Study of the $DKK$ and $DK\bar{K}$ systems

TL;DR

This study uses the Fixed Center Approximation to Faddeev equations to explore three-body systems involving D, K, and anti-K mesons, predicting a new state near certain thresholds.

Contribution

It provides the first evidence of a $DKar{K}$-generated state with specific quantum numbers and mass, while finding no such state in the $DKK$ system.

Findings

Identified a $I(J^P)=1/2(0^-)$ state around 2833-2858 MeV

State is primarily composed of $Df_0(980)$

No evidence of a state from $DKK$ interaction

Abstract

Using the Fixed Center Approximation to Faddeev equations we have investigated the $DKK$ and $DK\bar{K}$ three-body systems, considering that the $DK$ dynamically generates, through its $I=0$ component, the $D^*_{s0}(2317)$ molecule. According to our findings, for $DK\bar{K}$ interaction we have found an evidence of a state $I(J^P)=1/2(0^-)$ just above the $D^*_{s0}(2317)\bar{K}$ threshold and around the $Df_0(980)$ thresholds, with mass about $2833 - 2858$ MeV, made mostly of $Df_0(980)$. On the other hand, no evidence related to a state from the $DKK$ interaction is found. The state found could be seen in the $\pi \pi D$ invariant mass.

Tables

Table 1. Table 1: Comparison between position and intensity of the peaks found in the D ​ K ​ K ¯ 𝐷 𝐾 ¯ 𝐾 DK\bar{K} amplitude.

	Prescription I    Prescription II
	$\sqrt{s}$	${\left|T\right|}^2$	$\sqrt{s}$	${\left|T\right|}^2$
Total	$2833$	$6.8\times {10}^6$	$2858$	$1.8\times {\mathrm{\hspace{0.17em}10}}^7$
$I=1$ only	$2842$	$7.7\times {\mathrm{\hspace{0.17em}10}}^4$	$2886$	$7.8\times {\mathrm{\hspace{0.17em}10}}^4$

Equations

$$T_{11}^{\mathrm{FCA}}(s)=t_1+t_1{G}_0{T}_{41}^{\mathrm{FCA}}+t_2{G}_0{T}_{61}^{\mathrm{FCA}},$$

$$G_0(s)=\frac{1}{2M_{D_{s0}^{\ast }}}\int \frac{d^3\mathbf{q}}{(2\pi {)}^3}\frac{F_R(\mathbf{q})}{(q^0{)}^2-{\omega }_K^2(\mathbf{q})+iϵ},$$

$$q^0(s)=\frac{s-m_K^2-M_{D_{s0}^{\ast }}^2}{2M_{D_{s0}^{\ast }}}.$$

$$F_R(\mathbf{q})=\frac{1}{N}\underset{|\mathbf{p}|,|\mathbf{p}-\mathbf{q}|<\Lambda }{\int }{d}^3\mathbf{p}\frac{1}{M_{D_{s0}^{\ast }}-{\omega }_D(\mathbf{p})-{\omega }_K(\mathbf{p})}\frac{1}{M_{D_{s0}^{\ast }}-{\omega }_D(\mathbf{p}-\mathbf{q})-{\omega }_K(\mathbf{p}-\mathbf{q})},$$

$$\displaystyle N=\int\limits_{|{\bf p}|<\Lambda}\,d^{3}{\bf p}\,\Big{(}\,\frac{1}{M_{D^{*}_{s0}}-\omega_{D}({\bf p})-\omega_{K}({\bf p})}\Big{)}^{2}\,.$$

$$T_{ij}^{\mathrm{FCA}}(s)=V_{ij}^{\mathrm{FCA}}(s)+\sum _{l=1}^6{\tilde{V}}_{il}^{\mathrm{FCA}}(s)G_0(s)T_{lj}^{\mathrm{FCA}}(s),$$

$$V^{\rm FCA}=\left(\begin{array}[]{@{\,}cccccc@{\,}}t_{1}&0&t_{2}&0&0&0\\ 0&t_{3}&0&0&0&0\\ t_{2}&0&t_{4}&0&0&0\\ 0&0&0&t_{5}&0&0\\ 0&0&0&0&t_{6}&t_{7}\\ 0&0&0&0&t_{7}&t_{8}\\ \end{array}\right),\quad\tilde{V}^{\rm FCA}=\left(\begin{array}[]{@{\,}cccccc@{\,}}0&0&0&t_{1}&0&t_{2}\\ 0&0&0&0&t_{3}&0\\ 0&0&0&t_{2}&0&t_{4}\\ t_{5}&0&0&0&0&0\\ 0&t_{6}&t_{7}&0&0&0\\ 0&t_{7}&t_{8}&0&0&0\\ \end{array}\right).$$

$$\displaystyle T^{\rm FCA}_{ij}(s)=\sum\limits_{l=1}^{6}\Big{[}\,1-\tilde{V}^{\rm FCA}(s)\,G_{0}(s)\,\Big{]}^{-1}_{il}\,V_{lj}^{\rm FCA}(s)\,.$$

