Cosmological α'-corrections from the functional renormalization group
PPrepared for submission to JHEP
Cosmological α (cid:48) -corrections from the functionalrenormalization group Ivano Basile a Alessia Platania b a Service de Physique de l’Univers, Champs et Gravitation, Universit´e de Mons, Place du Parc 20,7000 Mons, Belgium b Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, ON N2L 2Y5, Canada
E-mail: [email protected] , [email protected] Abstract:
We employ the techniques of the Functional Renormalization Group in stringtheory, in order to derive an effective mini-superspace action for cosmological backgroundsto all orders in the string scale α (cid:48) . To this end, T-duality plays a crucial role, classifying allperturbative curvature corrections in terms of a single function of the Hubble parameter.The resulting renormalization group equations admit an exact, albeit non-analytic, solutionin any spacetime dimension D , which is however incompatible with Einstein gravity at lowenergies. Within an (cid:15) -expansion about D = 2, we also find an analytic solution whichexhibits a non-Gaussian ultraviolet fixed point with positive Newton coupling, as wellas an acceptable low-energy limit. Yet, within polynomial truncations of the full theoryspace, we find no evidence for an analog of this solution in D = 4. Finally, we commenton potential cosmological implications of our findings. a r X i v : . [ h e p - t h ] J a n ontents D dimensions 114.3 Analytic solution within an (cid:15) -expansion 154.4 Analytic fixed-point solutions within truncations 184.5 Cosmological implications 23 Despite the tremendous success of Einstein gravity in describing low-energy phenomena,its perturbative non-renormalizability, along with other deeply rooted subtleties, has longremained a remarkably puzzling challenge to a more complete understanding of the high-energy behavior of the theory and to the formulation of a consistent ultraviolet (UV)completion. While a number of proposals have been put forth, any candidate theory is todetermine infinitely many curvature corrections to the low-energy effective action in termsof finitely many parameters. – 1 –ithin the framework of asymptotically safe gravity, this would be achieved by asuitable fixed point of the renormalization group (RG) flow, whose critical surface is finite-dimensional [1]. One of the possible approaches to investigate this proposal is the studyof the RG flow of the so-called effective average action (EAA) [2], a scale-dependent ac-tion functional containing all possible operators compatible with the symmetries and fieldcontent of the theory. The RG flow of the EAA is typically studied using Functional Renor-malization Group (FRG) techniques [3, 4], which involve a functional integro-differentialequation for the EAA [2]. Although in principle these methods can capture the completephysics in a non-perturbative fashion, solving the FRG equation exactly is currently outof reach and the full theory space is typically restricted via truncated derivative or vertexexpansions of the EAA [3]. These methods have been successfully employed in the studiesof the RG flow of Quantum Chromodynamics and also in the context of Condensed-matterPhysics [4], obtaining results in agreement with those derived from conformal-bootstraptechniques [5]. In the context of gravity, the FRG has been extensively used to assess theexistence of the so-called Reuter fixed point and investigate its features in progressivelylarger truncations of the full theory space . Despite these studies provide strong evidencefor the existence of the Reuter fixed point (see [7, 8] and references therein), a completeproof is still lacking due to the practical necessity to truncate. To this end, frameworkssuch as tensor models [9, 10] and dynamical triangulations [11–13] could provide alterna-tive ways to seek asymptotic safety in quantum gravity [14–20], thus complementing theFRG and circumventing its limitations.On the other hand, string theory appears to provide a consistent UV completion, butthe resulting all-order corrections are best understood insofar as supersymmetry remainsunbroken. Aside from peculiar settings, in general only the computation of the first few cor-rections is currently feasible. In the context of (perturbative) string theory, corrections areorganized in a double expansion in the string coupling g s and the “Regge slope” α (cid:48) , whichdefines the string length scale (cid:96) s ≡ √ α (cid:48) . At least within the first few orders in g s , stringtheory provides a well-defined recipe to compute curvature corrections, weighed by α (cid:48) , ina systematic fashion, to arbitrarily high orders in α (cid:48) . In particular, to leading order in g s they can be determined from the conditions that the two-dimensional quantum field theorydescribing the string worldsheet embedded in a generic D -dimensional spacetime be freeof Weyl anomalies [21–23]. The simplest solutions to this consistency condition entail thatthe spacetime dimension D be the critical dimension, namely 26 for the bosonic string [24]and 10 for the superstring [25], while “non-critical” backgrounds involve the presence ofstrong curvatures and/or large fluxes. However, even in the best-understood critical casethe computation of all-order curvature corrections appears unfeasible at present, with theexception of particularly simple classes of backgrounds [26, 27].While these peculiar classes do not include cosmological backgrounds, which are amongthe most relevant for phenomenological applications, in these settings curvature corrections Let us remark that, although additional (ghost) poles appear upon truncating a derivative expansionof the effective action, they do not necessarily spoil unitarity, since they could be truncation artifacts [6]. – 2 –re dramatically constrained by T-duality [28–31], a genuinely stringy effect. The resultingmini-superspace effective actions involve a single function of the Hubble parameter, whoseperturbative expansion ought to match the α (cid:48) expansion order by order. Recent workshave investigated some phenomenological aspects of effective actions of this type [32–35].This remarkable result provides our starting point: combining the consistency constraintsof string theory with FRG methods can potentially shed light on string theory at largecurvature.In this paper we move the first step in this direction, studying α (cid:48) -corrections in cosmo-logical backgrounds via FRG techniques. We start with a brief overview of the necessarybackground on string theory (in Section 2) and on the FRG (in Section 3). In Section 4we impose invariance under T-duality and work directly with the constrained spacetimeeffective action within a mini-superspace ansatz, deriving flow equations and studying theirsolutions. In Section 4.2, we show the existence of an exact solution to the flow equationsin any spacetime dimension. Within this solution, the RG running of the Newton cou-pling matches precisely the one found in [36] in the context of asymptotically safe gravity.Nonetheless, due to the non-minimal coupling between the dilaton and the metric degreesof freedom appearing in the Hohm-Zwiebach effective action, this solution turns out tobe non-analytic and phenomenologically unviable, since it cannot reproduce the Einstein-Hilbert action at any scale. In order to investigate the existence of phenomenologicallyviable solutions, in Section 4.3 we study the existence of analytic solutions to the flowequations. To this end, we first perform an (cid:15) -expansion around two spacetime dimensions,along the lines of [37–44], finding a closed-form solution to leading order in (cid:15) withouttruncations, whose UV-behavior is governed by a non-Gaussian fixed point (NGFP) witha positive value g ∗ > α (cid:48) corrections to the effective action in terms of the finitely many relevant couplingsin the UV. Although the corresponding effective action affords a closed-form expression toall orders in α (cid:48) , the resulting cosmology does not appear to deviate qualitatively from itsclassical counterpart. In particular, the absence of de Sitter solutions appears to supportSwampland conjectures in this setting [45]. In Section 4.4 we study the extension of thissolution to higher spacetime dimensions, and in particular to D = 4. Within polynomialtruncations of the full theory space up to O ( H ) in the Hubble parameter H , it appearsthat the analytic solution found in (2 + (cid:15) ) dimensions does not persists in higher dimen-sions: the fixed-point value of the dimensionless Newton coupling g ∗ vanishes at a “critical”dimension D crit (cid:39) . D > D crit . If these results keep holding inhigher-order truncations, our findings might indicate at least one of the following: • The NGFP is highly non-perturbative in D = 4 and cannot be detected via a trun-cated derivative expansion of the EAA; • The mini-superspace approximation is unable to capture all the relevant physics;– 3 – α (cid:48) -corrections are insufficient to achieve both a well-defined continuum limit and aphenomenologically viable IR limit and thus g s -corrections might be crucial in thisrespect.In Section 4.5 we discuss some potential cosmological implications of our findings, and inparticular the possible existence of de Sitter solutions with large curvature. We concludein Section 5 with a summary and a discussion of our results. Let us begin with a brief overview of some results of string theory that we shall make use ofin the following sections. Specifically, in Section 2.1 we outline how spacetime gravitationaleffective actions emerge from the worldsheet formulation of string perturbation theory,and in Section 2.2 we describe how the resulting mini-superspace cosmological actions areheavily constrained by T-duality. In the following we shall use the “mostly-plus” signaturefor the spacetime metric, and we shall keep this convention in the remainder of this paper.
