# Multilinear Wavelet Compact T(1) Theorem

Anastasios Fragkos  ,  A. Walton Green  and  Brett D. Wick

###### Abstract.

We prove a wavelet $T(1)$ theorem for compactness of multilinear Calderón-
Zygmund (CZ) operators. Our approach characterizes compactness in terms of
testing conditions and yields a representation theorem for compact CZ forms in
terms of wavelet and paraproduct forms that reflect the compact nature of the
operator.

A. W. Green’s research supported in part by NSF grant NSF-DMS-2202813.

B. D. Wick’s research supported in part by NSF-DMS-2000510, NSF-DMS-2054863 as
well as ARC DP 220100285.

##  1\. Introduction

An $(m+1)$-linear form $\Lambda$ defined on the $(m+1)$-fold product of the
Schwarz space ${\mathcal{S}}(\mathbb{R}^{d})$ is a singular integral form if
its off-diagonal kernel satisfies the standard size and smoothness estimates
(see Definition 3.1 below). A singular integral form $\Lambda$ is bounded on
$L^{p_{0}}(\mathbb{R}^{d})\times\cdots\times L^{p_{m}}(\mathbb{R}^{d})$ for
all

(1.1) $1<p_{j}\leq\infty,\quad\sum_{j=0}^{m}\frac{1}{p_{j}}=1,$

if and only if $\Lambda$ is Calderón-Zygmund, which we take to mean that it
satisfies the weak boundedness property (see Definition 3.2 below) and the
following multilinear $T(1)$ condition: There exists
$\mathfrak{b}_{j}\in\mathrm{BMO}(\mathbb{R}^{d})$ such that for every $\phi$
in ${\mathcal{S}}(\mathbb{R}^{d})$ with mean zero,

(1.2) $\Lambda^{*,j}(\phi,1,\ldots,1)=\langle\phi,\mathfrak{b}_{j}\rangle,$

where $\Lambda^{*,j}$ permutes the $0$th and $j$th argument (see (1.3) below).
When $m=1$, this is the celebrated $T(1)$ theorem of David and Journé
[david84] which was extended to $m\geq 2$ by Grafakos and Torres [grafakos-
torres] (see also the earlier works [christ1987polynomial, coifman-meyer]) and
more recently developed in, for example, [DPLiMV2, DPLiMV1,
lerner2019intuitive, li2019bilinear, li2014sharp].

Our goal here is to prove a $T(1)$ theorem for compactness of multilinear
singular integral forms. The first distinction between the boundedness problem
and compactness problem is that compactness is a property of operators, while
boundedness in the reflexive range (1.1) can be equivalently stated in terms
of forms. Accordingly for each $j=0,1,\ldots,m$, we associate to $\Lambda$ the
$m$-linear adjoint operators $T^{*,j}$ and transposed forms $\Lambda^{*,j}$ by

(1.3) $\langle
f_{0},T^{*,j}(f_{1},\ldots,f_{m})\rangle=\Lambda^{*,j}(f_{0},f_{1},\ldots,f_{m})=\Lambda(f_{j},f_{1},\ldots,f_{j-1},f_{0},f_{j+1},\ldots,f_{m}).$

If $\Lambda$ is Calderón-Zygmund, then we say each $T^{*,j}$ is an $m$-linear
Calderón-Zygmund operator (CZO). Furthermore, define
$\mathsf{T}_{\Lambda}=\\{T^{*,j}\\}_{j=0}^{m}$.

In the linear case, Villaroya in [Vil15] gave a complete characterization of
compact Calderón-Zygmund operators on $L^{2}(\mathbb{R}^{d})$, which was
further developed in [OV, PPV, SVW, Vil19]. Recently, Mitkovski and Stockdale
in [MS23] gave a simplified formulation of the $T(1)$ theorem for compactness
of Villaroya. More precisely, they showed that a CZO $T$ is compact if and
only if $\mathfrak{b}_{0}$ and $\mathfrak{b}_{1}$ both belong to
$\mathrm{CMO}(\mathbb{R}^{d})$ and a vanishing version of the weak boundedness
property, called the weak compactness property, is satisfied. In contrast, a
multilinear version of these compactness testing theorems remains unexplored.
These $T(1)$ theorems for compactness are complemented by a compact Rubio De
Francia theory of extrapolation developed by Hytönen and Lappas in [HL2, HL1].
Subsequently, in [COY], Cao, Olivo, and Yabuta extended the bilinear results
of [HL2] to the multilinear setting and to the quasi-Banach range, in which
case the target space can be $L^{r}$ with $r>\frac{1}{m}$. However, due to the
difficulties of extrapolating to the upper endpoint in the multilinear setting
[li2021endpoint, nieraeth2019quantitative], the results of [COY, HL2] do not
consider the case where one (or more) input spaces is
$L^{\infty}(\mathbb{R}^{d})$. We point out that our results below do yield
compactness when an input space is $L^{\infty}(\mathbb{R}^{d})$.

To state our main result in the bilinear setting (see Theorem 2 below for the
full $m$-linear result), it remains only to introduce our bilinear version of
the weak compactness property. Given $\varphi\in
L^{1}_{\mathrm{loc}}(\mathbb{R}^{d})$, $w\in\mathbb{R}^{d}$, and $t>0$, define

$\varphi_{z}=\frac{1}{t^{d}}\varphi\left(\frac{\cdot-w}{t}\right),\quad
z=(w,t).$

Moreover, let $\mathbb{B}(z)$ denote a hyperbolic ball in the upper half space
of radius $1$ centered at $z=(w,t)$ (see (2.2) below for the precise
definition). Fixing three functions $\phi,\psi^{1},\psi^{2}$, define

(1.4)
$\mathsf{W}_{\Lambda}(z)=t^{2d}\cdot\sup_{z_{1},z_{2}\in\mathbb{B}(z)}\sup_{\begin{subarray}{c}\sigma\in\mathrm{S}_{3}\\\
i=1,2\end{subarray}}\left|\Lambda^{\sigma}\left(\phi_{z_{1}},\psi^{i}_{z_{2}},\phi_{z}\right)\right|,$

where for each $\sigma\in\mathrm{S}_{m+1}$, the permutation group on
$\\{0,1,\ldots,m\\}$, we define

(1.5)
$\Lambda^{\sigma}(f_{0},\ldots,f_{m})=\Lambda(f_{\sigma(0)},\ldots,f_{\sigma(m)}).$

######  Theorem 1.

Let $\Lambda$ be a trilinear Calderón-Zygmund form and $\phi$ be a mother
wavelet. Then, there exists
$\psi^{1},\psi^{2}\in{\mathcal{S}}(\mathbb{R}^{d})$ such that the following
are equivalent:

  * A.

Each $T\in\mathsf{T}_{\Lambda}$, $1<p_{1},p_{2}\leq\infty$, and
$\tfrac{1}{2}<r<\infty$ satisfying
$\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{r}$, $T$ extends to a compact
bilinear operator from $L^{p_{1}}(\mathbb{R}^{d})\times
L^{p_{2}}(\mathbb{R}^{d})$ to $L^{r}(\mathbb{R}^{d})$ and from
$L^{\infty}_{0}(\mathbb{R}^{d})\times L^{\infty}_{0}(\mathbb{R}^{d})$ to
$\mathrm{CMO}(\mathbb{R}^{d})$.

  * B.

For each $T\in\mathsf{T}_{\Lambda}$, $T(1,1)\in\mathrm{CMO}(\mathbb{R}^{d})$,
and $\lim\limits_{z\to\infty}\mathsf{W}_{\Lambda}(z)=0$.

If the testing conditions in B. of Theorem 1 are satisfied, then we say
$\Lambda$ is a compact Calderón-Zygmund form. Aside from having simple and
precise wavelet testing conditions that ensure compactness of the adjoint
operators, we also have a concrete description of compact Calderón-Zygmund
forms. We refer to Theorem 2 and Definitions 2.6 and 5.1 below for a precise
statement, but in words, we will show that a compact CZ form is a finite
linear combination of compact wavelet forms and compact paraproduct forms. The
strategy of representing CZOs as simpler model operators has been very
successful in understanding more refined properties of CZOs, most famously in
providing sharp quantitative weighted estimates. Dyadic representation
theorems, finding their roots in the works of Figiel [figiel90]; Nazarov,
Treil, and Volberg [NTV2]; and Petermichl [PetAJM] among others, have led to a
powerful and comprehensive theory of quantitative $L^{p}$ estimates for
singular integrals, culminating with Hytönen’s proof of the $A_{2}$ conjecture
[hytonen12]. See [m2022, HL2022, li2019bilinear,
martikainen2012representation] and references therein for more recent
extensions. Wavelet representations, which may be traced back to David and
Journé’s original proof of the $T(1)$ theorem, have recently been developed by
Di Plinio and some of the authors in a series of papers [DGW2, DGW, DWW]
primarily to extend the Lebesgue theory to Sobolev spaces, though some new
Lebesgue space results have also been obtained as a byproduct.

Our main result is another example of a new result in the Lebesgue setting
obtained through wavelet representations. This approach differs from the ones
in [MS23] and [Vil15] as we do not use the machinery regarding localization,
and neither do we require additional decay of the kernel, but rather we use a
wavelet averaging procedure to obtain the representation formula in §3 and
then use the Riesz-Kolmogorov criterion to obtain the precompactness of the
image of the unit ball under the adjoint operators to the wavelet and
paraproduct forms in §4 and §5.

##  2\. Wavelets

In this section we will review some preliminaries regarding wavelets and
introduce one of the building blocks of our representation, wavelet forms.

###  2.1. Analysis of the parameter space

Introduce the parameter space

$Z^{d}=\\{z=(w,t):w\in\mathbb{R}^{d},\ t>0\\},$

whose elements $z=(w,t)$ act on functions $f\in
L^{1}_{\mathrm{loc}}(\mathbb{R}^{d})$ functions by the formula

${\mathsf{Sy}}_{z}f=\frac{1}{t^{d}}f\left(\frac{\cdot-w}{t}\right).$

Let $\mu$ be the measure on $Z^{d}$ given by

$\int\displaylimits_{Z^{d}}F(z)\,{\rm
d}\mu(z)=\int_{0}^{\infty}\int_{\mathbb{R}^{d}}F(w,t)\,\frac{{\rm d}w\,{\rm
d}t}{t},\quad F\in{\mathrm{C}}_{0}(Z^{d}).$

Notice that $\mu$ is invariant under ${\mathsf{Sy}}_{z}$. To analyze
multilinear operators, we will use wavelets adapted to two parameters, one in
$Z^{md}$ and the other in $Z^{d}$. First, given $\mathbf{w}\in\mathbb{R}^{md}$
and $w_{0}\in\mathbb{R}^{d}$, define

$|\mathbf{w}-w_{0}|_{2}=\sqrt{\sum_{i=1}^{m}|w_{i}-w_{0}|^{2}},\quad\mathbf{w}=(w_{1},\ldots,w_{m}),\quad
w_{i}\in\mathbb{R}^{d}.$

For $\mathbf{z}=(\mathbf{w},s)\in\mathbb{Z}^{md}$ and
$\zeta=(w_{0},t)\in\mathbb{Z}^{d}$, define

(2.1)
$[\mathbf{z},\zeta]_{\delta}=\frac{\min\\{s,t\\}^{\delta}}{\max\\{t,s,|\mathbf{w}-w_{0}|_{2}\\}^{md+\delta}}.$

We will also say $\mathbf{z}\geq\zeta$ if $s\geq t$. Notice that if
$\delta\geq\delta^{\prime}$ then
$[\mathbf{z},\zeta]_{\delta}\leq[\mathbf{z},\zeta]_{\delta^{\prime}}.$ For
$M\geq 1$ and $\zeta\in Z^{d}$, introduce $\mathbb{B}_{M}^{m}(\zeta)$ which
are the following approximate balls in the hyperbolic metric,

(2.2) $\mathbb{B}_{M}^{m}(\zeta)=\left\\{\mathbf{z}\in Z^{md}:t2^{-M}\leq
s\leq t2^{M},\,\left|\mathbf{w}-w_{0}\right|_{2}\leq t2^{M}\right\\}$

when $\zeta=(0,1)$, $M=1$, or $m=1$, those parameters are omitted from the
notation. Given a function $F:Z^{d}\to\mathbb{C}$ we say
$\lim\limits_{z\to\infty}F(z)=L$ if

$\lim_{M\to\infty}\sup_{z\not\in\mathbb{B}_{M}}|F(z)-L|=0.$

###  2.2. Wavelet classes and forms

The building blocks of our representation theorem are wavelets for which we
are going to introduce notation and relevant classes as well as the averaging
lemmata from [DGW, DWW]. We denote the space of Schwartz functions by
$\mathcal{S}(\mathbb{R}^{d})$ and the mean-zero subspace

${\mathcal{S}}^{0}(\mathbb{R}^{d})=\left\\{\varphi\in{\mathcal{S}}(\mathbb{R}^{d}):\int\varphi(x)\,{\rm
d}x=0\right\\}.$

We fix a radial function $\phi\in{\mathcal{S}}^{0}(\mathbb{R}^{d})$, supported
in a ball and appropriately normalized which we will call the mother wavelet,
in which case the Calderón reproducing formula holds, namely

(2.3) $f=\int_{Z^{d}}\langle f,\phi_{z}\rangle\phi_{z}\,{\rm
d}\mu(z)\qquad\forall f\in\mathcal{S}(\mathbb{R}^{d}).$

For the convenience of the reader, we restate the setup from [DGW] on which we
will base our analysis. For $0<\delta\leq 1$ we introduce the norm on
functions $\varphi\in{\mathrm{C}}^{\delta}(\mathbb{R}^{md})$,

(2.4)
$\|\varphi\|_{\star,\delta}=\sup_{\begin{subarray}{c}x,h\in\mathbb{R}^{md}\\\
0<|h|\leq 1\end{subarray}}\langle
x\rangle^{md+\delta}\left(|\varphi(x)|+\frac{|\varphi(x+h)-\varphi(x)|}{|h|^{\delta}}\right),\quad\langle
x\rangle=\sqrt{1+|x|^{2}}.$

######  Definition 2.3.

For $z=(w,t)\in\mathbb{Z}^{d}$, the wavelet class $\Psi^{m,\delta}_{z}$ is
defined by

$\Psi^{m,\delta}_{z}=\left\\{\varphi\in\mathcal{C}^{\delta}(\mathbb{R}^{md}):\|({\mathsf{Sy}}_{\mathbf{z}})^{-1}\varphi\|_{\star,\delta}\leq
1\right\\},\quad\mathbf{z}=(w,\ldots,w,t)\in Z^{md},$

and its cancellative subclass, for $j=1,\ldots,m$ is denoted by
$\Psi^{m,\delta;j}_{z}$ and consists of $\varphi\in\Psi^{m,\delta}_{z}$ such
that

$\int_{\mathbb{R}^{d}}\varphi(x_{1},x_{2},\ldots,x_{m})\,{\rm d}x_{j}=0.$

Let $\chi_{z}$ denote the $L^{\infty}$-normalized decay factor adapted to the
parameter $z=(w,t)$,

$\chi_{z}(x)=\left\langle\frac{x-w}{t}\right\rangle^{-1}.$

With this notation, we can recast $\varphi\in\Psi^{m,\delta}_{z}$ as

(2.5)
$|\varphi|\leq\frac{1}{t^{md}}\chi_{\mathbf{z}}^{md+\delta},\quad|\varphi-\varphi(\cdot+h)|\leq\frac{|h|^{\delta}}{t^{md+\delta}}\chi_{\mathbf{z}}^{md+\delta}.$

Aiming to make this paper as self contained as possible, as well as to set the
stage for the representation in Proposition 3.9, we state the averaging
lemmata from [DGW] and a refinement of the averaging procedure in [DWW].

