$\Omega NN$ and $\Omega\Omega N$ states

TL;DR

This paper investigates three-body systems involving $\\Omega$ baryons using lattice QCD potentials, predicting bound and resonant states relevant for understanding hyperon interactions.

Contribution

It provides the first detailed analysis of $\\Omega NN$ and $\\Omega\\Omega N$ states using recent lattice QCD potentials, identifying bound and resonant states with specific quantum numbers.

Findings

The $\\Omega d$ system with $(I)J^P=0 5/2^+$ is bound with about 20 MeV energy.

The $(I)J^P=1 3/2^+$ $\\Omega nn$ state is a resonance decaying to $\\Lambda \\Xi n$ and $\\Sigma \\Xi n$.

The $(I)J^P=1/2 1/2^+$ $\\Omega\\Omega N$ state is a resonance decaying to $\\Lambda \\Xi \\Omega$ and $\\Sigma \\Xi \\Omega$.

Abstract

The lattice QCD analysis of the HAL QCD Collaboration has recently derived $\Omega N$ and $\Omega\Omega$ interacting potentials with nearly physical quark masses ($m_\pi \simeq $ 146 MeV and $m_K \simeq $ 525 MeV). They found an attractive interaction in the $\Omega N$ $^5S_2$ channel which supports a bound state with a central binding energy of 1.54 MeV. The $\Omega \Omega$ $^1S_0$ channel shows an overall attraction with a bound state with a central binding energy of 1.6 MeV. In this paper we looked closely at the $\Omega NN$ and $\Omega\Omega N$ three-body systems making use of the latest HAL QCD Collaboration $\Omega N$ and $\Omega\Omega$ interactions. Our results show that the $\Omega d$ system in the state with maximal spin $(I)J^P=(0)5/2^+$ is bound with a binding energy of about 20 MeV. The $(I)J^P=(1)3/2^+$ $\Omega nn$ state presents a resonance decaying to $\Lambda \Xi n$ and $\Sigma \Xi n$, with a separation energy of $\sim$ 1 MeV. The $(I)J^P=(1/2)1/2^+$ $\Omega \Omega N$ state also exhibits a resonance decaying to $\Lambda \Xi \Omega$ and $\Sigma \Xi \Omega$, with a separation energy of $\sim$ 4.6 MeV. We have calculated the contribution of the Coulomb potential to differentiate among the different charged states.

Tables

Table 1. Table 1: Low-energy data and parameters of the local central Yukawa-type potential given by Eq. ( 1 ) for the S 0 1 superscript subscript 𝑆 0 1 {}^{1}S_{0} and S 1 3 superscript subscript 𝑆 1 3 {}^{3}S_{1} N ​ N 𝑁 𝑁 NN interactions  Fri90 .

$(I,J)$	$a_0$ (fm)	$r_{\mathrm{eff}}$ (fm)	$C_1$ (MeV fm)	${\mu }_1$ (fm-1)	$C_2$ (MeV fm)	${\mu }_2$ (fm-1)
$(1,0)$	$-23.56$	$2.88$	$-513.968$	$1.55$	$1438.72$	$3.11$
$(0,1)$	$5.51$	$1.89$	$-626.885$	$1.55$	$1438.72$	$3.11$

Table 2. Table 2: Fitting parameters in Eq. ( 2 ) for different models, P i subscript 𝑃 𝑖 P_{i} , of the S 2 5 superscript subscript 𝑆 2 5 {}^{5}S_{2} Ω ​ N Ω 𝑁 \Omega N interaction  Iri18 .

	$P_1$	$P_2$	$P_3$	$P_4$
$b_1$ (MeV)	$-$306.5	$-$313.0	$-$316.7	$-$296
$b_2$ (fm-2)	73.9	81.7	81.9	64
$b_3$ (MeV fm2)	$-$266	$-$252	$-$237	$-$272
$b_4$ (fm-2)	0.78	0.85	0.91	0.76

