# Spectrum of $[8]_{[c\bar{s}]}\otimes[8]_{[q\bar{q^{\prime}}]}$ systems with
quantum numbers $J^{P}=0^{\pm}$ and $1^{\pm}$

Si-Nuo Li1 and Liang<EMAIL_ADDRESS>1 College of Physics and Hebei
Key Laboratory of Photophysics Research and Application, Hebei Normal
University, Shijiazhuang 050024, China

###### Abstract

Inspired by the recent experimental progress on the
$T_{c\bar{s}0}^{a}(2900)^{0/++}$, the fully open tetraquark spectrum with the
configuration of $[8]_{[c\bar{s}]}\otimes[8]_{[q\bar{q^{\prime}}]}$ is
systematically investigated by the QCD sum rules. In this article, we
concentrate on the quantum numbers $J^{P}=0^{+}/0^{-}/1^{+}/1^{-}$.

Firstly, we construct four scalar currents ($J^{P}=0^{+}$) in the form of
$[8]_{[c\bar{s}]}\otimes[8]_{[u\bar{d}]}$ type tetraquark structure and
perform the analysis of the QCD sum rules, where we consider the leading order
contributions up to dimension 11 in the operator product expansion and retain
contributions linear in the strange quark mass $m_{s}$. Our results,
$M_{0^{+}}^{A}=2.91_{-0.20}^{+0.20}~{}\text{GeV}$ and
$M_{0^{+}}^{C}=2.98_{-0.21}^{+0.20}~{}\text{GeV}$, are consistent with the
experimentally discovered $T_{c\bar{s}0}^{a}(2900)^{++}$ within error margins.
Thus, our calculation supports classifying $T_{c\bar{s}0}^{a}(2900)^{++}$ as a
tetraquark state with the $[8]_{[c\bar{s}]}\otimes[8]_{u\bar{d}}$ color
configuration.

Moreover, on the basis of $J^{P}=0^{+}$, we also study tetraquark states with
quantum numbers $J^{P}=0^{-}$, $1^{+}$, and $1^{-}$, predicting two new
hadronic states awaiting experimental verification for each quantum number.
Their masses are as follows:
$M_{0^{-}}^{A}=3.45_{-0.10}^{+0.12}~{}\text{GeV}$,
$M_{0^{-}}^{D}=3.20_{-0.19}^{+0.17}~{}\text{GeV}$,
$M_{1^{+}}^{A}=2.95_{-0.19}^{+0.19}~{}\text{GeV}$,
$M_{1^{+}}^{C}={2.95_{-0.19}^{+0.19}}~{}\text{GeV}$,
$M_{1^{-}}^{A}={3.23_{-0.10}^{+0.11}}~{}\text{GeV}$ and
$M_{1^{-}}^{D}={3.23_{-0.11}^{+0.09}}~{}\text{GeV}$. With advances in
experimental techniques and accumulation of new data, these predicted results
are hoped to be confirmed in future experiments.

###### pacs:

11.55.Hx, 12.38.Lg, 12.39.Mk

##  I Introduction

For a long time, it was believed that hadrons were classifiable into two
groups: mesons, comprising a quark and an antiquark ($q\bar{q}$), and baryons,
made up of three quarks ($qqq$) Gell-Mann:1964ewy ; Zweig:1964ruk .
Nonetheless, quantum chromodynamics (QCD), the foundational theory of strong
interactions, theoretically permits the existence of any color-singlet
configurations such as tetraquarks, pentaquarks, hybrids, and glueballs among
others. Over the past two decades, numerous potential exotic states have been
experimentally reported, beginning with the Belle Collaboration’s discovery of
the X(3872) in 2003 CDF:2003cab . These candidates have sparked significant
interest in theoretical research, leading to a substantial number of
publications investigating the nature and characteristics of these potential
exotic states. For comprehensive reviews on the current experimental and
theoretical research into exotic states, refer to References Ali:2017jda ;
Guo:2017jvc ; Esposito:2016noz ; Lebed:2016hpi ; Richard:2016eis ;
Chen:2016qju ; Liu:2019zoy ; Brambilla:2019esw .

In 2020, the LHCb Collaboration announced the discovery of two isosinglet
exotic states, $X_{0}(2900)$ and $X_{1}(2900)$, observed in the $D^{-}K^{+}$
invariant mass spectrum LHCb:2020pxc . These states are naturally
characterized by a $\bar{c}\bar{s}ud$ flavor structure. Recently, in 2022, a
doubly charged tetraquark and its neural partner were reported by the LHCb
Collaboration LHCb:2022sfr , which have spin-parity $J^{P}=0^{+}$ and quark
constituent $[c\bar{s}u\bar{d}]$ and $[c\bar{s}d\bar{u}]$, respectively. The
masses of these two new resonant states are $2.908\pm 0.011\pm 0.020$ GeV, and
widths are $0.136\pm 0.023\pm 0.013$ GeV. Therefore, they are nominated as
$T_{c\bar{s}0}^{a}(2900)^{++/0}$. From the perspective of isospin analysis,
they should be two components of an isospin vector, namely
$T_{c\bar{s}0}^{a}(2900)^{0}$ and $T_{c\bar{s}0}^{a}(2900)^{++}$ are the
components with $I_{3}=-1$ and $1$, respectively.

These states are composed of four distinct flavor quarks (or antiquarks),
setting them apart from exotic states confirmed previously. The
$X_{0/1}(2900)$ particle represents the inaugural discovery of an exotic
structure with entirely open flavor, sparking numerous research endeavors
aimed at unraveling its internal structure, including the tetraquark state
Lu:2020qmp ; Wang:2020xyc ; He:2020jna ; Karliner:2020vsi ; Zhang:2020oze ;
Agaev:2021jsz , molecular state Hu:2020mxp ; Liu:2020nil ; Chen:2020aos ;
Huang:2020ptc , triangle singularity Liu:2020orv , and cusp effect
Burns:2020epm . The fully open flavor tetraquark states
$T_{c\bar{s}0}^{a}(2900)^{++/0}$ could potentially be the isospin counterparts
of the tetraquark candidates made up of $[c\bar{s}u\bar{d}]$ and
$[c\bar{s}d\bar{u}]$, thereby garnering significant research interest aimed at
understanding their inner structures. Very recently, the
$T_{c\bar{s}0}^{a}(2900)^{++/0}$ have been investigated by various theoretical
methods, containing the potential quark model Liu:2022hbk , the color flux-
tube model Wei:2022wtr , the one-boson-exchange model Chen:2022svh , the
effective Lagrantian approach Yue:2022mnf , and the framework of the QCD sum
rules Agaev:2022duz ; Agaev:2022eyk ; Yang:2023evp . The authors in Ref.
Wei:2022wtr  carried out a systematical investigation on the properties of the
ground and P-wave states $[cs][\bar{u}\bar{d}]$ and $[cu][\bar{s}\bar{d}]$
with various spin, isospin or U-spin, and color combinations in a multiquark
color flux-tube model. In Ref. Chen:2022svh , based on the one-boson-exchange
model to study the $D^{(*)}K^{*}$ interactions, Chen and Huang could
simultaneously reproduced the masses of the $T_{c\bar{s}0}^{a}(2900)^{0/++}$
in the hadronic molecular picture, where the $T_{c\bar{s}0}^{a}(2900)^{0/++}$
are regarded as the $D^{*}K^{*}[1(0^{+})]$ charmed-strange molecular states.
In Ref. Agaev:2022duz , Agaev $\textit{et al}.$ investigated the new scalar
resonances $T_{c\bar{s}0}^{a}(2900)^{++}$ and $T_{c\bar{s}0}^{a}(2900)^{0}$ as
molecules $D_{s}^{*+}\rho^{+}$ and $D_{s}^{*+}\rho^{-}$ of conventional vector
mesons, respectively, within QCD sum rules. Basing on the analysis of the QCD
sum rules Yang:2023evp , Yang et al. assigned the
$T_{c\bar{s}0}^{a}(2900)^{++/0}$ as the $A-\bar{A}$-type tetraquark states in
the framework of the QCD sum rules, where the A denotes the axialvector
diquark state.

