Production mechanism of the hidden charm pentaquark states $P_{c\bar{c}}$

The discovery of the pentaquarks by the LHCb collaboration has invigorated research in heavy exotic baryon spectroscopy. Pentaquarks with the minimal quark content $uudc\bar{c}$ were first identified in the $J/\psi p$ invariant mass spectrum from $\Lambda_{b}^{0}\to J/\psi pK^{-}$ decays [1, 2]. To date, four states have been observed: three narrow states below the $\bar{D}^{*}\Sigma_{c}$ threshold and one broad state below the $\bar{D}\Sigma_{c}^{*}$ threshold. Recently, a new state designated as $P_{c\bar{c}}(4330)$ was identified in $B_{s}^{0}\to J/\psi p\bar{p}$ decays, while the previously observed $P_{c\bar{c}}(4312)$ was notably absent [3] ¹¹1The naming convention of heavy pentaquarks is not yet settled. While the LHCb Collaboration suggested a naming convention for exotic hadrons [4], the Particle Data Group (PDG) uses a different one [5]. In the current work, we will follow the PDG convention.. Intriguingly, the GlueX collaboration has found no evidence for these pentaquark states in recent $J/\psi$ photoproduction experiments on protons [6, 7]. Rather than invalidating previous findings, these seemingly conflicting results offer an opportunity to deepen our understanding of pentaquark nature. The LHCb Collaboration has also identified the strangeness partner of the pentaquark in the $J/\psi\Lambda$ invariant mass spectrum from $\Xi_{b}^{-}\to J/\psi\Lambda K^{-}$ decays [8]. Recently, they discovered the strangeness partner of $P_{c\bar{c}}$(4330) at a nearly identical mass, just 5 MeV higher [9]. Investigations into pentaquark spectra with strangeness $-2$ and $-3$ are ongoing, with the CMS collaboration recently observing the decay $\Lambda_{b}^{0}\to J/\psi\Xi^{-}K^{+}$ [10]. However, low yield and poor resolution precluded observation of a clear spectrum in the $J/\psi\Xi^{-}$ invariant mass.

Since the LHCb discovery of hidden charm pentaquarks $P_{c\bar{c}}$, a plethora of theoretical works has been proposed to explain their nature. Two distinct approaches suggest that $P_{c\bar{c}}$’s are molecular states, lying below the thresholds of various heavy mesons and singly heavy baryons. The first model posits bound states arising from quark potential models in configuration space [11, 12]. The second interprets them as poles in the lower Riemann sheet, generated by non-perturbatively produced scattering matrices [13, 14, 15]. Furthermore, constituent quark interpretations offer another potential scheme for explaining the pentaquark spectrum [16, 17, 18]. Additionally, alternative hypotheses suggest that the peak structure is a consequence of kinematic singularities [19, 20, 21, 22, 23]. It is not possible to distinguish these schemes from the LHCb data alone. However, results from the GlueX collaboration allow some constituent pentaquark spectra to be ruled out due to their absence in $J/\psi$ photoproduction. In contrast, the triangle singularity scheme offers a partial explanation for the disappearance of $P_{c\bar{c}}$ states observed by the GlueX collaboration. This is attributed to the inability of photoproduction to generate the double triangle diagram, as pointed out in Ref. [22]. On the other hand, in the context of the molecular picture, no mechanism has yet been identified that can explain this disappearance.

In the current work, we investigate hidden-charm pentaquark states using heavy meson and singly heavy baryon scattering in an off-shell coupled channel formalism. To facilitate understanding of experimental findings, we also include charmonium-nucleon scattering. The transition amplitudes are generated by solving coupled integral equations, with kernel amplitudes constructed from meson-exchange diagrams. These processes are governed by an effective Lagrangian that respects both heavy quark symmetry and hidden local symmetry. Our analysis yields seven distinct peaks are related to molecular states of the heavy mesons $\bar{D}$ ($\bar{D}^{*}$) and singly heavy baryons $\Sigma_{c}$ ($\Sigma_{c}^{*}$). Four of these peaks can be identified with the known $P_{c\bar{c}}$ states: $P_{c\bar{c}}(4312)$, $P_{c\bar{c}}(4380)$, $P_{c\bar{c}}(4440)$, and $P_{c\bar{c}}(4457)$. The other three resonances, with masses around 4.5 GeV, are predicted to be $\overline{D}^{*}\Sigma_{c}^{*}$ molecular states and are yet to be discovered. Notably, these peaks undergo significant modifications in the $J/\psi N$ elastic channel, with some even disappearing due to interference from the positive parity channel. This phenomenon may provide insight into the absence of pentaquark states in $J/\psi$ photoproduction, where elastic $J/\psi N$ scattering plays a crucial role. A more quantitative explanation requires comparison of our theoretical model with experimental data. This will be the subject of future studies, employing more rigorous fitting strategies to provide a more comprehensive and valid explanation of the observed phenomena.

