Spectrum of the charmed and b-flavored mesons in the relativistic potential model

We study the bound states of heavy-light quark-antiquark system in the relativistic potential model, where the potential includes the long-distance confinement term, the short-distance Coulomb term and spin-dependent term. The spectrum of $B$, $B^*$, $D$, $D^*$ and states with higher orbital quantum numbers are obtained. Compared with previous results predicted in the relativistic potential model, the predictions are improved and extended in this work, more theoretical masses are predicted which can be tested in experiment in the future.


I Introduction
Bound state of heavy-light quark-antiquark system Qq is of special interest. Weak decays of such heavy-light system can be used to determine the fundamental parameters such as the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements in the standard model (SM), and to explore the source of CP violation. Experimental data from B factories have confirmed the existence of CP violation in B meson weak decays [1,2]. Theoretically to treat the weak decays of heavy-flavor meson B or D, effects of strong interaction have to be considered. Strong interactions in B decays can be separated into two parts. One part can be calculated perturbatively in QCD. The other part is dynamically non-perturbative. The binding effect in the quark-antiquark system is one of the main source of the non-perturbative dynamics. How to treat the binding effect in QCD is still an open question at present. Before the non-perturbative problem in QCD being solved, using phenomenological method to treat the bound state is an effective way in practice. The bound state of quarkantiquark can be described by the wave equation [3,4] with an effective potential compatible with QCD. The potential shows a linear confining behavior at large distance and a Coulombic behavior at short distance. Since the light quark in the heavy-light system is relativistic, the wave equation is assumed to be with a relativistic kinematics.
The bound state effects in B and D mesons have been investigated with the relativistic potential model previously in Refs. [5][6][7][8][9][10]. In these works the spin-dependent interactions are not included. For the heavy-light quarkantiquark system, the heavy quark can be viewed as a static color source in the rest frame of the meson, and the light quark is bound around the heavy quark by an effective potential. In the heavy quark limit the spin of the heavy quark decouples from the interaction [11][12][13][14][15][16][17][18], therefore the spin-dependent interactions can be treated as perturbative corrections. In this work we will extend the work of Ref. [10], the spin-dependent interactions are included. The spectrum of B, B * , D, D * and other heavy-light bound states with higher orbital angular momentum and higher radial quantum number are obtained, which are consistent with experimental data. There are also some predictions for the bound states waiting for experimental test.
The paper is organized as follows. In section II, the relativistic wave equation for the heavy-light quarkantiquark system and the effective Hamiltonian are given. In section III the wave equation is solved. Section IV is for the numerical result and discussion. Section V is a brief summary.

II The wave equation for heavy-light system and the effective Hamiltonian
The heavy flavor mesons B and D contain light quarks, which requires the wave equation describing the heavylight system include relativistic kinematics. The equation is a relativistic generalization of Schrödinger equation The effective Hamiltonian can be written as with where r = x 2 − x 1 , and x 1 and x 2 are the coordinates of the heavy and light quarks, respectively. The operators ∇ 2 1 and ∇ 2 2 involve partial derivatives relevant to the coordinates x 1 and x 2 , respectively. m 1 is the mass of the heavy quark, and m 2 the mass of the light antiquark. V (r) is the effective potential of the strong interaction between the heavy and light quarks. It can be taken as a combination of a Coulomb term and a linear confining term, whose behavior is compatible with QCD at both short-and long-distance [4,19,20] The first term is contributed by one-gluon-exchange diagram calculated in perturbative QCD. The Coulomb term dominates the behavior of the potential at shortdistance. The second term is the linear confining term. The third term c is a phenomenological constant, which is adjusted to give the correct ground state energy level of the quark-antiquark system. The running coupling constant in coordinate space α s (r) can be obtained from the coupling constant in momentum space α s (Q 2 ) by Fourior transformation. It can be written in the following form [4] where α i are free parameters fitted to make the behavior of the running coupling constant at short distance be consistent with the coupling constant in momentum space predicted by QCD. The numerical values of these parameters are α 1 = 0.15, α 2 = 0.15, α 3 = 0.20, and γ 1 = 1/2, γ 2 = √ 10/2, γ 2 = √ 1000/2. The second term H ′ in eq. (2) is the spin-dependent part of the Hamiltonian where H hyp is the spin-spin hyperfine interaction, and H so is spin-orbit interaction. The spin-spin hyperfine interaction used in this work is where the parameter σ is taken as quark mass-dependent [4] σ = σ 2 here σ 0 and s 0 are phenomenological parameters.
The spin-orbit interaction is where L = r× P is the relative orbital angular momentum between the quark and antiquark. The spin-dependent interactions can be predicted by one-gluon-exchange forces in QCD [3,4]. The exact form of ∝ δ( r) for the spin-spin contact term s 1 · s 2 and 1/m q for the tensor term in H hyp and spin-orbit interaction are the predictions of one-gluon-exchange calculation in the non-relativistic approximation. It is reasonable that there might be contributions of non-perturbative dynamics in the bound state system and relativistic corrections for the light quark. In this work the form of the spin-spin contact hyperfine interaction is replaced by an interaction with the behavior of exponential suppression e −σ 2 r 2 as in Ref. [21], and the mass of the light quark m 2 in the denominator is replaced by a parameterm 2 , which includes the relativistic corrections and the bound-state effect in the heavy meson.

