The strong vertices of charmed mesons D, D^* and charmonia J/ψ, η_c
Jie Lu^1
Guo-Liang Yu^1
Zhi-Gang Wang^1

Department of Mathematics and Physics, North China
Electric power university, Baoding 071003, People's Republic of

In this work, the strong form factors and coupling constants of the vertices $DDJ/\psi$, $DD^{*}J/\psi$, $D^{*}D^{*}J/\psi$, $DD^{*}\eta_{c}$, $D^{*}D^{*}\eta_{c}$ are calculated within the framework of the QCD sum rule. For each vertex, we analyze the form factor considering all possible off-shell cases and the contributions of the vacuum condensate terms $\langle\overline{q}q\rangle$, $\langle\overline{q}g_{s}\sigma Gq\rangle$, $\langle g_{s}^{2}G^{2}\rangle$, $\langle f^{3}G^{3}\rangle$ and $\langle\overline{q}q\rangle\langle g_{s}^{2}G^{2}\rangle$. Then, the form factors are fitted into analytical functions $g(Q^2)$ and are extrapolated into time-like regions to get the strong coupling constants. Finally, the strong coupling constants are obtained by using on-shell cases of the intermediate mesons($Q^2=-m^2$). The results are as follows, $g_{DDJ/\psi}=5.33^{+0.57}_{-0.45}$, $g_{DD^{*}J/\psi}=4.02^{+0.36}_{-0.24}$GeV$^{-1}$, $g_{D^{*}D^{*}J/\psi}=5.98^{+0.61}_{-0.58}$, $g_{DD^{*}\eta_{c}}=4.05^{+0.42}_{-0.13}$ and $g_{D^{*}D^{*}\eta_{c}}=5.73^{+0.49}_{-0.47}$GeV$^{-1}$.

13.25.Ft; 14.40.Lb

§ INTRODUCTION

In recent years, many exotic particles which are beyond the usual quark-model were observed in experments[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Some of them have exotic quantum numbers and were interpreted as  tetraquark states, pentaquark states, hadron molecular states, quark-gluon hybrids, glueballs and many others[14, 15, 16, 17, 18, 19, 20, 21, 22].  The composition of these exotic particles can not be determined only by the mass spectrum. We need to further study their decay behaviours, where the form factor and coupling constant become particularly important. For example, the exotic states $D_{s0}^{*}$(2317) and $D_{s1}$(2460) were discovered in the decay processes $B\to\bar{D}^{(*)}D_{s0}^{*}\mathrm{(2317)}$ and $B\to\bar{D}^{(*)}D_{s1}\mathrm{(2460)}$[1, 23, 24], and their decay  branching fractions were also measured in the experiment[25].
If they are interpreted as $D^{(*)}K$ or $D_{s}^{(*)}\eta$ molecular states, their decay behaviors can be studied in the triangle mechanism[26], which are shown in Figs. <ref> and <ref>. Under this mechanism, the three meson vertices $DD^{*}\eta$, $D^{*}D^{*}\eta$, $DDJ/\psi$, $D^{*}D^{*}J/\psi$, $DD^{*}\eta_{c}$ and $D^{*}D^{*}\eta_{c}$ are especially important for us to analyze their strong decays.

Triangle diagrams accounting for the $B$ decays: (a)$B^{+(0)}\to D^{+}_{s}\bar{D}^{*0}(D^{*-})\to D_{s0}^{*}(2317)^{+}\bar{D}^{0}(D^{-})$, (b)$B^{+(0)}\to D^{+}_{s}\bar{D}^{0}(D^{-})\to D_{s0}^{*}(2317)^{+}\bar{D}^{*0}(D^{*-})$, (c)$B^{+(0)}\to \eta_{c}K^{+(0)}\to D_{s0}^{*}(2317)^{+}\bar{D}^{*0}(D^{*-})$ and (d)$B^{+(0)}\to J/\psi K^{+0}\to D_{s0}^{*}(2317)^{+}\bar{D}^{0}(D^{-})$.

Triangle diagrams accounting for the $B$ decays: (a)$B^{+(0)}\to J/\psi K^{+(0)} \to D_{s1}(2460)^{+}\bar{D}^{*0}(D^{*-})$, (b)$B^{+(0)}\to \eta_{c}K^{+(0)} \to D_{s1}(2460)^{+}\bar{D}^{*0}(D^{*-})$, (c)$B^{+(0)}\to \eta_{c}K^{+(0)}\to D_{s1}(2460)\bar{D}^{0}(D^{-})$, (d)$B^{+(0)}\to D_{s}^{*+}\bar{D}^{*-}(D^{*-}) \to D_{s1}(2460)^{+}\bar{D}^{0}(D^{-})$, (e)$B^{+(0)}\to D_{s}^{*+}\bar{D}^{*-}(D^{*-}) \to D_{s1}(2460)^{+}\bar{D}^{*0}(D^{*-})$ and (f)$B^{+(0)}\to D_{s}^{*+}\bar{D}^{0}(D^{-}) \to D_{s1}(2460)^{+}\bar{D}^{*0}(D^{*-})$.

The QCD is perturbative field theory which has been successfully applied in the high energy region. However, the strong coupling constant between the hadrons lies in the low energy region, which can not be studied by perturbative method. As for the mesons composed of u, d or s quark, the couplings between these mesons have been constrained by the SU(3) chiral symmetry and phenomenological analyses of low energy reactions[27, 28, 29]. For the charmed mesons, we can use SU(4) chiral symmetry and phenomenological analysis to get the interaction Lagrangion[30, 31, 32]. Because the mass of c quark is much heavier than u, d and s quark, the SU(4) chiral symmetry is badly broken. Therefore, the accurately calculation of the strong coupling constants is of great significance to the study of the destruction of SU(4) chiral symmetry.

Since the strong interaction is non-perturbative in the low energy region, it is difficult to calculate the strong form factors and coupling constants from the QCD first principle. Except for lattice caculation[33, 34], some phenomenological methods are usually employed to carry out this work such as the QCDSR[35, 36, 37, 38, 39, 40, 41, 42, 43, 44], LCSR[45, 46, 47, 48, 49, 50], VMD model[31, 32] and other methods[51, 52]. The QCDSR[53] is one of the most powerful non-perturbative methods to analyze the strong vertices. In our previous work, we have analyzed the vertices $D_{s}^{*}D_{s}\phi$, $D_{2}^{*}D^{*}\pi$, $D_{s2}^{*}D^{*}K$, $D_{2}^{*}D\rho$, $D_{2}^{*}D\omega$, $D_{s2}^{*}D_{s}\phi$ using the three-point correlation[42, 54, 55]. As a continuation of these tasks, we systematically analyze the strong vertices of the charmed mesons $D$, $D^{*}$ and the charmonia $J/\psi$, $\eta_{c}$ using the three-point QCDSR. Although some strong vertices $DDJ/\psi$, $DD^{*}J/\psi$, $D^{*}D^{*}J/\psi$ and $DD^{*}\eta_{c}$ were already analyzed by other collaborations in QCDSR[56, 57]. However, the high dimension condensate terms such as the $\langle\overline{q}g_{s}\sigma Gq\rangle$, $\langle f^{3}G^{3}\rangle$ and $\langle\bar{q}q\rangle\langle g_{s}^{2}G^{2}\rangle$ were neglected in their studies, which is significant to the accuracy of the final results.

This article is organized as follows. After the introduction in Sec. <ref>, we analyze the strong vertices $DDJ/\psi$, $DD^{*}J/\psi$, $D^{*}D^{*}J/\psi$, $DD^{*}\eta_{c}$ and $D^{*}D^{*}\eta_{c}$ with the QCDSR in Sec. <ref>. In these analyses, all off-shell cases of the intermediate mesons are considered. In the QCD side, we consider the perturbative contribution and vacuum condensate terms $\langle\overline{q}q\rangle$, $\langle\overline{q}g_{s}\sigma Gq\rangle$, $\langle g_{s}^{2}G^{2}\rangle$, $\langle f^{3}G^{3}\rangle$ and $\langle\overline{q}q\rangle\langle g_{s}^{2}G^{2}\rangle$. In Sec. <ref>, we present the numerical results and discussions. Sec. <ref> is reserved for our conclusions. Some important figures and formulas will be shown in the Appendixes A-C.

§ QCD SUM RULES

Firstly, we write down the three-point correlation function,

\begin{eqnarray}
\notag
\Pi (p,p') &&= {i^2}\int {{d^4}x} {d^4}y{e^{ip'x}}{e^{i(p - p')y}}\\
&&\times \left\langle 0 \right.| T\{ {J_C}(x){J_B}(y)J_A^ \dagger (0)\} |\left. 0 \right\rangle
\end{eqnarray}

where $T$ denotes the time ordered product and $J_{A}^{\dagger}, J_{B}$ and $J_{C}$ are the meson interpolating currents. The subscripts A, B, and C denote three mesons in each vertex, where B is the intermediate meson and is off-shell. Assignments of the mesons for each vertex are presented in Table <ref>.

The assignments of the mesons A, B and C for each vertex where B denotes the off-shell mesons.

Vertices   A   B(off-shell)   C
2*$DDJ/\psi$   $D$   $J/\psi$   $D$

$D$   $D$   $J/\psi$
3*$DD^{*}J/\psi$   $D^{*}$   $J/\psi$   $D$

$D^{*}$   $D$   $J/\psi$

$D$   $D^{*}$   $J/\psi$
2*$D^{*}D^{*}J/\psi$   $D^{*}$   $J/\psi$   $D^{*}$

$D^{*}$   $D^{*}$   $J/\psi$
3*$DD^{*}\eta_{c}$   $D$   $\eta_{c}$   $D^{*}$

$D^{*}$   $D$   $\eta_{c}$

$D$   $D^{*}$   $\eta_{c}$
2*$D^{*}D^{*}\eta_{c}$   $D^{*}$   $\eta_{c}$   $D^{*}$

$D^{*}$   $D^{*}$   $\eta_{c}$

The meson interpolating currents in these analyses are as follows,

\begin{eqnarray}
\notag
{J_D}(x) = \bar u(x)i{\gamma _5}c(x)\\
\notag
{J_{{D^*}}}(x) = \bar u(x){\gamma _\mu }c(x)\\
\notag
{J_{J/\psi }}(x) = \bar c(x){\gamma _\mu }c(x)\\
{J_{{\eta _c}}}(x) = \bar c(x)i{\gamma _5}c(x)
\end{eqnarray}

In the framework of QCD sum rule, the correlation function is firstly calculated in two levels: the hadron level which is called the phenomenological side, and the quark level which is called the QCD side. Then we use the quark hadron duality coordinate the calculation of these two levels. In the following section, we will obtain the sum rules about these strong vertices.

