AAPPS Bulletin

Research and Review

The two-body strong decays of the fully-charm tetraquark states

writerZhi-Gang Wang & Xiao-Song Yang

Vol.34 (Feb) 2024 | Article no.5 2024

Abstract

We study the hadronic coupling constants in the two-body strong decays of the fully-charm tetraquark states with the $J^{PC}=0^{++}$, $1^{+-}$, and $2^{++}$ via the QCD sum rules based on rigorous quark-hadron duality. Then we obtain the hadronic coupling constants and partial decay widths therefore total decay widths, which support assigning the X(6552) as the first radial excitation of the scalar tetraquark state. And other predictions can be used to diagnose exotic states in the future.

1 Introduction

In 2020, the LHCb collaboration explored the di-$J/\psi$ invariant mass spectrum using the pp collision data at center-of-mass energies of $\sqrt{s}=7$, 8 and $13\,\textrm{TeV}$, which corresponds to an integrated luminosity of $9\, \textrm{fb}^{-1}$ [1]. They observed a narrow structure at about 6.9 GeV and a broad structure above the di-$J/\psi$ threshold ranging from 6.2 to 6.8 GeV, and they also observed some vague structures around 7.2 GeV. It is the first time to observe the fully-heavy tetraquark candidates.

In 2023, the ATLAS collaboration explored the di-$J/\psi$ and $J/\psi \psi ^\prime$ invariant mass spectra and observed several resonances (R) using the pp collision data at center-of-mass energy of $\sqrt{s}=13\,\textrm{TeV}$, which corresponds to an integrated luminosity of $140\, \textrm{fb}^{-1}$ [2]. The fitted Breit-Wigner masses and widths are

$\begin{aligned} R_0 : M ={} & {} 6.41\pm 0.08_{-0.03}^{+0.08} \ \textrm{GeV}\, , \, \Gamma = 0.59\pm 0.35_{-0.20}^{+0.12} \ \textrm{GeV} \, , \nonumber \\ R_1 : M ={} & {} 6.63\pm 0.05_{-0.01}^{+0.08} \ \textrm{GeV}\, , \, \Gamma = 0.35\pm 0.11_{-0.04}^{+0.11} \ \textrm{GeV} \, , \nonumber \\ R_2 : M ={} & {} 6.86\pm 0.03_{-0.02}^{+0.01} \ \textrm{GeV}\, , \, \Gamma = 0.11\pm 0.05_{-0.01}^{+0.02} \ \textrm{GeV} \, , \end{aligned}$

(1)

$\begin{aligned} R_0 : M ={} & {} 6.65\pm 0.02_{-0.02}^{+0.03} \ \textrm{GeV}\, , \, \Gamma = 0.44\pm 0.05_{-0.05}^{+0.06} \ \textrm{GeV} \, , \nonumber \\ R_2 : M ={} & {} 6.91\pm 0.01\pm 0.01 \ \textrm{GeV}\, , \, \Gamma = 0.15\pm 0.03\pm 0.01 \ \textrm{GeV} \, , \end{aligned}$

(2)

in the di-$J/\psi$ mass spectrum, and

$\begin{aligned} R_3 : M = 7.22\pm 0.03_{-0.04}^{+0.01} \ \textrm{GeV}\, , \, \Gamma = 0.09\pm 0.06_{-0.05}^{+0.06} \ \textrm{GeV} \, , \end{aligned}$

(3)

$\begin{aligned} R_3 : M = 6.96\pm 0.05\pm 0.03 \ \textrm{GeV}\, , \, \Gamma = 0.51\pm 0.17_{-0.10}^{+0.11} \ \textrm{GeV} \, , \end{aligned}$

(4)

in the $J/\psi \psi ^\prime$ mass spectrum [2].

While at the ICHEP 2022 conference, the ATLAS collaboration reported evidences of several fully-charm tetraquark excesses in the di-$J/\psi$ and $J/\psi \psi ^\prime$ invariant mass spectra using the pp collision data at center-of-mass energy of $\sqrt{s}=13\,\textrm{TeV}$, which corresponds to an integrated luminosity of $139\,\textrm{fb}^{-1}$ [3]. The fitted Breit-Wigner masses and widths are

$\begin{aligned} R_0 : M ={} & {} 6.22 \pm 0.05 {}^{+0.04}_{-0.05} \ \textrm{GeV}\, , \, \Gamma = 0.31 \pm 0.12{}^{+0.07}_{-0.08} \ \textrm{GeV} \, , \nonumber \\ R_1 : M ={} & {} 6.62 \pm 0.03{}^{+0.02}_{-0.01} \ \textrm{GeV}\, , \, \Gamma = 0.31 \pm 0.09{}^{+0.06}_{-0.11} \ \textrm{GeV} \, , \nonumber \\ R_2 : M ={} & {} 6.87 \pm 0.03{}^{+0.06}_{-0.01} \ \textrm{GeV}\, , \, \Gamma = 0.12 \pm 0.04{}^{+0.03}_{-0.01} \ \textrm{GeV} \, , \end{aligned}$

(5)

in the di-$J/\psi$ mass spectrum [3].

Also in 2023, the CMS collaboration studied the di-$J/\psi$ mass spectrum using the pp collision data center-of-mass energies of $\sqrt{s} = 13\,\textrm{TeV}$, which corresponds to an integrated luminosity of $135\,\textrm{ fb}^{-1}$ [4]. They observed three new resonance structures, the fitted Breit-Wigner masses and widths are

$\begin{aligned} R_1 : M ={} & {} 6552\pm 10\pm 12 \ \textrm{MeV}\, , \, \Gamma = 124^{+32}_{-26}\pm 33 \ \textrm{MeV} \, , \nonumber \\ R_2 : M ={} & {} 6927\pm 9\pm 4 \ \textrm{MeV}\, , \, \Gamma = 122^{+24}_{-21}\pm 18 \ \textrm{MeV} \, , \nonumber \\ R_3 : M ={} & {} 7287^{+20}_{-18}\pm 5 \ \textrm{MeV}\, , \, \Gamma = 95^{+59}_{-40}\pm 19 \ \textrm{MeV} \, , \end{aligned}$

(6)

the local significances of the three peaks are 6.5, 9.4, and 4.1 standard deviations, respectively.

Up to now, only the X(6900) is established; however, the quantum numbers $J^{PC}$ have not been determined yet. The predicted masses of the fully-heavy tetraquark (molecular) states from the different theoretical approaches before and after the LHCb experiment [1] lie either above or below the $J/\psi J/\psi$ or $\Upsilon \Upsilon$ threshold, and vary at a large range [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32], and none of them are fully consistent with the more precise measurements of the ATLAS and CMS collaborations [2, 4]. It is certainly that the new ATLAS and CMS experimental data would provide more robust promises to decipher the novel peaking structures appeared in the di-$J/\psi$ invariant mass spectrum, and serve as more efficacious examinations of various theoretical models and would be used to diagnose the nature of the exotic X, Y, and Z states.

In Refs. [6, 14], we study the mass spectrum of the scalar, axialvector, vector and tensor fully-heavy tetraquark states in the framework of the QCD sum rules. Subsequently, we explore the mass spectrum of the first radial excitations of the fully-charm tetraquark states, and resort to the Regge trajectories to obtain the masses of the second radial excitations, and make possible assignments of the LHCb’s new resonances [16]. In Ref. [23], we re-study the mass spectrum of the ground state and first/second/third radial excited diquark-antidiquark type fully-charm tetraquark states, then we take account of the new CMS and ATLAS experimental data and try to make possible assignments of the X(6600), X(6900), and X(7300) consistently.

Because we cannot assign a hadron by the mass alone, we should explore the decays at least. In the present work, we explore the two-body strong decays of the ground states and first radial excitations of the scalar, axialvector and tensor fully-charm tetraquark states in the framework of the QCD sum rules based on rigorous quark-hadron duality, which works well in studying the hadronic coupling constants [33,34,35,36,37], and make more robust assignments based on the masses and widths together.

The article is organized as follows: we obtain the QCD sum rules for the hadronic coupling constants of the fully-charm tetraquark states in Section 2; in Section 3, we present the numerical results and discussions; and Section 4 is reserved for our conclusion.

2 QCD sum rules for the hadronic coupling constants

Let us write down the three-point correlation functions firstly,

$\begin{aligned} \Pi ^1(p,q) ={} & {} i^2 \int d^4 x d^4 y e^{ip\cdot x} e^{iq\cdot y} \left\langle 0| T\left\{ J^{\eta _c}(x) J^{\eta _c}(y) J^{0\dagger }(0)\right\} |0\right\rangle \, , \nonumber \\ \Pi ^2_{\alpha \beta }(p,q) ={} & {} i^2 \int d^4 x d^4 y e^{ip\cdot x} e^{iq\cdot y} \left\langle 0| T\left\{ J^{J/\psi }_{\alpha }(x) J^{J/\psi }_{\beta }(y) J^{0\dagger }(0)\right\} |0\right\rangle \, , \nonumber \\ \Pi ^3_{\alpha }(p,q) ={} & {} i^2 \int d^4 x d^4 y e^{ip\cdot x} e^{iq\cdot y} \left\langle 0| T\left\{ J^{\chi _c}_{\alpha }(x) J^{\eta _c}(y) J^{0\dagger }(0)\right\} |0\right\rangle \, , \end{aligned}$

(7)

$\begin{aligned} \Pi ^4_{\mu \alpha \beta }(p,q) ={} & {} i^2 \int d^4 x d^4 y e^{ip\cdot x} e^{iq\cdot y} \left\langle 0| T\left\{ J^{J/\psi }_{\mu }(x) J^{\eta _c}(y) J_{\alpha \beta }^{1\dagger }(0)\right\} |0\right\rangle \, , \nonumber \\ \Pi ^5_{\mu \nu \alpha \beta }(p,q) ={} & {} i^2 \int d^4 x d^4 y e^{ip\cdot x} e^{iq\cdot y} \left\langle 0| T\left\{ J^{h_c}_{\mu \nu }(x) J^{\eta _c}(y) J_{\alpha \beta }^{1\dagger }(0)\right\} |0\right\rangle \, , \end{aligned}$

(8)

$\begin{aligned} \Pi ^6_{\alpha \beta }(p,q) ={} & {} i^2 \int d^4 x d^4 y e^{ip\cdot x} e^{iq\cdot y} \left\langle 0| T\left\{ J^{\eta _c}(x) J^{\eta _c}(y) J_{\alpha \beta }^{2\dagger }(0) \right\} |0\right\rangle \, , \nonumber \\ \Pi ^7_{\mu \nu \alpha \beta }(p,q) ={} & {} i^2 \int d^4 x d^4 y e^{ip\cdot x} e^{iq\cdot y} \left\langle 0| T\left\{ J^{J/\psi }_{\mu }(x) J^{J/\psi }_{\nu }(y) J_{\alpha \beta }^{2\dagger }(0)\right\} |0\right\rangle \, , \end{aligned}$

(9)

where the currents

$\begin{aligned} J^{\eta _c}(x) ={} & {} \bar{c}(x)i\gamma _5 c(x)\, , \nonumber \\ J^{J/\psi }_{\alpha }(x) ={} & {} \bar{c}(x)\gamma _\alpha c(x)\, , \nonumber \\ J^{\chi _c}_{\alpha }(x) ={} & {} \bar{c}(x)\gamma _\alpha \gamma _5 c(x)\, , \nonumber \\ J^{h_c}_{\alpha \beta }(x) ={} & {} \bar{c}(x)\sigma _{\alpha \beta } c(x)\, , \end{aligned}$

(10)

interpolate the charmonia $\eta _c$, $J/\psi$, $\chi _{c1}$ and $h_c$, respectively, the currents

