Zooming in on fermions and quantum gravity
ZZooming in on fermions and quantum gravity
Astrid Eichhorn,
1, 2, ∗ Stefan Lippoldt, † and Marc Schiffer ‡ CP3-Origins, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark Institut f¨ur Theoretische Physik, Universit¨at Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
We zoom in on the microscopic dynamics for fermions and quantum gravity within the asymptotic-safety paradigm. A key finding of our study is the unavoidable presence of a nonminimal derivativecoupling between the curvature and fermion fields in the ultraviolet. Its backreaction on the prop-erties of the Reuter fixed point remains small for finite fermion numbers within a bounded range.This constitutes a nontrivial test of the asymptotic-safety scenario for gravity and fermionic matter,additionally supplemented by our studies of the momentum-dependent vertex flow which indicatethe subleading nature of higher-derivative couplings. Moreover our study provides further indica-tions that the critical surface of the Reuter fixed point has a low dimensionality even in the presenceof matter.
I. INTRODUCTION
In the search for a quantum theory of gravity that isviable in our universe, the existence of fermionic mattermust be accounted for. Our strategy to achieve this isbased on a quantum field theoretic framework that in-cludes the metric field and fermion fields at the micro-scopic level. Such a setting requires an ultraviolet com-pletion or extension of the effective field theory frame-work within which a joint description of gravity andmatter is possible up to energies close to the Planckscale. Asymptotic safety [1, 2] is the idea that scale-invariance provides a way to extend the dynamics toarbitrarily high momentum scales without running intoLandau poles which would indicate a triviality problem.Moreover, scale-invariance is a powerful dynamical prin-ciple, that is expected to fix all but a finite number of freeparameters in an infinite dimensional space of theories.It can be reached at a fixed point of the RenormalizationGroup (RG), which can be free (asymptotic freedom) orinteracting (asymptotic safety). Compelling indicationsfor the existence of the asymptotically safe Reuter fixedpoint in four-dimensional gravity have been found, e.g.,in [3–14]. For recent reviews and introductions includinga discussion of open questions, see [15–19].A central part of the interplay of the Standard Modelwith gravity is the impact of quantum gravity on themicroscopic dynamics for fermions as well as the cor-responding “backreaction” of fermionic matter on thequantum structure of spacetime. In line with the obser-vation that asymptotically safe quantum gravity couldbe near-perturbative [20, 21] or “as Gaussian as it gets”[14, 22–24], studies of fermion-gravity-systems follow atruncation scheme by canonical power counting. Fur-thermore, the chiral structure of the fermion sector ofthe Standard Model is a key guiding principle. Thus, theleading-order terms according to canonical power count- ∗ [email protected] † [email protected] ‡ m.schiff[email protected] ing have been explored in the sector of chirally symmet-ric fermion self interactions [25–27], and fermion-scalarinteraction sector [27, 28]. These are dimension-6 anddimension-8-operators, respectively. Explicitly chiral-symmetry breaking interactions, including a mass termand two dimension-5, nonminimal couplings of fermionsto gravity [29], have been studied. The effect of quan-tum gravity on a Yukawa coupling of fermions to scalarshas been studied in [27, 28, 30–34]. Conversely, the im-pact of fermionic fluctuations on the Reuter fixed pointhas been explored in [29, 35–40]. An asymptotically safefixed point exists in all of these studies, as long as thefermion number is sufficiently small. Moreover, all oper-ators that have been explored follow the pattern thatcanonical dimensionality is a robust predictor of rele-vance at the interacting fixed point. Further, they con-firm the conjecture that asymptotically safe quantumgravity could preserve global symmetries [27], at leastin the Euclidean regime. Thus, all symmetry-breakinginteractions can be set to zero consistently. Addition-ally, interacting fixed points could, but need not exist forthese, as in the case of the Yukawa coupling [27, 28, 30–34]. In contrast, the interacting nature of asymptoticallysafe gravity percolates into the symmetric sector, whereinteractions can typically not be set to zero consistently[27, 41]. Hence, their “backreaction” on the asymptoti-cally safe fixed point could be critical. Further, this sec-tor is a potential source of important constraints on themicroscopic gravitational parameter space: Strong grav-ity fluctuations could trigger new divergences in the mat-ter sector, manifesting themselves in complex fixed-pointvalues for matter interactions. The corresponding boundon the gravitational parameter space that separates theallowed, weakly coupled gravity regime from the forbid-den strongly-coupled regime, is called the weak-gravitybound [27, 28, 42].In Tab. we provide an overview over interactionsin the fermion sector that have been explored in anasymptotically safe context. The table contains a cru-cial gap, namely nonminimal, chirally symmetric inter-actions. This is the sector that we will begin to tackle inthis paper. For an analogous study in the scalar sectorsee [43]. a r X i v : . [ h e p - t h ] A p r ref. interaction dimension relevant symmetry free fixed point weak-gravity bound[29] ¯ ψψ χ sym. yes no[27] ¯ ψψ φ χ sym. yes no[29] ¯ ψ ∇ ψ χ sym. yes no[29] R ¯ ψψ χ sym. yes no this work R µν ¯ ψγ µ ∇ ν ψ χ sym. no no [25–27] ( ¯ ψγ µ γ ψ ) χ sym. no no[25–27]] ( ¯ ψγ µ ψ ) χ sym. yes no[27, 28] ¯ ψ ( / ∇ ψ )( ∂φ ) , ¯ ψγ µ ( ∇ ν ψ )( ∂ ν φ )( ∂ µ φ ) χ sym. no yes Table 1 . We list already investigated interactions in gravity-fermion systems in order of increasing canonical dimension, andspecify whether they are relevant at the Reuter fixed point. Further, we highlight that all but one interactions that respectchiral symmetry, i.e., allow a separate phase transformation of left-handed and right-handed fermions, are necessarily nonzeroat the free fixed point. A subset of these exhibits a weak-gravity bound, whereas interactions that feature a free fixed pointcannot give rise to a weak-gravity bound.
