∎

e1e-mail<EMAIL_ADDRESS>e2e-mail<EMAIL_ADDRESS>
11institutetext: Department of Physics, Sardar Vallabhbhai National Institute
of Technology, Surat, Gujarat 395007, India

# Study of $B$, $B_{s}$ mesons using heavy quark effective theory

Keval Gandhi1,a Ajay Kumar Rai1,b

(Received: date / Accepted: date)

###### Abstract

Inspired by the lower statistical information in the bottom sector, in this
paper, we calculate the masses and the strong decays of excited $B$ and
$B_{s}$ mesons in the framework of heavy quark effective theory (HQET). Using
an effective chiral Lagrangian approach based on heavy quark spin-flavor and
light quark chiral symmetry, we explore the flavor independent parameters
$\Delta_{F}^{(c)}=\Delta_{F}^{(b)}$ and $\lambda_{F}^{(c)}=\lambda_{F}^{(b)}$
to calculate the masses of experimentally unknown bottom mesons. Our
predictions are consistent with the available experimental results and other
theoretical studies. Their strong decay to the ground state bottom mesons plus
light pseudoscalar mesons is calculated in terms of the square of the
couplings $g_{H}$, $g_{S}$, $g_{T}$, $g_{X}$, $g_{Y}$, and $g_{R}$. The
weighted average value of the couplings $g_{H}$, $g_{S}$ and $g_{T}$ is
obtained in the charm sector [Phys. Rev. D 86, 054024 (2012)] by fitting the
calculated decay widths with experimental measurements, which will be used in
the present study to analyze the strong decays of excited open bottom mesons.
Moreover, the ratio of the decay rates is also predicted, which can be
countered with future experimental data.

###### Keywords:

Heavy Quark Effective Theory (HQET) Mass spectroscopy Strong decay analysis

††journal: Eur. Phys. J. C

##  1 Introduction

In the past decade, evidence of the charmed mesons has increased rapidly and
remarkably compared to the bottom mesons Zyla2020 . Due to large non-resonant
continuum contributions, however, experimentally the broad resonance states
are difficult to identify. To date only the ground state and the low-lying
excited states of $B$ and $B_{s}$ mesons ($B_{1}(5721)$, $B_{J}^{*}(5732)$,
$B_{2}^{*}(5747)$, $B_{s1}(5830)$, $B_{J}(5840)$, $B_{s2}^{*}(5840)$, and
$B_{J}(5970)$) are reported by various experimental groups such as LHCb
aaij2015precise ; aaij2013first , CDF aaltonen2014study ;
aaltonen2009measurement , and D0 abazov2007properties . In the bottom sector,
we believe that more experimental statistics may be reported in the coming
years. For that, the LHCb experiment working at the CERN will be in a unique
position.

At the latest, LHCb aaij2020  measured the masses $(M)$ and decay widths
$(\Gamma)$ of two new states of excited $B_{s}$ meson in the $B^{+}K^{-}$
decay mode as

$\displaystyle(M_{1},\Gamma_{1})={}$ $\displaystyle(6063.5\pm 1.2\pm
0.8,~{}26\pm 4\pm 4)~{}MeV,$ (1) $\displaystyle(M_{2},\Gamma_{2})={}$
$\displaystyle(6114\pm 3\pm 5,~{}66\pm 18\pm 21)~{}MeV,$

with the first statistical and second systematic uncertainties. Based on
predictions from various theoretical and phenomenological studies for the
masses of excited $B_{s}$ mesons godfrey2016b ; sun2014higher ; ebert2010heavy
; lu2016excited ; kher2017spectroscopy ; devlani2012spectroscopy ;
shah2016spectroscopy ; jia2019regge ; chen2018regge ; lahde2000spectra , these
newly observed states may result from the first orbitally excited states. The
$B_{s}(1^{3}D_{3})$ is predicted with a relatively narrow decay width of less
than 50 MeV sun2014higher ; Xiao2014 ; Asghar2018 . The states $1^{1}D_{2}$
and $1^{3}D_{2}$ can be a mix, depending on the mixing angle one of the states
from them is as narrow as 20 MeV. The spectroscopic notation, $\cal{N}$
${}^{2S+1}L_{J}$, is commonly used to describe these states; where $\cal{N}$
is the radial quantum number, $S$ is the sum of the intrinsic spin of the
quarks, $L$ is the orbital angular momentum of the quarks, and $J$ gives the
total angular momentum.

The heavy quark effective theory (HQET) is an important tool for calculating
properties of heavy-light systems where the heavy quark is assumed to have
infinite mass ($m_{Q}\rightarrow\infty$) eichten1993properties ;
mannel1996review ; falk1996excited . In which the total angular momentum of
the system is defined as $\vec{J}=\vec{s}_{Q}+\vec{s}_{l}$, where
$\vec{s}_{Q}$ is the spin of the heavy quark and
$\vec{s}_{l}=\vec{s}_{\bar{q}}+\vec{L}$ is the spin of the light degrees of
freedom, $s_{Q}$ and $s_{l}$ are conserved in the heavy quark mass limit.
Here, $\vec{s}_{\bar{q}}$ and $\vec{L}$ are the spin and the orbital angular
momentum of the light antiquark, respectively close1992effective ;
neubert1992heavy . For $P$-wave ($L$ = 1), two doublets
$\vec{s}_{l}={\frac{1}{2}}$ and $\vec{s}_{l}={\frac{3}{2}}$, containing four
states $J^{P}_{s_{l}}=(0^{+},1^{+})_{{\frac{1}{2}}}$ and
$J^{P}_{s_{l}}=(1^{+},2^{+})_{{\frac{3}{2}}}$. The first doublet has not been
filled experimentally to date, while the second doublet is filled with
$B_{s1}(5730)^{0}$ and $B_{s2}^{*}(5840)^{0}$, respectively.

In heavy quark limit, using available experimental information on open charm
mesons Ref. colangelo2012new  predicts the masses of some unobserved bottom
mesons. The same formalism used in Ref. gupta2019placing  by Gupta and
Upadhyay, and examine the states $B_{s1}(5730)^{0}$ and $B_{s2}^{*}(5840)^{0}$
as the strange partners of $B_{J}(5721)$ and $B_{s2}^{*}(5747)$, respectively.
In Ref. gupta2018analysis , they analyzed experimentally available excited
open charmed mesons to predict similar spectra in open bottom mesons. At the
latest, G.-L. Yu and Z.-G. Wang yu2019analysis  studied the strong decays of
the experimentally observed excited bottom and strange bottom mesons with the
${}^{3}P_{0}$ decay model. They identify $B_{1}(5721)$ as $1^{3}P_{1}$ and
$B_{2}^{*}(5747)$ as $1^{3}P_{2}$ and suggest the quantum states $2^{3}S_{1}$
and $1^{3}D_{3}$ for $B_{J}(5840)$ and $B(5970)$ mesons, respectively
yu2019analysis . These theoretical predictions are not consistent with each
other. Therefore, a further test of calculations against the experimental
measurements is required to identify the excited $B$ and $B_{s}$ mesons.

The main aim of this work is to discuss the possible spin-parity of
experimentally observed excited bottom mesons. We shall calculate masses and
strong decay behaviors of excited bottom mesons in the framework of leading
order approximation of heavy quark effective theory. The flavor symmetry of
heavy quark explores the flavor independent parameters
$\Delta_{F}^{(c)}=\Delta_{F}^{(b)}$ and $\lambda_{F}^{(c)}=\lambda_{F}^{(b)}$
to calculate the masses of excited excited $B$ and $B_{s}$ mesons. Their
strong two-body decays to ground state bottom mesons plus light pseudoscalar
mesons are calculated. The suppression factor arising from the breakdown of
isospin symmetry, which occurs when the mass difference between the parent
heavy-light meson with strangeness and the daughter nonstrange meson is
smaller than the kaon mass, is taken into account here. Our investigation
confirmed the spin-parity of experimentally observed open bottom mesons and we
believe that this study could provide valuable information in future
experimental studies.

This paper is organized as follows: in the framework of heavy quark effective
theory section 2 presents the mass parameters of excited heavy-light flavor
mesons and discusses their strong decays to a member of the lowest-lying
negative parity states plus light pseudoscalar mesons ($\pi$, $\eta$, and
$K$). In section 3, we explore the flavor independent parameters in heavy
quark mass limit and calculate the mass spectra of excited open bottom mesons.
Their two-body strong decays are analyzed in section 4. Finally, a summary is
given in section 5.

##  2 Theoretical framework

In the leading order approximation, the heavy quark effective theory is
developed by expanding the QCD Lagrangian in the power of $1/m_{Q}$ as

$\displaystyle{\cal{L}}{{}_{HQET}}={\cal{L}}{{}_{0}}+\frac{1}{m_{Q}}{\cal{L}}{{}_{1}}+\frac{1}{m_{Q}^{2}}{\cal{L}}{{}_{2}}+...,$
(2)

where finite heavy quark mass corrections are applied and the heavy quark
symmetry breaking terms are studied order by order chen2017review . Here the
leading Lagrangian ${\cal{L}}{{}_{0}}=\bar{h}_{\nu}(i\nu\cdot D)h_{\nu}$ has
an exact spin-flavor symmetry of heavy quark, therefore, only the first term
survives in the heavy quark mass limit (a detailed discussion is given in
manohar2000 ).

Table 1: Calculated values of spin-averaged masses $\bar{M}_{F}$ (in MeV), mass splitting $\Delta_{F}$ (in MeV), and the hyperfine splitting $\lambda_{F}$ (in MeV2) of excited $D$ and $D_{s}$ mesons. |  |  $D$ meson |  |  |  $D_{s}$ meson |
---|---|---|---|---|---|---
$\cal{N}$ $J^{P}_{s_{l}}$ | $\bar{M}_{F}$ | $\Delta_{F}$ | $\lambda_{F}$ | $\bar{M}_{F}$ | $\Delta_{F}$ | $\lambda_{F}$
1 $(0^{-},1^{-})_{\frac{1}{2}}$ | 1971.3 | $-$ | (262.2)2 | 2076.5 $\pm$ 0.4 | $-$ | (270.2 $\pm$ 0.9)2
1 $(0^{+},1^{+})_{\frac{1}{2}}$ | 2395.0 $\pm$ 7.3 | 424.0 $\pm$ 7.3 | (240 $\pm$ 24)2 | 2424.3 $\pm$ 0.7 | 347.8 $\pm$ 0.8 | (290.9 $\pm$ 0.8)2
1 $(1^{+},2^{+})_{\frac{3}{2}}$ | 2448.0 $\pm$ 9.7 | 477.0 $\pm$ 9.7 | (240 $\pm$ 69)2 | 2556.5 $\pm$ 0.2 | 480.0 $\pm$ 0.5 | (255.0 $\pm$ 2.4)2
2 $(0^{-},1^{-})_{\frac{1}{2}}$ | 2610 $\pm$ 10 | 640 $\pm$ 10 | (200 $\pm$ 28)2 | 2703.0 $\pm$ 2.8 | 627.0 $\pm$ 2.8 | (114.0 $\pm$ 7.9)2
1 $(1^{-},2^{-})_{\frac{3}{2}}$ | 2799.2 | 827.9 | (194.3)2 | 2924.2 | 847.7 $\pm$ 0.4 | (198.6)2
1 $(2^{-},3^{-})_{\frac{5}{2}}$ | 2857.6 | 886.6 | (166.9)2 | 2966.8 | 890.3 $\pm$ 0.4 | (149.2)2
2 $(0^{+},1^{+})_{\frac{1}{2}}$ | 2928.8 | 957.4 | (97.5)2 | 3063.8 | 987.4 $\pm$ 0.4 | (99.7)2
2 $(1^{+},2^{+})_{\frac{3}{2}}$ | 2995.0 $\pm$ 4.1 | 1024.0 $\pm$ 4.1 | (300 $\pm$ 25)2 | 3146.5 | 1070.0 $\pm$ 0.4 | (168.32)2
3 $(0^{-},1^{-})_{\frac{1}{2}}$ | 3087.5 | 1116.2 | (161.8)2 | 3236.2 | 1159.8 $\pm$ 0.4 | (136.3)2
1 $(2^{+},3^{+})_{\frac{5}{2}}$ | 3112.8 | 1141.4 | (389.3)2 | 3244 | 1167.5 $\pm$ 0.4 | (311.9)2
1 $(3^{+},4^{+})_{\frac{7}{2}}$ | 3169.5 | 1198.2 | (407.7)2 | 3285.8 | 1209.3 $\pm$ 0.4 | (373.5)2
2 $(1^{-},2^{-})_{\frac{3}{2}}$ | 3247.4 | 1276 | (274.6)2 | 3395.5 | 1319.0 $\pm$ 0.4 | (225.6)2
2 $(2^{-},3^{-})_{\frac{5}{2}}$ | 3323.3 | 1352.3 | (264.1)2 | 3463.6 | 1387.0 $\pm$ 0.4 | (183.74)2

With the help of fields presented in Eq.(10) in Ref. wang2013analysis , the
kinetic terms of the heavy meson doublets and the field $\Sigma=\xi^{2}$ of
light pseudoscalar mesons are defined in the effective Lagrangian as

