JAM mean-field update: mean-field effects on collective flow in high-energy heavy-ion collisions at $sqrt{s_{NN}}=2-20$ GeV energies

We consider different implementations of momentum-dependent hadronic mean-fields in the relativistic quantum molecular dynamics (RQMD) framework. First, Lorentz scalar implementation of Skyrme type potential is examined. Then, full implementation of Skyrme type potential as a Lorentz vector in the RQMD approach is proposed. We find that scalar implementation of Skyrme force is too weak to generate repulsion explaining observed data of sideward flows at $sqrt{s_{NN}}<10$ GeV, while vector implementation gives collective flows compatible with the data for a wide range of beam energies $2.7 <sqrt{s_{NN}}<20$ GeV. We show that our approach reproduces the negative proton directed flow at $sqrt{s_{NN}}>10$ GeV discovered by the experiments. We discuss the dynamical generation mechanisms of the directed flow within a conventional hadronic mean-field. A positive slope of proton directed flow is generated predominantly during compression stages of heavy-ion collisions by the strong repulsive interaction due to high baryon densities. In contrast, at the expansion stages of the collision, the negative directed flow is generated more strongly over the positive one by the tilted expansion and shadowing by the spectator matter. At lower collision energies $sqrt{s_{NN}}<10$ GeV, the positive flow wins against the negative flow because of a long compression time. On the other hand, at higher energies $sqrt{s_{NN}}>10$ GeV, negative flow wins because of shorter compression time and longer expansion time. A transition beam energy from positive to negative flow is highly sensitive to the strength of the interaction.

Replica evolution of classical field in 4+1 dimensional spacetime toward real time dynamics of quantum field

Real-time evolution of replicas of classical field is proposed as an approximate simulator of real-time quantum field dynamics at finite temperatures. We consider $N$ classical field configurations dubbed as replicas which interact with each other via the $tau$-derivative terms and evolve with the classical equation of motion. The partition function of replicas is found to be proportional to that of quantum field in the imaginary time formalism. As the replica index $tau$ can be regarded as the imaginary time index, the replica evolution is technically the same as the molecular dynamics part of the hybrid Monte-Carlo sampling and the replica configurations should reproduce the correct quantum equilibrium distribution after the long-time evolution. At the same time, evolution of the replica-index average of field variables is described by the classical equation of motion when the fluctuations are small. In order to examine the real-time propagation properties of replicas, we first discuss replica evolution in quantum mechanics. Statistical averages of observables are precisely obtained by the initial condition average of replica evolution, and the time evolution of the unequal-time correlation function, $langle x(t) x(t)rangle$, in a harmonic oscillator is also described well by the replica evolution in the range $T/omega > 0.5$. Next, we examine the statistical and dynamical properties of the $phi^4$ theory in the 4+1 dimensional spacetime, which contains three spatial, one replica index or the imaginary time, and one real-time. We note that the Rayleigh-Jeans divergence can be removed in replica evolution with $N geq 2$ when the mass counterterm is taken into account. We also find that the thermal mass obtained from the unequal-time correlation function at zero momentum grows as a function of the coupling as in the perturbative estimate in the small coupling region.

The path optimization for the sign problem of low dimensional QCD

The path optimization has been proposed to weaken the sign problem which appears in some field theories such as finite density QCD. In this method, we optimize the integration path in complex plain to enhance the average phase factor. In this study, we discuss the application of this method to low dimensional QCD as a first step of finite density QCD.

Evading the model sign problem in the PNJL model with repulsive vector-type interaction via path optimization

We discuss the sign problem in the Polyakov loop extended Nambu--Jona-Lasinio model with repulsive vector-type interaction by using the path optimization method. In this model, both of the Polyakov loop and the vector-type interaction cause the model sign problem, and several prescriptions have been utilized even in the mean field treatment. In the path optimization method, integration variables are complexified and the integration path (manifold) is optimized to evade the sign problem, or equivalently to enhance the average phase factor. Within the homogeneous field ansatz, the path is optimized by using the feedforward neural network. We find that the assumptions adopted in previous works, $mathrm{Re},A_8 simeq 0$ and $mathrm{Re},omega simeq 0$, can be justified from the Monte-Carlo configurations sampled on the optimized path. We also derive the Euler-Lagrange equation for the optimal path to satisfy. The two optimized paths, the solution of the Euler-Lagrange equation and the variationally optimized path, agree with each other in the region with large statistical weight.

