The renormalization of fluctuating branes, the Galileon and asymptotic safety
aa r X i v : . [ h e p - t h ] O c t Prepared for submission to JHEP
MZ-TH/12-54
The renormalization of fluctuating branes,the Galileon and asymptotic safety
A. Codello, N. Tetradis and O. Zanusso SISSA, Via Bonomea 265, 34136 Trieste, Italy Nuclear and Particle Physics Sector, Department of Physics,University of Athens, Zographou 15784, Greece PRISMA Cluster of Excellence and Institute of Physics (THEP),University of Mainz, Staudingerweg 7, D-55099 Mainz, Germany
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We consider the renormalization of d -dimensional hypersurfaces (branes)embedded in flat ( d + 1)-dimensional space. We parametrize the truncated effective actionin terms of geometric invariants built from the extrinsic and intrinsic curvatures. We studythe renormalization-group running of the couplings and explore the fixed-point structure.We find evidence for an ultraviolet fixed point similar to the one underlying the asymptotic-safety scenario of gravity. We also examine whether the structure of the Galileon theory,which can be reproduced in the nonrelativistic limit, is preserved at the quantum level. ontents β -functions 65 Fixed points and asymptotic safety 86 Conclusions 12 Scalar field theories are ubiquitous in physics, describing a plethora of classical and quan-tum systems. Because of their relative simplicity, they have often been used as a testingground for new ideas or techniques. The action of a scalar field is usually assumed tocontain a standard kinetic term, especially when quantum or statistical fluctuations of thesystem are studied. The inclusion of higher-derivative terms may lead to two pathologies:a) The presence of derivatives higher than the second in the equation of motion resultsin the appearance of modes with negative norm, characterized as ghosts. b) The higher-derivative terms are perturbatively nonrenormalizable and the theory loses predictivity.We are interested in the systematic study of quantum corrections in scalar field the-ories with higher-derivative terms. As we have mentioned, such theories are in generalnonrenormalizable in the perturbative sense. However, their scale dependence can be stud-ied through the exact renormalization group (ERG). Their renormalizability may resultfrom the presence of a nonperturbative fixed point. The main drawback of the ERG ap-proach is that the integration of the flow equation for the scale-dependent effective actioncan be achieved only for truncated versions of the action. However, it is still possible tocheck the reliability of the predictions by expanding the truncation scheme and examiningtheir stability. This procedure has been applied to scalar theories with a general potentialand a standard kinetic term, leading to an accurate determination of nontrivial quantities,such as critical exponents [1]. The precision can be improved by going to higher orders ofthe derivative expansion [2].The theories we consider in this work describe hypersurfaces, which we term branes,embedded in a higher-dimensional flat spacetime, to which we refer as bulk spacetime. Theleading contribution to the action is given by the volume swept by the brane, expressed interms of the induced metric. It is invariant under arbitrary changes of the brane world-volume coordinates. We can fix this gauge freedom by identifying the brane coordinates– 1 –ith certain bulk coordinates. This choice is usually characterized as the static gauge. Theremaining bulk coordinates become scalar fields of the worldvolume theory. More compli-cated terms can also be included in the effective action. The crucial property that makesthis class of theories interesting is that the effective action can be expressed in terms ofgeometric quantities, such as the intrinsic and extrinsic curvatures of the hypersurface. Inthe static gauge these can be written in terms of the scalar fields and their derivatives.In this way we obtain a higher-derivative scalar theory with a particular structure. TheERG flow of the scalar theory describing two-dimensional membranes has been consideredin ref. [3]. Here we present the generalization to a d -dimensional brane, embedded in a( d + 1)-dimensional bulk.Scalar field theories with derivative interactions have been considered extensively dur-ing the last years in the context of particle physics and cosmology under a variety of names,such as: k -essence [4], Dirac-Born-Infeld (DBI) inflation [5], the Dvali-Gabadadze-Porrati(DGP) model [6] in the decoupling limit and the Galileon [7], scalar-tensor models withkinetic gravity braiding [8], etc. All