We respond to the comment by Matteo Giordano [1] on our article [2]. We find – and [1] itself acknowledges – that the proposed counterexamples intended to refute our argument violate a crucial assumption of QCD at high temperatures, namely that every gluonic observable is an analytic function of the squared quark mass, m^2. We further point out a technical mistake found in [1]. We conclude that the arguments presented in [1] are not valid.

We investigate S-wave kaon-nucleon (KN) interactions with strangeness S 1/4 & thorn;1 in lattice QCD using the time-dependent HAL QCD method. Employing the (2 & thorn; 1)-flavor gauge configuration with m pi 137 MeV and mK 502 MeV, we calculate the KN potentials at the leading order in the derivative expansion. The potentials in both isospin channels (I 1/4 1 and I 1/4 0) exhibit repulsion at short distances, while only the I 1/4 0 potential has a small attractive pocket at intermediate distances. The phase shifts computed from these potentials show no signals corresponding to resonances or bound states in both isospin channels, suggesting the absence of the Theta & thorn;& eth;1540 & THORN; pentaquark in the S-wave KN systems. The scattering lengths result in aI1/41 01/4 -0.226 & eth;5 & THORN;& eth;& thorn;5-0 & THORN; fm and aI01/40 1/4 & thorn;0.028 & eth;61 & THORN;& eth;& thorn;3 I 1/4 1 are consistent with some of the experimental data within 2-3 sigma, while they deviate from others. The results for I 1/4 0, combined with recent studies on chiral perturbation theory, suggest that the scattering amplitudes in this channel are dominated by P-wave components rather than S-wave.

A systematic framework for constructing optimized interpolating operators strongly coupled to QCD two-particle states is developed, which is achieved by incorporating interhadron spatial wave functions. To efficiently implement these operators in lattice QCD, a novel quark smearing technique utilizing noise vectors is proposed. Applied to the Ω c c c Ω c c c system, these optimized operators prove superior to combinations of limited plane-wave operators, enabling the resolution of distinct eigenstates separated by only ∼ 5 MeV near the threshold 2 m Ω c c c ≃ 9700 MeV . This exceptional resolving power opens new possibilities for studies of a wide range of hadronic systems in QCD.

We report on the ongoing study of symmetry of N_f=2 QCD around the critical temperature. Our simulations of N_f = 2 QCD employ the Möbius domain-wall fermion action with residual mass ∼ 1 or less, maintaining a good chiral symmetry. Using the screening masses from the two point spatial correlators we compare the mass difference between channels connected through various symmetry transformations. Our analysis focuses on restoration of the SU(2)_L× SU(2)_R as well as anomalously broken axial U(1)_A. We also present additional study of a potential SU(2)_CS symmetry which may emerge at sufficiently high temperatures.

We present two analytical models of gravitational collapse toward the Schwarzschild black hole, starting from the interior of the revisited Schwarzschild solution recently reported [Phys. Rev. D 109, 104032 (2024)]. Both models satisfy some energy conditions at all times as long as the collapse is slower than some limit. While a singularity of the Schwarzschild black hole at the origin (R mu nu alpha beta R mu nu alpha beta similar to r-6) forms immediately after the start of the collapse in one model, such a singularity never appears at finite time during the collapse (except t -> infinity) in the other model. The scheme used shows great potential for studying in detail the appearance of singularities in general relativity.

