# Towards establishing an abundant $B$ and $B_{s}$ spectrum up to the second
orbital excitations

Qi Li, Ru-Hui Ni, Xian-Hui Zhong 111mail<EMAIL_ADDRESS>1) Department
of Physics, Hunan Normal University, Changsha 410081, China  2) Synergetic
Innovation Center for Quantum Effects and Applications (SICQEA), Changsha
410081,China  3) Key Laboratory of Low-Dimensional Quantum Structures and
Quantum Control of Ministry of Education, Changsha 410081, China

###### Abstract

Stimulated by the exciting progress in experiments, we carry out a combined
analysis of the masses, and strong and radiative decay properties of the $B$
and $B_{s}$-meson states up to the second orbital excitations. Based on our
good descriptions of the mass and decay properties for the low-lying well-
established states $B_{1}(5721)$, $B_{2}^{*}(5747)$, $B_{s1}(5830)$ and
$B_{s2}^{*}(5840)$, we give a quark model classification for the high mass
resonances observed in recent years. It is found that (i) the $B_{J}(5840)$
resonance may be explained as the low mass mixed state $B(|SD\rangle_{L})$ via
$2^{3}S_{1}$-$1^{3}D_{1}$ mixing, or the pure $B(2^{3}S_{1})$ state, or
$B(2^{1}S_{0})$. (ii) The $B_{J}(5970)$ resonance may be assigned as the
$1^{3}D_{3}$ state in the $B$ meson family, although it as a pure $2^{3}S_{1}$
state cannot be excluded. (iii) The narrow structure around 6064 MeV observed
in the $B^{+}K^{-}$ mass spectrum at LHCb may be mainly caused by the
$B_{sJ}(6109)$ resonance decaying into $B^{*+}K^{-}$, and favors the
assignment of the high mass $1D$-wave mixed state $B_{s}(1D^{\prime}_{2})$
with $J^{P}=2^{-}$, although it as the $1^{3}D_{3}$ state cannot be excluded.
(iv) The relatively broader $B_{sJ}(6114)$ structure observed at LHCb may be
explained with the mixed state $B_{s}(|SD\rangle_{H})$ via
$2^{3}S_{1}$-$1^{3}D_{1}$ mixing, or a pure $1^{3}D_{1}$ state. Most of the
missing $1P$-, $1D$-, and $2S$-wave $B$\- and $B_{s}$-meson states have a
relatively narrow width, they are most likely to be observed in their dominant
decay channels with a larger data sample at LHCb.

##  I Introduction

Since 2007, significant progress has been made in the observations of the
bottom and bottom-strange mesons Zyla:2020zbs . In 2007, two low-lying
orbitally excited narrow $B$ mesons $B_{1}(5721)^{0,+}$ and
$B_{2}^{*}(5747)^{0,+}$ were observed by the D0 experiment Abazov:2007vq , and
were confirmed by the CDF experiment one year later Aaltonen:2008aa . Their
strange analogues, $B_{s1}(5830)$ and $B_{s2}^{*}(5840)$, as the first
orbitally excited $B_{s}$ mesons, were also reported by the CDF Collaboration
in 2007 Aaltonen:2007ah . The $B_{s1}(5830)$ and $B_{s2}^{*}(5840)$ were
confirmed by the D0 and LHCb experiments Aaij:2012uva ; Abazov:2007af . In
2013, two higher resonances $B(5970)^{0,+}$ were observed in the $B\pi$ final
states by the CDF Collaboration Aaltonen:2013atp . In 2015, four higher
resonances $B_{J}(5840)^{0,+}$ and $B_{J}(5960)^{0,+}$ were observed in the
$B\pi$ final states by the LHCb Collaboration when they carried out precise
measurements of the properties of the $B_{1}(5721)^{0,+}$ and
$B_{2}^{*}(5747)^{0,+}$ states Aaij:2015qla . The properties of the
$B_{J}(5960)^{0,+}$ states are consistent with those of $B(5970)^{0,+}$
obtained by the CDF Collaboration. Recently, the LHCb Collaboration observed
two structures $B_{sJ}(6064)$ and $B_{sJ}(6114)$ in the $B^{+}K^{-}$ mass
spectrum Aaij:2020hcw . More experimental information about the excited bottom
and bottom-strange mesons is collected in Table 1. More and more excited
bottom and bottom-strange mesons are expected to be observed in future LHCb
experiments due to its huge production cross sections of beauty, together with
a good reconstruction efficiency, versatile trigger scheme and an excellent
momentum and mass resolution Belyaev:2021cyr .

Table 1: Summary of the experimental information for the excited $B$\- and $B_{s}$-meson states. The date for the $B_{1}(5721)^{+,0}$, $B_{2}(5747)^{+,0}$, $B_{s1}(5830)^{0}$, and $B_{s2}(5840)^{0}$ resonances are adopted the average values of the Review of Particle Physics (RPP) of Particle Data Group (PDG) Zyla:2020zbs . The N and UN stand for the natural spin parity $P=(-1)^{J}$ and unnatural spin parity $P=-(-1)^{J}$. |  Resonance |  $J^{P}$ |  Mass (MeV) |  Width (MeV) |  Observed channel | Experiment
---|---|---|---|---|---|---
| $B_{1}(5721)^{0}$ | $1^{+}$ | $5726.1\pm 1.3$ | $27.5\pm 3.4$ | $B^{*+}\pi^{-}$ | D0 Abazov:2007vq ,CDF Aaltonen:2008aa ,LHCb Aaij:2015qla
| $B_{1}(5721)^{+}$ | $1^{+}$ | $5725.9^{+2.5}_{-2.7}$ | $31\pm 6$ | $B^{*0}\pi^{+}$ | D0 Abazov:2007vq ,CDF Aaltonen:2008aa ,LHCb Aaij:2015qla
| $B_{2}(5747)^{0}$ | $2^{+}$ | $5739.5\pm 0.7$ | $24.2\pm 1.7$ |  $B^{+}\pi^{-}$,$B^{*+}\pi^{-}$ | D0 Abazov:2007vq ,CDF Aaltonen:2008aa ,LHCb Aaij:2015qla
| $B_{2}(5747)^{+}$ | $2^{+}$ | $5737.2\pm 0.7$ | $20\pm 5$ |  $B^{+}\pi^{-}$,$B^{*+}\pi^{-}$ | D0 Abazov:2007vq ,CDF Aaltonen:2008aa ,LHCb Aaij:2015qla
| $B_{J}(5970)^{+}$ | $?$ | $5961\pm 17$ | $60^{+30}_{-20}\pm 40$ |  $B^{0}\pi^{+}$ [or $B^{*0}\pi^{+}$] | CDF Aaltonen:2013atp
| $B_{J}(5970)^{0}$ | $?$ | $5978\pm 17$ | $70^{+30}_{-20}\pm 30$ |  $B^{+}\pi^{-}$ [or $B^{*+}\pi^{-}$] | CDF Aaltonen:2013atp
Case A | $B_{J}(5960)^{+}$ | UN | $5964.9\pm 6.8$ | $63.0\pm 31.7$ | $B^{*0}\pi^{+}$ | LHCb Aaij:2015qla
| $B_{J}(5960)^{0}$ | UN | $5969.2\pm 8.2$ | $82.3\pm 17.1$ | $B^{*+}\pi^{-}$ | LHCb Aaij:2015qla
| $B_{J}(5840)^{+}$ | UN | $5850.3\pm 26.6$ | $224.4\pm 103.7$ | $B^{*0}\pi^{+}$ | LHCb Aaij:2015qla
| $B_{J}(5840)^{0}$ | UN | $5862.9\pm 11.9$ | $127.4\pm 50.9$ | $B^{*+}\pi^{-}$ | LHCb Aaij:2015qla
Case B | $B_{J}(5960)^{+}$ | UN | $6010.6\pm 7.1$ | $61.4\pm 31.7$ | $B^{*0}\pi^{+}$ | LHCb Aaij:2015qla
| $B_{J}(5960)^{0}$ | UN | $6015.9\pm 9.2$ | $81.6\pm 19.3$ | $B^{*+}\pi^{-}$ | LHCb Aaij:2015qla
| $B_{J}(5840)^{+}$ | N | $5874.5\pm 39.6$ | $214.6\pm 106.5$ |  $B^{*0}\pi^{+}$, $B^{0}\pi^{+}$? | LHCb Aaij:2015qla
| $B_{J}(5840)^{0}$ | N | $5889.7\pm 29.1$ | $107.0\pm 53.8$ |  $B^{*+}\pi^{-}$, $B^{+}\pi^{-}$? | LHCb Aaij:2015qla
Case C | $B_{J}(5960)^{+}$ | N | $5966.4\pm 7.2$ | $60.8\pm 31.2$ |  $B^{*0}\pi^{+}$, $B^{0}\pi^{+}$? | LHCb Aaij:2015qla
| $B_{J}(5960)^{0}$ | N | $5993.6\pm 11.7$ | $55.9\pm 16.0$ |  $B^{*+}\pi^{-}$, $B^{+}\pi^{-}$? | LHCb Aaij:2015qla
| $B_{J}(5840)^{+}$ | UN | $5889.3\pm 29.3$ | $229.3\pm 104.7$ | $B^{*0}\pi^{+}$ | LHCb Aaij:2015qla
| $B_{J}(5840)^{0}$ | UN | $5907.8\pm 13.0$ | $119.4\pm 51.4$ | $B^{*+}\pi^{-}$ | LHCb Aaij:2015qla
| $B_{s1}(5830)^{0}$ | $1^{+}$ | $5828.70\pm 0.20$ | $0.5\pm 0.3\pm 0.3$ |  $B^{*+}K^{-}$, $B^{*0}K^{0}$ | CDF Aaltonen:2007ah , D0 Abazov:2007af , LHCb Aaij:2012uva
| $B_{s2}(5840)^{0}$ | $2^{+}$ | $5839.86\pm 0.12$ | $1.49\pm 0.27$ |  $B^{*}K$, $BK$ | CDF Aaltonen:2007ah , D0 Abazov:2007af ,LHCb Aaij:2012uva
| $B_{sJ}(6064)^{0}$ | $?$ | $6063.5\pm 2.0$ | $26\pm 4\pm 4$ | $B^{+}K^{-}$ | LHCb Aaij:2020hcw
| [or $B_{sJ}(6109)^{0}$] | $?$ | [or $6108.8\pm 1.8$] | [or $22\pm 5\pm 4$] | [or $B^{*+}K^{-}$] | LHCb Aaij:2020hcw
| $B_{sJ}(6114)^{0}$ | $?$ | $6114\pm 8$ | $66\pm 18\pm 21$ | $B^{+}K^{-}$ | LHCb Aaij:2020hcw
| [or $B_{sJ}(6158)^{0}$] | $?$ | [or $6158\pm 9$] | [or $72\pm 18\pm 25$] | [or $B^{*+}K^{-}$] | LHCb Aaij:2020hcw

In theory, many theoretical studies of the masses Godfrey:1985xj ; Zeng:1994vj
; DiPierro:2001dwf ; Jia:2018vwl ; Godfrey:2019cmi ; Gregory:2010gm ;
Lang:2015hza ; Shah:2016mgq ; Lu:2017hma ; Yazarloo:2016luc ; Alhakami:2020vil
; Ebert:2009ua ; Cheng:2017oqh ; Liu:2016efm ; Asghar:2018tha ; Lu:2016bbk ;
Sun:2014wea ; Godfrey:2016nwn ; Kher:2017mky , strong decays Zhu:1998wy ;
Orsland:1998de ; DiPierro:2001dwf ; Sun:2014wea ; Lu:2016bbk ; Godfrey:2016nwn
; Asghar:2018tha ; Gupta:2018xds ; Godfrey:2019cmi ; Yu:2019iwm ;
Gupta:2017bcm ; Zhang:2018ubo ; Alhendi:2015rka ; Ferretti:2015rsa ;
Wang:2014oca ; Xu:2014mka ; Wang:2014cta ; Zhang:2010iea ; Luo:2009wu ;
Yu:2019sqp , radiative decays Orsland:1998de ; Godfrey:2019cmi ; Shah:2016mgq
; Lu:2016bbk ; Yu:2019sqp ; Aliev:2018kry ; Godfrey:2016nwn ; Asghar:2018tha ,
and weak decays Gan:2010hw ; Shah:2016mgq ; Grinstein:2015aua  for the excited
bottom and bottom-strange meson states have been carried out with different
methods. For the well established states $B_{1}(5721)$, $B_{2}^{*}(5747)$,
$B_{s1}(5830)$ and $B_{s2}^{*}(5840)$, there are no puzzles to classify them
as the first orbital excitations (i.e. the $1P$-wave states) of quark models.
While for the newly observed resonances/structures $B_{J}(5840)^{0,+}$,
$B(5970)^{0,+}$, $B_{sJ}(6064)$, and $B_{sJ}(6114)$, although they are good
candidates of the $2S$ and $1D$-wave states according to the mass spectrum
predictions in various quark models, their quark model classification is not
clear. There are some theoretical interpretations of the newly observed
$B_{J}(5840)$ and $B_{J}(5970)$ based on the predicted masses and strong decay
properties, since they have been reported by the CDF and LHCb experiments. In
the literature, the $B_{J}(5840)$ resonance is explained with the
$B(2^{1}S_{0})$ Gupta:2017bcm ; Lu:2016bbk ; Asghar:2018tha , the
$B(2^{3}S_{1})$ Yu:2019iwm , or the $B(1^{3}D_{1})$ state Gupta:2018xds .
While for the resonance $B_{J}(5970)$, there are interpretations with the
radially excited state $B(2^{3}S_{1})$ Sun:2014wea ; Gupta:2017bcm ;
Liu:2016efm , or with the second orbitally excited $B$-meson states
Godfrey:2019cmi  either $B(1^{3}D_{3})$ Yu:2019iwm ; Lu:2016bbk  or
$B(1^{3}D_{1})$ Asghar:2018tha . It should be mentioned that in Ref.
Xiao:2014ura , our group assigned the $B_{J}(5970)$ resonance to be the
$B(1^{3}D_{3})$ state by analyzing the strong decay properties within a chiral
quark model. Based on this assignment, the authors further predicted that as
the partner of $B_{J}(5970)$, the mass and width of the $B_{s}(1^{3}D_{3})$
state might be $M\simeq 6.07$ GeV and $\Gamma\simeq 30$ MeV, respectively,
which are also in agreement with those predictions in Refs. Lu:2016bbk ;
Asghar:2018tha . If assigning the $B_{sJ}(6064)$ to be the $B_{s}(1^{3}D_{3})$
state, both the measured mass and width are consistent with quark model
predictions. It should be mentioned that there are no discussions about the
recently observed structures $B_{sJ}(6064)$ and $B_{sJ}(6114)$ in the
literature. More information about the status of the bottom and bottom-strange
meson study can be found in the recent review work Chen:2016spr .

The experimental progress provides us good opportunities to establish an
abundant $B$ and $B_{s}$-meson spectrum up to the second orbital ($L=2$)
excitations. In this work we deepen our study by carrying out a combined
analysis of the masses and decay properties of the $B$ and $B_{s}$-meson
states up to the $L=2$ excitations. First, we calculate the mass spectrum of
$B$ and $B_{s}$ mesons within a nonrelativistic potential model. With this
model the masses for the observed $B$ and $B_{s}$-meson states can be
described successfully. Then, with the available wave functions from the
potential model, we calculate the OZI-allowed two-body strong decays of the
excited $B$ and $B_{s}$ mesons with a chiral quark model Manohar:1983md ;
Li:1994cy ; Li:1997gd ; Zhao:2002id . This model has been successfully applied
to describe the strong decays of the heavy-light mesons and baryons
DiPierro:2001dwf ; Xiao:2014ura ; Zhong:2010vq ; Zhong:2008kd ; Zhong:2009sk ;
Liu:2012sj ; Zhong:2007gp ; Xiao:2013xi ; Nagahiro:2016nsx ; Yao:2018jmc ;
Wang:2017kfr ; Xiao:2017udy ; Wang:2017hej ; Liu:2019wdr . To provide more
knowledge for the excited $B$ and $B_{s}$ meson states, we also evaluate the
their electromagnetic (EM) transitions within a nonrelativistic constituent
quark model Brodsky:1968ea ; Close:1970kt ; Close:1989aj ; Li:1994cy ;
Li:1993zzb ; Li:1997gd ; Zhao:1998fn ; Zhao:2000tb ; Zhao:2001jw ; Zhao:2002id
. This model has also been successfully applied to describe the radiative
decays of baryon states Yao:2018jmc ; Wang:2017kfr ; Lu:2017meb ; Xiao:2017udy
; Wang:2017hej ; Liu:2019vtx ; Liu:2019wdr  and meson systems Deng:2016stx ;
Deng:2016ktl ; Li:2019tbn ; Close:2002ky ; Chen:2020jku . Based on our good
descriptions of the mass and decay properties for the low-lying well-
established states $B_{1}(5721)^{0,+}$, $B_{2}^{*}(5747)^{0,+}$,
$B_{s1}(5830)$ and $B_{s2}^{*}(5840)$, we give our quark model classifications
of the high mass resonances/structures $B_{J}(5840)^{0,+}$, $B(5970)^{0,+}$,
$B_{sJ}(6064)$, and $B_{sJ}(6114)$. Finally, according to our assignments for
the newly observed resonances, we attempt to predict the properties of the
missing resonances, which may be useful for future investigations in
experiments.

