Phase structure of NJL model with weak renormalization group

We analyze the chiral phase structure of the Nambu--Jona-Lasinio model at finite temperature and density by using the functional renormalization group (FRG). The renormalization group (RG) equation for the fermionic effective potential $V(\sigma;t)$ is given as a partial differential equation, where $\sigma:=\bar \psi\psi$ and $t$ is a dimensionless RG scale. When the dynamical chiral symmetry breaking (D$\chi$SB) occurs at a certain scale $t_c$, $V(\sigma;t)$ has singularities originated from the phase transitions, and then one cannot follow RG flows after $t_c$. In this study, we introduce the weak solution method to the RG equation in order to follow the RG flows after the D$\chi$SB and to evaluate the dynamical mass and the chiral condensate in low energy scales. It is shown that the weak solution of the RG equation correctly captures vacuum structures and critical phenomena within the pure fermionic system. We show the chiral phase diagram on temperature, chemical potential and the four-Fermi coupling constant.Comment: 32 pages, 12 figures; Version published in Nuclear Physics

Singularity Free Direct calculation of spontaneous mass generation

We propose a new iterative method to directly calculate the spontaneous mass generation. It is regarded as a new regularization method resembling the finite volume calculation which assures non-negative fluctuation property at every stage. We work with the Nambu–Jona-Lasinio model and the strong coupling gauge theory where the dynamical chiral symmetry breaking occurs. We are able to conclude the physical mass definitely without encountering any singularity nor recourse to any additional consideration like the free energy comparison. However in special case of the 1st order phase transition, we find that the iterative method has a chance to go wrong

Power Laws in Stochastic Processes for Social Phenomena: An Introductory Review

Many phenomena with power laws have been observed in various fields of the natural and social sciences, and these power laws are often interpreted as the macro behaviors of systems that consist of micro units. In this paper, we review some basic mathematical mechanisms that are known to generate power laws. In particular, we focus on stochastic processes including the Yule process and the Simon process as well as some recent models. The main purpose of this paper is to explain the mathematical details of their mechanisms in a self-contained manner