$N\Omega$ dibaryon from lattice QCD near the physical point

The nucleon($N$)-Omega($\Omega$) system in the S-wave and spin-2 channel ($^5$S$_2$) is studied from the (2+1)-flavor lattice QCD with nearly physical quark masses ($m_\pi \simeq 146$ MeV and $m_K \simeq 525$ MeV). The time-dependent HAL QCD method is employed to convert the lattice QCD data of the two-baryon correlation function to the baryon-baryon potential and eventually to the scattering observables. The $N\Omega$($^5$S$_2$) potential is found to be attractive in all distances and to produce a quasi-bound state near unitarity: In this channel, the scattering length, the effective range and the binding energy from QCD alone read $a_0= 5.30(0.44)(^{+0.16}_{-0.01})$ fm, $r_{\rm eff} = 1.26(0.01)(^{+0.02}_{-0.01})$ fm, $B = 1.54(0.30)(^{+0.04}_{-0.10})$ MeV, respectively. Including the extra Coulomb attraction, the binding energy of $p\Omega^-$($^5$S$_2$) becomes $B_{p\Omega^-} = 2.46(0.34)(^{+0.04}_{-0.11})$ MeV. Such a spin-2 $p\Omega^-$ state could be searched through two-particle correlations in $p$-$p$, $p$-nucleus and nucleus-nucleus collisions.


Introduction
Quest for dibaryons is a long-standing experimental and theoretical challenge in hadron physics [1,2]. Among various theoretical attempts to study dibaryons, one of the recent highlights is the (2+1)-flavor lattice QCD simulations near the physical point (m π 146 MeV and m K 525 MeV) by HAL QCD Collaboration. (For a recent summary, see Ref. [3].) This enables us to make model-independent investigations of the elusive H-dibaryon, originally proposed by the MIT bag model [4], on the basis of a coupled channel analysis of the lattice QCD data [5]. Also, the possible di-Omega (ΩΩ), originally proposed by the Skyrme model [6], has recently been examined in detail from the same lattice QCD data [7].
Another interesting candidate of the dibaryon is N Ω and (uudsss or uddsss) in the 5 S 2 channel. Since the Pauli exclusion does not operate among valence quarks and the color-magnetic interaction is attractive in the channel, it was predicted to be a resonance below the N Ω threshold in the constituent quark model [8,9]. Moreover, N Ω( 5 S 2 ) is expected to have relatively a small width since its strong decay into octet baryons such as ΛΞ and ΣΞ, which must have orbital D-wave, would be kinematically suppressed. A pilot (2+1)flavor lattice QCD simulations with a heavy pion mass (m π 875 MeV) [10] suggests a short-range attraction between N and Ω in the 5 S 2 channel. Subsequently, theoretical studies on the N Ω system [11,12,13,14,15] as well as experimental measurements in relativistic heavy ion collisions [16] have been reported.
The purpose of this Letter is to study N Ω( 5 S 2 ) on the basis of realistic (2+1)-flavor lattice QCD simulations near the physical point (m π 146 MeV and m K 525 MeV). As in the case of our previous pilot study [10], we employ the HAL QCD method [17,18,19] which allows us to extract the interaction between N and Ω from the spacetime dependence of the twobaryon correlation function on the lattice. This paper is organized as follows. In Sec. 2, we introduce the HAL QCD method to extract the hadron interaction from lattice QCD. In Sec. 3, we summarize the setup of our lattice QCD simulations near the physical point. In Sec. 4, we analyze the N Ω system in 5 S 2 channel in detail. Sec. 5 is devoted to summary and concluding remarks.

