The recoil correction and spinorbit force for the possible and states
Abstract
In the framework of the oneboson exchange model, we have calculated the effective potentials between two heavy mesons and from the t and uchannel , , ,  and meson exchanges. We keep the recoil corrections to the and systems up to , which turns out to be important for the very loosely bound molecular states. Our numerical results show that the momentumrelated corrections are favorable to the formation of the molecular states in the , in the and systems.
pacs:
13.75.n, 13.75.Cs, 14.20.GkI Introduction
A lot of charmoniumlike states have been reported in the past decade by the experiment collaborations such as Belle, , CDF, D0, LHCb, BESIII, and CLEOc. The underlying structures of many charmoniumlike states are not very clear. Sometimes they are called as XYZ states. They decay into conventional chromium, but not all of them can be accommodated into the quarkmodel charmonium spectrum. The neutral XYZ states include Choi:2003 , B. Aubert:2005 , C.Z. Yuan:2007 , B. Aubert:2007 , X.L. Wang:2007 , and G.Pakhlova:2008 etc. There are also many charged charmoniumlike states such as and R.Mizuk:2008 , Choi:2008 ; K.Chilikin:2013 , Ablikim:2013 ; Z.Q. Liu:20132 ; T.Xiao:2013 , M. Ablikim:2013 , Ablikim:2014 . The charged bottomoniumlike states and were observed by Belle Collaboration I.Adachi:2011 .
Theoretical speculations of these XYZ states include the hybrid meson ZHUPLB2005 , tetraquark states H.Hogaasen:2006 ; D.Ebert:2006 ; N.Barnea:2006 ; Y.Cui:2007 ; R.D.Matheus:2007 ; T.W.Chiu:2007 ; L.Zhao:2014 , dynamically generated resonance D.Gamermann:2007 and molecular states F.E.Close:2004 ; M.B.Voloshin:2004 ; C.Y.Wong:2004 ; E.S.Swanson:2004 ; N.A.Tornqvist:2004 ; Y.R.Liu:2010 ; N.Li:2012 ; mali ; liuxiaohai etc. Since many of these XYZ states are close to the thresholds of a pair of charmed or bottom mesons, the molecular hypothesis seems a natural picture for some of these states.
Within the framework of the molecular states, there exist extensive investigations of the charged and states Q.Wang:2013 ; Oset ; Y.R.Liu:2008 ; X.Liu:2009 ; A.E.Bondar:2011 ; L.Xiang:2011 ; W.Chen:2014 ; J.He:2013 ; Z.F.Sun:2012 ; C.D. Deng:2014 ; J.M. Dias:2014 . In our previous work L.Zhao:2013 , we explored the possibility of as the isovector molecule partner of and considered the recoil correction and the spinorbit force in the and system very carefully.
Although there exist quite a few literatures on the possibility of as the molecular state and as the molecular state, most of the available investigations are either based on the heavy quark spinflavor symmetry or derived in the . In other words, the recoil correction and the spinorbit interaction have not been investigated for the and systems. Since the binding energies of these system are very small, the high order recoil correction and the spinorbit interaction may lead to significant corrections.
In this work, we will go one step further. We will consider the recoil correction and the spinorbit force for the and systems. With the onebosonexchange model (OBE), we will derive the effective potential with the relativistic Lagrangian and keep the momentum related terms explicitly in order to derive the recoil correction and the spinorbit interaction up to , where is the mass of the heavy meson. We investigate the system with for , and the system with for . For completeness, we also investigate the system and system with other quantum numbers: , , and . Compared to the case, the expressions of the recoil corrections and spin orbit force are more complicated. There appear several new structures. For some systems, the numerical results show that the high order correction is important for the loosely bound heavymeson states.
Ii The effective potential
A. Wave function, Effective Lagrangian and Coupling constants
First, we construct the flavor wave functions of the isovector and isoscalar molecular states composed of the and as in Refs. Y.R.Liu:2008 ; X.Liu:2009 . The flavor wave function of the system reads
(1) 
(2) 
For the system
(3) 
(4) 
The meson exchange Feynman diagrams for the and systems at the tree level is shown in Fig. 1.
Based on the chiral symmetry, the Lagrangian for the pseudoscalar, scalar and vector meson interaction with the heavy flavor mesons reads
(5)  
(6)  
(7)  
(8)  
(9) 
(10) 
where the heavy flavor meson fields and represent or and or . Its corresponding heavy antimeson fields and represent or and or . , represent the the exchanged pseudoscalar and vector meson matrices. is the only scalar meson interacting with the heavy flavor meson.
(11) 
(12) 
According to the OBE model, five mesons ( , , , and ) contribute to the effective potential. For the and systems, the potentials are the same for the three isovector states in Eqs. (1)(4) with the exact isospin symmetry. Expanding the Lagrangian densities in Eqs. (5)(10) leads to each meson’s contribution for this channel. These channeldependent coefficients are listed in Table 1.
The pionic coupling constant is extracted from the width of S.Ahmed:2001 . =132 MeV is the pion decay constant. According the vector meson dominance mechanism, the parameters and can be determined as and . At the same time, by matching the form factor obtained from the light cone sum rule and that calculated from the lattice QCD, we can get GeV C.Isola:2003 ; M.Bando:1988 . The coupling constant related to the scalar meson exchange is with X.Liu:2009 ; A.F.Falk:1992 . All these parameters are listed in Table 2.
isospin  mesonexchange  

