s123 ( s12 m3 ) 3

2
singular, that the round of this point at s + ie and s - ie gives identical result. s = +
( )
1 2 3 s12 m
1 2 3 1 2 3 3


is the singular point, but in our case the integration contour can not pass through this point that the
region in consideration is situated below the production threshold of the four particles s < 1 6 2
m . The
similar situation for the integration over s in the function J is occurred. But the difference consists
1 3 3

of the given integration region that is conducted between the complex conjugate points (contour 2
Fig.4). In Fig.4, 5b, 6 the dotted lines define the square root cut of the Chew-Mandelstam functions.
They correspond to two-particles threshold and also three-particles threshold in Fig.5a. The integration
2
contour 1 (Fig.5a) is determined by s < s -
( , the contour 2 corresponds to the case
1 2 s34 )

2 2 2
( s - < < + . s = s -
( is not singular point, that the round of this point
1 2 s34 )
1 2 s34 ) s ( s12 s34 )

at s + ie and s - ie gives identical results. The integration contour 1 (Fig.5b) is determined by region
s < ( s - s )2 2
< ( - ) < ( - )2
12 34 and s s m
134 34 1 , the integration contour 2 corresponds to s s s
12 34

2 2 2 2
and ( s - m ) s < ( s + m ) - < < +
34 1 134 34 1 . The contour 3 is defined by ( s .
1 2 s34 ) s ( s12 s34 )

2
Here the singular point would be s = ( s + m )
134 34 1 . But in our case this point is not achievable, if one

has the condition s < 1 6 2
m . We have to consider the integration over s in the function J .
2 4 3

2 2 2
While s < + + 4 - +
/ ( ) the integration is conducted along the complex axis
1 2 4 s12 m4 m2 m4 s12 s12 m1 m2
2 2 2
(the contour 1, Fig.6). If we come to the point s = + + 4 - +
/ ( ) , that the
1 2 4 s12 m4 m2 m4 s12 s12 m1 m2

output into the square root cut of Chew-Mandelstam function (contour 2, Fig.6) 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 show that the
contribution of the integration over the nonphysical region is small.





9


Table 1. Cryptoexotic low-lying meson masses.
J PC masses MeV
I II III
0++ 1010 f0 (985) 1418 f0 (1370) 1575 f0 (1500)
1++ 1306 f1(1285) 1443 f1(1420) 1615 f1(1530)
2 ++ 1430 f2 (1430) 1520 f2 (1520) 1710 f2 (1710)
0-+ 958 h (958) 1295 h (1295) 1440 h (1440)
1-- 1416 r (1450) 1511 ( - ) 1697 ( - )
1+- 958 ( - ) 1295 ( - ) 1440 ( - )
Parameters of model: I) =32,7; g =0,1613; =0, II) =18; g =0,2535; =22,5 MeV,

III) =19,6; g =0,2354; =70 MeV. Experimental values of the cryptoexotic mesons are given in

parentheses [31].




Table 2. Contributions of glueball A + A + A subamplitudes to the cryptoexotic meson amplitude in %
5 6 7

(groups I-III).
J PC I II III
0++ 45,94 41,69 42,09
1++ 52,99 42,65 43,96
2 ++ 58,46 48,04 49,34
0-+ 45,89 35,51 36,98
1-- 57,59 47,17 48,48
1+- 45,89 35,51 36,98




Table 3. Vertex functions
J PC G 2
1

0++ - 8g/3
1++ 4g/3
2 ++ 4g/3
0-+ 8 / 3 - 4 + 2
g g(m m ) / (3s )
i k ik

1-- 4g/3
1+- 8 / 3 - 4 + 2
g g(m m ) / (3s )
i k ik
2
The vertex functions G ( ) = 3
1 correspond to colour singlet states. G s g
2 ik , here g is the gluon

constant. In the present paper the contribution of axial interaction to the states J PC = 0-+ , 1+- is taken
into account.





10


Table 4. Coefficient of Chew-Mandelstam functions.
J PC a (J PC ) b(J PC ) d (J PC )

0++ -1/2 1/2 0
1++ 1/2 -e/2 0
2 ++ 3/10 1/5-3e/2 -1/5
0-+ 1/2 -e/2 0
1-- 1/3 1/6 - e/3 -1/6
1+- 1/2 -e/2 0
2 2
Here is e = ( m - m ) / (m + m )
i k i k





Figure captions.


Fig.1. Graphic representation of the equations for the four-quark subamplitudes A (s, s ,
1 2 , s1 2 3 )
l


l = -
1 4 . The coefficients are determined by the permutation of particles.

Fig.2. Graphic representation of the equations for the glueball subamplitudes A (s, s , s ) , p = -
5 7 .
p 12 34


Fig.3. Diagram of gluonic exchange defines the short-range component of quark interactions a) and

box-diagram of meson M takes into account the long-range interaction component of the quark forces

b).

Fig.4. Contours of integration 1, 2 in the complex plane s13 for the functions J1 , J3 .

Fig.5. Contours of integration 1, 2, 3 in the complex plane s134 (a) and s13 (b) for the function J2 .

Fig.6. Contours of integration 1, 2 in the complex plane s24 for the function J3 .





11


References.


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12


u _ u s _ s





u _ u s _ s
_ u u _ s s





2 2


+ + s _ u d _ u



_ u _ s

u s
u s s _ u d _ u
_ s u _ d u

_ u _ s
_ u _ s
u s





2 2 2 2


+ + + +


u _ u s _ s s _ u d _ u


u s u u
_ u _ s _ s _ d





= = = =



u _ u s _ s s _ u d _ u


u s u u

_ u _ s _ s _ d 1


Fig.

=1 =2 =3 =4
l l l l


_ s s _ u u _ s u
s u s
_ s _ u _ u





_ u d _ u d _ u d


u _ d u _ d u _ d





4 4 4

_ s + s _ u + u _ s + u
s u s
_ s _ u _ u





_ u s _ u s _ u s


u _ s u _ s u _ s





4 4 4


_ s + s _ u + u _ s + u
s u s
_ s _ u _ u





_ u u _ u u _ u u


u _ u u _ u u _ u





4 4 4

_ s + s _ u + u _ s + u
s u s
_ s _ u _ u





_ s s _ s s _ s s


s _ s s _ s s _ s





4 4 4

+ + +
_ s s _ u u _ s u
s _ s u _ u s _ u





= = =

s _ s s _ s u _ u u _ u s _ s u _ u Fig.





= = =


q q


G
_ _
q q


Fig.





q M q


_ M _
q q


Fig.


s
13




s +
13




1 2 s
13



s _ s +
13 13


s _
13





Fig.





s s
134 13




s + + s +
s
134 13 13




1 2 s 1 2 3 s
134 13


_
s _ s + s s +
134 134 13 13

_ _
s _ s s
13 13
134





Fig.





s
24


s +
24
s +
24
2
1 s
24





s _
24
s _
24





Fig.