$\begin{aligned} J^0(x) ={} & {} \epsilon ^{ijk} \epsilon ^{imn} c^T_j(x) C \gamma _\alpha c_k(x) \bar{c}_m(x)\gamma ^\alpha C \bar{c}^T_n(x)\, , \nonumber \\ J_{\alpha \beta }^1(x) ={} & {} \epsilon ^{ijk} \epsilon ^{imn} \Big \{c^T_j(x) C \gamma _\alpha c_k(x) \bar{c}_m(x)\gamma _\beta C \bar{c}^T_n(x) -c^T_j(x) C \gamma _\beta c_k(x) \bar{c}_m(x)\gamma _\alpha C \bar{c}^T_n(x) \Big \}\, ,\nonumber \\ J_{\alpha \beta }^2(x) ={} & {} \frac{\epsilon ^{ijk} \epsilon ^{imn}}{\sqrt{2}} \Big \{c^T_j(x) C \gamma _\alpha c_k(x) \bar{c}_m(x)\gamma _\beta C \bar{c}^T_n(x) +c^T_j(x) C \gamma _\beta c_k(x) \bar{c}_m(x)\gamma _\alpha C \bar{c}^T_n(x) \Big \}\, , \end{aligned}$

(11)

interpolate the fully-charm tetraquark states with the spins 0, 1 and 2, respectively [14, 16, 23], the i, j, k, m, and n are color indexes. We take those correlation functions to study the hadronic coupling constants $G_{X_0^{(\prime )}\eta _c\eta _c}$, $G_{X_0^{(\prime )}J/\psi J/\psi }$, $G_{X_0^{(\prime )}\chi _c \eta _c}$, $G_{X_1^{(\prime )}J/\psi \eta _c}$, $G_{X_1^{(\prime )}h_c \eta _c}$, $G_{X_2^{(\prime )}\eta _c\eta _c}$, and $G_{X_2^{(\prime )}J/\psi J/\psi }$, respectively.

At the hadron side of the correlation functions, see Eqs. (7)–(9), we insert a complete set of intermediate hadronic states with the same quantum numbers as the currents, and isolate the ground state contributions in the charmonium channels, and the ground state plus first radial excited state contributions in the tetraquark channels explicitly [38, 39],

$\begin{aligned} \Pi ^1(p,q)={} & {} \frac{\lambda _{X_0} f_{\eta _c}^2 m_{\eta _c}^4 G_{X_0\eta _c \eta _c} }{4m_c^2\left(m_{X_0}^2-p^{\prime 2}\right)\left(m_{\eta _c}^2-p^2\right)\left(m_{\eta _c}^2-q^2\right)}\,p\cdot q \nonumber \\{} & {} +\frac{\lambda _{X^\prime _0} f_{\eta _c}^2 m_{\eta _c}^4 G_{X^\prime _0\eta _c \eta _c} }{4m_c^2\left(m_{X^\prime _0}^2-p^{\prime 2}\right)\left(m_{\eta _c}^2-p^2\right)\left(m_{\eta _c}^2-q^2\right)}\,p\cdot q + \cdots \, , \nonumber \\ ={} & {} \Pi _{1}(p^{\prime 2},p^2,q^2)\, p\cdot q + \cdots \, , \end{aligned}$

(12)

$\begin{aligned} \Pi ^2_{\alpha \beta }(p,q)={} & {} \frac{\lambda _{X_0} f_{J/\psi }^2 m_{J/\psi }^2 G_{X_0J/\psi J/\psi } }{\left(m_{X_0}^2-p^{\prime 2}\right)\left(m_{J/\psi }^2-p^2\right)\left(m_{J/\psi }^2-q^2\right)}\, g_{\alpha \beta } \nonumber \\{} & {} +\frac{\lambda _{X^\prime _0} f_{J/\psi }^2 m_{J/\psi }^2 G_{X^\prime _0J/\psi J/\psi } }{\left(m_{X^\prime _0}^2-p^{\prime 2}\right)\left(m_{J/\psi }^2-p^2\right)\left(m_{J/\psi }^2-q^2\right)}\, g_{\alpha \beta } + \cdots \, , \nonumber \\ ={} & {} \Pi _{2}\left(p^{\prime 2},p^2,q^2\right)\, g_{\alpha \beta } + \cdots \, , \end{aligned}$

(13)

$\begin{aligned} \Pi ^3_{\alpha }(p,q)={} & {} -\frac{\lambda _{X_0} f_{\chi _c} m_{\chi _c} f_{\eta _c}m_{\eta _c}^2 G_{X_0\chi _c \eta _c} }{2m_c\left(m_{X_0}^2-p^{\prime 2}\right)\left(m_{\chi _c}^2-p^2\right)\left(m_{\eta _c}^2-q^2\right)}\,iq_\alpha \nonumber \\{} & {} -\frac{\lambda _{X^\prime _0} f_{\chi _c} m_{\chi _c} f_{\eta _c}m_{\eta _c}^2 G_{X^\prime _0\chi _c \eta _c} }{2m_c\left(m_{X^\prime _0}^2-p^{\prime 2}\right)\left(m_{\chi _c}^2-p^2\right)\left(m_{\eta _c}^2-q^2\right)}\,iq_\alpha + \cdots \, , \nonumber \\ ={} & {} \Pi _{3}\left(p^{\prime 2},p^2,q^2\right)\, (-iq_\alpha ) + \cdots \, , \end{aligned}$

(14)

$\begin{aligned} P_{A}^{\alpha \beta \alpha ^\prime \beta ^\prime }(p^\prime )\,\epsilon _{\alpha ^\prime \beta ^\prime }{}^{\mu \tau }\,p_\tau \,\Pi ^4_{\mu \alpha \beta }(p,q)= \widetilde{\Pi }_4\left(p^{\prime 2},p^2,q^2\right)\left( p^2+p\cdot q\right) \, , \end{aligned}$

(15)

$\begin{aligned} \Pi _4\left(p^{\prime 2},p^2,q^2\right)={} & {} \widetilde{\Pi }_4(p^{\prime 2},p^2,q^2)\,p^2\, ,\nonumber \\ ={} & {} \frac{\lambda _{X_1} f_{J/\psi }m^3_{J/\psi }f_{\eta _c} m_{\eta _c}^2 G_{X_1J/\psi \eta _c} }{2m_cm_{X_1}\left(m_{X_1}^2-p^{\prime 2}\right)\left(m_{J/\psi }^2-p^2\right)\left(m_{\eta _c}^2-q^2\right)}\nonumber \\{} & {} +\frac{\lambda _{X_1^\prime } f_{J/\psi }m^3_{J/\psi }f_{\eta _c} m_{\eta _c}^2 G_{X^\prime _1J/\psi \eta _c} }{2m_cm_{X^\prime _1}\left(m_{X^\prime _1}^2-p^{\prime 2}\right)\left(m_{J/\psi }^2-p^2\right)\left(m_{\eta _c}^2-q^2\right)} + \cdots \, , \end{aligned}$

(16)

$\begin{aligned} P_{A}^{\mu \nu \mu ^\prime \nu ^\prime }(p)P_{A}^{\alpha \beta \alpha ^\prime \beta ^\prime }(p^\prime )\,\epsilon _{\mu ^\prime \nu ^\prime \alpha ^\prime \beta ^\prime }\, \Pi ^5_{\mu \nu \alpha \beta }(p,q)=i\, \widetilde{\Pi }_5(p^{\prime 2},p^2,q^2)\left( -p^2q^2+(p\cdot q)^2\right) \, , \end{aligned}$

(17)

$\begin{aligned} \Pi _5(p^{\prime 2},p^2,q^2)={} & {} \widetilde{\Pi }_5\left(p^{\prime 2},p^2,q^2\right)\,p^2q^2\, ,\nonumber \\ ={} & {} \frac{\lambda _{X_1} f_{h_c}m^2_{h_c}f_{\eta _c} m_{\eta _c}^4 G_{X_1h_c \eta _c} }{9m_cm_{X_1}\left(m_{X_1}^2-p^{\prime 2}\right)\left(m_{h_c}^2-p^2\right)\left(m_{\eta _c}^2-q^2\right)}\nonumber \\{} & {} +\frac{\lambda _{X^\prime _1} f_{h_c}m^2_{h_c}f_{\eta _c} m_{\eta _c}^4 G_{X^\prime _1h_c \eta _c} }{9m_cm_{X^\prime _1}\left(m_{X^\prime _1}^2-p^{\prime 2}\right)\left(m_{h_c}^2-p^2\right)\left(m_{\eta _c}^2-q^2\right)} + \cdots \, , \end{aligned}$

(18)

$\begin{aligned} \Pi ^6_{\alpha \beta }(p,q)={} & {} - \frac{\lambda _{X_2} f_{\eta _c}^2 m_{\eta _c}^4 \left(m_{X_2}^2-m_{\eta _c}^2\right)\, G_{X_2\eta _c \eta _c} }{6m_c^2m_{X_2}^2\left(m_{X_2}^2-p^{\prime 2}\right)\left(m_{\eta _c}^2-p^2\right)\left(m_{\eta _c}^2-q^2\right)}\,p_\alpha q_\beta \nonumber \\{} & {} - \frac{\lambda _{X^\prime _2} f_{\eta _c}^2 m_{\eta _c}^4 \left(m_{X^\prime _2}^2-m_{\eta _c}^2\right)\, G_{X_2^\prime \eta _c \eta _c} }{6m_c^2m_{X^\prime _2}^2\left(m_{X^\prime _2}^2-p^{\prime 2}\right)\left(m_{\eta _c}^2-p^2)(m_{\eta _c}^2-q^2\right)}\,p_\alpha q_\beta + \cdots \, , \nonumber \\ ={} & {} \Pi _{6}\left(p^{\prime 2},p^2,q^2\right)\left( -p_{\alpha }q_{\beta }\right) + \cdots \, , \end{aligned}$

(19)

$\begin{aligned} \Pi ^7_{\mu \nu \alpha \beta }(p,q)={} & {} - \frac{\lambda _{X_2} f_{J/\psi }^2 m_{J/\psi }^2 \, G_{X_2J/\psi J/\psi } }{2\left(m_{X_2}^2-p^{\prime 2}\right)\left(m_{J/\psi }^2-p^2\right)\left(m_{J/\psi }^2-q^2\right)}\, \left( g_{\mu \alpha }g_{\nu \beta }+g_{\mu \beta }g_{\nu \alpha }\right) \nonumber \\{} & {} - \frac{\lambda _{X^\prime _2} f_{J/\psi }^2 m_{J/\psi }^2 \, G_{X^\prime _2J/\psi J/\psi } }{2\left(m_{X^\prime _2}^2-p^{\prime 2}\right)\left(m_{J/\psi }^2-p^2\right)\left(m_{J/\psi }^2-q^2\right)}\, \left( g_{\mu \alpha }g_{\nu \beta }+g_{\mu \beta }g_{\nu \alpha }\right) + \cdots \, , \nonumber \\ ={} & {} \Pi _{7}\left(p^{\prime 2},p^2,q^2\right)\left( -g_{\mu \alpha }g_{\nu \beta }-g_{\mu \beta }g_{\nu \alpha }\right) + \cdots \, , \end{aligned}$

(20)

where

$\begin{aligned} P_{A}^{\mu \nu \alpha \beta }(p)={} & {} \frac{1}{6}\left( g^{\mu \alpha }-\frac{p^\mu p^\alpha }{p^2}\right) \left( g^{\nu \beta }-\frac{p^\nu p^\beta }{p^2}\right) \, , \end{aligned}$