these theories are constructed so that the equationof motion does not contain field derivatives higher than the second, even though a largeor infinite series of derivative terms can be present in the action. In this way, ghost fieldsdo not appear in the spectrum. The most general scalar-tensor theory with this propertywas constructed a long time ago [9], and rediscovered recently. It is characterized as thegeneralized Galileon (see ref. [10] and references therein).The absence of derivatives higher than the second in the equation of motion is notprotected by some underlying symmetry. For example, for the Galileon theory it is knownthat quantum corrections generate terms that destroy this property. The one-loop correc-tions computed through dimensional regularization induce a term φ (cid:3) φ in the effectiveaction [11]. It is still possible to consider the Galileon as a consistent quantum theory atlow energies, for which such a term is subleading. The main motivation for our study stemsfrom the wish to understand the issue of quantum corrections for such derivative theoriesthrough the ERG approach.The connection between the Galileon and the brane picture that we discussed aboveis provided by ref. [12], which shows that the Galileon theory can be reproduced in thenonrelativistic limit by considering the effective action for the position modulus of a probebrane within a five-dimensional bulk. Derivatives higher than the second can be avoidedby employing only Lovelock invariants in the geometric picture. In this work we consider atruncation of the brane effective action that takes into account the lowest-order geometricinvariants. Some of these reproduce the structure of the Galileon theory [7]. On the otherhand, our truncation includes a contribution involving the extrinsic curvature of the branethat does not have an analogue in the Galileon theory, as it would induce a field derivativehigher than the second in the equation of motion. We examine how this term scales underquantum corrections and whether it is consistent to assume that it does not appear in theeffective action.Our study has another very interesting spinoff. If the contributions from the extrinsiccurvature are omitted, the ERG flow can be expressed as the evolution of an effectiveNewton’s constant and the cosmological constant. The picture is similar to that obtained– 2 –n ERG studies of gravity, in which the metric is considered as the fundamental field.The β -functions display a fixed-point structure that is analogous to that associated withasymptotic safety [13]. Thus we obtain a very useful testing ground for a concept thatcould provide the UV completion of gravity.In the following section we establish our notation and summarize the correspondencebetween the brane and Galileon theories. In section 3 we introduce the effective action thatwe consider and the flow equation that describes its evolution. In section 4 we derive the β -functions for the couplings of the theory and discuss the effect of quantum correctionson the structure of the brane and Galileon theories. In section 5 we analyze the fixed-pointstructure for a consistent truncation that preserves only the cosmological-constant andEinstein terms. We discuss the analogy with the asymptotic-safety scenario of gravity. Inthe final section we present a summary and our conclusions. The ERG formalism has beendeveloped for field theories in Euclidean space. For this reason we assume the analyticcontinuation to imaginary time throughout the paper. Following ref. [12], we summarize briefly the connection between the dynamics of fluctuatingbranes and the Galileon theory. The connection has been established for a four-dimensionalbrane embedded in five-dimensional flat space. The induced metric in the static gauge is g µν = η µν + ∂ µ π ∂ ν π , where π denotes the extra coordinate of the bulk space. We preservethe notation η µν even though we use imaginary time and the bulk metric is Euclidean. Theinduced extrinsic curvature is K µν = − ∂ µ ∂ ν π/ p ∂π ) . We denote its trace by K . Theleading terms in the brane effective action are S µ = µ Z d x √ g = µ Z d x p ∂π ) (2.1) S ν = ν Z d x √ g K = − ν Z d x (cid:0) [Π] − γ [ φ ] (cid:1) (2.2) S ¯ κ = ¯ κ Z d x √ g R = ¯ κ Z d x γ (cid:0) [Π] − [Π ] + 2 γ ([ φ ] − [Π][ φ ]) (cid:1) , (2.3)where γ = 1 / √ g = 1 / p ∂π ) . We have adopted the notation of ref. [12], with Π µν = ∂ µ ∂ ν π and square brackets representing the trace (with respect to η µν ) of a tensor. Also,we denote [ φ n ] ≡ ∂π · Π n · ∂π , so that [ φ ] = ∂ µ π ∂ µ ∂ ν π ∂ ν π . The field π has mass dimension −