The current work is organized as follows: In Section II, we present the off-shell coupled-channel formalism used to study the hidden-charm pentaquark states. This includes the effective Lagrangian, the partial-wave expansion of the scattering amplitude, and the method for solving the coupled integral equations. Section III is devoted to the results and discussions, where we analyze the scattering matrices for both negative and positive parity states, identify the pole positions, and compare our findings with experimental data. We also address the apparent conflict between LHCb and GlueX results, proposing a qualitative explanation based on our molecular scheme. Finally, we summarize our findings and conclude in Section IV, discussing the implications of our results and outlining future directions.

The scattering amplitude is defined as

$\displaystyle\mathcal{S}_{fi}=\delta_{fi}-i(2\pi)^{4}\delta(P_{f}-P_{i})\mathcal{T}_{fi},$

(1)

where $P_{i}$ and $P_{f}$ stand for the total four momenta of the initial and final states, respectively. The transition amplitudes $\mathcal{T}_{fi}$ can be derived from the Bethe-Salpeter integral equation

$\displaystyle\mathcal{T}_{fi}(p^{\prime},p;s)=\,\mathcal{V}_{fi}(p^{\prime},p;s)+\frac{1}{(2\pi)^{4}}\sum_{k}\int d^{4}q\mathcal{V}_{fk}(p^{\prime},q;s)\mathcal{G}_{k}(q;s)\mathcal{T}_{ki}(q,p;s),$

(2)

where $p$ and $p^{\prime}$ denote the relative four-momenta of the initial and final states, respectively. $q$ is the off-mass-shell momentum for the intermediate states in the center of mass (CM) frame. $s$ represents the square of the total energy, which is just one of the Mandelstam variables, $s=P_{i}^{2}=P_{f}^{2}$. The coupled integral equations given in Eq. (2) can be depicted as in Fig. 1.

To avoid the complexity due to the four-dimensional integral equations, we make a three-dimensional reduction. While there are several different methods for the three-dimensional reduction, we employ the Blankenbecler-Sugar scheme [24, 25], which takes the two-body propagator in the form of

$\displaystyle\mathcal{G}_{k}(q)=\;\delta\left(q_{0}-\frac{E_{k1}(\bm{q})-E_{k2}(\bm{q})}{2}\right)\frac{\pi}{E_{k1}(\bm{q})E_{k2}(\bm{q})}\frac{E_{k}(\bm{q})}{s-E_{k}^{2}(\bm{q})},$

(3)

where $E_{k}$ represents the total on-mass-shell energy of the intermediate state, $E_{k}=E_{k1}+E_{k2}$, and $\bm{q}$ denotes the three-momentum of the intermediate state. Note that the spinor factors from the meson-baryon propagator $G_{k}$ have been absorbed to the matrix elements of $\mathcal{V}$ and $\mathcal{T}$. Utilizing Eq. (3), we obtain the following coupled integral equations

$\displaystyle\mathcal{T}_{fi}(\bm{p}^{\prime},\bm{p})=\,\mathcal{V}_{fi}(\bm{p}^{\prime},\bm{p})+\frac{1}{(2\pi)^{3}}\sum_{k}\int\frac{d^{3}q}{2E_{k1}(\bm{q})E_{k2}(\bm{q})}\mathcal{V}_{fk}(\bm{p}^{\prime},\bm{q})\frac{E_{k}(\bm{q})}{s-E_{k}^{2}(\bm{q})+i\varepsilon}\mathcal{T}_{ki}(\bm{q},\bm{p}),$

(4)

where $\bm{p}$ and $\bm{p}^{\prime}$ are the relative three-momenta of the initial and final states in the CM frame, respectively.

We construct two-body coupled channels by combining the charmed meson triplet and singly charmed baryon antitriplet and sextet with total strangeness number $S=0$ to study the pentaquark $P_{c\bar{c}}$. In addition, we also introduce the $J/\psi N$ channel, since $P_{c}$’s were experimentally known to decay into $J/\psi N$. Thus, we have the seven different channels as follows: $J/\psi N$, $\bar{D}\Lambda_{c}$, $\bar{D}^{*}\Lambda_{c}$, $\bar{D}\Sigma_{c}$, $\bar{D}\Sigma_{c}^{*}$, $\bar{D}^{*}\Sigma_{c}$ and $\bar{D}^{*}\Sigma_{c}^{*}$. Thus, we first construct the kernel matrix expressed as

