Examination of the $\phi-NN$ bound-state problem with lattice QCD $N-\phi$ potentials

In strangeness nuclear physics, the possibility of the formation of $\phi$-mesic bound states with nucleons (N) [1, 2, 3, 4] is one the most exciting and important fields due to the quark content of the $\phi$-meson as being $s\bar{s}$. One of the simplest candidates for $\phi$-mesic nuclei can be the $\phi{-NN}$ system [5, 6, 7, 8]. Accordingly, the binding energy of $\phi{-NN}$ state has been calculated using the folding method [5], solving the Faddeev equations in the coordinate space  [6] and in two-variable integro-differential equations on the ${D=}{3}\left({A-1}\right)$-dimensional space [7], where mostly attractive phenomenological ${N-}\phi$ interaction [9] by central binding energy of $9.47$ MeV and the semi-realistic Malfliet-Tjon (MT) ${NN}$ potential[10], are employed. They concluded that the $\phi{-NN}$ system is bound with a binding energy of about $40\left(23\right)$ MeV for triplet(singlet) ${NN}$ interaction [7].

On the other hand, from the experimental point of view, ALICE collaboration measured the correlation function of proton-$\phi$ in heavy-ion collisions  [11], together by indicating a $p{-}\phi$ bound state using two-particle correlation functions [12] request the proton${-}\phi$ bound state hypothesis. To go one step further, the femtoscopic analysis of hadron-deuteron ($hd$) correlation functions could play a crucial role in understanding the structure and dynamics of the atomic nuclei. Therefore the three-body calculations of the $hd$ correlation function have special importance [13, 14]. Accordingly, the $hd$ correlation functions are investigated for some hadron-deuteron systems like $pd$ [13, 15], ${K}^{-}d$ [14, 15], $\Lambda d$ [16], $\Xi d$ [17], and the production of $\Omega NN$ in ultra-relativistic heavy ion collisions are studied in [18, 19]. Hence, it is more desirable to study the properties of $\phi{-d}$ system by using the state-of-the-art few-body computational method .

Very recently, the first lattice QCD simulations of the ${N-}\phi$ potential in ${}^{4}S_{3/2}$ channel (by the notation ${}^{2s+1}L_{J}$ where $s,L$ and $J$ are the total spin, orbital angular momentum and total angular momentum, respectively) are published [20]. The simulation is performed by $\left(2+1\right)$-flavor with quark masses near the physical point $m_{\pi}\simeq 146.4$ MeV and $m_{K}\simeq 525$ MeV on a sufficient large lattice size of $\simeq\left(8.1\>{fm}\right)^{3}$. Here, we employ this ${N-}\phi$ potential to study $\phi{-d}$ in the highest spin $\left(I\right)J^{p}=\left(0\right)2^{-}$, this channel is selected because it cannot couple to the lower channels $\Lambda KN$ and $\Sigma KN$ with the $\Lambda K$ and $\Sigma K$ subsystem in $D$ waves. In addition, because of the small phase space, the decay to final states by four or more particles like $\Sigma\pi KN,\Lambda\pi KN$ and $\Lambda\pi\pi KN$ are supposed to be suppressed [20].

We should emphasize that since, some coupling constants like $\phi\rho\pi$ are known to be far from being OZI (Okubo-Zweig-Iizuka ) suppressed, the coupling to channels like $\rho N$ or $\pi\Delta$ could be sizable and, this has been all ignored in the lattice simulations [20]. Such a coupling would change the scenario that will be studied here considerably.

Motivated by the above discussion, the binding energies and matter radius of $\phi{-d}\left(0\right)2^{-}$ state in cluster model based on expansion in hyperspherical harmonics (HH method) [21, 22, 19] are calculated in this paper. For ${NN}$ interaction we have considered the semi-realistic Malfliet-Tjon ${NN}$ potential.

The paper is organized as follows. In Section II a brief sketch of the three-body hyperspherical basis formalism is given. The input two-body potentials are described in Section  III. In Section IV, the numerical results are discussed. And the last Section V, is devoted to a summary and conclusion.

The expansion on hyperspherical harmonics method is a developed version of the Faddeev equations in coordinate space to study weakly bound three-body systems. The method is well-described in [21, 23, 24], so we present an outline of it very briefly here.

The complete three-body wave function $\Psi$ is sum of three components $\Psi^{(i)}$, i.e, $\Psi=\sum_{i=1}^{3}\Psi^{\left(i\right)}$. The components $\Psi^{\left(i\right)}$ are function of the three different sets of Jacobi coordinates (One of the three sets is shown in the Fig. 1), and they satisfy the three Faddeev coupled equations,

where $T$ is the kinetic energy, $E$ is the total energy, $V_{jk}$ is the two-body interactions between the corresponding pair and the indexes $i,j,k$ is a cyclic permutation of $\left(1,2,3\right)$.