1, as it corresponds to a spatial coordinate. It can be given a more conventional massdimension through multiplication with the appropriate power of the fundamental energyscale M of the theory. We implicitly assume that all other scales are expressed in termsof M , which is effectively set equal to 1. The couplings µ , ν , ¯ κ correspond to the effectivefour-dimensional cosmological constant, the five-dimensional Planck scale M and the four-dimensional Planck scale M , respectively.The effective action of the Galileon theory can be obtained in the nonrelativistic limit( ∂π ) ≪
1. It must be noted, however, that terms with second derivatives of the field, suchas π (cid:3) π , are not assumed to be small (here (cid:3) = η µν ∂ µ ∂ ν ). If total derivatives are neglected,– 3 –he integrants of the leading terms in the expansion of (2.1)–(2.3) are proportional to ( ∂π ) .In this way, one obtains three of the terms appearing in the action of the Galileon theory[12]. The term of highest order in this theory can be obtained by including in the braneaction the Gibbons-Hawking-York term associated with the Gauss-Bonnet term of five-dimensional gravity. We omit this term in the truncated effective action that we consider,as it complicates significantly the study of the renormalization of the theory. Its effect willbe the focus of future work. In the context of asymptotic safety boundary terms have beenconsidered in [14].The first Gauss-Codazzi equation gives R = K − K µν K µν . This relation indicates thatthe truncation of the effective brane action that includes a term ∼ R should also includea term ∼ K . On the other hand, such a term must be excluded if the equation of motionis assumed not to contain field derivatives higher than the second. Its absence cannot beenforced by some underlying symmetry, and quantum corrections may introduce it even ifit is omitted in the tree-level action. In order to study its role in the renormalized theorywe include in our truncated action the contribution S κ = κ Z d x √ g K = κ Z d x γ (cid:0) [Π] − γ [ φ ] (cid:1) . (2.4)In the limit ( ∂π ) ≪
1, the contribution [Π] included in this term generates in the integranta leading contribution ∼ π (cid:3) π . A term K µν K µν in the Lagrangian density would producea contribution [Π ], which would again become ∼ π (cid:3) π in the nonrelativistic limit. Thetwo leading contributions cancel in R = K − K µν K µν , so that the structure of the Galileonis generated. On the other hand, if quantum corrections spoil the cancellation, the Galileontheory is not reproduced.It is worth pointing out that the term S ν can be omitted if we assume the discretesymmetry π → − π . The same symmetry would eliminate the higher-order contributionrelated to the Gauss-Bonnet term of the bulk theory. For a probe brane the presence in theaction of terms odd in the extrinsic curvature indicates an asymmetry in the fluctuations oneither side of the brane. The origin of such terms is not obvious, unless the bulk space is nothomogeneous or the brane is viewed as its boundary. These considerations indicate that itseems more natural to include the contributions (2.1), (2.3), (2.4) in a consistent quantumtheory than the ones that reproduce the Galileon theory. The terms (2.1), (2.3), (2.4)form the basis for the study of the renormalization of two-dimensional fluid membranes(see ref. [3] and references therein). The focus of our study is the evolution of the scale-dependent effective actionΓ k = Z d d x √ g (cid:16) µ k + ν k K + κ k K + ¯ κ k R (cid:17) , (3.1)with the various invariants expressed through the field π . We have included the contri-butions (2.1)-(2.4) discussed in the previous section, but we now assume that the variouscouplings depend on the running energy scale k . The action describes the dynamics of a– 4 – -dimensional brane embedded in ( d + 1)-dimensional flat space. We use imaginary time,so that the bulk metric is Euclidean.The formal treatment of the action (3.1) can be carried out through the ERG. Wefirst introduce the scale k by adding to the action a term R k ( q ) in momentum space,so that fluctuations of the field with characteristic momenta q < ∼ k are cut off [1, 15].We subsequently introduce sources and define the generating functional for the connectedGreen functions. Through a Legendre transformation we obtain the generating functionalfor the 1PI Green functions, from which we subtract the regulating term involving R k ( q ).In this way we obtain the scale-dependent effective action Γ k [ π ]. The procedure results inthe effective integration of the fluctuations with q > ∼ k . The theory is assumed to possessa fundamental high-energy cutoff M , so that Γ k is identified with