$$V^{\rm FCA}=\left(\begin{array}[]{@{\,}cccccc@{\,}}\bar{t}_{1}&0&0&0&0&0\\ 0&\bar{t}_{2}&\bar{t}_{3}&0&0&0\\ 0&\bar{t}_{3}&\bar{t}_{4}&0&0&0\\ 0&0&0&\bar{t}_{5}&0&\bar{t}_{5}\\ 0&0&0&0&\bar{t}_{6}&0\\ 0&0&0&\bar{t}_{5}&0&\bar{t}_{5}\\ \end{array}\right),\quad\tilde{V}^{\rm FCA}=\left(\begin{array}[]{@{\,}cccccc@{\,}}0&0&0&\bar{t}_{1}&0&0\\ 0&0&0&0&\bar{t}_{2}&\bar{t}_{3}\\ 0&0&0&0&\bar{t}_{3}&\bar{t}_{4}\\ \bar{t}_{5}&0&\bar{t}_{5}&0&0&0\\ 0&\bar{t}_{6}&0&0&0&0\\ \bar{t}_{5}&0&\bar{t}_{5}&0&0&0\\ \end{array}\right).$$

$$\begin{array}[]{@{\,}cc@{\,}}t_{1}=t_{D^{+}K^{-}\to D^{+}K^{-}}\,;&t_{4}=t_{D^{0}\bar{K}^{0}\to D^{0}\bar{K}^{0}}\,;\\ t_{2}=t_{D^{+}K^{-}\to D^{0}\bar{K}^{0}}\,;&t_{5}=t_{K^{0}K^{-}\to K^{0}K^{-}}\,;\\ t_{3}=t_{D^{0}K^{-}\to D^{0}K^{-}}\,;&t_{6}=t_{K^{+}K^{-}\to K^{+}K^{-}}\,;\\ \end{array}\quad\begin{array}[]{@{\,}c@{\,}}t_{7}=t_{K^{+}K^{-}\to K^{0}\bar{K}^{0}}\,;\\ t_{8}=t_{K^{0}\bar{K}^{0}\to K^{0}\bar{K}^{0}}\,,\\ \end{array}$$

$$\begin{array}[]{@{\,}cc@{\,}}\bar{t}_{1}=t_{D^{+}K^{+}\to D^{+}K^{+}}\,;&\bar{t}_{4}=t_{D^{+}K^{0}\to D^{+}K^{0}}\,;\\ \bar{t}_{2}=t_{D^{0}K^{+}\to D^{0}K^{+}}\,;&\bar{t}_{5}=t_{K^{+}K^{0}\to K^{+}K^{0}}\,;\\ \bar{t}_{3}=t_{D^{0}K^{+}\to D^{+}K^{0}}\,;&\bar{t}_{6}=t_{K^{+}K^{+}\to K^{+}K^{+}}\,,\\ \end{array}$$

$$T_{X-D_{s0}^{\ast }}$$

$$t=(1-vG{)}^{-1}v,$$

$$G(s_i)$$

$$G(s_l)=\int \frac{d^3\mathbf{q}}{(2\pi {)}^3}\frac{{\omega }_1(\mathbf{q})+{\omega }_2(\mathbf{q})}{2{\omega }_1(\mathbf{q}){\omega }_2(\mathbf{q})}\frac{1}{(P^0{)}^2-[{\omega }_1(\mathbf{q})+{\omega }_2(\mathbf{q}){]}^2+iϵ},$$

$${\mathcal{L}}_2=\frac{1}{12f_{\pi }^2}⟨({\partial }_{\mu }\Phi \Phi -\Phi {\partial }_{\mu }\Phi {)}^2+M{\Phi }^4⟩,$$

$$\Phi=\left(\begin{array}[]{@{\,}ccc@{\,}}\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta_{8}}{\sqrt{6}}&\pi^{+}&K^{+}\\ \pi^{-}&-\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta_{8}}{\sqrt{6}}&K^{0}\\ K^{-}&\bar{K}^{0}&-\frac{2\,\eta_{8}}{\sqrt{6}}\\ \end{array}\right)\,;\quad M=\left(\begin{array}[]{@{\,}ccc@{\,}}m^{2}_{\pi}&0&0\\ 0&m^{2}_{\pi}&0\\ 0&0&2\,m^{2}_{K}-m^{2}_{\pi}\\ \end{array}\right)\,,$$

$$\displaystyle\begin{aligned} v_{K^{+}K^{0}\,,\,K^{+}K^{0}}&=\displaystyle\frac{1}{2f^{2}_{\pi}}\Big{(}s_{KK}-2m^{2}_{K}\Big{)}\,;\\ \vspace{15pt}v_{K^{+}K^{+}\,,\,K^{+}K^{+}}&=\displaystyle\frac{1}{f^{2}_{\pi}}\Big{(}s_{KK}-2m^{2}_{K}\Big{)}\,,\\ \end{aligned}$$

$$s_{DK(D\bar{K})}=m_K^2+m_D^2+\frac{1}{2M_{D_{s0}^{\ast }}^2}(s-m_K^2-M_{D_{s0}^{\ast }}^2)(M_{D_{s0}^{\ast }}^2+m_D^2-m_K^2),$$

