Complex singularities around the QCD critical point at finite densities

Partition function zeros provide alternative approach to study phase structure of finite density QCD. The structure of the Lee-Yang edge singularities associated with the zeros in the complex chemical potential plane has a strong influence on the real axis of the chemical potential. In order to investigate what the singularities are like in a concrete form, we resort to an effective theory based on a mean field approach in the vicinity of the critical point. The crossover is identified as a real part of the singular point. We consider the complex effective potential and explicitly study the behavior of its extrema in the complex order parameter plane in order to see how the Stokes lines are associated with the singularity. Susceptibilities in the complex plane are also discussed.Comment: LaTeX, 27 pages with 15 figure

Lattice Field Theory with the Sign Problem and the Maximum Entropy Method

Although numerical simulation in lattice field theory is one of the most effective tools to study non-perturbative properties of field theories, it faces serious obstacles coming from the sign problem in some theories such as finite density QCD and lattice field theory with the $\theta$ term. We reconsider this problem from the point of view of the maximum entropy method.Comment: This is a contribution to the Proc. of the O'Raifeartaigh Symposium on Non-Perturbative and Symmetry Methods in Field Theory (June 2006, Budapest, Hungary), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Maximum Entropy Method Approach to θ\theta Term

In Monte Carlo simulations of lattice field theory with a $\theta$ term, one confronts the complex weight problem, or the sign problem. This is circumvented by performing the Fourier transform of the topological charge distribution $P(Q)$. This procedure, however, causes flattening phenomenon of the free energy $f(\theta)$, which makes study of the phase structure unfeasible. In order to treat this problem, we apply the maximum entropy method (MEM) to a Gaussian form of $P(Q)$, which serves as a good example to test whether the MEM can be applied effectively to the $\theta$ term. We study the case with flattening as well as that without flattening. In the latter case, the results of the MEM agree with those obtained from the direct application of the Fourier transform. For the former, the MEM gives a smoother $f(\theta)$ than that of the Fourier transform. Among various default models investigated, the images which yield the least error do not show flattening, although some others cannot be excluded given the uncertainty related to statistical error.Comment: PTPTEX , 25 pages with 11 figure

MEM study of true flattening of free energy and the θ\theta term

We study the sign problem in lattice field theory with a $\theta$ term, which reveals as flattening phenomenon of the free energy density $f(\theta)$. We report the result of the MEM analysis, where such mock data are used that `true' flattening of $f(\theta)$ occurs. This is regarded as a simple model for studying whether the MEM could correctly detect non trivial phase structure in $\theta$ space. We discuss how the MEM distinguishes fictitious and true flattening.Comment: Poster presented at Lattice2004(topology), Fermilab, June 21-26, 2004; 3 pages, 3 figure

Application of Maximum Entropy Method to Lattice Field Theory with a Topological Term

In Monte Carlo simulation, lattice field theory with a $\theta$ term suffers from the sign problem. This problem can be circumvented by Fourier-transforming the topological charge distribution $P(Q)$. Although this strategy works well for small lattice volume, effect of errors of $P(Q)$ becomes serious with increasing volume and prevents one from studying the phase structure. This is called flattening. As an alternative approach, we apply the maximum entropy method (MEM) to the Gaussian $P(Q)$. It is found that the flattening could be much improved by use of the MEM.Comment: talk at Lattice 2003 (topology), 3 pages with 3 figure

Sign problem and MEM in lattice field theory with the θ\theta term

Lattice field theory with the $\theta$ term suffers from the sign problem. The sign problem appears as flattening of the free energy. As an alternative to the conventional method, the Fourier transform method (FTM), we apply the maximum entropy method (MEM) to Monte Carlo data obtained using the CP$^3$ model with the $\theta$ term. For data without flattening, we obtain the most probable images of the partition function ${\hat{\cal Z}}(\theta)$ with rather small errors. The results are quantitatively close to the result obtained with the FTM. Motivated by this fact, we systematically investigate flattening in terms of the MEM. Obtained images ${\hat{\cal Z}}(\theta)$ are consistent with the FTM for small values of $\theta$, while the behavior of ${\hat{\cal Z}}(\theta)$ depends strongly on the default model for large values of $\theta$. This behavior of ${\hat{\cal Z}}(\theta)$ reflects the flattening phenomenon.Comment: PTPTEX, 20 pages with 15 figure