††thanks: Corresponding Author

# Heavy-light hybrid mesons with different spin-parities

B. Barsbaymissing<EMAIL_ADDRESS>Department of Physics, Doǧuş
University, Dudullu-Ümraniye, 34775 Istanbul, Turkey  K. Azizi
<EMAIL_ADDRESS>Department of Physics, University of Tehran, North
Karegar Ave. Tehran 14395-547, Iran  Department of Physics, Doǧuş University,
Dudullu-Ümraniye, 34775 Istanbul, Turkey  School of Particles and
Accelerators, Institute for Research in Fundamental Sciences (IPM) P.O. Box
19395-5531, Tehran, Iran  H. Sundu<EMAIL_ADDRESS>Department of
Physics, Kocaeli University, 41380 Izmit, Turkey

###### Abstract

The spectroscopic parameters of the heavy-light hybrid mesons with different
spin-parities and different quark contents are investigated in the framework
of the QCD sum rule method. The mass and current coupling of these states are
calculated by taking into account the quark, gluon and mixed vacuum
condensates up to dimension 10. The obtained results are compared with the
existing QCD Laplace sum rule predictions. The results of mass and current
coupling for all the considered channels are obtained to be stable and
reliable. Our results can be useful for future experimental searches as well
as theoretical studies of different parameters related to the hybrid mesons
and their decays and interactions with other particles.

##  I Introduction

The standard hadrons, i.e., mesons and baryons are respectively made of two
and three valence quarks, which strongly interact through gluon exchange. The
field theory of these interactions is the Quantum Chromodynamics (QCD) and
they are categorized using the quark model. Both the QCD and quark model do
not exclude existing of other states with different configurations. Hence, it
was already suggested that, in addition to the conventional hadrons, there may
exist particles composed of various combinations of quarks and gluons. Some of
such states were observed in numerous experiments. These states are particles
including four-quark XYZ tetraquarks, hadronic molecules composed of mesons
and baryons, pentaquarks, glueballs, hybrids, etc., which are nominated as
exotic states. Exploring the existence and properties of such exotic states is
one of the most interesting research topics of high energy physics. In the
past two decades, with the development of experimental facilities, starting
with the observation of the $X(3872)$ Choi:2003ue , we witness the
observations of many exotic hadrons (for more information see for instance
Refs. D0:2016mwd ; LHCb:2016axx ; LHCb:2016nsl ; Belle:2007umv ; Belle:2014wyt
; LHCb:2015yax ; LHCb:2016lve ; LHCb:2019kea ; Chen:2022asf ; Meng:2022ozq ).

An important category in the exotic hadrons is hybrid hadrons containing
valence gluon(s), besides valence quarks. The existence of hybrids was first
predicted by Jaffe and Johnson in 1976 Jaffe:1975fd . Understanding the inner
organizations and properties of these hadrons may support their future
investigations. Besides, they may carry new information on QCD. Therefore, the
hybrids hadrons have been studied in the framework of different theoretical
methods such as the constituent gluon model Horn:1977rq , the flux-tube model
Barnes:1995hc , lattice QCD Perantonis:1990dy ; Liu:2005rc ;
HadronSpectrum:2012gic ; Liu:2011rn ; Cheung:2016bym  and QCD sum rules
Govaerts:1984hc ; Govaerts:1985fx ; Govaerts:1986pp ; Zhu:1998ki ;
Narison:2009vj ; Qiao:2010zh ; Harnett:2012gs ; Berg:2012gd ; Chen:2013pya ;
Kleiv:2014kua ; Palameta:2017ols ; Palameta:2018yce . Unfortunately, the
predictions for the masses of the hybrids acquired within these approaches are
inconsistent with each other. Moreover, most of the studies have been carried
out on heavy quarkonium hybrids and there are less studies devoted to the
investigations of the properties of heavy-light hybrids, which make an
important category of these states. The first investigations of heavy-light
hybrids using QCD sum rule were performed by Govaerts, Reinders, and Weyers in
1985 Govaerts:1985fx . Thus, more studies are needed to clarify the physical
properties of this category of heavy-light hybrids, which may be in agenda of
future experiments.

Motivated by this situation, in this article, we are going to calculate the
mass and current coupling of the scalar, pseudoscalar, vector and axial-vector
heavy-light hybrid mesons with all possible quantum numbers and quark contents
in the framework of the Borel QCD sum rule method. This method is one of the
powerful and predictive non-perturbative approaches in hadron physics
Shifman:1978bx ; Shifman:1978by ; Reinders:1984sr ; Narison:1989aq . This
approach has been successfully applied to not only the standard hadrons but
also to the exotics (see for instance Refs. Wang:2018ntv ; Wu:2018xdi ;
Voloshin:2018vym ; Cao:2018vmv ; Agaev:2021vur ; Agaev:2020zad ; Agaev:2020mqq
; Sundu:2018uyi ; Agaev:2019qqn ; Wang:2021itn ; Azizi:2021utt ; Wang:2020rdh
; Wang:2018waa ; Wang:2019iaa ; Wang:2019hyc ; Azizi:2021pbh ) and the
obtained results well explain the existing experimental data.

The spectroscopic parameters of the heavy-light hybrid mesons with various
quark contents have already been calculated in Ref. Ho:2016owu  using QCD
Laplace sum rules. As a result of analyses, the states with
$J^{P(C)}\in\\{0^{+(+)},1^{-(-)},1^{+(-)}\\}$ were led to stable results on
the mass and current coupling constant, while for the states with the
$J^{P(C)}\in\\{0^{+(-)},1^{+(+)},1^{-(+)}\\}$ quantum numbers the analyses
were led to unstable predictions. In the present study we construct various
currents, interpolating the heavy-light hybrid mesons of different possible
spin-parities and quark contents, to investigate the spectroscopic properties
of different kinds of the heavy-light hybrid mesons. The scalar and vector
states as well as the pseudoscalar and axial-vector states couple to the same
currents. We separate contribution of these states and calculate the mass and
current coupling of all these hybrid mesons. We get stable and reliable
results in all channels considered in the present study.

This work is organized in the following manner. In sec. II, we derive QCD sum
rules for the mass and current coupling of the heavy-light hybrid mesons of
different spin-parities. In sec. III, we numerically analyze the obtained sum
rules and present our predictions for the mass and current coupling of the
considered states. Section IV is reserved for summary and conclusion.

##  II Mass and current coupling of the heavy-light hybrid mesons

To calculate the mass and current coupling of the heavy-light hybrid mesons,
we start with the two-point correlation function

$\displaystyle\Pi_{\mu\nu}(q)=$ $\displaystyle i\int d^{4}xe^{iqx}\langle
0|\mathcal{T}\\{J_{\mu}(x)J_{\nu}^{\dagger}(0)\\}|0\rangle$ $\displaystyle=$
$\displaystyle\frac{q_{\mu}q_{\nu}}{q^{2}}\Pi_{S(PS)}(q^{2})+\left(\frac{q_{\mu}q_{\nu}}{q^{2}}-g_{\mu\nu}\right)\Pi_{V(AV)}(q^{2}),$
(1)

where $J_{\mu}(x)$ is the interpolating current for a vector and an axial-
vector hybrid states. The currents are taken as

$\displaystyle
J_{\mu}^{1}(x)=\frac{g_{s}}{2}\overline{Q}_{a}(x)\gamma_{\theta}\gamma_{5}\frac{\lambda_{ab}^{n}}{2}\tilde{G}_{\mu\theta}^{n}(x)q_{b}(x),$
(2)

for $0^{+(-)}$ and $1^{-(-)}$ quantum numbers,

$\displaystyle
J_{\mu}^{2}(x)=g_{s}\overline{Q}_{a}(x)\gamma_{\theta}\gamma_{5}\frac{\lambda_{ab}^{n}}{2}G_{\mu\theta}^{n}(x)q_{b}(x),$
(3)

for $0^{-(-)}$ and $1^{+(-)}$ quantum numbers,

$\displaystyle
J_{\mu}^{3}(x)=\frac{g_{s}}{2}\overline{Q}_{a}(x)\gamma_{\theta}\frac{\lambda_{ab}^{n}}{2}\tilde{G}_{\mu\theta}^{n}(x)q_{b}(x),$
(4)

for $0^{-(+)}$ and $1^{+(+)}$ quantum numbers, and

$\displaystyle
J_{\mu}^{4}(x)=g_{s}\overline{Q}_{a}(x)\gamma_{\theta}\frac{\lambda_{ab}^{n}}{2}G_{\mu\theta}^{n}(x)q_{b}(x),$
(5)

for the states with $0^{+(+)}$ and $1^{-(+)}$ quantum numbers. In the above
currents, the $g_{s}$ is the strong coupling constant, $a,b=1,2,3$,
$\lambda^{n}$ with $n=1,2,\cdots,8$ are the Gell-Mann matrices;
$\tilde{G}_{\mu\theta}^{n}(x)=\epsilon_{\mu\theta\alpha\beta}G_{\alpha\beta}^{n}(x)/2$
is the dual field strength of $G_{\mu\theta}^{n}(x)$; and $q=u,d,s$ and
$Q=c,b$ are light and heavy quarks, respectively.