The worldsheet formulation of (perturbative) string theory rests on an action principle thatgoverns the classical dynamics of a string probing a background spacetime according to anembedding of the type j : ( σ , τ ) (cid:55)→ X µ ( σ, τ ) . (2.1)where ( τ, σ ) are local coordinates on the worldsheet. X µ ( τ, σ ) then sweeps the stringworldsheet in the background spacetime. The natural starting point is the Nambu-Gotoaction, which takes the simple form S NG = − πα (cid:48) (cid:90) dσ dτ (cid:112) − det j ∗ G (2.2)in terms of the background metric G and the pull-back j ∗ of the embedding. A perturbativequantization of this theory can be carried out introducing an auxiliary dynamical metric γ on the worldsheet, obtaining the Polyakov action [24] S P = − πα (cid:48) (cid:90) dσ dτ (cid:112) − det γ G µν ( X ) γ αβ ∂ α X µ ∂ β X ν . (2.3)The action of Eq. (2.3) takes the form of a non-linear σ -model whose target spacetimeis described by the metric G µν ( X ), with the important difference that γ is dynamical.Remarkably, the resulting spectrum in flat spacetime contains massless states associatedto the (quanta of the) gravitational field, a 2-form field strength B and the dilaton φ .Using the state-operator correspondence one can show that introducing the corresponding– 4 –oherent states is tantamount to modifying the spacetime metric G . More generally, co-herent states of this type are related to the couplings of the most general (renormalizable)two-dimensional σ -model, which is described by [22] S σ = − πα (cid:48) (cid:90) dσ dτ (cid:112) − det γ (cid:20) G µν ( X ) γ αβ ∂ α X µ ∂ β X ν + B µν ( X ) (cid:15) αβ ∂ α X µ ∂ β X ν + α (cid:48) R (2) φ ( X ) (cid:21) (2.4)up to the inclusion of fermions, where R (2) is the worldsheet Ricci scalar. Moreover, theintroduction of the auxiliary worldsheet metric γ entails the presence of Weyl invariance,which acts according to γ αβ (cid:55)→ Ω ( σ, τ ) γ αβ (2.5)and is generally broken by the trace anomaly √ γ (cid:104) T αα (cid:105) = − Ω δWδ Ω (cid:12)(cid:12)(cid:12)(cid:12) Ω=1 (2.6)in the quantum theory, where W = log Z is the generating functional of connected dia-grams. Preserving Weyl invariance is therefore paramount to consistency and to a sensiblegeometric interpretation of the theory. In addition, the cancellation of the trace anomalyentails effective spacetime equations of motion for the background fields. Remarkably, theseequations stem from spacetime effective actions that include gravity coupled to matter. In-cluding worldsheet fermions [25] leads to the appearance of additional degrees of freedomin spacetime, both bosonic and fermionic, and most importantly of the gravitino. However,in this paper we shall focus on the simpler bosonic case. Furthermore, since the Polyakovaction introduces worldsheet diffeomorphism invariance, one ought to take into account thecorresponding Faddeev-Popov ghosts, whose contribution to the trace anomaly of Eq. (2.6)is proportional to the worldsheet Ricci scalar R (2) . It combines with a similar contributionarising from the other worldsheet fields, leading to the background-independent term (cid:104) T αα (cid:105) gh = D − D c π R (2) , (2.7)where the critical dimension is D c = 26 in the bosonic case, while D c = 10 when oneincludes worldsheet fermions and, correspondingly, worldsheet supersymmetry. Therefore,since the additional contributions to (cid:104) T αα (cid:105) vanish in Minkowski backgrounds, the latter areonly allowed for D = D c , and thus string perturbation theory is anomaly-free in this case.In any dimension, the resulting tree-level spacetime effective action contains a “universal”gravitational sector which takes the form [22] S string ⊃ πG N (cid:90) d D x √− g e − φ (cid:18) R + 4 ( ∂φ ) − H − D − D c )3 α (cid:48) (cid:19) (2.8) Notice that Eq. (2.6) is written in Euclidean signature for later convenience. – 5 –t leading order in α (cid:48) in the “string frame”, with H ≡ dB . One can recast it inthe more conventional Einstein frame with a Weyl rescaling. It is worth noting that thefinal term in Eq. (2.8), pertaining to non-critical strings, would dominate the other termsin the α (cid:48) expansion, in the absence of an additional expansion parameter. Achievingperturbative control of non-critical strings is therefore especially challenging, but for specialbackgrounds, such as those with a linear dilaton or those studied in [26, 27], one canshow that the trace anomaly cancels exactly in α (cid:48) . However, in general one expects thatnon-critical strings require all-order α (cid:48) -corrections or the presence of additional expansionparameters .Higher-derivative α (cid:48) -corrections to Eq. (2.8) can be derived following the same pro-cedure, at least in principle, while string-loop corrections, which we shall neglect in thispaper, are subtler, and are weighed by (powers of the) local string coupling g s ≡ e φ . Let usremark that, in general, g s is not a free parameter in string theory, since it is determinedby the (vacuum value of the) dilaton. Consistently with more familiar notions of effectiveaction, scattering amplitudes computed from worldsheet correlators of vertex operators [48]match the ones computed from the spacetime effective action in the low-energy limit, andought to do so to all orders in α (cid:48) and g s . At the level of (perturbative) string theory, T-duality manifests itself whenever stringspropagate on backgrounds with non-trivial 1-cycles around which strings can wind. Theresulting massive excitations feature the usual Kaluza-Klein oscillations and novel windingmodes, and the spectrum exhibits a discrete symmetry under their exchange, provided thatthe size R of the relevant cycle can be “inverted” according to R ↔ α (cid:48) R . (2.9)More generally, in the presence of a Killing isometry the proper inversion is described indetail by Buscher’s rules [49, 50]. Since T-duality invariance is an inherently stringy notion,its presence cannot be captured by standard tree-level effective actions , which encompassthe low-energy dynamics in which winding modes decouple. However, in particularly simplebackgrounds, curvature corrections can be heavily constrained on grounds of T-dualityalone. Indeed, at the level of the low-energy effective action, taken in D = d + 1 spacetimedimensions for the sake of generality, T-duality appears as an internal O ( d, d, R ) symmetrythat acts on the universal low-energy field content, which comprises the dilaton φ , the In critical string models where supersymmetry is broken at the string scale, similar contributions tothe spacetime effective action arise. In these settings, large fluxes appear to provide suitable expansionparameters [46]. In the super-critical case,
