# Is Einstein-Cartan Theory Coupled to Light Fermions Asymptotically Safe?

Authors: Eckehard W. Mielke
Journal: Journal of Gravity
 (2013)

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## Abstract

The difference between Einstein's general relativity and its Cartan extension is analyzed within
the scenario ofasymptotic safety of quantum gravity. In particular, we focus on the four-fermion
interaction which distinguishes the Einstein-Cartan theory from its Riemannian limit.


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## Body

## 1. Introduction

In the coupling of gravity to Dirac type spinor fields [1], it is at times surmised that the Einstein-Cartan (EC) theory [2] is superior to standard General Relativity (GR), inasmuch as the involved torsion tensor of Cartan [3, 4] can accommodate the spin of fundamental Fermions of electrons and quarks in gravity.However, classically,the effects of spin and torsion cannot be detected by Lageos or Gravity Probe B [5] and would be significant only at densities of matter that are very high but nevertheless smaller than the Planck density at which quantum gravitational effects are believed to dominate. It was even claimed [6] that EC theory may avert the problem of singularities in cosmology, but for a coupling to Dirac fields, the opposite happens [7–9].Recently, it has been stressed by Weinberg [10–12] that the Riemann-Cartan (RC) connection Γ=Γ{}-K, a one-form, is just a deformation of the Christoffel connection Γ{} by the (con-)tortion tensor-valued one-form K=iKαβσαβ/4, at least from the field theoretical point of view. Although algebraically complying with [13], this argument has been refuted [14] on the basis of the special geometrical interpretation [15, 16] of Cartan’s torsion.It is well-known [17, 18] that EC theory coupled to the Dirac field is effectively GR with an additional four-fermion (FF) interaction. However, such contact interactions are perturbatively nonrenormalizable in D>2 without Chern-Simons (CS) terms [19], which was one of the reasons for giving up Fermi’s theory of the beta decay.Since GR with a cosmological constantΛ appears to be asymptotically safe, in the scenario [20] first devised by Weinberg [21], one may ask [22] what the situation in EC theory is, where Cartan’s algebraic equation relates torsion to spin, that is, to the axial current j5i in the case of Dirac fields, on dimensional grounds coupled with gravitational strength.


## 2. Dirac Fields in Riemann-Cartan Spacetime

In our notation [13, 23–25], a Dirac field is a bispinor-valued zero-form ψ for which ψ¯∶=ψ†γ0 denotes the Dirac adjoint and Dψ∶=dψ+Γ∧ψ is the exterior covariant derivative with respect to the RC connection one-form Γαβ=-Γβα, providing a minimal gravitational coupling.In the manifestlyHermitian formulation, the Dirac Lagrangian is given by the four-form
(1)LD=L(γ,ψ,Dψ)=i2{ψ¯γ*∧Dψ+Dψ¯∧γ*ψ}+mψ¯ψη,
where γ∶=γαϑα is the Clifford algebra-valued coframe, obeying Dγ=[γ,K]=γαTα, and
(2)Tα∶=Dϑα=dϑα+Γβα∧ϑβ=12Tijαdxi∧dxj
is the torsion two-form.SinceLD=L¯D=LD† even in a nonholonomic frame, the minimal coupling provides us automatically with the Hermitian charge current and standard axial current three-forms
(3)j∶=ψ¯γ*ψ=jμημ,j5∶=ψ¯γ*γ5ψ=j5μημ,
respectively, which are familiar with quantum electrodynamics (QED) in curved spacetime.Let us now separate the purely Riemannian part from spin-contortion pieces:(4)LD=L(γ,ψ,D{}ψ)-i2ψ¯(γ*∧K-K∧γ*)ψ=L(γ,ψ,D{}ψ)+14𝒜∧j5.Hence, in an RC spacetime, a massive Dirac spinor only feels theaxial torsion one-form
(5)𝒜∶=14Tr*(γ∧Dγ)=(ϑα∧Tα)*=12T[αβγ]ηαβγ=𝒜idxi.
The spin current of the Dirac field is given by the Hermitian three-form
(6)ταβ≔∂LD∂Γαβ=18Ψ¯(γ*σαβ+σαβγ*)Ψ=14ηαβγδΨ¯γδγ5Ψηγ=ταβγηγ,
with totally antisymmetric components ταβγ=τ[αβγ]. Equivalently,  torsion merely couples to the spin-energy potential  μα=ϑα∧j5*/4, that is; to a two-form that is proportional to the axial current j5 compare [25] for more details.


