# Strong decays of higher excited heavy-light mesons in a chiral quark model

Li-Ye Xiao and Xian-Hui Zhong<EMAIL_ADDRESS>Department of Physics,
Hunan Normal University, and Key Laboratory of Low-Dimensional Quantum
Structures and Quantum Control of Ministry of Education, Changsha 410081,
China

###### Abstract

The strong decay properties of the higher excited heavy-light mesons from the
first radially excited states up to the first $F$-wave states are studied in a
constituent quark model. It is found that many missing excitations have good
potentials to be found in future experiments for their narrow widths, and some
of them dominantly decay into the first orbital excitations rather than into
ground states. In future observations, one should focus on the decay processes
not only into the ground states, but also into the low-lying $P$-wave
excitations with $J^{P}=0^{+},1^{+}$. Furthermore, the nature of the newly
observed states $D_{J}(3000)$, $D_{J}^{*}(3000)$ and $B(5970)$ is discussed.
It is predicted that $D_{J}(3000)$ seems to be a partner of $D_{sJ}(3040)$,
which could be identified as the low-mass mixed state $|2{P_{1}}\rangle_{L}$
($J^{P}=1^{+}$) via the $2^{1}P_{1}$-$2^{3}P_{1}$ mixing. The
$D_{J}^{*}(3000)$ resonance seems to favor the $1^{3}F_{4}$ state, while the
quantum numbers $J^{P}=0^{+}$ and $2^{+}$ cannot be excluded complectly, more
experimental observations are needed to determine its $J^{P}$ values. The
$B(5970)$ resonance is most likely to be the $1^{3}D_{3}$ with $J^{P}=3^{-}$.

###### pacs:

12.39.Fe, 12.39.Jh, 13.25.Ft, 13.25.Hw

##  I Introduction

During the past several years, significant progress has been made in the
observation of the heavy-light mesons. More and more higher excitations have
been found in experiments. In the $D$-meson family, four new members, the
$D(2550)$, $D(2600)$, $D(2750)$ and $D(2760)$ were discovered by the $BABAR$
Collaboration delAmoSanchez:2010vq , which were confirmed by the LHCb
Collaboration with slightly different masses Aaij:2013sza . Furthermore, the
LHCb Collaboration observed two new higher $D$-meson excitations,
$D^{*}_{J}(3000)$ and $D_{J}(3000)$, with natural and unnatural parities,
respectively Aaij:2013sza . In the $D_{s}$-meson family, except for the
$D_{s1}(2700)$ and $D_{sJ}(2860)$ observed by $BABAR$ and Belle before
Aubert:2006mh ; jb:2007aa , recently a new broad higher $D_{s}$-meson
excitation $D_{sJ}(3040)$ was observed by $BABAR$ as well Aubert:2009ah . Very
recently, progress has been obtained in the search for the excited $B$ meson
as well. The CDF Collaboration observed some evidence of a new narrow excited
state $B(5970)$ in the $B\pi$ channel Aaltonen:2013atp .

About the newly observed states $D(2550)$, $D(2600)$, $D(2750)$, $D(2760)$,
$D_{s1}(2700)$, $D_{sJ}(2860)$ and $D_{sJ}(3040)$, their strong decays had
been analyzed in our previous papers Zhong:2008kd ; Zhong:2010vq ;
Zhong:2009sk . We found that the $D(2760)$ could be identified as the
$1^{3}D_{3}$ with $J^{P}=3^{-}$, while the $D(2750)$ is most likely to be the
mixed state $|1{D_{2}}^{\prime}\rangle_{H}$ ($J^{P}=2^{-}$) via the
$1^{1}D_{2}$-$1^{3}D_{2}$ mixing. The $D(2600)$ favors the mixed state
$|(SD)_{1}\rangle_{L}$ $(J^{P}=1^{-})$ via the $1^{3}D_{1}$-$2^{3}S_{1}$
mixing. The $D(2550)$ might be assigned as the $2^{1}S_{0}$ assignment
although its predicted width is obviously narrower than that given in
experiments. For the newly observed $D_{sJ}$ mesons, we found that the
$D_{s1}(2700)$ is most likely to be the strange partner of the $D(2600)$.
There might exist two-state scenario for the $D_{sJ}(2860)$, one resonance
corresponds to the strange partner of the $D(2760)$ [denoted by
$D_{sJ_{1}}(2860)$] and the other resonance is the strange partner of the
$D(2750)$ [denoted by $D_{sJ_{2}}(2860)$]. The $D_{sJ}(3040)$ could be
identified as the low-mass physical state $|2P_{1}\rangle_{L}$ via the
$2^{1}P_{1}$-$2^{3}P_{1}$ mixing. More discussions of these newly observed $D$
and $D_{s}$ mesons can be found in the literature Lu:2014zua ; Godfrey:2013aaa
; Zhen:2013dca ; Wang:2012wk ; Colangelo:2007ds ; Colangelo:2012xi ;
Mohler:2012na ; Badalian:2011tb ; Mohler:2011ke ; Chen:2011rr ; Wang:2010ydc ;
vanBeveren:2009jq ; Sun:2009tg ; Guo:2011dd ; Sun:2010pg ; Colangelo:2010te ;
Li:2009qu ; Close:2005se ; Close:2006gr ; Swanson:2006st ; Zhang:2006yj .

On the other hand, about the higher excitations $D^{*}_{J}(3000)$,
$D_{J}(3000)$ and $B(5970)$ observed very recently, a few studies can be found
in the literature Xu:2014mqa ; Yu:2014dda ; Sun:2014wea ; Sun:2013qca ;
Wang:2013tka ; Xu:2014mka ; Wang:2014cta . For example, the $D^{*}_{J}(3000)$
and $D_{J}(3000)$ states were explained as the first $P$-wave states in the
$D$ meson family with $J^{P}=0^{+}$ and $1^{+}$, respectively Sun:2010pg . As
a candidate of the $2^{3}S_{1}$ Xu:2014mka ; Wang:2014cta , $1^{3}D_{1}$ and
$1^{3}D_{3}$ states Wang:2014cta , the strong decay properties of the
$B(5970)$ were studied with the heavy meson effective theory.

These newly observed higher excitations provide us a good chance to establish
a fairly abundant $D$\- and $D_{s}$-meson spectroscopy, which has been
summarized in Tab. 1. For comparisons, some interested predictions of the
heavy-light meson spectroscopy in theory are also included Di Pierro:2001uu ;
Ebert:2009ua . From the table, it is seen that the knowledge about the $B$\-
and $B_{s}$-meson spectroscopy is very poor. Except for the ground states,
only two low-lying orbital excitations in the $P$ waves are found in the $B$\-
and $B_{s}$-meson spectroscopy. The discovery of the $B(5970)$ enhances our
confidence in the search for more higher excitations in the $B$\- and
$B_{s}$-meson families. From now on, we might have a golden time to study the
higher excitations of the heavy-light mesons.

To provide helpful information for the experimental search for more excited
heavy-light mesons and to gain a unified understanding of the newly observed
resonances, in this work we continue to study the strong decay properties of
the higher excitations of heavy-light mesons up to the $F$ waves (see Tab. 1)
with the chiral quark model as well. This model has been developed and
successfully used to deal with the strong decays of heavy-light mesons,
charmed and strange baryons Zhong:2008kd ; Zhong:2010vq ; Zhong:2009sk ;
Zhong:2007gp ; Liu:2012sj ; Xiao:2013xi .

This paper is organized as follows. In the subsequent section, a brief review
of the model is given. The numerical results are presented and discussed in
Sec. III. Finally, a summary is given in Sec. IV.

Table 1: The heavy-light meson spectroscopy predicted in theory compared with the observations in experiments. The predicted masses (MeV) are obtained from Di Pierro:2001uu ; Ebert:2009ua , while the observed states are obtained from the PDG Beringer:1900zz . The $1P_{1}$ and $1P_{1}^{\prime}$ stand for the low-mass and high-mass mixed states via the $1^{1}P_{1}$-$1^{3}P_{1}$ mixing, respectively, which have been defined in Zhong:2008kd . The $1D_{2}$ and $1D_{2}^{\prime}$ stand for the low-mass and high-mass mixed states via the $1^{1}D_{2}$-$1^{3}D_{2}$ mixing, respectively. The $2P_{1}$ and $2P_{1}^{\prime}$ stand for the low-mass and high-mass mixed states via the $2^{1}P_{1}$-$2^{3}P_{1}$ mixing, respectively.  States | $D$ mesons | $D_{s}$ mesons | $B$ mesons | $B_{s}$ mesons
---|---|---|---|---
$n^{2S+1}L_{J}$ | Predicted Mass | Observed state | Predicted Mass | Observed state | Predicted Mass | Observed state | Predicted Mass | Observed state
$1^{1}S_{0}$ | 1868/1871 | $D(1865)$ | 1965/1969 | $D_{s}(1969)$ | 5279/5280 | $B(5280)$ | 5373/5372 | $B_{s}(5367)$
$1^{3}S_{1}$ | 2005/2010 | $D^{*}(2007)$ | 2113/2111 | $D^{*}_{s}(2112)$ | 5324/5326 | $B^{*}(5325)$ | 5421/5414 | $B^{*}_{s}(5415)$
$1^{3}P_{0}$ | 2377/2406 | $D^{*}_{0}(2400)$ | 2487/2509 | $D_{s0}(2317)$ | 5706/5749 | ? | 5804/5833 | ?
$1P_{1}$ | 2417/2426 | $D_{1}(2430)$ | 2535/2536 | $D_{s1}(2460)$ | 5700/5723 |  $B_{J}^{*}(5732)$? | 5805/5831 | ?
$1P_{1}^{\prime}$ | 2490/2469 | $D_{1}(2420)$ | 2605/2574 | $D_{s1}(2536)$ | 5742/5774 | $B_{1}(5721)$ | 5842/5865 | $B_{s1}(5830)$
$1^{3}P_{2}$ | 2460/2460 | $D_{2}(2460)$ | 2581/2571 | $D_{s2}(2573)$ | 5714/5741 | $B_{2}(5747)$ | 5820/5842 | $B_{s2}(5840)$
$2^{1}S_{0}$ | 2589/2581 |  $D(2550)$? | 2700/2688 | ? | 5886/5890 | ? | 5985/5976 | ?
$2^{3}S_{1}$ | 2692/2632 |  $D(2600)$? | 2806/2731 |  $D_{s1}(2700)$? | 5920/5906 | ? | 6019/5992 | ?
$1^{3}D_{1}$ | 2795/2788 | ? | 2913/2913 | ? | 6025/6119 | ? | 6127/6209 | ?
$1D_{2}$ | 2775/2806 | ? | 2900/2931 | ? | 5985/6102 | ? | 6095/6189 | ?
$1D_{2}^{\prime}$ | 2883/2850 |  $D(2750)$? | 2953/2961 |  $D_{sJ_{2}}(2860)$? | 6037/6121 | ? | 6140/6218 | ?
$1^{3}D_{3}$ | 2799/2863 |  $D(2760)$? | 2925/2971 |  $D_{sJ_{1}}(2860)$? | 5993/6091 |  $B(5970)$? | 6103/6191 | ?
$2^{3}P_{0}$ | 2949/2919 | ? | 3067/3054 | ? | 6163/6221 | ? | 6264/6318 | ?
$2P_{1}$ | 2995/2932 |  $D_{J}(3000)$? | 3114/3067 |  $D_{sJ}(3040)$? | 6175/6209 | ? | 6278/6321 | ?
$2P_{1}^{\prime}$ | 3045/3021 | ? | 3165/3154 | ? | 6194/6281 | ? | 6296/6345 | ?
$2^{3}P_{2}$ | 3035/3012 | ? | 3157/3142 | ? | 6188/6260 | ? | 6292/6359 | ?
$1^{3}F_{2}$ | 3101/3090 | ? | 3224/3230 | ? | 6264/6412 | ? | 6369/6501 | ?
$1F_{3}$ | 3074/3129 | ? | 3203/3254 | ? | 6220/6391 | ? | 6332/6468 | ?
$1F_{3}^{\prime}$ | 3123/3145 | ? | 3247/3266 | ? | 6271/6420 | ? | 6376/6515 | ?
$1^{3}F_{4}$ | 3091/3187 | ? | 3220/3300 | ? | 6226/6380 | ? | 6337/6475 | ?

##  II The model

In this section, we give a brief review of the chiral quark model. The details
of this model can be found in our previous papers Zhong:2008kd ; Zhong:2010vq
; Zhong:2009sk . In this model, the low energy quark-pseudoscalar-meson and
quark-vector-meson interactions in the SU(3) flavor basis might be described
by the effective Lagrangians

$\displaystyle{\cal L}_{Pqq}$ $\displaystyle=$
$\displaystyle\sum_{j}\frac{1}{f_{m}}\bar{\psi}_{j}\gamma^{j}_{\mu}\gamma^{j}_{5}\psi_{j}\partial^{\mu}\phi_{m},$
(1) $\displaystyle{\cal L}_{Vqq}$ $\displaystyle=$
$\displaystyle\sum_{j}\bar{\psi}_{j}(a\gamma^{j}_{\mu}+\frac{ib}{2m_{j}}\sigma_{\mu\nu}q^{\nu})V^{\mu}\psi_{j},$
(2)

respectively, where $\psi_{j}$ represents the $j$th quark field in the hadron,
$\phi_{m}$ is the pseudoscalar meson field, $f_{m}$ is the pseudoscalar meson
decay constant, and $V^{\mu}$ represents the vector meson field. Parameters
$a$ and $b$ denote the vector and tensor coupling strength, respectively.