######  Lemma 2.4.

Let $\phi$ be the mother wavelet and $k\geq 0$. There exist functions
$\psi^{i}$, $i=1,2,3,4$, satisfying

  * (i)

${\mathrm{supp}\,}\psi^{i}\subset B(0,1)$;

  * (ii)

$\psi^{1},\psi^{3}\in{\mathrm{C}}^{k}(\mathbb{R}^{d})$;

  * (iii)

$\psi^{2},\psi^{4}\in{\mathcal{S}}^{0}(\mathbb{R}^{d})$;

  * (iv)

For any $s>0$ and $f\in{\mathcal{S}}(\mathbb{R}^{d})$,

$\int_{r\geq s}\int_{u\in\mathbb{R}^{d}}\langle
f,\phi_{u,r}\rangle\phi_{u,r}\dfrac{{\rm d}u\ {\rm
d}r}{r}=\int_{\mathbb{R}^{d}}\langle
f,\psi_{u,s}^{1}\rangle\psi_{u,s}^{2}+\langle
f,\psi_{u,s}^{3}\rangle\psi_{u,s}^{4}\,{\rm d}u.$

######  Lemma 2.5.

Let $\varphi_{j}\in{\mathcal{S}}(\mathbb{R}^{d})$ for $j=1,\ldots,m$ and
$0<\eta<\delta\leq 1$. There exists $C>0$ such that for any $H:Z^{md}\times
Z^{d}\to\mathbb{C}$ satisfying

$\left|H(\mathbf{z},\zeta)\right|\leq\left[\mathbf{z},\zeta\right]_{\delta},$

there holds

(2.6) $\nu_{\zeta}=\int_{\begin{subarray}{c}\mathbf{z}\in Z^{md}\\\
\mathbf{z}\geq\zeta\end{subarray}}H(\mathbf{z},\zeta)\left(\varphi_{1}\otimes\cdots\otimes\varphi_{m}\right)_{\mathbf{z}}{\rm
d}\mu(\mathbf{z})\in C\Psi^{m,\eta}_{\zeta}.$

Furthermore, if $\phi_{j}\in{\mathcal{S}}^{0}$, then $\nu_{\zeta}\in
C\Psi^{m,\eta;j}_{\zeta}$.

Finally, we end this section by introducing the wavelet forms, which will be
used to systematically study $m$-linear CZOs.

######  Definition 2.6.

Given a collection
$\\{\varepsilon_{z}\in\mathbb{C},\nu_{z}\in\Psi^{m,\delta;1}_{z}:z\in
Z^{d}\\}$ define the associated $(m+1)$-linear canonical wavelet form by

$\int_{Z^{d}}\varepsilon_{z}\left\langle
f_{0},\phi_{z}\right\rangle\left\langle f_{1}\otimes\ldots\otimes
f_{m},\nu_{z}\right\rangle\,{\rm d}\mu(z).$

More generally, we say $U$ is a wavelet form if $U^{\sigma}$ is a canonical
wavelet form for some $\sigma\in\mathrm{S}_{m+1}$. We say $U$ is a bounded
wavelet form if $\sup_{z\in Z^{d}}\left|\varepsilon_{z}\right|<\infty$.
Additionally, we say $U$ is a compact wavelet form if it is a bounded wavelet
form for which

$\lim_{z\to\infty}\varepsilon_{z}=0.$

Wavelet forms may be viewed as a generalization of the Calderón-Toeplitz
operators considered in [nowak1993calderon, rochberg1990toeplitz]. More
generally though, all cancellative CZ forms can be realized as wavelet forms
[DGW, DWW]. Bounded wavelet forms are bounded in the following sense. In fact,
they themselves are cancellative CZ forms so Proposition 2.8 below follows
from any number of results [grafakos-torres, lerner2019intuitive,
li2021endpoint, li2014sharp, nieraeth2019quantitative]. See also
[DGW]*Proposition 5.1 for a direct proof of the sparse $(1,\ldots,1)$ bound
for bounded wavelet forms. Let us introduce some bookkeeping to concisely
describe the full range of Lebesgue space estimates for CZOs.

######  Definition 2.7.

Let

$P=\left\\{(p_{1},\ldots,p_{m}):1<p_{j}\leq\infty\right\\}.$

We introduce the shorthand for $\vec{p}\in P$,

$L^{\vec{p}}(\mathbb{R}^{d})=\bigtimes_{j=1}^{m}L^{p_{j}}(\mathbb{R}^{d}),\quad\mathcal{B}^{\vec{p}}=\left\\{(f_{1},\ldots,f_{m})\in
L^{\vec{p}}:\left\|f_{j}\right\|_{L^{p_{j}}(\mathbb{R}^{d})}\leq
1\right\\},\quad r(\vec{p})=\left(\sum_{j=1}^{m}\frac{1}{p_{j}}\right)^{-1}.$

Then the admissible classes of Hölder tuples we consider are

$Q=\left\\{(\vec{p},r(\vec{p})):\vec{p}\in P,\ r(\vec{p})<\infty\right\\}.$

Given $(\vec{p},r)\in Q$, and an $m$-linear operator $T$, denote by
$\left\|T\right\|_{\vec{p},r}$ the operator norm from
$L^{\vec{p}}(\mathbb{R}^{d})\to L^{r}(\mathbb{R}^{d})$, i.e.

$\left\|T\right\|_{\vec{p},r}=\sup_{(f_{1},\ldots,f_{m})\in\mathcal{B}^{\vec{p}}}\left\|T(f_{1},\ldots,f_{m})\right\|_{L^{r}(\mathbb{R}^{d})}.$

Furthermore, we define the following modification at the endpoint,

$\left\|T\right\|_{\infty,\mathrm{BMO}}=\sup_{(f_{1},\ldots,f_{m})\in\mathcal{B}^{\vec{p}}}\left\|T(f_{1},\ldots,f_{m})\right\|_{\mathrm{BMO}(\mathbb{R}^{d})},\quad\vec{p}=(\infty,\ldots,\infty).$

######  Proposition 2.8.

For each $(\vec{p},r)\in Q$, there exists $C_{\vec{p},r}>0$ such that for any
bounded wavelet form $U$ and any $T\in\mathsf{T}_{U}$,

(2.7) $\left\|T\right\|_{\vec{p},r}\leq C_{\vec{p},r}\sup_{z\in
Z^{d}}\left|\varepsilon_{z}\right|.$

Furthermore, there exists $C_{\infty}>0$ such that

(2.8) $\left\|T\right\|_{\infty,\mathrm{BMO}}\leq C_{\infty}\sup_{z\in
Z^{d}}\left|\varepsilon_{z}\right|.$

##  3\. Wavelet representation of compact Calderón-Zygmund forms

Let us make rigorous the informal definitions given in the introduction.

######  Definition 3.1.

A function $K\in
L^{1}_{\mathrm{loc}}(\mathbb{R}^{(m+1)d}\setminus\\{x\in(\mathbb{R}^{d})^{m+1}:x_{0}=\cdots=x_{m}\\})$
is a $\delta$-singular integral kernel if there exist $C_{K},\delta>0$ such
that

$|K(x_{0},\ldots,x_{m})|\leq\dfrac{C_{K}}{(\sum_{j=1}^{m}|x_{0}-x_{j}|)^{md}},$
$\max_{j=0,\ldots,m}|\Delta^{j}_{h}K(x_{0},\ldots,x_{m})|\leq\dfrac{C_{K}|h|^{\delta}}{(\sum_{j=1}^{m}|x_{0}-x_{j}|)^{md+\delta}},$

were $\Delta_{h}^{j}$ denotes the difference operator in the $j$-th position.
We say $\Lambda$ is an $m$-linear singular integral form if there exists a
singular integral kernel $K$ such that for any
$f_{0},\ldots,f_{m}\in{\mathcal{S}}$ with
$\cap_{j=0}^{m}{\mathrm{supp}\,}f_{j}=\varnothing$ one has

$\Lambda(f_{0},\ldots,f_{m})=\int_{(\mathbb{R}^{d})^{m+1}}K(x)\prod_{j=0}^{m}f_{j}(x_{j})\,{\rm
d}x.$

When $m$ is understood, we simply say $\Lambda$ is a singular integral form.

We need the following function spaces in order to define Calderón-Zygmund
forms. For $f\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{d})$, define the
$\mathrm{BMO}(\mathbb{R}^{d})$ norm

$\left\|f\right\|_{\mathrm{BMO}(\mathbb{R}^{d})}=\sup_{Q\,\mathrm{cube}}\frac{1}{|Q|}\int_{Q}\left|f(x)-f_{Q}\right|\,{\rm
d}x,\quad f_{Q}=\frac{1}{|Q|}\int_{Q}f(y)\,{\rm d}y.$

Then, the functions of bounded mean oscillation (BMO) are those with finite
$\mathrm{BMO}(\mathbb{R}^{d})$ norm. Let
${\mathrm{C}}_{\mathrm{v}}(\mathbb{R}^{d})$ be the space of all continuous
functions $f$ on $\mathbb{R}^{d}$ for which $\lim_{|x|\to\infty}f(x)=0$. Then,
the Banach space of functions with continuous mean oscillation,
$\mathrm{CMO}(\mathbb{R}^{d})$, is defined to be the closure of
${\mathrm{C}}_{\mathrm{v}}(\mathbb{R}^{d})$ in the norm
$\left\|\cdot\right\|_{\mathrm{BMO}(\mathbb{R}^{d})}$. Furthermore,
$\mathrm{BMO}(\mathbb{R}^{d})$ becomes a a Banach space upon identifying
functions which differ by a constant.

######  Definition 3.2.

We say a singular integral form $\Lambda$ is a Calderón-Zygmund (CZ) form if
there exists $C_{\mathsf{W}}$ such that

$t^{md}\left|\Lambda({\mathsf{Sy}}_{z}\varphi_{0},\ldots,{\mathsf{Sy}}_{z}\varphi_{m})\right|\leq
C_{\mathsf{W}},\quad\forall\varphi_{j}\in{\mathrm{C}}^{\infty}_{0}(B(0,1))\cap\Psi^{1,1}_{(0,1)},$

and furthermore, there exist $\mathfrak{b}_{j}\in\mathrm{BMO}(\mathbb{R}^{d})$
such that (1.2) holds. The rigorous definition of (1.2) is as follows. Let
$\theta\in{\mathrm{C}}^{\infty}_{0}(\mathbb{R}^{d})$ with $\theta=1$ near the
origin. Then, for $t>0$, set $\theta_{t}=\theta(\cdot t)$. We say (1.2) holds,
and sometimes use the language $T^{*,j}(1,\ldots,1)=\mathfrak{b}_{j}$, if for
all $\varphi\in{\mathcal{S}}^{0}(\mathbb{R}^{d})$,

(3.1) $\lim_{t\to
0}\Lambda^{*,j}(\varphi,\theta_{t},\theta_{t},\ldots,\theta_{t})=\left\langle\varphi,\mathfrak{b}_{j}\right\rangle.$

Furthermore, let $\phi$ be a mother wavelet and $\psi^{2},\psi^{4}$ be the
Schwartz functions from Lemma 2.4. For $j=1,\ldots,m$ define

(3.2)
$\vec{\psi}_{j}=\left(\bigotimes_{i=1}^{j-1}\psi^{2}\right)\otimes\left(\bigotimes_{i=j}^{m-1}\psi^{4}\right)\otimes\phi.$

For each $M\geq 1$, define

(3.3)
$\mathsf{W}_{\Lambda}^{M}(\zeta)=\sup_{\mathbf{z}\in\mathbb{B}^{m}_{M}(\zeta)}\sup_{\begin{subarray}{c}\sigma\in\mathrm{S}_{m+1}\\\
j=1,\ldots,m\end{subarray}}\left|\Lambda^{\sigma}\left((\vec{\psi}_{j})_{\mathbf{z}},\phi_{\zeta}\right)\right|t^{md},\quad\zeta=(w,t).$

A CZ form $\Lambda$ is said to be a compact CZ form if for some (all) $M\geq
1$

$\mathfrak{b}_{j}\in\mathrm{CMO}(\mathbb{R}^{d}),\quad\lim\limits_{\zeta\to\infty}\mathsf{W}_{\Lambda}^{M}(\zeta)=0.$

Finally, we say a CZ form is cancellative if $\mathfrak{b}_{j}=0$.

######  Remark 3.3.

It is not too difficult to see that if $\mathsf{W}_{\Lambda}^{M}\to 0$ for
some $M$ then the same holds for each $M$. Furthermore, since
${\mathcal{S}}^{0}(\mathbb{R}^{d})$ is dense in the Hardy space
$H^{1}(\mathbb{R}^{d})$, which is the dual space of
$\mathrm{BMO}(\mathbb{R}^{d})$, (3.1) should be interpreted as a weak limit,
i.e. $T^{*,j}(\theta_{t},\ldots,\theta_{t})$ converges weakly to
$\mathfrak{b}_{j}$ in $\mathrm{BMO}(\mathbb{R}^{d})$.

Now we justify our description of such forms as compact.

######  Definition 3.4.

Let $T$ be an $m$-linear CZO. We say $T$ is a compact CZO if for each
$(\vec{p},r)\in Q$, $T(\mathcal{B}^{\vec{p}})$ is precompact in
$L^{r}(\mathbb{R}^{d})$, and at the upper endpoint,
$T(\mathcal{B}^{\infty}_{0})$ is precompact in $\mathrm{CMO}(\mathbb{R}^{d})$,
where

$\mathcal{B}^{\infty}_{0}=\left\\{(f_{1},\ldots,f_{m})\in\mathcal{B}^{(\infty,\ldots,\infty)}:f_{j}\
\mathrm{compactly}\ \mathrm{supported}\right\\}.$

######  Remark 3.5.

The classical definition of compactness of an abstract $m$-linear operator on
quasi-normed spaces [BT13, COY] is that it maps bounded sets to precompact
sets. In Definition 3.4, we are imposing this definition of compactness on $T$
acting from $L^{\vec{p}}(\mathbb{R}^{d})\to L^{r}(\mathbb{R}^{d})$ for all
$(\vec{p},r)$ in $Q$ and at the upper endpoint because our testing conditions
allow us to conclude compactness in this full range. It is tempting to only
require $T:L^{\vec{p}}(\mathbb{R}^{d})\to L^{r}(\mathbb{R}^{d})$ be compact
for a single $(\vec{p},r)$, but the current state of multilinear extrapolation
of compactness [COY, HL2] does not include the endpoints.

######  Proposition 3.6.

Let $\Lambda$ be an $(m+1)$-linear CZ form such that each
$T\in\mathsf{T}_{\Lambda}$ is a compact CZO. Then, each
$\mathfrak{b}_{j}\in\mathrm{CMO}(\mathbb{R}^{d})$, $j=0,\ldots,m$, and for
each $M\geq 1$,

(3.4) $\lim_{\zeta\to\infty}\mathsf{W}_{\Lambda}^{M}(\zeta)=0.$

###### Proof.

To prove the first conclusion, since $\Lambda$ is a CZ form, (3.1) holds. Let
us fix $j=0,\ldots,m$. Up to an absolute constant,
$(\theta_{t},\ldots,\theta_{t})\in\mathcal{B}^{\infty}_{0}$, so there exists
$t_{n}\to 0$ such that $T^{*,j}(\theta_{t_{n}},\ldots,\theta_{t_{n}})$
converges in $\mathrm{CMO}(\mathbb{R}^{d})$. However, the second statement in
Remark 3.3 requires this limit to coincide with $\mathfrak{b}_{j}$, and the
first part is proved. To prove the second conclusion, let us suppose, towards
a contradiction, that there exists $M\geq 1$ such that (3.4) does not hold.
Therefore we can find $\sigma\in\mathrm{S}_{m+1}$,
$\vec{\psi}\in{\mathcal{S}}(\mathbb{R}^{d})^{m}$, $\varepsilon>0$ and
sequences $\zeta_{n}=(w_{n},t_{n})\in Z^{d}$ and $\mathbf{z}_{n}\in Z^{md}$
such that

(3.5)
$\zeta_{n}\to\infty,\quad\mathbf{z}_{n}\in\mathbb{B}^{m}_{M}(\zeta_{n}),\quad\left|\Lambda^{\sigma}\left(\phi_{n},\vec{\psi}_{n}\right)\right|\geq\varepsilon,\quad\phi_{n}=t_{n}^{\frac{d}{r}}\phi_{\zeta_{n}},\quad\vec{\psi}_{n}=t_{n}^{d(m-\frac{1}{r})}\vec{\psi}_{\mathbf{z}_{n}},$

and $r>1$. Let us suppose, for simplicity that $\sigma$ is the identity. If
not, then in what follows we would replace $T\vec{\psi}_{n}$ by
$T^{*,\sigma(0)}(\vec{\psi}^{\sigma^{\prime}}_{n})$ where
$\vec{\psi}^{\sigma^{\prime}}_{n}$ is an appropriate permutation of the $m$
elements of $\vec{\psi}_{n}$. Since $\\{\vec{\psi}_{n}\\}_{n\in\mathbb{N}}$ is
a bounded sequence in $L^{\vec{p}}(\mathbb{R}^{d})$, for some $\vec{p}\in P$
with $r(\vec{p})=r$, by the compactness of $T$ we find a subsequence so that
$\\{T\vec{\psi}_{n_{k}}\\}_{k\in\mathbb{N}}$ is convergent in
$L^{r}(\mathbb{R}^{d})$. Finally, it is not hard to check that since
$\zeta_{n}\to\infty$, $\phi_{n}$ converges weakly to zero in
$L^{\frac{r}{r-1}}(\mathbb{R}^{d})$ which, since $r>1$, is the dual space of
$L^{r}(\mathbb{R}^{d})$. Therefore,

$\Lambda^{\sigma}(\phi_{n_{k}},\vec{\psi}_{n_{k}})=\left\langle\phi_{n_{k}},T\vec{\psi}_{n_{k}}\right\rangle\to
0,\,\mbox{as}\,k\to\infty,$

which contradicts (3.5). ∎

The main step in the representation theorem is given in Proposition 3.8 below.
There we will show that a compact cancellative CZ form enjoys additional decay
in the wavelet basis. For a general cancellative CZ form, we recall the
following lemma from [DGW]*Lemma 3.3 regarding its decay when applied to a
$(m+1)$-tuple of wavelets. When we want to specify the value of the smoothness
parameter $\delta>0$ from Definition 3.1, we say $\Lambda$ is a $\delta$-CZ
form.

