# Remarks on the $S$-wave masses of singly heavy mesons

Jisi Pan<EMAIL_ADDRESS>School of Mechanical and Electrical
Engineering, Guangxi Science $\&$ Technology Normal University, Laibin 546199,
China  Ji-Hai Pan111Corresponding author<EMAIL_ADDRESS>College of
Mathematics and Physics, Liuzhou Institute of Technology, Liuzhou 545000,
China

###### Abstract

Based on the study of the string model method of singly heavy mesons and
singly heavy baryons, we calculate the mass spectrum of $1S$-and $2S$-wave for
both charm and bottom mesons$(D/D_{s},B/B_{s})$. Experimentally, there are
most masses spectra of $1S$-wave have been found, while the masses part of the
$2S$-state is not determined. In this paper, we will use singly light quark or
diquark model images and Regge trajectory models, combined with perturbation
processing methods, to analyze and study the observed singly heavy mesons,
further predict the unobserved mesons masses and their corresponding spin-
parity quantum numbers.

##  I introduction

The $D/D_{s}$ mesons with charm quark and the $B/B_{s}$ mesons with bottom
quark are typical heavy-light mesons, which is structurally analogous to a
hydrogen-like atoms( the singly light antiquark and the heavy quark resemble
the extranuclear electron and the proton, respectively). In recent decades,
more and more singly heavy (SH) mesons states $Q\bar{q}$ $(Q=c,b;\
\bar{q}=\bar{n},\bar{s})$ have been discovered by BaBar, Belleale, CLEO, and
the LHCb experiment Tanabashi:PP88 , so the study of heavy-light mesons system
has been attracting great attention. It can be seen from PDG Olive:PP88  that
the experimental values of some low energy $D$, $D_{s}$ mesons Olive:PP888
have been basically determined. Since the experimental observation of the
singly charm mesons states $D(2550)$, $D^{\ast}(2600)$, $D(2750)$and
$D^{\ast}(2760)$, there have been different research methods Moir:PP88 ;
Lewis:PP88 ; Godfrey:PP88 ; Godfrey:PP888 ; JiaD:PP888 ; JiaDH:PP888 ;
Canonical:PP888  for the calculation and analysis of these charm mesons
states. For example, ${}^{3}P_{0}$ model Sun:PP88 ; GodfreyMoats:PP88 ;
TianZhang:PP88 , Chiral quark model Zhong:PP88 , lattice QCD model Moir:PP88 ;
Lewis:PP88 ; MoirRTh:PP88 , other models EbertFaustov:PP888 ; AsgharM:PP888 ;
GodfreySw:PP888 ; PanWang:PP888 ; MasudSwa:PP888 , etc. In addition, the high
energy $B$, $B_{s}$ mesons ZengVan:PP888 ; LahdeN:PP888 ; PierroE:PP888 ;
Chen:PP88 ; Chen:PP888  have been extensively studied, there are different
interpretations for the bottom meson states, such as $B_{J}(5840)$,
$B^{\ast}_{J}(5970)$ Zylae:PP888 ; Aaije:PP888 ; AsgharMasud:PP888 ;
YuWang:PP888 ; ChenLuo:PP888 ; FengHao:PP888 . So far, these quantum states
are still controversial and need to be further confirmed by experiments.

In this paper, The purpose of this work is to calculate the masses of $1S$-and
$2S$-wave $D/D_{s}$ and $B/B_{s}$ mesons states from the linear Regge
trajectory (spin independent mass) formula and the spin dependent potential
(spin dependent mass) with the corresponding spin-parity quantum numbers
$J^{P}=0^{-}$ or $1^{-}$, respectively. In Table I, we list the heavy quark
and singly light quark masses of charm and bottom mesons, and the string
tensions $\alpha(Q\bar{q})$.

Table 1: The effective masses(GeV) of quarks determined by Regge trajectory and the relativistic quark model are compared, with $\alpha$ in $GeV^{-2}$.  Parameters | ${\small M}_{c}$ | ${\small M}_{b}$ | ${\small m}_{n}$ | ${\small m}_{s}$ |  ${\small\alpha}$(${\small c\bar{n}}$) |  ${\small\alpha}$(${\small c\bar{s}}$) | ${\small\alpha(b\bar{n})}$ | ${\small\alpha(b\bar{s})}$
---|---|---|---|---|---|---|---|---
Ref.Jia:PP88  | ${\small 1.44}$ | ${\small 4.48}$ | ${\small 0.23}$ | ${\small 0.328}$ | ${\small 0.223}$ | ${\small 0.249}$ | ${\small 0.275}$ | ${\small 0.313}$
EFG EFG:C10  | ${\small 1.55}$ | ${\small 4.88}$ | ${\small 0.33}$ | ${\small 0.5}$ |  ${\small 0.64}$/${\small 0.58}$ | ${\small 0.68/0.64}$ | ${\small 1.25/1.21}$ | ${\small 1.28/1.23}$

This paper is organized as follows. We analyze the Regge trajectory formula to
give $S$-wave spin-average masses in Sec. II. In Sec. III, we review about the
spin-dependent Hamiltonian. In Sec. IV, we talk about the scaling relations.
In Sec. V, we mainly employ the Breit-Fermi spin interaction to calculate
their spin coupling parameters and wave functions . We present conclusions in
Sec. VI.

