In this article, we review the HAL QCD method to investigate baryon-baryon interactions such as nuclear forces in lattice QCD. We first explain our strategy in detail to investigate baryon-baryon interactions by defining potentials in field theories
such as QCD. We introduce the Nambu-Bethe-Salpeter (NBS) wave functions in QCD for two baryons below the inelastic threshold. We then define the potential from NBS wave functions in terms of the derivative expansion, which is shown to reproduce the scattering phase shifts correctly below the inelastic threshold. Using this definition, we formulate a method to extract the potential in lattice QCD. Secondly, we discuss pros and cons of the HAL QCD method, by comparing it with the conventional method, where one directly extracts the scattering phase shifts from the finite volume energies through the Luschers formula. We give several theoretical and numerical evidences that the conventional method combined with the naive plateau fitting for the finite volume energies in the literature so far fails to work on baryon-baryon interactions due to contaminations of elastic excited states. On the other hand, we show that such a serious problem can be avoided in the HAL QCD method by defining the potential in an energy-independent way. We also discuss systematics of the HAL QCD method, in particular errors associated with a truncation of the derivative expansion. Thirdly, we present several results obtained from the HAL QCD method, which include (central) nuclear force, tensor force, spin-orbital force, and three nucleon force. We finally show the latest results calculated at the nearly physical pion mass, $m_pi simeq 146$ MeV, including hyperon forces which lead to form $OmegaOmega$ and $NOmega$ dibaryons.
The $S$-wave $LambdaLambda$ and $N Xi$ interactions are studied on the basis of the (2+1)-flavor lattice QCD simulations close to the physical point ($m_pi simeq 146{rm{MeV}}$ and $m_K simeq 525{rm{MeV}}$). Lattice QCD potentials in four different sp
in-isospin channels are extracted by using the coupled-channel HAL QCD method and are parametrized by analytic functions to calculate the scattering phase shifts. The $Lambda Lambda$ interaction at low energies shows only a weak attraction, which does not provide a bound or resonant dihyperon. The $NXi$ interaction in the spin-singlet and isospin-singlet channel is most attractive and lead the $NXi$ system near unitarity. Relevance to the strangeness=$-2$ hypernuclei as well as to two-baryon correlations in proton-proton, proton-nucleus and nucleus-nucleus collisions is also discussed.
There exist two methods to study two-baryon systems in lattice QCD: the direct method which extracts eigenenergies from the plateaux of the temporal correlator and the HAL QCD method which extracts observables from the non-local potential associated
with the tempo-spatial correlator. Although the two methods should give the same results theoretically, qualitatively different results have been reported. Recently, we pointed out that the separation of the ground state from the excited states is crucial to obtain sensible results in the former, while both states provide useful signals in the latter. In this paper, we identify the contribution of each state in the direct method by decomposing the two-baryon correlators into the finite-volume eigenmodes obtained from the HAL QCD method. We consider the $XiXi$ system in the $^1$S$_0$ channel at $m_pi = 0.51$ GeV in 2+1 flavor lattice QCD using the wall and smeared quark sources. We demonstrate that the pseudo-plateau at early time slices (t = 1~2 fm) from the smeared source in the direct method indeed originates from the contamination of the excited states, and the true plateau with the ground state saturation is realized only at t > 5~15 fm corresponding to the inverse of the lowest excitation energy. We also demonstrate that the two-baryon operator can be optimized by utilizing the finite-volume eigenmodes, so that (i) the finite-volume energy spectra from the HAL QCD method agree with those from the optimized temporal correlator and (ii) the correct spectra would be accessed in the direct method only if highly optimized operators are employed. Thus we conclude that the long-standing issue on the consistency between the Luschers finite volume method and the HAL QCD method for two baryons is now resolved: They are consistent with each other quantitatively only if the excited contamination is properly removed in the former.
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 em
ployed to convert the lattice QCD data of the two-baryon correlation function to the baryon-baryon potential and eventually to the scattering observables. The $NOmega$($^5$S$_2$) potential, obtained under the assumption that its couplings to the D-wave octet-baryon pairs are small, 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 $pOmega^-$($^5$S$_2$) becomes $B_{pOmega^-} = 2.46(0.34)(^{+0.04}_{-0.11})$~MeV. Such a spin-2 $pOmega^-$ state could be searched through two-particle correlations in $p$-$p$, $p$-nucleus and nucleus-nucleus collisions.
The $XiXi$ interaction in the $^1$S$_0$ channel is studied to examine the convergence of the derivative expansion of the non-local HAL QCD potential at the next-to-next-to-leading order (N$^2$LO). We find that (i) the leading order potential from the
N$^2$LO analysis gives the scattering phase shifts accurately at low energies, (ii) the full N$^2$LO potential gives only small correction to the phase shifts even at higher energies below the inelastic threshold, and (iii) the potential determined from the wall quark source at the leading order analysis agrees with the one at the N$^2$LO analysis except at short distances, and thus, it gives correct phase shifts at low energies. We also study the possible systematic uncertainties in the HAL QCD potential such as the inelastic state contaminations and the finite volume artifact for the potential and find that they are well under control for this particular system.