D > D c , one can also consider large dimensions D [47]. Going beyond local field theory in spacetime, it is possible to capture the low-energy dynamics in aT-duality-invariant fashion via double field theory [51–57], albeit its ability to fully capture α (cid:48) -correctionsis still unsettled [58, 59]. – 6 –etric G and the Kalb-Ramond two-form B . It has been shown [28–31] that, for a time-dependent ansatz of the form ds = − n ( t ) dt + h ij ( t ) dx i dx j ,B ij = b ij ( t ) ,φ = φ ( t ) , (2.10)where n ( t ) is the lapse function and h is the metric on spatial slices, all curvature correc-tions can be captured by a single function of the O ( d, d, R )-covariant matrix ˙ S , where S ≡ (cid:32) b h − h − b h − bh − − h − b (cid:33) , (2.11)while the dilaton appears, along with the spatial determinant of the metric, in the O ( d, d, R )- invariant combination e − Φ ≡ (cid:113) det( h ij ) e − φ . (2.12)Leaving Φ( t ) as an independent field, we shall focus on the more specific cosmologicalansatz ds = − n ( t ) dt + e σ ( t ) d x ,B = 0 , Φ = Φ( t ) , (2.13)for which the tree-level action reduces to S red = Vol d πG N (cid:90) dt n e − Φ (cid:16) − ˙Φ + d ˙ σ (cid:17) , (2.14)where Vol d is the (unwarped) volume of the d = D − G N is the D -dimensional Newton constant, and H ≡ ˙ σ is the Hubble parameter. Within this mini-superspace framework, multi-trace and single-trace contributions in ˙ S combine into a singlefunction of ˙ σ , and T-duality acts changing the sign of σ . Therefore, the effective action isto be even in σ on consistency grounds. Including perturbative curvature corrections, theresulting action can then be written as an asymptotic series of the form S HD ∼ Vol d πG N (cid:90) dt n e − Φ (cid:32) − ˙Φ + d n ∞ (cid:88) m =1 a m α (cid:48) m − (cid:18) ˙ σn (cid:19) m (cid:33) , (2.15)up to integration by parts and terms that vanish on-shell. The coefficients a m are relatedto the coefficients c m of [29–31] according to a m = 8 ( − m c m . At present, the extent towhich the corrections of Eq. (2.15) can encode non-perturbative physics is unclear . Whileit is conceivable that worldsheet instantons could be included resumming the asymptoticseries, e.g. via FRG techniques, focusing on cosmological configurations could neglectcontributions arising from (functional) traces over degrees of freedom “orthogonal” to mini-superspace. At any rate, our considerations rest on parametrically small string couplings See [60] for a discussion in the context of de Sitter solutions. – 7 – s = e φ (cid:28)
1, a condition that is to be verified a posteriori within solutions of the effectiveequations of motion. As a final remark, let us emphasize that the corrections contained inEq. (2.15) also encode the contribution of higher-spin massive string modes – a key featureof string theory. These modes are integrated out to obtain the low-energy effective actionfor the massless modes.
In the spacetime effective action of Eq. (2.8), the sign in front of the dilaton kinetic termis positive. This is not problematic, since the corresponding Einstein-frame action reads S Einstein = 116 πG N (cid:90) d D x √− g (cid:18) R − D − ∂φ ) − e − D − φ H (cid:19) . (2.16)From the Einstein-frame action of Eq. (2.16) one can observe that the usual Wick rotation t = − i t E leads to a positive-definite kinetic term in the Euclidean action S E , which isdefined by iS Einstein = − S E . Performing this Wick rotation on the mini-superspace actionof Eq. (2.14), the (string-frame) Euclidean action defined by S E = − i S red takes the sameform and signs as S red , since in the Einstein frame all kinetic terms become positive. Thesame result is achieved performing the Wick rotation on the full gravitational action, sincethe Ricci scalar does not change sign. Upon specializing the Wick-rotated covariant actionto a cosmological ansatz, one obtains again Eq. (2.14).As a final remark, let us observe that, within the mini-superspace framework that wehave described, fixing n = 1 in the (Euclidean) path integral amounts to computing1Vol(gauge) (cid:90) D n D a D Φ e − S E ∆ FP δ ( n − , (2.17)where the gauge-fixing δ -functional yields a trivial Faddeev-Popov determinant ∆ FP , since n is a Lagrange multiplier. Hence, the corresponding ghosts would be non-dynamical anddecoupled, and we shall neglect them in the following. In this section we provide a brief review of the main features of the FRG equations. TheFRG is a mathematical tool to study the RG flow of quantum field theories and theiruniversality properties. The basic idea is to convert the Wilsonian shell-by-shell integrationof fluctuating modes, at the level of the path integral, into a functional integro-differentialequation for the so-called EAA Γ k , k being a RG scale. The action functional Γ k is ascale-dependent effective action, resulting from the integration of quantum fluctuationswith momenta p (cid:38) k . Accordingly, Γ k interpolates smoothly between the bare action ofthe theory (whereby k → ∞ ) and the ordinary effective action (whereby k → Let us recall that these equations are correct when written adhering to the “mostly-plus” convention. – 8 –nce one has specified the symmetries of the theory, along with its field content, theflow of the EAA in the theory space is determined by a flow equation. One of the mostcommonly used FRG equations is the Wetterich equation [2, 61, 62] k∂ k Γ k = 12 STr (cid:26)(cid:16) Γ (2) k + R k (cid:17) − k∂ k R k (cid:27) . (3.1)Here the supertrace STr entails an integral over continuous coordinates, as well as sumsover any additional internal index. The EAA Γ k [Φ] is a functional of all the fields in thetheory, and Γ (2) k is its second functional derivative with respect to these fields. In the case ofgauge theories, a gauge-fixing and Faddeev-Popov ghosts have to be added, and these termsalso contribute to the flow of Γ k . The