Table 3. Table 3: Binding energy of the S 2 5 superscript subscript 𝑆 2 5 {}^{5}S_{2} Ω ​ N Ω 𝑁 \Omega N state, B Ω ​ N subscript 𝐵 Ω 𝑁 B_{\Omega N} , and the ( I ) ​ J P = ( 0 ) ​ 5 / 2 + 𝐼 superscript 𝐽 𝑃 0 5 superscript 2 (I)J^{P}=(0)5/2^{+} Ω ​ d Ω 𝑑 \Omega d state, B Ω ​ d subscript 𝐵 Ω 𝑑 B_{\Omega d} , for the different models of the Ω ​ N Ω 𝑁 \Omega N interaction given in Table  2   Iri18 . The results have been obtained with the experimental masses of the N 𝑁 N and Ω Ω \Omega , 938.9 MeV/c 2 and 1672.45 MeV/c 2 respectively. We have indicated between parenthesis the results corresponding to the N and Ω Ω \Omega masses derived by the HAL QCD Collaboration, 954.7 MeV/c 2 and 1711.5 MeV/c 2 respectively  Gon18 ; Iri18 . All energies are in MeV.

	$P_1$	$P_2$	$P_3$	$P_4$
$B_{\Omega N}$	1.29 (1.52)	1.38 (1.61)	1.29 (1.44)	1.37 (1.60)
$B_{\Omega d}$	19.6 (20.6)	20.0 (21.1)	19.6 (20.5)	19.9 (20.9)

Table 4. Table 4: Binding energy, B Ω ​ n ​ n subscript 𝐵 Ω 𝑛 𝑛 B_{\Omega nn} , and separation energy, S Ω ​ n ​ n subscript 𝑆 Ω 𝑛 𝑛 S_{\Omega nn} , of the the ( I ) ​ J P = ( 1 ) ​ 3 / 2 + 𝐼 superscript 𝐽 𝑃 1 3 superscript 2 (I)J^{P}=(1)3/2^{+} state for the different models of the Ω ​ N Ω 𝑁 \Omega N interaction given in Table  2   Iri18 . All energies are in MeV. The numbers between parenthesis have the same meaning as in Table  3 .

	$P_1$	$P_2$	$P_3$	$P_4$
$B_{\Omega nn}$	2.25 (2.60)	2.35 (2.72)	2.14 (2.50)	2.34 (2.71)
$S_{\Omega nn}$	0.96 (1.08)	0.97 (1.11)	0.85 (1.06)	0.97 (1.11)

Table 5. Table 5: Binding energy, B Ω ​ Ω ​ N subscript 𝐵 Ω Ω 𝑁 B_{\Omega\Omega N} , and separation energy, S Ω ​ Ω ​ N subscript 𝑆 Ω Ω 𝑁 S_{\Omega\Omega N} , of the the ( I ) ​ J P = ( 1 / 2 ) ​ 1 / 2 + 𝐼 superscript 𝐽 𝑃 1 2 1 superscript 2 (I)J^{P}=(1/2)1/2^{+} Ω ​ Ω ​ N Ω Ω 𝑁 \Omega\Omega N state for the different models of the Ω ​ N Ω 𝑁 \Omega N interaction given in Table  2   Iri18 and the Ω ​ Ω Ω Ω \Omega\Omega interaction of Ref.  Gon18 . All energies are in MeV. The numbers between parenthesis have the same meaning as in Table  3 .

	$P_1$	$P_2$	$P_3$	$P_4$
$B_{\Omega \Omega N}$	6.0 (6.7)	6.2 (6.9)	5.9 (6.6)	6.1 (6.8)
$S_{\Omega \Omega N}$	4.6 (5.1)	4.8 (5.3)	4.5 (5.0)	4.7 (5.2)

Table 6. Table 6: Binding energy of the different Ω ​ N Ω 𝑁 \Omega N , Ω ​ N ​ N Ω 𝑁 𝑁 \Omega NN and Ω ​ Ω ​ N Ω Ω 𝑁 \Omega\Omega N charged states including the Coulomb potential given by Eq. ( 21 ). All energies are in MeV. The numbers between parenthesis have the same meaning as in Table  3 .

$\left(I\right)J^P$	System	$P_1$	$P_2$	$P_3$	$P_4$
$\left(1/2\right)2^{+}$	$\Omega n$	1.29 (1.52)	1.38 (1.61)	1.29 (1.44)	1.37 (1.60)
	$\Omega p$	2.19 (2.42)	2.30 (2.53)	2.19 (2.34)	2.28 (2.51)
$\left(1\right)3/2^{+}$	$\Omega nn$	2.25 (2.60)	2.35 (2.72)	2.14 (2.50)	2.34 (2.71)
	$\Omega pp$	2.91 (3.30)	3.04 (3.43)	2.81 (3.20)	3.02 (3.41)
$\left(0\right)5/2^{+}$	$\Omega d$	20.9 (22.0)	21.3 (22.4)	20.7 (21.8)	21.2 (22.3)
$\left(1/2\right)1/2^{+}$	$\Omega \Omega n$	5.0 (5.6)	5.1 (5.8)	4.9 (5.5)	5.1 (5.8)
	$\Omega \Omega p$	6.3 (7.0)	6.5 (7.2)	6.2 (6.9)	6.4 (7.2)