##  II The Regge trajectory and the spin average masses

Considering the color interaction between quarks and quarks in the heavy-light
mesons system, in order to estimate the masses splitting in orbitally excited
charm and bottom mesons , we use Regge-like masses relation ChenDLM:PP88  to
comprehensively analyze the whole singly heavy system of mesons,

$(\bar{M}_{L}-M_{Q})^{2}=\alpha\pi L+a_{0},$ (1)

where, $\bar{M}_{L}$, $M_{Q}$ are the spin-average mass and the heavy quark
mass of the singly heavy meson, respectively. $L$ is the orbital angular
momentum of the mesons system($L=0,1,2,\cdot\cdot\cdot$). $\alpha$ is the QCD
string tension coefficient between heavy quark and light antiquark. The
intercept factor $a_{0}$ depends on the light antiquark mass $m_{\bar{q}}$ and
the non-relativistic kinematic energy $P^{2}_{Q}/M_{Q}$ of the heavy quark,

$a_{0}=(m_{\bar{q}}+\frac{P^{2}_{Q}}{M_{Q}})^{2},$ (2)

note that non-relativistic kinematic $3-$momentum of heavy quark in Eq. (2)
has been associated with both $M_{Q}$ and $v_{Q}$,

$\displaystyle P_{Q}\equiv M_{Q}v_{Q}\ ,\ \ \ \ \ \ \ \
v_{Q}=(1-\frac{m_{bareQ}^{2}}{M_{Q}^{2}})^{\frac{1}{2}},$ (3)

here, $v_{Q}$ is the velocity of the heavy quark, and the $3-$momentum $P_{Q}$
is conserved in the heavy quark limit of $M_{Q}\rightarrow\infty$. Using Eqs.
(1),(2) and (3),one can obtain the spin-averaged masses Jia:PP88 ;
JiaPan:PP888  by

$\displaystyle\bar{M}_{L}$ $\displaystyle=$ $\displaystyle
M_{Q}+\sqrt{\alpha\pi
L+\left(m_{\bar{q}}+M_{Q}\left(1-\frac{m_{bareQ}^{2}}{M_{Q}^{2}}\right)\right)^{2}}\text{,}$
(4)

where, $m_{bareQ}$ and $m_{\bar{q}}$ are the heavy quark bare mass of singly
heavy meson and the singly light antiquark mass, respectively. The selection
of these parameters is listed in Table I.

For the singly heavy mesons, it is interesting to note that in JiaDH:PP888
this Regge trajectory is proposed in consideration that the trajectory slope
ratio between the radial and angular-momentum Regge trajectories to be
$\pi/2:1$. Then,$\pi\alpha L$ in Eq. (4) is replaced by
$\pi\alpha(L+\frac{\pi}{2}n)$,

$\displaystyle\bar{M}_{L}$ $\displaystyle=$ $\displaystyle
M_{Q}+\sqrt{\alpha\pi(L+\frac{\pi}{2}n)+\left(m_{\bar{q}}+M_{Q}\left(1-\frac{m_{bareQ}^{2}}{M_{Q}^{2}}\right)\right)^{2}}\text{,}$
(5)

where, $n$ is the radial quantum number. By analogy of the heavy meson system,
we estimate the S-wave the spin-average mass of the $Q(\bar{q})$ state using
Eq. (5). For instance, we get the $1S$-wave spin-average mass $(L=0,n=0)$ in
Eq. (5) is
$\bar{M}_{L}=M_{Q}+\left(m_{\bar{q}}+M_{Q}\left(1-\frac{m_{bareQ}^{2}}{M_{Q}^{2}}\right)\right)$,
and the $2S$-wave spin-average mass $(L=0,n=1)$ is
$\bar{M}_{L}=M_{Q}+\sqrt{\alpha\pi(\pi/2)+\left(m_{\bar{q}}+M_{Q}\left(1-\frac{m_{bareQ}^{2}}{M_{Q}^{2}}\right)\right)^{2}}$.
Given trajectory parameters in Table I, one can predict Regge trajectories of
the charm excited mesons $D/D_{s}$ in FIG. 1-2 and the bottom excited mesons
$B/B_{s}$ in FIG. 3-4 with the radial quantum number $n=0$, $1$, $2$, $3$ and
$4$ .

Figure 1: $D$ meson spin-average mass

Figure 2: $D_{s}$ meson spin-average mass

Figure 3: $B$ meson spin-average mass

Figure 4: $B_{s}$ meson spin-average mass

Regge trajectories of the heavy-light hadron systems relating the shifted
spin-averaged mass squared to the orbital angular momentum $L$ of the systems,
with the parameters in Table I corresponding to Eq. (5). The red solid circles
correspond to the observed (mean) masses, and the empty circles indicate the
predicted value in FIG. $1-4$.

##  III The spin-dependent potential

Due to the spin-spin interaction between singly light antiquark and heavy
quark of singly heavy mesons, in order to estimate the mass splitting, we
consider the spin-dependent Hamiltonian $H^{SD}$ EbertFGS:PP888 ;
KarlinerRP:PP888  ,

$H^{SD}=a_{1}\mathbf{L}\cdot\mathbf{S}_{\bar{q}}+a_{2}\mathbf{L}\cdot\mathbf{S}_{Q}+bS_{12}+c\mathbf{S}_{\bar{q}}\cdot\mathbf{S}_{Q},$
(6)
$S_{12}=3(\mathbf{S}_{\bar{q}}\cdot\mathbf{\hat{r}})(\mathbf{S}_{Q}\cdot\mathbf{\hat{r}})/r^{2}-\mathbf{S}_{\bar{q}}\cdot\mathbf{S}_{Q}\mathbf{,}$

where, the first two terms are spin-orbit interactions, the third is the
tensor energy, and the last is the contact interaction between the heavy quark
spin $\mathbf{S}_{Q}$ and the antiquark spin $\mathbf{S}_{\bar{q}}$. Here,
$a_{1}$, $a_{2}$, $b$, $c$ are spin coupling parameters.