the bare action S for k = M . For k = 0 we obtain the standard effective action. The means for calculating Γ k from S is provided by the exact flow equation [15] ∂ t Γ k [ π ] = 12 Tr ∂ t R k ( − (cid:3) )Γ (2) k [ π ] + R k ( − (cid:3) ) , (3.2)where we have reverted to position space and defined t = ln k . Here Γ (2) k [ π ] denotes thesecond functional derivative of the action with respect to the field. The rhs of the aboveequation receives contributions only from fluctuations with characteristic momenta q ≃ k .In this sense, the high-energy cutoff M is only a formal element in the definition of Γ k . Itcan be replaced by a UV fixed point in the flow of Γ k .When gauge symmetries, such as the reparametrization invariance of the brane world-volume theory, are present the definition of Γ k is more involved. We shall not presentthe details here, and we refer the reader to refs. [13] for the case of gravity, and toref. [3] for the case of brane reparametrization invariance. In the scale-dependent actionthe reparametrization invariance is implemented through the use of the background fieldmethod. The brane position is determined by the embedding function r = ( x µ , π ) and theinduced metric is given by g µν = ∂ µ r · ∂ ν r = η µν + ∂ µ π ∂ ν π . We parametrize the fluctuationsaround a background configuration r as r + δ r . In the static gauge that we have adopted,we have δ r = δπ n , where n is the unit vector normal to the brane and δπ is the fluctuatingfield. The cutoff R k (∆) is constructed by means of the operator ∆ = − g µν ∇ µ ∇ ν , where theinduced metric g µν and covariant derivatives ∇ µ compatible with it are expressed in termsof the (background) field π . The scale-dependent (background) effective action Γ k [ h δπ i ; π ]depends both on the (background) field π and the expectation value h δπ i of the fluctuatingfield. It is constructed so as to be invariant under reparametrizations of the background,while the expectation value h δπ i has to transform covariantly on a given background π .The exact flow equation has the form ∂ t Γ k [ h δπ i ; π ] = 12 Tr ∂ t R k (∆)Γ (2;0) k [ h δπ i ; π ] + R k (∆) . (3.3)In the limit h δπ i → k [ π ] ≡ Γ k [0; π ]. The difficulty we– 5 –ave to face is that the flow equation (3.3) is not a closed relation for Γ k [0; π ]. It becomesclosed if we make the ansatz Γ (2;0) k [0; π ] = Γ (0;2) k [0; π ] ≡ Γ (2) k [ π ]. We obtain eq. (3.2),where now the d’Alembertian − (cid:3) is replaced by the d’Alembertian ∆ constructed withthe full induced metric. As truncation ansatz for the effective action Γ k [ π ] we choose eq.(3.1), which is reparametrization invariant by construction. In this way, the invariance ispreserved by the evolution, even though the full dependence of the functional Γ k [ h δπ i ; π ]on the two fields π and h δπ i is not taken into account. (For example, this functional inprinciple includes a separate wavefunction renormalization for the fluctuation field h δπ i .)There is a more intuitive, albeit less rigorous, way to generate the flow equation. Theone-loop correction to a tree-level action of the form (3.1) is proportional to the logarithmof the fluctuation determinant around a given background. In order to compute it, weemploy the static gauge and expand the field as r + δπ n , keeping only the terms quadraticin δπ . The resulting expression depends on the various couplings appearing in (3.1). Theseare now the bare ones and have no k -dependence. The contribution of fluctuations withcharacteristic momenta below a given scale k can be excluded if we add to the Lagrangiandensity a term ∼ δπR k (∆) δπ . It must be kept in mind that the theory (3.1) has a geometricorigin, which must be preserved even when we employ the static gauge and express theaction in terms of the field π . The cutoff must be constructed in a way consistent withthis property. This can be achieved if we construct the d’Alembertian employing the fullinduced metric, expressed in terms of π . A “renormalization-group improvement” of theeffective action can be achieved by taking its logarithmic derivative with respect to k andsubstituting the running couplings, which are k -dependent, for the bare ones. The resultingexpression is the flow equation we discussed above. β -functions Extracting information from the flow equation requires an appropriate parametrizationand truncation of the scale-dependent effective action. For this purpose we employ thetruncation (3.1). In order to calculate the trace in the rhs of the flow equation we need thesecond functional derivative of (3.1) on an arbitrary background. We findΓ (2) k [ π ] = κ k ∆ + µ k ∆ + V µν ∇ µ ∇ ν + U + O ( K , ∇ K ) , (4.1)where V µν = 2 ν k ( K µν − Kg µν ) + κ k (cid:20) − (cid:0) K − K ρσ K ρσ (cid:1) g µν + 2 KK µν (cid:21) +¯ κ k (cid:20)(cid:18) R µν − R g µν (cid:19) − (cid:0) K − K ρσ K ρσ (cid:1) g µν + 2 KK µν + 2 K µσ K νσ (cid:21) (4.2) U = µ k (cid:0) K − K ρσ K ρσ (cid:1) (4.3)and the covariant derivatives are evaluated with the full induced metric. The first Gauss-Codazzi equation allows us to express K − K ρσ K ρσ in terms of R in the above expressions.A similar simplification can be carried for K µσ K νσ . However, we have preserved the ex-pression in a form similar to that given in ref. [3] for the two-dimensional brane.