§.§ The Phenomenological side

In phenomenological side, we insert complete sets of hadronic states with the same quantum numbers as the current $J_{A}^{\dagger}$, $J_{B}$ and $J_{C}$ into the correlation function,

\begin{eqnarray}\label{eq:3}
\notag
\Pi (p,p') &&= \frac{{\left\langle 0 \right.|{J_C}(0)|\left. {C(p')} \right\rangle \left\langle 0 \right.|{J_B}(0)|\left. {B(q)} \right\rangle }}{{(m_A^2 - {p^2})(m_C^2 - p{'^2})(m_B^2 - {q^2})}}\\
&&\times \left\langle {A(p)} \right.|J_A^ \dagger (0)|\left. 0 \right\rangle \left\langle {B(q)C(p')|\left. {A(p)} \right\rangle } \right. + h.c
\end{eqnarray}

where $h.c$ represents the contributions of higher resonances and continuum states of each meson. A, B, and C
stand for mesons in each vertex which are already shown in Table <ref>. The meson vacuum matrix elements appearing in this equation are substituted by the following parameterized equations,

\begin{eqnarray} \label{eq:4}
\notag
&&\langle0|J_{D}(0)|D\rangle =\frac{f_{D}m_{D}^{2}}{m_{c}}\\
\notag
&&\langle0|J_{D^{*}}(0)|D^{*}\rangle=f_{D^{*}}m_{D^{*}}\varepsilon _{\mu }\\
\notag
&&\langle0|J_{J/\psi }(0)|J/\psi\rangle=f_{J/\psi}m_{J/\psi}\xi_{\mu}\\
&&\langle0|J_{\eta _{c}}(0)|\eta _{c}\rangle=\frac{f_{\eta_{c}}m_{\eta_{c}}^{2}}{2m_{c}^{2}}
\end{eqnarray}

where $f_{D}$, $f_{D^{*}}$, $f_{J/\psi}$ and $f_{\eta_{c}}$ are the meson decay constants, $\varepsilon_{\mu}$ and $\xi_{\mu}$ are the polarization vectors of $D^{*}$ and $J/\psi$, respectively. All of the meson vertex matrix elements in our calculations can be obtained by using the effective Lagrangian in Eq. (<ref>) [58, 59],

\begin{eqnarray} \label{eq:5}
\notag
&&{\mathscr{L}_{DDJ/\psi}} = i{g_{DDJ/\psi }}{\psi _\mu }({\partial ^\mu }D\bar D  - D{\partial ^\mu }\bar D ) \\
\notag
&&{\mathscr{L}_{DD^{*}J/\psi}} =  - {g_{{D^*}DJ/\psi }}{\varepsilon ^{\alpha \beta \sigma \rho }}{\partial _\alpha }{\psi _\beta }({\partial _\sigma }D_\rho ^*\bar D  + D{\partial _\sigma }\bar D _\rho ^*) \\
\notag
&&{\mathscr{L}_{D^{*}D^{*}J/\psi}} = i{g_{{D^*}{D^*}J/\psi }}[{\psi ^\alpha }({\partial _\alpha }{D^{*\beta }}\bar {D_\beta ^*}  - {D^{*\beta }}{\partial _\alpha }\bar {D_\beta ^*} ) \\
\notag
&&+ ({\partial _\alpha }{\psi _\beta }{D^{*\beta }} - {\psi _\beta }{\partial _\alpha }{D^{*\beta }}){\bar D ^{*\alpha }} + {D^{*\alpha }}({\psi ^\beta }{\partial _\alpha }\bar {D_\beta ^*}  - {\partial _\alpha }{\psi _\beta }{\bar D ^{*\beta }})]  \\
\notag
&&{\mathscr{L}_{DD^{*}\eta_{c}}} = i{g_{{D^*}D{\eta _c}}}[{D^{*\alpha }}({\partial _\alpha }{\eta _c}\bar D  - {\eta _c}{\partial _\alpha }\bar D ) \\
\notag
&& + ({\partial _\alpha }{\eta _c}D - {\eta _c}{\partial _\alpha }D){\bar D ^{*\alpha }}]  \\
&&{\mathscr{L}_{D^{*}D^{*}\eta_{c}}} =  - {g_{{D^*}{D^*}{\eta _c}}}{\varepsilon ^{\alpha \beta \sigma \rho }}{\partial _\alpha }D_\beta ^*{\partial _\sigma }\bar D_\rho ^*{\eta _c}
\end{eqnarray}

Full expressions of the meson vertex elements for all vertices and off-shell cases are as follows,

\begin{eqnarray} \label{eq:6}
\notag
&&\left\langle {D(p')J/\psi (q)|\left. {D(p)} \right\rangle } \right. = g_{DDJ/\psi }^{J/\psi }({q^2})\xi _\nu ^*{(p + p')^\nu }\\
\notag
&&\left\langle {D(q)J/\psi (p')|\left. {D(p)} \right\rangle } \right. = g_{DDJ/\psi }^D({q^2}){\xi _\mu }{(p + q)^\mu }\\
\notag
&&\left\langle {D(p')J/\psi (q)|\left. {{D^*}(p)} \right\rangle } \right. =  - g_{D{D^*}J/\psi }^{J/\psi }({q^2}){\varepsilon ^{\sigma \alpha \beta \rho }}{p_\sigma }p{'_\alpha }{\xi _\beta }{\zeta _\rho }\\
\notag
&&\left\langle {D(q)J/\psi (p')|\left. {{D^*}(p)} \right\rangle } \right. =  - g_{D{D^*}J/\psi }^D({q^2}){\varepsilon ^{\sigma \alpha \beta \rho }}{p_\sigma }p{'_\alpha }{\xi _\beta }{\zeta _\rho }\\
\notag
&&\left\langle {{D^*}(q)J/\psi (p')|\left. {D(p)} \right\rangle } \right. =  - g_{D{D^*}J/\psi }^{{D^*}}({q^2}){\varepsilon ^{\alpha \beta \sigma \rho }}p{'_\alpha }{q_\sigma }{\xi _\beta }{\zeta _\rho }\\
\notag
&&\left\langle {{D^*}(p')J/\psi (q)|\left. {{D^*}(p)} \right\rangle } \right. = g_{{D^*}{D^*}J/\psi }^{J/\psi }[({p_\alpha } + p{'_\alpha }){\zeta ^\beta }\varepsilon _\beta ^*{\xi ^{*\alpha }}  \\
\notag
&&- ({p_\alpha } + {q_\alpha })\xi _\beta ^*{\zeta ^\beta }{\varepsilon ^{*\alpha }} - (p{'_\alpha } - {q_\alpha }){\zeta ^\alpha }{\xi ^{*\beta }}\varepsilon _\beta ^*]\\
\notag
&&\left\langle {{D^*}(q)J/\psi (p')|\left. {{D^*}(p)} \right\rangle } \right. = g_{{D^*}{D^*}J/\psi }^{{D^*}}[({p_\alpha } + {q_\alpha }){\zeta ^\beta }\varepsilon _\beta ^*{\xi ^{*\alpha }} \\
\notag
&&- ({p_\alpha } + p{'_\alpha })\xi _\beta ^*{\zeta ^\beta }{\varepsilon ^{*\alpha }} - ({q_\alpha } - p{'_\alpha }){\zeta ^\alpha }{\xi ^{*\beta }}\varepsilon _\beta ^*]\\
\notag
&&\left\langle {D(p'){\eta _c}(q)|\left. {{D^*}(p)} \right\rangle } \right. =  - g_{D{D^*}{\eta _c}}^{{\eta _c}}({q^2}){\zeta ^\alpha }({q_\alpha } - p{'_\alpha })\\
\notag
&&\left\langle {D(q){\eta _c}(p')|\left. {{D^*}(p)} \right\rangle } \right. =  - g_{D{D^*}{\eta _c}}^D({q^2}){\zeta ^\alpha }(p{'_\alpha } - {q_\alpha })\\
\notag
&&\left\langle {{D^*}(q){\eta _c}(p')|\left. {D(p)} \right\rangle } \right. =  - g_{D{D^*}{\eta _c}}^{{D^*}}({q^2})({p_\alpha } + p{'_\alpha }){\xi ^{*\alpha }}\\
\notag
&&\left\langle {{D^*}(p'){\eta _c}(q)|\left. {{D^*}(p)} \right\rangle } \right. =  - g_{{D^*}{D^*}{\eta _c}}^{{\eta _c}}({q^2}){\varepsilon ^{\alpha \beta \sigma \rho }}{p_\alpha }p{'_\sigma }{\zeta _\beta }\varepsilon _\rho ^*\\
&&\left\langle {{D^*}(q){\eta _c}(p')|\left. {{D^*}(p)} \right\rangle } \right. =  - g_{{D^*}{D^*}{\eta _c}}^{{D^*}}({q^2}){\varepsilon ^{\alpha \beta \sigma \rho }}{p_\alpha }{q_\sigma }{\varepsilon _\beta }\zeta _\rho ^*
\end{eqnarray}

where $\xi _{\mu},\zeta_{\mu}$ and $\varepsilon _{\mu}$ are the meson polarization vectors, and $q=p-p'$. The superscript of $g$ in these equations denotes the off-shell meson, and subscript denotes the vertex. For example, $g_{DDJ/\psi }^{J/\psi }$ represents the strong coupling constant of the vertex $DDJ/\psi$, where $J/\psi$ is the intermediate meson and is off-shell. From Eqs. (<ref>)-(<ref>), the correlation functions in hadron side will be obtained, and it can be expanded into different tensor structures. Theoretically, different tensor structures for a correlation function will lead to the same result, thus we can choose a certain tensor structure to obtain the sum rules.

§.§ The QCD side

In QCD side, we do the operator product expansion(OPE) of the correlation function by contracting the quark fields with Wick's theorem. The correlation functions in QCD side for vertices $DDJ/\psi$, $DD^{*}J/\psi$, $D^{*}D^{*}J/\psi$, $DD^{*}\eta_{c}$ and $D^{*}D^{*}\eta_{c}$ for all off-shell cases are shown in Eqs. (<ref>)-(<ref>) respectively,

\begin{eqnarray}\label{eq:7}
\notag
\Pi _\mu ^{J/\psi }(p,p') &&= \int {{d^4}x{d^4}y{e^{ip'x}}{e^{i(p - p')y}}} \\
\notag
&&\times Tr\{ {C^{nk}}(y){\gamma _5}{U^{km}}( - x){\gamma _5}{C^{mn}}(x - y){\gamma _\mu }\} \\
\notag
\Pi _\mu ^D(p,p') &&= \int {{d^4}x{d^4}y{e^{ip'x}}{e^{i(p - p')y}}} \\
&&\times Tr\{ {\gamma _\mu }{C^{nk}}(x){\gamma _5}{U^{km}}( - y){\gamma _5}{C^{mn}}(y - x)\}
\end{eqnarray}

\begin{eqnarray}\label{eq:8}
\notag
\Pi _{\mu \nu }^{J/\psi }(p,p') &&=  - i\int {{d^4}x{d^4}y{e^{ip'x}}{e^{i(p - p')y}}}\\
\notag
&&\times Tr\{ {C^{nk}}(y){\gamma _\nu }{U^{km}}( - x){\gamma _5}{C^{mn}}(x - y){\gamma _\mu }\} \\
\notag
\Pi _{\mu \nu }^D(p,p') &&=  - i\int {{d^4}x{d^4}y{e^{ip'x}}{e^{i(p - p')y}}} \\
\notag
&&\times Tr\{ {\gamma _\mu }{C^{nk}}(x){\gamma _\nu }{U^{km}}( - y){\gamma _5}{C^{mn}}(y - x)\}  \\
\notag
\Pi _{\mu \nu }^{{D^*}}(p,p') &&=  - i\int {{d^4}x{d^4}y{e^{ip'x}}{e^{i(p - p')y}}} \\
&&\times Tr\{ {\gamma _\mu }{C^{nk}}(x){\gamma _5}{U^{km}}( - y){\gamma _\nu }{C^{mn}}(y - x)\}
\end{eqnarray}