$\displaystyle\mathcal{V}$

$\displaystyle=\begin{pmatrix}\mathcal{V}_{J/\psi N\to J/\psi N}&\mathcal{V}_{\bar{D}\Lambda_{c}\to J/\psi N}&\mathcal{V}_{\bar{D}^{*}\Lambda_{c}\to J/\psi N}&\mathcal{V}_{\bar{D}\Sigma_{c}\to J/\psi N}&\mathcal{V}_{\bar{D}\Sigma_{c}^{*}\to J/\psi N}&\mathcal{V}_{\bar{D}^{*}\Sigma_{c}\to J/\psi N}&\mathcal{V}_{\bar{D}^{*}\Sigma_{c}^{*}\to J/\psi N}\\ \mathcal{V}_{J/\psi N\to\bar{D}\Lambda_{c}}&\mathcal{V}_{\bar{D}\Lambda_{c}\to\bar{D}\Lambda_{c}}&\mathcal{V}_{\bar{D}^{*}\Lambda_{c}\to\bar{D}\Lambda_{c}}&\mathcal{V}_{\bar{D}\Sigma_{c}\to\bar{D}\Lambda_{c}}&\mathcal{V}_{\bar{D}\Sigma_{c}^{*}\to\bar{D}\Lambda_{c}}&\mathcal{V}_{\bar{D}^{*}\Sigma_{c}\to\bar{D}\Lambda_{c}}&\mathcal{V}_{\bar{D}^{*}\Sigma_{c}^{*}\to\bar{D}\Lambda_{c}}\\ \mathcal{V}_{J/\psi N\to\bar{D}^{*}\Lambda_{c}}&\mathcal{V}_{\bar{D}\Lambda_{c}\to\bar{D}^{*}\Lambda_{c}}&\mathcal{V}_{\bar{D}^{*}\Lambda_{c}\to\bar{D}^{*}\Lambda_{c}}&\mathcal{V}_{\bar{D}\Sigma_{c}\to\bar{D}^{*}\Lambda_{c}}&\mathcal{V}_{\bar{D}\Sigma_{c}^{*}\to\bar{D}^{*}\Lambda_{c}}&\mathcal{V}_{\bar{D}^{*}\Sigma_{c}\to\bar{D}^{*}\Lambda_{c}}&\mathcal{V}_{\bar{D}^{*}\Sigma_{c}^{*}\to\bar{D}^{*}\Lambda_{c}}\\ \mathcal{V}_{J/\psi N\to\bar{D}\Sigma_{c}}&\mathcal{V}_{\bar{D}\Lambda_{c}\to\bar{D}\Sigma_{c}}&\mathcal{V}_{\bar{D}^{*}\Lambda_{c}\to\bar{D}\Sigma_{c}}&\mathcal{V}_{\bar{D}\Sigma_{c}\to\bar{D}\Sigma_{c}}&\mathcal{V}_{\bar{D}\Sigma_{c}^{*}\to\bar{D}\Sigma_{c}}&\mathcal{V}_{\bar{D}^{*}\Sigma_{c}\to\bar{D}\Sigma_{c}}&\mathcal{V}_{\bar{D}^{*}\Sigma_{c}^{*}\to\bar{D}\Sigma_{c}}\\ \mathcal{V}_{J/\psi N\to\bar{D}\Sigma_{c}^{*}}&\mathcal{V}_{\bar{D}\Lambda_{c}\to\bar{D}\Sigma_{c}^{*}}&\mathcal{V}_{\bar{D}^{*}\Lambda_{c}\to\bar{D}\Sigma_{c}^{*}}&\mathcal{V}_{\bar{D}\Sigma_{c}\to\bar{D}\Sigma_{c}^{*}}&\mathcal{V}_{\bar{D}\Sigma_{c}^{*}\to\bar{D}\Sigma_{c}^{*}}&\mathcal{V}_{\bar{D}^{*}\Sigma_{c}\to\bar{D}\Sigma_{c}^{*}}&\mathcal{V}_{\bar{D}^{*}\Sigma_{c}^{*}\to\bar{D}\Sigma_{c}^{*}}\\ \mathcal{V}_{J/\psi N\to\bar{D}^{*}\Sigma_{c}}&\mathcal{V}_{\bar{D}\Lambda_{c}\to\bar{D}^{*}\Sigma_{c}}&\mathcal{V}_{\bar{D}^{*}\Lambda_{c}\to\bar{D}^{*}\Sigma_{c}}&\mathcal{V}_{\bar{D}\Sigma_{c}\to\bar{D}^{*}\Sigma_{c}}&\mathcal{V}_{\bar{D}\Sigma_{c}^{*}\to\bar{D}^{*}\Sigma_{c}}&\mathcal{V}_{\bar{D}^{*}\Sigma_{c}\to\bar{D}^{*}\Sigma_{c}}&\mathcal{V}_{\bar{D}^{*}\Sigma_{c}^{*}\to\bar{D}^{*}\Sigma_{c}}\\ \mathcal{V}_{J/\psi N\to\bar{D}^{*}\Sigma_{c}^{*}}&\mathcal{V}_{\bar{D}\Lambda_{c}\to\bar{D}^{*}\Sigma_{c}^{*}}&\mathcal{V}_{\bar{D}^{*}\Lambda_{c}\to\bar{D}^{*}\Sigma_{c}^{*}}&\mathcal{V}_{\bar{D}\Sigma_{c}\to\bar{D}^{*}\Sigma_{c}^{*}}&\mathcal{V}_{\bar{D}\Sigma_{c}^{*}\to\bar{D}^{*}\Sigma_{c}^{*}}&\mathcal{V}_{\bar{D}^{*}\Sigma_{c}\to\bar{D}^{*}\Sigma_{c}^{*}}&\mathcal{V}_{\bar{D}^{*}\Sigma_{c}^{*}\to\bar{D}^{*}\Sigma_{c}^{*}}\\ \end{pmatrix}.$