In this model, the wave function of a heavy-light meson is adopted by the
nonrelativistic harmonic oscillator wave function, i.e.,
$\psi^{n}_{lm}=R_{nl}Y_{lm}$. To match the nonrelativistic wave functions of
the heavy-light mesons, we should adopt the nonrelativistic form of Eq. (1) in
the calculations, which is given by Li:1994cy ; Li:1997gda ; qk3

$\displaystyle
H_{m}=\sum_{j}\left[A\mbox{\boldmath$\sigma$\unboldmath}_{j}\cdot\textbf{q}+\frac{\omega_{m}}{2\mu_{q}}\mbox{\boldmath$\sigma$\unboldmath}_{j}\cdot\textbf{p}_{j}\right]I_{j}\varphi_{m},$
(3)

in the center-of-mass system of the initial meson, where we have defined
$A\equiv-(1+\frac{\omega_{m}}{E_{f}+M_{f}})$. On the other hand, from Eq. (2),
one can easily derive the nonrelativistic transition operators for the
emission of a transversely or longitudinally polarized vector meson, which are
given by zhao:1998fn ; Zhao:2000tb ; Zhao:2001jw

$\displaystyle
H_{m}^{T}=\sum_{j}\left\\{i\frac{b^{\prime}}{2m_{q}}\mbox{\boldmath$\sigma$\unboldmath}_{j}\cdot(\mathbf{q}\times\mathbf{\epsilon})+\frac{a}{2\mu_{q}}\mathbf{p}_{j}\cdot\mathbf{\epsilon}\right\\}I_{j}\varphi_{m},$
(4)

and

$\displaystyle H_{m}^{L}=\sum_{j}\frac{aM_{v}}{|\mathbf{q}|}I_{j}\varphi_{m}\
.$ (5)

In the above equations, q and $\omega_{m}$ are the three-vector momentum and
energy of the final-state light meson, respectively; $\textbf{p}_{j}$ is the
internal momentum operator of the $j$th quark in the heavy-light meson rest
frame; $\mbox{\boldmath$\sigma$\unboldmath}_{j}$ is the spin operator
corresponding to the $j$th quark of the heavy-light system; and $\mu_{q}$ is a
reduced mass given by $1/\mu_{q}=1/m_{j}+1/m^{\prime}_{j}$ with $m_{j}$ and
$m^{\prime}_{j}$ for the masses of the $j$th quark in the initial and final
mesons, respectively. $M_{v}$ is the mass of the emitted vector meson. The
plane wave part of the emitted light meson is
$\varphi_{m}=e^{-i\textbf{q}\cdot\textbf{r}_{j}}$, and $I_{j}$ is the flavor
operator defined for the transitions in the SU(3) flavor space Li:1997gda ;
qk3 ; zhao:1998fn ; Zhao:2000tb ; Zhao:2001jw . The parameter $b^{\prime}$ in
Eq. (4) is defined as $b^{\prime}\equiv b-a$.

For a light pseudoscalar meson emission in heavy-light meson strong decays,
the partial decay width can be calculated with Zhong:2008kd ; Zhong:2007gp

$\Gamma=\left(\frac{\delta}{f_{m}}\right)^{2}\frac{(E_{f}+M_{f})|\textbf{q}|}{4\pi
M_{i}(2J_{i}+1)}\sum_{J_{iz},J_{fz}}|\mathcal{M}_{J_{iz},J_{fz}}|^{2},$ (6)

where $\mathcal{M}_{J_{iz},J_{fz}}$ is the transition amplitude, and $J_{iz}$
and $J_{fz}$ stand for the third components of the total angular momenta of
the initial and final heavy-light mesons, respectively. $\delta$ as a global
parameter accounts for the strength of the quark-meson couplings. It has been
determined in our previous study of the strong decays of the charmed baryons
and heavy-light mesons Zhong:2007gp ; Zhong:2008kd . Here, we take the same
value as that determined in Refs. Zhong:2008kd ; Zhong:2007gp , i.e.,
$\delta=0.557$.

In the calculation, the standard quark model parameters are adopted. Namely,
we set $m_{u}=m_{d}=330$ MeV, $m_{s}=450$ MeV, $m_{c}=1700$ MeV, and
$m_{b}=5100$ MeV for the constituent quark masses. The harmonic oscillator
parameter $\beta$ in the wave functions of heavy-light mesons is taken as
$\beta=0.40$ GeV. The decay constants for $\pi$, $K$ and $\eta$ mesons are
taken as $f_{\pi}=132$ MeV, $f_{K}=f_{\eta}=160$ MeV, respectively. For the
quark-vector-meson coupling strength which still suffers relatively large
uncertainties, we adopt the values extracted from vector meson
photoproduction, i.e. $a\simeq-3$ and $b^{\prime}\simeq 5$ zhao:1998fn ;
Zhao:2000tb ; Zhao:2001jw . The masses of the mesons used in the calculations
are adopted from the PDG Beringer:1900zz . With these parameters, the strong
decay properties of the well-known heavy-light mesons and charmed baryons have
been described reasonably Zhong:2008kd ; Zhong:2010vq ; Zhong:2009sk ;
Zhong:2007gp ; Liu:2012sj .

##  III Calculations and Results

###  III.1 The $2^{1}S_{0}$ states

Recently, the $BABAR$ Collaboration observed an excited $D$-meson state
$D(2550)$ in the $D^{*}\pi$ channel with a width of $\Gamma\simeq 130$ MeV
delAmoSanchez:2010vq , which was also observed by the LHCb Collaboration
recently Aaij:2013sza . In theory, the first radially excited $D$-meson state
$2^{1}S_{0}$ has a mass $\sim 2.58$ GeV Godfrey:1985xj ; Di Pierro:2001uu ;
Ebert:2009ua . Furthermore, the decay mode and the _BABAR_ analysis of angle
distributions indicate that the $D(2550)$ should be classified as the radially
excited state $2^{1}S_{0}$. We have studied the strong decays of the $D(2550)$
as the first radially excited $D$-meson state $2^{1}S_{0}$ in Refs.
Zhong:2010vq ; Zhong:2008kd . The strong decays of $D(2^{1}S_{0})$ are
dominated by the $D^{*}\pi$ and $D_{0}(2400)\pi$ modes. In the present work we
have improved our predictions according to the new observations from the LHCb
Collaboration Aaij:2013sza . The results are listed in Tab. 2. The predicted
partial decay width ratio is

$\displaystyle\frac{\Gamma[D^{*}\pi]}{\Gamma[D_{0}(2400)\pi]}\simeq 0.28,$ (7)

which strongly depends on the mass of $D_{0}(2400)$. Our predicted width,
$\Gamma\simeq 68$ MeV, is about a factor of 2 narrower than the data. The
${}^{3}P_{0}$ model Sun:2010pg  and relativistic quark model Di Pierro:2001uu
calculations also obtained a narrow width for the $D(2^{1}S_{0})$ state.

If the $D(2550)$ is the $2^{1}S_{0}$ assignment indeed, the same excitation in
the $D_{s}$-, $B$-, and $B_{s}$-meson spectroscopy might be found in future
experiments. To obtain more information about these missing states, we further
study the strong decay properties of these radially excited states
$D_{s}(2^{1}S_{0})$, $B(2^{1}S_{0})$ and $B_{s}(2^{1}S_{0})$ in the following.

Table 2: The strong decay width (MeV) of the $D(2550)$ as the $2^{1}S_{0}$ with a mass of 2580 MeV. Mode | $D^{*}\pi$ | $D_{0}(2400)\pi$ | Total
---|---|---|---
Width | 15 | 53 | 68

In the $D_{s}$-meson family, the predicted mass of the first radially excited
state $2^{1}S_{0}$ is around $2.7$ GeV Godfrey:1985xj ; Di Pierro:2001uu ;
Ebert:2009ua . There is no experimental information about this state. The
strong decay properties of $D_{s}(2^{1}S_{0})$ have been studied in our
previous work Zhong:2008kd . Its strong decays are governed by the $D^{*}K$
channel. Based on our model, the $D_{s}(2^{1}S_{0})$ is a very narrow state
with a width of $\Gamma\simeq 10$ MeV. Such a narrow state should be observed
in the $D^{*}K$ final states if it indeed exists.

In the $B$-meson family, the predicted mass of $B(2^{1}S_{0})$ is around $5.9$
GeV Godfrey:1985xj ; Di Pierro:2001uu ; Ebert:2009ua . We calculate its strong
decay properties, which are listed in Tab. 3. It is seen that the strong
decays are dominated by the $B^{*}\pi$ and $B(1^{3}P_{0})\pi$ channels, and
the predicted partial width ratio is

$\displaystyle\frac{\Gamma[B^{*}\pi]}{\Gamma[B(1^{3}P_{0})\pi]}\simeq 1.3.$
(8)

The $B(2^{1}S_{0})$ is also a narrow state with a width of $\Gamma\simeq 40$
MeV, which is slightly narrower than that of $D(2^{1}S_{0})$. The decay width
predicted by us is roughly compatible with the recent calculations with
${}^{3}P_{0}$ model Sun:2014wea .

Table 3: The strong decay width (MeV) of the $B(2^{1}S_{0})$ with a mass of 5886 MeV. The mass of $B(1^{3}P_{0})$ is taken as 5706 MeV. Mode | $B^{*}\pi$ | $B(1^{3}P_{0})\pi$ | $B(1^{3}P_{2})\pi$ | $B^{*}\eta$ | Total
---|---|---|---|---|---
Width | 16 | 24 | 0.01 | 0.25 | 40

In the $B_{s}$-meson family, the predicted mass of $B_{s}(2^{1}S_{0})$ is
around 6.0 GeV Godfrey:1985xj ; Di Pierro:2001uu ; Ebert:2009ua . We calculate
the strong decay properties of this state with a mass of $M=5985$ MeV. This
state dominantly decays into the $B^{*}K$ channel. The total decay width is
$\Gamma\simeq 26$ MeV, which is a little narrower than that of the
${}^{3}P_{0}$ calculations Sun:2014wea .

As a whole, the first radially excited states $2^{1}S_{0}$ in the $D$-,
$D_{s}$-, $B$\- and $B_{s}$-meson spectroscopy might have a fairly narrow
width. However, up to now none of them has been established. It is still a
puzzle why such narrow radially excited states are still missing. More
theoretical and experimental studies are needed.

Figure 1:  The partial decay width and total decay width for the mixed states
via $2^{3}S_{1}$-$1^{3}D_{1}$ mixing in the $B$\- and $B_{s}$-meson families
as functions of the mixing angle $\phi$. The shaded bands correspond to the
possible mixing angle region derived from the strong decay properties of the
$D(2600)$ and $D_{s1}(2700)$. The masses for the mixed states in the $B$-meson
family, $B(|(SD)_{1}\rangle_{L})$ and $B(|(SD)_{1}^{\prime}\rangle_{H})$, are
adopted as $5920$ and $6025$ MeV, respectively, while, the masses for the
mixed states in the $B_{s}$-meson family, $B_{s}(|(SD)_{1}\rangle_{L})$ and
$B_{s}(|(SD)_{1}^{\prime}\rangle_{H})$, are adopted as $6000$ and $6100$ MeV,
respectively. The mass of $B(1P_{1})$ is taken as 5720 MeV. In the figure, we
have hidden some decay channels because of their small partial decay widths.

###  III.2 The $2^{3}S_{1}$-$1^{3}D_{1}$ mixing

In 2010, the _BABAR_ Collaboration discovered a new excited $D$-meson state
$D(2600)$ in $D\pi$ and $D^{*}\pi$ decay channels with a mass of $M=2612\pm 6$
MeV and a total decay width of $\Gamma=93\pm 6\pm 13$ MeV delAmoSanchez:2010vq
, which might be a partner of the $D_{s1}(2700)$ observed previously by
_BABAR_ and Belle Aubert:2006mh ; jb:2007aa . The $D(2600)$ may correspond to
the $D(2650)$ observed by LHCb very recently Aaij:2013sza .

In our previous work Zhong:2010vq ; Zhong:2009sk , we have carefully studied
the strong decay properties of the $D(2600)$ and $D_{s1}(2700)$. According to
our analysis, both $D(2600)$ and $D_{s1}(2700)$ could be explained as the
mixed state $|(SD)_{1}\rangle_{L}$ via the $2^{3}S_{1}$-$1^{3}D_{1}$ mixing:

$\left(\begin{array}[]{c}|(SD)_{1}\rangle_{L}\cr|(SD)^{\prime}_{1}\rangle_{H}\end{array}\right)=\left(\begin{array}[]{cc}\cos\phi&\sin\phi\cr-\sin\phi&\cos\phi\end{array}\right)\left(\begin{array}[]{c}|2^{3}S_{1}\rangle\cr|1^{3}D_{1}\rangle\end{array}\right).$
(9)

The mixing angle for the $D(2600)$ is $\phi\simeq-(36\pm 6)^{\circ}$, while
that for the $D_{s1}(2700)$ is $\phi=(-54\pm 7)^{\circ}$. To explain the
strong decay properties of the $D(2600)$ and/or $D_{s1}(2700)$, configuration
mixing between $2^{3}S_{1}$ and $1^{3}D_{1}$ is also suggested in the
literature Close:2006gr ; Chen:2011rr ; Li:2009qu .

Following the mixing scheme in Eq. (9), the strong decay properties of the
physical partners of the $D(2600)$ and $D_{s1}(2700)$, i.e.,
$|(SD)_{1}^{\prime}\rangle_{H}$, have been obtained as well. Taking the mass
of $D(|(SD)_{1}^{\prime}\rangle_{H})$ in the range of $(2.65\sim 2.80)$ GeV,
we predict that the $D(|(SD)_{1}^{\prime}\rangle_{H})$ is a broad state with a
width of $\Gamma\simeq(360\pm 120)$ MeV. Its strong decays are dominated by
$D\pi$ and $D_{1}(2420)\pi$. On the other hand, estimating the mass of
$D_{s}(|(SD)_{1}^{\prime}\rangle_{H})$ in the range of $(2.72\sim 2.88)$ GeV,
the total width is $\Gamma\simeq(120\pm 10)$ MeV Zhong:2009sk . The predicted
width for the $D_{s}(|(SD)_{1}^{\prime}\rangle_{H})$ is not very broad. This
state might be observed in the $DK$ channel. It is interestingly found that
the LHCb Collaboration has observed a new resonance $D_{s1}(2860)$ with a
width of $\Gamma=(159\pm 122)$ MeV very recently Aaij:2014xza ; Aaij:2014baa .
Their analysis shows that the spin-parity should be $J^{P}=1^{-}$. This newly
observed resonance $D_{s1}(2860)$ can be favorably assigned as the physical
partners of the $D_{s1}(2700)$, i.e., $D_{s}(|(SD)_{1}^{\prime}\rangle_{H})$,
predicted in our previous work Zhong:2009sk .

If both $D(2600)$ and $D_{s1}(2700)$ indeed correspond to the mixed state
$|(SD)_{1}\rangle_{L}$, the $2^{3}S_{1}$-$1^{3}D_{1}$ mixing might be a common
character in the heavy-light mesons. The future search for these missing
mixing states $|(SD)_{1}\rangle_{L}$ and $|(SD)_{1}^{\prime}\rangle_{H}$ in
the heavy-light meson spectroscopy becomes interesting and important.

Figure 2: The partial decay width and total decay width for the mixed states
via $2^{3}S_{1}$-$1^{3}D_{1}$ mixing in the $B$\- and $B_{s}$-meson families
as functions of the mass. The mixing angle is fixed with $\phi=-40^{\circ}$.
In the figure, we have hidden some decay channels because of their small
partial decay widths.