$$s_{KK(K\bar{K})}=2m_K^2+\frac{1}{2M_{D_{s0}^{\ast }}^2}(s-m_K^2-M_{D_{s0}^{\ast }}^2)(M_{D_{s0}^{\ast }}^2+m_K^2-m_D^2).$$

$$\displaystyle s_{DK(D\bar{K})}=\Big{(}\frac{\sqrt{s}}{M_{D^{*}_{s0}}+m_{K}}\Big{)}^{2}\,\Big{(}m_{K}+\frac{m_{D}\,M_{D^{*}_{s0}}}{m_{D}+m_{K}}\Big{)}^{2}-{\bf P}^{2}_{2}\,,$$

$$\displaystyle s_{KK(K\bar{K})}=\Big{(}\frac{\sqrt{s}}{M_{D^{*}_{s0}}+m_{K}}\Big{)}^{2}\,\Big{(}m_{K}+\frac{m_{K}\,M_{D^{*}_{s0}}}{m_{D}+m_{K}}\Big{)}^{2}-{\bf P}^{2}_{1}\,,$$

Full text

I Introduction

The study of three-body systems is one of the starting points in the study of nuclei and nuclear dynamics. The traditional Quantum Mechanical approach to this problem is based on the Faddeev equations faddeev and the main application was done for three nucleons systems. The simplicity of the Faddeev equations is deceiving since in practice its evaluation is very involved and one approximation or another is done to solve them. One popular choice is the use of separable potentials to construct the two-body scattering amplitudes via the Alt-Grassberger-Sandhas (AGS) form of the Faddeev equations Alt:1967fx . Incorporation of chiral symmetry into the scheme has lead to interesting developments Epelbaum:2005pn . Another way to tackle these three-body systems is using a variational method Hiyama:2003cu ; Dote:2008hw ; Kanada-Enyo:2008wsu . Gradually, other systems involving not only nucleons or hyperons but mesons were tackled. The interaction of $K^{-}d$ at threshold was thoroughly investigated using Faddeev equations Toker:1981zh ; Torres:1986mr , or approximations to it, basically the Fixed Center Approximation (FCA) Kamalov:2000iy . The investigation of a possible state of $K^{-}pp$ nature has also received much attention Dote:2008in ; Shevchenko:2006xy ; Ikeda:2007nz ; Bayar:2011qj ; Bayar:2012hn ; Uchino:2011jt ; bicudo and, according to the calculations done in Ref. sekiramos , the recent J-PARC experiment jparc has found support for this state.

Another step in this direction was the investigation of systems with two mesons and one baryon. Surprisingly it was found in Refs. alber1 ; alber2 ; alber3 that with such systems one could obtain the low energy baryon states of $J^{P}=1/2^{+}$. Work in this direction with different methods was also done in Ref. kbarkbarn for the $\bar{K}\bar{K}N$ system and in Ref. kkbarn for the $K\bar{K}N$ system. In this latter case a bound system developed, giving rise to a $N^{*}$ state around 1920 MeV, mostly made of a $Na_{0}(980)$, which was also predicted in Ref. alber3 .

Systems of three mesons also followed, and in Ref. albermeson the $\phi K\bar{K}$ system was studied and shown to reproduce the properties of the $\phi(2170)$. Similarly, in Ref. kkkbar the $KK\bar{K}$ system is studied and a bound cluster found is associated to the $K(1460)$. Another similar system, the $\pi K\bar{K}$ is studied in Ref. albereta and the state found is associated to the $\pi(1300)$. The $\eta K\bar{K}$ and $\eta^{\prime}K\bar{K}$ systems are also studied in Refs. albereta ; xiaoliang ; ollereta and they are revised in Ref. albernew with the full Faddeev equations and more solid results.

An important result was found in Refs. alber1 ; alber2 ; alber3 ; albermeson . In the Faddeev equations one uses input from the two-body amplitudes of the different components and the off-shell part of the amplitudes appears in the calculations. This off-shell part is unphysical and observables cannot depend upon it. The finding in those works was that the use of chiral Lagrangians provides three-body contact terms that cancel the off-shell two-body contributions. In other calculations empirical three-body forces are introduced which might have some genuine part, but an important part of it will serve the purpose of effectively cancelling these unphysical off-shell contributions. Rather than putting these terms empirically, and fitting them to some data, the message of those works is that to make predictions it is safer to use as input only on-shell two-body amplitudes, without extra three-body terms, and an example of it is given in Ref. alber3 .

Extension to the charm sector was also done. The $DNN$ system, analogous to the $\bar{K}NN$ system is studied in Ref. japocola , and the $NDK$, $\bar{K}DN$ and $ND\bar{D}$ molecules are studied in Ref. Xiao:2011rc . In the hidden charm sector a resonance is found for the $J/\psi K\bar{K}$ system which is associated to the $Y(4260)$ in Ref. alberdani . Closer to our work is the one of Ref. albermari where the $DK\bar{K}$ is studied using QCD sum rules and Faddeev equations and in both methods a state coupling strongly to $Df_{0}(980)$ is found. We will study this system with a different method, and in addition the $DKK$ system.