This paper is organized as follows. In Sec. II, the mass spectrum is
calculated within a non-relativistic linear potential model. In Sec. III, a
brief review of the chiral quark model is given. The numerical results are
presented and discussed in Sec. IV. Finally, a summary is given in Sec. V.

Table 2: The parameters of the nonrelativistic potential model. | $B$ | $B_{s}$
---|---|---
$m_{b}$ (GeV) | 4.852 | 4.852
$m_{u,d}$ (GeV) | 0.450 | $\cdots$
$m_{s}$ (GeV) | $\cdots$ | 0.600
$\alpha$ | 0.564 | 0.550
$\sigma$ (GeV) | 0.98 | 1.06
$b$ (GeV2) | 0.120 | 0.120
$C_{0}$ (GeV) | $-0.2537$ | $-0.2318$
$r_{c}$ (fm ) | 0.337 | 0.292
Table 3: The predicted bottom meson masses (MeV) compared with the data and some other model predictions. The mixing angle of $1^{3}P_{1}-1^{1}P_{1}$ and $1^{3}D_{2}-1^{1}D_{2}$ obtained in present work are $\theta_{1P}=-35.2^{\circ}$, and $\theta_{1D}=-39.5^{\circ}$. In the table, $\beta_{eff}^{NR}$ and $\beta_{eff}^{R}$ stand for the effective harmonic oscillator parameters (GeV) of our nonrelativistic quark model calculations and those with the relativized quark model calculations Godfrey:2016nwn , respectively, while $\beta_{eff}^{C}$ stands for our results including relativistic corrections of the length contraction effects.  | State | $J^{P}$ | $\beta_{eff}^{NR}$ | $\beta_{eff}^{C}$ |  $\beta_{eff}^{R}$ Godfrey:2016nwn  | Ours | KDR Kher:2017mky  | EFG Ebert:2009ua  | LPW Lu:2016bbk  | LL Liu:2016efm  | GI Godfrey:2016nwn  | AMS Asghar:2018tha  | PE DiPierro:2001dwf  | Exp. Zyla:2020zbs
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| $B(1^{1}S_{0})$ | $0^{-}$ | 0.444 | 0.628 | 0.580 | 5279(fitted) | 5287 | 5280 | 5280 | 5273 | 5312 | 5268 | 5279 | $5280$
| $B(1^{3}S_{1})$ | $1^{-}$ | 0.418 | 0.575 | 0.542 | 5325(fitted) | 5323 | 5326 | 5329 | 5329 | 5371 | 5329 | 5324 | $5325$
| $B(1^{3}P_{0})$ | $0^{+}$ | 0.367 | 0.525 | 0.536 | 5722 | 5730 | 5749 | 5683 | 5776 | 5756 | 5704 | 5706 | $\cdots$
| $B(1P_{1})$ | $1^{+}$ | 0.348 | 0.487 | 0.511 | 5716 | 5733 | 5723 | 5729 | 5719 | 5784 | 5739 | 5700 | $5698$
| $B(1P^{\prime}_{1})$ | $1^{+}$ | 0.348 | 0.487 | 0.499 | 5753(fitted) | 5752 | 5744 | 5754 | 5837 | 5777 | 5755 | 5742 | $5726$
| $B(1^{3}P_{2})$ | $2^{+}$ | 0.334 | 0.460 | 0.472 | 5727(fitted) | 5740 | 5741 | 5768 | 5739 | 5797 | 5769 | 5714 | $5740$
| $B(2^{1}S_{0})$ | $0^{-}$ | 0.327 | 0.472 | 0.477 | 5876 | 5926 | 5890 | 5910 | 5957 | 5904 | 5877 | 5886 | $5863$
| $B(2^{3}S_{1})$ | $1^{-}$ | 0.320 | 0.458 | 0.468 | 5899 | 5947 | 5906 | 5939 | 5997 | 5933 | 5905 | 5920 | $\cdots$
| $B(1^{3}D_{1})$ | $1^{-}$ | 0.316 | 0.453 | 0.488 | 6056 | 6016 | 6119 | 6095 | 6143 | 6110 | 6022 | 6025 | $\cdots$
| $B(1D_{2})$ | $2^{-}$ | 0.312 | 0.445 | 0.463 | 5973 | 6031 | 6103 | 6004 | 5993 | 6095 | 6026 | 5985 | $\cdots$
| $B(1D^{\prime}_{2})$ | $2^{-}$ | 0.312 | 0.445 | 0.469 | 6067 | 6065 | 6121 | 6113 | 6165 | 6124 | 6031 | 6037 | $\cdots$
| $B(1^{3}D_{3})$ | $3^{-}$ | 0.312 | 0.445 | 0.444 | 5979 | 6016 | 6091 | 6014 | 6004 | 6106 | 6031 | 5993 | $5971$
Table 4: The predicted bottom-strange meson masses (MeV) compared with the data and some other model predictions. The mixing angle of $1^{3}P_{1}-1^{1}P_{1}$ and $1^{3}D_{2}-1^{1}D_{2}$ obtained in present work are $\theta_{1P}=-39.6^{\circ}$, and $\theta_{1D}=-39.9^{\circ}$. In the table, $\beta_{eff}^{NR}$ and $\beta_{eff}^{R}$ stand for the effective harmonic oscillator parameters (GeV) obtained from our nonrelativistic quark model calculations and those with the relativized quark model calculations Godfrey:2016nwn , respectively, while $\beta_{eff}^{C}$ stands for our results including relativistic corrections of the length contraction effects.  | State | $J^{P}$ | $\beta_{eff}^{NR}$ | $\beta_{eff}^{C}$ |  $\beta_{eff}^{R}$ Godfrey:2016nwn  | Ours | KDR Kher:2017mky  | EFG Ebert:2009ua  | ZVR Zeng:1994vj  | LPW Lu:2016bbk  | AMS Asghar:2018tha  | GI Godfrey:2016nwn  | PE DiPierro:2001dwf  | Exp.
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| $B_{s}(1^{1}S_{0})$ | $0^{-}$ | 0.507 | 0.681 | 0.636 | 5367(fitted) | 5367 | 5372 | 5370 | 5362 | 5377 | 5394 | 5373 |  $5367$ Zyla:2020zbs
| $B_{s}(1^{3}S_{1})$ | $1^{-}$ | 0.475 | 0.621 | 0.595 | 5415(fitted) | 5413 | 5414 | 5430 | 5413 | 5422 | 5450 | 5421 |  $5416$ Zyla:2020zbs
| $B_{s}(2^{1}S_{0})$ | $0^{-}$ | 0.363 | 0.490 | 0.508 | 5944 | 6003 | 5976 | 5930 | 5977 | 5929 | 5984 | 5985 | $\cdots$
| $B_{s}(2^{3}S_{1})$ | $1^{-}$ | 0.356 | 0.465 | 0.497 | 5966 | 6029 | 5992 | 5970 | 6003 | 5949 | 6012 | 6019 | $\cdots$
| $B_{s}(1^{3}P_{0})$ | $0^{+}$ | 0.408 | 0.549 | 0.563 | 5788 | 5812 | 5833 | 5750 | 5756 | 5770 | 5831 | 5804 | $\cdots$
| $B_{s}(1P_{1})$ | $1^{+}$ | 0.387 | 0.509 | 0.538 | 5810 | 5828 | 5831 | 5790 | 5801 | 5801 | 5857 | 5805 | $\cdots$
| $B_{s}(1P^{\prime}_{1})$ | $1^{+}$ | 0.387 | 0.509 | 0.528 | 5821(fitted) | 5842 | 5865 | 5800 | 5836 | 5803 | 5861 | 5842 |  $5829$ Zyla:2020zbs
| $B_{s}(1^{3}P_{2})$ | $2^{+}$ | 0.369 | 0.477 | 0.504 | 5821(fitted) | 5840 | 5842 | 5820 | 5851 | 5822 | 5876 | 5820 |  $5840$ Zyla:2020zbs
| $B_{s}(1^{3}D_{1})$ | $1^{-}$ | 0.351 | 0.472 | 0.504 | 6101 | 6119 | 6209 | 6070 | 6142 | 6057 | 6182 | 6127 |  $6114$ Aaij:2020hcw
| $B_{s}(1D_{2})$ | $2^{-}$ | 0.344 | 0.459 | 0.487 | 6061 | 6128 | 6189 | 6070 | 6087 | 6059 | 6169 | 6095 | $\cdots$
| $B_{s}(1D^{\prime}_{2})$ | $2^{-}$ | 0.344 | 0.459 | 0.482 | 6113 | 6157 | 6218 | 6080 | 6159 | 6064 | 6196 | 6140 |  $6109$ Aaij:2020hcw
| $B_{s}(1^{3}D_{3})$ | $3^{-}$ | 0.340 | 0.451 | 0.467 | 6067 | 6172 | 6191 | 6080 | 6096 | 6063 | 6179 | 6103 |  $6064$ Aaij:2020hcw

##  II mass spectrum

To describe the bottom and bottom-strange meson spectra, we adopt a
nonrelativistic linear potential model. In this model, the effective potential
is adopted as Godfrey:1986wj ; Barnes:2005pb ; Eichten:1978tg ; Godfrey:2004ya

$\displaystyle V(r)=V_{0}(r)+V_{sd}(r),$ (1)

where

$\displaystyle V_{0}(r)=-\frac{4}{3}\frac{\alpha_{s}}{r}+br+C_{0}$ (2)

includes the standard color Coulomb interaction and linear confinement, and
zero point energy $C_{0}$. The spin-dependent part $V_{sd}(r)$ can be
expressed as Godfrey:1986wj ; Godfrey:2004ya ; Eichten:1980mw

$\displaystyle V_{sd}(r)=H_{SS}+H_{T}+H_{LS},$ (3)

where

$\displaystyle
H_{SS}=\frac{32\pi\alpha_{s}}{9m_{q}m_{\bar{q}}}\tilde{\delta}_{\sigma}(r)\mathbf{S}_{q}\cdot\mathbf{S}_{\bar{q}}$
(4)

is the spin-spin contact hyperfine potential. Here, we take
$\tilde{\delta}_{\sigma}(r)=(\sigma/\sqrt{\pi})^{3}e^{-\sigma^{2}r^{2}}$ as
suggested in Ref. Barnes:2005pb . The tensor potential $H_{T}$ is adopted as

$\displaystyle
H_{T}=\frac{4}{3}\frac{\alpha_{s}}{m_{q}m_{\bar{q}}}\frac{1}{r^{3}}S_{T},$ (5)

with
$S_{T}=\frac{3\mathbf{S}_{q}\cdot\mathbf{r}\mathbf{S}_{\bar{q}}\cdot\mathbf{r}}{r^{2}}-\mathbf{S}_{q}\cdot\mathbf{S}_{\bar{q}}$.

The spin-orbit interaction $H_{LS}$ can be decomposed into symmetric part
$H_{sym}$ and antisymmetric part $H_{anti}$:

$\displaystyle H_{LS}=H_{sym}+H_{anti},$ (6)

with

$\displaystyle H_{sym}=\frac{\mathbf{S_{+}\cdot
L}}{2}\left[\left(\frac{1}{2m_{\bar{q}}^{2}}+\frac{1}{2m_{q}^{2}}\right)\left(\frac{4}{3}\frac{\alpha_{s}}{r^{3}}-\frac{b}{r}\right)+\frac{8\alpha_{s}}{3m_{q}m_{\bar{q}}r^{3}}\right],$
(7) $\displaystyle H_{anti}=\frac{\mathbf{S_{-}\cdot
L}}{2}\left(\frac{1}{2m_{q}^{2}}-\frac{1}{2m_{\bar{q}}^{2}}\right)\left(\frac{4}{3}\frac{\alpha_{s}}{r^{3}}-\frac{b}{r}\right).\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ (8)

In these equations, $\mathbf{L}$ is the relative orbital angular momentum of
the $q\bar{q}$ system; $\mathbf{S}_{q}$ and $\mathbf{S}_{\bar{q}}$ are the
spins of the quark $q$ and antiquark $\bar{q}$, respectively, and
$\mathbf{S}_{\pm}\equiv\mathbf{S}_{q}\pm\mathbf{S}_{\bar{q}}$; $m_{q}$ and
$m_{\bar{q}}$ are the masses of quark $q$ and antiquark $\bar{q}$,
respectively; $\alpha_{s}$ is the running coupling constant of QCD; and $r$ is
the distance between the quark $q$ and antiquark $\bar{q}$. The six parameters
in the above potentials ($\alpha_{s}$, $b$, $\sigma$, $m_{q}$, $m_{\bar{q}}$,
$C_{0}$) are determined by fitting the mass spectrum.

It should be emphasized that when $m_{q}\neq m_{\bar{q}}$, the antisymmetric
part of the spin-orbit potential, $H_{anti}$, can cause a configuration mixing
between spin triplet $n^{3}L_{J}$ and spin singlet $n^{1}L_{J}$. Thus, the
physical states $nL_{J}$ and $nL^{\prime}_{J}$ are expressed as

$\left(\begin{array}[]{c}nL_{J}\\\ nL^{\prime}_{J}\\\
\end{array}\right)=\left(\begin{array}[]{cc}\cos\theta_{nL}&\sin\theta_{nL}\\\
-\sin\theta_{nL}&\cos\theta_{nL}\\\
\end{array}\right)\left(\begin{array}[]{c}n^{1}L_{J}\\\ n^{3}L_{J}\\\
\end{array}\right).$ (9)

where $J=L=1,2,3\cdots$, and the $\theta_{nL}$ is the mixing angle. In this
work $nL^{\prime}_{J}$ corresponds to the higher mass mixed state as often
adopted in the literature.

In this work, we solve the radial Schrödinger equation by using the three-
point difference central method Haicai  from central ($r=0$) towards outside
($r\to\infty$) point by point. This method was successfully to deal with the
spectroscopies of $c\bar{c}$, $b\bar{b}$, $b\bar{c}$ and $s\bar{s}$
Deng:2016ktl ; Deng:2016stx ; Li:2019tbn ; Li:2019qsg ; Li:2020xzs . To
overcome the singular behavior of $1/r^{3}$ in the spin-dependent potentials,
following the method of our previous works Deng:2016ktl ; Deng:2016stx ;
Li:2019tbn ; Li:2019qsg ; Li:2020xzs , we introduce a cutoff distance $r_{c}$
in the calculation. Within a small range $r\in(0,r_{c})$, we let
$1/r^{3}=1/r_{c}^{3}$. By introducing the cutoff distance $r_{c}$, we can
nonperturbatively include the corrections from these spin-dependent potentials
containing $1/r^{3}$ to both the mass and wave function of a meson state,
which are crucial for our predicting the decay properties.