Firstly, we consider the $2^{3}S_{1}$-$1^{3}D_{1}$ mixing in the $B$-meson
family. According to the theoretical calculations of the $B$-meson
spectroscopy, the mass of the low-mass state $|(SD)_{1}\rangle_{L}$ is $M\sim
5.9$ GeV Di Pierro:2001uu ; Ebert:2009ua . Thus, we set the mass of the low-
mass mixed state $|(SD)_{1}\rangle_{L}$ with $M=5.9$ GeV, and plot its partial
decay widths and total decay width as functions of the mixing angle $\phi$ in
Fig. 1. Theoretically, the mixing angle $\phi$ between $2^{3}S_{1}$ and
$1^{3}D_{1}$ in the $D$-, $D_{s}$-, $B$\- and $B_{s}$-meson families should be
the same in the heavy quark symmetry limit. Combining our previous analysis of
the strong decay properties of the $D(2600)$ and $D_{s1}(2700)$ Zhong:2010vq ;
Zhong:2009sk , the mixing angle might be $\phi\simeq-(45\pm 16)^{\circ}$. From
Fig. 1, it is seen that the mixed state $B(|(SD)_{1}\rangle_{L})$ has a width
of $\Gamma\simeq(150\pm 50)$ MeV. Its decays are dominated by the $B\pi$ and
$B^{*}\pi$; the partial width ratio between $B\pi$ and $B^{*}\pi$ is

$\displaystyle\frac{\Gamma[B\pi]}{\Gamma[B^{*}\pi]}\simeq 1.0\pm 0.3.$ (10)

For the high-mass mixed state $B(|(SD)_{1}^{\prime}\rangle_{H})$, we take its
mass with $M=6.0$ GeV, and plot its partial decay widths and total decay width
as functions of the mixing angle $\phi$ in Fig. 1 as well, where we can see
that this resonance is a broad state with a width of $\Gamma=(310\pm 90)$ MeV.
It decays mainly through the $B\pi$ and $B_{1}(5721)\pi$ channels.

Finally, we study the $2^{3}S_{1}$-$1^{3}D_{1}$ mixing in the $B_{s}$-meson
family. According to the theoretical predictions in various models, the masses
for the mixed states $|(SD)_{1}\rangle_{L}$ and
$|(SD)_{1}^{\prime}\rangle_{H}$ are around $6.0$ and $6.1$ GeV, respectively
Di Pierro:2001uu ; Ebert:2009ua . With these predicted masses, we plot the
partial decay widths and total decay widths for the mixed states
$|(SD)_{1}\rangle_{L}$ and $|(SD)_{1}^{\prime}\rangle_{H}$ as functions of the
mixing angle $\phi$ in Fig. 1, respectively. Adopting the mixing angle
$\phi\simeq-(45\pm 16)^{\circ}$, from Fig. 1 we see that the low-mass state
$|(SD)_{1}\rangle_{L}$ might have a width of $\Gamma=(130\pm 50)$ MeV. Its
strong decays are dominated by the $BK$ and $B^{*}K$ channels. The high-mass
state $|(SD)_{1}^{\prime}\rangle_{H}$ is a slightly broader state with a width
of $\Gamma=(140\pm 60)$ MeV. The decay widths for the $|(SD)_{1}\rangle_{L}$
and $|(SD)_{1}^{\prime}\rangle_{H}$ are sensitive to the mixing angle.

The predicted masses of these mixed states $|(SD)_{1}\rangle_{L}$ and
$|(SD)_{1}^{\prime}\rangle_{H}$ in $B$\- and $B_{s}$-meson spectroscopy have a
large uncertainty, which may bring uncertainties to the predicted decay
widths. To investigate this effect, we plot the decay width as a function of
the mass in Fig. 2 with the mixing angle $\phi=-40^{\circ}$. It shows that
there is an uncertainty of about $(50\sim 200)$ MeV in the total decay width
for the uncertainties of the mass. The sensitivity of different decay modes to
the mass can also be seen clearly in the plot.

In summary, configuration mixing might exist between $2^{3}S_{1}$ and
$1^{3}D_{1}$ according to our model prediction. The decay properties are
sensitive to the mixing angle. As flavor partners of the $D(2600)$ and
$D_{s1}(2700)$, the low-mass mixed states $B(|(SD)_{1}\rangle_{L})$ and
$B_{s}(|(SD)_{1}\rangle_{L})$ have a relatively narrow width and are most
likely to be observed in future experiments. In the high-mass mixed states,
the widths of charmed-strange state $D_{s}(|(SD)_{1}^{\prime}\rangle_{H})$ and
bottom-strange state $B_{s}(|(SD)_{1}^{\prime}\rangle_{H})$ are as comparable
as those of the low-mass mixed states, which are possibly to be found in
experiments as well, while the bottom state $B(|(SD)_{1}^{\prime}\rangle_{H})$
is a broad state, which might be hard to find in experiments. However, it
should be pointed out that in some Refs. Lu:2014zua ; Colangelo:2007ds ;
Colangelo:2012xi ; Godfrey:2013aaa ; Wang:2010ydc , one believes that
configuration mixing might be weak between $2^{3}S_{1}$ and $1^{3}D_{1}$.
Within their models, the strong decay properties of the $D(2600)$ and/or
$D_{s1}(2700)$ can be well explained by taking them as a pure $2^{3}S_{1}$
state. To uncover the puzzles in the $2^{3}S_{1}$ and $1^{3}D_{1}$ states,
more accurate measurements for the $D(2600)$ and $D_{s1}(2700)$ as well as a
further search for the missing resonances with $J^{P}=1^{-}$ in the heavy-
light meson spectroscopy in experiment is needed.

###  III.3 The $1^{1}D_{2}$-$1^{3}D_{2}$ mixing

The $D(2750)$ is a new excitation observed by _BABAR_ Collaboration in the
$D\pi$ channel with a narrow width of $\Gamma\simeq 71\pm 17$ MeV
delAmoSanchez:2010vq . This state might correspond to the $D(2740)$ observed
by the LHCb Collaboration in the same channel recently Aaij:2013sza . Their
helicity analysis indicates that this state should have an unnatural parity,
i.e., $J^{P}=0^{-},1^{+},2^{-},\cdot\cdot\cdot$. In our previous work
Zhong:2010vq , the new resonance $D(2750)$ was discussed carefully. We
concluded that the spin and parity numbers of the $D(2750)$ might be
$J^{P}=2^{-}$, this state is most likely to be the high-mass mixed state
$|1D_{2}^{\prime}\rangle_{H}$ via the $1^{1}D_{2}$-$1^{3}D_{2}$ mixing:

$\left(\begin{array}[]{c}|1D_{2}\rangle_{L}\cr|1D_{2}^{\prime}\rangle_{H}\end{array}\right)=\left(\begin{array}[]{cc}\cos\phi_{1D}&\sin\phi_{1D}\cr-\sin\phi_{1D}&\cos\phi_{1D}\end{array}\right)\left(\begin{array}[]{c}|1^{1}D_{2}\rangle\cr|1^{3}D_{2}\rangle\end{array}\right),$
(11)

with a mixing angle $\phi_{1D}=-(51\pm 18)^{\circ}$. The mixing angle is
consistent with that ($\phi_{1D}=-50.8^{\circ}$) obtained in the heavy quark
symmetry limit Close:2005se . Our predictions are in compatible with the
recent analysis in Ref. Lu:2014zua . It should be mentioned that with the
mixing angle $\phi_{1D}=-50.8^{\circ}$, the low-mass state
$|1D_{2}\rangle_{L}$ and high-mass state $|1D_{2}^{\prime}\rangle_{H}$ will
correspond to a broad state and a narrow state, respectively Close:2005se ;
Swanson:2006st .

The strong decay properties of the low-mass state $D(|1D_{2}\rangle_{L})$ have
been studied in Zhong:2010vq  as well. The mass of $D(|1D_{2}\rangle_{L})$ is
likely to be $\sim 2.7$ GeV. This state is a broad state with a width of
$\Gamma\simeq(250\sim 500)$ MeV, and its strong decays are dominated by the
$D^{*}\pi$ and $D_{2}(2460)\pi$ channels. The $D(|1D_{2}\rangle_{L})$ might be
hard to observe in experiments because of its broad width.

In our previous study Zhong:2009sk , we also found that the
$1^{1}D_{2}$-$1^{3}D_{2}$ mixing might be crucial to uncovering the
longstanding puzzles in the $D_{sJ}(2860)$. Many people believe that the
$D_{sJ}(2860)$ might be the $1^{3}D_{3}$ state. However, considering the
$D_{sJ}(2860)$ as the $1^{3}D_{3}$ state only, one can not understand the
important partial width ratio of $\Gamma(DK)/\Gamma(D^{*}K)$ measured by
_BABAR_ Aubert:2009ah  at all. Considering there are two $D_{s}$ states with
masses around $2.86$ GeV, one resonance corresponds to the $1^{3}D_{3}$
[denoted by $D_{sJ_{1}}(2860)$] and the other resonance is the mixed state
$|1D_{2}\rangle_{H}$ [denoted by $D_{sJ_{2}}(2860)$] via the
$1^{1}D_{2}$-$1^{3}D_{2}$ mixing with the same mixing angle, we can explain
the present strong decay properties of the $D_{sJ}(2860)$ observed in
experiments naturally Zhong:2009sk . Our conclusion is compatible with the
recent theoretical analysis in Ref. Godfrey:2013aaa .

If the $D_{sJ_{2}}(2860)$ could be assigned as the high-mass state
$|1D_{2}^{\prime}\rangle_{H}$ indeed, its low-mass partner
$|1D_{2}\rangle_{L}$ might be observed in experiments as well. The mass of
low-mass partner $|1D_{2}\rangle_{L}$ is estimated to be $30\sim 50$ MeV
lighter than that of the high-mass state Godfrey:1985xj ; Di Pierro:2001uu ;
Ebert:2009ua . Thus, the mass of $|1D_{2}\rangle_{L}$ might be in the range of
($2.80\sim 2.83$) GeV. Its strong decay properties have been analyzed in our
previous work Zhong:2010vq . It is found $|1D_{2}\rangle_{L}$ is a broad state
with a width of $\Gamma\simeq(260\pm 30)$ MeV. The strong decays are dominated
by the $D^{*}K$ channel.

It should be emphasized that if both $D(2750)$ and $D_{sJ_{2}}(2860)$ indeed
correspond to the mixed state $|1D_{2}^{\prime}\rangle_{H}$, the
$1^{1}D_{2}$-$1^{3}D_{2}$ mixing might exist in the $B$\- and $B_{s}$-meson
spectroscopy as well. To provide useful clues for the future experimental
search for these mixed states $|1D_{2}^{\prime}\rangle_{H}$ and
$|1D_{2}\rangle_{L}$ in $B$\- and $B_{s}$-meson spectroscopy, we study their
strong decay properties in the following.

Figure 3: The partial decay width and total decay width for the mixed states
via $1^{1}D_{2}$-$1^{3}D_{2}$ mixing in the $B$\- and $B_{s}$-meson families
as functions of the mixing angle $\phi_{1D}$. The shaded region corresponds to
possible mixing angle region derived from the strong decay properties of the
$D(2750)$. The masses of the low-mass mixed states $|1D_{2}\rangle_{L}$ in the
$B$\- and $B_{s}$-meson families are adopted as 5.95 GeV and 6.05 GeV,
respectively. The masses of the high-mass mixed states
$|1D_{2}^{\prime}\rangle_{H}$ in the $B$\- and $B_{s}$-meson families are
adopted as 5.98 GeV and 6.08 GeV, respectively. In the figure, we have hidden
some decay channels because of their small partial decay widths.

.

Figure 4: The partial decay width and total decay width for the mixed states
via $1^{1}D_{2}$-$1^{3}D_{2}$ mixing in the $B$\- and $B_{s}$-meson families
as functions of the mass. The mixing angle is adopted as
$\phi_{1D}=-50.8^{\circ}$. In the figure, we have hidden some decay channels
because of their small partial decay widths.

.

Firstly, we study the $1^{1}D_{2}$-$1^{3}D_{2}$ mixing in the $B$-meson
family. According to the predictions in Refs. Godfrey:1985xj ; Di
Pierro:2001uu ; Ebert:2009ua , the mass of the high-mass mixed state
$|1D_{2}^{\prime}\rangle_{H}$ is in the range of $(6.04\sim 6.12)$ GeV.
However, we find the masses of the $|1D_{2}^{\prime}\rangle_{H}$ resonances
might be systemically overestimated by $\sim 100$ MeV in theory, if the
$D(2750)$ and $D_{sJ_{2}}(2860)$ correspond to the mixed state
$|1D_{2}^{\prime}\rangle_{H}$. Thus, the mass of the
$|1D_{2}^{\prime}\rangle_{H}$ in the $B$-meson family might be
$M\simeq(5.94\sim 6.02)$ GeV.

Taking the mass of $B(|1D_{2}^{\prime}\rangle_{H})$ with $M=5.98$ GeV, we plot
the partial decay widths and total decay width as functions of the mixing
angle $\phi_{1D}$ in Fig. 3. It is shown that the decay width of the high-mass
state $|1D_{2}^{\prime}\rangle_{H}$ is about $\Gamma\simeq(40\sim 90)$ MeV,
when we take the mixing angle with $\phi_{1D}=(-51\pm 18)^{\circ}$. The main
decay channels are $B^{*}\pi$ and $B_{2}(5747)\pi$. The predicted branching
ratio is

$\displaystyle\frac{\Gamma[B^{*}\pi]}{\Gamma_{\mathbf{total}}}\simeq 70\%\sim
93\%.$ (12)

The partial widths for the $B(1^{3}P_{0})\pi$, $B(1P_{1})\pi$,
$B(1P_{1}^{\prime})\pi$, $B^{*}\eta$ and $B_{s}^{*}K$ channels are tiny and
are not shown in the figure. Such a narrow state,
$B(|1D_{2}^{\prime}\rangle_{H})$, is most likely to be observed in the
$B^{*}\pi$ channel.

For the low-mass mixed state $|1D_{2}\rangle_{L}$, its mass might be $(20\sim
50)$ MeV lighter than that of the $|1D_{2}\rangle_{H}$ Godfrey:1985xj ; Di
Pierro:2001uu ; Ebert:2009ua . Thus, the mass of $|1D_{2}\rangle_{L}$ might be
in the range $M=(5.9\sim 6.0)$ GeV. The dependence of the strong decay width
of the $|1D_{2}\rangle_{L}$ on the mass is shown in Fig. 4 as well. From the
figure, we can see that the low-mass state $B(|1D_{2}\rangle_{L})$ is a very
broad state with a width of $\Gamma\simeq(400\pm 20)$ MeV. Its strong decays
are governed by the $B^{*}\pi$ and $B_{2}(5747)\pi$ channels. This broad state
might be hard to observe in experiments.

Secondly, we study the $1^{1}D_{2}$-$1^{3}D_{2}$ mixing in the $B_{s}$-meson
family. According to the theoretical predictions, the mass gap between the
$B_{s}$ resonances and the $B$ resonances is about 100 MeV Godfrey:1985xj ; Di
Pierro:2001uu ; Ebert:2009ua . Thus, the high-mass state
$B_{s}(|1D_{2}^{\prime}\rangle_{H})$ might have a mass of $M\simeq(6.04\sim
6.12)$ GeV, while the mass of the low-mass state $B_{s}(|1D_{2}\rangle_{L})$
might be $M\simeq(5.99\sim 6.10)$ GeV.