The former review of work done shows a constant feature, which is that systems that add $K\bar{K}$ to another particle turn out to generate states in which the $K\bar{K}$ clusters around the $f_{0}(980)$ or the $a_{0}(980)$. The $DKK$ system benefits from the $DK$ attraction that forms the $D_{s0}^{*}(2317)$ according to works using chiral Lagrangians and unitary approach kolo ; hofmann ; chiang ; danids ; hanhart1 ; hanhart2 . It is also supported by analysis of lattice QCD data sasa . However, the $KK$ interaction is repulsive and the system might not bind. On the other hand, the $DK\bar{K}$ system has repulsion for $D\bar{K}$ in $I=1$, and attraction for $I=0$, and the $DK$ interaction is attractive, as it is also the $K\bar{K}$. Altogether this latter system could have more chances to bind than the $DKK$ system, but a detailed calculation is called for to find the answer, and this is the purpose of the present work.

The starting point of our approach is to use the FCA with a preexisting molecule, which is the $D_{s0}^{*}(2317)$, formed from the $DK$ interaction. On top of that, another $K$ (or $\bar{K}$) is introduced which is allowed to undergo multiple scattering with the $D$ and $K$ components of the molecule. The resulting thing, as we shall see, is that in the $DKK$ system we do not see a signal of a three-body bound state, but in the $DK\bar{K}$ system we find a peak which we interpret as the $K\bar{K}$ fusing to produce the $f_{0}(980)$ which gets then bound to the $D$ meson, and a narrow peak appears at an energy below the $Df_{0}(980)$ threshold. Such state could be seen in the $\pi\pi D$ invariant mass.

II Formalism

The Fixed Center Approximation (FCA) to Faddeev equations is useful when a light hadron $H_{3}$ interacts with a cluster $H$ composed of two other hadrons $H_{1}$ and $H_{2}$, $H[H_{1}\,H_{2}]$, which are heavier than the first one, i.e. $M_{(H[H_{1}\,H_{2}])}>M_{H_{3}}$. This cluster comes out from the two-body interaction between the hadrons $H_{1}$ and $H_{2}$ that can be described using a chiral unitary approach in coupled channels. Hence, the Faddeev equations in this approximation have as an input the two-body $t$ matrices for the different pairs of mesons which form the system and, in this way the generated bound states and resonances are encoded. In our case, we have $H_{1}=D$ and $H_{2}=K$ while $H_{3}=\bar{K}$ if we consider the $DK\bar{K}$ interaction or $H_{3}=K$ for the $DKK$ system. Both three-body interactions involve the $D^{*}_{s0}(2317)$ and $f_{0}(980)/a_{0}(980)$ molecules that, according to Refs. danids ; oller are dynamically generated through $DK$ and $K\bar{K}$ interactions, respectively, taking into account their associated coupled channels. Therefore, we shall have the following channels contributing to the three-body interaction systems we are concerned: (1) $K^{-}[D^{+}K^{0}]$, (2) $K^{-}[D^{0}K^{+}]$, (3) $\bar{K}^{0}[D^{0}K^{0}]$, (4) $[D^{+}K^{0}]K^{-}$, (5) $[D^{0}K^{+}]K^{-}$ and (6) $[D^{0}K^{0}]\bar{K}^{0}$ for the $DK\bar{K}$ interaction and (1) $K^{+}[D^{+}K^{0}]$, (2) $K^{+}[D^{0}K^{+}]$, (3) $K^{0}[D^{+}K^{+}]$, (4) $[D^{+}K^{0}]K^{+}$, (5) $[D^{0}K^{+}]K^{+}$ and (6) $[D^{+}K^{+}]K^{0}$ for $DKK$ system. Note that the states (1), (2) and (3) are the same as (4), (5) and (6), respectively. Their distinction is to signify that the interaction in the FCA formalism occurs with the particle outside the cluster, which is represented by the brackets $[\,.\,.\,.]$, and the particle of the cluster next to it. This allows for a compact formulation that describes all the charge exchange steps and distinguishes the interaction with the right or left component of the cluster sekiramos . These channels will contribute to the $T_{DK\bar{K}}$ and $T_{DKK}$ three-body scattering matrices and, if those interactions generate bound states or resonances, they will manifest as a pole in the solutions of the Faddeev equations. In what follows we shall discuss how to construct these three-body scattering matrices and its solution for both, the $DK\bar{K}$ and $DKK$ systems.

II.1 $DK\bar{K}$ and $DK\bar{K}$ three-body systems

In order to write the contributions to Faddeev equations of all the channels mentioned previously, we shall adopt the following procedure to construct the relevant amplitudes: for each channel the anti-kaon (kaon) meson to the left side in (1), (2) and (3) interacts with the hadron to its right side. Similarly, for the (4), (5) and (6) the $K$ or $\bar{K}$ to the right interacts with the particle to its left. In doing so, we can distinguish the order of the anti-kaon (kaon) and two other mesons with which the anti-kaon (kaon) interacts first and last. This procedure is similar to that used in Ref. sekiramos to study the $\bar{K}NN$ interaction. For instance, in the $DK\bar{K}$ system, the channel (1) $K^{-}[D^{+}K^{0}]$ in the initial state, means that the $K^{-}$ interacts with the $D^{+}$ meson to its right. The channel (4) $[D^{+}K^{0}]K^{-}$ indicates that the $K^{-}$ interacts with the $K^{0}$ to its left. This procedure allows us to divide the multiple anti-kaon (kaon) scattering process in such a way that the formulation of the multiple scattering becomes easier.