As our key result we find a continuation of the asymp-totically safe Reuter fixed point to finite fermion numbersthat passes a nontrivial test by remaining robust undera crucial extension of the approximation to the full dy-namics. Moreover, we find further indications that thecritical hypersurface of the Reuter fixed point has a lowdimensionality also in the presence of matter.This paper is structured as follows: In Sec. II, we pro-vide an overview of the setup, and specify the approxi-mation to the full dynamics that we will explore in thefollowing. In Sec. III we discuss in some detail how toderive the beta functions in our setting. In particular, wediscuss the relation of the derivative expansion to the pro-jection at finite momenta. Sec. IV provides an overviewof the fixed-point results for N f = 1, which are represen-tative for the results at small fermion numbers. We dis-cuss tests of the robustness of the fixed point, the impactof the newly included nonminimal derivative interactionon the fixed-point results in a smaller truncation, and thefeature of effective universality. Sec. V contains a discus-sion of structural aspects of the weak-gravity bound forcubic beta functions and highlights that no such boundexists for the nonminimal derivative interaction in theregime of gravitational parameter space where our trun-cation remains viable. In Sec. VI we extend our investi-gations to N f (cid:29)
1, and discuss the continuation of theReuter fixed point to larger fermion numbers. In Sec. VIIwe provide a short summary of our key results and high-light possible routes forward in gravity-matter systems inan outlook. App. A includes a general derivation of theform of the flow equation for the dimensionless effectiveaction. This form can be used to directly derive dimen-sionless beta functions, in contrast to the usual procedureof only introducing dimensionless quantities after a trun-cation has been specified.
II. SETUP
The system we analyze contains a gravitational sec-tor and a matter sector with chiral fermions. We aim atderiving the beta functions in this system, and will em- ploy the well-suited functional Renormalization Group.It is based on the flow equation for the scale-dependenteffective action, the Wetterich-equation [44–46],˙Γ k [Φ; ¯ g ] = 12 STr (cid:104)(cid:0) Γ (2) k [Φ; ¯ g ] + R k [¯ g ] (cid:1) − ˙ R k [¯ g ] (cid:105) . (1)The “superfield” Φ is simply a collection of all fields inour system,(Φ A ) = (cid:0) h µν ( x ) , ψ i ( x ) , ¯ ψ i ( x ) , c µ ( x ) , ¯ c µ ( x ) (cid:1) , (2)where Einsteins summation convention over the “su-perindex” A contains a summation over discrete space-time, spinor and flavor indices and an integration overthe continuous coordinates. Here R k is a scale-dependentregulator that implements a momentum-shell wise inte-gration of quantum fluctuations and the dot in ˙Γ k refersto a derivative with respect to t = ln k/k , the RG-“time”with k an arbitrary reference scale. The IR-regulator R k enters the generating functional in the form of a termthat is quadratic in the fluctuation fields and renders theWetterich equation UV and IR finite. Specifically, wechoose a Litim-type cutoff [47] with appropriate factorsof the wave-function renormalization for all fields. Nextto the gauge-fixing term for the metric fluctuations, it is asecond source of breaking of diffeomorphism invariance.It must be set up with respect to an auxiliary metricbackground ¯ g µν , which provides a notion of locality andthereby enables a local form of coarse graining. In themain part of this paper we focus on a flat background,¯ g µν = δ µν , (3)while in this section we will keep ¯ g µν arbitrary for ped-agogical reasons. For introductions and reviews of themethod, see, e.g., [48–51]; specifically for gauge theoriesand gravity, see, e.g., [15, 52, 53].The Wetterich equation provides a tower of coupleddifferential equations for the scale dependence of all in-finitely many couplings in theory space. In practice, thishas to be truncated to a (typically) finite-dimensionaltower. Let us briefly summarize how we proceed, beforeproviding more details. To construct our truncation, wedefine a diffeomorphism invariant “seed action”. Next,we expand the terms in this seed action to fifth order inmetric fluctuations, defined as h µν = ¯ g µν − g µν . (4)This corresponds to an expansion of the seed action invertices. At this point all terms in the seed action, exceptthose arising from the kinetic term for fermions, comewith one of the couplings of the seed action. We next takeinto account that in the presence of a regulator and gaugefixing, the beta functions for those couplings genericallydiffer, when extracted from different terms. Accordingly,we introduce a separate coupling in front of each term inthe expanded action. This provides the truncation whichwe analyze in the following. To close the truncation, thecouplings of higher-order vertices are partially identifiedwith those of lower-order ones.In more detail, these steps take the following form: Ourseed action reads S = S grav + S gh + S mat . (5)Classical gravity is described by the Einstein-Hilbert ac-tion S EH , S EH = − π ¯ G N (cid:90) d x √ g ( R − λ ) . (6)In order to tame the diffeomorphism symmetry of gravity,we choose a gauge-fixing condition F µ , F µ = (cid:18) ¯ g µκ ¯ D λ − β g κλ ¯ D µ (cid:19) h κλ , β = 0 . (7)The gauge choice is incorporated using the gauge fixingaction S gf , S gf = 132 π ¯ G N α (cid:90) d x √ ¯ g F µ ¯ g µν F ν , α → . (8)To take care of the resulting Faddeev-Popov determinant,we use ghost fields c µ and ¯ c ν with the appropriate ghostaction S gh , S gh = (cid:90) d x √ ¯ g ¯ c µ δF µ δh αβ L c g αβ , (9)where L c g αβ is the Lie derivative of the full metric g µν in ghost c µ direction, L c g αβ = 2¯ g ρ ( α ¯ D β ) c ρ + c ρ ¯ D ρ h αβ + 2 h ρ ( α ¯ D β ) c ρ . (10)In the following, we choose the Landau gauge, i.e., α → h µν we see thatthis choice of gauge-fixing parameters leads to contribu-tions from only a transverse-traceless (TT) mode h TT µν anda trace mode h Tr , h µν (cid:98) = h TT µν + 14 ¯ g µν h Tr , (11) where the TT-mode satisfies ¯ D µ h TT µν = 0 and ¯ g µν h TT µν = 0,while the trace mode is given