Taking the mass of the $B_{s}(|1D_{2}^{\prime}\rangle_{H})$ with $M=6.08$ GeV,
we plot the partial widths and total decay width as functions of the mixing
angle in Fig. 3. Adopting the physical mixing angle $\phi_{1D}=(-51\pm
18)^{\circ}$ determined before, we find the
$B_{s}(|1D_{2}^{\prime}\rangle_{H})$ is a fairly narrow state with a width of
$\Gamma\simeq(13\sim 50)$ MeV. Its decays are dominated by the $B^{*}K$
channel. The predicted branching ratio is

$\displaystyle\frac{\Gamma[B^{*}K]}{\Gamma_{\mathbf{total}}}\simeq 100\%.$
(13)

This narrow state is most likely to be discovered in the $B^{*}K$ channel in
future experiments.

Furthermore, we analyze the strong decay properties of the low-mass state
$B_{s}(|1D_{2}\rangle_{L})$. Its partial decay widths and total width as
functions of the mixing angle are plotted in Fig. 3 as well, where we have
fixed the mass with $M=6.05$ GeV. From the figure, we find that the low-mass
state $B_{s}(|1D_{2}\rangle_{L})$ has a broad width of $\Gamma\simeq 240$ MeV,
and its decays are governed by the $B^{*}K$ channel.

Finally, we study the effects of the mass uncertainties of the mixing states
$|1D_{2}\rangle_{L}$ and $|1D_{2}^{\prime}\rangle_{H}$ in $B$ and $B_{s}$
spectroscopy on the decay widths. To investigate this effect, we plot the
decay width as a function of the mass in Fig. 4 with the mixing angle fixed at
$\phi_{1D}=-50.8^{\circ}$. The sensitivity of different decay modes to the
mass can also be seen clearly in the plot.

As a whole, the high-mass mixed states $B(|1D_{2}^{\prime}\rangle_{H})$ and
$B_{s}(|1D_{2}^{\prime}\rangle_{H})$ might be narrow states, which might be
observed in the $B^{*}\pi$ and $B^{*}K$ channels, respectively. However, the
low-mass mixed state $|1D_{2}\rangle_{L}$ should be a broad state. The typical
total decay widths for the low-mass mixed states $D(|1D_{2}\rangle_{L})$ and
$B(|1D_{2}\rangle_{L})$ are usually larger than $300$ MeV, which might be too
broad to be observed in experiments. However, those low-mass mixed states
containing a strange quark, $D_{s}(|1D_{2}\rangle_{L})$ and
$B_{s}(|1D_{2}\rangle_{L})$, have a relatively narrower width, $\sim 250$ MeV,
which have some possibilities to be observed in future experiments.

###  III.4 The $1^{3}D_{3}$ states

The $D(2760)$ is a new excitation in the $D$-meson family observed by _BABAR_
in the $D\pi$ channel with a mass of $M=2763.3\pm 4.6$ MeV and a width of
$\Gamma=60.9\pm 8.7$ MeV. Although its mass and width are very close to those
of the $D(2750)$, they might be two different resonances due to several
reasons, which have been explained in Ref. Zhong:2010vq . Recently, this state
was confirmed by the LHCb Collaboration in the $D\pi$ and $D^{*}\pi$ channels,
and their helicity analysis indicates that this state should have a natural
parity, i.e., $J^{P}=0^{+},1^{-},2^{+},3^{-},\cdot\cdot\cdot$. In our previous
work Zhong:2010vq , this new resonance $D(2760)$ was studied carefully. It is
predicted that the $D(2760)$ could be assigned to the
$1^{3}D_{3}(J^{P}=3^{-})$ state. Our conclusion is consistent with that in
Refs. Wang:2010ydc ; Sun:2010pg ; Lu:2014zua . Taking the $D(2760)$ as the
assignment of the $1^{3}D_{3}$, the total decay width is $\Gamma\simeq 68$
MeV, which is consistent with the data. Its strong decays are dominated by the
$D\pi$ and $D^{*}\pi$ channels, and the partial width ratio is

$\displaystyle\frac{\Gamma[D^{*}\pi]}{\Gamma[D\pi]}\simeq 0.65,$ (14)

which can be tested in future experiments.

Figure 5: The partial decay widths and total decay width of the $1^{3}D_{3}$
excitations in the $B$\- and $B_{s}$-meson families as functions of the mass.
In the figure, we have hidden some decay channels because of their small
partial decay widths.

Furthermore, in our previous work Zhong:2009sk , we suggested there are two
largely overlapping resonances at 2.86 GeV in the $D_{s}$-meson family. One
state corresponds to the $1^{3}D_{3}$ $(J^{P}=3^{-})$ state [denoted by
$D_{sJ_{1}}(2860)$], which dominantly decays into the $DK$ channel, while the
other one corresponds to the mixed state $|1D_{2}^{\prime}\rangle_{H}$ with
$J^{P}=2^{-}$ [denoted by $D_{sJ_{2}}(2860)$], which dominantly decays into
the $D^{*}K$ channel. Considering the $D_{sJ_{1}}(2860)$ as the $1^{3}D_{3}$
state, we have studied its strong decay properties. The predicted decay width
is $\Gamma\simeq 45$ MeV, which is highly comparable with the experimental
data ($\Gamma=58\pm 11$ MeV). This state has two main decay channels $DK$ and
$D^{*}K$, and the predicted partial width ratio is

$\displaystyle\frac{\Gamma[D^{*}K]}{\Gamma[DK]}\simeq 0.5,$ (15)

which can be tested in future experiments. The present experimental
measurement of the ratio, $\Gamma[D^{*}K]/\Gamma[DK]=1.10\pm 0.15\pm 0.19$,
might include the contributions of both $D_{sJ_{1}}(2860)$ and
$D_{sJ_{2}}(2860)$. Thus, its value is obviously larger than the theoretical
predictions by assuming the $D_{sJ}(2860)$ as the $1^{3}D_{3}$ state only. If
both $D(2760)$ and $D_{sJ_{1}}(2860)$ correspond to the $1^{3}D_{3}$
excitation indeed, their flavor partners in $B$ and $B_{s}$ spectroscopy
should be observed in future experiments as well.

In the $B$-meson family, the predicted mass of $1^{3}D_{3}$ is around $6.0\sim
6.1$ GeV Godfrey:1985xj ; Di Pierro:2001uu ; Ebert:2009ua ; Sun:2014wea .
Considering the $D(2760)$ and $D_{sJ_{1}}(2860)$ as the $1^{3}D_{3}$
assignment, their mass is systemically overestimated by $50\sim 100$ MeV in
theory. Thus, we estimate the mass of $B(1^{3}D_{3})$ is in the range of
$(5.95\sim 6.05)$ GeV. We calculate the strong decay properties of
$B(1^{3}D_{3})$, which are shown in Fig. 5. From the figure, it is seen that
the $B(1^{3}D_{3})$ is a narrow state with a width of $\Gamma\simeq 50\sim
120$ MeV. Its decays are dominated by the $B\pi$ and $B^{*}\pi$ channels. The
partial width ratio between $B\pi$ and $B^{*}\pi$ is insensitive to the mass,
and its predicted value is

$\displaystyle\frac{\Gamma[B^{*}\pi]}{\Gamma[B\pi]}\simeq 0.9.$ (16)

Recently, the CDF Collaboration claimed that they found some evidence of a new
resonance $B(5970)$ in the $B\pi$ channel Aaltonen:2013atp . The central value
of the width is $\Gamma\sim 70$ MeV. From Fig. 5, it is interestingly found
that the $B(5970)$ is most likely to be the $1^{3}D_{3}$ assignment.
Considering the $B(5970)$ as the $1^{3}D_{3}$ state, the predicted decay width
is $\Gamma\simeq 60$ MeV, and the $B\pi$ and $B^{*}\pi$ decay channels are its
dominant decay modes. Our predictions are in a good agreement with the
observations. Although the mass of the $B(5970)$ is close to the estimated
masses of the mixed states $B(|(SD)_{1}\rangle_{L})$ and
$B(|1D_{2}^{\prime}\rangle_{H})$, they are not good candidates for the
$B(5970)$. For the $B(|(SD)_{1}\rangle_{L})$, when we set its mass with $\sim
5.97$ GeV, its theoretical width is $\sim 300$ MeV, which is too large to
compared with the width of the $B(5970)$, while for the
$B(|1D_{2}^{\prime}\rangle_{H})$, its strong decays are governed by the
$B^{*}\pi$, and the $B\pi$ channel is forbidden. Thus, it could not be
assigned to the $B(5970)$.

Finally, we study the strong decays of $1^{3}D_{3}$ in the $B_{s}$-meson
family. Usually, the mass of the $B_{s}$ resonances is about 100 MeV larger
than that of the $B$ resonances Godfrey:1985xj ; Di Pierro:2001uu ;
Ebert:2009ua . Thus, we estimate the mass of $B_{s}(1^{3}D_{3})$ might be in
the range of $(6.05\sim 6.15)$ GeV. We have studied the strong decay
properties of $B_{s}(1^{3}D_{3})$, which are shown in Fig. 5. It is found that
the $B_{s}(1^{3}D_{3})$ might be a narrow state with a width of
$\Gamma\simeq(25\sim 75)$ MeV, and its strong decays are dominated by the $BK$
and $B^{*}K$ channels. If the $B(5970)$ corresponds to the $B(1^{3}D_{3})$
indeed, the mass of the $B_{s}(1^{3}D_{3})$ might be $M\simeq 6.07$ GeV. With
this mass, the predicted width for the $B_{s}(1^{3}D_{3})$ is

$\displaystyle\Gamma\simeq 30\ \mathrm{MeV},$ (17)

and the predicted ratio between $BK$ and $B^{*}K$ is

$\displaystyle\frac{\Gamma[BK]}{\Gamma[B^{*}K]}\simeq 1.4.$ (18)

We hope the experimenters can carry out a search for the $B_{s}(1^{3}D_{3})$
in the $BK$ and $B^{*}K$ channels, which is also helpful to clarify the nature
of the newly observed state $B(5970)$.

In brief, the $D(2760)$, $D_{sJ_{1}}(2860)$ and $B(5970)$ might be classified
as the low-lying $D$-wave excitations with $J^{P}=3^{-}$ (i.e., $1^{3}D_{3}$).
The last unobserved one in the $B_{s}$-meson family should be a narrow state,
which is most likely to be found in the $BK$ and $B^{*}K$ channels.

Figure 6:  The partial decay width and total decay widths of the $2^{3}P_{0}$
states in the heavy-light mesons as functions of the mass. In the figure, we
have hidden some decay channels because of their small partial decay widths.
The mass of the $B(2^{1}S_{0})$ is adopted as $M=5886$ MeV.

###  III.5 The $2^{3}P_{0}$ states

In the $D$-meson family, the predicted mass for the $2^{3}P_{0}$ state is
$\sim 2.95$ GeV Di Pierro:2001uu ; Ebert:2009ua ; Sun:2013qca . For the
uncertainties of the predicted mass, we vary the mass of $D(2^{3}P_{0})$ in
the range of $M=(2.9\sim 3.0)$ GeV, and plot the strong decay properties in
Fig. 6. From the figure it is seen that the $D(2^{3}P_{0})$ might be a very
broad state with a width of $\Gamma\simeq 600\sim 800$ MeV. This state
dominantly decays into the $D(2550)\pi$, $D\pi$ and $D(2430)\pi$ channels. A
very broad width for the $D(2^{3}P_{0})$ was also predicted in a recent study
with the ${}^{3}P_{0}$ model Lu:2014zua . Such a broad resonance might be
difficult to observe in experiments.

Recently, the LHCb Collaboration observed a new excited $D$-meson state
$D_{J}^{*}(3000)$ in the $D\pi$ channel with a rather narrow width,
$\Gamma\simeq 110$ MeV Aaij:2013sza . Their further helicity analysis
indicates that this new state should have a natural parity, that is, the spin-
parity number of the $D_{J}^{*}(3000)$ should be
$J^{P}=0^{+},1^{-},2^{+},\cdot\cdot\cdot$. In Ref. Sun:2013qca , Sun _et al._
believed that the $D_{J}^{*}(3000)$ could be explained as the $D(2^{3}P_{0})$.
If we exclude the contribution of the $D(2550)\pi$ channel to the decay width,
our results agree with the predictions in Ref. Sun:2013qca . However,
including the contribution of the $D(2550)\pi$ decay channel, our calculations
show that the decay width, $\Gamma\simeq 800$ MeV, is too broad to compare
with the data. Thus, the $D_{J}^{*}(3000)$ as the $D(2^{3}P_{0})$ assignment
still bears controversies, although the parity and decay modes are consistent
with observations.

In the $D_{s}$-meson family, the predicted mass of the $2^{3}P_{0}$ is about
$3.05$ GeV, which is $\sim 100$ MeV heavier than that of $D(2^{3}P_{0})$ Di
Pierro:2001uu ; Ebert:2009ua . Considering the uncertainties of the predicted
mass, we plot the strong decay properties of $D_{s}(2^{3}P_{0})$ as functions
of the mass in Fig. 6. It is shown that if the mass of $D_{s}(2^{3}P_{0})$ is
less than $3.05$ GeV, the $D_{s}(2^{3}P_{0})$ has a moderate width of
$\Gamma\simeq 150\sim 200$ MeV, and its strong decays are governed by the $DK$
channel. In this case, the $D_{s}(2^{3}P_{0})$ is most likely to be observed
in the $DK$ channel. On the other hand, if the mass of $D_{s}(2^{3}P_{0})$ is
larger than $3.05$ GeV, the decay channel $D(2550)K$ will be opened, which
dominates the strong decays. Then, the $D_{s}(2^{3}P_{0})$ might be a broad
state with a width of $\Gamma\simeq 400\sim 800$ MeV. In this case, the
$D_{s}(2^{3}P_{0})$ might be too broad to observe in experiments.

In the $B$-meson family, the predicted mass of the $2^{3}P_{0}$ is about $6.2$
GeV Di Pierro:2001uu ; Ebert:2009ua ; Sun:2014wea . To know about the decay
properties of the $B(2^{3}P_{0})$, in Fig. 6 we plot its decay width as a
function of mass in the range of $M=(6.1\sim 6.25)$ GeV. From the figure it is
seen that the $B(2^{3}P_{0})$ might be a broad state with a width of
$\Gamma\simeq 400\sim 800$ MeV. This state dominantly decays into the
$B(1^{3}P_{0})\pi$, $B\pi$, and $B(1P_{1})\pi$ channels. Such a broad
resonance might be difficult to observe in experiments.