(5)

Each component of the kernel matrix is constructed by using one-meson exchange tree-level diagram which is shown in Fig. 2. We do not include pole diagrams in the $s$ channel. We consider the $t$-channel diagrams, which are essential for generating the $P_{c\bar{c}}$ states dynamically. The contributions of $u$-channel diagrams are very small, so we ignore them. The interactions at the vertex are governed by the effective Lagrangian that respects heavy-quark spin symmetry, hidden local gauge symmetry, and flavor SU(3) symmetry [26]. The mesonic vertices are then computed by the effective Lagrangian given as

	$\displaystyle\mathcal{L}_{PP\mathbb{V}}$	$\displaystyle=-i\frac{\beta g_{V}}{\sqrt{2}}\,P^{\dagger}_{a}\overleftrightarrow{\partial_{\mu}}P_{b}\,\mathbb{V}^{\mu}_{ba}+i\frac{\beta g_{V}}{\sqrt{2}}\,P^{\prime\dagger}_{a}\overleftrightarrow{\partial_{\mu}}P^{\prime}_{b}\,\mathbb{V}^{\mu}_{ab},$		(6)
	$\displaystyle\mathcal{L}_{PP\sigma}$	$\displaystyle=-2g_{\sigma}MP^{\dagger}_{a}\sigma P_{a}-2g_{\sigma}MP^{\prime\dagger}_{a}\sigma P^{\prime}_{a},$		(7)
	$\displaystyle\mathcal{L}_{P^{*}P^{*}\mathbb{P}}$	$\displaystyle=-\frac{g}{f_{\pi}}\epsilon^{\mu\nu\alpha\beta}P^{*\dagger}_{a\nu}\,\overleftrightarrow{\partial_{\mu}}\,P^{*}_{b\beta}\partial_{\alpha}\mathbb{P}_{ba}-\frac{g}{f_{\pi}}\epsilon^{\mu\nu\alpha\beta}P^{\prime*\dagger}_{a\nu}\,\overleftrightarrow{\partial_{\mu}}\,P^{\prime*}_{b\beta}\partial_{\alpha}\mathbb{P}_{ab},$		(8)
	$\displaystyle\mathcal{L}_{P^{*}P^{*}\mathbb{V}}$	$\displaystyle=i\frac{\beta g_{V}}{\sqrt{2}}\,P^{*\dagger}_{a\nu}\overleftrightarrow{\partial_{\mu}}P^{*\nu}_{b}\mathbb{V}_{ba}^{\mu}+i2\sqrt{2}\lambda g_{V}M^{*}P^{*\dagger}_{a\mu}P^{*}_{b\nu}\mathbb{V}_{ba}^{\mu\nu}$		(9)
		$\displaystyle\;\;\;\;-i\frac{\beta g_{V}}{\sqrt{2}}\,P^{\prime*\dagger}_{a\nu}\overleftrightarrow{\partial_{\mu}}P^{\prime*\nu}_{b}\mathbb{V}_{ab}^{\mu}-i2\sqrt{2}\lambda g_{V}M^{*}P^{\prime*\dagger}_{a\mu}P^{\prime*}_{b\nu}\mathbb{V}_{ab}^{\mu\nu},$		(10)
	$\displaystyle\mathcal{L}_{P^{*}P^{*}\sigma}$	$\displaystyle=2g_{\sigma}M^{*}P^{*\dagger}_{a\mu}\sigma P^{*\mu}_{a}+2g_{\sigma}M^{*}P^{\prime*\dagger}_{a\mu}\sigma P^{\prime*\mu}_{a},$		(11)
	$\displaystyle\mathcal{L}_{P^{*}P\mathbb{P}}$	$\displaystyle=-\frac{2g}{f_{\pi}}\sqrt{MM^{*}}\,\left(P^{\dagger}_{a}P^{*}_{b\mu}+P^{*\dagger}_{a\mu}P_{b}\right)\,\partial^{\mu}\mathbb{P}_{ba}+\frac{2g}{f_{\pi}}\sqrt{MM^{*}}\,\left(P^{\prime\dagger}_{a}P^{\prime*}_{b\mu}+P^{\prime*\dagger}_{a\mu}P^{\prime}_{b}\right)\,\partial^{\mu}\mathbb{P}_{ab},$		(12)
	$\displaystyle\mathcal{L}_{P^{*}P\mathbb{V}}$	$\displaystyle=-i\sqrt{2}\lambda g_{V}\,\epsilon^{\beta\alpha\mu\nu}\left(P^{\dagger}_{a}\overleftrightarrow{\partial_{\beta}}P^{*}_{b\alpha}+P^{*\dagger}_{a\alpha}\overleftrightarrow{\partial_{\beta}}P_{b}\right)\,\left(\partial_{\mu}\mathbb{V}_{\nu}\right)_{ba}$		(13)
		$\displaystyle\;\;\;\;-i\sqrt{2}\lambda g_{V}\,\epsilon^{\beta\alpha\mu\nu}\left(P^{\prime\dagger}_{a}\overleftrightarrow{\partial_{\beta}}P^{\prime*}_{b\alpha}+P^{\prime*\dagger}_{a\alpha}\overleftrightarrow{\partial_{\beta}}P^{\prime}_{b}\right)\,\left(\partial_{\mu}\mathbb{V}_{\nu}\right)_{ab}.$		(14)

