RReconstructing the graviton
Alfio Bonanno,
1, 2
Tobias Denz, Jan M. Pawlowski,
3, 4 and Manuel Reichert INAF, Osservatorio Astrofisico di Catania, via S. Sofia 78, 95123 Catania, Italy INFN, Sezione di Catania, via S. Sofia 64, 95123 Catania, Italy Institut f¨ur Theoretische Physik, Universit¨at Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum f¨urSchwerionenforschung mbH, Planckstr. 1, 64291 Darmstadt, Germany Department of Physics and Astronomy, University of Sussex, Brighton, BN1 9QH, U.K.
We reconstruct the Lorentzian graviton propagator in asymptotically safe quantum gravity fromEuclidean data. The reconstruction is applied to both the dynamical fluctuation graviton and thebackground graviton propagator. We prove that the spectral function of the latter necessarily hasnegative parts similar to, and for the same reasons, as the gluon spectral function. In turn, thespectral function of the dynamical graviton is positive. We argue that the latter enters cross sectionsand other observables in asymptotically safe quantum gravity. Hence, its positivity may hint at theunitarity of asymptotically safe quantum gravity.
I. INTRODUCTION
In the past two decades, asymptotically safe (AS) grav-ity has emerged as an interesting and solid contender for aquantum theory of gravity. By now a lot of non-trivial evi-dence has been collected for the existence of AS gravity asan ultraviolet (UV) complete quantum field theory. Mostof the investigations have been done with the functionalrenormalisation group (fRG), for a recent overview see[1]. For reviews on AS gravity including its applicationto high energy physics see [1–14]. However, there aresome pivotal challenges yet to be resolved, see [13]. Avery prominent one is the setup of a non-perturbativeLorentzian signature approach: most investigations so farhave been done within Euclidean quantum gravity. TheWick rotation to a Lorentzian version is one of the re-maining challenges yet to be met. For first steps towardsLorentzian flows see, e.g., [15–24], for discussions of ghostsand the Ostrogradsky instability see [25–29]. The properdefinition of the Wick rotation and the interpretation ofthe spectral properties of Lorentzian correlation functionsalso touch upon the question of unitarity of the approach.A first, but important, step in this direction is doneby the reconstruction of spectral functions from theirEuclidean counterparts. While being short of a full res-olution of the challenges mentioned about, it providesnon-trivial insight into the possible complex structure ofAS gravity. In the present work, we apply reconstruc-tion methods, already used successfully in non-Abeliangauge theories [30], to the fundamental correlation func-tion of AS gravity, the graviton two-point function orpropagator. In (non-Abelian) gauge theories, as in grav-ity, one has to face the fact that the gauge field is not aon-shell physical field. Hence, the standard derivation ofthe K¨all´en–Lehmann spectral representation fails, and thespectral function of the gauge field features non-positiveparts if it exists. This property follows already in the per-turbative high-momentum regime of the gluon from the Oehme-Zimmermann super-convergence property. Conse-quently, non-positive parts of spectral functions have butnothing to do with strongly-correlated physics such asconfinement, and, even more importantly, do not signalthe failure of unitarity of the theory. In turn, they alsoshould not be taken lightly.Similar properties may be present in AS quantum grav-ity. For a first discussion see [31], for related recentwork see [32–34]. In the present work, we formally showthat the spectral function of the background graviton hasnon-positive parts. We also compute the spectral func-tion of the dynamical fluctuation graviton, which turnsout to be positive. As the derivation of these spectralfunctions requires many steps and intermediate results,we present the final spectral functions already in Fig. 1.The knowledge of the structures visible there certainlyallows to better follow some of the technical steps, andalso understand their origin. In particular, the positivityof the spectral function of the fluctuation graviton inFig. 1 is a non-trivial and very exciting result, as it is thefluctuation graviton which propagates in the diagramsof cross-sections and other potential observables. Whilethe results leave much to be explained, the present workis a first rather non-trivial step towards a discussion ofunitarity in AS gravity.This work is structured as follows: In Sec. II, we givea brief overview of the setup of our approach to ASquantum gravity as well as the fRG approach used for thecomputation of Euclidean correlation functions. In Sec.III,we introduce the K¨all´en–Lehmann spectral representationand discuss the properties of the graviton spectral function.We also prove that the spectral function of the backgroundgraviton has negative parts. In Sec. IV, we present ourcomputation of Euclidean correlation functions, mainlybased on results in previous works. In Sec.V, we introduceour reconstruction framework, compute and discuss ourresults on graviton spectral functions. A summary andoutlook of our findings can be found in Sec. VI. a r X i v : . [ h e p - t h ] F e b ω [ M pl ] ρ h [ / M p l ] ρ h ( ω, ~p = 0) (a) Spectral function of the fluctuation graviton. − ω [ M pl ] ρ ¯ g [ / M p l ] ρ ¯ g ( ω, ~p = 0) (b) Spectral function of the background graviton. FIG. 1. Spectral functions of the fluctuation and background graviton. The shaded areas constitute the estimated error ofthe reconstruction. The fluctuation graviton spectral function is strictly positive, which may be important for the unitarity ofasymptotically safe gravity. In turn, we show that the background graviton spectral function has a vanishing spectral weightand hence positive and negative parts. For more details see Sec. V B and Sec. V C.
II. ASYMPTOTICALLY SAFE GRAVITY
Asymptotically safe quantum gravity [35, 36] is a vi-able and minimal UV closure of fundamental physics. Itis minimal in the sense that it only relies on standardquantum field theory, and its viability has been furtheredby many results in the past two decades following theseminal paper [37] within the fRG-approach, see [1–14].Most of these fRG-based investigations are done withina Euclidean setting, also commonly used for other non-perturbative investigations, being short of numericallyaccessible approaches with Lorentzian signature.The extraction of (timelike) physics from Euclideancorrelation function is already intricate and challengingwithin standard high energy physics, and in particular forstrongly correlated physics such as infrared (IR) QCD.However, in quantum gravity, a further, even conceptual,challenge is the very definition of a Wick rotation.While chiefly important, we will not touch upon thiscrucial subject here, and simply assume the existence of astandard Wick rotation at least for backgrounds close toflat ones. This allows us to use standard reconstructiontechniques for the computation of spectral propertiesof correlation functions in quantum gravity from theirEuclidean counterparts. Hence, below we introduce theEuclidean approach to AS gravity.
A. Euclidean quantum gravity and the fRG
In the present work, we utilise results for momentum-dependent correlation function in a Euclidean flat back-ground based on [38] within the fluctuation approach.This approach has been set-up in [38–41] and used, e.g.,for pure gravity investigations in [38–50], and for gravity-matter systems in [47–61]; for a recent review see [14]. Here we briefly describe the computation and approxima-tion scheme, for more details see these works.Central to the functional approach to asymptotic safetyis the effective action Γ, the quantum analogue of theclassical action. Importantly, the RG-approach to ASquantum gravity does not rely on a specific classical ac-tion, but on a non-trivial fixed-point action at the UVReuter fixed point. In turn, in the IR the effective actionapproaches the Einstein-Hilbert action, S EH [ g µν ] = 116 πG N (cid:90) d x √ g (cid:16) − R ( g µν ) (cid:17) , (1)with the (IR) classical Newton constant G N and the abbre-viation √ g = (cid:112) det g µν ( x ). The definition of a gravitonpropagator, a pivotal ingredient to the approach, requiresa gauge fixing. A standard linear gauge fixing requiresthe definition of a background metric, which also servesas the expansion point of the effective action. We use alinear split for the full metric, g µν = ¯ g µν + (cid:112) π G N h µν , (2)where h µν is the dynamical fluctuation field with massdimension 1. Importantly, the fluctuation field carriesthe quantum fluctuations. In the present work, we usethe flat O (4)-symmetric Euclidean metric for ¯ g µν . Thegauge fixing is done within this background, and we takea de-Donder type gauge fixing, see App. A.This formulation introduces a separate dependence ofthe effective action on the background metric and thefluctuation fields, Γ = Γ[¯ g µν , φ ], where φ is the fluctuationmulti-field including the ghosts, φ i = ( h αβ , ¯ c µ , c ν ) . (3)We consider a vertex expansion of the effective actionabout the given Euclidean background ¯ g ,Γ[¯ g µν , φ ] = ∞ (cid:88) n =0 n ! n (cid:89) l =1 (cid:90) d x l (cid:113) det ¯ g µν ( x l ) φ i l ( x l ) × Γ ( φ i ...φ in ) [¯ g µν ,
0] ( x ) , (4)where Γ ( φ i ...φ in ) are the n th derivatives of the effectiveaction with respect to the fluctuation fields φ i , ..., φ i n and x = ( x , . . . , x n ). We shall also use the abbreviationΓ ( n ) for the sake of simplicity.A suggestive choice for the background metric is a so-lution of the quantum equations of motion. There, back-ground independence is regained and we expect the mostrapid convergence of the vertex expansion, for a detaileddiscussion see e.g. [14]. Such an expansion is technicallyvery challenging, and in the present work we consider anexpansion about the flat Euclidean background, ¯ g µν = δ µν with the flat O (4)-metric δ = diag(1 , , , ( n ) ( p ) = n (cid:89) j =1 (cid:90) d x j e i x µj p µj Γ ( n ) [ δ µν , x ) . (5)Here, p = ( p , . . . , p n ). The vertices in (5) are computedwithin the fRG-approach to quantum gravity discussedbelow. Owing to the flat background the respective fRG-flow equations are standard momentum loops and hencecan be solved within the well-developed computationalmachinery of fRG-computations in quantum field theories.Moreover, it facilitates the discussion of the Wick rotationto Minkowski space as we can resort to standard spectralproperties. As already discussed before, this does notresolve the problem of a Wick rotation in the presence ofa dynamical metric. However, the results here may alsoshed some light into this intricate challenge.In the fRG approach to quantum gravity, the theory isregularised with an IR cutoff that suppresses quantumfluctuations with momenta p (cid:46) k . This cutoff is succes-sively lowered and finally removed. The respective flowequation for the IR regularised effective action Γ k is givenby [62–64] ∂ t Γ k [¯ g µν , φ ] = 12 Tr G k [¯ g µν , φ ] ∂ t R k [¯ g µν ] , (6)with G φ i φ i ,k [¯ g µν , φ ] = (cid:34) ( φφ ) k [¯ g µν , φ ] + R k [¯ g µν ] (cid:35) φ i φ i . (7) ∂ t Γ (2 h ) = −
12 + − ∂ t Γ (3 h ) = −
12 + 3 − FIG. 2. Diagrammatic representation of the flows of thefluctuation graviton two- and three-point functions, from whichwe extract the flow of the fluctuation graviton propagator andthe flow of the physical Newton coupling, respectively. Thelatter relates to the flow of the background graviton propagator.Double blue lines represent graviton propagators, red singlelines ghost propagators, and the cross stands for a regulatorinsertion.