It should be highlighted that QCD suggests the presence of a tetraquark
configuration, consisting of two color-octet components. This configuration,
due to QCD interactions, distinguishes itself from a molecular state comprised
of two color-singlet mesons, allowing for the possibility of decay into two
mesons via the exchange of one or more gluons. Thus, exploring the color
octet-octet tetraquark state holds significant importance for identifying
potential new hadronic states Wang:2006ri ; Yang:2020wkh ; Wang:2021mma . In
this work, we systematically construct the color octet-octet tetraquark
currents comprised of $[8]_{c\bar{s}}\otimes[8]_{q\bar{q^{\prime}}}$
($[q\bar{q^{\prime}}]=[u\bar{d}]$ or $[d\bar{u}]$) with quantum numbers
$J^{P}=0^{\pm}$ and $1^{\pm}$ and determine their masses using QCD sum rules.
Subsequently, we analysis the possibility of $T_{c\bar{s}0}^{a}(2900)^{0/++}$
as color octet-octet states and discuss the predictions with $0^{-}$ and
$1^{\pm}$.

Following the introduction, Sect. II outlines the fundamental equations of the
QCD sum rules. Sect. III is dedicated to numerical analyses and the
presentation of corresponding figures. Finally, we provide the conclusion and
discussion of the tetraquark states with fully open flavors in Sec. IV.

##  II Formalism

The research content of this work involves the study of the color octet-octet
tetraquark states with quantum numbers $J^{P}=0^{\pm}$ and $1^{\pm}$ using the
QCD sum rule method. The starting point of the QCD sum rules is the two-point
correlation function constructed from two hadronic currents. For the scalar
states considered in this paper, the form of the two-point correlation
function is as follows:

$\displaystyle\Pi(q^{2})$ $\displaystyle=$ $\displaystyle i\int
d^{4}xe^{iq\cdot x}\langle 0|T\\{j(x),j^{\dagger}(0)\\}|0\rangle.$ (1)

Here, $j(x)$and $j(0)$ are hadronic currents with the aforementioned scalar
quantum number. By consulting literature and conducting independent analyses,
we construct the following four types of color octet-octet structure scalar
currents:

$\displaystyle j_{A}^{0^{+}}(x)$ $\displaystyle=$
$\displaystyle[i\bar{s}^{j}(x)\gamma^{5}(t^{a})_{jk}c^{k}(x)][i\bar{d}^{m}(x)\gamma^{5}(t^{a})_{mn}u^{n}(x)],$
(2) $\displaystyle j_{B}^{0^{+}}(x)$ $\displaystyle=$
$\displaystyle[\bar{s}^{j}(x)(t^{a})_{jk}c^{k}(x)][\bar{d}^{m}(x)(t^{a})_{mn}u^{n}(x)],$
(3) $\displaystyle j_{C}^{0^{+}}(x)$ $\displaystyle=$
$\displaystyle[\bar{s}^{j}(x)\gamma^{{\mu}}(t^{a})_{jk}c^{k}(x)][\bar{d}^{m}(x)\gamma_{{\mu}}(t^{a})_{mn}u^{n}(x)],$
(4) $\displaystyle j_{D}^{0^{+}}(x)$ $\displaystyle=$
$\displaystyle[\bar{s}^{j}(x)\gamma^{{\mu}}\gamma^{5}(t^{a})_{jk}c^{k}(x)][\bar{d}^{m}(x)\gamma_{{\mu}}\gamma^{5}(t^{a})_{mn}u^{n}(x)].$
(5)

Here, $j$, $k$, $m$, and $n$ are color indices, $t^{a}=\frac{\lambda^{a}}{2}$,
where $\lambda^{a}$ are the Gell-Mann matrices, and $t^{a}$ are the generators
of the $SU(3)_{c}$ group; the subscripts $A$, $B$, $C$, and $D$ respectively
represent currents composed of two $0^{-}$, two $0^{+}$, two $1^{-}$, and two
$1^{+}$ color-octet components. In the construction process, by swapping
$u\leftrightarrow d$, we can obtain the corresponding currents within the same
isospin multiplet. In the isospin limit, the tetraquark states within the same
isospin multiplet have the same mass. It is noteworthy that between the two
color-octet components within each current, there is an S-wave interaction,
which is the reason we only study these four currents in this paper.

Similarly, we present the $[8]_{c\bar{s}}\otimes[8]_{u\bar{d}}$ currents for
quantum numbers $0^{-}$ and $1^{\pm}$ below, respectively:

$\displaystyle j_{A}^{0^{-}}(x)$ $\displaystyle=$
$\displaystyle[i\bar{s}^{j}(x)\gamma^{5}(t^{a})_{jk}c^{k}(x)][i\bar{d}^{m}(x)(t^{a})_{mn}u^{n}(x)],$
(6) $\displaystyle j_{B}^{0^{-}}(x)$ $\displaystyle=$
$\displaystyle[\bar{s}^{j}(x)(t^{a})_{jk}c^{k}(x)][\bar{d}^{m}(x)\gamma^{5}(t^{a})_{mn}u^{n}(x)],$
(7) $\displaystyle j_{C}^{0^{-}}(x)$ $\displaystyle=$
$\displaystyle[\bar{s}^{j}(x)\gamma^{{\mu}}(t^{a})_{jk}c^{k}(x)][\bar{d}^{m}(x)\gamma_{{\mu}}\gamma^{5}(t^{a})_{mn}u^{n}(x)],$
(8) $\displaystyle j_{D}^{0^{-}}(x)$ $\displaystyle=$
$\displaystyle[\bar{s}^{j}(x)\gamma^{{\mu}}\gamma^{5}(t^{a})_{jk}c^{k}(x)][\bar{d}^{m}(x)\gamma_{\mu}(t^{a})_{mn}u^{n}(x)],$
(9) $\displaystyle j_{A}^{1^{+}}(x)$ $\displaystyle=$
$\displaystyle[i\bar{s}^{j}(x)\gamma^{5}(t^{a})_{jk}c^{k}(x)][\bar{d}^{m}(x)\gamma_{{\mu}}(t^{a})_{mn}u^{n}(x)],$
(10) $\displaystyle j_{B}^{1^{+}}(x)$ $\displaystyle=$
$\displaystyle[\bar{s}^{j}(x)(t^{a})_{jk}c^{k}(x)][\bar{d}^{m}(x)\gamma_{{\mu}}\gamma^{5}(t^{a})_{mn}u^{n}(x)],$
(11) $\displaystyle j_{C}^{1^{+}}(x)$ $\displaystyle=$
$\displaystyle[\bar{s}^{j}(x)\gamma^{{\mu}}(t^{a})_{jk}c^{k}(x)][i\bar{d}^{m}(x)\gamma^{5}(t^{a})_{mn}u^{n}(x)],$
(12) $\displaystyle j_{D}^{1^{+}}(x)$ $\displaystyle=$
$\displaystyle[\bar{s}^{j}(x)\gamma^{{\mu}}\gamma^{5}(t^{a})_{jk}c^{k}(x)][\bar{d}^{m}(x)(t^{a})_{mn}u^{n}(x)],$
(13) $\displaystyle j_{A}^{1^{-}}(x)$ $\displaystyle=$
$\displaystyle[i\bar{s}^{j}(x)\gamma^{5}(t^{a})_{jk}c^{k}(x)][\bar{d}^{m}(x)\gamma_{{\mu}}\gamma^{5}(t^{a})_{mn}u^{n}(x)],$
(14) $\displaystyle j_{B}^{1^{-}}(x)$ $\displaystyle=$
$\displaystyle[\bar{s}^{j}(x)(t^{a})_{jk}c^{k}(x)][\bar{d}^{m}(x)\gamma_{{\mu}}(t^{a})_{mn}u^{n}(x)],$
(15) $\displaystyle j_{C}^{1^{-}}(x)$ $\displaystyle=$
$\displaystyle[\bar{s}^{j}(x)\gamma^{{\mu}}(t^{a})_{jk}c^{k}(x)][\bar{d}^{m}(x)(t^{a})_{mn}u^{n}(x)],$
(16) $\displaystyle j_{D}^{1^{-}}(x)$ $\displaystyle=$
$\displaystyle[\bar{s}^{j}(x)\gamma^{{\mu}}\gamma^{5}(t^{a})_{jk}c^{k}(x)][i\bar{d}^{m}(x)\gamma^{5}(t^{a})_{mn}u^{n}(x)].$
(17)