This short report is dedicated to the 40th anniversary of International Journal of Modern Physics A (IJMPA) and Modern Physics Letters A (MPLA). While the report is based on a series of papers [S. Aoki, T. Onogi and S. Yokoyama, Int. J. Mod. Phys. A 36, 2150098 (2021), doi:10.1142/S0217751X21500986; S. Aoki, T. Onogi and S. Yokoyama, Int. J. Mod. Phys. A 36, 2150201 (2021), doi:10.1142/S0217751X21502018; S. Aoki and T. Onogi, Int. J. Mod. Phys. A 37, 2250129 (2022), doi:10.1142/S0217751X22501299; S. Aoki, Prog. Theor. Exp. Phys. 2022, 123A02 (2022), doi:10.1093/ptep/ptac160; S. Aoki, Int. J. Mod. Phys. A 38, 2350120 (2023), doi:10.1142/S0217751X23501208; S. Aoki and K. Kawana, Int. J. Mod. Phys. A 38, 2350072 (2023), doi:10.1142/S0217751X23500720; S. Aoki, T. Onogi and T. Yamaoka, Int. J. Mod. Phys. A 39, 2450049 (2024), doi:10.1142/S0217751X24500490; S. Aoki, Y. Hidaka, K. Kawana and K. Shimada, Eur. Phys. J. C 85, 419 (2025), doi:10.1140/epjc/s10052-025-14138-5.], its content reflects my personal viewpoints. Therefore I am solely responsible for all the statements in the report. In this report we discuss conservation of energies in a curved spacetime including general relativity. We argue that the matter energy is not necessarily conserved in a curved space time due to a lack of time translational invariance, and adding energy of gravitational fields to recover the conservation law of energies fails due to Noether’s second theorem. We show that there exist conserved quantities associated with the matter energy momentum tensor other than the energy in a curved spacetime, and discuss consequences of the existence of such conserved charges.

I report recent developments on hadron interactions in lattice QCD by the HAL QCD method. As an introduction, I summarize the lattice community's consensus on the absence of the deeply bound dinucleons at heavier pion masses. We then present 4 results by the HAL QCD method using 2+1 flavor QCD gauge configurations at m_π≃ 146 MeV and a≃ 0.0846 fm, which are the ΛΛ-NΞ interactions, NΩ dibaryon, the tetra-quark state T_cc and the N-ϕ interaction with the 2-pion tail. A brief summary follows.

We investigate how the left-hand cut (LHC) problem is treated in the HAL QCD method. For this purpose, we first consider the effect of the LHC to the scattering problem in non-relativistic quantum mechanics with potentials. We show that the $S$-matrix or the scattering phase shift obtained from the potential including the Yukawa term ($e^{- m_\pi r}/r$) with the infra-red (IR) cutoff $R$ is well-defined even for the complex momentum $k$ as long as $R$ is finite, and they are compared with those obtained by the analytic continuation without the IR cutoff. In the $R\to\infty$ limit, the phase shift approaches the result from the analytic continuation at ${\rm Im}\, k < m_\pi/2$, while they differ at ${\rm Im}\, k > m_\pi/2$, except $k= k_b$, where $k_b$ is the binding momentum. We also observe that $k_b$ can be correctly obtained even at finite but large $R$. Using knowledge obtained in the non-relativistic quantum mechanics, we present how we should treat the LHC in the HAL QCD potential method.

Using the HAL QCD method, we investigate S-wave meson-baryon interactions in singlet and two octet channels in the flavor SU(3) limit, where the chiral unitary model predicts that a combination of bound-state poles in these channels corresponds to Λ(1405). To avoid the singular behavior of the leading-order potentials of these channels in the derivative expansion, we instead employ a separable potential in the time-dependent HAL QCD method. To calculate all-to-all propagators in the three-point correlation functions including Λ-baryon source operators with zero momentum, we employ the conventional stochastic estimation combined with the covariant approximation averaging. Separable potentials both in the singlet and octet channels show attraction without singular behavior. Our results of the corresponding phase shifts indicate that the attractive interaction in the singlet channel is stronger than that in the octet. Binding energies are consistent with the estimates from the two-point correlation function within one (two) sigma for the singlet (octet) channel, and this ordering of binding energies is consistent with the mass hierarchy suggested by the chiral unitary model.