\begin{eqnarray}\label{eq:9}
\notag
\Pi _{\mu \nu \sigma }^{J/\psi }(p,p') &&= \int {{d^4}x{d^4}y{e^{ip'x}}{e^{i(p - p')y}}}  \\
\notag
&&\times Tr\{ {C^{nk}}(y){\gamma _\nu }{U^{km}}( - x){\gamma _\sigma }{C^{mn}}(x - y){\gamma _\mu } \} \\
\notag
\Pi _{\mu \nu \sigma }^{{D^*}}(p,p') &&= \int {{d^4}x{d^4}y{e^{ip'x}}{e^{i(p - p')y}}} \\
&&\times Tr\{ {\gamma _\mu }{C^{nk}}(x){\gamma _\sigma }{U^{km}}( - y){\gamma _\nu }{C^{mn}}(y - x)\}
\end{eqnarray}

\begin{eqnarray}\label{eq:10}
\notag
\Pi _\mu ^{{\eta _c}}(p,p') &&= \int {{d^4}x{d^4}y{e^{ip'x}}{e^{i(p - p')y}}} \\
\notag
&&\times Tr\{ {C^{nk}}(y){\gamma _\mu }{U^{km}}( - x){\gamma _5}{C^{mn}}(x - y){\gamma _5}\} \\
\notag
\Pi _\mu ^D(p,p') &&= \int {{d^4}x{d^4}y{e^{ip'x}}{e^{i(p - p')y}}} \\
\notag
&&\times Tr\{ {\gamma _5}{C^{nk}}(x){\gamma _\mu }{U^{km}}( - y){\gamma _5}{C^{mn}}(y - x)\}  \\
\notag
\Pi _\mu ^{{D^*}}(p,p') &&= \int {{d^4}x{d^4}y{e^{ip'x}}{e^{i(p - p')y}}}  \\
&&\times Tr\{ {\gamma _5}{C^{nk}}(x){\gamma _5}{U^{km}}( - y){\gamma _\mu }{C^{mn}}(y - x)\}
\end{eqnarray}

\begin{eqnarray}\label{eq:11}
\notag
\Pi _{\mu \nu }^{{\eta _c}}(p,p') &&=  - i\int {{d^4}x{d^4}y{e^{ip'x}}{e^{i(p - p')y}}} \\
\notag
&&\times Tr\{ {C^{nk}}(y){\gamma _\nu }{U^{km}}( - x){\gamma _\mu }{C^{mn}}(x - y){\gamma _5}\}  \\
\notag
\Pi _{\mu \nu }^{{D^*}}(p,p') &&=  - i\int {{d^4}x{d^4}y{e^{ip'x}}{e^{i(p - p')y}}} \\
&&\times Tr\{ {\gamma _5}{C^{nk}}(x){\gamma _\mu }{U^{km}}( - y){\gamma _\nu }{C^{mn}}(y - x)\}
\end{eqnarray}

It is the same as $g$ in Eqs. (<ref>)-(<ref>), the superscript of $\Pi$ in these above equations denotes the meson which is off-shell. $U^{ij}(x)$ and $C^{ij}(x)$ are the full propagators of $u(d)$ and $c$ quark[60],

\begin{eqnarray}
\notag
{U^{ij}}(x) &&= \frac{{i{\delta ^{ij}}\slashed{x}}}{{2{\pi ^2}{x^4}}} - \frac{{{\delta ^{ij}}{m_q}}}{{4{\pi ^2}{x^4}}} - \frac{{{\delta ^{ij}}\left\langle {\bar qq} \right\rangle }}{{12}} + \frac{{i{\delta ^{ij}}\slashed{x}{m_q}\left\langle {\bar qq} \right\rangle }}{{48}}\\
\notag
&& - \frac{{{\delta ^{ij}}{x^2}\left\langle {\bar q{g_s}\sigma Gq} \right\rangle }}{{192}} + \frac{{i{\delta ^{ij}}{x^2}\slashed{x}{m_q}\left\langle {\bar q{g_s}\sigma Gq} \right\rangle }}{{1152}}\\
\notag
&& - \frac{{i{g_s}G_{\alpha \beta }^at_{ij}^a(\slashed{x}{\sigma ^{\alpha \beta }} + {\sigma ^{\alpha \beta }}\slashed x)}}{{32{\pi ^2}{x^2}}} - \frac{{i{\delta ^{ij}}{x^2}\slashed{x}g_s^2{{\left\langle {\bar qq} \right\rangle }^2}}}{{7776}}\\
\notag
&& - \frac{{{\delta ^{ij}}{x^4}\left\langle {\bar qq} \right\rangle \left\langle {g_s^2GG} \right\rangle }}{{27648}} - \frac{{\left\langle {{{\bar q}^j}{\sigma ^{\mu \nu }}{q^i}} \right\rangle {\sigma _{\mu \nu }}}}{8}\\
\notag
&& - \frac{{\left\langle {{{\bar q}^j}{\gamma ^\mu }{q^i}} \right\rangle {\gamma _\mu }}}{4} + ...\\
\notag
{C^{ij}}(x) &&= \frac{i}{{{{(2\pi )}^4}}}\int {{d^4}k} {e^{ - ik \cdot x}}\{ \frac{{{\delta ^{ij}}}}{{\slashed k - {m_c}}}\\
\notag
&& - \frac{{{g_s}G_{\alpha \beta }^nt_{ij}^n}}{4}\frac{{{\sigma ^{\alpha \beta }}(\slashed{k} + {m_c}) + (\slashed{k} + {m_c}){\sigma ^{\alpha \beta }}}}{{{{({k^2} - m_c^2)}^2}}}\\
\notag
&& + \frac{{{g_s}{D_\alpha }G_{\beta \lambda }^nt_{ij}^n({f^{\lambda \beta \alpha }} + {f^{\lambda \alpha \beta }})}}{{3{{({k^2} - m_c^2)}^4}}}\\
\notag
&& - \frac{{g_s^2{{({t^a}{t^b})}_{ij}}G_{\alpha \beta }^aG_{\mu \nu }^b({f^{\alpha \beta \mu \nu }} + {f^{\alpha \mu \beta \nu }} + {f^{\alpha \mu \nu \beta }})}}{{4{{({k^2} - m_c^2)}^5}}} \\
&&+ ...\}
\end{eqnarray}

where $\langle g_{s}^{2}G^{2}\rangle=\langle g_{s}^{2}G^{n}_{\alpha\beta}G^{n\alpha\beta}\rangle$, $D_{\alpha}=\partial_{\alpha}-ig_{s}G^{n}_{\alpha}t^{n}$, $t^{n}=\frac{\lambda^{n}}{2}$, $\lambda^{n}$ is the Gell-Mann matrix,  $i$, $j$ are color indices, $q=u(d)$, $\sigma_{\alpha\beta}=\frac{i}{2}[\gamma_{\alpha},\gamma_{\beta}]$ and $f^{\lambda\alpha\beta}$, $f^{\alpha\beta\mu\nu}$ have the following forms,

\begin{eqnarray}
{f^{\lambda \alpha \beta }} = (\slashed k + {m_c}){\gamma ^\lambda }(\slashed k + {m_c}){\gamma ^\alpha }(\slashed k + {m_c}){\gamma ^\beta }(\slashed k + {m_c})
\end{eqnarray}

\begin{eqnarray}
\notag
{f^{\alpha \beta \mu \nu }} = && (\slashed k + {m_c}){\gamma ^\alpha }(\slashed k + {m_c}){\gamma ^\beta }(\slashed k + {m_c})\\
&&{\gamma ^\mu }(\slashed k + {m_c}){\gamma ^\nu }(\slashed k + {m_c})
\end{eqnarray}

The correlation functions about $DDJ/\psi$ and $DD^{*}\eta_{c}$ in Eqs. (<ref>) and (<ref>) have the same tensor structures, which can be expanded into the following structures,

\begin{eqnarray}
\Pi_{\mu}(p,p')=\Pi_{1}(p^{2},p'^{2},q^{2})p_{\mu}+ \Pi_{2}(p^{2},p'^{2},q^{2})p'_{\mu}
\end{eqnarray}

We choose the scalar amplitude $\Pi_{1}^{J/\psi(D)}$ and $\Pi_{1}^{D^{*}}$ to obtain $g_{DDJ/\psi}^{J/\psi(D)}$ and $g_{DD^{*}\eta_{c}}^{D^{*}}$, and use $\Pi_{2}^{\eta_{c}(D)}$ to obtain $g_{DD^{*}\eta_{c}}^{\eta_{c}(D)}$.

For the vertices $DD^{*}J/\psi$ and $D^{*}D^{*}\eta_{c}$, their correlation functions in Eqs. (<ref>) and (<ref>) also have the same structures and can be written as follows,

\begin{eqnarray}
\Pi_{\mu\nu}(p,p')= \Pi(p^{2},p'^{2},q^{2})\varepsilon_{\mu\nu\alpha\beta}p^{\alpha}p'^{\beta}
\end{eqnarray}

where $\varepsilon_{\mu\nu\alpha\beta}$ is the 4-dimension Levi-Civita tensor. The invariant amplitude $\Pi^{J/\psi(D,D^{*})}$ and $\Pi^{\eta_{c}(D^{*})}$ are used to obtain $g_{DD^{*}J/\psi}^{J/\psi(D,D^{*})}$ and $g_{D^{*}D^{*}\eta_{c}}^{\eta_{c}(D^{*})}$, respectively.

There are three Lorentz indices in the correlation function about vertex $D^{*}D^{*}J/\psi$ in Eq. (<ref>), which leads to complicated tensor structures. For simplicity, we will contract the correlation functions with the metric tensor $g^{\mu\nu}$ to get the following reduced correlation function with only one Lorentz index,

\begin{eqnarray}
\notag
&&g^{\mu\nu}\Pi_{\mu\nu\sigma}(p,p') = \slashed{\Pi}_{\sigma}(p,p') \\
&&= \slashed{\Pi}_{1}(p^{2},p'^{2},q^{2})p_{\sigma}+\slashed{\Pi}_{2}(p^{2},p'^{2},q^{2})p'_{\sigma}
\end{eqnarray}

Here $\slashed{\Pi}_{\sigma}$ is the reduced correlation function which has above two structures. $\slashed{\Pi}_{1}^{J/\psi(D^{*})}$ is used to obtain  $g_{D^{*}D^{*}J/\psi}^{J/\psi(D^{*})}$ in this work.