1/2  1/2  1  
3/2  1/2  1  
1/2  1/2  1  
3/2  1/2  1 
mass(MeV)  coupling constants  
pseudoscalar  


vector  




scalar  


heavy flavor  
In order to include all the momentumrelated terms in our calculation, we introduce the polarization vector of the vector mesons. At the rest frame we have
(13) 
We make a lorentz boost to Eq. 13 to derive the polarization vector in the laboratory frame
(14) 
where is the particle’s 4momentum in the laboratory frame and is the mass of the particle.
B. Effective potential
With the wave function and Feynman diagram, we can derive the relativistic scattering amplitude at the tree level
(15) 
where the Tmatrix is the interaction part of the Smatrix and is defined as the invariant matrix element. After applying Bonn approximation to the LippmannSchwinger equation, the Smatrix reads
(16) 
with being the effective potential. Considering the different normalization conventions used for the scattering amplitude , matrix and , we have
(17) 
where denotes the four momentum of the final (initial) state.
During our calculation, and denote the four momenta of the initial states in the center mass system, while and denote the four momenta of the final states, respectively.
(18) 
is the transferred four momentum or the four momentum of the meson propagator. For convenience, we always use
(19) 
and
(20) 
instead of and in the practical calculation.
In the OBE model, a form factor is introduced at each vertex to suppress the high momentum contribution. We take the conventional form for the form factor as in the Bonn potential model.
(21) 
is the mass of the exchanged meson and is the mass of the heavy flavor meson or . So far, the effective potential is derived in the momentum space. In order to solve the time independent Schrödinger equation in the coordinate space, we need to make the Fourier transformation to . The details of the Fourier transformations are presented in the Appendix.
The expressions of the potential through exchanging the , mesons are
(22)  
(23)  
The and meson exchange potentials have the same form except that the meson mass and channeldependent coefficients are different.
The expression of the potential through exchanging the meson is
(24) 
where and have the form
(25) 
(26) 
Compared to the case, there appear several new interaction operators: , , and . These operator represent the new form of the tensor, spinspin and spinorbit interactions.
Similarly, the and meson exchange potential has the same form in the and system except the meson mass and channeldependent coefficients. The explicit forms of ,,, are shown in the Appendix.
In our calculation, we explicitly consider the external momentum of the initial and final states. Due to the recoil corrections, several new terms appear which were omitted in the heavy quark symmetry limit. These momentum dependent terms are related to the momentum :
(27) 
and
(28) 
The term in Eq. (28) is the wellknown spin orbit force. In short, all the terms in the effective potentials in the form of , , etc with the subindices arise from the recoil corrections and vanish when the heavy meson mass goes to infinity. The recoil correction and the spin orbit force appear at .
C. Schrödinger equation
With the effective potential in Eqs. (23) (27), we are able to study the binding property of the system by solving the Schrödinger Equation
(29) 
where is the total wave function of the system. The total spin of the system and the orbital angular momenta and . Thus the wave function should have the following form
(30) 
