Dynamical chiral symmetry breaking and weak nonperturbative   renormalization group equation in gauge theory
  We analyze the dynamical chiral symmetry breaking in gauge theory with the nonperturbative renormalization group equation (NPRGE), which is a first order nonlinear partial differential equation (PDE). In case that the spontaneous chiral symmetry breaking occurs, the NPRGE encounters some non-analytic singularities at the finite critical scale even though the initial function is continuous and smooth. Therefore there is no usual solution of the PDE beyond the critical scale. In this paper, we newly introduce the notion of a weak solution which is the global solution of the weak NPRGE. We show how to evaluate the physical quantities with the weak solution. 
Ken-Ichi Aoki, Shin-Ichiro Kumamoto, and Daisuke Sato
physics.gen-ph