For the singly heavy mesons, the singly light antiquark spin and the heavy
quark spin are ${S}_{\bar{q}}=\frac{1}{2}$, ${S}_{Q}=\frac{1}{2}$,
respectively. Therefore, there are two kinds of the total spin $S$, one is 0
and the other is 1. In the scheme of $LS$ coupling, note that the total
angular momentum $J=S+L$. Coupling of $L=0$ with the spin $S=0$ gives states
with the total angular momentum $J=0$, while coupling with $S=1$ leads to
states the angular momentum $J=1$.

We consider the $S$-wave($L=0$) states in $Q\bar{q}$ mesons case. Then, the
first three terms of in Eq. (6) are eliminated, only the last term survives,

$H^{SD}=c\mathbf{S}_{\bar{q}}\cdot\mathbf{S}_{Q}.$ (7)

It is very convenient to analyze spin-spin interaction into the non-trivial
terms for the mass splitting, the eigenvalues (two diagonal elements) of
$<\mathbf{S}_{\bar{q}}\cdot\mathbf{S}_{Q}>$ can be obtained,

$<\mathbf{S}_{\bar{q}}\cdot\mathbf{S}_{Q}>=[S(S+1)-S_{Q}(S_{Q}+1)-S_{\bar{q}}(S_{\bar{q}}+1)]/2,$
(8)
$<\mathbf{S}_{\bar{q}}\cdot\mathbf{S}_{Q}>=\left[\begin{array}[]{cc}-\frac{3}{4}&0\\\
0&\frac{1}{4}\end{array}\right],$ (9)

combining with Eqs. (5) and (9), the $S$-wave masses are,

$M(S)=\bar{M}_{L}+c\left[\begin{array}[]{cc}-\frac{3}{4}&0\\\
0&\frac{1}{4}\end{array}\right].$ (10)

Using the above Eq.(10), we can calculate the mass spectrum of the charm and
the bottom mesons states $(D/D_{s},B/B_{s})$. There is much discussion about
the hyperfine splitting terms, and here we mainly determine the parameter
values and then derive the effective mass of the mesons.

##  IV The hyperfine splitting parameters

Figure 5: $D$ meson radial wavefunction

Figure 6: $D_{s}$ meson radial wavefunction

Figure 7: $B$ meson radial wavefunction

Figure 8: $B_{s}$ meson radial wavefunction

It is possible to apply the charm and bottom mesons masses of
$M(D,1S)=1869.59MeV$, $M(D^{\ast},1S)=2010.26MeV$ and $M(D_{s},1S)=1968.3MeV$,
$M(D_{s}^{\ast},1S)=2112.2MeV$ Tanabashi:PP88 . A linear relationship between
them is their spin-weighted sum:
$\mathop{\sum}\limits_{J}(2J+1)\Delta\bar{M}=0$,

$\displaystyle\bar{M}_{D,1S}$ $\displaystyle=$
$\displaystyle\frac{1869.59MeV+3\times 2010.26MeV}{4}$ (11) $\displaystyle=$
$\displaystyle 1975.09MeV,$ $\displaystyle\bar{M}_{D_{s},1S}$ $\displaystyle=$
$\displaystyle\frac{1968.3MeV+3\times 2112.2MeV}{4}$ (12) $\displaystyle=$
$\displaystyle 2076.23MeV,$

the spin-averaged masses calculated by Eq. (5) are nearly equal to the results
of the experimental value, namely, Eq. (11) and Eq. (12), respectively. In
addition, the mass splitting are

$\displaystyle c(D,1S)$ $\displaystyle=$ $\displaystyle
M(1^{3}S_{1},1^{-})-M(1^{1}S_{0},0^{-})$ (13) $\displaystyle=$ $\displaystyle
140.6MeV,$ $\displaystyle c(D_{s},1S)$ $\displaystyle=$ $\displaystyle
M(1^{3}S_{1},1^{-})-M(1^{1}S_{0},0^{-})$ (14) $\displaystyle=$ $\displaystyle
143.9MeV.$

From Eqs. (13) and (14), it is assumed that $c(D,1S)\approx c(D_{s},1S)$ is
about $140MeV$ in the charm meson of the $1S$-wave. However, one can use a
similar approach in the $1S$-wave bottom mesons $(B/B_{s})$, corresponding to
the scaling relations from the charm meson $(c\bar{n})$ and then to the bottom
meson $(b\bar{s})$, as well as the following rough estimate of the parameter
$c$,

$c(b\bar{s})\approx c(b\bar{n}).$ (15)

To estimate the mass of the bottom flavor meson with vector singly light
quarks (spin $S_{\bar{q}}=1/2$), we use the spin coupling parameters obtained
from experimental measurements $D$ and $D_{s}$ mesons values, which should
theoretically have an approximate mapping with the following scaling
relationship,

$c(b\bar{q})=\frac{M_{c}}{M_{b}}c(c\bar{q}),$ (16)

with the heavy quarks relation $M_{c}/M_{b}$ for the parameter of
$c(B,1S),c(B_{s},1S)$ are about 45 MeV. We will show the results of our
calculations, as compared with the experimental values, with reasonable values
of the parameters achieved.