In the $B_{s}$-meson family, the predicted mass of the $2^{3}P_{0}$ is about
$M\simeq 6.3$ GeV Di Pierro:2001uu ; Ebert:2009ua ; Sun:2014wea . To know
about the decay properties of the $B_{s}(2^{3}P_{0})$, in Fig. 6 we plot its
decay width as a function of mass in the range of $M=(6.2\sim 6.35)$ GeV. From
the figure it is seen that the decay width of $B_{s}(2^{3}P_{0})$ is
$\Gamma\simeq 240$ MeV with $M\simeq 6.27$ GeV, which is much narrower than
that of $B(2^{3}P_{0})$. The strong decays of $B_{s}(2^{3}P_{0})$ are governed
by the $BK$ channel. The other decay modes $B(1P_{1})K$ and
$B(1P_{1}^{\prime})K$ also contribute to the decays obviously, if the mass of
$B_{s}(2^{3}P_{0})$ is larger than $6.25$ GeV. Since the predicted width of
$B_{s}(2^{3}P_{0})$ is not very broad, this resonance might be observed in the
$BK$ channel in future experiments.

As a whole, in the $2^{3}P_{0}$ states, the widths of $D(2^{3}P_{0})$ and
$B(2^{3}P_{0})$ might be too broad to observe in experiments. However, the
widths of the resonances $D_{s}(2^{3}P_{0})$ and $B_{s}(2^{3}P_{0})$ are
relatively narrower, thus, they might be observed in the $DK$ and $BK$
channels, respectively.

###  III.6 The $2^{1}P_{1}$-$2^{3}P_{1}$ mixing

The $D_{sJ}(3040)$ is a new broad state in the $D_{s}$-meson family observed
by $BABAR$ in the $D^{*}K$ channel Aubert:2009ah . It has a mass of $M=3044\pm
8^{+30}_{-5}$ MeV and a broad width of $\Gamma=239\pm 35^{+46}_{-42}$ MeV.
According to our previous study Zhong:2009sk , the $D_{sJ}(3040)$ can be
naturally explained as the low-mass mixed state $|2P_{1}\rangle_{L}$ via the
$2^{1}P_{1}$-$2^{3}P_{1}$ mixing:

$\left(\begin{array}[]{c}|2P_{1}\rangle_{L}\cr|2P_{1}^{\prime}\rangle_{H}\end{array}\right)=\left(\begin{array}[]{cc}\cos\phi_{2P}&\sin\phi_{2P}\cr-\sin\phi_{2P}&\cos\phi_{2P}\end{array}\right)\left(\begin{array}[]{c}|2^{1}P_{1}\rangle\cr|2^{3}P_{1}\rangle\end{array}\right),$
(19)

with a mixing angle $\phi_{2P}\simeq-(51\pm 27)^{\circ}$. This mixing angle is
consistent with that ($\phi_{2P}=-54.7^{\circ}$) derived in the heavy quark
symmetry limit Close:2005se . It should be mentioned that with the mixing
angle $\phi_{2P}=-54.7^{\circ}$, the low-mass state $|2P_{1}\rangle_{L}$
usually has a broad width, while the high-mass state
$|2P_{1}^{\prime}\rangle_{H}$ has a narrower width Close:2005se ;
Swanson:2006st . As a broad mixed state, the strong decays of the
$D_{sJ}(3040)$ are dominated by the $D^{*}K$ channel. The partial widths of
$D_{0}(2400)K$, $D_{1}(2430)K$, and $D_{2}(2460)K$ are sizable as well. Our
conclusion is consistent with that obtained in Refs. Sun:2009tg ; Ebert:2009ua
; Mohler:2011ke .

As the physical partner of $|2P_{1}\rangle_{L}$, the high-mass state
$|2P_{1}^{\prime}\rangle_{H}$ might be observed in experiments. Assuming the
mass of the $|2P_{1}^{\prime}\rangle_{H}$ is in the range of $(3.04\sim 3.2)$
GeV, its strong decay properties have been analyzed in our previous work
Zhong:2009sk . The decay width of $|2P_{1}^{\prime}\rangle_{H}$ is narrower
than that of the $D_{sJ}(3040)$, which increases fast with the increasing
mass. The main decay channels might be $D_{0}(2400)K$, $D_{1}(2430)K$ and
$DK^{*}$.

If the $D_{sJ}(3040)$ could indeed be assigned as a mixed state via the
$2^{1}P_{1}$-$2^{3}P_{1}$ mixing, the other mixed states between $2^{1}P_{1}$
and $2^{3}P_{1}$ in the $D$-, $B$\- and $B_{s}$-meson spectroscopy might be
found in future experiments.

Firstly, we study the mixed states of $2^{1}P_{1}$-$2^{3}P_{1}$ in the
$D$-meson family. According to the predictions in Refs. Godfrey:1985xj ; Di
Pierro:2001uu ; Ebert:2009ua , the mass of the low-mass state
$D(|2P_{1}\rangle_{L})$ might be in the range of $(2.9\sim 3.0)$ GeV, while
the mass of the high-mass state $D(|2P_{1}^{\prime}\rangle_{H})$ might be in
the range of $(3.0\sim 3.1)$ GeV. Adopting the mixing angle
$\phi_{2P}=-54.7^{\circ}$ predicted in the heavy quark symmetry limit, we have
plotted their partial decay widths and total decay width as functions of the
mass in Fig. 7. From the figure, it is seen that the low-mass state
$D(|2P_{1}\rangle_{L})$ is a broad state with a width of $\Gamma\simeq(370\pm
90)$ MeV, its strong decays are dominated by the $D^{*}\pi$ and $D(2600)\pi$
channels, and the partial decay widths of the $D_{0}(2400)\pi$,
$D_{1}(2430)\pi$ and $D^{*}_{s}K$ channels are sizable as well.

Although the high-mass state $D(|2P_{1}^{\prime}\rangle_{H})$ has a relatively
narrower width than the low-mass state $D(|2P_{1}\rangle_{L})$, the
$D(|2P_{1}^{\prime}\rangle_{H})$ is also a very broad state with a width of
$\Gamma\simeq(300\pm 90)$ MeV when we adopt its mass in the range of $(3.0\sim
3.1)$ GeV. Its strong decays are governed by the $D(2600)\pi$ channel. The
other channels $D\rho$, $D_{0}(2400)\pi$, $D_{s0}(2317)K$ and $D_{1}(2430)\pi$
also have sizable contributions to the decays.

Figure 7:  The partial decay widths and total decay width of the mixed states
via the $2^{1}P_{1}$-$2^{3}P_{1}$ mixing in the $D$-meson family as functions
of the mass. Here, the mixing angle $\phi_{2P}=-54.7^{\circ}$ is adopted. In
the figure, we have hidden some decay channels because of their small partial
decay widths.  Figure 8:  The variation the decay properties of the mixed
state $|2P_{1}\rangle_{L}$ as a candidate of $D_{J}(3000)$ with the mixing
angle $\phi_{2P}$. The left shaded band corresponds to the possible mixing
angle region derived from the strong decay properties of the $D_{sJ}(3040)$,
while the right shaded band corresponds to the mixing angle region derived
from the strong decay properties of the $D_{J}(3000)$. In the figure, we have
hidden some decay channels for their small partial decay widths.

Recently, a new resonance $D_{J}(3000)$ was observed by the LHCb. Its measured
mass and width are $M\simeq 2972$ MeV and $\Gamma\simeq 188\pm 45$ MeV,
respectively. Furthermore, the helicity analysis from LHCb indicates that this
state has an unnatural parity, that is, its spin-parity should be
$J^{P}=0^{-},1^{+},2^{-},\cdot\cdot\cdot$. According to the observed mass, the
observed $D^{*}\pi$ decay mode and possible $J^{P}$ numbers of the
$D_{J}(3000)$, we predict that the $D_{J}(3000)$ might be a good candidate for
the low-mass state $D(|2P_{1}\rangle_{L})$. As the $D(|2P_{1}\rangle_{L})$
candidate with the mixing angle $\phi_{2P}=-54.7^{\circ}$ derived in the heavy
quark symmetry limit, the $D_{J}(3000)$ might have a broad width of
$\Gamma\simeq 360$ MeV (see Fig. 7), which is about a factor 2 larger than the
data ($\Gamma\simeq 188\pm 45$ MeV). The predicted partial decay width ratio
of the main two decay channels $D^{*}\pi$ and $D(2600)\pi$ is

$\displaystyle\frac{\Gamma[D^{*}\pi]}{\Gamma[D(2600)\pi]}\simeq 0.7.$ (20)

However, we have noticed that the mixing angle $\phi_{2P}$ determined by the
strong decay properties of the $D_{sJ}(3040)$ bares a large uncertainty.
Taking the $D_{J}(3000)$ as a candidate of the $D(|2P_{1}\rangle_{L})$ state,
we have plotted the partial widths and total decay width as functions of the
mixing angle $\phi_{2P}$ in Fig. 8. From the figure, it is found that the
decay width is very sensitive to the mixing angle. If we adopt a mixing angle
in the range of $\phi_{2P}=-(3\sim 26)^{\circ}$, the decay width of the
$D_{J}(3000)$ can be explained. The mixing angle for $D_{J}(3000)$ has an
overlap with that for the $D_{sJ}(3040)$ in the range of $\phi_{2P}=-(20\sim
26)^{\circ}$. It is found that the mixing angle $\phi_{2P}=-(20\sim
26)^{\circ}$ obtained in present work is close to the mixing angle
$\phi_{2P}=-30.4^{\circ}$ suggested in Godfrey:2013aaa .

If both $D_{sJ}(3040)$ and $D_{J}(3000)$ can be assigned as the mixed state
$|2P_{1}\rangle_{L}$, they should share nearly the same mixing angle. Taking
the same mixing angle we predict that the width of the $D_{J}(3000)$ should be
a little larger than that of the $D_{sJ}(3040)$ for the larger decay phase
space and more decay channels of the $D_{J}(3000)$.

Recently, the strong decay properties of the $D_{J}(3000)$ were studied in
Ref. Sun:2013qca  as well. Considering the $D_{J}(3000)$ as the
$D(|2P_{1}\rangle_{L})$ with the mixing angle $\phi_{2P}=-54.7^{\circ}$, they
could well explain the measured width of the $D_{J}(3000)$. However, it should
be pointed out that they did not include the $D(2600)\pi$ decay mode in their
calculations. Excluding the $D(2600)\pi$, our prediction is in agreement with
that in Ref. Sun:2013qca . To understand the nature of the $D_{J}(3000)$, more
observations are needed in future experiments.

Figure 9: The decay width of the mixed states via the
$2^{1}P_{1}$-$2^{3}P_{1}$ mixing in the $B$\- and $B_{s}$-meson families as a
function of the mass. Here, the mixing angle $\phi_{2P}=-54.7^{\circ}$ is
adopted. The $B(5920)$ stands for the mixed state
$B(|(SD)_{1}\rangle_{L})\pi$. In the figure, we have hidden some decay
channels because of their small partial decay widths.

In the $B$-meson family, the predicted mass for the low-mass state
$B(|2P_{1}\rangle_{L})$ is $M\simeq 6.2$ GeV Godfrey:1985xj ; Di Pierro:2001uu
; Ebert:2009ua , the mass of the high-mass state
$B(|2P_{1}^{\prime}\rangle_{H})$ is about $(20\sim 80)$ MeV heavier than that
of the low-mass state. Considering the uncertainty of the predicted masses for
these states, in Fig. 9 we plot the partial decay widths and total decay width
of $B(|2P_{1}\rangle_{L})$ and $B(|2P_{1}^{\prime}\rangle_{H})$ as functions
of the mass in the possible range, where we adopt the mixing angle
$\phi_{2P}=-54.7^{\circ}$ derived in the heavy quark symmetry limit. From the
figure, we find that the low-mass state $B(|2P_{1}\rangle_{L})$ is a broad
state with a width of $\Gamma\simeq(450\pm 150)$ MeV. The $B^{*}\pi$ decay
mode governs its strong decays, while the other decay channels
$B(|(SD)_{1}\rangle_{L})\pi$, $B(1P_{1})\pi$, $B(1^{3}P_{2})\pi$,
$B_{s}^{*}K$, $B(1^{3}P_{0})\pi$ and $B^{*}\eta$ also have sizable partial
decay widths.

The width of the high-mass state $B(|2P_{1}^{\prime}\rangle_{H})$,
$\Gamma\simeq(50\sim 450)$ MeV, is much narrower than that of the low-mass
state $B(|2P_{1}\rangle_{L})$. The $B(|2P_{1}^{\prime}\rangle_{H})$ dominantly
decays into $B(|(SD)_{1}\rangle_{L})\pi$, $B(1^{3}P_{0})\pi$, $B(1P_{1})\pi$
and $B(1^{3}P_{2})\pi$. It might be a challenge to observe the
$B(|2P_{1}^{\prime}\rangle_{H})$ in these final states with higher excitations
of the $B$ meson.

Since the $D_{sJ}(3040)$ as a broad state has been observed in experiments,
the broad resonance $B(|2P_{1}\rangle_{L})$ might be observed in the
$B^{*}\pi$ as well. To further know about the effects of the uncertainties of
the mixing angle on the strong decays of $B(|2P_{1}\rangle_{L})$, we have
plotted the partial widths and total decay width of $B(|2P_{1}\rangle_{L})$ as
functions of the mixing angle $\phi_{2P}$ in Fig. 10, where we have fixed the
mass of $B(|2P_{1}\rangle_{L})$ with $M=6175$ MeV predicted in Di
Pierro:2001uu . From the figure, it is seen that even the mixing angle
$\phi_{2P}$ varies in a large range
$\phi_{2P}\simeq-20^{\circ}\sim-80^{\circ}$, the strong decays of
$B(|2P_{1}\rangle_{L})$ are still dominated by the $B^{*}\pi$ channel, and the
uncertainty of the decay width is no more than 60 MeV.

In the $B_{s}$-meson family, the predicted mass for the low-mass state
$B_{s}(|2P_{1}\rangle_{L})$ is $M\simeq(6.28\sim 6.32)$ GeV Godfrey:1985xj ;
Di Pierro:2001uu ; Ebert:2009ua , the mass of the high-mass state
$B_{s}(|2P_{1}^{\prime}\rangle_{H})$ is about $(20\sim 30)$ MeV heavier than
that of the low-mass state.

Figure 10: The partial decay width and total decay width for the mixed states
via $2^{1}P_{1}$-$2^{3}P_{1}$ mixing in the $B$\- and $B_{s}$-meson families
as functions of the mixing angle $\phi_{2P}$. The shaded region corresponds to
the possible mixing angle region derived from the strong decay properties of
the $D_{sJ}(3040)$ and $D_{J}(3000)$. The masses of the low-mass mixed states
$|2P_{1}\rangle_{L}$ in the $B$\- and $B_{s}$-meson families are adopted as
6.175 GeV and 6.275 GeV, respectively. The masses of the high-mass mixed
states $|2P_{1}^{\prime}\rangle_{H}$ in the $B$\- and $B_{s}$-meson families
are adopted as 6.225 GeV and 6.30 GeV, respectively. In the figure, we have
hidden some decay channels because of their small partial decay widths.