by h Tr = ¯ g µν h µν . All othermodes drop out of the flow equation once it is projectedonto monomials with nonvanishing powers of the field. Itis important to note that the TT-mode is present in anygauge and to linear order in h µν a gauge invariant quan-tity. Thus, for external metric fluctuations we exclusivelyconsider the TT-mode. For internal metric fluctuations,also the remaining trace mode is taken into account. Wesummarize the purely gravitational parts of the action as S grav , S grav = S EH + S gf . (12)Next we turn to the chiral fermions. Their minimalcoupling to gravity is via the kinetic term S kinmat , S kinmat = N f (cid:88) i =1 (cid:90) d x √ g ¯ ψ i / ∇ ψ i . (13)For the construction of the covariant derivative forfermions, we use the spin-base invariance formalism [54–56]. For our purposes, this is equivalent to using thevierbein formalism with a Lorentz symmetric gauge.Upon expansion in h µν , this minimal interaction betweenfermions and gravity gives rise to an invariant linear inderivatives. There are several invariants containing termsof third order in derivatives and canonical mass dimen-sion, namely: S ∇ mat = N f (cid:88) i =1 (cid:90) d x √ g (cid:0) ¯ κR ¯ ψ i / ∇ ψ i + ¯ τ ( D µ R ) ¯ ψ i γ µ ψ i (14)+ ¯ ξ ¯ ψ i / ∇ ψ i + ¯ σR µν ( ¯ ψ i γ µ ∇ ν ψ i − ( ∇ ν ¯ ψ i ) γ µ ψ i ) (cid:1) , where each of the invariants respects the Osterwalder-Schrader positivity of the Euclidean action. Out of thesefour invariants, the ones corresponding to ¯ κ and ¯ τ donot contribute linearly to an external h TT µν , as R does notcontain a transverse traceless part to linear order. Inthe following, we restrict ourselves to the nonminimalcoupling ¯ σ and neglect the ¯ ξ term. Thus, the kineticmatter action is complemented with S R mat = N f (cid:88) i =1 ¯ σ (cid:90) d x √ g R µν (cid:0) ¯ ψ i γ µ ∇ ν ψ i − ( ∇ ν ¯ ψ i ) γ µ ψ i (cid:1) . (15) S R mat has all the symmetries of the original action (5) andtherefore does not enlarge the theory space.The nonminimal coupling ¯ σ introduces an invariant ofcubic order in derivatives, capturing parts of the higher-derivative structure of the fermion-gravity interaction.Once expanded around a flat background, the interac-tion with h TT µν is given by S R mat = N f (cid:88) i =1 ¯ σ (cid:90) d x ( (cid:3) h TT µν ) ¯ ψ i γ µ ∂ ν ψ i + O ( h ) , (16) Couplings S EH S kinmat S R mat Γ (2) k ¯ λ , Z h Z ψ –Γ (3) k ¯ λ , ¯ G h ¯ G ψ ¯ σ Γ (4) k ¯ λ = ¯ λ , ¯ G h, = ¯ G h ¯ G ψ, = ¯ G ψ ¯ σ = ¯ σ Γ (5) k ¯ λ = ¯ λ , ¯ G h, = ¯ G h ¯ G ψ, = ¯ G ψ ¯ σ = ¯ σ Table 2 . We list the couplings and wave function renor-malizations appearing in the n -th functional derivative Γ ( n ) k of the effective action and indicate to which part of the seedaction, S EH , S kinmat or S R mat they are related. where (cid:3) = − δ µν ∂ µ ∂ ν is the d’Alambertian in flat Eu-clidean space. Eq. (16) is the unique invariant consistingof one h TT µν , ψ , ¯ ψ and γ µ together with two derivativesacting on the TT-mode and one derivative acting on the ψ . We summarize the matter parts of the action as S mat , S mat = S kinmat + S R mat . (17)After having specified our complete seed action, weexpand the scale-dependent effective action in powers ofthe fluctuation field,Γ k [Φ; ¯ g ] = ∞ (cid:88) n =0 n ! Γ ( n ) k A ...A n [0; ¯ g ]Φ A n . . . Φ A , (18)where Γ ( n ) k refers to functional derivatives with respectto the field Φ,Γ ( n ) k A ...A n [Φ; ¯ g ] = Γ k [Φ; ¯ g ] ←− δδ Φ A . . . ←− δδ Φ A n . (19)Note the order of the indices and fields, which is impor-tant to keep in mind for the Grassmann-valued quanti-ties.By using this vertex form, the flow of 5 individual cou-plings ¯ λ , ¯ λ , ¯ G h , ¯ G ψ and ¯ σ as well as the anomalousdimension of two wave-function renormalizations Z h and Z ψ is disentangled, cf. Tab. and see Sect. III for moredetails. Here the barred couplings, e.g., ¯ G ψ and ¯ G h , referto dimensionful couplings.For the gravity-fermion vertex the contributing dia-grams are shown in Fig. . This highlights the necessityto truncate the tower of vertices, as the flow of each n -point vertex depends on the ( n + 1)- and ( n + 2)-pointvertices. We use the seed action in Eq. (5) to parametrizethe vertices appearing in the diagrams. When generat-ing, e.g., a graviton three-point vertex or a graviton four-point vertex for the scale-dependent effective action fromthe seed action by expanding to the appropriate powerin h µν , both would depend on the same Newton coupling¯ G N and the same cosmological constant ¯ λ due to dif-feomorphism symmetry. However, the gauge fixing andthe regulator break diffeomorphism symmetry. Hence,the effective action is known to satisfy Slavnov-Tayloridentities instead, [2, 52, 57–60]. As these identities ingeneral are much more involved, there is no such simple ∂ t Γ ( h ¯ ψψ ) k = 12 ˜ ∂ t −
12 ˜ ∂ t − ∂ t + ˜ ∂ t + ˜ ∂ t Figure 1 . We show the diagrams contributing to the flow ofthe gravity-fermion system. Double lines denote metric fluc-tuations, single lines fermions. Each diagram is understood tocarry a regulator insertion on one of the propagators, that ˜ ∂ t acts upon. When ˜ ∂ t is evaluated, a diagram with n internalpropagators becomes the sum of n diagrams, such that theregulator insertion occurs on each of the internal propagatorsonce. relation between the three- and four-point vertex of theeffective action as there is for the seed action. In otherwords, the breaking of diffeomorphism symmetry leadsto an enlargement of theory space in which the couplingsparameterizing the vertices are independent. There aredifferent routes towards a truncation of this large theoryspace. In principle, one could pick some random tensorstructure and momentum-dependence in each n − pointfunction and parameterize this by some coupling. Then,the connection to the diffeomorphism-invariant seed ac-tion would be lost completely. Instead, we derive thetensor structures of the vertices from the seed-action, butalso take into account that the various couplings are nowindependent. Specifically, we proceed using the followingrecipe: The structure of the n -point vertex is drawn fromthe seed action,Γ ( n ) k A ...A n = Z B Φ A . . . Z B n Φ A n S ( n ) B ...B n (cid:12)(cid:12)(cid:12) ¯ λ → ¯ λ n , (20)where the replacement ¯ λ → ¯ λ n only affects pure gravityvertices. Furthermore in Eq. (20) the metric fluctuationsof the purely gravitational action S grav are rescaled ac-cording to S grav : ( h n ) µν → (16 π ) n ¯ G N ( ¯ G h ) n − ( h n ) µν , (21)whereas the graviton in S gh and S mat is rescaled to S gh : ( h n ) µν → (16 π ) n ( ¯ G h ) n ( h n ) µν , (22) S mat : ( h n ) µν → (16 π ) n ( ¯ G ψ ) n ( h n ) µν . (23)This rescaling breaks diffeomorphism symmetry andhelps us choosing a basis in the appropriate theory space.Note that the field-redefinitions in Eq. (21), (22) and (23)are not to be understood as actual field-redefinitions inthe effective action. They are just a way of arriving at aparameterization of the truncated effective action in theenlarged theory space.In the following we use the term “avatar”, when a singlecoupling in the seed action leads to various incarnationsin the effective action, e.g., ¯ G h and ¯ G ψ are avatars of theNewton coupling ¯ G N .In order to close the flow equation, we identify cou-plings of higher order n -point vertices with the corre-sponding couplings of the three-point vertex. This wasalready implicitly done with the rescaling in Eqs. (21),(22) and (23) and with the usage of one single cou-pling ¯ σ . Similarly, all n ≥ λ , i.e., ¯ λ → ¯ λ and¯ λ → ¯ λ . The relation between ¯ λ and the gravitationalmass-parameter ¯ µ h that is often used in the literature,reads − λ = ¯ µ h . In the next section we provide detailson how the beta functions are extracted from the sum ofthe diagrams in Fig. . III. HOW TO OBTAIN BETA FUNCTIONS
We now discuss in some detail how to derive betafunctions. We concentrate on the dimensionless cou-plings, which are obtained from their dimensionful coun-terparts by a multiplication with an appropriate powerof k . Dimensionful couplings are denoted with overbars,e.g., ¯ G ψ , ¯ λ etc., whereas their dimensionless counter-parts lack the overbar, e.g., G ψ , λ etc.A key goal of ours is to test the quality of our trun-cation. Thus, we place a main focus on the momentum-dependence of the flow, i.e., the dependence of the n -point vertices on the momenta of the fields. Higher-order momentum-dependencies than those included inthe truncation are in general present. This implies thatdifferent projection schemes might yield different resultswhen working in truncations. We will discuss these dif-ferent schemes and their relation to each other in thefollowing. A. Fermionic Example
As a concrete example let us consider the fermionic sec-tor. To arrive at beta functions, we have to take severalsteps. First we define a projector P (3) p ,p on the gravity-fermion vertex. Its form is motivated by the tensor struc-ture of the considered three-point function, S kinmat (cid:2) g = δ +(16 π ¯ G ψ ) h TT , ψ, ¯ ψ (cid:3) (24)= N f (cid:88) i =1 (cid:90) d x (cid:2) ¯ ψ i /∂ψ i − π ¯ G ψ h TT µν ¯ ψ i γ µ ∂ ν ψ i (cid:3) + O ( h ) . By taking the corresponding functional derivatives ofEq. (24) and evaluating in momentum space, while usingthe projector onto transverse traceless symmetric tensorsΠ TT , we find that, (cid:90) x,y,z e i ( p · x + p · y + p · z ) S kinmat ←− δδh TT µν ( x ) ←− δδψ i ( y ) ←− δδ ¯ ψ j ( z ) (25)= (2 π ) δ ( p + p + p )( − i π ¯ G ψ )Π TT µνρσ ( p ) γ ρ p σ δ ij . Of the three momenta, only two are independent, thethird can be eliminated by momentum conservation.Thus we define the projector P (3) p ,p on ¯ G ψ as P (3) ijp ,p µν ( x, y, z )= iγ ρ p σ π N f p Π TT ρσµν ( p ) e i ( p · y + p · z ) δ ( x ) δ ij , (26)which we evaluate at the symmetric point for the mo-menta, p = p = − p · p ) = p . The normalization of P (3) p ,p follows fromΠ TT µνµσ ( p ) p σ p ν = 53 (cid:16) p − ( p · p ) p (cid:17) . (27)Using P (3) p ,p we define the projected dimensionful vertex¯ V as¯ V ( p ) = (cid:90) x,y,z tr (cid:20) P (3) ijp ,p µν ( x, y, z )Γ k ←− δδh µν ( x ) ←− δδψ i ( y ) ←− δδ ¯ ψ j ( z ) (cid:21) Φ=0 , (28)where tr implies the trace over Dirac and flavor indices.This definition is independent of any truncation, while atruncation for ¯ V can be viewed as choosing a specificpoint in theory space. For instance, when evaluating¯ V for our chosen truncation we find that ¯ V is equal to (cid:112) ¯ G ψ (1 − σp ). Having defined ¯ V , we aim at derivingthe beta function for the dimensionless counterpart V , V ( p k ) = kZ h ( p ) Z ψ ( p ) ¯ V ( p ) . (29)Note that ¯ V carries a non-trivial dimension, as thegravity-fermion vertex contains an additional momentum p . The scale derivative of V reads β V ( p k ) = V ( p k ) (cid:0) η ψ ( p ) + η h ( p ) (cid:1) + 2 p k V (cid:48) ( p k )+ k ˙¯ V ( p ) , (30)where one has to take into account the scaling of themomentum, p ∼ k . Here η h and η ψ are the anomalousdimensions, η h ( p ) = − ˙ Z h ( p ) Z h ( p ) , η ψ ( p ) = − ˙ Z ψ ( p ) Z ψ ( p ) . (31)We can read off ˙¯ V by replacing Γ k with ˙Γ k in equation(28), ˙¯ V ( p ) = k Flow (3) ψ ( p ) , (32)where Flow (3) ψ is a short hand for the contributing dia-grams in Fig. ,Flow (3) ψ ( p ) = k (cid:90) x,y,z tr (cid:20) STr (cid:104) (2) k + R k ˙ R k (cid:105) ←− δδh µν ( x ) ←− δδψ i ( y ) ←− δδ ¯ ψ j ( z ) (cid:12)(cid:12) Φ=0 × P (3) ijp ,p µν ( x, y, z ) (cid:21) . (33)Here we made use of Eq. (1).By inserting the expression for ˙¯ V given in Eq. (32) intoEq. (30) for β V we finally arrive at the beta function β V for V , β V ( p k ) = Flow (3) ψ ( p ) (34)+ V ( p k ) (cid:0) η ψ ( p ) + η h ( p ) (cid:1) + 2 p k V (cid:48) ( p k ) . This equation will take center stage in our analysis ofthe momentum-dependence of the flow and tests of ro-bustness of the truncation. We highlight that in generalthe right-hand-side of the flow equation generates termsbeyond the chosen truncation. In Eq. (34) the conse-quence is, that our truncation does not capture the fullmomentum-dependence that is generated. Accordingly,the fixed-point equation β V ( p k ) = 0 cannot be satisfiedfor all momenta, but instead only at selected points. Wewill extensively test how large the deviations of β V fromzero are in order to judge the quality of different trunca-tions. B. Projection schemes
