Conformal dynamics of quantum gravity with torsion

arXiv:hep-th/9212140v1 23 Dec 1992CPTH-A213.1292HUPD-9216December 1992Conformal dynamics of quantum gravity with torsionI. AntoniadisCentre de Physique Th´eoriqueEcole Polytechnique91128 Palaiseau, FranceandS.D.

Odintsov⋆Department of PhysicsHiroshima UniversityHigashi-Hiroshima 724, JapanABSTRACTThe trace anomaly induced dynamics of the conformal factor is investigated infour-dimensional quantum gravity with torsion. The constraints for the couplingconstants of torsion matter interaction are obtained in the infrared stable fixedpoint of the effective scalar theory.⋆On leave from Pedagogical Institute, 634041 Tomsk, Russia

In a recent paper [1] the trace anomaly induced dynamics of the conformalfactor in four-dimensional quantum gravity has been investigated. The conformalsector was described by spacetime metrics of the form:gµν(x) = e2σ(x)ηµν ,where eσ is the conformal factor and ηµν is the flat metric.

The generalization tocurved fiducial metric was considered in refs.[2,3]. The effective action for σ canbe obtained by integrating the general form of trace anomaly in curved spacetime[4,3].

It turns out that the resulting action is local in σ and possesses a non-trivial,infrared stable critical point which can be computed to all orders in perturbationtheory [1]. In particular, the anomalous scaling dimension of the conformal fieldeσ was found using the same line of reasoning as in two-dimensional gravity [5].

Itwas argued that the effective σ theory describes four-dimensional quantum grav-ity at large distances and provides a framework for a dynamical solution of thecosmological constant problem [1].The purpose of this note is to generalize the results of ref. [1] in the case ofquantum gravity with torsion.Torsion interactions start becoming non-trivialin four dimensions.In fact in two dimensions there is no minimal coupling ofmatter with torsion.

Furthermore, in the non-minimal case [6] torsion appearsquadratically and, after functional integration, 2d induced gravity with torsion isequivalent to 2d gravity without torsion up to coupling constants redefinitions.In four dimensions minimal interactions of spinors with torsion are non-trivial.Moreover, the torsion dependence of trace anomaly is much more complicatedthan in two dimensions [7], so that one may expect to find some new properties ofthe conformal factor dynamics in this case.Let us start with the free actions for scalar and spinor fields in curved spacetimein the presence of torsion T αγβ. Recall that in a theory with torsion the connectionΓαβγ is not symmetric in the lower indices:Γαβγ −Γαγβ = T αγβ ̸= 0.

(1)1

This relation is the origin of torsion. Here, we consider the case where the onlynon-vanishing components of torsion are those which correspond to the totallyantisymmetric part, Sµ ≡iεαβνµTαβν, called pseudotrace.

This is the only part oftorsion which can interact minimally with matter (see ref. [7] for more details).The conformally invariant quadratic actions for scalars (S0) and spinors (S 12)are [7]:S0 = 12Zd4x√−g{gαβ∂αϕ∂βϕ + 16Rϕ2 + ξ0SµSµϕ2},(2)S 12 = iZd4x√−g{ ¯ψ(γµ∇µ −ξ 12γ5γµSµ)ψ},(3)where ξ0, ξ 12 are arbitrary constants.

These actions are invariant under the confor-mal transformations:gµν →e2σgµν,ϕ →e−σϕ,ψ →e−3σ2 ψ(4)for any values of ξ0, ξ 12. The choice ξ0 = 0, ξ 12 = 18 corresponds to the minimalinteraction of matter with torsion.

Here we do not consider torsion interactionwith gauge fields. Such interaction is non minimal, and it exists in general only inthe abelian case leading to terms which are first order in quantum fields [7].