where and are the wave and wave functions, respectively. We use the same matrix method in Ref. L.Zhao:2013 to solve this SD wave couplechannel equation.
We detach the terms related to the kineticenergyoperator from and rewrite Eq. (29) as
(31) 
with
(32) 
in which is
(33)  
The total Hamiltonian contains three angular momentum related operators , , , , , which corresponds to the spinspin interaction, spin orbit force and tensor force respectively. They act on the S and Dwave coupled wave functions and split the total effective potential into the subpotentials , , and . The matrix form reads
(34)  
(35) 
(36)  
(37) 
(38)  
Iii Numerical Results for the system
We diagonalize the Hamiltonian matrix to obtain the eigenvalue and eigenvector. If there exists a negative eigenvalue, there exists a bound state. The corresponding eigenvector is the wave function. We use the variation principle to solve the equation. We change the variable parameter to get the lowest eigenvalue. We also change the number of the basis functions to reach a stable result.
iii.1
Since the mass of the charged bottomoniumlike state is close to the system, we first consider the possibility of the molecule with , . In order to reflect the recoil correction of the momentumrelated terms, we plot the effective potential of the Swave and Dwave with or without the momentumrelated terms in Fig. 2. and are the effective potentials of the wave and wave interactions after adding the momentumrelated terms. and are the effective potentials of the wave and wave interactions without the momentumrelated terms. Fig. 2 C corresponds to the , system, where the curves of and , and and are almost overlapping. In other words, the recoil correction is small.
We collect the numerical results in Table 3. and are the eigenenergy of Hamiltonian with and without the momentumrelated terms, respectively. The fourth, fifth and sixth column represent the contribution of wave, wave, and spinorbit force components, respectively. The last column is the mass of as a molecular state of , . When the cut off lies within GeV, there exists a boundstate. The binding energy with the recoil correction is between MeV. The binding energy without the recoil correction is between MeV. When the cutoff parameter GeV, the binding energy is MeV, and the recoil correction is only MeV. The contribution from the spinorbit force is as small as MeV. When the cutoff parameter GeV, the binding energy is MeV, and the recoil correction is MeV. The correspondence spinorbit force contribution is MeV, which is also small compared with the binding energy. The recoil correction and the contribution of the spinorbit are very small. However the recoil correction is favorable to the formation of the molecular state with , .
(GeV)  Eigenvalue  Mass  

total  S  D  LS  (MeV)  
2.2  0.97  13.15  0.15  0.001  10649.03  
0.94  12.91  0.15    10649.06  
2.4  3.49  28.68  0.32  0.004  10646.51  
3.43  28.27  0.32    10646.57  
2.6  8.04  49.81  0.56  0.01  10641.96  
7.94  49.19  0.55    10642.06  
2.8  15.15  77.53  0.88  0.02  10634.85  
14.98  76.66  0.87    10635.02 
From Fig 3 C, it is clear that the exchange is much more important than the other mesonexchanges. Considering that the coupling constant is extracted from the decay width with some uncertainty, we multiply by a factor from to to check the dependence of the binding energy on this parameter. The numerical results are listed in Table 4. The binding energy with the recoil correction varies from MeV. The binding energy without the recoil correction varies from MeV. The binding energy is sensitive to the coupling constant.
(GeV)  Eigenvalue  Mass  

(MeV)  
  
  