The model parameters adopted in this work are listed in Table 2. To be
consistent with our previous study Li:2019tbn ; Li:2019qsg ; Li:2020xzs ;
Liu:2019wdr , the bottom quark mass $m_{b}$, the light up or down quark mass
$m_{u/d}$, the strange quark mass $m_{s}$ are taken from the determinations,
i.e., $m_{b}=4.852$ GeV, $m_{u/d}=0.45$ GeV, $m_{s}=0.60$ GeV. The other four
parameters ($\alpha_{s}$, $b$, $\sigma$, $C_{0}$) for the bottom meson sector,
they are determined by fitting the masses of the well established states
$B(5279)$, $B^{*}(5325)$, $B_{1}(5721)$ and $B^{*}_{2}(5747)$, while for the
bottom-strange meson sector, they are determined by fitting the masses of the
well established states $B_{s}(5367)$, $B^{*}_{s}(5415)$, $B_{s1}(5830)$ and
$B^{*}_{s2}(5840)$. It should be pointed out that the zero-point-energy
parameter $C_{0}$ is taken to be zero for the $c\bar{c}$, $b\bar{b}$,
$b\bar{c}$ heavy quarkonium systems in the literature Deng:2016ktl ;
Deng:2016stx ; Li:2019tbn ; Li:2019qsg . For these heavy quarkonium systems,
the zero point energy can be absorbed into the constituent quark masses
because it only affects the heavy quark masses slightly. However, if the zero
point energy is absorbed into the meson systems containing light quarks, it
can significantly change the light constituent quark masses, which play an
important role in the spin-dependent potentials. Thus, to obtain a good
description of both the masses and the hyperfine/fine splittings for the meson
systems containing light quarks, a zero-point-energy parameter $C_{0}$ is
usually adopted in the calculations.

Figure 1: The radial wave function $\psi(r)$ of the $B(1^{3}P_{0})$ state. The
solid and dashed lines stand for the wave functions with and without
corrections from the perturbation term $H^{\prime}$ containing $1/r^{3}$,
respectively. Figure 2: The predicted mass spectra of $B$ and $B_{s}$ mesons.
The solid circles stand for the measured masses obtained from the Particle
Data Group Zyla:2020zbs  and the recent LHCb measurements Aaij:2020hcw .

Finally, we need determine the cutoff distance $r_{c}$ in our calculations. It
is found that the masses of the $1^{3}P_{0}$ states are more sensitive to the
cutoff distance $r_{c}$. The reason is that the singular terms of $1/r^{3}$ in
the spin-dependent potentials have more effects on the mass of the
$1^{3}P_{0}$ state due to its relatively larger factors of
$\langle\mathbf{S_{+}\cdot L}\rangle$ and $\langle S_{T}\rangle$ than the
other excited meson states. Thus, in the present work, we determine the
$r_{c}$ by fitting the masses of the $B(1^{3}P_{0})$ and $B_{s}(1^{3}P_{0})$
states. Note that when the other parameters are well determined, the masses of
these $1^{3}P_{0}$ states can be reliably worked out with the perturbation
method without introducing the cutoff distance $r_{c}$, although the wave
functions obtain no corrections from the spin-dependent potentials containing
$1/r^{3}$. To calculate the masses of the $B(1^{3}P_{0})$ and
$B_{s}(1^{3}P_{0})$ states with the perturbation method, we let
$H=H_{0}+H^{\prime}$, where $H_{0}$ is the main part of the Hamiltonian
without singular $1/r^{3}$ term contributions, i.e.,

$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\frac{P^{2}}{2\mu}+V_{0}(r)+\frac{32\pi\alpha_{s}}{9m_{q}m_{\bar{q}}}\tilde{\delta}_{\sigma}(r)\langle\mathbf{S}_{q}\cdot\mathbf{S}_{\bar{q}}\rangle$
(10) $\displaystyle+$ $\displaystyle\frac{\langle\mathbf{S_{+}\cdot
L}\rangle}{2}\left[\left(\frac{1}{2m_{\bar{q}}^{2}}+\frac{1}{2m_{q}^{2}}\right)\left(-\frac{b}{r}\right)\right],$

where $\frac{P^{2}}{2\mu}$ is the kinetic energy term with a reduced mass
$\mu=m_{q}m_{\bar{q}}/(m_{q}+m_{\bar{q}})$; while $H^{\prime}$ is the
perturbation part containing $1/r^{3}$ terms, i.e.,

$\displaystyle H^{\prime}$ $\displaystyle=$
$\displaystyle\frac{\langle\mathbf{S_{+}\cdot
L}\rangle}{2}\left[\left(\frac{1}{2m_{\bar{q}}^{2}}+\frac{1}{2m_{q}^{2}}\right)\left(\frac{4}{3}\frac{\alpha_{s}}{r^{3}}\right)+\frac{8\alpha_{s}}{3m_{q}m_{\bar{q}}r^{3}}\right]$
(11) $\displaystyle+$
$\displaystyle\frac{4}{3}\frac{\alpha_{s}}{m_{q}m_{\bar{q}}}\frac{1}{r^{3}}\left\langle
S_{T}\right\rangle.$

For a $1^{3}P_{0}$ state, one can easily obtain the matrix elements
$\langle\mathbf{S}_{q}\cdot\mathbf{S}_{\bar{q}}\rangle=1/4$,
$\langle\mathbf{S_{+}\cdot L}\rangle=-2$, and $\langle S_{T}\rangle=-1$ by
using the relations
$\langle\mathbf{S}_{q}\cdot\mathbf{S}_{\bar{q}}\rangle=S(S+1)/2-3/4$,
$\langle\mathbf{S_{+}}\cdot\mathbf{L}\rangle=[J(J+1)-L(L+1)-S(S+1)]/2$, and
$\langle
S_{T}\rangle=-[6\langle\mathbf{S_{+}}\cdot\mathbf{L}\rangle^{2}+3\langle\mathbf{S_{+}}\cdot\mathbf{L}\rangle-2\mathbf{L}^{2}\mathbf{S_{+}}^{2}]/[2(2L-1)(2L+3)]$.
By solving the radial Schrödinger equation of
$H_{0}\psi^{(0)}(r)=E_{0}\psi^{(0)}(r)$ with the three-point difference
central method, we work out the eigen energies $E_{0}=508$, 415 MeV for the
$B(1^{3}P_{0})$ and $B_{s}(1^{3}P_{0})$ states as well as their zero order
radial wave functions $\psi^{(0)}(r)$. Using these zero order wave functions,
we further work out the perturbation energies
$\langle\psi^{(0)}(r)|H^{\prime}|\psi^{(0)}(r)\rangle=-88,-79$ MeV for the
$B(1^{3}P_{0})$ and $B_{s}(1^{3}P_{0})$ states, respectively. Finally, with
the relation
$M=m_{q}+m_{\bar{q}}+E_{0}+\langle\psi^{(0)}(r)|H^{\prime}|\psi^{(0)}(r)\rangle$
we predict the masses 5722 and 5788 MeV for $B(1^{3}P_{0})$ and
$B_{s}(1^{3}P_{0})$, respectively. These masses calculated with the
perturbation method are in good agreement with the predictions in the
literature Kher:2017mky ; Ebert:2009ua ; Asghar:2018tha ; DiPierro:2001dwf ;
Zeng:1994vj . Within a small range $r\in(0,r_{c})$, letting
$1/r^{3}=1/r_{c}^{3}$ in the perturbation Hamiltonian $H^{\prime}$, we solve
the radial Schrödinger equation $(H_{0}+H^{\prime})\psi(r)=E\psi(r)$ with the
three-point difference central method again. By reproducing the masses 5722
and 5788 MeV of the $B(1^{3}P_{0})$ and $B_{s}(1^{3}P_{0})$ states obtained
with the perturbation method, we determine the cutoff distance parameters
$r_{c}=0.337$ and 0.292 fm for the $B$ and $B_{s}$ mass spectra, respectively.
The corrections of the perturbation part $H^{\prime}$ to the radial wave
function $\psi(r)$ can be conveniently included by introducing the cutoff
distance $r_{c}$. As an example, in Fig. 1 we plot the radial wave functions
of the $B(1^{3}P_{0})$ state with and without the corrections from the
perturbation part $H^{\prime}$ containing $1/r^{3}$. It is seen that the
perturbation term $H^{\prime}$ has an obvious correction to the radial wave
function.

With the determined model parameters listed in Table 2, by solving the radial
Schrödinger equation we obtain the masses of the bottom and bottom-strange
meson states, which have been listed in Tab. 3 and Tab. 4, respectively. For
comparison, some other model predictions in Refs. Kher:2017mky ; Ebert:2009ua
; Lu:2016bbk ; Liu:2016efm ; Godfrey:2016nwn ; Asghar:2018tha ;
DiPierro:2001dwf ; Zeng:1994vj  and the data from the Review of Particle
Physics (RPP) of Particle Data Group (PDG) Zyla:2020zbs  are listed in the
same table as well. Furthermore, for a clarity, the spectra are also shown in
Fig. 2. It is shown that the $B_{J}(5970)$ resonance and the structures
$B_{sJ}(6064)$ and $B_{sJ}(6114)$ newly observed at LHCb can be explained as
the $1D$-wave states from the point of view of the mass, while the
$B_{J}(5840)$ may be a good candidate of the $2S$-wave state.

##  III strong and radiative decays

###  III.1 Models

We calculate the strong decays of the bottom and bottom-strange mesons with
the chiral quark model Manohar:1983md ; Li:1994cy ; Li:1997gd ; Zhao:2002id .
In this model, the light pseudoscalar mesons, i.e. $\pi$, $K$ and $\eta$, are
treated as fundamental states. The low energy quark-pseudoscalar-meson and
quark-vector-meson interactions in the SU(3) flavor basis are described by the
effective Lagrangian

$H_{m}=\sum_{j}\frac{1}{f_{m}}\bar{\psi}_{j}\gamma_{\mu}\gamma_{5}\psi_{j}\vec{\tau}\cdot\partial^{\mu}\vec{\phi}_{m},$
(12)

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. The nonrelativistic form of Eq. (12) is given by Li:1994cy ;
Li:1997gd ; Zhao:2002id

$\displaystyle
H_{m}=\sum_{j}\left[A\mbox{\boldmath$\sigma$\unboldmath}_{j}\cdot\textbf{q}+h\mbox{\boldmath$\sigma$\unboldmath}_{j}\cdot\textbf{p}_{j}\right]I_{j}\varphi_{m},$
(13)

in the center-of-mass system of the initial meson, where we have defined
$A\equiv-(1+\frac{\omega_{m}}{E_{f}+M_{f}})$ and
$h\equiv\frac{\omega_{m}}{2\mu_{q}}$. In Eq. (13), 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. 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. The
chiral quark model has been successfully applied to describe the strong decays
of the heavy-light mesons and baryons DiPierro:2001dwf ; Xiao:2014ura ;
Zhong:2010vq ; Zhong:2008kd ; Zhong:2009sk ; Liu:2012sj ; Zhong:2007gp ;
Xiao:2013xi ; Nagahiro:2016nsx . The details of this model can be found in
Refs. Zhong:2008kd ; Zhong:2007gp . It should be mentioned that the chiral
quark model is similar to the pseudoscalar emission model in the literature
Godfrey:1985xj ; Koniuk:1979vy ; Capstick:2000qj . The nonrelativistic form of
quark-pseudoscalar-meson interactions expressed in Eq. (13) is similar to that
of the pseudoscalar emission model, except that the factors
$A\equiv-(1+\frac{\omega_{m}}{E_{f}+M_{f}})$ and
$h\equiv\frac{\omega_{m}}{2\mu_{q}}$ in this work have an explicit dependence
on the energies of final hadrons.

Meanwhile, to treat the radiative decay of a hadron we apply the constituent
quark model Brodsky:1968ea ; Close:1970kt ; Close:1989aj ; Li:1994cy ;
Li:1993zzb ; Li:1997gd ; Zhao:1998fn ; Zhao:2000tb ; Zhao:2001jw ; Zhao:2002id
. In this model, the quark-photon EM coupling at the tree level is adopted as

$\displaystyle
H_{e}=-\sum_{j}e_{j}\bar{\psi}_{j}\gamma^{j}_{\mu}A^{\mu}(\mathbf{k},\mathbf{r}_{j})\psi_{j},$
(14)

where $A^{\mu}$ represents the photon field with three-momentum $\mathbf{k}$.
$e_{j}$ and $\mathbf{r}_{j}$ stand for the charge and coordinate of the
constituent quark $\psi_{j}$, respectively. In the initial-hadron-rest system,
including the effects of the binding potential between quarks, the
nonrelativistic form of the quark-photon EM coupling can be written as
Brodsky:1968ea ; Li:1994cy ; Li:1997gd ; Zhao:2002id ; Zhao:1998fn ;
Zhao:2000tb ; Zhao:2001jw ; Li:1993zzb

$H_{e}^{nr}=\sum_{j}\left[e_{j}\mathbf{r}_{j}\cdot\mbox{\boldmath$\epsilon$\unboldmath}-\frac{e_{j}}{2m_{j}}\mbox{\boldmath$\sigma$\unboldmath}_{j}\cdot(\mbox{\boldmath$\epsilon$\unboldmath}\times\hat{\mathbf{k}})\right]\varphi_{\gamma},$
(15)

where $\mbox{\boldmath$\sigma$\unboldmath}_{j}$ stand for the Pauli spin
vector for the $j$th quark. The vector $\epsilon$ is the polarization vector
of the photon. The plane wave part of the emitted light meson is
$\varphi_{\gamma}=e^{-i\textbf{k}\cdot\textbf{r}_{j}}$. The first and second
terms in Eq.(15) are responsible for the electric and magnetic transitions,
respectively. The second term in Eq.(15) is the same as that used in Refs.
Godfrey:1985xj ; Koniuk:1979vy ; Capstick:2000qj ; Copley:1969ft ;
Sartor:1986sf , while the first term in Eq.(15) is different from
$(1/m_{j})\mathbf{p}_{j}\cdot\mbox{\boldmath$\epsilon$\unboldmath}$ of these
works due to the binding potential effects are included Brodsky:1968ea . This
model has been successfully applied to describe the radiative decays of baryon
states Yao:2018jmc ; Wang:2017kfr ; Lu:2017meb ; Xiao:2017udy ; Wang:2017hej ;
Liu:2019vtx ; Liu:2019wdr  and meson systems Deng:2016stx ; Deng:2016ktl ;
Li:2019tbn ; Close:2002ky ; Chen:2020jku .

For a strong decay process, the partial decay width is calculated with
Zhong:2008kd ; Zhong:2007gp

$\Gamma_{m}=\left(\frac{\delta}{f_{m}}\right)^{2}\frac{(E_{f}+M_{f})|\textbf{q}|}{4\pi
M_{i}(2J_{i}+1)}\sum_{J_{fz},J_{iz}}|\mathcal{M}_{J_{fz},J_{iz}}|^{2},$ (16)

while for a radiative decay process, the partial decay width is calculated
with Deng:2016stx ; Deng:2016ktl

$\Gamma_{\gamma}=\frac{|\mathbf{k}|^{2}}{\pi}\frac{2}{2J_{i}+1}\frac{M_{f}}{M_{i}}\sum_{J_{fz},J_{iz}}|\mathcal{A}_{J_{fz},J_{iz}}|^{2},$
(17)

where $\mathcal{M}_{J_{fz},J_{iz}}$ and $\mathcal{A}_{J_{fz},J_{iz}}$
correspond to the strong and radiative transition amplitudes, respectively.
The quantum numbers $J_{iz}$ and $J_{fz}$ stand for the third components of
the total angular momenta of the initial and final hadron states,
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 fix its value the same as that in Refs. Zhong:2008kd ;
Zhong:2007gp , i.e. $\delta=0.557$.

###  III.2 Parameters

In the calculation, the constituent quark masses for the $u$, $d$, and $s$
quarks are taken with $m_{u}=m_{d}=450$ MeV and $m_{s}=600$ MeV to be
consistent with the spectrum study in Sec. II. The decay constants for $\pi$,
$K$ and $\eta$ mesons are taken as $f_{\pi}=132$ MeV, $f_{K}=f_{\eta}=160$
MeV, respectively. The masses of the well established hadrons involving in the
calculations are adopted from the PDG Zyla:2020zbs . The masses of the missing
$B$\- and $B_{s}$-meson states are adopted our determinations by solving the
Schröinger equation in Sec. II.