Here, R k is a regulator that implements the suppressionof IR modes and t = log k/k ref is the (negative) RG-timewith the reference scale k ref that is at our disposal. Notethat the second derivative of the effective action withrespect to the fluctuation field enters in (6). The flowequations for Γ ( n ) k are obtained by n -derivatives w.r.t. thefluctuation field φ . For more details on the fRG-approachto quantum gravity see [1–14].We extract the momentum-dependent fluctuation gravi-ton propagator from the flow of the two-point function,and the momentum-dependent flow of the fluctuation New-ton coupling from the flow of the three-point function.The corresponding flow equations are diagrammaticallydepicted in Fig. 2. The regulator used for the numericalcomputations is a Litim-type regulator, see App. B. B. Projection on momentum-dependent couplings
We now describe the fRG setup for the solution of themomentum-dependent vertices as defined in (4) and (5)within a flat background. These numerical computationsrequire a truncation of the effective action to a finite setof vertices. The present work builds on results and flowsin [38], where the momentum dependence of the two-,there-, and four-point graviton vertices have been takeninto account, as well as that of the graviton-ghost sector.Further works including momentum dependences can befound in [39–41, 48–51, 65–68], for a review see [14].The general tensor structure of the vertices is further-more reduced to the Einstein-Hilbert tensor structures, T ( φ i ...φ in )EH ( p ; Λ n ) = S ( φ i ...φ in )EH ( p ; Λ n ) (cid:12)(cid:12) G N → . (8)In (8), S ( φ i ...φ in )EH is the n th derivative of the Einstein-Hilbert action (1). We send G N → G N . The Λ n are the (running) coeffi-cients of the tensor structure arising from the cosmological-constant term. These coefficients can be understood asavatars of the cosmological constant. Guided by the re-sults in [38], we use the further approximation Λ n ≈ n -point vertices, the above tensor structures aremultiplied by a scalar vertex dressing that depends on p .Here we only consider the dressing with a dependence onthe average momentum ¯ p ,¯ p = p n . (9)As it is multiplying the Einstein-Hilbert tensor structure,it can be understood as a power of an avatar G n (¯ p ) of theNewton coupling, multiplied with (cid:112) Z φ i ,k ( p i ) for each leg.Here, the Z φ i ,k ( p i ) are the momentum-dependent wave-function renormalisations of the fields φ i . They relate tothe anomalous dimensions of the fields φ i via η φ i ( p ) = − ∂ t ln Z φ i ,k ( p ) . (10)The anomalous dimension η φ i are k -dependent just asthe wave-function renormalisations Z φ i ,k but we chooseto suppress the index k for convenience of notation. Insummary, this leads us an ansatz for the n -point functionsof the effective action given byΓ ( φ i ...φ in ) k ( p ) = n (cid:89) j =1 Z φ ij ,k ( p j ) G n − n (¯ p ) (11) × ( S EH + S gf + S gh ) ( φ i ...φ in ) (cid:12)(cid:12) G N → . The first term in the second line is precisely the tensorstructure defined in (8). For n >
2, there are no contri-butions from the gauge-fixing term, and for n >
3, alsoghost-graviton contributions are absent.The G n (¯ p ) are running avatars of the Newton couplingfor each n -point function. The flows of Γ ( n ) or that of the G n (¯ p ) are obtained within an evaluation of ∂ t Γ ( n ) at asymmetric point, p i = ¯ p , where for the 3-point functionwe choose p i · p j = 12 (3 δ ij −
1) ¯ p . (12)We work with the dimensionless versions of G n and Λ n ,which are given by g n ( p ) = k G n ( p ) , λ n = Λ n k . (13)Here and in the following, we drop the bar on the momen-tum argument of g n but it is understood that we considerthe average momentum flow through the vertex, see (9).More details on the projection procedure (contractionof the tensor structure) and the results can be found in[14, 38].For the analysis of the spectral properties of the gravi-ton presented here, we consider additional approximationsthat are guided by the results in [38]. There, the UV-fixedpoint as well as full UV-IR trajectories with momentumdependences for two-, three- and four-point functions havebeen considered. While not being identical, the avatars g n (¯ p ) of the Newton couplings showed similar ¯ p - and k -dependences. We assume a vanishing cosmological constant, whichin the present approximation entails λ n = 0 at vanishingcutoff scale, k = 0. We know from [38] that the flowis not dominantly driven by the λ n and for the sake ofsimplicity we use λ n ( k ) = 0. Similarly, we use that themomentum-dependence of the ghost, while present, is onlyof quantitative interest. We use a vanishing ghost anoma-lous dimension, η c ( p ) ≈
0. In summary, we compute theflows of Z h,k ( p ) , g ,k ( p ) = g k ( p ) , (14a)with vanishing λ n and η c . Furthermore, we identify allavatars of the Newton coupling with g ,k , g n,k ( p ) = g k ( p ) , (14b)which leaves us with a unique momentum- and cutoff-dependent Newton coupling G k ( p ), and a unique physicalNewton coupling G N ( p ) with G N ( p ) = G k =0 ( p ) , where G k ( p ) = g k ( p ) k . (14c)We emphasise that physical simply refers to the physicallimit k → G N = G N (0), which also defines the Planck mass, M = 1 G N ( p = 0) . (15)Within the approximation described above, we can ac-cess the full momentum- and cutoff-dependent fluctuationpropagator via Z h,k ( p ) as well as the three-graviton cou-pling g k ( p ). Here, p = ¯ p is the average momentum flowthrough the vertex, see (9). The respective flow is evalu-ated at the symmetric point (12). The flows are integratedfrom the initial condition close to the UV fixed point tothe physical theory for k →
0. The diagrams contributingto the respective two- and three-point function flows aredisplayed in Fig. 2.
III. THE GRAVITON SPECTRAL FUNCTION
The Euclidean fluctuation approach in Sec. II withinthe approximations discussed in Sec. II B provides us withresults for the momentum- and cutoff-dependent fluctua-tion field propagator and Newton coupling. For k → k = 0 and cutoff dependences at p = 0 is at least work-ing qualitatively. Such an identification underlies manyphysics studies in asymptotic safety, most of which onlyprovide cutoff dependences and not momentum depen-dences. While not being at the heart of the current work,the Euclidean momentum dependences provided here arehence very important for the physics interpretation ofthese cutoff scale studies. Even more importantly, thecurrent results, as well as those already provided in [14, 38–41] can be used as input for the direct computation ofscattering vertices for general momentum configurations, S -matrix elements, and asymptotically safe cosmology.These interesting applications are left to future work.Here we aim at the reconstruction of the graviton spec-tral function from the numerical Euclidean data of thegraviton propagator, for our results see Fig. 1. Such recon-structions based on numerical data with statistical andsystematic errors are typically ill-conditioned problems.Moreover, for (unphysical) gauge fields they also requirethe additional key assumption that such a spectral repre-sentation exists. In gravity, this is further complicated bythe intricacies of the Wick rotation. The considerationsand numerical reconstructions here are based on theseassumptions, a detailed investigation of the difficulties ofthe reconstruction for numerical data in the context ofQCD can be found in [30]. Here, we follow the discussionthere and extend it to positivity and normalisability ofspectral functions in the presence of anomalous UV andIR momentum scalings. While most of the respectiveproperties, in particular the UV ones, are well-known, thepresent work is to our knowledge the first comprehensiveapplication to quantum gravity.Time-ordered propagators G F ( x, y ) = (cid:104) T φ ( x ) φ ( y ) (cid:105) −(cid:104) φ ( x ) (cid:105)(cid:104) φ ( y ) (cid:105) of physical fields (asymptotic states) inMinkowski space have a K¨all´en–Lehmann (KL) spectralrepresentation. In momentum space, it is given by G F ( p ) = i ∞ (cid:90) d λπ λ ρ ( λ ) p − λ + i (cid:15) , (16)with the spectral function ρ ( λ ). In (16), the restrictionto positive frequencies in the integral follows from theantisymmetry of the spectral function ρ ( λ ) = − ρ ( − λ ) . (17)A simply example is provided by the classical spectralfunction ρ cl of a particle with pole mass m pol , ρ cl ( λ ) = πλ (cid:104) δ ( λ − m pol ) − δ ( λ + m pol ) (cid:105) . (18)Inserting (18) in (16) leads to the classical Feynman prop-agator G F ( p ) = i / ( p − m +i (cid:15) ). The KL-representationin (16) constructs the full propagator in terms of a spec-tral integral over ’on-shell’ propagators 1 / ( p − λ + i (cid:15) )of states with pole masses λ . If properly normalised, thetotal spectral weight of all states is unity, ∞ (cid:90) d λπ λ ρ ( λ ) = 1 . (19) The spectral sum rule in (19) holds for the spectral func-tion of asymptotic states and also encodes the unitarityof the theory. In general, the propagator of the funda-mental fields in the theory is subject to renormalisationand its amplitude can be changed by an RG equation. Inthis case, one first has to define renormalisation groupinvariant fields to apply the above arguments. For gaugefields, the discussion is even more intricate as detailedbelow.The condition (19) entails that the decay of the spectralfunction for asymptotically large spectral values has tobe faster than 1 /λ . The latter decay is the canonicalone, as the momentum-dimension of the spectral functionis that of the propagator: −