We have plotted the partial decay widths and total decay width of
$B_{s}(|2P_{1}\rangle_{L})$ and $B_{s}(|2P_{1}^{\prime}\rangle_{H})$ as
functions of the mass with the mixing angle $\phi_{2P}=-54.7^{\circ}$ in Fig.
9. From the figure, we find that the low-mass state
$B_{s}(|2P_{1}\rangle_{L})$ has a broad width of $\Gamma\simeq(220\pm 70)$ MeV
within the predicted mass region $M\simeq(6.28\sim 6.32)$ GeV. The $B^{*}K$
decay channel governs its strong decays, while the other decay channels
$B(1P_{1})K$, $B(1^{3}P_{2})K$, $B(1^{3}P_{0})K$ also have sizable partial
decay widths. The width of the high-mass state
$B_{s}(|2P_{1}^{\prime}\rangle_{H})$ is $\Gamma\simeq(95\pm 33)$ MeV within
the possible mass region $M\simeq(6.30\sim 6.35)$ GeV, which is much narrower
than that of $B_{s}(|2P_{1}\rangle_{L})$. The
$B_{s}(|2P_{1}^{\prime}\rangle_{H})$ dominantly decays into the
$B(1^{3}P_{0})K$, $B(1P_{1})K$, and $BK^{*}$ channels. Although the
$B_{s}(|2P_{1}^{\prime}\rangle_{H})$ has a relatively narrow width, it might
be a great challenge to observe this resonance in these final states with
higher $B_{s}$ resonances.

The decay width of $B_{s}(|2P_{1}\rangle_{L})$ is comparable with that of the
$D_{sJ}(3040)$. Thus, the $B_{s}(|2P_{1}\rangle_{L})$ is most likely to be
observed in the $B^{*}K$ channel. To further know about the effects of the
uncertainties of the mixing angle on the strong decays of
$B_{s}(|2P_{1}\rangle_{L})$, we have plotted the partial widths and total
decay width of $B_{s}(|2P_{1}\rangle_{L})$ as functions of the mixing angle
$\phi_{2P}$ in Fig. 10, where we have fixed the mass of
$B_{s}(|2P_{1}\rangle_{L})$ with $M=6275$ MeV. From the figure, it is seen
that even the mixing angle $\phi_{2P}$ is changed in a large range
$\phi_{2P}\simeq-20^{\circ}\sim-80^{\circ}$, the strong decays of
$B_{s}(|2P_{1}\rangle_{L})$ are still dominated by the $B^{*}K$ channel, and
the uncertainty of the decay width is no more than 50 MeV.

In summary, the strong decay properties of both $D_{sJ}(3040)$ and
$D_{J}(3000)$ can be explained by assigning them as the low-mass mixed state
$|2P_{1}\rangle_{L}$ via the $2^{1}P_{1}$-$2^{3}P_{1}$ with a mixing angle in
the range of $\phi_{2P}=-(20\sim 26)^{\circ}$. The mixed state
$B_{s}(|2P_{1}\rangle_{L})$ is a narrow state, which is most likely to be
observed in the $B^{*}K$ channel. The mixed state $B(|2P_{1}\rangle_{L})$
might have a broad width; thus, its discovery potentials might be small. The
high-mass mixed states $D(|2P_{1}^{\prime}\rangle_{H})$,
$D_{s}(|2P_{1}^{\prime}\rangle_{H})$, $B(|2P_{1}^{\prime}\rangle_{H})$, and
$B_{s}(|2P_{1}^{\prime}\rangle_{H})$ are usually narrower than those of low-
mass states. These states dominantly decay into the first $P$-wave states by
emitting a light pseudoscalar meson. Observations in the final states
containing a low-lying $P$-wave state are expected to be carried out in future
experiments.

Figure 11: The partial decay width and total decay widths of the $2^{3}P_{2}$
states in the heavy-light mesons as functions of the mass. In the figure, we
have hidden some decay channels because of their small partial decay widths.
The mass of the $B(1^{3}D_{3})$ is adopted as $M=5970$ MeV.

###  III.7 The $2^{3}P_{2}$ states

In the $D$-meson family, the predicted mass of the $2^{3}P_{2}$ is about
$3.02$ GeV Di Pierro:2001uu ; Ebert:2009ua ; Sun:2014wea . To know about the
decay properties of the $D(2^{3}P_{2})$, in Fig. 11 we plot its decay width as
a function of mass in the range of $M=(2.95\sim 3.05)$ GeV. From the figure,
it is found that the $D(2^{3}P_{2})$ mainly decays into the $D_{1}(2430)\pi$,
$D(2550)\pi$ and $D(2600)\pi$ channels. The predicted total decay width of
$D(2^{3}P_{2})$ is $\Gamma\simeq 150$ MeV, if we adopt the mass $\sim 3.01$
GeV as predicted in Ref. Ebert:2009ua .

Considering the newly observed state $D_{J}^{*}(3000)$ as a candidate of
$D(2^{3}P_{2})$, we find that the observed mass, width ($\Gamma\simeq 110$
MeV) and natural parity of the $D_{J}^{*}(3000)$ Aaij:2013sza  can be well
explained. However, the $D\pi$ decay channel is not the main decay channel of
$D(2^{3}P_{2})$, which is tiny compared with the dominant decay mode
$D_{1}(2430)\pi$. The predicted branching fraction is

$\displaystyle\frac{\Gamma[D\pi]}{\Gamma_{\mathbf{total}}}\simeq 1\sim 2\%,$
(21)

which is compatible with the predictions in Refs. Sun:2013qca ; Lu:2014zua .
To clarify the nature of the $D_{J}^{*}(3000)$, it is suggested to further
observe this state in the $D_{1}(2430)\pi$ channel. If the $D_{J}^{*}(3000)$
corresponds to the $D(2^{3}P_{2})$ state, it should be found in the
$D_{1}(2430)\pi$ channel as well.

In the $D_{s}$-meson family, the predicted mass of the $2^{3}P_{2}$ is about
$3.15$ GeV Di Pierro:2001uu ; Ebert:2009ua ; Sun:2014wea . To know about the
decay properties of the $D_{s}(2^{3}P_{2})$, in Fig. 11 we plot its decay
width as a function of mass in the range of $M=(3.1\sim 3.2)$ GeV. From the
figure, it is found that the $D_{s}(2^{3}P_{2})$ mainly decays into the
$D_{1}(2430)K$ channel. The total decay width of $D_{s}(2^{3}P_{2})$ is
$\Gamma\simeq 120\pm 60$ MeV. This resonance might be observed in the
$D_{1}(2430)K$ final states.

In the $B$-meson family, the predicted mass of the $2^{3}P_{2}$ is about
$6.19\sim 6.26$ GeV Di Pierro:2001uu ; Ebert:2009ua . In Fig. 11, we have
plotted its decay width as a function of mass in the range of $M=(6.15\sim
6.3)$ GeV. From the figure it is found that the total decay width of
$B(2^{3}P_{2})$ is $\Gamma\simeq(50\sim 300)$ MeV. The $B(2^{3}P_{2})$ mainly
decays into the $B_{1}(5721)\pi$ channel.

In the $B_{s}$-meson family, the predicted mass of the $2^{3}P_{2}$ is about
$6.29\sim 6.36$ GeV Di Pierro:2001uu ; Ebert:2009ua ; Sun:2014wea . The decay
properties of the $B(2^{3}P_{2})$ have been shown in Fig. 11. From the figure,
it is found that the $B_{s}(2^{3}P_{2})$ mainly decays into $B_{1}(5721)K$.
The total decay width of $B_{s}(2^{3}P_{2})$ is $\Gamma\simeq(30\sim 180)$
MeV. This state might be observed in the $B_{1}(5721)K$ channel.

In a brief, the decay widths of the $2^{3}P_{2}$ excitations in the heavy-
light spectroscopy are not very broad, they have good potentials to be
observed in their dominant decay channels.

Figure 12: The partial decay width and total decay widths of the $1^{3}F_{2}$
states in the heavy-light mesons as functions of the mass. In the figure, we
have hidden some decay channels because of their small partial decay widths.
The mass of the $B(1D_{2}^{\prime})$ is adopted as $M=6025$ MeV.

###  III.8 The $1^{3}F_{2}$ states

In the $D$-meson family, the predicted mass for the $1^{3}F_{2}$ state is
$\sim 3.1$ GeV Di Pierro:2001uu ; Ebert:2009ua ; Sun:2014wea . Considering the
uncertainties of the mass, in Fig. 12 we plot the partial decay widths and
total width as functions of mass in the range of $M=(3.05\sim 3.15)$ GeV. It
is found that the $1^{3}F_{2}$ state in the $D$-meson family is a very broad
state with a width of $\Gamma\simeq 900\pm 200$ MeV. This state dominantly
decays into the $D(2420)\pi$, $D(2750)\pi$ and $D\pi$ channels. The partial
widths for the $D_{s}(2460)\pi$, $D^{*}\pi$ and $D(2760)\pi$ are also sizable.

In the $D_{s}$-meson family, the predicted mass for the $1^{3}F_{2}$ state is
$\sim 3.23$ GeV Di Pierro:2001uu ; Ebert:2009ua ; Sun:2014wea . Considering
the uncertainties of the mass, we vary the mass in the range of $M=(3.15\sim
3.30)$ GeV, the strong decay properties of the $1^{3}F_{2}$ state have been
shown in Fig. 12 as well. From the figure it is seen that the decay width of
the $D_{s}(1^{3}F_{2})$ state is $\Gamma\sim 500$ MeV. The main decay channels
are $D(2420)K$, $DK$ and $D^{*}K$. If the $D(2750)K$ channel is opened, the
$D(2750)K$ mode will dominate the decays, the decay width of the
$D_{s}(1^{3}F_{2})$ state will become very broad with a width of $\Gamma>700$
MeV.

In the $B$-meson family, the predicted mass for the $1^{3}F_{2}$ state is
$6.26\sim 6.41$ GeV Di Pierro:2001uu ; Ebert:2009ua ; Sun:2014wea . In this
mass region, we show the strong decay properties of the $B(1^{3}F_{2})$ state
in Fig. 12. It is found that this state is also a very broad state with a
width of $\Gamma\simeq 1200\pm 400$ MeV. It mainly decays into the
$B(1P_{1})\pi$, $B(1D_{2}^{\prime})\pi$, $B\pi$ and $B^{*}\pi$ channels.

In the $B_{s}$-meson family, the predicted mass for the $1^{3}F_{2}$ state is
$6.37\sim 6.50$ GeV Di Pierro:2001uu ; Ebert:2009ua . In this mass region, we
study the strong decays of the $B_{s}(1^{3}F_{2})$ state, our results are
shown in Fig. 12 as well. It is seen that this state is a very broad state
with a width of $\Gamma\simeq 400\sim 900$ MeV. The strong decays are governed
by the $B(1P_{1})K$. The other main decay channels are $BK$, $B^{*}K$ and
$B(1^{3}P_{2})K$.

As a whole, the $1^{3}F_{2}$ excitations in the heavy-light mesons are very
broad resonances, whose width is larger than 400 MeV. Such broad states might
be difficult to observe in experiments, which might explain why these states
are still missing in the heavy-light meson spectroscopy.

Figure 13:  The partial decay width and total decay width for the mixed states
via the $1^{1}F_{3}$-$1^{3}F_{3}$ mixing as functions of the mass. The mixing
angle is adopted as $\phi_{1F}=-49.1^{\circ}$. In the figure, we have hidden
some decay channels because of their small partial decay widths.

###  III.9 The $1^{3}F_{3}$-$1^{3}F_{3}$ mixing

Since the heavy-light mesons are not charge conjugation eigenstates, state
mixing between spin $\mathbf{S}=0$ and $\mathbf{S}=1$ states with the same
$J^{P}$ can occur via the spin-orbit interactions Godfrey:1986wj ;
Close:2005se ; Swanson:2006st . The physical states with $J^{P}=3^{+}$ can
then be described as

$\left(\begin{array}[]{c}|1F_{3}\rangle_{L}\cr|1F_{3}^{\prime}\rangle_{H}\end{array}\right)=\left(\begin{array}[]{cc}\cos\phi_{1F}&\sin\phi_{1F}\cr-\sin\phi_{1F}&\cos\phi_{1F}\end{array}\right)\left(\begin{array}[]{c}|1^{1}F_{3}\rangle\cr|1^{3}F_{3}\rangle\end{array}\right),$
(22)

where the subscriptions $L$ and $H$ stand for the low mass and high mass of
the physical states after the mixing. In the heavy quark symmetry limit, the
mixing angle $\phi_{1F}$ is predicted to be $\phi_{1F}\simeq-49.1^{\circ}$
Ebert:2009ua . It should be mentioned that with the mixing angle
$\phi_{1F}\simeq-49.1^{\circ}$, the low-mass state $|1F_{3}\rangle_{L}$
usually has a broad width, while the high-mass state
$|1F_{3}^{\prime}\rangle_{H}$ has a narrower width.

In the $D$-meson family, the predicted masses for the mixed states
$|1F_{3}\rangle_{L}$ and $|1F_{3}\rangle_{H}$ are about $3.07\sim 3.13$ GeV
and $3.12\sim 3.15$ GeV, respectively Di Pierro:2001uu ; Ebert:2009ua . With
the mixing angle $\phi_{1F}\simeq-49.1^{\circ}$ derived in the heavy quark
symmetry limit, we predicted the strong decay properties of the mixed states
$D(|1F_{3}\rangle_{L})$ and $D(|1F_{3}^{\prime}\rangle_{H})$ in their possible
mass region. The results are shown in Fig. 13. It is found that the
$D(|1F_{3}\rangle_{L})$ is a very broad state with a width of $\Gamma\simeq
600\sim 900$ MeV. Its strong decays are dominated by the $D(2760)\pi$,
$D(2460)\pi$ and $D^{*}\pi$ channels. The width of the
$D(|1F_{3}^{\prime}\rangle_{H})$ is $\Gamma\simeq 200\sim 300$ MeV, which is
much narrower than that of the low-mass state. The
$D(|1F_{3}^{\prime}\rangle_{H})$ mainly decays into the $D(2460)\pi$,
$D^{*}\pi$, $D\rho$, $D(2420)\pi$ and $D(2750)\pi$ channels.

