The phase structure of Einstein-Cartan theory

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 {\it low-energy limit}.

Einstein-Cartan theory. In the Palatini framework, the basic gravitational variables constitute a pair of tetrat and spin-connection fields (e a µ , ω ab µ ). They are 1-form fields on M the 4-dimensional Euclidean space-time manifold, taking values, respectively, in the vector space V and in the Lie algebra so(η) of the group SO(η) of the linear transformations of V preserving η ab = (+, +, +, +). The 2-form curvature associating with the spin-connection is The Palatini action for gravitational field is given by, where k ≡ 8πG. The relationship between spin-connection ω ab µ and the tetrat e a µ is determined by δS P (e, ω)/δω = 0, Cartan's structure equation, which gives the torsion-free spin-connection: ω = ω(e). Replacing ω in Eq. (2) by ω = ω(e), the Palatini action S P [e, ω(e)] reduces to the Einstein-Hilbert action and its variation with respect to the tetrad field e a µ leads to the Einstein field equation, Adding the Host modification with the Immirzi parameter γ, one has S H (e, ω) = S P (e, ω) − 1 2kγ M d 4 x det(e)e a ∧ e b ∧ R ab .
The Palatini action (2) and fermion action (6) give the Einstein-Cartan action, Analogously to Eq. (3), δS EC (e, ω)/δω = 0 gives Cartan's structure equation, where the non-vanishing torsion field T a = kβe b ∧e c J ab,c , relating to the fermion spin-current and {σ ab , γ c } = iǫ abcd γ 5 γ d . The hermitian γ 5 -matrix γ 5 = γ 5 = γ 0 γ 1 γ 2 γ 3 = γ 0 γ 1 γ 2 γ 3 , γ † 5 = γ 5 and γ 2 5 = 1. The totally antisymmetric tensor ǫ µνρσ = ǫ abcd e a µ e b ν e c ρ e d σ . The solution to Eq. (9) is where the connection ω ab µ (e) obeys Eq. (3) for torsion-free case. The fermion spin-current (10) contributes only to the pseudo-trace axial vector of torsion tensor, which is one of irreducible parts of torsion tensor [12]. Replacing the spin-connection ω in the Einstein-Cartan action In the case of the Host action (5), the four-fermion interaction term is given by [9] − 3 16 As we can see from Eqs. (6) to (14), the bilinear term (6) of massless fermion fields coupled to the spin-connection (7) is bound to yield a non-vanishing torsion field T a (9), which is local and static. As a result, the spin-connection ω is no longer torsion-free and acquires a torsion-related spin-connectionω ab µ (11), in addition to the torsion-free spin-connection ω ab µ (e). The torsion-related spin-connectionω ab µ is related to the fermion spin-current (10). The quadratic term of the spin-connection field ω in Eq. (1) and the coupling between the spin-connection field and fermion spin-current in Eqs. (6,7) lead to the quadrilinear terms of fermion fields in Eqs. (12) and (13). Another way to see this is to treat the static torsionrelated spin-connectionω ab µ (11) as a static auxiliary field, which has its quadratic term and linear coupling to the spin-current of fermion fields. Performing the Gaussian integral of the static auxiliary field, we exactly obtain the quadrilinear term (14), in addition to the torsion-free action.
A postulation and fermion-mass generation. The gauge principle requires the action of gravitational and fermion fields be invariant under the diffeomorphisms of the manifold M and local Lorentz transformations. This leads to a pseudo-trace axial vector of static (nondynamics) torsion field, which is related to the spin-current of fermion fields. The interaction between the pseudo-trace axial vector of torsion fields and the spin-current of fermion fields results in the four-fermion interaction (quadrilinear terms in massless fermion fields). As a consequence, the gauge-invariant action consists of the torsion-free action of gravitational and fermion fields and four-fermion interaction. We thus postulate that it is impossible to have any gauge-invariant theories made by the bilinear terms of massless fermion fields coupled to torsion-free gravitational field, and quadrilinear terms (or high-order terms) of massless fermion fields must be present [13].
The quadrilinear term (15) is a dimension-6 operator, and four-fermion couping is in terms of the Immirzi parameter γ, gravitational-coupling k and gauge-coupling β. If quantum gravity is taken into account, we expect non-local and high-dimensional operators (d > 6), which contain high-order derivatives. In this case, the torsion-related spin-connectionω ab µ (11) is not completely static, rather has a mass of the order of the Planck mass, mediating in a few Planck length to form effective high-dimensional operators of massless fermion fields.
The fundamental fields e, ω, ψ, and operators O(e, ω, ψ, k, β) are functions of the couplings k, β, γ, depending on the energy-scale E. First we should adopt appropriate vacuum expec-tational value (e.v.e) of operators O(e, ω, ψ, k, β) as order parameters, to describe different phases and phase transitions in the space of couplings k, β, and parameter γ. Second, we try to identify the scaling-invariant regime (ultraviolet fix points) for the low-energy limit (E/m p → 0), where the variation of fundamental fields, couplings and operators as functions of the energy-scale E is govern by renormalization group equations. Third, in such scaling-invariant regime we try to determine the relevant and renormalizable operators that are effective dimension-4 operators, to obtain an effective low-energy theory for the present Universe.
In this Letter, we are interested in the one-particle-irreducible (1PI) two-point functions of fermion fields ψ(0)ψ(x) , since they contribute to effective operators for the energymomentum tensor entering the right-hand side of the Einstein equation (4) for classical gravity. Our goal is limited to find non-trivial fermion-mass operators ( ψψ ≡ 0) in terms of the four-fermion interacting strength. For convenience in calculations, using the Planck mass m p , we rescale fermion fields ψ → ψ/m p and rewrite four-fermion interaction (15) as where the four-fermion coupling g has dimension [m 2 p ]. We assume the gauge-coupling β to be perturbatively small.
The non-vanishing mass operator (17) obeys the NJL gap-equation, where momentum q and couplingg = gN f /m 2 p are dimensionless. The critical pointg c = 8π 2 , which can be obtained by Σ → 0 + , separates the "broken phase" (Σ = 0,g >g c ) from the "weak-coupling symmetric phase" (Σ ≡ 0,g <g c ). Σ(p) ∼ O(m p ) forg >g c . The inverse propagators of these fermions can then be written as, The SU L (2) ⊗ SU R (2) chiral symmetry is realized to be SU(2) with three Goldstone modes and a massive Higgs mode that are not presented here. Eq. (20) corresponds to the bilinear term of massive fermion fields in the effective action, which does not preserves chiral symmetries.