In order to illustrate the structure of the multiple scattering in the fixed center approximation we define the partition functions $T^{\rm FCA}_{ij}$, which contain all possible intermediate multiple steps, where the first index refers to the initial $\bar{K}[DK]$, (1), (2) and (3) or $[DK]\bar{K}$ (4), (5) and (6) states and the second index to the final state. If we consider the $K^{-}[D^{+}K^{0}]\to K^{-}[D^{+}K^{0}]$ amplitude denoted by $T^{\rm FCA}_{11}$, which is diagramatically represented in Fig. 1, it provides the following expression Roca:2010tf ; sekiramos

[TABLE]

which tells us that the transition between the $K^{-}[D^{+}K^{0}]$ to itself is given in terms of a single and double scattering, coupled to the amplitudes $T^{\rm FCA}_{ij}$ related to the other channels. As a result, the three-body problem is given in terms of the $T^{\rm FCA}_{ij}$ partitions, where the $i,\,j$ indices run from 1 to 6 and stand for the initial and final channels, respectively, and as we will discuss later, can be displayed in a matrix form.

In Eq. (1), $s$ is the Mandelstam variable that is equal to the squared of the three-body energy system, while $t_{1}$ and $t_{2}$ are, respectively, the $D^{+}K^{-}\to D^{+}K^{-}$ and $D^{+}K^{-}\to D^{0}\bar{K}^{0}$ two-body scattering amplitudes studied in Ref. danids , in which the authors have applied the chiral unitary approach in coupled channels to investigate the $D\bar{K}$ and $DK$ two-body interaction. $G_{0}$ is the kaon propagator yamafix between the particles of the cluster, which is evaluated through the equation below

[TABLE]

with $\omega^{2}_{K}({\bf q})\equiv{\bf q}^{2}+m^{2}_{K}$ and $q^{0}$ is the energy carried by kaon meson in the cluster rest frame where $F({\bf q})$ is calculated, which corresponds to the following expression

[TABLE]

In this work, we are using the isospin symmetric masses such that $m_{D}$ and $m_{K}$ are the $D$ and $K$ mesons average masses, respectively, while $M_{D^{*}_{s0}}$ is the $D^{*}_{s0}$ molecule mass. This molecule dynamics does not come into play explicitly in our formalism. The information on the molecule is encoded in the function $F_{R}({\bf q})$ appearing in Eq. (2), the form factor, which is related to the cluster wave function by a Fourier transform, as discussed in Refs. Roca:2010tf ; YamagataSekihara:2010pj . According to these works, for the form factor to be used consistently, the theory that generates the bound states and resonances (clusters), the chiral unitary approach, which is developed for scattering amplitudes, has to be extended to wave functions. This was done in those references for $s$-wave bound states, $s$-wave resonant states as well as in states with arbitrary angular momentum acetiwf . In our work we need the form factor expression only for $s$-wave bound states, which is given by Roca:2010tf

[TABLE]

where $\omega_{D}({\bf p})\equiv\sqrt{{\bf p}^{2}+m^{2}_{D}}$ and the normalization factor $N$ is

[TABLE]

The upper integration limit $\Lambda$ has the same value of the cut-off used to regularize the loop $DK$, adjusted in order to get the $D^{*}_{s0}(2317)$ molecule from the $DK$ interaction.

Analogously to $T^{\rm FCA}_{11}$ expressed in Eq. (1), we can calculate all the relevant multiple scattering amplitudes, the partitions $T^{\rm FCA}_{ij}$, using similar diagrams like the one in Fig. 1. As a result, they can be written as

[TABLE]

where $V_{ij}$ and $\tilde{V}_{il}$ are the elements of the matrices below

[TABLE]

Therefore, according to Eq. (6), in our case we can solve the three-body problem in terms of the multiple scattering amplitudes given by partitions $T^{\rm FCA}_{ij}$, which contain only the $D\bar{K}$ and $K\bar{K}$ two-body amplitudes. Thus, for the $DK\bar{K}$ system the solution of the scattering equation, Eq. (6), will be

[TABLE]

Analogously, for the $DKK$ system, we will have the same solution as in Eq. (8). However, in this case, the $\tilde{V}^{\rm FCA}$ and $V^{\rm FCA}$ matrices, in terms of the $DK$ and $KK$ two-body amplitudes, are now given by

[TABLE]

The elements of the matrices in Eqs. (7) and (9), i.e. $t_{1},\,t_{2},\,.\,.\,.\,,t_{8}$ and $\bar{t}_{1},\,.\,.\,.,\,\bar{t}_{6}$ related to the three-body interaction $DK\bar{K}$ and $DKK$ systems are the two-body scattering matrices elements, respectively, given by

[TABLE]

and

[TABLE]

which we shall discuss in the next subsection.

