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

Abstract

We analyze the dynamical chiral symmetry breaking (DχSB) 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 multifermion effective interactions V (x, t) where x is ψψ operator and t is the logarithm of the renormalization scale. The DχSB occurs due to the quantum corrections, which means it emerges at some finite t c in the mid of integrating the equation with respect to t. At t 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 c . We newly introduce the notion of weak solution to get the global solution including the infrared limit t → ∞ 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.

Fixed-point structure of low-dimensional relativistic fermion field theories: Universality classes and emergent symmetry

“…In the framework of the functional RG this can be done, e.g., by means of partial [23,42] or dynamical [3,43,44,54] bosonization techniques, or by various decomposition schemes in momentum space [40,41]. A third option is offered by working with full potentials for fermion bilinears and determining the solution of the flow on all scales on a larger function space including weak solutions [55], as has been used for 3d models in [48]. This will be left for future work, and we will here confine ourselves to the study of the RG flow in the limit of pointlike (momentum-independent) four-fermion couplings.…”

“…Within the functional RG, this is, for instance, possible by suitable partial or dynamical bosonization techniques [43,44]. The nature of the interacting phases expected at large coupling can also be investigated by computing the flow of the order-parameter susceptibilities [65], or the flows of full potentials for fermion bilinears [55]. Here, we content ourselves with an outlook on possible symmetry breakings associated with the critical fixed points we have found and leave a more detailed analysis for future work. )…”

Non-Gaussian fixed points in fermionic field theories without auxiliary Bose fields

“…In a recent paper we have proposed a scheme where one can explore the fixed point structure of fermionic models in the framework of the Wetterich-equation without introducing auxiliary variables [19]. Such approach has been introduced earlier in connection with the dynamical breakdown of chiral symmetry in gauged Nambu-Jona-Lasinio models [20,21,22,23]. Fermionic evolution equations were developed for QCD by Meggiolaro and Wetterich [24] truncated at the 4-Fermi level.…”

“…Although there exists a considerable number of works in the literature dealing with the local fermionic potential approximation [19,20,21,22,23], and in our earlier publication [19] we have already worked out a rather general framework for the treatment of this formalism, it is worth to describe in a mathematically more accurate and less intuitive way how the Local Potential Approximation (LPA) is introduced in the fermionic case and what kind of approximations lie in the background.…”

“…[8] around the non-Gaussian fixed point. This correspondence gives some support to treat ψ0 ψ 0 = ρ as a true bosonic field [23], but it is cleaner to think in terms of the "average" correspondence where ∆V is the volume defined by the compositeness scale (cf. the discussion of the fermionic effective potential in the introduction).…”