The quark-hadron duality principle is a fundamental assumption of the QCD sum
rule method: on one hand, the correlation function $\Pi(q^{2})$ can be
computed at the quark-gluon level using the Operator Product Expansion (OPE);
on the other hand, it can be represented at the hadron level by introducing
hadronic coupling constant and mass; the two representations are equivalent
under an appropriate energy scale, which is the essence of the quark-hadron
duality principle. For convenience, we refer to the calculation at the OPE
level as the OPE side of the correlation function, and the representation at
the hadron level as the phenomenological side of the correlation function.

To calculate the spectral density on the OPE side, we utilize the full
propagators for heavy quarks ($Q=c,b$) and light quarks ($q=u,d,s$), denoted
as $S_{ij}^{Q}(p)$ and $S_{ij}^{q}(x)$, respectively:

$\displaystyle S^{Q}_{ij}(p)\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{i\delta_{ij}(p\\!\\!\\!/+m_{Q})}{p^{2}-m_{Q}^{2}}-\frac{i}{4}\frac{t^{a}_{ij}G^{a}_{\alpha\beta}}{(p^{2}-m_{Q}^{2})^{2}}[\sigma^{\alpha\beta}(p\\!\\!\\!/+m_{Q})+(p\\!\\!\\!/+m_{Q})\sigma^{\alpha\beta}]$
(18) $\displaystyle+$ $\displaystyle\\!\\!\\!\frac{i\delta_{ij}m_{Q}\langle
g_{s}^{2}G^{2}\rangle}{12(p^{2}-m_{Q}^{2})^{3}}\bigg{[}1+\frac{m_{Q}(p\\!\\!\\!/+m_{Q})}{p^{2}-m_{Q}^{2}}\bigg{]}$
$\displaystyle+$
$\displaystyle\\!\\!\\!\frac{i\delta_{ij}}{48}\bigg{\\{}\frac{(p\\!\\!\\!/+m_{Q})[p\\!\\!\\!/(p^{2}-3m_{Q}^{2})+2m_{Q}(2p^{2}-m_{Q}^{2})]}{(p^{2}-m_{Q}^{2})^{6}}\times(p\\!\\!\\!/+m_{Q})\bigg{\\}}\langle
g_{s}^{3}G^{3}\rangle\;,$ $\displaystyle S_{ij}^{q}(x)\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\frac{i\delta_{ij}x\\!\\!\\!/}{2\pi^{2}x^{4}}-\frac{m_{q}\delta_{ij}}{4\pi^{2}x^{2}}-\frac{it^{N}_{ij}}{32\pi^{2}x^{2}}(\sigma_{\alpha\beta}x\\!\\!\\!/+x\\!\\!\\!/\sigma_{\alpha\beta})$
(19) $\displaystyle-$
$\displaystyle\frac{\delta_{ij}}{12}\langle\bar{q}q\rangle+\frac{i\delta_{ij}x\\!\\!\\!/}{48}m_{q}\langle\bar{q}q\rangle-\frac{\delta_{ij}x^{2}}{192}\langle\bar{q}Gq\rangle$
$\displaystyle+$
$\displaystyle\frac{i\delta_{ij}x^{2}x\\!\\!\\!/}{1152}m_{q}\langle\bar{q}Gq\rangle-\frac{t^{N}_{ij}\sigma_{\alpha\beta}}{192}\langle\bar{q}Gq\rangle$
$\displaystyle+$
$\displaystyle\frac{it^{N}_{ij}}{768}(\sigma_{\alpha\beta}x\\!\\!\\!/+x\\!\\!\\!/\sigma_{\alpha\beta})m_{q}\langle\bar{q}Gq\rangle\;,$

where the vacuum condensates of QCD operators are clearly displayed. The
complete propagators for both light and heavy quarks have been extensively
derived and explained in the literature Reinders:1984sr . Therefore, once we
obtain the spectral density $\rho_{i}^{\text{OPE}}(s)$ using the full
propagators, we can determine the correlation function $\Pi(q^{2})$ at the
quark-gluon level based on the dispersion relation:

$\displaystyle\Pi^{\text{OPE}}_{i}(q^{2})=\int_{(m_{s}+m_{c})^{2}}^{s_{0}}ds\frac{\rho_{i}^{\text{OPE}}(s)}{s-q^{2}}+\Pi_{i}^{\text{cond}}(q^{2}),$
(20)

where, the subscript $i$ ranges from $A$ to $D$, with the first term
$\rho_{i}^{\text{OPE}}(s)=\text{Im}[\Pi_{i}^{\text{OPE}}(s)]/\pi$, and the
second term $\Pi_{i}^{\text{cond}}(q^{2})$ represents the terms that can be
directly transformed by the Borel transformation without the need for a
dispersion relation. $\rho_{i}^{\text{OPE}}(s)$ can be expanded as follows:

$\displaystyle\rho^{\text{OPE}}_{i}(s)$ $\displaystyle=$
$\displaystyle\rho^{\text{pert}}_{i}(s)+\rho^{\langle\bar{s}s\rangle}_{i}(s)+\rho^{\langle
G^{2}\rangle}_{i}(s)+\rho^{\langle\bar{s}Gs\rangle}+\rho^{\langle
G^{3}\rangle}_{i}(s)+\rho_{i}^{\langle\bar{q}q\rangle^{2}}(s)$ (21)
$\displaystyle+$
$\displaystyle\rho_{i}^{\langle\bar{s}Gs\rangle}(s)+\rho_{i}^{\langle\bar{s}s\rangle\langle
G^{2}\rangle}(s)+\rho_{i}^{\langle\bar{s}s\rangle\langle
G^{2}\rangle}(s)+\rho_{i}^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}(s)+\rho_{i}^{\langle
G^{2}\rangle^{2}}(s).$

Applying the Borel transformation to the OPE side, namely equation (20), we
have:

$\displaystyle\Pi^{\text{OPE}}_{i}(M^{2}_{B})$ $\displaystyle=$
$\displaystyle\int_{(m_{s}+m_{c})^{2}}^{\infty}ds\rho^{\text{OPE}}(s)e^{-s/M_{B}^{2}}+\Pi_{i}^{\langle
G^{3}\rangle}(M^{2}_{B})+\Pi_{i}^{\langle
G^{2}\rangle\langle\bar{s}s\rangle}(M^{2}_{B})$ (22)
$\displaystyle+\Pi_{i}^{\langle\bar{s}s\rangle\langle\bar{q}Gq\rangle}(M^{2}_{B})+\cdots.$

To account for the effect induced by the mass of the strange quark, we retain
the linear terms of the strange quark mass in our subsequent calculations.
Taking the current defined by equation (2) as an example, the spectral density
obtained through analytical calculation is presented as follows:

$\displaystyle\rho_{A}^{\text{pert}}(s)$ $\displaystyle=$
$\displaystyle-\frac{1}{18432\pi^{6}}\int^{\lambda}_{0}dx\frac{H_{x}^{3}(H_{x}-4m_{c}m_{s}x)}{(x-1)^{3}x},$
(23) $\displaystyle\rho_{A}^{\langle\bar{s}s\rangle}(s)$ $\displaystyle=$
$\displaystyle-\frac{\langle\bar{s}s\rangle}{384\pi^{4}}\int^{\lambda}_{0}dx\frac{H_{x}^{2}(m_{s}(x-1)+m_{c}x)}{(x-1)^{2}x},$
(24) $\displaystyle\rho_{A}^{\langle G^{2}\rangle}(s)$ $\displaystyle=$
$\displaystyle\frac{\langle
g_{s}^{2}G^{2}\rangle}{55296\pi^{6}}\int^{\lambda}_{0}dx\frac{m_{c}x^{2}(-H_{x}m_{c}+3H_{x}m_{s}+m_{c}^{2}m_{s}x)}{(x-1)^{3}}$
(25) $\displaystyle-\frac{\langle
g_{s}^{2}G^{2}\rangle}{98304\pi^{6}}\int^{\lambda}_{0}dx\frac{H_{x}^{2}}{(x-1)^{2}}+\frac{\langle
g_{s}^{2}G^{2}\rangle}{49152\pi^{6}}\int^{\lambda}_{0}dx\frac{H_{x}(H_{x}-2m_{c}m_{s})}{(x-1)x},$
$\displaystyle\rho_{A}^{\langle\bar{s}Gs\rangle}(s)$ $\displaystyle=$
$\displaystyle\frac{\langle\bar{s}g_{s}\sigma\cdot
Gs\rangle}{1152\pi^{4}}\int^{\lambda}_{0}dx\frac{H_{x}(m_{s}(x-1)+3m_{c}x)}{(x-1)x}$
$\displaystyle-$ $\displaystyle\frac{\langle\bar{s}g_{s}\sigma\cdot
Gs\rangle}{3072\pi^{4}}\int^{\lambda}_{0}dx\frac{H_{x}(m_{s}(x-1)+m_{c}x)}{(x-1)^{2}},$
$\displaystyle\rho_{A}^{\langle G^{3}\rangle}(s)$ $\displaystyle=$
$\displaystyle\frac{\langle
g_{s}^{3}G^{3}\rangle}{221184\pi^{6}}\int^{\lambda}_{0}dx\frac{x^{2}(H_{x}+2m_{c}(m_{c}-3m_{s})x)}{(x-1)^{3}}$
$\displaystyle+$ $\displaystyle\frac{\langle
g_{s}^{3}G^{3}\rangle}{589824\pi^{6}}\int^{\lambda}_{0}dx\frac{x(3H_{x}+2m_{c}^{2}x)}{(x-1)^{2}},$
$\displaystyle\rho_{A}^{\langle\bar{q}q\rangle^{2}}(s)$ $\displaystyle=$
$\displaystyle-\frac{\langle\bar{q}q\rangle^{2}}{72\pi^{2}}\int^{\lambda}_{0}dx\frac{-H_{x}+m_{c}m_{s}x}{x},$
(28) $\displaystyle\rho_{A}^{\langle\bar{s}s\rangle\langle G^{2}\rangle}(s)$
$\displaystyle=$ $\displaystyle-\frac{\langle\bar{s}s\rangle\langle
g_{s}^{2}G^{2}\rangle}{4608\pi^{4}}\int^{\lambda}_{0}dx\frac{m_{c}x^{2}}{(x-1)^{2}}$
$\displaystyle-$ $\displaystyle\frac{\langle\bar{s}s\rangle\langle
g_{s}^{2}G^{2}\rangle}{36864\pi^{4}}(m_{c}-2m_{s}),$
$\displaystyle\rho_{A}^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}(s)$
$\displaystyle=$
$\displaystyle-\frac{\langle\bar{q}q\rangle\langle\bar{q}g_{s}\sigma\cdot
Gq\rangle}{144\pi^{2}},$ (30) $\displaystyle\rho_{A}^{\langle
G^{2}\rangle^{2}}(s)$ $\displaystyle=$ $\displaystyle\frac{\langle
g_{s}^{2}G^{2}\rangle^{2}}{10616832\pi^{6}}\int^{\lambda}_{0}dx\frac{H_{x}^{3}(2-5x)+6x^{4}}{(x-1)x^{3}}+\frac{\langle
g_{s}^{2}G^{2}\rangle^{2}}{196608\pi^{6}},$ (31)

where $M_{B}^{2}$ is the Borel parameter, $H_{x}=m_{c}^{2}x-x(1-x)s$, and
$\lambda=1-m_{c}^{2}/s$.

The results of the direct Borel transformation applied to equation (20),
without going through the dispersion relation, are as follows:

$\displaystyle\Pi_{A}^{\langle G^{3}\rangle}(M^{2}_{B})$ $\displaystyle=$
$\displaystyle\frac{\langle
g_{s}^{3}G^{3}\rangle}{110592\pi^{6}}\int^{1}_{0}dx\frac{m_{c}^{3}m_{s}x^{3}}{(x-1)}e^{\frac{m_{c}^{2}}{(x-1)M_{B}^{2}}},$
(32) $\displaystyle\Pi_{A}^{\langle
G^{2}\rangle\langle\bar{s}s\rangle}(M^{2}_{B})$ $\displaystyle=$
$\displaystyle-\frac{\langle
g_{s}^{2}G^{2}\rangle\langle\bar{s}s\rangle}{13824\pi^{4}}\int^{1}_{0}dx\frac{m_{c}^{2}x(m_{s}(x-1)+m_{c}x)}{(x-1)^{3}}e^{\frac{m_{c}^{2}}{(x-1)M_{B}^{2}}},$
(33)
$\displaystyle\Pi_{A}^{\langle\bar{s}s\rangle\langle\bar{q}Gq\rangle}(M^{2}_{B})$
$\displaystyle=$
$\displaystyle-\frac{\langle\bar{s}s\rangle\langle\bar{q}g_{s}\sigma\cdot
Gq\rangle}{144\pi^{2}}\int^{1}_{0}dxm_{c}m_{s}e^{\frac{m_{c}^{2}}{(x-1)M_{B}^{2}}},$
(34) $\displaystyle\Pi_{A}^{\langle G^{2}\rangle^{2}}(M^{2}_{B})$
$\displaystyle=$ $\displaystyle\frac{\langle
g_{s}^{2}G^{2}\rangle^{2}}{10616832\pi^{6}}\int^{1}_{0}dx\\{\frac{-8m_{c}^{3}m_{s}x}{M_{B}^{2}(x-1)^{3}}$
(35)
$\displaystyle+\frac{M_{B}^{2}m_{c}(x-1)(8m_{c}x-m_{s}(2+21x))}{M_{B}^{2}(x-1)^{3}}\\}e^{\frac{m_{c}^{2}}{(x-1)M_{B}^{2}}},$
$\displaystyle\Pi_{A}^{\langle G^{3}\rangle\langle\bar{s}s\rangle}(M^{2}_{B})$
$\displaystyle=$ $\displaystyle-\frac{\langle
g_{s}^{3}G^{3}\rangle\langle\bar{s}s\rangle}{55296\pi^{4}}\int^{1}_{0}dx\\{2m_{c}^{2}\frac{m_{s}(x-1)+m_{c}x}{M_{B}^{2}(x-1)^{4}}$
(36)
$\displaystyle+2m_{c}^{2}\frac{M_{B}^{2}(x-1)m_{s}(x-1)}{M_{B}^{2}(x-1)^{4}}\\}e^{\frac{m_{c}^{2}}{(x-1)M_{B}^{2}}},$
$\displaystyle\Pi_{A}^{\langle
G^{2}\rangle\langle\bar{s}Gs\rangle}(M^{2}_{B})$ $\displaystyle=$
$\displaystyle\frac{\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}g_{s}\sigma\cdot
Gq\rangle}{82944\pi^{4}}\int^{1}_{0}dx\frac{m_{c}(m_{s}m_{c}(x-1))+3m_{c}^{2}x+9M_{B}^{2}(x-1)x}{M_{B}^{2}(x-1)^{3}}e^{\frac{m_{c}^{2}}{(x-1)M_{B}^{2}}}$
(37) $\displaystyle+$ $\displaystyle\frac{\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}g_{s}\sigma\cdot
Gq\rangle}{221184\pi^{4}}\int^{1}_{0}dx\frac{m_{c}^{2}m_{s}+M_{B}^{2}(-3m_{c}+m_{s})}{M_{B}^{2}}e^{\frac{m_{c}^{2}}{(x-1)M_{B}^{2}}},$
$\displaystyle\Pi_{A}^{\langle G^{3}\rangle\langle G^{2}\rangle}(M^{2}_{B})$
$\displaystyle=$ $\displaystyle\frac{\langle g_{s}^{3}G^{3}\rangle\langle
g_{s}^{2}G^{2}\rangle}{7077888\pi^{6}}\int^{1}_{0}dx\\{\frac{(-2m_{c}^{3}m_{s}x+2M_{B}^{2}m_{c}x(-3m_{s}+m_{c})(x-1)}{M_{B}^{2}(x-1)^{4}}$
(38)
$\displaystyle\frac{m_{c}(x-1)+M_{B}^{2}(x-1)^{2})x}{M_{B}^{2}(x-1)^{4}}\\}e^{\frac{m_{c}^{2}}{(x-1)M_{B}^{2}}},$
$\displaystyle\Pi_{A}^{\langle\bar{q}Gq\rangle^{2}}(M^{2}_{B})$
$\displaystyle=$ $\displaystyle\frac{\langle\bar{q}g_{s}\sigma\cdot
Gq\rangle^{2}}{1152\pi^{2}}\int^{1}_{0}dx\frac{M_{B}^{4}+M_{B}^{2}m_{c}^{2}+m_{c}^{3}m_{s}}{M_{B}^{2}}e^{\frac{m_{c}^{2}}{(x-1)M_{B}^{2}}},$
(39) $\displaystyle\Pi_{A}^{\langle
G^{2}\rangle\langle\bar{q}q\rangle^{2}}(M^{2}_{B})$ $\displaystyle=$
$\displaystyle-\frac{\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle^{2}}{5184\pi^{2}}\int^{1}_{0}dx\frac{m_{c}(-m_{c}^{3}m_{s}+M_{B}^{4}m_{c}(m_{c}-3m_{s})(x-1))}{M_{B}^{2}(x-1)^{3}}e^{\frac{m_{c}^{2}}{(x-1)M_{B}^{2}}}$
(40) $\displaystyle-\frac{\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle^{2}}{13824\pi^{2}}\int^{1}_{0}dxe^{\frac{m_{c}^{2}}{(x-1)M_{B}^{2}}},$
$\displaystyle\Pi_{A}^{\langle
G^{3}\rangle\langle\bar{s}Gs\rangle}(M^{2}_{B})$ $\displaystyle=$
$\displaystyle\frac{\langle
g_{s}^{3}G^{3}\rangle\langle\bar{s}g_{s}\sigma\cdot
Gs\rangle}{331776\pi^{4}}\int^{1}_{0}dx\\{\frac{2m_{c}^{2}(m_{s}(x-1))+3m_{c}x}{M_{B}^{4}(x-1)^{4}}$
(41)
$\displaystyle+\frac{M_{B}^{2}(x-1)(m_{s}(x-1)+18m_{c}x)}{M_{B}^{4}(x-1)^{4}}\\}e^{\frac{m_{c}^{2}}{(x-1)M_{B}^{2}}},$
$\displaystyle\Pi_{A}^{\langle\bar{s}s\rangle\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}(M^{2}_{B})$
$\displaystyle=$
$\displaystyle\frac{\langle\bar{s}s\rangle\langle\bar{q}q\rangle\langle\bar{q}g_{s}\sigma\cdot
Gq\rangle}{216}\int^{1}_{0}\text{d}x\frac{m_{c}^{3}(2M_{B}^{2}-m_{c}m_{s})}{M_{B}^{6}}e^{\frac{m_{c}^{2}}{(x-1)M_{B}^{2}}},$
(42)
$\displaystyle\Pi_{A}^{\langle\bar{q}q\rangle^{2}\langle\bar{s}Gs\rangle}(M^{2}_{B})$
$\displaystyle=$
$\displaystyle\frac{\langle\bar{q}q\rangle^{2}\langle\bar{s}g_{s}\sigma\cdot
Gs\rangle}{648}\int^{1}_{0}\text{d}x\frac{m_{c}^{3}(3M_{B}^{2}-m_{c}m_{s})}{M_{B}^{3}}e^{\frac{m_{c}^{2}}{(x-1)M_{B}^{6}}}.$
(43)

For the sake of brevity, in this paper, we only provide the analytic results
of the OPE side for the scalar case A, omitting the analytic results for other
scalar cases and other quantum numbers.

On the phenomenological side, after isolating the ground state contribution of
the tetraquark states, we obtain the phenomenological representation of the
correlation function $\Pi_{i}(q^{2})$, which is expressed as a dispersion
integral over the hadronic spectrum,

$\displaystyle\Pi_{i}^{\text{phen}}(q^{2})=\frac{(\lambda_{X}^{2})^{2}}{(M_{X}^{i})^{2}-q^{2}}+\int_{s_{0}}^{\infty}\text{d}s\frac{\rho_{X}^{i}(s)}{s-q^{2}},$
(44)

where, $M_{X}^{i}$ represents the mass of the $J^{P}=0^{+}$ tetraquark state,
$\rho_{X}^{i}(s)$ denotes its spectral density that includes contributions
from higher excited states and continuum states, and $s_{0}$ is the threshold
for the higher excited states and continuum states. The coupling constant
$\lambda_{X}$ is defined by $\langle 0|j_{X}^{i}|X\rangle=\lambda_{X}^{i}$,
where $X$ is the lowest-lying tetraquark state.