We perform our analysis in several different projectionschemes, as a comparison between the fixed-point struc-ture of the different truncations provides indications foror against the robustness of the fixed point. We nowmotivate the use and explain the details of these threeprojection schemes.Using Eq. (34), the momentum-dependent fixed-pointvertex V ∗ ( p k ) could be found by demanding β V ( p k ) = 0and solving Eq. (34). In practice, we choose an ansatz V trunc ( p k ) for V ( p k ), which is part of choosing a trun-cation. At a point in theory space defined by V trunc ,Eq. (34) holds, but indicates that terms not yet capturedby V trunc are generated. These are present in Eq. (34), sothat we need to truncate the beta function β V → β trunc V in order to close the system. For example, in our setup,we restrict V trunc to a polynomial up to first order in p k ,i.e., V trunc ( p k ) = (cid:112) G ψ − σ mod p k , (35)and β trunc V ( p k ) = 12 (cid:112) G ψ β G ψ − β σ mod p k . (36)Here we introduced a modified version of the coupling σ , σ mod = (cid:112) G ψ σ, (37)where G ψ and σ are the dimensionless counterparts of¯ G ψ and ¯ σ , G ψ = k ¯ G ψ , σ = k ¯ σ. (38) However, this specific ansatz does not satisfy Eq. (34)for all values of p k . Accordingly, the right-hand side ofEq. (34) differs from Eq. (36). This is simply an examplefor the general fact that, plugging a truncation into theright-hand-side of the Wetterich equation, terms beyondthe truncation are generated and therefore the truncationis not closed.As β trunc V is not equal to β V for all momenta, we canchoose selected points in the interval p k ∈ [0 ,
1] for whichwe demand that β trunc V is exactly equal to β V at thesepoints, see, e.g., Eqs. (40) and (41). However, we canalso choose superpositions of more values for p k , see, e.g.,Eq. (42) for σ = 0. Even though this superposition mightlead to β trunc V being not exactly equal to β V at any point,it can still lead to an overall better description of the fullmomentum dependence, by being almost equal in a largerregion. The values of the coefficients in the ansatz, i.e., (cid:112) G ψ and σ mod , depend on this choice.Let us now compare two popular choices, namely thederivative expansion about p k = 0, and a projection atvarious values for p k . Working within a derivative ex-pansion about p k = 0, one extracts the flow of the n -thcoefficient of the polynomial by the n -th derivative ofEq. (34), evaluated at p k = 0. Specifically, for the cho-sen ansatz in Eq. (35) together with Eq. (36), this yields: β DE G ψ = 2 (cid:112) G ψ β V (0) , β DE σ mod = − β (cid:48) V (0) . (39)This expansion ensures that β V and its derivative areequal to β trunc V and its derivative at p k = 0. However,the derivative expansion to this order does not satisfythis equality away from p k = 0. This simply means thathigher-order terms in the derivative expansion around p k = 0 are generated by the flow. By the evaluation ata single point in p k , this scheme is very sensitive to localfluctuations at p k = 0, which might cause deviations forlarger momenta.Alternatively, we can choose finite momenta, e.g., p k =1, to extract one of the beta functions. Equating β V and β trunc V at p k = and p k = 1, and solving for the betafunctions yields β (0 , G ψ = 2 (cid:112) G ψ (cid:0) β V ( ) − β V (1) (cid:1) , (40) β (0 , σ mod = β V ( ) − β V (1) . (41)In this scheme, the beta functions β V and β trunc V by con-struction are equal at p k = and p k = 1. Thus, itprovides an interpolation between these momenta, whilethe derivative expansion provides an extrapolation from p k = 0 onwards. The same projection schemes can anal-ogously be applied to other n -point functions, includingthe anomalous dimensions. We will refer to the projec-tion at n different values for the momentum p k by n -sample-point projection in the following. More specifi-cally, starting from Eq. (40), the beta functions for G ψ and σ take the following form β G ψ = 2 (cid:112) G ψ (cid:2) C ( k ) V trunc ( ) − C ( k ) V trunc (1)+ 2Flow (3) ψ ( k ) − Flow (3) ψ ( k ) (cid:3) , (42) β σ = 2 σ + 12 (cid:112) G ψ (cid:32) V trunc (1) (cid:18) − C ( k ) + β G ψ G ψ (cid:19) − Flow (3) ψ ( k ) (cid:33) , (43)with C ( p ) = 1 + η h ( p ) + η ψ ( p ) . (44)In practice, the ingredients to evaluate the beta func-tions are the following: Flow (3) ψ is given by the sum ofdiagrams in Fig. which uses xAct [61–64] as well as theFORM-tracer [65], V trunc is given by Eq. (36) and theanomalous dimensions are extracted from a projection ofthe corresponding two-point functions at p k = 1, as in[12, 37, 66, 67].We now provide our motivations for using projectionswith n = 1 and n = 2 sampling points. The deriva-tive expansion at p k = 0 for the gravity-matter avatarsof the Newton coupling does not capture all propertiesof the flow in a quantitatively reliable way cf. the dis-cussion in [20]. In particular, a derivative expansion ofthe Einstein-Hilbert truncation at p = 0, together witha momentum-independent anomalous dimension for thegraviton results in a slightly screening property of grav-ity fluctuations on the Newton coupling. We expect thatat higher orders in the truncation, the derivative expan-sion becomes quantitatively reliable, but in our trunca-tion projections at finite momenta are preferable.Instead, an expansion at finite momenta is expectedto be more stable in small truncations. This is easiest toappreciate when thinking of the flow equation in terms ofa vertex expansion: The n -point functions that enter theflow depend on n − q ≈ k . Accordingly,the flow depends on the vertex at a finite momentum,not vanishing momentum, cf. Fig. .Accordingly, a good approximation of the full flowmight require higher orders in the derivative expansionaround p k = 0 than in projection schemes at finite mo-mentum. For technical simplicity, a symmetric pointwhere the magnitudes of all momenta at the vertex arechosen to be the same nonzero value is preferable, al-though the example in Fig. showcases that a non-symmetric point is likely to most accurately capture themomentum-dependence of the vertex as it is relevant forthe feedback into the flow equation.We point out that for this type of projections, a one-to-one mapping between the couplings extracted in thisway and the couplings of the action written in a deriva-tive expansion in terms of curvature invariants, as it is p