######  Lemma 3.7.

Let $\Lambda$ be a cancellative $\delta$-CZ form, $\eta\in(0,\delta)$, and
$\psi_{j}\in{\mathrm{C}}^{\infty}_{0}(B(0,1))$, $j=1,\ldots,m-1$. Then, there
exists $C>0$ such that for all $\sigma\in\mathrm{S}_{m+1}$, $\zeta\in Z^{d}$,
and $\mathbf{z}\in Z^{md}$ with $\mathbf{z}\geq\zeta$,

$\left|\Lambda^{\sigma}\left(\vec{\psi}_{\mathbf{z}},\phi_{\zeta}\right)\right|\leq
C\left[\mathbf{z},\zeta\right]_{\eta},\quad\vec{\psi}=\psi_{1}\otimes\cdots\otimes\psi_{m-1}\otimes\phi.$

######  Proposition 3.8.

Let $\Lambda$ be a cancellative compact $\delta$-CZ form and
$\eta\in(0,\delta)$. Let $\phi$ be the mother wavelet and let $\vec{\psi}_{j}$
be defined as in (3.2). Then there exists $C>0$ and
$\left\\{\varepsilon_{\zeta}\in\mathbb{C}:\zeta\in Z^{d}\right\\}$ with the
properties that

(3.6) $\sup_{\zeta\in Z^{d}}\left|\varepsilon_{\zeta}\right|\leq
C,\quad\lim_{\zeta\to\infty}\left|\varepsilon_{\zeta}\right|=0,\quad\max_{\begin{subarray}{c}\sigma\in\mathrm{S}_{m+1}\\\
j=1,\ldots,m\end{subarray}}\left|\Lambda^{\sigma}\left((\vec{\psi}_{j})_{\mathbf{z}},\phi_{\zeta}\right)\right|\leq\left|\varepsilon_{\zeta}\right|\left[\mathbf{z},\zeta\right]_{\eta},\quad\mathbf{z}\geq\zeta.$

###### Proof.

By symmetry, we can assume $\sigma$ is the identity element and $j=1$.
Furthermore, set $\vec{\psi}=\vec{\psi}_{1}$ to declutter the notation. For
each $n\in\mathbb{N}$, we will construct
$\left\\{\varepsilon_{\zeta}^{n}\in\mathbb{C}:\zeta\in Z^{d}\right\\}$ and
$\rho>0$ with the properties that

$\displaystyle\sup_{\zeta\in Z^{d}}|\varepsilon_{\zeta}^{n}|\lesssim
1,\quad\lim_{\zeta\to\infty}|\varepsilon_{\zeta}^{n}|=0,$
$\displaystyle\left|\Lambda\left(\vec{\psi}_{\mathbf{z}},\phi_{\zeta}\right)\right|\lesssim
2^{-\rho
n}\left|\varepsilon^{n}_{\zeta}\right|\left[\mathbf{z},\zeta\right]_{\eta},\quad\mathbf{z}\in\left\\{\begin{array}[]{ll}\mathbb{B}_{1}^{m}(\zeta)&n=1,\\\
\mathbb{B}_{n}^{m}(\zeta)\setminus\mathbb{B}_{{n-1}}^{m}(\zeta)&n\geq
2,\end{array}\right.\quad\mathbf{z}\geq\zeta.$

Assuming we have such $\varepsilon_{\zeta}^{n}$, define

$\varepsilon_{\zeta}=\sum_{n=1}^{\infty}2^{-\rho n}\varepsilon^{n}_{\zeta}.$

The first and third properties of $\varepsilon_{\zeta}$ in (3.6) are immediate
and the second follows by Lebesgue’s dominated convergence theorem since each
$\varepsilon^{n}_{\zeta}$ approaches zero. Now, to construct
$\varepsilon_{\zeta}^{n}$, let $\delta_{j}>0$ such that
$\eta<\delta_{1}<\delta_{2}<\delta$. By Lemma 3.7 and the definition of
$\mathsf{W}_{\Lambda}^{M}$ from (3.3), we have for any $\theta\in(0,1)$,

$\left|\Lambda\left(\vec{\psi}_{\mathbf{z}},\phi_{\zeta}\right)\right|\lesssim\left(t^{-md}\right)^{1-\theta}\left[\mathbf{z},\zeta\right]_{\delta_{2}}^{\theta}\mathsf{W}_{\Lambda}^{n}(\zeta)^{1-\theta},\quad\mathbf{z}\in\mathbb{B}^{m}_{n}(\zeta),\
\mathbf{z}\geq\zeta.$

We choose $\theta$ close enough to $1$ that $md\theta+\delta_{2}\theta-
md=\delta_{1}$. With this specific choice of $\theta$, one can easily verify
that
$\left(t^{-md}\right)^{1-\theta}[\mathbf{z},\zeta]_{\delta_{2}}^{\theta}=[\mathbf{z},\zeta]_{\delta_{1}}$.
Setting $\varepsilon_{z}^{1}=\mathsf{W}^{1}_{\Lambda}(\zeta)^{1-\theta}$
handles the case $n=1$. For $n\geq 2$, we factor, with $\rho=\delta_{1}-\eta$,

$\left[\mathbf{z},\zeta\right]_{\delta_{1}}=\left(\frac{t}{\max\left\\{s,|\mathbf{w}-w_{0}|\right\\}}\right)^{\rho}\left[\mathbf{z},\zeta\right]_{\eta}.$

Therefore, one only needs to verify that for
$\mathbf{z}\in\mathbb{B}^{m}_{n}(\zeta)\setminus\mathbb{B}^{m}_{{n-1}}(\zeta)$,
the first factor is comparable to $2^{-\rho n}$. The proof is concluded by
setting $\varepsilon^{n}_{\zeta}=\mathsf{W}^{n}_{\Lambda}(\zeta)$. ∎

######  Proposition 3.9.

Every compact cancellative CZ form is a finite sum of compact wavelet forms.

###### Proof.

We recycle the proof of the representation theorem in [DGW] relying on Lemma
2.4 and Lemma 2.5. First expand each $f_{j}$ using (2.3) to obtain

$\Lambda(f_{0},\ldots,f_{m})=\int_{(Z^{d})^{m+1}}\Lambda(\phi_{z_{0}},\ldots,\phi_{z_{m}})\left\langle
f_{0},\phi_{z_{0}}\right\rangle\,{\rm d}\mu(z_{0})\ldots\left\langle
f_{m},\phi_{z_{m}}\right\rangle\,{\rm d}\mu(z_{m}).$

Split the integration region into $m(m+1)$ components defined by

$\\{(z_{0},\ldots,z_{m})\in(Z^{d})^{m+1}:z_{i}\geq z_{j}\geq z_{k},\,i\neq
j,k\\},\quad k=0,\ldots,m,\,j=0,\ldots,k-1,k+1,\ldots,m.$

For each $j,k$ and each $i\neq j,k$, we apply Lemma 2.4 to $\left\langle
f_{i},\phi_{z_{i}}\right\rangle\phi_{z_{i}}$. Furthermore, setting
$f^{\sigma}=\bigotimes_{j=0}^{m}f_{\sigma(j)}$ and relabelling the variables,
we obtain

$\Lambda(f_{0},\ldots,f_{m})=\sum\int_{\zeta\in
Z^{d}}\int_{\begin{subarray}{c}\mathbf{z}\in Z^{md}\\\
\mathbf{z}\geq\zeta\end{subarray}}\Lambda^{\sigma}((\vec{\psi}_{\mathrm{e}})_{\mathbf{z}},\phi_{\zeta})\left\langle
f^{\sigma},(\vec{\psi}_{\mathrm{o}})_{\mathbf{z}}\otimes\phi_{\zeta}\right\rangle{\rm
d}\mu(\mathbf{z})\,{\rm d}\mu(\zeta),$

where the sum is taken over all $\sigma\in\mathrm{S}_{m+1}$ and over the
combinations $\vec{\psi}_{\mathrm{e}}$ is of the form (3.2) for some
$j=1,\ldots,m$ and $\vec{\psi}_{\mathrm{o}}$ is of the same form but with
$\psi^{2}$ and $\psi^{4}$ replaced by $\psi^{1}$ and $\psi^{3}$ from Lemma
2.4. Each of these summands (of which there are only finitely many depending
on $m$), will now be converted to a compact wavelet form by Proposition 3.8
and Lemma 2.5. Fix now one $\sigma$, $\vec{\psi}_{\mathrm{e}}$, and
$\vec{\psi}_{\mathrm{o}}$. We define

$\vartheta_{\zeta}=\int_{\begin{subarray}{c}\mathbf{z}\in Z^{md}\\\
\mathbf{z}\geq\zeta\end{subarray}}\Lambda^{\sigma}((\vec{\psi}_{\mathrm{e}})_{\mathbf{z}},\phi_{\zeta})\vec{\psi}_{\mathrm{o}}\,d\mu(\mathbf{z})$

and the result will be proved if we can show
$\vartheta_{\zeta}=\varepsilon_{\zeta}\nu_{\zeta}$ for some
$\nu_{\zeta}\in\Psi^{m,\delta;m}_{\zeta}$ and $\varepsilon_{\zeta}$
approaching zero as $\zeta\to\infty$. Let $\varepsilon_{\zeta}$ be the
collection provided by Proposition 3.8. In particular, the third property in
(3.6) guarantees

$\Lambda^{\sigma}((\vec{\psi}_{\mathrm{e}})_{\mathbf{z}},\phi_{\zeta})=\varepsilon_{\zeta}H(\mathbf{z},\zeta),\quad\left|H(\mathbf{z},\zeta)\right|\leq[\mathbf{z},\zeta]_{\eta},$

whence the proof is concluded by Lemma 2.5 and recalling that the $m$-th
component of $\vec{\psi}_{\mathrm{o}}$ is the mother wavelet and thus belongs
to ${\mathcal{S}}^{0}(\mathbb{R}^{d})$. ∎

##  4\. Compact wavelet forms

In this section we will prove the compact analogue of Proposition 2.8.

######  Proposition 4.1.

Let $U$ be a compact wavelet form. Then each $T\in\mathsf{T}_{U}$ is a compact
CZO.

###### Proof.

Let us fix a compact wavelet form $U$, $T\in\mathsf{T}_{U}$, and
$(\vec{p},r)\in Q$. $T(f_{1},\ldots,f_{m})(x)$ is either of the form

(4.1) $\int_{Z^{d}}\varepsilon_{z}\left\langle f_{1}\otimes\cdots\otimes
f_{m},\nu_{z}\right\rangle\phi_{z}(x)\,{\rm
d}\mu(z)\quad\mathrm{or}\,\int_{Z^{d}}\varepsilon_{z}\left\langle
f_{\sigma(1)}\otimes\cdots\otimes
f_{\sigma(m)},\nu_{z}(x,\cdot)\otimes\phi_{z}\right\rangle\,{\rm d}\mu(z)$

for some $\sigma\in\mathrm{S}_{m}$ and $\nu_{z}\in\Psi^{m,\delta;j}_{z}$; by
$\nu_{z}(x,\cdot)$ we mean for each $x$ it returns the function
$(x_{2},\ldots,x_{m})\mapsto\nu_{z}(x,x_{2},\ldots,x_{m})$. We will only
handle the second case, and we will reduce, by symmetry, to the case where
$\sigma$ is the identity. The first case in (4.1) is simpler, though in fact
they are handled in exactly the same fashion. Let $\rho>0$ and split the
integral defining $T$ over $\mathbb{B}_{M}$ and
$\mathbb{Z}^{d}\setminus\mathbb{B}_{M}$ where $M$ is chosen large enough that
$|\varepsilon_{z}|\leq\rho$ for $z\not\in\mathbb{B}_{M}$. Therefore, the
operator norm of the second component is, by (2.7) controlled by $\rho$. The
proof will be concluded if we can show $R_{\rho}$ defined by

$R_{\rho}\mathbf{f}(x)=\int_{\mathbb{B}_{M}}\left\langle
f_{1}\otimes\cdots\otimes
f_{m},\nu_{z}(x,\cdot)\otimes\phi_{z}\right\rangle\,{\rm
d}\mu(z),\quad\mathbf{f}=(f_{1},\ldots,f_{m}),$

is compact. By the Riesz-Kolmogorov compactness criteria (see e.g. [MSWW] and
[Tsuji] for the case $0<r<1$), we need to prove that

(4.2)
$\displaystyle\lim_{N\to\infty}\sup_{\mathbf{f}\in\mathcal{B}^{\vec{p}}}\int_{|x|>N}|R_{\rho}\mathbf{f}(x)|^{r}{\rm
d}x=0,$ (4.3) $\displaystyle\lim_{h\to
0}\sup_{\mathbf{f}\in\mathcal{B}^{\vec{p}}}\int_{\mathbb{R}^{d}}|R_{\rho}\mathbf{f}(x+h)-R_{\rho}\mathbf{f}(x)|^{r}{\rm
d}x=0.$

To this end we will give suitable pointwise bounds on the operator $R_{\rho}$
and the differences induced by it. Since $M$ is fixed, we will crucially use
that $z=(w,t)\in\mathbb{B}_{M}$ satisfies

(4.4) $t\sim 1,\quad|w|\lesssim 1,\quad\chi_{z}\lesssim\chi,$

where $\chi=\chi_{(0,1)}$ while $\sim$ and $\lesssim$ now denote comparability
with constants depending on $M$. Let us now note the preliminary trivial
bounds that can be obtained via applying Hölder’s inequality and (4.4). To
this end, introduce, for $j=1,\ldots,m-1$,

$\lambda_{0}=\frac{d}{r}+\eta,\quad\lambda_{j}=\frac{d}{p_{j}^{\prime}}+\eta,\quad\eta=\frac{1}{m}\left(\frac{d}{p_{m}^{\prime}}+\delta\right)>0.$

It is easy to check that $md+\delta=\sum_{j=0}^{m-1}\lambda_{j}$,
$\lambda_{0}r>d$, and $\lambda_{j}p_{j}^{\prime}>d$, thus for
$z\in\mathbb{B}_{M}$,

