We give a formula to calculate a matrix element of a conserved current in the effective quantum mechanics defined by the wave function equivalent potentials proposed by HAL QCD collaboration. As a first step, a non-relativistic field theory with two channel coupling is considered as the original theory, with which a wave function equivalent HAL QCD potential is obtained in a closed analytic form. The external field method is used to derive the formula by demanding that the result should agree with the original theory. With this formula, the matrix element is obtained by sandwiching the effective current operator between the left and the right eigen functions of the effective Hamiltonian associated with the HAL QCD potential. In addition to the naive one-body current, the effective current operator contains an additional two-body term emerging from the degrees of freedom which has been integrated out.
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 …
In this paper, employing an all-to-all quark propagator technique, we investigate the kaon-nucleon interactions in lattice QCD. We calculate the S-wave kaon-nucleon potentials at the leading order in the derivative expansion in the time-dependent HAL QCD method, using (2+1)-flavor gauge configurations at the lattice spacing $a \approx 0.09$ fm on $32^3 \times 64$ lattices and the pion mass $m_π \approx 570$ MeV. We take the one-end trick for all-to-all propagators, which allows us to put the zero momentum hadron operators at both source and sink and to smear quark operators at the source. We f…
We study decuplet baryons from meson-baryon interactions, in particular, $Δ$ and $Ω$ baryons from P-wave $I=3/2$ $Nπ$ and $I=0$ $Ξ\bar{K}$ interactions, respectively. The interaction potentials are calculated in the HAL QCD method using 3-quark-type source operators at $m_π \approx 410~\textrm{MeV}$. We use the conventional stochastic estimation of all-to-all propagators combined with the all-mode averaging to reduce statistical fluctuations. We have found that two potentials have quite similar behaviors, suggesting that a mass difference between $Δ$ and $Ω$ comes mainly from a difference of k…
In this article, we report the $ρ$ resonance study using the HAL QCD method. We calculate the $I=1$ $ππ$ potential at $m_π \approx 0.41$ GeV by a combination of the one-end trick, sequential propagator and covariant approximation averaging (CAA). Thanks to those techniques, we determine the non-local $I=1$ $ππ$ potential at the next-to-next-to-leading order (N$^2$LO) of the derivative expansion for the first time and obtain the pole of the S-matrix corresponding to the $ρ$ resonance. We also discuss the comparison between our result and a previous calculation, done by Lüscher's method.
We consider the HAL QCD method in the system with non-zero total momentum (laboratory frame). We derive a relation between the NBS wave function in the laboratory frame and the energy-independent non-local potential (HAL QCD potential), and propose the time-dependent method to extract the potential from correlation functions in the laboratory frame. We then apply this formulation to the $I=2$ $ππ$ system to calculate the corresponding potential in the laboratory frame, employing the 2+1 flavor gauge configuration on a $32^3\times 64$ lattice at the lattice spacing $a\simeq 0.091$ fm and $m_π\s…
We report our preliminary progress in the calculation of the interquark potential of bottomonium at non-zero temperature using the HAL QCD method. We use NRQCD correlation functions of non-local mesonic S-wave states to obtain the central potential as a function of temperature. These results have been obtained using our anisotropic 2+1 flavour "Generation 2" FASTSUM ensembles.
We study the convergence of the derivative expansion in HAL QCD method from the finite volume analysis. Employing the (2+1)-flavor lattice QCD data obtained at nearly physical light quark masses $(m_π, m_K) \simeq (146, 525)$ MeV and the physical charm quark mass, we study two representative systems, $ΩΩ$ and $Ω_{ccc}Ω_{ccc}$ in the $^1S_0$ channel, where both systems were found to have a shallow bound state in our previous studies. The HAL QCD potentials are determined at the leading-order in the derivative expansion, from which finite-volume eigenmodes are obtained. Utilizing the eigenmode p…
We report progress in the calculation of the thermal interquark potential of bottomonium using the HAL QCD method applied to bottom quarks in the non-relativistic approximation (i.e. NRQCD). We exploit the fast Fourier transform algorithm, using a momentum space representation, to efficiently calculate NRQCD correlation functions of non-local mesonic S-wave states, and thus obtain the potential for temperatures in both the hadronic and plasma phases. This work was performed on our anisotropic 2+1 flavour ''Generation 2'' FASTSUM ensembles.
We investigate the $I=1$ $ππ$ interaction using the HAL QCD method in lattice QCD. We employ the (2+1)-flavor gauge configurations on $32^3 \times 64$ lattice at the lattice spacing $a \approx 0.0907$ fm and $m_π \approx 411$ MeV, in which the $ρ$ meson appears as a resonance state. We find that all-to-all propagators necessary in this calculation can be obtained with reasonable precision by a combination of three techniques, the one-end trick, the sequential propagator, and the covariant approximation averaging (CAA). The non-local $I=1$ $ππ$ potential is determined at the next-to-next-to-lea…
We investigate a doubly-bottomed tetraquark state $T_{bb}$ $(bb \bar{u}\bar{d})$ with quantum number $I(J^P)=0(1^+)$ in $(2+1)$-flavor lattice QCD. Using the Non-Relativistic QCD (NRQCD) quark action for $b$ quarks, we have extracted the coupled channel potential between $\bar{B}\bar{B}^*$ and $\bar{B}^* \bar{B}^*$ in the HAL QCD method at $a \approx 0.09$ {fm} on $32^3\times 64$ lattices. The potential predicts an existence of a bound $T_{bb}$ below the $\bar{B}\bar{B}^*$ threshold. At the physical pion mass $m_π\approx140$ {MeV} extrapolated from $m_π\approx 410,\, 570,\, 700$ {MeV}, a bindi…
We investigate how the derivative expansion in the HAL QCD method works to extract physical observables, using a separable potential in quantum mechanics, which is solvable but highly non-local in the coordinate system. We consider three cases for inputs to determine the HAL QCD potential in the derivative expansion, (1) energy eigenfunctions (2) time-dependent wave functions as solutions to the time dependent Schrödinger equation with some boundary conditions (3) time-dependent wave function made by a linear combination of finite number of eigenfunctions at low energy to mimic the finite volu…
A set of optimized interpolating operators which are dominantly coupled to each eigenstate of two baryons on the lattice is constructed by the HAL QCD method. To test its validity, we consider heavy dibaryons $Ω_{3Q}Ω_{3Q}$ ($Q=s,c$) calculated by (2+1)-flavor lattice QCD simulations with nearly physical pion mass. The optimized two-baryon operators are shown to provide effective energies of the ground and excited states separately stable as a function of the Euclidean time. Also they agree to the eigenenergies in a finite lattice box obtained from the leading-order HAL QCD potential $V({\bold…
In this study, we extend the HAL QCD method to a case where a total momentum of a two-particle system is non-zero and apply it to the $I=2$ S-wave $ππ$ scattering in order to confirm its validity. We derive a fundamental relation of an energy-independent non-local potential defined in the center of mass frame with NBS wave functions in a laboratory frame. Based on the relation, we propose the time-dependent method to extract potentials, often used in practice for the HALQCD method in the center of mass frame. For numerical simulations in the $I=2$ $ππ$ system, we employ (2+1)-flavor gauge conf…