$\displaystyle\cal{L}=$ $\displaystyle
iTr[\bar{H}_{b}\nu^{\mu}D_{{\mu}ba}H_{a}]+\frac{f_{\pi}^{2}}{8}Tr[\partial^{\mu}\Sigma\partial_{\mu}\Sigma^{{\dagger}}]$
(3)
$\displaystyle+Tr[\bar{S}_{b}(i\nu^{\mu}D_{{\mu}ba}-\delta_{ba}\Delta_{S})S_{a}]$
$\displaystyle+Tr[\bar{T}_{b}^{\alpha}(i\nu^{\mu}D_{{\mu}ba}-\delta_{ba}\Delta_{T})T_{a\alpha}]$
$\displaystyle+Tr[\bar{X}_{b}^{\alpha}(i\nu^{\mu}D_{{\mu}ba}-\delta_{ba}\Delta_{X})X_{a\alpha}]$
$\displaystyle+Tr[\bar{Y}_{b}^{\alpha\beta}(i\nu^{\mu}D_{{\mu}ba}-\delta_{ba}\Delta_{Y})Y_{a\alpha\beta}]$
$\displaystyle+Tr[\bar{Z}_{b}^{\alpha\beta}(i\nu^{\mu}D_{{\mu}ba}-\delta_{ba}\Delta_{Z})Z_{a\alpha\beta}]$
$\displaystyle+Tr[\bar{R}_{b}^{\alpha\beta}(i\nu^{\mu}D_{{\mu}ba}-\delta_{ba}\Delta_{R})R_{a\alpha\beta}],$

where $\Delta_{F}$ (with $F=S,T,X,Y,Z,R)$ is the mass parameter that gives the
mass splittings between the excited and the low-lying negative parity doublets
colangelo2012new ; gupta2019placing . That is written in the form of spin-
averaged masses as

$\Delta_{F}=\bar{M}_{F}-\bar{M}_{H},$ (4)

with

$\displaystyle\bar{M}_{H}=\frac{3M_{P^{*}}+M_{P}}{4}$ (5)
$\displaystyle\bar{M}_{S}=\frac{3M_{P_{1}^{\prime}}+M_{P_{0}^{*}}}{4}\hskip
28.45274pt\bar{M}_{T}=\frac{5M_{P_{2}^{*}}+3M_{P_{1}}}{8}$
$\displaystyle\bar{M}_{X}=\frac{5M_{P_{2}}+3M_{P_{1}^{*}}}{8}\hskip
28.45274pt\bar{M}_{Y}=\frac{7M_{P_{3}^{*}}+5M_{P_{2}^{\prime}}}{12}$
$\displaystyle\bar{M}_{Z}=\frac{7M_{P_{3}}+5M_{P_{2}^{\prime*}}}{12}\hskip
28.45274pt\bar{M}_{R}=\frac{7M_{P_{4}^{*}}+5M_{P_{3}^{\prime}}}{12},$

where the ground state doublet $J^{P}_{s_{l}}=(0^{-},1^{-})_{{\frac{1}{2}}}$
represented by $(P,P^{*})$; $P$-wave doublets
$J^{P}_{s_{l}}=(0^{+},1^{+})_{{\frac{1}{2}}}$ and
$J^{P}_{s_{l}}=(1^{+},2^{+})_{{\frac{3}{2}}}$ represented by
$(P^{*}_{0},P^{\prime}_{1})$ and $(P_{1},P_{2}^{*})$, respectively; $D$-wave
doublets $J^{P}_{s_{l}}=(1^{-},2^{-})_{{\frac{3}{2}}}$ and
$J^{P}_{s_{l}}=(2^{-},3^{-})_{{\frac{5}{2}}}$ are represented by
$(P^{*}_{1},P_{2})$ and $(P^{\prime}_{2},P^{*}_{3})$, respectively; and the
$F$-wave doublets $J^{P}_{s_{l}}=(2^{+},3^{+})_{{\frac{5}{2}}}$ and
$J^{P}_{s_{l}}=(3^{+},4^{+})_{{\frac{7}{2}}}$ are represented by
$(P^{\prime*}_{2},P_{3})$ and $(P^{\prime}_{3},P^{*}_{4})$, respectively. The
mass degeneracy between the members of the meson doublets is broken by the
Lagrangian as

$\displaystyle{\cal{L}}_{{1}/{m_{Q}}}=$
$\displaystyle\frac{1}{2m_{Q}}\\{\lambda_{H}Tr[\bar{H}_{a}\sigma^{\mu\nu}H_{a}\sigma_{\mu\nu}]-\lambda_{S}Tr[\bar{S}_{a}\sigma^{\mu\nu}S_{a}\sigma_{\mu\nu}]$
(6)
$\displaystyle+\lambda_{T}Tr[\bar{T}_{a}^{\alpha}\sigma^{\mu\nu}T_{a}^{\alpha}\sigma_{\mu\nu}]-\lambda_{X}Tr[\bar{X}_{a}^{\alpha}\sigma^{\mu\nu}X_{a}^{\alpha}\sigma_{\mu\nu}]$
$\displaystyle+\lambda_{Y}Tr[\bar{Y}_{a}^{\alpha\beta}\sigma^{\mu\nu}Y_{a}^{\alpha\beta}\sigma_{\mu\nu}]-\lambda_{Z}Tr[\bar{Z}_{a}^{\alpha\beta}\sigma^{\mu\nu}Z_{a}^{\alpha\beta}\sigma_{\mu\nu}]$
$\displaystyle+\lambda_{R}Tr[\bar{R}_{a}^{\alpha\beta}\sigma^{\mu\nu}R_{a}^{\alpha\beta}\sigma_{\mu\nu}]\\}.$

The induced symmetry breaking terms are suppressed by increasing the powers of
inverse heavy quark mass colangelo2012new ; gupta2019placing . Here the
constants $\lambda_{H}$, $\lambda_{S}$, $\lambda_{T}$, $\lambda_{X}$,
$\lambda_{Y}$, $\lambda_{Z}$, and $\lambda_{R}$ represents the hyperfine mass
splitting between the members of the doublets, given by,

$\displaystyle\lambda_{H}=\frac{1}{8}(M^{2}_{P^{*}}-M^{2}_{P})$ (7)
$\displaystyle\lambda_{S}=\frac{1}{8}(M^{2}_{P_{1}^{\prime}}-M^{2}_{P_{0}^{*}})\hskip
28.45274pt\lambda_{T}=\frac{3}{8}(M^{2}_{P_{2}^{*}}-M^{2}_{P_{1}})$
$\displaystyle\lambda_{X}=\frac{3}{8}(M^{2}_{P_{2}}-M^{2}_{P_{1}^{*}})\hskip
28.45274pt\lambda_{Y}=\frac{3}{8}(M^{2}_{P_{3}^{*}}-M^{2}_{P_{2}^{\prime}})$
$\displaystyle\lambda_{Z}=\frac{5}{8}(M^{2}_{P_{3}}-M^{2}_{P_{2}^{\prime*}})\hskip
28.45274pt\lambda_{R}=\frac{5}{8}(M^{2}_{P_{4}^{*}}-M^{2}_{P_{3}^{\prime}}).$

Flavor symmetry implies that the mass splitting $\Delta_{F}$ between doublets
and the mass splitting $\lambda_{F}$ between spin-partners in doublets are
free from the heavy quark flavor, $i.e.$,

$\Delta_{F}^{(c)}=\Delta_{F}^{(b)}\hskip
28.45274pt\lambda_{F}^{(c)}=\lambda_{F}^{(b)};$ (8)

where $F$ stands for $H,S,T,X,Y,Z,$ and $R$.

Table 2:  Predicted masses of excited $B$ and $B_{s}$ mesons (in MeV). |  |  $B$ meson |  |  |  |  $B_{s}$ meson |  |
---|---|---|---|---|---|---|---|---
$\cal{N}$ ${}^{2S+1}L_{J}$ | Present | Exp. Zyla2020  | Ref. godfrey2016b  | Ref. sun2014higher  | Present | Exp. Zyla2020  | Ref. godfrey2016b  | Ref. sun2014higher
$1^{3}P_{0}$ | 5700 $\pm$ 10 | 5698 $\pm$ 8 | 5756 | 5756 | 5709.0 $\pm$ 4.4 |  | 5831 | 5830
|  | $B_{J}^{*}(5732)$ |  |  |  |  |  |
$1^{1}P_{1}$ | 5740 $\pm$ 11 | 5726.1 $\pm$ 1.3 | 5777 | 5779 | 5768.0 $\pm$ 3.1 | 5828.7 $\pm$ 0.2 | 5857 | 5858
|  | $B_{1}(5721)^{0}$ |  |  |  | $B_{s1}(5730)^{0}$ |  |
$1^{3}P_{1}$ | 5780 $\pm$ 20 |  | 5784 | 5782 | 5875.0 $\pm$ 2.4 |  | 5861 | 5859
$1^{3}P_{2}$ | 5800 $\pm$ 16 | 5739.5 $\pm$ 0.7 | 5797 | 5796 | 5890.0 $\pm$ 1.7 | 5839.9 $\pm$ 0.1 | 5876 | 5875
|  | $B_{2}^{*}(5747)^{0}$ |  |  |  | $B_{s2}^{*}(5840)^{0}$ |  |
$2^{1}S_{0}$ | 5930 $\pm$ 12 | 5863 $\pm$ 9 | 5904 | 5904 | 6025.0 $\pm$ 3.2 |  | 5984 | 5985
|  | $B_{J}(5840)^{0}$ |  |  |  |  |  |
$2^{3}S_{1}$ | 5960 $\pm$ 11 | 5971 $\pm$ 5 | 5933 | 5934 | 6033.0 $\pm$ 2.4 | 6063.5 $\pm$ 1.2 | 6012 | 6013
|  | $B_{J}(5970)^{0}$ |  |  |  | $B_{s}(6063)$ |  |
$1^{3}D_{1}$ | 6136.3 $\pm$ 0.4 |  | 6110 | 6110 | 6247.0 $\pm$ 2.4 |  | 6182 | 6181
$1^{1}D_{2}$ | 6144.5 $\pm$ 0.3 |  | 6095 | 6108 | 6256.0 $\pm$ 1.7 |  | 6169 | 6180
$1^{3}D_{2}$ | 6196.3 $\pm$ 0.6 |  | 6124 | 6113 | 6292.0 $\pm$ 3.5 |  | 6196 | 6185
$1^{3}D_{3}$ | 6202.3 $\pm$ 0.4 |  | 6106 | 6105 | 6297.0 $\pm$ 2.5 |  | 6179 | 6178
$2^{3}P_{0}$ | 6266.3 $\pm$ 0.2 |  | 6213 | 6214 | 6387.0 $\pm$ 1.3 |  | 6279 | 6279
$2^{1}P_{1}$ | 6272.4 $\pm$ 0.2 |  | 6197 | 6206 | 6393.0 $\pm$ 0.9 |  | 6279 | 6284
$2^{3}P_{1}$ | 6325.0 $\pm$ 8.1 |  | 6228 | 6219 | 6470.0 $\pm$ 2.4 |  | 6296 | 6291
$2^{3}P_{2}$ | 6344.0 $\pm$ 6.5 |  | 6213 | 6213 | 6476.0 $\pm$ 1.7 |  | 6295 | 6295
$3^{1}S_{0}$ | 6417.4 $\pm$ 0.2 |  | 6335 | 6334 | 6556.0 $\pm$ 1.3 |  | 6410 | 6409
$3^{3}S_{1}$ | 6433.7 $\pm$ 0.2 |  | 6335 | 6355 | 6567.3 $\pm$ 0.9 |  | 6429 | 6429
$1^{3}F_{2}$ | 6436.5 $\pm$ 0.6 |  | 6387 | 6387 | 6560.0 $\pm$ 3.5 |  | 6454 | 6453
$1^{1}F_{3}$ | 6467.8 $\pm$ 0.4 |  | 6358 | 6375 | 6580.0 $\pm$ 2.4 |  | 6425 | 6441
$1^{3}F_{3}$ | 6499.8 $\pm$ 0.6 |  | 6396 | 6380 | 6613.0 $\pm$ 3.5 |  | 6462 | 6446
$1^{3}F_{4}$ | 6520.2 $\pm$ 0.4 |  | 6364 | 6364 | 6629.0 $\pm$ 2.5 |  | 6432 | 6431
$2^{3}D_{1}$ | 6580.0 $\pm$ 0.4 |  | 6475 | 6475 | 6717.0 $\pm$ 2.3 |  | 6542 | 6542
$2^{1}D_{2}$ | 6595.2 $\pm$ 0.3 |  | 6450 | 6464 | 6728.0 $\pm$ 1.6 |  | 6526 | 6536
$2^{3}D_{2}$ | 6657.3 $\pm$ 0.6 |  | 6486 | 6472 | 6790.0 $\pm$ 3.5 |  | 6553 | 6542
$2^{3}D_{3}$ | 6671.3 $\pm$ 0.4 |  | 6460 | 6459 | 6796.0 $\pm$ 2.5 |  | 6535 | 6534

The effective heavy meson chiral Lagrangians (define in Eq.(11) in Ref.
wang2013analysis ) determine the expressions of strong decays of heavy-light
mesons to a member of lowest-lying heavy-light spin doublet plus light
pseudoscalar mesons ($\pi$, $\eta$, and $K$),

$\Gamma=\frac{1}{2J+1}\sum\frac{\vec{P}_{\cal{P}}}{8\pi
P_{a}^{2}}|{\cal{M}}|^{2};$ (9)

For $B_{2}^{*}(5747)$ as $1^{3}P_{2}$, the two-body strong decays
$B_{2}^{*}(5747)\rightarrow B^{*}\pi,B\pi$; having the $\pi$ meson three
momenta ${\vec{P}_{\cal{\pi}}}=$ 374 and 418 MeV, respectively. The decay
widths

$\Gamma(B_{2}^{*}(5747)\rightarrow
B^{*}\pi,B\pi)\propto{\vec{P}_{\cal{\pi}}^{5}},$ (10)

where ${\vec{P}_{\cal{\pi}}^{5}}=$ 7.3 $\times$ 1012 and 1.3 $\times$ 1013
MeV5 in the decay to the final states $B^{*}\pi$ and $B\pi$, respectively. A
small difference in ${\vec{P}_{\cal{\pi}}}$ can lead to a significant
difference in ${\vec{P}_{\cal{\pi}}^{5}}$. So, we have to take into account
the heavy quark symmetry breaking corrections and chiral symmetry breaking
corrections to make robust predictions. The higher-order corrections for spin
and flavor violation of an order ${\cal{O}}(\frac{1}{m_{Q}})$ are not taking
into consideration to avoid introducing new unknown coupling constants
wang2015 . We expect that the corrections would not be larger than (or as
large as) the leading order contributions. At the hadronic level, the
$\frac{1}{m_{Q}}$ corrections can be crudely estimated to be of the order
$\frac{{\vec{P}_{\cal{\pi}}}}{M_{B}}\left(\frac{{\vec{P}_{K}}}{M_{B_{s}}}\right)\approx$
0.1-0.2.