Finally, we use $\Pi^{\mathrm{OPE}}$ to represent all the scalar invariant amplitudes in the QCD side, and it can be divided into two parts,

\begin{eqnarray}
\Pi^{\mathrm{OPE}}=\Pi^{\mathrm{pert}}+\Pi^{\mathrm{non-pert}}
\end{eqnarray}

where $\Pi^{\mathrm{pert}}$ is the perturbative invariant amplitude and $\Pi^{\mathrm{non-pert}}$ is the non-perturbative contributions including $\langle\bar{q}q\rangle$, $\langle g_{s}^{2}G^{2}\rangle$, $\langle\bar{q}g_{s}\sigma Gq\rangle$, $\langle f^{3}G^{3}\rangle$ and $\langle\bar{q}q\rangle\langle g_{s}^{2}G^{2}\rangle$.  The perturbative part, $\langle g_{s}^{2}G^{2}\rangle$ and $\langle f^{3}G^{3}\rangle$ terms can be written in the following form by using the dispersion relation,

\begin{eqnarray}
\notag
\Pi }(p,p') =  - \int\limits_0^\infty  {\int\limits_0^\infty  {\frac{{{\rho }(s,u,{q^2})}}{{(s - {p^2})(u - p{'^2})}}dsdu}
\end{eqnarray}
\begin{eqnarray}
\notag
\rho (s,u,{Q^2}) &&= {\rho ^{\mathrm{pert}}}(s,u,{Q^2}) + {\rho ^{\left\langle {g_s^2{G^2}} \right\rangle }}(s,u,{Q^2}) \\
&&+ {\rho ^{\left\langle {{f^3}{G^3}} \right\rangle }}(s,u,{Q^2})
\end{eqnarray}
Here $s=p^{2}$, $u=p'^{2}$ and $q=p-p'$. The QCD spectral density $\rho(s,u,q^{2})$ will be obtained by using the Cutkosky's rules[41, 61](see Fig. \ref{fig:free}). Besides of these above contributions, we also take into account the contributions from $\langle\overline{q}q\rangle$, $\langle\overline{q}g_{s}\sigma Gq\rangle$, and $\langle\overline{q}q\rangle\langle g_{s}^{2}G^{2}\rangle$. The feynman diagrams for all of the condensate terms can be classified into two groups which can be seen in Figs. \ref{fig:FM1} and \ref{fig:FM2}.
As for the spectral density and contributions of $\langle\overline{q}q\rangle$, $\langle\overline{q}g_{s}\sigma Gq\rangle$, and $\langle\overline{q}q\rangle\langle g_{s}^{2}G^{2}\rangle$, only expressions for the vertex $DDJ/\psi$ are shown in Appendixes A and B for simplicity.

\begin{figure}[htbp]
\centering
\includegraphics[width=9cm]{Feynmandiagramfree.eps}
\caption{The perturbative contributions for $J/\psi(\eta_{c})$ (a) and $D(D^{*})$ (b) off-shell. The dashed lines denote the Cutkosky cuts.}
\label{fig:free}
\end{figure}

\begin{figure*}[htbp]
\centering
\includegraphics[width=18cm]{Feynmandiagram1.eps}
\caption{Contributions of the non-perturbative parts for $J/\psi(\eta_{c})$ off-shell.}
\label{fig:FM1}
\end{figure*}

\begin{figure*}[htbp]
\centering
\includegraphics[width=18cm]{Feynmandiagram2.eps}
\caption{Contributions of the non-perturbative parts for $D(D^{*})$ off-shell.}
\label{fig:FM2}
\end{figure*}

We take the change of variables $p^{2}\to-P^{2}$, $p'^{2}\to-P'^{2}$ and $q^{2}\to-Q^{2}$ and perform the double Borel transform[62, 63] to the both phenomenological and QCD sides. The variables $P^{2}$ and $P'^{2}$ will be replaced by $T_{1}^{2}$ and $T_{2}^{2}$, where $T_{1}$ and $T_{2}$ are the Borel parameters. In this article we take $T^{2}=T_{1}^{2}$ and $T_{2}^{2}=kT_{1}^{2}=kT^{2}$, where $k$ is a constant related to meson mass. It takes different values for different vertices and off-shell cases, and these values are represented in Table \ref{kE}. Then we match the phenomenological and the QCD sides using the quark-hardon duality. Finally, the sum rules for the strong form factor are obtained.

In order to eliminate the contributions of higher resonances and continuum states in Eq. (\ref{eq:3}), the threshold parameters $s_{0}$ and $u_{0}$ in dispersion integral will be introduced(see Eq. (\ref{eq:20})). They fulfill the relations, $m_{\mathrm{i}}^{2}<s_{0}<m'^{2}_{\mathrm{i}}$ and $m_{\mathrm{o}}^{2}<u_{0}<m'^{2}_{\mathrm{o}}$, where subscript i and o represent incoming and outcoming mesons respectively. $m$ and $m'$ are the masses of the ground and first excited state of the mesons. There are relations $m'=m+\Delta$ for them, where $\Delta$ is usually taken as a value of $0.4-0.6$ GeV[56].

After these performing processes, the strong form factor can be written as,
\begin{align}\label{eq:20}
g({Q^2}) = \frac{{ - \int\limits_{{s_1}}^{{s_0}} {\int\limits_{{u_1}}^{{u_0}} {\rho (s,u,{Q^2}){e^{ - s/{T^2}}}{e^{ - u/k{T^2}}}} dsdu + \mathscr B\mathscr B[{{\Pi }^{\mathrm{non - pert}}}]} }}{{\frac{E}{{(m_B^2 + {Q^2})}}{e^{ - m_A^2/{T^2}}}{e^{ - m_C^2/k{T^2}}}}}
\end{align}
where the factors $k$ and $E$ have different expressions and are shown in Table. \ref{kE}. Besides, $\mathscr{BB}[~]$ stands for the double Borel transform. However, in order to obtain the strong coupling constants, it is necessary to extrapolate the results into time-like regions $(Q^{2}<0)$. This process is realized by fitting the form factors with appropriate analytical functions and by setting the intermediate meson on-shell($Q^2=-m_{\mathrm{on-shell}}^{2}$).

\section{Numerical results and Discussions}\label{sec3}

The hadronic parameters and meson decay constants in our calculation are
taken as $m_{D}=1.86$ GeV[64], $m_{D^{*}}=2.01$ GeV[64], $m_{J/\psi}=3.09$ GeV[64], $m_{\eta_{c}}=2.98$ GeV[64], $m_{u(d)}=0.006\pm0.001$ GeV[64], $m_{c}=1.275\pm0.025$ GeV[64], $f_{D}=0.18\pm0.02$ GeV[64], $f_{D_{*}}=0.24\pm0.02$ GeV[64], $f_{J/\psi}=0.405\pm0.015$ GeV[64], $f_{\eta_{c}}=0.395\pm0.015$ GeV[64]. The vacuum condensates are taken as $\langle\overline{q}q\rangle=-(0.23\pm0.01)^{3}$ GeV$^{3}$[64], $\langle\overline{q}g_{s}\sigma Gq\rangle=m_{0}^{2}\langle\overline{q}q\rangle$[64], $m_{0}^{2}=0.8\pm0.1$ GeV$^2$[65, 66, 67], $\langle g_{s}^{2}G^{2}\rangle=0.88\pm0.15$ GeV$^{4}$[65, 66, 67], $\langle f^{3}G^{3}\rangle=(8.8\pm5.5)$ GeV$^{2}\langle g_{s}^{2}G^{2}\rangle$[65, 66, 67]. The continuum parameters in Eq. (\ref{eq:20}) are defined as $s_{0}=(m_{\mathrm{i}}+\Delta_{\mathrm{i}})^{2}$ and $u_{0}=(m_{\mathrm{o}}+\Delta_{\mathrm{o}})^{2}$. In this article, we uniformly take $\Delta_{\mathrm{i}}=\Delta_{\mathrm{o}}=$0.4, 0.5 and 0.6 GeV for lower, intermediate, and upper values of the strong form factors, respectively.

The pole and continuum contributions play important roles for reliability of the final results. We can write,
\begin{eqnarray}
\notag
\Pi^{\mathrm{OPE}}_{\mathrm{pole}}(T^{2})=-\int_{s_{1}}^{s_{0}}\int_{u_{1}}^{u_{0}}\rho^{\mathrm{OPE}}(s,u,Q^2)e^{-\frac{s}{T^{2}}}e^{-\frac{u}{kT^{2}}}dsdu \\
\Pi^{\mathrm{OPE}}_{\mathrm{cont}}(T^{2})=-\int_{s_{0}}^{\infty}\int_{u_{0}}^{\infty}\rho^{\mathrm{OPE}}(s,u,Q^2)e^{-\frac{s}{T^{2}}}e^{-\frac{u}{kT^{2}}}dsdu
\end{eqnarray}
Then the pole and continuum contribution are defined as[56],
\begin{eqnarray}
\notag
\mathrm{Pole}=\frac{\Pi^{\mathrm{OPE}}_{\mathrm{pole}}(T^{2})}{\Pi^{\mathrm{OPE}}_{\mathrm{pole}}(T^{2})+\Pi^{\mathrm{OPE}}_{\mathrm{cont}}(T^{2})} \\
\mathrm{Continuum}=\frac{\Pi^{\mathrm{OPE}}_{\mathrm{cont}}(T^{2})}{\Pi^{\mathrm{OPE}}_{\mathrm{pole}}(T^{2})+\Pi^{\mathrm{OPE}}_{\mathrm{cont}}(T^{2})}
\end{eqnarray}
The contributions of pole and continuum for vertices $DDJ/\psi$, $DD^{*}J/\psi$, $D^{*}D^{*}J/\psi$, $DD^{*}\eta_{c}$ and $D^{*}D^{*}\eta_{c}$ are shown in Fig. \ref{PC}. It is noted that QCD sum rule requires the pole contribution to be larger than $40\%$.

