Spectra of heavy–light mesons in a relativistic model
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Abstract
The spectra and wave functions of heavy–light mesons are calculated within a relativistic quark model which is based on a heavyquark expansion of the instantaneous Bethe–Salpeter equation by applying the Foldy–Wouthuysen transformation. The kernel we choose is the standard combination of linear scalar and Coulombic vector. The effective Hamiltonian for heavy–light quark–antiquark system is calculated up to order \(1/m_Q^2\). Our results are in good agreement with available experimental data except for the anomalous \(D_{s0}^*(2317)\) and \(D_{s1}(2460)\) states. The newly observed heavy–light meson states can be accommodated successfully in the relativistic quark model with their assignments presented. The \(D_{sJ}^*(2860)\) can be interpreted as the \(1^{3/2}D_1\rangle \) and \(1^{5/2}D_3\rangle \) states being members of the 1D family with \(J^P=1^\) and \(3^\).
1 Introduction
Great experimental progress has been achieved in studying the spectroscopy of heavy–light mesons in the last decades [1, 2, 3, 4, 5, 6, 7, 8]. In the charm sector several new excited charmed meson states were discovered in addition to the lowlying states. For \(D_J\) mesons, the excited resonances \(D(2740)^0\), \(D^*(2760)\) [1], \(D_J(2580)^0\), \(D^*(2650)\) and \(D^*(3000)\) [2] were found in the \(D^{(*)}\pi \) invariant mass spectrum by the BaBar and LHCb Collaborations. For \(D_{sJ}\) mesons, besides the wellestablished 1S and 1P charmedstrange states, the excited resonances \(D_{sJ}(2632)\) [3], \(D_{sJ}(2860)\) [4], \(D_{sJ}(2700)\) [5] and \(D_{sJ}(3040)\) [6] were observed in the \(D^{(*)}K\) invariant mass distribution by the two collaborations. In the bflavored meson sector, several excited states were studied in experiment as well as the ground B and \(B_s\) meson states [9]. The strangeless resonances \(B_J(5840)^0\) and \(B(5970)^0\) were found in the \(B\pi \) invariant mass spectrum by the LHCb and CDF Collaborations, respectively [10, 11]. The stranged \(B^*_{sJ}(5850)\) were observed in the \(B^{(*)}K\) invariant mass distribution by the OPAL Collaboration [12].
The heavy–light meson spectroscopy plays an important role in understanding the strong interactions between quark and antiquark. Meanwhile, it provides a powerful test of the various phenomenological quark models inspired by QCD. Heavy–light mesons have been investigated extensively in relativistic quark models [13, 14, 15, 16, 17, 18, 19], where many relativistic potential models are constructed by modifying or relativizing nonrelativistic quark potential models and additional phenomenological parameters are employed. For the heavy–light system one needs a model that can retain the relativistic effects of the light quark. In this work we resort to the originally relativistic Bethe–Salpeter equation [20]. The Bethe–Salpeter approach was widely used in studying mesons so as to embody the relativistic dynamics [21, 22, 23, 24, 25, 26]. It is rather difficult to solve the Bethe–Salpeter equation for meson states, especially when considering states with large angular momentum quantum number. In order to study the spectrum of heavy–light mesons systematically, we choose to reduce the Bethe–Salpeter equation in the first place.
In our previous work [27], we apply the instantaneous approximation and obtain an equation equivalent to the Bethe–Salpeter equation. The Hamiltonian for the heavy–light quark–antiquark system is expanded to order \(1/m_Q\) by applying the Foldy–Wouthuysen transformation to the equivalent equation. We find that the leading Hamiltonian is actually not Diraclike. The interaction we derive is essentially different from the Breit interaction [28, 29, 30]. In this paper we extend and improve our study of the spectrum of the heavy–light mesons D, \(D_{s}\), B and \(B_{s}\). The running of the coupling constant is considered. Moreover, the \(1/m_Q^2\) correction is calculated. Many papers have only considered the leading \(1/m_Q\) term in the heavyquark expansion [27, 31, 32, 33, 34]. Our calculation shows that the \(1/m_Q^2\) corrections to the masses of the mesons are around 50 MeV, which is too large to be neglected. The parameters in the equations are determined by fitting the masses of the 1S and 1P meson states presented by particle data group (PDG) [9], while the states beyond 1P are calculated as a prediction. We find that in the Bethe–Salpeter formalism the linear confining parameter, i.e. the string tension, actually depends on the masses of the constituent quark and antiquark in mesons. The large discrepancy between experimental data and our previous work is decreased in this work. The newly observed heavy–light meson states can be accommodated successfully in our predicted spectra.
