$N^{*}$ states with hidden charm and a three-body nature

Recently, the LHCb collaboration announced the existence of possible hidden charm pentaquarks in the $J/\psi p$ invariant mass distribution of the process $\Lambda^{0}_{b}\to J/\psi pK^{-}$ Aaij:2015tga ; Aaij:2016phn ; Aaij:2019vzc . The observation of such states, denoted as $P^{+}_{c}(4312)$, $P^{+}_{c}(4440)$ and $P^{+}_{c}(4457)$, constitutes a turning point in the experimental search for signals related to exotic baryons, which had gradually reduced after the lack of evidence for the existence of $\Theta^{+}(1540)$ was reported in higher statistics experiments Battaglieri:2005er ; Miwa:2006if ; Moritsu:2014bht ; Shen:2016csu  ¹¹1See also Refs. Torres:2010jh ; MartinezTorres:2010zzb for a possible theoretical explanation of the signal observed by the LEPS collaboration for $\Theta^{+}(1540)$.. Indeed, continuing with the hunting of exotic baryons, more recently, the LHCb collaboration has claimed the existence of yet another pentaquark with hidden charm, similar to the above mentioned $P_{c}$ states, but with nonzero strangeness, whose mass is around 4459 MeV wang . Coming back to the discussion on the $P_{c}$’s, we must recall that two states were initially claimed in Ref. Aaij:2015tga , the so called $P^{+}_{c}(4380)$, with a width of $205\pm 18\pm 86$ MeV, and $P^{+}_{c}(4450)$, with a width of $39\pm 5\pm 19$. Out of the two states, the existence of the former was based more on a requirement for a good description of the data than on a direct observation Aaij:2015tga . The best fit to the data in Ref. Aaij:2015tga was obtained by considering spin-parity $J^{P}=3/2^{-}$ for $P^{+}_{c}(4380)$ and $5/2^{+}$ for $P^{+}_{c}(4450)$, although acceptable solutions were also obtained for the combinations $J^{P}=3/2^{+}$ and $5/2^{-}$ and $J^{P}=5/2^{+}$ and $3/2^{-}$, respectively. The updated analysis of the same processes as in Ref. Aaij:2015tga , but with much larger statistical significance, was made in Ref. Aaij:2019vzc , which revealed the presence of two states, $P^{+}_{c}(4440)$ (mass $M=4440.3\pm 1.3^{+4.1}_{-4.7}$ MeV, width $\Gamma=20.6\pm 4.9^{+8.7}_{-10.1}$ MeV) and $P^{+}_{c}(4457)$ ($M=4457.3\pm 0.6^{+4.1}_{-1.7}$ MeV, $\Gamma=6.4\pm 2.0^{+5.7}_{-1.9}$ MeV), instead of $P^{+}_{c}(4450)$, and a narrow peak at 4312 MeV, $P^{+}_{c}(4312)$ ($M=4311.9\pm 0.7^{+6.8}_{-0.6}$ MeV, $\Gamma=9.8\pm 2.7^{+3.7}_{-4.5}$), though no definite spin-parity is assigned to the three states yet. The fits performed in Ref. Aaij:2019vzc with and without $P^{+}_{c}(4380)$ describe well the data, and its existence could not be ascertained. Several theoretical descriptions, as well as different spin-parity assignments, have been proposed for the nature of these $P_{c}$ states, like pentaquarks Maiani:2015vwa ; Lebed:2015tna ; Ghosh:2015ksa ; Weng:2019ynv ; Wang:2019got , meson-baryon states Roca:2015dva ; Xiao:2019aya ; Liu:2019tjn ; Burns:2019iih ; He:2019ify ; Xiao:2019mst ; Guo:2019kdc ; Du:2019pij ; Xu:2020gjl , peaks arising from triangle singularities Guo:2015umn ; Liu:2015fea ; Nakamura:2021qvy (which seems to be more plausible for $P^{+}_{c}(4457)$, although further investigations are necessary Aaij:2019vzc ) or a virtual state for $P^{+}_{c}(4312)$ Fernandez-Ramirez:2019koa . In case of the meson-baryon molecular attribution, the predominant interpretation is to consider the $P_{c}$’s as states arising from the $\Sigma_{c}\bar{D}^{*}$, $\Sigma^{*}_{c}\bar{D}^{*}$ and coupled channel dynamics in s-wave. Within such a description, the $P_{c}$ states, although there is no consensus on the spin assignment, would have negative parity. However, in Ref. Burns:2019iih , by introducing $\Lambda_{c}(2595)\bar{D}$ coupled to $\Sigma_{c}\bar{D}^{*}$, the authors determined a positive parity assignment for $P_{c}(4457)$.

With all the experimental and theoretical efforts being made to understand the properties of the $P_{c}$ states claimed in Ref. Aaij:2019vzc , and with the debate on their spin-parity assignments still under way, it is important to investigate the possible formation of $N^{*}$ states with hidden charm and positive parity. Such states may naturally arise as a consequence of the dynamics involved in the three-body systems $ND\bar{D}^{*}$ and $N\bar{D}D^{*}$ since the interactions between the different subsystems are attractive in s-wave, forming states like $X(3872)$, $Z_{c}(3900)$, $\Lambda_{c}(2595)$ Braaten:2003he ; AlFiky:2005jd ; Gamermann:2007fi ; Gamermann:2010zz ; Aceti:2014uea ; Ortega:2018cnm ; He:2017lhy ; Hofmann:2005sw ; Mizutani:2006vq ; GarciaRecio:2008dp ; Romanets:2012hm ; Liang:2014kra ; Nieves:2019nol . Further, the $ND\bar{D}^{*}$ ($N\bar{D}D^{*}$) threshold lies around 4814 MeV, and due to the strong attraction present in the $ND$ and $ND^{*}$ coupled channel system in s-wave, where $\Lambda_{c}(2595)$ (among other $\Lambda_{c}$ and $\Sigma_{c}$ states) has been found to get generated around 210 (352) MeV below the $DN$ ($D^{*}N$) threshold, we can expect large binding energies in the three-body system considered, of the order of 200-300 MeV. In this way, there exists a possibility of finding positive parity state(s) in the energy region 4400-4600 MeV, precisely where the $P_{c}$’s have been observed in Refs. Aaij:2015tga ; Aaij:2019vzc .

