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TIT-HEP-275 KEK-TH-423 UT-Komaba/94-22 December 1994 5 9 Two–loop Renormalization in Quantum Gravity near 9 1 Two Dimensions n a J 5 1 2 v Toshiaki Aida1)∗, Yoshihisa Kitazawa1)†, Jun Nishimura2)‡and Asato Tsuchiya3)§ 6 5 0 1 1) Department of Physics, Tokyo Institute of Technology, 0 5 Oh-okayama, Meguro-ku, Tokyo 152, Japan 9 / h 2) National Laboratory for High Energy Physics (KEK), t - p Tsukuba, Ibaraki 305, Japan e h 3) Institute of Physics, University of Tokyo, : v Komaba, Meguro-ku, Tokyo 153, Japan i X r a Abstract We study two–loop renormalization in (2 + ǫ)–dimensional quantum gravity. As a first step towards the full calculation, we concentrate on the divergences which are proportionaltothenumberofmatterfields. Wecalculatetheβ functionsandshowhow thenonlocaldivergences aswellastheinfrareddivergences cancelamongthediagrams. Although the formalism includes a subtlety concerning the general covariance due to the dynamics of the conformal mode, we find that the renormalization group allows the existence of a fixed point which possesses the general covariance. Our results strongly suggest that we can construct a consistent theory of quantum gravity by the ǫ expansion around two dimensions. ∗E-mail address : [email protected] †E-mail address : [email protected] ‡E-mail address : [email protected], JSPS Research Fellow. §E-mail address : [email protected] 1 Introduction Quantum gravity beyond two dimensions may be renormalizable in the 2 + ǫ expansion approach. The remarkable point is that we find the short distance fixed point in the renormalization group for proper matter contents. Therefore the gravitational interaction may not become uncontrollably strong at short distance in quantum theory [1, 2, 3, 4, 5]. Let us consider the matter scattering due to gravitation. Since the gravitational coupling constant (Newton’s constant) has a dimension, the cross section grows at short distance and ultimately exceeds the unitarity bound. On the other hand, we can define the dimensionless gravitational coupling constant G by introducing the renormalization scale µ in quantum theory. If the dimensionless gravitational coupling constant G possesses a short distance fixed point, the unitarity problem can be overcome since the cross section at the momentum scale p2 = µ2 is f(G(µ))/µ2 where f is a calculable function of G, on dimensional grounds. Needlesstosay,unitaritymightbebrokenbyothersourcessuchasblackholesinquantum gravity and there might be a real paradox here. We hope to address these questions also in the 2+ǫexpansion of quantum gravity. Although a consistent quantum theory of gravitation may require more than local field theory such as superstring in four dimensions, we should not forget this simpler possibility. At least we can learn lessons of quantum gravity in this simple setting in low dimensions. This approach is also useful to study two–dimensional quantum gravity and string theory [6, 7, 8]. As is widely perceived, the renormalization group is one of the most powerful tool to study quantum field theory. In quantum gravity, we need to examine the meaning of the renormalization groupcarefully since the spacetime distance itself fluctuates. Let usconsider the Einstein action in D = 2+ǫ dimensions 1 dDx √gR, (1.1) G Z 0 where G is the gravitational coupling constant. 