with $\overleftrightarrow{\partial}=\overrightarrow{\partial}-\overleftarrow{\partial}$. The lowest isoscalar-scalar meson is denoted by $\sigma$. The heavy meson and anti heavy meson matrices $P^{(*)}$ and $P^{\prime(*)}$ are given by

$\displaystyle P=\left(D^{0},D^{+},D_{s}^{+}\right),\hskip 14.22636ptP^{*}_{\mu}=\left(D^{*0}_{\mu},D^{*+}_{\mu},D_{s\mu}^{*+}\right),\hskip 14.22636ptP^{\prime}=(\bar{D}^{0},\,D^{-},\,D_{s}^{-}),\hskip 14.22636ptP^{\prime*}_{\mu}=(\bar{D}^{*0}_{\mu},\,D^{*-}_{\mu},\,D^{*-}_{s\mu}),$

(15)

while the light pseudoscalar and vector meson matrices are

$\displaystyle\mathbb{P}=\begin{pmatrix}\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&\pi^{+}&K^{+}\\ \pi^{-}&-\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta&K^{0}\\ K^{-}&\bar{K}^{0}&-\frac{2}{\sqrt{6}}\eta\end{pmatrix},\;\;\;\;\mathbb{V}_{\mu}=\begin{pmatrix}\frac{1}{\sqrt{2}}\rho^{0}_{\mu}+\frac{1}{\sqrt{2}}\omega_{\mu}&\rho_{\mu}^{+}&K_{\mu}^{*+}\\ \rho_{\mu}^{-}&-\frac{1}{\sqrt{2}}\rho_{\mu}^{0}+\frac{1}{\sqrt{2}}\omega_{\mu}&K_{\mu}^{*0}\\ K_{\mu}^{*-}&\bar{K}^{*0}_{\mu}&\phi_{\mu}\end{pmatrix}.$

(16)

The coupling constants in the Lagrangian are obtained from Ref. [27], i.e., $g=0.59\pm 0.07\pm 0.01$ from experimental results of $D^{*+}$ full width, $g_{V}=m_{\rho}/f_{\pi}\approx 5.8$ by using the KSRF relation with $f_{\pi}=132$ MeV, $\beta\approx 0.9$ by assuming vector meson dominance in the radiative decay of heavy mesons, and $\lambda=-0.56\,\mathrm{GeV}^{-1}$ by using light-cone sum rules and lattice QCD. Notice that we use a different sign of $\lambda$ from Ref. [27], since we use the same phase of heavy vector meson as in Ref. [26]. The coupling constant for the sigma meson is utilized to calculate the $2\pi$ transition of $D_{s}(1^{+})$ in Ref. [28]. The lowest isoscalar-scalar meson coupling is $g_{\sigma}=g_{\pi}/2\sqrt{6}$ with $g_{\pi}=3.73$.

As for the effective Lagrangian for the heavy baryon, we take it from Ref. [29], where a more general form of the Lagrangian was considered [30]. The interaction vertices for the baryonic sector in the tree level diagram of the meson-exchange diagram are governed by the following effective Lagrangian