– 6 –e substitute the above expressions in the rhs of the flow equation and expand thedenominator in powers of the curvatures. The trace of the resulting terms can be computedthrough the heat kernel expansion, as described in [16]. The details of this procedure havebeen presented in ref. [3] for the case d = 2 and we do not repeat them here. We insert thetruncation (3.1) in the lhs of the flow equation and match the contributions that involvethe same curvature invariants on both sides of the equation. In this way we obtain the β -functions for the various couplings. They are ∂ t µ k = 1(4 π ) d/ Q d [ G k ∂ t R k ] (4.4) ∂ t ν k = − π ) d/ d − Q d +1 (cid:2) G k ∂ t R k (cid:3) ν k (4.5) ∂ t κ k = 1(4 π ) d/ ( d + 44 Q d +1 (cid:2) G k ∂ t R k (cid:3) κ k + ( d − Q d +2 (cid:2) G k ∂ t R k (cid:3) ν k ) (4.6) ∂ t ¯ κ k = 1(4 π ) d/ ( Q d − [ G k ∂ t R k ] − Q d (cid:2) G k ∂ t R k (cid:3) µ k − Q d +2 (cid:2) G k ∂ t R k (cid:3) ν k − d Q d +1 (cid:2) G k ∂ t R k (cid:3) κ k − d − Q d +1 (cid:2) G k ∂ t R k (cid:3) ¯ κ k ) . (4.7)The regularized propagators are G k ( z ) = 1 κ k z + µ k z + R k ( z ) , (4.8)while the Q -functionals are defined as Q n [ f ] = 1Γ( n ) Z ∞ dz z n − f ( z ) n > Q n [ f ] = ( − n f ( n ) (0) n ≤ . (4.9)Some qualitative properties of the evolution are immediately apparent. The β -functionof ν k vanishes for ν k = 0. This is an expected result, as setting ν k = 0 in the tree-level actioninduces the discrete symmetry π → − π , which protects this value at the quantum levelas well. For ν k = 0, which is a necessary assumption in order to reproduce the Galileontheory in the nonrelativistic limit, the β -function of κ k does not vanish. It is apparentfrom eq. (4.6) that a contribution ∼ ν k K is induced through quantum fluctuations. Inthe nonrelativistic limit a term ∼ ν k π (cid:3) π will be generated, which is not present in theGalileon theory. A similar phenomenon occurs for a scalar field coupled to gravity [17]. Onthe other hand, the analysis of the one-loop corrections to the Galileon through the use ofdimensional regularization shows that the lowest-order induced term is ∼ ν k π (cid:3) π [11].In order to understand this point we need to make contact with perturbation theory.With the appropriate approximations, the β -functions (4.4)-(4.7) can reproduce standardperturbative results. For ν k = κ k = 0 the scale-dependent effective action (3.1) has thesame structure as Einstein gravity with a cosmological constant [13]. The one-loop contri-bution to the cosmological constant can be obtained if we set µ k = 1 in the rhs of eq. (4.4).– 7 –his is the bare value of this parameter that leads to a canonically normalized kinetic termwhen √ g is expanded in powers of ( ∂π ) and the leading term is retained. Independentlyof the choice of cutoff function R k ( q ), we obtain (with z = q ) ∂ t µ k = ∂ t (cid:20) Z d d q (2 π ) d ln( q + R k ( q )) (cid:21) . (4.10)The trivial integration of this equation for k in the range [0 , M ] reproduces the one-loopcontribution to the vacuum energy arising from the quantum fluctuations of a single mass-less mode in a theory with a fundamental high-energy cutoff ∼ M . It must be noted thatin the brane theory the renormalization of the cosmological constant is the same as that ofthe leading kinetic term at low energies. This is obvious from the form of the propagator(4.8), in which z = q is multiplied by µ k . As a result the field π has a large anomalousdimension.We can obtain the one-loop correction to κ k in a similar fashion, by substituting thebare couplings for the running ones in the rhs of eq. (4.6). We assume that the bare theorydoes not contain a term ∼ K and the kinetic term is canonically normalized. With theseassumptions we can set µ k = 1 and κ k = 0 in the rhs of (4.6) and replace ν k by a constantvalue ν M . We obtain ∂ t κ k = − d − d ( d + 2) ν M ∂ t (cid:20)Z d d q (2 π ) d q ( q + R k ( q )) (cid:21) . (4.11)The integration of this equation for k in the range [0 , M ] results in