Performing the Borel transformation on the phenomenological side of the
correlation function, we obtain:

$\displaystyle\Pi_{i}^{\text{phen}}(s_{0},M_{B}^{2})=(\lambda^{i}_{X})^{2}e^{-(M_{X}^{i})^{2}/M_{B}^{2}}+\int_{s_{0}}^{\infty}\text{d}s\rho^{i}_{X}(s)e^{-s/M_{B}^{2}}.$
(45)

According to the quark-hadron duality principle, we have the equality between
equation (22) and equation (45), thus:

$\displaystyle(\lambda^{i}_{X})^{2}e^{-(M_{X}^{i})^{2}/M_{B}^{2}}+\int_{s_{0}}^{\infty}\text{d}s\rho^{i}_{X}(s)e^{-s/M_{B}^{2}}$
(46) $\displaystyle=$
$\displaystyle\int_{(m_{s}+m_{c})^{2}}^{\infty}\text{d}s\rho^{\text{OPE}}_{i}(s)e^{-s/M_{B}^{2}}+\Pi_{i}^{\langle
G^{3}\rangle}(M^{2}_{B})+\Pi_{i}^{\langle
G^{2}\rangle\langle\bar{s}s\rangle}(M^{2}_{B})+\Pi_{i}^{\langle\bar{s}s\rangle\langle\bar{q}Gq\rangle}(M^{2}_{B})+\cdots.$

Under the quark-hadron duality approximation, for the equation (46), the
integral on the left side is approximately equal to the integral of the first
term on the right side in the interval $s_{0}<s<\infty$. Therefore, both sides
can simultaneously remove this integral part, ultimately resulting in the
relationship between the ground state and the OPE calculation results:

$\displaystyle(\lambda^{i}_{X})^{2}e^{-(M_{X}^{i})^{2}/M_{B}^{2}}$
$\displaystyle=$
$\displaystyle\int_{(m_{s}+m_{c})^{2}}^{s_{0}}\text{d}s\rho^{\text{OPE}}_{i}(s)e^{-s/M_{B}^{2}}+\Pi_{i}^{\langle
G^{3}\rangle}(M_{B}^{2})$ (47) $\displaystyle+\Pi_{i}^{\langle
G^{2}\rangle\langle\bar{s}s\rangle}(M^{2}_{B})+\Pi_{i}^{\langle\bar{s}s\rangle\langle\bar{q}Gq\rangle}(M^{2}_{B})+\cdots.$

For convenience, we define the zeroth moment in the QCD sum rules, which
equals the right side of the aforementioned equation, i.e.,

$\displaystyle L_{0}^{i}(s_{0},M_{B}^{2})$ $\displaystyle=$
$\displaystyle\int_{m_{s}^{2}}^{\infty}\text{d}s\,\rho^{\text{OPE}}(s)e^{-s/M_{B}^{2}}$
(48) $\displaystyle+$ $\displaystyle\Pi_{i}^{\langle
G^{3}\rangle}(M_{B}^{2})+\Pi_{i}^{\langle
G^{2}\rangle\langle\bar{s}s\rangle}(M_{B}^{2})+\Pi_{i}^{\langle\bar{s}s\rangle\langle\bar{q}Gq\rangle}(M_{B}^{2})+\cdots\,.$

Taking the derivative of the zeroth moment $L_{0}^{i}(s_{0},M_{B}^{2})$ with
respect to $1/M_{B}^{2}$, we obtain the first moment
$L_{1}^{i}(s_{0},M_{B}^{2})$:

$\displaystyle
L_{1}^{i}(s_{0},M_{B}^{2})\\!\\!=\\!\\!\frac{\partial}{\partial(M_{B}^{2})^{-1}}L_{0}^{i}(s_{0},M_{B}^{2}).$
(49)

From the left side expression of equation (47), it is evident that
$-(M^{i}_{X})^{2}=L_{1}(s_{0},M_{B}^{2})/L_{0}(s_{0},M_{B}^{2})$, thus the
mass of the studied hadron is:

$\displaystyle
M_{X}^{i}(s_{0},M_{B}^{2})=\sqrt{-\frac{L_{1}^{\text{i}}(s_{0},M_{B}^{2})}{L_{0}^{i}(s_{0},M_{B}^{2})}}.$
(50)

In this paper, we will determine the mass of the studied tetraquark states by
analyzing equation (50).

##  III Numerical analysis

In this section, we take the current $j_{A}^{0^{+}}$ as an example and perform
the numerical analysis. In our analysis, we utilize the quark masses and
various QCD parameters as provided in Table 1 Reinders:1984sr ; Shifman:1978bx
; Shifman:1978by ; Narison:1989aq ; Colangelo:2000dp ;
ParticleDataGroup:2022pth , where $\overline{m_{c}}$ represents the running
mass of the charm quark in the $\overline{\text{MS}}$ scheme.

Table 1:  The QCD input parameters required for the numerical calculations in this paper Reinders:1984sr ; Shifman:1978bx ; Shifman:1978by ; Narison:1989aq ; Colangelo:2000dp ; ParticleDataGroup:2022pth . Parameter Name | Value
---|---
$m_{u},m_{d}$ | $0$
$m_{c}(m_{c})=\overline{m_{c}}$ | $(1.27\pm 0.02)~{}\text{GeV}$
$m_{s}$ | $(0.13\pm 0.03)~{}\text{GeV}$
$\langle q\bar{q}\rangle$ | $-(0.24\pm 0.01)^{3}~{}\text{GeV}^{3}$
$\langle g_{s}^{2}G^{2}\rangle$ | $0.88~{}\text{GeV}^{4}$
$\langle g_{s}^{3}G^{3}\rangle$ | $0.045~{}\text{GeV}^{6}$
$\langle s\bar{s}\rangle/\langle q\bar{q}\rangle$ | $(0.8\pm 0.1)$
$m_{0}^{2}\equiv\langle qG\bar{q}\rangle/\langle q\bar{q}\rangle$ | $(0.8\pm 0.2)~{}\text{GeV}^{2}$

As mentioned earlier (50), the extracted hadron mass $M_{X}$ is a function of
the threshold parameter $s_{0}$ and the Borel parameter $M_{B}^{2}$, which are
two important parameters in QCD sum rule analysis. If the final result $M_{X}$
does not depend on these two free parameters, then the QCD sum rule method
will have perfect predictive power. However, in reality, reliable mass
predictions can be obtained when these parameters exhibit weak dependencies
within a reasonable working region. Determining the Borel window (a reasonable
working region for $M_{B}^{2}$) primarily involves two criteria: the
requirement of OPE convergence provides the lower limit of the parameters
$(M_{B}^{2})_{\text{min}}$, while the dominance of the pole contribution
determines the upper limit $(M_{B}^{2})_{\text{max}}$. Additionally, we need
to analyze the variation of the hadron mass $M_{X}$ with the threshold
parameter $s_{0}$. To minimize the dependence of the extracted hadron mass
$M_{X}$ on the Borel mass $M_{B}^{2}$, an optimal threshold parameter $s_{0}$
needs to be chosen, where the $M_{X}-M_{B}^{2}$ curve exhibits a good plateau.