In this work, we will reference each of the $\Pi_{\mathrm{S(PS)}}(q^{2})$ and
$\Pi_{\mathrm{V(AV)}}(q^{2})$ according to the chosen quantum numbers of the
heavy-light hybrid meson. In order to derive desired sum rules, we write down
the correlation function $\Pi_{\mu\nu}(q)$ using the mass and current coupling
of the heavy-light hybrid meson. Saturating the correlation function with
complete sets of states with quantum numbers of hybrid state and performing
the integral over $x$ in Eq. (1), we get the following result:

$\displaystyle\Pi_{\mu\nu}^{\mathrm{Phys}}(q)=\frac{\langle
0|J_{\mu}|H_{S(PS)}(q)\rangle\langle
H_{S(PS)}(q)|J_{\nu}^{\dagger}|0\rangle}{m_{H_{S(PS)}}^{2}-q^{2}}$
$\displaystyle+\frac{\langle 0|J_{\mu}|H_{V(AV)}(q)\rangle\langle
H_{V(AV)}(q)|J_{\nu}^{\dagger}|0\rangle}{m_{H_{V(AV)}}^{2}-q^{2}}+\ldots,$ (6)

where $m_{H_{S(PS)}}$ and $m_{H_{V(AV)}}$ are the masses of the scalar
(pseudoscalar) and vector (axial-vector) hybrid states. As is seen, the states
$S$ and $V$ mix with each other and $PS$ and $AV$ come together in the
physical representation. The function can be simplified by introducing the
matrix elements

$\langle 0|J_{\mu}|H_{S(PS)}(q)\rangle=q_{\mu}f_{H_{S(PS)}}\,$ (7)

and

$\langle
0|J_{\mu}|H_{V(AV)}(q)\rangle=m_{H_{V(AV)}}f_{H_{V(AV)}}\varepsilon_{\mu},$
(8)

where $\varepsilon_{\mu}$ is the polarization vector of the vector (axial-
vector) hybrid state. $f_{H_{S(PS)}}$ and $f_{H_{V(AV)}}$ are the current
couplings of the scalar (pseudoscalar) and vector (axial-vector) mesons. In
terms of the parameters entered, the function
$\Pi_{\mu\nu}^{\mathrm{Phys}}(q)$ takes the form

$\displaystyle\Pi_{\mu\nu}^{\mathrm{Phys}}(q)=$
$\displaystyle\frac{f_{H_{S(PS)}}^{2}}{m_{H_{S(PS)}}^{2}-q^{2}}q_{\mu}q_{\nu}+\frac{m_{H_{V(AV)}}^{2}f_{H_{V(AV)}}^{2}}{m_{H_{V(AV)}}^{2}-q^{2}}$
$\displaystyle\times\left(-g_{\mu\nu}+\frac{q_{\mu}q_{\nu}}{q^{2}}\right)+\cdots.$
(9)

As we previously mentioned, the correlator $\Pi_{\mu\nu}^{\mathrm{Phys}}(q)$
includes $\mathrm{S}$ and $\mathrm{V}$ contributions as well as $\mathrm{PS}$
and $\mathrm{AV}$ contributions, simultaneously. To separate the $\mathrm{S}$
($\mathrm{PS}$) contribution, Eq. (9) is multiplied by
$q_{\mu}q_{\nu}/{q^{2}}$ which leads to

$\frac{q_{\mu}q_{\nu}}{q^{2}}\Pi_{\mu\nu}^{\mathrm{Phys}}(q)=-f_{H_{S(PS)}}^{2}+\frac{m_{H_{S(PS)}}^{2}f_{H_{S(PS)}}^{2}}{m_{H_{S(PS)}}^{2}-q^{2}}+\cdots.$
(10)

To find the contributions of the vector and axial-vector states, it is enough
to chose the structure $g_{\mu\nu}$ which contains only the contribution of
$\mathrm{V(AV)}$ particle and it is free of the contamination by
$\mathrm{S(PS)}$ state:

$\Pi_{\mu\nu}^{\prime\mathrm{Phys}}(q)=\frac{m_{H_{V(AV)}}^{2}f_{H_{V(AV)}}^{2}}{m_{H_{V(AV)}}^{2}-q^{2}}\times\left(-g_{\mu\nu}\right)+\cdots.$
(11)

The Borel transformation with respect to $q^{2}$ applied to
$\Pi_{\mu\nu}^{\prime\mathrm{Phys}}(q)$ and
$q_{\mu}q_{\nu}/{q^{2}}\Pi_{\mu\nu}^{\mathrm{Phys}}(q)$ leads to the final
forms of the physical sides:

$\mathcal{B}_{q^{2}}\Pi_{\mu\nu}^{\prime\mathrm{Phys}}(q)=m_{H_{V(AV)}}^{2}f_{H_{V(AV)}}^{2}e^{-m_{H_{V(AV)}}^{2}/M^{2}}\left(-g_{\mu\nu}\right)+\cdots,$
(12)

and

$\mathcal{B}_{q^{2}}\frac{q_{\mu}q_{\nu}}{q^{2}}\Pi_{\mu\nu}^{\mathrm{Phys}}(q)=m_{H_{S(PS)}}^{2}f_{H_{S(PS)}}^{2}e^{-m_{H_{S(PS)}}^{2}/M^{2}}\
+\cdots.$ (13)

The QCD side of the sum rules, $\Pi_{\mu\nu}^{\mathrm{OPE}}(q)$, has to be
calculated in the operator product expansion (OPE) with certain accuracy. To
get $\Pi_{\mu\nu}^{\mathrm{OPE}}(q)$, we calculate the correlation function
using explicit forms of the currents $J_{\mu}^{1}(x)$, $J_{\mu}^{2}(x)$,
$J_{\mu}^{3}(x)$ and $J_{\mu}^{4}(x)$. As a result, we express
$\Pi_{\mu\nu}^{\mathrm{OPE}}(q)$ for each current in terms of the heavy and
light quark propagators as well as the gluon propagator/condensate,

$\displaystyle\Pi_{\mu\nu}^{\mathrm{OPE}_{(1,1)}}(q)=\frac{g_{s}^{2}\epsilon_{\mu\theta\alpha\beta}\epsilon_{\nu\delta\alpha^{\prime}\beta^{\prime}}}{4}i\int
d^{4}xe^{iqx}$ $\displaystyle\times\langle
0|G^{n}_{\alpha\beta}(x)G^{m}_{\alpha^{\prime}\beta^{\prime}}(0)|0\rangle\mathrm{t}^{n}[a,b]\mathrm{t}^{m}[a^{\prime},b^{\prime}]$
$\displaystyle\times\mathrm{Tr}\left[S_{Q}^{a^{\prime}a}(-x)\gamma_{\theta}\gamma_{5}S_{q}^{bb^{\prime}}(x)\gamma_{\delta}\right],$
(14) $\displaystyle\Pi_{\mu\nu}^{\mathrm{OPE}_{(2,2)}}(q)=g_{s}^{2}i\int
d^{4}xe^{iqx}\langle 0|G^{n}_{\mu\theta}(x)G^{m}_{\nu\delta}(0)|0\rangle$
$\displaystyle\times\mathrm{t}^{n}[a,b]\mathrm{t}^{m}[a^{\prime},b^{\prime}]\mathrm{Tr}\left[S_{Q}^{a^{\prime}a}(-x)\gamma_{\theta}\gamma_{5}S_{q}^{bb^{\prime}}(x)\gamma_{5}\gamma_{\delta}\right],$
(15)
$\displaystyle\Pi_{\mu\nu}^{\mathrm{OPE}_{(3,3)}}(q)=-\frac{g_{s}^{2}\epsilon_{\mu\theta\alpha\beta}\epsilon_{\nu\delta\alpha^{\prime}\beta^{\prime}}}{4}i\int
d^{4}xe^{iqx}$ $\displaystyle\times\langle
0|G^{n}_{\alpha\beta}(x)G^{m}_{\alpha^{\prime}\beta^{\prime}}(0)|0\rangle\mathrm{t}^{n}[a,b]\mathrm{t}^{m}[a^{\prime},b^{\prime}]$
$\displaystyle\times\mathrm{Tr}\left[S_{Q}^{a^{\prime}a}(-x)\gamma_{\theta}S_{q}^{bb^{\prime}}(x)\gamma_{\delta}\right],$
(16)

and

$\displaystyle\Pi_{\mu\nu}^{\mathrm{OPE}_{(4,4)}}(q)=-g_{s}^{2}i\int
d^{4}xe^{iqx}\langle 0|G^{n}_{\mu\theta}(x)G^{m}_{\nu\delta}(0)|0\rangle$
$\displaystyle\times\mathrm{t}^{n}[a,b]\mathrm{t}^{m}[a^{\prime},b^{\prime}]\mathrm{Tr}\left[S_{Q}^{a^{\prime}a}(-x)\gamma_{\theta}S_{q}^{bb^{\prime}}(x)\gamma_{\delta}\right],$
(17)

where $t^{n}={\lambda^{n}}/2$. In the above formulas, $S_{q}^{ab}(x)$ and
$S_{Q}^{ab}(x)$ are propagators of $q(u,d,s)$ and $c,b$-quarks. In the present
work, for the light quark propagator, $S_{q}^{ab}(x)$, we employ the following
expression

$\displaystyle
S_{q}^{ab}(x)=i\frac{\not{x}\delta_{ab}}{2\pi^{2}x^{4}}-\frac{m_{q}\delta_{ab}}{4\pi^{2}x^{2}}-\frac{\langle\overline{q}q\rangle}{12}\left(1-i\frac{m_{q}}{4}\not{x}\right)\delta_{ab}$
$\displaystyle-\frac{x^{2}}{192}\langle\overline{q}g_{s}\sigma
Gq\rangle\left(1-i\frac{m_{q}}{6}\not{x}\right)\delta_{ab}-\frac{\not{x}x^{2}g_{s}^{2}}{7776}\langle\overline{q}q\rangle^{2}\delta_{ab}$
$\displaystyle-\frac{x^{4}\langle\overline{q}q\rangle\langle
g_{s}^{2}G^{2}\rangle}{27648}\delta_{ab}+\cdots.$ (18)

For the heavy quark $Q$, we use the propagator $S_{Q}^{ab}(x)$ as

$\displaystyle
S_{Q}^{ab}(x)=i\int\frac{d^{4}k}{(2\pi)^{4}}e^{-ikx}\Bigg{\\{}\frac{\delta_{ab}\left({\not{k}}+m_{Q}\right)}{k^{2}-m_{Q}^{2}}$
$\displaystyle+\frac{\langle
g_{s}^{2}G^{2}\rangle}{12}\delta_{ab}m_{Q}\frac{k^{2}+m_{Q}{\not{k}}}{(k^{2}-m_{Q}^{2})^{4}}+\frac{\langle
g_{s}^{3}G^{3}\rangle}{48}\delta_{ab}\frac{\left({\not{k}}+m_{Q}\right)}{(k^{2}-m_{Q}^{2})^{6}}$
$\displaystyle\times\left[{\not{k}}\left(k^{2}-3m_{Q}^{2}\right)+2m_{Q}\left(2k^{2}-m_{Q}^{2}\right)\right]\left({\not{k}}+m_{Q}\right)+\cdots\Bigg{\\}}.$
(19)