(4.5) $\displaystyle\left|\langle
f_{m},\phi_{z}\rangle\right|\leq\left\|f_{m}\right\|_{p_{m}}\left\|\phi_{z}\right\|_{p_{m}^{\prime}}\lesssim
1$ $\displaystyle\left|\langle v_{z}(x,\cdot),f_{1}\otimes\ldots\otimes
f_{m-1}\rangle\right|\leq\frac{1}{t^{d}}\chi_{z}(x)^{\lambda_{0}}\prod_{j=1}^{m-1}\left\|f_{j}\right\|_{p_{j}}\left\|\frac{1}{t^{d}}\chi_{z}^{\lambda_{j}}\right\|_{p_{j}^{\prime}}\lesssim\chi(x)^{\lambda_{0}}.$

With these estimates in hand we have that
$|R_{\rho}\mathbf{f}|\lesssim\chi^{\lambda_{0}}$ and therefore (4.2) holds
since $\lambda_{0}r>d$. In the same way, we have that

$\left|\langle v_{z}(x,\cdot),f_{1}\otimes\ldots\otimes f_{m-1}\rangle-\langle
v_{z}(x+h,\cdot),f_{1}\otimes\ldots\otimes
f_{m-1}\rangle\right|\lesssim|h|^{\delta}\chi(x)^{\lambda_{0}}$

and therefore
$|R_{\rho}\mathbf{f}(x)-R_{\rho}\mathbf{f}(x-h)|\lesssim|h|^{\delta}\chi(x)^{\lambda_{0}}$
from which (4.3) now follows. It remains to handle the endpoint case. We claim
it suffices to establish that for each sequence
$\\{\mathbf{f}_{n}\\}_{n\in\mathbb{N}}\subset{\mathcal{B}}^{\infty}_{0}$,
$R_{\rho}\mathbf{f}_{n}$ has a convergent subsequence in
$\mathrm{CMO}(\mathbb{R}^{d})$. Indeed, given such an
$\\{\mathbf{f}_{n}\\}_{n\in\mathbb{N}}$, by a diagonalization argument, we may
extract a subsequence $\\{\mathbf{f}_{n_{k}}\\}_{k\in\mathbb{N}}$ such that
for each $\rho_{n}=\frac{1}{n}$,
$\\{R_{\rho_{n}}\mathbf{f}_{n_{k}}\\}_{k\in\mathbb{N}}$ is Cauchy in
$\mathrm{CMO}(\mathbb{R}^{d})$. Then, for any $\epsilon>0$ pick $n$ large
enough that

$\left\|T-R_{\rho_{n}}\right\|_{\infty,\mathrm{BMO}}<\frac{\epsilon}{3}.$

Such an $n$ exists by the condition $\varepsilon_{z}\to 0$ and (2.8). Then,
pick $N$ large enough that for all $i,k\geq N$,
$\left\|R_{\rho_{n}}\mathbf{f}_{n_{i}}-R_{\rho_{n}}\mathbf{f}_{n_{k}}\right\|_{\mathrm{BMO}(\mathbb{R}^{d})}<\frac{\epsilon}{3}$.
Therefore, by the triangle inequality,

$\left\|T\mathbf{f}_{n_{k}}-T\mathbf{f}_{n_{i}}\right\|_{\mathrm{BMO}(\mathbb{R}^{d})}<\epsilon$

whence $\\{T\mathbf{f}_{n_{k}}\\}_{k\in\mathbb{N}}$ is Cauchy and has a limit
in $\mathrm{CMO}(\mathbb{R}^{d})$. Now it remains to show each $R_{\rho}$ is
compact. Applying (4.5) with all $p_{j}=\infty$ implies the same pointwise
estimates as above, which implies
$R_{\rho}:{\mathcal{B}}^{\infty}_{0}\to{\mathrm{C}}_{\mathrm{v}}(\mathbb{R}^{d})$
and that the family
$\\{R_{\rho}\mathbf{f}:\mathbf{f}\in{\mathcal{B}}^{\infty}_{0}\\}$ is
equicontinuous. Therefore, by the Arzela-Ascoli theorem and a diagonalization
argument, given a sequence
$\\{\mathbf{f}_{n}\\}_{n\in\mathbb{N}}\in{\mathcal{B}}^{\infty}_{0}$, we can
obtain a subsequence such that
$\\{R_{\rho}\mathbf{f}_{n_{k}}\\}_{k\in\mathbb{N}}$ is Cauchy in
$\left\|\cdot\right\|_{L^{\infty}([-n,n]^{d})}$ for each $n\in\mathbb{N}$.
However, the pointwise estimate for $R_{\rho}\mathbf{f}$ shows that given
$\epsilon>0$ we can find $n$ large enough that
$\left|R_{\rho}\mathbf{f}_{n_{k}}(x)\right|<\frac{\epsilon}{3}$ for $x$
outside $[-n,n]^{d}$. Combining these two facts with the triangle inequality
shows that $\\{R_{\rho}\mathbf{f}_{n_{k}}\\}_{k\in\mathbb{N}}$ is Cauchy in
$\left\|\cdot\right\|_{L^{\infty}}$ which is a stronger norm than
$\left\|\cdot\right\|_{\mathrm{BMO}(\mathbb{R}^{d})}$. Finally, recalling that
$R_{\rho}\mathbf{f}_{n_{k}}\in{\mathrm{C}}_{\mathrm{v}}(\mathbb{R}^{d})$
establishes that the limit belongs to $\mathrm{CMO}(\mathbb{R}^{d})$. ∎

##  5\. Compactness of paraproducts

In this section we will deal with the compactness of the paraproducts that
arise from our representation theorem. Specifically, we will prove that the
membership of the symbols in $\mathrm{CMO}(\mathbb{R}^{d})$ is sufficient for
the compactness of the associated paraproduct by giving an essential norm
estimate. Given $(\vec{p},r)\in Q$, and an $m$-linear operator $T$, define the
essential norm

$\left\|T\right\|_{\mathrm{ess}(\vec{p},r)}=\inf_{K\,\mathrm{compact}}\left\|T-K\right\|_{\vec{p},r},$

and the natural modification at the endpoint which we denote by
$\left\|\cdot\right\|_{\mathrm{ess}(\infty,\mathrm{BMO})}$ where the operator
norm of $T-K$ is measured in $\left\|\cdot\right\|_{\infty,\mathrm{BMO}}$.

######  Definition 5.1.

Let $\vartheta\in C^{\infty}_{0}(B(0,1))$ with $\int\vartheta(x)\,{\rm d}x=1$.
For each $z\in Z^{d}$, set $\vartheta_{z}={\mathsf{Sy}}_{z}\vartheta$. Given
$\mathfrak{b}\in\mathrm{BMO}(\mathbb{R}^{d})$, define the $(m+1)$-linear form
$\Pi_{\mathfrak{b}}$ by

$\Pi_{\mathfrak{b}}(f_{0},f_{1},\ldots,f_{m})=\int_{Z^{d}}\left\langle\mathfrak{b},\phi_{z}\right\rangle\left\langle
f_{0},\phi_{z}\right\rangle\prod_{j=1}^{m}\left\langle
f_{j},\vartheta_{z}\right\rangle\,{\rm d}\mu(z).$

$\Pi_{b}$ is called a paraproduct form with symbol $\mathfrak{b}$. Any
$m$-linear operator in $\mathsf{T}_{\Pi_{b}}$ is called a paraproduct with
symbol $\mathfrak{b}$, which we denote by $S_{\mathfrak{b}}$.

The main goal of this section is to prove the following.

######  Proposition 5.2.

If $\mathfrak{b}\in\mathrm{CMO}(\mathbb{R}^{d})$, then any paraproduct
$S_{\mathfrak{b}}$ is a compact CZO.

Before proving this, let us review the standard boundedness theory of
paraproducts, analogous to Proposition 2.8 for wavelet forms. To do so, it is
convenient to view $\Pi_{\mathfrak{b}}$ as an $(m+2)$-linear form, where the
extra input function is $\mathfrak{b}$ itself. In fact, in this way
$\Pi_{\mathfrak{b}}(f_{0},\ldots,f_{m})=U(\mathfrak{b},f_{0},\ldots,f_{m})$
where $U$ is a canonical $(m+2)$-linear wavelet form with
$\left|\varepsilon_{z}\right|\lesssim 1$. Such forms are cancellative in the
first and second positions, for which a slight strengthening of (2.7) and
(2.8) holds:

(5.1) $\displaystyle T\in\mathsf{T}_{V_{\mathfrak{b}}},\quad
V_{\mathfrak{b}}(f_{0},\ldots,f_{m})=U(\mathfrak{b},f_{0},\ldots,f_{m}),$
$\displaystyle\left\|T\right\|_{\vec{p},r}\leq
C_{\vec{p},r}\left\|\mathfrak{b}\right\|_{\mathrm{BMO}(\mathbb{R}^{d})},\quad\left\|T\right\|_{\infty,\mathrm{BMO}}\leq
C_{\infty}\left\|\mathfrak{b}\right\|_{\mathrm{BMO}(\mathbb{R}^{d})},\quad(\vec{p},r)\in
Q.$

In particular, (5.1) applies to $T=S_{\mathfrak{b}}$. A proof of (5.1) is
omitted since it follows from standard considerations; see e.g. the proofs and
comments following Propositions 2.5 and 2.7 in [DWW].

Now, to give a description of $\mathrm{CMO}(\mathbb{R}^{d})$ which is more
amenable to $S_{\mathfrak{b}}$, let us introduce an orthonormal wavelet system
$\left\\{\psi_{I}\right\\}_{I\in{\mathcal{D}}}$. Here ${\mathcal{D}}$ is a
dyadic grid on $\mathbb{R}^{d}$ and for each $I\in{\mathcal{D}}$, set
$\zeta(I)=(c(I),\ell(I))\in Z^{d}$, where $c(I)$ is center of the cube $I$ and
$\ell(I)$ the side length. Then $\psi_{I}={\mathsf{Sy}}_{\zeta(I)}\psi$ for a
specific $\psi\in C\Psi^{1,1;1}_{(0,1)}$. We have kept $\psi_{I}$ to be
$L^{1}$-normalized, so the reproducing formula is

$f=\sum_{I\in{\mathcal{D}}}|I|\left\langle f,\psi_{I}\right\rangle\psi_{I}.$

Now introduce the family of orthogonal projections for $M\geq 1$,

$P_{M}f=\sum_{I\in{\mathcal{D}}_{M}}|I|\langle
f,\psi_{I}\rangle\psi_{I},\quad\mathcal{D}_{M}\coloneqq\left\\{I\in{\mathcal{D}}:\zeta(I)\in\mathbb{B}_{M}\right\\},\quad
P_{M}^{\perp}=\mathrm{Id}-P_{M}.$

From [Vil15]*Lemma 2.20, an equivalent characterization of
$\mathfrak{b}\in\mathrm{CMO}(\mathbb{R}^{d})$ is that

(5.2)
$\lim_{M\to\infty}\left\|P^{\perp}_{M}\mathfrak{b}\right\|_{\mathrm{BMO}(\mathbb{R}^{d})}=0.$

Therefore, Proposition 5.2 will be a consequence of the following essential
norm estimate.

######  Proposition 5.3.

Let $S_{\mathfrak{b}}$ be a paraproduct with symbol
$\mathfrak{b}\in\mathrm{BMO}(\mathbb{R}^{d})$. Then for each $(\vec{p},r)\in
Q\cup\\{(\infty,\mathrm{BMO})\\}$,

(5.3)
$\left\|S_{\mathfrak{b}}\right\|_{\mathrm{ess}(\vec{p},r)}\lesssim\liminf_{M\to\infty}\left\|P_{M}^{\perp}\mathfrak{b}\right\|_{\mathrm{BMO}(\mathbb{R}^{d})}.$

###### Proof.

Let $M\geq 1$ large, and perform the splitting $S_{\mathfrak{b}}=R_{M}+T_{M}$
where

$T_{M}(f_{1},\ldots,f_{m})=\int_{Z^{d}\setminus\mathbb{B}_{100M}}\left\langle\mathfrak{b},\phi_{z}\right\rangle\prod_{j=1}^{m}\left\langle
f_{j},\vartheta_{z}\right\rangle\phi_{z}\,{\rm d}\mu(z),$

and $R_{M}$ is same but the integration is taken over $\mathbb{B}_{100M}$. The
operator $R_{M}$ is clearly compact by repeating the discussion made in §4 to
show that $R_{\rho}$ there was compact. Therefore the essential norm of
$S_{\mathfrak{b}}$ is controlled by the operator norm of $T_{M}.$ Now, for
$z\not\in\mathbb{B}_{100M},$ we can calculate the pairings appearing in the
above equation by expanding $\mathfrak{b}$ with the aid of the wavelet basis
$\left\\{\psi_{I}\right\\}_{I\in{\mathcal{D}}}$:

$\langle\mathfrak{b},\phi_{z}\rangle=\langle
P_{M}\mathfrak{b},\phi_{z}\rangle+\langle
P_{M}^{\perp}\mathfrak{b},\phi_{z}\rangle.$

For the first term we will employ the linear wavelet averaging process from
Lemma 2.5. We expand

$\langle
P_{M}\mathfrak{b},\phi_{z}\rangle=\sum_{I\in{\mathcal{D}}_{M}}|I|\langle\mathfrak{b},\psi_{I}\rangle\langle\psi_{I},\phi_{z}\rangle=\left\langle\mathfrak{b},\sum_{I\in{\mathcal{D}}_{M}}|I|\langle\psi_{I}^{1},\phi_{z}\rangle\psi_{I}^{1}\right\rangle.$

Since $\psi_{I}\in\Psi^{1,1;1}_{\zeta(I)}$ and $\phi_{z}\in\Psi^{1,1;1}_{z}$,
one can compute
$\left|\langle\psi_{I},\phi_{z}\rangle\right|\lesssim[z,\zeta(I)]_{\frac{1}{2}}$
(see e.g. [DWW]*Lemma 2.3 or [FJW]*Appendix, Lemmata 2 and 4). Furthermore,
since $\zeta(I)\in\mathbb{B}_{M}$ and $z\not\in\mathbb{B}_{100M}$,
$[z,\zeta]_{\frac{1}{2}}\lesssim M^{-\frac{1}{4}}[z,\zeta]_{\frac{1}{4}}$.
Now, to apply Lemma 2.5, rewrite

$\sum_{I\in{\mathcal{D}}_{M}}|I|\langle\psi_{I},\phi_{z}\rangle\psi_{I}=\int_{Z^{d}}H(\zeta,z)\tilde{\psi}_{\zeta}\,{\rm
d}\mu(\zeta),\quad$

where for each $I\in{\mathcal{D}}_{M}$ and $\zeta=(w,t)\in
I\times(\tfrac{\ell(I)}{2},\ell(I)]$, we define

$\tilde{\psi}_{\zeta}=\psi_{I},\quad
H(\zeta,z)=\frac{|I|}{\mu(I\times(\tfrac{\ell(I)}{2},\ell(I)])}\langle\psi_{I}^{1},\phi_{z}\rangle,$

and $H(\zeta,z)=0$ if
$\zeta\not\in\cup_{I\in{\mathcal{D}}_{M}}I\times(\tfrac{\ell(I)}{2},\ell(I)]$.
Since $\left|H(\zeta,z)\right|\lesssim
M^{-\frac{1}{4}}[z,\zeta]_{\frac{1}{4}}$, Lemma 2.5 provides a universal
constant $C$ and $\lambda_{z}\in C\Psi_{z}^{1,\frac{1}{8};1}$ such that

$T_{M}(f_{1},\ldots,f_{m})=\int_{Z^{d}\setminus\mathbb{B}_{100M}}\left(M^{-\frac{1}{4}}\left\langle\mathfrak{b},\lambda_{z}\right\rangle+\left\langle
P_{M}^{\perp}\mathfrak{b},\phi_{z}\right\rangle\right)\prod_{j=1}^{m}\left\langle
f_{j},\vartheta_{z}\right\rangle\phi_{z}\,d\mu(z).$

Therefore, (5.3) follows by the triangle inequality and (5.1). ∎

##  6\. Main result and proof

Let us now put together the pieces from the previous sections to prove the
following compact $T(1)$ wavelet representation theorem.

######  Theorem 2.

Suppose $\Lambda$ is an $(m+1)$-linear CZ form, with associated symbols
$\mathfrak{b}_{j}\in\mathrm{BMO}(\mathbb{R}^{d})$, $j=0,1,\ldots,m$ satisfying
(1.2). The following are equivalent.

  * A.