Strong four-fermion coupling.
We turn to the strong-coupling region, where fourfermion coupling g in (14) is sufficiently larger than a certain critical value g crit , bound states of three fermions (three-fermion states) are formed which can be understood as a bound state of one fermion and one composite boson (ψ · ψ).
These three-fermion states (21) carry the appropriate quantum numbers of the gauge group that accommodates ψ. The fermion-mass operator isψΨ and thus massive fermion spectrum is consistent with the chiral symmetry SU L (2) ⊗ SU R (2).
For the purpose of understanding three-fermion states and their spectra, we henceforth focus on the strong-coupling region (g/m 2 p ≫ 1). We make a rescaling of fermion fields, and rewrite the fermion action in terms of the new fermion fields where the gauge coupling β is assumed to be weak. For the limit of strong coupling g/m 2 p → ∞, the kinetic terms S f (x) can be dropped and we calculate the partition function Z (18) in this strong-coupling limit. With S i (x) given in Eq. (24), the integral of e −S i (x) is calculated by Grassmann anticommuting algebra, which shows a non-trivial strong-coupling limit. About this strong-coupling limit (25), we now can perform the strong-coupling expansion of e −S f (x) in powers of 1/g to calculate Green-functions of fermion fields ψ(x 1 )ψ(x 2 ) · · · ψ(x n ) [15,16]. In order to do integral of Grassmann anticommuting algebra, we rewrite the kinetic term S f (x) (23) as a hopping term in the Planck spacing a µ , |a µ | = a = 1/m p , We consider the following two-point functions that form the propagator of the composite Dirac particle In the lowest non-trivial order O(1/g), we obtain the following recursion relations [17] S LL (x) = 1 g 1 2a where for an arbitrary function Transforming these two-point functions (27,28,29) into momentum space, where X = LL, ML, MM respectively, we obtain three recursion relations in momentum We solve these recursion relations (35,36,37) and obtain where the chiral-invariant mass is In addition, the two-point function, As a result, in the lowest non-trivial order of the strong-coupling expansion we obtain the massive propagator of the composite Dirac fermions, for modes p µ a ≪ 1. Eq. (43) corresponds to the bilinear term of massive fermion fields preserving chiral symmetries in the effective action, which can be written as, and its variation with respect to the tetrat field e a µ gives rise to the energy-momentum tensor that contributes to the right-handed side of the Einstein equation (4). This is the "strong-coupling symmetric phase", where fermion fields are massive.
The critical value g crit that separates the " strong-coupling symmetric phase" from the "broken phase" can be qualitatively determined by considering the complex composite scalar field, and its propagator, i.e., the two-point function: Analogously, using the strong-coupling expansion in powers of 1/g (g/m 2 p ≫ 1), we obtain the following recursion relation in the lowest order, Going to momentum space, where q is the momentum of the composite scalar A, we obtain the recursion relation (47) in momentum space As a result, we find the propagator of the massive composite scalar field A, (50) where the factor 4 is due to the four components of the composite scalar field A. Thus, µ 2 AA † gives the mass term of the composite scalar field A in the effective Lagrangian. We assume that the 1PI vertex AA † AA † is positive and the energy of ground states of the theory is bound from the bellow. Then, we can qualitatively discuss the second order phase transition (threshold) from the "strong-coupling symmetric phase" to the "broken phase" by examining the mass term of these composite scalars µ 2 AA † . Spontaneous symmetry breaking SU(2) → U(1) occurs, where µ 2 > 0 turns to µ 2 < 0. Eq. (51) for µ 2 = 0 gives rise to the critical value g crit : where a phase transition takes place between the "strong-coupling symmetric phase" and the "broken phase".
Some discussions. As already mentioned, high-dimensional operators of massless fermion fields containing high-order derivatives are expected if the quantum gravity is included.
In this case the four-fermion coupling (16) and fermion-mass (41) should be functions of fermion's momentum p µ . Both the phase-structure and critical points for phase-transition characterized by the coupling g (16) depend clearly also on the Immirzi parameter γ. Although three different phases have been differentiated, we have not been able to identify the scaling-invariant region for the low-energy limit where some of high-dimensional operators receive anomalous dimensions become relevant operators of effective dimension-4, others are non-relevant and suppressed. We expect that the scaling-invariant regime be probably near to the critical point (52) so that the low-energy effective theory preserves chiral-gauge symmetry in high-energies and has a soft symmetry-breaking for fermion masses in low-energies.
In this Letter we discuss the phase-structure of the Einstein-Cartan theory and a theoretical possibility to understand how fermion fields become massive and couple to torsion-less gravitational field.