(21)

the decay constants or pole residues are defined by,

$\begin{aligned} \langle 0|J^{\eta _c}(0)|\eta _c(p)\rangle ={} & {} \frac{f_{\eta _c} m_{\eta _c}^2}{2m_c} \,\, , \nonumber \\ \langle 0|J_{\mu }^{J/\psi }(0)|J/\psi (p)\rangle ={} & {} f_{J/\psi } m_{J/\psi } \,\xi _\mu \,\, , \nonumber \\ \langle 0|J_{\mu \nu }^{h_c}(0)|h_c(p)\rangle ={} & {} f_{h_c} \epsilon _{\mu \nu \alpha \beta }\, p^\alpha \xi ^\beta \,\, , \nonumber \\ \langle 0|J_{\mu }^{\chi _c}(0)|\chi _c(p)\rangle ={} & {} f_{\chi _c} m_{\chi _c}\, \zeta _\mu \,\, , \end{aligned}$

(22)

$\begin{aligned} \langle 0|J^0 (0)|X^{(\prime )}_0 (p)\rangle ={} & {} \lambda _{X^{(\prime )}_0} \, , \nonumber \\ \langle 0|J^1_{\mu \nu }(0)|X^{(\prime )}_1(p)\rangle ={} & {} \tilde{\lambda }_{X^{(\prime )}_1} \, \epsilon _{\mu \nu \alpha \beta } \, \varepsilon ^{\alpha }p^{\beta }\, , \nonumber \\ \langle 0|J^2_{\mu \nu }(0)|X^{(\prime )}_2 (p)\rangle ={} & {} \lambda _{X^{(\prime )}_2} \, \varepsilon _{\mu \nu } \, , \end{aligned}$

(23)

$\tilde{\lambda }_{X^{(\prime )}_1}m_{X^{(\prime )}_1}=\lambda _{X^{(\prime )}_1}$, the hadronic coupling constants are defined by,

$\begin{aligned} \langle \eta _c(p)\eta _c(q)|X_0^{(\prime )}(p^\prime )\rangle ={} & {} i p\cdot q \,G_{X_0^{(\prime )}\eta _c\eta _c}\, ,\nonumber \\ \langle J/\psi (p)J/\psi (q)|X_0^{(\prime )}(p^\prime )\rangle ={} & {} i \xi ^* \cdot \xi ^* \, G_{X_0^{(\prime )}J/\psi J/\psi }\, ,\nonumber \\ \langle \chi _c(p)\eta _c(q)|X_0^{(\prime )}(p^\prime )\rangle ={} & {} -\zeta ^* \cdot q \,G_{X_0^{(\prime )}\chi _c \eta _c}\, , \end{aligned}$

(24)

$\begin{aligned} \langle J/\psi (p)\eta _c(q)|X_1^{(\prime )}(p^\prime )\rangle ={} & {} i\xi ^* \cdot \varepsilon \,G_{X_1^{(\prime )}J/\psi \eta _c}\, ,\nonumber \\ \langle h_c(p)\eta _c(q)|X_1^{(\prime )}(p^\prime )\rangle ={} & {} \epsilon ^{\lambda \tau \rho \sigma }p_{\lambda }\xi ^*_{\tau }p^\prime _\rho \varepsilon _\sigma \,G_{X_1^{(\prime )}h_c \eta _c}\, , \end{aligned}$

(25)

$\begin{aligned} \langle \eta _c(p)\eta _c(q)|X_2^{(\prime )}(p^\prime )\rangle ={} & {} -i \varepsilon _{\mu \nu }p^{\mu }q^{\nu } \,G_{X_2^{(\prime )}\eta _c\eta _c}\, ,\nonumber \\ \langle J/\psi (p)J/\psi (q)|X_2^{(\prime )}(p^\prime )\rangle ={} & {} -i \varepsilon ^{\alpha \beta } \xi ^*_\alpha \xi ^*_\beta \, G_{X_2^{(\prime )}J/\psi J/\psi }\, , \end{aligned}$

(26)

the $\xi _\mu$, $\zeta _\mu$, $\varepsilon _{\mu }$, and $\varepsilon _{\mu \nu }$ are the polarization vectors of the corresponding charmonium/tetraquark states. As the currents $J_{\alpha \beta }^{h_c}(x)$ and $J^1_{\alpha \beta }(x)$ couple potentially to the charmonia/tetraquarks with both the $J^{PC}=1^{+-}$ and $1^{--}$, we introduce the projector $P_{A}^{\mu \nu \alpha \beta }(p)$ to project out the states with the $J^{PC}=1^{+-}$ [14, 16, 23]. In this work, we explore the hadronic coupling constants with the components $\Pi _{i}\left(p^{\prime 2},p^2,q^2\right)$ with $i=1$, $\cdots$, 7 to avoid contaminations.

Generally speaking, if a quark current and a hadron have the same quantum numbers, then they have non-vanishing couplings with each other. For example, the currents $J^0(x)$, $J^1_{\alpha \beta }(x)$, and $J^2_{\alpha \beta }(x)$ couple potentially to the 1S, 2S, $\cdots$ states of the fully-charm tetraquarks, we take into account the 1S and 2S states in the two-point QCD sum rules, and obtain the masses and poles residues by solving an univariate quadratic equation. For the technical details, one can consult Refs. [16, 23].

Now, we obtain the hadronic spectral densities $\rho _H(s^\prime ,s,u)$ through triple dispersion relation,

$\begin{aligned} \Pi _{H}(p^{\prime 2},p^2,q^2)=\int _{16m_c^2}^\infty ds^{\prime } \int _{4m_c^2}^\infty ds \int _{4m_c^2}^\infty du \frac{\rho _{H}(s^\prime ,s,u)}{(s^\prime -p^{\prime 2})(s-p^2)(u-q^2)}\, , \end{aligned}$

(27)

where

$\begin{aligned} \rho _{H}(s^\prime ,s,u)=\underset{\epsilon _3\rightarrow 0}{\textrm{lim}}\,\,\underset{\epsilon _2\rightarrow 0}{\textrm{lim}} \,\,\underset{\epsilon _1\rightarrow 0}{\textrm{lim}}\,\,\frac{ \textrm{Im}_{s^\prime }\, \textrm{Im}_{s}\,\textrm{Im}_{u}\,\Pi _{H}(s^\prime +i\epsilon _3,s+i\epsilon _2,u+i\epsilon _1) }{\pi ^3} \, , \end{aligned}$

(28)

we add the subscript H to stand for the components $\Pi _{i}(p^{\prime 2},p^2,q^2)$ with $i=1-7$ at the hadron side. Although the variables $p^\prime$, p and q have the relation $p^\prime =p+q$, it is feasible to take the $p^{\prime 2}$, $p^2$, and $q^2$ as free parameters to determine the spectral densities, and we can obtain a nonzero imaginary part indeed for all the variables $p^{\prime 2}$, $p^2$, and $q^2$.

At the QCD side, we take account of the perturbative terms and gluon condensate contributions, and obtain the QCD spectral densities of the components $\Pi _{i}(p^{\prime 2},p^2,q^2)$ through double dispersion relation,

$\begin{aligned} \Pi _{QCD}(p^{\prime 2},p^2,q^2)= & {} \int _{4m_c^2}^\infty ds \int _{4m_c^2}^\infty du \frac{\rho _{QCD}(p^{\prime 2},s,u)}{(s-p^2)(u-q^2)}\, , \end{aligned}$

(29)

$\begin{aligned} \textrm{lim}_{\epsilon \rightarrow 0}\textrm{Im}\,\Pi _{QCD}\left(s^\prime +i\epsilon _3,p^2,q^2\right)= & {} 0\, , \end{aligned}$

(30)

we cannot obtain a nonzero imaginary part for the variable $p^{\prime 2}$. In Fig. 1, we plot the lowest order Feynman diagrams for the hadronic coupling constants $G_{X_0\eta _c \eta _c}$ and $G_{\chi _{c0}\eta _c\eta _c}$ as an example. We calculate those diagrams with the Cutkosky’s rules, although the left diagram for the tetraquark state $X_0$ does not allow a nonzero imaginary part with respect to the $s^\prime +i\epsilon _3$, it does not forbid putting the $s^\prime$ on the mass-shell at the hadron side; while the right diagram for the traditional charmonium $\chi _{c0}$ does forbid putting the $s^\prime$ on the mass-shell at the hadron side. Thus, in the QCD sum rules for the traditional mesons, we have to put one of the variables $p^{\prime 2}$, $p^2$, and $q^2$ off mass-shell.

At the hadron side, there is triple dispersion relation, see Eq. (27), while at the QCD side, there is only double dispersion relation, see Eq. (29), they do not match with each other channel by channel without performing some trick. We accomplish the integral over $ds^\prime$ firstly at the hadron side, then match the hadron side with the QCD side of the components $\Pi _{i}(p^{\prime 2},p^2,q^2)$ of the correlation functions bellow the continuum thresholds $s_0$ and $u_0$ to obtain rigorous quark-hadron duality [33, 34],

$\begin{aligned} \int _{4m_c^2}^{s_0}ds \int _{4m_c^2}^{u_0}du \left[ \int _{16m_c^2}^{\infty }ds^\prime \frac{\rho _H(s^\prime ,s,u)}{\left(s^\prime -p^{\prime 2}\right)\left(s-p^2\right)\left(u-q^2\right)} \right] =\int _{4m_c^2}^{s_{0}}ds \int _{4m_c^2}^{u_0}du \frac{\rho _{QCD}(s,u)}{\left(s-p^2\right)\left(u-q^2\right)} \, . \nonumber \\ \end{aligned}$

(31)

Now we write down the hadron representation explicitly,

$\begin{aligned} \Pi _{1}(p^{\prime 2},p^2,q^2)={} & {} \frac{\lambda _{X_0} f_{\eta _c}^2 m_{\eta _c}^4 G_{X_0\eta _c \eta _c} }{4m_c^2\left(m_{X_0}^2-p^{\prime 2}\right)\left(m_{\eta _c}^2-p^2\right)\left(m_{\eta _c}^2-q^2\right)}\nonumber \\{} & {} +\frac{\lambda _{X^\prime _0} f_{\eta _c}^2 m_{\eta _c}^4 G_{X^\prime _0\eta _c \eta _c} }{4m_c^2\left(m_{X^\prime _0}^2-p^{\prime 2}\right)\left(m_{\eta _c}^2-p^2\right)\left(m_{\eta _c}^2-q^2\right)} +\frac{C_1 }{\left(m_{\eta _c}^2-p^2\right)\left(m_{\eta _c}^2-q^2\right)} \, , \end{aligned}$

(32)

$\begin{aligned} \Pi _{2}(p^{\prime 2},p^2,q^2)={} & {} \frac{\lambda _{X_0} f_{J/\psi }^2 m_{J/\psi }^2 G_{X_0J/\psi J/\psi } }{\left(m_{X_0}^2-p^{\prime 2}\right)\left(m_{J/\psi }^2-p^2\right)\left(m_{J/\psi }^2-q^2\right)}\nonumber \\{} & {} +\frac{\lambda _{X^\prime _0} f_{J/\psi }^2 m_{J/\psi }^2 G_{X^\prime _0J/\psi J/\psi } }{\left(m_{X^\prime _0}^2-p^{\prime 2}\right)\left(m_{J/\psi }^2-p^2\right)\left(m_{J/\psi }^2-q^2\right)} +\frac{C_2 }{\left(m_{J/\psi }^2-p^2\right)\left(m_{J/\psi }^2-q^2\right)} \, , \end{aligned}$

(33)

$\begin{aligned} \Pi _{3}(p^{\prime 2},p^2,q^2)={} & {} \frac{\lambda _{X_0} f_{\chi _c} m_{\chi _c} f_{\eta _c}m_{\eta _c}^2 G_{X_0\chi _c \eta _c} }{2m_c\left(m_{X_0}^2-p^{\prime 2}\right)\left(m_{\chi _c}^2-p^2\right)\left(m_{\eta _c}^2-q^2\right)}\nonumber \\{} & {} +\frac{\lambda _{X^\prime _0} f_{\chi _c} m_{\chi _c} f_{\eta _c}m_{\eta _c}^2 G_{X^\prime _0\chi _c \eta _c} }{2m_c\left(m_{X^\prime _0}^2-p^{\prime 2}\right)\left(m_{\chi _c}^2-p^2\right)\left(m_{\eta _c}^2-q^2\right)} + \frac{C_3}{\left(m_{\chi _c}^2-p^2\right)\left(m_{\eta _c}^2-q^2\right)}\, , \end{aligned}$