Motivated by this reasoning, in this work, we study the $NX(3872)-NZ_{c}(3900)$ coupled configurations of the $ND\bar{D}^{*}-N\bar{D}D^{*}$ system. To do this, we solve the Faddeev equations within the fixed center approximation (FCA) Foldy:1945zz ; Brueckner:1953zz ; Deloff:1999gc ; MartinezTorres:2020hus assuming that $D\bar{D}^{*}-\bar{D}D^{*}$ clusters as $X(3872)/Z_{c}(3900)$. As we will show, $N^{*}$ states with hidden charm, spin-parity $J^{P}=1/2^{+}$, $3/2^{+}$ and masses in the energy region 4400-4600 MeV are found to arise due to the underlying three-body dynamics. We discuss different processes where such positive parity $P_{c}$’s can be studied experimentally. We also investigate the possible existence of isospin 3/2 states with hidden charm. Finally we test the applicability of FCA to the current system by calculating diagrams beyond the approximation and show that such contributions are negligible.

In the FCA to the Faddeev equations, the scattering between the three particles forming the system is described in terms of a particle interacting with a scattering center. Such a treatment is relevant when two of the particles cluster as a state whose mass is heavier than the third one, which rescatters with those forming the cluster through a series of interactions (for a review on the topic see Ref. MartinezTorres:2020hus ). In this way, the cluster plays the role of the scattering center, which stays unaltered in the scattering process. In this work we study the $NX(3872)/NZ_{c}(3900)$ configurations of the $ND\bar{D}^{*}-N\bar{D}D^{*}$ system. We consider then that the $D\bar{D}^{*}-\bar{D}D^{*}$ system clusters as $X(3872)$ [$Z_{c}(3900)$] in isospin $0$ ($1$) and the nucleon rescatters, successively, off the $D$ ($\bar{D}$) and $\bar{D}^{*}$ ($D^{*}$) mesons. Since,

	$\displaystyle|X(3872)\rangle$	$\displaystyle=\frac{1}{\sqrt{2}}\Big{[}|D\bar{D}^{*};I_{D\bar{D}^{*}}=0,I_{zD\bar{D}^{*}}=0\rangle$
		$\displaystyle\quad+|\bar{D}D^{*};I_{\bar{D}D^{*}}=0,I_{z\bar{D}D^{*}}=0\rangle\Big{]},$
	$\displaystyle|Z_{c}(3900)\rangle$	$\displaystyle=\frac{1}{\sqrt{2}}\Big{[}|D\bar{D}^{*};I_{D\bar{D}^{*}}=1,I_{zD\bar{D}^{*}}=1\rangle$
		$\displaystyle\quad+|\bar{D}D^{*};I_{\bar{D}D^{*}}=1,I_{z\bar{D}D^{*}}=1\rangle\big{]},$

when adding a nucleon, we can write the three-body state as,

	$\displaystyle|N\mathbb{C}_{a};I,I_{z}\rangle$	$\displaystyle=\frac{1}{\sqrt{2}}\Big{[}|N(D\bar{D}^{*})_{I_{a}};I,I_{z}\rangle$
		$\displaystyle\quad+|N(\bar{D}D^{*})_{I_{a}};I,I_{z}\rangle\Big{]},$		(1)

where $\mathbb{C}_{a}$ represents the cluster, whose isospin is $I_{a}$, and $I$ ($I_{z}$) is the isospin (and its third component) of the three-body system. In this way, to describe the interaction between a $N$ and the cluster $\mathbb{C}_{a}$ we need to calculate the scattering $T$-matrix

	$\displaystyle\langle N\mathbb{C}_{a};I,I_{z}|T|N\mathbb{C}_{a};I,I_{z}\rangle$
	$\displaystyle\quad=\frac{1}{2}\Big{[}\langle N(D\bar{D}^{*})_{I_{a}};I,I_{z}|T|N(D\bar{D}^{*})_{I_{a}};I,I_{z}\rangle$
	$\displaystyle\quad\quad+\langle N(\bar{D}D^{*})_{I_{a}};I,I_{z}|T|N(\bar{D}D^{*})_{I_{a}};I,I_{z}\rangle\Big{]}.$		(2)

Note that the transition

$\displaystyle\langle N(D\bar{D}^{*})_{I_{a}};I,I_{z}|T|N(\bar{D}D^{*})_{I_{a}};I,I_{z}\rangle$

requires that the scattering of the nucleon on $D\bar{D}^{*}$ converts the cluster to $D^{*}\bar{D}^{*}$ or $D\bar{D}$ in the intermediate states (see Fig. 1). Since our purpose is to identify the cluster with $X(3872)$ or $Z_{c}(3900)$ in all intermediate states, both of which are understood as $D\bar{D}^{*}-\bar{D}D^{*}$ “molecules”, other intermediate processes are expected to be small and are not considered in the formalism.