##  3 Mass Spectroscopy: Results and Discussion

In our previous study, we analyzed the strong decay of experimentally observed
excited $D$ and $D_{s}$ mesons gandhi2019strong ; gandhi2020identifying . The
ratio of the strong decay rates has identified the doublets:
$(D_{1}(2420),D_{2}^{*}(2460))$ as $(1^{3}P_{1},1^{3}P_{2})$,
$(D_{s0}^{*}(2317),D_{s1}(2460))$ as $(1^{3}P_{0},1^{1}P_{1})$,
$(D_{s1}(2536),\\\ D_{s2}^{*}(2573))$ as $(1^{3}P_{1},1^{3}P_{2})$
$(D(2550),D_{J}^{*}(2600))$ as
$(2^{1}S_{0},2^{3}S_{1})$ and $(D(2740),D_{J}^{*}(2750))$ as
$(1^{3}D_{2},1^{3}D_{3})$. The resonances $D{{}_{J}^{*}}(3000)$ and
$D_{2}^{*}(3000)$ are identified with $2^{3}P_{2}$ and $1^{3}F_{2}$ quantum
states, respectively. The $D_{J}(3000)$ and $D_{sJ}(3040)$ are interpreted as
the mixing of $2^{1}P_{1}-2^{3}P_{1}$ states.

The calculated values of spin-averaged masses $\bar{M}_{F}$, the mass
splitting between the doublets $\Delta_{F}$, and the mass splitting between
spin-partner in a doublet $\lambda_{F}$ are listed in Table (LABEL:tab1).
Further, we compute their propagation in the determination of uncertainty
using the generic method given in the Appendix. Uncertainty arises in
presented results because of the uncertainties in the various experimental
inputs. Here, the masses of $D^{0}(D_{s}^{\pm})$ as $(1^{1}S_{0})$,
$D^{*}(2007)^{0}(D_{s}^{*\pm})$ as $(1^{3}S_{0})$,
$D_{0}^{*}(2400)^{0}(D_{s0}^{*}(2317)^{\pm})$ as $(1^{3}P_{0})$,
$D_{1}(2430)^{0}(D_{s1}(2460)^{\pm})$ as $(1^{1}P_{1})$, $D_{1}(2420)^{0}\\\
(D_{s1}(2536))$ as $(1^{3}P_{1})$, $D_{2}^{*}(2460)^{0}(D_{s2}^{*}(2573))$ as
$(1^{3}P_{2})$, $D(2550)^{0}$ as $(2^{1}S_{0})$,
$D_{J}^{*}(2600)(D_{s1}^{*}(2700))$ as $(2^{3}S_{1})$, $D_{J}(3000)$ as
$(2^{3}P_{1})$, and $D_{J}^{*}(3000)$ as $(2^{3}P_{2})$ are taken from PDG
Zyla2020 . Experimentally the $1D$, $2P$, $1F$, and $2D$ states of excited $D$
and $D_{s}$ mesons are missing (or PDG Zyla2020  need more confirmation),
$i.e.$ the predictions of $D(2740)$ as $(1^{3}D_{2})$, $D_{J}^{*}(2750)$ as
$(1^{3}D_{3})$, $D_{s1}^{*}(2860)$ as $(1^{3}D_{1})$, $D_{s3}^{*}(2860)$ as
$(1^{3}D_{3})$, $D_{2}^{*}(3000)$ as $(1^{3}F_{2})$, and $D_{sJ}(3040)^{\pm}$
as a mixture of $2P(J=1)$, are still uncertain. Also, $D_{s}(2^{1}S_{0})$, the
spin-partner of $D_{s}(2^{3}S_{1})$ is still missing experimentally. So their
masses are taken from the predictions of Ebert, Faustov, and Galkin (EFG)
ebert2010heavy .

The charm data can be exploited to make predictions of excited bottom mesons.
Our calculated masses of the excited $B$ and $B_{s}$ mesons are listed in
Table (2) with available experimental observations and theoretical
predictions. Note that the ground state masses of the $B$ and $B_{s}$ mesons
are fixed from PDG Zyla2020 . For $1P$-wave, our calculated mass 5700 $\pm$ 10
MeV of $B(1^{3}P_{0})$ is in good agreement with experimental observed mass
5698 $\pm$ 8 MeV Zyla2020  of $B_{J}^{*}(5732)$. Experimentally
$B_{1}(5721)^{0}$ is measured with spin-parity $J^{P}=1^{+}$. That is
consistent with our prediction of $B(1^{1}P_{1})$ by a mass difference of
$\approx$ 14 MeV. The mass 5768.0 $\pm$ 3.1 MeV of $B_{s}(1^{1}P_{1})$ and
5875.0 $\pm$ 2.4 MeV of $B_{s}(1^{3}P_{1})$ are underestimated to the
experimental measurement 5828.7 $\pm$ 0.2 MeV of $B_{s1}(5730)^{0}$ Zyla2020 .
And, the mass of $B_{s}(1^{3}P_{1})$ is consistent with the results of
godfrey2016b ; sun2014higher  with a mass difference of 14-15 MeV. Our result
5800 $\pm$ 16 MeV of $B(1^{3}P_{2})$ is overestimated to the PDG fit value
5739.5 $\pm$ 0.7 MeV Zyla2020  and is in good agreement with the predictions
of godfrey2016b ; sun2014higher . The mass 5930 $\pm$ 12 MeV of
$B(2^{1}S_{0})$ is overestimated to the theoretical and experimental results
Zyla2020 ; godfrey2016b ; sun2014higher  with a mass difference of 26-67 MeV.
Our predicted mass 5960 $\pm$ 11 MeV of its spin-partner $B(2^{3}S_{1})$ is
within the errorbar of PDG fit value 5971 $\pm$ 5 MeV Zyla2020 . The
calculated masses of $B_{s}(2S)$ states are underestimated to the LHCb
measurement 6063.5 $\pm$ MeV aaij2020  with a mass difference of 30.5-38.5 MeV
and overestimated by the theoretical predictions godfrey2016b ; sun2014higher
with a mass difference of 20-49 MeV. Here, our predicted masses of $2S$ and
$1D$ states are inconsistent with the observed mass 6114 $\pm$ 33 MeV aaij2020
of excited $B_{s}$ meson. Our results of 1$D$, 2$P$, $3S$, 1$F$, and 2$D$
states of bottom mesons are overestimated to other theoretical outcomes
godfrey2016b ; sun2014higher  (shown in Table (2)). The masses of $1F$ states
are close to $3S$ multiplets and overlap. Experimentally $3S$ states are more
likely to be observed first rather than $1F$ states, although, it is
challenging.

##  4 Strong Decay: Numerical Analysis

###  4.1 $B$ meson

####  4.1.1 $1P$ and $2P$ states

Table 3: Strong decay widths of excited $B$ and $B_{s}$ mesons (in MeV).