2. The classical spectralis ultralocal, and vanishes identically for λ > m pol . Inturn, scattering events for λ > m pol induce a spectraltail, which indeed decays faster than 1 /λ . However, ifthe propagator shows an anomalous momentum scalingfor large momenta, this analysis is more intricate and isdetailed below. This case with anomalous scaling appliesto gauge theories, and in particular to the graviton.The reconstruction of the spectral function is done withthe Euclidean propagator G ( p ) for Euclidean momenta p . In the Euclidean branch this spectral representationof G ( p ) = i G F (i p ) is given by G ( p ) = ∞ (cid:90) d λπ λ ρ ( λ ) λ + p . (20)Equivalently, the spectral function can be obtained fromthe Euclidean propagator by means of an analytic contin-uation, ρ ( ω ) = 2 Im G ( − i( ω + i0 + )) , (21)i.e. from the discontinuity of the propagator. Insertingthe limit on the right-hand side of (21) in (20) leads to a δ -function from the KL kernel, and the spectral integralcan readily performed, leading to the left-hand side of(21). This concludes the brief introduction to the KLrepresentation. A. Properties of the graviton spectral function
Gauge fields such as the graviton and the gluon are notdirectly linked to asymptotic states. Therefore, they donot necessarily enjoy a spectral representation. While thephoton in QED is believed to have a spectral represen-tation, this is currently a debated subject in QCD, seee.g. [30] and references therein. In QCD, it is the confin-ing nature at large distances that complicates the matter,and in particular the relation to asymptotic states. Inturn, in AS quantum gravity, it is the strongly-correlatedUV fixed-point regime that complicates spectral consid-erations, even if leaving aside the intricacies of spectralrepresentation in the presence of dynamical metrics.As discussed in Sec. II, we consider AS gravity withinan expansion about a flat background. In this setup,gravity is classical for large distances since the UV-IRtrajectories approach classical scaling in the IR [38]. Inthis limit, similarly to QED, gravity is weakly coupledand may enjoy a spectral representation.The full propagator can be decomposed in its differ-ent components, leaving us with a traceless-transversetensor as well as vector and scalar components. In thepresent approximation based on the Einstein-Hilbert ten-sor structure, all components are related and it suffices todiscuss the spectral representation of one of them. Here,we concentrate on the spectral function of the traceless-transverse part G hh, TT of the graviton propagators in aflat background, also considered in [38]. We parametrise G hh, TT with the TT-projection operator Π TT ( p ) in App.C G hh, TT ( p ) = G hh ( p )Π TT ( p ) , G ¯ g ¯ g, TT ( p ) = G ¯ g ¯ g ( p )Π TT ( p ) , (22)with the scalar parts G hh ( p ) and G ¯ g ¯ g ( p ) of the fluctuationand background graviton respectively. Both scalar propa-gators are assumed to have a KL representation (20) withspectral functions ρ h ( λ ) and ρ ¯ g ( λ ).Importantly, both the analytic IR and the UV tail ofthe Euclidean propagators can be used to analyticallydetermine the spectral functions ρ ( λ ) for the asymptoticregimes λ → λ → ∞ , see [30]. In the UV thisis related to the well-known Oehme-Zimmermann super-convergence relation [69, 70]. We recall the argument here,adapted to the AS graviton. We consider dimensionlesspropagators and momenta, which are rescaled by appro-priate powers of the Planck mass, see (15). This leads usto dimensionless momenta and spectral parameters,ˆ p = p M , ˆ λ = λM pl , (23a)and dimensionless propagators and spectral functions,ˆ G (ˆ p ) = M G ( p ) , ˆ ρ (ˆ λ ) = M ρ ( λ ) . (23b)With (23) the UV limit of the dimensionless propagatorsreads lim ˆ p →∞ ˆ G (ˆ p ) = Z UV ˆ p ( − η ) 1(log ˆ p ) ¯ γ , (24)where the Z UV ’s are dimensionless normalisations andthe η ’s are the anomalous dimensions of the fluctuationgraviton h µν and the background graviton ¯ g µν , and ¯ γ isnon-vanishing for marginal scalings. Eq.(24) is the generalasymptotic form that includes a monomial behaviour aswell as a logarithmic cut.In the current approximation, they take the values η h ≈ . , Z UV h = 0 . ,η ¯ g = − , Z UV¯ g = 18 . . (25)with ¯ γ = 0. While the anomalous dimension of the fluctu-ation graviton η h is a dynamical quantity and depends on the approximation, the background anomalous dimensionis uniquely fixed by asymptotic safety: It is linked tothe β -function β N of the background Newton couplingwith η ¯ g = β N −
2. At the fixed point β N is vanishing bydefinition and hence η ¯ g = − η h isapproximation-dependent but one finds η h > η h and the underlying approxima-tion is explained later, and is done in the de-Donder typegauge (A2) with α = 0 and β = 1, given in App. A. Thelarge momentum value in (25) has been computed in [38]within a rather elaborate approximation. The approxi-mation here is a variant of that put forward in [38], andutilises the flows derived there.For the general discussion as well as the considerationof subleading UV- and IR-momentum dependences in thegraviton propagators we also take into account a poten-tial logarithmic running. This is known from resummedperturbation theory, where ¯ γ is given by the ratio of theanomalous dimension and the β -function of the runningcoupling, ¯ γ = ηβ . (26)In summary, (24) allows us to discuss the UV-asymptoticsof the spectral function of a given propagator. Moreover,it also can be used for the IR asymptotics, p →
0, where itgives access to the IR asymptotics of the spectral function.
1. Spectral function of the background graviton
We first discuss the UV limit of the background gravi-ton. The argument follows closely that for the Oehme-Zimmermann super-convergence relation in QCD. Weshow that the spectral function of the background gravi-ton is negative for large spectral values and its totalspectral weight vanishes, ∞ (cid:90) d λ λ ρ ¯ g ( λ ) = 0 . (27)Hence, in contradistinction to the spectral sum rule (19)related to unitarity of the theory, the spectral sum rule(27) enforces a vanishing spectral sum. Note also that(27) necessitates a spectral function ρ ¯ g ( λ ) that is bothpositive and negative for some λ .For proving (27), we consider asymptotically large Eu-clidean momenta as compared to the Planck mass. It isconvenient to study the asymptotic properties in termsof the dimensionless quantities defined in (23) within thelimit ˆ p = p/M pl → ∞ . In this limit the propagator of thebackground graviton decays with,lim ˆ p →∞ ˆ G ¯ g ¯ g (ˆ p ) = Z UV¯ g ˆ p ( − η ¯ g ) , (28)with η ¯ g → −
2, see (24) and (25). The UV asymptoticsin (28) allows us to determine the spectral representationfor ˆ λ → ∞ . With the definition (21) we getlim ˆ λ →∞ ˆ ρ ¯ g (ˆ λ ) = 2 sin (cid:104) π η ¯ g (cid:105) Z UV¯ g ˆ λ − η ¯ g . (29)Evidently, for η ¯ g <
0, the spectral function decays morerapidly as 1 / ˆ λ . Moreover, if we approach the UV-fixedpoint scaling with η ¯ g → −
2, the right-hand side in (29)vanishes. Then, the spectral function ρ ¯ g decays eithermore rapidly than 1 / ˆ λ or does vanish identically. Insummary, (29) guarantees that all spectral integrals inthe following are finite.Now we split the spectral integral in (20) for the back-ground graviton propagator into an asymptotic UV partwith spectral values ˆ λ ≥ √ ˆ p , and the respective IR part,ˆ G ¯ g ¯ g (ˆ p ) = √ ˆ p (cid:90) dˆ λπ ˆ λ ˆ ρ ¯ g (ˆ λ )ˆ λ + ˆ p + ∞ (cid:90) √ ˆ p dˆ λπ ˆ λ ˆ ρ ¯ g (ˆ λ )ˆ λ + ˆ p . (30)Let us first discuss the second term on the right-handside of (30): For √ ˆ p → ∞ only the asymptotic limit ofthe spectral function in (29) enters, and the term decaysfaster than 1 / ˆ p . We find,lim ˆ p →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ (cid:90) √ ˆ p dˆ λπ ˆ λ ˆ ρ ¯ g (ˆ λ )ˆ λ + ˆ p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ˆ p − η ¯ g . (31)Accordingly, also the first term on the right-hand side in(30) has to decay at least with 1 / ˆ p (2 − η ¯ g / , in order toguarantee the limit (28) in combination with (31). Forexample, for the fixed point scaling with η ¯ g = − / ˆ p .The first term in (30) can be rewritten as √ ˆ p (cid:90) dˆ λ ˆ λ ˆ ρ ¯ g (ˆ λ )ˆ λ + ˆ p = 1ˆ p √ ˆ p (cid:90) dˆ λ ˆ λ ˆ ρ ¯ g (ˆ λ )1 + ˆ λ ˆ p , (32)where we have dropped the 1 /π -term, as the total nor-malisation is not relevant for the present discussion. Nowwe use ˆ λ ≤ ˆ p in (32) due to the upper bound of theintegration. Hence, in the limit ˆ p → ∞ , the ˆ λ -part inthe denominator in (32) can be dropped to leading order.Accordingly, in this limit, we are led to1ˆ p √ ˆ p (cid:90) dˆ λ ˆ λ ˆ ρ ¯ g (ˆ λ )1 + ˆ λ ˆ p ˆ p →∞ −−−→ p √ ˆ p (cid:90) dˆ λ ˆ λ ˆ ρ ¯ g (ˆ λ ) . (33) The prefactor only decays with 1 / ˆ p for ˆ p → ∞ . Thisentails that for η ¯ g <
0, the spectral integral in (33) hasto decay at least as ˆ p η ¯ g / in order to be compatible with(28) for the background propagator. This leads us tolim ˆ p →∞ √ ˆ p (cid:90) dˆ λ ˆ λ ˆ ρ ¯ g ( λ ) = 0 for η ¯ g < . (34)Eq. (34) is nothing but the Oehme-Zimmermann super-convergence sum rule (27).This property holds for any field with a UV scaling with η <
0. In particular, we conclude that, if the backgroundgraviton admits a spectral representation, its spectralfunction, ρ ¯ g , has a vanishing total spectral weight, see(27). This also implies negative parts for ρ ¯ g , which isconfirmed in the explicit computation, see Fig. 1.We emphasise again that the property (27) does notentail unitary violations. The same property holds truefor the background gluon ¯ A µ in QCD, where the fullgauge field A µ is split into a background field ¯ A µ and afluctuation field a µ with the linear split A µ = ¯ A µ + a µ . Ifit has a spectral representation, it has the property (27)due to its anomalous dimension being negative, η ¯ A = β g s = − g s π < , (35)with the running strong coupling g s ( p/ Λ QCD ). For largemomenta the coupling tends to zero due to asymptoticfreedom and hence η ¯ A →
0. The ratio ¯ γ of anomalousdimension η ¯ A and β -function β g s is unity, ¯ γ ¯ A = 1 and weare left with a logarithmic running 1 /p / (log p/ Λ QCD )of the propagator. In this case, the decay in (31) solelyarises from the respective logarithms for ¯ γ > β -function of the coupling. Note that in QCD this propertyis even more peculiar: the spectral function is negative ina regime, where the theory is asymptotically free. In anycase, this analogy makes clear that a negative spectralfunction for an unphysical gauge boson does not entail alack of unitarity for the theory. While unitary of QCDhas not been proven rigorously, it is commonly assumedthat it is present. However, let us also add that negativespectral functions do not facilitate unitarity proofs orarguments either.