Indeed it was shown in Refs. MartinezTorres:2010ax ; MartinezTorres:2020hus that such considerations lead to small contributions to the dominant process for center of mass energies below the threshold. We thus limit ourselves to investigating the formation of three-body states below the threshold. Besides, to test the validity of the formalism, we have made explicit evaluation of the diagrams beyond FCA and confirm that such contributions are indeed negligible for the system considered in this work, at least for energies below the threshold (consistently with the findings of Ref. MartinezTorres:2010ax ). Details of these calculations are given in a subsequent section.

Continuing here with the discussions on the formalism, we can say that we need to study the $ND\bar{D}^{*}$ and $N\bar{D}D^{*}$ systems where $D\bar{D}^{*}$ ($\bar{D}D^{*}$) clusters either as $X(3872)$ or $Z_{c}(3900)$. Note that the interactions in the two kind of three-body systems are different. While one case involves a series of $ND$ and $N\bar{D}^{*}$ interactions, the other consists of $N\bar{D}$ and $ND^{*}$ interactions. Within the FCA to the Faddeev equations, the $T$-matrix for the $ND\bar{D}^{*}$ ($N\bar{D}D^{*}$) system is obtained from the resummation of two series which consider the rescattering of the nucleon on $D$ ($\bar{D}$) and $\bar{D}^{*}$ ($D^{*}$), as shown in Fig. 2. In this way,

$\displaystyle T=T_{1}+T_{2},$

(3)

where the partitions $T_{1}$ and $T_{2}$ are obtained by solving the coupled equations

	$\displaystyle T_{1}$	$\displaystyle=t_{1}+t_{1}G_{0}T_{2},$
	$\displaystyle T_{2}$	$\displaystyle=t_{2}+t_{2}G_{0}T_{1},$		(4)

for a given isospin $I$ and angular momentum $J$ of the three-body system. Since we consider s-wave interactions, the value of $J$ for the $ND\bar{D}^{*}$ ($N\bar{D}D^{*}$) system is given by the spin of the $N\bar{D}^{*}$ ($ND^{*}$) subsystem. The $t_{1}$ and $t_{2}$ in Eq. (4) are two-body $t$-matrices related to the $ND$ ($N\bar{D}$) and $N\bar{D}^{*}$ ($ND^{*}$) systems, respectively, and $G_{0}$ represents the propagator of a nucleon in the cluster formed. For total isospin $I=1/2$ of the $ND\bar{D}^{*}$ ($N\bar{D}D^{*}$) system, we treat $N(D\bar{D}^{*})_{0}$ and $N(D\bar{D}^{*})_{1}$ [$N(\bar{D}D^{*})_{0}$ and $N(\bar{D}D^{*})_{1}$] as coupled channels, since both configurations can lead to isospin 1/2.

We must emphasize here that both $X(3872)$ and $Z(3900)$ are interpreted as $D\bar{D}^{*}-\bar{D}D^{*}$ quasi-bound states with the same mass, although a different isospin and width Gamermann:2007fi ; Aceti:2014uea ; Gamermann:2009uq . Thus, FCA can be used to take into account transitions between the two systems as in the study of $Nf_{0}(980)-Na_{0}(980)$ Xie:2010ig . In case a reader is interested in an analysis of the effects of considering FCA for coupled channels with the clusters having different masses, we refer the reader to Ref. Roca:2011br , where effective systems like, $\pi K_{2}^{*}(1430)$ and $Ka_{2}(1320)$ have been studied.

In this way, $t_{1}$, $t_{2}$ and $G_{0}$ are matrices in the coupled channel space,

	$\displaystyle t_{1}$	$\displaystyle=\left(\begin{array}[]{cc}{(t_{1})}_{00}&{(t_{1})}_{01}\\ {(t_{1})}_{10}&{(t_{1})}_{11}\end{array}\right),~{}\quad t_{2}=\left(\begin{array}[]{cc}{(t_{2})}_{00}&{(t_{2})}_{01}\\ {(t_{2})}_{10}&{(t_{2})}_{11}\end{array}\right),$		(9)
	$\displaystyle G_{0}$	$\displaystyle=\left(\begin{array}[]{cc}(G_{0})_{00}&0\\ 0&(G_{0})_{11}\end{array}\right),$		(12)

where for a given isospin $I$

$\displaystyle\mathcal{O}_{ab}$

$\displaystyle=\langle N(\mathbb{C}_{1}\mathbb{C}_{2})_{I_{b}};I,I_{z}|\mathcal{O}|N(\mathbb{C}_{1}\mathbb{C}_{2})_{I_{a}};I,I_{z}\rangle,$

(13)

with $\mathcal{O}=t_{1}$, $t_{2}$, $G_{0}$ and $\mathbb{C}_{1}$, $\mathbb{C}_{2}$ being the two particles forming a cluster with a defined isospin. To see how the elements of Eq. (13) can be determined, let us consider, for example, the $N(D\bar{D}^{*})_{0}$ system. We can write,

	$\displaystyle|N(D\bar{D}^{*})_{0};I=1/2,I_{z}=1/2\rangle$
	$\displaystyle\quad=|N;I_{N}=1/2,I_{zN}=1/2\rangle$
	$\displaystyle\quad\quad\otimes|D\bar{D}^{*};I_{D\bar{D}^{*}}=0,I_{zD\bar{D}^{*}}=0\rangle.$		(14)