It is important to mention that, in this work, we are using the Mandl and Shaw normalization, which has different weight factors for the particle fields. In order to use these factors in a consistent manner in our problem, we should take into account how they appear in the single-scattering and double-scattering as well as in the full amplitude. The detailed calculation on how to do this can be found in Refs. Roca:2010tf ; YamagataSekihara:2010pj ; yamafix . According to these works, this is done multiplying the two-body amplitudes by the factor $M_{c}/M_{1(2)}$, where $M_{c}$ is the cluster mass while $M_{1(2)}$ is associated with the mass of the hadrons $H_{1}$ and $H_{2}$. In our case, we have $M_{c}/M_{D}$ for the two-body amplitudes related to the $D\bar{K}(DK)$ and $M_{c}/M_{K}$ for the one related to the $K\bar{K}(KK)$ appearing in Eqs. (10) and (11).

Once we solve the Faddeev equations for the systems we are concerned, we have to write this solution in such a way that it represents the amplitude of a $\bar{K}\,(K)$ meson interacting with the $D^{*}_{s0}$ molecule, which is the $DK$ cluster written into an $I=0$ combination. Taking into account that $|DK(I=0)\,\rangle=(1/\sqrt{2})\,|\,D^{+}K^{0}+D^{0}K^{+}\,\rangle$ (recall $(D^{+},-D^{0})$ is the isospin doublet), and summing the cases where the odd $\bar{K}\,(K)$ interacts first to the left (right) of the cluster, and finishes interacting at the left (right) we obtain the following combination for both $DK\bar{K}$ and $DKK$ system,

[TABLE]

where $X$ denotes a $\bar{K}$ in the $DK\bar{K}$ case and a $K$ meson for $DKK$ interaction.

II.2 Two-body amplitudes

In order to solve the Faddeev equations using the FCA for the systems we are concerned, we need to know the two-body scattering amplitudes appearing in Eqs. (10) and (11). They were studied in Refs. danids ; oller . These amplitudes are calculated using the chiral unitary approach (for a review see Oller:2000ma ). In this model, the transition amplitudes between the different pairs of mesons are extracted from Lagrangians based on symmetries as chiral and heavy quark symmetries. Then, they are unitarized using them as the kernels of the Bethe-Salpeter equation, which in its on-shell factorization form is given by

[TABLE]

where $G$ is the two meson loop function and its expression in dimensional regularization method is

[TABLE]

with $m_{1}$ and $m_{2}$ standing for the $i$-channel meson masses in the loop and $p$ is the three-momentum in the two meson center-of-mass energy, $\sqrt{s_{i}}$. In Eq. (14) $\mu$ is a scale fixed a priori and the subtraction constant $\alpha(\mu)$ is a free parameter. In Ref. danids , $\mu$ is considered to be equal to $1500$ MeV for the $D\bar{K}$ system, corresponding to $\alpha_{D\bar{K}}=-1.15$. On the other hand, since the amount of $DK$ content in $D^{*}_{s0}(2317)$ is about $70\%$ sasa , we consider just one channel, with $\alpha_{DK}=-0.925$, adjusted to provide the $D^{*}_{s0}(2317)$ peak, corresponding to a cut-off value equal to $650$ MeV. This value also has to be used as the upper limit in the integrals given by Eqs. (4) and (5). For the $f_{0}(980)/a_{0}(980)$ we consider the same channels as Refs. xiedai ; Dias:2016gou where a cut-off equal to $600$ MeV was used to regularize the loops, given by

[TABLE]

where $(P^{0})^{2}=s_{l}$, the two-body center-of-mass energy squared. The index $l$ stands for the following channels: 1) $\pi^{+}\pi^{-}$, 2) $\pi^{0}\pi^{0}$, 3) $K^{+}K^{-}$, 4) $K^{0}\bar{K}^{0}$, 5) $\eta\eta$ and 6) $\pi\eta$. In each channel $\omega_{1(2)}({\bf q})=\sqrt{{\bf q}^{2}+m_{1(2)}^{2}}\,$, where $m_{1(2)}$ is the mass of the mesons inside the loop.

In order to get the scattering amplitude for the $KK$ interaction, we follow Ref. oller . First, we have to find the kernel $v$ to be used in Eq. (13). This kernel is the lowest order amplitude describing the $KK$ interaction and it is calculated using the chiral Lagrangian

[TABLE]

where $\langle\,.\,.\,.\,\rangle$ means the trace in the flavour space of the $SU(3)$ matrices appearing in $\Phi$ and $M$ while $f_{\pi}$ is the pion decay constant. The matrices $\Phi$ and $M$ are given by

[TABLE]

where in $M$ we have taken the isospin limit ($m_{u}=m_{d}$). Hence, from Eqs. (16) and (17) we can calculate the tree level amplitudes for $K^{+}K^{0}$ and $K^{+}K^{+}$, which after projection in $s$-wave read as