This paper is organized as follows. In the next section, we have a brief review of the relativistic quark model. Section 3 is for the solution of the wave equation and the perturbative corrections. In Sect. 4 we have numerical results and discussions. The last section is for a brief summary.
2 The model
The instantaneous Bethe–Salpeter equation as an integral equation is equivalent to a less complicated differential equation shown in Eq. (7) but it is still difficult to solve. For heavy–light systems, the heavyquark effective theory is applied. It is reasonable to consider the heavyquark expansion, i.e. the \(1/m_Q\) expansion. One can reduce the equivalent eigenequation by calculating the interactions of the heavy–light quark–antiquark meson order by order.
3 Solution of the wave equation
In this section we solve the eigenequation of the leading order Hamiltonian \(H_0\) in Eq. (26). Before doing this we would like to discuss the properties of the solution of the eigenequation associated with \(H_0\).
Here we turn to discussing the perturbative corrections of \(H^\prime \) defined in Eq. (27). The perturbative term \(H^\prime \) does not commute with the standard operators introduced for the free Dirac Hamiltonian, but it still commutes with the total angular momentum operator \(\varvec{J}=\varvec{j}+\varvec{S}\) and the parity operator \(\mathcal {P}\) of the bound state.
4 Numerical results and discussions

When \(m_q\) is taken large enough the three schemes tend to give the same value for the energy gap \(\Delta E\). It indicates the equivalence of the three schemes when dealing with doubleheavy mesons.

In the region \(m_q < 1\, \mathrm{GeV}\), which is the case for heavy–light mesons, the three schemes give quite different values for the energy gap. It has the pattern \(\Delta E^{\mathrm{Schr}}> \Delta E^{\mathrm{Dirac}} > \Delta E^{BS}\). In order to give the same energy gap for a specific meson the confinement parameter should be chosen as \(b^{\mathrm{Schr}}< b^{\mathrm{Dirac}} < b^{BS}\). The literature supports this sequence. For instance, \(b^{\mathrm{Schr}}\) is taken as \(0.175\;\mathrm{GeV}^2\) [41], \(0.180\;\mathrm{GeV}^2\) [13], \(b^{\mathrm{Dirac}}\) is taken as \(0.257\;\mathrm{GeV}^2\) [34], \(0.309\;\mathrm{GeV}^2\) [28], while \(b^{BS}\) can be taken up to \(0.400\;\mathrm{GeV}^2\) in this work.

In the Schrödinger and Dirac schemes the energy gap changes slowly over \(m_q\). This is especially true when \(m_q\) is less than 1 GeV. \(\Delta E^{\mathrm{Schr}}\) and \(\Delta E^{\mathrm{Dirac}}\) can be viewed as constants. In the Bethe–Salpeter scheme, \(\Delta E^{BS}\) changes drastically over \(m_q\). From the experimental data we know that the \(\Delta E\) are not sensitive to their light quark masses. For example, the \(\Delta E\) for both D and \(D_s\) mesons are around \(0.7\;\mathrm{GeV}\). Thus \(b^{\mathrm{Schr}}\) and \(b^{\mathrm{Dirac}}\) can be taken as a constant, while \(b^{BS}\) varies with the quark mass.
Theoretical deviations from experimental data mainly occur in the \(D_{s}\) meson sector, specifically, the \(D_{{s0}}^*(2317)\) and \(D_{{s1}}(2460)\) resonances. Our calculations for the two resonances are about \(100\;\mathrm{GeV}\) higher than their masses measured in the experiment. The discrepancy may be ascribed to the instantaneous approximation, the naive assumption of the kernel or the \(\alpha ^2_\mathrm{s}(r)\) contributions, i.e. the loop corrections. However, it is more likely to find an explanation beyond the naive quark model [42]. The masses of the two resonances predicted by the constituentquark model are generally 100–200 MeV higher than experiments [33, 34, 43, 44, 45]. The mass of \(D_0^*\), \(2318 \pm 29\;\mathrm{MeV}\) is almost identical to the mass of \(D_{{s0}}^*\), \(2317.8\pm 0.6\;\mathrm{MeV}\). It cannot be explained in the conventional quark model if the difference between the two anomalous resonances in the model is merely their lightquark masses \(m_s\) and \(m_{u,d}\). In this work the confinement parameter b takes different values for different systems but it still is not capable to explain the small mass difference of the two resonances.