$\cal{N}$ ${}^{2S+1}L_{J}$ | Decay | Width | Decay | Width
---|---|---|---|---
| mode |  | mode |
$1^{3}P_{0}$ | $B^{+}\pi^{-}$ | (589.6 $\pm$ 15.6)$g_{S}^{2}$ | $B^{+}K^{-}$ | $-$
| $B^{0}\pi^{0}$ | (295.0 $\pm$ 7.8)$g_{S}^{2}$ | $B^{0}K^{0}$ | $-$
| $B_{s}K$ | $-$ | $B_{s}\pi^{0}$ | (149.6 $\pm$ 3.9)$g_{S}^{2}$ $\times$ $10^{-4}$
| $B^{0}\eta$ | $-$ | $B_{s}\eta$ | $-$
$1^{1}P_{1}$ | $B^{*}\pi^{+}$ | (568.9 $\pm$ 15.2)$g_{S}^{2}$ | $B^{+}K^{-}$ | $-$
| $B^{*}\pi^{0}$ | (285.4 $\pm$ 7.6)$g_{S}^{2}$ | $B^{0}K^{0}$ | $-$
| $B{{}_{s}^{*}}K$ | $-$ | $B_{s}\pi^{0}$ | (163.6 $\pm$ 4.3)$g_{S}^{2}$ $\times$ $10^{-4}$
| $B^{*}\eta$ | $-$ | $B_{s}\eta$ | $-$
$1^{3}P_{1}$ | $B^{*}\pi^{+}$ | (144.2 $\pm$ 34.0)$g_{T}^{2}$ | $B^{+}K^{-}$ | (7.5 $\pm$ 0.9)$g_{T}^{2}$
| $B^{*}\pi^{0}$ | (73.3 $\pm$ 17.2)$g_{T}^{2}$ | $B^{0}K^{0}$ | (6.4 $\pm$ 0.8)$g_{T}^{2}$
| $B{{}_{s}^{*}}K$ | $-$ | $B_{s}\pi^{0}$ | (77.0 $\pm$ 3.2)$g_{T}^{2}$ $\times$ $10^{-4}$
| $B^{*}\eta$ | $-$ | $B_{s}\eta$ | $-$
$1^{3}P_{2}$ | $B^{*}\pi^{+}$ | (108.3 $\pm$ 9.4)$g_{T}^{2}$ | $B^{*+}K^{-}$ | (4.5 $\pm$ 0.4)$g_{T}^{2}$
| $B^{*}\pi^{0}$ | (55 $\pm$ 10)$g_{T}^{2}$ | $B^{*0}K^{0}$ | (3.8 $\pm$ 0.4)$g_{T}^{2}$
| $B{{}_{s}^{*}}K$ | $-$ | $B{{}_{s}^{*}}\pi^{0}$ | (46.2 $\pm$ 1.9)$g_{T}^{2}$ $\times$ $10^{-4}$
| $B^{*}\eta$ | $-$ | $B_{s}^{*}\eta$ | $-$
| $B^{+}\pi^{-}$ | (115.1 $\pm$ 18.5)$g_{T}^{2}$ | $B^{+}K^{-}$ | (14.2 $\pm$ 0.9)$g_{T}^{2}$
| $B^{0}\pi^{0}$ | (57.3 $\pm$ 9.2)$g_{T}^{2}$ | $B^{0}K^{0}$ | (12.9 $\pm$ 0.8)$g_{T}^{2}$
| $B_{s}K$ | $-$ | $B_{s}\pi^{0}$ | (51.7 $\pm$ 1.8)$g_{T}^{2}$ $\times$ $10^{-4}$
| $B^{0}\eta$ | $-$ | $B_{s}\eta$ | $-$
$2^{1}S_{0}$ | $B^{*}\pi^{+}$ | (1471.7 $\pm$ 96.3)$g_{H}^{2}$ | $B^{*+}K^{-}$ | (847.1 $\pm$ 31.2)$g_{H}^{2}$
| $B^{*}\pi^{0}$ | (739.9 $\pm$ 48.2)$g_{H}^{2}$ | $B^{*0}K^{0}$ | (827.1 $\pm$ 31.1)$g_{H}^{2}$
| $B{{}_{s}^{*}}K$ | (20.4 $\pm$ 6.7)$g_{H}^{2}$ | $B_{s}^{*}\pi^{0}$ | (757.0 $\pm$ 23.8)$g_{H}^{2}$ $\times$ $10^{-4}$
| $B^{*}\eta$ | (16.3 $\pm$ 18.0)$g_{H}^{2}$ | $B_{s}^{*}\eta$ | (90.7 $\pm$ 8.2)$g_{H}^{2}$
$2^{3}S_{1}$ | $B^{*}\pi^{+}$ | (1129.3 $\pm$ 65.7)$g_{H}^{2}$ | $B^{*+}K^{-}$ | (602.0 $\pm$ 11.6)$g_{H}^{2}$
| $B^{*}\pi^{0}$ | (567.4 $\pm$ 32.9)$g_{H}^{2}$ | $B^{*0}K^{0}$ | (588.3 $\pm$ 11.5)$g_{H}^{2}$
| $B{{}_{s}^{*}}K$ | (52.7 $\pm$ 19.6)$g_{H}^{2}$ | $B_{s}^{*}\pi^{0}$ | (524.2 $\pm$ 7.2) $g_{H}^{2}$ $\times$ $10^{-4}$
| $B^{*}\eta$ | (26.2 $\pm$ 5.3)$g_{H}^{2}$ | $B_{s}^{*}\eta$ | (73.2 $\pm$ 4.7)$g_{H}^{2}$
| $B^{+}\pi^{-}$ | (687.2 $\pm$ 37.6)$g_{H}^{2}$ | $B^{+}K^{-}$ | (415.8 $\pm$ 6.8)$g_{H}^{2}$
| $B^{0}\pi^{0}$ | (344.5 $\pm$ 18.8)$g_{H}^{2}$ | $B^{0}K^{0}$ | (407.3 $\pm$ 6.7)$g_{H}^{2}$
| $B_{s}K$ | (81.1 $\pm$ 15.2)$g_{H}^{2}$ | $B_{s}\pi^{0}$ | (325.8 $\pm$ 3.5)$g_{H}^{2}$ $\times$ $10^{-4}$
| $B^{0}\eta$ | (25.3 $\pm$ 3.5)$g_{H}^{2}$ | $B_{s}\eta$ | (84.9 $\pm$ 2.8)$g_{H}^{2}$
$1^{3}D_{1}$ | $B^{*}\pi^{+}$ | (869.2 $\pm$ 22.7)$g_{X}^{2}$ | $B^{*+}K^{-}$ | (1017.2 $\pm$ 26.6)$g_{X}^{2}$
| $B^{*}\pi^{0}$ | (435.8 $\pm$ 11.4)$g_{X}^{2}$ | $B^{*0}K^{0}$ | (1008.04 $\pm$ 26.3)$g_{X}^{2}$
| $B{{}_{s}^{*}}K$ | (211.3 $\pm$ 5.5)$g_{X}^{2}$ | $B_{s}^{*}\pi^{0}$ | (490.4 $\pm$ 12.8)$g_{X}^{2}$ $\times$ $10^{-4}$
| $B^{*}\eta$ | (64.4 $\pm$ 1.7)$g_{X}^{2}$ | $B_{s}^{*}\eta$ | (305.6 $\pm$ 8.0)$g_{X}^{2}$
| $B^{+}\pi^{-}$ | (2228.8 $\pm$ 58.2)$g_{X}^{2}$ | $B^{+}K^{-}$ | (2650.4 $\pm$ 69.2)$g_{X}^{2}$
| $B^{0}\pi^{0}$ | (1114.8 $\pm$ 29.1)$g_{X}^{2}$ | $B^{0}K^{0}$ | (2624.2 $\pm$ 68.5)$g_{X}^{2}$
| $B_{s}K$ | (665.8 $\pm$ 17.4)$g_{X}^{2}$ | $B_{s}\pi^{0}$ | (1272.1 $\pm$ 33.2)$g_{X}^{2}$ $\times$ $10^{-4}$
| $B^{0}\eta$ | (185.0 $\pm$ 4.8)$g_{X}^{2}$ | $B_{s}\eta$ | (945.9 $\pm$ 24.7)$g_{X}^{2}$
$1^{1}D_{2}$ | $B^{*}\pi^{+}$ | (2732.2 $\pm$ 71.4)$g_{X}^{2}$ | $B^{*+}K^{-}$ | (3223.6 $\pm$ 84.2)$g_{X}^{2}$
| $B^{*}\pi^{0}$ | (1369.7 $\pm$ 35.8)$g_{X}^{2}$ | $B^{*0}K^{0}$ | (3195.5 $\pm$ 83.4)$g_{X}^{2}$
| $B{{}_{s}^{*}}K$ | (688 $\pm$ 18)$g_{X}^{2}$ | $B_{s}^{*}\pi^{0}$ | (1546.4 $\pm$ 40.4)$g_{X}^{2}$ $\times$ $10^{-4}$
| $B^{*}\eta$ | (207.1 $\pm$ 5.4)$g_{X}^{2}$ | $B_{s}^{*}\eta$ | (985.9 $\pm$ 25.8)$g_{X}^{2}$
$1^{3}D_{2}$ | $B^{*}\pi^{+}$ | (900.7 $\pm$ 23.9)$g_{Y}^{2}$ | $B^{*+}K^{-}$ | (331.5 $\pm$ 13.8)$g_{Y}^{2}$
| $B^{*}\pi^{0}$ | (453 $\pm$ 12)$g_{Y}^{2}$ | $B^{*0}K^{0}$ | (324.9 $\pm$ 13.6)$g_{Y}^{2}$
| $B{{}_{s}^{*}}K$ | (78.4 $\pm$ 2.8)$g_{Y}^{2}$ | $B_{s}^{*}\pi^{0}$ | (236.9 $\pm$ 9.3)$g_{Y}^{2}$ $\times$ $10^{-4}$
| $B^{*}\eta$ | (28.1 $\pm$ 0.8)$g_{Y}^{2}$ | $B_{s}^{*}\eta$ | (60.1 $\pm$ 3.3)$g_{Y}^{2}$
$1^{3}D_{3}$ | $B^{*}\pi^{+}$ | (538.4 $\pm$ 14.2)$g_{Y}^{2}$ | $B^{*+}K^{-}$ | (396.4 $\pm$ 13.8)$g_{Y}^{2}$
| $B^{*}\pi^{0}$ | (270.8 $\pm$ 7.1)$g_{Y}^{2}$ | $B^{*0}K^{0}$ | (388.7 $\pm$ 13.6)$g_{Y}^{2}$
| $B{{}_{s}^{*}}K$ | (48.8 $\pm$ 1.7)$g_{Y}^{2}$ | $B_{s}^{*}\pi^{0}$ | (281.0 $\pm$ 9.6)$g_{Y}^{2}$ $\times$ $10^{-4}$
| $B^{*}\eta$ | (17.3 $\pm$ 0.5)$g_{Y}^{2}$ | $B_{s}^{*}\eta$ | (73.1 $\pm$ 3.3)$g_{Y}^{2}$

continued…

Table 3: Strong decay widths of excited $B$ and $B_{s}$ mesons (in MeV).

$\cal{N}$ ${}^{2S+1}L_{J}$ | Decay | Width | Decay | Width
---|---|---|---|---
| mode |  | mode |
| $B^{+}\pi^{-}$ | (559.4 $\pm$ 14.7)$g_{Y}^{2}$ | $B^{+}K^{-}$ | (438.4 $\pm$ 14.8)$g_{Y}^{2}$
| $B^{0}\pi^{0}$ | (280.5 $\pm$ 7.4)$g_{Y}^{2}$ | $B^{0}K^{0}$ | (429.5 $\pm$ 14.5)$g_{Y}^{2}$
| $B_{s}K$ | (69.4 $\pm$ 1.9)$g_{Y}^{2}$ | $B_{s}\pi^{0}$ | (298.6 $\pm$ 9.5)$g_{Y}^{2}$ $\times$ $10^{-4}$
| $B^{0}\eta$ | (21.9 $\pm$ 0.6)$g_{Y}^{2}$ | $B_{s}\eta$ | (95.9 $\pm$ 3.6)$g_{Y}^{2}$
$2^{3}P_{0}$ | $B^{+}\pi^{-}$ | (5907.0 $\pm$ 154.2)$g_{S}^{2}$ | $B^{+}K^{-}$ | (7460.6 $\pm$ 194.8)$g_{S}^{2}$
| $B^{0}\pi^{0}$ | (2951.86 $\pm$ 77.1)$g_{S}^{2}$ | $B^{0}K^{0}$ | (7443.11 $\pm$ 194.4)$g_{S}^{2}$
| $B_{s}K$ | (4080.7 $\pm$ 106.6)$g_{S}^{2}$ | $B_{s}\pi^{0}$ | (3243.1 $\pm$ 84.6)$g_{S}^{2}$ $\times$ $10^{-4}$
| $B^{0}\eta$ | (866.5 $\pm$ 22.6)$g_{S}^{2}$ | $B_{s}\eta$ | (3850.9 $\pm$ 100.6)$g_{S}^{2}$
$2^{1}P_{1}$ | $B^{*}\pi^{+}$ | (5319.5 $\pm$ 138.9)$g_{S}^{2}$ | $B^{*+}K^{-}$ | (6757.6 $\pm$ 176.5)$g_{S}^{2}$
| $B^{*}\pi^{0}$ | (2661.1 $\pm$ 69.5)$g_{S}^{2}$ | $B^{*0}K^{0}$ | (6746.9 $\pm$ 176.2)$g_{S}^{2}$
| $B{{}_{s}^{*}}K$ | (3519.2 $\pm$ 91.9)$g_{S}^{2}$ | $B_{s}^{*}\pi^{0}$ | (2904.4 $\pm$ 75.9)$g_{S}^{2}$ $\times$ $10^{-4}$
| $B^{*}\eta$ | (768.4 $\pm$ 20.1)$g_{S}^{2}$ | $B_{s}^{*}\eta$ | (3394.1 $\pm$ 88.6)$g_{S}^{2}$
$2^{3}P_{1}$ | $B^{*}\pi^{+}$ | (6653.4 $\pm$ 309.0)$g_{T}^{2}$ | $B^{*+}K^{-}$ | (7606.9 $\pm$ 218.9)$g_{T}^{2}$
| $B^{*}\pi^{0}$ | (3337.5 $\pm$ 154.8)$g_{T}^{2}$ | $B^{*0}K^{0}$ | (7537.2 $\pm$ 217.0) $g_{T}^{2}$
| $B{{}_{s}^{*}}K$ | (1874.9 $\pm$ 125.6)$g_{T}^{2}$ | $B_{s}^{*}\pi^{0}$ | (4278.2 $\pm$ 124.0)$g_{T}^{2}$ $\times$ $10^{-4}$
| $B^{*}\eta$ | (475.1 $\pm$ 28.9)$g_{T}^{2}$ | $B_{s}^{*}\eta$ | (2692.6 $\pm$ 84.4)$g_{T}^{2}$
$2^{3}P_{2}$ | $B^{*}\pi^{+}$ | (4350.8 $\pm$ 173.7)$g_{T}^{2}$ | $B^{*+}K^{-}$ | (4698.6 $\pm$ 129.1)$g_{T}^{2}$
| $B^{*}\pi^{0}$ | (2182.2 $\pm$ 87.0)$g_{T}^{2}$ | $B^{*0}K^{0}$ | (4656.1 $\pm$ 127.9)$g_{T}^{2}$
| $B{{}_{s}^{*}}K$ | (1289.4 $\pm$ 70.4)$g_{T}^{2}$ | $B_{s}^{*}\pi^{0}$ | (2634.0 $\pm$ 73.7)$g_{T}^{2}$ $\times$ $10^{-4}$
| $B^{*}\eta$ | (322.5 $\pm$ 16.1)$g_{T}^{2}$ | $B_{s}^{*}\eta$ | (1674.4 $\pm$ 49.4)$g_{T}^{2}$
| $B^{+}\pi^{-}$ | (3521.6 $\pm$ 136.8)$g_{T}^{2}$ | $B^{+}K^{-}$ | (3854.0 $\pm$ 105.3)$g_{T}^{2}$
| $B^{0}\pi^{0}$ | (1763.0 $\pm$ 68.5)$g_{T}^{2}$ | $B^{0}K^{0}$ | (3815.8 $\pm$ 104.3)$g_{T}^{2}$
| $B_{s}K$ | (1187.8 $\pm$ 59.3)$g_{T}^{2}$ | $B_{s}\pi^{0}$ | (2143.6 $\pm$ 58.1)$g_{T}^{2}$ $\times$ $10^{-4}$
| $B^{0}\eta$ | (282.0 $\pm$ 13.3)$g_{T}^{2}$ | $B_{s}\eta$ | (1466.9 $\pm$ 40.8)$g_{T}^{2}$
$3^{1}S_{0}$ | $B^{*}\pi^{+}$ | (7591.0 $\pm$ 198.3)$g_{H}^{2}$ | $B^{*+}K^{-}$ | (8120.4 $\pm$ 214.0) $g_{H}^{2}$
| $B^{*}\pi^{0}$ | (3801.7 $\pm$ 99.3)$g_{H}^{2}$ | $B^{*0}K^{0}$ | (8082.7 $\pm$ 213.0)$g_{H}^{2}$
| $B{{}_{s}^{*}}K$ | (4067.4 $\pm$ 108.7)$g_{H}^{2}$ | $B_{s}^{*}\pi^{0}$ | (4282.8 $\pm$ 113.8)$g_{H}^{2}$ $\times$ $10^{-4}$
| $B^{*}\eta$ | (837.4 $\pm$ 21.9)$g_{H}^{2}$ | $B_{s}^{*}\eta$ | (3922.6 $\pm$ 24.9)$g_{H}^{2}$
$3^{3}S_{1}$ | $B^{*}\pi^{+}$ | (5262.79 $\pm$ 137.5)$g_{H}^{2}$ | $B^{*+}K^{-}$ | (5570.4 $\pm$ 13.9)$g_{H}^{2}$
| $B^{*}\pi^{0}$ | (2635.5 $\pm$ 68.8)$g_{H}^{2}$ | $B^{*0}K^{0}$ | (5545.2 $\pm$ 13.9)$g_{H}^{2}$
| $B{{}_{s}^{*}}K$ | (2873.4 $\pm$ 76.7)$g_{H}^{2}$ | $B_{s}^{*}\pi^{0}$ | (2930.3 $\pm$ 12.4)$g_{H}^{2}$ $\times$ $10^{-4}$
| $B^{*}\eta$ | (588.7 $\pm$ 15.4)$g_{H}^{2}$ | $B_{s}^{*}\eta$ | (2706.3 $\pm$ 14.9)$g_{H}^{2}$
| $B^{+}\pi^{-}$ | (2914.44 $\pm$ 76.1)$g_{H}^{2}$ | $B^{+}K^{-}$ | (3099.9 $\pm$ 7.5)$g_{H}^{2}$
| $B^{0}\pi^{0}$ | (1458.1 $\pm$ 38.1)$g_{H}^{2}$ | $B^{0}K^{0}$ | (3084.3 $\pm$ 7.3)$g_{H}^{2}$
| $B_{s}K$ | (1687.6 $\pm$ 44.1)$g_{H}^{2}$ | $B_{s}\pi^{0}$ | (1627.6 $\pm$ 3.5)$g_{H}^{2}$ $\times$ $10^{-4}$
| $B^{0}\eta$ | (337.3 $\pm$ 8.8)$g_{H}^{2}$ | $B_{s}\eta$ | (1553.5 $\pm$ 4.2)$g_{H}^{2}$
$1^{3}F_{2}$ | $B^{*}\pi^{+}$ | (1781.5 $\pm$ 46.5)$g_{Z}^{2}$ | $B^{*+}K^{-}$ | (2378.7 $\pm$ 62.1)$g_{Z}^{2}$
| $B^{*}\pi^{0}$ | (892.9 $\pm$ 23.3)$g_{Z}^{2}$ | $B^{*0}K^{0}$ | (2361.7 $\pm$ 61.7)$g_{Z}^{2}$
| $B{{}_{s}^{*}}K$ | (573.9 $\pm$ 15.0)$g_{Z}^{2}$ | $B_{s}^{*}\pi^{0}$ | (1084.7 $\pm$ 28.3)$g_{Z}^{2}$ $\times$ $10^{-4}$
| $B^{*}\eta$ | (160.0 $\pm$ 4.1)$g_{Z}^{2}$ | $B_{s}^{*}\eta$ | (809.5 $\pm$ 21.1)$g_{Z}^{2}$
| $B^{+}\pi^{-}$ | (3423.6 $\pm$ 89.4)$g_{Z}^{2}$ | $B^{+}K^{-}$ | (4578.5 $\pm$ 118.6)$g_{Z}^{2}$
| $B^{0}\pi^{0}$ | (1712.1 $\pm$ 44.7)$g_{Z}^{2}$ | $B^{0}K^{0}$ | (4539.3 $\pm$ 119.6)$g_{Z}^{2}$
| $B_{s}K$ | (1229.8 $\pm$ 32.1)$g_{Z}^{2}$ | $B_{s}\pi^{0}$ | (2106.0 $\pm$ 55.0)$g_{Z}^{2}$ $\times$ $10^{-4}$
| $B^{0}\eta$ | (324.5 $\pm$ 8.5)$g_{Z}^{2}$ | $B_{s}\eta$ | (1660.2 $\pm$ 43.4)$g_{Z}^{2}$
$1^{1}F_{3}$ | $B^{*}\pi^{+}$ | (5312.0 $\pm$ 138.7)$g_{Z}^{2}$ | $B^{*+}K^{-}$ | (6666.4 $\pm$ 174.1)$g_{Z}^{2}$
| $B^{*}\pi^{0}$ | (2662 $\pm$ 70)$g_{Z}^{2}$ | $B^{*0}K^{0}$ | (6620.7 $\pm$ 172.9)$g_{Z}^{2}$
| $B{{}_{s}^{*}}K$ | (1815.0 $\pm$ 47.4)$g_{Z}^{2}$ | $B_{s}^{*}\pi^{0}$ | (3026.3 $\pm$ 79.0)$g_{Z}^{2}$ $\times$ $10^{-4}$
| $B^{*}\eta$ | (496.3 $\pm$ 13.0)$g_{Z}^{2}$ | $B_{s}^{*}\eta$ | (2311.9 $\pm$ 60.4)$g_{Z}^{2}$
$1^{3}F_{3}$ | $B^{*}\pi^{+}$ | (27012.7 $\pm$ 716.0)$g_{R}^{2}$ | $B^{*+}K^{-}$ | (29351.3 $\pm$ 1093.9)$g_{R}^{2}$
| $B^{*}\pi^{0}$ | (13563.3 $\pm$ 359.5)$g_{R}^{2}$ | $B^{*0}K^{0}$ | (28985.8 $\pm$ 1081.9)$g_{R}^{2}$
| $B{{}_{s}^{*}}K$ | (5196.0 $\pm$ 158.6)$g_{R}^{2}$ | $B_{s}^{*}\pi^{0}$ | (16050.7 $\pm$ 598.2)$g_{R}^{2}$ $\times$ $10^{-4}$
| $B^{*}\eta$ | (1578.6 $\pm$ 42.2)$g_{R}^{2}$ | $B_{s}^{*}\eta$ | (7805.3 $\pm$ 333.4)$g_{R}^{2}$

