Relativistic quark model and lowest hybrid mesons.




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



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





Abstract.



The relativistic four-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 mass spectrum values of lowest hybrid mesons are

calculated.





* Present address: Department of Physics, LTA, Institutski Per. 5, St. Petersburg

194021, Russia


I. Introduction.



The present understanding of strong interactions is that they are described by QCD. This non-Abelian

field theory not only describes how quarks and antiquarks interact but also predicts that the gluons

which are the quanta of the field will themselves interact to form mesons. If the object formed is

composed entirely of constituent gluons ( gg ) the meson is called a glueball, however if it is composed
of a mixture of constituent quarks, antiquarks and gluons ( qqg ) it is called hybrid. In addition, qqqq
states are also predicted. An unambiguous confirmation of these states would be an important test of

QCD and give fundamental information on the behavior of this theory in the confinement region.

Historically, there have been two approaches to the consideration of hybrids. The first assumes that

hybrids are predominantly quark-antiquark states with an additional constituent gluon [1 - 4] and that

decays proceed via constituent gluon dissociation [5 - 7]. The second assumes that hybrids are

predominantly quark-antiquark states moving on an adiabatic surface generated by an exited flux tube

configuration of glue [8]. Decays then proceed by a phenomenological pair production mechanism (the

 3 P model [9 - 13]) coupled with a flux tube overlap [14, 15].
0
In the recent papers [16, 17] the relativistic four-quark equations are represented in the form of the

dispersion relation over the two-body subenergy. The behavior of the low-energy four-particle

amplitude is determined by its leading singularities in the pair invariant masses. The suggested method of

approximate solutions of the relativistic four-quark equations was verified on the example of the lowest

cryptoexotic mesons [18]. The calculated mass values of cryptoexotic mesons are in good agreement

with the experimental ones.

In the present paper we calculated the masses of the lowest hybrid mesons using the method based on

the extraction of leading singularities of the amplitude. The interesting result of this model is the

calculation of hybrid meson amplitudes, which contain the contributions of two subamplitudes: four-

quark amplitude and hybrid amplitude. The contributions of these subamplitudes with different quantum

numbers are given in Table I. One can see that the main contribution corresponds to the four-quark

amplitude. The hybrid amplitude give rise to only less 40 % of the hybrid meson contributions.

In Section II 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

based on the extraction of leading singularities of the amplitude are obtained. The quark amplitudes of

hybrid mesons are calculated.

Section III is devoted to the calculation results for the lowest hybrid meson mass spectrum (Table I).

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 hybrid mesons

are given (Tables II, III) respectively.

In the Appendix B we search the integration contours of functions J , , , which are determined
1 J 2 J3
by the interaction of the four quarks.




II. Quark amplitudes of the hybrid mesons.



In the present paper we investigate scattering amplitudes of the constituent quarks of two flavours (u,

d). The poles of these amplitudes determined the masses of the lowest hybrid mesons. The constituent

quark is color triplet and quark amplitudes obey the global color symmetry. One uses the results of the

bootstrap quark model [19, 20] and introduce the qq amplitude in the colour octet channel c = 8 with
c
J PC = 1-- . This bound state should be identified as constituent gluon with mass of the order of 0.7 GeV.

In our consideration we take into account the colour octet state with J PC = 1-- and isospin I=1, which


determines with the constituent gluon the hybrid state qqg . Now we have the mixing between hybrid
and qqqq states. This state is called the hybrid meson.
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). The diagrams in Fig.1 allows

to graphically present the equations for the four-quark amplitudes. However, the correct equations for

the amplitude are obtained at the account of all possible subamplitudes. This corresponds to split

complete system into 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 . This defines the
12 13 14 23 24 34

division of the diagrams into 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 A . One
12

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 A are divided in three group A ( s, s , s ) , A ( s, s , s ) and A ( s, s , s ) (moreover the
12 1 12 123 1 12 124 2 12 34

subamplitudes A ( s, s , s ) and A ( s, s , s ) are analogous).
1 12 123 1 12 124

The equations for the four-quark amplitudes in the graphic form are presented (Fig.1). The

coefficients are determined by the permutation of quarks [21].