It should be mentioned that, we do not directly adopt the numerical wave
functions of $B$\- and $B_{s}$-meson states calculated by solving the
Schröinger equation. For simplicity, we first fit them with a simple harmonic
oscillator wave function by reproducing the root-mean-square radius
$\sqrt{\langle r^{2}\rangle}$. The obtained effective harmonic oscillator
parameters $\beta_{eff}$ for the meson states are listed in Tab. 3 and Tab. 4.
It is found that the effective harmonic oscillator parameters
$\beta_{eff}^{NR}$ obtained from our nonrelativistic quark model calculations
are obviously smaller than the parameters $\beta_{eff}^{R}$ obtained from the
relativized quark models Godfrey:2016nwn . It indicates that the relativistic
effects of the length contraction on the wave function may be important. To
take into account the relativistic effects, as that suggested in Ref.
Licht:1970pe  we introduce the Lorentz boost factor $\gamma$ in the spatial
wave function, i.e.,

$\psi_{nlm}(\mathbf{r})\to\psi_{nlm}(\gamma\mathbf{r}),$ (18)

where $\gamma=M_{q}/E_{q}$. $M_{q}$ and $E_{q}$ correspond to the effective
mass and energy of the light quark, respectively. According to Ref.
Jaczko:1998uj , the effective mass $M_{q}$ can be estimated by
$M_{q}=\sqrt{\langle p^{2}\rangle+m_{q}^{2}}$, while the energy $E_{q}$ is
estimated by $E_{q}=\langle p\rangle^{2}/(2M_{q})+M_{q}$. To realize this
transformation, we only need replace $\beta_{eff}^{NR}$ in the harmonic
oscillator wave function with $\beta_{eff}^{C}=\gamma\beta_{eff}^{NR}$. The
effective harmonic oscillator parameters $\beta_{eff}^{C}$ including
relativistic corrections of the length contraction are given in Tab. 3 and
Tab. 4 as well. It is found that the $\beta_{eff}^{C}$ values are comparable
with those obtained with the relativized quark models Godfrey:2016nwn . In
Refs.Zhong:2008kd ; Xiao:2014ura , the strong decays of the heavy-light meson
states are studied with the chiral quark model by using the simple harmonic
oscillator wave functions with fixed harmonic oscillator parameters
$\beta=468,466$ MeV for the $B$ and $B_{s}$ spectra, respectively, which are
close to the parameters $\beta_{eff}^{C}$ determined for the $1P$-, $1D$\- and
$2S$-wave states in present work. The parameters $\beta_{eff}^{C}$ of the
ground states $1^{1}S_{0}$ and $1^{3}S_{1}$ are notably lager than that of the
excited states, this effect is mainly caused by the strong color Coulomb
interaction at the small distance $r$ between two quarks.

The effective parameters $\beta_{eff}^{C}$ of the ground states
$B(1^{1}S_{0})$ and $B^{*}(1^{3}S_{1})$ are crucial for understanding the
decay properties of the excited $B$ and $B_{s}$ states because all of the
excited states should decay into these ground states. Considering the
uncertainty of the parameters $\beta_{eff}^{C}$ of the ground states
$B(1^{1}S_{0})$ and $B^{*}(1^{3}S_{1})$, we properly adjust their
$\beta_{eff}^{C}$ parameters to more reasonably describe the strong decay
properties of the well established $1P$-wave states $B_{2}^{*}(5747)^{0}$ and
$B_{s2}^{*}(5840)$. In this work, we determine them to be
$\beta_{B(1^{1}S_{0})}=0.537$ GeV and $\beta_{B^{*}(1^{3}S_{1})}=0.510$ GeV
for $B(1^{1}S_{0})$ and $B^{*}(1^{3}S_{1})$, respectively. There is about a
$10\%$ correction to the effective parameters $\beta_{eff}^{C}$.

With above parameters, our calculated decay properties for the $1P$-, $2S$-,
and $1D$-wave states are listed in Tables 5, 6, and 7, respectively. From
Table 5, it is found that the decay properties of the well-established
$1P$-wave state can be successfully described.

##  IV discussion

Table 5: Partial and total decay widths (MeV) for $1P$-wave bottom and bottom-strange mesons compared with the the data and some recent model predictions. It should be mention that some masses for the initial states adopted in the literature are slightly different. The total widths inside the square brackets are estimated with the mixing angle $\theta_{1P}=-(55\pm 5)^{\circ}$.  $n^{2S+1}L_{J}$ | State | Channel |  Ours | XZ Zhong:2008kd  | SSC Sun:2014wea  |  LPW Lu:2016bbk  |  GI Godfrey:2016nwn  |  AMS Asghar:2018tha  |  $\Gamma_{exp}$ Zyla:2020zbs
---|---|---|---|---|---|---|---|---|---
$1^{3}P_{0}$ | $B^{*}_{0}(5722)^{+}$ | $B^{+}\pi^{0}+B^{0}\pi^{+}$ | 97.5+195.5 |  |  |  |  | 141.5 |
|  | $B^{*+}\gamma$ | 477$\times 10^{-3}$ |  |  |  | 325$\times 10^{-3}$ | 575$\times 10^{-3}$ |
|  | Total | 293 |  |  |  |  | 142.08 |
| $B^{*}_{0}(5722)^{0}$ | $B^{0}\pi^{0}+B^{+}\pi^{-}$ | 97.5+195.3 | 272 | 225 | 230.43 | 154 | 141.5 |
|  | $B^{*0}\gamma$ | 149$\times 10^{-3}$ |  |  | 116.9$\times 10^{-3}$ | 92.7$\times 10^{-3}$ | 175$\times 10^{-3}$ |
|  | Total | 292.8 | 272 | 225 | 230.43 | 154 | 141.7 |
$1^{3}P_{2}$ | $B^{*}_{2}(5747)^{+}$ | $B^{+}\pi^{0}+B^{0}\pi^{+}$ | 5.3+10.6 |  |  |  |  | 9.77 |
|  | $B^{*+}\pi^{0}+B^{*0}\pi^{+}$ | 5.1+9.9 |  |  |  |  | 9.79 |
|  | $B^{*+}\gamma$ | 146$\times 10^{-3}$ |  |  |  | 444$\times 10^{-3}$ | 761$\times 10^{-3}$ |
|  | Total | 31 |  |  |  | 11.71 | 20.3 | $20\pm 5$
| $B^{*}_{2}(5747)^{0}$ | $B^{0}\pi^{0}+B^{+}\pi^{-}$ | 5.5+10.8 | 25 | 1.9 | 12.62 | 6.23 | 9.77 |
|  | $B^{*0}\pi^{0}+B^{*+}\pi^{-}$ | 5.2+10.2 | 22 | 1.8 | 11.89 | 5.04 | 9.79 |
|  | $B^{*0}\gamma$ | 51$\times 10^{-3}$ |  |  | 177.7$\times 10^{-3}$ | 126$\times 10^{-3}$ | 232$\times 10^{-3}$ |
|  | Total |  32 (fitted) | 47 | 3.7 | 24.51 | 11.40 | 19.8 | $24.2\pm 1.7$
$1P$ | $B_{1}(5680)^{+}$ | $B^{*+}\pi^{0}+B^{*0}\pi^{+}$ | 46.8+93.9 |  |  |  | 163 | 125.53 |
|  | $B^{*+}\gamma$ | 75$\times 10^{-3}$ |  |  |  | 300 | 448 |
|  | $B^{+}\gamma$ | 111$\times 10^{-3}$ |  |  |  | 132$\times 10^{-3}$ | 415$\times 10^{-3}$ |
|  | Total | 140.9 |  |  |  | 163 | 126.4 | $128\pm 18$
| $B_{1}(5680)^{0}$ | $B^{*0}\pi^{0}+B^{*+}\pi^{-}$ | 46.8+93.9 | 219 | 200 | 199.4 | 163 | 125.53 |
|  | $B^{*0}\gamma$ | 24$\times 10^{-3}$ |  |  | 53.1$\times 10^{-3}$ | 85.5$\times 10^{-3}$ | 137$\times 10^{-3}$ |
|  | $B^{0}\gamma$ | 38$\times 10^{-3}$ |  |  | 130.2$\times 10^{-3}$ | 37.8$\times 10^{-3}$ | 127$\times 10^{-3}$ |
|  | Total | 140.8 | 219 | 200 | 199.4 | 163 | 125.8 | $128\pm 18$
$1P^{\prime}$ | $B_{1}(5721)^{+}$ | $B^{*+}\pi^{0}+B^{*0}\pi^{+}$ | 13.8+27.4 |  |  |  | 6.80 | 15.62 |
|  | $B^{*+}\gamma$ | 206$\times 10^{-3}$ |  |  |  | 300$\times 10^{-3}$ | 339$\times 10^{-3}$ |
|  | $B^{+}\gamma$ | 69$\times 10^{-3}$ |  |  |  | 132$\times 10^{-3}$ | 448$\times 10^{-3}$ |
|  | Total | 41.4 [$24.5\pm 2.5$] |  |  |  | 7.27 | 16.4 | $31\pm 6$
| $B_{1}(5721)^{0}$ | $B^{*0}\pi^{0}+B^{*+}\pi^{-}$ | 13.8+27.4 | 30 | 10 | 40.63 | 6.80 | 15.62 |
|  | $B^{*0}\gamma$ | 66$\times 10^{-3}$ |  |  | 108.5$\times 10^{-3}$ | 27.8$\times 10^{-3}$ | 103$\times 10^{-3}$ |
|  | $B^{0}\gamma$ | 24$\times 10^{-3}$ |  |  | 60.4$\times 10^{-3}$ | 106$\times 10^{-3}$ | 137$\times 10^{-3}$ |
|  | Total | 41.3 [$24.5\pm 2.5$] | 30 | 10 | 40.63 | 6.93 | 15.9 | $27.5\pm 3.4$
$1^{3}P_{0}$ | $B_{s0}^{*}(5788)^{0}$ | $B^{+}K^{-}+B^{0}\bar{K^{0}}$ | 86.7+76.7 | 227 | 225 |  | 138 | 135.66 |
|  | $B_{s}^{*0}\gamma$ | 102$\times 10^{-3}$ |  |  | 84.7$\times 10^{-3}$ | 76$\times 10^{-3}$ | 133$\times 10^{-3}$ |
|  | Total | 163.4 | 227 | 225 |  | 138 | 135.8 |
$1^{3}P_{2}$ | $B_{s2}^{*}(5840)^{0}$ | $B^{+}K^{-}+B^{0}\bar{K^{0}}$ | 0.69+0.61 | 2 |  | 1.9 | 0.663 | 1.55 |
|  | $B^{*+}K^{-}+B^{*0}\bar{K^{0}}$ | 0.06+0.04 | 0.12 |  | 0.14 | 0.00799 | 0.13 |
|  | $B_{s}^{*0}\gamma$ | 51$\times 10^{-3}$ |  |  | 159$\times 10^{-3}$ | 106$\times 10^{-3}$ | 225$\times 10^{-3}$ |
|  | Total |  1.31 (fitted) | 2 | 0.26 | 1.66 | 0.777 | 1.9 | 1.49$\pm$0.27
$1P$ | $B_{s1}(5820)^{0}$ | $B^{*+}K^{-}+B^{*0}\bar{K^{0}}$ | $\cdots$ | 149 | 120 |  |  |  |
|  | $B_{s}^{*0}\gamma$ | 56$\times 10^{-3}$ |  |  | 39.5$\times 10^{-3}$ | 57.3$\times 10^{-3}$ |  |
|  | $B_{s}^{0}\gamma$ | 37$\times 10^{-3}$ |  |  | 97.7$\times 10^{-3}$ | 47.8$\times 10^{-3}$ |  |
|  | Total | $0.093$ | 149 | 120 | 160 | 0.1051 |  |
$1P^{\prime}$ | $B_{s1}(5830)^{0}$ | $B^{*+}K^{-}+B^{*0}\bar{K^{0}}$ | 4.3+3.3 | $0.4-1$ | $\sim 0$ |  |  |  |
|  | $B_{s}^{*0}\gamma$ | 53$\times 10^{-3}$ |  |  | 98.8$\times 10^{-3}$ | 36.9$\times 10^{-3}$ |  |
|  | $B_{s}^{0}\gamma$ | 27$\times 10^{-3}$ |  |  | 56.6$\times 10^{-3}$ | 70.6$\times 10^{-3}$ |  |
|  | Total | 7.6 [$0.1-0.8$] | $0.4-1$ | $\sim 0$ | 20 | 0.1075 |  | 0.5$\pm$0.3$\pm$0.3

###  IV.1 $1P$-wave states

Two $1P$-wave excited $B$ meson states $B_{1}(5721)$ and
$B_{2}^{*}(5747)^{+,0}$ together with their flavor partners $B_{s1}(5830)$ and
$B_{s2}^{*}(5840)^{0}$ in the $B_{s}$-meson family have been well-established
in experiments Zyla:2020zbs . However, two resonances with $J^{P}=0^{+}$,
$B(1^{3}P_{0})$ and $B_{s}(1^{3}P_{0})$, and two resonances with
$J^{P}=1^{+}$, $B(P_{1})$ and $B_{s}(P_{1})$, predicted in the quark model are
still missing.

####  IV.1.1 $1^{3}P_{2}$ states

There are no puzzles to assign the $B_{2}^{*}(5747)^{+,0}$ and
$B_{s2}^{*}(5840)^{0}$ resonances to the $1^{3}P_{2}$ states in the $B$ and
$B_{s}$ families, respectively.

In the $B$ meson sector, as the $1^{3}P_{2}$ state both the mass and width of
$B_{2}^{*}(5747)$ can be resonantly reproduced in the quark model. Our fitted
mass $M\simeq 5727$ MeV and width $\Gamma\simeq 31$ MeV are compatible with
the measurements $M_{exp}=5737$ MeV and width $\Gamma_{exp}=(20\pm 5)$ MeV for
$B_{2}^{*}(5747)^{+}$. This state dominantly decays into $B\pi$ and $B^{*}\pi$
channels with comparable partial widths. The ideal partial width ratios
between $B\pi$ and $B^{*}\pi$ channels for the $B_{2}^{*}(5747)^{+,0}$ are
fitted to be

$\displaystyle R_{1}=\frac{\Gamma[B_{2}^{*}(5747)^{0}\to
B^{*+}\pi^{-}]}{\Gamma[B_{2}^{*}(5747)^{0}\to B^{+}\pi^{-}]}\approx 0.94,$
(19) $\displaystyle R_{2}=\frac{\Gamma[B_{2}^{*}(5747)^{+}\to
B^{*0}\pi^{+}]}{\Gamma[B_{2}^{*}(5747)^{+}\to B^{0}\pi^{+}]}\approx 0.93,$
(20)

which are also consistent with the recent LHCb measurements
$R_{1}^{exp}=0.71\pm 0.14\pm 0.30$ and $R_{2}^{exp}=1.0\pm 0.5\pm 0.8$
Aaij:2015qla  and the predictions in Refs. Alhendi:2015rka ; Sun:2014wea ;
Orsland:1998de ; Ferretti:2015rsa ; Gupta:2018xds ; Asghar:2018tha ;
Yu:2019iwm ; Lu:2016bbk .

We further study the radiative decay processes of $B_{2}^{*}(5747)^{+,0}\to
B^{*+,0}\gamma$. Their partial decay widths are predicted to be

$\displaystyle\Gamma[B_{2}^{*}(5747)^{0}\to B^{*0}\gamma]=51\ \mathrm{keV},$
(21) $\displaystyle\Gamma[B_{2}^{*}(5747)^{+}\to B^{*+}\gamma]=146\
\mathrm{keV},$ (22)

which are in good agreement with the predictions in Ref. Orsland:1998de ,
however, notably smaller than the predictions in Refs. Asghar:2018tha ;
Godfrey:2019cmi ; Yu:2019sqp ; Godfrey:2016nwn ; Lu:2016bbk . The radiative
decay branching fractions can reach up to $\mathcal{O}(10^{-3})$, thus
radiative decays of $B_{2}^{*}(5747)^{+,0}\to B^{*+,0}\gamma$ might be
observed in future experiments.