III The solution of the wave equation
Without the spin-dependent interaction, the solutions of the wave equation for pseudoscalar and vector states of the quark-antiquark system shall be degenerate. The prediction to the masses of B and B * , D and D * will be the same. For the heavy quark and light antiquark system, the interaction decouples to the heavy quark spin in the heavy quark limit [11][12][13][14][15][16][17][18]. Quark-spin dependent interaction can be treated as perturbation. The masses of B and B * , D and D * measured in experiment support this treatment, the mass differences of B and B * , D and D * are only at the order of a few percent [22].
We solve the eigen equation of H 0 at first, then treat the spin-dependent Hamiltonian H ′ in the perturbation theory. The effect of H ′ will be considered to the first order in the perturbative expansion. Denote the eigenfunction and eigenvalue of the Hamiltonian H 0 by ψ (0) ( r) and E (0) , respectively, then the eigen equation To solve the above equation, we express the wave function in terms of spectrum integration With the above expression, the wave equation becomes The exponential e i k· r/hc can be decomposed in terms of spherical harmonics where j l is the spherical Bessel function, Y ln (r) is the spherical harmonics, andr the unit vector along the direction of r. The spherical harmonics satisfies the normalization condition Using eq. (14) and factorize the wave function into radial and angular parts then equation (13) is transformed to be Define a reduced radial wave function u l (r) by then the wave equation becomes As explained in Ref. [10], for a bound state of quark and antiquark, when the distance between them is large enough, the wave function will drop seriously. Eventually the wave function will effectively vanish at a typically large distance. We assume such a typical distance is L, then the quark and antiquark in bound state can be viewed as if they are restricted in a limited space, 0 < r < L. In the limited space the reduced wave function u l (r) for angular momentum l can be expanded in terms of the spherical Bessel function where c n 's are the expansion coefficients, a n the n-th root of the spherical Bessel function j l (a n ) = 0. In practice the above summation can be truncated to a large enough integer N u l (r) = N n=1 c n a n r L j l ( a n r L ).
In the limited space, the momentum k will be discrete. From the argument of j l ( anr L ) in eq. (20), one can see the relevance a n r L ⇐⇒ kr hc .
Then the momentum is discretized, and the integration over k in eq. (19) should be replaced by a summation where ∆a n = a n − a n−1 .
Considering the limited space 0 < r, r ′ < L, the discrete momentum of eq.(23), and substituting eq.(21) into eq. (19), and simplify it, one can finally obtain the equation for the coefficients c n 's N n=1 a n N 2 where N m is the module of the spherical Bessel function Equation (24) is the eigenstate equation in the matrix form. It can be reduced to eq.(17) in Ref. [10] for the case l = 0. It is not difficult to solve this equation numerically. The solution only slightly depends on the values of N and L if they are large enough. We find that when N > 50, L > 5 fm, the solution of the wave equation will be stationary.
Next we shall discuss the contribution of the spindependent interaction.
The spin-dependent interaction is considered perturbatively in the basis of the |JM, sl sectors. |JM, sl is the eigenvector of spin-independent Hamiltonian H 0 , where J is the total angular momentum of the bound state, M the magnetic quantum number, s the total spin of the quark and antiquark, l the relevant orbital angular momentum between them. The tensor part of the hyperfine interaction H hyp in eq.(7) does not conserve the orbital angular momentum, it causes mixing between the states with different orbital angular momenta 3 L J ↔ 3 L ′ J , while the spin-orbit interaction H so in eq.(10) does not conserve the total quark and antiquark spin, it can cause the mixing between the states with different total spin quantum numbers 1 L J ↔ 3 L J . The mass matrix elements are calculated perturbatively in the basis of |JM, sl . The matrix is then diagonalized to get the mixing eigenstates. The perturbative contribution of the spin-dependent Hamiltonian H ′ to the eigenvalues of the bound states are given below.
