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

Abstract

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 $t_{\mathrm{c}}$ in the mid of integrating the equation with respect to $t$. At $t_{\mathrm{c}}$ some singularities suddenly appear in $V$ which is compulsory in the spontaneous symmetry breakdown. Therefore there is no solution of the equation beyond $t_{\mathrm{c}}$. We newly introduce the notion of weak solution to get the global solution including the infrared limit $t\to \infty$ 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.

29 pages, 18 figures

References in corpus (1)

• Fermion Interactions and Universal Behavior in Strongly Interacting Theories

Cited by in corpus (13)

• The nonperturbative functional renormalization group and its applications
• Functional renormalization group analysis of the soft mode at the QCD critical point
• Renormalization Group Studies of Dense Relativistic Systems
• Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases. II. Entropy production and irreversibility of RG flows
• Tachyonic instability of the scalar mode prior to QCD critical point based on Functional renormalization-group method
• Gravitational instantons and anomalous chiral symmetry breaking
• Disorder-driven quantum transition in relativistic semimetals: functional renormalization via the porous medium equation
• Functional renormalization group study of the critical region of the quark-meson model with vector interactions
• Scalar mass stability bound in a simple Yukawa-theory from renormalisation group equations
• The RG flow of Nambu--Jona-Lasinio model at finite temperature and density
• One-meson-loop NJL model: Effect of collective and noncollective excitations on the quark condensate at finite temperature
• Functional renormalization group study of a four-fermion model with $CP$ violation
• Fermion Mass Generation without Symmetry Breaking