[TABLE]

where $s_{KK}$ is the Mandelstam variable $s$ in the $KK$ center-of-mass frame. From these equations one finds that $v^{I=0}_{KK}=0$ (and $t^{I=0}_{KK}=0$) and taking the unitary normalization appropriate for identical particles $|K^{+}K^{+},I=1\rangle=|K^{+}K^{+}\rangle/\sqrt{2}$, we find $v^{I=1}_{KK}=\frac{1}{2}v_{K^{+}K^{+}\,,\,K^{+}K^{+}}$. The $t$ matrix will be $t^{I=1}_{KK}=(1-v^{I=1}_{KK}\,G_{KK})^{-1}\,v^{I=1}_{KK}$, and then $t^{I=1}_{KK}$ has to be multiplied by two to restore the good normalization. Therefore, using these expressions we obtain the $KK$ scattering amplitudes $\bar{t}_{5}$ and $\bar{t}_{6}$ present in Eq. (11) ($\bar{t}_{6}=t^{I=1}_{KK}$, $\bar{t}_{5}=\frac{1}{2}t^{I=1}_{KK}$, with $t^{I=1}_{KK}$ with the good normalization), where we have used a cut-off of $600$ MeV to regularize the $KK$ loops, the same that was used in the $K\bar{K}$ and coupled channels system. After these considerations we are able to determine all the two-body amplitudes in Eqs. (10) and (11).

It is worth mentioning that the arguments of the partitions $T^{\rm FCA}_{ij}(s)$ and the $t_{i}(s_{i})$ two-body amplitudes are different. While the former is written into the three-body center-of-mass energy $\sqrt{s}$, the latter is given in the two-body one. In order to write the $\sqrt{s_{i}}$’s in terms of $\sqrt{s}$, we are going to use the same transformations used in Ref. pedro ; yamafix , which are

[TABLE]

where the subscript $DK(D\bar{K})$ stands for the two-body channels associated with the energy in the center-of-mass of $DK(D\bar{K})$. Analogously, for the energy in the $KK(K\bar{K}$) center-of-mass, we have

[TABLE]

In this work, we are going to call this set of transformations Prescription I. In order to estimate the uncertainties in our calculations, we will also use another set of transformations, which we are going call Prescription II, given by

[TABLE]

and

[TABLE]

where ${\bf P}_{1}$ and ${\bf P}_{2}$ stand for the momenta of the $D$ and $K$ mesons in the cluster, which we take equal and such that the kinetic energy in the $DK$ cluster is of the order of the binding energy, hence ${\bf P}^{2}_{1}={\bf P}^{2}_{2}=2\tilde{\mu}B_{D^{*}_{s0}}=2\tilde{\mu}\,(m_{D}+m_{K}-M_{D^{*}_{s0}})$, with $\tilde{\mu}$ the reduced mass of $DK$. This prescription is based on another one discussed in Refs. pedro ; bayarren , which shares the binding energy among the three-particles proportionally to their respective masses.

III Results

In all our calculations we use $m_{K}=495$ MeV, $m_{D}=1865$ MeV, $m_{D_{s0}^{*}(2317)}=2317$ MeV, $m_{\pi}=138$ MeV, $m_{\eta}=548$ MeV and $f_{\pi}=93$ MeV. In Fig. 2 we plot the energies in the center-of-mass of each of the two-body systems as a function of the energy of the center-of-mass of the three-body system, according to Eqs. (19), (20), (21) and (22). Both prescriptions map the energy range around 2812 MeV, the threshold of $D_{s0}^{*}(2317)K$ (or $D_{s0}^{*}(2317)\bar{K}$), to an energy range around each of the thresholds of the two-body interactions, i. e. the $KK$ system (or $K\bar{K}$) interact in the energy range around 990 MeV in their center-of-mass, which corresponds to $2\,m_{K}$, and the $DK$ (or $D\bar{K}$) interact in the energy range around 2360 MeV, which corresponds to $m_{K}+m_{D}$.

The main uncertainty in our calculation is the difference between these two ways of mapping the total energy into the center-of-mass of each two-body system. This feature was also found in other works using FCA, for instance in Ref. pedro .

III.1 The $DK\bar{K}$ system

In Fig. 3a we show the result of the total Faddeev amplitude squared from Eq. (12) using Prescription I. We can see a strong peak around 2833 MeV, which could be interpreted as a $D[f_{0}(980)/a_{0}(980)]$ bound state, since it is below the $D[f_{0}(980)/a_{0}(980)]$ threshold of $2855$ MeV. On the other hand, using Prescription II we observe a peak around 2858 MeV, as can seen in Fig. 3b, and now could be interpreted as a $D[f_{0}(980)/a_{0}(980)]$ resonance since it is above its threshold.