### 2.1. Axial Anomaly in Riemann-Cartan Spacetime

In quantum field theory (QFT), however, the axial current is not conserved, rather there arises in RC spacetime theaxial anomaly(7)〈dj5〉=2im〈ψ¯γ5ψ〉η-196π2[2Rαβ{}∧R{}αβ+d𝒜∧d𝒜],
for its vacuum expectation value, which involves the topological Pontrjagin term quadratic in the curvature. This result [26], which can easily be transferred to the chiral current j±, is based on the Pauli-Villars regularization schem; compare also [27].Since the axial torsion𝒜 is not a gauge field, it is legitimate to absorb [28] its contribution to the anomaly (7) into the redefined current
(8)j^5∶=j5+196π2𝒜∧d𝒜,
such that
(9)〈dj^5〉=2im〈ψ¯γ5ψ〉η+148π2Rαβ{}∧R{}αβ
is the same result as in the Riemannian spacetime of GR.One way to avoid such anomalies is to employ curvature constraints likeRαβ≡0 typical for teleparallel models [29]. Another approach, inspired by the BF schemes [30, 31] of Topological Quantum Field Theory (TQFT), is to start from a minimalists SL(5,R) gauge model which includes only a “bare” Pontrjagin type four-form as its own counterterm. However, then a tiny symmetry breaking is mandatory, in order to recover the classical metrical background of GR.


## 2.1. Axial Anomaly in Riemann-Cartan Spacetime

In quantum field theory (QFT), however, the axial current is not conserved, rather there arises in RC spacetime theaxial anomaly(7)〈dj5〉=2im〈ψ¯γ5ψ〉η-196π2[2Rαβ{}∧R{}αβ+d𝒜∧d𝒜],
for its vacuum expectation value, which involves the topological Pontrjagin term quadratic in the curvature. This result [26], which can easily be transferred to the chiral current j±, is based on the Pauli-Villars regularization schem; compare also [27].Since the axial torsion𝒜 is not a gauge field, it is legitimate to absorb [28] its contribution to the anomaly (7) into the redefined current
(8)j^5∶=j5+196π2𝒜∧d𝒜,
such that
(9)〈dj^5〉=2im〈ψ¯γ5ψ〉η+148π2Rαβ{}∧R{}αβ
is the same result as in the Riemannian spacetime of GR.One way to avoid such anomalies is to employ curvature constraints likeRαβ≡0 typical for teleparallel models [29]. Another approach, inspired by the BF schemes [30, 31] of Topological Quantum Field Theory (TQFT), is to start from a minimalists SL(5,R) gauge model which includes only a “bare” Pontrjagin type four-form as its own counterterm. However, then a tiny symmetry breaking is mandatory, in order to recover the classical metrical background of GR.


## 3. Effective Einstein-Cartan Theory

The Einstein-Cartan Lagrangian(10)LEC∶=-12κRαβ∧ηαβ=LHE+112κ𝒜∧𝒜*,
where κ=8πGN is the gravitational constant in natural units, generalizes the metrical Hilbert-Einstein Lagrangian LHE to an RC spacetime with torsion, where only the axial torsion 𝒜 enters algebraically. (Adding torsion squared terms [32, 33] is not an unambiguous procedure, since the particular combination Tα∧(*Tα(1)-2Tα(2)-(1/2)Tα(3)) of irreducible pieces is related to a fourth boundary term derived from the dual CS term CTT*∶=ϑα∧Tα*; cf. [34]. In the space of gravity theories, the nontopological boundary term dCTT* is interrelating GR with its teleparallelism equivalent [35]. Exactly the previous teleparallel “nucleus” leaves its traces in the controversies [36, 37] about the well posedness of the classical Cauchy problem and the particle content of the (broken) Poincaré gauge theory.)The Einstein-Cartan equation [2]
(11)Gα∶=12Rβγ∧ηαβγ=κΣα,
coupled to the canonical energy-momentum current Σα of matter, is obtained by varying for the coframe ϑα. Likewise, the EC three-form
(12)Gα≔12Rβγ∧ηαβγ=Gα{}+(-1)s12(eα⌋𝒜∧𝒜*-13𝒜∧eα⌋𝒜*)+(-1)s6ϑα∧d𝒜
can be decomposed into the Einstein three-form Gα{}=Gαβηβ with respect to the Riemannian connection Γ{} and additional axial torsion pieces [17, 18]. It satisfies the first Noether identity
(13)D⌢Gα≡12(eα⌋Rβγ)∧ηβγμ∧Tμ
with respect to the transposed connection Γ⌢αβ∶=Γαβ+eα⌋Tβ; compare equation (5.4.13) of [13]. (Observe that the three-form (12) is not covariantly conserved in RC spacetime. Only for vanishing torsion, it reduces to the conservation law D{}Gα{}≡0 familiar with GR as a consequence of the contracted second Bianchi identity.)By varying with respect to the linear connectionΓαβ, we obtain the second field equation of EC theory, that is, Cartan’s algebraic relation:
(14)ηαβγ∧Tγ=2κταβ
between torsion and the canonical spin of matter. Due to (15), in the case of Dirac fields, this is equivalent to
(15)𝒜*=κ2j5,
coupled via the “bare” fundamental length ℓ=κ. Then, “on shell,” EC theory deviates from GR merely via
(16)ΔL~∶=κ(LEC-LHE)≃κ248j5∧j5*=4f2j5∧j5*.


