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 Lüscher's 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, mπ ≃ 146 MeV, including hyperon forces which lead to form ΩΩ and NΩ dibaryons.
The nucleon(N)-Omega(Ω) system in the S-wave and spin-2 channel (S25) is studied from the (2+1)-flavor lattice QCD with nearly physical quark masses (mπ≃146MeV and mK≃525MeV). 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 NΩ(S25) 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 a0=5.30(0.44)(−0.01+0.16)fm, reff=1.26(0.01)(−0.01+0.02)fm, B=1.54(0.30)(−0.10+0.04)MeV, respectively. Including the extra Coulomb attraction, the binding energy of pΩ−(S25) becomes BpΩ−=2.46(0.34)(−0.11+0.04)MeV. Such a spin-2 pΩ− state could be searched through two-particle correlations in p-p, p-nucleus and nucleus-nucleus collisions.
We propose a novel algorithm for calculating multi-baryon correlation functions on the lattice. By considering the permutation of quarks (Wick contractions) and color/spinor contractions ...simultaneously, we construct a unified index list for the contraction where the redundancies in the original contraction are eliminated. We find that a significant reduction in the computational cost of correlators is achieved, e.g., by a factor of 192 for 3H and 3He nuclei, and a factor of 20736 for the 4He nucleus, without assuming isospin symmetry. A further reduction is possible by exploiting isospin symmetry, and/or interchange symmetries associated with sink baryons, if such symmetries exist. Extensions for systems with hyperons are presented as well.
A novel quantum-classical hybrid scheme is proposed to efficiently solve large-scale combinatorial optimization problems. The key concept is to introduce a Hamiltonian dynamics of the classical flux ...variables associated with the quantum spins of the transverse-field Ising model. Molecular dynamics of the classical fluxes can be used as a powerful preconditioner to sort out the frozen and ambivalent spins for quantum annealers. The performance and accuracy of our smooth hybridization in comparison to the standard classical algorithms (the tabu search and the simulated annealing) are demonstrated by employing the MAX-CUT and Ising spin-glass problems.
Baryon–baryon potentials are obtained from 3-flavor QCD simulations with the lattice volume L≃4 fm, the lattice spacing a≃0.12 fm, and the pseudo-scalar-meson mass Mps=469–1171 MeV. The NN scattering ...phase-shifts and the mass of H-dibaryon in the flavor SU(3) limit are extracted from the resultant potentials by solving the Schrödinger equation. The NN phase-shifts in the SU(3) limit is shown to have qualitatively similar behavior as the experimental data. A bound H-dibaryon in the SU(3) limit is found to exist in the flavor-singlet JP=0+ channel with the binding energy of about 26 MeV for the lightest quark mass Mps=469 MeV. Effect of flavor SU(3) symmetry breaking on the H-dibaryon is estimated by solving the coupled-channel Schrödinger equation for ΛΛ–NΞ–ΣΣ with the physical baryon masses and the potential matrix obtained in the SU(3) limit: a resonant H-dibaryon is found between ΛΛ and NΞ thresholds in this treatment.
Imaginary-time Nambu–Bethe–Salpeter (NBS) wave function is introduced to extend our previous approach for hadron–hadron interactions on the lattice. Scattering states of hadrons with different ...energies encoded in the NBS wave function are utilized to extract non-local hadron–hadron potential. “The ground state saturation”, which is commonly used in lattice QCD but is hard to be achieved for multi-baryons, is not required. We demonstrate that the present method works efficiently for the nucleon–nucleon interaction (the potential and the phase shift) in the S01 channel.
A
bstract
There exist two methods to study two-baryon systems in lattice QCD: the direct method which extracts eigenenergies from the plateaux of the temporal correlation function and the HAL QCD ...method which extracts observables from the non-local potential associated with the tempo-spatial correlation function. Although the two methods should give the same results theoretically, there have been reported qualitative difference for observables from lattice QCD simulations. Recently, we pointed out in
1
,
2
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 for observables in the latter. In this paper, we identify the contribution of each state in the direct method by decomposing the two-baryon correlation functions into the finite-volume eigenmodes obtained from the HAL QCD method. As in our previous studies, we consider the ΞΞ system in the
1
S
0
channel at
m
π
= 0.51 GeV in (2+1)-flavor lattice QCD using the wall and smeared quark sources with spatial extents,
La
= 3.6, 4.3, 5.8 fm. 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 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 temporal correlation function with the optimized operators and (ii) the correct finite-volume 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 Lüscher’s finite volume method and the HAL QCD method for two baryons is now resolved at least for this particular system considered here: they are consistent with each other quantitatively only if the excited contamination is properly removed in the former.
The existence of a nucleon–ϕ (N–ϕ) bound state has been subject of theoretical and experimental investigations for decades. In this letter, indication of a p–ϕ bound state is found, using for the ...first time two-particle correlation functions as alternative to invariant mass spectra. Newly available lattice calculations for the spin 3/2 N–ϕ interaction by the HAL QCD collaboration are used to constrain the spin 1/2 counterpart from the fit of the experimental p–ϕ correlation function measured by ALICE. The corresponding scattering length and effective range are f0(1/2)=(−1.54−0.53+0.53(stat.)−0.09+0.16(syst.)+i⋅0.00−0.00+0.35(stat.)−0.00+0.16(syst.)) fm and d0(1/2)=(0.39−0.09+0.09(stat.)−0.03+0.02(syst.)+i⋅0.00−0.04+0.00(stat.)−0.02+0.00(syst.)) fm, respectively. The results imply the appearance of a p–ϕ bound state with an estimated binding energy in the range of 12.8−56.1 MeV.
The ΔΔ dibaryon resonance d⁎(2380) with (JP,I)=(3+,0) is studied theoretically on the basis of the 3-flavor lattice QCD simulation with heavy pion masses (mπ=679,841 and 1018 MeV). By using the HAL ...QCD method, the central Δ-Δ potential in the S37 channel is obtained from the lattice data with the lattice spacing a≃0.121 fm and the lattice size L≃3.87 fm. The resultant potential shows a strong short-range attraction, so that a quasi-bound state corresponding to d⁎(2380) is formed with the binding energy 25-40 MeV below the ΔΔ threshold for the heavy pion masses. The tensor part of the transition potential from ΔΔ to NN is also extracted to investigate the coupling strength between the S-wave ΔΔ system with JP=3+ and the D-wave NN system. Although the transition potential is strong at short distances, the decay width of d⁎(2380) to NN in the D-wave is kinematically suppressed, which justifies our single-channel analysis at the range of the pion mass explored in this study.
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