Since $t_{1}$ ($t_{2}$) is related to the $ND$ ($N\bar{D}^{*}$) interaction, to evaluate $(t_{1})_{00}$ from Eq. (13) it is convenient to write Eq. (14) in terms of the isospin of the $ND$ system, while to determine $(t_{2})_{00}$ it is useful to express Eq. (14) in terms of the isospin of the $N\bar{D}^{*}$ system. For the former case, using Clebsch-Gordan coefficients, we get

	$\displaystyle|N(D\bar{D}^{*})_{0};I=1/2,I_{z}=1/2\rangle$
	$\displaystyle\quad=\frac{1}{\sqrt{2}}\Big{[}|I_{ND}=1,I_{zND}=1\rangle\otimes\Big{|}I_{\bar{D}^{*}}=\frac{1}{2},I_{z\bar{D}^{*}}=-\frac{1}{2}\Big{\rangle}$
	$\displaystyle\quad\quad-\frac{1}{\sqrt{2}}\Big{(}|I_{ND}=1,I_{zND}=0\rangle+|I_{ND}=0,I_{zND}=0\rangle\Big{)}$
	$\displaystyle\quad\quad\otimes\Big{|}I_{\bar{D}^{*}}=\frac{1}{2},I_{z\bar{D}^{*}}=\frac{1}{2}\Big{\rangle}\Big{]},$		(15)

and using now Eq. (13),

	$\displaystyle(t_{1})_{00}=\langle N(D\bar{D}^{*})_{0};I=1/2,I_{z}=1/2|t_{1}$
	$\displaystyle\quad\quad\quad\quad\times|N(D\bar{D}^{*})_{0};I=1/2,I_{z}=1/2\rangle$
	$\displaystyle\quad=\frac{1}{4}(3t^{1}_{ND}+t^{0}_{ND}).$		(16)

In Eq. (16), $t^{0}_{ND}$ ($t^{1}_{ND}$) corresponds to the two-body $t$-matrix for the $ND\to ND$ transition in isospin 0 (isospin 1). This process has been investigated within different models by solving the Bethe-Salpeter equation in a coupled channel approach (see, for example, Refs. Hofmann:2005sw ; Mizutani:2006vq ; GarciaRecio:2008dp ; Romanets:2012hm ; Liang:2014kra ). All studies point to a common finding: the $DN$, $\pi\Sigma_{c}$ and coupled channel dynamics is attractive and gives rise to the formation of $\Lambda_{c}(2595)$ ($J^{P}=1/2^{-}$). In Refs. Hofmann:2005sw ; Mizutani:2006vq , the pseudoscalar-nucleon interactions with charm $+1$ are deduced from a Lagrangian based on the SU(4) symmetry. In Refs. GarciaRecio:2008dp ; Romanets:2012hm , by using a model based on the SU(8) spin-flavor symmetry, which is compatible with the heavy-quark symmetry, pseudoscalar-baryon as well as vector-baryon channels are considered as coupled systems and generation of several $J^{P}=1/2^{-}$, $3/2^{-}$ $\Lambda_{c}$ and $\Sigma_{c}$ states is reported. In the latter references, $\Lambda_{c}(2595)$ is found to have large couplings to the $DN$ as well as to the $D^{*}N$ channel. The studies in Refs. GarciaRecio:2008dp ; Romanets:2012hm have been updated more recently in Ref. Nieves:2019nol for the $\Lambda_{c}$ sector and it is again concluded that $\Lambda_{c}(2595)$ has a predominant molecular nature. However, it is suggested that $\Lambda_{c}(2625)$ ($J^{P}=3/2^{-}$) should be viewed mostly as a dressed three-quark state. In Ref. Liang:2014kra , within a different formalism based on arguments of heavy-quark and SU(4) symmetries, the $DN$, $\pi\Sigma_{c}$ and other coupled pseudoscalar-baryon channels and the $D^{*}N$, $\rho\Sigma_{c}$, and other coupled vector-baryon channels systems are studied. In this latter work, box diagrams are considered to construct a transition amplitude for the process $DN\leftrightarrow D^{*}N$. Several $\Lambda_{c}$ and $\Sigma_{c}$ states with $J^{P}=1/2^{-}$ and $3/2^{-}$, including $\Lambda_{c}(2595)$, are found. A large coupling of $\Lambda_{c}(2595)$ to $DN$ and $D^{*}N$ is also obtained as in Refs. GarciaRecio:2008dp ; Romanets:2012hm . In the present work, we have considered the models of Refs. GarciaRecio:2008dp ; Liang:2014kra , since in both cases the $DN$ and $D^{*}N$ channels are coupled, which is compatible with the heavy-quark symmetry. However, since the $\Lambda_{c}$ and $\Sigma_{c}$ states obtained in Refs. GarciaRecio:2008dp ; Liang:2014kra are not all same, and many of them have not been observed experimentally yet, when investigating the $ND\bar{D}^{*}$ ($N\bar{D}D^{*}$) system, we focus mainly on the energy region in which $\Lambda_{c}(2595)$ is generated, since all models find similar properties and attribute a molecular nature to it.