In the $D_{s}$-meson family, the predicted masses for the mixed states
$|1F_{3}\rangle_{L}$ and $|1F_{3}^{\prime}\rangle_{H}$ are about $3.20\sim
3.26$ GeV and $3.25\sim 3.27$ GeV, respectively Di Pierro:2001uu ;
Ebert:2009ua . The strong decay properties of these mixed states are also
studied in their possible mass region. Our predictions are shown in Fig. 13.
It is seen that the width of the low-mass mixed state
$D_{s}(|1F_{3}\rangle_{L})$ is very broad, which is in the range of
$\Gamma\simeq 400\sim 800$ MeV. Its strong decays are governed by the
$D(2460)K$ and $D^{*}K$ channels. The high-mass state
$D_{s}(|1F_{3}^{\prime}\rangle_{H})$ has a much narrower decay width, i.e.,
$\Gamma\simeq 100\sim 250$ MeV. This high-mass state dominantly decays into
the $D(2460)K$, $D^{*}K$, and $DK^{*}$ channels.

Figure 14: The partial decay width and total decay width for the mixed states
via the $1^{1}F_{3}$-$1^{3}F_{3}$ mixing as functions of the mixing angle. The
masses for the low-mass mixed states $D(|1F_{3}\rangle_{L})$,
$D_{s}(|1F_{3}\rangle_{L})$, $B(|1F_{3}\rangle_{L})$ and
$B_{s}(|1F_{3}\rangle_{L})$ are adopted as 3100, 3230, 6305, and 6400 MeV,
respectively. The masses for the high-mass mixed states
$D(|1F_{3}^{\prime}\rangle_{H})$, $D_{s}(|1F_{3}^{\prime}\rangle_{H})$,
$B(|1F_{3}^{\prime}\rangle_{H})$ and $B_{s}(|1F_{3}^{\prime}\rangle_{H})$ are
adopted as 3130, 3260, 6335, and 6450 MeV, respectively. In the figure, we
have hidden some decay channels for their small partial decay widths.

In the $B$-meson family, the predicted masses for the mixed states
$|1F_{3}\rangle_{L}$ and $|1F_{3}^{\prime}\rangle_{H}$ are about $6.22\sim
6.39$ GeV and $6.27\sim 6.42$ GeV, respectively Di Pierro:2001uu ;
Ebert:2009ua . In the possible mass region, we study the strong decay
properties of these mixed states in the $B$-meson family. The results are
shown in Fig. 13. It is found that the low-mass mixed state
$B(|1F_{3}\rangle_{L})$ is a very broad state with a width of $\Gamma\simeq
650\sim 1400$ MeV. This state dominantly decays into the $B_{2}(5747)\pi$,
$B(1^{3}D_{3})\pi$, and $B^{*}\pi$ channels. While for the
$B(|1F_{3}^{\prime}\rangle_{H})$, the decay width is $\Gamma\simeq 200\sim
650$ MeV, which is sensitive to the mass. If the
$B(|1F_{3}^{\prime}\rangle_{H})$ state has a smaller mass as predicted in Ref.
Di Pierro:2001uu , it might be observed in its main decay channels $B^{*}\pi$
and $B_{2}(5747)\pi$.

In the $B_{s}$-meson family, the predicted masses for the mixed states
$|1F_{3}\rangle_{L}$ and $|1F_{3}^{\prime}\rangle_{H}$ are about $6.33\sim
6.47$ GeV and $6.38\sim 6.52$ GeV, respectively Di Pierro:2001uu ;
Ebert:2009ua . Considering the uncertainties of the mass, we have plotted the
strong decay properties of these mixed states as functions of the mass in Fig.
13. From the figure, it is seen that the $|1F_{3}\rangle_{L}$ state in the
$B_{s}$-meson family has a width of $\Gamma\simeq 200\sim 700$ MeV, whose
strong decays are dominated by the $B_{2}(5747)K$ and $B^{*}K$ channels. The
high-mass state $|1F_{3}\rangle_{H}$ has a relatively smaller width, which is
$\Gamma\simeq 100\sim 500$ MeV. Its strong decays are governed by the
$B^{*}K$, $BK^{*}$, and $B_{2}(5747)K$ channels. These mixed states
$|1F_{3}\rangle_{L}$ and $|1F_{3}^{\prime}\rangle_{H}$ in the $B_{s}$-meson
family might be observed in the $B^{*}K$ channel, if they have a smaller mass
as predicted in Ref. Di Pierro:2001uu .

In the calculations we have adopted the ideal mixing angle
$\phi_{1F}\simeq-49.1^{\circ}$ extracted from the heavy quark symmetry limit.
This ideal mixing angle might have some uncertainties. To see the effects of
the uncertainties of the mixing angle on the strong decay properties, we have
plotted the partial decay widths and total decay width as functions of the
mixing angle in Fig. 14. From the figure, it is seen that strong decay
properties of the mixed states do not change obviously, if we consider an
uncertainty $\pm 20^{\circ}$ around the ideal mixing angle
$\phi_{1F}\simeq-49.1^{\circ}$.

In summary, the mixed states $|1F_{3}\rangle_{L}$ in the heavy-light mesons
are usually broad states, they should be difficult to find in experiments. The
decay width of the high-mass mixed states $|1F_{3}^{\prime}\rangle_{H}$ is
relatively narrower than that of the low-mass states. These high-mass states
$|1F_{3}^{\prime}\rangle_{H}$ might be observed in future experiments if they
have a smaller mass as predicted in Ref. Di Pierro:2001uu . However, these
high-mass states $|1F_{3}^{\prime}\rangle_{H}$ might be very broad states if
their masses are as large as those predicted in Ref. Ebert:2009ua .

Figure 15: The partial decay width and total decay width for the $1^{3}F_{4}$
excitations as functions of the mass. In the figure, we have hidden some decay
channels because of their small partial decay widths.

###  III.10 The $1^{3}F_{4}$ states

In the $D$-meson family, the predicted mass for the $1^{3}F_{4}$ state is
$3.09\sim 3.19$ GeV Di Pierro:2001uu ; Ebert:2009ua ; Sun:2014wea . In this
mass region, the predicted strong properties are shown in Fig. 15. The width
of the $D(1^{3}F_{4})$ state is $\Gamma\simeq 230\pm 70$ MeV. It dominantly
decays into the $D^{*}\rho$, $D\pi$, $D(2430)\pi$, $D(2460)\pi$, $D^{*}\pi$
and $D(2760)\pi$ channels. The branching ratios for $D^{*}\pi$ and $D\pi$ are
10% and 11%, respectively.

We find that if taking the newly observed natural parity resonance
$D^{*}_{J}(3000)$ as a candidate of the $D(1^{3}F_{4})$, the strong decay
properties of $D^{*}_{J}(3000)$ could be well explained (see Tab. 4). However,
the predicted mass is about $100\sim 200$ MeV larger than the data. If the
$D^{*}_{J}(3000)$ corresponds to the $D(1^{3}F_{4})$ state indeed, it should
be observed in both $D\pi$ and $D^{*}\pi$ channels. To clarify the puzzles in
the $D^{*}_{J}(3000)$, more observations are needed.

Table 4: The strong partial decay widths (MeV) and total width (MeV) of $D^{*}(3000)$ as a candidate of $D(1^{3}F_{4})$. Mode | $D\pi$ | $D^{*}\pi$ | $D(2420)\pi$ | $D(2430)\pi$ | $D(2460)\pi$ | $D(2550)\pi$ | $D(2600)\pi$ | $D(2750)\pi$ | $D(2760)\pi$
---|---|---|---|---|---|---|---|---|---
Width | 14.9 | 11.0 | 2.3 | 9.2 | 7.0 | 0.9 | $5.9\times 10^{-2}$ | 4.6 | 11.4
Mode | $D\eta$ | $D\eta^{\prime}$ | $D^{*}\eta$ | $D^{*}\eta^{\prime}$ | $D(2420)\eta$ | $D(2430)\eta$ | $D(2460)\eta$ | $D_{s}(2460)K$ | $D_{s}^{*}K$
Width | 1.0 | $3.1\times 10^{-2}$ | 0.5 | $8.9\times 10^{-5}$ | $5.6\times 10^{-4}$ | $1.0\times 10^{-3}$ | $8.1\times 10^{-12}$ | $3.6\times 10^{-3}$ | 0.8
Mode | $D_{s}K$ | $D\omega$ | $D^{*}\omega$ | $D\rho$ | $D^{*}\rho$ | $D_{s}K^{*}$ | $D_{s}^{*}K^{*}$ | Total |
Width | 2.1 | $9.2\times 10^{-6}$ | 4.5 | 1.6 | 14.5 | $5.9\times 10^{-7}$ | $6.4\times 10^{-4}$ | 86.4 |

In the $D_{s}$-meson family, the predicted mass for the $1^{3}F_{4}$ state is
$3.22\sim 3.30$ GeV Di Pierro:2001uu ; Ebert:2009ua ; Sun:2014wea . The strong
decay properties are shown in Fig. 15 as well. In the possible mass region,
the predicted width of the $D_{s}(1^{3}F_{4})$ state is $\Gamma\simeq 200\pm
100$ MeV. The main decay channels are $D^{*}K^{*}$, $DK$, $D^{*}K$,
$D(2430)K$, and $D(2460)K$. If the mass of the $D_{s}(1^{3}F_{4})$ state is
larger than the threshold of $D(2760)K$, the $D(2760)K$ channel will become a
dominant decay channel. The predicted partial width ratio of $DK$ and $D^{*}K$
is

$\displaystyle\frac{\Gamma[DK]}{\Gamma[D^{*}K]}\simeq 1.2.$ (23)

This state might be observed in both $DK$ and $D^{*}K$ channels.

In the $B$-meson family, the predicted mass for the $1^{3}F_{4}$ state is
$6.22\sim 6.38$ GeV Di Pierro:2001uu ; Ebert:2009ua ; Sun:2014wea . In this
mass region, we study the strong decay properties of the $B(1^{3}F_{4})$
state, which are shown in Fig. 15. From the figure, it is seen that the strong
decays of the $B(1^{3}F_{4})$ state are dominated by the $B\pi$ and
$B^{*}\pi$, their partial width ratio is

$\displaystyle\frac{\Gamma[B\pi]}{\Gamma[B^{*}\pi]}\simeq 1.1,$ (24)

which is less sensitive to the mass. Furthermore, the other decay channels
$B^{*}\rho$, $B(1^{3}D_{3})\pi$, $B(1^{2}P_{3})\pi$ and $B(1P_{1})\pi$ also
have obvious contributions to the strong decays. The decay width is sensitive
to the mass. In the possible mass region, the predicted width of the
$B(1^{3}F_{4})$ state is $\Gamma\simeq 120\sim 480$ MeV. If this state has a
mass of $\sim 6.2$ GeV as predicted in Ref. Di Pierro:2001uu , it might be
observed in future experiments. However, if the mass is close to $\sim 6.4$
GeV as predicted in Ref. Ebert:2009ua , the $B(1^{3}F_{4})$ state might be
difficult to be found for its broad width.

In the $B_{s}$-meson family, the predicted mass for the $1^{3}F_{4}$ state is
$6.33\sim 6.48$ GeV Di Pierro:2001uu ; Ebert:2009ua ; Sun:2014wea . In this
mass region, we have shown the strong decay properties in Fig. 15. The decay
width of the $B_{s}(1^{3}F_{4})$ is sensitive to the mass. The total decay
width is $\Gamma\simeq 60\sim 400$ MeV, which bares a large uncertainty. Its
strong decays are dominated by the $BK$, $B^{*}K$ and $B^{*}K^{*}$ channels.
The predicted partial width ratio of the $BK$ and $B^{*}K$ is

$\displaystyle\frac{\Gamma[BK]}{\Gamma[B^{*}K]}\simeq 1.2,$ (25)

which is insensitive to the mass. If this state has a smaller mass of $\sim
6.34$ GeV as predicted in Ref. Di Pierro:2001uu , it should be a narrow state
with a width of $\Gamma\simeq 80$ MeV. In this case, the $B_{s}(1^{3}F_{4})$
might be observed in both $BK$ and $B^{*}K$ channels.

As a whole the decay widths of the $1^{3}F_{4}$ excitations in the heavy-light
mesons are sensitive to the mass. If these states have a smaller mass as
predicted in Ref. Di Pierro:2001uu , they might have relatively narrow width
$\Gamma\sim 100$ MeV. In this case, these excitations have good discovery
potentials in future experiments. However, if these states have a larger mass
as predicted in Ref. Ebert:2009ua , they might be difficult to be found in
experiments for their broad widths. Finally, it should be pointed out that the
newly observed natural parity resonance $D^{*}_{J}(3000)$ seems to be a good
candidate of the $D(1^{3}F_{4})$, although its mass is about $100\sim 200$ MeV
less than the model predictions.

##  IV Summary

In the chiral quark model framework, we systematically study the strong decays
of the higher excited heavy-light mesons from the first radially excited
states up to the first $F$-wave states. We summarize our major results as
follows.

The first radially excited states $2^{1}S_{0}$ in the $D$-, $D_{s}$-, $B$\-
and $B_{s}$-meson spectroscopy are narrow states, and their predicted width
might be less than 100 MeV. If the $D(2550)$ corresponds to this excitation
indeed, the other radially excited states $D_{s}(2^{1}S_{0})$, $B(2^{1}S_{0})$
and $B_{s}(2^{1}S_{0})$ are most likely to be also observed in the $D^{*}K$,
$B^{*}\pi$ and $B^{*}K$ final states, respectively, for their narrower widths.

Configuration mixing might exist between the low-lying $J^{P}=1^{-}$ states
$2^{3}S_{1}$ and $1^{3}D_{1}$. Both $D(2600)$ and $D_{s1}(2700)$ could be
assigned as the low-mass mixed state $|(SD)_{1}\rangle_{L}$ via the
$2^{3}S_{1}$-$1^{3}D_{1}$ mixing. As flavor partners of the $D(2600)$ and
$D_{s1}(2700)$, the low-mass mixed states in the $B$\- and $B_{s}$-meson
families, $B(|(SD)_{1}\rangle_{L})$ and $B_{s}(|(SD)_{1}\rangle_{L})$, should
have a relatively narrow width. They are most likely to be observed in future
experiments. In the high-mass mixed states, the widths of the charmed-strange
state $D_{s}(|(SD)_{1}^{\prime}\rangle_{H})$ and the bottom-strange state
$B_{s}(|(SD)_{1}^{\prime}\rangle_{H})$ are as comparable as the low-mass mixed
states, which might be observed in the $DK$ and $BK$ channels, respectively.
While, the bottom state $B(|(SD)_{1}^{\prime}\rangle_{H})$ is a broad state,
which might be difficult to be found in experiments. It should be pointed out
that the newly observed resonance $D_{s1}(2860)$ by LHCb Collaboration might
correspond to the physical partners of the $D_{s1}(2700)$, i.e., the high-mass
state $D_{s}(|(SD)_{1}^{\prime}\rangle_{H})$.