Equations

$$V_{NN}(r)=\sum _{n=1}^2{C}_n\frac{e^{-{\mu }_{n}r}}{r},$$

$$V_{\Omega N}(r)=b_1{e}^{-b_2{r}^2}+b_3\left(1-e^{-b_4{r}^2}\right){\left(\frac{e^{-m_{\pi }r}}{r}\right)}^2.$$

$$V_{\Omega \Omega }(r)=\sum _{j=1}^3{c}_j{e}^{-(r/d_j{)}^2},$$

$$T_{i;IJ}^{i_i{j}_i}(p_i{q}_i)=$$

$${\mu }_i$$

$${\nu }_i$$

$$p_i^{\prime }$$

$$p_j$$

$$h_{ij;IJ}^{i_i{j}_i;i_j{j}_j}=$$

$$x_i=\frac{p_i-b}{p_i+b},$$

$$T_{i;IJ}^{i_i{j}_i}(x_i{q}_i)=$$

$$t_{i;i_i{j}_i}(x_i,x_i^{\prime };e)=\sum _{nr}{P}_n(x_i){\tau }_{i;i_i{j}_i}^{nr}(e)P_r(x_i^{\prime }),$$

$${\tau }_{i;i_i{j}_i}^{nr}(e)=\frac{2n+1}{2}\frac{2r+1}{2}{\int }_{-1}^{1}d{x}_i{\int }_{-1}^{1}d{x}_i^{\prime }{P}_n(x_i)t_{i;i_i{j}_i}(x_i,x_i^{\prime };e)P_r(x_i^{\prime }).$$

$$T_{i;IJ}^{i_i{j}_i}(x_i{q}_i)=\sum _n{P}_n(x_i)T_{i;IJ}^{n{i}_i{j}_i}(q_i),$$

$$T_{i;IJ}^{n{i}_i{j}_i}(q_i)=\sum _{j\ne i}\sum _{m{i}_j{j}_j}{\int }_0^{\infty }d{q}_j{A}_{ij;IJ}^{n{i}_i{j}_i;m{i}_j{j}_j}(q_i,q_j;E)T_{j;IJ}^{m{i}_j{j}_j}(q_j),$$

$$A_{ij;IJ}^{n{i}_i{j}_i;m{i}_j{j}_j}(q_i,q_j;E)=$$

$$T_{1;IJ}^{r{i}_1{j}_1}(q_1)=2\sum _{m{i}_2{j}_2}{\int }_0^{\infty }d{q}_3{A}_{13;IJ}^{r{i}_1{j}_1;m{i}_2{j}_2}(q_1,q_3;E)T_{2;IJ}^{m{i}_2{j}_2}(q_3),$$

$$T_{2;IJ}^{n{i}_2{j}_2}(q_2)=$$

$$g=(-{)}^{1+{\sigma }_1+\sigma 3-j_2+{\tau }_1+{\tau }_3-i_2},$$

$$T_{2;IJ}^{n{i}_2{j}_2}(q_2)=\sum _{m{i}_3{j}_3}{\int }_0^{\infty }d{q}_3{K}_{IJ}^{n{i}_2{j}_2;m{i}_3{j}_3}(q_2,q_3;E)T_{2;IJ}^{m{i}_3{j}_3}(q_3),$$

$$K_{IJ}^{n{i}_2{j}_2;m{i}_3{j}_3}(q_2,q_3;E)=$$

$$t_i(p_i,p_i^{\prime };e)=V_i(p_i,p_i^{\prime })+{\int }_0^{\infty }{p_i^{\prime \prime }}^{2}d{p}_i^{\prime \prime }{V}_i(p_i,p_i^{\prime \prime })\frac{1}{e-{p_i^{\prime \prime }}^2/2{\eta }_i+iϵ}{t}_i(p_i^{\prime \prime },p_i^{\prime };e).$$