$\Lambda$ is a compact CZ form, i.e. $\mathsf{W}_{\Lambda}^{M}(z)\to 0$ as
$z\to\infty$ and $\mathfrak{b}_{j}\in\mathrm{CMO}(\mathbb{R}^{d})$.

  * B.

There exist compact wavelet forms $\\{U_{k}\\}_{k=1}^{K_{m}}$ and compact
paraproduct forms $\Pi_{\mathfrak{b}_{j}}$ such that for all
$f_{j}\in{\mathcal{S}}(\mathbb{R}^{d})$,

$\Lambda(\mathbf{f})=\sum_{k=1}^{K_{m}}U_{k}(\mathbf{f})+\sum_{j=0}^{m}\Pi^{*,j}_{\mathfrak{b}_{j}}(\mathbf{f}),\quad\mathbf{f}=(f_{0},\ldots,f_{m}).$

  * C.

Each element of $\ \mathsf{T}_{\Lambda}$ is a compact CZO.

###### Proof.

To show A. implies B. we isolate the cancellative part of $\Lambda$, namely

$\Lambda_{\mathsf{c}}=\Lambda-\Pi_{\Lambda},\quad\Pi_{\Lambda}=\sum_{j=0}^{m}\Pi^{*,j}_{\mathfrak{b}_{j}}.$

$\Lambda_{\mathsf{c}}$ is definitely a CZ form, and to verify that it is
cancellative, simply note that since $\left\langle\vartheta,1\right\rangle=1$,
by the reproducing formula (2.3), for
$\varphi\in{\mathcal{S}}^{0}(\mathbb{R}^{d})$,

$\Pi_{\mathfrak{b}}(\varphi,1,\ldots,1)=\int_{Z^{d}}\left\langle\mathfrak{b},\phi_{z}\right\rangle\left\langle\phi_{z},\varphi\right\rangle\,{\rm
d}\mu(z)=\left\langle\mathfrak{b},\varphi\right\rangle,$

and since $\phi$ is cancellative,
$\Pi^{*,j}_{\mathfrak{b}}(\varphi,1,\ldots,1)=0$ for $j=1,\ldots,m$. We want
to apply Proposition 3.9 to $\Lambda_{\mathsf{c}}$ so we must establish that
it is a compact CZ form. Since
$\mathfrak{b}_{j}\in\mathrm{CMO}(\mathbb{R}^{d})$, each
$S\in\mathsf{T}_{\Pi_{\Lambda}}$ is compact by Proposition 5.2, so by
Proposition 3.6, $\Pi_{\Lambda}$ is a compact CZ form. Since we also know that
$\Lambda$ is a compact CZ form, $\Lambda_{\mathsf{c}}$ must indeed be compact,
and B. follows by applying Proposition 3.9 to $\Lambda_{\mathsf{c}}$. B.
implies C. is a consequence of Propositions 4.1 and 5.2. Finally, C. implies
A. is the content of Proposition 3.6. ∎

A fourth equivalent condition in Theorem 2 would be the compactness of the
adjoint operators on weighted Lebesgue spaces for weights belonging to the
multilinear Muckenhoupt classes [lerner-et-al, nieraeth2019quantitative]. If
one is only concerned with weights corresponding to $p_{j}<\infty$, then this
follows immediately from [COY]. Alternatively, to handle the full range of
$Q$, including when some $p_{j}=\infty$, one can verify the conditions of the
weighted Riesz-Kolmogorov theorem [COY]*Proposition 2.9 as we did in
Propositions 4.1 and 5.2 by hand. Furthermore, in [COY], there are provided
examples of multilinear operators which are compact on these weighted spaces.

We note that Theorem 2 applies to the linear case as well and a few
simplifications can be made due to the greater symmetry enjoyed in this
setting. Let us restate Theorem 2 when $m=1$ for additional clarity.

######  Theorem 3.

Let $T$ be a linear CZO. The following are equivalent.

  * A.

$T(1),T^{*}(1)\in\mathrm{CMO}(\mathbb{R}^{d})$ and

$\lim_{\zeta\to\infty}\sup_{z\in\mathbb{B}(\zeta)}t^{d}\left|\left\langle
T\phi_{z},\phi_{\zeta}\right\rangle\right|=0.$

  * B.

There exists a compact wavelet form $U$ such that for all
$f,g\in{\mathcal{S}}(\mathbb{R}^{d})$,

$\left\langle Tf,g\right\rangle=U(f,g)+\Pi_{T(1)}(f,g)+\Pi_{T^{*}(1)}(g,f).$

  * C.

$T$ and $T^{*}$ are compact CZOs.

  * D.

$T(1),T^{*}(1)\in\mathrm{CMO}(\mathbb{R}^{d})$ and

$\lim_{z\to\infty}t^{\frac{d}{2}}\left\|T\phi_{z}\right\|_{L^{2}(\mathbb{R}^{d})}=0.$

###### Proof.

To compare A. with its $m$-linear analogue, one only needs to note that
$z\in\mathbb{B}_{M}(\zeta)$ is equivalent to
$\zeta\in\mathbb{B}_{M^{\prime}}(z)$ as long as $M\sim M^{\prime}$. Therefore,
A. implies B. follows from Theorem 2. B. implies C. is a special case of
Theorem 2. C. implies D. since $t^{\frac{d}{2}}\phi_{z}\to 0$ weakly in
$L^{2}(\mathbb{R}^{d})$ as $z\to\infty$. Finally, D. implies A. by Cauchy-
Schwarz. ∎

## References


# Placing of the recently observed bottom strange state $B_{sJ}(6063)$ and
$B_{sJ}(6114)$ in bottom spectra

Ritu Garg, Pallavi Gupta, A. Upadhyay
School of Physics and Material Science,
Thapar University, Patiala, Punjab-147004,
Mehr Chand Mahajan DAV college for Women, Chandigarh
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  1 Abstract

We have employed HQET to give the spin-parity quantum numbers for recently
observed bottom strange states $B_{sJ}(6063)$ and $B_{sJ}(6114)$ by LHCb
collaborations. By exploring flavour independent parameters
$\Delta_{F}^{(c)}=\Delta_{F}^{(b)}$ and $\lambda_{F}^{(c)}=\lambda_{F}^{(b)}$,
we calculated masses of experimentally missing bottom strange meson states
$2S,1P,1D$. We have also analyzed these bottom strange masses by taking
${1/m_{Q}}$ corrections which lead modifications of parameter terms as
$\Delta_{F}^{(b)}=\Delta_{F}^{(c)}+\delta\Delta_{F}$ and
$\lambda_{F}^{(b)}=\lambda_{F}^{(c)}\delta\lambda_{F}$. Further, we have
analyzed their two-body decays, couplings, and branching ratios via the
emission of light pseudoscalar mesons. Based on predicted masses and decay
widths, we tentatively identified the states $B_{sJ}(6063)$ as $2^{3}S_{1}$
and $B_{sJ}(6114)$ as $1^{3}D_{1}$. Our predictions provide crucial
information for future experimental studies.

##  2 Introduction

During the last decades, different experimental facilities like LHCb, BABAR,
BESIII, FOCUS, SLAC, etc., have been on a discovery spree for stimulating the
spectrum of heavy-light mesons. Based on flavour of the heavy quark, heavy-
light mesons can be cataloged into the charm and bottom mesons. In the charm
meson sector, observations of some ground and excited states like
$D_{0}(2550)$, $D_{1}^{*}(2600)$, $D_{2}(2740)$, $D_{3}^{*}(2750)$,
$D^{0}(3000)$, $D_{J}^{*}(3000)$ and strange states $D_{s1}(2860)$,
$D_{sJ}(3040)$, $D_{s0}(2590)$ [1, 2, 3, 4, 5, 6, 7] have not only broaden the
spectra but also help us in exploring their properties through decay studies.
However, experimental growth toward establishing the bottom sector is still
lacking. Only ground states $B^{0,\pm}(5279)$, $B^{*}(5324)$, $B_{s}(5366)$,
$B_{s}^{*}(5415)$ and some low lying states $B_{1}(5721)$, $B_{J}^{*}(5732)$,
$B_{2}^{*}(5747)$, $B_{s1}(5830)$, $B_{s2}^{*}(5840)$, $B_{sJ}^{*}(5850)$,
$B_{J}(5840)$, $B_{J}(5970)$ are observed experimentally and listed in PDG
[8]. But apart from these states, the whole bottom meson spectra are unknown.
To fill this gap, experimentalists, and theoretical models are trying to
predict new states that could fill this gap. In this process, recently, LHCb
collaborations discovered two new states $B_{sJ}(6063)$ and $B_{sJ}(6114)$ in
$B^{+}K^{-}$ mass spectrum [9]. The measured masses and decay widths are given
below:

$M(B_{sJ}(6063))=6063.5\pm 1.2(stat)\pm 0.8(syst)MeV$
$\Gamma(B_{sJ}(6063))=26\pm 4\pm 4$ $MeV/c^{2}$

$M(B_{sJ}(6114))=6114\pm 3(stat)\pm 5(syst)MeV$
$\Gamma(B_{sJ}(6114))=66\pm 18\pm 21$ $MeV/c^{2}$

The successes of the observations of these radially excited states by LHCb
have demonstrated that more excited bottom meson states will be discovered in
future LHC experiments. Exploring higher excited B mesons is not continuing.
LEP (Large electron-positron collider) in 1999 reported an orbitally excited
bottom meson state in the hadronic Z decays process [10]. This bottom meson’s
measured mass, and decay width is $5937\pm 21\pm 4MeV$ and $50\pm 22\pm 5MeV$,
respectively. However, this state was never reconfirmed by other experimental
facilities. After many years, CDF collaboration in 2013 observed new resonance
$B(5970)$ in decay modes $B^{0}\pi^{+}$ and $B^{+}\pi^{-}$ simultaneously
[11]. Two years after LHCb collaboration in 2015, four resonances
$B_{J}(5840)^{0,+}$ and $B_{J}(5960)^{0,+}$ were announced [12]. Despite these
observations of mesons, the spectrum of excited bottom mesons is not much
explored. In strange bottom mesons family, only some states $B_{s}(5366)$,
$B^{*}_{s}(5415)$, $B_{s1}(5830)$, $B^{*}_{s2}(5840)$ are well established and
collected in PDG.[8]. Among them, states $B_{s}(5366)$, $B^{*}_{s}(5415)$ are
classified as $1S$ states and states $B_{s1}(5830)$, $B^{*}_{s2}(5840)$ are
assigned as $1P$ ($1^{+},2^{+}$) states. This shows experimentalists continue
trying to establish the bottom strange meson spectrum.
In theory, various theoretical studies have performed different analyses for
higher excited bottom non-strange and bottom strange meson states [13, 14, 15,
16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. With
the help of theoretical models, states $B^{0,\pm}(5279)$, $B^{*}(5324)$,
$B_{s}(5366)$, $B_{s}^{*}(5415)$ are assigned as $1S$ state which very well
matches with experimental data. Further the states $B_{1}(5721)$,
$B_{2}^{*}(5747)$ are also well established experiemntally and classified as
$1P(1^{+},2^{+})$ respectively. But theoretically, state $B_{1}(5721)$ is
still a disputed candidate because some of the theoretical work with heavy
meson effective theory favors it as $1P(1^{+})$ state [25, 32], while other
work using relativistic quark model and non-relativistic quark model explained
this state as a mixture of ${}^{3}P_{1}$ and ${}^{1}P_{1}$ states [14, 16,
33]. The $J^{P}$ of state $B_{J}(5840)$ is still ambiguous as different models
suggest different $J^{P}$’s state for it. Authors in [16, 17] explained
$B_{J}(5840)$ with the quark model and suggested the assignment as
$2^{1}S_{0}$ while G. L. Yu and Z. G. Wang with ${}^{3}P_{0}$ decay model
analysis favors the assignments of state $B_{J}(5840)$ as $2^{3}S_{1}$ [20].
But Heavy quark effective theory explained resonances $B_{J}(5840)$ as
$1^{3}D_{1}$ state [28]. The state $B_{J}(5960)^{0,+}$ is assigned as
$2^{3}S_{1}$ or $1^{3}D_{3}$ state or $1^{3}D_{1}$ state with different
theoretical models [14, 15, 16, 22, 23, 24, 34]. But its $J^{P}$ value is
still a question mark in PDG, which only mention its mass and decay width. We
have discussed here a brief literature on these non-strange bottom states. The
assignments of these states ($B_{1}(5721)$, $B_{2}^{*}(5747)$, $B_{J}(5970)$)
are also suggested in our previous work [28]. In the case of the strange
bottom sector, only a few states have been observed, out of which
$B_{s1}(5830)$, $B_{s2}(5840)$ states are well observed by CDF [11, 35], D0
[36], and LHCb [37] collaborations and are identified as $1P(1^{+},2^{+})$
respectively. But, there is ambiguity for recently observed strange bottom
meson states $B_{sJ}(6063)$ and $B_{sJ}(6114)$. The states $B_{sJ}(6063)$ and
$B_{sJ}(6114)$ in non-relativistic quark potential model identified as
$1^{3}D_{1}$ and $1^{3}D_{3}$ states respectively [38, 39]. While authors in
Ref. [40] assign these states as $1^{3}D_{1}$ and $2^{3}S_{1}$ states,
respectively. Theoretical analysis for these newly observed states is thus
limited in the literature, indicating it needs more attention. As a
continuation of previous work [28], we analyzed observed strange bottom mesons
states $B_{sJ}(6063)$ and $B_{sJ}(6114)$ and give their $J^{P}$ values within
this framework.
The paper is arranged as follows: Section II briefly describes the model
“Heavy Quark Effective Theory.” Section III represents the numerical analysis
of $B_{sJ}(6063)$ and $B_{sJ}(6114)$ based on predicted masses and decay
widths. Section IV presents our final conclusion.

##  3 Framework

We apply heavy quark effective theory to assign spin-parity quantum numbers
for recently observed heavy-light strange bottom meson states. This theory is
simple and powerful, which provides precise calculations of masses, and decay
behavior of heavy-light bottom mesons [41]. HQET assumes an infinite mass of
heavy quarks (Q = c, b) and most of the momentum of the bottom meson is
carried by heavy quark. In this heavy quark limit ($m_{Q}\rightarrow\infty$),
spin of the heavy quark $s_{Q}$ decouples from the light d.o.f.(degree of
freedom), which incorporates the light antiquark and the gluons. The total
angular momentum of light d.o.f is $s_{l}=s_{q}+l$, where $s_{q}$ = 1/2, the
spin of light quark and $l$ is the total orbital momentum of light quarks. In
heavy quark limit, mesons are categorized in doublets based on the total
angular momentum of light quarks. For $l=0$, $s_{l}=1/2$ associate with spin
of heavy quark $s_{Q}$ = 1/2 and resulted with doublet $(0^{-},1^{-})$. This
doublet is denoted by $(P,P^{*})$. While $l=1$ forming two doublets are given
by $(P^{*}_{0},P_{1}^{{}^{\prime}})$ and $(P_{1},P_{2}^{*})$ with
$J_{s_{l}}^{P}=(0^{+},1^{+})_{1/2}$ and $J_{s_{l}}^{P}=(1^{+},2^{+})_{3/2}$
respectively. For $l=2$, two doublets are expressed by $(P^{*}_{1},P_{2})$ and
$(P_{2}^{{}^{\prime}},P_{3}^{*})$ with $J_{s_{l}}^{P}=(1^{-},2^{-})_{3/2}$ and
$J_{s_{l}}^{P}=(2^{-},3^{-})_{5/2}$ respectively. These doublets are
introduced in terms of super effective fields
$H_{a},S_{a},T_{a},X^{\mu}_{a},Y^{\mu\nu}_{a}$ and described for fields are
given below [25, 42]:

$\displaystyle
H_{a}=\frac{1+\not{v}}{2}\\{P^{*}_{a\mu}\gamma^{\mu}-P_{a}\gamma_{5}\\}$ (1)
$\displaystyle
S_{a}=\frac{1+\not{v}}{2}[{P^{{}^{\prime}\mu}_{1a}\gamma_{\mu}\gamma_{5}}-{P_{0a}^{*}}]$
(2) $\displaystyle
T^{\mu}_{a}=\frac{1+\not{v}}{2}\\{P^{*\mu\nu}_{2a}\gamma_{\nu}-P_{1a\nu}\sqrt{\frac{3}{2}}\gamma_{5}[g^{\mu\nu}-\frac{\gamma^{\nu}(\gamma^{\mu}-\upsilon^{\mu})}{3}]\\}$
(3) $\displaystyle
X^{\mu}_{a}=\frac{1+\not{v}}{2}\\{P^{\mu\nu}_{2a}\gamma_{5}\gamma_{\nu}-P^{*}_{1a\nu}\sqrt{\frac{3}{2}}[g^{\mu\nu}-\frac{\gamma_{\nu}(\gamma^{\mu}+v^{\mu})}{3}]\\}$
(4)

$Y^{\mu\nu}_{a}=\frac{1+\not{v}}{2}\\{P^{*\mu\nu\sigma}_{3a}\gamma_{\sigma}-P^{{}^{\prime}\alpha\beta}_{2a}\sqrt{\frac{5}{3}}\gamma_{5}[g^{\mu}_{\alpha}g^{\nu}_{\beta}-\frac{g^{\nu}_{\beta}\gamma_{\alpha}(\gamma^{\mu}-v^{\mu})}{5}-\frac{g^{\mu}_{\alpha}\gamma_{\beta}(\gamma^{\nu}-v^{\nu})}{5}]\\}$
(5)

The field $H_{a}$ describes doublets of $S$-wave for $J^{P}=(0^{-},1^{-})$.
The fields $S_{a}$ and $T_{a}$ represent doublets of $P$-wave for
$J^{P}=(0^{+},1^{+})$ and $(1^{+},2^{+})$ respectively. $D$-wave doublets for
$J^{P}=(1^{-},2^{-})$ and $(2^{-},3^{-})$ belong to fields $X^{\mu}_{a}$ and
$Y^{\mu\nu}_{a}$ respectively. $a$ in above expressions is light quark
($u,d,s$) flavour index. $v$ is heavy quark velocity, unchanged in strong
interactions. The approximate chiral symmetry $SU(3)_{L}\times SU(3)_{R}$ is
incorporated with fields of pseudoscalar mesons $\pi$, K, and $\eta$ that are
the lightest strongly interacting bosons. They are considered as approximate
Goldstone bosons of this chiral symmetry and can be expressed by the matrix
field $\xi=e^{\frac{i\mathcal{M}}{f_{\pi}}}$ and $\Sigma=\xi^{2}$, where
$\mathcal{M}$ is given by

$\mathcal{M}=\begin{pmatrix}\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&\pi^{+}&K^{+}\\\
\pi^{-}&-\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&K^{0}\\\
K^{-}&\overline{K}^{0}&-\sqrt{\frac{2}{3}}\eta\end{pmatrix}$ (6)

$f_{\pi}$ is pion decay constant, and its value is taken 130 $MeV$. Fields of
heavy meson doublets is given in eqn.(1-5) interact with pseudoscalar
goldstone bosons through covariant derivative $D_{\mu
ab}=-\delta_{ab}\partial_{\mu}+\mathcal{V}_{\mu
ab}=-\delta_{ab}\partial_{\mu}+\frac{1}{2}(\xi^{+}\partial_{\mu}\xi+\xi\partial_{\mu}\xi^{+})_{ab}$
and axial vector field $A_{\mu
ab}=\frac{i}{2}(\xi\partial_{\mu}\xi^{{\dagger}}-\xi^{{\dagger}}\partial_{\mu}\xi)_{ab}$.
By comprising all meson doublet fields and goldstone fields, effective
lagrangian is written as:

$\mathcal{L}=iTr[\overline{H}_{b}v^{\mu}D_{\mu
ba}H_{a}]+\frac{f_{\pi}^{2}}{8}Tr[\partial^{\mu}\Sigma\partial_{\mu}\Sigma^{+}]+Tr[\overline{S_{b}}(iv^{\mu}D_{\mu
ba}-\delta_{ba}\Delta_{S})S_{a}]+Tr[\overline{T_{b}^{\alpha}}(iv^{\mu}D_{\mu
ba}-\delta_{ba}\Delta_{T})T_{a\alpha}\\\
+Tr[\overline{X_{b}^{\alpha}}(iv^{\mu}D_{\mu
ba}-\delta_{ba}\Delta_{X})X_{a\alpha}+Tr[\overline{Y_{b}^{\alpha\beta}}(iv^{\mu}D_{\mu
ba}-\delta_{ba}\Delta_{Y})Y_{a\alpha\beta}$ (7)

The mass parameter $\Delta_{F}$ in equation (7) gives the mass difference
between excited mass doublets ($F$) and ground mass doublet ($H$) in the form
of spin average masses of these doublets with the same principle quantum
number ( n ). The mass parameters are described by:

$\displaystyle\Delta_{F}=\overline{M_{F}}$
$\displaystyle-\overline{M_{H}},~{}~{}F=S,T,X,Y$ (8)
$\displaystyle\text{where, }~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\overline{M_{H}}$
$\displaystyle=(3m^{Q}_{P^{*}}+m^{Q}_{P})/4$ (9)
$\displaystyle\overline{M_{S}}$
$\displaystyle=(3m^{Q}_{P_{1}^{{}^{\prime}}}+m^{Q}_{P_{0}^{*}})/4$ (10)
$\displaystyle\overline{M_{T}}$
$\displaystyle=(5m^{Q}_{P_{2}^{*}}+3m^{Q}_{P_{1}})/8$ (11)
$\displaystyle\overline{M_{X}}$
$\displaystyle=(5m^{Q}_{P_{2}}+3m^{Q}_{P_{1}^{*}})/8$ (12)
$\displaystyle\overline{M_{Y}}$
$\displaystyle=(7m^{Q}_{P_{3}^{*}}+5m^{Q}_{P_{2}^{{}^{\prime}}})/12$ (13)

The $1/m_{Q}$ corrections to heavy quark limit are given by symmetry-breaking
terms. The corrections are form of:

$\mathcal{L}_{1/m_{Q}}=\frac{1}{2m_{Q}}[\lambda_{H}Tr(\overline{H}_{a}\sigma^{\mu\nu}{H_{a}}\sigma_{\mu\nu})-\lambda_{S}Tr(\overline{S}_{a}\sigma^{\mu\nu}S_{a}\sigma_{\mu\nu})+\lambda_{T}Tr(\overline{T}_{a}^{\alpha}\sigma^{\mu\nu}{T_{a}^{\alpha}}\sigma_{\mu\nu})-\lambda_{X}Tr(\overline{X}_{a}^{\alpha}\sigma^{\mu\nu}{X_{a}^{\alpha}}\sigma_{\mu\nu})\\\
+\lambda_{Y}Tr(\overline{Y}_{a}^{\alpha\beta}\sigma^{\mu\nu}{Y_{a}^{\alpha\beta}}\sigma_{\mu\nu})$
(14)

Here parameters $\lambda_{H}$, $\lambda_{S}$, $\lambda_{T}$, $\lambda_{X}$,
$\lambda_{Y}$ are analogous with hyperfine splittings and expressed in Eq.(15
- 19). These mass terms in lagrangian give only first order in $1/m_{Q}$
terms, but higher order terms may also be present otherwise. We are limited to
the first order corrections in $1/m_{Q}$.

$\displaystyle\lambda_{H}=\frac{1}{8}(M^{2}_{P^{*}}-M^{2}_{P})$ (15)
$\displaystyle\lambda_{S}=\frac{1}{8}({M^{2}_{P_{1}^{{}^{\prime}}}}-{M^{2}_{P_{0}^{*}}})$
(16)
$\displaystyle\lambda_{T}=\frac{3}{8}({M^{2}_{P_{2}^{*}}}-{M^{2}_{P_{1}}})$
(17)
$\displaystyle\lambda_{X}=\frac{3}{8}({M^{2}_{P_{2}}}-{M^{2}_{P_{1}^{*}}})$
(18)
$\displaystyle\lambda_{Y}=\frac{3}{8}({M^{2}_{P_{3}}}-{M^{2}_{P_{2}^{{}^{\prime}*}}})$
(19)

In HQET, at the scale of 1 $GeV$, flavour symmetry spontaneously arises for b
(bottom quark) and c (charm quark) and hence beauty of flavour symmetry
implies to

$\displaystyle\Delta_{F}^{(c)}=\Delta_{F}^{(b)}$ (20)
$\displaystyle\lambda_{F}^{(c)}=\lambda_{F}^{(b)}$ (21)

This symmetry is broken by the higher order terms in the HQET Lagrangian
involving terms of factor $1/m_{Q}$ and the parameters $\Delta_{F}$ and
$\lambda_{F}$ are modified by extra terms $\delta\Delta_{F}$ and
$\delta\lambda_{F}$. The hyperfine splitting term $\lambda_{F}$, which
originates from the chromomagnetic interaction, is dominated by the QCD
corrections and the $1/m_{Q}$ effect is neglected. QCD corrections change the
$\lambda_{F}$ relation to
$\lambda_{F}^{(b)}=\lambda_{F}^{(c)}{{(\frac{\alpha_{s}(m_{b})}{\alpha_{s}(m_{c})})}^{9/25}}$.
The difference of the spin averaged masses at $1/m_{Q}$ order modifies the
parameter $\Delta_{F}$ by $\delta\Delta_{F}$ given by

$\Delta_{F}^{(b)}=\Delta_{F}^{(c)}+\delta\Delta_{F}$

,

The decays $F\rightarrow H+M$ (F = $H,S,T,X,Y$ and $M$ presents a light
pseudoscalar meson) can be described by effective Lagrangians explained in
terms of the fields introduced in (9-14) that are valid at leading order in
the heavy quark mass and light meson momentum expansion:

$\displaystyle
L_{HH}=g_{HH}Tr\\{\overline{H}_{a}H_{b}\gamma_{\mu}\gamma_{5}A^{\mu}_{ba}\\}$
(22) $\displaystyle
L_{SH}=g_{SH}Tr\\{\overline{H}_{a}S_{b}\gamma_{\mu}\gamma_{5}A^{\mu}_{ba}\\}+h.c.$
(23) $\displaystyle
L_{TH}=\frac{g_{TH}}{\Lambda}Tr\\{\overline{H}_{a}T^{\mu}_{b}(iD_{\mu}\not{A}+i\not{D}A_{\mu})_{ba}\gamma_{5}\\}+h.c.$
(24) $\displaystyle
L_{XH}=\frac{g_{XH}}{\Lambda}Tr\\{\overline{H}_{a}X^{\mu}_{b}(iD_{\mu}\not{A}+i\not{D}A_{\mu})_{ba}\gamma_{5}\\}+h.c.$
(25)
$L_{YH}=\frac{1}{\Lambda^{2}}Tr\\{\overline{H}_{a}Y^{\mu\nu}_{b}[k^{Y}_{1}\\{D_{\mu},D_{\nu}\\}A_{\lambda}+k^{Y}_{2}(D_{\mu}D_{\lambda}A_{\nu}+D_{\nu}D_{\lambda}A_{\mu})]_{ba}\gamma^{\lambda}\gamma_{5}\\}+h.c.$
(26)

In these equations $D_{\mu}=\partial_{\mu}+V_{\mu}$,
$\\{D_{\mu},D_{\nu}\\}=D_{\mu}D_{\nu}+D_{\nu}D_{\mu}$ and
$\\{D_{\mu},D_{\nu}D_{\rho}\\}=D_{\mu}D_{\nu}D_{\rho}+D_{\mu}D_{\rho}D_{\nu}+D_{\nu}D_{\mu}D_{\rho}+D_{\nu}D_{\rho}D_{\mu}+D_{\rho}D_{\mu}D_{\nu}+D_{\rho}D_{\nu}D_{\mu}$.
$\Lambda$ is the chiral symmetry breaking scale taken as 1 $GeV$. $g_{HH}$,
$g_{SH}$, $g_{TH}$, $g_{YH}=k^{Y}_{1}+k^{Y}_{2}$ are the strong coupling
constants involved. Using the lagrangians $L_{HH},L_{SH},L_{TH},L_{YH}$, the
two body strong decays of $Q\overline{q}$ heavy-light bottom mesons are given
as [43, 44, 45, 46]

$(0^{-},1^{-})\rightarrow(0^{-},1^{-})+M$

$\displaystyle\Gamma(1^{-}\rightarrow
1^{-})=C_{M}\frac{g_{HH}^{2}M_{f}p_{M}^{3}}{3\pi f_{\pi}^{2}M_{i}}$ (27)
$\displaystyle\Gamma(1^{-}\rightarrow
0^{-})=C_{M}\frac{g_{HH}^{2}M_{f}p_{M}^{3}}{6\pi f_{\pi}^{2}M_{i}}$ (28)
$\displaystyle\Gamma(0^{-}\rightarrow
1^{-})=C_{M}\frac{g_{HH}^{2}M_{f}p_{M}^{3}}{2\pi f_{\pi}^{2}M_{i}}$ (29)

$(0^{+},1^{+})\rightarrow(0^{-},1^{-})+M$

$\displaystyle\Gamma(1^{+}\rightarrow
1^{-})=C_{M}\frac{g_{SH}^{2}M_{f}(p^{2}_{M}+m^{2}_{M})p_{M}}{2\pi
f_{\pi}^{2}M_{i}}$ (30) $\displaystyle\Gamma(0^{+}\rightarrow
0^{-})=C_{M}\frac{g_{SH}^{2}M_{f}(p^{2}_{M}+m^{2}_{M})p_{M}}{2\pi
f_{\pi}^{2}M_{i}}$ (31)

$(1^{+},2^{+})\rightarrow(0^{-},1^{-})+M$

$\displaystyle\Gamma(2^{+}\rightarrow
1^{-})=C_{M}\frac{2g_{TH}^{2}M_{f}p_{M}^{5}}{5\pi
f_{\pi}^{2}\Lambda^{2}M_{i}}$ (32) $\displaystyle\Gamma(2^{+}\rightarrow
0^{-})=C_{M}\frac{4g_{TH}^{2}M_{f}p_{M}^{5}}{15\pi
f_{\pi}^{2}\Lambda^{2}M_{i}}$ (33) $\displaystyle\Gamma(1^{+}\rightarrow
1^{-})=C_{M}\frac{2g_{TH}^{2}M_{f}p_{M}^{5}}{3\pi
f_{\pi}^{2}\Lambda^{2}M_{i}}$ (34)

$(1^{-},2^{-})\rightarrow(0^{-},1^{-})+M$

$\displaystyle\Gamma(1^{-}\rightarrow 0^{-})=C_{M}\frac{4g_{XH}^{2}}{9\pi
f_{\pi}^{2}\Lambda^{2}}\frac{M_{f}}{M_{i}}[p_{M}^{3}(m_{M}^{2}+p_{M}^{2})]$
(35) $\displaystyle\Gamma(1^{-}\rightarrow 1^{-})=C_{M}\frac{2g_{XH}^{2}}{9\pi
f_{\pi}^{2}\Lambda^{2}}\frac{M_{f}}{M_{i}}[p_{M}^{3}(m_{M}^{2}+p_{M}^{2})]$
(36) $\displaystyle\Gamma(2^{-}\rightarrow 1^{-})=C_{M}\frac{2g_{XH}^{2}}{3\pi
f_{\pi}^{2}\Lambda^{2}}\frac{M_{f}}{M_{i}}[p_{M}^{3}(m_{M}^{2}+p_{M}^{2})]$
(37)

$(2^{-},3^{-})\rightarrow(0^{-},1^{-})+M$

$\displaystyle\Gamma(2^{-}\rightarrow 1^{-})=C_{M}\frac{4g_{YH}^{2}}{15\pi
f_{\pi}^{2}\Lambda^{4}}\frac{M_{f}}{M_{i}}[p_{M}^{7}]$ (38)
$\displaystyle\Gamma(3^{-}\rightarrow 0^{-})=C_{M}\frac{4g_{YH}^{2}}{35\pi
f_{\pi}^{2}\Lambda^{4}}\frac{M_{f}}{M_{i}}[p_{M}^{7}]$ (39)
$\displaystyle\Gamma(3^{-}\rightarrow 1^{-})=C_{M}\frac{16g_{YH}^{2}}{105\pi
f_{\pi}^{2}\Lambda^{4}}\frac{M_{f}}{M_{i}}[p_{M}^{7}]$ (40)

Here $M_{i}$, $M_{f}$ gives initial and final momentum, $\Lambda$ is chiral
symmetry breaking scale of 1 GeV. $p_{M}$, $m_{M}$ denotes the final momentum
and mass of light pseudoscalar meson. The coupling constant plays a key role
in the phenomenology study of heavy-light mesons. These dimensionless coupling
constants describe the strength of transition between H-H field (negative-
negative parity), S-H field (positive-negative parity), T-H field (positive-
negative parity). These couling constants are notated as $g_{HH}$, $g_{SH}$,
$g_{TH}$, $g_{XH}$, $g_{YH}$ etc. The coefficient $C_{M}$ for different
pseudoscalar particles are: $C_{\pi^{\pm}}$, $C_{K^{\pm}}$, $C_{K^{0}}$,
$C_{\overline{K}^{0}}=1$, $C_{\pi^{0}}=\frac{1}{2}$ and
$C_{\eta}=\frac{2}{3}(c\bar{u},c\bar{d})$ or $\frac{1}{6}(c\bar{s})$. In our
paper, we are not including higher order corrections of $\frac{1}{m_{Q}}$ to
introduce new couplings. We also expect that higher corrections give a small
contribution in comparison the leading order contributions. The numerical
values of various meson masses used in the calculations are listed in Table 1.

