Flow equations for chiral problem in QCD

We analyze the chiral symmetry breaking of QCD in the Coulomb gauge. Using flow equations, we derive the renormalized gap equation and the Bethe-Salpeter equation and show that they are finite in both UV and IR regions. No additional UV renormalization is required in the chiral limit. We take into account the hyperfine interaction as well as chiral symmetry breaking and obtain the $\pi -\rho$ mass splitting caused by the instantaneous and dynamical interactions.