$$V_C(r)=\pm \alpha \frac{e^{-r/r_0}}{r},$$

Full text

baryon-baryon interactions, Faddeev equations

I Introduction

Few-body systems containing nucleons may enhance the binding of two-body resonances or bound states. We have simple examples in nature. Whereas there is no evidence for strangeness $-1$ dibaryon states, the hypertriton ${}^{3}_{\Lambda}$H, $(I)J^{P}=(0)1/2^{+}$, is bound with a separation energy of $130\pm 50$ keV, and the ${}^{4}_{\Lambda}$H, $(I)J^{P}=(0)0^{+}$, is bound with a separation energy of $2.04\pm 0.04$ MeV. Similarly, in the nonstrange sector the binding per nucleon, $B/A$, increases from $1:3:7$ for increasing number of nucleons, $A=2,3,4$. Thus, the study of few-body systems could help in the search for signals of two-body bound states or resonances in the strange sector.

The possible existence of dibaryons is a challenge since a long time ago Dys64 ; Jaf77 ; Mul80 . Their occurrence would clearly help us to better understand Quantum Chromodinamycs in the nonperturbative regime. Unfortunately, despite innumerable experimental searches, the deuteron has been the only known dibaryon state up to very recently. The increasing quality of experimental data together with exclusive measurements in hadronic reactions Adl11 , direct measurements in relativistic heavy-ion collisions Cho17 , or the development of new methods for nuclear emulsion experiments Nak15 , has renewed the interest in dibaryons. Thus, the so–called WASA dibaryon, a resonance 90 MeV below the nominal $\Delta\Delta$ threshold with a width of 70 MeV, is now firmly established although its nature is still under debate Cle17 . Moreover, the so–called KISO event represents the first clear evidence of a deeply bound state of $\Xi^{-}-^{14}$N Nak15 , which combined with other indications of emulsion data suggest an average attractive $\Xi N$ interaction Rij16 . In this latter case, the increase in the binding energy per baryon within few-body systems with additional nucleons has already been noticed. If the $(I)J^{P}=(1)1^{+}$ $\Xi N$ state would be bound by 1.56 MeV Nak15 ; Rij16 , the $\Xi d$ $(I)J^{P}=(1/2)3/2^{+}$ state becomes bound by 17.2 MeV Gac16 ; Fil17 .

In addition to improvements in the experimental data and techniques, the theoretical efforts of the lattice HAL QCD Collaboration Aok12 have reached the point of deriving baryon-baryon interactions near the physical pion mass. Their most recent results hint to the existence of shallow bound states in the $\Omega N$ and $\Omega\Omega$ systems Gon18 ; Iri18 . Two-body systems containing an $\Omega$ baryon seem to be specially suited to lodge a two-body bound state or resonance. Thus, the $\Omega N$ interaction is expected to lack a repulsive core since the quark flavors of the nucleon are different from those of the $\Omega$, so that the Pauli exclusion principle can not act Eti14 ; Gol87 . Moreover, the color magnetic term of the one-gluon exchange interaction is attractive in some particular channels, one example being the ${}^{5}S_{2}$, what led to the prediction of an $\Omega N$ dibaryon in constituent quark models Gol87 ; Hua15 . The recent lattice QCD results by the HAL QCD Collaboration hint toward the existence of such $\Omega N$ bound state Iri18 . Another two-body system containing $\Omega$ baryons, $\Omega\Omega$, is also interesting because it would be the only possible strong interaction stable state made of two decuplet baryons. The adequacy of such two-body system for having an overall attractive interaction has become apparent on phenomenological quark models predicting an $\Omega\Omega$ ${}^{1}S_{0}$ bound state Zha00 ; Dai06 . The recent near to physical pion mass results of the lattice HAL QCD Collaboration corroborate such finding Gon18 . It is worth to notice that the theoretical analysis Mor16 of the first measurement of the proton-$\Omega$ correlation function in heavy-ion collisions by the STAR experiment Ada18 at the Relativistic Heavy-Ion Collider (RHIC) favors the proton-$\Omega$ bound state hypothesis 111Note, however, that the first measurement of the proton-$\Omega$ correlation function drives to a proton-$\Omega$ bound state with a binding energy of $\sim$ 27 MeV Ada18 , far from the present lattice QCD results of about 1.54 MeV Iri18 that will be used in this work..