\(n^jL_J \)  Meson  \(\mathrm {This\, work}\)  Previous work [27]  \(\mathrm {Ref.}\) [39]  \(\mathrm {Ref.}\) [34]  

\(1^{1/2}S_0\)  D  \(1869.62\pm 0.15\)  1871  1859  1871  1868 
\(1^{1/2}S_1\)  \(D^*\)  \(2010.28\pm 0.13\)  2008  2026  2010  2005 
\(1^{1/2}P_0\)  \(D_0^*(2400)^0\)  \(2318 \pm 29 \)  2364  2357  2406  2377 
\(1^{1/2}P_1\)  2507  2529  2469  2490  
\(1^{3/2}P_1\)  \(D_1(2420)\)  \(2421.3\pm 0.6 \)  2415  2434  2426  2417 
\(1^{3/2}P_2\)  \(D_2^*(2460)\)  \(2464.4 \pm 1.9 \)  2460  2482  2460  2460 
\(1^{3/2}D_1\)  2836  2852  2788  2795  
\(1^{3/2}D_2\)  2881  2900  2850  2833  
\(1^{5/2}D_2\)  \(D_J(2740)^0\)  \(2737.0\pm 3.5\pm 11.2\)  2737  2728  2806  2775 
\(1^{5/2}D_3\)  \(D_J^*(2760)^0\)  \(2760.1\pm 1.1\pm 3.7\)  2753  2753  2863  2799 
\(1^{5/2}F_2\)  3122  3107  3090  3101  
\(1^{5/2}F_3\)  3139  3134  3145  3123  
\(1^{7/2}F_3\)  \(D^*_J(3000)^0\)  \(3008.1\pm 4.0\)  2980  2942  3129  3074 
\(2^{1/2}S_0\)  \(D_J(2580)^0\)  \(2579.5\pm 3.4\pm 5.5\)  2594  2575  2581  2589 
\(2^{1/2}S_1\)  \(D_J^*(2650)^0\)  \(2649.2\pm 3.5\pm 3.5\)  2672  2686  2632  2692 
\(2^{1/2}P_0\)  2895  2902  2919  2949  
\(2^{1/2}P_1\)  2983  2999  3021  3045  
\(2^{3/2}P_1\)  2926  2932  2932  2995  
\(2^{3/2}P_2\)  \(D_J(3000)^0\)  \(2971.8\pm 8.7\)  2965  2969  3012  3035 
\(2^{3/2}D_1\)  3230  3228  3228  
\(2^{3/2}D_2\)  3259  3260  3307  
\(2^{5/2}D_2\)  3159  3139  3259  
\(2^{5/2}D_3\)  3176  3160  3335  
\(2^{5/2}F_2\)  3455  3425  
\(2^{5/2}F_3\)  3465  3444  3551  
\(2^{7/2}F_3\)  3346  3301  
\(1^{1/2}S_0\)  \(D_s^\pm \)  \(1968.49\pm 0.32\)  1964  1949  1969  1965 
\(1^{1/2}S_1\)  \(D_s^{*\pm }\)  \(2112.3\pm 0.5\)  2107  2110  2111  2113 
\(1^{1/2}P_0\)  \(D_{s0}^*(2317)\)  \(2317.8\pm 0.6\)  2437  2412  2509  2487 
\(1^{1/2}P_1\)  \(D_{s1}(2536)\)  \(2535.12\pm 0.13\)  2558  2562  2574  2605 
\(1^{3/2}P_1\)  \(D_{s1}(2460)\)  \(2459.6\pm 0.6\)  2524  2528  2536  2535 
\(1^{3/2}P_2\)  \(D_{s2}^*(2573)\)  \(2571.9\pm 0.8\)  2570  2575  2571  2581 
\(1^{3/2}D_1\)  \(D_{s1}^*(2860)^\)  \(2859\pm 12\pm 6\pm 23\) [7]  2885  2873  2913  2913 
\(1^{3/2}D_2\)  2923  2916  2961  2953  
\(1^{5/2}D_2\)  2857  2829  2931  2900  
\(1^{5/2}D_3\)  \(D_{s3}^*(2860)^\)  \(2860.5\pm 2.6\pm 2.5\pm 6.0\) [7]  2871  2852  2971  2925 
\(1^{5/2}F_2\)  3172  3128  3230  3224  
\(1^{5/2}F_3\)  3184  3152  3266  3247  
\(1^{7/2}F_3\)  3107  3049  3254  3203  
\(2^{1/2}S_0\)  \(D_{sJ}(2632)\)  \(2632.5\pm 1.7\) [3]  2647  2624  2688  2700 
\(2^{1/2}S_1\)  \(D_{s1}^*(2710)\)  \(2708\pm 9^{+11}_{10}\) [5]  2734  2729  2731  2806 
\(2^{1/2}P_0\)  2945  2918  3054  3067  
\(2^{1/2}P_1\)  \(D_{sJ}(3040)\)  \(3044\pm 8^{+30}_{5}\) [6]  3028  3017  3154  3165 
\(2^{3/2}P_1\)  3009  2994  3067  3114  
\(2^{3/2}P_2\)  3047  3031  3142  3157  
\(2^{3/2}D_1\)  3277  3247  3383  