Phenomenologically, fermion-gravity systems with N f = 24 are of most interest, as this is the num-ber of Dirac fermions in the Standard Model, extendedby three right-handed neutrinos. There are indications[29, 35–37, 39, 40] that such a fermion-gravity systemwith N f > N f = 1 fermions.In this section, we aim at answering three key ques-tions:1. Is there a fixed point in the fermion-gravity systemthat is robust under extensions of the truncationand changes of the projection scheme?2. Is the nonminimal coupling nonzero at the fixedpoint, and how large is its “backreaction” onto theminimally coupled system? 3. Do the avatars of the Newton coupling exhibit ef-fective universality at this fixed point? A. A fermion-gravity fixed point and tests of itsrobustness
The n =1 truncation, i.e., σ = 0, exhibits an interactingfixed point providing further evidence for the existence ofthe Reuter fixed point in gravity-fermion systems. Thefixed-point results in the first two lines of Tab. are well-compatible with those of [21], calculated for a differentvalue of the gauge-parameter β , for a similar system. TheReuter fixed point is characterized by three relevant andone irrelevant direction in this truncation, cf. Tab. . Dueto the large overlap of the corresponding eigendirectionwith λ , the negative critical exponent can be associatedwith this direction in theory space. Thus, while λ re-mains relevant, λ is shifted into irrelevance at the UVfixed point. The most relevant direction, correspondingto the critical exponent θ in Tab. has largest over-lap with G ψ , while the remaining two directions form acomplex pair. We highlight that only a subset of thesecritical exponents is physical, namely those in the diffeo-morphism invariant theory space. Here, we are explor-ing a larger theory space that accounts for symmetry-breaking through the regulator and gauge-fixing terms,and therefore includes different avatars of the Newtoncoupling, as well as the two couplings λ , λ . Of the fourcritical exponents, only two are therefore physical. Theobservation that θ is essentially associated with G ψ canbe interpreted as a hint that diffeomorphism-symmetryrestoration in the flow towards the physical point k = 0 is Trunc σ ∗ G ∗ ψ G ∗ h λ ∗ λ ∗ θ θ / θ θ η h (0) η h ( k ) η ψ (0) η ψ ( k ) A G h A f ,G h A G ψ A f ,G ψ n =1, p = k − .
79 0 .
78 0 .
26 0 .
080 2 . . ± . − . − .
87 0 . − . − . − . . − . . n =2, p = k , k − .
90 0 .
75 0 .
26 0 .
083 3 . . ± . − . − .
86 0 . − . − . − . . − . . n =2, p = k , k − .
063 0 .
93 0 .
75 0 .
26 0 .
081 3 . . ± . − . − . .
83 0 . − . − . − . . − . − . Table 3 . Fixed-point values as well as coefficients of the beta functions in the three schemes. The classification of differenttruncations refers to the order in the n -sample point projection. Here the θ i are the critical exponents, i.e., the eigenvalues ofthe stability matrix multiplied by an additional negative sign. For the definition of the A ’s see Eq. (46). possible, as a relevant coupling can be chosen arbitrarilyin the IR.The robustness of these results can be tested by chang-ing the projection scheme for G ψ from n =1 to n =2 for thenumber of sample-points, while staying within the sametruncation. As shown in the second line of Tab. thisleads to a change of about 14% for the fixed-point valueof G ψ , but more importantly a change of about 21% atthe level of the universal critical exponent θ . This in-dicates the necessity of extensions of the truncation. Westress that the change of the other couplings and criticalexponents stays small, as the feedback of G ψ into theother beta functions is with a very small prefactor. Therelative contribution of G ψ versus that of G h in β G h isfor instance approximately given by A f ,G h /A G h . Here, A G i is the N f -independent and A f ,G i the prefactor forthe N f -dependent quadratic coefficient of β G i once bothavatars are equated, β G i (cid:12)(cid:12) G h = G ψ = G = 2 G + G ( A G i + N f A f ,G i )+ O ( G ) . (46)Accordingly, a change of the projection scheme for G ψ ,accompanied by an unchanged projection for the othercouplings, is not expected to result in significant changesin any values except for G ∗ ψ and θ . Hence, the smallrelative changes in G ∗ h , λ ∗ , λ ∗ and θ , , should not betaken as an indication of quantitative convergence in thesystem.A further aspect providing information on the qual-ity of our truncation is the momentum dependence ofthe vertex flow. In Sec. III, we derived a formal ex-pression Eq. (34) for the beta function of the fully mo-mentum dependent vertex function V (cid:0) p k (cid:1) . Evaluatedat the full fixed point V ∗ in untruncated theory space,the right-hand side of Eq. (34) would vanish for all mo-menta – assuming that such a fixed point indeed exists.In truncations this is not the case. Within a given trun-cation scheme, we can use the truncated beta functions,cf. Eqs. (36), to find a fixed point V ∗ trunc of the trun-cated RG flow projected onto the subspace defined bythe truncation. However, this truncated fixed point will With universality at the level of critical exponents we refer tothe insensitivity of critical exponents to unphysical details likegauge, regulator and scheme. We refer to universality in thelarger theory space, accordingly a universal critical exponent isnot automatically associated to physics. not satisfy the fixed-point equation for all momenta, i.e., β V (cid:0) p k (cid:1) | V = V ∗ trunc (cid:54) = 0. This is a consequence of the factthat the RG flow in truncations is not closed. Terms be-yond the truncation are generated. Accordingly, the RGflow features additional, higher-order momentum depen-dence than what is captured by the truncation. Thus weintroduce the quantity V , which allows us to estimate thedeviation of the truncated fixed point V ∗ trunc from the fullfixed point V ∗ , V (cid:0) p k (cid:1) = β V (cid:0) p k (cid:1) | V = V ∗ trunc + V ∗ trunc (cid:0) p k (cid:1) . (47)Specifically, the idea is simply that if we can satisfy β V (cid:0) p k (cid:1) | V = V ∗ trunc ≈ V