Heavy dibaryons
Abstract
The relativistic sixquark 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 heavy hexaquark amplitudes are obtained. The relativistic sixquark amplitudes of dibaryons including the light quarks , and heavy quarks , are calculated. The poles of these amplitudes determine the masses of charmed and bottom dibaryons with the isospins , , .
pacs:
11.55.Fv, 12.39.Ki, 12.39.Mk, 12.40.Yx.I Introduction.
Hadron spectroscopy has always played an important role in the revealing mechanisms underlying the dynamic of strong interactions.
The heavy hadron containing a single heavy quark is particularly interesting. The light degrees of freedom (quarks and gluons) circle around the nearby static heavy quark. Such a system behaves as the QCD analog of familar hydrogen bound by the electromagnetic interaction.
The heavy quark expansion provides a systematic tool for heavy hadrons. When the heavy quark mass , the angular momentum of the light degree of freedom is a good quantum number. Therefore, heavy hadrons form doublets. For example, and will be degenerate in the heavy quark limit. Their mass splitting is caused by the chromomagnetic interaction at the order , which can be taken into account systematically in the framework of heavy quark effective field theory (HQET) 1 ; 2 ; 3 .
In 1977, Jaffe 4 studied the colormagnetic interaction of the onegluonexchange potential in the multiquark system and found that the most attractive channel is the flavor singlet with quark content . The same symmetry analysis of the chiral boson exchange potential leads to the similar result 5 .
The particle, state and di may be strong interaction stable. Up to now, these three interesting candidates of dibaryons are still not found or confirmed by experiments. It seems that one should go beyond these candidates and should search the possible candidates in a wider region, expecially the systems with heavy quarks, in terms of a more reliable model.
There were a number of theoretical predictions by using various models: the quark cluster model 6 ; 7 , the quarkdelocation model 8 ; 9 , the chiral quark model 10 , the flavor skyrmion model 11 . Lomon predicted a deuteronlike dibaryon resonance using Rmatrix theory 12 . By employing the chiral quark model Zhang and Yu studied and states 13 ; 14 .
In a series of papers 15 ; 16 ; 17 ; 18 ; 19 a method has been developed which is convenient for analyzing relativistic threehadron systems. The physics of the threehadron system can be described by means of a pair interaction between the particles. There are three isobar channels, each of which consists of a twoparticle isobar and the third particle. The presence of the isobar representation together with the condition of unitarity in the pair energies and of analyticity leads to a system of integral equations in a single variable. Their solution makes it possible to describe the interaction of the produced particles in threehadron systems.
In Ref. 20 a representation of the Faddeev equation in the form of a dispersion relation in the pair energy in the two interacting particles was used. This was found to be convenient in order to obtain an approximate solution of the Faddeev equation by a method based on extraction of the leading singularities of the amplitude. With a rather crude approximation of the lowenergy interaction a relatively good description of the form factor of tritium (helium3) at low was obtained.
In our papers 21 ; 22 ; 23 relativistic generalization of the threebody Faddeev equations was obtained in the form of dispersion relations in the pair energy of two interacting quarks. The mass spectrum of wave baryons including , , quarks was calculated by a method based on isolating the leading singularities in the amplitude. We searched for the approximate solution of integral threequark equations by taking into account twoparticle and triangle singularities, all the weaker ones being neglected. If we considered such approximation, which corresponds to taking into account twobody and triangle singularities, and defined all the smooth functions of the subenergy variables (as compared with the singular part of the amplitude) in the middle point of the physical region of Dalitzplot, then the problem was reduced to the one of solving a system of simple algebraic equations.
In the present paper the relativistic sixquark equations are found in the framework of coupledchannel formalism. We use only planar diagrams; the other diagrams due to the rules of expansion 24 ; 25 ; 26 are neglected.
The sixquark amplitudes of dibaryons are calculated. The poles of these amplitudes determine the masses of dibaryons. We calculated the contribution of sixquark subamplitudes to the hexaquark amplitudes. In Sec. II, we briefly discuss the relativistic Faddeev approach. The relativistic threequark equations are constructed in the form of the dispersion relation over the twobody subenergy. The approximate solution of these equations using the method based on the extraction of leading singularities of the amplitude are obtained. We calculated the mass spectrum of wave bottom baryons with , (Tables 1, 2). In Sec. III, the sixquark amplitudes of hexaquarks are constructed. The dynamical mixing between the subamplitudes of dibaryons are considered. The relativistic sixquark equations are constructed in the form of the dispersion relation over the twobody subenergy. The approximate solutions of these equations using the method based on the extraction of leading singularities of the amplitude are obtained. Sec. IV is devoted to the calculation results for the dibaryon mass spectra and the contributions of subamplitudes to the hexaquark amplitude (Tables 4, 5, 6, 7). In conclusion, the status of the considered model is discussed.