Preliminary results by the HAL QCD Collaboration Doi18 had already shown the attractive character of the ${}^{5}S_{2}$ $\Omega N$ interaction. These exploratory results were parametrized in Ref. Sek18 by means of an equivalent local potential reproducing the $\Omega N$ ${}^{5}S_{2}$ scattering amplitude of Ref. Doi18 . Making use of the $\Omega N$ equivalent local potential of Ref. Sek18 together with the Malfliet-Tjon ${}^{3}S_{1}$ $NN$ potential of Ref. Fri90 , we studied the $\Omega d$ system in the maximal spin channel, $(I)J^{P}=(0)5/2^{+}$. We concluded that the system is bound, with a binding energy of about 17 MeV Gar18 . The latest results of the HAL QCD Collaboration nearly physical quark masses ($m_{\pi}\simeq$ 146 MeV and $m_{K}\simeq$ 525 MeV) have been recently made public Iri18 , validating the attractive nature of the $\Omega N$ ${}^{5}S_{2}$ state and providing their own parametrization of the lattice QCD interaction. Moreover, the HAL QCD Collaboration has also recently published the $\Omega\Omega$ ${}^{1}S_{0}$ interaction nearly physical pion mass, $m_{\pi}=$ 146 MeV, showing its overall attractive character Gon18 . Thus, we have all the necessary ingredients to explore possible implications of the attractive nature of the $\Omega N$ and $\Omega\Omega$ interactions on few-body systems on the basis of first-principle lattice QCD-based interactions.

Our purpose in this work is to study the three-body systems $\Omega NN$ and $\Omega\Omega N$ looking for deeply bound states or resonances which may be sought experimentally. The paper is organized as follows. In Sec. II we summarize the two-body interactions. In Sec. III we outline the solution of the three-body bound-state Faddeev equations for the case of two identical particles. In Sec. IV we present and discuss our results. The most important conclusions of our work are summarized in Sec. V.

II Two-body interactions

In the case of the $NN$ subsystem the two-body potential in configuration space is given by Fri90 ,

[TABLE]

with the parameters specified in Table 1. The predicted binding energy of the deuteron is $B=$ 2.2307 MeV.

For the $\Omega N$ system the HAL QCD Collaboration has recently published the $S$-wave and spin 2 interaction with nearly physical quark masses Iri18 . The lattice results are fitted by an analytic function composed of an attractive Gaussian core plus a long range $($Yukawa$)^{2}$ attraction with a form factor Eti14 ,

[TABLE]

The $($Yukawa$)^{2}$ form at long distance is motivated by the picture of two-pion exchange between $N$ and $\Omega$ with an OZI violating vertex Sek18 . The pion mass in Eq. (2) is taken from the simulation, $m_{\pi}=$ 146 MeV Iri18 . The lattice results are fitted reasonably well, $\chi^{2}/$d.o.f $\simeq$ 1, with four different set of parameters given in Table 2. The above interactions drive to the following central values of the low-energy data: a scattering length $a_{0}^{\Omega N}=$ 5.30 fm, an effective range $r_{\rm eff}^{\Omega N}=$ 1.26 fm, and a binding energy $B_{\Omega N}$ = 1.54 MeV

Finally, the $\Omega\Omega$ interaction in the ${}^{1}S_{0}$ channel has also been recently studied by the HAL QCD Collaboration with a large volume and nearly the physical pion mass Gon18 . The results indicate that the $\Omega\Omega$ interaction has an overall attraction. In particular, the potential derived has been fitted by means of an analytic function composed of three Gaussians,

[TABLE]

where $(c_{1},c_{2},c_{3})=(914,305,-112)$ in MeV and $(d_{1},d_{2},d_{3})=(0.143,0.305,0.949)$ in fm with $\chi^{2}/$d.o.f $\simeq$ 1.3. A functional form similar to the one used for the $\Omega N$ interaction, i.e., two Gaussians plus a long range $($Yukawa function$)^{2}$ provides an equally good fit and does not affect the results. The low-energy data obtained with this interaction are the following: a scattering length $a_{0}^{\Omega\Omega}=$ 4.6 fm, an effective range $r_{\rm eff}^{\Omega\Omega}=$ 1.27 fm, and a binding energy $B_{\Omega\Omega}$ = 1.6 MeV.