The Jacobi coordinates $\left\{\vec{x},\vec{y}\right\}$, as depicted in Fig. 1, are employed to define the framework of three-body systems. The Jacobi-T coordinate set is the most suitable one for the $\phi{-NN}$ system because the antisymmetrization of the wave function should be preserved under exchange of nucleons linked by the $\vec{x}$ coordinate. The variable $\vec{x}$ represents the relative coordinates between two of the particles and $\vec{y}$ is between their center of mass and the third particle, both with a scaling mass factor. From the Jacobi coordinates, we can define the hyperspherical coordinates $\{\rho,\alpha,\Omega_{x},\Omega_{y}\}$, with hyperradius (generalized radial coordinate) $\rho^{2}=x^{2}+y^{2}$ and the hyperangle (generalized angle) $\alpha=\arctan(x/y)$. The $\Omega_{x}$ and $\Omega_{y}$ are the angles defining the direction of $\vec{x}$ and $\vec{y}$, respectively. For the sake of simplicity, we describe all angular dependencies by $\phi=(\alpha,\Omega_{x},\Omega_{y})$.

The Hamiltonian of a three-body system in hyperspherical coordinates is defined as follows:

the $\hat{V}(\rho,\phi)$ is potential operator which is the summation of pair interactions and the $\hat{T}(\rho,\phi)$ is the free Hamiltonian operator [25],

where $\hat{K}$ is hyperangular momentum (generalized angular momentum) operator and $m$ is a normalization mass for which we choose $m=m_{N}$.

Employing the hyperspherical coordinates, the solutions of the Schrödinger equation with the three-body Hamiltonian of Eq. (2) by total angular momentum $j$ can be expanded for each $\rho$ as

where the functions $\mathcal{Y}_{\beta}^{j\mu}(\phi)$ are a complete set of hyperangular functions that can be expanded in hyperspherical harmonics [21, 22, 23], and $R_{i\beta}\left(\rho\right)$ are the hyperradial wave functions, where the subscript $i$ denotes the hyperradial excitation (for the purpose of solving the coupled equations (6) the hyper-radial functions, $\mathcal{R}_{\beta}^{j}(\rho)$ are expanded in terms of orthonormal discrete basis up to $i_{max}$ [24]). Moreover, the $\beta\equiv\{K,l_{x},l_{y},l,S_{x},j_{ab}\}$ is a set of quantum numbers coupled to $j$. $l_{x}$ and $l_{y}$ are the orbital angular momenta related to the Jacobi coordinates $\vec{x}$ and $\vec{y}$, respectively. $l=l_{x}+l_{y}$ represents the total orbital angular momentum, $S_{x}$ gives the total spin of pair particles related by $\vec{x}$, and $j_{ab}=l+S_{x}$. The total angular momentum is $j=j_{ab}+I$ where $I$ indicates the spin of the third particle.

Therefore wave function of the system is defined by

where $C_{i\beta}^{j}$ are the diagonalization coefficients that can be calculated by diagonalizing the three-body Hamiltonian for $i=0,...,i_{max}$ basis functions. The hyperradial wave functions $\mathcal{R}_{\beta}(\rho)$ are solutions to the coupled set of differential equations,

the term $V_{\beta^{\prime}\beta}^{j\mu}(\rho)$ is related to the two-body potentials between each pair of particles ($V_{{ij}}$), by

In this section, we introduce the two-body interaction of ${NN}$ and different analytical form of extracted lattice ${N-}\phi$ potential which we used in our calculations for $\phi{-NN}$ system.

Here, we present and discuss our numerical results for the ground state binding energy $B_{3}$ and nuclear matter radius $r_{mat}$ for three-body $\phi{-NN}$ state systems. For this purpose, the coupled equations (6) are solved by applying the FaCE computational toolkit [24] employing the two-body interactions described in Sec. III.

In the HH method, the final results depend on a maximum value of hypermomentum $K_{max}$ due to the expansion of the total three-body wave function in hypermomentum components, which are truncated by $K_{max}$, therefore, it is necessary to investigate the convergence of results as a function of $K_{max}$ and the maximum number of hyperradial excitations $i_{max}$ (see Eq. (II)). The results converged quite well with the $K_{max}=70$ and $i_{max}=25$ for $(I)J^{\pi}=(0)2^{-}$ $\phi{-d}$ state.