(34)

$\begin{aligned} \Pi _4(p^{\prime 2},p^2,q^2)={} & {} \frac{\tilde{\lambda }_{X_1} f_{J/\psi }m^3_{J/\psi }f_{\eta _c} m_{\eta _c}^2 G_{X_1J/\psi \eta _c} }{2m_c\left(m_{X_1}^2-p^{\prime 2}\right)\left(m_{J/\psi }^2-p^2\right)\left(m_{\eta _c}^2-q^2\right)}\nonumber \\{} & {} +\frac{\tilde{\lambda }_{X_1^\prime } f_{J/\psi }m^3_{J/\psi }f_{\eta _c} m_{\eta _c}^2 G_{X^\prime _1J/\psi \eta _c} }{2m_c\left(m_{X^\prime _1}^2-p^{\prime 2}\right)\left(m_{J/\psi }^2-p^2\right)\left(m_{\eta _c}^2-q^2\right)} + \frac{C_4}{(m_{J/\psi }^2-p^2)\left(m_{\eta _c}^2-q^2\right)}\, , \end{aligned}$

(35)

$\begin{aligned} \Pi _5(p^{\prime 2},p^2,q^2)={} & {} \frac{\tilde{\lambda }_{X_1} f_{h_c}m^2_{h_c}f_{\eta _c} m_{\eta _c}^4 G_{X_1h_c \eta _c} }{9m_c\left(m_{X_1}^2-p^{\prime 2}\right)\left(m_{h_c}^2-p^2\right)\left(m_{\eta _c}^2-q^2\right)}\nonumber \\{} & {} +\frac{\tilde{\lambda }_{X^\prime _1} f_{h_c}m^2_{h_c}f_{\eta _c} m_{\eta _c}^4 G_{X^\prime _1h_c \eta _c} }{9m_c\left(m_{X^\prime _1}^2-p^{\prime 2}\right)\left(m_{h_c}^2-p^2\right)\left(m_{\eta _c}^2-q^2\right)} +\frac{C_5}{\left(m_{h_c}^2-p^2\right)\left(m_{\eta _c}^2-q^2\right)} \, , \end{aligned}$

(36)

$\begin{aligned} \Pi _{6}(p^{\prime 2},p^2,q^2)={} & {} \frac{\lambda _{X_2} f_{\eta _c}^2 m_{\eta _c}^4 \left(m_{X_2}^2-m_{\eta _c}^2\right)\, G_{X_2\eta _c \eta _c} }{6m_c^2m_{X_2}^2\left(m_{X_2}^2-p^{\prime 2}\right)\left(m_{\eta _c}^2-p^2\right)\left(m_{\eta _c}^2-q^2\right)} \nonumber \\{} & {} + \frac{\lambda _{X^\prime _2} f_{\eta _c}^2 m_{\eta _c}^4 (m_{X^\prime _2}^2-m_{\eta _c}^2)\, G_{X_2^\prime \eta _c \eta _c} }{6m_c^2m_{X^\prime _2}^2\left(m_{X^\prime _2}^2-p^{\prime 2}\right)\left(m_{\eta _c}^2-p^2\right)\left(m_{\eta _c}^2-q^2\right)} + \frac{C_6}{\left(m_{\eta _c}^2-p^2\right)\left(m_{\eta _c}^2-q^2\right)} \, , \end{aligned}$

(37)

$\begin{aligned} \Pi _{7}(p^{\prime 2},p^2,q^2)={} & {} \frac{\lambda _{X_2} f_{J/\psi }^2 m_{J/\psi }^2 \, G_{X_2J/\psi J/\psi } }{2\left(m_{X_2}^2-p^{\prime 2}\right)\left(m_{J/\psi }^2-p^2\right)\left(m_{J/\psi }^2-q^2\right)} \nonumber \\{} & {} + \frac{\lambda _{X^\prime _2} f_{J/\psi }^2 m_{J/\psi }^2 \, G_{X^\prime _2J/\psi J/\psi } }{2\left(m_{X^\prime _2}^2-p^{\prime 2}\right)\left(m_{J/\psi }^2-p^2\right)\left(m_{J/\psi }^2-q^2\right)} + \frac{C_7}{\left(m_{J/\psi }^2-p^2\right)\left(m_{J/\psi }^2-q^2\right)}\, , \end{aligned}$

(38)

where we introduce the parameters $C_{i}$ with $i=1-7$ to represent all the contributions concerning the higher resonances in the $s^\prime$ channel.

We set $p^{\prime 2}=2p^2$ in the components $\Pi _H(p^{\prime 2},p^2,q^2)$, and perform double Borel transform in regard to $P^2=-p^2$ and $Q^2=-q^2$, respectively. The spectral densities $\rho _{H}(s^\prime ,s,u)$ and $\rho _{QCD}(s,u)$ are physical, while the variables $p^{\prime 2}$, $p^2$, and $q^2$ in Eq. (31) are free variables, as we perform the operator product expansion at the large space-like regions $-p^2 \rightarrow \infty$ and $-q^2 \rightarrow \infty$. Generally speaking, we can set $p^{\prime 2}=\alpha p^2$ or $\alpha q^2$ with $\alpha$ to be a finite quantity. Considering the mass poles at $s^\prime =m^2_{X^{(\prime )}_{0,1,2}}$, $s=m^2_{\eta _c,J/\psi ,\chi _c,h_c}$ and $u=m^2_{\eta _c,J/\psi ,\chi _c,h_c}$ have the relations $s^\prime =4s=4u$ approximately, we can set $\alpha =1-4$. In calculations, we observe that the optimal value $\alpha =2$.

Then we set $T_1^2=T_2^2=T^2$ to obtain seven QCD sum rules,

\(\begin{aligned}{} & {} \frac{\lambda _{X_0\eta _c \eta _c} G_{X_0\eta _c\eta _c}}{2 \left(\widetilde{m}^2_{X_0}-m^2_{\eta _c}\right)}\left[ \exp \left( -\frac{m^2_{\eta _c}}{T^2}\right) -\exp \left( -\frac{\widetilde{m}^2_{X_0}}{T^2}\right) \right] \exp \left( -\frac{m^2_{\eta _c}}{T^2}\right) + \nonumber \\{} & {} \frac{\lambda _{X^\prime _0\eta _c \eta _c} G_{X^\prime _0\eta _c\eta _c}}{2 \left(\widetilde{m}^2_{X^\prime _0}-m^2_{\eta _c}\right)}\left[ \exp \left( -\frac{m^2_{\eta _c}}{T^2}\right) -\exp \left( -\frac{\widetilde{m}^2_{X^\prime _0}}{T^2}\right) \right] \exp \left( -\frac{m^2_{\eta _c}}{T^2}\right) +C_{1} \exp \left( -\frac{2m^2_{\eta _c}}{T^2}\right) \nonumber \\ ={} & {} \frac{3m^2_c}{4\pi ^4} \int ^{s_{\eta _c}^0}_{4m^2_c} ds \int ^{s_{\eta _c}^0}_{4m^2_c} du \sqrt{1-\frac{4m^2_c}{s}} \sqrt{1-\frac{4m^2_c}{u}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} +\frac{ m^2_c }{4\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s_{\eta _c}^0}_{4m^2_c} ds \int ^{s_{\eta _c}^0}_{4m^2_c} du \frac{s \left( s^2-10s m^2_c+20m^4_c\right) }{\sqrt{s\left( s-4m^2_c\right) }^5} \sqrt{1-\frac{4m^2_c}{u}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} +\frac{ m^2_c }{4\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s_{\eta _c}^0}_{4m^2_c} ds \int ^{s_{\eta _c}^0}_{4m^2_c} du \frac{u \left( u^2-10u m^2_c+20m^4_c\right) }{\sqrt{u\left( u-4m^2_c\right) }^5} \sqrt{1-\frac{4m^2_c}{s}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} -\frac{ m^2_c }{4\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s_{\eta _c}^0}_{4m^2_c} ds \int ^{s_{\eta _c}^0}_{4m^2_c} du \frac{1}{\sqrt{s\left( s-4m^2_c\right) } \sqrt{u\left( u-4m^2_c\right) }} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} -\frac{ m^2_c }{4\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s_{\eta _c}^0}_{4m^2_c} ds \int ^{s_{\eta _c}^0}_{4m^2_c} du \frac{ \left( u-2m^2_c\right) }{ \sqrt{u\left( u-4m^2_c\right) }^3} \sqrt{1-\frac{4m^2_c}{s}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} -\frac{ m^2_c }{4\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s_{\eta _c}^0}_{4m^2_c} ds \int ^{s_{\eta _c}^0}_{4m^2_c} du\frac{ \left( s-2m^2_c\right) }{ \sqrt{s\left( s-4m^2_c\right) }^3} \sqrt{1-\frac{4m^2_c}{u}} \exp \left( -\frac{s+u}{T^2}\right) \, , \end{aligned}\)

(39)

where we introduce the notations,

$\begin{aligned} \lambda _{X^{(\prime )}_0\eta _c\eta _c}=\frac{\lambda _{X^{(\prime )}_0} f^2_{\eta _c} m^4_{\eta _c}}{4m^2_c}\, , \end{aligned}$

(40)

the other six QCD sum rules are given explicitly in the Appendix. There are three unknown parameters in each QCD sum rules in Eq. (39) and Appendix; we set the hadronic coupling constants $G_{X_0\eta _c\eta _c}=G_{X_0^\prime \eta _c \eta _c}$, $G_{X_0J/\psi J/\psi }=G_{X_0^\prime J/\psi J/\psi }$, $G_{X_0\chi _c\eta _c}=G_{X_0^\prime \chi _c \eta _c}$, $G_{X_1J/\psi \eta _c}=G_{X_1^\prime J/\psi \eta _c}$, $G_{X_1h_c \eta _c}=G_{X_1^\prime h_c \eta _c}$, $G_{X_2\eta _c\eta _c}=G_{X_2^\prime \eta _c \eta _c}$, and $G_{X_2J/\psi J/\psi }=G_{X_2^\prime J/\psi J/\psi }$ and take the $C_{i}$ as free parameters and adjust the suitable values to obtain flat Borel platforms for the hadronic coupling constants [33,34,35,36,37]. In calculations, we observe that there appear endpoint divergences at the thresholds $s=4m_c^2$ and $u=4m_c^2$ due to powers of $s-4m_c^2$ and $u-4m_c^2$ in the denominators, we make the replacements $s-4m_c^2\rightarrow s-4m_c^2+\Delta ^2$ and $u-4m_c^2\rightarrow u-4m_c^2+\Delta ^2$ with $\Delta ^2=m_c^2$ to regulate the divergences [40, 41]. As the gluon condensates make tiny contributions, such regulations work well.

