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 in
puts to determine the HAL QCD potential in the derivative expansion, (1) energy eigenfunctions (2) time-dependent wave functions as solutions to the time dependent Schrodinger 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 volume effect. We have found that, for all three cases, the potentials provide reasonable scattering phase shifts even at the leading order of the derivative expansion, and they give more accurate results as the order of the expansion increases. By comparing the above results with those from the formal derivative expansion for the separable potential, we conclude that the derivative expansion is not a way to obtain the potential but a method to extract physical observables such as phase shifts and binding energies, and that the scattering phase shifts from the derivative expansion in the HAL QCD method converge to the exact ones much faster than those from the formal derivative expansion of the separable potential.
We investigate the $I=1$ $pi pi$ 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 $m_{pi} approx 411$ MeV, in which the $rh
o$ 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 $I=1$ $pi pi$ potential is determined at the next-to-next-to-leading order (N$^2$LO) of the derivative expansion for the first time, and the resonance parameters of the $rho$ meson are extracted. The obtained $rho$ meson mass is found to be consistent with the value in the literature, while the value of the coupling $g_{rho pi pi}$ turns out to be somewhat larger. The latter observation is most likely attributed to the lack of low-energy information in our lattice setup with the center-of-mass frame. Such a limitation may appear in other P-wave resonant systems and we discuss possible improvement in future. With this caution in mind, we positively conclude that we can reasonably extract the N$^2$LO potential and resonance parameters even in the system requiring the all-to-all propagators in the HAL QCD method, which opens up new possibilities for the study of resonances in lattice QCD.
We propose a new class of vector fields to construct a conserved charge in a general field theory whose energy momentum tensor is covariantly conserved. We show that there always exists such a vector field in a given field theory even without global
symmetry. We also argue that the conserved current constructed from the (asymptotically) time-like vector field can be identified with the entropy current of the system. As a piece of evidence we show that the conserved charge defined therefrom satisfies the first law of thermodynamics for an isotropic system with a suitable definition of temperature. We apply our formulation to several gravitational systems such as the expanding universe, Schwarzschild and BTZ black holes, and gravitational plane waves. We confirm the conservation of the proposed entropy density under any homogeneous and isotropic expansion of the universe, the precise reproduction of the Bekenstein-Hawking entropy incorporating the first law of thermodynamics, and the existence of gravitational plane wave carrying no charge, respectively. We also comment on the energy conservation during gravitational collapse in simple models.
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 $m_{pi} approx 570$ 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 find the stronger repulsive interaction in the $I=1$ channel than in the $I=0$. The phase shifts obtained by solving the Schr{o}dinger equations with the potentials qualitatively reproduce the energy dependence of the experimental phase shifts, and have the similar behavior to the previous results from lattice QCD without all-to-all propagators. Our study demonstrates that the all-to-all quark propagator technique with the one-end trick is useful to study interactions for meson-baryon systems in the HAL QCD method, so that we will apply it to meson-baryon systems which contain quark-antiquark creation/annihilation processes in our future studies.
We present a precise definition of a conserved quantity from an arbitrary covariantly conserved current available in a general curved spacetime with Killing vectors. This definition enables us to define energy and momentum for matter by the volume in
tegral. As a result we can compute charges of Schwarzschild and BTZ black holes by the volume integration of a delta function singularity. Employing the definition we also compute the total energy of a static compact star. It contains both the gravitational mass known as the Misner-Sharp mass in the Oppenheimer-Volkoff equation and the gravitational binding energy. We show that the gravitational binding energy has the negative contribution at maximum by 68% of the gravitational mass in the case of a constant density. We finally comment on a definition of generators associated with a vector field on a general curved manifold.
We take a first step towards a holographic description of a black hole by means of a flow equation. We consider a free theory of multiple scalar fields at finite temperature and study its holographic geometry defined through a free flow of the scalar
fields. We find that the holographic metric has the following properties: i) It is an asymptotic Anti-de Sitter (AdS) black brane metric with some unknown matter contribution. ii) It has no coordinate singularity and milder curvature singularity. iii) Its time component decays exponentially at a certain AdS radial slice. We find that the matter spreads all over the space, which we speculate to be due to thermal excitation of infinitely many massless higher spin fields. We conjecture that the above three are generic features of a black hole holographically realized by the flow equation method.
In this paper, we investigate the HAL QCD potential in the $I=1$ $pi pi$ scattering using the hybrid method for all-to-all propagators, in which a propagator is approximated by low-eigenmodes and the remaining high-eigenmode part is stochastically es
timated. To verify the applicability of the hybrid method to systems containing quark creation$/$annihilation contributions such as the $rho$ meson, we calculate the $I=1$ $pipi$ potential with the 2+1 flavor gauge configurations on $16^3 times 32$ lattice at the lattice spacing $a approx 0.12$ fm and $(m_{pi},m_{rho}) approx (870, 1230)$ MeV, in which the $rho$ meson appears as a deeply-bound state. While we find that the naive stochastic evaluations for quark creation$/$annihilation contributions lead to extremely large statistical fluctuations, additional noise reduction methods enable us to obtain a sufficiently precise potential, which shows a strong attractive force. We also confirm that the binding energy and $k^3 cot delta$ obtained from our potential are roughly consistent with an existing $rho$ meson bound state, within a large systematic error associated with our calculation, whose possible origin is also discussed.
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.
We present the recent study on dibaryons at the almost physical pion mass in lattice QCD by the HAL QCD potential method.
We investigate the high-temperature phase of QCD using lattice QCD simulations with $N_f = 2$ dynamical Mobius domain-wall fermions. On generated configurations, we study the axial $U(1)$ symmetry, overlap-Dirac spectra, screening masses from mesonic
correlators, and topological susceptibility. We find that some of the observables are quite sensitive to lattice artifacts due to a small violation of the chiral symmetry. For those observables, we reweight the Mobius domain-wall fermion determinant by that of the overlap fermion. We also check the volume dependence of observables. Our data near the chiral limit indicates a strong suppression of the axial $U(1)$ anomaly at temperatures $geq$ 220 MeV.