Ii Brief introduction of relativistic Faddeev equations.
In our papers, 21 ; 22 ; 23 ; 27 ; 28 relativistic generalization of the threebody Faddeev equations was obtained in the form of dispersion relations in the pair energy of two interacting particles. The mass spectra of wave baryons including , , , quarks were calculated by a method based on isolating of the leading singularities in the amplitude.
We searched for the approximate solution of integral threequark equations by taking into account twoparticle and triangle singularities, the weaker ones being neglected. If we considered such an approximation, which corresponds to taking into account twobody and triangle singularities, and defined all the smooth functions at the middle point of the physical region of Dalitzplot, then the problem was reduced to the one of solving a system of simple algebraic equations.
In the paper 27 the relativistic threeparticle amplitudes in the coupledchannels formalism are considered. We take into account the , , , , quarks and construct the flavorspin functions for the baryons with the spinparity and :

In the paper 28 , the relativistic equations were obtained and the mass spectrum of wave charmed baryons was calculated.
In the present paper, we will be able to use the similar method. In this case, we consider baryons with the spinparity and , which include one, two and three bottom quarks. We have considered the baryons with different masses (Tables 1, 2).
We calculate the masses of the bottom baryons in a relativistic approach using the dispersion relation technique. The relativistic threequark integral equations are constructed in the form of the dispersion relations over the twobody subenergy.
We use the graphical equations for the functions . In order to represent the amplitude in the form of dispersion relations, it is necessary to define the amplitudes of quarkquark interaction . The pair amplitudes are calculated in the framework of the dispersion method with the input fourfermion interaction with quantum numbers of the gluon. We use results of our relativistic quark model 29 and write down the pair quark amplitudes in the following form:
(1) 
(2) 
(3)  
Here is the vertex function of a diquark, which can be expressed in terms of the function of the bootstrap method as , is the ChewMandelstam function 30 , and is the phase space for a diquark. is the pair energy squared of diquark, the index determines the spinparity of diquark. The coefficients of ChewMandelstam function , and in Table 3 are given. is the pair energy cutoff. In the case under discussion the interacting pairs of quarks do not form bound states. Therefore, the integration in the dispersion integral (2) is carried out from to (). Including all possible rescatterings of each pair of quarks and grouping the terms according to the final states of the particles, we obtained the coupled systems of integral equations. For instance, for the with the wave function is
(4) 
Here the function has the form
(5) 
The integral operator is
(6) 
The function is the truncated function of ChewMandelstam:
(7) 
is the cosine of the angle between the relative momentum of particles and in the intermediate state and the momentum of particle in the final state, taken in the c.m. of the particles and . Let some current produces three quarks with the vertex constant . This constant do not affect to the spectra mass of bottom baryons. By analogy with the state, we obtain the rescattering amplitudes of the three various quarks for the all bottom states.
Let us extract twoparticle singularities in :
(8) 
is the reduced amplitude. Accordingly to this, all integral equations can be rewritten using the reduced amplitudes. The function is the smooth function of as compared with the singular part of the amplitude. We do not extract the threebody singularities, because they are weaker than the twoparticle singularities. For instance, one considers the first equation of system for the with :
(9)  
The connection between and is 21 :
(10)  
The formula for is similar to (10) with replaced by . Thus must be replaced by . is the cutoff at the large value of , which determines the contribution from small distances.
The construction of the approximate solution of coupled systemequations is based on the extraction of the leading singularities which are close to the region 31 .
We consider the approximation, which corresponds to the single interaction of the all three particles (twoparticle and triangle singularities) and neglecting all the weaker ones.
The functions are the smooth functions of as compared with the singular part of the amplitude, hence it can be expanded in a series at the singulary point and only the first term of this series should be employed further. As it is convenient to take the middle point of physical region of the Dalitzplot in which . In this case, we get from (10) , where . We define functions and at the point . Such a choice of point allows us to replace integral equations (4) by the algebraic couple equations for the state :
(11) 
The function takes into account singularity which corresponds to the simultaneous vanishing of all propagators in the triangle diagram.