## 4. Asymptotic Safety of EC Theory

For the Hilbert-Einstein Lagrangian(17)LHEΛ∶=LHE+Λκη
with cosmological term, one can define the dimensionless running coupling constants
(18)gN∶=κk2,λ∶=Λk2,
where k is the renormalization group (RG) scale in momentum space and Λ the cosmological constant related to dark energy (DE) of density ρΛ; see also [38].Asymptotic safety amounts to the requirement that dimensionless coupling constants remain bounded in the ultraviolet limitk→∞. In 4D, this is controlled by the renormalization group equations(19)k∂∂kgN=β1(gN,λ)=(2+dN)gN,k∂∂kλ=β2(gN,λ),
where dN is the anomalous dimension of the running Newton coupling gN. According to the Asymptotic Safety (AS) scenario [20], they run into some nontrivial fixed points gN* and λ*, depending on the specific truncation of the effective Lagrangian to the celebrated Hilbert-Einstein one (10) without torsion. This can be extended [39] to high-order polynomials Rn of the Ricci scalar similarly as in the classically bifurcating f(R) models [40], but then the issue of physical ghosts or nonunitarity known [41, 42] from Stelle-type higher-derivative models needs to be seen.Quite generally, the dimensionless product(20)μ=43κΛ=13(2κ)2ρΛ≤43gN*λ*≃0.2.
exhibits a universal bound independent of the particular truncation.


### 4.1. The Issue of the Four-Fermion Interaction

Interesting enough, the EC induced FF interaction (16) with its tiny “bare” coupling constant
(21)f2=1192κ2=2-10(μΛ)2=2-8μρΛ
also scales with the gravitational constant κ but is inversely compared to the Hilbert-Einstein and cosmological terms.If the renormalization flow starts to the right from the non-Gaussian fixed point, the coupling actually diverges [43] at a finite RG scale. When the contact- or point-like truncation breaks down, a boson-like description of fermion bilinears is mandatory, including the 1/k4 dependence in the functional integral. Then, the FF interaction becomes nonlocal [44], and the corresponding dimensionless renormalized running coupling f*2 becomes asymptotically safe or even free. In a nonlinear σ model [45], nonrenormalizable FF interactions may be instrumental for restoring asymptotic safety.In view of these problems, the EC theory has been amended [32, 46] by the pseudocurvature scalar term of Hojman et al. [47] (the infamous “Holst” term, cf. [34]), or even nonminimally coupled Dirac fields [48]. Unfortunately, many of these extensions [49–51] are ignoring a possible running of the gravitational couplings and therefore appear not to be conclusive.Apparently, the search for a Quantum Theory of Gravity (QG) which is free of anomalies and is leaving Einstein’s GR as a well-established macroscopic sign post has produced rather contradictory partial results, to some extent resembling a Babylonian confusion; compare [52].


## 4.1. The Issue of the Four-Fermion Interaction

Interesting enough, the EC induced FF interaction (16) with its tiny “bare” coupling constant
(21)f2=1192κ2=2-10(μΛ)2=2-8μρΛ
also scales with the gravitational constant κ but is inversely compared to the Hilbert-Einstein and cosmological terms.If the renormalization flow starts to the right from the non-Gaussian fixed point, the coupling actually diverges [43] at a finite RG scale. When the contact- or point-like truncation breaks down, a boson-like description of fermion bilinears is mandatory, including the 1/k4 dependence in the functional integral. Then, the FF interaction becomes nonlocal [44], and the corresponding dimensionless renormalized running coupling f*2 becomes asymptotically safe or even free. In a nonlinear σ model [45], nonrenormalizable FF interactions may be instrumental for restoring asymptotic safety.In view of these problems, the EC theory has been amended [32, 46] by the pseudocurvature scalar term of Hojman et al. [47] (the infamous “Holst” term, cf. [34]), or even nonminimally coupled Dirac fields [48]. Unfortunately, many of these extensions [49–51] are ignoring a possible running of the gravitational couplings and therefore appear not to be conclusive.Apparently, the search for a Quantum Theory of Gravity (QG) which is free of anomalies and is leaving Einstein’s GR as a well-established macroscopic sign post has produced rather contradictory partial results, to some extent resembling a Babylonian confusion; compare [52].



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