Table 1: Numerical value of the meson masses used in this work [8]. States | $B^{0}$ | $B^{\pm}$ | $B^{*}$ | $B_{s}$ | $B^{*}_{s}$
---|---|---|---|---|---
Masses(MeV) | 5279.58 | 5279.25 | 5325.20 | 5366.77 | 5415.40
States | $\pi^{\pm}$ | $\pi^{0}$ | $\eta$ | $K^{+}$ | $K^{0}$
Masses(MeV) | 139.57 | 134.97 | 547.85 | 493.67 | 497.61

##  4 Numerical analysis

Assigning a particular $J^{P}$ (spin-parity quantum number) to the
experimentally predicted excited states is very crucial. A specific position
of the state in its mass spectra can help in redeeming many other important
strong interaction parameters such as hadronic coupling constants, branching
ratios, spins and mass splittings, decay widths and many more. Therefore, in
this analysis, we aim to suggest a particular $J^{P}$ state to the recently
strange bottom states observed by LHCb. Moreover, we also completed the empty
spaces of this strange mass spectra by predicting the masses and other
parameters of the missing bottom spectra in the framework of HQET.
In the bottom strange sector, only ground states 1S ($0^{-},1^{-}$) states are
confirmed both experimentally and theoretically. The rest of the spectra is
still unknown. Although many other theoretical models, such as the chiral
perturbation theory, the chiral unitary approach, the Regge trajectory, the
${}^{3}P_{0}$ model, the relativistic flux tube model, QCD sum rules, and the
lattice QCD, etc tried to fill the gaps but yet they are not verified
experimentally [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31]. In addition to this, such theoretically calculated masses are
unreliable because all these theoretical models depend on certain unknown
parameters. The HQET framework chosen by us is free from such parameters and
uses the most important relations of spin and flavour symmetry present in
heavy-light mesons.

###  4.1 Mass spectroscopy:

To study the behavior of heavy-light mesons, masses are the prime property
that determines many other properties of the mesons. Therefore, we start our
calculations by predicting the masses of unavailable strange bottom states
using the flavour and spin symmetry property
$\Delta_{F}^{(c)}=\Delta_{F}^{(b)}$ and $\lambda_{F}^{(c)}=\lambda_{F}^{(b)}$.
As discussed in Section II, the parameter $\Delta_{F}$ is defined in terms of
the spin averaged mass splittings between the higher state doublet and ground
state doublet, whereas the other parameter $\lambda_{F}$ is the mass
splittings between the spin partners of the doublets.
The main aim of our calculation is to incorporate the $J^{P}$ to recently
observed strange states of the bottom flavour $B_{sJ}(6063)$ and
$B_{sJ}(6114)$ and to fill the gaps near these states so that our calculations
can provide motivation and reference to the yet to be predicted experimental
information. The details of the numerical analysis on the mass spectrum are as
follows:

Table 2:  Input values used in this work. All values are in units of $MeV$. State | $J^{P}$ | $c\overline{s}$ | $b\overline{s}$
---|---|---|---
$1^{1}S_{0}$ | $0^{-}$ | 1968.35 [6] | 5366.92 [6]
$1^{3}S_{1}$ | $1^{-}$ | 2112.20 [6] | 5415.40 [6]
$1^{3}P_{0}$ | $0^{+}$ | 2317.80 | -
$1^{1}P_{1}$ | $1^{+}$ | 2459.50 | -
$1^{3}P_{1}$ | $1^{+}$ | 2535.11 | -
$1^{3}P_{2}$ | $2^{+}$ | 2569.10 | -
$1^{3}D_{1}$ | $1^{-}$ | 2859.00 | -
$1^{1}D_{2}$ | $2^{-}$ | 2902.40 | -
$1^{3}D_{2}$ | $2^{-}$ | 2895.40 | -
$1^{3}D_{3}$ | $3^{-}$ | 2860.50 | -
$2^{1}S_{0}$ | $0^{-}$ | 2676.20 | -
$2^{3}S_{1}$ | $1^{-}$ | 2714.00 | -

Using the masses of charm and bottom states tabulated in Table 2, the spin
averaged mass splittings $\Delta_{F}^{(c)}$ and the hyperfine splittings
$\lambda_{F}^{(c)}$ for $1S,1P,1D$ and $2S$ comes out to be

$\Delta_{\tilde{H}}^{(c)}$ = 628.313 $MeV$

$\Delta_{S}^{(c)}$ = 347.838 $MeV$

$\Delta_{T}^{(c)}$ = 480.116 $MeV$

$\Delta_{X}^{(c)}$ = 763.513 $MeV$

$\Delta_{Y}^{(c)}$ = 798.804 $MeV$

$\lambda_{\tilde{H}}^{(c)}$ = $(159.589)^{2}MeV^{2}$

$\lambda_{S}^{(c)}$ = $(290.892)^{2}MeV^{2}$

$\lambda_{T}^{(c)}$ = $(180.360)^{2}MeV^{2}$

$\lambda_{X}^{(c)}$ = $(181.228)^{2}MeV^{2}$

$\lambda_{Y}^{(c)}$ = $(204.573)^{2}MeV^{2}$

Table 3: Predicted values of the radially excited strange $2S,1P$ and $1D$ bottom meson states. All the masses are in $MeV$ units | Without corrections | Corrections in $\lambda_{F}$ | Corrections in $\Delta_{F}$ | Correction in both parameters $\lambda_{F}$ and $\Delta_{F}$
---|---|---|---|---
$0^{-}(2^{1}S_{0})$ | 6018.92 | 6020.98 | 6017.74 | 6019.80
$1^{-}(2^{3}S_{1})$ | 6035.82 | 6035.13 | 6034.65 | 6033.96
$0^{+}(1^{3}P_{0})$ | 5706.86 | 5714.07 | 5705.35 | 5712.56
$1^{+}(1^{1}P_{1})$ | 5765.87 | 5763.47 | 5764.37 | 5761.97
$1^{+}(1^{3}P_{1})$ | 5874.18 | 5875.68 | 5867.30 | 5868.80
$2^{+}(1^{3}P_{2})$ | 5888.93 | 5888.03 | 5882.06 | 5881.16
$1^{-}(1^{3}D_{1})$ | 6175.67 | 6174.23 | 6180.17 | 6178.73
$2^{-}(1^{1}D_{2})$ | 6161.47 | 6162.33 | 6165.98 | 6166.84
$2^{-}(1^{3}D_{2})$ | 6211.53 | 6205.10 | 6210.70 | 6209.16
$3^{-}(1^{3}D_{3})$ | 6195.34 | 6194.50 | 6194.51 | 6195.60

The charm mesons with $J^{P}=2^{-}$ of D-wave with j = 3/2 and j = 5/2 for
$1^{1}D_{2}$ and $1^{3}D_{2}$ are experimentally unavailable, so we have taken
the average of the theoretical masses [13, 47, 48, 49] for them.
Masses obtained for $B_{s}$ mesons with the help of the symmetries are listed
in the $2^{nd}$ column of Table 3. Masses predicted for P-wave j = 3/2 strange
bottom states $1^{3}P_{1}$ and $1^{3}P_{2}$ are in very good agreement with
the experimental masses predicted for these states by LHCb [37], CDF [11, 35]
and D0 [36]. Our calculated masses for $1^{3}P_{1}$ and $1^{3}P_{2}$ are only
0.78% , 0.84% deviating from their experimental values respectively. In
comparison with predictions of other theoretical models shown in Table 4, our
calculated masses are in good agreement. Note that predictions of Ref.[13, 51]
are smaller by 100 $MeV$ than our results. While data in [40] is larger than
our masses by order of 100 $MeV$. Also, the calculated mass for
$J^{P}=(1^{-},2^{-}),3^{-}$ belonging to D-wave with j = 3/2, j = 5/2 and
$J^{P}=1^{-}$ of radially excited S-wave have also beautifully matched with
the LHCb states $B_{sJ}(6114)$ and $B_{sJ}(6063)$ respectively. Since masses
of these two states $B_{sJ}(6114)$ and $B_{sJ}(6063)$ deviates only by 61, 47,
81 $MeV$ and 28 $MeV$ from our predicted values, so one easily conclude that
$B_{sJ}(6063)$ state belong to $J^{P}$ $2S1^{-}$ while the other state
$B_{sJ}(6114)$ belong to one of the $J^{P}$ of 1 D-wave. Authors in Ref.[40]
predicted the $J^{P}$ for $B_{sJ}(6114)$ as $1D1^{-}$ for j = 3/2. But, to
prove convincingly, we computed the masses by taking higher order corrections
as splitting parameters ($\Delta_{F}$ and $\lambda_{F}$) can drastically
change in the presence of QCD and higher order ($\tilde{}{1/m_{Q}}$)
corrections in the HQET lagrangian. Also, the calculated masses should be able
to predict other parameters, such as decay width, which should match with
experimental data. Therefore, in the next part, we present the analysis of
such corrections to these splittings and calculate the two-body strong decay
widths of these bottom states.
QCD and higher order ($\tilde{1/m_{Q}}$) corrections are applied to a scale of
$\Lambda_{QCD}/m_{Q}$, where they can significantly decide the level of
symmetry breaking. The corrections to $\Delta_{F}$ and $\lambda_{F}$
parameters change heavy quark symmetry relations to
$\Delta_{F}^{(b)}=\Delta_{F}^{(c)}+\delta\Delta_{F}$ and
$\lambda_{F}^{(b)}=\lambda_{F}^{(c)}\delta\lambda_{F}$. We apply such
corrections to these splittings one by one and check the effect of these
corrections on the bottom masses followed by a step in which both corrections
would be applied simultaneously. In the case of $\lambda_{F}$ parameter, QCD
corrections are dominant over $1/m_{Q}$ corrections because these mass
splitting parameters $\lambda_{F}$ originate from the chromomagnetic
interactions. The leading QCD corrections to the $\lambda_{F}$ are in the form
of
$\lambda_{F}^{(b)}=\lambda_{F}^{(c)}{{(\frac{\alpha_{s}(m_{b})}{\alpha_{s}(m_{c})})}}^{9/25}$.
The parameters $\alpha_{s}(m_{b})$ and $\alpha_{s}(m_{c})$ for applying the
QCD corrections to this splitting parameters are taken as 0.22 and 0.36 [50].
The corrections in $\lambda_{F}$ parameters modify the value to:

$\lambda_{\tilde{H}}^{(c)}$ = $(133.662)^{2}MeV^{2}$

$\lambda_{S}^{(c)}$ = $(243.632)^{2}MeV^{2}$

$\lambda_{T}^{(c)}$ = $(151.058)^{2}MeV^{2}$

$\lambda_{X}^{(c)}$ = $(151.785)^{2}MeV^{2}$

$\lambda_{Y}^{(c)}$ = $(171.33)^{2}MeV^{2}$

The $B_{s}$ masses inherited using these corrections are tabulated in the
$3^{rd}$ column of Table 3. The resulting masses are deflected upto 7.5 $MeV$
from their initial masses and have resulted in reducing the gap between our
and LHCb masses. Now we analyze the effect of $1/m_{Q}$ and QCD corrections to
our other parameter $\Delta_{F}$, which depicts the mass splittings between
the higher mass doublet and the ground state H field states. The corrections
to these parameters are in the form of $\delta\Delta_{F}$, where
$F=S,T,X,Y,\tilde{H}$. The best-fitted values of these corrections come to be
$\mathcal{O}$ (0.63) $GeV$, $\mathcal{O}$(0.35) $GeV$, $\mathcal{O}$(0.49)
$GeV$, $\mathcal{O}$(0.77) $GeV$ and $\mathcal{O}$(0.80) $GeV$ for doublets
$2S(0^{-},1^{-})$, $1P(0^{+},1^{+})$, $1P(1^{+},2^{+})$, $1D(1^{-},2^{-})$ and
$1D(2^{-},3^{-})$ respectively. Masses are calculated using these corrections
are tabulated in $4^{th}$ column of Table 3. A comparison of masses concluded
that the deviation lies in the range of 0.83 - 6.90 $MeV$, which again shows
that masses are not affected much by corrections too but again resulted in
narrowing the gap between our and experimental masses. Lastly, applying both
corrections simultaneously the masses obtained are listed in the last column
of Table 3. It can be summarized that the effect of such corrections is very
small in the higher excited states. Because of the higher angular momentum,
such states do not remain in their stable state for much time and thus do not
explicitly show chromomagnetic effects. Although such corrections have
resulted in narrowing the gap between the predicted and experimentally
predicted mass values but have not adversely affected the masses. To suggest a
particular $J^{P}$ value for LHCb observed strange bottom states
$B_{sJ}(6063)$ and $B_{sJ}(6114)$, we explore the second utmost important
property of heavy-light mesons i.e. decay widths.

