and Analogous Heavy Flavor States
The proximity of to the threshold suggests that it may be a molecular state. The system has been studied dynamically from quark model, and state mixing effect is taken into account by solving the multichannel Schrdinger equation numerically. We suggest the most favorable quantum number is , if future experiments confirm as a loosely bound molecule state. More precise measurements of mass and width, partial wave analysis are helpful to understand its structure. The analogous heavy flavor mesons and are studied as well, and the masses predicted in our model are in agreement with the predictions from potential model and QCD sum rule. We further apply our model to the and system. We find the exotic bound molecule doesn’t exist, while the bound state solution can be found only if the screening mass is smaller than 0.17 GeV. The state mixing effect between the molecular state and the conventional charmonium should be considered to understand the nature of X(3872).
PACS numbers: 12.39.Jh, 12.40.Yx,13.75.Lb
In the past years many new mesons have been discovered through B meson decays. Recently the Belle Collaboration has reported a narrow peak in the invariant mass spectrum in with statistical significance greater than 2007wga (). This structure is denoted as . The Breit Wigner fit for this resonance yields the peak mass MeV and the width MeV. The product branching fraction is determined to be . Since the G-parity of both and is negative, is a isovector with positive G-parity. However, is far from being established, no significant evidence for has been observed neither in the total or mass distribution by the Babar Collaboration :2008nk ().
There are already many theoretical investigations for the possible structures and the properties of Ding:2007ar (); Rosner:2007mu (); Maiani:2007wz (); Meng:2007fu (); Cheung:2007wf (); Gershtein:2007vi (); Qiao:2007ce (); Lee:2007gs (); Liu:2007bf (); Li:2007bh (); Braaten:2007xw (); Bugg:2008wu (); Liu:2008qx (); Liu:2008xz (); Liu:2008yy (); Cardoso:2008dd (). Because it is very close to the threshold of , and the width of is approximately the same as that of , it is suggested that could be a molecular state Rosner:2007mu (); Meng:2007fu (); Ding:2007ar (); Liu:2007bf (); Liu:2008xz (). Other interpretations such as tetraquark state Maiani:2007wz (); Gershtein:2007vi () or a cusp in the channel Bugg:2008wu () are proposed as well. In Ref. Ding:2007ar (), we suggested how to distinguish the molecule and the tetraquark hypothesis, and as a molecule was studied from the effective field theory. In Ref. Liu:2007bf (); Liu:2008xz (), the authors investigated dynamically whether could be a S-wave or molecular state by one-pion exchange and exchange.
In principle, nothing in QCD prevents the formation of nuclear-like bound states of mesons and speculation on the existence of such states dates back thirty years molecule (). Trnqvist suggested that two open flavor heavy mesons can form deuteron-like states due to the strong exchange interaction Tornqvist:1993ng (), and the monopole form factor is introduced to regularize the interaction potential at short distance. In Ref. Swanson:2003tb (), the author investigated the possible heavy flavor molecules base on long distance one pion exchange and short distance quark interchange model. However, the dynamics of hadronic molecule is still unclear so far. In this work, we will dynamically study and analogous heavy flavor states and from quark model. We shall discuss the interaction between two hadrons at the quark level instead of at the hadron level. The effective interactions between quarks including the screened color-Coulomb, screened linear confinement and spin-spin interactions are employed to describe the interactions between the components of the interacting hadrons.
This paper is organized as follows. In section II, the canonical coordinate system and the effective interactions are introduced. We give the details of the evaluation of the matrix element in section III. In section IV, the system coupled with is studied, and the possible structure of is discussed. In section V, the analogous heavy flavor states and , and systems are investigated, the static properties such as the mass and the root of mean square radius etc. are calculated. We present our conclusions and some relevant discussions in section VI. Finally the spatial matrix elements involved are given in the Appendix.