To present the amplitudes A (s, s , s ) and A (s, s , s ) in form of the dispersion relation it is
1 12 123 2 12 34
necessary to define the amplitude of two-quark interaction a ( s ) . One uses the results of the bootstrap
j ik

quark model [19, 20] and writes down the pair quarks amplitude in the form:



G 2 (s )
j ik
a (s ) = , (1)
j ik -
1 B (s )
j ik



r ' 2 '
ds' (s )G (s )
j ik j ik
B (s = ik
) . (2)
j ik p
2 s' - s
m ik ik



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

function [23] and the phase space respectively. We introduced the cut-off parameter . There j=1

corresponds to pair of quarks qq with isospin I=1 and J PC = 0++ , 1++ , 2++ , 0-+ , 1-- (colour singlet

S U (3) ) and j=2 defines the quark pair with J PC = 1-- in colour channel 8 (constituent gluon). j=3
c c

defines the quark pair with J PC = 1-- (isospin I=1) in colour channel 8 (for instance,
c ud state). The
vertex functions are shown in the Table II, the functions r ( s ) are given in the Appendix A (see Table
j ik

III). In the case in question the interacting quarks do not produce bound state, then the integration in (3)

- (4) is carried out from the threshold 4 2
m to the cut-off . The integral equation systems,
corresponding to Fig 1, have the following form:


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


l B (s )B (s ) G
8 (s G
) (s ) $
A (s, s , s = 2 2 12 3 34 + 2 12 3 34
) J A (s, s' , s' ) . (4)
2 12 34 [ - B (s )][ - B (s )] 1
[ - B (s 1
)][ - B (s 2 1 13 134
1 1 )]
2 12 3 34 2 12 3 34
l are the current constants. Here we introduce the integral operators:
i


'
ds r ( '
s ) G ( '
s ) +1 dz
$
J ( ,
s m ) 12 1 12 1 12 1
= , (5)
1 '
p s - s 2
4m 2 12 12 1
-




+ +
' r ' ' ' r ' ' 1 1
$ ds (s ) G ( s ) ds (s ) G (s ) dz dz
J ( ,
s m) = 12 2 12 2 12 34 2 34 3 34 3 4
, (6)
2 p ' - p ' -
- 2 - 2
2 s s 2 s s
m 12 12
4 m 34 34
4 1 1



+ + +
' r ' ' 1 1 z2
$ 1 ds (s ) G (s ) dz
J ( ,
s m) = 12 1 12 1 12 1
dz dz 
3 p p '
4 -
2 s s 2
- 2 -
m 12 12
4 1 1
- z2 . (7)

1


1 - z 2 - z 2 - z 2 + 2zz z
1 2 1 2



there m are the masses of nonstrange quarks. In the equations (5) and (7) z is the cosine of the angle
1
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 (7) 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 momentum of particles 1 and 2 in the
2
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 (6) we have defined: z is the cosine of the angle between relative
3
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
4
angle between the relative momentum of the particles 3 and 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

(8) - (12) 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
1 J 2 J3
[18] (see Appendix B).



s - s' - m2 z s' - 4m2
s' = m2 + 123 12 + 1 12 s - s' - m2 2 - 4s'
[( ) m2
2 ] , (8)
13 2 2 s' 123 12 12
12



s' - s' - m2 z s' - 4m2
s' = m2 + 124 12 + 2 12 s' - s' - m2 2 - 4s'
[( ) m2
2 ] , (9)
24 2 2 s' 124 12 12
12



2s' (s + s' - s - s' -
) (s - s' - m2 )(s' - s' - m2 )
z = 12 12 123 124 123 12 124 12 , (10)
[(s - s' - m2 2 -
) 4m2s' ][(s' - s' - m2 2 -
) 4m2s' ]
123 12 12 124 12 12




s - s' - s' z s' - 4m2
s' = m2 + s' + 12 34 + 3 12 s - s' - s' 2 - , (11)
4s' s'
[( ) ]
134 34 s' 12 34 12 34
2 2 12



s' - s' - m2 z s' - 4m2
' = 2 + 134 34 + 4 34 [ ' - ' - 2 2 - 2 ' ]
s 2m (s s m ) 4m s . (12)
13 2 2 s' 134 34 34
34
Let us extract two-particle singularities in the amplitudes A (s, s , s ) and A (s, s , s ) :
1 12 123 2 12 34


a (s, s , s )B (s )
A (s, s , s = 1 12 123 1 12
) , (13)
1 12 123 1 - B (s )
1 12


a (s, s , s )B (s )B (s )
A (s, s , s = 2 12 34 2 12 3 34
) . (14)
2 12 34 [ - ][ - ]
1 B (s ) 1 B (s )
2 12 3 34



In the amplitude A (s, s , s ) we do not extract three-particle singularity, because it is weaker than
1 12 123
two-particle and taking into account in the function a ( ,
s s , s ) .
1 12 123
We used the classification of singularities, which was proposed in papers [17]. The construction of

approximate solution of the (13) and (14) 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
ik
all 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 and
ik
on the second sheet in case of virtual bound state. The diagrams Fig.1 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 (13) and (14) 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 ) and a ( ,
s s , s ) are the
1 12 123 2 12 34
smooth functions of s , s as compared with the singular part of the amplitudes, hence they can be

ik ijk

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 ) and a ( ,
s s , s ) as well as the
1 12 123 2 12 34
B-functions in the middle point of the physical region of Dalitz-plot at the point s :
0



s + m2
8
s = , (15)
0 6

s = s - m2
3 3 . (16)
123 0


Such a choice of points s allows as to replace the integral equations (3) and (4) by the algebraic
0
equations (17) - (18) respectively:

a = l + a + a B (s )B (s )
2 0 3 0
2 J 2 J , (17)
1 1 1 1 2 3 B (s )
1 0

a = l + a B (s )
1 0
8 J . (18)
2 2 1 2 B (s )B (s )
2 0 3 0
Here we introduce following functions:



+
ds' r (s' 1
) dz 1
J (s, m) = G 2 12 1 12 1
, (19)
1 1 p ' -
- 2 1 - ( ' )
2 s s B s
4m 12 0 1 1 13



+ +
ds' r (s' ) ds' r (s' 1
) dz 1 dz 1
J (s, m) = G 2G 2 12 2 12 34 2 34 3 4
, (20)
2 2 3 p ' - p ' -
- 2 - 2 1 - ( ' )
2 s s 2 s s B s
m 12 0
4 m 34 0
4 1 1 1 13


~ +
- + +
' r ' 1 1 z
1 B (s , ) 1 ds (s ) dz 2
J (s, m) = G 2 1 0 12 1 12 1

~ dz dz
3 1 1 - B (s , ) 4p p ' -
2 s s 2
- 2 -
1 0 m 12 0
4 1 1
- z2 . (21)

1 1

' '
1 - z 2 - z 2 - z 2 + - -
2zz z 1
[ B (s 1
)][ B (s )]
1 2 1 2 2 13 3 24



In our approximation the vertex functions (Table II) are constants. As the integration region the physical

region of the reaction should be chosen, therefore -
1 z 1 ( i=1,2,3,4 ). From these conditions we
i
can define the regions of the integration over s' s' s' s'
, , , . Let us consider the integration region
13 24 134 124

over s' . For this purpose we use equation (10). This condition corresponds to 0 2
z 1. By
124
consideration of these inequalities one can obtain:



- - + ' -
 (s s m2 )(s s m2 )
s = s' + m2 + 123 123 12 
124 12 2s123 (22)
1
 [(s - s' - m2 2 -
) 4m2s' ][(s - s - m2 2 -
) 4m2s ]
2s 123 12 12 123 123
123



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



if + 2
~ , ( s m)
= 123 (23)
( +

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



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


a ( s ) = F ( s, l ) / ( s ) (24)
i i i




There ( s ) is the determinant:



(s) = - J -
1 2 16J J (25)
1 2 3


Right-hand sides of (25) might have a pole in s which corresponds to the bound state of the four
quarks. The poles of rescattering amplitudes for the lowest hybrid mesons with J PC = 0++ , 1++ , 2++ ,

0-+ , 1-- (I=1) correspond to the bound state and determine the masses of the hybrid mesons.




III. Calculation results.



In the bootstrap quark of model [19, 20] there is a bound state in the gluon channel with mass of the

order 0.7 GeV. This bound state should be identified as a constituent gluon. In our consideration we

take into account the colour octet state (like ud ) with J PC = --
1 and I=1, which determines with the
constituent gluon the hybrid state qqg . The calculated values of mass lowest hybrid mesons are shown
in the Table I. The results are in the agreement with the papers [4, 24]. This perturbative calculation


corresponds to mixing hybrid and qqqq states. The absence of experimental data for these states does
not to allow to verify the detailed coincidence. In the considered calculation the quark masses m is not
1
fixed. In order to fix anyhow m , we assume m = 570 MeV (m m (2230))
. The model under
4 a$2

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. 2(a)). But for the hybrid 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 pair. This means that in the transition qq qq the forces appear
which are connected with the process of the Fig. 2(b) type. These box-diagrams can be important in the

formation of hadron spectra [25]. 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 h
, , KK , K and so on. We suggest that the parameter takes into
account the confinement potential effectively: m = m + , m = 0,385 GeV [19,20], = 0,185 GeV. It
0 0
changes the behavior pair quark amplitude (1). It allows us to construct the hybrid mesons amplitudes

and calculate the mass spectrum hybrid mesons by analogy with the calculation of mass values of the

+ +
1 3
lowest baryons ( J P = , ) in the bootstrap quark model [26]. The model in consideration have two
2 2
parameters: cut-off parameter and gluon constant. The subenergy cut-off and the vertex function

g can be determined by mean of fixing of lowest hybrid meson mass values ( J PC = ++ ++
0 , 2 ). The
vertex functions of various types of the interactions are given in Table II. The mass of the hybrid meson

with J PC = -+
0 is obtained smaller as compared paper [27 - 29]. It may be determine by them, that the
main contribution to state J PC = -+
0 is given by the four-particle state. The contribution of hybrid
amplitude is only 22 % of the hybrid meson amplitude.