A systematic way to constructing optimized interpolating operators for two-hadron systems is developed by incorporating inter-hadron spatial wavefunctions. The wavefunctions can be obtained from an iterative process with an appropriate initial guess. To implement these operators, a novel quark smearing technique utilizing Z_3 noise vectors is proposed, which allows for effectively incorporating inter-hadron spatial wavefunctions at the source without using all-to-all quark propagators. Proof-of-principle application to the Ω_cccΩ_ccc system using physical-point lattice configurations with a large size La≃8.1 fm shows that optimized operators enables clear identification of states around 2m_Ω_ccc≃ 9700 MeV with the energy gap as narrow as ∼ 5 MeV. A comparison on correlation functions, effective energies, and HAL QCD potentials between unoptimized operators and optimized operators is given, with a special emphasis on the effects from nearby elastic scattering states. Potential applicability of the optimized operator to various two-hadron systems and its relation to the variational method are also discussed.

We derive the GKP-Witten relation in terms of correlation functions by symmetry without referring to a Lagrangian or the large N expansion. By constructing bulk operators from boundary operators in conformal field theory (CFT) by the conformal smearing, we first determine bulk-boundary 2-pt functions for an arbitrary spin using both conformal and bulk symmetries, then evaluate their small z behaviors, where z is the (d+1)-th coordinate in the bulk. Next, we explicitly determine small z behaviors of bulk-boundary-boundary 3-pt functions also by the symmetries, while small z behaviors of correlation functions among one bulk and n boundary operators with n≥ 3 are fixed by the operator product expansion (OPE). Combining all results, we construct the GKP-Witten relation in terms of these correlation functions at all orders in an external source J. We compare our non-Lagrangian approach with the standard approach employing the bulk action. Our results indicate that the GKP-Witten relation holds not only for holographic CFTs but also for generic CFTs as long as certain conditions are satisfied.

We provide an improved definition of new conserved quantities derived from the energy–momentum tensor in curved spacetime by introducing an additional scalar function. We find that the conserved current and the associated conserved charge become geometric under a certain initial condition of the scalar function, and show that such a conserved geometric current generally exists in curved spacetime. Furthermore, we demonstrate that the geometric conserved current agrees with the entropy current in an effective theory of a perfect fluid, thus the conserved charge is the total entropy of the system. While the geometric charge can be regarded as the entropy for a nondissipative fluid, its physical meaning should be investigated for more general cases.

In this paper, we investigate relations or differences among various conserved quantities which involve the matter Energy Momentum Tensor (EMT) in general relativity. These quantities include the energy with Einstein's pseudo EMT, the generalized Komar integral, or the ADM energy, all of which can be derived from Noether's second theorem, as well as an extra conserved charge recently proposed in general relativity. For detailed analyses, we apply definitions of these charges to a system of free massive particles. We employ the post-Newtonian (PN) expansion to make physical interpretations. We find that the generalized Komar integral is not conserved at the first non-trivial order in the PN expansion due to non-zero contributions at spatial boundaries, while the energy with Einstein's pseudo EMT at this order agrees with a total energy of massive particles with gravitational interactions through the Newtonian potential, and thus is conserved. In addition, this total energy is shown to be identical to the ADM energy not only at this order but also all orders in the PN expansion. We next calculate an extra conserved charge for the system of massive particles, at all orders in the PN expansion, which turns out to be a total number of particles. We call it a gravitational charge, since it is clearly different from the total energy. We finally discuss an implication from a fact that there exist two conserved quantities, energy and gravitational charge, in general relativity.

We study the $U(1)_A$ anomaly at high temperatures of $N_f=2+1$ lattice QCD with chiral fermions. Gauge ensembles are generated with M\"obius domain-wall (MDW) fermions, and the measurements are reweighted to those with overlap fermions. We report on the results for the Dirac spectra, the $U(1)_A$ susceptibility, and the topological susceptibility in the temperature range of $T=136$, $153$, $175$, and $204$ MeV, where the up and down quark masses are set to be near the physical point as well as at lighter or heavier masses.