3 Numerical results and discussions

At the QCD side, we take the standard gluon condensate $\langle \frac{\alpha _s GG}{\pi }\rangle =0.012\pm 0.004\,\textrm{GeV}^4$ [38, 39, 42] and take the $\overline{MS}$ mass $m_{c}(m_c)=(1.275\pm 0.025)\,\textrm{GeV}$ from the Particle Data Group [43]. We take account of the energy-scale dependence of the $\overline{MS}$ mass,

$\begin{aligned} m_c(\mu )={} & {} m_c(m_c)\left[ \frac{\alpha _{s}(\mu )}{\alpha _{s}(m_c)}\right] ^{\frac{12}{33-2n_f}} \, ,\nonumber \\ \alpha _s(\mu )={} & {} \frac{1}{b_0t}\left[ 1-\frac{b_1}{b_0^2}\frac{\log t}{t} +\frac{b_1^2(\log ^2{t}-\log {t}-1)+b_0b_2}{b_0^4t^2}\right] \, , \end{aligned}$

(41)

where $t=\log \frac{\mu ^2}{\Lambda ^2}$, $b_0=\frac{33-2n_f}{12\pi }$, $b_1=\frac{153-19n_f}{24\pi ^2}$, $b_2=\frac{2857-\frac{5033}{9}n_f+\frac{325}{27}n_f^2}{128\pi ^3}$, $\Lambda =213\,\textrm{MeV}$, $296\,\textrm{MeV}$, and $339\,\textrm{MeV}$ for the flavors $n_f=5$, 4, and 3, respectively [43]. As we study the fully-charm mesons, so we choose the flavor numbers $n_f=4$.

At the hadron side, we take $m_{\eta _c}=2.9834\,\textrm{GeV}$, $m_{J/\psi }=3.0969\,\textrm{GeV}$, $m_{h_c}=3.525\,\textrm{GeV}$, $m_{\chi _{c1}}=3.51067\,\textrm{GeV}$ from the Particle Data Group [43], $s^0_{h_c}=(3.9\,\textrm{GeV})^2$, $s^0_{\chi _{c1}}=(3.9\,\textrm{GeV})^2$, $s^0_{J/\psi }=(3.6\,\textrm{GeV})^2$, $s^0_{\eta _c}=(3.5\,\textrm{GeV})^2$, $f_{h_c}=0.235\,\textrm{GeV}$, $f_{J/\psi }=0.418 \,\textrm{GeV}$, $f_{\eta _c}=0.387 \,\textrm{GeV}$ [44], $f_{\chi _{c1}}=0.338\,\textrm{GeV}$ [45], $M_{X_0}=6.20\,\textrm{GeV}$, $\lambda _{X_0}=2.68\times 10^{-1}\,\textrm{GeV}^5$, $M_{X_1}=6.24\,\textrm{GeV}$, $\lambda _{X_1}=2.18\times 10^{-1}\,\textrm{GeV}^5$, $M_{X_2}=6.27\,\textrm{GeV}$, $\lambda _{X_2}=2.35\times 10^{-1}\,\textrm{GeV}^5$, $M_{X^\prime _0}=6.57\,\textrm{GeV}$, $\lambda _{X^\prime _0}=8.12\times 10^{-1}\,\textrm{GeV}^5$, $M_{X^\prime _1}=6.64\,\textrm{GeV}$, $\lambda _{X^\prime _1}=6.43\times 10^{-1}\,\textrm{GeV}^5$, $M_{X^\prime _2}=6.69\,\textrm{GeV}$, and $\lambda _{X^\prime _2}=6.84\times 10^{-1}\,\textrm{GeV}^5$ from the QCD sum rules [23]. In general, we take the hadron masses, decay constants or pole residues from the two-point QCD sum rules. As it is easy to reproduce the experimental values of the charmonium masses with the QCD sum rules, we take the precise masses from the Particle Data Group for simplicity. For other hadron parameters, we take the central values, as the uncertainties originate from the input parameters at the QCD side, and the uncertainties of the continuum threshold parameters can be absorbed into the decay constants or pole residues to a large extent in the two-point QCD sum rules. Taking the central values can avoid doubly counting the uncertainties.

In calculations, we fit the free parameters to be $C_{1}=0.007\,(T^2-1.0\,\textrm{GeV}^2)\,\textrm{GeV}^4$, $C_{2}=0.097\,(T^2-1.5\,\textrm{GeV}^2)\,\textrm{GeV}^6$, $C_{3}=0.0096\,(T^2-2.0\,\textrm{GeV}^2)\,\textrm{GeV}^5$, $C_4=0.29\,(T^2-1.1\textrm{GeV}^2)\,\textrm{GeV}^7$, $C_5=0.0017\,(T^2-2.2\textrm{GeV}^2)\,\textrm{GeV}^6$, $C_6=0.0086\,(T^2-0.9\textrm{GeV}^2)\,\textrm{GeV}^4$, and $C_7=0.066\,(T^2-1.5\textrm{GeV}^2)\,\textrm{GeV}^6$, and obtain the Borel platforms $T^2_{X_0\eta _c\eta _c}=(1.9-2.9)\,\textrm{GeV}^2$, $T^2_{X_0J/\psi J/\psi }=(2.3-3.3)\,\textrm{GeV}^2$, $T^2_{X_0\chi _c \eta _c}=(3.1-4.1)\,\textrm{GeV}^2$, $T^2_{X_1 J/\psi \eta _c}=(2.0-3.0)\,\textrm{GeV}^2$, $T^2_{X_1 h_c \eta _c}=(2.0-3.0)\,\textrm{GeV}^2$, $T^2_{X_2 \eta _c \eta _c}=(2.4-3.4)\,\textrm{GeV}^2$, and $T^2_{X_2 J/\psi J/\psi }=(2.3-3.3)\,\textrm{GeV}^2$, where we add the subscripts $X_0\eta _c \eta _c$, $X_0J/\psi J/\psi$, $X_0\chi _c \eta _c$, $X_1 J/\psi \eta _c$, $X_1 h_c\eta _c$, $X_2\eta _c \eta _c$, and $X_2J/\psi J/\psi$ to denote the corresponding channels. We obtain uniform flat platforms $T^2_{max}-T^2_{min}=1\,\textrm{GeV}^2$, where the max and min denote the maximum and minimum, respectively, just like what have been done in our previous works [33,34,35,36,37]. For example, in Fig. 2, we plot the hadronic coupling constants $G_{X_0\eta _c\eta _c}$ and $G_{X_0J/\psi J/\psi }$ with variations of the Borel parameters at large intervals. In the Borel windows, there appear very flat platforms indeed, it is reliable to extract the hadronic coupling constants.

Now, we estimate the uncertainties in the same way. For example, the uncertainties of an input parameter $\xi$, $\xi = \bar{\xi } +\delta \xi$, result in the uncertainties $\lambda _{X_0}f_{J/\psi }^2G_{X_0J/\psi J/\psi } = \bar{\lambda }_{X_0}\bar{f}_{J/\psi }^2\bar{G}_{X_0J/\psi J/\psi } +\delta \,\lambda _{X_0}f_{J/\psi }^2G_{X_0J/\psi J/\psi }$, $C_{2} = \bar{C}_{2}+\delta C_{2}$,

$\begin{aligned} \delta \,\lambda _{X_0}f_{J/\psi }^2G_{X_0J/\psi J/\psi } =\bar{\lambda }_{X_0}\bar{f}_{J/\psi }^2\bar{G}_{X_0J/\psi J/\psi }\left( 2\frac{\delta f_{J/\psi }}{\bar{f}_{J/\psi }} +\frac{\delta \lambda _{X_0}}{\bar{\lambda }_{X_0}}+\frac{\delta G_{X_0J/\psi J/\psi }}{\bar{G}_{X_0J/\psi J/\psi }}\right) \, , \end{aligned}$

(42)

we can set $\delta C_{2}=0$ and $\frac{\delta f_{J/\psi }}{\bar{f}_{J/\psi }} = \frac{\delta \lambda _{X_0}}{\bar{\lambda }_{X_0}}=\frac{\delta G_{X_0J/\psi J/\psi }}{\bar{G}_{XJ/\psi J/\psi }}$ approximately. Finally, we obtain the values of the hadronic coupling constants,

$\begin{aligned} G_{X_0\eta _c \eta _c} ={} & {} 0.19^{+0.04}_{-0.03}\,\textrm{GeV}^{-1}\, , \nonumber \\ G_{X_0J/\psi J/\psi } ={} & {} 5.54^{+0.68}_{-0.53}\,\textrm{GeV}\, , \nonumber \\ G_{X_0\chi _c \eta _c} ={} & {} 0.82^{+0.09}_{-0.08} \, , \nonumber \\ G_{X_1 J/\psi \eta _c}={} & {} 8.47^{+1.32}_{-1.15}\,\textrm{GeV},\nonumber \\ G_{X_1 h_c \eta _c}={} & {} 0.30^{+0.04}_{-0.03}\,\textrm{GeV}^{-1},\nonumber \\ G_{X_2 \eta _c \eta _c}={} & {} 0.94^{+0.17}_{-0.16}\,\textrm{GeV}^{-1},\nonumber \\ G_{X_2 J/\psi J/\psi }={} & {} 9.69^{+1.25}_{-1.02}\,\textrm{GeV}. \end{aligned}$

(43)

Now we set $m_{X_0}=6.20\,\textrm{GeV}$, $m_{X^\prime _0}=6.57\,\textrm{GeV}$, $m_{X_1}=6.24\,\textrm{GeV}$, $m_{X^\prime _1}=6.64\,\textrm{GeV}$, $m_{X_2}=6.27\,\textrm{GeV}$, and $m_{X^\prime _2}=6.69\,\textrm{GeV}$ based on the calculations of the QCD sum rules [23]. Then we obtain the partial decay widths directly,

$\begin{aligned} \Gamma (X_0\rightarrow \eta _c \eta _c)={} & {} 3.3^{+1.6}_{-1.2} \,\textrm{MeV}\, , \nonumber \\ \Gamma (X_0\rightarrow J/\psi J/\psi )={} & {} 13.2^{+3.5}_{-2.4} \,\textrm{MeV}\, , \nonumber \\ \Gamma (X^\prime _0\rightarrow \eta _c \eta _c)={} & {} 7.3^{+3.5}_{-2.1} \,\textrm{MeV}\, , \nonumber \\ \Gamma (X^\prime _0\rightarrow J/\psi J/\psi )={} & {} 110.5^{+28.7}_{-20.2} \,\textrm{MeV}\, , \nonumber \\ \Gamma (X^\prime _0\rightarrow \chi _{c1} \eta _c)={} & {} 0.26^{+0.06}_{-0.05} \,\textrm{MeV}\, , \end{aligned}$

(44)

$\begin{aligned} \Gamma (X_1 \rightarrow J/\psi \eta _c)={} & {} 52.2^{+17.5}_{-13.2}\, \textrm{MeV}\, ,\nonumber \\ \Gamma (X^\prime _1 \rightarrow J/\psi \eta _c)={} & {} 91.7^{+30.8}_{-23.2}\,\textrm{MeV}\, ,\nonumber \\ \Gamma (X^\prime _1 \rightarrow h_c \eta _c)={} & {} 0.67^{+0.19}_{-0.13}\,\textrm{MeV}\, , \end{aligned}$

(45)

$\begin{aligned} \Gamma (X_2 \rightarrow \eta _c \eta _c)={} & {} 0.10^{+0.04}_{-0.03}\, \textrm{MeV}\, ,\nonumber \\ \Gamma (X_2 \rightarrow J/\psi J/\psi )={} & {} 47.1^{+12.9}_{-9.4}\, \textrm{MeV}\, ,\nonumber \\ \Gamma (X^\prime _2 \rightarrow \eta _c \eta _c)={} & {} 0.83^{+0.33}_{-0.26}\, \textrm{MeV}\, ,\nonumber \\ \Gamma (X^\prime _2 \rightarrow J/\psi J/\psi )={} & {} 117.6^{+32.3}_{-23.5}\, \textrm{MeV}\, . \end{aligned}$

(46)

Then we obtain the total decay widths,

$\begin{aligned} \Gamma _{X_0}={} & {} 16.6^{+6.0}_{-2.4}\,\textrm{MeV}\, ,\nonumber \\ \Gamma _{X^\prime _0}={} & {} 118.1^{+32.2}_{-22.3}\,\textrm{MeV}\, , \end{aligned}$

(47)

$\begin{aligned} \Gamma _{X_1}={} & {} 52.2^{+17.5}_{-13.2}\, \textrm{MeV}\, ,\nonumber \\ \Gamma _{X^\prime _1}={} & {} 92.3^{+31.0}_{-23.3}\, \textrm{MeV}\, , \end{aligned}$

(48)

$\begin{aligned} \Gamma _{X_2}={} & {} 47.2^{+13.2}_{-9.4}\, \textrm{MeV}\, ,\nonumber \\ \Gamma _{X^\prime _2}={} & {} 118.5^{+32.6}_{-23.7}\, \textrm{MeV}\, . \end{aligned}$

(49)

by saturating the total widths with only the supposed dominant decays, which take place through the Okubo-Zweig-Iizuka super-allowed fall-apart mechanism. In fact, such approximations would underestimate the total widths slightly, as there maybe exist other non-dominant decays, for example, the two-body strong decays $X_0 \rightarrow D\bar{D}$ and $D^*\bar{D}^*$ can take place through annihilating a $c\bar{c}$ pair and creating a $q\bar{q}$ pair, which are Okubo-Zweig-Iizuka forbidden.