IV. Conclusion.



In the present paper in the framework of approximate method of solution four-particle relativistic

problem the mass spectrum of hybrid mesons, including u, d - quarks, are calculated. The interesting

result of this model is the calculation of hybrid meson amplitudes, which contain the contributions of

two subamplitudes: four four-quark amplitudes A and hybrid amplitudes A . The contributions of these
1 2
subamplitudes with different quantum numbers are given in Table I. One can see that the main

contributions correspond to the four-quark amplitudes A . The hybrid contribution corresponds to only
1
less 40 %. We obtained that the small contributions of hybrid give rise to the smaller mass of lowest

hybrid meson with J PC = -+
0 as compared paper [27 - 29]. The decay width of hybrid mesons can be
calculated in the framework this model. The suggested approximate method allows to construct the

hybrid meson amplitudes, including heavy quarks Q = s, c, b and calculate the mass spectrum of heavy

hybrid mesons.


APPENDIX A



The two-particle phase space for the equal quark masses is defined as:



- 2

r PC = a s
PC ik + b s 4m
(s , J ) ( J ) ( J PC ) ik ,
1 ik 2
4m sik

r = r = r - -
(s ) (s ) (s 1
, ) .
2 ik 3 ik 1 ik



The vertex functions are shown in Table II. The coefficients a (J PC ) and b (J PC ) are given in Table

III.




APPENDIX B



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

defined by the connection ( s - )
m 2 < s < ( s + )
m 2 . The point s = s -
( m 2
) is not singular,
12 123 12 123 12

that the round of this point at s + ie and s - ie gives identical result. = + 2
123 123 s ( s m) is the
123 12
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 < 16m2 . The similar
situation for the integration over s in the function J is occurred. But the difference consists of the
13 3
given integration region that is conducted between the complex conjugate points (contour 2 Fig. 3). In

Fig. 3, 4b, 5 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. 4(a). The integration

contour 1 (Fig. 4(a)) is determined by 2
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 ( s1 2 s34 )
at s + ie and s - ie gives identical results. The integration contour 1 (Fig. 4(b)) is determined by region

s < ( s - s )2 and < - 2 < ( - )2
12 34 s ( s m) , the integration contour 2 corresponds to s s s
134 34 12 34

and 2 2
( s - )
m 2 s < ( s + )
m 2 . The contour 3 is defined by ( s - < < + .
1 2 s34 ) s ( s1 2 s34 )
34 134 34

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

has the condition s < 16m2 . We have to consider the integration over s in the function .
24 J 3
While s < s + m2
5 the integration is conducted along the complex axis (the contour 1, Fig. 5). If we
124 12
come to the point s = s + m2
5 , that the output into the square root cut of Chew-Mandelstam
124 12
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 show that the contribution of the integration over the

nonphysical region is small [18].


Table I. Low-lying hybrid meson masses with the isospin I=1 and contributions

of four-quark subamplitude A and hybrid subamplitude A subamplitudes to
1 2
the hybrid meson amplitude in %.



J PC Masses (MeV) A A
1 2
0++ $
a 1800 (1800) 71,48 28,52
0
1++ $
a 1997 (1940) 67,29 32,71
1
2++ $
a 2230 (2230) 58,20 41,80
2
0-+ $
p 1364 (1610) 77,89 22,11

1-- $
r 2200 (2020) 59,84 40,16

Parameters of model: cut-off parameter =17,9; gluon constant g =0,373;

effective mass m = 570 MeV. The hybrid meson mass values of paper [4, 24]
are given in parentheses.





Table II. 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

The vertex functions G correspond to colour singlet states. 2 and
1 G (s ) = 2g
2 ik

G 2 (s ) = g , correspond to the constituent gluon and ud state in colour channel
3 ik

8 with J PC = --
1 and I=1 respectively. Here g is the gluon constant. In the
c
present paper the contribution of axial interaction to the state J PC = 0-+ is taken

into account.





Table III. Coefficient of Chew-Mandelstam functions.



J PC a (J PC ) b(J PC )

0++ -1/2 1/2

1++ 1/2 0

2++ 3/10 1/5

0-+ 1/2 0

1-- 1/3 1/6


Figure captions.



Fig. 1. Graphic representation of the equations for the four-quark subamplitude A (s, s , s ) (a) and
1 12 123
the hybrid subamplitude A (s, s , s ) (b). The bold line corresponds to the constituent gluon
2 12 34
contribution.

Fig. 2. 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. 3. Contours of integration 1, 2 in the complex plane s for the functions J , J .
13 1 3

Fig. 4. Contours of integration 1, 2, 3 in the complex plane s (a) and s (b) for the function J .
134 13 2

Fig. 5. Contours of integration 1, 2 in the complex plane s for the function J .
24 3





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