\subsection{The form factors}\label{sec3.1}

The form factors will be obtained by Eq. (\ref{eq:20}), where the parameters $k$ and $E$ for all vertices and off-shell cases are shown in Table~\ref{kE}.
\begin{table}[htbp]
\caption{The parameters $k$ and $E$ for different vertices and off-shell cases.}
\label{kE}
\begin{tabular}{c c c c}
\hline\hline
Vertices&off-shell&$k$&$E$ \\ \hline
\specialrule{0em}{1pt}{1pt}
\multirow{2}*{$DDJ/\psi$}&$J/\psi$ &$1$&$\frac{f_{D}^{2}m_{D}^{4}f_{J/\psi}m_{J/\psi}}{m_{c}^{2}}$  \\
\specialrule{0em}{1pt}{1pt}
~&$D$ &$\frac{m_{J/\psi}^{2}}{m_{D^{*}}^{2}}$&$\frac{2f_{D}^{2}m_{D}^{4}f_{J/\psi}m_{J/\psi}}{m_{c}^{2}}$\\
\specialrule{0em}{1pt}{1pt} \hline
\specialrule{0em}{1pt}{1pt}
\multirow{3}*{$DD^{*}J/\psi$}&$J/\psi$ &$\frac{m_{D^{*}}^{2}}{m_{D}^{2}}$&\multirow{3}*{$-\frac{f_{D}m_{D}^{2}f_{D^{*}}m_{D^{*}}f_{J/\psi}m_{J/\psi}}{m_{c}}$}  \\
\specialrule{0em}{1pt}{1pt}
~&$D$ &$\frac{m_{J/\psi}^{2}}{m_{D^{*}}^{2}}$&~   \\
\specialrule{0em}{1pt}{1pt}
~&$D^{*}$ &$\frac{m_{J/\psi}^{2}}{m_{D}^{2}}$&~   \\
\specialrule{0em}{1pt}{1pt} \hline
\specialrule{0em}{1pt}{1pt}
\multirow{2}*{$D^{*}D^{*}J/\psi$}&$J/\psi$ &$1$&$f_{D^{*}}^{2}m_{D^{*}}^{2}f_{J/\psi}m_{J/\psi}(5-\frac{Q^{2}}{2m_{D^{*}}^{2}})$  \\
\specialrule{0em}{1pt}{1pt}
~&$D^{*}$ &$\frac{m_{J/\psi}^{2}}{m_{D}^{2}}$&$f_{D^{*}}^{2}m_{D^{*}}^{2}f_{J/\psi}m_{J/\psi}\Lambda(m_{J/\psi}^{2},m_{D^{*}}^{2},Q^{2})$  \\
\specialrule{0em}{1pt}{1pt} \hline
\specialrule{0em}{1pt}{1pt}
\multirow{3}*{$DD^{*}\eta_{c}$}&$\eta_{c}$ &$\frac{m_{D^{*}}^{2}}{m_{D}^{2}}$&\multirow{2}*{$-\frac{f_{D}m_{D}^{2}f_{D^{*}}m_{D^{*}}f_{\eta_{c}}m_{\eta_{c}}^{2}}{m_{c}^{2}}$}  \\
\specialrule{0em}{1pt}{1pt}
~&$D$ &$\frac{m_{\eta_{c}}^{2}}{m_{D^{*}}^{2}}$&~   \\
\specialrule{0em}{1pt}{1pt}
~&$D^{*}$ &$\frac{m_{\eta_{c}}^{2}}{m_{D}^{2}}$&$\frac{f_{D}m_{D}^{2}f_{D^{*}}m_{D^{*}}f_{\eta_{c}}m_{\eta_{c}}^{2}(m_{\eta_{c}}^{2}+m_{D^{*}}^{2}-m_{D}^{2})}{2m_{c}^{2}m_{D^{*}}^{2}}$   \\
\specialrule{0em}{1pt}{1pt} \hline
\specialrule{0em}{1pt}{1pt}
\multirow{2}*{$D^{*}D^{*}\eta_{c}$}&$\eta_{c}$ &$1$&\multirow{2}*{$-\frac{f_{D^{*}}^{2}m_{D^{*}}^{2}f_{\eta_{c}}m_{\eta_{c}}^{2}}{2m_{c}^{2}}$}  \\
\specialrule{0em}{1pt}{1pt}
~&$D^{*}$ &$\frac{m_{\eta_{c}}^{2}}{m_{D^{*}}^{2}}$&~   \\
\specialrule{0em}{1pt}{1pt} \hline\hline
\end{tabular}
\end{table}
In this table, the function $\Lambda(m_{J/\psi}^{2},m_{D^{*}}^{2},Q^{2})$ has the following form,
\begin{eqnarray}
\notag
\Lambda (m_{J/\psi }^2,m_{{D^*}}^2,{Q^2}) &&= \frac{{m_{{D^*}}^6 + m_{{D^*}}^4(m_{J/\psi }^2 + {Q^2})}}{{2m_{{D^*}}^2m_{J/\psi }^2{Q^2}}}- \frac{{{Q^4}}}{{2m_{{D^*}}^2m_{J/\psi }^2}}\\
\notag
&& - \frac{{m_{{D^*}}^2(m_{J/\psi }^4 - 10m_{J/\psi }^2{Q^2} + {Q^4})}}{{2m_{{D^*}}^2m_{J/\psi }^2{Q^2}}}\\
&& - \frac{{m_{J/\psi }^6 - 9m_{J/\psi }^4{Q^2} - 9m_{J/\psi }^2{Q^4}}}{{2m_{{D^*}}^2m_{J/\psi }^2{Q^2}}}
\end{eqnarray}

Fixing $Q^{2}=3$ GeV$^{2}$ in Eq. (\ref{eq:20}), we find good stability of the results for these form factors. This   stable region is called the Borel platform. In the Borel platform, the contributions of the total, pertubative and all vacuum condensate terms are shown in Fig. \ref{BW}. Then we choose suitable $T_{0}^{2}$ in the Borel platform to meet the conditions of pole dominance($>40\%$) and get the form factors in $Q^{2}=3$ GeV$^{2}$. Finally, by taking different values of $Q^{2}$, we can obtain the form factors $g(Q^{2})$. The Borel platform, $T_{0}^{2}$, pole contributions at $T_{0}^{2}$ and the range of $Q^{2}$ which are used to get form factors, are shown in Table~\ref{BTI}.

\begin{table}[htbp]
\begin{ruledtabular}\caption{The Borel platform, $T_{0}^{2}$, the pole contributions and the range of $Q^{2}$ for all vertices.}
\label{BTI}
\begin{tabular}{c c c c c c }
Vertices&off-shell&Borel platform&$T_{0}^{2}($GeV$^{2})$&Pole$(T_{0}^{2})$&$Q^{2}($GeV$^{2})$ \\ \hline
\multirow{2}*{$DDJ/\psi$}&$J/\psi$ &$4.5-6.5$&$5.5$&$40.51\%$&$1-8$  \\
~&$D$ &$4-6$&$4.8$&$40.2\%$&$4-10$   \\  \hline
\multirow{3}*{$DD^{*}J/\psi$}&$J/\psi$ &$5-7$&$6$&$40.21\%$&$3.5-9.5$  \\
~&$D$ &$5.5-7.5$&$6.5$&$42.5\%$&$3-9$   \\
~&$D^{*}$ &$4.5-6.5$&$5.5$&$42.16\%$&$3-9$   \\ \hline
\multirow{2}*{$D^{*}D^{*}J/\psi$}&$J/\psi$ &$4.5-6.5$&$5.8$&$40.2\%$&$1-7$  \\
~&$D^{*}$ &$4-6$&$5$&$42.63\%$&$5-11$   \\  \hline
\multirow{3}*{$DD^{*}\eta_{c}$}&$\eta_{c}$ &$4-6$&$4.8$&$40.51\%$&$2-8$  \\
~&$D$ &$3.5-5.5$&$4.5$&$40.03\%$&$3-9$   \\
~&$D^{*}$ &$2-4$&$2.5$&$40.11\%$&$3-9$   \\ \hline
\multirow{2}*{$D^{*}D^{*}\eta_{c}$}&$\eta_{c}$ &$6-8$&$7$&$41.05\%$&$3-9$  \\
~&$D^{*}$ &$5-7$&$6$&$43.23\%$&$3-9$   \\
\end{tabular}
\end{ruledtabular}
\end{table}

\subsection{The coupling constants}\label{sec3.2}

The numerical results of form factors can be fitted uniformly by following analytical functions,
\begin{eqnarray}\label{eq:26}
\end{eqnarray}
where the parameters $A_{1},B_{1},C_{1}$ and $C_{2}$ are shown in Table~\ref{FFSC}. The fitting diagrams of strong form factors for each vertex and off-shell case are shown in Figs. \ref{fig:FF1}-\ref{fig:FF5}.
Then the strong form factors $g(Q^{2})$ are extrapolated into the time-like region $(Q^{2}<0)$ by Eq. \ref{eq:26}, and on-shell condition is satisfied by setting $Q^{2}=-m_{\mathrm{on-shell}}^{2}$. For each vertex, we finally obtain the strong coupling constants of different off-shell cases which are shown in the last column of Table~\ref{FFSC}.
\begin{table}[htbp]
\begin{ruledtabular}\caption{The parameters for the analysis function and the strong coupling constants for different vertices and off-shell cases.}
\label{FFSC}
\begin{tabular}{l l l l l l l }
Vertex&off-shell&$A_{1}$&$B_{1}$&$C_{1}$&$C_{2}$&$g(Q^{2}=-m_{\mathrm{on-shell}}^{2})$ \\ \hline
\multirow{2}*{$DDJ/\psi$}&$J/\psi$ &$2.83$&$0.06$&$0$&$0.011$&$6.16^{+0.74}_{-0.76}$  \\
~&$D$ &$2.53$&$0.17$&$0$&$0$&$4.50^{+0.40}_{-0.15}$   \\  \hline
\multirow{3}*{$DD^{*}J/\psi$}&$J/\psi$ &$2.40$&$0.04$&$0$&$0.005$&$4.00^{+0.62}_{-0.64}$ GeV$^{-1}$  \\
~&$D$ &$2.67$&$0.12$&$0$&$0$&$3.97^{+0.16}_{-0.05}$ GeV$^{-1}$   \\
~&$D^{*}$ &$2.21$&$0.15$&$0$&$0$&$4.08^{+0.31}_{-0.02}$ GeV$^{-1}$   \\ \hline
\multirow{2}*{$D^{*}D^{*}J/\psi$}&$J/\psi$ &$3.10$&$0.05$&$0$&$0.012$&$6.12^{+0.78}_{-0.85}$  \\
~&$D^{*}$ &$3.11$&$0.16$&$-0.05$&$0$&$5.84^{+0.43}_{-0.32}$   \\  \hline
\multirow{3}*{$DD^{*}\eta_{c}$}&$\eta_{c}$ &$0.33$&$0.24$&$1.31$&$0$&$4.17^{+0.93}_{-0.16}$  \\
~&$D$ &$1.77$&$0.16$&$0$&$0$&$3.03^{+0.08}_{-0.13}$   \\
~&$D^{*}$ &$1.90$&$0.24$&$-0.02$&$0$&$4.94^{+0.27}_{-0.09}$   \\ \hline
\multirow{2}*{$D^{*}D^{*}\eta_{c}$}&$\eta_{c}$ &$3.84$&$0.043$&$0$&$0.008$&$6.21^{+0.86}_{-0.88}$ GeV$^{-1}$  \\
~&$D^{*}$ &$2.94$&$0.14$&$0$&$0$&$5.24^{+0.13}_{-0.06}$ GeV$^{-1}$   \\
\end{tabular}
\end{ruledtabular}
\end{table}

\begin{figure}[hbp]
\centering
\includegraphics[width=5cm]{DDJpsi.eps}
\caption{The fitting curves of form factors for $DDJ/\psi$ vertex in each off-shell case.}
\label{fig:FF1}
\end{figure}
\begin{figure}[hbp]
\centering
\includegraphics[width=5cm]{DDstarJpsi.eps}
\caption{The fitting curves of form factors for $DD^{*}J/\psi$ vertex in each off-shell case.}
\label{fig:FF2}
\end{figure}
\begin{figure}[hbp]
\centering
\includegraphics[width=5cm]{DstarDstarJpsi.eps}
\caption{The fitting curves of form factors for $D^{*}D^{*}J/\psi$ vertex in each off-shell case.}
\label{fig:FF3}
\end{figure}
\begin{figure}[hbp]
\centering
\includegraphics[width=5cm]{DDstaretac.eps}
\caption{The fitting curves of form factors for $DD^{*}\eta_{c}$ vertex in each off-shell case.}
\label{fig:FF4}
\end{figure}
\begin{figure}[hbp]
\centering
\includegraphics[width=5cm]{DstarDstaretac.eps}
\caption{The fitting curves of form factors for $D^{*}D^{*}\eta_{c}$ vertex in each off-shell case.}
\label{fig:FF5}
\end{figure}