a momentum integralwith a quartic divergence for d = 4, which is cut off by a high-energy scale ∼ M . Thisquantum correction would not be visible if dimensional regularization was used. On theother hand, the regularization with an explicit cutoff, such as the one employed in thecontext of the ERG, picks up corrections with possible quadratic or quartic divergences.The correction of eq. (4.11) induces a term ∼ ν M K , which in the nonrelativistic limitbecomes ∼ ν M π (cid:3) π . This term is not present in the Galileon theory, and is of a lower orderthan the term ∼ ν M π (cid:3) π expected from an analysis based on dimensional regularization.We emphasize that this conclusion does not require a specific choice of the cutoff function R k ( z ), and thus is ERG-scheme independent.A cross-check of the β -functions (4.4)-(4.7) can be obtained if we set µ k = ν k = 0.For d = 2 the resulting theory can describe two-dimensional fluid membranes in three-dimensional space. The couplings κ k and ¯ κ k correspond to the bending and Gaussianrigidities. The β -functions of these couplings were computed in ref. [3]. They agree withthose derived through perturbation theory [18] if the anomalous dimension of the fluctu-ating field is set to zero. Explicit expressions for the β -functions can be obtained for specific forms of the cutofffunction R k ( z ). The results are particularly simple for the choice R k ( z ) = (cid:2) κ k ( k − z ) + µ k ( k − z ) (cid:3) θ ( k − z ) . (5.1)– 8 –espite its unconventional form, the cutoff function generates the required behavior for theeffective propagator 1 /G k ( q ): For z = q > k the effective propagator is the perturbativeone (1 /G k ( z ) = κ k z + µ k z ), so that the corresponding fluctuations remain unaffected by thepresence of the cutoff. For z < k , 1 /G k ( z ) is finite and constant (1 /G k ( z ) = κ k k + µ k k )and the low-energy fluctuations are suppressed. It has been verified through several studiesthat, when these criteria are fulfilled, the predictions obtained in the limit k → R k ( z ) [1].For R k ( z ) given by eq. (5.1) the function f ( z ) in eqs. (4.4)-(4.7) has the general form f ( z ) = [ G k ( z )] m ∂ t R k ( z ), with m a positive integer. Our cutoff choice leads to the appear-ance of terms ∼ ∂ t κ k , ∂ t µ k in ∂ t R k ( z ). For κ k = 0, the term ∂ t µ k would correspond tothe anomalous dimension of the field. We shall neglect these contributions in our analysis,as they are not expected to affect the qualitative features of the evolution. They must beincluded, however, if quantitative precision is required.For n ≥
0, the Q -functionals become Q n [ f ] = k n Γ( n + 1) f (0) , (5.2)with f (0) = (cid:18) κ k k + µ k k (cid:19) m (cid:0) κ k k + 2 µ k k (cid:1) . (5.3)The evolution equations (4.4)-(4.7) take the form ∂ t µ k = k d (4 π ) d/ Γ (cid:0) d + 1 (cid:1) κ k k + µ k κ k k + µ k (5.4) ∂ t ν k = − k d (4 π ) d/ Γ (cid:0) d + 2 (cid:1) ( d −
1) (2 κ k k + µ k ) ν k ( κ k k + µ k ) (5.5) ∂ t κ k = 2 k d (4 π ) d/ Γ (cid:0) d + 2 (cid:1) ( d + 44 (2 κ k k + µ k ) κ k ( κ k k + µ k ) + 4( d − d + 4 (2 κ k k + µ k ) ν k ( κ k k + µ k ) ) (5.6) ∂ t ¯ κ k = k d (4 π ) d/ Γ (cid:0) d + 2 (cid:1) ( d ( d + 2)12 2 κ k k + µ k ( κ k k + µ k ) k − d + 4 (2 κ k k + µ k ) ν k ( κ k k + µ k ) − (cid:20) ( d + 2) µ k k + 2 dκ k + 3( d − κ k (cid:21) κ k k + µ k ( κ k k + µ k ) ) . (5.7)The structure of the above equations is typical of the ERG, with terms involving variouspowers of the effective propagator appearing in the β -functions. The class of theories thatwe are considering involves only generalized kinetic terms. For this reason couplings such as κ k , µ k that multiply the leading terms appear often in the denominator in the β -functions.As the theory has a geometric origin, the fundamental field π has mass dimension − M of the theory. Throughout the paper we assumethat all scales are expressed in terms of M . The scaling dimensions of the various couplings– 9 –an be deduced from eqs. (5.4)-(5.7) if we remove the explicit factors of k through theappropriate redefinitions. If we define µ k = k d ˜ µ k , ν k = k d − ˜ ν k , κ k = k d − ˜ κ k , ¯ κ k = k d − ˜¯ κ k , (5.8)eqs. (5.4)-(5.7) become ∂ t ˜ µ k = − d ˜ µ k + 1(4 π ) d/ Γ (cid:0) d + 1 (cid:1) κ k + ˜ µ k ˜ κ k + ˜ µ k (5.9) ∂ t ˜ ν k = − ( d − ν k − π ) d/ Γ (cid:0) d + 2 (cid:1) ( d −
