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The rotation effect on the QCD properties is an open question. We study the dynamic gluon mass in a dense QCD matter, the rotation is introduced by taking a covariant transformation between the flat and curved spaces. The law of causality which restricts the rotation strength of the system is carefully considered in the calculation. we find that the rotation effect is not monotonous. Overall, it behaves like an anti-screening effect, reflecting in the decreasing gluon mass, but the strength changes with the rotation. For a QCD matter with low baryon density, the screening effect in the flat space can be completely canceled by the rotation, and gluons are confined in a strongly rotating matter. When the rotation is extremely high, the matter approaches to a weakly interacting gas., Comment: 9 pages, 2 figures

In this work a rotating SU(2) gluon system have been studied with the dyon ensemble in dilute limit. By solving the rotation-modified Yang-Mills equation we have obtained rotational corrections to the so-called dyon solutions with arbitrary centers to $\mathcal{O}(\omega^2)$ order and the corresponding semi-classical potential. The radial position dependent deconfinement temperature have been obtained by minimizing the semi-classical potential in both real and imaginary angular velocity cases. Although without the $\omega$-dependent coupling constant the critical temperature behaves different from the lattice simulation at each radial position as the rotation goes faster, its radial dependence is qualitatively the same as the lattice. That is in the real velocity case the outer layer will deconfine more difficult while the reverse is true in the imaginary velocity case., Comment: 9 pages, 3 figures

In this study, we introduce a novel approach in quantum field theories to estimate the action using the artificial neural networks (ANNs). The estimation is achieved by learning on system configurations governed by the Boltzmann factor, $e^{-S}$ at different temperatures within the imaginary time formalism of thermal field theory. We focus on 0+1 dimensional quantum field with kink/anti-kink configurations to demonstrate the feasibility of the method. The continuous-mixture autoregressive networks (CANs) enable the construction of accurate effective actions with tractable probability density estimation. Our numerical results demonstrate that this methodology not only facilitates the construction of effective actions at specified temperatures but also adeptly estimates the action at intermediate temperatures using data from both lower and higher temperature ensembles. This capability is especially valuable for the detailed exploration of phase diagrams., Comment: 8 pages, 4 figures. Accepted at Chinese Phys. C

In this work the non-abelian gauge theory is reformulated in a local inertial frame with the presence of a background rotation. With this new formalism the influence of the background rotation on the color deconfinement transition for a SU(2) pure gluon system. The KvBLL caloron, which is a color neutral and asymptotically nontrivial solution of Yang-Mills equation at finite temperature, is adopted to confine the color charges. With new solutions of the caloron's constituent particles, i.e. dyons, the semi-classical potential, which confines color charges, and the perturbative potential, induced by the Gaussian fluctuation, have been obtained for this system under rotation. By solving the critical temperature of confinement-deconfinement phase transition in different computation schemes, it is found that neither the rotational semi-classical potential nor Gaussian fluctuations can confine color charges more tightly when the rotation becomes faster. While only a stronger coupling constant is able to make the critical temperature increasing with angular velocity, as that indicated in lattice simulations. And it is also found with some particular sets of parameters, a non-monotonic dependence of the critical temperature will be obtained in the most physically realistic case, in which all the three factors are taken into account., Comment: 9 pages, 4 figures

Gluon interaction introduces remarkable corrections to the magnetic polarization effects on the chiral fermions, which is known as the inverse magnetic catalysis. It is a natural speculation that the vorticity, which has many similar properties as magnetic field, would bring non-negligible contribution to the chiral rotational suppression. Using the intuitive semi-classical background field method we studied the rotation dependence of the effective strong interaction coupling. Contrary to the magnetic field case the rotation increases the effective coupling which leads to slowing down the condensate melting procedure with temperature. This could be named as the chiral vortical catalysis or inverse rotation suppression. Imposing such dependence to the coupling in the NJL model, we numerically checked this analysis qualitatively. The pseudo critical temperature is shown to rise with the rotation and approach saturation eventually which may be induced by the model cutoff., Comment: 6 pages, 4 figures

In the paper \cite{yau1974convex}, Yau proved that: There is no non-trivial continuous concave function on a complete manifold with finite volume. We prove analogue theorems for several metric spaces, including Alexandrov spaces with curvature bounded below/above, $C^{\alpha}$-H\"older Riemannian manifolds., Comment: 19pages

A strong background field will change the vacuum structure and the proper basis of a system drastically in both classical and quantum mechanics, e.g. the Landau levels in a background magnetic field. The situation is the same for the rotating case. In such a system the usual set of plane-wave states would no longer be suitable as a starting point of perturbation. Alternatively and straightforwardly in a rapidly and globally rotating system, it is better to reformulate the perturbation computation in principle. In this work we will complete the first step for the spin-1 field, which includes solving the Proca equation in present of a background rotation and complete its canonical quantization. It will be shown that because of the symmetry the eigen states are actually the same as the ones of Maxwell equations in cylindrical coordinate. The propagator as well as the near-central approximation will be obtained by considering the vorticity areas are so small in the relativistic QGP., Comment: 7 pages, 0 figures