For each vertex, the strong coupling constants which are obtained by considering different off-shell cases, should be equal to each other. For $g_{DD^{*}J/\psi}$ as an example, it is shown by Table~\ref{FFSC} that the results of $g_{DD^{*}J/\psi}^{D}$, $g_{DD^{*}J/\psi}^{D^{*}}$ and $g_{DD^{*}J/\psi}^{J/\psi}$ for different off-shell cases are consistent with each other.
Thus, by taking average value of the results for different off-shell cases, we can obtain the strong coupling constants for each vertex. All of the final results are collected in Table~\ref{FFFF}, where we can see that there is a significant difference between the results of QCDSR and those of other methods. Although the same method is employed in Refs.[56, 57], their results are not consistent well with ours. This difference may be due to the negligence of the contributions of high-dimension condensate terms in their studies.
\begin{widetext}
\begin{table*}[htbp]
\begin{ruledtabular}\caption{The strong coupling constants for all vertices.}
\label{FFFF}
\begin{tabular}{c c c c c }
Vertices&Present work&QCDSR &VMD&Other models \\  \hline
$DD^{*}J/\psi$&$4.02^{+0.36}_{-0.24}$ GeV$^{-1}$&$4.0\pm0.6$ GeV$^{-1}$[56]&$8.0\pm0.6$ GeV$^{-1}$[32]&$4.05\pm0.25$ GeV$^{-1}$[52] \\
$D^{*}D^{*}J/\psi$&$5.98^{+0.61}_{-0.58}$&$6.2\pm0.9$[56]&7.64[31]&$8.0\pm0.5$[52] \\
$DD^{*}\eta_{c}$&$4.05^{+0.42}_{-0.13}$&$5.23^{+1.80}_{-1.38}$[57]&7.68[69]&$15.51\pm0.45$[70] \\
$D^{*}D^{*}\eta_{c}$&$5.73^{+0.49}_{-0.47}$ GeV$^{-1}$&-&-&$9.76\pm0.32$ GeV$^{-1}$[70] \\
\end{tabular}
\end{ruledtabular}
\end{table*}
\end{widetext}

According to SU(4) symmetry, the strong coupling constants of vertice $DDJ/\psi$ and $D^{*}D^{*}J/\psi$ statisfy the relation, $\frac{g_{DDJ/\psi}}{g_{D^{*}D^{*}J/\psi}}=1$[56]. And there are following relations, $\frac{g_{DDJ/\psi}}{g_{DD^{*}J/\psi}}=m_{D}$[56], $\frac{g_{DD^{*}\eta_{c}}}{g_{D^{*}D^{*}\eta_{c}}}=\frac{m_{D}}{2}$, $g_{DD^{*}\eta_{c}}=g_{2}\sqrt{m_{\eta_{c}}}m_{D}$, and $g_{2}=2.36$GeV$^{-3/2}$ in the heavy quark efficiency theory[68]. However, in charmed sector, the SU(4) and heavy-quark spin symmetry is only approximate, these relations are violated[56]. In our calculations, this violation is proved by the following  relations about strong coupling constants, which are  $\frac{g_{DDJ/\psi}}{g_{D^{*}D^{*}J/\psi}}\approx0.98<1$, $\frac{g_{DDJ/\psi}}{g_{DD^{*}J/\psi}}\approx1.33<m_{D}$, $\frac{g_{DD^{*}\eta_{c}}}{g_{D^{*}D^{*}\eta_{c}}}\approx0.71<\frac{m_{D}}{2}$ and $g_{DD^{*}\eta_{c}}=4.05<g_{2}\sqrt{m_{\eta_{c}}}m_{D}$.

\section{Conclusions}\label{sec4}

In this work, we analyze the strong vertices $DDJ/\psi$, $DD^{*}J/\psi$, $D^{*}D^{*}J/\psi$, $DD^{*}\eta_{c}$, $D^{*}D^{*}\eta_{c}$ by different QCD sum rules, where all off-shell cases are considered for each vertex. Under this physical sketch, the strong form factors are fristly calculated in the space-like ($Q^{2}>0$) regions, and then are fitted into appropriate functions. By extrapolating these functions into the time-like ($Q^{2}<0$) regions and taking $Q^{2}=-m^{2}_{\mathrm{on-shell}}$, we obtain the strong coupling constants. For each vertex, we take the average value of the strong coupling constants for all off-shell cases as the final results. These strong form factors and coupling constants are important in describing the dynamic behavior of hadrons, which can be used to analyze the strong decay processes of the exotic particles.

\begin{widetext}

\begin{large}
\textbf{Appendix A: Full expressions of the perturbative, $\langle g_{s}^{2}G^{2}\rangle$ and $\langle f^{3}G^{3}\rangle$ spectral density for $J/\psi$ and $D$ off-shell cases.}\label{B}
\end{large}

\begin{eqnarray}
\notag
{\rho ^{pert(J/\psi )}}&& =  - \frac{3}{{4{\pi ^2}{{[{{({Q^2} + s + u)}^2} - 4su]}^{5/2}}}}\{  - m_c^6{Q^2}({Q^2} + s - u) + m_c^4{Q^2}[ - 2{Q^4}- {Q^2}(4s + u)\\
\notag
&& - 2{s^2} + su + {u^2}] + m_c^2[{Q^6}(s + u) + {Q^4}(3{s^2} + 2{u^2}) + {Q^2}(3{s^3} - 4{s^2}u + {u^3})\\
&& + s{(s - u)^3}] + su[{Q^4}s + {Q^2}(2{s^2} - su - {u^2}) + {(s - u)^3}]\}
\end{eqnarray}
\begin{eqnarray}
\notag
{\rho ^{\left\langle {g_s^2{G^2}} \right\rangle (J/\psi )}}&& = \frac{{\left\langle {g_s^2{G^2}} \right\rangle }}{{96{\pi ^2}{{[{{({Q^2} + s + u)}^2} - 4su]}^{7/2}}}}\{ 12m_c^4[2{Q^4}s - {Q^2}({s^2} - 4su + 5{u^2}) - 3{s^3}\\
\notag
&& + {s^2}u - 3s{u^2} + 5{u^3}] + 6m_c^2[2{Q^8} + 2{Q^6}(4s + u) + {Q^4}(7{s^2} + 2su - 7{u^2})\\
\notag
&& - 2{Q^2}({s^3} + {s^2}u - s{u^2} + {u^3}) - 3{s^4} - 2{s^3}u - 2{s^2}{u^2} + 2s{u^3} + 5{u^4}] + 5{Q^{10}} \\
\notag
&&+ {Q^8}(13s + 15u) + 2{Q^6}({s^2} + 3su + 14{u^2}) - 2{Q^4}(11{s^3} + 6{s^2}u - 5s{u^2} - 22{u^3})\\
&& + {Q^2}( - 23{s^4} + 18{s^3}u - 18s{u^3} + 39{u^4}) - {(s - u)^2}(7{s^3} - 7{s^2}u + 9s{u^2} - 13{u^3})\}
\end{eqnarray}
\begin{eqnarray}
\notag
{\rho ^{\left\langle {{f^3}{G^2}} \right\rangle (J/\psi )}} &&=  - \frac{{\left\langle {{f^3}{G^2}} \right\rangle }}{{192{\pi ^2}{{[{{({Q^2} + s + u)}^2} - 4su]}^{9/2}}}}\{ 126m_c^5{Q^6}({Q^2} + s - u) - 9m_c^4[35{Q^8}\\
\notag
&& + {Q^6}(36s - 35u) - 3{Q^4}({s^2} - su + 6{u^2}) - {Q^2}(2{s^3} + 3{s^2}u + 33s{u^2} - 48{u^3})\\
\notag
&& + {(s - u)^2}(2{s^2} + su - 18{u^2})] + 18m_c^3{Q^4}[{Q^6} - 2{Q^4}(2s + 3u) - {Q^2}(11{s^2} - 12su + {u^2})\\
\notag
&& - 6{(s - u)^3}] + m_c^2[ - 142{Q^{10}} - 16{Q^8}(4s - 7u) + 3{Q^6}(126{s^2} - 134su + 89{u^2})\\
\notag
&& + {Q^4}(386{s^3} - 558{s^2}u + 1359s{u^2} - 527{u^3}) + {Q^2}(92{s^4} + 100{s^3}u + 873{s^2}{u^2}\\
\notag
&& - 874s{u^3} - 571{u^4}) + {(s - u)^2}(6{s^3} + 12{s^2}u + 233s{u^2} - 31{u^3})] - 9{m_c}{Q^2}[{Q^8}(s + u)\\
\notag
&& + {Q^6}({s^2} - 6su - {u^2}) - 3{Q^4}({s^3} + {s^2}u - 3s{u^2} + {u^3}) - {Q^2}{(s - u)^3}(5s + u) - 2{(s - u)^5}]\\
\notag
&& + {Q^{10}}(77s + 52u) + {Q^8}(229{s^2} - 35su + 124{u^2}) + {Q^6}(218{s^3} - 115{s^2}u + 405s{u^2} + 25{u^3})\\
\notag
&& + {Q^4}(50{s^4} + 125{s^3}u + 357{s^2}{u^2} + 58s{u^3} - 122{u^4}) - {Q^2}(23{s^5} - 159{s^4}u + 97{s^3}{u^2}\\
&& - 311{s^2}{u^3} + 317s{u^4} + 83{u^5}) - {(s - u)^2}(7{s^4} + 8{s^3}u - 24{s^2}{u^2} - 144s{u^3} + 8{u^4})\}
\end{eqnarray}
\begin{eqnarray}
\notag
{\rho ^{pert(D)}} &&=  - \frac{3}{{4{\pi ^2}{{[{{({Q^2} + s + u)}^2} - 4su]}^{5/2}}}}\{  - 2m_c^6{u^2} + m_c^4u[{Q^4} + {Q^2}(2s + 5u) + {s^2} - 5su + 4{u^2}]\\
\notag
&& + m_c^2[{Q^8} + 4{Q^6}(s + u) + {Q^4}(6{s^2} + 4su + 5{u^2}) + 2{Q^2}(2{s^3} - 2{s^2}u + {u^3}) + s(s - 2u){(s - u)^2}]\\
&& + {Q^2}su[{Q^4} + {Q^2}(2s + u) + s(s - u)]\}
\end{eqnarray}
\begin{eqnarray}
\notag
{\rho ^{\left\langle {g_s^2{G^2}} \right\rangle (D)}}&& = \frac{{\left\langle {g_s^2{G^2}} \right\rangle }}{{48{\pi ^2}{{[{{({Q^2} + s + u)}^2} - 4su]}^{7/2}}}}\{ 12m_c^4[{Q^6} + {Q^4}(s - 3u) + {Q^2}( - {s^2} - 4su + {u^2})\\
\notag
&& - s({s^2} + 3su + {u^2})] - 6m_c^2[{Q^8} - 2{Q^6}u - 2{Q^4}{s^2} + {Q^2}{u^2}(5u - 2s) + {s^4} + 2{s^3}u - 5s{u^3} \\
\notag
&& + 2{u^4}]+ 2{Q^{10}} + 3{Q^8}(2s + u) + 2{Q^6}(2{s^2} + {u^2}) + {Q^4}( - 4{s^3} - 2{s^2}u + 6s{u^2} + 8{u^3}) \\
&& - 2{Q^2}(3{s^4} - 4{s^3}u + 5{s^2}{u^2} + 2s{u^3} - 6{u^4}) - {(s - u)^3}(2{s^2} - su + 5{u^2})\}
\end{eqnarray}
\begin{eqnarray}
\notag
{\rho ^{\left\langle {{f^3}{G^2}} \right\rangle (D)}}&& = \frac{{\left\langle {{f^3}{G^2}} \right\rangle }}{{192{\pi ^2}{{[{{({Q^2} + s + u)}^2} - 4su]}^{9/2}}}}\{ 9m_c^4[7{Q^8} + {Q^6}(23s - 32u) + 3{Q^4}(9{s^2} - 9su\\
\notag
&& + 14{u^2}) + {Q^2}(13{s^3} + 27{s^2}u + 57s{u^2} - 2{u^3}) + 2{s^4} + 22{s^3}u + 42{s^2}{u^2} + 17s{u^3} - 83{u^4}]\\
\notag
&& + 2m_c^2[15{Q^{10}} + {Q^8}(91s + 109u) + {Q^6}(174{s^2} - 325su - 209{u^2}) + {Q^4}(126{s^3} - 717{s^2}u\\
\notag
&& + 132s{u^2} - 545{u^3}) + {Q^2}(19{s^4} - 178{s^3}u - 90{s^2}{u^2} - 915s{u^3} - 102{u^4}) - {(s - u)^2}(9{s^3}\\
\notag
&& - 87{s^2}u - 253s{u^2} - 140{u^3})] - 3{Q^{12}} - {Q^{10}}(51s + 32u) + {Q^8}( - 115{s^2} + 189su - 7{u^2})\\
\notag
&& + {Q^6}( - 75{s^3} + 115{s^2}u - 290s{u^2} + 242{u^3}) + 2{Q^4}u( - 192{s^3} + 18{s^2}u + 6s{u^2} + 179{u^3})\\
\notag
&& + 2{Q^2}({s^5} - 126{s^4}u - 27{s^3}{u^2} - 225{s^2}{u^3} + 325s{u^4} + 52{u^5})\\
&& - 2{(s - u)^3}(3{s^3} - 4{s^2}u - 42s{u^2} - 17{u^3})\}
\end{eqnarray}