1) (2˜ κ k + ˜ µ k )˜ ν k (˜ κ k + ˜ µ k ) (5.10) ∂ t ˜ κ k = − ( d − κ k + 2(4 π ) d/ Γ (cid:0) d + 2 (cid:1) ( d + 44 (2˜ κ k + ˜ µ k )˜ κ k (˜ κ k + ˜ µ k ) + 4( d − d + 4 (2˜ κ k + ˜ µ k )˜ ν k (˜ κ k + ˜ µ k ) ) (5.11) ∂ t ˜¯ κ k = − ( d − κ k + 1(4 π ) d/ Γ (cid:0) d + 2 (cid:1) ( d ( d + 2)12 2˜ κ k + ˜ µ k ˜ κ k + ˜ µ k − d + 4 (2˜ κ k + ˜ µ k )˜ ν k (˜ κ k + ˜ µ k ) − (cid:20) ( d + 2)˜ µ k + 2 d ˜ κ k + 3( d − κ k (cid:21) κ k + ˜ µ k (˜ κ k + ˜ µ k ) ) . (5.12)This is the most convenient form of the evolution equations for the determination of theirfixed points.As a first check we can compute the β -functions of κ k , ¯ κ k for two-dimensional fluidmembranes. In the membrane theory the volume (now area) term is considered subleading.This means that we can get the relevant equations by setting d = 2, µ k = ν k = 0 ineqs. (5.6), (5.7). We obtain ∂ t κ k = 34 π , ∂ t ¯ κ k = − π . (5.13)These expressions reproduce the results of refs. [3, 18] for the renormalization of the bendingand Gaussian rigidities of fluctuating membranes in a three-dimensional bulk space. It mustbe pointed out, however, that the relation µ k = 0 is not consistent with eq. (5.4), whichbecomes ∂ t µ k = k π (5.14)for d = 2, µ k = 0. Neglecting the area term can be viewed only as a low-energy approxi-mation. Setting µ k = ν k = 0 in eqs. (5.6), (5.7) provides a generalization of the evolutionfor branes of arbitrary dimensionality.An important point, which we have already discussed in the previous section, is thestability of the conditions ν k = 0 and κ k = 0 under quantum corrections. The first oneis expected to be stable, as it is protected by the symmetry π → − π . The evolutionequation (5.5) explicitly demonstrates that ∂ t ν k vanishes for ν k = 0. On the other handthe condition κ k = 0 does not enhance the symmetry of the action and is not expected tosurvive at the quantum level. Eq. (5.6) indicates that corrections ∼ ν k are generated for κ k under renormalization. It is noteworthy that, if we set ν k = 0, we have ∂ t κ k = 0 for– 10 – k = 0 and µ k = 0. We believe that this is an accidental feature. Notice also that the β -function does not vanish if we first set µ k = 0 and then take the limit κ k → µ k , ¯ κ k = 0 we can consistently assume that ν k = κ k = 0, as then the associated β -functions vanish. The reduced action (3.1) contains only the Einstein and cosmological-constant terms. It must be emphasized that the theory we are considering is not dynamicalgravity. The action (3.1) involves only one fluctuating scalar degree of freedom that hasgeometric origin. Despite the different nature of the theory, we find that the flows displaystriking similarity with what has been observed in the analysis of gravitational theories.The evolution of the couplings is described by eqs. (5.9), (5.12) with ˜ κ k = 0. Weconcentrate on the case d = 4 which is closest to four-dimensional gravity. In order tomake the analogy with gravity more apparent we define the dimensionless cosmologicaland Newton’s constants through the relations˜ µ k = Λ k πG k , ˜¯ κ k = − πG k . (5.15)Their evolution is given by ∂ t Λ k = − k + 16 π G k (3 − k ) (5.16) ∂ t G k = 2 G k + 112 π G k Λ k (3 − k ) . (5.17)This system of equations has two fixed points at which the β -functions vanish: a) theGaussian one, at Λ k = G k = 0, and b) a nontrivial one, at Λ k = 9 / G k = 18 π .The evolution of the couplings is depicted in fig. 1 for increasing k . For Λ k > k → ∞ to be taken. The flows in the region Λ k < k , are disconnected, as the β -function of G k diverges on the lineΛ k = 0, while the flows are in opposite directions on either side of this line.The presence of the nontrivial fixed point and the form of the flows around it display astrong similarity with the corresponding flows for gravity in the Einstein-Hilbert truncation,in which only the cosmological and Newton’s constants are retained [13]. In gravity theflows for Λ k < k can display a strong sensitivity to the choice of thecutoff function. However, their qualitative form in the vicinity of the fixed points is stableand provides support for the asymptotic safety scenario, which assumes a UV completionof gravity through a nontrivial fixed point. A nice feature of our flows is that they displaystream lines connecting the region near the UV fixed point with the physical IR region inthe limit k → - - Λ k G k Figure 1 . The flows predicted by the evolution equations (5.16), (5.17). this sense these results are similar to those obtained by considering the gravitational flowsinduced by matter fields in the large N limit [19]. It must be emphasized, however, thatthe theory we are considering has an underlying gauge symmetry, the reparametrizationinvariance of the worldvolume, which must be preserved in the cutoff theory. In this senseit poses difficulties analogous to those encountered when trying to preserve the generalcovariance of gravity. The focus of this work has been on