2. Spectral functions for large spectral values &normalisation
Now we use the UV-leading term of the propagatorsfor both, the fluctuation graviton and the backgroundgraviton in (24) for determining the asymptotic form forboth spectral functions. Here we expect a qualitativedifference between gravity and QCD. In the latter theory,the fluctuation gluon has the same vanishing spectralweight property of (27) (in the Landau-DeWitt gauge) dueto the negative anomalous dimension η a of the fluctuationgluon a µ , η a = − g s π (cid:18)
103 + (1 − ξ ) (cid:19) < , (36)for ξ < /
3. As for the background propagator we have η a → p/ Λ QCD → ∞ . The resummed logarithmicrunning has a power ¯ γ a = 1322 , (37)for the Landau gauge with ξ = 0. For more detailsand the discussions of general covariant gauges we referthe reader to e.g. [71] and references therein. As forthe background gluon, the fluctuation gluon propagatordecays more rapidly as 1 /p and we arrive at (27): bothgluon propagators obey the sum rule (27), their totalspectral weight vanishes.In turn, in AS gravity the anomalous dimension of thefluctuation graviton is positive, η h >
0, see (25). Therespective computation in [38] as well as the present onesare done in the de-Donder type gauge (A2) with α = 0and β = 1. The choice α = 0 enforces the gauge strictlysimilar to the Landau-DeWitt gauge in QCD with gaugefixing parameter ξ = 0.We now proceed with the analytic computation of theUV asymptotics of the spectral functions. Using (21) oneobtains the asymptotic behaviour of the smooth part ofthe dimensionless spectral functions ˆ ρ in (23) with,lim ˆ ω →∞ ˆ ρ (ˆ ω ) = 2 Z UV ˆ ω ( − η ) 1(log ˆ ω ) ¯ γ × (cid:18) sin (cid:104) π η (cid:105) − cos (cid:104) π η (cid:105) π ¯ γ log ˆ ω (cid:19) , (38)valid for the η ∈ ( η h , η ¯ g ) considered here, η ∈ ( − , . (39)In (38) we have dropped subleading terms in the loga-rithms with (cid:112) π / ω ) → log ˆ ω .The lower limit in (39) comes from the IR constraint,that the propagator has a Fourier representation in thefirst place. For η < − p = 0. The boundary value η = − η >
2, see [48]. Again the boundary valuerequires special attention.Importantly, we can already conclude from our analysisthat fields with η (cid:54) = 0 cannot describe asymptotic states:for η < η > / ˆ λ . For η = 0 we are left with the dependence on ¯ γ . In thiscase, as for η = ±
2, the first term in (38) vanishes. For¯ γ (cid:54) = 0 we are left with the second term, triggered by thelogarithmic running of the UV-asymptotics. This part,with η = 0 and ¯ γ >
0, covers the QCD-behaviour. TheUV asymptotics is given bylim ˆ ω →∞ ˆ ρ (ˆ ω ) = − Z UV ˆ ω π ¯ γ (log ˆ ω ) (1+¯ γ ) . (40)With (40), the total spectral weight is finite for ¯ γ > γ ≤
0, the spectral weightis UV-divergent, and the spectral function cannot benormalised.In summary, we have found that a spectral functionhas a vanishing total spectral weight for η <
0, see (27).In this case ¯ γ can be general. This property also holdstrue for η = 0 and ¯ γ > { η < η = 0 ∧ ¯ γ > } : ∞ (cid:90) d λ λ ρ ( λ ) = 0 . (41)Then, the spectral function also has negative parts, andin particular, its UV asymptotics is negative in the range(39), see (38). This case applies to the spectral func-tion of the background graviton, ρ ¯ g . We emphasise thatthis property is not at odds with unitarity for two rea-sons. First of all, it is a well-known property of the gluonin QCD (assuming the existence of a spectral represen-tation). Secondly, the background graviton is not thegraviton propagating in loop diagrams that contribute tothe (unitary) S-matrix.The graviton that is relevant for the latter processesin the S-matrix, is the fluctuation graviton. For thefluctuation graviton, the case η > η h > ρ h is positive for all spectral values. However,we shall see later that this is indeed the case withinthe reconstruction, see also Fig. 1. This leads us with apositive, though not normalisable, spectral function ρ h >
0. With the latter property of the fluctuation graviton oneof the necessary condition for applying Cutkosky cuttingrules, [72], is satisfied. This brings us closer to a reliablediscussion of unitarity in asymptotic safety.
IV. EUCLIDEAN CORRELATION FUNCTIONS
With the setup discussed in Sec.II, we now compute theEuclidean fluctuation graviton propagator, see Sec. IV A,as well as the Euclidean coupling of the fluctuation three-point function for all momenta and cutoff scales, seeSec. IV B. This is based on momentum-dependent results p [ M pl ] G hh [ / M p l ] G hh ( p ) p − Z UV h p − η h − − − (a) Momentum dependence of the Euclidean fluctuation gravitonpropagator (blue, solid) with its IR and UV asymptotics. TheUV asymptotic (violet, dotted) is given by Z UV h p − η h with η h = 1 .
03 and Z UV h = 0 .
64, see (25). The IR asymptotic (red,dashed) is simply the classical dispersion 1 /p . − − − − − − p [ M pl ] G hh [ / M p l ] G hh ( p )∆ G (1) hh ( p )∆ G (2) hh ( p ) (b) Subleading contributions at small momenta in comparison tothe full propagator (blue, solid). ∆ G (1) hh (magenta, dashed) carries asubleading log-like contribution, while ∆ G (2) hh (cyan, dotted) carriesa constant contribution for small momenta. FIG. 3. Momentum dependence of the Euclidean fluctuation graviton propagator G hh ( p ). for the anomalous dimensions η h ( p ) and the β -function β g ( p ) in [38]. Here we provide, for the first time, thefull physical momentum-dependence of the graviton two-and three-point function at vanishing cutoff scale. Thisalso allows us to explicitly check the reliability of theidentification of cutoff and momentum scales.For the sake of simplicity, we use an analytical flowequation for the zero-momentum cutoff-dependent New-ton coupling g k = g k ( p = 0). This analytic flow equationis based on [38] with the approximations from (14) and η h = 0. It takes the simple form, ∂ t g k = 2 g k (cid:18) − g k g ∗ (cid:19) = 2 g k − g k π . (42)where g ∗ = 570 π/
833 is the UV fixed-point value. Thisflow equation has the solution g k = g ∗ k g ∗ M + k . (43)This solution is consistent with the fixed-point value inthe UV, g k →∞ = g ∗ , and has the physical IR behaviour, g k → = k /M . This allows us to express the RG scalein units of the Planck mass.On the trajectory (43), we evaluate the momentum-dependent graviton anomalous dimension η h ( p ) as well asthe momentum-dependent three-point Newton coupling g k ( p ). We emphasise that both quantities are evaluatedconsistently on the trajectory (43) but we neglect anyfeedback on the trajectory itself. The impact of this ap-proximation is subleading since the full trajectory, whichcan be obtained in an iterative procedure, exhibits thesame qualitative features as (43). A. Fluctuation graviton propagator
In this section, we present the Euclidean results forthe propagator of the fluctuation graviton. We first dis-cuss the details of the computation and the numericalresults while we discuss analytic fits for the IR asymp-totics in Sec. IV A 1. The latter are important for thereconstruction of the spectral function.The momentum-dependence of the Euclidean propaga-tor is incorporated in the momentum-dependence of theanomalous dimension already computed in [38]. For theEuclidean scalar part G hh,k ( p ) of the transverse-tracelessmode (22) we parametrise the cutoff-dependent gravitonpropagator with G hh,k ( p ) = 1 Z h,k ( p ) p . (44)The wave-function renormalisation is readily computedfrom the anomalous dimension η h ( p ), defined in (10).We emphasise that the anomalous dimension naturallyalso depends on the graviton couplings via the diagrams,see Fig.2. With the definition (10), we obtain the physicalwave-function renormalisation Z h ( p ), in the double limit k → → ∞ , Z h ( p ) = lim k → →∞ Z h, Λ ( p ) exp − k (cid:90) Λ d k (cid:48) k (cid:48) η h,k (cid:48) ( p ) , (45)where we set Z h, Λ = 1 at a large cutoff scale. The compu-tation of the fluctuation graviton anomalous dimension isdetailed in App. E.The result for the physical full momentum-dependentEuclidean graviton propagator G hh ( p ) is presented inFig. 3a. The leading asymptotics of G hh ( p ) are propor-tional to 1 /p for small momenta, and p η h − for large0momenta, where η h ≈ .
03 is the graviton anomalousdimension at the UV fixed point and p = 0, see (25).
1. IR asymptotics
The low-momentum asymptotic of 1 /p captures theclassical IR-regime: the theory approaches classical grav-ity with the Einstein-Hilbert action in (1). This does notexclude the presence of subleading features which maycarry important physics. We access the subleading IRbehaviour by subtracting the 1 /p -pole and introduce thedifference propagator ∆ ˆ G (1) hh ,∆ ˆ G (1) hh ( p ) = ˆ G hh (ˆ p ) − p . (46)This difference propagator is displayed with a red-dashedline in Fig. 3b. As expected, the Euclidean propagatordoes indeed show non-trivial subleading behaviour intro-duced by scatterings. It exhibits a log-like contribution,and for small momenta we find,lim ˆ p → ∆ ˆ G (1) hh (ˆ p ) = − A h ln ˆ p + C h , (47)with A h ≈ .
11 and C h ≈ . / ˆ p in (46) satisfies these prop-erties as it is subleading in the UV due to η h >
0. Inturn, we cannot use the logarithmic and constant terms in(47) as an analytic IR fit for the subleading IR behaviour.Instead, we use the confluent hypergeometric function U a,b (ˆ p ), whose leading large-momentum asymptotic is1 / ˆ p a . For b = 1 and small momenta, it approacheslim ˆ p → U a, (ˆ p ) = − a ) (cid:18) γ + Γ (cid:48) ( a )Γ( a ) + ln (cid:0) ˆ p (cid:1)(cid:19) , (48)where γ is the Euler–Mascheroni constant and Γ( z ) thegamma function. Hence, it shows the subleading IR-asymptotics in (47) with a cut at Re( p ) = 0. Moreover,for a > − η h / ≈ .
