Weak solution of the non-perturbative renormalization group equation to describe the dynamical chiral symmetry breaking

We analyze the dynamical chiral symmetry breaking (D$\chi$SB) in the Nambu-Jona-Lasinio (NJL) model by using the non-perturbative renormalization group (NPRG) equation. The equation takes a form of two-dimensional partial differential equation for the multi-fermion effective interactions $V(x,t)$ where $x$ is $\bar\psi\psi$ operator and $t$ is the logarithm of the renormalization scale. The D$\chi$SB occurs due to the quantum corrections, which means it emerges at some finite $t_{\rm c}$ in the mid of integrating the equation with respect to $t$. At $t_{\rm c}$ some singularities suddenly appear in $V$ which is compulsory in the spontaneous symmetry breakdown. Therefore there is no solution of the equation beyond $t_{\rm c}$. We newly introduce the notion of weak solution to get the global solution including the infrared limit $t\rightarrow \infty$ and investigate its properties. The obtained weak solution is global and unique, and it perfectly describes the physically correct vacuum even in case of the first order phase transition appearing in finite density medium. The key logic of deduction is that the weak solution we defined automatically convexifies the effective potential when treating the singularities.