continued…

Table 3: Strong decay widths of excited $B$ and $B_{s}$ mesons (in MeV). $\cal{N}$ ${}^{2S+1}L_{J}$ | Decay | Width | Decay | Width
---|---|---|---|---
| mode |  | mode |
$1^{3}F_{4}$ | $B^{*}\pi^{+}$ | (17284.0 $\pm$ 454.9)$g_{R}^{2}$ | $B^{*+}K^{-}$ | (18345.7 $\pm$ 589.2)$g_{R}^{2}$
| $B^{*}\pi^{0}$ | (8677.2 $\pm$ 228.4)$g_{R}^{2}$ | $B^{*0}K^{0}$ | (18123.6 $\pm$ 582.6)$g_{R}^{2}$
| $B{{}_{s}^{*}}K$ | (3510.5 $\pm$ 105.3)$g_{R}^{2}$ | $B_{s}^{*}\pi^{0}$ | (9942.0 $\pm$ 327.1)$g_{R}^{2}$ $\times$ $10^{-4}$
| $B^{*}\eta$ | (1051.2 $\pm$ 27.8)$g_{R}^{2}$ | $B_{s}^{*}\eta$ | (4980.1 $\pm$ 181.0)$g_{R}^{2}$
| $B^{+}\pi^{-}$ | (18610.8 $\pm$ 491.5)$g_{R}^{2}$ | $B^{+}K^{-}$ | (20108.9 $\pm$ 635.9)$g_{R}^{2}$
| $B^{0}\pi^{0}$ | (9282.8 $\pm$ 244.1)$g_{R}^{2}$ | $B^{0}K^{0}$ | (19832.6 $\pm$ 627.4)$g_{R}^{2}$
| $B_{s}K$ | (4352.6 $\pm$ 115.0)$g_{R}^{2}$ | $B_{s}\pi^{0}$ | (10891.5 $\pm$ 336.7)$g_{R}^{2}$ $\times$ $10^{-4}$
| $B^{0}\eta$ | (1223.3 $\pm$ 32.3)$g_{R}^{2}$ | $B_{s}\eta$ | (5924.5 $\pm$ 196.6)$g_{R}^{2}$
$2^{3}D_{1}$ | $B^{*}\pi^{+}$ | (6272.98 $\pm$ 163.8)$g_{X}^{2}$ | $B^{*+}K^{-}$ | (8404.7 $\pm$ 219.5)$g_{X}^{2}$
| $B^{*}\pi^{0}$ | (3139.82 $\pm$ 82.0)$g_{X}^{2}$ | $B^{*0}K^{0}$ | (8379.2 $\pm$ 218.8)$g_{X}^{2}$
| $B{{}_{s}^{*}}K$ | (3528.4 $\pm$ 92.1)$g_{X}^{2}$ | $B_{s}^{*}\pi^{0}$ | (3717.4 $\pm$ 97.0)$g_{X}^{2}$ $\times$ $10^{-4}$
| $B^{*}\eta$ | (802.8 $\pm$ 21.0)$g_{X}^{2}$ | $B_{s}^{*}\eta$ | (3883.5 $\pm$ 101.4)$g_{X}^{2}$
| $B^{+}\pi^{-}$ | (14595.9 $\pm$ 381.1)$g_{X}^{2}$ | $B^{+}K^{-}$ | (19445.5 $\pm$ 507.8)$g_{X}^{2}$
| $B^{0}\pi^{0}$ | (7296.1 $\pm$ 190.5)$g_{X}^{2}$ | $B^{0}K^{0}$ | (19367.9 $\pm$ 505.8)$g_{X}^{2}$
| $B_{s}K$ | (8630.2 $\pm$ 225.4)$g_{X}^{2}$ | $B_{s}\pi^{0}$ | (8693.1 $\pm$ 227.0)$g_{X}^{2}$ $\times$ $10^{-4}$
| $B^{0}\eta$ | (1906.0 $\pm$ 49.8)$g_{X}^{2}$ | $B_{s}\eta$ | (9269.46 $\pm$ 242.1)$g_{X}^{2}$
$2^{1}D_{2}$ | $B^{*}\pi^{+}$ | (19845.8 $\pm$ 518.2)$g_{X}^{2}$ | $B^{*+}K^{-}$ | (26170.2 $\pm$ 683.4)$g_{X}^{2}$
| $B^{*}\pi^{0}$ | (9933.2 $\pm$ 259.4)$g_{X}^{2}$ | $B^{*0}K^{0}$ | (26092.3 $\pm$ 681.4)$g_{X}^{2}$
| $B{{}_{s}^{*}}K$ | (11300.9 $\pm$ 295.1)$g_{X}^{2}$ | $B_{s}^{*}\pi^{0}$ | (11573.4 $\pm$ 302.2)$g_{X}^{2}$ $\times$ $10^{-4}$
| $B^{*}\eta$ | (2558.9 $\pm$ 66.8)$g_{X}^{2}$ | $B_{s}^{*}\eta$ | (12148.3 $\pm$ 317.3)$g_{X}^{2}$
$2^{3}D_{2}$ | $B^{*}\pi^{+}$ | (13738.2 $\pm$ 361.2)$g_{Y}^{2}$ | $B^{*+}K^{-}$ | (8350.3 $\pm$ 261.6)$g_{Y}^{2}$
| $B^{*}\pi^{0}$ | (6886.6 $\pm$ 181.1)$g_{Y}^{2}$ | $B^{*0}K^{0}$ | (8289.8 $\pm$ 259.9)$g_{Y}^{2}$
| $B{{}_{s}^{*}}K$ | (5026.2 $\pm$ 140.4)$g_{Y}^{2}$ | $B_{s}^{*}\pi^{0}$ | (4230.8 $\pm$ 133.7)$g_{Y}^{2}$ $\times$ $10^{-4}$
| $B^{*}\eta$ | (1236.3 $\pm$ 32.6)$g_{Y}^{2}$ | $B_{s}^{*}\eta$ | (3163.0 $\pm$ 106.5)$g_{Y}^{2}$
$2^{3}D_{3}$ | $B^{*}\pi^{+}$ | (8380.4 $\pm$ 219.6)$g_{Y}^{2}$ | $B^{*+}K^{-}$ | (9820.6 $\pm$ 283.6)$g_{Y}^{2}$
| $B^{*}\pi^{0}$ | (4200.6 $\pm$ 110.1)$g_{Y}^{2}$ | $B^{*0}K^{0}$ | (9750.1 $\pm$ 281.6)$g_{Y}^{2}$
| $B{{}_{s}^{*}}K$ | (3125.8 $\pm$ 86.8)$g_{Y}^{2}$ | $B_{s}^{*}\pi^{0}$ | (4968.4 $\pm$ 146.1)$g_{Y}^{2}$ $\times$ $10^{-4}$
| $B^{*}\eta$ | (764.7 $\pm$ 20.1)$g_{Y}^{2}$ | $B_{s}^{*}\eta$ | (3734.8 $\pm$ 114.7)$g_{Y}^{2}$
| $B^{+}\pi^{-}$ | (7674.2 $\pm$ 201.2)$g_{Y}^{2}$ | $B^{+}K^{-}$ | (9033.7 $\pm$ 259.0)$g_{Y}^{2}$
| $B^{0}\pi^{0}$ | (3839.8 $\pm$ 100.6)$g_{Y}^{2}$ | $B^{0}K^{0}$ | (8958.1 $\pm$ 256.9)$g_{Y}^{2}$
| $B_{s}K$ | (3094.9 $\pm$ 81.2)$g_{Y}^{2}$ | $B_{s}\pi^{0}$ | (4592.0 $\pm$ 130.6)$g_{Y}^{2}$ $\times$ $10^{-4}$
| $B^{0}\eta$ | (728.8 $\pm$ 19.1)$g_{Y}^{2}$ | $B_{s}\eta$ | (3597.9 $\pm$ 105.4)$g_{Y}^{2}$

Our predicted mass of $B(1^{3}P_{0})$ is above the $B\pi$ and $B^{*}\pi$
threshold, but due to the parity conservation, only decay in $B\pi$ mode is
possible. Using the weighted average value of coupling $g_{S}$ = 0.56
(obtained in the strong decay analysis of the $1P$-wave doublet
$(D_{0}^{*}(2400),\\\ D_{1}(2420))=(1^{3}P_{0},1^{1}P_{1})$) from
colangelo2012new , we get the partial decay width of $B(1^{3}P_{0})\rightarrow
B\pi$ = 278 $\pm$ 3 MeV. That is slightly overestimated to the QCD sum rule
prediction in HQET 250 MeV zhu1999effect , and consistent with the chiral
quark model value 272 MeV zhong2008strong , and larger than the results of
quark pair creation (QPC) model 230 MeV lu2016excited  and 225 MeV
sun2014higher . Such a broad decay width is the fundamental reason for the
experimental unavailability of this state. For its spin-partner
$B(1^{1}P_{1})$, taking a sum of its partial decay width listed in Table (3),
which reach up to 267.0 $\pm$ 5.3 MeV. This state is very broad compared to
the $B_{1}(5721)$ measurements 30.1 $\pm$ 1.5 MeV of LHCb aaij2015precise  and
23 $\pm$ 3 MeV of CDF aaltonen2014study . From the strong decay analysis of
$B(1^{1}P_{1})$, we conclude that $B_{1}(5721)$ is not a member of doublet
$(0^{+},1^{+})_{{\frac{1}{2}}^{+}}$.