Coming back to the discussions on the three-body formalism, proceeding in a similar way as in Eq. (16), we can get the rest of the elements in Eq. (12), for which we need the two-body $t$-matrices for the $N\bar{D}$ and $N\bar{D}^{*}$ systems in isospins 0 and 1. Within the SU(8) model of Refs. GarciaRecio:2008dp ; Romanets:2012hm , in Ref. Gamermann:2010zz the $N\bar{D}$ and $N\bar{D}^{*}$ coupled system has been studied and several $\Lambda_{\bar{c}}$ and $\Sigma_{\bar{c}}$ with $J^{P}=1/2^{-}$ and $3/2^{-}$ have been claimed to get generated with masses in the energy region $2800-3000$ MeV. But, so far, no clear experimental evidence on the existence of such states is available, although the existence of an anticharm baryon with mass around 3000 MeV was claimed in Ref. Aktas:2004qf . However, subsequent experimental investigations have failed to confirm the former finding Aubert:2006qu . Within the model of Ref. Mizutani:2006vq , where $\bar{D}N$ and $\bar{D}^{*}N$ are not considered as coupled channels, we do not find formation of any state since the interaction is repulsive for the charm $-1$ sector. If we extend the model of Ref. Liang:2014kra to the charm $-1$ sector, we find formation of a $\Lambda_{\bar{c}}$ with a mass around 2950 MeV. Due to the different predictions within the models of Refs. Mizutani:2006vq ; Liang:2014kra ; Gamermann:2010zz , and the absence of conclusive experimental evidences in favor of the existence of any states with charm $-1$, we adopt the same strategy as in the charm $+1$ sector. We thus consider $\bar{D}N$ and $\bar{D}^{*}N$ as coupled channels within the models of Refs. Gamermann:2010zz ; Liang:2014kra and restrict ourselves to an energy region in which the findings of the models, including the one of Ref. Mizutani:2006vq , are compatible. In this way, our predictions for three-body states would be consistent with different input two-body amplitudes.

Proceeding further with the discussion of the FCA, in Eq. (12), the propagator $G_{0}$ is given by Xie:2010ig

$\displaystyle(G_{0})_{aa}=\frac{1}{2M_{a}}\int\frac{d^{3}q}{(2\pi)^{3}}\frac{m_{N}}{\omega_{N}(\vec{q})}\frac{F_{a}(\vec{q})}{q^{0}-\omega_{N}(\vec{q})+i\epsilon},$

(17)

with $M_{a}$ being the mass of the cluster formed by $D\bar{D}^{*}$ ($\bar{D}D^{*}$), $m_{N}$ [$\omega_{N}(\vec{q})=\sqrt{\vec{q}^{\,2}+m^{2}_{N}}$] represents the mass (energy) of the nucleon which rescatters off the components of the cluster and $q^{0}$ is the nucleon on-shell energy in the nucleon-cluster rest frame, i.e.,

$\displaystyle q^{0}=\frac{s+m^{2}_{N}-M^{2}_{a}}{2\sqrt{s}},$

(18)

where $\sqrt{s}$ is the center-of-mass energy of the three-body system. The function $F_{a}(\vec{q})$ in Eq. (17) is a form factor related to the wave function of the particles of the cluster and it is given by MartinezTorres:2020hus ; MartinezTorres:2010ax

	$\displaystyle F_{a}(\vec{q})$	$\displaystyle=\frac{1}{\mathbb{N}}\int\limits_{|\vec{p}|,|\vec{p}-\vec{q}|<\Lambda}d^{3}pf_{a}(\vec{p})f_{a}(\vec{p}-\vec{q}),$
	$\displaystyle f_{a}(\vec{p})$	$\displaystyle=\frac{1}{\omega_{a1}(\vec{p})\omega_{a2}(\vec{p})}\frac{1}{M_{a}-\omega_{a1}(\vec{p})-\omega_{a2}(\vec{p})+i\epsilon},$		(19)

where $\omega_{a1(a2)}(\vec{p})=\sqrt{\vec{p}+M^{2}_{a1(a2)}}$ are the energies of the particles in the cluster and $\mathbb{N}$ is a normalization factor, $\mathbb{N}=F_{a}(\vec{q}=0)$. The upper limit of the integration in Eq. (19) is related to the cut-off used when regularizing the two-body loops in the Bethe-Salpeter equation to generate $X(3872)$ or $Z_{c}(3900)$ from the coupled channel interactions. We consider a value for $\Lambda\sim 700$ MeV as in Refs. Aceti:2014uea ; Gamermann:2006nm ; Aceti:2012cb and vary it in a reasonable region to determine the uncertainties involved in the results. The unstable character of $Z_{c}(3900)$ is implemented in the formalism by substituting $M_{a}\to M_{a}-i\frac{\Gamma_{a}}{2}$ in Eq. (19) (a width of 28 MeV is considered in this case). Note that the $1/(2M_{a})$ present in Eq. (17) is a normalization factor whose origin lies in the normalization of the fields when comparing the scattering matrix $S$ for a process in which a particle, in this case, a nucleon, rescatters off particles $\mathbb{C}_{1}$ and $\mathbb{C}_{2}$ of a cluster $\mathbb{C}$ and the $S$-matrix for an effective two-body particle-cluster scattering MartinezTorres:2020hus ; Ren:2018pcd . As a consequence of the normalization of those $S$-matrices,

$\displaystyle(G_{0})_{aa}\to\frac{1}{2M_{a}}(G_{0})_{aa},$

(20)

and a normalization factor needs to be included in the two-body $t$-matrices $t_{1}$ and $t_{2}$ too,

	$\displaystyle(t_{1})_{ab}\to\sqrt{\frac{M_{a}}{M_{1a}}}\sqrt{\frac{M_{b}}{M_{1b}}}(t_{1})_{ab},$		(21)
	$\displaystyle(t_{2})_{ab}\to\sqrt{\frac{M_{a}}{M_{2a}}}\sqrt{\frac{M_{b}}{M_{2b}}}(t_{2})_{ab}.$		(22)

With these ingredients we first solve Eq. (4) and determine the $T$-matrix from Eq. (3) as a function of the center-of-mass energy $\sqrt{s}$ for the $N(D\bar{D}^{*})$ and $N(\bar{D}D^{*})$ systems. We then construct the $T$-matrix of Eq. (2) and search for peaks in $|T|^{2}$, which are identified with three-body states generated from the $NX(3872)/NZ_{c}(3900)$ coupled channel dynamics.