Ii canonical coordinate system and the effective interactions
The coordinate shown in Fig. 1 is taken as the canonical coordinate system, which defines the asymptotic states. The relevant coordinates of this system can be expressed in terms of , and as follows,
where , , and are respectively the masses of constituents 1, 2, 3 and 4. The relative position between the center of mass of the two mesons is
As is shown in Eq.(1), we can compactly represent the coordinate in terms of , and as follows
The parameters and are listed in Table 1.
In the above canonical coordinate, the Hamiltonian for this system, including the relative motion and the interaction between two mesons, is split into
where and are respectively the Hamiltonian for the two mesons and , which contains the kinetic term and all the interactions within each meson. is the reduced mass . The third term is the kinetic energy operator of the relative motion. The interaction potential is the sum of two-body interaction between quarks in the mesons and ,
The phenomenological interaction between a quark and an antiquark in a single meson ( e.g. and ) is reasonably well known, it is described by the short distance one-gluon exchange interaction and the long distance phenomenological confinement interaction De Rujula:1975ge (); Godfrey:1985xj ()
where is the strong coupling constant, is the string tension, and are the masses of the interacting constituents. For an antiquark, the generator is replaced by -.
Since we mainly concentrate on the molecular states comprising two heavy flavor mesons in this work, and the molecule is generally weakly bound. Therefore the separation between the two mesons in the molecule is rather larger than the average radius of the individual meson, and the two mesons interact mainly through two gluons exchange processes Appelquist:1977es (); Peskin:1979va (), which results in the color van der Waals interaction. By comparing with the van der Waals interaction between electric dipoles in QED, the author in Ref. Wong:2003xk () introduced the effective charges for quarks and antiquarks to describe the color van der Waals interaction between two mesons. The effective charges for quark and antiquark respectively are and , here is the number of color with in QCD. It is remarkable that the effective charge correctly describes the interaction between quark and antiquark in an individual meson as well. The effective charge is also consistent with the Lattice QCD results, which found the nonperturbative potential between a quark and an antiquark in different representations is proportional to the eigenvalue of the quadratic Casimir operator Bali:2000un ().
Different from the interactions between quarks in a single meson, as the interaction between the constituents in a molecule takes place at large distances, we are well advised to use a screened potential to represent the effects of dynamical quarks and gluon Lipkin:1982ta (). A simple way to incorporate the screening effect is to replace in the Fourier transformation of the interaction potential by Wong:2003xk (); Zhang:2003zb (), where is the screening mass. With the effective charge and the screening effect in mind, in momentum space, the effective interaction potential between two quarks in the mesons and is
The effective interaction in coordinate space is the Fourier transformation of
Therefore in coordinate space the effective interaction is
In this work, we will use the above effective interaction to study the possible heavy flavor molecules dynamically. Comparing with Ref. Wong:2003xk (), we have introduced the spin-spin interaction in addition to the screened color-Coulomb and the screened linear confinement interactions. In the light quark hadrons, the spin-spin hyperfine interaction makes the dominant contribution to the hadron-hadron interactions Barnes:1991em (). Whereas, for the heavy flavor mesons, the hyperfine interaction contribution is smaller due to the large heavy quark mass Barnes:1999hs (); Wong:2001td (). Therefore we expect that the contribution of spin-spin interaction should be smaller than those of the screened color-Coulomb and screened linear confinement interactions. However, the spin-spin hyperfine interaction may play an important role when we study the dynamics of molecular state, since the binding energy of molecular state is usually rather small. On the other hand, if we neglect the spin-spin interaction, the static properties of the molecule, such as the binding energy and the root of mean square radius (rms) etc., would be independent of the spin of the molecular state, which contradict with the experimental observations for deuteron.