Statistical double $Lambda$ hypernuclear formation from $Xi^-$ absorption at rest in light nuclei

We investigate double $Lambda$ hyperfragment formation from the statistical decay of double $Lambda$ compound nuclei produced in the $Xi^-$ absorption at rest in light nuclei, $^{12}mathrm{C}$, $^{14}mathrm{N}$ and $^{16}mathrm{O}$. We examine the target and the $LambdaLambda$ bond energy dependence of the double $Lambda$ hyperfragment formation probabilities, especially of those double hypernuclei observed in experiments. For the $^{12}mathrm{C}$ ($^{14}mathrm{N}$) target, the formation probabilities of $^{6}_{LambdaLambda}mathrm{He}$ and $^{10}_{LambdaLambda}mathrm{Be}$ ($^{13}_{LambdaLambda}mathrm{B}$) are found to be reasonably large as they are observed in the KEK-E373 (KEK-E176) experiment. By comparison, for $^{16}mathrm{O}$ target, the formation probability of $^{11}_{LambdaLambda}mathrm{Be}$ is calculated to be small with $Delta B_{LambdaLambda}$ consistent with the Nagara event. We also evaluate the formation probability of ${}^{5}_{LambdaLambda}mathrm{H}$ from a $Xi^-$-${}^{6}mathrm{He}$ bound state, ${}^{7}_{Xi}mathrm{H}$.

Path optimization in 0+1 dimensional QCD at finite density

We investigate the sign problem in 0+1 dimensional QCD at finite chemical potential by using the path optimization method. The SU(3) link variable is complexified to the SL(3,$mathbb{C}$) link variable, and the integral path is represented by a feedforward neural network. The integral path is then optimized to weaken the sign problem. The average phase factor is enhanced to be greater than 0.99 on the optimized path. Results with and without diagonalized gauge fixing are compared and proven to be consistent. This is the first step of applying the path optimization method to gauge theories.

Application of the path optimization method to the sign problem in an effective model of QCD with a repulsive vector-type interaction

The path optimization method is applied to a QCD effective model with the Polyakov loop and the repulsive vector-type interaction at finite temperature and density to circumvent the model sign problem. We show how the path optimization method can increase the average phase factor and control the model sign problem. This is the first study which correctly treats the repulsive vector-type interaction in the QCD effective model with the Polyakov-loop via the Markov-chain Monte-Carlo approach. It is shown that the complexification of the temporal component of the gluon field and also the vector-type auxiliary field are necessary to evade the model sign problem within the standard path-integral formulation.

Path optimization method with use of neural network for the sign problem in field theories

We investigate the sign problem in field theories by using the path optimization method with use of the neural network. For theories with the sign problem, integral in the complexified variable space is a promising approach to obtain a finite (non-zero) average phase factor. In the path optimization method, the imaginary part of variables are given as functions of the real part, $y_i=y_i({x})$, and are optimized to enhance the average phase factor. The feedforward neural network can be used to give and to optimize functions with many variables. The combined framework, the path optimization with use of the neural network, is applied to the complex $phi^4$ theory at finite density, the 0+1 dimensional QCD at finite density, and the Polyakov loop extended Nambu-Jona-Lasinio (PNJL) model, all of which have the sign problem. In these cases, the average phase factor is found to be enhanced significantly. In the complex $phi^4$ theory, it is demonstrated that the number density is calculated at a high precision. On the optimized path, the imaginary part is found to have strong correlation with the real part on the temporal nearest neighbor site. In the 0+1 dimensional QCD, we compare the results in two different treatments of the link variable: optimization after the diagonal gauge fixing and optimization without the diagonal gauge fixing. These two methods show consistent eigenvalue distribution of the link variables. In the PNJL model with homogeneous field ansatz, finite volume results approach the mean field results as expected, and the phase transition behavior can be described.

Control the model sign problem via path optimization method: Monte-Carlo approach to QCD effective model with Polyakov loop

We apply the path optimization method to a QCD effective model with the Polyakov loop at finite density to circumvent the model sign problem. The Polyakov-loop extended Nambu--Jona-Lasinio model is employed as the typical QCD effective model and then the hybrid Monte-Carlo method is used to perform the path integration. To control the sign problem, the path optimization method is used with complexification of temporal gluon fields to modify the integral path in the complex space. We show that the average phase factor is well improved on the modified integral-path compared with that on the original one. This indicates that the complexification of temporal gluon fields may be enough to control the sign problem of QCD in the path optimization method.

Investigation of confinement-deconfinement transition via probability distributions

We investigate the confinement-deconfinement transition at finite temperature in terms of the probability distribution of Polyakov-loop complex-phase via the Jensen-Shannon divergence. The Jensen-Shannon divergence quantifies the difference of two probability distributions, namely the target and reference probability distributions. We adopt the complex-phase distributions of the spatially averaged Polyakov loop at $mu/T=0$ and $mu/T=ipi/3$ as the target and reference distributions, respectively. It is shown that the Jensen-Shannon divergence has the inflection point when the target system approaches the Roberge-Weiss endpoint temperature even in the finite-volume system. This means that we can detect the confinement-deconfinement transition from the structural change of probability distributions when we suitably set the reference probability distribution. It is also shown that we can pick up the information of the confinement-deconfinement transition from the quark number density by using the Fourier decomposition; Fourier coefficients have a long tail at around the transition temperature and show a divergent series in calculating the normalized kurtosis.