According to HQET, there are two $1^{+}$ states in $P$-wave doublets
$(0^{+},1^{+})_{{\frac{1}{2}}^{+}}$ and $(1^{+},2^{+})_{{\frac{3}{2}}^{+}}$
with $\vec{s}_{l}^{P}={\frac{1}{2}}^{+}$ and
$\vec{s}_{l}^{P}={\frac{3}{2}}^{+}$, respectively. For the second doublet
$(1^{+},2^{+})_{{\frac{3}{2}}^{+}}$, the decay widths 41 $\pm$ 7 MeV
$B(1^{3}P_{1})$ and 63 $\pm$ 4.6 MeV $B(1^{3}P_{2})$ are calculated by taking
a sum of its partial decay widths listed in Table (3) and using the weighted
average value of coupling $g_{T}$ = 0.43 (obtained in the strong decay
analysis of the $1P$-wave doublet
$(D_{1}(2430),D_{2}^{*}(2460))=(1^{3}P_{1},1^{3}P_{2})$ and
$(D_{s2}^{*}(2573))=(1^{3}P_{2})$) from colangelo2012new . For
$B(1^{3}P_{1})$, the decay width 41 $\pm$ 7 MeV is slightly overestimated to
LHCb aaij2015precise  and CDF aaltonen2014study  measurements 30.1 $\pm$ 1.5
MeV and 23 $\pm$ 3 MeV, respectively. For natural parity state
$B(1^{3}P_{2})$, the decay width 63 $\pm$ 4.6 MeV is overestimated to the
experimentally observed narrow decay width of $B_{2}^{*}(5747)$: 24.5 $\pm$
1.0 MeV of LHCb aaij2015precise  and 22${}^{+3}_{-2}$ MeV of CDF
aaltonen2014study , and the theoretical result 47 MeV of zhong2008strong .

The ratio of the decay rates is independent of the unknown hadronic couplings
and compare with the experimental results where available. The ratio,

$\frac{\Gamma(B(1^{3}P_{2})\rightarrow
B^{*}\pi)}{\Gamma(B(1^{3}P_{2})\rightarrow B\pi)}=1.0\pm 1.1,$ (11)

is within the errorbar of LHCb aaij2015precise  measurements 0.71 $\pm$ 0.14
$\pm$ 0.30 and 0.91 $\pm$ 0.13 $\pm$ 0.12 for neutral and charged states
respectively, and the result of $D0$ experiment 1.10 $\pm$ 0.42 $\pm$ 0.31
abazov2007properties . The best estimate value 1.0 (of Eq. (11)) is close to
the other theoretical result 0.9 of sun2014higher ; lu2016excited ;
gupta2019placing ; zhong2008strong . Also, we find out the ratio

$\displaystyle R_{1}$ $\displaystyle=\frac{\Gamma(B(1^{3}P_{1})\rightarrow
B^{*}\pi))}{\Gamma(B(1^{3}P_{1})\rightarrow
B^{*}\pi))+\Gamma(B(1^{3}P_{2})\rightarrow B^{*}\pi))}$ (12)
$\displaystyle=0.6\pm 0.9,$ (13)

and

$\displaystyle R_{2}$ $\displaystyle=\frac{\Gamma(B(1^{3}P_{2})\rightarrow
B^{*}\pi))}{\Gamma(B(1^{3}P_{2})\rightarrow
B^{*}\pi))+\Gamma(B(1^{3}P_{2})\rightarrow B\pi))}$ (14) $\displaystyle=0.6\pm
0.7,$ (15)

which are consistent with the $D0$ abazov2007properties  measurements $R_{1}$
= 0.47 $\pm$ 0.06 and $R_{2}$ = 0.47 $\pm$ 0.09, and underestimated to the
theoretical prediction $R_{1}$ = 0.3 and consistent with $R_{2}$ = 0.5 of
zhong2008strong , and close to $R_{1}$ = 0.6 and $R_{2}$ = 0.5 of
gupta2019placing . So, we write

$B_{J}^{*}(5732)=(0^{+})_{{\frac{1}{2}}^{+}}=\big{(}1^{3}P_{0}\big{)}.$ (16)

Its spin-partner $B(1^{1}P_{1})$ is still not found experimentally. Its
identification is an important task in future experimental studies. The
doublet,

$\big{(}B_{1}(5721),B_{2}^{*}(5747)\big{)}=(1^{+},2^{+})_{{\frac{3}{2}}^{+}}=\big{(}1^{3}P_{1},1^{3}P_{2}\big{)},$
(17)

is identified very well in the present study. Since the experimental
information of the $2P$ states is missing. For the $2P$ states, more decay
channels are open. Suppose the strong decays of excited $B$ meson (listed in
Table (3)) are contributing dominantly to the total decay width, then we can
give an information of fundamental decay modes of the particular decay state.
The $B(2^{3}P_{0})$ dominantly decays in $B^{+}\pi^{-}$ and $B_{s}K$ modes,
and $B^{*}\pi^{+}$ and $B^{+}\pi^{-}$ are the main decay channels of
$B(2^{3}P_{2})$. The $B(2^{1}P_{1})$ and $B(2^{3}P_{1})$ states can be found
in $B^{*}\pi^{+}$ decay mode.

####  4.1.2 $2S$ and $3S$ states

The experimentally observed mass of $B_{J}(5840)$ is close to the
$B(2^{1}S_{0})$ and $B(2^{3}S_{1})$ assignments. The $B_{J}(5840)$ was
observed in $B\pi$ decay mode, i.e. the member of natural parity states.
Therefore, we can exclude the $B(2^{1}S_{0})$ assignment of $B_{J}(5840)$.
Using the weighted average value of coupling $g_{H}$ = 0.28 (obtained in the
strong decay analysis of the $1P$-wave doublet $(D_{1}(2550),\\\
D_{J}^{*}(2600))=(2^{1}S_{0},2^{3}S_{1})$ and
$(D_{s1}^{*}(2700))=(2^{3}S_{1})$) from colangelo2012new  and taking the sum
of the partial decay widths of $B(2^{3}S_{1})$ presented in Table (3), which
reach up to 229.0 $\pm$ 6.9 MeV. This state is very broad and in good
agreement with the LHCb aaij2015precise  measurement 224.4 $\pm$ 23.9 MeV of
$B_{J}(5840)$ and overestimated to the theoretical results 107.8 MeV of
godfrey2016b , 106.1 MeV of lu2016excited , and 121.9 MeV of yu2019analysis .
Therefore, the strong decay analysis suggests $B(2^{3}S_{1})$ for
$B_{J}(5840)$, but still uncertain. It can also be a candidate of
$B(1^{3}D_{1})$ and $B(1^{3}D_{3})$ states. The calculated mass of the
$B(2^{3}S_{1})$ state is close to $B(1^{3}D_{1})$ and $B(1^{3}D_{3})$ states.
However, experimentally the $B(2^{3}S_{1})$ states might be more likely to be
observed than the $1D$ states. So, we write

$\big{(}B_{J}(5840)\big{)}=(1^{-})_{{\frac{1}{2}}^{-}}=\big{(}2^{3}S_{1}\big{)}.$
(18)

Our predicted decay width 229.0 $\pm$ 6.9 MeV of $B(2^{3}S_{1})$ state with
coupling $g_{H}$ = 0.28 is much larger compare to LHCb aaij2015precise
measurement 82 $\pm$ 8 MeV of $B_{J}(5970)$. Also, the mass difference
$M_{B_{J}(5970)}-M_{B_{J}(5840)}$ $\approx$ 100 MeV. So that they may not be
members of the same wave family. The $B^{*}\pi$ is the main decay channel of
$B(2^{3}S_{1})$, its branching ratio relative to $B\pi$ decay mode is

$\frac{\Gamma(B(2^{3}S_{1})\rightarrow
B^{*}\pi)}{\Gamma(B(2^{3}S_{1})\rightarrow B\pi)}=1.6\pm 0.1,$ (19)

calculated from Table (3), which is underestimated to the predictions 1.9 of
godfrey2016b , 2.5 of sun2014higher , and 2.3 of lu2016excited . From Table
(3), the $3S$ states are dominantly decaying in $B^{*}\pi^{+}$ mode compared
to $B^{+}\pi^{-}$. For $B(3^{3}S_{1})$, the branching ratio

$\frac{\Gamma(B(3^{3}S_{1})\rightarrow
B^{*}\pi)}{\Gamma(B(3^{3}S_{1})\rightarrow B\pi)}=1.8\pm 0.1,$ (20)

is predicted in the range of other theoretical estimations 1.4 of godfrey2016b
and 2.4 of sun2014higher .

####  4.1.3 $1D$, $2D$, and $1F$ states

There are four $1D$ states, from which an unnatural parity states
$B(1^{1}D_{2})$ and $B(1^{3}D_{2})$ are mainly to decay in $B^{*}\pi$ mode
(see Table (3)), and in accordance with the results of Refs. godfrey2016b ;
sun2014higher ; lu2016excited . For the natural parity states $B(1^{3}D_{1})$
and $B(1^{3}D_{3})$, the branching ratio

$R_{3}=\frac{\Gamma(B(1^{3}D_{1})\rightarrow
B\pi)}{\Gamma(B(1^{3}D_{1})\rightarrow B^{*}\pi)}=2.6\pm 0.1,$ (21)

and

$R_{4}=\frac{\Gamma(B(1^{3}D_{3})\rightarrow
B\pi)}{\Gamma(B(1^{3}D_{3})\rightarrow B^{*}\pi)}\approx 1.0,$ (22)

are calculated from Table (3). The value of $R_{3}$ is overestimated to 2.0 of
godfrey2016b ; sun2014higher , 1.7 of lu2016excited ; while $R_{4}$ is
consistent with 1.0 of godfrey2016b ; sun2014higher ; lu2016excited .
Therefore, the $B\pi$ mode is more dominant over $B^{*}\pi$ for
$B(1^{3}D_{1})$ state than $B(1^{3}D_{3})$. $B_{J}(5970)$ is observed in the
decay of two pseudoscalar mesons. Thus, we suggest

$\big{(}B_{J}(5970)\big{)}=(1^{-})_{{\frac{3}{2}}^{-}}=\big{(}1^{3}D_{1}\big{)}.$
(23)

We obtain the coupling $g_{X}$ $\sim$ 0.1 by equating the calculated decay
width of $B(1^{3}D_{1})$ from Table (3) with LHCb aaij2015precise  and CDF
aaltonen2014study  measurements for $B_{J}(5970)$. No experimental information
is available for $2D$ and $1F$ states. From the results listed in Table (3),
the $B(2^{3}D_{1})$ is mainly to decay in $B\pi$, and the $2^{3}D_{3}$ state
is equally found in $B\pi$ and $B^{*}\pi$ modes. The branching ratio,

$R_{5}=\frac{\Gamma(B(2^{3}D_{1})\rightarrow
B\pi)}{\Gamma(B(2^{3}D_{1})\rightarrow B^{*}\pi)}=2.3\pm 0.9,$ (24)

and

$R_{6}=\frac{\Gamma(B(2^{3}D_{3})\rightarrow
B\pi)}{\Gamma(B(2^{3}D_{3})\rightarrow B^{*}\pi)}\approx 1.0.$ (25)

The $R_{5}$ is consistent with 2.5 of godfrey2016b  and 2.2 of sun2014higher ,
and for $R_{6}$ our prediction is larger than 0.5 of godfrey2016b . The decay
behavior of the $1F$ states is shown in Table (3). The $BK$ mode is dominant
over $B^{*}K$, for the $B(1^{3}F_{2})$ and $B(1^{3}F_{4})$ states. The
branching ratio,

$R_{7}=\frac{\Gamma(B(1^{3}F_{2})\rightarrow
B\pi)}{\Gamma(B(1^{3}F_{2})\rightarrow B^{*}\pi)}=1.9\pm 0.1,$ (26)

and

$R_{8}=\frac{\Gamma(B(1^{3}F_{4})\rightarrow
B\pi)}{\Gamma(B(1^{3}F_{4})\rightarrow B^{*}\pi)}\approx 1.1.$ (27)

Here, the $R_{7}$ is quite larger than 1.4 of godfrey2016b ; sun2014higher ,
and $R_{8}$ is consistent with 1.0 of godfrey2016b . $B^{*}\pi$ is the main
decay channel for $B(2^{1}D_{2})$, $B(2^{3}D_{2})$, $B(1^{1}F_{3})$,
$B(1^{3}F_{3})$ states and in accordance with the results of Ref. godfrey2016b
and sun2014higher . The ratios of the strong decay rates will be helpful in
searching these states experimentally.