485 it is subleading in the UV. Forsimplicity, we choose a = 1 and arrive at,∆ ˆ G (2) hh (ˆ p ) = ∆ ˆ G (1) hh (ˆ p ) − A h U , (ˆ p ) , (49a)where U , (ˆ p ) = e ˆ p Γ(0 , ˆ p ) , (49b) with the upper incomplete gamma function Γ( a, z ) = (cid:82) ∞ z d t t a − e − t . The subleading IR-asymptotics in (49)is depicted as the dotted cyan line in Fig. 3b. In sum-mary, the two IR subtractions leave us with a constantcontribution remaining for small momenta.The spectral function of these asymptotic IR fits isreadily computed, which leaves us only with a reconstruc-tion task of the remaining part of the propagator, ∆ G (2) hh .This is done by a fit of Breit-Wigner (BW) structures aswell as an analytic UV-asymptotic ρ UV h . This is detailedin Sec. V, the resulting spectral function is discussed inSec. V B. B. Newton coupling
In this section, we present the Euclidean results for thephysical momentum-dependent Newton coupling G N ( p ) = G k =0 ( p ), which is derived from the transverse-tracelesspart of the fluctuation three-graviton vertex.In our approximation, we only retain the dependence ofthe Newton coupling on the average momentum flowingthrough the vertex, see (9), and we evaluate the flowof the graviton three-point function at the momentumsymmetric point, see (12). We feed the dependence ofthe average momentum back on the right-hand side ofthe flow, see Fig. 2. Note that different combinationsof external and loop momenta run through the verticesin the diagrams but the coupling only depends on theaverage momentum. Furthermore, we use that the loopmomentum q is bounded by the cutoff scale, q (cid:46) k ,and that the diagrams give subleading contributions for p i (cid:29) k . This implies that through all vertices, wehave an average momentum flow of the order of ¯ p =1 / p + p + p ) and we approximate g k ( p i , q ) ≈ g k (¯ p ).More details can be found in [14, 38]. In summary, thisleads us to ∂ t g k ( p ) = (cid:16) η g ( p ) (cid:17) g k ( p ) , (50)with anomalous dimension η g ( p ) of the flow of the gravitonthree-point function, see App. D for details. Eq. (50) is in-tegrated for given data of η g ( p ). The resulting momentum-dependent Newton coupling at vanishing cutoff scale isgiven by the blue solid line in Fig. 4. Together with theexplicit depiction of the momentum dependence of thefluctuation propagator, it is a key Euclidean result ofthe present work. It encodes, for the first time, the fullmomentum-dependence of the scattering coupling of threegravitons in the physical cutoff limit k → G N ( p ) = G k =0 ( p ) . (51)The coupling G N ( p ) shows a flat classical IR regime, andexhibits a slight increase in strength between about 1 and2 Planck masses, before decaying with 1 /p . Whetheror not this increase about the Planck scale is a physicsfeature or a truncation artefact remains to be seen withinimproved approximations.1 − − − − p [ M pl ] g N g RG g ∗ . FIG. 4. Physical Newton coupling g N ( p ) = G N ( p ) M as afunction of momentum in units of the Planck mass (blue,solid). For comparison, we also show the scale-dependentNewton coupling g RG = g k ( p = 0) (red, dashed) as a functionof k = p , and the fixed-point coupling g ∗ ( p/k ) as a functionof a dimensionless momentum variable (green, dotted).
1. Physical Newton coupling
This novel result of a momentum-dependent couplingat k = 0 allows us to evaluate the standard approximationof identifying the physical momentum-dependent New-ton coupling at vanishing cutoff with the k -running ofthe Newton coupling at vanishing momentum, and thatof the fixed-point coupling. In Fig. 4, we depicted themomentum-dependent fixed-point coupling g ∗ ( p/k ), andthe k -dependent coupling at p = 0, g RG ( k ), for compari-son:(i) cutoff-momentum identification ( p − k ): While g RG ( k ) trivially agrees with the physical coupling g N ( p ) for small momenta, it turns over towards theasymptotically safe fixed point running at smallermomentum scales. Indeed this happens nearly anorder of magnitude earlier. Moreover, the UV cou-pling is also far smaller, and it does not show theintermediate rise of the coupling.(ii) Fixed-point identification:
The fixed-point coupling g ∗ ( p FP ), normalised with the cutoff scale k lacks adetermination of the (IR) Planck mass. Here, p FP indicates that the momentum in g ∗ is measuredin the cutoff scale. This normalisation of both,the Newton coupling and the momentum, with thecutoff scale, leads to the deviation of its ’IR’ valuefrom the physical one. If rescaled to fit the IRcoupling, it turns towards the asymptotically saferegime even earlier than g RG ( k ). Also the UV valueof the coupling is even smaller than that of g RG ( k ).In summary, we conclude that both procedures, (i) and (ii) , mimic the qualitative aspects of the physical Newtoncoupling. Moreover, the common p − k –identification, (i) ,works considerably better than the fixed-point identifi-cation. However, the comparison also shows that both procedures cannot be used for quantitative statements.This concerns in particular physics that covers both theasymptotically safe UV regime and the classical IR regime.In both procedures, (i) and (ii) , the relative momentumscales in the two regimes are off by one or more orders ofmagnitude. We emphasise that while this has been shownhere for the scattering coupling of three gravitons, thisreadily translates to other observables: the three-gravitoncoupling is at the root of all scattering processes.Finally, the results of the present work can also be usedto improve upon the procedures (i) and (ii) used in theliterature: the comparison of g RG ( k ), g ∗ ( p FP ) with thephysical coupling g N ( p ) allows us to establish identifica-tions k → p and p FP → p for phenomenological use. Still,for more quantitative statements and scattering observ-ables with several momentum scales this is bound to fail,and one has to resort to the full computation within thepresent fluctuation approach, see [14, 38–51, 61] for puregravity and [47, 48, 50–61] for gravity-matter systems. C. Euclidean background propagator
In this section, we relate the Newton coupling obtainedin Sec. IV B from the fluctuation three-graviton vertexto the background graviton propagator. Similarly to thefluctuation graviton propagator in (44), the backgroundgraviton propagator is parametrised as G ¯ g ¯ g ( p ) = 1 Z ¯ g ( p ) p , (52)with the (inverse) dressing or wave-function renormal-isation Z ¯ g ( p ) = Z ¯ g,k → ( p ). The latter is related withbackground diffeomorphism invariance to the β -functionof the Newton coupling. With (50), this leads us to therelation η ¯ g ( p ) = − ∂ t Z ¯ g ( p ) Z ¯ g ( p ) = η g ( p ) . (53)In (53), we have used that we have identified all avatars ofthe Newton coupling with g k ( p ), including the backgroundcoupling. This (symmetric-point) approximation has beenproven to hold true semi-quantitatively, for more detailssee e.g. [14, 38]. Note also that it is at the root of thebackground-field approximation used in the literature.Given later applications to cross-sections and otherobservables, it is instructive to relate the definition ofthe background propagator to the tree-level scattering FIG. 5. Tree-level graviton-graviton scattering diagram. p [ M pl ] G ¯ g ¯ g [ / M p l ] G ¯ g ¯ g ( p ) p − Z UV¯ g p − − − − − (a) Momentum dependence of the Euclidean background gravitonpropagator (blue, solid) with its IR and UV asymptotics. TheUV asymptotic (violet, dotted) is given by p − with Z UV¯ g ≈ . /p . − − − − − − − − p [ M pl ] G ¯ g ¯ g [ / M p l ] G ¯ g ¯ g ( p ) (cid:12)(cid:12)(cid:12) ∆ G (1)¯ g ¯ g ( p ) (cid:12)(cid:12)(cid:12) − ∆ G (2)¯ g ¯ g ( p ) (b) Subleading contributions at small momenta in comparison tothe full propagator (blue, solid). ∆ G (1)¯ g ¯ g (magenta, dashed) carries asubleading log-like contribution, while ∆ G (2)¯ g ¯ g (cyan, dotted) carriesa constant contribution for small momenta. FIG. 6. Momentum dependence of the Euclidean background graviton propagator G ¯ g ¯ g ( p ). of fluctuation gravitons with a one-graviton exchange.Such a scattering process in the s -channel is displayeddiagrammatically in Fig. 5. To relate it to the backgroundpropagator, we need to contract the external legs withtwo further fluctuation graviton propagators. This readsschematically G ¯ g ¯ g ( p ) (cid:39) G hh ( p ) (cid:104) Γ ( hhh ) ( p ) G hh ( p )Γ ( hhh ) ( p ) (cid:105) . (54)Here, we have implicitly projected on transverse-tracelesspart of the scattering process. Note that in (54), all fluc-tuation wave-function renormalisations cancel out and weare left with a g k ( p ) /p behaviour in the high-momentumregime, as expected. This way of defining a backgroundpropagator or running coupling has a straightforwardanalogy in QCD, where the analogous tree-level processof gluon-gluon scattering can be linked to the backgroundpropagator. While not identical, they share both qualita-tive as well as quantitative features.We emphasise that while (54) as well as its gluon ana-logue G ¯ A ¯ A are reminiscent of an s -channel contributionto 2-to-2–scattering of particles, they do not describe anon-shell physical process: for the present case of grav-ity the initial and final ’states’ are fluctuation gravitons,which are not diffeomorphism-invariant. For QCD thefinal ’states’ are fluctuation gluons, which are not gaugeinvariant.Note also that considering a 2-to-2–scattering of back-ground gravitons does not improve on this situation. De-spite their rˆole for the construction of a diffeomorphism-invariant effective action, they are no on-shell physicalparticles. We recall the fact that the background gluonshares all these gauge-covariant properties with the back-ground graviton. Still, its spectral function has negativeparts.With (53), we readily compute the Euclidean back- ground graviton propagator or s -channel scattering ofgravitons on the symmetric point. The result is depictedwith the blue solid line in Fig. 6. The leading asymptoticsof G ¯ g ¯ g ( p ) are proportional to 1 /p for small momenta,and p η ¯ g − = 1 /p for large momenta. Asymptotic safetyrequires η ¯ g = −
1. IR asymptotics
The low-momentum asymptotic of 1 /p captures theclassical IR-regime: the theory approaches classical grav-ity with the Einstein-Hilbert action in (1). This doesnot exclude the presence of subleading features that maycarry important physics.To access the subleading IR behaviour, we follow thesame procedure as for the fluctuation graviton propagatorin Sec. IV C 1 and subtract the 1 /p -pole. This leads usto the difference propagator ∆ ˆ G (1)¯ g ¯ g ,∆ ˆ G (1)¯ g ¯ g (ˆ p ) = ˆ G ¯ g ¯ g (ˆ p ) − p . (55)In contradistinction to the fluctuation graviton, ∆ ˆ G (1)¯ g ¯ g isnot subleading in the UV: the subtraction introduces a1 /p -dependence that dominates the 1 /p asymptotics.This is seen in Fig. 6b, where the magenta dashed linedepicts the resulting subleading contribution.Similarly to the fluctuation graviton propagator, weobserve a subleading log-like contribution for small mo-menta, lim ˆ p → ∆ ˆ G (1)¯ g ¯ g (ˆ p ) = − A ¯ g ln ˆ p + C ¯ g , (56)with A ¯ g ≈ .