Beside the above expressions, which contain the perturbative or free light and
heavy propagators and various vacuum condensates, we have extra terms
containing one gluon field strength tensor, as well. These terms are
$-\frac{ig_{s}G_{ab}^{\mu\nu}}{32\pi^{2}x^{2}}\left[\not{x}\sigma_{\mu\nu}+\sigma_{\mu\nu}\not{x}\right]$
and
$-\frac{g_{s}G_{ab}^{\alpha\beta}}{4}\frac{\sigma_{\alpha\beta}\left({\not{k}}+m_{Q}\right)+\left({\not{k}}+m_{Q}\right)\sigma_{\alpha\beta}}{(k^{2}-m_{Q}^{2})^{2}}$
to be replaced in Eqs. (14) -(17) instead of the light and heavy quark
propagators, respectively. These terms, when multiplied to each other in the
presence of vacuum, make another two-gluon condensates, which we take into
account their contributions and they will be discussed later. Here, we use the
short-hand notations

$\displaystyle G_{ab}^{\alpha\beta}$ $\displaystyle=$ $\displaystyle
G_{A}^{\alpha\beta}\lambda_{ab}^{A}/2,\,\,~{}~{}G^{2}=G_{\alpha\beta}^{A}G_{\alpha\beta}^{A},$
$\displaystyle G^{3}$ $\displaystyle=$
$\displaystyle\,\,f^{ABC}G_{\mu\nu}^{A}G_{\nu\delta}^{B}G_{\delta\mu}^{C},$
(20)

where $f^{ABC}$ are the structure constants of the color group $SU_{c}(3)$.

We will treat $\langle
0|G^{n}_{\alpha\beta}(x)G^{m}_{\alpha^{\prime}\beta^{\prime}}(0)|0\rangle$ in
Eqs. (14) -(17) in two different forms: first we displace it by the full
propagator of the gluon in coordinate space,

$\displaystyle\langle
0|G^{n}_{\alpha\beta}(x)G^{m}_{\alpha^{\prime}\beta^{\prime}}(0)||0\rangle=\frac{\delta^{mn}}{2\pi^{2}x^{4}}[g_{\beta\beta^{\prime}}(g_{\alpha\alpha^{\prime}}-\frac{4x_{\alpha}x_{\alpha^{\prime}}}{x^{2}})$
$\displaystyle+(\beta,\beta^{\prime})\leftrightarrow(\alpha,\alpha^{\prime})-\beta\leftrightarrow\alpha-\beta^{\prime}\leftrightarrow\alpha^{\prime}],$

and perform all the computations. This is equivalent to the diagrams at which
the valence-gluon is a full propagator. At the next step, we consider $\langle
0|G^{n}_{\alpha\beta}(x)G^{m}_{\alpha^{\prime}\beta^{\prime}}(0)|0\rangle$ as
the gluon condensate and take the first term of the Taylor expansion at $x=0$
, i.e.

$\displaystyle\langle
0|G^{n}_{\alpha\beta}(0)G^{m}_{\alpha^{\prime}\beta^{\prime}}(0)|0\rangle=\frac{\langle
G^{2}\rangle}{96}\delta^{mn}[g_{\alpha\alpha^{\prime}}g_{\beta\beta^{\prime}}$
$\displaystyle-g_{\alpha\beta^{\prime}}g_{\alpha^{\prime}\beta}],$ (22)

which stands for the diagrams with the gluon interacting with the QCD vacuum.

By substituting Eq. (II) or (II) in Eqs. (14) -(17), we need to use

$\displaystyle\mathrm{t}^{n}[a,b]\mathrm{t}^{n}[a^{\prime},b^{\prime}]=\frac{1}{2}\left(\delta^{ab^{\prime}}\delta^{a^{\prime}b}-\frac{1}{3}\delta^{ab}\delta^{a^{\prime}b^{\prime}}\right).$
(23)

The desired QCD sum rules for the physical observables can be obtained using
the same Lorentz structures in physical and OPE sides. In the case under
consideration these structures are $\sim g_{\mu\nu}$ and $\sim I$. We choose
the terms $\sim g_{\mu\nu}$ and $\sim I$ which represent contributions of
V(AV) and S(PC) particles, respectively. The invariant amplitudes in
$\Pi_{\mu\nu}^{\prime\mathrm{OPE}}(q)$ and $\Pi_{\mu\nu}^{\mathrm{OPE}}(q)$
corresponding to the structure $g_{\mu\nu}$ and $I$ in our following analysis
will be denoted by $\Pi_{\mu\nu}^{\prime\mathrm{OPE}}(q^{2})$ and
$\Pi_{\mu\nu}^{\mathrm{OPE}}(q^{2})$.To derive sum rules, we equate invariant
amplitudes $\Pi^{\mathrm{Phys}}(q^{2})$ and $\Pi^{\mathrm{OPE}}(q^{2})$
corresponding to these structures, and apply the Borel transformation to both
sides of the obtained sum rules. The last operation is necessary to suppress
contributions stemming from the higher resonances and continuum states. At the
following phase of manipulations, we make use of an assumption about the
quark-hadron duality as well, and subtract from the physical side
contributions of higher resonances and continuum terms. After these
manipulations, the final sum rule equality acquires a dependence on the Borel
$M^{2}$ and continuum threshold $s_{0}$ parameters for each case. This
equality, and second expression obtained by applying the operator
$d/d(-1/M^{2})$ to its both sides, form a system which is solved to find sum
rules for the mass $m_{H}$ and current coupling $f_{H}$

$m_{H}^{2}=\frac{d/d(-1/M^{2})[\Pi(M^{2},s_{0})]}{\Pi(M^{2},s_{0})},$ (24)
$f_{H}^{2}=\frac{e^{m_{H}^{2}/M^{2}}}{m_{H}^{2}}\Pi(M^{2},s_{0}),$ (25)

for each case.

In Eqs. (24) and (25) the function $\Pi(M^{2},s_{0})$ is the Borel transformed
and continuum subtracted invariant amplitude $\Pi^{\mathrm{OPE}}(q^{2})$ for
each channel. In the present work, we calculate $\Pi(M^{2},s_{0})$ by taking
into account quark, gluon and mixed vacuum condensates up to dimension $10$.
It has the following form

$\Pi(M^{2},s_{0})=\int_{m_{Q}^{2}}^{s_{0}}ds\rho^{\mathrm{OPE}}(s)e^{-s/M^{2}}+\Pi(M^{2}),$
(26)

where $\rho^{\mathrm{OPE}}(s)$ is the two-point spectral density. The second
component of the invariant amplitude $\Pi(M^{2})$ contains nonperturbative
contributions calculated directly from the OPE side of each case. The explicit
expressions of the functions $\rho^{\mathrm{OPE}}(s)$ and $\Pi(M^{2})$ are
shown in the Appendix, as examples, for the charm-nonstrange hybrid meson with
the quantum numbers $J^{P(C)}=1^{-(-)}$.

##  III Numerical Results

Now, we proceed to numerically analyze the obtained sum rules to get the
values of the physical quantities for different channels. The quark, gluon and
mixed condensates which enter to the sum rules (24) and (25) are universal
parameters of the computations:

$\displaystyle\langle\overline{q}q\rangle=-(0.24\pm
0.01)^{3}~{}\mathrm{GeV}^{3},\ $
$\displaystyle\langle\overline{s}s\rangle=0.8~{}\langle\overline{q}q\rangle,\
$ $\displaystyle\langle\overline{q}g_{s}\sigma
Gq\rangle=m_{0}^{2}\langle\overline{q}q\rangle,\ m_{0}^{2}=(0.8\pm
0.1)~{}\mathrm{GeV}^{2},\ $
$\displaystyle\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle=(0.012\pm
0.004)~{}\mathrm{GeV}^{4},$ $\displaystyle\langle
g_{s}^{3}G^{3}\rangle=(0.57\pm 0.29)~{}\mathrm{GeV}^{6},$ $\displaystyle
m_{c}=1.275\pm 0.025~{}\mathrm{GeV},$ $\displaystyle
m_{b}=4.18^{+0.03}_{-0.02}~{}\mathrm{GeV},$ $\displaystyle
m_{s}=93^{+11}_{-5}~{}\mathrm{MeV}.$ (27)

The vacuum condensates have fixed numerical values, whereas the Borel and
continuum threshold parameters $M^{2}$ and $s_{0}$ can be varied within some
regions, which have to satisfy the standard constraints of the sum rules
computations. The continuum threshold at each channel depends on the energy of
the first excited state and it is chosen like that the integrals do not
receive contributions from the excited states when the ground state is
studied. Unfortunately, we do not have experimental information on the
spectrum of the hybrid mesons under study. Hence, we apply some conditions to
fix their working windows. The main restrictions that simultaneously are
applied are reaching to maximum possible pole contribution ($\mathrm{PC}$),
convergence of the OPE and relatively weak dependence of the results on
auxiliary parameters. The $\mathrm{PC}$ is defined as

$\mathrm{PC}=\frac{\Pi(M^{2},\ s_{0})}{\Pi(M^{2},\ \infty)}.$ (28)

For the OPE, we impose the condition of the higher the dimension of the
perturbative and non-perturbative operators entering the calculations, the
lower their contributions. For the higher dimensional terms, i.e.,
$\mathrm{DimN=Dim(8+9+10)}$ we demand

$R(M_{\mathrm{min}}^{2})=\frac{\Pi^{\mathrm{DimN}}(M_{\mathrm{min}}^{2},\
s_{0})}{\Pi(M_{\mathrm{min}}^{2},\ s_{0})}<0.01,$ (29)

which helps us to fix $M_{\mathrm{min}}^{2}$. As we mentioned above, the last
condition is the relatively weak dependence of the physical quantities to the
helping parameters $M^{2}$ and $s_{0}$. By applying all these constraints we
fix the working windows of the $M^{2}$ and $s_{0}$ that their full intervals
will be shown in the tables containing the values of the mass and current
coupling for all the channels under consideration. But before that, let us
present the values of $\mathrm{PC}$ for different $J^{P(C)}$ states of heavy-
light hybrid mesons at average value of $s_{0}$ and
$[M_{\mathrm{min}}^{2},M_{\mathrm{max}}^{2}]$ in Tables 1-4. From these
tables, we observe that the values of $\mathrm{PC}$ at lower and higher limits
of $M^{2}$ are obtained to be very high at bottom channels compared to the
charmed ones. However, at the worst case which belongs to the charm-nonstrange
hybrid mesons with $1^{-(-)}$ the $\mathrm{PC}$ is obtained to be $20\%$ at
$M_{\mathrm{max}}^{2}$, which is acceptable for the exotic hadrons.