In the analytic calculations of this paper, we include non-perturbative terms,
i.e., condensates, up to the dimension-11 terms. Through numerical analysis,
we find that the contributions of the two-quark condensate $\langle
q\bar{q}\rangle$ and the four-quark condensate $\langle q\bar{q}\rangle^{2}$
are the dominant non-perturbative contributions, while the contributions of
other condensates are much smaller. For the additional parameters $M_{B}^{2}$
and $s_{0}$ introduced by QCD sum rules, we will constrain these parameters
according to the two fundamental criteria introduced earlier. For instance, in
the references Shifman:1978bx ; Shifman:1978by ; Reinders:1984sr ;
Colangelo:2000dp , the authors elaborate on these two criteria. We outline
these criteria as follows in this paper: The first criterion is OPE
convergence, which requires comparing the ratio of each condensate
contribution to the total contribution and selecting a reliable $M_{B}^{2}$
region to maintain their convergence. In practical applications, the degree of
OPE convergence can be expressed by the following formula:

$\displaystyle
R_{i}^{\text{OPE}}(s_{0},M_{B}^{2})=\frac{L_{0}^{\text{cond}}(s_{0},M_{B}^{2})}{L_{0}(s_{0},M_{B}^{2})}\,.$
(51)

The second criterion is the pole contribution (also known as the ground state
contribution) Reinders:1984sr ; Colangelo:2000dp , which compares the ratio of
the ground state contribution to the total hadron spectrum contribution.
Generally, this ratio should be greater than $50\%$, but sometimes, depending
on the specific problem, this percentage can be slightly relaxed, but it still
needs to be greater than $40\%$, and can be expressed as:

$\displaystyle
R_{i}^{\text{PC}}(s_{0},M_{B}^{2})=\frac{L_{0}(s_{0},M_{B}^{2})}{L_{0}(\infty,M_{B}^{2})}\;.$
(52)

By constraining these parameters with these two criteria, we can reasonably
eliminate contributions from not only higher-order condensates but also higher
excited states and continuum states.

At the same time, in order to find a suitable value for $\sqrt{s_{0}}$, we
conducted a scan. Since the threshold parameter $s_{0}$ is associated with the
mass of the ground state through $\sqrt{s_{0}}\sim(M_{X}+\delta)$ GeV, where
the range of $\delta$ is between 0.4 and 0.8 GeV, various values of
$\sqrt{s_{0}}$ that satisfy this constraint should be considered in the
numerical analysis. Among these values, we aim to identify one that offers an
optimal window for the Borel parameter $M_{B}^{2}$, or in other words, where
the mass of the tetraquark state $M_{X}$ is as independent of the Borel
parameter $M_{B}^{2}$ as possible. Ultimately, the value of $\sqrt{s_{0}}$
corresponding to the optimal mass curve becomes the central value of
$\sqrt{s_{0}}$. In practical applications, that is, in the calculations of QCD
sum rules, a variation of $\sqrt{s_{0}}$ by 0.2 GeV is acceptable, thus
setting the upper and lower limits for $\sqrt{s_{0}}$. The fluctuation of the
parameter $\sqrt{s_{0}}$ then corresponds to its range of error.

In Fig.1, we plot the pole contribution $R^{\text{PC}}$ curves for currents
with quantum numbers $J^{P}=0^{+}$, specifically for currents (2) and (4). The
upper limit constraints on the Borel parameter $M_{B}^{2}$ obtained were
$M_{B}^{2}=2.8~{}\text{GeV}^{2}$ and $2.6~{}\text{GeV}^{2}$, respectively,
where the threshold parameter $s_{0}$ used was the optimal value of
$3.6^{2}~{}\text{GeV}^{2}$. The corresponding OPE convergence curves for the
two currents are shown in Fig.2, where the lower limits for $M^{2}_{B}$ were
found to be $M^{2}_{B}=2.2~{}\text{GeV}^{2}$ and $2.0~{}\text{GeV}^{2}$, again
with the threshold parameter $s_{0}$ set to the optimal value of
$3.6^{2}~{}\text{GeV}^{2}$. It’s important to note that the constraints on the
upper and lower limits of $M^{2}_{B}$ also depend on the threshold parameter
$s_{0}$. That is, for different values of $s_{0}$, $M^{2}_{B}$ will have
different upper and lower limits. To determine the optimal $s_{0}$ value, we
performed an analysis similar to that found in references Shifman:1978bx ;
Shifman:1978by ; Reinders:1984sr ; Colangelo:2000dp . For different values of
$s_{0}$, the masses $M_{X}^{A}$ and $M_{X}^{C}$ as functions of the Borel
parameter $M^{2}_{B}$ are shown in Fig.3. In analyzing the cases of currents
(3) and (5), we find that no reasonable Borel parameter $M_{B}^{2}$ window
could be obtained regardless of the value of $s_{0}$. Therefore, we conclude
that currents (2) and (4) correspond to reasonable tetraquark states, with
specific numerical results summarized in Table 2; currents (3) and (5), due to
the lack of a reasonable parameter window, do not correspond to reasonable
tetraquark states.

Figure 1:  For the case of $J^{P}=0^{+}$, graphs showing the variation of the
pole contribution ratio with the Borel parameter $M_{B}^{2}$. Figure (a) shows
the $R_{A}^{\text{PC}}-M_{B}^{2}$ curve; figure (b) shows the
$R_{C}^{\text{PC}}-M_{B}^{2}$ curve, where $s_{0}$ corresponds to the optimal
threshold parameter obtained after scanning, with $s_{0}=3.6^{2}$ GeV2, and
the x-axis represents the Borel parameter $M^{2}_{B}$.

Figure 2:  For $J^{P}=0^{+}$, the OPE convergence ratio $R_{A/C}^{\text{OPE}}$
as a function of the parameter $M_{B}^{2}$, where $s_{0}$ corresponds to the
optimal threshold parameter obtained after scanning, $s_{0}=3.6^{2}$GeV2.
Figure (a) corresponds to current (2), and figure (b) corresponds to current
(4). In the graphs, the solid line with black squares represents the ratio of
perturbations, the red solid line with red circles represents the ratio of
two-quark condensates $\langle q\bar{q}\rangle$, and the blue solid line with
blue triangles represents the ratio of four-quark condensates $\langle
q\bar{q}\rangle^{2}$. These three ratios are shown because, apart from
perturbations, only the two-quark and four-quark condensates make significant
contributions, with other contributions being very small, hence not marked on
the graph with their contribution ratios.

Figure 3:  For $J^{P}=0^{+}$ under different threshold parameters $s_{0}$, the mass $M^{A/C}_{X}$ as a function of $M^{2}_{B}$, with figure (a) corresponding to current (2), and figure (b) corresponding to current (4). In each graph, the two vertical dashed lines represent the upper and lower bounds of the Borel window, with the central value of $s_{0}$ as the input. Table 2:  The numerical results for the parameter $M_{B}^{2}$, threshold $\sqrt{s_{0}}$, PC ratio, and mass associated with currents $j_{A}^{0^{+}}$ and $j_{C}^{0^{+}}$. $j^{J^{P}}_{X}$ | $M_{B}^{2}\,(\rm{GeV^{2}})$ | $\sqrt{s_{0}}\,(\rm{GeV})$ | PC | Mass $(\rm{GeV})$
---|---|---|---|---
$j^{0^{+}}_{A}$ | ${2.20\\!-\\!2.80}$ | $3.6\pm 0.2$ | ${(65\\!-\\!40)\%}$ | ${2.91_{-0.20}^{+0.20}}$
$j^{0^{+}}_{C}$ | ${2.00\\!-\\!2.60}$ | $3.6\pm 0.2$ | ${(66\\!-\\!40)\%}$ | ${2.98_{-0.21}^{+0.20}}$

Ultimately, the masses and uncertainties for the tetraquark states with color
octet-octet structure corresponding to currents $j_{A}^{0^{+}}$ and
$j_{C}^{0+}$ are as follows:

$\displaystyle M_{X}^{A}$ $\displaystyle=$ $\displaystyle
2.91_{-0.20}^{+0.20}~{}\text{GeV},$ (53) $\displaystyle M_{X}^{C}$
$\displaystyle=$ $\displaystyle 2.98_{-0.21}^{+0.20}~{}\text{GeV},$ (54)

where, the central values of the masses $M_{X}^{A}$ and $M_{X}^{C}$ correspond
to the most stable positions on the mass curve, with errors arising from
various condensates, quark masses, the threshold parameter $\sqrt{s_{0}}$, and
the Borel parameter $M_{B}^{2}$.