\(2^{3/2}D_2\)  3305  3278  3456  
\(2^{5/2}D_2\)  3260  3217  3403  
\(2^{5/2}D_3\)  3274  3237  3469  
\(2^{5/2}F_2\)  3508  3449  
\(2^{5/2}F_3\)  3517  3468  3710  
\(2^{7/2}F_3\)  3459  3390 
\(n^jL_J \)  Meson  \(E_\mathrm {expt.}\) [9]  \(\mathrm {This \,work}\)  Previous work [27]  \(\mathrm {Ref.}\) [39]  \(\mathrm {Ref.}\) [34] 

\(1^{1/2}S_0\)  B  \(5279.25\pm 0.17\)  5273  5262  5280  5279 
\(1^{1/2}S_1\)  \(B^*\)  \(5325.2\pm 0.4\)  5329  5330  5326  5324 
\(1^{1/2}P_0\)  5776  5740  5749  5706  
\(1^{1/2}P_1\)  5837  5812  5774  5742  
\(1^{3/2}P_1\)  \(B_1(5721)\)  \(5723.5\pm 2.0\)  5719  5736  5723  5700 
\(1^{3/2}P_2\)  \(B_2^*(5747)\)  \(5743\pm 5\)  5739  5754  5741  5714 
\(1^{3/2}D_1\)  6143  6128  6119  6025  
\(1^{3/2}D_2\)  6165  6147  6121  6037  
\(1^{5/2}D_2\)  5993  5989  6103  5985  
\(1^{5/2}D_3\)  6004  5998  6091  5993  
\(1^{5/2}F_2\)  6379  6344  6412  6264  
\(1^{5/2}F_3\)  6391  6354  6420  6271  
\(1^{7/2}F_3\)  6202  6175  6391  6220  
\(2^{1/2}S_0\)  5957  5915  5890  5886  
\(2^{1/2}S_1\)  5997  5959  5906  5920  
\(2^{1/2}P_0\)  6270  6211  6221  6163  
\(2^{1/2}P_1\)  6301  6249  6281  6194  
\(2^{3/2}P_1\)  6216  6189  6209  6175  
\(2^{3/2}P_2\)  6232  6200  6260  6188  
\(2^{3/2}D_1\)  6514  6458  6534  
\(2^{3/2}D_2\)  6527  6471  6554  
\(2^{5/2}D_2\)  6401  6357  6528  
\(2^{5/2}D_3\)  6411  6365  6542  
\(2^{5/2}F_2\)  6692  6621  
\(2^{5/2}F_3\)  6700  6629  6786  
\(2^{7/2}F_3\)  6553  6493  
\(1^{1/2}S_0\)  \(B_s\)  \(5366.77\pm 0.24\)  5363  5337  5372  5373 
\(1^{1/2}S_1\)  \(B_s^*\)  \(5415.4^{+2.4}_{2.1} \)  5419  5405  5414  5421 
\(1^{1/2}P_0\)  5811  5776  5833  5804  
\(1^{1/2}P_1\)  5864  5841  5865  5842  
\(1^{3/2}P_1\)  \(B_{s1}(5830)\)  \(5829.4\pm 0.7\)  5819  5824  5831  5805 
\(1^{3/2}P_2\)  \(B_{s2}^*(5840)\)  \(5839.7\pm 0.6\)  5838  5843  5842  5820 
\(1^{3/2}D_1\)  6167  6146  6209  6127  
\(1^{3/2}D_2\)  6186  6163  6218  6140  
\(1^{5/2}D_2\)  6098  6085  6189  6095  
\(1^{5/2}D_3\)  6109  6094  6191  6103  
\(1^{5/2}F_2\)  6405  6363  6501  6369  
\(1^{5/2}F_3\)  6416  6373  6515  6376  
\(1^{7/2}F_3\)  6313  6276  6468  6332  
\(2^{1/2}S_0\)  6010  5961  5976  5985  
\(2^{1/2}S_1\)  6048  6003  5992  6019  
\(2^{1/2}P_0\)  6291  6227  6318  6264  
\(2^{1/2}P_1\)  6323  6266  6345  6296  
\(2^{3/2}P_1\)  6288  6249  6321  6278  
\(2^{3/2}P_2\)  6304  6263  6359  6292  
\(2^{3/2}D_1\)  6540  6478  6629  
\(2^{3/2}D_2\)  6553  6491  6651  
\(2^{5/2}D_2\)  6487  6434  6625  
\(2^{5/2}D_3\)  6496  6441  6637  
\(2^{5/2}F_2\)  6723  6647  
\(2^{5/2}F_3\)  6731  6654  6880  
\(2^{7/2}F_3\)  6650  6580 