Table 1: Values of PC for different quantum numbers of the ground state charm-nonstrange hybrid mesons.  $J^{P(C)}$ | $s_{0}(average)\ (GeV^{2})$ | $[M_{\mathrm{min}}^{2},M_{\mathrm{max}}^{2}]\ (GeV^{2})$ | $\mathrm{PC}$
---|---|---|---
$0^{+(+)}$ | $26.0$ | $[6.0,7.0]$ | $[0.36,0.22]$
$0^{+(-)}$ | $22.0$ | $[5.0,7.0]$ | $[0.64,0.37]$
$0^{-(-)}$ | $32.0$ | $[6.0,8.0]$ | $[0.50,0.35]$
$0^{-(+)}$ | $22.0$ | $[5.0,7.0]$ | $[0.66,0.39]$
$1^{+(+)}$ | $22.0$ | $[5.0,7.0]$ | $[0.41,0.21]$
$1^{+(-)}$ | $22.0$ | $[5.0,7.0]$ | $[0.55,0.30]$
$1^{-(-)}$ | $22.0$ | $[5.0,7.0]$ | $[0.38,0.20]$
$1^{-(+)}$ | $22.0$ | $[5.0,7.0]$ | $[0.55,0.30]$
Table 2: Values of PC for different quantum numbers of the ground state charm-strange hybrid mesons.  $J^{P(C)}$ | $s_{0}(average)\ (GeV^{2})$ | $[M_{\mathrm{min}}^{2},M_{\mathrm{max}}^{2}]\ (GeV^{2})$ | $\mathrm{PC}$
---|---|---|---
$0^{+(+)}$ | $28.0$ | $[6.0,8.0]$ | $[0.46,0.29]$
$0^{+(-)}$ | $24.0$ | $[5.0,7.0]$ | $[0.69,0.42]$
$0^{-(-)}$ | $34.0$ | $[6.0,8.0]$ | $[0.55,0.40]$
$0^{-(+)}$ | $24.0$ | $[5.0,7.0]$ | $[0.72,0.45]$
$1^{+(+)}$ | $24.0$ | $[5.0,7.0]$ | $[0.49,0.28]$
$1^{+(-)}$ | $24.0$ | $[5.0,7.0]$ | $[0.62,0.35]$
$1^{-(-)}$ | $24.0$ | $[5.0,7.0]$ | $[0.47,0.26]$
$1^{-(+)}$ | $24.0$ | $[5.0,7.0]$ | $[0.63,0.36]$
Table 3: Values of PC for different quantum numbers of the ground state bottom-nonstrange hybrid mesons.  $J^{P(C)}$ | $s_{0}(average)\ (GeV^{2})$ | $[M_{\mathrm{min}}^{2},M_{\mathrm{max}}^{2}]\ (GeV^{2})$ | $\mathrm{PC}$
---|---|---|---
$0^{+(+)}$ | $100.0$ | $[10.0,14.0]$ | $[0.91,0.72]$
$0^{+(-)}$ | $95.0$ | $[10.0,14.0]$ | $[0.91,0.70]$
$0^{-(-)}$ | $75.0$ | $[10.0,14.0]$ | $[0.53,0.33]$
$0^{-(+)}$ | $95.0$ | $[10.0,14.0]$ | $[0.92,0.72]$
$1^{+(+)}$ | $95.0$ | $[10.0,14.0]$ | $[0.88,0.65]$
$1^{+(-)}$ | $95.0$ | $[10.0,14.0]$ | $[0.89,0.67]$
$1^{-(-)}$ | $95.0$ | $[10.0,14.0]$ | $[0.87,0.64]$
$1^{-(+)}$ | $95.0$ | $[10.0,14.0]$ | $[0.89,0.67]$
Table 4: Values of PC for different quantum numbers of the ground state bottom-strange hybrid mesons.  $J^{P(C)}$ | $s_{0}(average)\ (GeV^{2})$ | $[M_{\mathrm{min}}^{2},M_{\mathrm{max}}^{2}]\ (GeV^{2})$ | $\mathrm{PC}$
---|---|---|---
$0^{+(+)}$ | $105.0$ | $[10.0,14.0]$ | $[0.94,0.77]$
$0^{+(-)}$ | $100.0$ | $[10.0,14.0]$ | $[0.93,0.74]$
$0^{-(-)}$ | $80.0$ | $[10.0,14.0]$ | $[0.61,0.40]$
$0^{-(+)}$ | $100.0$ | $[10.0,14.0]$ | $[0.94,0.77]$
$1^{+(+)}$ | $100.0$ | $[10.0,14.0]$ | $[0.91,0.71]$
$1^{+(-)}$ | $100.0$ | $[10.0,14.0]$ | $[0.92,0.72]$
$1^{-(-)}$ | $100.0$ | $[10.0,14.0]$ | $[0.90,0.70]$
$1^{-(+)}$ | $100.0$ | $[10.0,14.0]$ | $[0.92,0.73]$

The mass $m_{H}$ and current coupling $f_{H}$ are demonstrated as functions of
the Borel and continuum threshold parameters in Figs. 1 and 2. We observe that
the results depict good stability against the changes in the value of the
auxiliary parameter $M^{2}$ in its working interval. The residual dependence
of the mass and current coupling on $s_{0}$ constitutes the main source of the
uncertainties that remain below the limits allowed by the method.

Figure 1:  The mass of the charm-nonstrange hybrid meson with the quantum
numbers $J^{P(C)}=1^{-(-)}$ as a function of the Borel parameter $M^{2}$ at
fixed values of $s_{0}$ (left panel), and as a function of the continuum
threshold $s_{0}$ at fixed values of $M^{2}$ (right panel).

Figure 2:  The current coupling of the charm-nonstrange hybrid meson with the
quantum numbers $J^{P(C)}=1^{-(-)}$ as a function of the Borel parameter
$M^{2}$ at fixed values of $s_{0}$ (left panel), and as a function of the
continuum threshold $s_{0}$ at fixed values of $M^{2}$ (right panel).

Obtained from our analyses, the QCD sum-rules results for the mass and current
coupling of the heavy-light hybrid mesons are depicted in Tables 5-8. The
presented uncertainties are related to the errors in the calculations of the
working regions for the auxiliary parameters as well as those exist in the
values of other input parameters. Comparing our results with those of the
existing predictions in Ref. Ho:2016owu , made by Laplace sum rules, i.e.,
states with $J^{P(C)}\in\\{0^{+(+)},1^{-(-)},1^{+(-)}\\}$, our results on the
masses are well consistent with the predictions of Ref. Ho:2016owu . However,
our predictions for the current couplings differ considerably with the
predictions of Ref. Ho:2016owu  using Laplace sum rules for the mentioned
quantum numbers. Let us compare the predictions of two studies for example for
the $1^{-(-)}$ quantum numbers and different quark contents in Table 9. As is
seen from this table, the values of the masses are in nice consistencies
between the present work and Ref. Ho:2016owu  within the presented errors. In
the case of current couplings, the results of Ref. Ho:2016owu  were presented
dimensionless and without uncertainties, but when we change them to $GeV^{3}$,
the results of Ref. Ho:2016owu  are about (2-4) times greater than our
predictions. As we also previously mentioned, Ref. Ho:2016owu  finds unstable
and non-reliable results for the states with
$J^{P(C)}\in\\{0^{+(-)},1^{+(+)},1^{-(+)}\\}$. The current couplings are among
main input parameters to investigate the interactions of heavy-light hybrid
mesons with other particles as well as their strong, electromagnetic and weak
decays.