In the low-lying $D$-wave states with $J^{P}=2^{-}$, $1^{1}D_{2}$ and
$1^{3}D_{2}$, there might be configuration mixing as well. The narrow
resonances $D(2750)$ and $D_{sJ_{2}}(2860)$ might be classified as the high-
mass mixed state $|1D_{2}^{\prime}\rangle_{H}$ via the
$1^{1}D_{2}$-$1^{3}D_{2}$ mixing. The other high-mass mixed states in the
$B$\- and $B_{s}$-meson families, $B(|1D_{2}^{\prime}\rangle_{H})$ and
$B_{s}(|1D_{2}^{\prime}\rangle_{H})$ have a narrow width as well. These two
narrow states might be observed in the $B^{*}\pi$ and $B^{*}K$, respectively.
However, the low-mass mixed state $|1D_{2}\rangle_{L}$ should be a broad
state. The typical total decay widths for the mixed states
$D(|1D_{2}\rangle_{L})$ and $B(|1D_{2}\rangle_{L})$ in the $D$\- and $B$-meson
families are usually larger than $300$ MeV, which might be too broad to be
observed in experiments. While the widths for the mixed states
$D_{s}(|1D_{2}\rangle_{L})$ and $B_{s}(|1D_{2}\rangle_{L})$ containing a
strange quark are $\sim 250$ MeV, which still have some possibilities to be
observed in future experiments.

For the low-lying $D$-wave excitations with $J^{P}=3^{-}$, $1^{3}D_{3}$, in
the $D$-, $D_{s}$-, $B$\- and $B_{s}$-meson spectroscopy, we might have
observed three excitations $D(2760)$, $D_{sJ_{1}}(2860)$ and $B(5970)$ in
recent experiments. The last unobserved one in the $B_{s}$-meson family should
be a narrow state with a width of $\Gamma\simeq(25\sim 75)$ MeV, which is most
likely to be found in the $BK$ and $B^{*}K$ channels.

No evidence of the second $P$-wave excitations with $J^{P}=0^{+}$ is found in
experiments. In the $D$\- and $B$-meson families, the $D(2^{3}P_{0})$ and
$B(2^{3}P_{0})$ excitations should be very broad states, whose widths are
larger than 400 MeV. It might be difficult to be found in experiments.
However, the widths of the resonances $D_{s}(2^{3}P_{0})$ and
$B_{s}(2^{3}P_{0})$ in the $D_{s}$\- and $B_{s}$-meson spectroscopy are
relatively narrower, which might be observed in the $DK$ and $BK$ channels,
respectively.

The physical states of the second $P$-wave states with $J^{P}=1^{+}$ might be
mixed states between the $2^{1}P_{1}$ and $2^{3}P_{1}$. The strong decay
properties of both $D_{sJ}(3040)$ and $D_{J}(3000)$ can be explained by
assigning them as the low-mass mixed state $|2P_{1}\rangle_{L}$ via the
$2^{1}P_{1}$-$2^{3}P_{1}$ mixing with a mixing angle in the range of
$\phi_{2P}=-(20\sim 26)^{\circ}$. The mixed state $B_{s}(|2P_{1}\rangle_{L})$
in the $B_{s}$-meson family is the narrowest state in these low-mass mixed
states, thus, it is most likely to be observed in the $B^{*}K$ channel. On the
contrary, the mixed state $B(|2P_{1}\rangle_{L})$ in the $B$-meson family
might have the broadest width in these low-mass mixed states, thus, its
discovery potentials might be small. The high-mass mixed states
$D(|2P_{1}^{\prime}\rangle_{H})$, $D_{s}(|2P_{1}^{\prime}\rangle_{H})$,
$B(|2P_{1}^{\prime}\rangle_{H})$ and $B_{s}(|2P_{1}^{\prime}\rangle_{H})$ are
usually narrower than those of low-mass states. These states dominantly decay
into the first $P$-wave states by emitting a light pseudoscalar meson.
However, experimental analysis of these decay channels is still absent, which
might explain why these broader mixed states $D_{sJ}(3040)$ and $D_{J}(3000)$
have been first found in experiments. It is strongly suggested to carry out
experimental analysis of the final states containing a low-lying $P$-wave
state with $J^{P}=0^{+},1^{+}$.

No evidence of the second $P$-wave excitations with $J^{P}=2^{+}$ in the $D$-,
$D_{s}$-, $B$\- and $B_{s}$-meson spectroscopy is found in experiments. Our
calculations indicate that these $P$-wave excitations $D(2^{3}P_{2})$,
$D_{s}(2^{3}P_{2})$, $B(2^{3}P_{2})$ and $B_{s}(2^{3}P_{2})$ have a relatively
narrow width, their strong decays are governed by the $D_{1}(2430)\pi$,
$D_{1}(2430)K$, $B(1P_{1})\pi$ and $B(1P_{1})K$, respectively. Thus, it is
might be a good chance for us to find the $D(2^{3}P_{2})$ and
$D_{s}(2^{3}P_{2})$ excitations by analyzing the data in the $D_{1}(2430)\pi$
and $D_{1}(2430)K$ final states, respectively.

The $1^{3}F_{2}$ excitations in the heavy-light mesons are most likely to be
very broad resonances, whose widths are larger than 400 MeV. Such broad states
might be difficult to be observed in experiments, which might explain why
these states are still missing in the heavy-light meson spectroscopy.

Considering configuration mixing in the first $F$-wave states with
$J^{P}=3^{+}$, we predict that the low-mass mixed states $|1F_{3}\rangle_{L}$
in the heavy-light mesons are usually broad states, they should be difficult
to be found in experiments. While the decay widths of the high-mass mixed
states $|1F_{3}^{\prime}\rangle_{H}$ are relatively narrower than those of the
low-mass states. These high-mass states $|1F_{3}^{\prime}\rangle_{H}$ might be
observed in future experiments if they have a smaller mass as predicted in
Ref. Di Pierro:2001uu . The optimal observed channels for the
$D(|1F_{3}^{\prime}\rangle_{H})$, $D_{s}(|1F_{3}^{\prime}\rangle_{H})$,
$B(|1F_{3}^{\prime}\rangle_{H})$ and $B_{s}(|1F_{3}^{\prime}\rangle_{H})$ are
$D^{*}\pi$, $D^{*}K$, $B^{*}\pi$ and $B^{*}K$, respectively. However, these
high-mass states $|1F_{3}^{\prime}\rangle_{H}$ might be very broad states if
their mass is as large as that predicted in Ref. Ebert:2009ua .

For the $1^{3}F_{4}$ excitations, their decay widths are sensitive to the
mass. If these states have a smaller mass as predicted in Ref. Di
Pierro:2001uu , they might have relatively narrow width $\Gamma\sim 100$ MeV.
In this case, these excitations have good observation potentials in future
experiments. The optimal observed channels for the $D(1^{3}F_{4})$,
$D_{s}(1^{3}F_{4})$, $B(1^{3}F_{4})$ and $B_{s}(1^{3}F_{4})$ are
$D\pi$/$D^{*}\pi$, $DK$/$D^{*}K$, $B\pi$/$B^{*}\pi$ and $BK$/$B^{*}K$,
respectively. However, if these states have a larger mass as predicted in Ref.
Ebert:2009ua , they might be difficult to be found in experiments for their
broad widths. Taking the newly observed natural parity resonance
$D^{*}_{J}(3000)$ as a candidate of the $D(1^{3}F_{4})$, we find that the
theoretical predictions of the strong decay properties are consistent with the
observation. However, the predicted mass is inconsistent with the observation,
which is about $100\sim 200$ MeV larger than the data. If the
$D^{*}_{J}(3000)$ corresponds to the $D(1^{3}F_{4})$ state indeed, it should
be observed in both $D\pi$ and $D^{*}\pi$ channels. To clarify the puzzles in
the $D^{*}_{J}(3000)$, more observations are needed.

Finally, it should be pointed out that the chiral quark model still has some
limitations. For example, the simple harmonic oscillator wave functions of the
resonances have been adopted in the calculations, which should be different
from the realistic wave functions more or less. Thus, the wave functions might
bring some uncertainties to our predictions. Furthermore, our model is a
nonrelativistic model, the relativistic effects could bring some uncertainties
to the decay widths as well. Thus, we might only give a qualitative prediction
of the strong decay properties for some resonances.

##  Acknowledgements

This work is supported, in part, by the National Natural Science Foundation of
China (Grants No. 11075051 and No. 11375061), and the Hunan Provincial Natural
Science Foundation (13JJ1018).

## References

  * (1) P. del Amo Sanchez et al. [BaBar Collaboration], Phys. Rev. D 82, 111101 (2010).
  * (2) RAaij et al. [LHCb Collaboration], JHEP 1309, 145 (2013).
  * (3) B. Aubert [BABAR Collaboration], Phys. Rev. Lett. 97, 222001 (2006).
  * (4) J. Brodzicka et al. [Belle Collaboration], Phys. Rev. Lett. 100, 092001 (2008).
  * (5) B. Aubert et al. [BaBar Collaboration], Phys. Rev. D 80, 092003 (2009).
  * (6) T. A. Aaltonen et al. [CDF Collaboration], Phys. Rev. D 90, 012013 (2014) [arXiv:1309.5961 [hep-ex]].
  * (7) X. H. Zhong and Q. Zhao, Phys. Rev. D 78, 014029 (2008).
  * (8) X. H. Zhong and Q. Zhao, Phys. Rev. D 81, 014031 (2010).
  * (9) X. -H. Zhong, Phys. Rev. D 82, 114014 (2010).
  * (10) Q. -F. L and D. -M. Li, arXiv:1407.3092 [hep-ph].
  * (11) S. Godfrey and I. T. Jardine, Phys. Rev. D 89, 074023 (2014).
  * (12) Q. Zhen, D. Xin-Ping and W. Ke-Wei, Chin. Phys. C 37, 053102 (2013).
  * (13) Z. -H. Wang, G. -L. Wang, J. -M. Zhang and T. -H. Wang, J. Phys. G 39, 085006 (2012).
  * (14) D. Mohler, S. Prelovsek and R. M. Woloshyn, Phys. Rev. D 87, 034501 (2013) [arXiv:1208.4059 [hep-lat]].
  * (15) A. M. Badalian and B. L. G. Bakker, Phys. Rev. D 84, 034006 (2011).
  * (16) D. Mohler and R. M. Woloshyn, Phys. Rev. D 84, 054505 (2011).
  * (17) B. Chen, L. Yuan and A. Zhang, Phys. Rev. D 83, 114025 (2011).
  * (18) Z. -G. Wang, Phys. Rev. D 83, 014009 (2011).
  * (19) P. Colangelo, F. De Fazio, S. Nicotri and M. Rizzi, Phys. Rev. D 77, 014012 (2008) [arXiv:0710.3068 [hep-ph]].
  * (20) P. Colangelo, F. De Fazio, F. Giannuzzi and S. Nicotri, Phys. Rev. D 86, 054024 (2012) [arXiv:1207.6940 [hep-ph]].
  * (21) Z. -F. Sun and X. Liu, Phys. Rev. D 80, 074037 (2009).
  * (22) F. -K. Guo and U. -G. Meissner, Phys. Rev. D 84, 014013 (2011).
  * (23) Z. -F. Sun, J. -S. Yu, X. Liu and T. Matsuki, Phys. Rev. D 82, 111501 (2010). Colangelo:2010te
  * (24) P. Colangelo and F. De Fazio, Phys. Rev. D 81, 094001 (2010).
  * (25) E. van Beveren and G. Rupp, Phys. Rev. D 81, 118101 (2010).
  * (26) D. -M. Li and B. Ma, Phys. Rev. D 81, 014021 (2010).
  * (27) F. E. Close and E. S. Swanson, Phys. Rev. D 72, 094004 (2005).
  * (28) F. E. Close, C. E. Thomas, O. Lakhina and E. S. Swanson, Phys. Lett. B 647, 159 (2007) [hep-ph/0608139].
  * (29) E. S. Swanson, Phys. Rept. 429, 243 (2006).
  * (30) B. Zhang, X. Liu, W. Z. Deng and S. L. Zhu, Eur. Phys. J. C 50, 617 (2007) [hep-ph/0609013].
  * (31) H. Xu, Q. Huang, H. -W. Ke and X. Liu, arXiv:1406.5796 [hep-ph].
  * (32) G. Yu, Z. -G. Wang and Z. Li, arXiv:1402.5955 [hep-ph].
  * (33) Y. Sun, Q. -T. Song, D. -Y. Chen, X. Liu and S. -L. Zhu, Phys. Rev. D 89, 054026 (2014).
  * (34) Y. Sun, X. Liu and T. Matsuki, Phys. Rev. D 88, 094020 (2013).
  * (35) Z. -G. Wang, Phys. Rev. D 88, 114003 (2013).
  * (36) Z. -G. Wang, arXiv:1401.7580 [hep-ph].
  * (37) H. Xu, X. Liu and T. Matsuki, Phys. Rev. D 89, 097502 (2014).
  * (38) J. Beringer et al. [Particle Data Group Collaboration], Phys. Rev. D 86, 010001 (2012).
  * (39) M. Di Pierro and E. Eichten, Phys. Rev. D 64, 114004 (2001).
  * (40) D. Ebert, R. N. Faustov and V. O. Galkin, Eur. Phys. J. C 66, 197 (2010).
  * (41) X. -H. Zhong and Q. Zhao, Phys. Rev. D 77, 074008 (2008).
  * (42) L. -H. Liu, L. -Y. Xiao and X. -H. Zhong, Phys. Rev. D 86, 034024 (2012).
  * (43) L. -Y. Xiao and X. -H. Zhong, Phys. Rev. D 87, 094002 (2013).
  * (44) S. Godfrey and N. Isgur, Phys. Rev. D 32, 189 (1985).
  * (45) S. Godfrey and R. Kokoski, Phys. Rev. D 43, 1679 (1991).
  * (46) Z. P. Li, Phys. Rev. D 50, 5639 (1994).
  * (47) Z. P. Li, H. X. Ye and M. H. Lu, Phys. Rev. C 56, 1099 (1997).
  * (48) Q. Zhao, J. S. Al-Khalili, Z. P. Li and R. L. Workman, Phys. Rev. C 65, 065204 (2002).
  * (49) Q. Zhao, Z. P. Li and C. Bennhold, Phys. Rev. C 58, 2393 (1998).
  * (50) Q. Zhao, Phys. Rev. C 63, 025203 (2001).
  * (51) Q. Zhao, J. S. Al-Khalili and C. Bennhold, Phys. Rev. C 64, 052201 (2001).
  * (52) R. Aaij et al. [LHCb Collaboration], arXiv:1407.7574 [hep-ex].
  * (53) R. Aaij et al. [LHCb Collaboration], arXiv:1407.7712 [hep-ex].