We construct a bulk spacetime from a boundary CFT, $O(N)$ free scalar model, at finite temperature using a smearing technique, called a conformal flow. The bulk metric is constructed as an information metric associated with the boundary thermal state. Near the boundary (UV region), an asymptotically AdS spacetime is obtained with a leading order perturbation of scalar mode. Based on the falloff behavior of the perturbations and the $O(N)$ symmetry in the CFT, we argue that the corresponding bulk theory is a modified gravity with scalar mode such as $f(R)$ gravity rather than Einstein's general relativity coupled minimally to matter fields. Moving to Einstein frame, we show that the metric is asymptotically the same as the AdS black brane solution. On the other hand, deep in the bulk (IR region), the spacetime turns out to be conformally equivalent to the near horizon limit of AdS extremal black brane, though it is no longer a solution of $f(R)$ gravity, and hence more general classes of modified gravity need to be considered.

We investigate a structure of a 4-dimensional bulk space constructed from the $O(N)$ invariant critical $\varphi^4$ model in 3-dimension using the conformal smearing. We calculate a bulk metric corresponding to the information metric and the bulk-to-boundary propagator for a composite scalar field $\varphi^2$ in the large $N$ expansion. We show that the bulk metric describes an asymptotic AdS space at both UV (near boundary) and IR (deep in the bulk) limits, which correspond to the asymptotic free UV fixed point and the Wilson-Fisher IR fixed point of the 3-dimensional $\varphi^4$ model, respectively. The bulk-to-boundary scalar propagator, on the other hand, encodes $\Delta_{\varphi^2}$ (the conformal dimension of $\varphi^2$) into its $z$ (a coordinate in the extra direction of the AdS space) dependence. Namely it correctly reproduces not only $\Delta_{\varphi^2}=1$ at UV fixed point but also $\Delta_{\varphi^2}=2$ at the IR fixed point for the boundary theory. Moreover, we confirm consistency with the GKP-Witten relation in the interacting theory that the coefficient of the $z^{\Delta_{\varphi^2}}$ term in $z\to 0$ limit agrees exactly with the two-point function of $\varphi^2$ including an effect of the $\varphi^4$ interaction. }

Based on simulations of 2+1 flavor lattice QCD with M\"obius domain wall fermions at high temperatures, we compute a series of spatial correlation functions to study the screening masses in mesonic states. We compare these masses with the symmetry relations for various quark masses and lattice sizes at temperatures above the critical point. Using these spatial correlation functions we examine the $SU(2)_L \times SU(2)_R$ symmetry as well as the anomalously broken axial $U(1)_A$ symmetry. Additionally we explore a possible and emergent chiral-spin symmetry $SU(2)_{CS}$.

We study the doubly charmed tetraquark state T_cc^+ by the HAL QCD method applied to the D^*D system in (2+1) flavor lattice QCD at nearly physical pion mass, m_π= 146 MeV. We obtain the attractive potential at all distances in the S-wave of the isoscalar D^* D system, whose long distance behavior is well described by the two-pion exchange (TPE), and it generates a virtual pole near D^* D threshold with a pole position E_ pole = -59 (^+53_-99) (^+2_-67) keV and an inverse scattering length 1/a_0=0.05(5)(^+2_-2) fm^-1. The virtual pole turns into a loosely bound state pole if the pion mass in the TPE potential is extrapolated to the physical value, m_π =135 MeV. The potential at the physical pion mass is shown to give a semi-quantitative description of the D^0 D^0π^+ mass spectrum at the LHCb.

We explain our proposal for an alternative bulk reconstruction of AdS/CFT correspondences from a scalar field by the flow method. By smearing and then normalizing a primary field in a d dimensional CFT, we construct a bulk field, through which a d+1 dimensional AdS space emerges. The content of this proceeding is based on a talk given by S. Aoki at the 14th International Workshop “Lie Theory and its Applications in Physics" (LT-14), 21-25 June 2021, Sofia, Bulgaria (on-line).