In lattice QCD, both direct method and HAL QCD method are used to investigate the two-baryon systems. We show that due to the contamination of the scattering excited states, it is challenging to measure the eigenenergy from the temporal correlation i
n the direct method, while the HAL QCD method can extract the information of the interaction from both scattering states and ground state by using the spatial correlation. We examine the systematic uncertainty of the derivative expansion in the HAL QCD method, which is found to be well under control at the low energies. By using the time-dependent HAL QCD method, we study the nucleon($N$)-Omega($Omega$) system in the $^5$S$_2$ channel with almost physical quark masses at $m_pi simeq 146$ MeV. We find the interaction is attractive at all distances, which produces a quasi-bound state with the binding energy 1.54(0.30)($^{+0.04}_{-0.10}$) MeV. We also consider the extra Coulomb interaction in the $pOmega^{-}$($^5$S$_2$) system, whose binding energy becomes 2.46(0.34)($^{+0.04}_{-0.01}$) MeV. $NOmega$($^5$S$_2$) dibaryon could be searched through two-particle correlations in the heavy ion collision experiments.
Both direct and HAL QCD methods are currently used to study the hadron interactions in lattice QCD. In the direct method, the eigen-energy of two-particle is measured from the temporal correlation. Due to the contamination of excited states, however,
the direct method suffers from the fake eigen-energy problem, which we call the mirage problem, while the HAL QCD method can extract information from all elastic states by using the spatial correlation. In this work, we further investigate systematic uncertainties of the HAL QCD method such as the quark source operator dependence, the convergence of the derivative expansion of the non-local interaction kernel, and the single baryon saturation, which are found to be well controlled. We also confirm the consistency between the HAL QCD method and the Luschers finite volume formula. Based on the HAL QCD potential, we quantitatively confirm that the mirage plateau in the direct method is indeed caused by the contamination of excited states.
We make a detailed comparison between the direct method and the HAL QCD potential method for the baryon-baryon interactions, taking the $XiXi$ system at $m_pi= 0.51$ GeV in 2+1 flavor QCD and using both smeared and wall quark sources. The energy shif
t $Delta E_mathrm{eff}(t)$ in the direct method shows the strong dependence on the choice of quark source operators, which means that the results with either (or both) source are false. The time-dependent HAL QCD method, on the other hand, gives the quark source independent $XiXi$ potential, thanks to the derivative expansion of the potential, which absorbs the source dependence to the next leading order correction. The HAL QCD potential predicts the absence of the bound state in the $XiXi$($^1$S$_0$) channel at $m_pi= 0.51$ GeV, which is also confirmed by the volume dependence of finite volume energy from the potential. We also demonstrate that the origin of the fake plateau in the effective energy shift $Delta E_mathrm{eff}(t)$ at $t sim 1$ fm can be clarified by a few low-lying eigenfunctions and eigenvalues on the finite volume derived from the HAL QCD potential, which implies that the ground state saturation of $XiXi$($^1$S$_0$) requires $t sim 10$ fm in the direct method for the smeared source on $(4.3 mathrm{fm})^3$ lattice, while the HAL QCD method does not suffer from such a problem.
A comparative study between the Luschers finite volume method and the time-dependent HAL QCD method is given for the $XiXi$($^1mathrm{S}_0$) interaction as an illustrative example. By employing the smeared source and the wall source for the interpola
ting operators, we show that the effective energy shifts $Delta E_{rm eff} (t)$ in Luschers method do not agree between different sources, yet both exhibit fake plateaux. On the other hand, the interaction kernels $V(vec{r})$ obtained from the two sources in the HAL QCD method agree with each other already for modest values of $t$. We show that the energy eigenvalues $Delta E(L)$ in finite lattice volumes ($L^3$) calculated by $V(vec{r})$ indicate that there is no bound state in the $XiXi(^1mathrm{S}_0)$ channel at $m_{pi}=0.51$ GeV in 2+1 flavor QCD.
We present a lattice QCD study for the cutoff effects on nuclear forces. Two-nucleon forces are determined from Nambu-Bethe-Salpeter (NBS) wave functions using the HAL QCD method. Lattice QCD simulations are performed employing N_f = 2 clover fermion
configurations at three lattice spacings of a = 0.108, 0.156, 0.215 fm on a fixed physical volume of L^3 x T = (2.5 fm)^3 x 5 fm with a large quark mass corresponding to m_pi = 1.1 GeV. We observe that while the discretization artifact appears at the short range part of potentials, it is suppressed at the long distance region. The cutoff dependence of the phase shifts and scattering length is also presented.