Table 5: Values of the mass and current coupling for different quantum numbers of the ground state charm-nonstrange hybrid mesons.  $J^{P(C)}$ | $M^{2}\pm\delta M^{2}\ (GeV^{2})$ | $s_{0}\pm\delta s_{0}\ (GeV^{2})$ | $m_{H}\pm\delta m_{H}\ (GeV)$ | $(f_{H}\pm\delta f_{H})\times 10\ (GeV^{3})$
---|---|---|---|---
$0^{+(+)}$ | $7.0\pm 1.0$ | $26.0\pm 2.0$ | $4.86\pm 0.15$ | $0.68\pm 0.13$
$0^{+(-)}$ | $6.0\pm 1.0$ | $22.0\pm 2.0$ | $3.73\pm 0.25$ | $0.62\pm 0.10$
$0^{-(-)}$ | $7.0\pm 1.0$ | $32.0\pm 2.0$ | $5.50\pm 0.26$ | $1.13\pm 0.28$
$0^{-(+)}$ | $6.0\pm 1.0$ | $22.0\pm 2.0$ | $3.73\pm 0.23$ | $0.66\pm 0.10$
$1^{+(+)}$ | $6.0\pm 1.0$ | $22.0\pm 2.0$ | $4.22\pm 0.15$ | $0.64\pm 0.13$
$1^{+(-)}$ | $6.0\pm 1.0$ | $22.0\pm 2.0$ | $3.88\pm 0.23$ | $0.73\pm 0.14$
$1^{-(-)}$ | $6.0\pm 1.0$ | $22.0\pm 2.0$ | $4.36\pm 0.15$ | $0.65\pm 0.14$
$1^{-(+)}$ | $6.0\pm 1.0$ | $22.0\pm 2.0$ | $3.91\pm 0.22$ | $0.74\pm 0.13$
Table 6: Values of the mass and current coupling for different quantum numbers of the ground state charm-strange hybrid mesons.  $J^{P(C)}$ | $M^{2}\pm\delta M^{2}\ (GeV^{2})$ | $s_{0}\pm\delta s_{0}\ (GeV^{2})$ | $m_{H}\pm\delta m_{H}\ (GeV)$ | $(f_{H}\pm\delta f_{H})\times 10\ (GeV^{3})$
---|---|---|---|---
$0^{+(+)}$ | $7.0\pm 1.0$ | $28.0\pm 2.0$ | $4.91\pm 0.14$ | $0.82\pm 0.13$
$0^{+(-)}$ | $6.0\pm 1.0$ | $24.0\pm 2.0$ | $3.88\pm 0.25$ | $0.67\pm 0.11$
$0^{-(-)}$ | $7.0\pm 1.0$ | $34.0\pm 2.0$ | $5.61\pm 0.25$ | $1.23\pm 0.31$
$0^{-(+)}$ | $6.0\pm 1.0$ | $24.0\pm 2.0$ | $3.85\pm 0.24$ | $0.76\pm 0.11$
$1^{+(+)}$ | $6.0\pm 1.0$ | $24.0\pm 2.0$ | $4.38\pm 0.13$ | $0.77\pm 0.14$
$1^{+(-)}$ | $6.0\pm 1.0$ | $24.0\pm 2.0$ | $4.05\pm 0.23$ | $0.83\pm 0.15$
$1^{-(-)}$ | $6.0\pm 1.0$ | $24.0\pm 2.0$ | $4.44\pm 0.13$ | $0.76\pm 0.13$
$1^{-(+)}$ | $6.0\pm 1.0$ | $24.0\pm 2.0$ | $4.03\pm 0.22$ | $0.86\pm 0.15$
Table 7: Values of the mass and current coupling for different quantum numbers of the ground state bottom-nonstrange hybrid mesons.  $J^{P(C)}$ | $M^{2}\pm\delta M^{2}\ (GeV^{2})$ | $s_{0}\pm\delta s_{0}\ (GeV^{2})$ | $m_{H}\pm\delta m_{H}\ (GeV)$ | $(f_{H}\pm\delta f_{H})\times 10\ (GeV^{3})$
---|---|---|---|---
$0^{+(+)}$ | $12.0\pm 2.0$ | $100.0\pm 5.0$ | $8.44\pm 0.25$ | $2.48\pm 0.45$
$0^{+(-)}$ | $12.0\pm 2.0$ | $95.0\pm 5.0$ | $7.92\pm 0.38$ | $2.07\pm 0.51$
$0^{-(-)}$ | $12.0\pm 2.0$ | $75.0\pm 5.0$ | $8.31\pm 0.28$ | $1.58\pm 0.45$
$0^{-(+)}$ | $12.0\pm 2.0$ | $95.0\pm 5.0$ | $7.84\pm 0.38$ | $2.10\pm 0.51$
$1^{+(+)}$ | $12.0\pm 2.0$ | $95.0\pm 5.0$ | $8.24\pm 0.30$ | $2.65\pm 0.57$
$1^{+(-)}$ | $12.0\pm 2.0$ | $95.0\pm 5.0$ | $8.08\pm 0.35$ | $2.56\pm 0.61$
$1^{-(-)}$ | $12.0\pm 2.0$ | $95.0\pm 5.0$ | $8.32\pm 0.26$ | $2.70\pm 0.54$
$1^{-(+)}$ | $12.0\pm 2.0$ | $95.0\pm 5.0$ | $8.07\pm 0.34$ | $2.58\pm 0.35$
Table 8: Values of the mass and current coupling for different quantum numbers of the ground state bottom-strange hybrid mesons.  $J^{P(C)}$ | $M^{2}\pm\delta M^{2}\ (GeV^{2})$ | $s_{0}\pm\delta s_{0}\ (GeV^{2})$ | $m_{H}\pm\delta m_{H}\ (GeV)$ | $(f_{H}\pm\delta f_{H})\times 10\ (GeV^{3})$
---|---|---|---|---
$0^{+(+)}$ | $12.0\pm 2.0$ | $105.0\pm 5.0$ | $8.50\pm 0.26$ | $2.74\pm 0.51$
$0^{+(-)}$ | $12.0\pm 2.0$ | $100.0\pm 5.0$ | $8.04\pm 0.39$ | $2.18\pm 0.55$
$0^{-(-)}$ | $12.0\pm 2.0$ | $80.0\pm 5.0$ | $8.46\pm 0.27$ | $1.80\pm 0.48$
$0^{-(+)}$ | $12.0\pm 2.0$ | $100.0\pm 5.0$ | $7.92\pm 0.39$ | $2.31\pm 0.57$
$1^{+(+)}$ | $12.0\pm 2.0$ | $100.0\pm 5.0$ | $8.35\pm 0.30$ | $2.92\pm 0.62$
$1^{+(-)}$ | $12.0\pm 2.0$ | $100.0\pm 5.0$ | $8.20\pm 0.35$ | $2.78\pm 0.67$
$1^{-(-)}$ | $12.0\pm 2.0$ | $100.0\pm 5.0$ | $8.41\pm 0.28$ | $2.92\pm 0.59$
$1^{-(+)}$ | $12.0\pm 2.0$ | $100.0\pm 5.0$ | $8.16\pm 0.36$ | $2.81\pm 0.67$
Table 9: Comparison of the values for the mass and current coupling for quantum numbers $J^{P(C)}=1^{-(-)}$ of the different heavy-light hybrid mesons between the present work (PW) and Ref. Ho:2016owu . The masses are in $GeV$ and the current couplings are in $GeV^{3}$. $J^{P(C)}=1^{-(-)}$ |  $m_{H}$ (PW) |  $f_{H}\times 10$ (PW) |  $m_{H}$ Ho:2016owu  |  $f_{H}\times 10$ Ho:2016owu
---|---|---|---|---
charm-nonstrange | $4.36\pm 0.15$ | $0.65\pm 0.14$ | $4.40\pm 0.19$ | $2.99$
charm-strange | $4.44\pm 0.13$ | $0.76\pm 0.13$ | $4.28\pm 0.19$ | $2.60$
bottom-nonstrange | $8.32\pm 0.26$ | $2.70\pm 0.54$ | $8.74\pm 0.25$ | $8.87$
bottom-strange | $8.41\pm 0.28$ | $2.92\pm 0.59$ | $8.46\pm 0.32$ | $6.72$
Table 10: Comparison of the values for the mass and current coupling for states with quantum numbers $J^{P(C)}=1^{-(-)}$ of the different heavy-light hybrid mesons obtained by including non-perturbative effects up to six and ten dimensions. The primed masses and decay constants belong to the results up to dimension six and those without prime show the results up to ten mass dimensions. The masses are in $GeV$ and the current couplings are in $GeV^{3}$. $J^{P(C)}=1^{-(-)}$ |  $m_{H^{\prime}}$ (PW) |  $f_{H^{\prime}}\times 10$ (PW) |  $m_{H}$ (PW) |  $f_{H}\times 10$ (PW)
---|---|---|---|---
charm-nonstrange | $4.32\pm 0.14$ | $0.64\pm 0.13$ | $4.36\pm 0.15$ | $0.65\pm 0.14$
charm-strange | $4.40\pm 0.13$ | $0.75\pm 0.13$ | $4.44\pm 0.13$ | $0.76\pm 0.13$
bottom-nonstrange | $8.30\pm 0.28$ | $2.69\pm 0.55$ | $8.32\pm 0.26$ | $2.70\pm 0.54$
bottom-strange | $8.39\pm 0.29$ | $2.90\pm 0.61$ | $8.41\pm 0.28$ | $2.92\pm 0.59$

Figure 3: The perturbative and non-perturbative contributions in units of
$GeV^{8}$ to the sum rule of the charm-nonstrange hybrid meson with the
quantum numbers $J^{P(C)}=1^{-(-)}$: Perturbative contribution (left panel)
and absolute values of different non-perturbative contributions (right panel)
as functions of $M^{2}$ at average $s_{0}$.

Regarding the comparison of our results with those of Ref. Ho:2016owu  and our
reason for making improvements on the subject, we would like to present more
details. As we previously mentioned, Ref. Ho:2016owu  consider the analyses up
to six-dimension non-perturbative operators, while we make our analyses
considering these effects up to ten mass dimensions. Despite, our results on
the masses of some states are in nice consistency with those of Ref.
Ho:2016owu  , the values of decay constants obtained in these two studies are
very different and Ref. Ho:2016owu  finds the results of masses and decay
constants for some states to be unstable. Now, a question is raised: Do these
conclusions are still valid when we consider the analyses up to six
dimensional non-perturbative operators? What are the contributions of higher
dimensional operators? To answer this questions let us first compare our
results for instance for $J^{P(C)}=1^{-(-)}$ states obtained by including non-
perturbative effects up to six and ten dimensions in Table 10. As is seen from
this table, the higher dimensions (seven to ten non-pertubative) show very
small contributions and the results with six and ten dimensions do not differ
considerably. As we previously mentioned, we have a nice convergence of the
OPE and it is also clear from figure 3, which depicts the perturbative and
different non-perturbative contributions for the sum rules of the charm-
nonstrange hybrid meson with the quantum numbers $J^{P(C)}=1^{-(-)}$, as an
example. This figure shows that the main contribution comes from the
perturbative part and the higher dimensional non-perturbative operators have
very small contributions referring to the nice convergence of the OPE in the
analyses. Considering these evidences, the differences between our results and
those of Ref. Ho:2016owu  can be attributed to different methods of analyses
that these two studies use.

##  IV Summary and Conclusion

In this article, we investigated the the mass and current coupling of the
scalar, pseodoscalar, vector and axial-vector heavy-light hybrid mesons with
different quantum numbers and quark contents employing the QCD two-point Borel
sum rule method by choosing appropriate interpolating currents. We restricted
the auxiliary parameters entered the sum rules based on the standard
prescriptions of the method and included into analyses the quark-gluon
condensations up to dimension 10. We got stable and reliable results for the
mass and current couplings for all the considered quantum numbers and quark
contents with respect to the variations of the auxiliary parameters in their
working windows. We extracted the values of the mass and current coupling for
various $J^{P(C)}$ and quark contents and presented them in Tables 5-8. Our
predictions for the mass of the states with
$J^{P(C)}\in\\{0^{+(+)},1^{-(-)},1^{+(-)}\\}$ are in good agreements with the
existing predictions of the Ref. Ho:2016owu  made using Laplace sum rules. The
obtained results for the current couplings of the hybrid mesons with
$J^{P(C)}\in\\{0^{+(+)},1^{-(-)},1^{+(-)}\\}$ differ considerably with those
of the Ref. Ho:2016owu  when they are written in the same dimensions.
Comparison of the theoretical results on various parameters of the hybrid
states obtained in the present study with possible future experimental data
will shed light on the nature, quark-gluon organization and quantum numbers of
these states. The obtained values for the spectroscopic parameters of the
heavy-light hybrid mesons with different spin-parities can be used in
investigation of their strong, electromagnetic and weak decays as well as
their various interactions with other particles.