0 We parametrize the metric g as g = g˜ e−φ where φ is the conformal mode of the µν µν µν metric. g˜ can be parametrized by a traceless symmetric tensor h as g˜ = (eh) . We µν µν µν µν further introduce the dimensionless gravitational coupling constant G by introducing the renormalization scale µ by 1 = µǫ. In this parametrization, our action becomes G0 G µǫ ǫ(D 1) dDx e−2ǫφ R˜ − g˜µν∂ φ∂ φ , (1.2) µ ν G − 4 ! Z 1 where R˜ is the scalar curvature made out of g˜ . We note the similarity of this action to the µν nonlinear sigma model µǫ dDx e−2ǫφ∂ ~n ∂µ~n. (1.3) µ F · Z Since R˜ involves two derivatives, it is analogous to the kinetic term of the nonlinear sigma model. F is the coupling constant for the ~n field and G is that for the h field. The µν physical length scale is set by the line element ds2 = e−φdx dxµ. If we scale the length as µ ds2 λ2ds2, the coupling constant changes as 1 λǫ and 1 λǫ respectively. In this → F → F G → G sense the coupling constants grow canonically at short distance in boththeories. However we can choose the renormalization scale µ such that µλ = 1 and consider the running coupling constants. If the running coupling constants possess the short distance fixed points, the theory is under control. The novel feature of quantum gravity is that the zero mode of the φ field sets the scale of the metric and hence the scale of the length. In fact the zero mode is determined by the classical solution of the theory. For example the scale of the metric expands with time in our universe. Therefore we can take the constant mode of φ to be the present scale factor of the metric. Since the definite combination µǫe−2ǫφ appears in the action, it is most advantageous to choose the renormalization scale µ to compensate the scale factor of the metric (or the constant mode of φ). It is analogous to choose the renormalization scale to match the momentum scale of the relevant scattering in the conventional field theory problem. In this way the renormalization scale of the dimensionless gravitational coupling constant G(µ) is related to the scale factor of the metric. In particular, large renormalization scale is relevant at short distance physics. Weneedtoconsiderallpossiblevaluesoftheconstantmodeofφforthewholetheorysince we are integrating over it. Therefore we consider the whole renormalization group trajectory as the whole quantum theory of gravitation. Such an idea satisfies the independence of the theory from the scale factor of a particular metric. In our universe, the scale factor of the metric can be identified with time. In this interpretation of the renormalization group in quantum gravity, we may say that the renormalization scale is identified with time. The renormalization group evolution is hence naturally related to the time evolution in quantum gravity. In this paper we study the two–loop renormalization of Einstein gravity coupled to c 2 copies of scalar fields in the conformally invariant way with the action µǫ ǫ 1 dDx√g R 1 ϕ2 + gµν∂ ϕ ∂ ϕ , (1.4) G ( − 8(D 1) i! 2 µ i ν i) Z − where i runs from 1 to c. This action can be rewritten as µǫ 1 ǫ 2 ǫ 1 1 dDx gˆ R˜ 1+ ψ ϕ2 g˜µν∂ ψ∂ ψ + g˜µν∂ ϕ ∂ ϕ , G Z q   2s2(D −1) ! − 8(D −1) i− 2 µ ν 2 µ i ν i  (1.5)   where we have reparametrized the conformal mode as e−ǫ4φ = λ2ǫ 1+ 1 ǫ ψ in order 2 2(D−1) (cid:18) (cid:19) to make the kinetic term of ψ canonical. The λ factor can be cancellqed by choosing the appropriate renormalization scale µ. In this way we can get rid of the 1 pole of the conformal ǫ mode propagator. Note that the conformal mode ψ can be viewed as another conformally coupled scalar field in this parametrization. Therefore we can quantize the theory treating the conformal mode as a matter field coupled in the conformally invariant way. In such a quantization procedure it is important to keep the conformal invariance. Since it is well known that the conformal anomaly arises in quantum field theory, we need to modify the tree action to cancel the quantum conformal anomaly. It has been proposed to generalize the action in the following form which possesses the manifest