HAL QCD method
Let us consider the N Ω( 5 S 2 ) characterized by the following two-baryon correlation function, with J N Ω being the wall-type quark source. The interpolating operators for the nucleon and the Ω-baryon are where α and β are Dirac indices, is a spatial label of gamma matrices, a, b, c are the color indices and C ≡ γ 4 γ 2 and Dirac indices are restricted to the upper two components. The summation over the cubic group element R ∈ O leads to a projection onto the S-wave state 1 . On the other hand, the projection operator onto the spin-2 state is chosen to be P (s=2) αβ, Here we neglect the coupling of N Ω( 5 S 2 ) to the D-wave octet-octet channels, ΛΞ and ΣΞ, below the N Ω threshold. In the present paper, we assume the coupling between the S-wave and the D-wave is kinematically suppressed and focus only on the single channel. 2 It is convenient to define the following ratio which we call the "R-correlator", where C N (t) and C Ω (t) are single-baryon correlators. Below the inelastic threshold, R N Ω ( r, t) can be shown to satisfy the integro-differential equation 1 Strictly speaking, this operation projects onto the A + 1 state which contains not only L = 0 state but also L = 4, 6, · · · states in the continuum theory.
2 A recent phenomenological study with such S-wave -D-wave coupling [14] indicates that the contribution from the D-wave channels to the volume integral of the N Ω( 5 S 2 ) interaction is ∼ 1% and is negligible. Nevertheless, further analysis with the coupled-channel HAL QCD method [21] would be necessary in the future to validate our assumption. Note that such an assumption is not justified for N Ω( 3 S 1 ) which can decay into S-wave octetbaryons.
with a non-local and energy-independent kernel U ( r, r ) [19], The central potential in the leading-order (LO) analysis under the derivative expansion U ( r, r ) = n V n ( r)∇ n δ( r − r ) is given by 3 up to O(δ 2 ∂ 3 t )-terms in the right hand side. Spatial and temporal derivatives on the lattice at ( r, t) are calculated in central difference scheme using nearest neighbour points. If R N Ω ( r, t) is dominated by a single state (ground state) at large t, each term in the r.h.s. of Eq. (5) should have no t-dependence. Such a ground state saturation, however, is not necessary to obtain V C (r) in the time-dependent HAL QCD method as long as R N Ω ( r, t) is dominated by the elastic states. In general, each term in the r.h.s. receives t-dependence which provides "signal" instead of "noise" for V C (r). (If there remains residual tdependence in V C (r), the next-to-leading order of the derivative expansion must be taken into account [20].) This is why the data at moderate values of t ∼ 1 fm are sufficient to extract the baryon-baryon interaction in HAL QCD method. 4

Lattice Setup
Gauge configurations are generated by using the (2+1)-flavor lattice QCD with the Iwasaki gauge action at β = 1.82 and the non-perturbatively O(a)improved Wilson quark action with the six APE stout smearing with the smearing parameter ρ = 0.1 at nearly physical quark masses (m π 146 MeV and m K 525 MeV) [25]. The lattice cutoff is a −1 2.333 GeV (a 0.0846 fm) and the lattice volume L 4 is 96 4 , corresponding to La 8.1 fm. This is sufficiently large volume to accommodate two baryons. We employ the wall-type quark source with the Coulomb gauge fixing. The periodic (Dirichlet) boundary condition for the spatial (temporal) direction is imposed for quarks. The quark propagators are obtained by using the domaindecomposed solver [26,27,28,29], and the unified contraction algorithm is employed to calculate the correlation functions [30].
The forward and backward propagations are averaged and the hypercubic symmetry on the lattice (4 rotations) are utilized for each configuration.  Fig. 1 is the R-correlator defined by Eq. (3) in the range t/a = 10 − 15, which are rescaled by the value of r = 3 fm. At large r, the Rcorrelator approaches a constant. This implies that V C (r) in Eq. (5) becomes a constant at long distance. At small r, the R-correlator increases with the second-order derivative in r being always positive, which implies that there is an attractive potential at short distances. The weak t-dependence at small r indicates contributions from the elastic scattering states. As mentioned before, this t-dependence provides signal instead of noise.