Before ending the discussions, we would like to analyze the uncertainties involved in our findings from additional diagrams beyond those involved in the FCA. In the current study, we consider that $D$ and $\bar{D}^{*}$ form a cluster which stays unperturbed during the scattering. The meaning of such a consideration is that the third hadrons scatters off the constituents of the cluster which act like static sources. Within such an approximation, at the level of one loop, the diagrams besides those shown in Fig. 8 are considered to be suppressed.

In the present case, where the mass of the nucleon is about half of the mass of $D$/$\bar{D}^{*}$, one might wonder if the additional diagrams shown in Fig. 9 can be neglected and if the fixed center approximation is appropriate for the current system. To test the applicability of the approximation, we make explicit calculations of the diagrams in Fig. 9 in this section and show that the resulting contributions turn out to be negligible. We also discuss the reason behind such findings.

Before going to the details of the calculations, we would like to remind the reader that the fixed center or the static approximation has been applied to studies of anti-kaon deuteron scattering in Refs. Kamalov:2000iy ; Chand:1962ec , where the kaon is about half as heavy as the nucleon. Still, results on the scattering length were obtained in good agreement with the experimental data. Indeed contributions from recoil effects were scrutinized in detail in Ref. Baru:2009tx by considering a perturbative expansion in terms of the ratio of the masses $M_{K}/m_{N}$ and corrections of the order of 10-15$\%$ were found. Such a small correction was attributed to cancellations among different terms in the perturbative series. The mentioned cancellations were attributed, in Ref.Baru:2009tx , to the Pauli principle, or to the orthogonality of the deuteron wave function and the $NN$ continuum, depending on the isospin of the $\bar{K}N$ interactions. Similar conclusions were obtained in a later study in Ref. Mai:2014uma too. It should be mentioned that besides the anti-kaon deuteron case, cancellations have been found in the case of the $\pi d$ scattering too Faldt:1974sm , where even though the static approximation may be expected to work well, corrections (from binding energy) turn out to be large when considering each term of the scattering series separately. However, corrections to the different terms end up canceling with each other when the series is summed Faldt:1974sm , rendering the FCA applicable to the system. Interestingly, validity of the FCA was also discussed in Ref. MartinezTorres:2010ax in case of the $\phi K\bar{K}$ system. It was found that the FCA amplitude is not reliable in the former case, as expected, except for energies below the cluster-particle threshold, thereby limiting the prospects of the excitation of the constituents of the cluster. Thus, the static approximation has been found to work in a series of unexpected systems due to different reasons.

It might also be useful to cite examples of some three-hadron systems which have been studied by solving the Faddeev equations with and without the consideration of the static approximation for one of the subsystems. For example, the $NK\bar{K}$ system and coupled channels were studied in Refs. MartinezTorres:2008kh ; MartinezTorres:2010zv by solving the Faddeev equations without invoking the static approximation for any of the pairs. In this case a $1/2^{+}$ state, with mass around 1924 MeV, was found with the width ranging between 20-30 MeV. Similar results have been reported in Ref. Jia:2011zzd , where Faddeev equations were solved using an effective potential for each of the pairs. A state with a mass around $1880-1920$ MeV is obtained in the former work. Further, the same system was studied by treating $\bar{K}N$ and $\bar{K}K$ as fixed scattering centers in Ref. Xie:2010ig and results compatible with those of Refs. MartinezTorres:2008kh ; MartinezTorres:2010zv ; Jia:2011zzd (mass $\sim 1915-1925$ MeV, width $\sim 30-80$ MeV) were found. As shown in Ref. MartinezTorres:2010ax , the condition in which FCA seems to works well is when a three-body system is studied at energies below the threshold, besides having a two-body cluster which is heavier than the third one. Yet another system, $DK\bar{K}$, was studied by solving full Faddeev equations MartinezTorres:2012jr as well as by introducing the FCA Debastiani:2017vhv . A $D$-meson with spin parity $1/2^{-}$, mass around 2900 MeV and with of 55 MeV was found to arise from the three-body interactions in Ref. MartinezTorres:2012jr , whose decay to two mesons has been studied in Ref. Malabarba:2021gyq . Indeed a state with same quantum numbers was found in Ref. Debastiani:2017vhv but with the values of mass (and, consequently, width) about 50 MeV lower than those determined with full Faddeev calculations MartinezTorres:2012jr . However, it must be mentioned that the state found in Ref. MartinezTorres:2012jr appeared in the $Df_{0}(980)$ configuration, while a clear signal was not found in the $D_{s}(2317)\bar{K}$ arrangement of the three-body system. The latter is precisely the configuration studied in Ref. Debastiani:2017vhv in order to use the FCA (although the authors of Ref. Debastiani:2017vhv arrive to the conclusion that $Df_{0}(980)$ is the dominant configuration). From the above discussion one can see that, as far as the energy region studied is below the three-body threshold, the order of uncertainties in the results obtained by using FCA is similar to that found within other methods for solving the Faddeev equations where no static approximations are considered.