In the $B_{s}$ meson sector, as the $1^{3}P_{2}$ state both the mass and width
of $B_{s2}^{*}(5840)$ can be well reproduced in the quark model as well. Our
fitted mass $M\simeq 5821$ MeV and width $\Gamma\simeq 1.3$ MeV are compatible
with the measured mass $M_{exp}=5840$ MeV and width $\Gamma_{exp}=(1.49\pm
0.27)$ MeV. There are two OZI allowed two-body strong decay channels $BK$ and
$B^{*}K$. The $BK$ mode governs the decays of $B_{s2}^{*}(5840)$. The ideal
partial width ratio between $B^{*}K$ and $BK$ is fitted to be

$\displaystyle R=\frac{\Gamma[B_{s2}^{*}(5840)\to
B^{*+}K^{-}]}{\Gamma[B_{s2}^{*}(5840)\to B^{+}K^{-}]}\approx 8.7\%,$ (23)

which is in good agreement with the recent LHCb measured one $R^{exp}=(9.3\pm
2.5)\%$ Aaij:2012uva  and the predictions in the literature Luo:2009wu ;
Sun:2014wea ; Godfrey:2019cmi ; Asghar:2018tha ; Lu:2016bbk .

Furthermore, it is found that $B_{s2}^{*}(5840)^{0}$ has a large decay rate
into $B_{s}^{*}\gamma$, the partial width and branching fraction are predicted
to be

$\displaystyle\Gamma[B_{s2}^{*}(5840)^{0}\to B^{*0}_{s}\gamma]$
$\displaystyle\simeq$ $\displaystyle 51\ \mathrm{keV},$ (24) $\displaystyle
Br[B_{s2}^{*}(5840)^{0}\to B^{*0}_{s}\gamma]$ $\displaystyle\simeq$
$\displaystyle 3.4\%.$ (25)

Our predicted radiative partial decay width is consistent with those
predictions in Refs. Orsland:1998de ; Yu:2019sqp . In the literature, a larger
partial width $\Gamma[B_{s2}^{*}(5840)^{0}\to B^{*0}_{s}\gamma]\simeq 100-230$
keV is predicted Lu:2016bbk ; Godfrey:2016nwn ; Asghar:2018tha . The
$B^{*0}_{s}\gamma$ decay channel of $B_{s2}^{*}(5840)^{0}$ may have good
potentials to be observed in future experiments.

####  IV.1.2 $1^{3}P_{0}$ states

The $1^{3}P_{0}$ states in the $B$ and $B_{s}$ families are still missing
experimentally. In the $B$ meson sector, we predict that the mass of the
$B(1^{3}P_{0})$ state is $5722$ MeV, which is consistent with those
predictions in Refs. Gregory:2010gm ; Kher:2017mky ; Ebert:2009ua ;
Godfrey:2016nwn ; DiPierro:2001dwf ; Asghar:2018tha . The $B\pi$ channel is
the only OZI-allowed two body decay channel. Taking the estimated mass
$M=5722$ MeV, we obtain a broad width $\Gamma\simeq 290$ MeV for the
$B(1^{3}P_{0})$ state, which is compatible with our previous result
$\Gamma\simeq 270$ MeV predicted with the SHO wave functions in Ref.
Zhong:2008kd . The $B(1^{3}P_{0})$ state is also predicted to be a broad state
with a width of $\sim 150-250$ MeV in the other models Lu:2016bbk ;
Godfrey:2019cmi ; Godfrey:2016nwn ; Yu:2019iwm ; Sun:2014wea ; Zhu:1998wy . We
also study the radiative decays of $B(1^{3}P_{0})^{+,0}$, our results have
been listed in Table 5, the predicted partial width for
$\Gamma[B(1^{3}P_{0})^{+}\to B^{*+}\gamma]\simeq 477$ keV is about a factor 3
larger than that for $\Gamma[B(1^{3}P_{0})^{0}\to B^{*0}\gamma]\simeq 149$
keV. Our predictions are comparable with those predicted in Refs.
Asghar:2018tha ; Godfrey:2016nwn . It should be mentioned that the radiative
decays of $B(1^{3}P_{0})^{+,0}$ are governed by the $E1$ transitions. For a
heavy-light meson system, the $E1$ transition amplitude $\mathcal{A}$ is
mainly contributed by the light quark, it is proportional to $e_{q}/m_{q}$,
where $e_{q}$ and $m_{q}$ stands for the charge and mass of the light quark,
respectively. The heavy quark contributions are strongly suppressed by the
heavy quark mass $m_{Q}$ (i.e., the factor $1/m_{Q}$). If neglecting the heavy
quark contributions, one has $\Gamma[B(1^{3}P_{0})^{+}\to B^{*+}\gamma]:$
$\Gamma[B(1^{3}P_{0})^{0}\to B^{*0}\gamma]=4:1$, which is slightly larger than
the ratio $\sim 3$ including heavy quark contributions.

In the $B_{s}$ meson sector, the mass of $B_{s}(1^{3}P_{0})$ is expected to be
around $5788$ MeV, which is compatible with those predictions in Refs.
Gregory:2010gm ; Godfrey:2016nwn ; Kher:2017mky ; Ebert:2009ua ; Zeng:1994vj ;
Lu:2016bbk ; Asghar:2018tha ; DiPierro:2001dwf . The $BK$ channel is the only
OZI-allowed two body strong decay channel. Taking the estimated mass $M=5788$
MeV, we obtain a broad width $\Gamma\simeq 270$ MeV for the
$B_{s}(1^{3}P_{0})$ state. This prediction is compatible with our previous
result $\Gamma\simeq 227$ MeV predicted with the SHO wave functions in Ref.
Zhong:2008kd . The $B_{s}(1^{3}P_{0})$ state is also predicted to be a broad
state with a width of $\sim 130-230$ MeV in the other models Godfrey:2019cmi ;
Godfrey:2016nwn ; Sun:2014wea . We also study the radiative decay of
$B_{s}(1^{3}P_{0})$, the predicted partial width for
$\Gamma[B_{s}(1^{3}P_{0})\to B_{s}^{0*}\gamma]\simeq 100$ keV is close to the
predictions in Refs. Godfrey:2016nwn ; Asghar:2018tha . Finally, it should be
mentioned that in some works Lu:2016bbk ; Zeng:1994vj ; Asghar:2018tha , the
predicted mass of $B_{s}(1^{3}P_{0})$ is below the $BK$ mass threshold, which
will lead to a very narrow width for the $B_{s}(1^{3}P_{0})$ state.

Figure 3: Decay widths of the $1P$-wave mixed states $B_{s}(1P_{1})$ and
$B(1P_{1})$ as a function of mass. The mixing angles for the $B_{s}(1P_{1})$
and $B(1P_{1})$ states are adopted the potential model predictions
$\theta_{1P}=-39.6^{\circ}$ and $-35.2^{\circ}$, respectively.

####  IV.1.3  $1^{1}P_{1}$-$1^{3}P_{1}$ mixing

The spin-orbit potential causes a strong configuration mixing between
$1^{3}P_{1}$ and $1^{1}P_{1}$. It is generally believed that $B_{1}(5721)$ and
$B_{s1}(5830)$ correspond to the mixed states $|1P^{\prime}_{1}\rangle$ via
the $1^{1}P_{1}$-$1^{3}P_{1}$ mixing in the $B$ and $B_{s}$ families,
respectively. The other two mixed states $B_{1}(1P_{1})$ and $B_{s1}(1P_{1})$
in the the $B$ and $B_{s}$ families are waiting to be established in future
experiments.

Considering $B_{1}(5721)$ as the mixed state $|1P^{\prime}_{1}\rangle$ defined
in Eq.(9), our fitted mass $M=5753$ MeV is reasonably comparable with the
measured value $M_{exp}=5726$ MeV. With the mixing angle
$\theta_{1P}=-35.2^{\circ}$ determined from our quark model, the width of
$B_{1}(5721)^{0}$ is predicted to be $\Gamma\simeq 41$ MeV, which is in good
agreement with the predictions in Refs. Lu:2016bbk ; Yu:2019iwm , while
slightly larger than the observed width $\Gamma_{exp}\simeq 30$ MeV at LHC
Aaij:2015qla  and those predictions in Refs. Zhang:2018ubo ; Godfrey:2016nwn ;
Asghar:2018tha ; Sun:2014wea . The decay width is nearly saturated by the
$B^{*}\pi$ channel. If we taking the mixing angle around the value obtained in
the heavy-quark symmetry limit, i.e. $\theta_{1P}=-(55\pm 5)^{\circ}$
Isgur:1989vq ; Isgur:1989ed ; Close:2005se ; Zhong:2008kd ; Matsuki:2010zy ;
Barnes:2002mu , the decay width is predicted to be in the range of
$\Gamma=(24.5\pm 2.5)$ MeV, which seems to be more comparable with the LHCb
observations Aaij:2015qla . We also study the radiative decay processes of
$B_{1}(5721)\to B^{*}\gamma,B\gamma$, their partial decay widths are predicted
to be

$\displaystyle\Gamma[B_{1}(5721)^{0}\to B^{*0}\gamma/B^{0}\gamma]$
$\displaystyle=$ $\displaystyle 66/24\ \mathrm{keV},$ (26)
$\displaystyle\Gamma[B_{1}(5721)^{+}\to B^{*+}\gamma/B^{+}\gamma]$
$\displaystyle=$ $\displaystyle 206/69\ \mathrm{keV}.$ (27)

Our predictions are comparable with those in Ref. Orsland:1998de , however,
most of our predictions are notably smaller than the predictions in Refs.
Lu:2016bbk ; Godfrey:2016nwn ; Asghar:2018tha . The radiative decay modes
$B^{*}\gamma$ and $B\gamma$ of $B_{1}(5721)$ may be observed in future
experiments since their branching fractions can reach up to the order of
$O(10^{-3})$.

In the $B$-meson family, the mass for the other mixed state $B_{1}(1P_{1})$ is
about 40 MeV lower than that of $B_{1}(5721)$ according to our potential model
calculations, which is consistent with the prediction in Ref. DiPierro:2001dwf
. A slightly smaller mass splitting, $\sim(10-30)$ MeV, between $B_{1}(5721)$
and $B_{1}(1P_{1})$ is given in Refs. Kher:2017mky ; Ebert:2009ua ; Lu:2016bbk
; Godfrey:2016nwn ; Asghar:2018tha . Thus, the mass of $B_{1}(1P_{1})$ might
be in the range of $(5700\pm 15)$ MeV. Considering the mass uncertainties,
with the mixing angle $\theta_{1P}=-35.2^{\circ}$ we plot the decay width of
$B_{1}(1P_{1})$ as a function of its mass in Fig. 3. It is found that
$B_{1}(1P_{1})$ is a broad state with a width of $\Gamma\simeq(155\pm 10)$
MeV. The $B^{*}\pi$ channel is the only OZI-allowed two body strong decay
channel. The $B_{J}(5732)$ listed in the RPP Zyla:2020zbs  is a good candidate
for the $B_{1}(1P_{1})$. With this assignment, both the measured mass
$M_{exp}=5698$ MeV and width $\Gamma_{exp}=(128\pm 18)$ MeV for the
$B_{J}(5732)$ are in good agreement with the quark model predictions. However,
in Ref. Shah:2016mgq  the $B_{J}(5732)$ is assigned to the $B(1^{3}P_{0})$
state according to the mass spectrum analysis.

As the mixed state $|1P^{\prime}_{1}\rangle$, the mass of $B_{s1}(5830)$ is
consistent with our fitted value $M=5821$ MeV and other determinations in the
literature (See Table 4). It dominantly decays into the $B^{*}K$ channel.
Taking the mixing angle $\theta_{1P}=-39.6^{\circ}$ determined by our
potential model, we find that the theoretical width, $\Gamma\simeq 7.6$ MeV,
is too large to be comparable with the measured one $\Gamma_{exp}\simeq(0.5\pm
0.3\pm 0.3)$ MeV. However, if we taking the mixing angle around the value
obtained in the heavy-quark symmetry limit, i.e. $\theta_{1P}=-(55\pm
5)^{\circ}$, as that adopted in Ref. Zhong:2008kd , the decay width is
predicted to be in the range of $\Gamma=0.08-0.8$ MeV, which is comparable
with the data. It indicates that the mixing angle between $1^{1}P_{1}$ and
$1^{3}P_{1}$ may be close to the value $\theta_{1P}=-55^{\circ}$ obtained in
the heavy-quark symmetry limit. The $B_{s1}(5830)$ has large decay rates into
$B_{s}^{*}\gamma$ and $B_{s}\gamma$ channels. Their partial decay widths are
predicted to be

$\displaystyle\Gamma[B_{s1}(5830)\to B^{*}_{s}\gamma]$ $\displaystyle=$
$\displaystyle 53\ \mathrm{keV},$ (28) $\displaystyle\Gamma[B_{s1}(5830)\to
B_{s}\gamma]$ $\displaystyle=$ $\displaystyle 27\ \mathrm{keV}.$ (29)

The branching fractions of these radiative decays may reach up to
$O(10^{-2})$. Our predictions are comparable with those predicted in Refs.
Lu:2016bbk ; Godfrey:2016nwn ; Orsland:1998de . The $B_{s}^{*}\gamma$ and
$B_{s}\gamma$ decay channels of $B_{s1}(5830)$ may have good potentials to be
observed in future experiments.

In the $B_{s}$ meson sector, the mass of the other mixed state
$B_{s1}(1P_{1})$ is predicted to be about $2-40$ MeV lower than that of
$B_{s1}(5830)$ in various quark models Godfrey:2016nwn ; Kher:2017mky ;
Ebert:2009ua ; Zeng:1994vj ; Lu:2016bbk ; Asghar:2018tha ; DiPierro:2001dwf .
Thus, the mass of $B_{s1}(1P_{1})$ is estimated to be $\sim(5808\pm 19)$ MeV,
which is just around the $B^{*+}K^{-}$ and $B^{*0}K^{0}$ mass thresholds. From
Fig. 3, it is seen that the strong decay properties of $B_{s1}(1P_{1})$ are
very sensitive to the mass threshold. There are three cases to be considered.
(i) If the mass of $B_{s1}(1P_{1})$ is below the $B^{*+}K^{-}$ mass threshold
5818 MeV, the radiative decay modes $B^{*}\gamma$ and $B\gamma$ may play
crucial roles in the decays. The with partial decay widths are estimated to be

$\displaystyle\Gamma[B_{s1}(1P_{1})\to B^{*}_{s}\gamma]$ $\displaystyle\simeq$
$\displaystyle 60\ \mathrm{keV},$ (30) $\displaystyle\Gamma[B_{s1}(1P_{1})\to
B_{s}\gamma]$ $\displaystyle\simeq$ $\displaystyle 40\ \mathrm{keV}.$ (31)

Then, the $B_{s1}(1P_{1})$ should has a very narrow width of
$\Gamma\sim\mathcal{O}(100)$ keV. (ii) If the mass of $B_{s1}(1P)$ lies
between the $B^{*+}K^{-}$ mass threshold 5818 MeV and the $B^{*0}K^{0}$ mass
threshold 5822 MeV, the $B_{s1}(1P_{1})$ dominantly decays into $B^{*0}K^{0}$
mode, and has a narrow width of $\Gamma\simeq(20\pm 15)$ MeV. (iii) If the
mass of $B_{s1}(1P_{1})$ is above the $B^{*0}K^{0}$ mass threshold 5822 MeV,
the $B_{s1}(1P_{1})$ dominantly decays into $B^{*0}K^{0}$ and $B^{*+}K^{-}$
channels, and has a relatively broad width of $\Gamma\simeq(70\pm 30)$ MeV. It
should be mentioned that the OPAL Collaboration observed some signals of a
resonance denoted by $B_{sJ}(5850)$ with a mass of $M_{exp}=(5853\pm 15)$ MeV
and a width of $\Gamma\simeq(47\pm 22)$ MeV many year ago Akers:1994fz . As a
candidate of $B_{s1}(1P_{1})$, both the measured mass and width of
$B_{sJ}(5850)$ are consistent with the predictions. To confirm the
$B_{sJ}(5850)$ resonance and established the $B_{s1}(1P_{1})$ state, more
observations of the $B^{*0}K^{0}$ and $B^{*+}K^{-}$ final states are suggested
to be carried out in future experiments.