volume–preserving diffeomorphism invariance [4, 5] µǫ 1 1 dDx gˆ R˜L(ψ,ϕ ) g˜µν∂ ψ∂ ψ + g˜µν∂ ϕ ∂ ϕ , (1.6) i µ ν µ i ν i G − 2 2 Z q (cid:26) (cid:27) where L = 1+aψ+b(ψ2 ϕ2). It has been shown that the theory is renormalizable to the − i one–loop level and the β functions of the couplings are found to be AG β = ǫG AG2, β = a, β = 0, (1.7) G a b − − 2 where A = 25−c. The Einstein action is the infrared fixed point with G = 0,a = ǫ and 24π 2(D−1) b = ǫ . The theory possesses the short distance fixed point with G = ǫ,a =q0 and b = 8(D−1) A ǫ . The conformal anomaly is shown to be cancelled out on the whole renormalization 8(D−1) group trajectory. It is important to perform the two–loop renormalization of the theory. It serves to es- tablish the validity of the 2 +ǫ expansion in quantum gravity by showing that the higher order corrections can be computed systematically. However the two–loop calculations in quantum gravity is a formidable task due to the proliferation of diagrams and tensor indices. Therefore we have decided to calculate the two–loop counterterms which are proportional 3 to the number of matter fields (the central charge) first. In this paper we report the result of such a calculation. Since the number of scalar fields we couple to gravity is a free param- eter, the counterterms must be of the renormalizable form. They further must satisfy the requirement from the general covariance. Therefore this calculation serves as a check of the 2+ǫ expansion approach. This paper is organized as follows. In section 2, we briefly review the one–loop renormalization of the action (1.6) . In section 3, we explain our two–loop calculation of the counterterms. In section 4, we state the results of our calculation. In section 5, we compute the β functions and check the general covariance at the ultraviolet fixed point. We discuss the physical implications and draw conclusions in section 6. 2 Brief Review on the One–loop Renormalization We utilize the background field method to compute the effective action. The generating functional for the connected Green’s functions in the field theory is e−W[J] = ϕ exp( S J ϕ), (2.1) D − − · Z where S is the action and J ϕ dDx J(x)ϕ(x). ϕ denotes a collection of fields in the · ≡ theory. In this paper the metric is tRaken to be Euclidean since the Euclidean rotation from the Minkowski metric is straightforward within the perturbation theory. The effective action is obtained by the Legendre transform Γ[ ϕ ] = W[J] J ϕ , (2.2) h i − ·h i where ϕ(x) = δW[J]. Therefore the effective action is h i δJ(x) δΓ[ ϕ ] e−Γ[hϕi] = ϕ˜ exp S[ ϕ +ϕ˜]+ h i ϕ˜ , (2.3) D − h i δ ϕ · ! Z h i where ϕ˜ = ϕ ϕ since J = δΓ . The effective action can be expanded in terms of h¯ as −h i −δhϕi Γ = S +h¯Γ(1) +h¯2Γ(2) + . (2.4) ··· Hence we cancompute theeffective actionby expanding theaction S aroundthebackground ϕ and dropping the linear terms in ϕ˜. Namely the effective action is the sum of the one– h i particle irreducible diagrams with respect to ϕ˜. 4 In our context, we decompose the fields into the backgrounds and the quantum fields as ϕ ϕˆ +ϕ , ψ ψˆ+ψ and g˜ = gˆ (eh)ρ , where hµ is a traceless symmetric tensor. The i → i i → µν µρ ν ν effective action can be computed by summing the one–particle irreducible diagrams with respect to the quantum fields ϕ , ψ and hµ . i ν Thecruciallocalgaugeinvariance(generalcovariance)oftheaction(1.6)inthisparametriza- tion is 2 δg˜ = ∂ ǫρg˜ +g˜ ∂ ǫρ +ǫρ∂ g˜ ∂ ǫρg˜ , µν µ ρν µρ ν ρ µν ρ µν − D ∂L 2 δψ = ǫρ∂ ψ +(D 1) ∂ ǫρ, ρ ρ − ∂ψD ∂L 2 δϕ = ǫρ∂ ϕ (D 1) ∂ ǫρ. (2.5) i ρ i ρ − − ∂ϕ D i In addition to expanding the action around the background fields, we need to fix the gauge invariance (2.5) in order to perform the functional integration. We adopt the following background gauge. 