We present the latest lattice QCD results for baryon interactions obtained at nearly physical quark masses. $N_f = 2+1$ nonperturbatively ${cal O}(a)$-improved Wilson quark action with stout smearing and Iwasaki gauge action are employed on the latti
ce of $(96a)^4 simeq (8.1mbox{fm})^4$ with $a^{-1} simeq 2.3$ GeV, where $m_pi simeq 146$ MeV and $m_K simeq 525$ MeV. In this report, we study the two-nucleon systems and two-$Xi$ systems in $^1S_0$ channel and $^3S_1$-$^3D_1$ coupled channel, and extract central and tensor interactions by the HAL QCD method. We also present the results for the $NOmega$ interaction in $^5S_2$ channel which is relevant to the $NOmega$ pair-momentum correlation in heavy-ion collision experiments.
A sanity check rules out certain types of obviously false results, but does not catch every possible error. After reviewing such a sanity check for $NN$ bound states with the Luschers finite volume formula[1-3], we give further evidences for the oper
ator dependence of plateaux, a symptom of the fake plateau problem, against the claim in [4]. We then present our critical comments on [5] by NPLQCD: (i) Operator dependences of plateaux in NPL2013[6,7] exist with the $P$-values of 4--5%. (ii) The volume independence of plateaux in NPL2013 does not prove their correctness. (iii) Effective range expansion (ERE) fits in NPL2013 violate the physical pole condition. (iv) Ref.[5] is partly based on new data and analysis different from the original ones[6,7]. (v) A new ERE in Refs.[5,8] does not satisfy the Luschers finite volume formula. [1] T. Iritani et al., JHEP 10 (2016) 101. [2] S. Aoki et al., PoS (LATTICE2016) 109. [3] T. Iritani et al., 1703.0720. [4] T. Yamazaki et al., PoS (LATTICE2017) 108. [5] S.R. Beane et al., 1705.09239. [6] S.R. Beane et al., PRD87 (2013) 034506. [7] S.R. Beane et al., PRC88 (2013) 024003. [8] M.L. Wagman et al., 1706.06550.
On the basis of the Luschers finite volume formula, a simple test (consistency check or sanity check) is introduced and applied to inspect the recent claims of the existence of the nucleon-nucleon ($NN$) bound state(s) for heavy quark masses in latti
ce QCD. We show that the consistency between the scattering phase shifts at $k^2 > 0$ and/or $k^2 < 0$ obtained from the lattice data and the behavior of phase shifts from the effective range expansion (ERE) around $k^2=0$ exposes the validity of the original lattice data, otherwise such information is hidden in the energy shift $Delta E$ of the two nucleons on the lattice. We carry out this sanity check for all the lattice results in the literature claiming the existence of the $NN$ bound state(s) for heavy quark masses, and find that (i) some of the $NN$ data show clear inconsistency between the behavior of ERE at $k^2 > 0$ and that at $k^2 < 0$, (ii) some of the $NN$ data exhibit singular behavior of the low energy parameter (such as the divergent effective range) at $k^2<0$, (iii) some of the $NN$ data have the unphysical residue for the bound state pole in S-matrix, and (iv) the rest of the $NN$ data are inconsistent among themselves. Furthermore, we raise a caution of using the ERE in the case of the multiple bound states. Our finding, together with the fake plateau problem previously pointed out by the present authors, brings a serious doubt on the existence of the $NN$ bound states for pion masses heavier than 300 MeV in the previous studies.
The strangeness $S=-2$ baryon-baryon interaction is investigated directly from the fundamental theory of the strong interaction, QCD. The HAL QCD method enables us to extract baryon interactions from the Nambu-Bethe-Salpeter wave functions without us
ing any experimental information. We present our latest result on the $S = -2$ baryon interactions and discuss the H-dibaryon state using potentials which are calculated by using the (almost) physical point gauge configurations with large lattice volume of$(8.1{rm{fm}})^4$ generated on the K-computer.
We present lattice QCD results of baryon-baryon potentials in S=-3 sector, i.e., XiSigma (I=3/2) potentials and XiLambda-XiSigma coupled channel potentials (I=1/2) by using the 2+1 flavor gauge configurations with almost the physical quark masses gen
erated on 96^4 lattice with 1/a simeq 2.3 GeV and L = 96a simeq 8.1 fm where m_pi simeq 146 MeV and m_K simeq 525 MeV. These potentials are obtained based on the time-dependent HAL QCD method with a non-relativistic approximation. Qualitative behaviors of the results are found to be consistent with those in the flavor SU(3) limit.