As a whole the high mass mixed state $|1P^{\prime}_{1}\rangle$ via the
$1^{1}P_{1}$-$1^{3}P_{1}$ mixing in the $B$ and $B_{s}$ families have been
well-established, they correspond to two narrow states $B_{1}(5721)$ and
$B_{s1}(5830)$ observed in experiments. Some evidence for the low mass mixed
states $|1P_{1}\rangle$ with broad widths predicted in theory may have been
observed in experiments. The $B_{J}(5732)$ and $B_{sJ}(5850)$ resonances
listed by the PDG Zyla:2020zbs  may good candidates for the missing
$|1P_{1}\rangle$ in the $B$ and $B_{s}$ families, respectively.

Table 6: Partial and total decay widths (MeV) for the $2S$-wave $B$ and $B_{s}$ mesons.  $n^{2S+1}L_{J}$ | State | Channel |  $\Gamma_{th}$ (MeV)  | State | Channel | $\Gamma_{th}$ (MeV)
---|---|---|---|---|---|---
($b\bar{u}$ / $b\bar{d}$)
$2^{1}S_{0}$ | $B_{0}(5876)$ | $B^{*}\pi$ | 59 | $B_{s0}(5944)$ | $B^{*}K$ | 55
|  | $B^{*}\eta$ | 0.05 |  | $B_{s}^{*}\eta$ | $\cdots$
|  | $B(1^{3}P_{0})\pi$ | 4.8 |  | $B^{*}_{s}\gamma$ | 0.1$\times 10^{-3}$
|  | $B^{*}\gamma$ | 0.006/0.85$\times 10^{-3}$ |  | $B_{s}(1P)\gamma$ | 0.009
|  | $B(1P)\gamma$ | 0.1/0.034 |  | $B_{s1}(5830)\gamma$ | 4.9$\times 10^{-3}$
|  | $B_{1}(5721)\gamma$ | 0.026/8.8$\times 10^{-3}$ |  | Total | 55
|  | Total | 63 |  |  |
$2^{3}S_{1}$ | $B_{1}(5899)$ | $B\pi$ | 5.1 | $B_{s1}(5966)$ | $BK$ | 15
|  | $B\eta$ | 0.8 |  | $B_{s}\eta$ | 0.1
|  | $B_{s}K$ | 0.02 |  | $B^{*}K$ | 32
|  | $B^{*}\pi$ | 22 |  | $B_{s}^{*}\eta$ | 0.01
|  | $B^{*}\eta$ | 0.6 |  | $B_{s}\gamma$ | 3.6$\times 10^{-3}$
|  | $B_{2}(5747)\pi$ | $<0.01$ |  | $B_{s}(2^{1}S_{0})\gamma$ | 1.0$\times 10^{-6}$
|  | $B(1P)\pi$ | 11 |  | $B_{s2}(5840)\gamma$ | 9$\times 10^{-3}$
|  | $B_{1}(5721)\pi$ | 1 |  | $B_{s}(1P)\gamma$ | 2.2$\times 10^{-3}$
|  | $B\gamma$ | 0.029/0.008 |  | $B_{s1}(5830)\gamma$ | 2.7$\times 10^{-3}$
|  | $B(2^{1}S_{0})\gamma$ | 6$\times 10^{-5}$/2$\times 10^{-5}$ |  | $B(1^{3}P_{0})\gamma$ | 1.4$\times 10^{-3}$
|  | $B_{2}(5747)\gamma$ | 0.06/0.019 |  | Total | 47
|  | $B(1P)\gamma$ | 0.015/5.2$\times 10^{-3}$ |  |  |
|  | $B_{1}(5721)\gamma$ | 0.018/6.2$\times 10^{-3}$ |  |  |
|  | $B(1^{3}P_{0})\gamma$ | 0.004/1.6$\times 10^{-3}$ |  |  |
|  | Total | 41 |  |  |

###  IV.2 $2S$-wave states

####  IV.2.1 $2^{1}S_{0}$ states

The $2^{1}S_{0}$ states in the $B$ and $B_{s}$ families are still not
established. In the $B$ meson sector, our predicted mass for the
$B(2^{1}S_{0})$ state is $M=5876$ MeV, which is in agreement with the
predictions in Refs. Ebert:2009ua ; Godfrey:2016nwn ; Asghar:2018tha ;
DiPierro:2001dwf . Taking the mass $M=5876$ MeV, we calculate the strong and
radiative decay properties of $B(2^{1}S_{0})$, our results are listed in Table
6. The decays of $B(2^{1}S_{0})$ are governed by the $B^{*}\pi$ mode with a
fairly large branching fraction $\sim 94\%$. The width of $B(2^{1}S_{0})$ is
predicted to be $\Gamma\simeq 63$ MeV, which is about a factor $1.5-4$ larger
than the predictions in Refs. Xiao:2014ura ; DiPierro:2001dwf ; Sun:2014wea ;
Asghar:2018tha , while about a factor $1.5-2$ smaller than the predictions in
Refs. Lu:2016bbk ; Godfrey:2016nwn ; Yu:2019iwm ; Ferretti:2015rsa .

In 2015, the LHCb Collaboration observed two new resonances
$B_{J}(5840)^{0,+}$ Aaij:2015qla . Considering $B_{J}(5840)^{0,+}$ as
unnatural parity states, the relatively accurate measurements of the mass and
width for the neutral one are $M_{exp}=(5863\pm 9)$ MeV and
$\Gamma_{exp}=(127\pm 51)$ MeV, respectively Aaij:2015qla . In this case,
signals of $B_{J}(5840)^{0,+}$ should come from the $B^{*}\pi$ decay mode
other than $B\pi$. In Refs. Godfrey:2016nwn ; Jia:2018vwl ; Gupta:2017bcm ;
Lu:2016bbk ; Asghar:2018tha , the $B_{J}(5840)^{0,+}$ resonances were
suggested to be the $B(2^{1}S_{0})$ assignment. If $B_{J}(5840)^{0,+}$ have an
unnatural parity indeed, they strongly favor the $B(2^{1}S_{0})$ assignment.
The measured mass and width together with the decay modes of $B_{J}(5840)$ are
consistent with the our theoretical predictions. However, a natural parity for
$B_{J}(5840)^{0,+}$ is also possible according to the LHCb analysis (see Case
B listed in Table 1). The $B_{J}(5840)^{0,+}$ may possibly decay into both the
$B^{*}\pi$ and $B\pi$ channels. If the $B\pi$ decay mode is confirmed in
future experiments, the $B_{J}(5840)$ resonance should be other assignments
since the $B\pi$ mode is forbidden for the $B(2^{1}S_{0})$ state.

In the $B_{s}$ meson sector, our predicted mass for the $B_{s}(2^{1}S_{0})$
state is $M=5944$ MeV, which is comparable with the other quark model
predictions Kher:2017mky ; Ebert:2009ua ; Zeng:1994vj ; Lu:2016bbk ;
Asghar:2018tha ; Godfrey:2016nwn ; DiPierro:2001dwf . Taking $M=5944$ MeV, we
calculate the strong and radiative decay properties of $B_{s}(2^{1}S_{0})$,
our results are listed in Table 6. The $B^{*}K$ channel is the only OZI-
allowed two body strong decay channel for $B_{s}(2^{1}S_{0})$. Its decay width
is predicted to be $\Gamma\simeq 55$ MeV, which is comparable with those
predictions in Refs. Xiao:2014ura ; Godfrey:2016nwn ; Asghar:2018tha ;
Sun:2014wea ; Ferretti:2015rsa . The $B_{s}(2^{1}S_{0})$ state should have
large potentials to be seen in the $B^{*+}K^{-}$ channel since it has a fairly
narrow width.

####  IV.2.2 $2^{3}S_{1}$ states

In the $B$ meson sector, our predicted mass for $B(2^{3}S_{1})$ is $M=5899$
MeV, which is in good agreement with the predictions in Refs. Ebert:2009ua ;
Asghar:2018tha . Taking the mass $M=5899$ MeV, we calculate the strong and
radiative decay properties of $B(2^{3}S_{1})$, our results are listed in Table
6. It is found that the $B(2^{3}S_{1})$ state is a fairly narrow state with a
width of $\Gamma\simeq 40$ MeV, which is consistent with the prediction in
Refs. Xiao:2014ura ; Sun:2014wea , however, a factor $\sim 2-4$ smaller than
the predictions in Refs. Ferretti:2015rsa ; Godfrey:2016nwn ; Yu:2019iwm ;
Lu:2016bbk . This state dominantly decays into the $B^{*}\pi$ channel with a
branching fraction $\sim 54\%$. The partial width ratio between the two
typical channels $B\pi$ and $B^{*}\pi$ is predicted to be

$\displaystyle\frac{\Gamma[B\pi]}{\Gamma[B^{*}\pi]}\simeq 0.23,$ (32)

which can be used to identify the $B(2^{3}S_{1})$ from its possible candidates
observed in future experiments.

It should be mentioned that the $B_{J}(5840)$ resonance may be a candidate of
the $B(2^{3}S_{1})$ state as suggested in Refs. Godfrey:2019cmi ; Yu:2019iwm .
As this assignment, both the mass and typical decay modes $B^{*}\pi$ and
$B\pi$ predicted in theory are consistent with the LHCb observations
Aaij:2015qla . Our predicted width $\Gamma\simeq 41$ MeV is close to the lower
limit of the measured width $\Gamma_{exp}=(107\pm 54)$ MeV assuming a natural
parity for $B_{J}(5840)^{0}$ Aaij:2015qla . Furthermore, the $B_{J}(5970)$
resonance was also suggested to be the $B(2^{3}S_{1})$ state in the literature
Xu:2014mka ; Jia:2018vwl ; Sun:2014wea ; Lu:2016bbk ; Godfrey:2016nwn ;
Ferretti:2015rsa . The $B_{J}(5970)^{0,+}$ resonances were first observed by
the CDF Collaboration in the $B\pi$ final states in 2013 Aaltonen:2013atp ,
and confirmed by the LHCb Collaboration two years later Aaij:2015qla . The
central value of the measured width is $\Gamma_{exp}\simeq 60-70$ MeV with
large uncertainties (see Table 1). In the LHCb observations, the $B^{*}\pi$
decay mode has been seen, while the $B\pi$ mode may be possibly seen
Zyla:2020zbs . If assigning $B_{J}(5970)$ resonance to $B(2^{3}S_{1})$, our
predicted mass, $M\simeq 5899$ MeV, is about 70 MeV larger than the
observation, while the predicted width, $\Gamma\simeq 54$ MeV, is consistent
with the data. As a conclusion, we cannot exclude the possibilities of the
$B_{J}(5840)$ and $B_{J}(5970)$ resonances as candidates of the
$B(2^{3}S_{1})$ state based on the present experimental information.

In the $B_{s}$ meson sector, our predicted mass for the $B_{s}(2^{3}S_{1})$
state is $M=5966$ MeV, which is in good agreement with the predictions in
Refs. Zeng:1994vj ; Ebert:2009ua ; Asghar:2018tha . Taking $M=5966$ MeV, we
calculate the strong and radiative decay properties of $B_{s}(2^{3}S_{1})$,
our results are listed in Table 6. It is found that the $B_{s}(2^{3}S_{1})$
state is also a narrow state with a width of $\Gamma\simeq 50$ MeV, which is
consistent with the predictions in Refs. Xiao:2014ura ; Sun:2014wea , however,
a factor $\sim 2-4$ smaller than the predictions in Refs. Godfrey:2016nwn ;
Lu:2016bbk ; Ferretti:2015rsa . This state mainly decays into the $B^{*}K$ and
$BK$ channel with large branching fractions $\sim 68\%$ and $\sim 32\%$,
respectively. The partial width ratio between the two main channels $BK$ and
$B^{*}K$ is predicted to be

$\displaystyle\frac{\Gamma[BK]}{\Gamma[B^{*}K]}\simeq 0.47,$ (33)

which may be an important criterion for establishing the $B_{s}(2^{3}S_{1})$.

There may exist a strong configuration mixing between the $2^{3}S_{1}$ and
$1^{3}D_{1}$ states, which is to be discussed in the last part of this
section.

Table 7: Partial and total decay widths (MeV) for the $1D$-wave $B$ and $B_{s}$ mesons. $n^{2S+1}L_{J}$ |  Observed State | Channel |  $\Gamma_{th}$ (MeV)  |  Observed State | Channel | $\Gamma_{th}$ (MeV)
---|---|---|---|---|---|---
$B$ meson | ($b\bar{u}$ / $b\bar{d}$) |  $B_{s}$ meson
$1^{3}D_{3}$ | $B^{*}_{3}(5979)$ | $B\pi$ | 17 | $B^{*}_{s3}(6067)$ | $BK$ | 7.0
|  | $B\eta$ | 0.2 |  | $B_{s}\eta$ | 0.1
|  | $B_{s}K$ | 0.1 |  | $B^{*}K$ | 5.8
|  | $B^{*}\pi$ | 17 |  | $B_{s}^{*}\eta$ | 0.03
|  | $B^{*}\eta$ | 0.1 |  | $B_{s2}(5840)\gamma$ | 0.038
|  | $B_{s}^{*}K$ | 0.01 |  | $B_{s1}(5830)\gamma$ | 2.0$\times 10^{-4}$
|  | $B_{2}(5747)\pi$ | 0.9 |  | $B_{s}(1P_{1})\gamma$ | 2.0$\times 10^{-4}$
|  | $B_{1}(5721)\pi$ | 0.01 |  | $B_{s}(1^{3}P_{0})\gamma$ | 1.0$\times 10^{-4}$
|  | $B(1P_{1})\pi$ | 3.6 |  | Total | 13
|  | $B_{2}(5747)\gamma$ | 0.125/0.043 |  |  |
|  | $B_{1}(5721)\gamma$ | 1.9$\times 10^{-3}$/0.5$\times 10^{-3}$ |  |  |
|  | $B(1P_{1})\gamma$ | 1.9$\times 10^{-3}$/0.5$\times 10^{-3}$ |  |  |
|  | $B(1^{3}P_{0})\gamma$ | 0.7$\times 10^{-3}$/0.2$\times 10^{-3}$ |  |  |
|  | Total | 39 |  |  |
$1^{3}D_{1}$ | $B^{*}_{1}(6056)$ | $B\pi$ | 92 | $B^{*}_{s1}(6101)$ | $BK$ | 87
|  | $B\eta$ | 14 |  | $B_{s}\eta$ | 7.3
|  | $B_{s}K$ | 21 |  | $B^{*}K$ | 37
|  | $B^{*}\pi$ | 41 |  | $B_{s}^{*}\eta$ | 2.6
|  | $B^{*}\eta$ | 5.3 |  | $B_{s2}(5840)\gamma$ | 3.9$\times 10^{-3}$
|  | $B_{s}^{*}K$ | 7.3 |  | $B_{s}(1P_{1})\gamma$ | 0.017
|  | $B_{2}(5747)\pi$ | 2.1 |  | $B_{s1}(5830)\gamma$ | 0.023
|  | $B_{1}(5721)\pi$ | 148 |  | $B_{s}(1^{3}P_{0})\gamma$ | 0.044
|  | $B(1P)\pi$ | 20 |  | Total | 133
|  | $B_{2}(5747)\gamma$ | 0.035/0.01 |  |  |
|  | $B(1P)\gamma$ | 0.122/0.039 |  |  |
|  | $B_{1}(5721)\gamma$ | 0.178 /0.056 |  |  |
|  | $B(1^{3}P_{0})\gamma$ | 0.201/0.066 |  |  |
|  | Total | 350 |  |  |
$1D^{\prime}_{2}$ | $B_{2}(6067)$ | $B^{*}\pi$ | 66 | $B_{s2}(6113)$ | $B^{*}K$ | 22
|  | $B^{*}\eta$ | 1.7 |  | $B_{S}^{*}\eta$ | 0.5
|  | $B_{s}^{*}K$ | 1.4 |  | $B_{s2}(5840)\gamma$ | 0.016
|  | $B(1^{3}P_{0})\pi$ | 3.5 |  | $B_{s}(1P)\gamma$ | 1.3$\times 10^{-3}$
|  | $B_{2}(5747)\pi$ | 6.2 |  | $B_{s1}(5830)\gamma$ | 0.064
|  | $B_{1}(5721)\pi$ | 10 |  | $B_{s}(1^{3}P_{0})\gamma$ | 3.0$\times 10^{-4}$
|  | $B(1P)\pi$ | 0.4 |  | Total | 23
|  | $B_{2}(5747)\gamma$ | 0.106/0.033 |  |  |
|  | $B(1P)\gamma$ | 0.022/6.5$\times 10^{-3}$ |  |  |
|  | $B_{1}(5721)\gamma$ | 0.302/0.104 |  |  |
|  | $B(1^{3}P_{0})\gamma$ | 3.3$\times 10^{-3}$/1.0$\times 10^{-3}$ |  |  |
|  | Total | 90 |  |  |
$1D_{2}$ | $B_{2}(5973)$ | $B^{*}\pi$ | 92 | $B_{s2}(6061)$ | $B^{*}K$ | 85
|  | $B^{*}\eta$ | 6.7 |  | $B_{S}^{*}\eta$ | 4.2
|  | $B_{s}^{*}K$ | 5.2 |  | $B_{s2}(5840)\gamma$ | 6.0$\times 10^{-3}$
|  | $B(1^{3}P_{0})\pi$ | 0.002 |  | $B_{s}(1P)\gamma$ | 0.045
|  | $B_{2}(5747)\pi$ | 64 |  | $B_{s1}(5830)\gamma$ | 0.5$\times 10^{-3}$
|  | $B(1P)\pi$ | 0.4 |  | $B_{s}(1^{3}P_{0})\gamma$ | 0.1$\times 10^{-3}$
|  | $B_{1}(5721)\pi$ | 0.04 |  | Total | 90
|  | $B_{2}(5747)\gamma$ | 0.026/0.008 |  |  |
|  | $B(1P)\gamma$ | 0.223/0.076 |  |  |
|  | $B_{1}(5721)\gamma$ | 1.2$\times 10^{-3}$/0.3$\times 10^{-3}$ |  |  |
|  | $B(1^{3}P_{0})\gamma$ | 0.5$\times 10^{-3}$/0.2$\times 10^{-3}$ |  |  |
|  | Total | 169 |  |  |

###  IV.3 $1D$-wave states

Some evidence of the $1D$-wave $B$ and $B_{s}$ states may have been observed
in experiments. The $B(5970)^{0,+}$ together with the new resonances
$B_{sJ}(6064)$ and $B_{sJ}(6114)$ observed at LHCb may be good candidates of
the $1D$-wave states according to the mass spectrum predictions in various
quark models.