(1) The eigenvalue of pseudoscalar state The quantum number of the pseudoscalar state is J P = 0 − , the total spin and orbital angular momentum are s = 0, l = 0, i.e., it is 1 S 0 state. The eigenvalue of the pseudoscalar state is calculated to be (2) The mass matrix of the vector state, J P = 1 − Both s = 1, l = 0 and s = 1, l = 2 can construct J P = 1 − state. The 3 S 1 and 3 D 1 states can mix through the spin-orbit interaction. The basis for the mixing is denoted to be |ψ 1 = | 3 S 1 , and |ψ 2 = | 3 D 1 . The mass matrix can be written as the results of the matrix elements are Diagonalizing the matrix H, one can get the eigenvalues of the two mixing states and the mixing angle. With the matrix elements given in eq.s (28) ∼ (31), the above mixing matrix (27) can be easily extended to the cases with more | 3 S 1 and | 3 D 1 states mixing.
(3) The eigenvalue of the scalar state, J P = 0 + For the scalar state, J P = 0 + , the spin and orbital angular momentum are s = 1, l = 1. It is the 3 P 0 state. The eigenvalue of the scalar state is (4) The mass matrix of the axial-vector state, J P = 1 + The J P = 1 + state is mixture of 1 P 1 and 3 P 1 states, both states with s = 0, l = 1 and s = 1, l = 1 can construct the J P = 1 + state. The basis for the mixing is |ψ 1 = | 1 P 1 , and |ψ 2 = | 3 P 1 . The matrix elements of the mass matrix are (5) The mass matrix of the tensor state, J P = 2 + The J P = 2 + state is mixture of 3 P 2 and 3 F 2 states, both states with s = 1, l = 1 and s = 1, l = 2 can construct the J P = 2 + state. The basis for the mixing is |ψ 1 = | 3 P 2 , and |ψ 2 = | 3 F 2 . The matrix elements of the mass matrix are f (r) − 1 10 g(r) In the above equations, the functions f (r), g(r), h 1 (r) and h 2 (r) are defined as The value ofm 2 depends on the quark-antiquark system, which ism 75 GeV for (bq) system, 0.70 GeV for (bs) system, 0.40 GeV for (cq) system, 0.35 GeV for (cs) system, here q is the light quark u or d.
The solution of the wave equation does not depend on the values of L and N if they are taken large enough. Numerical calculation shows that the solution is stable when L > 5 fm, N > 50. Here we take L = 10 fm, N = 100.
The numerical results for (bq), (bs), (cq) and (cs) bound states with the component of radial quantum number n = 1 dominant are given in Table I. The theoreti-cal calculation can accommodate the experimental data well. The quantum states relevant to each meson is also given in this table. The vector meson states are generally mixing states of | 3 S 1 and | 3 D 1 . The components of |1 3 S 1 in B * , B * s , D * and D * s are overwhelmingly dominant, while the components of | 3 D 1 states are tiny. The masses of vector states with |1 3 D 1 component dominant have also been predicted, which are shown in Table I. TABLE I: Theoretical spectrum of (bq), (bs), (cq) and (cs) bound states mainly with the radial quantum number n = 1, and the comparison with experimental data. The data is quoted from the Particle Data Group [22].
The axial vector states with J P = 1 + found in experiment, such as B 1 (5721), B s1 (5830), D 1 (2420), can be explained as mixing states of | 1 P 1 and | 3 P 1 states. B 1 (5721) is dominantly |1 1 P 1 states, B s1 (5830) and D 1 (2420) are dominantly |1 3 P 2 . The predictions to the masses of D s1 (2460) and D s1 (2536) are slightly far away from experimental measurement, the deviation can be up to 70 ∼ 80 MeV. Within the error of 70 ∼ 80 MeV, D s1 (2460) can be explained as dominantly |1 1 P 1 state, and D s1 (2536) is dominantly |1 3 P 1 state. Finally the 2 + states can be explained as mixing states of | 3 P 2 and | 3 F 2 . The details can be found in Table I.
The radial excited states with the quantum number up to n = 2 are also predicted, they are given in Table II.
The wave function of each bound state can be obtained simultaneously when solving the wave equation, which is not given here explicitly. But it is easy to get the wave function when it is needed.

V Summary
The bound states of heavy-light quark and antiquark system are studied in the relativistic potential model. The dynamics of the light quark in the system requires the wave equation describing the bound state include relativistic kinematics. The potential is compatible with QCD, it shows the behavior of Coulomb potential at short distance, and a linear confining behavior at large distance. The spin-dependent interactions are also considered. The spectrums of B and D system are obtained. The masses of the bound states with the radial quantum number n = 1 are well consistent with the experimental measurement. The masses of bound states with n = 2 are predicted, which can be tested by experiment in the future. The wave function of each bound state can be also obtained by solving the wave equation. II: Theoretical spectrum of (bq), (bs), (cq) and (cs) bound states with the radial quantum number mainly n = 2, and the mixing.