\begin{large}
\textbf{Appendix B: Full expressions about the condensate terms $\langle\overline{q}q\rangle,\langle\overline{q}g_{s}\sigma Gq\rangle$ and $\langle\overline{q}q\rangle\langle g^{2}_{s}G^{2}\rangle$ for $J/\psi$ off-shell.}\label{C}
\end{large}

\begin{eqnarray}
{\Pi ^{\left\langle {\bar qq} \right\rangle (J/\psi )}}&& = \left\langle {\bar qq} \right\rangle \frac{{{m_c}}}{{({p^2} - m_c^2)(p{'^2} - m_c^2)}}\\
\notag
{\Pi ^{\left\langle {\bar q{g_s}\sigma Gq} \right\rangle (J/\psi )}}&& =  - \left\langle {\bar q{g_s}\sigma Gq} \right\rangle [\frac{{{m_c}}}{{4{{({p^2} - m_c^2)}^2}(p{'^2} - m_c^2)}}\\
&& + \frac{{2m_c^4}}{{({p^2} - m_c^2){{(p{'^2} - m_c^2)}^3}}} + \frac{{{m_c}}}{{2({p^2} - m_c^2){{(p{'^2} - m_c^2)}^2}}}]\\
\notag
{\Pi ^{\left\langle {\bar qq} \right\rangle \left\langle {g_s^2{G^2}} \right\rangle (J/\psi )}}&& = \left\langle {\bar qq} \right\rangle \left\langle {g_s^2{G^2}} \right\rangle [\frac{{m_c^3}}{{6({p^2} - m_c^2){{(p{'^2} - m_c^2)}^4}}}\\
\notag
&& + \frac{{m_c^5}}{{6({p^2} - m_c^2){{(p{'^2} - m_c^2)}^5}}} + \frac{{m_c^3}}{{12{{({p^2} - m_c^2)}^4}(p{'^2} - m_c^2)}}\\
&& + \frac{{{m_c}}}{{12({p^2} - m_c^2){{(p{'^2} - m_c^2)}^3}}} + \frac{{{m_c}}}{{24{{({p^2} - m_c^2)}^2}{{(p{'^2} - m_c^2)}^2}}}]
\end{eqnarray}

\begin{large}
\textbf{Appendix C: The pole and continuum contributions, and the contributions of different condensate terms for all vertices.}\label{A}
\end{large}

\begin{figure*}[htbp]
\centering
\includegraphics[width=17.5cm]{PC.eps}
\caption{The pole and continuum contributions with variation of the Borel parameter $T^{2}$ for $DDJ/\psi$ (a-b), $DD^{*}J/\psi$ (c-e), $D^{*}D^{*}J/\psi$ (f-g), $DD^{*}\eta_{c}$ (h-j), $D^{*}D^{*}\eta_{c}$ (k-l).}
\label{PC}
\end{figure*}

\begin{figure*}[htbp]
\centering
\includegraphics[width=17.5cm]{BW.eps}
\caption{The contributions of different vacuum condensate terms with variation of the Borel parameter $T^{2}$ for $DDJ/\psi$ (a-b), $DD^{*}J/\psi$ (c-e), $D^{*}D^{*}J/\psi$ (f-g), $DD^{*}\eta_{c}$ (h-j), $D^{*}D^{*}\eta_{c}$ (k-l), where A-G denote the total, perturbative term, $\langle g_{s}^{2}G^{2}\rangle,\langle f^{3}G^{3}\rangle,\langle\overline{q}q\rangle,\langle\overline{q}g_{s}\sigma G q\rangle$, and $\langle\overline{q}q\rangle\langle g_{s}^{2}G^{2}\rangle$ contributions.}
\label{BW}
\end{figure*}
\end{widetext}

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\end{thebibliography}

\end{document}


symmetry resolved entanglement in a geometric way. In particular, this method
may be useful to gain a new interpretation for the equipartition of
entanglement between sectors of different charge Xavier:2018kqb . The
equipartition of entanglement refers to the fact that systems in the
thermodynamic limit, to leading order in the UV cutoff the symmetry-resolved
entanglement entropy is independent of the charge. A recent study showed that,
for a two-dimensional CFT with a global $\text{U}(1)$ symmetry, when resolving
w.r.t. the $\text{U}(1)$ symmetry the equipartition of entanglement is related
to the form of the $\text{U}(1)$ characters DiGiulio:2022jjd . In a different
study it was shown that symmetry resolving a two-dimensional CFT w.r.t.
irreducible representations of the Virasoro group, equipartition of
entanglement is related to the form of the conformal characters, or
alternatively, the quantum dimension of the irreducible representation
Northe:2023khz . Finally in DiGiulio:2023nvz , after establishing the symmetry
resolution of elements of Tomita-Takesaki theory, it was found that also the
modular correlation function of the charge density for a massless Dirac theory
in two dimensions shows an equipartition when symmetry resolving w.r.t. a
global $\text{U}(1)$ symmetry. In general however, at the time of writing, the
origin of equipartition is not fully understood and requires further study. We
expect that an analysis of equipartition in terms of geometric phases will put
a new perspective on this question and in this way contribute to further
demystify the equipartition of entanglement.

###### Acknowledgements.

We thank Vijay Balasubramanian, Pablo Basteiro, Arpan Bhattacharyya, Saurya
Das, Giuseppe Di Giulio, Ro Jefferson, René Meyer, Djordje Minic, Flavio
Nogueira, Joris Raeymaekers, Shubho Roy, Eric Sharpe and Gideon Vos for useful
discussions. We acknowledge support by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Germany’s Excellence Strategy through
the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in
Quantum Matter - ct.qmat (EXC 2147, project-id 390858490), via the SFB 1170
ToCoTronics (project-id 258499086) and via the German-Israeli Project
Cooperation (DIP) grant ‘Holography and the Swampland’. This research was also
supported in part by Perimeter Institute for Theoretical Physics. Research at
Perimeter Institute is supported by the Government of Canada through the
Department of Innovation, Science and Economic Development and by the Province
of Ontario through the Ministry of Research, Innovation and Science.

##  Appendix A Some Essential Ingredients of von Neumann Algebras

Here we briefly review the basic notions of vN algebras relevant for the
discussion in the present paper. For more detailed reviews, we refer the
reader to Witten:2018zxz ; Witten:2021jzq ; Sorce:2023fdx .

The set of states describing a quantum system is called the Hilbert space.
With these states, measurements of observables are performed by calculating
expectation values of Hermitian operators representing the observables. Each
such operator acts on the Hilbert space. The algebra satisfied by these
operators is a vN algebra, i.e. a unital weakly closed ∗-algebra of bounded
operators. Equivalently, due to the bicommutant theorem vonNeumann1930algebra
, a vN algebra is a subalgebra of all bounded operators that is closed under
the ∗-operation and is its own double commutant. There exist three types of
operator algebras, which we will briefly describe in the following.

##### Type I:

Algebras of type I are those typically encountered in quantum mechanics. They
always allow for an irreducible representation. Equivalently, they allow for
defining pure states on the algebra.202020See e.g. Witten:2021jzq  for an
explanation why this is equivalent. Note also that a state on the algebra is
not the same as a state(vector) in the Hilbert space, the latter notion of a
state being more common in quantum mechanics. Since in this paper, we will not
explicitly need the concept of a state on the algebra, we will refer to the
elements of the Hilbert space as states. A trace on the algebra can be
naturally defined consistently, as we will discuss in an explicit example
shortly in sec. 2.2.1. Utilising the trace, also density matrices and
entanglement entropy are naturally defined. Type I algebras fall into two
subclasses. For finite dimensional, i.e. $d$-level quantum systems such as
qudits, the algebra is of type Id, $d<\infty$. Those can always be understood
as matrix algebras. For infinite dimensional quantum systems such as an
infinite collection of qubits, the algebra is of type I∞. In this case, the
trace on the algebra is not defined for every operator since e.g. the trace of
the identity diverges.

##### Type II:

Algebras of type II only appear for infinite-dimensional systems. They do not
have an irreducible representation. This can be understood by the fact that an
algebra of type II always has a commutant of the same type. The absence of an
irreducible representation is equivalent to such algebras not allowing for
pure states on the algebra (c.f. fn. 20). However, a trace on the algebra, and
therefore also density matrices, can be consistently defined. Again, there are
two subclasses: type II1 allows to define a trace for every operator. The
other case is denoted as type II∞ and can be thought of as a tensor product of
II1 and I∞. For the same reason as with type I∞, the trace is not defined for
every element of II∞.

