We present a spectroscopy scheme for the lattice field theory by using tensor renormalization group method combining with the transfer matrix formalism. By using the scheme, we can not only compute the energy spectrum for the lattice theory but also determine quantum numbers of the energy eigenstates. Furthermore, wave function of the corresponding eigenstate can also be computed. The first step of the scheme is to coarse-grain the tensor network of a given lattice model by using the higher order tensor renormalization group, and then after making a matrix corresponding to a transfer matrix from the coarse-grained tensors, its eigenvalues are evaluated to extract the energy spectrum. Secondly, the quantum number of the eigenstates can be identified by a selection rule that requires to compute matrix elements of an associated insertion operator. The matrix elements can be represented by an impurity tensor network and computed by the coarse-graining scheme. Moreover, we can compute the wave function of the energy eigenstate by putting the impurity tensor at each point in space direction of the network. Additionally, the momentum of the eigenstate can also be identified by computing an appropriate matrix elements represented by tensor network. As a demonstration of the new scheme, we show the spectroscopy of $(1+1)$d Ising model and compare it with exact results. We also present a scattering phase shift obtained from two-particle state energy using L\"uscher's formula.

The 4D compact U(1) gauge theory has a well-established phase transition between a confining and a Coulomb phase. In this paper, we revisit this model using state-of-the-art Monte Carlo simulations on anisotropic lattices. We map out the coupling-temperature phase diagram, and determine the location of the tricritical point, $T/K_0 \simeq 0.19$, below which the first-order transition is observed. We find the critical exponents of the high-temperature second-order transition to be compatible with those of the 3-dimensional $O(2)$ model. Our results at higher temperatures can be compared with literature results and are consistent with them. Surprisingly, below $T/K_0 \simeq 0.05$ we find strong indications of a second tricritical point where the first-order transition becomes continuous. These results suggest an unexpected second-order phase transition extending down to zero temperature, contrary to the prevailing consensus. If confirmed, these findings reopen the question of the detailed characterization of the transition including a suitable field theory description.

The spatial $z$-correlators of meson operators in $N_f=2+1+1$ lattice QCD with optimal domain-wall quarks at the physical point are studied for seven temperatures in the range of 190-1540 MeV. The meson operators include a complete set of Dirac bilinears (scalar, pseudoscalar, vector, axial vector, tensor vector, and axial-tensor vector), and each for six flavor combinations ($\bar u d$, $\bar u s$, $\bar s s$, $\bar u c$, $\bar s c$, and $\bar c c$). In Ref. \cite{Chiu:2023hnm}, we focused on the meson correlators of $u$ and $d$ quarks, and discussed their implications for the effective restoration of $SU(2)_L \times SU(2)_R$ and $U(1)_A$ chiral symmetries, as well as the emergence of approximate $SU(2)_{CS}$ chiral spin symmetry. In this work, we extend our study to meson correlators of six flavor contents, and first observe the hierarchical restoration of chiral symmetries in QCD, from $SU(2)_L \times SU(2)_R \times U(1)_A $ to $SU(3)_L \times SU(3)_R \times U(1)_A $, and to $SU(4)_L \times SU(4)_R \times U(1)_A $, as the temperature is increased from 190 MeV to 1540 MeV. Moreover, we compare the temperature windows for the emergence of the approximate $SU(2)_{CS}$ symmetry in light and heavy vector mesons, and find that the temperature windows are dominated by the $(\bar u c, \bar s c, \bar c c)$ sectors.

We discuss matters related to the point that topological quantization in the strong interaction is a consequence of an infinite spacetime volume. Because of the ensuing order of limits, i.e. infinite volume prior to summing over topological sectors, CP is conserved. Here, we show that this reasoning is consistent with the construction of the path integral from steepest-descent contours. We reply to some objections that aim to support the case for CP violation in the strong interactions that are based on the role of the CP-odd theta-parameter in three-form effective theories, the correct sampling of all configurations in the dilute instanton gas approximation and the volume dependence of the partition function. We also show that the chiral effective field theory derived from taking the volume to infinity first is in no contradiction with analyses based on partially conserved axial currents.