function R k is an infrared (IR) regulator, whichdepends on the ratio between the physical momentum p and the RG scale k . It entersΓ k as an effective, k -dependent mass term such that modes with p (cid:46) k are suppressed,while those with momenta p (cid:38) k are integrated out. Since R k acts as an effective, scale-dependent mass, it also enters the modified-inverse propagator in the right-hand-side ofthe Wetterich equation. Its derivative k∂ k R k induces the flow of Γ k . The most standardchoice for the R k -function is the Litim regulator R k ( p ) = ( k − p ) θ ( k − p ) [63], andwe shall adopt this regulator throughout this work.Since Γ k interpolates between the UV and the IR, the fixed points of its RG flow,where k∂ k Γ k = 0 , (3.2)provide all possible UV completions of the theory. In most cases such fixed points aresaddles of the RG flow: only a specific subset of RG trajectories – those lying on the UVcritical manifold of a fixed point – will attain it in the UV limit. The IR and UV behavior ofthe RG trajectories is thus determined by the initial conditions of the flow which, in turn,are specified by observations at low energies. Accordingly, a theory is UV-complete if its RGtrajectory, uniquely identified by observations in the IR, reaches a fixed point as k → ∞ .The theory is said to be asymptotically free if its RG trajectory ends in a Gaussian fixedpoint (GFP) in the UV. If instead its UV-completion is a NGFP, i.e. , an interacting theory,the theory is said to be asymptotically safe. A candidate UV-completion brings along keyinformation, namely the number of IR-relevant directions associated with the correspondingfixed point, i.e. , the co-dimension of the corresponding UV-critical manifold. Indeed, thisrepresents the number of free parameters of the theory and thus provides a measure ofits predictivity. Denoting the couplings of a theory by g i and the corresponding betafunctions by β i ( g , g , . . . ), the number of relevant directions coincides with the number ofpositive “critical exponents” θ i , defined as (minus) the eigenvalues of the stability matrix, S ij ≡ ∂ g i β j .The Wetterich equation, Eq. (3.1), cannot be solved exactly at present, and one isforced to “truncate” the theory space specifying an ansatz for the EAA Γ k . Specifically, Γ k is generally written in terms of a derivative or vertex expansion and truncated to a certainorder. The flow equation is then employed to extract the beta functions for a finite number– 9 –f couplings. It is worth mentioning that the derivative expansion of the effective actionrelies on a “natural” ordering of the operators, based on their canonical mass dimensions.Thus, insofar as the flow stays perturbative, i.e. the scaling of the couplings is well-approximated by the canonical scaling, the flow of the truncated derivative expansionought to provide a reliable approximation to the (projection of the) exact solution to (3.1)on the corresponding theory sub-space. In the following sections we shall employ the FRGequations and the concepts reviewed above to compute cosmological α (cid:48) -corrections in stringtheory. In practice, this amounts to promoting the coefficients a m in Eq. (2.15) to functionsof the RG scale k and determining their IR behavior in terms of the relevant parameters inthe UV. To this end, it is more convenient to introduce a function F k whose power seriesexpansion encodes the coefficients a m and thus α (cid:48) corrections to all orders. Let us now focus on the spacetime effective action of string theory in D = ( d +1) dimensions,with d the dimension of spatial slices. As we have discussed in Section 2.2, for the time-dependent backgrounds of Eq. (2.10) T-duality constrains α (cid:48) -corrections to a large extent,and for cosmological backgrounds they are encoded in a single function of the Hubbleparameters, at least perturbatively to all orders in α (cid:48) . Motivated by this remarkable result,in this section we apply FRG methods to the mini-superspace effective action, taking intoaccount the constraints of T-duality. In general, one would expect that, along the flow,the EAA deviate from exact T-duality, due to the presence of the regulator in Eq. (3.1).However, since the regulator vanishes in the IR, T-duality ought to be recovered in thislimit, and thus it appears reasonable that truncation errors be milder compared to moretraditional computations. In order to try to capture all-order α (cid:48) corrections, while neglecting string-loop corrections ,let us consider effective actions of the form of Eq. (2.15). According to the general procedureoutlined in Section 3, one ought to promote each parameter in the action to a functionof the RG scale k . To this end, we define the dimensionless running Newton constantaccording to 16 πG k ≡ g k k − d , and thus our ansatz for the EAA takes the formΓ k = Vol d (cid:90) dt n e − Φ (cid:20) − k d − ζ k g k ˙Φ + k d +1 n F k (cid:18) ˙ σn k (cid:19)(cid:21) , (4.1)where F k is an even dimensionless function of the dimensionless ratio x = H/k of the Hub-ble parameter H = ˙ σ to the RG scale k . We shall henceforth gauge-fix n = 1. Functionaltraces are to be defined according to the correct inner product on the space of fields, which Let us stress that this can only be self-consistent if the resulting RG-improved solutions that one studiesare such that e φ (cid:28) – 10 –s inherited from the kinetic terms of the classical action. Namely, letting χ ≡ ( σ , Φ),given an operator K on field space defined by a kernel K according to( K χ ) ( t ) ≡ (cid:90) dt (cid:48) K ( t, t (cid:48) ) χ ( t (cid:48) ) , (4.2)its trace is given by STr K = (cid:90) dt e − Φ( t ) K ( t, t ) . (4.3)The analog of a polynomial truncation in this setting would be F k ( x ) = dg k N (cid:88) n =0 ˆ f n ( k ) x n , ˆ f ( k ) = 1 , (4.4)and within truncations with N ≥ ζ k is marginal. This suggests that setting ζ k = 1 in the EAA (4.1) should be sensible: at the level of Eq. (4.1), the ratio ζ k g k is a singleindependent coupling which can be renamed g k without loss of generality, and the physicalNewton constant can still be extracted from the small- x behaviour of F k . Moreover, setting ζ k = ζ ∗ simplifies the flow . As we shall see, within the (cid:15) -expansion, ζ k is again marginal,and thus setting it to