##  V Spin-spin couplings in relativized quark model

Table 2: Model Parameters(in GeV).  $\begin{array}[]{ccccccccc}\hline\cr\hline\cr{\small State}&\alpha_{s}&\sigma&c_{1}&c_{2}&c_{3}&c_{4}&c_{5}\\\ \hline\cr\small D&0.564&0.230&0.143&0.089&0.049&0.022&0.005\\\ \small D_{s}&0.537&0.260&0.144&0.100&0.048&0.012&0.003\\\ \small B&0.532&0.216&0.041&0.027&0.016&0.007&0.002\\\ \small B_{s}&0.526&0.200&0.048&0.038&0.022&0.006&0.001\\\ \hline\cr\hline\cr\end{array}$ Table 3: The S-wave mass spectrum (MeV) of the charmed and bottomed mesons are given and compared with the different quark models. State $J^{P}$ | Meson | Mass | Ours | GI GodfreySw:PP888 ; GodfreyMo:D85  | EFG EFG:C10  | Meson | Mass | Ours | GI GodfreySw:PP888 ; GodfreyMo:D85  | EFG EFG:C10
---|---|---|---|---|---|---|---|---|---|---
$\begin{array}[]{rr}{\small 1}^{1}{\small S}_{0}&{\small 0}^{-}\\\ {\small 1}^{3}{\small S}_{1}&{\small 1}^{-}\end{array}$ | $\begin{array}[]{r}{\small D}^{\pm}\\\ {\small D}^{\ast}\end{array}$ | $\begin{array}[]{r}{\small 1869.59}\\\ {\small 2010.26}\end{array}$ | $\begin{array}[]{r}{\small 1870.86}\\\ {\small 2014.13}\end{array}$ | $\begin{array}[]{r}{\small 1877}\\\ {\small 2041}\end{array}$ | $\begin{array}[]{r}{\small 1871}\\\ {\small 2010}\end{array}$ | $\begin{array}[]{r}{\small D}_{s}\\\ {\small D}_{s}^{\ast}\end{array}$ | $\begin{array}[]{r}{\small 1968.3}\\\ {\small 2112.2}\end{array}$ | $\begin{array}[]{r}{\small 1968.18}\\\ {\small 2112.35}\end{array}$ | $\begin{array}[]{r}{\small 1979}\\\ {\small 2129}\end{array}$ | $\begin{array}[]{r}{\small 1969}\\\ {\small 2111}\end{array}$
$\par\begin{array}[]{rr}{\small 2}^{1}{\small S}_{0}&{\small 0}^{-}\\\ {\small 2}^{3}{\small S}_{1}&{\small 1}^{-}\end{array}$ | $\begin{array}[]{r}{\small D}_{0}\\\ {\small D}^{\ast}\end{array}$ | $\begin{array}[]{r}{\small 2549\pm 16}\\\ {\small 2637\pm 2}\end{array}$ | $\begin{array}[]{r}{\small 2551.58}\\\ {\small 2640.86}\end{array}$ | $\begin{array}[]{r}{\small 2581}\\\ {\small 2643}\end{array}$ | $\begin{array}[]{r}{\small 2581}\\\ {\small 2632}\end{array}$ | $\begin{array}[]{r}{\small}\hfil\\\ {\small D}^{\ast}_{s1}\end{array}$ | $\begin{array}[]{r}{\small}\hfil\\\ {\small 2708.3}\end{array}$ | $\begin{array}[]{r}{\small 2642.42}\\\ {\small 2742.72}\end{array}$ | $\begin{array}[]{r}{\small 2673}\\\ {\small 2732}\end{array}$ | $\begin{array}[]{r}{\small 2688}\\\ {\small 2731}\end{array}$
$\par\begin{array}[]{rr}{\small 3}^{1}{\small S}_{0}&{\small 0}^{-}\\\ {\small 3}^{3}{\small S}_{1}&{\small 1}^{-}\end{array}$ | $\begin{array}[]{r}{\small}\hfil\\\ {\small}\hfil\end{array}$ | $\begin{array}[]{r}{\small}\hfil\\\ {\small}\hfil\end{array}$ | $\begin{array}[]{r}{\small 2981.16}\\\ {\small 3029.66}\end{array}$ | $\begin{array}[]{r}{\small 3068}\\\ {\small 3110}\end{array}$ | $\begin{array}[]{r}{\small 3062}\\\ {\small 3096}\end{array}$ | $\begin{array}[]{r}{\small}\hfil\\\ {\small}\hfil\end{array}$ | $\begin{array}[]{r}{\small}\hfil\\\ {\small}\hfil\end{array}$ | $\begin{array}[]{r}{\small 3094.93}\\\ {\small 3143.35}\end{array}$ | $\begin{array}[]{r}{\small 3154}\\\ {\small 3193}\end{array}$ | $\begin{array}[]{r}{\small 3219}\\\ {\small 3242}\end{array}$
$\par\begin{array}[]{rr}{\small 4}^{1}{\small S}_{0}&{\small 0}^{-}\\\ {\small 4}^{3}{\small S}_{1}&{\small 1}^{-}\end{array}$ | $\begin{array}[]{r}{\small}\hfil\\\ {\small}\hfil\end{array}$ | $\begin{array}[]{r}{\small}\hfil\\\ {\small}\hfil\end{array}$ | $\begin{array}[]{r}{\small 3318.03}\\\ {\small 3339.75}\end{array}$ | $\begin{array}[]{r}{\small 3468}\\\ {\small 3497}\end{array}$ | $\begin{array}[]{r}{\small 3452}\\\ {\small 3482}\end{array}$ | $\begin{array}[]{r}{\small}\hfil\\\ {\small}\hfil\end{array}$ | $\begin{array}[]{r}{\small}\hfil\\\ {\small}\hfil\end{array}$ | $\begin{array}[]{r}{\small 3453.16}\\\ {\small 3464.92}\end{array}$ | $\begin{array}[]{r}{\small 3547}\\\ {\small 3575}\end{array}$ | $\begin{array}[]{r}{\small 3652}\\\ {\small 3669}\end{array}$