III The three-body bound-state Faddeev equations

The $\Omega NN$ and $\Omega\Omega N$ three-body problems are studied using the method of Ref. Gar16 , expanding the two-body amplitudes in terms of Legendre polynomials. We restrict ourselves to configurations where all three particles are in $S$-wave states so that the Faddeev equations for the bound-state problem in the case of three baryons with total isospin $I$ and spin $J$ are,

[TABLE]

where $t_{i;i_{i}j_{i}}$ stands for the two-body amplitudes with isospin $i_{i}$ and spin $j_{i}$. $p_{i}$ is the momentum of the pair $jk$ (with $ijk$ an even permutation of $123$) and $q_{i}$ the momentum of particle $i$ with respect to the pair $jk$. $\mu_{i}$ and $\nu_{i}$ are the corresponding reduced masses,

[TABLE]

and the momenta $p_{i}^{\prime}$ and $p_{j}$ in Eq. (4) are given by,

[TABLE]

$h_{ij;IJ}^{i_{i}j_{i};i_{j}j_{j}}$ are the spin–isospin coefficients,

[TABLE]

where $W$ is the Racah coefficient and $\tau_{i}$, $i_{i}$, and $I$ ($\sigma_{i}$, $j_{i}$, and $J$) are the isospins (spins) of particle $i$, of the pair $jk$, and of the three–body system.

Since the variable $p_{i}$ in Eq. (4) runs from 0 to $\infty$, it is convenient to make the transformation

[TABLE]

where the new variable $x_{i}$ runs from $-1$ to $1$ and $b$ is a scale parameter that has no effect on the solution. With this transformation Eq. (4) takes the form,

[TABLE]

Since in the amplitude $t_{i;i_{i}j_{i}}(x_{i},x_{i}^{\prime};e)$ the variables $x_{i}$ and $x_{i}^{\prime}$ run from $-1$ to $1$, one can expand this amplitude in terms of Legendre polynomials as,

[TABLE]

where the expansion coefficients are given by,

[TABLE]

Applying expansion (10) in Eq. (9) one gets,

[TABLE]

where $T_{i;IJ}^{ni_{i}j_{i}}(q_{i})$ satisfies the one-dimensional integral equation,

[TABLE]

with

[TABLE]

The three amplitudes $T_{1;IJ}^{ri_{1}j_{1}}(q_{1})$, $T_{2;IJ}^{mi_{2}j_{2}}(q_{2})$, and $T_{3;IJ}^{ni_{3}j_{3}}(q_{3})$ in Eq. (13) are coupled together. The number of coupled equations can be reduced, however, when two of the particles are identical, which is currently the case, $\Omega NN$ and $\Omega\Omega N$. The procedure for the case of two identical fermions has been described before Afn74 ; Gar90 and will not be repeated here. With the assumption that particles 2 and 3 are identical and particle 1 is the different one, only the amplitudes $T_{1;IJ}^{ri_{1}j_{1}}(q_{1})$ and $T_{2;IJ}^{mi_{2}j_{2}}(q_{2})$ are independent from each other and they satisfy the coupled integral equations,

[TABLE]

[TABLE]

with the identical–particle factor

[TABLE]

where $\sigma_{1}$ ($\tau_{1}$) stand for the spin (isospin) of the different particle and $\sigma_{3}$ ($\tau_{3}$) for those of the identical ones.

Substitution of Eq. (15) into Eq. (16) yields an equation with only the amplitude $T_{2}$,

[TABLE]

where

[TABLE]

The off-shell two-body $t-$matrices are obtained by solving the Lippmann-Schwinger equation,

[TABLE]

with the two-body interactions $V_{i}$, $i=NN,\Omega N,\Omega\Omega$, described in section II.

Finally, in order to separate the binding energies of the different charged states we included the Coulomb interaction as,

[TABLE]

where $\alpha$ is the fine structure constant and $r_{0}$ a screening radius taken to be $r_{0}=50$ fm.