## ACKNOWLEDGMENTS

K. Azizi is thankful to Iran Science Elites Federation (Saramadan) for the
partial financial support provided under the grant number ISEF/M/400150.

*

##  Appendix A  Invariant amplitude $\Pi(M^{2},s_{0})$ for the charm-
nonstrange hybrid meson with the quantum numbers $J^{P(C)}=1^{-(-)}$

The invariant amplitude $\Pi(M^{2},s_{0})$ obtained after the Borel
transformation and subtraction procedures is given by Eq. (26). Here, we
present the explicit expressions of the spectral density
$\rho^{\mathrm{OPE}}(s)$ and the function $\Pi(M^{2})$ as an example for the
charm-nonstrange hybrid meson with the quantum numbers $J^{P(C)}=1^{-(-)}$,
which are written as

$\rho^{\mathrm{OPE}}(s)=\rho^{\mathrm{pert.}}(s)+\sum_{N=3}^{6}\rho^{\mathrm{DimN}}(s),\
\ \Pi(M^{2})=\sum_{N=6}^{10}\Pi^{\mathrm{DimN}}(M^{2}),$ (A.30)

respectively. The components of $\rho^{\mathrm{OPE}}(s)$ and $\Pi(M^{2})$ are
given by the expressions

$\rho^{\mathrm{DimN}}(s)=\int_{0}^{1}d\alpha\rho^{\mathrm{DimN}}(s,\alpha),$
(A.31)

and

$\Pi^{\mathrm{DimN}}(M^{2})=\int_{0}^{1}d\alpha\Pi^{\mathrm{DimN}}(M^{2},\alpha).$
(A.32)

depending on whether $\rho$ and $\Pi(M^{2})$ are functions of $\alpha$. In
Eqs. (A.31) and (A.32) the variable $\alpha$ is the Feynman parameter.

The perturbative and nonperturbative components of the spectral density
$\rho^{\mathrm{pert.}}(s,\alpha)$ and $\rho^{\mathrm{Dim3(4,5,6)}}(s,\alpha)$
have the forms:

$\displaystyle\rho^{\mathrm{pert.}}(s,\alpha)=\frac{g_{s}^{2}(m_{Q}^{2}+sL)^{2}\alpha^{2}[6m_{q}m_{Q}(-3+2\alpha)+\alpha[m_{Q}^{2}(-10+3\alpha)+2s(14-17\alpha+3\alpha^{2})]]}{2^{9}\cdot
3^{2}\pi^{4}L^{2}},$ (A.33)
$\rho^{\mathrm{Dim3}}(s,\alpha)=\frac{g_{s}^{2}\langle\overline{q}q\rangle\alpha[-2m_{Q}(m_{Q}^{2}+sL)+m_{Q}[m_{Q}^{2}(-5+3\alpha)+4s(2-3\alpha+\alpha^{2})]]}{2^{5}\cdot
3\pi^{2}},$ (A.34)
$\displaystyle\rho^{\mathrm{Dim4}}(s,\alpha)=-\frac{\langle\alpha_{s}G^{2}/\pi\rangle
g_{s}^{2}\alpha^{2}[-16m_{q}m_{Q}L\alpha+sL^{2}(3+8\alpha)+m_{Q}^{2}(-3+37\alpha-6\alpha^{2})]}{2^{11}\cdot
3^{3}\pi^{2}L^{2}}$ $\displaystyle+\frac{1}{2^{5}\cdot
3}\langle\alpha_{s}G^{2}/\pi\rangle[3m_{q}m_{Q}+(3m_{Q}^{2}+4sL)\alpha],$
(A.35)
$\displaystyle\rho^{\mathrm{Dim5}}(s,\alpha)=\frac{g_{s}^{2}m_{o}^{2}\langle\overline{q}q\rangle[3m_{Q}L+m_{q}(-5+11\alpha-6\alpha^{2})]}{2^{6}\cdot
3^{2}\pi^{2}},$ (A.36)
$\displaystyle\rho^{\mathrm{Dim6}}(s,\alpha)=\frac{g_{s}^{4}\langle\overline{q}q\rangle^{2}(5-11\alpha+6\alpha^{2})}{2^{4}\cdot
3^{5}\pi^{2}},$ (A.37)

Components of the function $\Pi(M^{2})$ are:

$\displaystyle\Pi^{\mathrm{Dim6}}(M^{2},\alpha)=\frac{g_{s}^{2}\langle
g_{s}^{3}G^{3}\rangle\alpha^{2}m_{Q}^{2}}{2^{5}\cdot
3L^{5}\pi^{4}}\exp\left[\frac{m_{Q}^{2}}{M^{2}L}\right]\Bigg{\\{}-\frac{m_{q}m_{Q}^{3}}{2^{3}\cdot
3\cdot 5M^{4}L}-\frac{m_{q}m_{Q}^{3}}{2^{4}\cdot 3\cdot
5M^{4}}-\frac{m_{q}m_{Q}}{2^{2}\cdot 3\cdot
5M^{2}}-\frac{m_{Q}^{4}\alpha}{2^{5}\cdot 3\cdot 5M^{4}L^{2}}$
$\displaystyle+\frac{m_{q}m_{Q}^{3}\alpha}{3^{2}\cdot
5M^{4}L}-\frac{m_{Q}^{4}\alpha}{2^{5}\cdot
5M^{4}L}-\frac{m_{Q}^{2}\alpha}{2^{5}\cdot 3\cdot
5M^{2}L}+\frac{\alpha}{2^{5}\cdot 3\cdot
5}+\frac{m_{q}m_{Q}^{3}\alpha}{2^{3}\cdot 3^{2}\cdot
5M^{4}}+\frac{7m_{q}m_{Q}\alpha}{2^{2}\cdot 3^{2}\cdot
5M^{2}}-\frac{m_{Q}^{2}\alpha}{2^{5}\cdot 3\cdot 5M^{2}}$
$\displaystyle+\frac{19m_{Q}^{4}\alpha^{2}}{2^{6}\cdot 3^{2}\cdot
5M^{4}L^{2}}-\frac{7m_{q}m_{Q}^{3}\alpha^{2}}{2^{3}\cdot 3^{2}\cdot
5M^{4}L}+\frac{7m_{Q}^{4}\alpha^{2}}{2^{6}\cdot 3\cdot
5M^{4}L}+\frac{17m_{Q}^{2}\alpha^{2}}{2^{6}\cdot 3^{2}\cdot
5M^{2}L}-\frac{\alpha^{2}}{2^{5}\cdot
5}-\frac{m_{q}m_{Q}\alpha^{2}}{2^{2}\cdot
3^{2}M^{2}}+\frac{m_{Q}^{2}\alpha^{2}}{2^{6}\cdot 3\cdot 5M^{2}}$
$\displaystyle-\frac{7m_{Q}^{4}\alpha^{3}}{2^{6}\cdot 3\cdot
5M^{4}L^{2}}+\frac{m_{q}m_{Q}^{3}\alpha^{3}}{2^{2}\cdot 3^{2}\cdot
5M^{4}L}-\frac{m_{Q}^{4}\alpha^{3}}{2^{6}\cdot 3\cdot
5M^{4}L}-\frac{m_{Q}^{2}\alpha^{3}}{2^{6}\cdot
3M^{2}L}+\frac{\alpha^{3}}{2^{5}\cdot
5}+\frac{m_{q}m_{Q}\alpha^{3}}{2^{2}\cdot 3^{2}\cdot
5M^{2}}+\frac{m_{Q}^{2}\alpha^{3}}{2^{6}\cdot 3\cdot 5M^{2}}$
$\displaystyle+\frac{m_{Q}^{4}\alpha^{4}}{2^{6}\cdot
5M^{4}L^{2}}+\frac{m_{Q}^{2}\alpha^{4}}{2^{6}\cdot 3\cdot
5M^{2}L}-\frac{\alpha^{4}}{2^{5}\cdot 3\cdot
5}-\frac{m_{Q}^{4}\alpha^{5}}{2^{6}\cdot 3^{2}\cdot
5M^{4}L^{2}}+\frac{m_{Q}^{2}\alpha^{5}}{2^{6}\cdot 3^{2}\cdot
5M^{2}L}\Bigg{\\}},$ (A.38)

$\displaystyle\Pi^{\mathrm{Dim7}}(M^{2},\alpha)=\frac{\langle\alpha_{s}G^{2}/\pi\rangle\langle\overline{q}q\rangle}{2^{3}\cdot
3}\exp\left[-\frac{m_{Q}^{2}}{M^{2}}\right]\left\\{-\frac{g_{s}^{2}m_{Q}}{2^{3}\cdot
3^{2}\pi}+\pi^{2}(\frac{m_{q}}{3}+m_{Q}+\frac{m_{q}m_{Q}^{2}}{2\cdot
3M^{2}})\right\\},$ (A.39)