34 and C ¯ g ≈ − .
25. As in Sec. IV A 1,we capture this contribution with the hypergeometric3function U , . We define∆ ˆ G (2)¯ g ¯ g (ˆ p ) = ∆ ˆ G (1)¯ g ¯ g (ˆ p ) − A ¯ g U , (ˆ p ) + 1 + A ¯ g p + ∆Γ ) + 2 (1 + A ¯ g )∆Γ (1 + (ˆ p + ∆Γ ) ) . (57)In (57), we have employed a combination of hypergeo-metric functions and BW type structures, see Sec. V A:our specific choice ensures that ∆ ˆ G (2)¯ g ¯ g (ˆ p ) is negative forall momenta, and that its UV asymptotics is given by − / ˆ p . The lack of sign changes in ∆ ˆ G (2)¯ g ¯ g facilitates thereconstruction, though it is not necessary. This require-ment is satisfied for ∆Γ i (cid:38) .
4. We have also checkedthat the specific values of the ∆Γ i have no impact onthe reconstruction result, and the values used here are∆Γ = ∆Γ = 5.We depict − ∆ ˆ G (2)¯ g ¯ g for this choice with the cyan dottedline in Fig. 6b. It shows some smooth substructures thatare related to the analytic subtractions in (57), impor-tantly, these subtractions do not introduce cuts and poles.These structures could be smoothed out, but since thisdoes not have an impact on the resulting systematic errorof the reconstruction, we refrain from doing so.As for the fluctuation graviton, the spectral function ofthe asymptotic IR fits is readily computed, which leavesus only with a reconstruction task of the remaining partof the propagator, ∆ ˆ G (2)¯ g ¯ g . This is done by a fit of BWstructures. This is detailed in Sec.V, the resulting spectralfunction is discussed in Sec. V C. V. GRAVITON SPECTRAL FUNCTIONS
In this section, we compute the spectral functions of thefluctuation and background propagator with reconstruc-tion methods from the Euclidean propagators computedin Sec. IV. We discuss the results for the spectral functionof the fluctuation graviton ρ h in Sec. V B and the onefor the background graviton ρ ¯ g in Sec. V C. These resultsprovide an important first step towards a comprehensiveunderstanding of the spectral properties of asymptoticallysafe gravity including unitarity. A. Spectral reconstruction
As described in Sec. IV A 1 and Sec. IV C 1, the IRasymptotics of the Euclidean graviton propagators can betaken into account analytically in the form of a 1 / ˆ p poleand a hypergeometric function encompassing a subleadinglog-like pole. The remaining contributions, ∆ G (2) , areconstant for small momenta and tend towards ˆ p − η forlarge momenta. The respective anomalous dimension isdynamical for the fluctuation graviton η h ≈ .
03, whileasymptotic safety dictates η ¯ g = − G (2) is treatedwith the reconstruction method described in [30], for anassessment of other reconstruction methods see also thedetailed discussion there. We proceed by choosing anansatz of a combination of BW-like structures,ˆ G BW (ˆ p ) = K N ps (cid:88) k =1 N ( k )pp (cid:89) j =1 (cid:32) ˆ N k (ˆ p + ˆΓ k,j ) + ˆ M k,j (cid:33) δ k,j . (58)Here, N ps is linked to the number of BW-structuresneeded to describe the propagator, N ( k )pp , δ k,j are linked tothe shape and decay of the single structures, and K is anoverall normalisation of the propagator, for more detailssee [30]. As mentioned before, we could also describe thehigh momentum asymptotics analytically, and only fitthe remaining structures with the ansatz (58). However,since the exponents δ k,j naturally lead to the same typeof high momentum asymptotics, this does not improvethe convergence of the reconstruction.The fit of ∆ G (2) with BW-like structures leads us to afully analytical description of the propagators, see (61) forthe fluctuation propagator, and (64) for the backgroundpropagator. This allows us to readily compute the spectralfunction ρ and also to reconstruct the graviton propagatorfrom the obtained spectral function. The reconstructedgraviton propagator G rec is defined just as in (20) with G rec ( p ) = ∞ (cid:90) d λπ λ ρ ( λ ) λ + p . (59)The spectral function is now fixed by minimising theaveraged deviation or error E rel between the Euclideandata and its reconstruction, E rel = 1 N (cid:88) i (cid:18) G ( p i ) − G rec ( p i ) G ( p i ) (cid:19) , (60)where the index i runs over the N data points consideredfor the fit. The relative error E rel in (60) is measured interms of the values G ( p i ) of the Euclidean propagator onthe data points. This definition can be further optimisedas after the subtraction of the asymptotics the data pointsin the vicinity of the Planck scale (several orders of mag-nitude) are most relevant. Improved reconstructions onrecently developed methods based on machine learningas well as further structural insights, see [73], will bepresented elsewhere.To minimise bias w.r.t. the choice of BW structures,we use various fits with different N ps and N ( k )pp . Thenwe select the best fits by their relative error (60) and anadditional smoothness constraint: the error defined in(60) does not punish oscillations. This introduces a well-known instability towards smaller E rel at the expenseof oscillations, for a discussion see again [30, 73] andreferences therein.This finalises the set-up of our reconstruction proce-dure. The resulting spectral functions are discussed inthe following sections Sec. V B and Sec. V C.4 ω [ M pl ] ρ h [ / M p l ] ρ h ( ω, ~p = 0) ρ IR h ( ω ) ρ BW h ( ω ) ρ UV h ( ω ) (a) Spectral function of the fluctuation graviton (blue, solid line). Itfeatures a δ -function at ω = 0 (massless graviton), and an ensuingsmooth multi-particle continuum, ρ cont h ( ω ). We also depict theanalytic IR- and UV-asymptotics (red, dashed and violet, dash-dotted), and the Breit-Wigner part (cyan, dotted). p [ M pl ] G hh [ / M p l ] Euclidean datareconstruction − − − − − (b) Euclidean fluctuation propagator reconstructed from the spectralfunction presented in the left panel. The reconstructed and originalEuclidean data agree very well on all data points corresponding toa reconstruction error of E rel < − . FIG. 7. Spectral function of the fluctuation graviton and the reconstructed Euclidean propagator. The reconstruction, thedefinition of its error, and the error band in Fig. 7a are described in Sec. V A and Sec. V B.
B. Spectral function of the fluctuation graviton
The reconstruction method explained in Sec. V A pro-vides us with a spectral function, which is derived fromthe fluctuation graviton propagator,ˆ G hh (ˆ p ) = 1ˆ p + A h U , (ˆ p ) + ˆ G BW hh (ˆ p ) , (61)with ˆ G BW hh defined in (58). The parameters in (61) aresummarised in App. F in Tab. I, and the resulting spectralfunction is shown in Fig.7a. The reconstructed propagatoris in quantitative agreement with the Euclidean inputdata, see Fig. 7b. The best fit for the spectral function ρ h is given by the blue, solid line, and further reconstructionswithin the error E rel < − , see (60), are indicated bythe blue-shaded area. The latter provides our systematicerror estimate.We split the spectral function into two parts,ˆ ρ h (ˆ ω ) = π ˆ ω δ (ˆ ω ) + ˆ ρ cont h (ˆ ω ) , (62)where the δ -function at vanishing frequency comprises a’classical’ massless graviton and ˆ ρ cont h (ˆ ω ) comprises theensuing smooth multi-particle continuum and the UV-asymptotics. The spectral function of the fluctuationgraviton shows several well-understood properties:(i) Classical gravity:
It has a δ -function contribution atvanishing frequency due to the 1 /p IR asymptoticsof the Euclidean propagator. This contributionis simply that of a classical graviton propagatorthat arises from the curvature term in the Einstein-Hilbert action. We remind the reader in this contextthat we have set the cosmological constant to zero for the sake of simplicity. It can be resurrectedwithin the computation.(ii)
Perturbative low energy scattering spectrum:
Themassless pole contribution also leads to scatteringevents with arbitrarily small momenta. Hence themulti-particle scattering continuum leads to a (sub-leading) cut mirrored in the log-like divergence ofthe propagator at small momenta. In terms ofCutkosky rules, these would correspond to 1-to-2and 2-to-2 scattering events. However, we shouldkeep in mind that the fluctuation graviton is nota physical field, and hence these are not physicalscattering events. The logarithmic cut leads to afinite IR part, ˆ ρ cont h (0) = 2 πA h ≈ .
71, see Fig. 7a.The scattering events from perturbative low energygravity dominate roughly up to the Planck scale.(iii)
IR-UV transition regime at the Planck-scale:
Asexpected, in the regime about the Planck scale, theBW contributions take over, and facilitate a smoothtransition towards the large frequency asymptoticsin the asymptotically safe UV regime. We alsoemphasise that this regime does not feature anypronounced structure such as an additional peak.(iv)
Asymptotically safe regime:
The IR-UV transitionin (iii) tends towards the UV-asymptotics in theasymptotically safe UV-regime. This asymptoticsis given analytically by,lim ˆ ω →∞ ˆ ρ UV h (ˆ ω ) = 2 Z UV h sin (cid:16) η h π (cid:17) ω − η h , (63)with η h ≈ .
03 and Z UV h ≈ .