Comparing with experimental result, our findings are consistent within the
error range with the experimentally observed $T_{c\bar{s}0}^{a}(2900)^{++}$.
Therefore, our calculations support classifying $T_{c\bar{s}0}^{a}(2900)^{++}$
as a tetraquark state with the color octet-octet configuration.

In the following, we present the PC ratio plots, OPE ratio plots, and mass
curve plots for the quantum numbers $0^{-}$, $1^{+}$, and $1^{-}$. The
following Figs.4-6, 7-9, and 10-12 display the numerical results for cases
$0^{-}$, $1^{+}$, and $1^{-}$, respectively.

Figure 4:  The same caption as in Fig.1, but for case $0^{-}$.

Figure 5:  The same caption as in Fig.2, but for case $0^{-}$.

Figure 6:  The same caption as in Fig.3, but for case $0^{-}$.

Figure 7:  The same caption as in Fig.1, but for case $1^{+}$.

Figure 8:  The same caption as in Fig.2, but for case $1^{+}$.

Figure 9:  The same caption as in Fig.3, but for case $1^{+}$.

Figure 10:  The same caption as in Fig.1, but for case $1^{-}$.

Figure 11:  The same caption as in Fig.2, but for case $1^{-}$.

Figure 12:  The same caption as in Fig.3, but for case $1^{-}$.

Using the same numerical analysis method, we can apply the above analysis
process to cases with quantum numbers $0^{-}$, $1^{+}$ and $1^{-}$. We
summarize the numerical results for all four cases in Table 3.

Table 3:  Summary of the numerical results for tetraquark states in color octet-octet configuration, with quantum numbers $0^{+}$, $0^{-}$, $1^{+}$, and $1^{-}$. $j^{J^{P}}_{X}$ | $M_{B}^{2}\,(\rm{GeV}^{2})$ | $\sqrt{s_{0}}\,(\rm{GeV})$ | PC | Mass $(\rm{GeV})$
---|---|---|---|---
$j^{0^{+}}_{A}$ | ${2.20\\!-\\!2.80}$ | $3.6\pm 0.2$ | ${(65\\!-\\!40)\%}$ | ${2.91_{-0.20}^{+0.20}}$
$j^{0^{+}}_{C}$ | ${2.00\\!-\\!2.60}$ | $3.6\pm 0.2$ | ${(66\\!-\\!40)\%}$ | ${2.98_{-0.21}^{+0.20}}$
$j^{0^{-}}_{A}$ | ${2.20\\!-\\!2.65}$ | $3.8\pm 0.2$ | ${(54\\!-\\!40)\%}$ | ${3.45_{-0.10}^{+0.12}}$
$j^{0^{-}}_{D}$ | ${2.35\\!-\\!2.00}$ | $3.8\pm 0.2$ | ${(54\\!-\\!40)\%}$ | ${3.20_{-0.19}^{+0.17}}$
$j^{1^{+}}_{A}$ | ${2.20\\!-\\!2.80}$ | $3.6\pm 0.2$ | ${(64\\!-\\!40)\%}$ | ${2.95_{-0.19}^{+0.19}}$
$j^{1^{+}}_{C}$ | ${2.20\\!-\\!2.80}$ | $3.6\pm 0.2$ | ${(64\\!-\\!40)\%}$ | ${2.95_{-0.19}^{+0.19}}$
$j^{1^{-}}_{A}$ | ${1.80\\!-\\!2.40}$ | $3.6\pm 0.2$ | ${(62\\!-\\!40)\%}$ | ${3.23_{-0.10}^{+0.11}}$
$j^{1^{-}}_{D}$ | ${1.85\\!-\\!2.45}$ | $3.6\pm 0.2$ | ${(63\\!-\\!40)\%}$ | ${3.23_{-0.11}^{+0.09}}$

##  IV Discussion and conclusion

In this work, we have investigated tetraquark states composed of
$c\bar{s}u\bar{d}$ with a color octet-octet structure and quantum numbers
$J^{P}=0^{\pm}$ and $1^{\pm}$ by applying the QCD sum rule method. For the
$0^{+}$ case, we first constructed currents corresponding to the lowest
orbital angular momentum states (i.e., S-wave) between color octets, resulting
in four such currents as shown in currents (2)-(5). Then, using the QCD sum
rule method, we carried out the analytical calculation of the Operator Product
Expansion (OPE) up to dimension 11 condensates, including contributions linear
in the strange quark mass $m_{s}$. Finally, a numerical analysis was
performed.

The numerical analysis indicates that currents (3) and (5) correspond to
reasonable tetraquark states, with their masses being
$M_{X}^{A}=2.91_{-0.20}^{+0.20}~{}\text{GeV}$ and
$M_{X}^{C}=2.98_{-0.21}^{+0.20}~{}\text{GeV}$ respectively; currents (2) and
(4) do not correspond to reasonable tetraquark states due to the lack of a
reasonable parameter window.

By comparing with experimental results, our findings are consistent within the
error range with the experimentally observed $T_{c\bar{s}0}^{a}(2900)^{0/++}$.
Therefore, our calculations support classifying
$T_{c\bar{s}0}^{a}(2900)^{0/++}$ as a tetraquark state with a color octet-
octet structure.

Furthermore, based on the $J^{P}=0^{+}$ case, we also investigated tetraquark
states made up of $c\bar{s}u\bar{d}$ with color octet-octet structure and
quantum numbers $J^{P}=0^{-}$, $1^{+}$, and $1^{-}$. For each quantum number,
we predicted two new hadron states awaiting experimental verification, with
their masses as follows: $M_{0^{-}}^{A}=3.45_{-0.10}^{+0.12}~{}\text{GeV}$,
$M_{0^{-}}^{D}=3.20_{-0.19}^{+0.17}~{}\text{GeV}$,
$M_{1^{+}}^{A}=2.95_{-0.19}^{+0.19}~{}\text{GeV}$,
$M_{1^{+}}^{C}=2.95_{-0.19}^{+0.19}~{}\text{GeV}$,
$M_{1^{-}}^{A}=3.23_{-0.10}^{+0.11}~{}\text{GeV}$ and
$M_{1^{-}}^{D}=3.23_{-0.11}^{+0.09}~{}\text{GeV}$. With the advancement of
experimental techniques and the accumulation of new data, these predicted
results are expected to be confirmed in future experiments.

Our discovery not only provides theoretical support for the existence and
internal structure of $T_{c\bar{s}0}^{a}(2900)^{0/++}$ but also highlights the
QCD sum rules as a powerful tool for exploring the hadron spectrum of multi-
quark states. This consistency also indicates the great potential of QCD sum
rules in predicting the properties of exotic hadrons. This research and its
conclusions further reinforce the significance of multi-quark states in
particle physics, offering new directions for future experimental and
theoretical studies.

Acknowledgments

This work was supported by the Natural Science Foundation of Hebei Province
under Grant No. A2023205038.

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