###  4.2 $B_{s}$ meson

####  4.2.1 $1P$ and $2P$ states

The predicted mass of $B_{s}(1^{3}P_{0})$ is below $BK$ threshold
lu2016excited ; zeng1995heavy ; cheng2014near ; di2001excited . A similar
situation arises, like in the strange charmed partner $D_{s0}^{*}(2317)$.
Hence, only the strong decay to $B_{s}\pi$ mode (called isospin-breaking mode
gross1979 ) is allowed kinematically. To account for isospin violation, the
square of the suppression factor, given by,

$\epsilon^{2}=\frac{3}{16}\left(\frac{m_{d}-m_{u}}{m_{s}-\frac{m_{u}+m_{d}}{2}}\right)^{2}\simeq
10^{-4},$ (28)

is multiplied with the decay width formula colangelo2003understanding ;
matsuki2012 . Here $m_{u}$, $m_{d}$, and $m_{s}$ are the current quark masses.
Using the weighted average value of coupling $g_{S}$ = 0.56 (obtained in the
strong decay analysis of the $1P$-wave doublet
$(D_{0}^{*}(2400),D_{1}(2420))=(1^{3}P_{0},1^{1}P_{1})$) from colangelo2012new
, we get the partial decay width of $B_{s}(1^{3}P_{0})\rightarrow
B_{s}\pi^{0}$ = (47.0 $\pm$ 1.2) $\times$ $10^{-4}$ MeV. And, its doublet
partner $B_{s}(1^{1}P_{1})\rightarrow B_{s}\pi^{0}$ = (51.0 $\pm$ 1.3)
$\times$ $10^{-4}$ MeV. Here the decay $B_{s}(1^{3}P_{0})$ $\rightarrow$ $BK$
is kinematically forbidden. S. Godfrey et al. godfrey2016b  used the QPC model
and determined the partial decay width 138 MeV of $B_{s}(1^{3}P_{0})$ decay to
$BK$ mode. Such a broad decay width of $B_{s}(1^{3}P_{0})$ following some
other QPC model predictions 225 MeV of sun2014higher  and 217 MeV of
yu2019analysis , and the result from the chiral quark model 227 MeV
zhong2008strong . Such a broad resonance state is difficult to identify in
experimental studies.

The ratio among the decay rates can be used to confirm or reject the quantum
number assignments of the observed states. The ratio,

$\frac{\Gamma(B_{s}(1^{3}P_{2})\rightarrow
B^{*+}K^{-})}{\Gamma(B_{s}(1^{3}P_{2})\rightarrow B^{+}K^{-})}\approx 0.3,$
(29)

is calculated using the results of Table (3). It is larger than the
experimental measurements 0.081 $\pm$ 0.021 $\pm$ 0.015 of CMS
collaboration2018studies , 0.093 $\pm$ 0.013 $\pm$ 0.012 of LHCb aaij2013first
and 0.10 $\pm$ 0.03 $\pm$ 0.02 of CDF aaltonen2008observation , and the
theoretical prediction 0.1 of sun2014higher ; lu2016excited ; yu2019analysis ;
zhong2008strong . The $B_{s2}^{*}(5840)$ as $B_{s}(1^{3}P_{2})$, with PDG
Zyla2020  world average mass 5839.92 $\pm$ 0.14 MeV, our calculated partial
decay rates are (0.38 $\pm$ 0.02)$g_{T}^{2}$ MeV and (4.63 $\pm$
0.13)$g_{T}^{2}$ MeV for the decay states $B^{*+}K^{-}$ and $B^{+}K^{-}$,
respectively. That gives a ratio
$\frac{\Gamma({B^{*+}K^{-}})}{\Gamma({B^{+}K^{-}})}$ $\approx$ 0.082 $\pm$
0.005, which is within the errorbar of CMS collaboration2018studies
measurement, and in agreement with the results of other experimental and
theoretical studies aaij2013first ; sun2014higher ; yu2019analysis ;
zhong2008strong ; aaltonen2008observation .

The decay width 6.7 $\pm$ 0.3 MeV of $B_{s}(1^{3}P_{2})$ is calculated by
taking a sum of its partial decay width listed in Table (3) and using the
weighted average value of coupling $g_{T}$ = 0.43 (obtained in the strong
decay analysis of the $1P$-wave doublet
$(D_{1}(2430),D_{2}^{*}(2460))=(1^{3}P_{1},1^{3}P_{2})$ and
$(D_{s2}^{*}(2573))=(1^{3}P_{2})$) from colangelo2012new . Such a narrow decay
width of $B_{s}(1^{3}P_{2})$ is quite larger than the experimental observed
narrow resonance state $B_{s2}^{*}(5840)$: 1.52 $\pm$ 0.34 $\pm$ 30 MeV of CMS
collaboration2018studies , 1.4 $\pm$ 0.4 $\pm$ 0.2 MeV of LHCb aaij2013first ,
and 1.56 $\pm$ 0.13 $\pm$ 0.47 MeV of CDF aaltonen2008observation , and the
theoretical prediction 2 MeV of lu2016excited ; zhong2008strong . Therefore,
$B_{s2}^{*}(5840)$ can be a strong candidate for $B_{s}(1^{3}P_{2})$. For its
doublet partner $B_{s}(1^{3}P_{1})$, the decay width 2.6 $\pm$ 0.2 MeV is
obtained by summing up the partial decay width from Table (3) with $g_{T}$ =
0.43. Such a state is narrow and is underestimated to the prediction 21.4 MeV
of lu2016excited  and overestimated to the CDF measurement 0.5 $\pm$ 0.3 $\pm$
0.3 MeV aaltonen2014study . In $P$-wave, the states $1^{1}P_{1}$ and
$1^{3}P_{1}$ can be a mix, and as we said in the introduction, one of the
states may narrow. Experimentally $B_{s1}(5830)$ is observed as a narrow state
aaltonen2014study . So it is more reliable to predict $B_{s}(1^{3}P_{1})$.
Therefore, we write

$\big{(}B_{s1}(5830),B_{s2}^{*}(5840)\big{)}=(1^{+},2^{+})_{{\frac{3}{2}}^{+}}=\big{(}1^{3}P_{1},1^{3}P_{2}\big{)}.$
(30)

For the $2P$ states, more decay channels are open. But there has been no
experimental observation to date. Suppose the strong decays of excited
strange-bottom mesons (listed in Table (3)) are contributing dominantly to the
total decay width, we can give information of the fundamental decay modes of
the particular decay state. The $BK$ is the main decay mode of
$B_{s}(2^{3}P_{0})$, and the states $B_{s}(2^{1}P_{1})$ and
$B_{s}(2^{3}P_{1})$ are dominantly decaying in $B^{*}K$ mode; and $B^{*}K$ and
$BK$ are the main decay channels of $B_{s}(2^{3}P_{2})$. Our results are
consistent with the predictions of godfrey2016b ; sun2014higher .

####  4.2.2 $2S$ and $3S$ states

The decay behavior of the $B_{s}(2^{1}S_{0})$ and $B_{s}(2^{3}S_{1})$ states
are shown in Table (3). It shows that the $B^{*}K$ is the main decay mode of
$B_{s}(2^{1}S_{0})$, and for $B_{s}(2^{3}S_{1})$ state $B^{*}K$ is dominant
over $BK$. The branching ratio,

$\frac{\Gamma(B_{s}(2^{3}S_{1})\rightarrow
B^{*}K)}{\Gamma(B_{s}(2^{3}S_{1})\rightarrow BK)}\approx 1.4,$ (31)

which is in good agreement with 1.5 of godfrey2016b , and overestimated to 2.0
of sun2014higher ; lu2016excited . Like $2S$ states, the $B^{*}K$ is the main
decay mode of $B_{s}(3^{1}S_{0})$ state. Moreover, for $B_{s}(3^{3}S_{1})$,
the $B^{*}K$ mode is dominant over $BK$. The branching ratio,

$\frac{\Gamma(B_{s}(3^{3}S_{1})\rightarrow
B^{*}K)}{\Gamma(B_{s}(3^{3}S_{1})\rightarrow BK)}\approx 1.8,$ (32)

is predicted in the range of the theoretical results 1.2 of godfrey2016b  and
2.0 of sun2014higher . It is interesting to study the strong decays of a
strange-bottom meson with the help of couplings obtained in the charm sector.
Using the weighted average value of coupling $g_{H}$ = 0.28 (obtained in the
strong decay analysis of the $1P$-wave doublet
$(D_{1}(2550),D_{J}^{*}(2600))=(2^{1}S_{0},2^{3}S_{1})$ and
$(D_{s1}^{*}(2700))=(2^{3}S_{1})$) from colangelo2012new  and taking the sum
of the partial decay widths of $B_{s}(2^{1}S_{0})$ and $B_{s}(2^{3}S_{1})$
presented in Table (3), which reach up to 138.0 $\pm$ 3.5 MeV and 170.0 $\pm$
1.5 MeV, respectively. Such states are predicted as a broad resonance state,
which is underestimated to the predictions 213.4 MeV $B_{s}(2^{1}S_{0})$ and
221.9 MeV $B_{s}(2^{3}S_{1})$ of Ref. lu2016excited , and overestimated to
75.8 MeV $B_{s}(2^{1}S_{0})$ and 114 MeV $B_{s}(2^{3}S_{1})$ of Ref.
godfrey2016b , and 44 MeV $B_{s}(2^{1}S_{0})$ and 51 MeV $B_{s}(2^{3}S_{1})$
of Ref. sun2014higher .

####  4.2.3 $1D$, $2D$, and $1F$ states

The natural parity $1D$ states $1^{3}D_{1}$ and $1^{3}D_{3}$ are dominant to
decay in $BK$ (see Table (3)). Their calculated branching ratio of the decay
rates relative to $B^{*}K$ decay mode are,

$R_{1}=\frac{\Gamma(B_{s}(1^{3}D_{1})\rightarrow
BK)}{\Gamma(B_{s}(1^{3}D_{1})\rightarrow B^{*}K)}=2.6\pm 0.1,$ (33)

and

$R_{2}=\frac{\Gamma(B_{s}(1^{3}D_{3})\rightarrow
BK)}{\Gamma(B_{s}(1^{3}D_{3})\rightarrow B^{*}K)}\approx 1.1.$ (34)

Our calculated value of $R_{1}$ is overestimated by the theoretical results
2.0 of godfrey2016b , 2.2 of sun2014higher , and 1.8 of lu2016excited ; and
$R_{2}$ is consistent with 1.2 of godfrey2016b , 0.9 of sun2014higher , and
1.1 of lu2016excited . The $B^{*}K$ is the main channel of $B_{s}(1^{1}D_{2})$
and $B_{s}(1^{3}D_{2})$, which are in accordance with the results of Refs.
godfrey2016b ; sun2014higher ; lu2016excited . For $2^{3}D_{1}$ states, $BK$
is the fundamental decay mode; and for $2^{3}D_{3}$, $B^{*}K$ is dominant over
$BK$ (see Table (3)). The branching ratio,

$R_{3}=\frac{\Gamma(B_{s}(2^{3}D_{1})\rightarrow
BK)}{\Gamma(B_{s}(2^{3}D_{1})\rightarrow B^{*}K)}=2.3\pm 0.1,$ (35)

and

$R_{4}=\frac{\Gamma(B_{s}(2^{3}D_{3})\rightarrow
BK)}{\Gamma(B_{s}(2^{3}D_{3})\rightarrow B^{*}K)}\approx 0.9.$ (36)

Here, $R_{3}$ is consistent with 2.5 of godfrey2016b  and 2.3 of sun2014higher
, while for $R_{4}$ our prediction is quite larger than 0.5 of godfrey2016b
and 0.4 of sun2014higher . For $B_{s}(2^{1}D_{2})$ and $B_{s}(2^{3}D_{2})$
states, $B^{*}K$ is the main decay channel. The decay behavior of the $1F$
states is shown in Table (3). The $BK$ mode is dominant over $B^{*}K$, for the
states $B_{s}(1^{3}F_{2})$ and $B_{s}(1^{3}F_{4})$. The branching ratio,

$R_{5}=\frac{\Gamma(B_{s}(1^{3}F_{2})\rightarrow
BK)}{\Gamma(B_{s}(1^{3}F_{2})\rightarrow B^{*}K)}=1.9\pm 0.1,$ (37)

and

$R_{6}=\frac{\Gamma(B_{s}(1^{3}F_{4})\rightarrow
BK)}{\Gamma(B_{s}(1^{3}F_{4})\rightarrow B^{*}K)}\approx 1.1.$ (38)

Here, the $R_{5}$ is overestimated to 1.4 of godfrey2016b ; sun2014higher ,
and $R_{6}$ is consistent with 1.0 of godfrey2016b . The branching ratio
distinguishes the fundamental decay mode, which is valuable to further
experimental search. Using $g_{X}$ $\sim$ 0.1 (as we extract in sec 4.1.3 for
$B_{J}(5970)$ as $B(1^{3}D_{1})$) and taking summation of partial decay rates
of $B_{s}(1^{3}D_{1})$ and $B_{s}(1^{1}D_{2})$ listed in Table (3), we
obtained 82.0 $\pm$ 1.0 MeV and 74.0 $\pm$ 1.2 MeV, respectively. These are
underestimated to the results 213.4 MeV $B_{s}(1^{3}D_{1})$ and 198.6 MeV
$B_{s}(1^{1}D_{2})$ of lu2016excited , 183 MeV $B_{s}(1^{3}D_{1})$ of
godfrey2016b , and 191 MeV $B_{s}(1^{3}D_{1})$ of sun2014higher . Our
calculated decay width 82.0 $\pm$ 1.0 MeV of natural parity state
$B_{s}(1^{3}D_{1})$ is within the errorbar of experimental measurement 66
$\pm$ 18 $\pm$ 21 MeV of $B_{s}(6114)$. We expect more experimental efforts in
the future to identify its nature properly.