$\displaystyle\Pi^{\mathrm{Dim8}}(M^{2},\alpha)=\frac{\langle\alpha_{s}G^{2}/\pi\rangle^{2}\pi^{2}}{2^{4}\cdot
3^{2}L}\exp\left[\frac{m_{Q}^{2}}{M^{2}L}\right]\Bigg{\\{}\frac{m_{q}m_{Q}^{3}}{2^{2}M^{4}L^{3}}+\frac{m_{q}m_{Q}}{2M^{2}L^{2}}+\frac{5m_{Q}^{2}}{2^{5}\cdot
3M^{2}L}-\frac{1}{2^{3}\cdot 3}$
$\displaystyle-\frac{m_{q}m_{Q}^{3}\alpha}{2M^{4}L^{3}}-\frac{m_{Q}^{4}\alpha}{2^{2}\cdot
3M^{4}L^{3}}-\frac{m_{q}m_{Q}\alpha}{M^{2}L^{2}}+\frac{m_{Q}^{2}\alpha}{2^{2}\cdot
3M^{2}L^{2}}-\frac{5m_{Q}^{2}\alpha}{2^{5}\cdot
3M^{2}L}+\frac{\alpha}{2^{3}\cdot
3}+\frac{m_{q}m_{Q}^{3}\alpha^{2}}{2^{2}M^{4}L^{3}}$
$\displaystyle+\frac{m_{Q}^{4}\alpha^{2}}{2^{2}\cdot
3M^{4}L^{3}}+\frac{m_{q}m_{Q}\alpha^{2}}{2M^{2}L^{2}}-\frac{m_{Q}^{2}\alpha^{2}}{2^{2}\cdot
3M^{2}L^{2}}\Bigg{\\}},$ (A.40)

$\displaystyle\Pi^{\mathrm{Dim9}}(M^{2},\alpha)=\frac{\langle\alpha_{s}G^{2}/\pi\rangle\langle\overline{q}q\rangle
m_{o}^{2}}{2^{3}\cdot
3M^{2}}\exp\left[\frac{m_{Q}^{2}}{M^{2}L}\right]\Bigg{\\{}\frac{g_{s}^{2}}{L^{2}}(-\frac{m_{q}m_{Q}^{4}}{2^{4}\cdot
3^{2}M^{4}L^{2}}-\frac{m_{Q}^{3}}{2^{4}\cdot
3^{2}M^{2}L^{2}}-\frac{m_{q}m_{Q}^{2}}{2^{2}\cdot
3^{3}M^{2}L}-\frac{m_{Q}}{2^{3}\cdot 3^{2}L}$
$\displaystyle+\frac{5m_{q}m_{Q}^{4}\alpha}{2^{4}\cdot
3^{3}M^{4}L^{2}}+\frac{m_{Q}^{3}\alpha}{2^{2}\cdot
3^{2}M^{2}L^{2}}+\frac{m_{q}m_{Q}^{2}\alpha}{2^{2}\cdot
3^{3}M^{2}L}+\frac{m_{Q}\alpha}{2^{3}\cdot 3L}+\frac{m_{Q}\alpha}{2^{3}\cdot
3^{2}}-\frac{m_{q}m_{Q}^{4}\alpha^{2}}{2^{3}\cdot
3^{3}M^{4}L^{2}}-\frac{5m_{Q}^{3}\alpha^{2}}{2^{4}\cdot 3^{2}M^{2}L^{2}}$
$\displaystyle-\frac{m_{Q}\alpha^{2}}{2^{3}\cdot
3L}+\frac{m_{Q}\alpha^{2}}{2^{2}\cdot
3^{2}}+\frac{m_{Q}^{3}\alpha^{3}}{2^{3}\cdot
3^{2}M^{2}L^{2}}+\frac{m_{Q}\alpha^{3}}{2^{3}\cdot
3^{2}L})-\pi^{2}(\frac{m_{q}m_{Q}^{4}}{2^{2}\cdot
3^{2}M^{4}}-\frac{m_{Q}}{M^{2}}+\frac{13m_{q}m_{Q}^{2}}{2^{2}\cdot
3^{2}M^{2}}-\frac{29m_{q}}{2^{2}\cdot 3^{2}})\Bigg{\\}},$ (A.41)

$\displaystyle\Pi^{\mathrm{Dim10}}(M^{2},\alpha)=\frac{\langle\alpha_{s}G^{2}/\pi\rangle}{2^{3}\cdot
3^{3}M^{2}}\exp\left[\frac{m_{Q}^{2}}{M^{2}L}\right]\Bigg{\\{}\dfrac{\langle\overline{q}q\rangle^{2}g_{s}^{4}}{3^{4}L^{3}}(\frac{m_{Q}^{4}}{2^{2}M^{4}L}+\frac{m_{Q}^{2}}{3M^{2}}-\frac{5m_{Q}^{4}\alpha}{2^{2}\cdot
3M^{4}L}-\frac{m_{Q}^{2}\alpha}{3M^{2}}$
$\displaystyle+\frac{m_{Q}^{4}\alpha^{2}}{2\cdot
3M^{4}L})+\frac{\langle\overline{q}q\rangle^{2}\pi^{2}g_{s}^{2}}{3^{4}}(\frac{m_{Q}^{4}}{M^{4}}-\frac{13m_{Q}^{2}}{M^{2}}+29)+\frac{g_{s}^{3}G^{3}}{M^{4}L^{5}}(\frac{m_{q}m_{Q}^{5}}{2^{3}\cdot
5M^{2}L}+\frac{m_{q}m_{Q}^{5}}{2^{4}\cdot
5M^{2}L}+\frac{m_{q}m_{Q}^{3}}{2\cdot 5}+\frac{m_{Q}^{6}\alpha}{2^{5}\cdot
3\cdot 5M^{2}L^{2}}$ $\displaystyle-\frac{m_{q}m_{Q}^{5}\alpha}{2^{2}\cdot
5M^{2}L}+\frac{m_{Q}^{6}\alpha}{2^{5}\cdot
5M^{2}L}+\frac{m_{Q}^{4}\alpha}{2^{3}\cdot 3\cdot
5L}-\frac{m_{q}m_{Q}^{3}\alpha}{5}+\frac{m_{Q}^{4}\alpha}{2^{5}\cdot
5}-\frac{m_{Q}^{6}\alpha^{2}}{2^{5}\cdot
5M^{2}L^{2}}+\frac{m_{q}m_{Q}^{5}\alpha^{2}}{2^{3}\cdot
5M^{2}L}-\frac{m_{Q}^{6}\alpha^{2}}{2^{5}\cdot
5M^{2}L}-\frac{m_{Q}^{4}\alpha^{2}}{2^{3}\cdot 5L}$
$\displaystyle+\frac{m_{q}m_{Q}^{3}\alpha^{2}}{2\cdot
5}-\frac{m_{Q}^{4}\alpha^{2}}{2^{5}\cdot
5}+\frac{m_{Q}^{6}\alpha^{3}}{2^{5}\cdot
5M^{2}L^{2}}+\frac{m_{Q}^{4}\alpha^{3}}{2^{3}\cdot
5L}-\frac{m_{Q}^{6}\alpha^{4}}{2^{5}\cdot 3\cdot
5M^{2}L^{2}}-\frac{m_{Q}^{4}\alpha^{4}}{2^{3}\cdot 3\cdot 5L})\Bigg{\\}},$
(A.42)

where have used the following short-hand notation:

$\displaystyle L=\alpha-1.\ $ (A.43)

## References

  * (1) S. K. Choi et al. [Belle], “Observation of a narrow charmonium-like state in exclusive $B^{\pm}\to K^{\pm}\pi^{+}\pi^{-}J/\psi$ decays,”Phys. Rev. Lett. 91, 262001 (2003), [arXiv:hep-ex/0309032 [hep-ex]].
  * (2) V. M. Abazov et al. [D0], “Evidence for a $B_{s}^{0}\pi^{\pm}$ state,”Phys. Rev. Lett. 117, no.2, 022003 (2016), [arXiv:1602.07588 [hep-ex]].
  * (3) R. Aaij et al. [LHCb],
“Observation of $J/\psi\phi$ structures consistent with exotic states from
amplitude analysis of $B^{+}\to J/\psi\phi K^{+}$ decays,” Phys. Rev. Lett.
118, no.2, 022003 (2017) [arXiv:1606.07895 [hep-ex]].