	$\displaystyle\mathcal{L}_{B_{\bar{3}}B_{\bar{3}}\mathbb{V}}$	$\displaystyle=\frac{i\beta_{\bar{3}}g_{V}}{2\sqrt{2}M_{\bar{3}}}\left(\bar{B}_{\bar{3}}\overleftrightarrow{\partial_{\mu}}\mathbb{V}^{\mu}B_{\bar{3}}\right),$		(17)
	$\displaystyle\mathcal{L}_{B_{\bar{3}}B_{\bar{3}}\sigma}$	$\displaystyle=l_{\bar{3}}\left(\bar{B}_{\bar{3}}\sigma B_{\bar{3}}\right),$		(18)
	$\displaystyle\mathcal{L}_{B_{6}B_{6}\mathbb{P}}$	$\displaystyle=i\frac{g_{1}}{2f_{\pi}M_{6}}\bar{B}_{6}\gamma_{5}\left(\gamma^{\alpha}\gamma^{\beta}-g^{\alpha\beta}\right)\overleftrightarrow{\partial_{\alpha}}\partial_{\beta}\mathbb{P}B_{6},$		(19)
	$\displaystyle\mathcal{L}_{B_{6}B_{6}\mathbb{V}}$	$\displaystyle=-i\frac{\beta_{6}g_{V}}{2\sqrt{2}M_{6}}\left(\bar{B}_{6}\overleftrightarrow{\partial_{\alpha}}\mathbb{V}^{\alpha}B_{6}\right)-\frac{i\lambda_{6}g_{V}}{3\sqrt{2}}\left(\bar{B}_{6}\gamma_{\mu}\gamma_{\nu}\mathbb{V}^{\mu\nu}B_{6}\right),$		(20)
	$\displaystyle\mathcal{L}_{B_{6}B_{6}\sigma}$	$\displaystyle=-l_{6}\left(\bar{B}_{6}\sigma B_{6}\right),$		(21)
	$\displaystyle\mathcal{L}_{B_{6}^{*}B_{6}^{*}\mathbb{P}}$	$\displaystyle=\frac{3g_{1}}{4f_{\pi}M_{6}^{*}}\epsilon^{\mu\nu\alpha\beta}\left(\bar{B}_{6\mu}^{*}\overleftrightarrow{\partial_{\nu}}\partial_{\alpha}\mathbb{P}B_{6\beta}^{*}\right),$		(22)
	$\displaystyle\mathcal{L}_{B_{6}^{*}B_{6}^{*}\mathbb{V}}$	$\displaystyle=i\frac{\beta_{6}g_{V}}{2\sqrt{2}M_{6}^{*}}\left(\bar{B}_{6\mu}^{*}\overleftrightarrow{\partial_{\alpha}}\mathbb{V}^{\alpha}B_{6}^{*\mu}\right)+\frac{i\lambda_{6}g_{V}}{\sqrt{2}}\left(\bar{B}^{*}_{6\mu}\mathbb{V}^{\mu\nu}B^{*}_{6\nu}\right),$		(23)
	$\displaystyle\mathcal{L}_{B_{6}^{*}B_{6}^{*}\sigma}$	$\displaystyle=l_{6}\left(\bar{B}^{*}_{6\mu}\sigma B^{*\mu}_{6}\right),$		(24)
	$\displaystyle\mathcal{L}_{B_{6}B_{6}^{*}\mathbb{P}}$	$\displaystyle=\frac{g_{1}}{4f_{\pi}}\sqrt{\frac{3}{M_{6}^{*}M_{6}}}\epsilon^{\mu\nu\alpha\beta}\left[\left(\bar{B}_{6}\gamma_{5}\gamma_{\mu}\overleftrightarrow{\partial_{\nu}}\partial_{\alpha}\mathbb{P}B_{6\beta}^{*}\right)+\left(\bar{B}_{6\mu}^{*}\gamma_{5}\gamma_{\nu}\overleftrightarrow{\partial_{\alpha}}\partial_{\beta}\mathbb{P}B_{6}\right)\right],$		(25)
	$\displaystyle\mathcal{L}_{B_{6}B_{6}^{*}\mathbb{V}}$	$\displaystyle=\frac{i\lambda_{6}g_{V}}{\sqrt{6}}\left[\bar{B}_{6}\gamma_{5}\left(\gamma_{\mu}+\frac{i\overleftrightarrow{\partial_{\mu}}}{2\sqrt{M_{6}^{*}M_{6}}}\right)\mathbb{V}^{\mu\nu}B^{*}_{6\nu}+\bar{B}_{6\mu}^{*}\gamma_{5}\left(\gamma_{\nu}-\frac{i\overleftrightarrow{\partial_{\nu}}}{2\sqrt{M_{6}^{*}M_{6}}}\right)\mathbb{V}^{\mu\nu}B_{6}\right],$		(26)
	$\displaystyle\mathcal{L}_{B_{6}B_{\bar{3}}\mathbb{P}}$	$\displaystyle=-\frac{g_{4}}{\sqrt{3}f_{\pi}}\left[\bar{B}_{6}\gamma_{5}\left(\gamma_{\mu}+\frac{i\overleftrightarrow{\partial_{\mu}}}{2\sqrt{M_{6}M_{\bar{3}}}}\right)\partial^{\mu}\mathbb{P}B_{\bar{3}}+\bar{B}_{\bar{3}}\gamma_{5}\left(\gamma_{\mu}-\frac{i\overleftrightarrow{\partial_{\mu}}}{2\sqrt{M_{6}M_{\bar{3}}}}\right)\partial^{\mu}\mathbb{P}\,B_{6}\right],$		(27)
	$\displaystyle\mathcal{L}_{B_{6}B_{\bar{3}}\mathbb{V}}$	$\displaystyle=i\frac{\lambda_{6\bar{3}}\,g_{V}}{\sqrt{6M_{6}M_{\bar{3}}}}\epsilon^{\mu\nu\alpha\beta}\left[\left(\bar{B}_{6}\gamma_{5}\gamma_{\mu}\overleftrightarrow{\partial_{\nu}}\partial_{\alpha}\mathbb{V}_{\beta}B_{\bar{3}}\right)+\left(\bar{B}_{\bar{3}}\gamma_{5}\gamma_{\mu}\overleftrightarrow{\partial_{\nu}}\partial_{\alpha}\mathbb{V}_{\beta}B_{6}\right)\right],$		(28)
	$\displaystyle\mathcal{L}_{B_{6}^{*}B_{\bar{3}}\mathbb{P}}$	$\displaystyle=-\frac{g_{4}}{f_{\pi}}\left[\left(\bar{B}^{*}_{6\mu}\partial^{\mu}\mathbb{P}B_{\bar{3}}\right)+\left(\bar{B}_{\bar{3}}\partial^{\mu}\mathbb{P}B^{*}_{6\mu}\right)\right],$		(29)
	$\displaystyle\mathcal{L}_{B_{6}^{*}B_{\bar{3}}\mathbb{V}}$	$\displaystyle=i\frac{\lambda_{6\bar{3}}\,g_{V}}{\sqrt{2M_{6}^{*}M_{\bar{3}}}}\epsilon^{\mu\nu\alpha\beta}\left[\left(\bar{B}^{*}_{6\mu}\overleftrightarrow{\partial_{\nu}}\partial_{\alpha}\mathbb{V}_{\beta}B_{\bar{3}}\right)+\left(\bar{B}_{\bar{3}}\overleftrightarrow{\partial_{\nu}}\partial_{\alpha}\mathbb{V}_{\beta}B^{*}_{6\mu}\right)\right],$		(30)