$\par\begin{array}[]{rr}{\small 5}^{1}{\small S}_{0}&{\small 0}^{-}\\\ {\small 5}^{3}{\small S}_{1}&{\small 1}^{-}\end{array}$ | $\begin{array}[]{r}{\small}\hfil\\\ {\small}\hfil\end{array}$ | $\begin{array}[]{r}{\small}\hfil\\\ {\small}\hfil\end{array}$ | $\begin{array}[]{r}{\small 3601.57}\\\ {\small 3606.49}\end{array}$ | $\begin{array}[]{r}{\small 3814}\\\ {\small 3837}\end{array}$ | $\begin{array}[]{r}{\small 3793}\\\ {\small 3822}\end{array}$ | $\begin{array}[]{r}{\small}\hfil\\\ {\small}\hfil\end{array}$ | $\begin{array}[]{r}{\small}\hfil\\\ {\small}\hfil\end{array}$ | $\begin{array}[]{r}{\small 3743.86}\\\ {\small 3746.42}\end{array}$ | $\begin{array}[]{r}{\small 3894}\\\ {\small 3912}\end{array}$ | $\begin{array}[]{r}{\small 4033}\\\ {\small 4048}\end{array}$
$\begin{array}[]{rr}{\small 1}^{1}{\small S}_{0}&{\small 0}^{-}\\\ {\small 1}^{3}{\small S}_{1}&{\small 1}^{-}\end{array}$ | $\begin{array}[]{r}B^{0}\\\ B^{\ast}\end{array}$ | $\begin{array}[]{r}{\small 5279.6}\\\ {\small 5324.7}\end{array}$ | $\begin{array}[]{r}{\small 5279.17}\\\ {\small 5320.27}\end{array}$ | $\begin{array}[]{r}{\small 5312}\\\ {\small 5371}\end{array}$ | $\begin{array}[]{r}{\small 5280}\\\ {\small 5326}\end{array}$ | $\begin{array}[]{r}{\small B}_{s}\\\ {\small B}_{s}^{\ast}\end{array}$ | $\begin{array}[]{r}{\small 5366.9}\\\ {\small 5414.4}\end{array}$ | $\begin{array}[]{r}{\small 5366.58}\\\ {\small 5414.15}\end{array}$ | $\begin{array}[]{r}{\small 5394}\\\ {\small 5450}\end{array}$ | $\begin{array}[]{r}{\small 5372}\\\ {\small 5414}\end{array}$
$\par\begin{array}[]{rr}{\small 2}^{1}{\small S}_{0}&{\small 0}^{-}\\\ {\small 2}^{3}{\small S}_{1}&{\small 1}^{-}\end{array}$ | $\begin{array}[]{r}B_{J}\\\ B_{J}\end{array}$ | $\begin{array}[]{r}{\small 5851\pm 19}\\\ {\small 5964\pm 5}\end{array}$ | $\begin{array}[]{r}{\small 5892.94}\\\ {\small 5920.12}\end{array}$ | $\begin{array}[]{r}{\small 5904}\\\ {\small 5933}\end{array}$ | $\begin{array}[]{r}{\small 5890}\\\ {\small 5906}\end{array}$ |  | $\begin{array}[]{r}{\small}\hfil\\\ {\small}\hfil\end{array}$ | $\begin{array}[]{r}{\small 6000.83}\\\ {\small 6039.26}\end{array}$ | $\begin{array}[]{r}{\small 5984}\\\ {\small 6012}\end{array}$ | $\begin{array}[]{r}{\small 5976}\\\ {\small 5992}\end{array}$
$\par\begin{array}[]{rr}{\small 3}^{1}{\small S}_{0}&{\small 0}^{-}\\\ {\small 3}^{3}{\small S}_{1}&{\small 1}^{-}\end{array}$ |  | $\begin{array}[]{r}{\small}\hfil\\\ {\small}\hfil\end{array}$ | $\begin{array}[]{r}{\small 6316.81}\\\ {\small 6332.52}\end{array}$ | $\begin{array}[]{r}{\small 6335}\\\ {\small 6355}\end{array}$ | $\begin{array}[]{r}{\small 6379}\\\ {\small 6387}\end{array}$ |  | $\begin{array}[]{r}{\small}\hfil\\\ {\small}\hfil\end{array}$ | $\begin{array}[]{r}{\small 6450.78}\\\ {\small 6473.15}\end{array}$ | $\begin{array}[]{r}{\small 6410}\\\ {\small 6429}\end{array}$ | $\begin{array}[]{r}{\small 6467}\\\ {\small 6475}\end{array}\par$
$\par\begin{array}[]{rr}{\small 4}^{1}{\small S}_{0}&{\small 0}^{-}\\\ {\small 4}^{3}{\small S}_{1}&{\small 1}^{-}\end{array}$ |  | $\begin{array}[]{r}{\small}\hfil\\\ {\small}\hfil\end{array}$ | $\begin{array}[]{r}{\small 6661.00}\\\ {\small 6667.82}\end{array}$ | $\begin{array}[]{r}{\small 6689}\\\ {\small 6703}\end{array}$ | $\begin{array}[]{r}{\small 6781}\\\ {\small 6786}\end{array}$ |  | $\begin{array}[]{r}{\small}\hfil\\\ {\small}\hfil\end{array}$ | $\begin{array}[]{r}{\small 6820.55}\\\ {\small 6826.35}\end{array}$ | $\begin{array}[]{r}{\small 6759}\\\ {\small 6773}\end{array}$ | $\begin{array}[]{r}{\small 6874}\\\ {\small 6879}\end{array}$
$\par\begin{array}[]{rr}{\small 5}^{1}{\small S}_{0}&{\small 0}^{-}\\\ {\small 5}^{3}{\small S}_{1}&{\small 1}^{-}\end{array}$ |  | $\begin{array}[]{r}{\small}\hfil\\\ {\small}\hfil\end{array}$ | $\begin{array}[]{r}{\small 6956.72}\\\ {\small 6958.36}\end{array}$ | $\begin{array}[]{r}{\small 6997}\\\ {\small 7008}\end{array}$ | $\begin{array}[]{r}{\small 7129}\\\ {\small 7133}\end{array}$ |  | $\begin{array}[]{r}{\small}\hfil\\\ {\small}\hfil\end{array}$ | $\begin{array}[]{r}{\small 7133.48}\\\ {\small 7134.81}\end{array}$ | $\begin{array}[]{r}{\small 7063}\\\ {\small 7076}\end{array}$ | $\begin{array}[]{r}{\small 7231}\\\ {\small 7235}\end{array}$