understanding the effect of quantum corrections on thestructure of higher-derivative theories. Such theories are in general nonrenormalizable inthe perturbative sense. For this reason we employed the ERG, which has the potential toreveal nonperturbative features, such as fixed points not easily accessible to perturbativemethods. On the other hand, it must be kept in mind that the ERG approach reliesheavily on the use of truncated versions of the effective action, which may not capture allthe physics. – 12 –he analysis of a general higher-derivative theory would involve too many parameters.For this reason we limited our discussion to the class of theories that describe d -dimensionalfluctuating branes within a bulk space of d + 1 dimensions. The physical degree of free-dom is the position modulus π of the brane, which can be viewed as a scalar field ofthe worldvolume theory. The structure of the Lagrangian density is constrained by thereparametrization invariance of the brane worldvolume. The various terms correspond togeometric invariants, involving the extrinsic and intrinsic curvatures of the brane expressedin terms of π .In the nonrelativistic limit the classical brane theory can reproduce the structure ofthe Galileon theory [12]. An important question is whether this feature remains validat the quantum level as well. We found evidence that quantum corrections spoil thecorrespondence. They generate a geometric term in the brane theory ∼ K , where K denotes the trace of the extrinsic curvature. Even if the term is absent at the classical level,it will appear upon renormalization. In the nonrelativistic limit this term becomes ∼ π (cid:3) π ,a contribution not present in the Galileon theory. On the other hand, the analysis of thequantum corrections to the Galileon theory through the use of dimensional regularizationindicates that the lowest-order correction is ∼ π (cid:3) π [11]. The discrepancy can be resolvedby noting that the ERG analysis employs an explicit cutoff as a regulator of momentumintegrals. For this reason it is sensitive to corrections with quadratic or quartic divergences.The term ∼ π (cid:3) π is induced by a correction with a quartic divergence, which is not visiblethrough dimensional regularization. It must be noted that our conclusion does not dependon the specific choice of the infrared cutoff that we employ in the context of the ERG, andis, therefore, ERG-scheme independent.We considered the action of eq. (3.1), written in terms of geometric invariants. Thesecan be expressed through the position modulus π according to eqs. (2.1)-(2.4). The β -functions for the couplings of the theory are given by eqs. (4.4)-(4.7). They form the mainresult of this work. For the particular choice (5.1) for the cutoff function, the β -functionscan be written in the form (5.9)-(5.12), without an explicit reference to the running scale k . In an approximation consistent with these equations, we considered a truncation ofthe action that preserves only the cosmological-constant and Einstein terms. Despite thesimilarity with dynamical gravity, the theory has only one fluctuating scalar degree offreedom. It is remarkable, therefore, that the most prominent feature of the flow diagramis qualitatively similar to the one in the asymptotic-safety scenario for gravity. There is anattractive UV fixed point, which can be employed in order to obtain a UV completion ofthe theory.The fixed points and the related flows predicted by eqs. (4.4)-(4.7) for various valuesof d will be the focus of future research. The coupling ν k can be consistently set to zero ifwe assume a symmetry in the fluctuations on either side of the brane. The reduced systeminvolves three couplings ( µ k , κ k , ¯ κ k ) and possesses novel fixed points. It forms a consistentframework in which to study the renormalization-group evolution of a higher-derivativetheory with nontrivial features. The analogy with the evolution of d -dimensional gravityis a very interesting issue. – 13 – cknowledgments We would like to thank S. Abel, J. Rizos and R. Percacci for useful discussions. Thisresearch of N.T. has been supported in part by the ITN network “UNILHC” (PITN-GA-2009-237920). The research of N.T. has also been co-financed by the European Union(European Social Fund ESF) and Greek national funds through the Operational Program“Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)- Research Funding Program: “THALIS. Investing in the society of knowledge through theEuropean Social Fund”. The research of O.Z. is supported by the DFG within the Emmy-Noether program (Grant SA/1975 1-1).
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