64. This is a directconsequence of the UV asymptotics of the Euclidean5 − ω [ M pl ] ρ ¯ g [ / M p l ] ρ ¯ g ( ω, ~p = 0) ρ IR¯ g ( ω ) ρ BW¯ g ( ω ) (a) Spectral function of the background graviton (blue, solid line).It features a δ -function at ω = 0 (massless graviton), and an ensuingsmooth multi-particle continuum, ρ cont¯ g ( ω ). We also depict the IRasymptotics (red, dashed), and the Breit-Wigner part (cyan, dotted). p [ M pl ] G ¯ g ¯ g [ / M p l ] Euclidean datareconstruction − − − − − − (b) Euclidean background propagator reconstructed from the spec-tral function presented in the left panel. The reconstructed andoriginal Euclidean data agree very well on all data points corre-sponding to a reconstruction error of E rel < − . FIG. 8. Spectral function of the background graviton and reconstructed Euclidean graviton propagator. The reconstruction, thedefinition of its error, and the error band in Fig. 8a are described in Sec. V A and Sec. V C. propagator, see Sec. IV A 1. The positive value ofthe anomalous dimension η h implies that the totalweight of the spectral function diverges, see thediscussion in Sec. III A 2.In summary, the δ -function at vanishing frequency, andconsequently also the scattering cut, as well as the high-frequency asymptotics, all follow directly from analyticproperties of the Euclidean propagator. They do not de-pend on the details of the chosen reconstruction method.Another important and stable property is the positivityof the spectral function, which holds for all reconstruc-tions. In conclusion, these are the ’physics’ propertiesof the spectral function ρ h ( p ): while it is a gauge-fixedcorrelation function, it is, together with the Newton cou-pling g N ( p ), the pivotal building block of asymptoticallysafe gravity. In particular, the gravity contributions ofscattering elements are constructed from it. Hence, thefluctuation graviton satisfies one of the necessary condi-tion for applying Cutkosky cutting rules, see [72]. C. Spectral function of the background graviton
The reconstruction method explained in Sec. V A alsoprovides us with a spectral function, which is derivedfrom the background graviton propagator,ˆ G ¯ g ¯ g (ˆ p ) = 1ˆ p + A ¯ g U , (ˆ p ) − A ¯ g p + ∆Γ ) − A ¯ g )∆Γ (cid:104) p + ∆Γ ) (cid:105) + ˆ G BW¯ g ¯ g (ˆ p ) , (64)with ˆ G BW¯ g ¯ g defined in (58). The parameters in (64) aresummarised in App.F in Tab.II, and the resulting spectral function is shown in Fig. 8a. The reconstructed propa-gator is in quantitative agreement with the Euclideaninput data, see Fig. 8b. The best fit for the spectralfunction ρ ¯ g ( p ) is given by the blue, solid, line and furtherreconstructions within the error E rel < − , see (60), areindicated by the blue-shaded area. The latter providesour systematic error estimate.We again split the spectral function into two parts,ˆ ρ ¯ g (ˆ ω ) = π ˆ ω δ (ˆ ω ) + ˆ ρ cont¯ g (ˆ ω ) , (65)where the δ -function at vanishing frequency comprises a’classical’ massless graviton and ˆ ρ cont¯ g (ˆ ω ) comprises theensuing smooth multi-particle continuum and the UV-asymptotics. The spectral function shows several well-understood properties:(i) Classical gravity: it has a δ -function contributionat vanishing frequency due to the 1 /p IR asymp-totics of the Euclidean propagator. This contribu-tion is simply that of a classical graviton propa-gator that arises from the curvature term in theEinstein-Hilbert action. In this regime, classicaldiffeomorphism invariance holds up to small per-turbative corrections. Hence, the normalisation ofthe δ -function is the same as for the fluctuationgraviton.(ii) Perturbative low energy scattering spectrum:
Thismassless pole contribution of the fluctuation gravi-ton also leads to scattering events for the back-ground graviton with arbitrarily small momenta,and hence the multi-particle scattering continuumleads to a (subleading) cut mirrored in the log-likedivergence at small momenta. However, these eventsdiffer from those in the fluctuation graviton: while6the scattering events for the background gravitoncan be understood as tree-level ( s -channel) 2-to-2scatterings of gravitons, those of the fluctuationgraviton are loop contributions, which may be in-terpreted via cutting rules also as 1-to-2 scatterings.This can lead to significant differences: while ρ cont¯ g has a positive finite value of ˆ ρ cont¯ g (0) = 2 πA ¯ g ≈ . IR-UV transition regime in the Planck-scale regime:
In contradistinction to the spectral function of thefluctuation graviton, that of the background gravi-ton shows distinct peaks at frequencies about thePlanck scale. In this regime, the systematic errorof the reconstruction grows large. We remark thatif dropping the anomalous dimension of the fluctu-ation graviton in the diagrams for the propagatorof the background graviton, the spectral function ρ ¯ g only shows the positive δ -function at vanishingfrequency and a negative one with the same ampli-tude at around the Planck scale. The anomalousdimension η h carries rescattering events and softensthe negative δ -function, leading to the pronouncedpeak structure in Fig. 8a.(iv) Asymptotically safe regime:
The IR-UV transitionin (iii) tends towards the UV asymptotics in theasymptotically safe UV regime. The asymptotics isgiven analytically by,lim ˆ ω →∞ ˆ ρ UV¯ g (ˆ ω ) = 2 Z UV¯ g ˆ ω , (66)with Z UV¯ g ≈ .
5. Eq.(66) is dictated by asymptoticsafety, see Sec. III A 2. It also implies that the totalspectral weight vanishes, see (41). The sum of thethree terms in first line on the right-hand side of (64)has already analytically the property (66), while thetwo terms in the second line individually enjoy thisproperty. Accordingly the sum rule (41) is satisfiedanalytically, and we find ∞ (cid:90) d λ λ ρ ¯ g ( λ ) = 0 . (67)Eq. (67) is analogous to the Oehme-Zimmermannsuper-convergence relation [69, 70] for the gluonspectral function.In summary, the spectral function of the backgroundgraviton shows the required δ -function at vanishing fre-quency, identical to that of the fluctuation graviton, aswell as a finite low energy part that originates in pertur-bative scattering events. At large frequency is shows asoften negative peak that is dictated by asymptotic safety,and leads to a vanishing total spectral weight. We closethis part with an important comment on the physicalinterpretation of background correlation functions. To that end, we use the single-graviton exchange process inFig. 5 and its QCD analogue as a ’telling’ example. InQCD, we may define a ’background’ propagator G ¯ A ¯ A ( p )with tree-level gluon-gluon scattering or quark–anti-quarkscattering: (54) and Fig. 5, where the gravitons are sub-stituted by gluons. Then, an IR singular behaviour with G ¯ A ¯ A ( p ) ∝ /p for small momenta would give rise toconfinement with a linear potential within a single gluon-exchange picture. Assuming the resulting IR dominanceof gluons it has been shown that functional equations inthe Landau gauge indeed admit such a solution (the Man-delstam solution) [74]. What makes this self-consistentpicture even more appealing is the direct physics inter-pretation of the respective propagator G ¯ A ¯ A . However, afull computation reveals that the propagator G ¯ A ¯ A is evenIR suppressed, p G ¯ A ¯ A ( p ) → G ¯ A ¯ A contains negativeparts both for asymptotically small and large spectral val-ues [30]: that for large spectral values is triggered by the perturbative tail. In turn, that for small spectral values istriggered by the ghost. Moreover, the confining potentialis only revealed by an all order gauge-invariant compu-tation, see [76]. In conclusion, while the direct physicsinterpretation of the gluon-gluon scattering is very appeal-ing and is even confirmed within approximations based onthis assumption (self-consistency), it turns out to be evenqualitatively wrong. Note that this statement includesthe perturbative part.The analysis in QCD leads to an important lesson forasymptotically safe gravity: there is no evidence for the va-lidity of a direct physics interpretation of diffeomorphism- variant scattering events such as graviton-graviton scat-terings, in particular in the strongly correlated asymptoti-cally safe UV regime. In turn, while such an interpretationfails in QCD even for perturbative UV momenta, the IRscattering of gravitons simply describes the scattering ofmassless particles, see Fig. 1b or Fig. 8a.
VI. CONCLUSIONS
In the present work, we have reported on first, butimportant steps towards an understanding of asymptoticsafety with Lorentzian signature. In particular, we havecomputed the spectral functions of the fluctuation andbackground graviton, see Fig. 1, with reconstructionsmethods from the full momentum-dependent Euclideanpropagators at vanishing cutoff scale. The Euclideanresults also encompass a full momentum-dependent avatarof the Newton coupling at vanishing cutoff scale. Inparticular, they allow the discussion of scattering eventsas well as the benchmarking of standard phenomenologicalscale identifications in the literature, see Sec. IV B 1.The results for the spectral functions have been pre-sented and discussed in detail in Sec. V. For details on thefluctuation graviton we refer to Sec. V B, and for detailson the background graviton we refer to Sec. V C. Thespectral function of the fluctuation graviton is positive7and hence the fluctuation graviton in the Landau-DeWittgauge satisfies one of the necessary condition for applyingCutkosky cutting rules, see [72]. However, the spectralfunction is not normalisable, which signals the fact thatthe graviton is not directly related to an asymptotic state.In turn, that of the background graviton has positiveand negative parts, and has a vanishing total spectralweight. Its properties are reminiscent of that of the back-ground gluon, see the analysis at the end of the lastsection, Sec. V C. This analogy suggests in particular thatthe background graviton does not carry any signature ofunitarity violation or preservation, which is a far moreintricate matter.Next steps include the application of the spectral func-tions to the computation and analysis of observables suchas scattering events or the cosmological evolution inducedby asymptotically safe gravity. Moreover, we currentlycorroborate the results obtained in the present work witha direct computation of the Minkowski propagators withinthe spectral setup put forward in [77]. With all theseadvances we aim at an investigation of unitarity in asymp-totically safe gravity within the present approach.
Acknowledgements
We thank A. Barvinsky, J. Horak, U. Moschella, andN. Wink for discussions. This work is supported by theDFG, Project-ID 273811115, SFB 1225 (ISOQUANT), aswell as by the DFG under Germany’s Excellence StrategyEXC - 2181/1 - 390900948 (the Heidelberg ExcellenceCluster STRUCTURES). MR acknowledges funding bythe Science and Technology Research Council (STFC)under the Consolidated Grant ST/T00102X/1.