As for the \(D_s\) mesons, several states beyond the 1P state have been observed. Their masses and identifications are presented in the lower part of Table 1. Recently, the LHCb Collaboration identified \(D_{sJ}^*(2860)\) as an admixture of two resonances: \(D_{s3}^*(2860)^\) and \(D_{s1}^*(2860)^\) [7, 8], with their masses measured as \(2859\pm 12\pm 6\pm 23\;\text{ MeV }\) and \(2860.5\pm 2.6\pm 2.5\pm 6.0\;\text{ MeV }\), respectively. In Refs. [34, 39] cited in Table 1, their predictions do not favor this identification, with their calculations generally 60 MeV higher than the measured masses. While our results for both \(1^{3/2}D_1\rangle \) and \(1^{5/2}D_3\rangle \) are around 2860 MeV, the two resonances can be interpreted as members of the 1D family with \(J^P=1^\) and \(3^\). The resonances \(D_{sJ}(2632)\) , \(D_{s1}^*(2710)\) and \(D_{sJ}(3040)\) are identified as radially exited states with \(n=2\) in our model. The \(D_{sJ}(2632)\) was firstly observed by SELEX Collaboration at a mass of \(2632.5\pm 1.7\) MeV, it can be assigned as the \(2^{1/2}S_0\rangle \). The assignment for \(D_{s1}^*(2710)\) is proposed as \(J^P=1^\) in Refs. [54, 55], which agree with our prediction as our calculated mass for \(2^{1/2}S_1\rangle \) is close to its experimental mass \(2708\pm 9^{+11}_{10}\) MeV [5]. The \(D_{sJ}(3040)\) resonance is observed in the \(D^*K\) mass spectrum at a mass of \(3044\pm 8_{5}^{+30}\) MeV by the BABAR Collaboration [6]. Here we assign it as \(2^{1/2}P_1\rangle \) in our predicted \(D_s\) meson spectrum.
5 Summary
The spectra of heavy–light mesons are restudied in a relativistic model, which is derived by reducing the instantaneous Bethe–Salpeter equation. The kernel is chosen to be the standard combination of linear scalar and Coulombic vector. By applying the Foldy–Wouthuysen transformation on the heavy quark, the Hamiltonian for the heavy–light quark–antiquark system is calculated up to order \(1/m_Q^2\). We find that in the framework of an instantaneous Bethe–Salpeter equation the string tension b in the confinement potential is sensitive to the masses of the constituent quarks in the meson. The spectra of the D, \(D_s\), B and \(B_s\) mesons are calculated in the relativistic model. Most of the heavy–light meson states can be accommodated successfully in our model except for the anomalous \(D_{s0}^*(2317)\) and \(D_{s1}(2460)\) resonances. In the Bethe–Salpeter formalism, the assumption of the interaction kernel for mesons is rather a priori; kernels with other spin structures can also be studied. In this work, we only restrict our calculations to the spectra of heavy–light mesons. With the wave functions obtained when solving the wave equation, B and D decays can be studied in further research.
Notes
Acknowledgements
This work is supported in part by the National Natural Science Foundation of China (Grants No. 11375208, No. 11521505, No. 11235005 and No. 11621131001).
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