1 ∂ L ∂νL L µh ν hρν . (2.6) µν ρ 2 ∇ − L ! ∇ − L ! Throughout this paper the tensor indices are raised and lowered by the background metric and the covariant derivatives are taken with respect to the background metric. We also add the corresponding ghost terms ∂ L η¯ µην +Rˆµη¯ ην ν ( µη¯ )ην + . (2.7) ∇µ ν∇ ν µ − L ∇ µ ··· The background gauge has the advantage to keep the manifest general covariance with respect to the background metric. The one–loop counterterm of this theory is evaluated to be A dDx gˆR˜. (2.8) − ǫ Z q The resulting β functions are quoted in (1.7). The conformal anomaly of the theory is found to be 2 2 ∂L ∂L ǫL AG 2(D 1) R˜  − − −  ∂ψ! − ∂ϕi!    1 ∂2L  ǫ 4(D 1) ∂ ψ∂µψ  µ − 4 ( − − ∂ψ2) 1 ∂2L + ǫ+4(D 1) ∂ ϕ ∂µϕi. (2.9) 4 ( − ∂ϕ2i ) µ i 5 It has been shown that the conformal anomaly vanishes along the renormalization group trajectory. Note that the conformal invariance is crucial to restore the general covariance from the action which possesses only the volume–preserving diffeomorphism invariance. Therefore the general covariance is also maintained along the renormalization group trajectory. We are particularly interested in the short distance fixed point of the renormalization group. At the fixed point the tree action is µǫ 1 dDx gˆ R˜L(X )+ g˜µν∂ X ∂ Xi , (2.10) i µ i ν G 2 Z q (cid:26) (cid:27) where L = 1 bX Xi. X denotes the ψ and ϕ fields (X = ψ) and X Xi = ψ2+ c ϕ 2. − i i i 0 i − j=1 j This action possesses the Z symmetry X X . The enhancement of the sPymmetry 2 i i → − may be due to the fact that we are expanding the theory around the symmetric vacuum in which the expectation value of the metric vanishes. In this paper, we compute the two– loop counterterms which are proportional to the number of scalar fields c. We compute the counterterms at the fixed point where further simplification takes place due to the Z 2 symmetry. Note that the conformal mode ψ is just another matter field at the fixed point. Therefore the conformal mode contribution can be included by the replacement c c+1. → One of the difficulties of the 2 + ǫ expansion is the treatment of the conformal mode due to the kinematical (1) pole in the propagator before the reformulation of the theory. Our ǫ calculation certainly addresses this question. However it turns out that this problem is no more difficult than to quantize matter fields. In the renormalization program, the counterterms at the n-th loop level are local if the theory is renormalized at the (n 1)-th loop level, since all the subdivergences are − already subtracted. Therefore the two–loop level renormalizability of the theory at the fixed point is already guaranteed by the previous work [5], where all the one–loop subdiagrams which appear in the two–loop diagrams are subtracted. In fact we show that the theory is renormalizable by adding local counterterms in the leading order of c to the two–loop level. We further demonstrate that these counterterms can be chosen to respect the general covariance by having the conformal anomaly vanish at the fixed point. 6 3 Calculation of Two–loop Counterterms As is seen in the previous section, the one–loop bare action is µǫ 1 AG dDx gˆ R˜L(X)+ ηijg˜µν∂ X ∂ X R˜ , (3.1) µ i ν j G 2 − ǫ Z q (cid:26) (cid:27) where X = (ψ,ϕ ) and ηij = diag( 1,1, ,1). Paying attention to the ultraviolet fixed i − ··· point, we set 1 L(X) = 1 ǫbηijX X , (3.2) i j − 2 wherewe havereplacedboftheprevious sectionwithǫb/2toshowtheǫfactorexplicitly. The Z symmetryofthefixedpointactionispreserved alsointhetwo–loopcalculations. Asafirst 2 step towards the complete two–loop renormalization, we evaluate only those counterterms which are proportional to the number of matter fields in this paper. As