where the heavy baryon fields are given by

$\displaystyle B_{\bar{3}}=\begin{pmatrix}0&\Lambda_{c}^{+}&\Xi_{c}^{+}\\ -\Lambda_{c}^{+}&0&\Xi_{c}^{0}\\ -\Xi_{c}^{+}&-\Xi_{c}^{0}&0\end{pmatrix},\;\;B_{6}=\begin{pmatrix}\Sigma_{c}^{++}&\frac{1}{\sqrt{2}}\Sigma_{c}^{+}&\frac{1}{\sqrt{2}}\Xi{{}^{\prime}}_{c}^{+}\\ \frac{1}{\sqrt{2}}\Sigma_{c}^{+}&\Sigma_{c}^{0}&\frac{1}{\sqrt{2}}\Xi{{}^{\prime}}_{c}^{0}\\ \frac{1}{\sqrt{2}}\Xi{{}^{\prime}}_{c}^{+}&\frac{1}{\sqrt{2}}\Xi{{}^{\prime}}_{c}^{0}&\Omega_{c}^{0}\end{pmatrix},\;\;B_{6}^{*}=\begin{pmatrix}\Sigma_{c}^{*++}&\frac{1}{\sqrt{2}}\Sigma_{c}^{*+}&\frac{1}{\sqrt{2}}\Xi_{c}^{*+}\\ \frac{1}{\sqrt{2}}\Sigma_{c}^{*+}&\Sigma_{c}^{*0}&\frac{1}{\sqrt{2}}\Xi_{c}^{*0}\\ \frac{1}{\sqrt{2}}\Xi_{c}^{*+}&\frac{1}{\sqrt{2}}\Xi_{c}^{*0}&\Omega_{c}^{*0}\end{pmatrix}.$

(31)

$B_{\mu}$ denotes the spin 3/2 Rarita-Schwinger field, which satisfies the following constraint

$\displaystyle p^{\mu}B_{\mu}=0\hskip 14.22636pt{\rm and}\hskip 14.22636pt\gamma^{\mu}B_{\mu}=0.$

(32)

The coupling constants in the effective Lagrangian are given as follows [29, 31]: $\beta_{\bar{3}}=6/g_{V}$, $\beta_{6}=-2\beta_{\bar{3}}$, $\lambda_{6}=-3.31\,\mathrm{GeV}^{-1}$, $\lambda_{6\bar{3}}=-\lambda_{6}/\sqrt{8}$, $g_{1}=0.942$ and $g_{4}=0.999$. The different signs used above are taken from Refs. [31, 32].