The predicted width $\Gamma _{X_0}=16.6^{+6.0}_{-2.4}\,\textrm{MeV}$ is much smaller than the preliminary data $\Gamma _{X(6220)}=0.31 \pm 0.12{}^{+0.07}_{-0.08} \ \textrm{GeV}$ from the ATLAS collaboration [3]. The predicted width $\Gamma _{X^\prime _0}=118.1^{+32.2}_{-22.3}\,\textrm{MeV}$ is in very good agreement with the experimental data $\Gamma _{X(6552)} = 124^{+32}_{-26}\pm 33 \ \textrm{MeV}$ from the CMS collaboration [4], which indicates that the X(6552) can be assigned as the first radial excitation of the scalar tetraquark state. The resonances from the LHCb, ATLAS, and CMS collaborations are observed in the $J/\psi J/\psi$ or $J/\psi \psi ^\prime$ invariant mass spectrum, the charge conjugation should be positive, the assignments as the tetraquark states with the $J^{PC}=1^{+-}$ are not favored.

In Refs. [31, 32], Agaev et al take the X(6600) and X(6900) as the ground state $A\bar{A}$ and $P\bar{P}$ type fully-charm scalar tetraquark states, respectively, and study the masses and two-body strong decays with the QCD sum rules, where the A and P denote the axialvector and pseudoscalar diquarks, respectively. While in Ref. [23], the X(6220) and X(6552) are assigned as the ground state and first radial excited state of the $A\bar{A}$ type scalar tetraquark states, respectively. By studying the two-body strong decays in the present work, we obtain further support to assign the X(6552) as the first radial excitation of the scalar tetraquark state.

All the predictions can be confronted to the experimental data in the future and serve as valuable guides for the high energy experiments.

4 Conclusion

In this work, we study the hadronic coupling constants in the two-body strong decays of the fully-charm tetraquark states with the $J^{PC}=0^{++}$, $1^{+-}$, and $2^{++}$ via the three-point correlation functions. We carry out the operator product expansion up to the gluon condensate and obtain the spectral representation through double-dispersion, then match the hadron side with the QCD side to obtain rigorous quark-hadron duality. We obtain the hadronic coupling constants and acquire the partial decay widths therefore the total decay widths of the fully-charm tetraquark states with the $J^{PC}=0^{++}$, $1^{+-}$, and $2^{++}$, respectively. Our predictions support assigning the X(6552) as the first radial excitation of the scalar tetraquark state. Other predictions play a valuable role in diagnosing the X, Y, and Z states and can be confronted to the experimental data in the future.

Availability of data and materials

The data are available via contacting the corresponding author upon request.

References

R. Aaij et al., Observation of structure in the J/ψ -pair mass spectrum, Sci. Bull. 65, 1983 (2020)
G. Aad et al., Observation of an Excess of Dicharmonium Events in the Four-Muon Final State with the ATLAS Detector, Phys. Rev. Lett. 131, 151902 (2023)
E. Bouhova-Thacker on behalf of the ATLAS Collaboration, Proceedings at ICHEP 2022. ATLAS results on exotic hadronic resonances. https://agenda.infn.it/event/28874/contributions/170298/. Accessed 9 July 2022
A. Hayrapetyan, et al., Observation of new structure in the J/ψJ/ψ mass spectrum in proton-proton collisions at $\sqrt{s}$ = 13 TeV. arXiv:2306.07164 [hep-ex]
M. Karliner, J.L. Rosner, S. Nussinov, $QQ\bar{Q}\bar{Q}$ states: masses, production, and decays, Phys. Rev. D 95, 034011 (2017)
Z.G. Wang, Analysis of the $QQ\bar{Q}\bar{Q}$ tetraquark states with QCD sum rules, Eur. Phys. J. C 77, 432 (2017)
W. Chen, H.X. Chen, X. Liu, T.G. Steele, S.L. Zhu, Hunting for exotic doubly hidden-charm/bottom tetraquark states, Phys. Lett. B 773, 247 (2017)
J. Wu, Y.R. Liu, K. Chen, X. Liu, S.L. Zhu, Heavy-flavored tetraquark states with the $QQ\bar{Q}\bar{Q}$ configuration, Phys. Rev. D 97, 094015 (2018)
M.N. Anwar, J. Ferretti, F.K. Guo, E. Santopinto, B.S. Zou, Spectroscopy and decays of the fully-heavy tetraquarks, Eur. Phys. J. C 78, 647 (2018)
A. Esposito, A.D. Polosa, A $bb\bar{b}\bar{b}$ di-bottomonium at the LHC?, Eur. Phys. J. C 78, 782 (2018)
C. Hughes, E. Eichten, C.T.H. Davies, Searching for beauty-fully bound tetraquarks using lattice nonrelativistic QCD, Phys. Rev. D 97, 054505 (2018)
V.R. Debastiani, F.S. Navarra, A non-relativistic model for the $[cc][\bar{c}\bar{c}]$ tetraquark, Chin. Phys. C 43, 013105 (2019)
Y. Bai, S. Lu, J. Osborne, Beauty-full Tetraquarks, Phys. Lett. B 798, 134930 (2019)
Z.G. Wang, Z.Y. Di, Analysis of the vector and axialvector $QQ\bar{Q}\bar{Q}$ tetraquark states with QCD sum rules, Acta Phys. Polon. B 50, 1335 (2019)
M.A. Bedolla, J. Ferretti, C.D. Roberts, E. Santopinto, Spectrum of fully-heavy tetraquarks from a diquark+antidiquark perspective, Eur. Phys. J. C 80, 1004 (2020)
Z.G. Wang, Tetraquark candidates in the LHCb's di-J/ψ mass spectrum, Chin. Phys. C 44, 113106 (2020)
X. Jin, Y. Xue, H. Huang, J. Ping, Full-heavy tetraquarks in constituent quark models, Eur. Phys. J. C 80, 1083 (2020)
M.S. Liu, Q.F. Lu, X.H. Zhong, Q. Zhao, All-heavy tetraquarks, Phys. Rev. D 100, 016006 (2019)
J.Z. Wang, D.Y. Chen, X. Liu, T. Matsuki, Producing fully charm structures in the J/ψ-pair invariant mass spectrum, Phys. Rev. D 103, 071503 (2021)
F.X. Liu, M.S. Liu, X.H. Zhong, Q. Zhao, Higher mass spectra of the fully-charmed and fully-bottom tetraquarks, Phys. Rev. D 104, 116029 (2021)
R. Zhu, Fully-heavy tetraquark spectra and production at hadron colliders, Nucl. Phys. B 966, 115393 (2021)
Z.G. Wang, Revisit the tetraquark candidates in the J/ψJ/ψ mass spectrum, Int. J. Mod. Phys. A 36, 2150014 (2021)
Z.G. Wang, Analysis of the X(6600), X(6900), X(7300) and related tetraquark states with the QCD sum rules, Nucl. Phys. B 985, 115983 (2022)
J.R. Zhang, 0⁺ fully-charmed tetraquark states, Phys. Rev. D 103, 014018 (2021)
B.C. Yang, L. Tang, C.F. Qiao, Scalar fully-heavy tetraquark states $QQ′\bar{Q}\bar{Q}′$ in QCD sum rules, Eur. Phys. J. C 81, 324 (2021)
X.K. Dong, V. Baru, F.K. Guo, C. Hanhart, A. Nefediev, Coupled-Channel Interpretation of the LHCb Double-J/ψ Spectrum and Hints of a New State Near the J/ψJ/ψ Threshold, Phys. Rev. Lett. 127, 119901 (2021)
W.C. Dong, Z.G. Wang, Going in quest of potential tetraquark interpretations for the newly observed T_ψψ states in light of the diquark-antidiquark scenarios, Phys. Rev. D 107, 074010 (2023)
G.L. Yu, Z.Y. Li, Z.G. Wang, J. Lu, M. Yan, The S- and P-wave fully charmed tetraquark states and their radial excitations, Eur. Phys. J. C 83, 416 (2023)
X.K. Dong, V. Baru, F.K. Guo, C. Hanhart, A. Nefediev, Is the existence of a J/ψJ/ψ bound state plausible?, Sci. Bull. 66, 2462 (2021)
C. Gong, M.C. Du, Q. Zhao, X.H. Zhong, B. Zhou, Nature of X(6900) and its production mechanism at LHCb, Phys. Lett. B 824, 136794 (2022)
S.S. Agaev, K. Azizi, B. Barsbay, H. Sundu, Exploring fully heavy scalar tetraquarks $QQ\bar{Q}\bar{Q}$, Phys. Lett. B 844, 138089 (2023)
S.S. Agaev, K. Azizi, B. Barsbay, H. Sundu, Fully charmed resonance X(6900) and its beauty counterpart, Nucl. Phys. A 1041, 122768 (2024)
Z.G. Wang, J.X. Zhang, The decay width of the Z_c(3900) as an axialvector tetraquark state in solid quark–hadron duality, Eur. Phys. J. C 78, 14 (2018)
Z.G. Wang, Strong decays of the Y(4660) as a vector tetraquark state in solid quark-hadron duality, Eur. Phys. J. C 79, 184 (2019)
Z.G. Wang, Decay widths of Z_cs(3985/4000) based on rigorous quark-hadron duality, Chin. Phys. C 46, 103106 (2022)
Z.G. Wang, Strange cousin of Z_c(4020/4025) as a tetraquark state Chin. Phys. C 46, 123106 (2022)
Z.G. Wang, Analysis of the decay $Y(4500) \to D^* \bar{D}^* π$ with the light-cone QCD sum rules Nucl. Phys. B 993, 116265 (2023)
M. A. Shifman, A. I. Vainshtein, V. I. Zakharov, QCD and Resonance Physics. Theoretical Foundations, Nucl. Phys. B147, 385 (1979). QCD and Resonance Physics: Applications, Nucl. Phys. B147, 448 (1979)
L.J. Reinders, H. Rubinstein, S. Yazaki, Hadron Properties from QCD Sum Rules, Phys. Rept. 127, 1 (1985)
Z.G. Wang, Fully heavy hexaquark states via the QCD sum rules, Int. J. Mod. Phys. A 37, 2250166 (2022)
Z.G. Wang, Analysis of the fully-heavy pentaquark states via the QCD sum rules, Nucl. Phys. B 973, 115579 (2021)
P. Colangelo, A. Khodjamirian, QCD Sum Rules, a Modern Perspective, arXiv: hep-ph/0010175
R. L. Workman et al, Review of Particle Physics, Prog. Theor. Exp. Phys. 2022, 083C01 (2022)
D. Becirevic, G. Duplancic, B. Klajn, B. Melic, F. Sanfilippo, Lattice QCD and QCD sum rule determination of the decay constants of η_c, J/ψ and h_c states, Nucl. Phys. B 883, 306 (2014)
V.A. Novikov, L.B. Okun, M.A. Shifman, A.I. Vainshtein, M.B. Voloshin, V.I. Zakharov, Charmonium and Gluons: Basic Experimental Facts and Theoretical Introduction, Phys. Rept. 41, 1 (1978)

Acknowledgements

This work is supported by National Natural Science Foundation, Grant Number 12175068.