As the COVID-19 pandemic continues to ravage the world, it is of critical significance to provide a timely risk prediction of the COVID-19 in multi-level. To implement it and evaluate the public health policies, we develop a framework with machine learning assisted to extract epidemic dynamics from the infection data, in which contains a county-level spatiotemporal epidemiological model that combines a spatial Cellular Automaton (CA) with a temporal Susceptible-Undiagnosed-Infected-Removed (SUIR) model. Compared with the existing time risk prediction models, the proposed CA-SUIR model shows the multi-level risk of the county to the government and coronavirus transmission patterns under different policies. This new toolbox is first utilized to the projection of the multi-level COVID-19 prevalence over 412 Landkreis (counties) in Germany, including t-day-ahead risk forecast and the risk assessment to the travel restriction policy. As a practical illustration, we predict the situation at Christmas where the worst fatalities are 34.5 thousand, effective policies could contain it to below 21 thousand. Such intervenable evaluation system could help decide on economic restarting and public health policies making in pandemic., Comment: 17 pages, 13 figures, comments welcome!

In the present paper, we study the effect of the rotation on the masses of scalar meson as well as vector meson in the framework of 2-flavor Nambu--Jona-Lasinio model. The existence of rotation causes a tedious quark propagator and corresponding polarization function. Applying the random phase approximation, the meson mass is calculated numerically. It is found that the behavior of scalar and pseudoscalar meson masses under the angular velocity $\omega$ is similar to that at finite chemical potential, both rely on the behavior of constituent quark mass and reflect the property related to the chiral symmetry. However, masses of vector meson $\rho$ have more profound relation with rotation. After tedious calculation, it turns out that at low temperature and small chemical potenial, the mass for spin component $s_z=0,\pm 1$ of vector meson under rotation shows very simple mass splitting relation $m_{\rho}^{s_z}(\omega)=m_\rho(\omega=0)-\omega s_z$, similar to the Zeeman splitting of charged meson under magnetic fields. Especially it is noticed that the mass of spin component $s_z=1$ vector meson $\rho$ decreases linearly with $\omega$ and reaches zero at $\omega_c=m_\rho(\omega=0)$, this indicates the system will develop $s_z=1$ vector meson condensation and the system will be spontaneously spin polarized under rotation., Comment: 15 pages, 3 figures

We propose a general framework to extract microscopic interactions from raw configurations with deep neural networks. The approach replaces the modeling Hamiltonian by the neural networks, in which the interaction is encoded. It can be trained with data collected from Ab initio computations or experiments. The well-trained neural networks give an accurate estimation of the possibility distribution of the configurations at fixed external parameters. It can be spontaneously extrapolated to detect the phase structures since classical statistical mechanics as prior knowledge here. We apply the approach to a 2D spin system, training at a fixed temperature, and reproducing the phase structure. Scaling the configuration on lattice exhibits the interaction changes with the degree of freedom, which can be naturally applied to the experimental measurements. Our approach bridges the gap between the real configurations and the microscopic dynamics with an autoregressive neural network., Comment: fix expressions and results; 6 pages, 4 figures, comments are welcome!

We develop deep autoregressive networks with multi channels to compute many-body systems with \emph{continuous} spin degrees of freedom directly. As a concrete example, we embed the two-dimensional XY model into the continuous-mixture networks and rediscover the Kosterlitz-Thouless (KT) phase transition on a periodic square lattice. Vortices characterizing the quasi-long range order are accurately detected by the autoregressive neural networks. By learning the microscopic probability distributions from the macroscopic thermal distribution, the neural networks compute the free energy directly and find that free vortices and anti-vortices emerge in the high-temperature regime. As a more precise evaluation, we compute the helicity modulus to determine the KT transition temperature. Although the training process becomes more time-consuming with larger lattice sizes, the training time remains unchanged around the KT transition temperature. The continuous-mixture autoregressive networks we developed thus can be potentially used to study other many-body systems with continuous degrees of freedom., Comment: rewrite the whole manuscript. 6 pages, 4 figures, comments welcome

In this work the existence of Borromean states has been discussed for bosonic and fermionic cases in both the relativistic and non-relativistic limits from the 3-momentum shell renormalization. With the linear bosonic model we checked the existence of Efimov-like states in the bosonic system. In both limits a geometric series of singularities are found in the 3-boson interaction vertex, while the energy ratio is reduced by around 70\% in the relativistic limit because of the anti-particle contribution. Motivated by the quark-diquark model in heavy baryon studies, we have carefully examined the p-wave quark-diquark interaction and found an isolated Borromean pole at finite energy scale. This may indicate a special baryonic state of light quarks in high energy quark matters. In other cases trivial results are obtained as expected. In relativistic limit, for both bosonic and fermionic cases, potential Borromean states are independent of the mass, which means the results would be valid even in zero-mass limit as well., Comment: 5 pages, 3 figures