In this section, we mainly analyze the mass splitting of the singly heavy
meson in a relativistic quark model, in which the Breit-Fermi spin interaction
is used to calculate the spin coupling parameter $c$ and compare it with the
experimentally matched ones.

For this, we consider the quasi-static potential of the Brett-Fermi spin
interaction Godfrey:PP88 ; Yndurain83  is

$\begin{array}[]{r}V^{\text{quasi-
static}}_{L}=V(r)+S(r)+\left(\frac{V^{\prime}-S^{\prime}}{r}\right)\mathbf{L}\cdot\left(\frac{\mathbf{S}_{1}}{2m_{\bar{q}}^{2}}+\frac{\mathbf{S}_{2}}{2M_{Q}^{2}}\right)\\\
+\left(\frac{V^{\prime}}{r}\right)\mathbf{L}\cdot\left(\frac{\mathbf{S}_{1}+\mathbf{S}_{2}}{m_{\bar{q}}M_{Q}}\right)+\frac{1}{3m_{\bar{q}}M_{Q}}\left(\frac{V^{\prime}}{r}-V^{\prime\prime}\right)S_{12}\\\
+\frac{2}{3m_{\bar{q}}M_{Q}}\left(\mathbf{\nabla}^{2}V(r)\right)\mathbf{S}_{1}\cdot\mathbf{S}_{2},\end{array}$
(17)

here, $V$ and $S$ are the respective vector and scalar potentials, and
$V^{\prime}$, $S^{\prime}$ and $V^{\prime\prime}$ their derivatives. In
$S$-wave$(L=0)$, the $\mathbf{L}\cdot\mathbf{S}$ terms will be eliminated, the
spin-spin interaction of the last term contains a delta function in
configuration space and is treated nonperturbatively in Eq. (17),

$\displaystyle V^{\text{quasi-
static}}_{L=0}=-\frac{4\alpha_{s}}{3r}+br+\frac{32\alpha_{s}\pi}{9m_{\overline{q}}M_{Q}}\tilde{\delta}(r)\mathbf{S}_{1}\cdot\mathbf{S}_{2},$
(18)

where $\alpha_{s}$ is the strong coupling and $b$ is a parameter, one finds

$c=\frac{32\alpha_{s}\pi}{9m_{\overline{q}}M_{Q}}\langle\tilde{\delta}(r)\rangle.$
(19)

In Ref. Barnes:D85 ,
$\tilde{\delta}(r)=(\frac{\sigma}{\sqrt{\pi}})^{3}e^{-\sigma^{2}r^{2}}$, which
is a Gaussian-smeared term with a parameter $\sigma$ and the quantum average
$\langle\rangle$ is made over the S-wave wavefunction $R_{nL}(r)$ of the
meson($Q\bar{q}$) system. Then, the S-wave wavefunction $R_{nL}(r)$ with
radial quantum number $n=0,1,2,3,4$ and $L=0$ of the mesons are shown in
Figure $5-8$. The S-wave radial wavefunction $R_{nL}(r)$ has nodes.

In order to get the masses of the S-wave meson system, there are the second
parameters $(\alpha_{s},\sigma)$ are determined by fitting the mass spectrum,
and the values of the parameters $c_{1}$, $c_{2}$, $c_{3}$, $c_{4}$ and
$c_{5}$ obtained by fitting with Eq. (19) correspond to $1S$, $2S$, $3S$, $4S$
and $5S$ waves in Table II, respectively. The fitting result of parameters $c$
is consistent with the value calculated by Eqs. (15) and (16).