Let us now discuss the case of the $ND\bar{D}^{*}$ system by evaluating the amplitudes for the diagrams shown in Fig. 9, where the $D$ and $\bar{D}^{*}$ can propagate as free particles in the loop. Following the same normalization as in Ref. MartinezTorres:2010ax , we can write the contribution to the $S$ matrix, to start with, for the diagram in Fig. 9a as

	$\displaystyle S_{\ref{morediagrams}a}=$	$\displaystyle\sqrt{\frac{2M_{N}}{2k^{0}}}\sqrt{\frac{2M_{N}}{2k^{\prime 0}}}\sqrt{\frac{1}{2p_{1}^{0}}}\sqrt{\frac{1}{2p_{2}^{0}}}\sqrt{\frac{1}{2p_{1}^{\prime 0}}}\sqrt{\frac{1}{2p_{2}^{\prime 0}}}$
		$\displaystyle\quad\times\int d^{4}x_{1}\int d^{4}x_{2}\int d^{4}x_{3}\int\frac{d^{4}q}{\left(2\pi\right)^{4}}$
		$\displaystyle\quad\times\int\frac{d^{4}q^{\prime}}{\left(2\pi\right)^{4}}\int\frac{d^{4}p}{\left(2\pi\right)^{4}}\Bigl{[}-it_{DN}\left(k+p_{1}\right)\Bigr{]}$
		$\displaystyle\quad\times\Bigl{[}-it_{D\bar{D}^{*}}\left(\sqrt{s_{D\bar{D}^{*}}}\right)\Bigr{]}\Bigl{[}-it_{DN}\left(k^{\prime}+p_{1}^{\prime}\right)\Bigr{]}$
		$\displaystyle\quad\times\frac{ie^{iq\left(x_{1}-x_{2}\right)}}{q^{2}-m_{D}^{2}+i\epsilon}\frac{ie^{iq^{\prime}\left(x_{2}-x_{3}\right)}}{q^{\prime 2}-m_{\bar{D}^{*}}^{2}+i\epsilon}$
		$\displaystyle\quad\times\frac{ie^{ip\left(x_{1}-x_{3}\right)}\bar{u}(k^{\prime})\left(\not{p}+m_{N}\right)u(k)}{p^{2}-m_{N}^{2}+i\epsilon}$
		$\displaystyle\quad\times e^{ik^{\prime 0}x_{3}^{0}}e^{ip_{1}^{\prime 0}x_{3}^{0}}e^{ip_{2}^{\prime 0}x_{2}^{0}}e^{-ik^{0}x_{1}^{0}}e^{-ip_{1}^{0}x_{1}^{0}}e^{-ip_{2}^{0}x_{2}^{0}}$
		$\displaystyle\quad\times\frac{1}{\sqrt{V}}e^{-i\vec{k}^{\prime}\cdot\vec{x}_{3}}\phi_{1}(x_{3})\phi_{2}(x_{2})\frac{1}{\sqrt{V}}e^{i\vec{k}\cdot\vec{x}_{1}}\phi_{1}(x_{1})\phi_{2}(x_{2}),$		(23)

where $\phi_{i}(x_{j})$ represent the wave functions of the particles of the cluster in the initial/final state. We refer the reader to Fig. 9a to identify the momenta assigned to each particle. The invariant mass of the $D\bar{D}^{*}$ system, in Eq. (23), depends on a loop variable through

$\displaystyle s_{D\bar{D}^{*}}=s+m_{N}^{2}-2\sqrt{s}~{}\omega_{N}(\vec{p}).$

(24)

Integrating on the zero component of the six variables in Eq. (23) and defining

	$\displaystyle\vec{x}_{1}-\vec{x}_{2}\equiv\vec{r},$		(25)
	$\displaystyle\vec{x}_{3}-\vec{x}_{2}\equiv\vec{r}^{\,\prime},$
	$\displaystyle\vec{R}\equiv\frac{1}{2}\left(\vec{x}_{1}+\vec{x}_{2}\right),$

we can write

$\displaystyle\frac{1}{2}\left(\vec{x}_{3}+\vec{x}_{2}\right)=\vec{R}+\frac{\vec{r}}{2}+\frac{\vec{r}^{\,\prime}}{2}.$

(26)

Such change of variables allows us to write

$\displaystyle\phi_{1}(x_{1})\phi_{2}(x_{2})=\frac{1}{\sqrt{V}}e^{i\vec{P}_{12}\cdot\vec{R}}\phi\left(\,\vec{r}\,\right),$

(27)

and

$\displaystyle\phi_{1}(x_{3})\phi_{2}(x_{2})=\frac{1}{\sqrt{V}}e^{-i\vec{P}^{\prime}_{12}\cdot\vec{R}}\phi\left(\,\vec{r}\,\right)e^{-i\vec{P}^{\prime}_{12}\cdot\vec{r}^{\,\prime}/2}e^{i\vec{P}^{\prime}_{12}\cdot\vec{r}/2}\phi\left(\vec{r}^{\,\prime}\right),$

(28)

where $P_{12}\left(P^{\prime}_{12}\right)$ denotes the momentum of the cluster in the initial (final) state. Finally, integrating on $\vec{r}$, $\vec{r}^{\,\prime}$ and $\vec{R}$ we get