is an auxiliary quantity and has no directphysical meaning. For a fixed point in untruncated the-ory space, the function β V (cid:0) p k (cid:1) | V = V ∗ vanishes, such that V = V ∗ . This is a general self-consistency equation forthe fixed point of the system at any value of p . Our V is similar to what has been investigated in [12, 20]. To further explain the meaning of Eq. (47), we now spe-cialize to our truncation where V trunc (cid:0) p k (cid:1) is a polynomialin p k only up to first order, cf. Eq. (35), while Flow (3) ψ clearly contains higher powers. Therefore, V can at bestbe approximately equal to V ∗ trunc . Thus, the difference ofour choice for V ∗ trunc (cid:0) p k (cid:1) given in Eq. (35) from V (cid:0) p k (cid:1) shows how well V ∗ trunc (cid:0) p k (cid:1) approximates the momentumdependence of the full fixed point V ∗ (cid:0) p k (cid:1) .In summary, Eq. (47) can be read as follows: V (cid:0) p k (cid:1) captures the momentum-dependence as generated by theflow of the fermion-gravity vertex, i.e., by the diagrams inFig. . In untruncated theory space, the full momentumdependence would be captured by V ∗ (cid:0) p k (cid:1) , leading to β V (cid:0) p k (cid:1) | V = V ∗ = 0. In truncations, the ansatz V ∗ trunc (cid:0) p k (cid:1) differs from the momentum dependence V (cid:0) p k (cid:1) generatedby the flow, such that β V (cid:0) p k (cid:1) | V = V ∗ trunc vanishes only at There, the auxiliary quantity was defined as V Lit (cid:0) p /k (cid:1) = − β V ( p /k )2+ η h ( p )+2 η ψ ( p ) (cid:12)(cid:12) V = V ∗ trunc + V ∗ trunc ( p /k ), which only differsfrom V by a normalization in front of β V . We chose the defi-nition (47) to avoid artificial poles that might arise due to thedenominator in V Lit . . . . . . . p / k . . . . . . q p k q G ∗ ψ q p k V ( p k ) (cid:12)(cid:12) n =1 Figure 3 . Momentum dependence of the graviton-fermionthree-point vertex, evaluated with n = 1 sample-points, atthe corresponding fixed point. The dashed blue line corre-sponds to the truncated vertex V ∗ trunc , which is equal to (cid:112) G ∗ ψ in the n = 1 scheme. Note that V trunc always enters the flowequation with an additional factor pk . Thus we plot pk V ∗ trunc .The solid red line corresponds to the full momentum depen-dence of the auxiliary quantity pk V = pk ( β V | V = V ∗ trunc + V ∗ trunc ),cf. Eq. (47) that is generated from within our truncation, butgoes beyond the momentum-dependence of the term in ourtruncation. The difference between the two lines indicatesthe need for an extension of the truncation, as both wouldagree if evaluated at an untruncated fixed point V ∗ due tothe vanishing of β V | V = V ∗ . selected values of p k . Accordingly, the comparison of V (cid:0) p k (cid:1) and V ∗ trunc (cid:0) p k (cid:1) is well suited to check whetherhigher-order momentum dependences beyond the trun-cation are generated by the flow equation. Moreover,the magnitude of higher-order momentum dependencescan be estimated. Finally, the flow can of course show adifferent momentum dependence at large and small mo-menta; e.g., being well-approximated by a simple low-order polynomial at p k ≈
1, and exhibiting a more in-tricate momentum-dependence for p k ≈
0. Comparing V (cid:0) p k (cid:1) and V ∗ trunc (cid:0) p k (cid:1) at all values of p k provides informa-tion on such cases. This provides another guiding prin-ciple on how to extend the truncation: The momentum-dependence of the flow, evaluated at the fixed-point val-ues in the n =1 and n =2-sample-point projection with σ = 0 both indicate the presence of a p k term in the flowof the graviton-fermion vertex, cf. Fig. and Fig. .There, we plot the comparison of both sides of Eq. (47),multiplied by pk . This is motivated by the classical struc-ture of the graviton-fermion vertex, which takes the form pk V trunc (cid:0) p k (cid:1) . The fact that Eq. (35) does not carry thisfactor of pk is due to the specific choice of the projectorEq. (26). Therefore, Fig. and Fig. show the ver-tex V ∗ trunc (cid:0) p k (cid:1) weighted by how it contributes in the di-agrams. The presence of a p contribution in Fig. andFig. motivates our extension of the truncation by σ that we explore below. This extension allows us to feed . . . . . . p / k . . . . . . q p k q G ∗ ψ q p k V ∗ trunc (cid:0) p k (cid:1) q p k V (cid:0) p k (cid:1)(cid:12)(cid:12) n =2 σ =0 q p k V (cid:0) p k (cid:1)(cid:12)(cid:12) n =2 σ =0 Figure 4 . Momentum dependence of the graviton-fermionthree-point vertex evaluated at n =2 sample-points, at the cor-responding fixed point. The dashed blue and the dot-dashedorange lines correspond to the ansatz for the vertex pk V ∗ trunc (35), setting σ = 0 for the blue dashed line and including σ ∗ for the orange dot-dashed one. The solid red and dotted greenlines corresponds to the full momentum dependence given by pk V = pk (cid:0) β V | V = V ∗ trunc + V ∗ trunc (cid:1) in Eq. (47). the higher momentum dependence, which is already seenin the projections with σ = 0, back into the diagramsand therefore consider this information. B. Generation and backreaction of σ Motivated by the observation that higher-momentumdependence is clearly generated by the flow equation, wemake the next step in extensions of the truncation follow-ing canonical power-counting. In the present truncation,the next-to-leading-order coupling that contributes tothe graviton-fermion vertex is the nonminimal coupling σ . The interaction itself is dimension-6, accordingly thecanonical dimension of the coupling is −
2. Symmetry-arguments elaborated on in [27] indicate that it shouldnot be possible to realize σ ∗ = 0, once G ∗ ψ (cid:54) = 0. Here wewill check whether this is indeed the case in the corre-sponding truncation and will further explore the backre-action of σ on the system of couplings in the smaller trun-cation. For the latter, we also focus on the momentum-dependence of the flow, to find whether it is capturedmore adequately once the next-to-leading term beyond G ψ is included.At the fixed point for the nonminimally coupledfermion-gravity system in the n =2 projection, the flow ofthe nonminimal coupling σ is given by: β σ (cid:12)(cid:12)(cid:12)(cid:12) λ ∗ ,λ ∗ ,G ∗ h ,G ∗ ψ = 0 .