With the help of the relation Eq. (19) for fitting the $S$-wave masses, the
results are shown in the Table III. One can find that our results for excited
charm mesons $D_{0}(2550)$ and $D^{\ast}(2640)$ are considered as the
candidate of the 2S-wave, while our model calculations for the experimentally
determined $M(D_{s1}^{\ast}(2700)=2708.3MeV$ meson is slightly $40MeV$ higher.
On the other hand, according to the LHCb Collaboration’s suggestion
Aaije:PP888 , the $B_{J}(5840)$ and $B_{J}(5970)$ can be ${\small
2}^{1}{\small S}_{0}$ and ${\small 2}^{3}{\small S}_{1}$ states, respectively,
the mass splitting is about $110MeV$, which appears relatively large compared
to our value of $27MeV$ in Table II.

##  VI Summary

In this paper, one can based on studying the singly heavy mesons $(D/D_{s})$
and $(B/B_{s})$ of the quark and antiquark bound state system. We mainly
combine the linear Regge trajectories and the relativistic quark model,
discuss the similarity scaling relationship of the spin coupling parameters,
and then revisit the 2S-wave and 3S-wave mass spectrums of the spin parity
$J^{P}$ = $0^{-}$ or $1^{-}$, and predict the higher energy $S$-wave states
mass spectrum of mesons.

Therefore, comparing the experimental data of heavy-light mesons with the
predictions of existing theoretical models will provide favorable evidence for
the internal interaction and the internal structure of the mesons.