a constant ζ ∗ appears consistent. In order to focus on the flow of F k and g k , we shall work in backgrounds where ˙Φ and the Hubble parameter ˙ σ ≡ H ≡ k x areconstant.As a final remark, note that we shall interchangeably employ the notation F k ( x ) and F k ( H ) ≡ F k (cid:18) x = Hk (cid:19) , (4.5)since the latter is more suited to derive the flow equations that we shall present shortly,as well as for the study of truncations presented in Section 4.4. D dimensions The flow equations for the Newton coupling and the function F k are highly non-linear,and thereby several solutions might exist. In this section we derive an exact solution tothe flow equations, which actually involves a non-analytic function of H , i.e. of the Ricciscalar in the mini-superspace approximation, and is valid in any spacetime dimension.Substituting the ansatz (4.1) for Γ k in the Wetterich equation (3.1) yields the following In practice, this amounts to setting f = ζ ∗ . – 11 –ow equations for the (dimensionless) Newton coupling g k and the function F k ( H ) k∂ k g k = ( d − g k − − d π − − d d Γ (cid:0) d (cid:1) (cid:0) d + D k (cid:1) g k , (4.6) k∂ k F k = − ( d + 1) F k + 2 − d π − − d d Γ (cid:0) d (cid:1) E k k∂ k C k − C k (cid:34) − d π − − d (cid:0) d + D k (cid:1) d Γ (cid:0) d (cid:1) (cid:0) F (cid:48) k (cid:1) − − d π − − d ( d + 1) d Γ (cid:0) d (cid:1) E k (cid:35) − S k (cid:34) − d E k k∂ k C / k + 83 d (cid:32) g k F k − g k (cid:0) F (cid:48) k (cid:1) − g k (cid:0) F (cid:48) k (cid:1) E k (cid:33) k∂ k C / k (cid:35) (4.7) − S k C / k (cid:34) E k d (cid:0) F (cid:48) k (cid:1) (cid:32) dg k + 2 − d π − − d (cid:0) d + D k (cid:1) Γ (cid:0) d (cid:1) (cid:33) − d + 3 d E k (cid:35) − S k C / k (cid:34) − d (cid:0) F (cid:48) k (cid:1) E k (cid:32) d ( d + 1) g k + 2 − d π − − d (cid:0) d + D k (cid:1) Γ (cid:0) d (cid:1) (cid:33) + 4 ( d + 1) E k d (cid:32) g k F k − g k (cid:0) F (cid:48) k (cid:1) (cid:33)(cid:35) , where C k ( H ) = F (cid:48) k ( H ) + 2 H F (cid:48)(cid:48) k ( H ) H , (4.8) D k = k∂ k F (cid:48) k (0) F (cid:48) k (0) = k∂ k log (cid:0) F (cid:48) k (0) (cid:1) , (4.9) E k ( H ) = (cid:115) g k ( F (cid:48) k ) g k F k ( H ) − , (4.10) S k ( H ) = 2 − d π − − d sign( F (cid:48) k ) sign( H )Γ( d/
2) sign( E k ) arctanh (cid:32) |E k ( H ) | sign( F (cid:48) k ) sign( H ) (cid:112) C k ( H ) (cid:33) , (4.11)and we have dropped the dependence of various functions on H for convenience. At thispoint, it is easy to see that a particularly simple solution can be found imposing C k = 0,while maintaining all the other functions defined above finite. The differential equation C k ( H ) = 0 has indeed a very simple solution, namely F k ( H ) = f ( k ) + f ( k ) √ H . (4.12)When evaluated on this solution, all functionals defined above are finite: one can thussafely take the limit C k →
0. In this limit the flow equations take the simple form k∂ k g k = g k ( d − − − d π − − d/ g k d Γ( d/ (cid:20) (4 + d ) + (cid:18) k∂ k f ( k ) f ( k ) − (cid:19)(cid:21) , (4.13) k∂ k (cid:16) f ( k ) + f ( k ) √ H (cid:17) = − ( d + 1) (cid:16) f ( k ) + f ( k ) √ H (cid:17) , (4.14)– 12 – igure 1 . Position of the NGFP for g k as function of the spatial dimension d . and therefore one can also find an exact solution for the RG-scale dependence of thecouplings ( g k , f ( k ) , f ( k )). The final solution reads g k = g g ∗ g ∗ k d − + g ( k d − − k d − ) k d − , (4.15) F k ( H ) = k − d − (cid:16) c + c √ H (cid:17) , (4.16)where c and c are integration constants, g is the value of the dimensionless Newtoncoupling at the reference scale k , and the position g ∗ of the NGFP for g k is given by g ∗ = 2 d d ( d − π d/ Γ( d/ . (4.17)This NGFP for g k yields a critical exponent θ g = d −
1. As expected, g ∗ = 0 for d = 1,where θ g = 0. For d >
1, the NGFP emerges from the GFP and its critical exponent θ g becomes positive, indicating that for d > g k match those of theReuter fixed point encountered in the context of asymptotically safe gravity [36]. Moreover,at least in D = 4, the RG-running in Eq. (4.15) matches precisely the one found in thecontext of asymptotically safe gravity studying the beta functions for g k in the absence ofa cosmological constant [36]. The corresponding dimensionful Newton coupling G k = G g − ∗ G ( k d − − k d − ) , (4.18)with G = g k d − , is indeed the same scale-dependent Newton coupling in Eq. (2.24) of [36].A set of possible RG trajectories for g k and G k = g k k − d is shown in Fig. 2 for d = 3. Asone can observe from the figure, in d = 3 there exist trajectories which depart from theNGFP in the UV and reach the GFP in the IR (magenta lines). These trajectories areseparatrix lines which connect the two fixed points. In the present context only this type oftrajectories is physical, since they give a positive and finite Newton coupling in the IR. All– 13 – igure 2 . Running dimensionless (left panel) and dimensionful (right panel) Newton coupling in d = 3, for various initial conditions. The black-dashed lines indicate the position of the NGFP g ∗ .There are three types of RG trajectories. Trajectories with g k > g ∗ are attracted towards theNGFP at high energies but diverge in the IR. Trajectories with g k < < g k < g ∗ interpolate betweenthe NGFP in the UV and the GFP in the IR. Clearly, the first two classes of trajectories (depictedas blue lines in both plots) are unphysical, while the only phenomenologically viable trajectories arethose connecting the NGFP with the GFP (magenta lines), since they allow for a finite, non-zerodimensionful Newton coupling in the IR limit. Figure 3 . The function F k ( H ) for c = c = 1 and various values of the RG scale k , in units ofthe Planck mass, in d = 3 space dimensions. other trajectories (blue lines) correspond to Newton couplings which are either negative ordivergent in the IR. The flow of F k ( H ) of Eq. (4.12) takes a simple form. This function isnon-analytic in H , and the couplings f i ( k ) exhibit canonical scaling for all values of theRG scale k . The evolution of F k is shown in Fig. 3 for c = c = 1 and d = 3, while theprojection of the full RG flow of F k and g k on various two-dimensional theory sub-spacesis depicted in Fig. 4. – 14 – igure 4 . Projection of the RG flow of F k and g k on the theory sub-spaces spanned by ( g, f ) (leftpanel), ( g, f ) (central panel) and ( f , f ) (right panel), for d = 3. The Gaussian and non-Gaussianfixed points are denoted by black dots. As one can observe from the first two figures from the left,the NGFP for g ∗ is UV-attractive and comes with three relevant directions, similarly to the Reuterfixed-point of asymptotically safe gravity. The GFP is instead a saddle point: only RG trajectorieslying on the hypersurface { g k = 0 } are attracted towards the GFP in the UV. Let us finally remark that the solution in Eqs. (4.15) and (4.16) is unphysical, since itsIR behavior is not analytic in the curvature invariants. However, due to the non-linearityof the flow equations for g k and F k , the solution we found in this section might not beunique. Indeed, in the following we shall seek (analytic) solutions, first via an (cid:15) -expansionabout d = 1 and then via polynomial truncations. Let us note that computations basedon a truncated derivative expansion of F k , i.e. on a Taylor expansion of the function F k about H = 0, can only provide approximations to scaling solutions for F k which are botheven and analytic in H = 0. In other words, since the solution found in this section isnon-analytic in H , it is not accessible by polynomial truncations. (cid:15) -expansion In this section we investigate the flow equations via an (cid:15) -expansion. To this end, forconvenience we shall write all functions in terms of x = H/k . As we have discussed in thepreceding section, the flow equation for F k is considerably complicated. However, the flowequation for g k is simpler, and expressing the functional dependence on H in terms of x it reads k∂ k g k = ( d − g k − d + 3 + D k [ F k ]3 · d +1 π d +1 Γ( d + 1) g k , (4.19)where k is an arbitrary reference scale, and we have defined the functional D k [ F k ] ≡ k∂ k F (cid:48) k F (cid:48) k (cid:12)(cid:12)(cid:12)(cid:12) x =0 (4.20)for later convenience. Notice that here, and in the ensuing discussion, primes denotederivatives with respect to x . – 15 –ssuming that at a fixed point D k [ F k ] is well-defined , in general one finds two fixedpoints for g k . Since for (cid:15) ≡ d − (cid:28) O ( (cid:15) ), this motivates studying theRG flow within an (cid:15) -expansion, where the flow equation for F k dramatically simplifies.While the solution that we have described in the preceding section is exact, its low-energy behavior is phenomenologically not viable. Hence, motivated by the fact thatfor d = 1 the Newton coupling is classically marginal, in this section we investigate the RGflow of F k and g k at leading order in an (cid:15) -expansion, setting d = 1 + (cid:15) .In order to properly account for the scaling of each term in the flow equations as (cid:15) → + ,one must specify an ansatz for g k and F k . A consistent ansatz for g k is simply a function g k = (cid:15) γ k , with γ k = O (1), which flows from a positive fixed point γ ∗ in the UV (if D k isregular in the UV) to zero in the IR. A consistent ansatz for F k is then F k ( x ) = v k ( x ) (cid:15) γ k + w k ( x ) , (4.21)with v k ( x ) , w k ( x ) = O (1). This ansatz is motivated by the fact that in the IR regime F k is expected to adhere to the perturbative result, while the corrections are expected to besub-leading in (cid:15) .Using the aforementioned ansatz for g k and F k , the flow equations simplify considerablyand, assuming that D k [ F k ] is non-singular , at leading order they take the form k∂ k γ k = (cid:15) γ k γ ∗ (cid:18) γ ∗ − γ k − γ k D k [ v k ] (cid:19) ,k∂ k v k − x v (cid:48) k + 2 v k (cid:15) γ k = O (1) , (4.22)where γ ∗ = π is the fixed-point value of γ k . Correspondingly, the (dimensionful) Newtoncoupling scales as G k ∼ π (cid:15) k − (cid:15) (4.23)in the UV. The second line of Eq. (4.22) then implies that v k ( x ) = V ( k x ) k , where werequire that the arbitrary function V be analytic at the origin in order to be consistentwith perturbative α (cid:48) -corrections. Furthermore, in order to allow for a UV fixed point, thefunction v k ( x ) must be at most quadratic, v k ( x ) = Λ k + x ζ ∗ . (4.24)Here the coefficient of x is the marginal deformation corresponding to the wave-functionrenormalization of φ , as discussed in the preceding section. In addition, this solution has This assumption can be self-consistently checked, at least within an (cid:15) -expansion or truncations. We shall verify this assumption a posteriori . – 16 – k [ v k ] = 0, so that D k [ F k ] = O ( (cid:15) ) upon including the sub-leading term w k ( x ). The flowequation for γ k can then be solved exactly. Its solution reads γ k = γ ∗ c k − (cid:15) , (4.25)where c is an integration constant. Interestingly, this solution matches the result inEq. (4.15) that we have found in Section 4.2 for d = 1 + (cid:15) . In particular, for c > γ ∗ = π in the UV and zero in the IR, where γ k scales as γ k ∼ γ ∗ c k (cid:15) and, correspondingly, G k ∼ π (cid:15)c . (4.26)Since γ k flows very slowly, with ˙ γ k ∼ (cid:15) γ k , one can neglect contributions involving ˙ γ k in thesub-leading flow equation for w k , since the latter involves also O (1) terms which dominateover those proportional to k∂ k γ k . Using these results and observations, the resulting flowequation for w k can be solved analytically, yielding γ ∗ w k ( x ) = 12 + (cid:18) − γ ∗ γ k (cid:19) Λ k log (cid:18) kk (cid:19) + 3Λ8 k log (cid:18) − k − x ζ ∗ (cid:19) + (cid:18)(cid:18) γ ∗ γ − (cid:19) log ( x ) + 14 log (cid:18) − k − x ζ ∗ (cid:19)(cid:19) x ζ ∗ + (cid:18) k + x ζ ∗ (cid:19) arctanh ( B k ( x )) B k ( x )+ γ ∗ W ( k x ) k , (4.27)where we have defined the combination B k ( x ) ≡ x √ ζ ∗ (cid:113) k − x ζ ∗ (4.28)and, once more, W is an arbitrary function which we choose according to W ( k x ) k = w k + w x , (4.29)with w , w constants. The arbitrary scale k can thus be shifted by modifying w . In turn, w can be fixed by requiring that the quadratic terms reproduce the classical contribution d x ζ ∗ g k in the IR.The large- x behavior of the resulting solution F k = v k (cid:15) γ k + w k is consistent on dimen-sional grounds: it corresponds, at fixed ˙ σ = k x , to the small- k regime, which is perturbativeand ought to reconstruct the classically marginal operator ∝ x (cid:15) ∝ R D . Indeed, F k ( x ) ∼ x ζ ∗ g k (1 + (cid:15) log ( x )) ∼ x (cid:15) ζ ∗ g k . (4.30)Using these results, one can derive the leading-order IR coefficients for higher-derivativecorrections to the classical Lagrangian, which have the correct classical scaling. As a result,– 17 –he dependence on k disappears once the dimensionless variable x is replaced by x = Hk .In detail, expanding F k ( x ) in powers of x √ ζ ∗ and setting ζ ∗ = 1, the first few corrections tothe classical Lagrangian read e Φ L HD ∼ π Λ H − π Λ H + 1591241920 π Λ H . (4.31)These corrections are O ( (cid:15) ) with respect to the classical terms, whose leading-order IRcontribution to the Lagrangian takes the form e Φ L tree ∼ πG N (cid:16) − ˙Φ + Λ + H (cid:17) (4.32)at a suitable IR scale k = k IR such that g k IR = 16 πG N + O ( (cid:15) ). This identifies the relevantdeformation Λ as the leading contribution to the low-energy cosmological constant in the string-frame . The size of the quartic correction approaches that of the classical curvatureterm when H ≈ Λ , (4.33)from which one could be tempted to identify Λ ≈ α (cid:48) as an UV cutoff scale. The coefficientsin Eq. (4.31) can also be obtained by expanding the leading-order (in (cid:15) ) IR result e Φ L HD ∼ Λ6 π L (cid:18) H Λ (cid:19) (4.34)in powers of H , where L ( s ) ≡ − − s + (cid:18)
32 + s (cid:19) log (cid:16) s (cid:17) + (1 + s ) (cid:114) s arctanh (cid:18)(cid:114) s s ) (cid:19) . (4.35)All in all, reinstating the lapse n , the leading-order (in (cid:15) ) IR string-frame effective actionfor cosmological ansatze readsΓ string = Vol d πG N (cid:90) dt n e − Φ (cid:34) Λ − ˙Φ n + ˙ σ n + 83 π G N Λ L (cid:18) ˙ σ n Λ (cid:19)(cid:35) . (4.36)In Section 4.5 we shall comment on some potential cosmological consequences of theseresults, assuming that similar solutions and effective actions extend to the phenomenolog-ically relevant case of D = 4 spacetime dimensions. The flow equations for g k and F k admit an exact non-analytic solution valid for any dimen-sion D (cf. Section 4.2) and an analytic solution in D = (2+ (cid:15) ) dimensions, characterized by In the Einstein frame, it takes the form of an exponential potential for the dilaton V ( φ ) = Λ e D − φ . – 18 – g ∗ f ∗ f ∗ f ∗ f ∗ f ∗ Critical Exponents2 0 π − ∀ f π − ∀ f −
24 0 π − ∀ f − −
45 0 π − ∀ f − − − Table 1 . GFP line in D = 2. an UV fixed point with g ∗ > D = 4, the theory would be characterized by a UV fixedpoint with g ∗ > D = (2 + (cid:15) ) dimensions can indeed extend to D = 4.Since an analytic exact solution in arbitrary dimensions d cannot be found in closedform, in order to investigate the extension of the fixed point found in the previous sectionto d = 3 we shall truncate the theory space choosing an ansatz for F k of the form F k ( H ) = N (cid:88) n =1 f n − ( k ) (cid:0) k − H (cid:1) n − , (4.37)with N finite. For a given truncation order N , one can determine the beta functions forall couplings associated with (even) powers of H up to the order H N − . Specifically,once N is chosen, the ansatz for F k is to be replaced into Eq. (4.7). The beta functions β n ≡ k∂ k f n ( k ) for the couplings { f ( k ) , . . . , f N − ( k ) } can then be computed expandingthe flow equation for F k about H = 0 up to order N and requiring that the coefficientsof the expansion vanish. For a given N , one can thus investigate the fixed-point structureof the flow equations as a function of d . One can then investigate the robustness of theresults progressively enlarging the truncation order N . In fact, if a fixed-point functionalexists in the full theory space and is analytic, it should be possible to detect it withintruncations, and its critical exponents should converge to stable values upon increasing thetruncation order. Vice versa, if a fixed point disappears increasing the truncation order,or if its critical exponents do not converge, it is likely that the fixed point is spurious, i.e. ,that it is a truncation artifact rather than a genuine feature of the theory. In the followingwe shall apply these techniques, investigating the fixed-point structure of the truncatedflow of F k up to N = 5, namely up to O ( H ).For any dimension D there is a line of GFPs. The universality properties of this freefixed point are summarized in Tab. 1 and Tab. 2 for D = 2 and D = 4 respectively.Studying the fixed-point structure of the beta functions for increasing values of thetruncation order N and spatial dimension d , it is in principle possible to understand whether The condition that F k be even in H simplifies the computation of beta functions, and is required byT-duality. – 19 – g ∗ f ∗ f ∗ f ∗ f ∗ f ∗ Critical Exponents2 0 (12 π ) − − π ) − ∀ f − π ) − ∀ f − −
25 0 (12 π ) − ∀ f − − − Table 2 . GFP line in D = 4, for various values of N . the fixed point found analytically in d = (1+ (cid:15) ) is also present in d = 3. For N = 2, the onlycouplings involved are g k , f ( k ) and f ( k ) and all Gaussian and non-Gaussian fixed pointsare characterized by f , ∗ = 0 for all d . One can thus set f ( k ) = 0 and visualize how f , ∗ and g ∗ for the NGFP vary with the spatial dimension d . The result is depicted in Fig. 5.In d = 1 there is only the GFP. Increasing the value of d , a NGFP emerges from the GFP.For d (cid:38)
1, the NGFP is located at positive values of f , ∗ and g ∗ . Increasing d , this value of g ∗ rapidly decreases, vanishes, and then becomes negative. At the same time, f , ∗ movestowards arbitrary large and positive values. This is shown in Fig. 6, depicting the variationof g ∗ and f , ∗ with d . As is apparent from the figure, there exists a critical dimension d crit (cid:39) . g ∗ vanishes, while f , ∗ diverges. For d > d crit , there is only a NGFPlocated in the unphysical part of the theory space, g ( k ) <
0. Thus, for N = 2, the NGFPfound in d = 1 + (cid:15) seems not to survive in higher dimensions: it disappears at d = d crit and it is replaced by another UV NGFP with g ∗ < d for N = 2is shown in Fig. 7. For 1 < d (cid:46) .
55, The NGFP emerging from the GFP is a saddlepoint with two relevant directions. For 1 . (cid:46) d < d crit , the NGFP has three IR-irrelevantdirections, i.e. it is IR-attractive in the theory sub-space { g k , f ( k ) , f ( k ) } . Two of thecritical exponents are complex conjugates for 1 . (cid:46) d (cid:46) .