Appendix A: Gauge fixing
The gauge-fixing action is given by S gf [¯ g, h ] = 12 α (cid:90) d x √ ¯ g ¯ g µν F µ F ν , (A1)with the de-Donder type gauge fixing condition F µ , F µ = ¯ ∇ ν h µν − β ∇ µ h νν . (A2)This leads to the ghost action S gh [¯ g, h, ¯ c, c ] = (cid:90) d x √ ¯ g ¯ c µ M µν c ν (A3)with the Faddeev-Popov operator M following from adiffeomorphism variation of the gauge-fixing condition, M µν = ¯ ∇ ρ ( g µν ∇ ρ + g ρν ∇ µ ) − β g ρσ ¯ ∇ µ ( g νρ ∇ σ ) . (A4)Throughout this work, we use β = 1 and the Landau limit α → Appendix B: Regulator
We choose the regulator R k proportional to the two-point function at vanishing cosmological constant R k = Γ (2) k (Λ = 0) · r ( p /k ) . (B1)Here, r is the shape function for which we use the Litim-type cutoff function [78–80] r ( x ) = (cid:18) x − (cid:19) Θ(1 − x ) . (B2) Appendix C: Transverse-traceless projectionoperator
We project the two- and three-point graviton correlationfunctions on their transverse-traceless part. The TT-projection operator is given byΠ µνρσ TT ( p ) = 12 (Π µρ T ( p )Π νσ T ( p ) + Π µσ T ( p )Π νρ T ( p )) −
13 Π µν T ( p )Π ρσ T ( p ) , (C1)where Π µν T ( p ) = δ µν − p µ p ν p . (C2) Appendix D: Flow of the three-graviton vertex
The flow of the dimensionless three-point Newton cou-pling is obtained by the complete contraction of the flowof the three-graviton vertex with three TT-projectionoperators. We writeFlow ( hhh )TT = ∂ t Γ ( hhh ) k ( p , p , p ) (cid:81) i =1 (cid:112) Z h,k ( p i ) ∗ (cid:89) i =1 Π TT ( p i ) , (D1)where ’ ∗ ‘ stands for the complete contraction of the pro-jection operators with the vertex. The denominator in(D1) takes care of the wave-function renormalisations in(11). The flow (D1) is evaluated at the momentum sym-metric point, see (12), which makes is a function of onesingle momentum. This leads to flow equation for theNewton coupling of the three-point vertex as given in (50)with the anomalous dimension given by η g ( p ) = 3 η h ( p ) + g k (0) Flow ( hhh )TT ( p ) − Flow ( hhh )TT (0) p . (D2)Here, we have also substituted g k ( p ) → g k (0) in front ofthe flow term in (D2). This additional approximationweakens the decay of the flow for large momenta, andhence only influences subleading contributions.8 Appendix E: Computation of the anomalousdimension
The fluctuation graviton anomalous dimension η h ( p ) isderived from the flow of the graviton two-point function,(with λ n = 0 and g n,k = g ,k ) η h ( p ) = − π ( hh )TT ( p ) − Flow ( hh )TT (0) p , (E1)where Flow ( hh )TT ( p ) = ∂ t Γ ( hh ) k,µνρσ ( p ) Z h,k ( p ) Π µνρσ TT ( p ) , (E2)with the TT-projection operator as provided in App. C.Eq. (E2) is the projection of the full fluctuation two-pointflow on its TT-part. Eq. (E1) is an integral equation for η h ( p ) since the momentum-loop integrals of the flow con-tain the anomalous dimension via the regulator insertion ∂ t R k . We solve (E1) in a two-step procedure: we firstsolve for its asymptotics, i.e., we expand for small andlarge p . Then we solve the full equation for η h ( p ) by aniterative procedure.
1. Asymptotics of the anomalous dimension
The spectral reconstruction requires good understand-ing of the asymptotics of the Euclidean propagators, whichdirectly relate to the asymptotics of the anomalous di-mension. Here, we provide the explicit expressions of theanomalous dimension at large and small momenta.For p →
0, the definition of the anomalous dimension(E1) turns into a derivative with respect to p and weobtain η h (0) = − π ∂ p (cid:16) Flow ( hh )TT ( p ) (cid:17)(cid:12)(cid:12)(cid:12) p → (E3)= g k π (cid:18) − (cid:90) d q (3 q − q + 4 q ) η h ( q ) (cid:19) . Similarly, we obtain an equation for η h ( p ) at large mo-menta p (cid:29) k from the definition (E1), η h ( p (cid:29) k ) ≈ g k π (cid:32) k p (cid:18)
12 + 23 (cid:90) d q q ( q − q − η h ( q ) (cid:19) + k p (cid:18)
14 + 13 (cid:90) d q q ( q − η h ( q ) (cid:19) + k p (cid:18) − − (cid:90) d q q ( q − η h ( q ) (cid:19) + k p (cid:18) − − (cid:90) d q q ( q − η h ( q ) (cid:19)(cid:33) . (E4)
2. Symmetrisation of the anomalous dimension
For k →
0, the graviton anomalous dimension is afunction of momentum-squared, η h = η h ( p ): it can beexpanded in p , i.e., no terms proportional to ( p ) n p arepresent, with p = (cid:112) p . For finite cutoff-scale, k (cid:54) = 0,non-analyticities of the regulator function can break thisproperty. This is well-known for the sharp cutoff as wellas the Litim-type cutoff. The latter leads to contributionsof the form l p + l p + O ( p ) to the flow of the gravitontwo-point function, while the former would even introduceterms linear in p . This entails that for the Litim-typecutoff, η h ( p ) as defined in (E1), contain terms linear in p ,which have to integrate to zero for k →
0. This implies afine-tuning condition for the anomalous dimension at thelarge initial cutoff scale k/M pl → ∞ . This fine-tuning isstraightforward but tedious.Since these terms are subleading, we refrain from thefine-tuning and only consider the even part of the anoma-lous dimension at each cutoff step. This is done via aPad´e fit, η sym h ( p ) = η Pad´e h ( p ) + η Pad´e h ( − p )2 . (E5)The Pad´e fit, or any other fit, introduces an analyticbias for our reconstruction. Note however that this biasonly is an issue for small momenta, and not for the largemomentum regime we are mainly interested in. Moreover,the flows do allow for Taylor expansions in the asymptoticregimes, which further stabilises Pad´e approximants. Appendix F: Parameters of the spectral functions
In this appendix, we provide the parameters of theBW fits used for construction of the fluctuation and back-ground spectral functions presented in Sec. V.
1. Fluctuation spectral function
We describe the Euclidean fluctuation propagator, usedto reconstruct the spectral function shown in Fig. 7a,by a combination of a p − peak, a hypergeometric func-tion, and the average of four BW fits to the remainingstructures,ˆ G hh (ˆ p ) = 1ˆ p + A h U , (ˆ p ) + 14 (cid:88) i =1 ˆ G BW ( i ) hh (ˆ p ) , (F1a)with A h ≈ . . (F1b)The four BW fits used were selected from a range of BWstructures with different N ps and N ( k )pp , c.f. (58), and allexhibit relative errors E rel < − . We present their fitparameters in Tab. I.9 TABLE I. Parameters for the reconstruction of the fluctuationgraviton spectral function (central line in Fig. 7a).parameter ˆ G BW (1) hh ˆ G BW (2) hh ˆ G BW (3) hh ˆ G BW (4) hh K N .
517 0 .
408 0 .
552 0 . , .
222 1 .
20 1 . M , . · − . · − .
237 0 . δ , − . − .
52 2 .
31 1 . , .
43 0 .
196 0 .
628 0 . M , .
562 0 . .
68 4 . · − δ , .
06 1 .
36 1 . − . , .
48 1 . M , .
30 0 . δ , .
670 0 . N .
409 0 . .
410 0 . , .
75 1 .
15 1 .
30 0 . M , .
546 1 .
15 0 .
657 2 . δ , . .
57 0 .
632 1 . , .
63 0 .
389 0 .
298 1 . M , .
64 1 .
17 0 1 . δ , .
470 0 . − .
149 1 . , .
37 1 . M , .
487 1 . δ , .
18 1 . N .
335 1 . · − . . · − ˆΓ , .
615 2 .
48 1 .
90 2 . M , .
710 1 .
16 1 .
11 0 . δ , .
42 2 .
06 0 .
940 2 . , .
82 1 .
18 1 .
55 1 . M , .
50 1 .
08 1 .
08 1 . δ , .
21 0 .
618 0 .
758 1 . , .
329 2 . M , .
22 0 . δ , .
28 1 . E rel . · − . · − . · − . · −
2. Background spectral function
For the reconstruction of the background spectral func-tion shown in Fig. 8a, we introduced two additional termsfacilitating an easier fit of the remaining structures,ˆ G ¯ g ¯ g (ˆ p ) = 1ˆ p + A ¯ g U , (ˆ p ) − A ¯ g p + ∆Γ ) − A ¯ g )∆Γ (cid:104) p + ∆Γ ) (cid:105) + 13 (cid:88) i =1 ˆ G BW ( i )¯ g ¯ g (ˆ p ) , (F2a)with A ¯ g ≈ . , ∆Γ = ∆Γ = 5 . (F2b) TABLE II. Parameters for the reconstruction of the back-ground graviton spectral function (central line in Fig. 8a).parameter ˆ G BW (1)¯ g ¯ g ˆ G BW (2)¯ g ¯ g ˆ G BW (3)¯ g ¯ g K − − − N .
37 7 .
08 6 . , . . . M , .
10 7 .
98 7 . δ , .
97 1 .
96 1 . N .
66 3 .
28 3 . , .
790 0 .
709 0 . M , .
61 1 .
54 1 . δ , .
51 5 .
32 5 . N .
233 0 .
686 0 . , .
29 1 .
28 1 . M , .
69 1 .
18 0 . δ , .
907 4 .
24 2 . , . M , . δ , . , . M , . δ , . N . . · − ˆΓ , .
09 0 . M , .
11 1 . δ , .
15 2 . N . . · − ˆΓ , .
23 1 . M , .
336 1 . δ , .
03 3 . N . , . M , . δ , . N . · − ˆΓ , . M , . δ , . E rel . · − . · − . · − The three fits were chosen as the best fits with E rel < − ,and we show their parameters in Tab. II. Appendix G: Background spectral function forvanishing anomalous dimensions
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