As a result of the residual interaction between two mesons, at short distance the mesons may excite as they interact, and they could be virtually whatever the dynamics requires. This means that we need to consider the state mixing effect. It has been shown that the state mixing effect plays an important role in obtaining the phenomenologically required potential, when we study the nucleon-nucleon and nucleon-antinucleon interactions from the chiral soliton model Walet:1992gw (); Ding:2007xi (). The eigenvalue equation for the system is
where and are respectively the eigenvalue and the corresponding eigenfunction. If there is no residual interactions between and , the eigenfunction of the total system would simply be the product of meson’s wavefunction and meson’s. Consequently it is natural to expand the eigenfunction in terms of the model wavefunctions
where is the relative wavefunction between the mesons and , and denotes the intrinsic state of the two mesons, which will be mixed under the interaction . The wavefunction satisfies the Schrdinger equation , where depends on the relative coordinate , and similarly for . Inserting wavefunction into the eigenequation Eq.(10), multiplying by and integrating over the internal coordinates, we obtain
Where is the energy eigenvalue of channel . is the matrix element of the interaction potential , it is a function of the relative coordinate , the intrinsic coordinates and have been integrated out. There is clearly one equation for each state , and they are coupled each other by the terms on the right-hand side. It is important to notice that all the transitions represented by the right hand of Eq.(12) contribute coherently. If with , then the coupled channel Schrdinger equation Eq.(12) is reduced to the single channel Schrdinger equation
where is the effective interaction potential
Eq.(13) and Eq.(14) are exactly the results of the second order perturbation theory to deal with the state mixing effect, and this simplification is widely used Walet:1992gw (); Wong:2003xk (); Ding:2007xi (). However, if is rather small or of the same order comparing with , we have to solve the coupled channel Schrdinger equation exactly. Although in principle we should solve the infinite set of equations implied by Eq.(12), in practice we only need to concentrate on the nearly degenerate channels, which is a good approximation.
Iii Evaluation of the matrix element
For a system consisting of two mesons and with total angular momentum J and the third component , its wavefunction is written as
where , is the spin wavefunction, and represents the spatial wavefunction. , and denote respectively the spin, the orbital angular momentum and the total angular momentum of meson with similar notations for the meson . From Eq.(5) and Eq.(9), it is obvious that each term of can be factorized into the spatial and spin relevant part, consequently the interaction potential can be re-written as
where the superscript represents respectively the screened color Coulomb, screened linear, and spin-spin interactions for 1, 2, 3. Concretely, , , and the spatial part can be read from Eq.(9) straightforwardly. Therefore the matrix element is the sum of twelve terms, and each term is of the form
It is obvious that both the spatial matrix element and the spin matrix element are needed. Firstly we consider the spatial matrix element
In this work, the spatial wavefunctions are taken as the simple harmonic oscillator wavefunctions, which is a widely used approximation in the quark model calculations. The integral in Eq.(19) can be evaluated analytically in coordinate space following the procedures in Ref. Swanson:1992ec (). On the other hand this integration can be performed in momentum space as well, then the calculation will be greatly simplified Wong:2003xk (); Wong:2001td (),
Note that can be read from Eq.(7) directly. For the given quantum numbers , etc, the above integral can be straightforwardly calculated although it is somewhat lengthy, and the matrix elements involved in our calculation are listed in the Appendix.
Next we turn to the spin matrix element . We denote the spin of the constituents , , and by , , and respectively. In the present work, the constituent is quark or antiquark, consequently we have . We would like to recouple the constituents so that the spin operator (1,2,3) matrix elements can be easily calculated. We have
It is obvious that the matrix element of is
The matrix element of can be derived straightforwardly by using the recoupling formula Eq.(22). For or , the matrix element is given by
For or , the matrix element of is
Iv and molecular state
Because the mass of is close to the threshold and its width roughly is the same as that of , it is very likely that is a loosely bound molecular state. In this section we will dynamically study whether there exists molecule state consistent with . Since GeV, GeV and GeV pdg (), the masses of , and are close to each other. Under the residual interaction in Eq.(5) and Eq.(9), these three channels would be coupled with each other. However, the width of is very large MeV pdg (), consequently there should be very small component in the molecular state, otherwise it would decay so quickly that a weakly bound molecule can not form. As a result, we shall consider both and channels here, the effective interaction potential is induced by the pairwise interactions between quarks or antiquarks. Then we solve the corresponding two channels coupled Schrdinger equation to find whether there is bound state solutions, where we only concentrate on the lowest mass state.