####  IV.3.1 $1^{3}D_{3}$ states

In Ref. Xiao:2014ura , by analyzing the decay properties within the chiral
quark model our group found that the $B_{J}(5970)$ is most likely to be the
$1^{3}D_{3}$ assignment in the $B$-meson family. The $B_{J}(5970)$ as a
candidate of $B(1^{3}D_{3})$ is also suggested in Refs. Yu:2019iwm ;
Lu:2016bbk . In present work, we restudy the $B_{J}(5970)$ by combining the
decay properties with the mass spectrum. It is found that as the
$B(1^{3}D_{3})$ assignment the mass of $B(5970)$ can be well explained with
the potential model. Our predicted mass $M=5979$ MeV is in good agreement with
the observed value $M_{exp}=(5971\pm 5)$ MeV for the neutral state
$B(5970)^{0}$ Zyla:2020zbs . By using the wave function of $B(1^{3}D_{3})$
calculated from the potential model, we further study the decay properties,
our results are listed in Table 7. It is found that the predicted decay width,
$\Gamma\simeq 39$ MeV, is also consistent the measured width
$\Gamma_{exp}\simeq(56\pm 16)$ MeV of $B(5970)^{0}$ by assuming $P=(-1)^{J}$
and using three relativistic Breit-Wigner functions in the fit for mass
difference at LHCb Aaij:2015qla . The partial width ratio between $B\pi$ and
$B^{*}\pi$ is predicted to be

$\displaystyle\frac{\Gamma[B\pi]}{\Gamma[B^{*}\pi]}\simeq 1.0,$ (34)

which is waiting to be tested in future experiments. If the
$B_{J}(5970)^{0,+}$ resonances correspond to the $B(1^{3}D_{3})$ assignment
indeed, the charged state $B_{J}(5970)^{+}$ should have a large radiative
decay rate into $B_{2}^{*}(5747)^{+}\gamma$, the partial width and branching
fraction are predicted to be

$\displaystyle\Gamma[B_{J}(5970)^{+}\to B_{2}^{*}(5747)^{+}\gamma]$
$\displaystyle\simeq$ $\displaystyle 125\ \mathrm{keV},$ (35) $\displaystyle
Br[B_{J}(5970)^{+}\to B_{2}^{*}(5747)^{+}\gamma]$ $\displaystyle\simeq$
$\displaystyle 3\times 10^{-3}.$ (36)

The radiative decay mode $B_{2}^{*}(5747)^{+}\gamma$ may be observed in future
experiments.

In Ref. Xiao:2014ura , considering the $B_{J}(5970)$ as the $B(1^{3}D_{3})$
assignment, our group further predicted that the mass and width of the
$B_{s}(1^{3}D_{3})$ state, as a flavor partner of $B_{J}(5970)$, might be
$M\simeq 6.07$ GeV and $\Gamma\simeq 30$ MeV, respectively. The predicted mass
of $B_{s}(1^{3}D_{3})$ is consistent with the those predicted in Refs.
Zeng:1994vj ; Lu:2016bbk ; Asghar:2018tha , while a relatively narrow width is
also predicted by other works Asghar:2018tha ; Xu:2014mka ; Sun:2014wea ;
Lu:2016bbk ; Godfrey:2016nwn ; Ferretti:2015rsa . It is interestingly found
that the new bottom-strange structure $B_{sJ}(6064)$ with a mass of
$M_{exp}=(6063.5\pm 2.0)$ MeV and a very narrow width of $\Gamma_{exp}=(26\pm
8)$ MeV observed at LHCb Aaij:2020hcw  is consistent with the predictions. In
present work, from the aspects of both mass spectrum and decay properties we
further discuss the possibility of the $B_{sJ}(6064)$ structure as the
$B_{s}(1^{3}D_{3})$ assignment in the $B_{s}$-meson family. With this
assignment, it is found that the measured mass for the $B_{sJ}(6064)$
structure is consistent with the theoretical mass $M=6067$ MeV. Furthermore,
the narrow width of $B_{sJ}(6064)$ can also be explained within our chiral
quark model. From our predicted decay properties listed in Table 7, it is
found that the theoretical width $\Gamma\simeq 13$ MeV is close to lower limit
of the measured value $\Gamma_{exp}=(26\pm 8)$ MeV from LHCb Aaij:2020hcw .
The partial width ratio between $BK$ and $B^{*}K$ is predicted to be

$\displaystyle\frac{\Gamma[BK]}{\Gamma[B^{*}K]}\simeq 1.2.$ (37)

Our predicted mass and decay properties of $B_{s}(1^{3}D_{3})$ are compatible
with those predicted in Refs. Xiao:2014ura ; Lu:2016bbk ; Asghar:2018tha . It
should be pointed out that the large branching fraction for
$Br[B_{sJ}(6064)\to B^{*}K]\simeq 45\%$ seems to be not consistent with the
observations naturally. Since $B_{sJ}(6064)$ causes a clear bump structure
around $6064$ MeV in the $B^{+}K^{-}$ mass spectrum through the $B^{+}K^{-}$
decay, it should also cause another narrow bump structure around $6019$ MeV
through the $B^{*+}K^{-}$ decay with a missing photon from $B^{*+}\to
B^{+}\gamma$, however, this structure was not observed at LHCb. Thus, it
indicates that the $B_{s}(1^{3}D_{3})$ may not be the main contributor to the
$B_{sJ}(6064)$ structure observed in the $B^{+}K^{-}$ mass spectrum.

As a whole, the $B_{J}(5970)$ may be assigned as the $1^{3}D_{3}$ assignment,
which can be tested by the partial width ratio between $B\pi$ and $B^{*}\pi$.
It may be a flavor partner of the $D_{3}^{*}(2750)$ and $D_{s3}(2860)$
resonances listed in RPP Zyla:2020zbs . Their masses might be systematically
overestimated by a value of $\sim 100$ MeV in some quark models Ebert:2009ua ;
Godfrey:2016nwn ; Godfrey:2015dva ; Godfrey:1985xj . There still exists a
puzzle to identify the $B_{sJ}(6064)$ structure as the $B_{s}(1^{3}D_{3})$
state, although both the predicted mass and width seem to be consistent with
the observations. To establish the narrow $B_{s}(1^{3}D_{3})$ state finally,
the observations of both the $B^{+}K^{-}$ and $B^{*+}K^{-}$ decays and their
partial width ratio are crucial in future experiments.

####  IV.3.2 $1^{3}D_{1}$ states

In the $B$ meson sector, the mass for the $B(1^{3}D_{1})$ state is predicted
to be $M=6056$ MeV in our potential model calculations, which is comparable
with the predictions in Refs. Kher:2017mky ; Lu:2016bbk ; Asghar:2018tha ;
DiPierro:2001dwf . Our predicted mass for $B(1^{3}D_{1})$ is about $80$ MeV
larger than that for $B(1^{3}D_{3})$. Taking the mass $M=6056$ MeV, we
calculate the strong and radiative decay properties of $B(1^{3}D_{1})$, our
results are listed in Table 7. It is found that the $B(1^{3}D_{1})$ state is a
broad state with a width of $\Gamma\simeq 350$ MeV, which is consistent with
the predictions in Refs. Sun:2014wea ; Lu:2016bbk . This state mainly decays
into the $B\pi$, $B^{*}\pi$, and $B_{1}(5721)\pi$ channels with branching
fractions $\sim 26\%$, $12\%$, and $42\%$, respectively. The partial width
ratio between the two typical channels $B\pi$ and $B^{*}\pi$ is predicted to
be

$\displaystyle\frac{\Gamma[B\pi]}{\Gamma[B^{*}\pi]}\simeq 2.2,$ (38)

which may be helpful to identify the $B(1^{3}D_{1})$ state from future
observations. In Refs. Godfrey:2019cmi ; Asghar:2018tha , $B_{J}(5970)$ was
suggested to be a candidate for $B(1^{3}D_{1})$. With this assignment we find
that the theoretical width $\Gamma\simeq 230$ MeV is too broad to be
comparable with the measured value measured value $\Gamma_{exp}\simeq 60-70$
MeV (see Table 1). Thus, $B_{J}(5970)$ may not be a good candidate of the
$1^{3}D_{1}$ state.

In the $B_{s}$ meson sector, the mass for the $B_{s}(1^{3}D_{1})$ state is
predicted to be $M=6101$ MeV in our potential model calculations, which is
comparable with the predictions in Refs. Kher:2017mky ; Lu:2016bbk ;
DiPierro:2001dwf . Our predicted mass for $B_{s}(1^{3}D_{1})$ is about $30$
MeV larger than that of $B_{s}(1^{3}D_{3})$. There are large uncertainties in
the predictions of the mass splitting between $B_{s}(1^{3}D_{1})$ and
$B_{s}(1^{3}D_{3})$. In some works Kher:2017mky ; Zeng:1994vj ; Asghar:2018tha
, the $B_{s}(1^{3}D_{1})$ mass is even predicted to be smaller than that of
$B_{s}(1^{3}D_{3})$. Taking the mass $M=6101$ MeV, we calculate the strong and
radiative decay properties of $B_{s}(1^{3}D_{1})$, our results are listed in
Table 7. It is found that the $B_{s}(1^{3}D_{1})$ state has a width of
$\Gamma\simeq 130$ MeV, and mainly decays into the $BK$ and $B^{*}K$ channels
with branching fractions $\sim 65\%$ and $27\%$, respectively. The partial
width ratio between the two typical channels $BK$ and $B^{*}K$ is predicted to
be

$\displaystyle\frac{\Gamma[BK]}{\Gamma[B^{*}K]}\simeq 2.4,$ (39)

which is comparable with the predictions in Refs. Sun:2014wea ; Lu:2016bbk .
From the point of view of mass, the observed resonance $B_{sJ}(6114)$ by the
LHCb Collaboration Aaij:2020hcw  is a good candidate for the
$B_{s}(1^{3}D_{1})$ state. While, the theoretical width $\Gamma\simeq 130$ MeV
is also close the upper limit of the measured width $\Gamma_{exp}=(66\pm 39)$
MeV. There may exist a configuration mixing between the $2^{3}S_{1}$ and
$1^{3}D_{1}$ states, which will be discussed in the last part of this section.

####  IV.3.3 $1^{1}D_{2}$-$1^{3}D_{2}$ mixing

There is a strong configuration mixing between the $1^{3}D_{2}$ and
$1^{1}D_{2}$ states for the heavy-light mesons predicted in the potential
models. For the $B$ meson sector, with the mixing scheme defined in Eq. (9)
the mixing angle is predicted to be $\theta_{1D}=-39.5^{\circ}$, which is
close to the value $-50.8^{\circ}$ extracted in the heavy quark symmetry limit
Matsuki:2010zy ; Close:2005se ; Sun:2014wea . Our predicted masses for the
$B(1D_{2})$ and $B(1D^{\prime}_{2})$ states are about 5973 and 6067 MeV,
respectively, which are close to the predictions in Refs. Kher:2017mky ;
DiPierro:2001dwf . A fairly large mass splitting between $B(1D_{2})$ and
$B(1D^{\prime}_{2})$, $\Delta M=94$ MeV, is obtained in present work. It is
comparable with the predictions of $\Delta M=110-130$ MeV in Refs.
Godfrey:2016nwn ; Lu:2016bbk . With the masses and wave functions obtained
from our potential model calculations, the decay properties for these two
mixed states $B(1D_{2})$ and $B(1D^{\prime}_{2})$ are estimated, the results
are listed in Table 7. It is found that the low mass state $B(1D_{2})$ has a
broad width of $\Gamma\simeq 170$ MeV, and dominantly decays into $B^{*}\pi$
and $B^{*}_{2}(5747)\pi$ channels with branching fractions about $54\%$ and
$38\%$, respectively. While the high mass state $B(1D^{\prime}_{2})$ has a
relatively narrow width of $\Gamma\simeq 90$ MeV, and dominantly decays into
$B^{*}\pi$, $B_{1}(5721)\pi$ and $B^{*}_{2}(5747)\pi$ channels with branching
fractions about $73\%$, $11\%$ and $7\%$, respectively. Our predicted decay
properties are roughly comparable with those predictions in Refs. Xiao:2014ura
; Sun:2014wea ; Godfrey:2016nwn .

For the $B_{s}$ meson sector, we predict the mixing angle
$\theta_{1D}=-39.9^{\circ}$. Our predicted masses for the $B_{s}(1D_{2})$ and
$B_{s}(1D^{\prime}_{2})$ states are about 6061 and 6113 MeV, respectively,
which are close to the predictions in Refs. Lu:2016bbk ; Zeng:1994vj ;
DiPierro:2001dwf . An intermediate mass splitting between $B_{s}(1D_{2})$ and
$B_{s}(1D^{\prime}_{2})$, $\Delta M=52$ MeV, is obtained in present work,
which is comparable with the predictions of $\Delta M=45-70$ MeV in Refs.
Lu:2016bbk ; DiPierro:2001dwf . With the masses and wave functions obtained
from our potential model calculations, the decay properties for these two
mixed states $B_{s}(1D_{2})$ and $B_{s}(1D^{\prime}_{2})$ are estimated, the
results are listed in Table 7. It is found that the low mass state
$B_{s}(1D_{2})$(6061) has a intermediate width of

$\displaystyle\Gamma\simeq 90\ \ \mathrm{MeV},$ (40)

and dominantly decays into $B^{*}K$ channel with a branching fraction about
$94\%$. Our predicted width is about a factor $1.5-2$ smaller than those
predictions in Refs. Asghar:2018tha ; Lu:2016bbk ; Sun:2014wea ;
Godfrey:2016nwn . The $B_{s}(1D_{2})$ state may be observed around $6016$ MeV
in the $B^{+}K^{-}$ mass spectrum through the $B^{*+}K^{-}$ decay with a
missing photon from $B^{*+}\to B^{+}\gamma$.