We report the recent progress on the determination of three-nucleon forces (3NF) in lattice QCD. We utilize the Nambu-Bethe-Salpeter (NBS) wave function to define the potential in quantum field theory, and extract two-nucleon forces (2NF) and 3NF on
equal footing. The enormous computational cost for calculating multi-baryon correlators on the lattice is drastically reduced by developing a novel contraction algorithm (the unified contraction algorithm). Quantum numbers of the three-nucleon (3N) system are chosen to be (I, J^P)=(1/2,1/2^+) (the triton channel), and we extract 3NF in which three nucleons are aligned linearly with an equal spacing. Lattice QCD simulations are performed using N_f=2 dynamical clover fermion configurations at the lattice spacing of a = 0.156 fm on a 16^3 x 32 lattice with a large quark mass corresponding to m(pi)= 1.13 GeV. Repulsive 3NF is found at short distance.
We review recent lattice QCD activities with emphasis on the impact on nuclear physics. In particular, the progress toward the determination of nuclear and baryonic forces (potentials) using Nambu-Bethe-Salpeter (NBS) wave functions is presented. We
discuss major challenges for multi-baryon systems on the lattice: (i) signal to noise issue and (ii) computational cost issue. We argue that the former issue can be avoided by extracting energy-independent (non-local) potentials from time-dependent NBS wave functions without relying on the ground state saturation, and the latter cost is drastically reduced by developing a novel unified contraction algorithm. The lattice QCD results for nuclear forces, hyperon forces and three-nucleon forces are presented, and physical insights are discussed. Comparison to results from the traditional Lueschers method is given, and open issues to be resolved are addressed as well.
We explore three-nucleon forces (3NF) from lattice QCD simulations. Utilizing the Nambu-Bethe-Salpeter (NBS) wave function, two-nucleon forces (2NF) and 3NF are determined on the same footing. Quantum numbers of the three-nucleon (3N) system are chos
en to be (I, J^P)=(1/2,1/2^+) (the triton channel). The enormous computational cost is reduced by employing the simplest geometrical configuration, where 3N are aligned linearly with an equal spacing. We perform lattice QCD simulations using Nf=2 dynamical clover fermion configurations generated by CP-PACS Collaboration, at the lattice spacing of a = 0.156 fm on a 16^3 x 32 lattice with a large quark mass corresponding to m(pi) = 1.13 GeV. Repulsive 3NF is found at short distance.
We investigate three-nucleon forces (3NF) from lattice QCD simulations, utilizing the Nambu-Bethe-Salpeter (NBS) wave function to determine two-nucleon forces (2NF) and 3NF on the same footing. Quantum numbers of the three-nucleon (3N) system are cho
sen to be (I, J^P)=(1/2, 1/2^+) (the triton channel). We consider the simplest geometrical configuration where 3N are aligned linearly with an equal spacing, to reduce the enormous computational cost. Lattice QCD simulations are performed using Nf=2 dynamical clover fermion configurations at the lattice spacing of a = 0.156 fm on a 16^3 x 32 lattice with a large quark mass corresponding to m(pi) = 1.13 GeV. We find repulsive 3NF at short distance.
We investigate baryon-baryon interactions with strangeness $S=-2$ and isospin I=0 system from Lattice QCD. In order to solve this system, we prepare three types of baryon-baryon operators ($Lambda-Lambda$, $N-Xi$ and $Sigma-Sigma$) for the sink and c
onstruct three source operators diagonalizing the $3times3$ correlation matrix. Combining of the prepared sink operators with the diagonalized source operators, we obtain nine effective Nambu-Bethe-Salpeter (NBS) wave functions. The $3times3$ potential matrix is calculated by solving the coupled-channel Schrodinger equation. The flavor SU(3) breaking effects of the potential matrix are also discussed by comparing with the results of the SU(3) limit calculation. Our numerical results are obtained from three sets of 2+1 flavor QCD gauge configurations provided by the CP-PACS/JLQCD Collaborations.