in the one–loop calculation, we expand the fields around the backgrounds as g˜ = µν gˆ (eh)ρ , X Xˆ +X and employ the background gauge. We adopt the same gauge fixing µρ ν i → i i term (2.6) as in the one–loop case, which is not renormalized at the one–loop level. The ghost action (2.7) is used also in this case. In the two–loop calculations, we have to expand the action, in general, up to the fourth order of the quantum fields (h , X and ghosts). µν i However, since we compute the counterterms proportional to the number of matter fields, we need only the three– and four–point vertices which are quadratic with respect to the X i fields. We expand the one–loop bare action (3.1), the gauge fixing term (2.6) and the ghost action (2.7) around the background fields to a sufficient order as is explained above. Here we exploit the formula 1 R˜ = Rˆ hµ Rˆν hµν + hµ ρhν − ν µ −∇µ∇ν 4∇ρ ν∇ µ 1 1 + Rˆσ hρ hµν hν hρµ + (hµ ρhν )+O(h3). (3.3) 2 µνρ σ − 2∇ν µ∇ρ ∇µ ν∇ ρ The background metric is expanded around the flat one as ˆ gˆ = δ +h , (3.4) µν µν µν where δµνhˆ = 0 can be assumed for simplicity without loss of generality. The propagators µν of the quantum fields are defined on the flat metric. One finds that the kinetic term for the h field is given by 1L(Xˆ) hµ ρhν . In order to make it canonical, we have to divide µν 4 ∇ρ ν∇ µ 7 the h field by L(Xˆ). Further, we redefine the h field as a traceless symmetric tensor µν µν q on the flat metric. Thus we are lead to define the H field through µν 1 hµν = TµνλρH , (3.5) λρ L(Xˆ) δµνH q= 0, (3.6) µν where Tµνλρ is defined as 1 2 Tµνλρ = gˆµλgˆνρ +gˆµρgˆνλ gˆµνgˆλρ . (3.7) 2 − D (cid:18) (cid:19) After this prescription, we obtain the propagators and the vertices for the H , X and ghost µν i fields which are required in our calculation, as follows. propagators 1 H (p)H ( p) = P (3.8) µν λρ µνλρ h − i p2 η ij X (p)X ( p) = (3.9) i j h − i p2 δ µν η (p)η¯( p) = (3.10) µ ν h − i p2 Here P is defined as µνλρ 2 P = δ δ +δ δ δ δ . (3.11) µνλρ µλ νρ µρ νλ µν λρ − D In the following, the index i of the X –field is omitted and L is equal to 1 1ǫbXˆ Xî. i − 2 i two–point vertices Kµνλραβ∂ H ∂ H : 1 α µν β λρ 1 1 Kµνλραβ = gˆTµνλρgˆαβ Pµνλρδαβ (3.12) 1 4 − 8 q KµνλραH ∂ H : 2 µν α λρ i Kµνλρα = ǫb gˆXˆ∂ XˆTµνλρgˆαβ +i gˆgˆαβTγνλρΓˆµ 2 −4L β βγ q q 1 i ǫb gˆXˆ∂ XˆT γµνTβαλρ (3.13) − L γ β q µνλρ K H H : 3 µν λρ 1 1 Kµνλρ = ǫb gˆXˆ∂ XˆTµναρgˆβγΓˆλ + ǫb gˆXˆ∂ XˆT δµνTβγαρΓˆλ 3 4L γ αβ L δ γ αβ q q 1 1 + ǫb gˆXˆ∂ XˆTλρανgˆβγΓˆµ + ǫb gˆXˆ∂ XˆT δλρTβγανΓˆµ 4L γ αβ L δ γ αβ q q 1 gˆTανδρgˆβγΓˆµ Γˆλ gˆRˆα T δµνTβγλρ − αβ γδ − 2 βγδ α q q 8 1 gˆ∂ Xˆ∂ XˆT αµνTβγλρ (3.14) −4L α β γ q KµναH ∂ X : 4 µν α 1 Kµνα = i gˆ∂ XˆTαβµν (3.15) 4 √L β q µν K H X : 5 µν 1 Kµν = ǫb gˆXˆRˆ Tαβµν (3.16) 5 −√L αβ q Kαβ∂ X∂ X : 6 α β 1 Kαβ = ( gˆgˆαβ δαβ) (3.17) 6 2 − q K X2 : 7 1 ˆ K = ǫb gˆR (3.18) 7 2 q K˜µνλραβ∂ H ∂ H : 1 α µν β λρ AG 1 1 K˜µνλραβ = gˆTµνλρgˆαβ gˆT αµνTγβλρ (3.19) 1 − ǫ 4L − 2L γ (cid:18) q q (cid:19) K˜µνλραH ∂ H : 2 µν α λρ AG i 1 K˜µνλρα = ǫb gˆXˆ∂ XˆTµνλρgˆαβ +i gˆTγνλρgˆαβΓˆµ 2 − ǫ −4L2 β L βγ (cid:18) q q 2 i i gˆTγβδνT αλρΓˆµ + ǫb gˆXˆ∂ XˆT βµνTαγλρ (3.20) − L β γδ 2L2 β γ q q (cid:19) ˜µνλρ K H H : 3 µν λρ AG 1 1 K˜µνλρ = ǫb gˆXˆ∂ XˆTµναρgˆβγΓˆλ ǫb gˆXˆ∂ XˆT δµνTγβαρΓˆλ 3 − ǫ 4L2 γ αβ − 2L2 δ γ αβ (cid:18) q q 1 1 + ǫb gˆXˆ∂ XˆTλρανgˆβγΓˆµ ǫb gˆXˆ∂ XˆT δλρTγβανΓˆµ 4L2 γ αβ − 2L2 δ γ αβ q q 1 1 gˆTανδρgˆβγΓˆµ Γˆλ gˆRˆα T δµνTβγλρ −L αβ γδ − 2L βγδ α q q 2 + gˆT βγνTδαηρΓˆµ Γˆλ (3.21) L α βγ δη q (cid:19) K¯µναβ∂ η¯ ∂ η : 1 α µ β ν K¯µναβ = gˆgˆµνgˆαβ δµνδαβ (3.22) 1 − q K¯µνα∂ η¯ η : 2 α µ ν K¯µνα = i gˆgˆαβgˆµγΓˆν (3.23) 2 βγ q K¯µναη¯ ∂ η : 3 µ α ν K¯µνα = i gˆgˆαβgˆνγΓˆµ (3.24) 3 βγ q ¯µν K η¯ η : 4 µ ν K¯µν = gˆgˆαβgˆγδΓˆµ Γˆν gˆRˆµν (3.25) 4 − αγ βδ − q q 9