Within the context of gravity, algebras of type II1 and II∞ where found to
appear for empty de Sitter spacetime Chandrasekaran:2022cip  and black holes
in spacetimes of different (constant) curvature Witten:2021unn ;
Chandrasekaran:2022cip ; Chandrasekaran:2022eqq , respectively.

States corresponding to a type I algebra may contain a finite amount of
entanglement, even for infinite dimensional quantum systems with a type I∞
description. This is not true for the case of type II algebras: in this case,
entanglement is universally divergent since each state contains an infinite
amount of entanglement. The same is true for algebras of type III, whose
properties are stated in the following.

##### Type III:

As for type II, algebras of type III only appear in infinite dimensional
systems and do not allow for an irreducible representation. Moreover, they do
not allow for a consistent definition of a trace on the algebra.
Correspondingly, density matrices and in particular the entanglement entropy
are not defined. For this type, there exist three subclasses which can be
characterised by the spectrum of the modular operator $\Delta$. If the only
accumulation points of the spectrum of $\Delta$ are $0$ and $1$, the algebra
is said to be of type III0. If instead, the accumulation points are $0$ and
the integer powers of some $\lambda^{\ast}<1$, one has type
III${}_{\lambda^{\ast}}$. If there are (at least) two such values
$\lambda_{1}^{\ast},\lambda_{2}^{\ast}\neq 0$, the accumulation points are
given by
$(\lambda_{1}^{\ast})^{n}(\lambda_{2}^{\ast})^{m},\leavevmode\nobreak\
n,m\in\mathds{Z}$. Therefore, the accumulation points can approximate any real
number212121This is true except for the case where
$(\lambda_{1}^{\ast})^{n}=\lambda^{\prime},(\lambda_{2}^{\ast})^{m}=\lambda^{\prime},n,m\in\mathds{Z}$.
If such a $\lambda^{\prime}$ exists, the algebra is of type
III${}_{\lambda^{\prime}}$. See e.g. Witten:2018zxz  for a more detailed
discussion. and the algebra is said to be of type III1. It is the third case
which is the typical scenario for quantum field theory and is the primary
focus for the discussion in sec. 2.2.2. Type III1 also appears for the eternal
black hole in AdS spacetime in the large $N$ limit Jefferson:2018ksk ;
Leutheusser:2021frk ; Leutheusser:2021qhd .

##### Factor:

An algebra ${\cal A}$ is referred to as a factor if its centre consists only
of complex scalars, i.e. if the operators $\mathcal{O}_{C}=z\mathds{1}$,
$z\in\mathds{C}$ are the only operators satisfying $[a,\mathcal{O}_{C}]=0$ for
every $a\in{\cal A}$. In this case, the centre is said to be trivial.

##  Appendix B Entanglement Orbit Structure of Pure States

Here we provide details on the SZK construction Sinolecka2002manifolds .

Consider a generic quantum system in a Hilbert space $\cal H$ of size $d^{2}$.
The pure states form the projective Hilbert space ${\cal
H}_{P}=P(\mathds{C}^{d^{2}})=\mathds{C}\text{P}^{d^{2}-1}\subset{\cal H}$.
Every state $|\psi\rangle$ can be written in the Schmidt decomposition

$\displaystyle|\psi\rangle=\sum_{i=1}^{d}\kappa_{i}|i,\tilde{i}\rangle,$ (85)

where the two bases $|i\rangle$, $|\tilde{i}\rangle$ imply some partition of
the Hilbert space into sub-Hilbert spaces ${\cal H}\otimes\tilde{\cal H}$. The
numbers $0\leq\kappa_{i}\leq 1$ are known as Schmidt coefficients and, as the
square roots of the diagonal entries of the reduced density matrix
$\rho_{\text{red}}$ of either subsystem, uniquely fix the entanglement between
the subsystems by

$\displaystyle
S_{\text{EE}}(|\psi\rangle)=-\tr(\rho_{\text{red}}\ln\rho_{\text{red}})=-\sum_{i=1}^{d}\kappa_{i}^{2}\ln\kappa_{i}^{2}.$
(86)

The three most interesting cases are

  * •

$\kappa_{i}=0\leavevmode\nobreak\ \forall i$ except for $\kappa_{i^{\ast}}=1$:
vanishing entanglement,

  * •

$\kappa_{i}=\frac{1}{\sqrt{d}}\leavevmode\nobreak\ \forall i$: maximal
entanglement,

  * •

$0\leq\kappa_{1}\leq\kappa_{2}\leq...\leq\kappa_{d}<1$: intermediate
entanglement.

Note that the ordering of $\kappa_{i}$ in the third case is arbitrary. By a
convenient labeling of the base vectors, the reduced density matrix can be
written in the form

$\displaystyle\rho_{\text{red}}=\text{diag}(\underbrace{0,...,0}_{m_{0}},\underbrace{\kappa_{1},...,\kappa_{1}}_{m_{1}},...,\underbrace{\kappa_{d},...,\kappa_{d}}_{m_{d}}),$
(87)

such that the coefficients increase from the upper left to the lower right.
Generically, several Schmidt coefficients may have the same value; above, this
is denoted by the degeneracies $m_{l}$ for each Schmidt coefficient, i.e. the
lowest non-vanishing Schmidt coefficient $\kappa_{1}$ appears $m_{1}$ times,
the second lowest $\kappa_{2}$ appears $m_{2}$ times and so on. Note that this
also includes $m_{0}$, which always indicates how many Schmidt coefficients
are vanishing. By definition, $\sum_{l=0}^{d}m_{l}=d$. After finishing the
general discussion of this construction, below we will give a minimal example
using a two spin system for illustration.

The reduced density matrix $\rho_{\text{red}}$ is also understood as the
square of the coefficient matrix of the Schmidt decomposition of
$|\psi\rangle$,

$\displaystyle|\psi\rangle=\sum_{i,j=1}^{d}\sqrt{\rho_{\text{red}}}_{ij}|i,j\rangle.$
(88)

We can now ask for unitary matrices $U,\leavevmode\nobreak\ V\in\text{U}(d)$
that, up to an overall phase, leave the Schmidt decomposition invariant,

$\displaystyle U\otimes
V^{T}|\psi\rangle=\sum_{i,j=1}^{d}(U\sqrt{\rho_{\text{red}}}V)_{ij}|i,j\rangle\sim|\psi\rangle.$
(89)

It has been shown Sinolecka2002manifolds  that such matrices have the form

$\displaystyle U=\begin{bmatrix}U_{0}&0&0&\dots&0\\\ 0&U_{1}&0&&\\\
0&0&U_{2}&&\vdots\\\ \vdots&&&\ddots&\\\ 0&&\dots&&U_{d}\end{bmatrix},\quad
V=\begin{bmatrix}V_{0}&0&0&\dots&0\\\ 0&U_{1}^{\dagger}&0&&\\\
0&0&U_{2}^{\dagger}&&\vdots\\\ \vdots&&&\ddots&\\\
0&&\dots&&U_{d}^{\dagger}\end{bmatrix}.$ (90)

Here, $\dim(U_{i})=\dim(V_{i})=m_{i}$. This determines the orbit structure

$\displaystyle\mathcal{O}_{\psi}=\frac{\text{U}(d)}{\text{U}(m_{0})\times\text{U}(m_{1})\times...\times\text{U}(m_{d})}\times\frac{\text{U}(d)}{\text{U}(m_{0})\times\text{U}(1)},$
(91)

which we give reason for in the following. The two factors of $\text{U}(d)$
follow as the transformations consisting of local unitaries acting on the
$|\psi\rangle$. Due to the matrices $U$ and $V$ leaving the Schmidt
decomposition invariant, the first factor has the stabilisers
$\text{U}(m_{i})$. Since $V_{0}$ is not determined by any $U_{i}$, the second
factor also has the stabiliser $\text{U}(m_{0})$. Moreover, since
$\rho_{\text{red}}$ has to be invariant only up to an overall phase, there is
a factor $\text{U}(1)$. This construction can be understood as a fibre bundle.
The base space is formed by the reduced density matrices of the same spectrum.
The fibre is formed by all pure states related by the partial trace to density
matrices with a particular spectrum.

As a minimal example, consider two spins in the state

$\displaystyle|\chi\rangle=\sin\sigma|\\!\uparrow\downarrow\rangle+\cos\sigma|\\!\downarrow\uparrow\rangle.$
(92)

The projective Hilbert space for $|\chi\rangle$ is $\mathds{C}\text{P}^{3}$,
so $d=2$ and there are only two Schmidt coefficients. Comparing with (85)
shows that $\kappa_{1}=\sin\sigma$ and $\kappa_{2}=\cos\sigma$. For particular
values of $\sigma$, this leads to the following geometric classification:

$\displaystyle\sigma=0:$ $\displaystyle\kappa_{1}=0,\leavevmode\nobreak\
\kappa_{2}=1$ $\displaystyle\to S_{\text{EE}}=0\hskip 179.25244pt$
$\displaystyle m_{0}=1,\leavevmode\nobreak\ m_{1}=1$
$\displaystyle\mathcal{O}_{\chi}=\frac{\text{U}(2)}{\text{U}(1)\times\text{U}(1)}\times\frac{\text{U}(2)}{\text{U}(1)\times\text{U}(1)}=\mathds{C}\text{P}^{1}\times\mathds{C}\text{P}^{1}$
$\displaystyle\sigma=\frac{\pi}{4}:$
$\displaystyle\kappa_{1}=\kappa_{2}=\frac{1}{\sqrt{2}}$ $\displaystyle\to
S_{\text{EE}}=\ln 2\hskip 167.87108pt$ $\displaystyle
m_{0}=0,\leavevmode\nobreak\ m_{1}=2$
$\displaystyle\mathcal{O}_{\chi}=\frac{\text{U}(2)}{\text{U}(2)}\times\frac{\text{U}(2)}{\text{U}(1)}=\mathds{1}\times\mathds{R}\text{P}^{3}\hskip
82.51282pt$ $\displaystyle 0<\sigma<\frac{\pi}{4}:$ $\displaystyle
0<\kappa_{1}\neq\kappa_{2}<1$ $\displaystyle\to 0<S_{\text{EE}}<\ln 2\hskip
146.53151pt$ $\displaystyle m_{0}=0,\leavevmode\nobreak\ m_{1}=m_{2}=1$
$\displaystyle\mathcal{O}\chi=\frac{\text{U}(2)}{\text{U}(1)\times\text{U}(1)}\times\frac{\text{U}(2)}{\text{U}(1)}=\mathds{C}\text{P}^{1}\times\mathds{R}\text{P}^{3}\hskip
31.2982pt$

Note that for $\frac{\pi}{4}<\sigma<\frac{\pi}{2}$, the same situation as for
$0<\sigma<\frac{\pi}{4}$ arises, just with the roles of $\kappa_{1}$ and
$\kappa_{2}$ interchanged. Furthermore, $\sigma=\frac{\pi}{2}$ is the same as
$\sigma=0$, except that $\kappa_{1}=1$ and $\kappa_{2}=0$. This reflects that
there are two ways to embed
$\mathds{C}\text{P}^{1}\times\mathds{C}\text{P}^{1}$ into
$\mathds{C}\text{P}^{3}$, as mentioned below (4) for bipartite quantum systems
with general $d$.

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