Shown in
To extract V C (r) from the R-correlator, we choose t/a = 11 − 14 in order to reduce the systematic uncertainties 5 : For smaller values of t, the inelastic contribution starts to appear so that V C (r) remains non-vanishing even for large r. For larger values of t, it is difficult to control the systematic uncertainties of the fitting of the potential due to the large statistical errors. Note that we take relatively larger values of t/a to make accurate determination of m N and m Ω , whose values agree with the effective masses at t/a = 12 in 1%.
In Fig. 2, V C (r) as well as its breakdown into different components are shown for t/a = 12 as an example. First of all, V C (r) (red squares) is attractive everywhere. This is qualitatively consistent with the result in our pilot study with heavy pion mass (m π 875 MeV) [10]. Also, we found that the H 0 -term (blue circles) is dominant, yet the ∂/∂t-term (green triangles) gives non-negligible r-dependent contribution. On the other hand, the ∂ 2 /∂t 2 -term (orange diamonds) is consistent with zero.
We summarize the central potential V C (r) in Fig. 3(a) for t/a = 11 − 14. These potentials are consistent with each other within statistical errors, which is an indirect evidence of the small coupling with the D-wave octetoctet states below the N Ω threshold in the spin-2 channel as suggested by [14]. (Such a stability of the potential in the same range of t is not found for the spin-1 N Ω system which can couple to the S-wave octet-octet states below threshold.) To obtain observables such as the scattering phase shifts and binding energy, we fit the lattice QCD potential by Gaussian + (Yukawa) 2 with a form factor [10]: The (Yukawa) 2 form at long distance is motivated by the picture of two-pion exchange between N and Ω with an OZI violating vertex [14]. The pion mass in Eq. (6) is taken from our simulation, m π = 146 MeV. After trying both n = 1 and 2 in the form factor, we found that only n = 1 can reproduce the short distance behavior of the lattice potential, so that we will focus on the n = 1 case below. The results of the fit and the corresponding parameters are summarized in Fig. 3(b,c,d,e) and Table 1, respectively. Shown in Fig. 4 (Left) is the S-wave scattering phase shift δ 0 as a function of the kinetic energy. The values of k cot δ 0 are also shown in Fig. 4 (Right). These results for t/a = 11, 12, 13 and 14 are consistent with each other within the statistical errors. In the k → 0 limit, the phase shift approaches to 180 • , and the scattering length, 6 a 0 ≡ − lim k→0 tan δ 0 /k, becomes positive. This implies that the existence of a quasi-bound state of N Ω in the 5 S 2 channel.   The effective range expansion (ERE) of the phase shifts up to the nextleading-order (NLO) reads with r eff being the effective range. The ERE parameters (a 0 , r eff ) obtained from our phase shifts are found to be where the central values and the statistical errors are estimated at t/a = 12, while the systematic errors in the last parentheses are estimated from the central values for t/a = 11, 13 and 14.
In Fig. 5, the ratio r eff /a 0 as a function of r eff for N Ω( 5 S 2 ) is plotted together with the experimental values for N N ( 3 S 1 ) (deuteron) and N N ( 1 S 0 ) (di-neutron) as well as lattice QCD value for ΩΩ( 1 S 0 ) (di-Omega) [7]. Small  values of |r eff /a 0 | in all these cases indicate that these systems are located close to the unitary limit. 7 The binding energy B and the root mean square distance ( r 2 ) of N Ω( 5 S 2 ) are obtained by solving the Schrödinger equation with the potential fitted to our lattice results: B = 1.54(0.30)( +0.04 −0.10 ) MeV, r 2 = 3.77(0.31)( +0.11 −0.01 ) fm. (9) Although the N -Ω is attractive everywhere, the binding energy is as small as ∼1 MeV due to the short range nature of the potential. Accordingly, the root mean square distance is comparable to the scattering length, indicating that the system is loosely bound like the deuteron and the di-Omega. In our pilot study [10], we found B = 18.9(5.0)( +12.1 −1.8 ) MeV for heavy pion mass m π = 875 MeV. The larger magnitude of B than the present result in Eq. (9) originates partly from the heavy masses of N and Ω in [10]  reduce the kinetic energy and thus increase the binding energy. Another reason is that the short-range attraction for heavy pion is relatively stronger.
So far, we have not considered extra attraction in the pΩ − system due to Coulomb attraction. By taking into account the correction V C (r) → V C (r) − α/r with α ≡ e 2 /(4π) = 1/137.036, we obtain the observables, These results for pΩ − ( 5 S 2 ) are summarized in Fig. 6 together with nΩ − ( 5 S 2 ) without Coulomb correction. Before ending this section, let us briefly discuss other possible systematic errors in Eqs. (8), (9) and (10). The first one is the finite volume effect whose typical error would be exp(−2m π (L/2)) exp(−6) 0.25% and is much smaller than the statistical errors in our simulation. The second one is the finite cutoff effect, which is also expected to be small assuming the naive order estimate (Λa) 2 ≤ 1% with the non-perturbative O(a) improvement. The third systematic error is due to the slightly heavy hadron masses (m π = 146 MeV, m N = 955 MeV and m Ω = 1712 MeV). By using the same parameter set for t/a = 12 in Table 1 with m π = 146 MeV kept fixed but with physical baryon masses (m p = 938 MeV and m Ω − = 1672 MeV), we find less binding than Eq. (10) as expected: B pΩ − 2.18(32) MeV and r 2 pΩ − 3.45 (22) fm. On the other hand, if we additionally employ m ± π = 140 MeV for the potential (see Eq. (6)), we find more bounding than Eq. (10) due to smaller pion mass: B pΩ − 3.00(39) MeV and r 2 pΩ − 3.01(16) fm.

Summary
In this paper, we have studied the N -Ω system in the 5 S 2 channel, which is one of the promising candidates for quasi-stable dibaryon, from the (2+1)flavor lattice QCD simulations with nearly physical quark masses (m π 146 MeV and m K 525 MeV). The N -Ω central potential in the 5 S 2 channel obtained by the time-dependent HAL QCD method is found to be attractive in all distances. The scattering length and the effective range obtained by solving the Schrödinger equation using the resultant potential show that N Ω( 5 S 2 ) is close to unitarity similar to the cases of the deuteron (pn) and di-Omega (ΩΩ). The binding energy of pΩ − without (with) the Coulomb attraction is about 1.5 MeV (2.5 MeV), which indicates the existence of a shallow quasi-bound state below the N Ω threshold. In our simulation, we did not find a signature of the strong coupling between N Ω( 5 S 2 ) and ΛΞ or ΣΞ in the D-wave state.
The N Ω( 5 S 2 ) in the unitary regime can be studied in the two-particle correlation measurements in p-p and p-nucleus and nucleus-nucleus collisions as suggested theoretically in [12] and experimentally reported by the STAR Collaboration at RHIC [16]. Phenomenological analyses along this line on the basis of the results in the present paper will be reported elsewhere [32].