It istherefore irrelevant for our purposes.The general form of the trace anomaly for free conformally invariant matter incurved spacetime with torsion is:T µµ =b(F + 23R) + b′G + b′′R + a1FµνF µν+ a2(SµSµ)2 + a3(SµSµ) + a4∇µ(Sν∇νSµ −Sµ∇νSν),(5)whereF ≡CµναβCµναβ,G = R2µναβ −4R2µν + R2,Fµν = ∇µSν −∇νSµ,2

b =1120(4π)2(N0 + 6N 12 + 12N1),b′ = −1360(4π)2(N0 + 11N 12 + 62N1),a1 = −23(4π)2Xξ212,a2 =12(4π)2Xξ20,a3 =13(4π)2X(2ξ212 −12ξ20),a4 = −23(4π)2Xξ212 ,with N0, N 12 and N1 the number of scalars, Dirac fermions and vectors.Thecoefficients b and b′ can be found in ref. [8], b′′ is arbitrary since it can be changedby adding a local R2 term in the action, while the coefficients a1, a2, a3 and a4 aregiven in ref.

[7] for the case of minimal interaction with torsion (ξ0 = 0, ξ 12 = 18)a2 = 0.We now choose the conformal parametrization gµν = e2σ(x)¯gµν and Sµ = ¯Sµ,where ¯gµν and ¯Sµ are the fiducial metric and torsion, respectively. The conformalfactor acquires dynamics by an effective action whose σ-variation reproduces theanomalous trace (5):T µµ =bF + b′(G −23R) + [b′′ + 23(b + b′′)]R + a1FµνF µν+a2(SµSµ)2 + a3(SµSµ) + a4∇µ(Sν∇νSµ −Sµ∇νSν)≡1√−gδδσ(x)Sanom(σ)(6)Integrating eq.

(6) we get (see refs. [1,5] for Sµ = 0, and ref.

[7] for Sµ ̸= 0) :Sanom[σ] =Zd4x√−¯g{σ[b ¯F + b′( ¯G −23¯R) + a1 ¯F 2µν + a2 ¯S4]+ 2b′σ ¯∆σ −a3 ¯∇µσ ¯∇µ ¯S2 −a4 ¯∇µσ( ¯Sν ¯∇ν ¯Sµ −¯Sµ ¯∇ν ¯Sν)+ (a3 + 12a4) ¯S2 ¯∇µσ∇µσ + a4 ¯Sµ ¯Sν ¯∇µσ ¯∇νσ−112[b′′ + 23(b + b′)]Zd4x√−¯g[ ¯R −6σ −6( ¯∇µσ)( ¯∇µσ)]2},(7)where ¯S2 = ¯Sµ ¯Sµ, and ∆is the conformally covariant fourth-order operator actingon scalars,∆=2 + 2Rµν∇µ∇ν −23R+ 13∇µR∇µ. (8)In eq.

(7) we have omitted σ-independent terms.3

Now as in two dimensions [5] we add Sanom to the classical gravity action:Scl = 12κZd4x√−g(R + hSµSµ −2Λ)= 12κZd4x√−¯g{e2σ[R + 6( ¯∇µσ)( ¯∇µσ)] −2Λe4σ + he2σ ¯S2},(9)where h is a constant parameter (e.g. for the Einstein-Cartan theory h = −124).Thus, the total effective action is:Seff = Sanom + Scl(10)When the conformal factor is quantized, the requirement of general covariancein the vacuum state of σ-theory implies the vanishing of the total trace at thequantum level [3].

Therefore, the β-functions of all couplings appearing in theeffective σ-action must vanish. Without loss of generality, let us consider the effec-tive theory (10) with flat fiducial metric and arbitrary constant torsion backgroundSµ:Seff = −Q2(4π)2Zd4x(σ)2 −ζZd4x[2α(∂µσ)2σ + α2(∂µσ)4]+ γZd4xe2ασ(∂µσ)2 −λα2Zd4xe4ασ +Zd4x{(a3 + 12a4)S2(∂µσ)2+ a4SµSν∂µσ∂νσ +h2κα2e2ασS2 + a2σS4},(11)where the transformations σ →σα, Seff →α−2Seff are made, and the notationsof ref.

[1] are introduced,Q2(4π)2 = 2b + 3b′,ζ = 2b + 2b′ + 3b′′,γ = 3κ,λ = Λκ . (12)α is the anomalous scaling dimension of the conformal factor at the non-trivialinfrared critical point we want to determine.4

A simple power counting argument shows that the effective action (11) is renor-malizable in σ-perturbation theory if we add the action of the external torsion-field:Sext = ηα2Zd4x(SµSµ)2. (13)The ζ coupling renormalization does not depend on the remaining interactions of(11), and a straight-forward one-loop calculation of its β-function shows that ζ = 0is an infrared stable fixed point [1].