Funding

This research was supported by the National Natural Science Foundation of China through Grant No.12175068.

Author information

Authors and Affiliations

Department of Physics, North China Electric Power University, Baoding, 071003, People’s Republic of China
Zhi-Gang Wang & Xiao-Song Yang

School of Nuclear Science and Engineering, North China Electric Power University, Beijing, 102206, People’s Republic of China
Xiao-Song Yang

Contributions

All authors contributed to the study conception and design. All authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Zhi-Gang Wang.

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Appendix

The analytical expressions of the other QCD sum rules,

\(\begin{aligned}{} & {} \frac{\lambda _{X_0J/\psi J/\psi } G_{X_0J/\psi J/\psi }}{2\left(\widetilde{m}^2_{X_0}-m^2_{J/\psi }\right)}\left[ \exp \left( -\frac{m^2_{J/\psi }}{T^2}\right) -\exp \left( -\frac{\widetilde{m}^2_{X_0}}{T^2}\right) \right] \exp \left( -\frac{m^2_{J/\psi }}{T^2}\right) +\nonumber \\{} & {} \frac{\lambda _{X^\prime _0J/\psi J/\psi } G_{X^\prime _0J/\psi J/\psi }}{2\left(\widetilde{m}^2_{X^\prime _0}-m^2_{J/\psi }\right)}\left[ \exp \left( -\frac{m^2_{J/\psi }}{T^2}\right) -\exp \left( -\frac{\widetilde{m}^2_{X^\prime _0}}{T^2}\right) \right] \exp \left( -\frac{m^2_{J/\psi }}{T^2}\right) +C_{2} \exp \left( -\frac{2m^2_{J/\psi }}{T^2}\right) \nonumber \\ ={} & {} \frac{1}{12\pi ^4} \int ^{s^0_{J/\psi }}_{4m^2_c} ds \int ^{s^0_{J/\psi }}_{4m^2_c} du \left( s+2m^2_c\right) \left( u+2m^2_c\right) \sqrt{1-\frac{4m^2_c}{s}} \sqrt{1-\frac{4m^2_c}{u}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} +\frac{m^2_c }{36\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{J/\psi }}_{4m^2_c} ds \int ^{s^0_{J/\psi }}_{4m^2_c} du \frac{s \left( s^2-22sm^2_c+48m^4_c\right) \left( u+2m^2_c\right) }{\sqrt{s\left( s-4m^2_c\right) }^5} \sqrt{1-\frac{4m^2_c}{u}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} +\frac{m^2_c }{36\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{J/\psi }}_{4m^2_c} ds \int ^{s^0_{J/\psi }}_{4m^2_c} du \frac{u \left( u^2-22um^2_c+48m^4_c\right) \left( s+2m^2_c\right) }{\sqrt{u\left( u-4m^2_c\right) }^5} \sqrt{1-\frac{4m^2_c}{s}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} -\frac{5m^4_c}{12\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{J/\psi }}_{4m^2_c} ds \int ^{s^0_{J/\psi }}_{4m^2_c} du \frac{1}{\sqrt{s\left( s-4m^2_c\right) } \sqrt{u\left( u-4m^2_c\right) }} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} -\frac{m^2_c }{36\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{J/\psi }}_{4m^2_c} ds \int ^{s^0_{J/\psi }}_{4m^2_c} du \frac{ \left( u-6m^2_c\right) \left( s+2m^2_c\right) }{ \sqrt{u\left( u-4m^2_c\right) }^3} \sqrt{1-\frac{4m^2_c}{s}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} -\frac{m^2_c }{36\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{J/\psi }}_{4m^2_c} ds \int ^{s^0_{J/\psi }}_{4m^2_c} du \frac{ \left( s-6m^2_c\right) \left( u+2m^2_c\right) }{ \sqrt{s\left( s-4m^2_c\right) }^3} \sqrt{1-\frac{4m^2_c}{u}} \exp \left( -\frac{s+u}{T^2}\right) \, , \end{aligned}\)

(50)

\(\begin{aligned}{} & {} \frac{\lambda _{X_0\chi _c\eta _c} G_{X_0\chi _c\eta _c}}{2 (\widetilde{m}^2_{X_0}-m^2_{\chi _c})}\left[ \exp \left( -\frac{m^2_{\chi _c}}{T^2}\right) -\exp \left( -\frac{\widetilde{m}^2_{X_0}}{T^2}\right) \right] \exp \left( -\frac{m^2_{\eta _c}}{T^2}\right) +\nonumber \\{} & {} \frac{\lambda _{X^\prime _0\chi _c\eta _c} G_{X^\prime _0\chi _c\eta _c}}{2 (\widetilde{m}^2_{X^\prime _0}-m^2_{\chi _c})}\left[ \exp \left( -\frac{m^2_{\chi _c}}{T^2}\right) -\exp \left( -\frac{\widetilde{m}^2_{X^\prime _0}}{T^2}\right) \right] \exp \left( -\frac{m^2_{\eta _c}}{T^2}\right) +C_{3} \exp \left( -\frac{m^2_{\chi _c}+m^2_{\eta _c}}{T^2}\right) \nonumber \\ ={} & {} \frac{m_c}{4\pi ^4} \int ^{s^0_{\chi _c}}_{4m^2_c} ds \int ^{s^0_{\eta _c}}_{4m^2_c} du \left( s-4m^2_c\right) \sqrt{1-\frac{4m^2_c}{s}} \sqrt{1-\frac{4m^2_c}{u}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} +\frac{m^3_c }{12\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{\chi _c}}_{4m^2_c} ds \int ^{s^0_{\eta _c}}_{4m^2_c} du \frac{\left( s+6m^2_c\right) }{\sqrt{s\left( s-4m^2_c\right) }^3} \sqrt{1-\frac{4m^2_c}{u}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} +\frac{ m_c }{12\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{\chi _c}}_{4m^2_c} ds \int ^{s^0_{\eta _c}}_{4m^2_c} du \frac{u \left( s-4m^2_c\right) \left( u^2-10u m^2_c+20m^4_c\right) }{\sqrt{u\left( u-4m^2_c\right) }^5} \sqrt{1-\frac{4m^2_c}{s}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} +\frac{m^3_c}{6\pi ^2} \left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{\chi _c}}_{4m^2_c} ds \int ^{s^0_{\eta _c}}_{4m^2_c} du \frac{1}{\sqrt{s\left( s-4m^2_c\right) } \sqrt{u\left( u-4m^2_c\right) }} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} -\frac{m_c }{12\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{\chi _c}}_{4m^2_c} ds \int ^{s^0_{\eta _c}}_{4m^2_c} du \frac{ \left( s-4m^2_c\right) \left( u-2m^2_c\right) }{ \sqrt{u\left( u-4m^2_c\right) }^3} \sqrt{1-\frac{4m^2_c}{s}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} -\frac{m^3_c }{12\pi ^2 }\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{\chi _c}}_{4m^2_c} ds \int ^{s^0_{\eta _c}}_{4m^2_c} du \frac{ \left( s-6m^2_c\right) }{ \sqrt{s\left( s-4m^2_c\right) }^3} \sqrt{1-\frac{4m^2_c}{u}} \exp \left( -\frac{s+u}{T^2}\right) \, , \end{aligned}\)

(51)

\(\begin{aligned}{} & {} \frac{\lambda _{X_1J/\psi \eta _c} G_{X_1J/\psi \eta _c}}{2 \left(\widetilde{m}^2_{X_1}-m^2_{J/\psi }\right)}\left[ \exp \left( -\frac{m^2_{J/\psi }}{T^2}\right) -\exp \left( -\frac{\widetilde{m}^2_{X_1}}{T^2}\right) \right] \exp \left( -\frac{m^2_{\eta _c}}{T^2}\right) +\nonumber \\{} & {} \frac{\lambda _{X^\prime _1J/\psi \eta _c} G_{X^\prime _1J/\psi \eta _c}}{2 \left(\widetilde{m}^2_{X^\prime _1}-m^2_{J/\psi }\right)}\left[ \exp \left( -\frac{m^2_{J/\psi }}{T^2}\right) -\exp \left( -\frac{\widetilde{m}^2_{X^\prime _1}}{T^2}\right) \right] \exp \left( -\frac{m^2_{\eta _c}}{T^2}\right) +C_{4} \exp \left( -\frac{m^2_{J/\psi }+m^2_{\eta _c}}{T^2}\right) \nonumber \\ ={} & {} \frac{ m_c }{6\pi ^4}\int ^{s^0_{J/\psi }}_{4m^2_c} ds \int ^{s^0_{\eta _c}}_{4m^2_c} du \,u \left( 2s+m^2_c\right) \sqrt{1-\frac{4m^2_c}{s}} \sqrt{1-\frac{4m^2_c}{u}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} +\frac{ m_c }{36\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{J/\psi }}_{4m^2_c} ds \int ^{s^0_{\eta _c}}_{4m^2_c} du \frac{s u \left( 48m^6_c+38s m^4_c-29s^2 m^2_c+3s^3 \right) }{\sqrt{s\left( s-4m^2_c\right) }^5} \sqrt{1-\frac{4m^2_c}{u}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} +\frac{ m_c }{36\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{J/\psi }}_{4m^2_c} ds \int ^{s^0_{\eta _c}}_{4m^2_c} du \frac{u \left[ 8m^6_c\left( 6s+5u\right) -4um^4_c\left( s+5u\right) +2u^2 m^2_c\left( u-5s\right) +su^3 \right] }{\sqrt{u\left( u-4m^2_c\right) }^5}\nonumber \\{} & {} \sqrt{1-\frac{4m^2_c}{s}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} +\frac{ m_c^3 }{36\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{J/\psi }}_{4m^2_c} ds \int ^{s^0_{\eta _c}}_{4m^2_c} du \frac{m^2_c\left( s+u\right) +s \left( s-4u\right) }{36\pi ^2 s \sqrt{s\left( s-4m^2_c\right) } \sqrt{u\left( u-4m^2_c\right) }} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} +\frac{ m_c }{36\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{J/\psi }}_{4m^2_c} ds \int ^{s^0_{\eta _c}}_{4m^2_c} du \frac{4m^4_c\left( 3s+u\right) -2um^2_c\left( 5s+u\right) -su^2}{ \sqrt{u\left( u-4m^2_c\right) }^3} \sqrt{1-\frac{4m^2_c}{s}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} +\frac{ m_c }{36\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{J/\psi }}_{4m^2_c} ds \int ^{s^0_{\eta _c}}_{4m^2_c} du \frac{s^2 -3sm^2_c +6m^4_c}{\sqrt{s\left( s-4m^2_c\right) }^3} \sqrt{u\left( u-4m^2_c\right) } \exp \left( -\frac{s+u}{T^2}\right) \, , \end{aligned}\)

(52)

