Solving the QCD non-perturbative flow equation as a partial differential   equation and its application to the dynamical chiral symmetry breaking
  Non-perturbative renormalization group approach to the dynamical chiral symmetry breaking is an effective method which can accommodate beyond the ladder (mean filed) approximation. The usual method relying on the field operator expansion suffers explosive behaviors of the 4-fermi coupling constant, which prevent us from evaluating the physical quantities in the broken phase. In order to overcome this difficulty, we solve the flow equation directly as a partial differential equation and calculate the dynamical mass and the chiral condensates. Also we formulate a beyond the ladder equation and it gives almost gauge independent results for the chiral condensates. 
Ken-Ichi Aoki and Daisuke Sato
hep-th hep-ph