IV Results

Let us first of all present and discuss our results for the corresponding $\Omega NN$ states222We have not considered the channel coupling to lower inelastic channels and, therefore, we have not estimated the width of the states having a lower decay channel different from those made of $N$’s and $\Omega$’s.. We have first studied the $\Omega d$ state. We show in Table 3 the binding energies of the state with maximal spin, $(I)J^{P}=(0)5/2^{+}$, for the different models of the $\Omega N$ interaction reported in Ref. Iri18 and summarized in Table 2. For completeness we also include the binding energy of the ${}^{5}S_{2}$ $\Omega N$ state. As indicated in the caption, the numbers between parenthesis correspond to the results using the N and $\Omega$ masses derived by the HAL QCD Collaboration Gon18 ; Iri18 , that are somewhat larger than the experimental masses. If the masses increase, the repulsive kinetic energy contribution decreases, resulting in slightly larger binding energies. As it can be seen, the binding energy of the $\Omega N$ system is larger than in the preliminary results of the $\Omega N$ interaction presented in Ref. Doi18 and parametrized in Ref. Sek18 , where the $\Omega N$ binding energy was 0.3 MeV Sek18 ; Gar18 for the $N$ and $\Omega$ physical masses. As a consequence, the binding of the $\Omega d$ state with maximal spin $(I)J^{P}=(0)5/2^{+}$ also increases from the 16.34 MeV measured with respect to the $\Omega np$ threshold Gar18 to about 20 MeV. Let us emphasize that the $\Omega d$ in the maximal spin channel $(I)J^{P}=(0)5/2^{+}$ cannot couple to the lower channels $\Lambda\Xi N$ and $\Sigma\Xi N$ with the $\Lambda\Xi$ and $\Sigma\Xi$ subsystems in $S$ waves, so that the width of a $\Omega d$ bound state is expected to be small.

Regarding the $\Omega nn$ ($\Omega pp$) system, with the $\Omega N$ interaction derived in Ref. Iri18 one can construct a $\Omega nn$ ($\Omega pp$) state with quantum numbers $(I)J^{P}=(1)3/2^{+}$. The results obtained for this state with the different models of the $\Omega N$ interaction given in Table 2 are shown in Table 4. Such $\Omega nn$ state could decay to $\Lambda\Xi n$ and $\Sigma\Xi n$, thus it would appear as a resonance.

We present now the results obtained for the $\Omega\Omega N$ system using the $\Omega N$ potential derived in Ref. Iri18 and the $\Omega\Omega$ interaction reported in Ref. Gon18 , both for the same pion mass. The two partial waves analyzed by the HAL QCD Collaboration generate an $\Omega\Omega N$ three-body system with quantum numbers $(I)J^{P}=(1/2)1/2^{+}$. This state is bound for all parametrizations of the $\Omega N$ interaction, with the binding energy, $B_{\Omega\Omega N}$, and separation energy, $S_{\Omega\Omega N}$, given in Table 5. Note that the binding energy of the ${}^{1}S_{0}$ $\Omega\Omega$ state is 1.4 MeV for the physical mass of the $\Omega$ baryon, and 1.6 MeV for the HAL QCD mass of the $\Omega$ baryon. This state would also appear as a resonance decaying to $\Lambda\Xi\Omega$ and $\Sigma\Xi\Omega$.

We have calculated the contribution of the Coulomb potential exactly as dictated by Eq. (21), in order to differentiate among the different charged states of the $\Omega N$, $\Omega NN$ and $\Omega\Omega N$ systems. The Coulomb increases the binding for systems containing a proton compared to those with a neutron, due to the attractive $\Omega^{-}p$ interaction. The Coulomb effects on the binding energy of the ${}^{5}S_{2}$ $\Omega N$ state were estimated in Ref. Iri18 , obtaining a difference of 0.92 MeV between the $\Omega p$ and $\Omega n$ states. We have checked this result for the different models, $P_{i}$, of the $\Omega N$ interaction by means of the potential given in Eq. (21). We have obtained corrections of: 0.90, 0.92, 0.90 and 0.91 MeV respectively, in agreement with the lattice QCD results. In the $(I)J^{P}=(1)3/2^{+}$ $\Omega NN$ system the Coulomb potential induces an extra binding of 0.7 MeV for the $\Omega pp$ state as compared to the $\Omega nn$ one. In the $(I)J^{P}=(0)5/2^{+}$ $\Omega d$ state the Coulomb interaction generates an additional binding of about 1.3 MeV. For the $\Omega\Omega$ ${}^{1}S_{0}$ state the Coulomb interaction reduces the binding energy from the 1.6 MeV reported in Ref. Gon18 to 0.9 MeV. Finally, in the $\Omega\Omega N$ system the Coulomb potential induces an extra binding of 0.3 MeV for the $\Omega\Omega p$ state whereas it penalizes the $\Omega\Omega n$ state by 1 MeV, generating a splitting of 1.3 MeV between these two states. The final results are summarized in Table 6.