  * (4) R. Aaij et al. [LHCb], “ Amplitude analysis of $B^{+}\to J/\psi\phi K^{+}$ decays,”Phys. Rev. D 95, no.1, 012002 (2017), [arXiv:1606.07898 [hep-ex]].
  * (5) X. L. Wang et al. [Belle], “Observation of Two Resonant Structures in $e^{+}e^{-}\to\pi^{+}\pi^{-}\psi(2S)$ via Initial State Radiation at Belle,”Phys. Rev. Lett. 99, 142002 (2007), [arXiv:0707.3699 [hep-ex]].
  * (6) X. L. Wang et al. [Belle], “Measurement of $e^{+}e^{-}\to\pi^{+}\pi^{-}\psi(2S)$ via Initial State Radiation at Belle,”Phys. Rev. D 91, 112007 (2015), [arXiv:1410.7641 [hep-ex]].
  * (7) R. Aaij et al. [LHCb], “Observation of $J/\psi p$ Resonances Consistent with Pentaquark States in $\Lambda_{b}^{0}\to J/\psi K^{-}p$ Decays,”Phys. Rev. Lett. 115, 072001 (2015), [arXiv:1507.03414 [hep-ex]].
  * (8) R. Aaij et al. [LHCb], “Evidence for exotic hadron contributions to $\Lambda_{b}^{0}\to J/\psi p\pi^{-}$ decays,”Phys. Rev. Lett. 117, no.8, 082003 (2016), [arXiv:1606.06999 [hep-ex]].
  * (9) R. Aaij et al. [LHCb], “Observation of a narrow pentaquark state, $P_{c}(4312)^{+}$, and of two-peak structure of the $P_{c}(4450)^{+}$,”Phys. Rev. Lett. 122, no.22, 222001 (2019), [arXiv:1904.03947 [hep-ex]].
  * (10) H. X. Chen, W. Chen, X. Liu, Y. R. Liu and S. L. Zhu, “An updated review of the new hadron states,”[arXiv:2204.02649 [hep-ph]].
  * (11) L. Meng, B. Wang, G. J. Wang and S. L. Zhu, “Chiral perturbation theory for heavy hadrons and chiral effective field theory for heavy hadronic molecules,” [arXiv:2204.08716 [hep-ph]].
  * (12) R. L. Jaffe and K. Johnson, “Unconventional States of Confined Quarks and Gluons,”Phys. Lett. B 60, 201-204 (1976).
  * (13) D. Horn and J. Mandula, “A Model of Mesons with Constituent Gluons,”Phys. Rev. D 17, 898 (1978).
  * (14) T. Barnes, F. E. Close and E. S. Swanson, “Hybrid and conventional mesons in the flux tube model: Numerical studies and their phenomenological implications,”Phys. Rev. D 52, 5242-5256 (1995), [arXiv:hep-ph/9501405 [hep-ph]].
  * (15) S. Perantonis and C. Michael, “Static potentials and hybrid mesons from pure SU(3) lattice gauge theory,”Nucl. Phys. B 347, 854-868 (1990)].
  * (16) Y. Liu and X. Q. Luo, “Estimate of the charmed 0– hybrid meson spectrum from quenched lattice QCD,”Phys. Rev. D 73, 054510 (2006), [arXiv:hep-lat/0511015 [hep-lat]].
  * (17) L. Liu et al. [Hadron Spectrum], “Excited and exotic charmonium spectroscopy from lattice QCD,”JHEP 07, 126 (2012), [arXiv:1204.5425 [hep-ph]].
  * (18) L. Liu, S. M. Ryan, M. Peardon, G. Moir and P. Vilaseca, “Charmonium spectroscopy from an anisotropic lattice study,”PoS LATTICE2011, 140 (2011), [arXiv:1112.1358 [hep-lat]].
  * (19) G. K. C. Cheung et al. [Hadron Spectrum], “Excited and exotic charmonium, $D_{s}$ and $D$ meson spectra for two light quark masses from lattice QCD,”JHEP 12, 089 (2016), [arXiv:1610.01073 [hep-lat]].
  * (20) J. Govaerts, L. J. Reinders, H. R. Rubinstein and J. Weyers, “Hybrid Quarkonia From QCD Sum Rules,”Nucl. Phys. B 258, 215-229 (1985).
  * (21) J. Govaerts, L. J. Reinders and J. Weyers, “Radial Excitations and Exotic Mesons via QCD Sum Rules,”Nucl. Phys. B 262, 575-592 (1985).
  * (22) J. Govaerts, L. J. Reinders, P. Francken, X. Gonze and J. Weyers, “Coupled QCD Sum Rules for Hybrid Mesons,”Nucl. Phys. B 284, 674 (1987).
  * (23) S. L. Zhu, “Hybrid quarkonium masses up to the order of $O(1/m_{Q})$,”Phys. Rev. D 60, 031501 (1999), [arXiv:hep-ph/9812469 [hep-ph]].
  * (24) S. Narison, “$1^{-+}$ light exotic mesons in QCD,”Phys. Lett. B 675, 319-325 (2009), [arXiv:0903.2266 [hep-ph]].
  * (25) C. F. Qiao, L. Tang, G. Hao and X. Q. Li, “Determining $1^{--}$ Heavy Hybrid Masses via QCD Sum Rules,”J. Phys. G 39, 015005 (2012), [arXiv:1012.2614 [hep-ph]].
  * (26) D. Harnett, R. T. Kleiv, T. G. Steele and H. y. Jin, “Axial Vector $J^{PC}=1^{++}$ Charmonium and Bottomonium Hybrid Mass Predictions with QCD Sum-Rules,”J. Phys. G 39, 125003 (2012), [arXiv:1206.6776 [hep-ph]].
  * (27) R. Berg, D. Harnett, R. T. Kleiv and T. G. Steele, “Mass Predictions for Pseudoscalar $J^{PC}=0^{-+}$ Charmonium and Bottomonium Hybrids in QCD Sum-Rules,”Phys. Rev. D 86, 034002 (2012), [arXiv:1204.0049 [hep-ph]].
  * (28) W. Chen, H. y. Jin, R. T. Kleiv, T. G. Steele, M. Wang and Q. Xu, “QCD sum-rule interpretation of X(3872) with $J^{PC}=1^{++}$ mixtures of hybrid charmonium and $\overline{D}D^{*}$ molecular currents,”Phys. Rev. D 88, no.4, 045027 (2013), [arXiv:1305.0244 [hep-ph]].
  * (29) R. T. Kleiv, B. Bulthuis, D. Harnett, T. Richards, W. Chen, J. Ho, T. G. Steele and S. L. Zhu, “Exploring the spectrum of heavy quarkonium hybrids with QCD sum rules,”Can. J. Phys. 93, no.9, 952-955 (2015), [arXiv:1410.6259 [hep-ph]].
  * (30) A. Palameta, J. Ho, D. Harnett and T. G. Steele, “QCD sum-rules analysis of vector ($1^{--}$) heavy quarkonium meson-hybrid mixing,”Phys. Rev. D 97, no.3, 034001 (2018), [arXiv:1707.00063 [hep-ph]].
  * (31) A. Palameta, D. Harnett and T. G. Steele, “Meson-Hybrid Mixing in $J^{PC}=1^{++}$ Heavy Quarkonium from QCD Sum-Rules,”Phys. Rev. D 98, no.7, 074014 (2018), [arXiv:1805.04230 [hep-ph]].
  * (32) M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, “QCD and Resonance Physics. Theoretical Foundations,”Nucl. Phys. B 147, 385-447 (1979).
  * (33) M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, “QCD and Resonance Physics: Applications,”Nucl. Phys. B 147, 448-518 (1979).
  * (34) L. J. Reinders, H. Rubinstein and S. Yazaki, “Hadron Properties from QCD Sum Rules,”Phys. Rept. 127, 1 (1985).
  * (35) S. Narison, “QCD spectral sum rules,” World Sci. Lect. Notes Phys. 26, 1-527 (1989).
  * (36) Z. G. Wang, “Lowest vector tetraquark states: $Y(4260/4220)$ or $Z_{c}(4100)$,”Eur. Phys. J. C 78, no.11, 933 (2018), [arXiv:1809.10299 [hep-ph]].
  * (37) J. Wu, X. Liu, Y. R. Liu and S. L. Zhu, “Systematic studies of charmonium-, bottomonium-, and $B_{c}$-like tetraquark states,”Phys. Rev. D 99, no.1, 014037 (2019), [arXiv:1810.06886 [hep-ph]].
  * (38) M. B. Voloshin, “$Z_{c}(4100)$ and $Z_{c}(4200)$ as hadrocharmonium,”Phys. Rev. D 98, no.9, 094028 (2018), [arXiv:1810.08146 [hep-ph]].
  * (39) X. Cao and J. P. Dai, “Spin parity of $Z_{c}^{-}$(4100), $Z_{1}^{+}$(4050) and $Z_{2}^{+}$(4250),”Phys. Rev. D 100, no.5, 054004 (2019), [arXiv:1811.06434 [hep-ph]].
  * (40) S. S. Agaev, K. Azizi and H. Sundu, “Newly observed exotic doubly charmed meson Tcc+,”Nucl. Phys. B 975, 115650 (2022), [arXiv:2108.00188 [hep-ph]].
  * (41) S. Agaev, K. Azizi and H. Sundu, “Four-quark exotic mesons,”Turk. J. Phys. 44, no.2, 95-173 (2020), [arXiv:2004.12079 [hep-ph]].
  * (42) S. S. Agaev, K. Azizi, B. Barsbay and H. Sundu, “Semileptonic and nonleptonic decays of the axial-vector tetraquark $T_{bb;\overline{u}\overline{d}}^{-}$,”Eur. Phys. J. A 57, no.3, 106 (2021), [arXiv:2008.02049 [hep-ph]].
  * (43) H. Sundu, B. Barsbay, S. S. Agaev and K. Azizi, “Probing an axial-vector tetraquark Zs via its semileptonic decay Zs $\rightarrow$ X(4274) $\overline{{l}}$ $\nu_{l}$,”Eur. Phys. J. A 54, no.7, 124 (2018), [arXiv:1804.04525 [hep-ph]].
  * (44) S. S. Agaev, K. Azizi and H. Sundu, “Strong decays of double-charmed pseudoscalar and scalar $cc\overline{u}\overline{d}$ tetraquarks,”Phys. Rev. D 99, no.11, 114016 (2019), [arXiv:1903.11975 [hep-ph]].
  * (45) Z. G. Wang and Q. Xin, “Analysis of hidden-charm pentaquark molecular states with and without strangeness via the QCD sum rules *,”Chin. Phys. C 45, no.12, 123105 (2021), [arXiv:2103.08239 [hep-ph]].
  * (46) K. Azizi, Y. Sarac and H. Sundu, “Investigation of $P_{cs}(4459)^{0}$ pentaquark via its strong decay to $\Lambda J/\Psi$,”Phys. Rev. D 103, no.9, 094033 (2021), [arXiv:2101.07850 [hep-ph]].
  * (47) Z. G. Wang, H. J. Wang and Q. Xin, “The hadronic coupling constants of the lowest hidden-charm pentaquark state with the QCD sum rules in rigorous quark-hadron duality,”Chin. Phys. C 45, 063104 (2021), [arXiv:2005.00535 [hep-ph]].
  * (48) Z. G. Wang, “Analysis of the $\bar{D}\Sigma_{c}$, $\bar{D}\Sigma_{c}^{*}$, $\bar{D}^{*}\Sigma_{c}$ and $\bar{D}^{*}\Sigma_{c}^{*}$ pentaquark molecular states with QCD sum rules,”Int. J. Mod. Phys. A 34, no.19, 1950097 (2019), [arXiv:1806.10384 [hep-ph]].
  * (49) Z. G. Wang, “Strong decays of the $Y(4660)$ as a vector tetraquark state in solid quark-hadron duality,” Eur. Phys. J. C 79, no.3, 184 (2019), [arXiv:1901.02177 [hep-ph]].
  * (50) Z. G. Wang and X. Wang, “Analysis of the strong decays of the $P_{c}(4312)$ as a pentaquark molecular state with QCD sum rules,”Chin. Phys. C 44, 103102 (2020), [arXiv:1907.04582 [hep-ph]].
  * (51) K. Azizi, Y. Sarac and H. Sundu, “Investigation of hidden-charm double strange pentaquark candidate $P_{css}$ via its mass and strong decay to $\Xi^{0}J/\psi$,”[arXiv:2112.15543 [hep-ph]].
  * (52) J. Ho, D. Harnett and T. G. Steele, “Masses of Open-Flavour Heavy-Light Hybrids from QCD Sum-Rules,”JHEP 05, 149 (2017), [arXiv:1609.06750 [hep-ph]].