Weak solution of renormalization group equation
Ken-Ichi Aoki, Shin-Ichiro Kumamoto and Daisuke Sato
∗, Kanazawa University
(September, 2013)
The Wegner–Houghton (WH) equation with the local potential approximation is a partial differential equation (PDE) of the Wilsonian effective action Seff[ψ,ψ; Λ], and it allows us¯ to analyze the spontaneous mass generation of the fermion field ψ, so-called the dynamical chiral symmetry breaking. In this talk, we consider the discrete chiral Nambu–Jona-Lasinio model being a chiral effective model of quantum chromodynamics. Limiting the operator space corresponding to the large-N leading, whereN is the number of the fermion species, the Wilsonian effective potential is given by Seff[ψ,ψ;¯ t] =∫
d4x{ψ¯∂ψ/ −V(σ;t)}, where σ = ¯ψψ is the scalar bilinear-fermion field. Then the WH equation is reduced to the following PDE of M(σ;t) :=∂σV(σ;t):
∂tM(σ;t) = 1 2π2
M
1 +e−2tM2∂σM, (1)
with an appropriate initial condition at t= 0 determined by the bare action of the system.
Here we call M(σ;t) the “mass function” since it corresponds to the effective mass of the fermion at σ= 0.
The solution of Eq. (1) corresponding to the dynamical chiral symmetry breaking must be non-analytic at the origin σ = 0 after some finitet. The slope of it at the origin diverges at a critical scale, beyond which it has a finite jump. This is nothing but the spontaneous mass generation. Such a non-analytic solution is not mathematically authorized as a classical solution of the PDE. In this talk, we show that the symmetry-breakdown solution is a weak solution of the PDE. Actually, Eq. (1) belongs to Burgers’ type equation without viscosity, which can be evaluated by the method of characteristics.
Moreover, this weak solution method automatically “convexizes” the effective potential of the chiral condensates ⟨ψψ¯ ⟩. Especially, with finite chemical potential where the first order phase transition occurs, this “auto-convexization” works perfectly as well and correctly describes the renormalization evolution of the chiral condensates with a finite jump, while the renormalization flow is continuous in the usual sense. This finite jump reflects to the fact that multi local minima of the effective potential appear just as in the other methods, e.g., the large-N leading self-consistency equation, and our method of adopting the weak solution picks up the global minimum among those local minima at all renormalization scale t, exactly and automatically.
We also show the method can be generalized to analyze gauge theories, and present the well-defined calculational method of non-perturbative physical quantities in the dynamical chiral symmetry breaking.
The authors greatly appreciate helpful comments and lectures by Prof. Akitaka Mat- sumura who told us initially how to construct the weak solution for our target renormaliza- tion group equation.
∗Speaker. E-mail address: [email protected].
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