The model parameters employed are GeV, GeV, GeV, , which is a set of fairly conventional quark model parameters. In Ref. Wong:2003xk () the screening mass is taken to be 0.28 GeV, which was found to be consistent with the string breaking mechanism and meanwhile give a good description of the charmonium masses Wong:1999ur (). The uncertainty of screening parameter would be considered in the following. Moreover, we use a running coupling constant , which is given by
with and GeV. Theoretical estimates for the harmonic oscillator parameter scatter in a relative large region 0.3-0.7 GeV. Many recent quark model studies of meson and baryon decays use a value of GeV Ackleh:1996yt (); Capstick:1993kb (), therefore we assume GeV in this work.
In the limit that the heavy quark mass becomes infinite, the properties of the meson are determined by the light quark. The light quark is characterized by their total angular momentum, , where is the light quark spin and is its orbital angular momentum. The prime superscript ( or ) is used for the state with , it is very broad. The unprimed state ( or ) is used for the state, it is rather narrow. Heavy-light mesons are not charge conjugation eigenstates and so mixing can occur among the states with the same . The two states and are coherent superposition of the quark model and states
Little is known about the mixing angle at present. In the heavy quark limit, the mixing angle is predicted to be or if the expectation value of the heavy quark spin-orbit interaction is positive or negative Godfrey:1986wj (). Since the former implies that the state mass is larger than that of the state, and this agrees with the current experiment data, we shall employ in the following. The above analysis applies to and mixing as well.
There are various methods of integrating the multichannel Schrdinger equation numerically. In this work we shall employ two packages MATSCS matlab () and FESSDE2.2 fessde () to perform the numerical calculations so that the results obtained by one program can be checked by another. The first package is a Matlab software, and the second is written in Fortran 77. Both packages can fastly and accurately solve the eigenvalue problem for systems of the coupled Schrdinger equations, and the results obtained by two codes are exactly the same within error.
Calculating the relevant matrix elements of the residual interaction following the methods outlined in section III, then we solve the coupled channel schrdinger equation numerically. The numerical results for the lowest energy states are listed in Table 2. We find that the bound state could exist for reasonable screening mass . The binding energy is found to decrease with , since a smaller gives a stronger potential at short distance, which is displayed in Fig. 2. With 0.28 GeV and 0.33 GeV, the bound state mass is about 4411.839 MeV and 4419.014 MeV respectively, and the root of mean square radius is 0.971 fm and 1.183 fm respectively, which are widely extended in space. Because the total angular momentum of S wave is 1, 2 or 3, it can not be 0. Hence will not mix with for state, then we only need to solve single channel Schrdinger equation in this case. Similar bound state solutions have been found for , and the binding energy is approximately the same as that of the case for the same value. Both the and bound states are widely extended, it is a good feature of molecular states. The wavefunctions for the two states with GeV are shown in Fig. 3. For , bound state solutions could be found only if the screening mass is smaller than 0.16 GeV, which is quite different from the favored value 0.28 GeV. Therefore we tend to conclude the molecule doesn’t exist. In short, both and bound states are predicted to exist in our model, whereas only molecule may exist in the and exchange model from the heavy quark effective theory Liu:2008xz ().
Since the production of is highly suppressed in B meson decay for , the quantum is favored if future experiments confirm as a loosely molecular state. Experimentally the mass and width of are fitted to be MeV and MeV respectively. Considering the large error in the width measurement and the theoretical uncertainties from the screening mass , as a molecular state can not be excluded. More precise measurements of its mass and width, partial wave analysis are important to understand the nature of . As is suggested in Ref. Bugg:2008wu (), it is highly desirable to use the full amplitude including both the production and the decay processes, in performing partial wave analysis to determine the spin-parity of . If or is favored by future partial wave analysis, the molecule hypothesis is strongly supported, otherwise it is not appropriate to interpret as a molecule.