Table 4: The predicted values of bottom-strange meson masses ($MeV$) compared with some other model predictions. | Ours | Ref.[51] | Ref.[13] | Ref.[40]
---|---|---|---|---
$0^{-}(2^{1}S_{0})$ | 6018.92 | 6003 | 5985 | 6025
$1^{-}(2^{3}S_{1})$ | 6035.82 | 6029 | 6019 | 6033
$0^{+}(1^{3}P_{0})$ | 5706.86 | 5812 | 5804 | 5709
$1^{+}(1^{1}P_{1})$ | 5765.87 | 5828 | 5805 | 5768
$1^{+}(1^{3}P_{1})$ | 5874.18 | 5842 | 5842 | 5875
$2^{+}(1^{3}P_{2})$ | 5888.93 | 5840 | 5820 | 5890
$1^{-}(1^{3}D_{1})$ | 6175.67 | 6119 | 6127 | 6247
$2^{-}(1^{1}D_{2})$ | 6161.47 | 6128 | 6095 | 6256
$2^{-}(1^{3}D_{2})$ | 6211.53 | 6157 | 6140 | 6292
$3^{-}(1^{3}D_{3})$ | 6195.34 | 6172 | 6103 | 6297

###  4.2 Strong Decays

We apply the effective Lagrangian approach discussed in Sec II to calculate
the OZI allowed two body strong decay widths and the various branching ratios
involved with the bottom states $B_{s}(2S)$, $B_{s}(1P)$ and $B_{s}(1D)$. The
numerical value of the partial and total decay widths of these states are
given in Table 6 and 5. Here, we need to emphasize that calculated total decay
widths for the above states do not include the contribution of decays with the
emission of vector mesons ($\omega,\rho,K^{*},\phi$). Since the contribution
of vector mesons to total decay widths is small compared to pseudoscalar
mesons. They give the contribution of $\pm 10MeV$ [15] to total decay widths
for above analyzed states.
To choose the possible $J^{P}$ for the LHCb bottom state $B_{sJ}(6114)$, we
calculated the total decay width for all the possible $J^{P}$’s $1^{3}D_{1}$,
$1^{1}D_{2}$ and $1^{3}D_{3}$ in terms of strong coupling constants. Values
come out to be

$\Gamma(1^{3}D_{1})$ = 6052.34${g}^{2}_{XH}$

$\Gamma(1^{1}D_{2})$ = 4416.36${g}^{2}_{XH}$

$\Gamma(1^{3}D_{3})$ = 933.75${g}^{2}_{YH}$

On comparing these calculated decay widths with the experimental value of
decay width of 66 $MeV$, the coupling constant values come out to be

${g}_{XH}=0.104$

${g}_{XH}=0.12$

${g}_{YH}=0.26$

The available theoretical values of $g_{YH}$ are 0.61 [54], 0.53 [25], 0.42
[53]. Our computed value of $g_{YH}$ is much smaller, ruling out $1^{3}D_{3}$
state from possible $J^{P}$’s for state $B_{sJ}(6114)$. Now, we are left with
two available $J^{P}$ ($1^{3}D_{1}$ and $1^{1}D_{2}$). In literature, the
available theoretical values of $g_{XH}$ are 0.41 [28], 0.45 [32], 0.53 [52],
0.19 [25]. In Ref.[25], the coupling constant $g_{XH}$ is calculated for state
$B(5970)^{0}$ is $0.19\pm 0.049$. They assigned $1^{3}D_{1}$ for this state
and computed coupling constant by comparing their theoretical decay width with
the experimental value. We are also following the same procedure, and obtained
value of $g_{XH}$ enables us to assign the $J^{P}$ state of $B_{sJ}(6114)$.
The coupling value for $1^{3}D_{1}$ and $1^{1}D_{2}$ obtained are consistent
with $g_{XH}=0.19$. We suggest these two $J^{P}$ $1^{3}D_{1}$ and $1^{1}D_{2}$
as most favorable for state $B_{sJ}(6114)$. But $B_{sJ}(6114)$ were observed
in BK modes, so it cannot have $J^{P}=2^{-}$. Since it does not satisfy the
conservation of parity and angular momentum simultaneously for $J^{P}=2^{-}$.
So, we are assigning a particular $J^{P}=2^{-}(1^{3}D_{1})$ to state
$B_{sJ}(6114)$. Further experimental investigations like branching ratios may
provide the necessary conclusions.

####  4.2.1 $1D$ state

The natural parity D states $1^{3}D_{1}$ and $1^{3}D_{3}$ both are dominant in
BK decay mode with branching fractions of 56.73% and 36.67% respectively,
while the unnatural parity states $1^{1}D_{2}$ and $1^{3}D_{2}$ shows
dominance in $B^{*}K$ decay channel with branching fractions of 77.51% and
67.02% respectively. Column 4 of Table 6 gives the ratio of the partial decay
widths for $1D$ bottom states with respect to its partial decay width
$B^{*+}K^{-}$. Apart from the decay channels listed in this Table 6, these
bottom states also decay to P-wave bottom meson states, which occur via D-wave
and thus due to the small phase space, these decay modes are suppressed when
compared to decays to ground state S-wave mesons and hence are not shown in
Table 6. The calculated value of $g_{XH}=0.12$ can be beneficial in finding
the total and partial decay width of unobserved bottom state $1^{3}D_{1}$.
Thus the calculated total decay width for this state is 87.15 $MeV$ which
deviates by 22.87% from the result of Ref.[39]

####  4.2.2 $1P$ state

We have also analyzed the strong decay widths $1P$ bottom states and
calculated the various branching ratios involved. The calculated value of the
partial decay widths for the bottom states $1^{3}P_{0}$, $1^{1}P_{1}$,
$1^{3}P_{1}$ and $1^{3}P_{2}$ are listed in Table 5. The state $1^{3}P_{0}$
decaying to BK decay channel is kinematically suppressed as the predicted mass
for this state is lower than the BK threshold value. This is a similar
situation as seen in the charm sector for the same $J^{P}$ state
$D^{*}_{s0}(2317)$, thus reflecting the flavour symmetry in heavy hadrons. The
only kinematically allowed decay channel for this state is $B_{s}\pi$. Its
spin partner $1^{1}P_{1}$ is also decaying to $B^{*}_{s}\pi^{0}$ decay mode
only while all other modes are suppressed. Using the available value of
coupling constant $g_{SH}=0.56$ [25, 32, 40], predicted decay widths for
$(0^{+},1^{+})$ doublet are calculated as

$\Gamma(1^{3}P_{0})$ = 46.25 $MeV$

$\Gamma(1^{1}P_{1})$ = 50.51 $MeV$

Our estimated values are highly overestimated with the results of Ref.[40] and
underestimated with the data of QPC model, and chiral quark model [28, 36,
40].
The study of the doublet $(1^{+},2^{+})$ shows that $B^{*}_{s}\pi^{0}$ and
$B_{s}\pi^{0}$ are the dominant decay modes for $1^{3}P_{1}$ and $1^{3}P_{2}$
states with branching ratio of 85.33% and 32.30% respectively. Total decay
width calculated by taking a sum of its partial decay widths listed in Table 5

$\Gamma(1^{3}P_{1})$ = 14.33 $MeV$

$\Gamma(1^{3}P_{2})$ = 26.74 $MeV$

These decay values depict these states to be narrower but are still
overestimated with their experimental values measured by LHCb and CDF
collaborations [11, 37] but are in good agreement with the theoretical data
[11, 40] where authors used mixing angle to calculate the width of
$1^{3}P_{1}$ state.

####  4.2.3 $2S$ state

For the radially ground state S-wave bottom states, Table 5 reveals
$B^{*}K^{+}$ mode to be the dominant decay mode both for
$\widetilde{B}_{1s}^{*}$ and $\widetilde{B}_{0s}$ bottom states with branching
fraction of 22.53$\%$ and 33.46$\%$, respectively. Hence the decay modes
$B^{*}K^{+}$ is suitable for the experimental search for the missing strange
$2S$ bottom meson states. Using the strong coupling value
$\widetilde{g_{HH}}=0.31$ [53], total decay widths for bottom state
$\widetilde{B}_{1s}^{*}$ and its spin partner $\widetilde{B}_{0s}$ are
predicted as 207.44 $MeV$ and 231.11 $MeV$ respectively. These speculated
values indicate these radial bottom states to have broad resonance, which is
in good agreement with the results of Ref.[11, 40]. Apart from the mentioned
partial decay widths, these bottom states also decay to D-wave bottom mesons.
But these decays are suppressed in our calculations because of their small
contributions.

Table 5: Strong decay width of strange bottom mesons $B_{s}(2S0^{-})$, $B_{s}(2S1^{-})$, $B_{s}(1P0^{+})$, $B_{s}(1P1^{+})$, $B_{s}(1P1^{+})$, $B_{s}(1P2^{+})$. Ratio in 5th column represents the $\widehat{{\bf\Gamma}}=\frac{\Gamma}{\Gamma(B_{sJ}^{*}\rightarrow B^{*0}K^{+})}$. Fraction gives the percentage of the partial decay width with respect to the total decay width.  $nLs_{l}J^{P}$ | Decay channel | Decay Width($MeV$) | Ratio | Fraction
---|---|---|---|---
2$S_{s1/2}0^{-}$ | $B^{*0}K^{0}$ | 785.160$\widetilde{g}^{2}_{HH}$ | 1 | 32.64
| $B^{*+}K^{-}$ | 804.921$\widetilde{g}^{2}_{HH}$ | 1.02 | 33.46
| $B^{*}_{s}\pi^{0}$ | 737.438$\widetilde{g}^{2}_{HH}$ | 0.93 | 30.66
| $B^{*}_{s}\eta$ | 77.419$\widetilde{g}^{2}_{HH}$ | 0.09 | 3.21
| Total | 2404.940$\widetilde{g}^{2}_{HH}$ |  |
$2S_{s1/2}1^{-}$ | $B^{0}K^{0}$ | 342.716$\widetilde{g}^{2}_{HH}$ | 0.72 | 15.87
| $B^{+}K^{-}$ | 350.669$\widetilde{g}^{2}_{HH}$ | 0.73 | 16.24
| $B_{s}\pi^{0}$ | 292.676$\widetilde{g}^{2}_{HH}$ | 0.61 | 13.55
| $B_{s}\eta$ | 58.29$\widetilde{g}^{2}_{HH}$ | 0.12 | 2.70
| $B^{*0}K^{0}$ | 474.96$\widetilde{g}^{2}_{HH}$ | 1 | 22.00
| $B^{*+}K^{-}$ | 486.477$\widetilde{g}^{2}_{HH}$ | 1.2 | 22.53
| $B^{*}_{s}\pi^{0}$ | 466.530$\widetilde{g}^{2}_{HH}$ | 0.98 | 21.61
| $B^{*}_{s}\eta$ | 36.988$\widetilde{g}^{2}_{HH}$ | 0.07 | 1.71
| Total | 2158.650$\widetilde{g}^{2}_{HH}$ |  |
$1P_{s1/2}0^{+}$ | $B_{s}\pi^{0}$ | 147.500${g}^{2}_{SH}$ | - | 100
| $B_{s}\eta$ | - |  | -
| $B^{+}K^{0}$ | - | - | -
| $B^{-}K^{+}$ | - | - | -
| Total | 147.500${g}^{2}_{SH}$ |  |
1$P_{s1/2}1^{+}$ | $B^{*}_{s}\pi^{0}$ | 161.080${g}^{2}_{SH}$ | - | -
| $B_{s}^{*}\eta$ | - | - | -
| $B^{*+}K^{0}$ | - | - | -
| $B^{*-}K^{+}$ | - | - | -
| Total | 1161.080${g}^{2}_{SH}$ |  |
1$P_{s13/2}1^{+}$ | $B^{*0}K^{0}$ | 5.970${g}^{2}_{TH}$ | 1 | 6.66
| $B^{*+}K^{-}$ | 7.12${g}^{2}_{TH}$ | 1.19 | 7.93
| $B^{*}_{s}\pi^{0}$ | 76.49${g}^{2}_{TH}$ | 12.81 | 85.37
| $B^{*}_{s}\eta$ | - | - | -
| Total | 89.59${g}^{2}_{TH}$ |  |
1$P_{s13/2}2^{+}$ | $B^{0}K^{0}$ | 18.48${g}^{2}_{TH}$ | 22.67 | 11.05
| $B^{+}K^{-}$ | 20.13${g}^{2}_{TH}$ | 2.91 | 12.04
| $B_{s}\pi^{0}$ | 59.590${g}^{2}_{TH}$ | 8.62 | 35.65
| $B_{s}\eta$ | - | - | -
| $B^{*0}K^{0}$ | 6.92${g}^{2}_{TH}$ | 1 | 4.13
| $B^{*+}K^{-}$ | 7.92${g}^{2}_{TH}$ | 1.14 | 4.73
| $B^{*}_{s}\pi^{0}$ | 54.09${g}^{2}_{TH}$ | 7.82 | 32.36
| $B^{*}_{s}\eta$ | - | - | -
| Total | 167.15${g}^{2}_{TH}$ |  |
Table 6: Strong decay width of strange bottom mesons $B_{s}(1D1^{-})$, $B_{s}(1D2^{-})$, $B_{s}(1D2^{-})$ and $B_{s}(1D3^{-})$. Ratio in 5th column represents the $\widehat{{\bf\Gamma}}=\frac{\Gamma}{\Gamma(B_{sJ}^{*}\rightarrow B^{*0}K^{+})}$. Fraction gives the percentage of the partial decay width with respect to the total decay width.  $nLs_{l}J^{P}$ | Decay channel | Decay Width($MeV$) | Ratio | Fraction
---|---|---|---|---
1$D_{s3/2}1^{-}$ | $B^{0}K^{0}$ | 1706.81${g}^{2}_{XH}$ | 2.71 | 28.20
| $B^{+}K^{-}$ | 1727.24${g}^{2}_{XH}$ | 2.74 | 28.53
| $B_{s}\pi^{0}$ | 863.737${g}^{2}_{XH}$ | 1.37 | 14.27
| $B_{s}\eta$ | 126.531${g}^{2}_{XH}$ | 0.20 | 2.09
| $B^{*0}K^{0}$ | 628.449${g}^{2}_{XH}$ | 0.51 | 5.35
| $B^{*+}K^{-}$ | 635.727${g}^{2}_{XH}$ | 0.06 | 0.66
| $B^{*}_{s}\pi^{0}$ | 323.847${g}^{2}_{XH}$ | 1 | 10.38
| $B^{*}_{s}\eta$ | 40.0077${g}^{2}_{XH}$ | 1.01 | 10.50
| Total | 6052.34${g}^{2}_{XH}$ |  |
1$D_{s3/2}2^{-}$ | $B^{*0}K^{0}$ | 1701.43${g}^{2}_{XH}$ | 1 | 38.52
| $B^{*+}K^{-}$ | 1722.22${g}^{2}_{XH}$ | 1.01 | 38.99
| $B^{*}_{s}\pi^{0}$ | 889.31${g}^{2}_{XH}$ | 0.52 | 20.13
| $B^{*}_{s}\eta$ | 103.31${g}^{2}_{XH}$ | 0.06 | 2.34
| Total | 4416.36${g}^{2}_{XH}$ |  |
1$D_{s5/2}2^{-}$ | $B^{*0}K^{0}$ | 290.97${g}^{2}_{YH}$ | 1 | 33.08
| $B^{*+}K^{-}$ | 298.45${g}^{2}_{YH}$ | 1.02 | 33.93
| $B^{*}_{s}\pi^{0}$ | 251.13${g}^{2}_{YH}$ | 0.868 | 28.55
| $B^{*}_{s}\eta$ | 38.85${g}^{2}_{YH}$ | 0.13 | 4.41
| Total | 879.41${g}^{2}_{XH}$ |  |
1$D_{s5/2}3^{-}$ | $B^{0}K^{0}$ | 168.95${g}^{2}_{YH}$ | 1.21 | 18.09
| $B^{+}K^{-}$ | 173.47${g}^{2}_{YH}$ | 1.24 | 18.57
| $B_{s}\pi^{0}$ | 140.16${g}^{2}_{YH}$ | 18 | 15.01
| $B_{s}\eta$ | 27.27${g}^{2}_{YH}$ | 0.19 | 2.92
| $B^{*0}K^{0}$ | 139.01${g}^{2}_{YH}$ | 1 | 14.88
| $B^{*+}K^{-}$ | 142.78${g}^{2}_{YH}$ | 1.02 | 15.29
| $B^{*}_{s}\pi^{0}$ | 125.11${g}^{2}_{YH}$ | 0.90 | 13.39
| $B^{*}_{s}\eta$ | 16.98${g}^{2}_{YH}$ | 0.12 | 1.81
| Total | 933.75${g}^{2}_{YH}$ |  |

##  5 Conclusion

In this paper, we have applied the Heavy Quark Effective Theory to examine the
recently observed strange bottom mesons, $B_{sJ}(6063)$ and $B_{sJ}(6114)$ by
LHCb collaborations. In this framework, we have calculated masses of strange
bottom meson $2S,1P,1D$ with the use of available experimental and theoretical
data on charm mesons and including non-perturbative parameters ($\Delta_{F}$
and $\lambda_{F}$). These predicted masses for the above-said states have
beautifully matched with other models predictions. Also, by taking ${1/m_{Q}}$
corrections in terms of $\delta\Delta_{F}$ and $\delta\lambda_{F}$, we
estimated masses of strange bottom mesons $2S,1P,1D$, which narrow the gap
between our results and experimental data. On the basis of computed masses, we
have identified strange bottom states $B_{sJ}(6063)$ as $2^{3}S_{1}$ and give
three possible $J^{P}$’s ( $1^{-},2^{-},3^{-}$) belonging D-wave to
$B_{sJ}(6114)$ state. We have analyzed strong decay widths for possible
$J^{P}$’s state for $B_{sJ}(6114)$ and concluded that most favourable states
for $B_{sJ}(6114)$ are $1^{3}D_{1}$. In addition to this, we have predicted
the branching ratios and the coupling constants for the above states, which
can provide crucial information for future experimental searches.

##  6 Acknowledgment

The authors thankfully acknowledge the financial support by the Department of
Science and Technology (SERB/F/9119/2020), New Delhi.

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