While the high mass state $B_{s}(1D^{\prime}_{2})$(6113) has a very narrow
width of

$\displaystyle\Gamma\simeq 23\ \ \mathrm{MeV},$ (41)

and dominantly decays into $B^{*}K$ channel with a branching fraction about
$95\%$. The decay properties predicted in present work are in good agreement
with those predictions in Refs. Asghar:2018tha ; Sun:2014wea ; Xiao:2014ura ;
Godfrey:2016nwn . The narrow mixed state $B_{s}(1D^{\prime}_{2})$ with a mass
of $M=6113$ MeV may be the main contributor to the $B_{sJ}(6064)$ structure
observed in the $B^{+}K^{-}$ mass spectrum at LHCb Aaij:2020hcw . In this
case, the signal in the $B^{+}K^{-}$ mass spectrum may mainly come from the
$B^{*+}K^{-}$ decay with a missing photon from $B^{*+}\to B^{+}\gamma$.
Including the energy of the missing photon, the mass and width are determined
to be $M_{exp}=(6109\pm 1.8)$ MeV and $\Gamma_{exp}=(22\pm 9)$ MeV for the
resonance $B_{sJ}(6109)$ Aaij:2020hcw . It is interestingly found that the
$B_{sJ}(6109)$ resonance favors the assignment of the $1D$-wave mixed state
$B_{s}(1D^{\prime}_{2})$. The predicted mass, width, decay mode are in good
agreement with the observations. Finally, it should be mentioned that the
$B_{s}(1^{3}D_{3})$ state may have a few contributions to the $B_{sJ}(6064)$
structure through the $B^{+}K^{-}$ decay as well, since this state with a mass
of $M\simeq 6067$ MeV lies just around the peak position.

Figure 4: The partial decay widths and total decay widths for the mixed states
via $2^{3}S_{1}-1^{3}D_{1}$ mixing in the $B$-meson family as functions of the
mixing angle $\theta$. In the horizontal direction, the shaded region
represents the possible range of the measured width from LHCb. In the vertical
direction, shaded region represents the possible range of the mixing angle
$\theta\simeq-(45\pm 16)^{\circ}$ suggested in Refs. Zhong:2010vq ;
Zhong:2009sk . The masses for $B_{J}(5840)$ and $B(|SD\rangle_{H})$ are taken
to be 5890 and 6040 MeV, respectively. Figure 5: The partial decay widths and
total decay widths for the mixed states via $2^{3}S_{1}-1^{3}D_{1}$ mixing in
the $B_{s}$-meson family as functions of the mixing angle $\theta$. In the
horizontal direction, the shaded region represents the possible range of the
measured width from LHCb. In the vertical direction, shaded region represents
the possible range of the mixing angle $\theta\simeq-(45\pm 16)^{\circ}$
suggested in Refs. Zhong:2010vq ; Zhong:2009sk . The masses for $B_{sJ}(6114)$
and $B_{s}(|SD\rangle_{L})$ are taken to be 6114 and 5964 MeV, respectively.

###  IV.4 $2^{3}S_{1}$-$1^{3}D_{1}$ mixing

It should be mentioned that there may exist a configuration mixing between the
$2^{3}S_{1}$ and $1^{3}D_{1}$ states. To explain the strong decay properties
of the $D_{J}^{*}(2600)$ and/or $D_{s1}(2700)$, configuration mixing between
$2^{3}S_{1}$ and $1^{3}D_{1}$ is suggested in the literature Close:2006gr ;
Chen:2011rr ; Li:2009qu ; Yu:2020khh ; Chen:2015lpa . In Refs. Zhong:2010vq ;
Zhong:2009sk , our group also carefully studied the strong decay properties of
the $D_{J}^{*}(2600)$ and $D_{s1}(2700)$. According the analysis, both
$D_{J}^{*}(2600)$ and $D_{s1}(2700)$ could be explained as the mixed state
$|SD\rangle_{L}$ via the $2^{3}S_{1}$-$1^{3}D_{1}$ mixing with the following
mixing scheme:

$\left(\begin{array}[]{c}|SD\rangle_{L}\\\ |SD\rangle_{H}\\\
\end{array}\right)=\left(\begin{array}[]{cc}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\\\
\end{array}\right)\left(\begin{array}[]{c}2^{3}S_{1}\\\ 1^{3}D_{1}\\\
\end{array}\right),$ (42)

where the mixed angle is estimated to be $\theta\simeq-(45\pm 16)^{\circ}$.
The $D_{s1}(2860)$ resonance observed in the $BK$ final state at LHCb
Aaij:2014xza ; Aaij:2014baa  seems to be the high mass mixed state
$|SD\rangle_{H}$ as the partner of the low mass state $D_{s1}(2700)$
Xiao:2014ura , from which one can estimate a mass splitting $\Delta M\simeq
150$ MeV between the high and low mass mixed states. Similarly, the
$2^{3}S_{1}$-$1^{3}D_{1}$ mixing might also exist in the $B$ and $B_{s}$ meson
families.

The strong decay properties for the mixed states $|SD\rangle_{L}$ and
$|SD\rangle_{H}$ in the $B$ and $B_{s}$ meson families were studied in another
work of our group Xiao:2014ura . It is interestingly found that the newly
observed resonances $B_{J}(5840)$ and $B_{sJ}(6114)$ at LHCb Aaij:2015qla ;
Aaij:2020hcw  are most likely to be the mixed states $B(|SD\rangle_{L})$ and
$B_{s}(|SD\rangle_{H})$, respectively, by comparing the measured masses and
widths with the theoretical predictions (see Figs. 1 and 2 in Ref.
Xiao:2014ura ).

Considering the $B_{J}(5840)$ resonance as the low mass mixed state
$B(|SD\rangle_{L})$, we revise the strong decay properties by using the wave
functions obtained from our quark potential model calculations. Our results
are shown in Fig. 4. With the mixing angle $\theta\simeq-(45\pm 16)^{\circ}$
determined in Refs. Zhong:2010vq ; Zhong:2009sk  and the mass $M_{exp}\simeq
5890$ MeV measured at LHCb Aaij:2015qla , the $B(|SD\rangle_{L})$ state has a
width of $\Gamma\simeq(76\pm 20)$ MeV, and dominantly decays into $B^{*}\pi$
channel. There may be a sizeable decay rate into the $B\pi$ channel. The
partial width ratio between $B\pi$ and $B^{*}\pi$ is predicted to be

$\displaystyle\frac{\Gamma[B\pi]}{\Gamma[B^{*}\pi]}\simeq 0.1-0.7,$ (43)

which is sensitive to the mixing angle. The predicted width is consistent with
the measured width $\Gamma_{exp}=(107\pm 54)$ MeV by assuming $P=(-1)^{J}$.
Moreover, the predicted decay modes are also consistent with the observations
of $B_{J}(5840)$. To better understand the nature of $B_{J}(5840)$, more
accurate measurements of the width together with the partial width ratio are
expected to be carried out in future experiments.

If $B_{J}(5840)$ corresponds to the low mass state $B(|SD\rangle_{L})$ indeed,
the mass of $B(|SD\rangle_{H})$ may be about 150 MeV larger than that of
$B_{J}(5840)$. Taking a mass of $M\simeq 6040$ MeV for $B(|SD\rangle_{H})$, we
show its decay properties in Fig. 4 as well. Within the mixing angle range
$\theta\simeq-(45\pm 16)^{\circ}$ suggested in Zhong:2010vq ; Zhong:2009sk ,
the $B(|SD\rangle_{H})$ has a width of $\Gamma\simeq(197\pm 47)$ MeV, and
mainly decay into $B\pi$ and $B_{1}(5721)\pi$ channels. The partial width
ratio between $B\pi$ and $B_{1}(5721)\pi$ is predicted to be

$\displaystyle\frac{\Gamma[B\pi]}{\Gamma[B_{1}(5721)\pi]}\sim 1,$ (44)

which is insensitive to the mixing angle. Future observations in the $B\pi$
channel with a larger data sample at LHCb may have a potential to discover
this high mass mixed state $B_{1}(|SD\rangle_{H})$.

In the $B_{s}$ meson sector, considering the $B_{sJ}(6114)$ resonance observed
in the $B^{+}K^{-}$ mass spectrum Aaij:2020hcw  as the high mass mixed state
$B_{s}(|SD\rangle_{H})$, we revise the strong decay properties by using the
wave functions obtained from our quark potential model calculations. Our
results are shown in Fig. 5. It is found that with the mixing angle
$\theta\simeq-(45\pm 16)^{\circ}$ determined in Refs. Zhong:2010vq ;
Zhong:2009sk , the $B_{s}(|SD\rangle_{H})$ state has a width of
$\Gamma\simeq(95\pm 15)$ MeV, and dominantly decays into $BK$ channel with a
branching fraction $\sim 90\%$. Both the decay mode $B^{+}K^{-}$ and width
$\Gamma_{exp}=(66\pm 39)$ MeV observed for $B_{sJ}(6114)$ at LHCb Aaij:2020hcw
can be well understood in our quark model calculations. Thus, the
$B_{sJ}(6114)$ may favor the mixed state $B_{s}(|SD\rangle_{H})$.

The mass for the low mass state $B_{s1}(|SD\rangle_{L})$ may be about 150 MeV
smaller than that of $B_{sJ}(6114)$. Taking a mass of $M\simeq 5960$ MeV for
the low mass state $B_{s1}(|SD\rangle_{L})$, we show its decay properties in
Fig. 5 as well. In the mixing angle range $\theta\simeq-(45\pm 16)^{\circ}$,
the $B_{s1}(|SD\rangle_{L})$ has a width of $\Gamma\simeq(70\pm 10)$ MeV, and
mainly decay into $B^{*}K$ channel. The partial width ratio between $BK$ and
$B^{*}K$,

$\displaystyle\frac{\Gamma[BK]}{\Gamma[B^{*}K]}<0.5,$ (45)

is sensitive to the mixing angle. The $B_{s1}(|SD\rangle_{L})$ is most likely
to be observed in the $B^{+}K^{-}$ final state with a larger data sample at
LHCb.

As a whole the $B_{J}(5840)$ and $B_{sJ}(6114)$ may favor the mixed states
$B(|SD\rangle_{L})$ and $B_{s}(|SD\rangle_{H})$ via $2^{3}S_{1}$-$1^{3}D_{1}$
mixing, respectively. Their partners $B(|SD\rangle_{H})$ and
$B_{s}(|SD\rangle_{L})$ are expected to be observed in their dominant decay
channels with a larger data sample at LHCb.

##  V summary

The experimental progress provides us good opportunities to establish an
abundant $B$ and $B_{s}$-meson spectrum up to the second orbital excitations.
In this work, combining the newest experimental progress, we carry out a
systematical study of the mass spectrum, strong decays and radiative decays of
the $1P$-, $1D$-, and $2S$-wave excited $B$ and $B_{s}$ states in the
constitute quark model. The mass and strong decay properties for the well
established $1P$-wave resonances $B_{1}(5721)^{+,0}$, $B_{2}^{*}(5747)^{+,0}$,
$B_{s1}(5830)$ and $B_{s2}^{*}(5840)$ can be consistently explained. The
possible assignments for the high mass resonances/structures
$B_{J}(5840)^{0,+}$, $B(5970)^{0,+}$, $B_{sJ}(6064)$ and $B_{sJ}(6114)$ are
discussed. We hope that our study can provide some useful information towards
establishing an abundant $B$ and $B_{s}$-meson spectrum. Our main results are
summarized as follows.

For the $P$-wave states, several points should be emphasized. (i) Some
radiative decay processes, such as $B_{s1}(5830)\to B_{s}^{(*)}\gamma$ and
$B_{s2}^{*}(5840)\to B_{s}^{*}\gamma$, have good potentials to be found in
future experiments due to their fairly branching fractions of
$\mathcal{O}(10^{-2})$. (ii) The $B_{J}(5732)$ and $B_{sJ}(5850)$ resonances
listed by the PDG Zyla:2020zbs  may good candidates for the missing
$|1P_{1}\rangle$ state in the $B$ and $B_{s}$ families, respectively. (iii)
Both $B(1^{3}P_{0})$ and $B_{s}(1^{3}P_{0})$ may hardly be observed in
experiments due to their very broad width of $\Gamma\sim 300$ MeV.

The $B_{J}(5840)$ resonance and the new $B_{sJ}(6114)$ structure observed in
the $B^{+}K^{-}$ mass spectrum may be explained with the mixed states
$B(|SD\rangle_{L})$ and $B_{s}(|SD\rangle_{H})$ via $2^{3}S_{1}$-$1^{3}D_{1}$
mixing, respectively. To confirm the nature of the $B_{J}(5840)$ and
$B_{sJ}(6114)$, the typical ratios $\Gamma(B\pi)/\Gamma(B^{*}\pi)$ and
$\Gamma(B^{*}K)/\Gamma(BK)$ are suggested to be measured in future
experiments. The other two missing states, $B(|SD\rangle_{H})$ with a mass of
$M\simeq 6010$ MeV and $B_{s}(|SD\rangle_{L})$ with a mass of $M\simeq 5960$
MeV, may be observed in the $B\pi$ and $B^{*}K$ final states. On the other
hand, if there is little mixing between $2^{3}S_{1}$-$1^{3}D_{1}$, the
$B_{J}(5840)$ and $B_{sJ}(6114)$ resonances may be candidates for the
$B(2^{3}S_{1})$ and $B_{s}(1^{3}D_{1})$ states, respectively.

The $B_{J}(5970)$ resonance may be assigned as the $1^{3}D_{3}$ state in the
$B$ meson family, although it as a pure $2^{3}S_{1}$ state cannot be excluded
according to present experimental information. To clarify the nature of
$B_{J}(5970)$, further observations of the $B\pi$ and $B^{*}\pi$ channels and
a measurement of their partial width ratio are necessary. In the $B_{s}$
family, the predicted mass $M\simeq 6067$ MeV and width $\Gamma\simeq 13$ MeV
for the $B_{s}(1^{3}D_{3})$ are consistent with the $B_{sJ}(6064)$ structure
observed in the $B^{+}K^{-}$ mass spectrum. However, the $B_{s}(1^{3}D_{3})$
may not be the main contributor to the $B_{sJ}(6064)$ structure. If
$B_{sJ}(6064)$ corresponds to $B_{s}(1^{3}D_{3})$, another narrow structure
around $6019$ MeV coming from the $B^{*+}K^{-}$ decay should be observed in
the $B^{+}K^{-}$ mass spectrum, however, it was not seen at LHCb.

The narrow $B_{sJ}(6064)$ structure observed in the $B^{+}K^{-}$ mass spectrum
may mainly come from the resonance $B_{sJ}(6109)$ decaying into $B^{*+}K^{-}$.
The $B_{sJ}(6109)$ resonance favors the assignment of the high mass $1D$-wave
mixed state $B_{s}(1D^{\prime}_{2})$ with $J^{P}=2^{-}$. This state dominantly
decays into $B^{*}K$ channel with branching fraction about $95\%$, and the
$BK$ decay is forbidden. The other missing mixed state $B_{s}(1D_{2})$ has a
mass of $M\simeq 5973$ MeV and a width of $\Gamma\simeq 90$ MeV. It is most
likely to be established in the $B^{+}K^{-}$ mass spectrum through the
$B^{*+}K^{-}$ decay with a missing photon from $B^{*+}\to B^{+}\gamma$. In the
$B$-meson sector, two relatively broad mixed states, $B(1D_{2})$ with $M\simeq
5973$ MeV and $B(1D^{\prime}_{2})$ with $M\simeq 6067$ MeV, may be observed
their main decay channel $B^{*}\pi$ with a larger data sample at LHCb.

Finally, it should be mentioned that the $B_{J}(5840)$ resonance may be a
candidate of the $2S$-wave state $B(2^{1}S_{0})$ as well. In this case, the
$B\pi$ mode of $B_{J}(5840)$ is forbidden, which should be further confirmed
in future experiments. In the $B_{s}$ meson sector, our predicted mass and
width for the $B_{s}(2^{1}S_{0})$ state are $M=5944$ MeV and $\Gamma\simeq 55$
MeV, respectively. The $B^{*}K$ channel is the only OZI-allowed two body
strong decay mode. The $B_{s}(2^{1}S_{0})$ state should have large potentials
to be seen in the $B^{*+}K^{-}$ channel since it has a fairly narrow width.

## Acknowledgement

Helpful discussions with Qi-Fang Lü and Ming-Sheng Liu are greatly
appreciated. This work is supported by the National Natural Science Foundation
of China (Grants Nos. U1832173, 11775078).

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