(12) 
The functions have the smooth dependence from energy 27 , therefore we suggest them as constants. The parameters of model: vertex constant and cutoff parameter are chosen dimensionless;
(13) 
Here and are quark masses in the intermediate state of the quark loop. We calculate the coupled system of equations and can determine the mass values of the state. We calculate a pole in which corresponds to the bound state of the three quarks.
By analogy with hyperon we obtain the systems of equations for the reduced amplitudes of all bottom baryons.
The solutions of the coupled system of equations are considered as:
(14) 
where the zeros of the determinate the masses of bound states of baryons. are the functions of and . The functions determine the contributions of subamplitudes to the baryon amplitude.
In quark models, which describe rather well the masses and static properties of hadrons, the masses of the quarks usually have the similar values for the spectra of light and heavy baryons. However, this is achieved at the expense of some difference in the characteristics of the confinement potential. It should be borne in mind that for a fixed hadron mass the masses of the dressed quarks which enter into the composition of the hadron will become smaller when the slope of the confinement potential increases or its radius decreases. Therefore, conversely, we can change the masses of the dressed quarks when going from the spectrum of light baryons to the heavy baryons, while keeping the characteristics of the confinement potential unchanged. We can effectively take into account the contribution of the confinement potential in obtaining the spectrum of wave heavy baryons.
In the case of quark we have used two new parameters: the cutoff of the diquark and the coupling constant . These values have been determined by the baryon masses: and . In order to fix we use the baryon mass . We represent the masses of all wave bottom baryons in the Tables 1, 2. The calculated mass value is equal to the experimental data 32 , the mass value is close to the experimental one. We neglect with the mass distinction of and quarks. The estimation of the theoretical error on the bottom baryon masses is . This result was obtained by the choice of model parameters.
In our model the spinaveraged mass of the states and is predicted to lie around to above . The relativistic corrections are particularly important for the splitting between and baryons.
In the context of and masses, it is worth mentioning two relations among bottom baryons which incorporate the effects of breaking:
(15) 
(16) 
The sign in our prediction is
(17) 
Iii Sixquark amplitudes of the hexaquarks.
We derive the relativistic sixquark equations in the framework of the dispersion relation technique. We use only planar diagrams; the other diagrams due to the rules of expansion 24 ; 25 ; 26 are neglected. The current generates a sixquark system. The correct equations for the amplitude are obtained by taking into account all possible subamplitudes. Then one should represent a sixparticle amplitude as a sum of 15 subamplitudes:
(18) 
This defines the division of the diagrams into groups 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 . We shall consider the derivation of the relativistic generalization of the FaddeevYakubovsky approach. In our case, the lowlying dibaryons are considered. We take into account the pairwise interaction of all six quarks in the hexaquark.
For instance, we consider the state with , and quark content . The set of diagrams associated with the amplitude can further be broken down into eight groups corresponding to subamplitudes: , , , , , , , .
The amplitude consists of the three color substructures: the diquark in the color state , the quarks 3, 4 in the color state , and the quarks 5, 6 in the color state . Then we consider the total color singlet: . The dibaryon amplitude contains the following substructures: the two diquark and in the color state and the two quarks in the color state . Then the dibaryon amplitude is the total color singlet. The amplitude consists of the three diquark structures in the color state . Therefore the total color singlet can be constructed. For the others amplitudes color singlet also can be found.
The system of graphical equations (see for example equation for the amplitude for the state with and at the Fig. 1) is determined by the subamplitudes using the selfconsistent method. The coefficients are determined by the permutation of quarks.
In order to represent the subamplitudes , , , , , , , in the form of a dispersion relation, it is necessary to define the amplitude of and interactions. We use the results of our relativistic quark model 29 and write down the pair quark amplitudes in the form:
(19) 
(20) 
(21)  
The coefficients , and are given in Table 8. Here coresponds to and pairs with , corresponds to and pairs with .
The coupled integral equations correspond to Fig. 1 can be described similar to 36 . Then we can go from the integration of the cosine of the angles to the integration over the subenergies.
Let us extract two and threeparticle singularities in the amplitudes , , , , , , , :
We used the classification of singularities, which was proposed in paper 31 . Using this classification, one defines the reduced amplitudes , , as well as the functions in the middle point of physical region of Dalitzplot at the point .
Such choice of point allows us to replace integral equations (, , ) by the algebraic equations (III) – (37):