63 and real elsewhere. Finally,when d exceeds the critical dimension d crit , the NGFP at g ∗ > g ∗ <
0, which has three IR-relevant directions and is therefore UV-attractive in the theorysub-space under consideration.The existence of a critical dimension and the related disappearance of the NGFP at g ∗ > N .In higher-order truncations, up to the order N = 5, the beta functions become very involvedand highly singular. In particular, the denominators of all beta functions are proportionalto the couplings f i with i ≥
2, making them singular on the hypersurface f i = 0. Theexistence of fixed-point solutions and their connection with the NGFP found in d = 1 + (cid:15) can nevertheless be studied numerically. The extension of the NGFP at g ∗ < < N ≤
5, is a point with coordinates f i, ∗ = 0 for all f i with i ≥
2. In other words, increasing the truncation-order N , the NGFP at g ∗ < β i are singular. Thus, for N > g ∗ < β i = 0, rather it corresponds to– 20 – igure 5 . Position of the GFP and NGFP for increasing values of the spatial dimension d and N = 2. For d = 1 the two fixed points coincide (green dot). Increasing d , the GFP and NGFP splitand move apart from each other. In d = 2 the two fixed points are depicted by magenta dots, whilein d = 3 they are depicted by red dots. The smaller black dots in between are the two fixed pointsfor non-integer dimensions. The trajectories drawn by the Gaussian and non-Gaussian fixed pointsfor increasing values of d are shown as black and blue lines respectively. As can be seen from thisfigure, for N = 2 the NGFP in d = 1 + (cid:15) and d = 3 dimensions are not continuously connected:there exists a critical dimension d crit < g ∗ vanishes and becomes negative, while f , ∗ diverges at d crit and re-emerges at d > d crit with arbitrarily large and negative values. Figure 6 . Variation of g ∗ and f , ∗ with the spatial dimension d (blue lines), for N = 2. The NGFPemerging from the GFP in d = 1 + (cid:15) dimensions disappears at a critical dimension d crit < g ∗ < d = 3.Both fixed points are characterized by f , ∗ = 0 for any d . The variation of the position of the GFP(black line) is also shown for comparison. – 21 – igure 7 . Variation of the real part of the critical exponents with the spatial dimension d , for N = 2. The figures show two different zooms. Figure 8 . Position of the g ∗ -coordinate of the NGFP (left panel) and critical dimension d crit asfunctions of the truncation order N . For N >
2, the curves describing the variation of the g ∗ -coordinate of the NGFP/QFP overlap and are almost indistinguishable. Accordingly, for N > d crit does not undergo significant changes. N = 2 N = 3 N = 4 N = 5 d crit Table 3 . Value of the critical dimension as function of the truncation order N . a point in theory space where the beta functions are of the form 0 /
0. Depending on howthe couplings f i ( k ) with i ≥ H n with n >
1, the NGFP at d > d crit found for N = 2 becomesa quasi-fixed-point (QFP). Additional fixed points at g ∗ > N = 2, there exists a critical dimension d crit (cid:39) . g ∗ -coordinate of the NGFP found in d = 1 + (cid:15) vanishes (cf. Fig. 8). The value ofthe critical dimension d crit appears very stable: as shown in the right panel of Fig. 8 andin Table 3, for N > N = 5, the value of the critical dimension changes only tothe fourth decimal digit. – 22 – igure 9 . Scaling of the first time derivatives of the dilaton Φ( t ) (left panel) and of the Hubbleparameter H ( t ) (right panel). All quantities are in units of Λ. The classical (blue line) and quantum-corrected (orange line) curves share a qualitatively similar shape. The quantum-corrected curve,which arises from the effective action in Eq. (4.36), also appears to be shifted in time with respectto the classical one. Let us now briefly comment on some potential implications of our findings. Specifically,we discuss quantum-corrected cosmological solutions to the field equations for the dilatonand the scale factor stemming from the quantum effective action in Eq. (4.36).To begin with, let us observe that T-duality implies that α (cid:48) corrections only involvepowers of first derivatives, and thus yield second-order equations of motion, avoidingOstrogradsky-like instabilities. Moreover, let us recall that the IR parameter G N com-bines with the dilaton vacuum expectation value φ to build the observed Newton constant G obs ≡ g s G N , where the string coupling g s = e φ (cid:28) < H and is generally complex-valued,and thus unphysical. If Λ > In this paper we have investigated the application of FRG techniques within the frameworkof string theory. While the former appear to offer, at least in principle, quantitative toolsto understand non-perturbative physics, in practice they offer only approximate pictures– 23 –ithin truncations of the full theory space. On the other hand, the latter provides recipesto systematically derive curvature corrections, but the resulting computations are highly in-volved. Furthermore, genuinely stringy effects such as T-duality provide strong constraintson curvature corrections, to the extent that, in suitable time-dependent backgrounds, theyare entirely classified by a single function. Therefore, it is conceivable that applying FRGmethods to these settings could circumvent these technical issues, at least partially.Motivated by this enticing prospect, in Section 4 we have studied the flow of mini-superspace effective actions of the type derived in [28–31] via FRG methods, determining α (cid:48) -corrections to all-orders. In particular, we have obtained an asymptotically safe exactsolution in general spacetime dimension D , which however features unphysical low-energyproperties, and a phenomenologically viable solution within an (cid:15) -expansion for D = 2 + (cid:15) .However, within polynomial truncations of the theory space, the latter solution seemsto disappear in higher dimensions. This could be due, among other possibilities, to atruncation artifact, a limitation of our mini-superspace approach, or to the importance of g s -corrections, which we have neglected. Furthermore, we have investigated cosmologicalsolutions stemming from the EAA found within the (cid:15) -expansion, finding no qualitativedeviation from the classical case. In particular, the absence of de Sitter solutions garnersfurther evidence for some Swampland conjectures [45].All in all, determining curvature corrections in string theory is paramount to shed lighton a number of aspects of high-energy physics, including singularities whose resolutionwould presumably involve all-order effects. While still lacking a complete non-perturbativeformulation, the theory affords regimes in which some corrections can be systematicallycomputed to arbitrarily high orders in a well-defined fashion. While earlier efforts in thisrespect were crucial in establishing string theory as a candidate for a quantum theory ofgravity, a number of technical obstacles led a large portion of the community to focus onsupersymmetric constructions, whose remarkable non-renormalization properties simplifymatters dramatically. However, the phenomenological necessity to formulate models wheresupersymmetry is either broken or absent entails numerous subtleties, and the completepicture is still to be unveiled. Based on this state of affairs, we are therefore compelled toseek instructive lessons and methods beyond those provided by effective field theory, andto this end an approach based on stringy formulations and symmetries appears potentiallyfruitful. The results that we have presented in this paper constitute a first step in thisdirection, and we would like to explore these intriguing ideas further in future work. Acknowledgements
The authors would like to thank A. Sagnotti and F. Saueressig for discussions and com-ments on the manuscript. IB is grateful to F. Bascone and D. Bufalini for sharing someuseful references. During the development of this work, IB was supported in part byScuola Normale Superiore and by INFN (IS CSN4-GSS-PI), while AP was supported by– 24 –he Alexander von Humboldt Foundation. AP acknowledges support by Perimeter Insti-tute for Theoretical Physics. Research at Perimeter Institute is supported in part by theGovernment of Canada through the Department of Innovation, Science and Economic De-velopment Canada and by the Province of Ontario through the Ministry of Colleges andUniversities. AP is also grateful to Scuola Normale Superiore for hospitality during theearly stages of development of this work.
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