Abstract
In the Einstein–Cartan theory of torsion-free gravity coupling to massless fermions, the four-fermion interaction is induced and its strength is a function of the gravitational and gauge couplings, as well as the Immirzi parameter. We study the dynamics of the four-fermion interaction to determine whether effective bilinear terms of massive fermion fields are generated. Calculating one-particle-irreducible two-point functions of fermion fields, we identify three different phases and two critical points for phase transitions characterized by the strength of four-fermion interaction: (1) chiral symmetric phase for massive fermions in strong coupling regime; (2) chiral symmetric broken phase for massive fermions in intermediate coupling regime; (3) chiral symmetric phase for massless fermions in weak coupling regime. We discuss the scaling-invariant region for an effective theory of massive fermions coupled to torsion-free gravity in the low-energy limit.
Highlights
The self-dual connection of a Yang-Mills gauge theory introduced in the Ashtekar formalism for General Relativity [1] is crucial for the canonical quantization procedure, leading to the non-perturbative quantum theory of gravity, Loop Quantum Gravity [2, 3, 4]
The complex Ashtekar’s connection with reality condition and the real Barbero real connection [5] are linked by a canonical transformation of the connection with the Immirzi parameter γ = 0 [6], which has crucial effects on quantum gravity [7] at the Planck energy scale, but does not affect the classical dynamics of torsion-free gravity
Refs. [9] show that the four-fermion interacting strength in the EinsteinCartan theory is related to the Immirzi parameter, which can possibly lead to physical effects observable
Summary
The self-dual connection of a Yang-Mills gauge theory introduced in the Ashtekar formalism for General Relativity [1] is crucial for the canonical quantization procedure, leading to the non-perturbative quantum theory of gravity, Loop Quantum Gravity [2, 3, 4]. 1 2kγ d4x det(e)ea ∧ eb ∧ Rab. Introducing massless Dirac fermions ψ coupled to the gravitational field described by (eaμ, ωμab), we adopt the fermion action of Ashtekar-Romano-Tate type [10], SF (e, ω, ψ, ψ) d4x det(e) ψeμDμψ + h.c. Where the non-vanishing torsion field T a = kβeb ∧ecJab,c, relating to the fermion spin-current
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