In order to investigate if this strong peak in the $DK\bar{K}$ system comes mostly from $K\bar{K}$ merging into $a_{0}(980)$ or $f_{0}(980)$, we have separated the $K\bar{K}$ amplitudes (that enter in the Faddeev equations) in isospin basis and selected only one contribution at a time. In Fig. 4 we show the results where the $I=0$ component of $K\bar{K}$ was removed, therefore there is no $f_{0}(980)$ contribution. In this figure we can see clearly the shape of the $a_{0}(980)$ in the three-body amplitude, that peaks around 2842 MeV in Prescription I (and 2886 MeV in Prescription II), which according to Fig. 2, correspond to 990 MeV in the $K\bar{K}$ center-of-mass, exactly where the $a_{0}(980)$ peak results from the $I=1$ $K\bar{K}$ two-body amplitude. Notice that when we removed the $I=0$ isospin component from the $K\bar{K}$ amplitude the strength of the peaks in $|T_{DD\bar{K}}|^{2}$ have decreased more than two orders of magnitude in both prescriptions, pointing out that the $f_{0}(980)$ is indeed the most important contribution coming from $K\bar{K}$. It is interesting to recall that the same conclusion was obtained in albermari , where no apparent signal for $Da_{0}(980)$ was found. Furthermore, the small cusps seen in both prescriptions at 2812 MeV in Fig. 4 correspond to the $D^{*}_{s0}(2317)\bar{K}$ threshold. In Table 1 we compile the results of both prescriptions.

The results for the $DK\bar{K}$ system point out to the formation of a three-body state: the $D[f_{0}(980)/a_{0}(980)]$, in which the $Df_{0}(980)$ is the strongest contribution in both prescriptions. Specifically, in Prescription I the $Df_{0}(980)$ state would be bound by about $20$ MeV, while in Prescription II it would correspond to a resonance. This latter result would be similar to the findings of Ref. albermari where a peak is seen at higher energy, forming a $Df_{0}(980)$ resonant state at 2890 MeV.

As mentioned previously, the difference between the results of prescriptions I and II should be interpreted as the main uncertainty in our approach, but what emerges from both pictures is that a $Df_{0}(980)$ state is formed, slightly bound or unbound.

We would like to note that the theoretical uncertainty of the present method is of the order of $25$ MeV. To put this number in a proper context we can recall that the uncertainty in the QCD sum rules method in Ref. albermari is far larger, with a mass given by $m_{Df_{0}}=(2926\pm 237)$ MeV (the uncertainty for the mass in the Faddeev method of Ref. albermari is not given).

III.2 The $DKK$ system

In Fig. 5 we show the $DKK$ total amplitude squared from Eq. (12) using prescriptions I and II. We can see that in both prescriptions the amplitude decreases around 2812 MeV which corresponds to the $D_{s0}^{*}(2317)K$ threshold, and both have a maximum below this threshold, while Prescription II also develops a broad structure above threshold, but no clear peak which could indicate that a bound state or a resonance is found.

As a physical interpretation we could say that, even though the $DK$ interaction is attractive and responsible for the strong binding that generates the $D_{s0}^{*}(2317)$, the repulsion between $KK$ seems to be of the same magnitude and prevents the $DKK$ system to form a bound state.

One might be tempted to associate the peak below threshold to a physical state, but this is not the case. Indeed, one should note that the strength of $|T_{DKK}|^{2}$ in Fig. 5 is about three orders of magnitude smaller than for $|T_{DK\bar{K}}|^{2}$ in Fig. 3b, which simply indicates that no special hadron structure has been formed in this case.

IV Conclusions

In this work, we have used the FCA to Faddeev equations in order to look for bound states or resonances generated from $DK\bar{K}$ and $DKK$ three-body interactions. The cluster $DK$ in the $I=0$ component is the well known $D^{*}_{s0}(2317)$ bound state studied by means of the chiral unitary approach. From the $DK\bar{K}$ interaction we found a $I(J^{P})=1/2(0^{-})$ state with mass about $2833-2858$ MeV, where the uncertainties were estimated taking into account two different prescriptions to obtain $\sqrt{s_{DK}}$ and $\sqrt{s_{KK}}$ from the total energy of the system $\sqrt{s}$. Our findings corroborated those of Ref. albermari , where the authors studied the $DK\bar{K}$ interaction using two different nonperturbative calculation tools, the QCD sum rules and the Faddeev equations without FCA. They found a state around $2890$ MeV which is above the $Df_{0}(980)$ threshold. As we have pointed out before, this state could be seen in the $\pi\,\pi\,D$ invariant mass distribution. Therefore, as in Ref. albermari , we also suggest the search for such a state in future experiments. On the other hand, for the $DKK$ system we found an enhancement effect, but with a very small strength compared to the $DK\bar{K}$ system and should not be related to a physical bound state. In this case, the repulsion between $KK$ seems to be of the same magnitude as the attraction on the $DK$ interaction, preventing the formation of the three-body molecular state.

Acknowledgments

V. R. Debastiani wishes to acknowledge the support from the Programa Santiago Grisolia of Generalitat Valenciana (Exp. GRISOLIA/2015/005). J. M. Dias would like to thank the Brazilian funding agency FAPESP for the financial support. This work is also partly supported by the Spanish Ministerio de Economia y Competitividad and European FEDER funds under the contract number FIS2014-57026-REDT, FIS2014-51948-C2-1-P, and FIS2014-51948-C2-2-P, and the Generalitat Valenciana in the program Prometeo II-2014/068.