## References

  * (1) M. Tanabashi et al. [Particle Data Group], Review of Particle Physics, Phys. Rev. D 98, no. 3, 030001(2018).
  * (2) K. A. Olive et al (Particle Data Group Collaboration). Chin. Phys. C40, 100001(2016).
  * (3) K. A. Olive et al (Particle Data Group Collaboration). Review of Particle Physics[J]. Chin. Phys. C, 38: 090001(2014).
  * (4) G. Moir, M. Peardon, S. M. Ryan, C. E. Thomas and L. Liu, Excited spectroscopy of charmed mesons from lattice QCD, JHEP 1305, 021 (2013). [arXiv:1301.7670 [hep-ph]].
  * (5) R. Lewis and R. M. Woloshyn, $S$ and $P$-wave heavy-light mesons in lattice NRQCD Phys. Rev. D 62, 114507 (2000). [arXiv:0003011 [hep-ph]].
  * (6) S. Godfrey and N. Isgur, Mesons in a Relativized Quark Model with Chromodynamics, Phys. Rev. D 32, 189-231(1985).
  * (7) S. Godfrey, I. T. Jardine, Comment on the nature of the $D_{s1}(2710)$ and $D_{sJ}(2860)$ mesons, Phys Rev D, 89(7): 074023(2014). [arXiv:1312.6181 [hep-ph]].
  * (8) Duojie Jia and W-C Dong, Regge-like spectra of excited singly heavy mesons, Eur. Phys. J. Plus 134 (2019)123. [arXiv:1811.04214[hep-ph]].
  * (9) Duojie Jia, W-C. Dong and Atsushi Hosaka, Regge-like spectra of singly heavy hadrons, in Proc. 8th Int. Conf. Quarks and Nuclear Physics (QNP2018), JPS Conf. Proc. 26, 022021 (2019).
  * (10) O. Lakhina and E. S. Swanson, A Canonical $D_{s}(2317)?$, Phys.Lett.B650:159-165(2007). [arXiv:0608011 [hep-ph]].
  * (11) Z. F. Sun, J. S. Yu, X. Liu, and T. Matsuki, Newly observed $D(2550)$, $D(2610)$, and $D(2760)$ as $2S$ and $1D$ charmed mesons Phys. Rev. D 82, 111501(R) (2010). [arXiv:1008.3120 [hep-ph]].
  * (12) S. Godfrey and K. Moats, Properties of excited charm and charm-strange mesons, Phys. Rev. D 93, 034035(2016). [arXiv:1510.08305 [hep-ph]].
  * (13) Yu Tian, Ze Zhao, Ailin Zhang, Study of radially excited $D_{s}(2^{1}S_{0})$ and $D^{+}_{sJ}(2632)$, Chinese Phys. C 41 (2017)083107. [arXiv:1701.05678 [hep-ph]].
  * (14) X. H. Zhong, Strong decays of the newly observed $D(2550)$, $D(2600)$, $D(2750)$, and $D(2760)$, Phys. Rev. D 82, 114014 (2010). [arXiv:1009.0359 [hep-ph]].
  * (15) G. Moir, M. Peardon, S. M. Ryan, C. E. Thomas. Excited $D$ and $D_{s}$ meson spectroscopy from lattice QCD. [arXiv:1301.3397 [hep-lat]].
  * (16) D. Ebert, R. N. Faustov, V. O. Galkin, Strong decays of vector mesons to pseudoscalar mesons in the relativistic quark model, Physics Letters, B 1, 744(2015). [arXiv:1412.4534 [hep-ph]].
  * (17) I. Asghar, B. Masud, E. S. Swanson, F. Akram and M. Atif Sultan, Decays and spectrum of bottom and bottom strange mesons, Eur. Phys. J. A 54 no.7, 127(2018). [arXiv:1804.08802 [hep-ph]].
  * (18) S. Godfrey, K. Moats and E. S. Swanson, $B$ and $B_{s}$ Meson Spectroscopy, Phys. Rev. D 94, 054025(2016). [arXiv:1607.02169 [hep-ph]].
  * (19) Q. F. Lü, T. T. Pan, Y. Y. Wang, E. Wang, and D. M. Li, Excited bottom and bottom-strange mesons in the quark model, Phys. Rev. D 94, 074012 (2016). [arXiv:1607.02812 [hep-ph]].
  * (20) I. Asghar, B. Masud, E. S. Swanson, F. Akram and M. Atif Sultan, Decays and spectrum of bottom and bottom strange mesons, Eur. Phys. J. A 54 no.7, 127(2018). [arXiv:1804.08802 [hep-ph]].
  * (21) J. Zeng, J. W. Van Orden and W. Roberts, Heavy mesons in a relativistic model, Phys. Rev. D 52, 5229-5241(1995). [arXiv:9412269 [hep-ph]].
  * (22) T. A. Lahde, C. J. Nyfalt and D. O. Riska, Spectra and M1 decay widths of heavy light mesons, Nucl. Phys. A 674, 141-167(2000). [arXiv:9908485 [hep-ph]].
  * (23) M. Di Pierro and E. Eichten, Excited Heavy-Light Systems and Hadronic Transitions, Phys. Rev. D 64, 114004(2001). [arXiv:0104208 [hep-ph]].
  * (24) B. Chen, Deng-Xia. Wang and Ailin. Zhang, Interpretation of $D_{sJ}(2632)^{+}$, $D_{s1}(2700)^{\pm}$, $D_{sJ}(2860)^{+}$ and $D_{sJ}(3040)^{+}$, Phys. Rev. D80, 071502(2009). [arXiv:0908.3261 [hep-ph]].
  * (25) B. Chen, L. Yuan, Ailin. Zhang. Possible $2S$ and $1D$ charmed and charmed-strange mesons. Phys Rev D, 83.114025(2011). [arXiv:1102.4142 [hep-ph]].
  * (26) P. A. Zyla et al. [Particle Data Group], Review of Particle Physics, PTEP 2020 no.8, 083C01(2020).
  * (27) R. Aaij et al. [LHCb], Precise measurements of the properties of the $B_{1}(5721)^{0,+}$ and $B_{2}^{\ast}(5747)^{0,+}$ states and observation of $B^{+,0}$, $\pi^{-,+}$ mass structures, JHEP 04, 024(2015). arXiv:1502.02638 [hep-ph].
  * (28) I. Asghar, B. Masud, E. S. Swanson, F. Akram and M. Atif Sultan, Decays and spectrum of bottom and bottom strange mesons, Eur. Phys. J. A 54 (2018) no.7, 127. [arXiv:1804.08802 [hep-ph]].
  * (29) G. L. Yu and Z. G. Wang, The analysis of the excited bottom and bottom-strange states $B_{1}(5721)$, $B_{2}^{\ast}(5747)$, $B_{s1}(5830)$, $B_{2}^{\ast}(5840)$, $B_{J}(5840)$ and $B_{J}(5970)$ in $B$ meson family, Chin. Phys. C 44, 033103(2020). [arXiv:1907.08760 [hep-ph]].
  * (30) Bing Chen, Si-Qiang Luo, Ke-Wei Wei, Xiang Liu, $b$-hadron spectroscopy study based on the similarity of double bottom baryon and bottom meson, Phys. Rev. D 105, 074014 (2022). [arXiv:2201.05728 [hep-ph]].
  * (31) Xue-Chao Feng, Wei-Hao, li-Juan Liu, The assignments of the bottom mesons within the screened potential model and ${}^{3}P_{0}$ model. [arXiv:2205.07169 [hep-ph]].
  * (32) Duojie Jia, Wennian Liu and A. Hosaka, Regge behaviors in orbitally excited spectroscopy of charmed and bottom baryons, Phys.Rev. D101, 034016(2020). [arXiv:1907.04958 [hep-ph]].
  * (33) D. Ebert, R. N. Faustov and V. O. Galkin, Heavy-light meson spectroscopy and Regge trajectories in the relativistic quark model, Eur. Phys. J. C 66 (2010) 197. [arXiv:0910.5612 [hep-ph]].
  * (34) K. Chen, Y. Dong, X. Liu, Q. F. Lu and T. Matsuki, Regge-like relation and a universal description of heavy-light systems, Eur. Phys. J. C 78, no. 1, 20 (2018). [arXiv:1709.07196 [hep-ph]].
  * (35) Duojie Jia, Ji-Hai Pan, C-Q Pang. A mixing coupling scheme for spectra of singly heavy baryons with spin-1 diquarks in P-waves, Euro. Phys. J. C 81, 434(2021). [arXiv:2007.01545 [hep-ph]].
  * (36) D. Ebert, R. N. Faustov and V. O. Galkin, Spectroscopy and Regge trajectories of heavy baryons in the relativistic quark-diquark picture, Phys. Rev. D 84 (2011) 014025. [arXiv:1105.0583[hep-ph]].
  * (37) M. Karliner and J. L. Rosner, Prospects for observing the lowest-lying odd-parity $\Sigma_{c}$ and $\Sigma_{b}$ baryons, Phys. Rev. D92, 074026 (2015). [arXiv:1506.01702 [hep-ph]].
  * (38) F. J. Ynduráin, The theory of quark and gluon interactions, (Springer-Verlag, NY,1983).
  * (39) T. Barnes, S. Godfrey and E. S. Swanson, Higher charmonia, Phys. Rev. D 72, 054026 (2005). [arXiv:hep-ph/0505002 [hep-ph]].
  * (40) S. Godfrey, K. Moats, Properties of Excited Charm and Charm-Strange Mesons, Phys. Rev. D 93, 034035 (2016). [arXiv:1510.08305 [hep-ph]].