\(\begin{aligned}{} & {} \frac{\lambda _{X_1h_c\eta _c} G_{X_1h_c\eta _c}}{2 \left(\widetilde{m}^2_{X_1}-m^2_{h_c}\right)}\left[ \exp \left( -\frac{m^2_{h_c}}{T^2}\right) -\exp \left( -\frac{\widetilde{m}^2_{X_1}}{T^2}\right) \right] \exp \left( -\frac{m^2_{\eta _c}}{T^2}\right) +\nonumber \\{} & {} \frac{\lambda _{X^\prime _1h_c\eta _c} G_{X^\prime _1h_c\eta _c}}{2 \left(\widetilde{m}^2_{X^\prime _1}-m^2_{h_c}\right)}\left[ \exp \left( -\frac{m^2_{h_c}}{T^2}\right) -\exp \left( -\frac{\widetilde{m}^2_{X^\prime _1}}{T^2}\right) \right] \exp \left( -\frac{m^2_{\eta _c}}{T^2}\right) +C_{5} \exp \left( -\frac{m^2_{h_c}+m^2_{\eta _c}}{T^2}\right) \nonumber \\ ={} & {} \frac{1}{144\pi ^4} \int ^{s^0_{h_c}}_{4m^2_c} ds \int ^{s^0_{\eta _c}}_{4m^2_c} du u\left( s-4m^2_c\right) \sqrt{1-\frac{4m^2_c}{s}} \sqrt{1-\frac{4m^2_c}{u}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} +\frac{ m^2_c}{108\pi ^2 }\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{h_c}}_{4m^2_c} ds \int ^{s^0_{\eta _c}}_{4m^2_c} du \frac{s u }{ \sqrt{s\left( s-4m^2_c\right) }^3} \sqrt{1-\frac{4m^2_c}{u}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} -\frac{ m^4_c}{54\pi ^2 } \left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{h_c}}_{4m^2_c} ds \int ^{s^0_{\eta _c}}_{4m^2_c} du \frac{u \left( s-4m^2_c\right) \left( u-2m^2_c\right) }{ \sqrt{u\left( u-4m^2_c\right) }^5} \sqrt{1-\frac{4m^2_c}{s}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} +\frac{1}{216\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{h_c}}_{4m^2_c} ds \int ^{s^0_{\eta _c}}_{4m^2_c} du \frac{m^4_c \left( 3s-11u\right) +sm^2_c \left( u-s\right) +s^2u}{ s\sqrt{s\left( s-4m^2_c\right) } \sqrt{u\left( u-4m^2_c\right) }} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} -\frac{m^2_c }{108\pi ^2 }\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{h_c}}_{4m^2_c} ds \int ^{s^0_{\eta _c}}_{4m^2_c} du \frac{ \left( s-4m^2_c\right) \left( u-m^2_c\right) }{ \sqrt{u\left( u-4m^2_c\right) }^3} \sqrt{1-\frac{4m^2_c}{s}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} -\frac{m^4_c}{108\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{h_c}}_{4m^2_c} ds \int ^{s^0_{\eta _c}}_{4m^2_c} du \frac{\sqrt{u\left( u-4m^2_c\right) }}{\sqrt{s\left( s-4m^2_c\right) }^3} \exp \left( -\frac{s+u}{T^2}\right) \, , \end{aligned}\)

(53)

\(\begin{aligned}{} & {} \frac{\lambda _{X_2\eta _c\eta _c} G_{X_2\eta _c\eta _c}}{2 \left(\widetilde{m}^2_{X_2}-m^2_{\eta _c}\right)}\left[ \exp \left( -\frac{m^2_{\eta _c}}{T^2}\right) -\exp \left( -\frac{\widetilde{m}^2_{X_2}}{T^2}\right) \right] \exp \left( -\frac{m^2_{\eta _c}}{T^2}\right) +\nonumber \\{} & {} \frac{\lambda _{X^\prime _2\eta _c\eta _c} G_{X^\prime _2\eta _c\eta _c}}{2 \left(\widetilde{m}^2_{X^\prime _2}-m^2_{\eta _c}\right)}\left[ \exp \left( -\frac{m^2_{\eta _c}}{T^2}\right) -\exp \left( -\frac{\widetilde{m}^2_{X^\prime _2}}{T^2}\right) \right] \exp \left( -\frac{m^2_{\eta _c}}{T^2}\right) +C_{6} \exp \left( -\frac{2m^2_{\eta _c}}{T^2}\right) \nonumber \\ ={} & {} \frac{3m^2_c}{2\sqrt{2}\pi ^4}\int ^{s^0_{\eta _c}}_{4m^2_c} ds \int ^{s^0_{\eta _c}}_{4m^2_c} du \sqrt{1-\frac{4m^2_c}{s}} \sqrt{1-\frac{4m^2_c}{u}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} +\frac{m^2_c }{2\sqrt{2}\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{\eta _c}}_{4m^2_c} ds \int ^{s^0_{\eta _c}}_{4m^2_c} du \frac{s \left( s^2-10sm^2_c+20m^4_c\right) }{ \sqrt{s\left( s-4m^2_c\right) }^5} \sqrt{1-\frac{4m^2_c}{u}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} +\frac{m^2_c}{2\sqrt{2}\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{\eta _c}}_{4m^2_c} ds \int ^{s^0_{\eta _c}}_{4m^2_c} du \frac{u \left( u^2-10um^2_c+20m^4_c\right) }{\sqrt{u\left( u-4m^2_c\right) }^5} \sqrt{1-\frac{4m^2_c}{s}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} +\frac{m^2_c}{3\sqrt{2}\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{\eta _c}}_{4m^2_c} ds \int ^{s^0_{\eta _c}}_{4m^2_c} du \frac{1}{\sqrt{s\left( s-4m^2_c\right) } \sqrt{u\left( u-4m^2_c\right) }} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} +\frac{m^2_c }{2\sqrt{2}\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{\eta _c}}_{4m^2_c} ds \int ^{s^0_{\eta _c}}_{4m^2_c} du \frac{ u-2m^2_c}{ \sqrt{u\left( u-4m^2_c\right) }^3} \sqrt{1-\frac{4m^2_c}{s}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} +\frac{m^2_c }{2\sqrt{2}\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{\eta _c}}_{4m^2_c} ds \int ^{s^0_{\eta _c}}_{4m^2_c} du \frac{s-2m^2_c}{ \sqrt{s\left( s-4m^2_c\right) }^3} \sqrt{1-\frac{4m^2_c}{u}} \exp \left( -\frac{s+u}{T^2}\right) \, , \end{aligned}\)

(54)

\(\begin{aligned}{} & {} \frac{\lambda _{X_2J/\psi J/\psi } G_{X_2J/\psi J/\psi }}{2 \left(\widetilde{m}^2_{X_2}-m^2_{J/\psi }\right)}\left[ \exp \left( -\frac{m^2_{J/\psi }}{T^2}\right) -\exp \left( -\frac{\widetilde{m}^2_{X_2}}{T^2}\right) \right] \exp \left( -\frac{m^2_{J/\psi }}{T^2}\right) +\nonumber \\{} & {} \frac{\lambda _{X^\prime _2J/\psi J/\psi } G_{X^\prime _2J/\psi J/\psi }}{2 \left(\widetilde{m}^2_{X^\prime _2}-m^2_{J/\psi }\right)}\left[ \exp \left( -\frac{m^2_{J/\psi }}{T^2}\right) -\exp \left( -\frac{\widetilde{m}^2_{X^\prime _2}}{T^2}\right) \right] \exp \left( -\frac{m^2_{J/\psi }}{T^2}\right) +C_{7} \exp \left( -\frac{2m^2_{J/\psi }}{T^2}\right) \nonumber \\ ={} & {} \frac{1}{12\sqrt{2}\pi ^4}\int ^{s^0_{J/\psi }}_{4m^2_c} ds \int ^{s^0_{J/\psi }}_{4m^2_c} du \left( s+2m^2_c\right) \left( u+2m^2_c\right) \sqrt{1-\frac{4m^2_c}{s}} \sqrt{1-\frac{4m^2_c}{u}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} +\frac{m^2_c}{36\sqrt{2}\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{J/\psi }}_{4m^2_c} ds \int ^{s^0_{J/\psi }}_{4m^2_c} du \frac{s \left( u+2m^2_c\right) \left( s^2-22sm^2_c+48m^4_c\right) }{\sqrt{s\left( s-4m^2_c\right) }^5} \sqrt{1-\frac{4m^2_c}{u}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} +\frac{m^2_c}{36\sqrt{2}\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{J/\psi }}_{4m^2_c} ds \int ^{s^0_{J/\psi }}_{4m^2_c} du \frac{u \left( s+2m^2_c\right) \left( u^2-22um^2_c+48m^4_c\right) }{\sqrt{u\left( u-4m^2_c\right) }^5} \sqrt{1-\frac{4m^2_c}{s}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} +\frac{m^4_c}{12\sqrt{2}\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{J/\psi }}_{4m^2_c} ds \int ^{s^0_{J/\psi }}_{4m^2_c} du \frac{1}{\sqrt{s\left( s-4m^2_c\right) } \sqrt{u\left( u-4m^2_c\right) }} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} -\frac{m^2_c}{36\sqrt{2}\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{J/\psi }}_{4m^2_c} ds \int ^{s^0_{J/\psi }}_{4m^2_c} du \frac{\left( u-6m^2_c\right) \left( s+2m^2_c\right) }{ \sqrt{u\left( u-4m^2_c\right) }^3} \sqrt{1-\frac{4m^2_c}{s}} \exp \left( -\frac{s+u}{T^2}\right) \nonumber \\{} & {} -\frac{m^2_c}{36\sqrt{2}\pi ^2}\left\langle \frac{\alpha _{s}GG}{\pi }\right\rangle \int ^{s^0_{J/\psi }}_{4m^2_c} ds \int ^{s^0_{J/\psi }}_{4m^2_c} du \frac{ \left( s-6m^2_c\right) \left( u+2m^2_c\right) }{ \sqrt{s\left( s-4m^2_c\right) }^3} \sqrt{1-\frac{4m^2_c}{u}} \exp \left( -\frac{s+u}{T^2}\right) \, , \end{aligned}\)

(55)

where we introduce the notations,

$\begin{aligned} \lambda _{X^{(\prime )}_0J/\psi J/\psi }={} & {} \lambda _{X^{(\prime )}_0} f^2_{J/\psi } m^2_{J/\psi }\, , \nonumber \\ \lambda _{X^{(\prime )}_0\chi _c\eta _c}={} & {} \frac{\lambda _{X^{(\prime )}_0} f_{\chi _c} m_{\chi _c} f_{\eta _c} m^2_{\eta _c}}{2m_c}\, , \end{aligned}$

(56)

$\begin{aligned} \lambda _{X^{(\prime )}_1J/\psi \eta _c}={} & {} \frac{\lambda _{X^{(\prime )}_1} f_{J/\psi } m_{J/\psi }^3 f_{\eta _c} m^2_{\eta _c}}{2m_cm_{X^{(\prime )}_1}}\, , \nonumber \\ \lambda _{X^{(\prime )}_1h_c\eta _c}={} & {} \frac{\lambda _{X^{(\prime )}_1} f_{h_c} m_{h_c}^2 f_{\eta _c} m^4_{\eta _c}}{9m_cm_{X^{(\prime )}_1}}\, , \end{aligned}$

(57)

$\begin{aligned} \lambda _{X^{(\prime )}_2\eta _c\eta _c}={} & {} \frac{\lambda _{X^{(\prime )}_2} f_{\eta _c}^2 m^4_{\eta _c} \left(m_{X^{(\prime )}_2}^2-m_{\eta _c}^2\right)}{6m_c^2m_{X^{(\prime )}_2}^2}\, ,\nonumber \\ \lambda _{X^{(\prime )}_2J/\psi J/\psi }={} & {} \frac{\lambda _{X^{(\prime )}_2} f_{J/\psi }^2 m^2_{J/\psi }}{2} \, . \end{aligned}$

(58)

The two-body strong decays of the fully-charm tetraquark states

Abstract

1 Introduction

2 QCD sum rules for the hadronic coupling constants

3 Numerical results and discussions

4 Conclusion

Availability of data and materials

References

Acknowledgements

Funding

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Publisher’s Note

Appendix

Appendix

[Source: https://link.springer.com/article/10.1007/s43673-023-00112-4]