	$\displaystyle S_{\ref{morediagrams}a}=-i\frac{\left(2\pi\right)^{4}\delta^{4}\left(P_{tot}-P^{\prime}_{tot}\right)}{V^{2}}\sqrt{\frac{2M_{N}}{2k^{0}}}\sqrt{\frac{2M_{N}}{2k^{\prime 0}}}$
	$\displaystyle\quad\times\sqrt{\frac{1}{16p_{1}^{0}p_{2}^{0}p_{1}^{\prime 0}p_{2}^{\prime 0}}}\sqrt{\frac{k^{0}+m_{N}}{2m_{N}}}\sqrt{\frac{k^{\prime 0}+m_{N}}{2m_{N}}}$
	$\displaystyle\quad\times t_{DN}\left(k+p_{1}\right)t_{DN}\left(k^{\prime}+p_{1}^{\prime}\right)\int\frac{d^{4}q}{\left(2\pi\right)^{4}}\int\frac{d^{4}q^{\prime}}{\left(2\pi\right)^{4}}$
	$\displaystyle\quad\times\int\frac{d^{4}p}{\left(2\pi\right)^{4}}t_{D\bar{D}^{*}}\left(\sqrt{s_{D\bar{D}^{*}}}\right)\frac{\omega_{N}\left(\vec{p}\right)+m_{N}}{2\omega_{N}\left(\vec{p}\right)}$
	$\displaystyle\quad\times\frac{1}{\left[k^{0}+p_{1}^{0}-\omega_{N}\left(\vec{p}\right)\right]^{2}-\omega\left(\vec{q}\right)^{2}+i\epsilon}$
	$\displaystyle\quad\times\frac{1}{\left[k^{\prime 0}+p_{1}^{\prime 0}-\omega_{N}\left(\vec{p}\right)\right]^{2}-\omega\left(\vec{q}^{\,\prime}\right)^{2}+i\epsilon}$
	$\displaystyle\quad\times\phi_{1}\left(\vec{p}+\vec{q}-\frac{\vec{k}}{2}\right)\phi_{2}\left(\vec{p}+\vec{q}^{\,\prime}-\frac{\vec{k}^{\,\prime}}{2}\right),$		(29)

where $P_{tot}\left(P^{\prime}_{tot}\right)$ represents the total four momentum of the three body system in the initial (final) state and $\phi_{1}$, $\phi_{2}$ are calculated following Ref. Gamermann:2009uq . It must be emphasized here that the different two-body amplitudes get contributions from different isospin with different weights (as explained in section 2). Following the same procedure, we can obtain the amplitudes for the remaining diagrams in Fig. 9 as

	$\displaystyle t_{\ref{morediagrams}b}=t_{\ref{morediagrams}d}=\sqrt{\frac{k^{0}+m_{N}}{2m_{N}}}\sqrt{\frac{k^{\prime 0}+m_{N}}{2m_{N}}}t_{DN}\left(k+p_{1}\right)$
	$\displaystyle\quad\times t_{\bar{D}^{*}N}\left(k^{\prime}+p_{2}^{\prime}\right)\int\frac{d^{4}q}{\left(2\pi\right)^{4}}\int\frac{d^{4}q^{\prime}}{\left(2\pi\right)^{4}}\int\frac{d^{4}p}{\left(2\pi\right)^{4}}$
	$\displaystyle\quad\times\frac{\omega_{N}\left(\vec{p}\right)+m_{N}}{2\omega_{N}\left(\vec{p}\right)}\frac{t_{D\bar{D}^{*}}\left(\sqrt{s_{D\bar{D}^{*}}}\right)}{\left[k^{0}+p_{1}^{0}-\omega_{N}\left(\vec{p}\right)\right]^{2}-\omega\left(\vec{q}\right)^{2}+i\epsilon}$
	$\displaystyle\quad\times\frac{1}{\left[k^{\prime 0}+p_{2}^{\prime 0}-\omega_{N}\left(\vec{p}\right)\right]^{2}-\omega\left(\vec{q}^{\,\prime}\right)^{2}+i\epsilon}$
	$\displaystyle\quad\times\phi_{1}\left(\vec{p}+\vec{q}-\frac{\vec{k}}{2}\right)\phi_{2}\left(\vec{p}+\vec{q}^{\,\prime}-\frac{\vec{k}^{\,\prime}}{2}\right),$		(30)

and

	$\displaystyle t_{\ref{morediagrams}c}=\sqrt{\frac{k^{0}+m_{N}}{2m_{N}}}\sqrt{\frac{k^{\prime 0}+m_{N}}{2m_{N}}}t_{\bar{D}^{*}N}\left(k+p_{2}\right)t_{\bar{D}^{*}N}\left(k^{\prime}+p_{2}^{\prime}\right)$
	$\displaystyle\quad\times\int\frac{d^{4}q}{\left(2\pi\right)^{4}}\int\frac{d^{4}q^{\prime}}{\left(2\pi\right)^{4}}\int\frac{d^{4}p}{\left(2\pi\right)^{4}}t_{D\bar{D}^{*}}\left(\sqrt{s_{D\bar{D}^{*}}}\right)$
	$\displaystyle\quad\times\frac{\omega_{N}\left(\vec{p}\right)+m_{N}}{2\omega_{N}\left(\vec{p}\right)}\frac{1}{\left[k^{0}+p_{2}^{0}-\omega_{N}\left(\vec{p}\right)\right]^{2}-\omega\left(\vec{q}\right)^{2}+i\epsilon}$
	$\displaystyle\quad\times\frac{1}{\left[k^{\prime 0}+p_{2}^{\prime 0}-\omega_{N}\left(\vec{p}\right)\right]^{2}-\omega\left(\vec{q}^{\,\prime}\right)^{2}+i\epsilon}$		(31)
	$\displaystyle\quad\times\phi_{1}\left(\vec{p}+\vec{q}-\frac{\vec{k}}{2}\right)\phi_{2}\left(\vec{p}+\vec{q}^{\,\prime}-\frac{\vec{k}^{\,\prime}}{2}\right).$		(32)