Let us finally note that to draw definite conclusion about the width of states with a lower decay channel, one should have done a coupled-channel calculation. However, for this purpose one would need the transition potentials to the inelastic channels that have still not been derived from lattice QCD and thus the conclusions would be speculative. The width of a three-body resonance in a coupled two-channel system has been recently estimated in Ref. Gar17 , presenting a plausible argument to explain the small width of a three-body resonance lying close to the upper channel in spite of being open the lower one. The analysis of the width of a two-body resonance in a coupled-channel system has also been recently presented in Ref. Gac18 . It has been demonstrated how the width does not come only determined by the available phase space for its decay to the detection channel, but it greatly depends on the relative position of the mass of the resonance with respect to the masses of the coupled-channels generating the state. As seen in Fig. 1 of this reference, the resonance may still be narrow being close to the upper threshold if there is a little overlap with the wave function of the lower inelastic channel. Hence, in the region where the dynamics is dominated by the attraction in the upper channel, the resonance could still be narrow and the lower channel would be mainly a tool for the detection. This mechanism is somewhat related to the ’synchronization of resonances’ proposed by D. Bugg Bug08 . Thus, our studies about the width of two- and three-body resonances suggest the possibility of the experimental observation of narrow resonances lying well above their lowest decay threshold.

V Outlook

The lattice QCD analysis of the HAL QCD Collaboration has recently derived $\Omega N$ and $\Omega\Omega$ interacting potentials with nearly physical quark masses ($m_{\pi}\simeq$ 146 MeV and $m_{K}\simeq$ 525 MeV). They found an attractive potential in the $\Omega N$ ${}^{5}S_{2}$ channel which supports a bound state with a central binding energy of 1.54 MeV. The $\Omega\Omega$ ${}^{1}S_{0}$ channel shows an overall attraction with a bound state with a central binding energy of 1.6 MeV. On the basis of our current understanding of the nonstrange sector, where the binding energy may increase with the number of baryons, in this paper we have examined carefully the $\Omega NN$ and $\Omega\Omega N$ three-body systems making use of the latest HAL QCD Collaboration $\Omega N$ and $\Omega\Omega$ interactions. We have looked for deeply bound states or resonances which may be sought experimentally. Our results show that the $\Omega d$ system in the state with maximal spin $(I,J^{P})=(0,5/2^{+})$ is bound with a binding energy of about 20 MeV. The $\Omega d$ in the maximal spin channel $(I)J^{P}=(0)5/2^{+}$ cannot couple to the lower channels $\Lambda\Xi N$ and $\Sigma\Xi N$ with the $\Lambda\Xi$ and $\Sigma\Xi$ subsystems in $S$ waves, so that the width of a $\Omega d$ bound state is expected to be small. The $(I,J^{P})=(1,3/2^{+})$ $\Omega nn$ state presents a resonance decaying to $\Lambda\Xi n$ and $\Sigma\Xi n$, with a separation energy of $\sim$ 1 MeV. The $(I,J^{P})=(1/2,1/2^{+})$ $\Omega\Omega N$ state also exhibits a resonance decaying to $\Lambda\Xi\Omega$ and $\Sigma\Xi\Omega$, with a separation energy of $\sim$ 4.6 MeV. The Coulomb potential increases the binding for systems containing a proton compared to those with a neutron, due to the attractive $\Omega^{-}p$ interaction. It also penalizes the binding of the $\Omega\Omega$ state. The overall effect is of the order of 1 MeV, as noticed in the lattice QCD calculations of the two-body systems.

The latest baryon-baryon interactions in the strange sector with nearly physical pion mass based on lattice QCD hint toward the existence of bound states or sharp resonances. These states could be observed by hadron beam experiments at J-PARC and FAIR, or by relativistic heavy-ion collisions at RHIC and LHC. In Ref. Mor16 it has been discussed how the two-particle momentum correlation between the proton and the $\Omega$ baryon in high-energy heavy ion collisions may unveil the existence of these states. The first measurement of the proton-$\Omega$ correlation function in heavy-ion collisions by the STAR experiment Ada18 at RHIC favors the proton-$\Omega$ bound state hypothesis. We hope our theoretical studies could help to design experiments where these lattice QCD-based predictions could be tested.

Acknowledgements.

This work has been partially funded by COFAA-IPN (México) and by Ministerio de Economía, Industria y Competitividad and EU FEDER under Contract No. FPA2016-77177-C2-2-P.