At ζ = 0, the contribution to the anomalycoefficients b and b′ of the σ-field itself is coming from the quartic operator ∆(8) and was computed in ref.[3]. Moreover, due to the quartic kinetic operator,the effective action (11) becomes superrenormalizable and all β-functions can becomputed exactly.

The presence of torsion background does not modify the β-functions of γ and λ couplings. Their vanishing imply [1]:α± = 1 ±p1 −4/Q22/Q2,λγ2 = 2π2Q2 [1 + 4α2Q2 + 6α4Q4 ].(15)Eq.

(15) determines at the fixed point the anomalous scaling dimension α interms of the “central charge” Q2, and relates the cosmological with the Newtoniancouplings. Now we can calculate the additional β-functions connected with thetorsion.We first perform a one-loop computation using the standard algorithm [9]and then we will show that the one-loop result is exact.Expanding σ arounda classical background solution and keeping only terms which are quadratic inquantum fluctuations, we can write the resulting kinetic operator corresponding tothe action (11) in the following form:ˆH =2 + Dµν∇µ∇ν + Hµ∇µ + p(16)where Hµ does not give any contribution to the one-loop divergences and can be5

dropped,Dµν = (4π)2Q2 [γe2ασgµν + (a3 + 12a4)S2gµν + a4SµSν],p = −(4π)2Q2κ he2ασS2 −2(4πQ )2γα2e2ασ(∂µσ)2 + 8λ(4πQ )2e4ασ .The corresponding a2 Schwinger-De Witt coefficient in the heat kernel expansionof ˆH is:a2( ˆH) =1(4π)2{−p + 148D2 + 124DµνDµν}= e2ασS2[ hκQ2 + γ(4π)2Q4(a3 + 34a4)] + S4(4π)2Q4 [12a23 + 34a3a4 + 516a24],(17)where we neglected total derivatives, likeD and ∂µ∂νDµν, as well as S inde-pendent terms which have been already discussed. Taking also into account the“classical” contribution due to the change of the scaling dimension of the conformalfactor eσ we get:βh = (2 −2α + 2α2Q2 ) h2κ + α2γ(4π)2Q4(a3 + 34a4),βη = α2(4π)2Q4[12a23 + 34a3a4 + 516a24].

(18)One can now show by a Feynman graph analysis similar to the one done inref. [1] that βh is the exact β-function and there are no higher order corrections.In fact, all additional terms in (11) involving the torsion background amount onlyto a change of σ-propagator, with the exception of the h-term which generatesnew vertices.

A simple power counting argument then shows that the only graphswith primitive divergences must contain at most two γ-vertices or exactly one λ orh-vertex. It follows that the S-dependent part of divergences come from tadpolediagrams with one h or γ-vertex and lead to the expression for βh in (18).

βηis associated to vacuum (σ-independent) terms which can be made zero by anappropriate σ-translation, due to the presence of the last linear term in σ in (11).6

The gravitational (spin-two and reparametrization ghost) contribution remainsan open question. In ref.

[3], it was argued that in the infrared this amounts only toa change of the trace-anomaly coefficients (5) in the effective σ-theory. Futhermore,this contribution was calculated at the one-loop level using the Einstein action, aswell as the conformally invariant but four-derivative Weyl tensor squared action.

Inthe presence of non-vanishing torsion the computation becomes more complicated.A potentially interesting case could be the example of higher derivative gravitywith torsion which seems to avoid the usual unitarity problem for some values ofthe parameters at the classical level [10].Focussing in the σ-field sector, and using eq. (15), the vanishing of βh givesa3 + 34a4 = 0.

(19)This condition constraints the torsion parameters of the theory at the infraredstable fixed point. In the case of pure matter it implies ξ0 = ξ 12, which for minimalinteraction is satisfied only for zero torsion.Acknowledgements: This work was supported in part by the EEC contracts SC1-0394-C and SC1-915053.

SDO wishes to thank Particle Theory Group at HiroshimaUniversity for kind hospitality and JSPS for financial support. We also thank I.L.Shapiro for discussions on the point that βh is given by the one-loop result.7

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