However, such a comparison is not necessarily valid because heavier excited
meson resonances might decay to alternative final states, such as those
involving higher light pseudoscalar multiplicities, light vector mesons, and
lower-lying excited heavy mesons godfrey2016b ; sun2014higher ; lu2016excited
; yu2019analysis ; zhong2008strong ; campanella2018excited . The effective
heavy meson chiral Lagrangians reported in Ref. gandhi2019strong  do not
contain such extra decay modes.

##  5 Summary

We have carried out a systematic study of the excited $B$ and $B_{s}$ mesons
in the framework of heavy quark effective theory. The masses and the strong
two-body decays of excited $B$ and $B_{s}$ mesons are calculated. Our
predictions listed with other results obtained from the various potential
model as well as available experimental measurements give the mass range of
these mesons. With these predictions, we assign the possible quantum states
for the experimentally observed excited $B$ and $B_{s}$ mesons such as
$B_{J}^{*}(5732)$, $B_{1}(5721)$, $B_{s1}(5830)$, $B_{2}^{*}(5747)$,
$B_{J}(5840)$, $B_{s2}^{*}(5840)$, $B_{J}(5970)$, and $B_{s}(6063)$.
Furthermore, the strong decay behaviors of the excited $B$ and $B_{s}$ mesons
are investigated with the help of couplings $g_{H}$, $g_{S}$, and $g_{T}$
obtained by P. Colangelo et al. colangelo2012new  in the strong decay analysis
of excited charmed mesons. The $B_{J}^{*}(5732)$ is identified as
$B(1^{3}P_{0})$. The $B_{1}(5721)$ and $B_{2}^{*}(5747)$ are classified into
$P$-wave spin doublet $(1^{3}P_{1},1^{3}P_{2})$, and the $B_{J}(5840)$ is
interpreted as $2^{3}S_{1}$. The $B_{s1}(5830)$ and $B_{s2}^{*}(5840)$ is
found to be a strange partners of $B_{1}(5721)$ and $B_{2}^{*}(5747)$. The
branching ratio $\frac{B\pi}{B^{*}\pi}$ can explain $B_{J}(5970)$ as
$1^{3}D_{1}$. We tentatively identify $B_{s}(6063)$ as $B_{s}(2^{3}S_{1})$
from the spectroscopy study and $B_{s}(6114)$ as $B_{s}(1^{3}D_{1})$ from the
strong decay analysis. In this regard, we expect more information from future
experimental studies. The properties of other excited $B$ and $B_{s}$ mesons
are also predicted, which will be useful in future experimental searches.

## Acknowledgment

We thank Prof. T. Matsuki for the continuous support throughout this work and
for providing critical remarks for the improvement of the paper. Dr. K. Gandhi
thank Dr. N. R. Soni for some help in the uncertainty computation.

## Appendix: Uncertainty computation

Uncertainty arises in presented results are because of the uncertainties in
the various experimental inputs. For the computation of uncertainties in our
results, we use the most general technique. For instance, the uncertainty in
the computation of spin-averaged mass $\bar{M}(a,b)$ because of the
uncertainty in the experimental inputs $a$ and $b$ can be written as

$\Delta\bar{M}(a,b)=\sqrt{{\left(\frac{\partial\bar{M}(a,b)}{\partial
a}{\delta a}\right)}^{2}+{\left(\frac{\partial\bar{M}(a,b)}{\partial b}{\delta
b}\right)}^{2}},$ (39)

with $\delta a$ and $\delta b$ are the uncertainties in the $a$ and $b$
respectively. The uncertainties in the mass splitting between the excited and
the low-lying negative parity doublets, hyperfine mass splitting between the
members of the doublets, and the strong decay rates are also extracted using
the same method.

## References

  * (1) P.A. Zyla et al. (Particle Data Group), _Review of particle physics_ , Prog. Theor. Exp. Phys. 2020, 083C01 (2020).
  * (2) R. Aaij et al. (LHCb Collaboration), _Precise measurements of the properties of the $B_{1}(5721)^{0,+}$ and $B_{2}^{*}(5747)^{0,+}$ states and observation of $B^{+,0}$ $\pi^{-,+}$ mass structures_, JHEP 2015, 24 (2015).
  * (3) R. Aaij et al. (LHCb Collaboration), _First Observation of the Decay $B_{s2}^{*}\rightarrow B^{*+}k^{-}$ and Studies of Excited $B_{s}^{0}$ Mesons_, Phys. Rev. lett. 110, 151803 (2013).
  * (4) T. Aaltonen et al. (CDF Collaboration), _Study of orbitally excited B mesons and evidence for a new B $\pi$ resonance_, Phys. Rev. D 90, 012013 (2014).
  * (5) T. Aaltonen et al. (CDF Collaboration), _Measurement of resonance parameters of orbitally excited narrow $B^{0}$ mesons_, Phys. Rev. lett. 102, 102003 (2009).
  * (6) V. Abazov et al. (D0 Collaboration), _Properties of $L=1$ $B_{1}$ and $B_{2}^{*}$ mesons_, Phys. Rev. lett. 99, 172001 (2007).
  * (7) R. Aaij et al. (LHCb Collaboration), _Observation of new excited $B_{s}^{0}$ states_, arXiv:2010.15931v1 [hep-ex] (2020).
  * (8) S. Godfrey, K. Moats and E.S. Swanson, _B and $B_{s}$ meson spectroscopy_, Phys. Rev. D 94, 054025 (2016).
  * (9) Y. Sun et al., _Higher bottom and bottom-strange mesons_ , Phys. Rev. D 89, 054026 (2014).
  * (10) D. Ebert, R. Faustov and V. Galkin, _Heavy-light meson spectroscopy and Regge trajectories in the relativistic quark model_ , Eur. Phys. J. C 66, 197 (2010).
  * (11) Q.F. Lü et al., _Excited bottom and bottom-strange mesons in the quark model_ , Phys. Rev. D 94, 074012 (2016).
  * (12) V. Kher, N. Devlani and A.K. Rai, _Spectroscopy, decay properties and Regge trajectories of the B and $B_{s}$ mesons_, Chin. Phys. C 41, 093101 (2017).
  * (13) N. Devlani and A.K. Rai, _Spectroscopy and decay properties of B and $B_{s}$ mesons_, Eur. Phys. J. A 48, 104 (2012).
  * (14) M. Shah, B. Patel and P.C. Vinodkumar, _Spectroscopy and flavor changing decays of B, $B_{s}$ mesons in a Dirac formalism_, Phys. Rev. D 93, 094028 (2016).
  * (15) D. Jia and W.C. Dong, _Regge-like spectra of excited singly heavy mesons_ , Eur. Phys. J. Plus 134, 1 (2019).
  * (16) J.K. Chen, _Regge trajectories for the mesons consisting of different quarks_ , Eur. Phys. J. C 78, 648 (2018).
  * (17) T. Lähde, C. Nyfält and D. Riska, _Spectra and M1 decay widths of heavy-light mesons_ , Nucl. Phys. A 674, 141 (2000).
  * (18) L.-Y. Xiao and X.-H. Zhong, Strong decays of higher excited heavy-light mesons in a chiral quark model, Phys. Rev. D 90, 074029 (2014).
  * (19) I. Asghar et al., Decays and spectrum of bottom and bottom strange mesons, Eur. Phys. J. A 54, 127 (2018).
  * (20) E.J. Eichten, C.T. Hill and C. Quigg, _Properties of orbitally excited heavy-light $(Q\bar{q})$ mesons_, Phys. Rev. Lett. 71, 4116 (1993).
  * (21) T. Mannel, _Review of heavy quark effective theory_ , arXiv: 9611411 [hep-ph] (1996).
  * (22) A.F. Falk and T. Mehen, _Excited heavy mesons beyond leading order in the heavy quark expansion_ , Phys. Rev. D 53, 231 (1996).
  * (23) M. Neubert, _Heavy-meson form factors from QCD sum rules_ , Phys. Rev. D 45, 2451 (1992).
  * (24) F.E. Close and Z. Li, _Effective heavy quark theory_ , Phys. Lett. B 289, 143 (1992).
  * (25) P. Colangelo, F De. Fazio, F. Giannuzzi and S. Nicotri, _New meson spectroscopy with open charm and beauty_ , Phys. Rev. D 86, 054024 (2012).
  * (26) P. Gupta and A. Upadhyay, _Placing the newly observed state $B_{J}(5840)$ in bottom spectra along with states $B_{1}(5721)$, $B_{2}^{*}(5747)$, $B_{s1}(5830)$, $B_{s2}^{*}(5840)$, and $B_{J}(5970)$_, Phys. Rev. D 99, 094043 (2019).
  * (27) P. Gupta and A. Upadhyay, _Analysis of strong decays of charmed mesons $D_{2}^{*}(2460)$, $D_{0}(2560)$, $D_{2}(2740)$, $D_{1}(3000)$, $D_{2}^{*}(3000)$, and their spin partners $D_{1}^{*}(2680)$, $D_{3}^{*}(2760)$, and $D_{0}^{*}(3000)$_, Phys. Rev. D 97, 014015 (2018).
  * (28) G.L. Yu and Z.G. Wang, _The analysis of the excited bottom and bottom strange states $B_{s1}(5721)$, $B_{2}^{*}(5747)$, $B_{s1}(5830)$, $B_{s2}^{*}(5840)$, $B_{J}(5840)$, and $B_{J}(5970)$ in B meson family_, Chin. Phys. C 44, 033103 (2019).
  * (29) H.X. Chen et al., _A review of the open charm and open bottom systems_ , Rep. on Prog. in Phys. 80, 076201 (2017).
  * (30) A.V. Manohar and M.B. Wise, _Heavy Quark Physics_ (Cambridge University Press, Cambridge, 2000).
  * (31) Z.G. Wang, _Analysis of strong decays of the charmed mesons $D_{J}(2580)$, $D_{J}^{*}(2650)$, $D_{J}(2740)$, $D_{J}^{*}(2760)$, $D_{J}(3000)$, $D_{J}^{*}(3000)$_, Phys. Rev. D 88, 114003 (2013).
  * (32) Z.G. Wang, _$D_{s3}^{*}(2860)$ and $D_{s3}^{*}(2860)$ as the $1D$ $C\bar{s}$ states_, Eur. Phys. J. C 75, 25 (2015).
  * (33) K. Gandhi and A.K. Rai, _Strong decay analysis of excited nonstrange charmed mesons: Implications for spectroscopy_ , Eur. Phys. J. A 57, 23 (2021).
  * (34) K. Gandhi, A.K. Rai and T. Matsuki, _Identifying $D_{sJ}^{*}(2860)$ as four resonance states through strong decay analysis_, arXiv:2010.15931v1 [hep-ph] (2020).
  * (35) S.L. Zhu and Y.B. Dai, _THE EFFECT OF $B\pi$ CONTINUUM IN THE QCD SUM RULES FOR THE $(0^{+},1^{+})$ HEAVY MESON DOUBLET IN HQET_, Mod. Phys. Lett. A 14, 2367 (1999).
  * (36) X.H. Zhong and Q. Zhao, _Strong decays of heavy-light mesons in a chiral quark model_ , Phys. Rev. D 78, 014029 (2008).
  * (37) J. Zeng, J. Van Orden and W. Roberts, _Heavy mesons in a relativistic model_ , Phys. Rev. D 52, 5229 (1995).
  * (38) H.Y. Cheng and F.S. Yu, _Near mass degeneracy in the scalar meson sector: Implications for $B_{(s)0}^{*}$ and $B_{(S)1}^{\prime}$ mesons_, Phys. Rev. D 89, 114017 (2014).
  * (39) M. Di Pierro and E. Eichten, _Excited heavy-light systems and hadronic transitions_ , Phys. Rev. D 64, 114004 (2001).
  * (40) D. J. Gross, S. B. Treiman and F. Wilczek, _Light-quark masses and isospin violation_ , Phys. Rev. D 19, 2188 (1977).
  * (41) P. Colangelo and F. De Fazio, _Understanding $D_{sJ}(2317)$_, Phys. Lett. B 570, 180 (2003).
  * (42) T. Matsuki1 and K. Seo, _Chiral particle decay of heavy-light mesons in a relativistic potential model_ , Phys. Rev. D 85, 014036 (2012).
  * (43) A. M. Sirunyan et al. (CMS Collaboration), _Studies of $B_{s2}^{*}(5840)^{0}$ and $B_{s1}(5830)^{0}$ mesons including the observation of the $B_{s2}^{*}(5840)^{0}$ $\rightarrow$ $B^{0}K_{S}^{0}$ decay in proton-proton collisions at $\sqrt{s}=8$ TeV_, Eur. Phys. J. C 78, 939 (2018).
  * (44) R. Aaij et al. (LHCb Collaboration), _Observation of orbitally excited $B_{s}$ mesons_, Phys. Rev. Lett. 100, 082001 (2008).
  * (45) S. Campanella, P. Colangelo and F. De Fazio, _Excited heavy meson decays to light vector mesons: Implications for spectroscopy_ , Phys. Rev. D 98, 114028 (2018).