The ${N-}\phi$ two-body system in ${}^{4}S_{3/2}$ channel( Fig. 2) by parametrizations of all three model ${A}_{{erfc}}$, ${A}_{{exp}}$, and ${B-3G}$ does not form a bound state as previously predicted in [31, 32] by using the hidden gauge theory with unitary coupled-channel calculations within ${SU}(3)$ symmetry, whereas within chiral ${SU}(3)$ quark model [33] and unitary coupled-channel approach [34] the existence of a bound state is reported. In older studies, as mentioned in the introduction, in Refs. [5, 6, 7] by using the attractive ${N-}\phi$ interaction [9], it is predicted that the binding energy of $\phi{-NN}$ state could be about $40\left(23\right)$ MeV for triplet(singlet) ${NN}$ interaction.

Nevertheless, in this case, as expected no bound state found for $(I)J^{\pi}=(1)1^{-}$ $\phi{-NN}$ state, i.e. $\phi{-nn}$ and $\phi{-pp}$. While the $(I)J^{\pi}=(0)2^{-}$ $\phi{-d}$ state in the maximal spin channel is bound for all parametrizations of the ${N-}\phi$ potentials, i.e. ${A}_{{erfc}}$, ${A}_{{exp}}$, and ${B-3G}$. The ground state binding energy and nuclear matter radius of $\phi{-d}$ state are given in Table 3. To calculate the r.m.s. matter radius of the $\phi{-d}$ system, the strong interaction radius of proton, neutron and $\phi$-meson by the values $0.82,0.80$ fm and $0.46$ fm [8, 35], respectively, are have been used as input.

Moreover, the results with the $m_{N}$ and $m_{\phi}$ masses derived from $\left(2+1\right)$-flavor lattice QCD simulations [20], presented in Table 3. The values of these masses are slightly bigger than the experimental value. As it can be seen from the Table 3, $B_{3}$ by lattice masses are a bit larger than $B_{3}$ by the experimental masses. This is because by increasing the masses, repulsive kinetic energy contribution will decrease which in turn leads to an increment in binding energies [36].

In this work, the binding energy of the three-body system $\phi-NN$ is examined using the first lattice QCD $N-\phi$ potential in the ${}^{4}S_{3/2}$ channel and semi-realistic Malfliet-Tjon ${NN}$ interactions. The $N-\phi$ potential is obtained from QCD on a sufficiently large lattice at almost physical quark masses ($m_{\pi}\simeq 146.4$ MeV and $m_{K}\simeq 525$ MeV) and parametrized in three different analytical forms, i.e. ${A}_{{erfc}}$, ${A}_{{exp}}$ and ${B-3G}$, where the concrete parameterizations of these models, are taken straight from Ref. [20].

Then, by having a two-body potential between each pair of particles, the coupled Faddeev equations in the coordinate space are solved within the hyperspherical harmonics expansion method.

We have tried to find bound states or resonances that can be observed in future experiments. The numerical results suggest that no bound state or resonances found for $(I)J^{\pi}=(1)1^{-}$ $\phi{-nn}$ and $\phi{-pp}$ systems. The $(I)J^{\pi}=(0)2^{-}$ $\phi{-d}$ system in the maximal spin presents a bound state with a binding energy of about $7$ MeV and a nuclear matter radius of about $8$ fm. The $\phi{-d}$ system in the maximal spin channel cannot couple to the lower three-body open channels $\Lambda KN$ and $\Sigma KN$ because $D$ wave subsystems $\Lambda K$ and $\Sigma K$ are kinematically suppressed at low energies. In addition, because of the small phase space, the decay to final states by four or more particles e.g. $\Sigma\pi KN,\Lambda\pi KN$, and $\Lambda\pi\pi KN$ are supposed to be suppressed [20]. Last but not least, in Ref. [20] they have not consider the OZI violating $s\bar{s}$ annihilation in their simulations. Nevertheless, considering the coupling to channels like $\rho N$ or $\pi\Delta$ could change the results obtained here significantly.

In conclusion, adding a $\phi$-meson to the deuteron could enhance its binding energy. These bound states or resonances could be explored in hadron beam experiments. Recently, for the first time, ALICE collaboration measured the correlation function of proton-$\phi$ in heavy-ion collisions [11], and the existence of $p{-}\phi$ bound state has been discussed and explored in [12]. And very recently, the $K^{+}{-d}$ and $p{-d}$ femtoscopic correlations measured by the ALICE Collaboration in proton-proton (pp) collisions [14, 15], analogously, as the next step, these measurements could be done for $\phi{-d}$ system. We desire that our numerical results could help to plan experiments in the future.