Since we include the hidden-charm channel, we need effective Lagrangian to describe the coupling between heavy mesons and quarkonium. Here we use the Lagrangian from Ref. [33], i.e.,

	$\displaystyle\mathcal{L}_{PPJ/\psi}$	$\displaystyle=-ig_{\psi}M\sqrt{m_{J}}\left(J/\psi^{\mu}P^{\dagger}\overleftrightarrow{\partial_{\mu}}P{{}^{\prime}}^{\dagger}\right)+\mathrm{h.c.,}$		(33)
	$\displaystyle\mathcal{L}_{P^{*}PJ/\psi}$	$\displaystyle=ig_{\psi}\sqrt{\frac{MM^{*}}{m_{J}}}\epsilon^{\mu\nu\alpha\beta}\partial_{\mu}J/\psi_{\nu}\left(P^{\dagger}\overleftrightarrow{\partial_{\alpha}}P^{*}{{}^{\prime}}^{\dagger}_{\beta}+P_{\beta}^{*\dagger}\overleftrightarrow{\partial_{\alpha}}P{{}^{\prime}}^{\dagger}\right)+\mathrm{h.c.,}$		(34)
	$\displaystyle\mathcal{L}_{P^{*}P^{*}J/\psi}$	$\displaystyle=ig_{\psi}M^{*}\sqrt{m_{J}}(g^{\mu\nu}g^{\alpha\beta}-g^{\mu\alpha}g^{\nu\beta}+g^{\mu\beta}g^{\nu\alpha})\left(J/\psi_{\mu}P_{\nu}^{*\dagger}\overleftrightarrow{\partial_{\alpha}}P^{*}{{}^{\prime}}^{\dagger}_{\beta}\right)+\mathrm{h.c.}$		(35)

In this work, we only consider the vector quarkonia since it is directly related to experiments. However, the extension to the pseudoscalar state is straightforward since we assume the heavy quark spin symmetry to the quarkonia state as well [34]. Since there is no experimental data on the $J/\psi\to D\bar{D}$ decay, Shimizu et al. [35] estimated the value of the coupling constant $g_{\psi}$ as follows: the coupling constant $g_{\phi K\bar{K}}$ can be determined from the experimental decay width for the $\phi\to K\bar{K}$ decay. Assuming that the decay of $J/\psi$ is similar to that of $\phi$ apart from their masses, Shimizu et al. estimated $g_{\psi}$ to be $g_{\psi}=0.679\,\mathrm{GeV}^{-3/2}$. The coupling constants of the heavy baryons and heavy mesons are expressed [35] as

	$\displaystyle\mathcal{L}_{B_{8}B_{3}P}$	$\displaystyle=-g_{I{\bar{3}}}\sqrt{M}\bar{B}_{\bar{3}}\gamma_{5}PN+\mathrm{h.c.},$		(36)
	$\displaystyle\mathcal{L}_{B_{8}B_{3}P^{*}}$	$\displaystyle=g_{I{\bar{3}}}\sqrt{M^{*}}\bar{B}_{\bar{3}}\gamma^{\mu}P_{\mu}^{*}N+\mathrm{h.c.,}$		(37)
	$\displaystyle\mathcal{L}_{B_{8}B_{6}P}$	$\displaystyle=g_{I6}\sqrt{3M}\bar{B}_{6}\gamma_{5}B_{8}P+\mathrm{h.c.,}$		(38)
	$\displaystyle\mathcal{L}_{B_{8}B_{6}P^{*}}$	$\displaystyle=g_{I6}\sqrt{\frac{M^{*}}{3}}\bar{B}_{6}\gamma^{\nu}B_{8}P^{*}_{\nu}+\mathrm{h.c.,}$		(39)
	$\displaystyle\mathcal{L}_{B_{8}B_{6}^{*}P^{*}}$	$\displaystyle=2g_{I6}\sqrt{M^{*}}\bar{B}_{6}^{\mu}\gamma_{5}B_{8}P^{*}_{\mu}+\mathrm{h.c.}.$		(40)

We employ the coupling constants $g_{I\bar{3}}=-9.88\,\mathrm{GeV}^{-1/2}$ and $g_{I6}=1.14\,\mathrm{GeV}^{-1/2}$ taken from Ref. [35]. It is important to note that the coupling to the hidden charm channels have only a marginal effect to the production mechanism of the resonance. The current calculation implies that though these values of the coupling constants are taken from the rough estimation, the predicted masses of the hidden charm pentaquarks almost do not vary. This already indicates that the $J/\psi N$ channel has a tiny effect on the production of the heavy pentaquarks.

The Feynman amplitude for one-meson exchange diagram can be written as

$\displaystyle\mathcal{A}_{\lambda^{\prime}_{1}\lambda^{\prime}_{2},\lambda_{1}\lambda_{2}}=\mathrm{IS}\,F^{2}(q^{2})\,\Gamma_{\lambda^{\prime}_{1}\lambda^{\prime}_{2}}(p^{\prime}_{1},p^{\prime}_{2})\mathcal{P}(q)\Gamma_{\lambda_{1}\lambda_{2}}(p_{1},p_{2}),$

(41)

where $\lambda_{i}$ and $p_{i}$ denote the helicity and momentum of the corresponding particle respectively, while $q$ is the momentum of the exchange particle. The IS factor is related to the SU(3) Clebsch-Gordan coefficient and isospin factor. The IS factor for each exchanged diagram is listed in Table 1.