13 + 2 . σ − . σ + 1 . σ + O ( σ ) . (48)The existence of the σ -independent term confirms thatthe nonminimal coupling is indeed induced at the UV1fixed-point for gravity. The same conclusion can bedrawn in the derivative expansion, as indicated by vari-ous σ -independent terms in Eq. (B3) and Eq. (B5). Asa consequence of such terms, σ ∗ = 0 is not a solution ofthe flow equation and the nonminimal coupling cannotbe consistently set to zero. The fixed-point value of σ is indeed finite, cf. Tab. . Further, the inclusion of σ gives rise to an additional irrelevant direction, cf. Tab. .Due to the negative canonical mass dimension, this isexpected. It provides yet another indication that thecritical hypersurface of the Reuter fixed point is finitedimensional. The shift between the canonical dimension d ¯ σ = − | θ − d ¯ σ | (cid:28) σ on the minimally coupled system. This provides an indi-cation about the state of convergence of the truncation,at least in this particular direction in theory space andthereby it provides guidance about the setup of futuretruncations. In [27], it was shown that in the part of thegravitational coupling-space where fixed-point values lieat small numbers of matter fields, the “backreaction” ofinduced matter self-interactions onto the remaining sys-tem is subleading compared to the direct gravity contri-bution. Here, we make a similar observation for the non-minimal matter-gravity coupling. Due to the small fixed-point value for the nonminimal coupling σ ∗ = − . σ on the minimally coupled system is small.In fact, the smallness of σ ∗ is crucial in view of its neg-ative sign (see also Fig. ): For a sufficiently negativefixed-point value, σ can alter the effect of fermionic fluc-tuations on the Newton couplings from screening to an-tiscreening, see also Eq. (B2) and Eq. (B1) in App. B.Indeed for N f = 1, the fixed-point value for σ leads toa slightly negative A f ,G ψ . Yet one should keep in mindthat within the systematic error of A f ,G ψ , estimated, e.g.,by the difference of A f ,G ψ in the two projection schemes,cf. first two lines in Tab. , A f ,G ψ is compatible withbeing positive also in the presence of σ . For N f > . A f ,G ψ >
0. Takentogether, we view this as tentative evidence to considerthe sign of A f ,G ψ at N f = 1 as an artifact of unphysicalchoices, such as gauge, regulator and projection scheme.Due to the smallness of σ ∗ , the approximation of themomentum dependence does not differ significantly froma straight line, cf. Fig. , as the p contribution intro-duced by σ is small at the fixed point. However, theinclusion of σ nevertheless leads to a better approxima-tion of the momentum structure at p > . k , comparedto the n =2 projection where σ was neglected, cf. Fig. .To provide a more intuitive comparison of how well . . . . . . p / k . . . β V (cid:0) p k (cid:1)(cid:12)(cid:12) V = V ∗ trunc n = 1 n = 2 , σ = 0 n = 2 , σ = 0 Figure 5 . As a key result of this section, we compare alldifferent approximations of the momentum dependent flow.Here β V (cid:12)(cid:12) V = V ∗ trunc refers to the evaluation of the right-handside of Eq. (34) for a given truncation at the correspondingfixed point V ∗ trunc . The deviation of this expression from zeroencodes the accuracy of the projection. We highlight that the n = 1 scheme and n = 2 scheme including σ capture the fullmomentum-dependence at p = k by construction. Addi-tionally, the n =2 scheme including σ leads to a lower value of β V | V = V ∗ trunc , i.e., a better approximation of the full flow, alsoat lower values of the momentum. different truncations capture the full momentum depen-dence of the vertex flow, we directly evaluate β V givenby Eq. (34) for the different truncations. Evaluated atthe fixed point, this expression should vanish for all val-ues of p . Thus, the deviation of the right-hand side,which we denote by β V (cid:12)(cid:12) V = V ∗ trunc , encodes the deviationof a given truncation from the full momentum struc-ture. It describes how accurately the truncation ap-proximates the fixed-point Eq. (34) for all values of p .One can interpret β V (cid:12)(cid:12) V = V ∗ trunc simply as a test, whetherthe momentum-dependence of the right-hand-side of theWetterich equation is described accurately by a constantand a p term with k -dependent but p -independent cou-plings. If our truncation was exact, such that upon in-put of G ψ and σ no additional terms were generated onthe right-hand-side, then β V (cid:12)(cid:12) V = V trunc would vanish for allmomenta once the fixed-point values for the couplings areinserted. Compared to our previous tests shown in Fig. and Fig. , this allows for a more direct comparison ofdifferent truncations. As a drawback, the informationon which powers of p k are induced in each truncation issomewhat less obvious.Fig. shows this quantity evaluated at the correspond-ing fixed point in all different truncations. We cautionthat the three curves in Fig. have a systematic error due Note that β V (cid:12)(cid:12) V = V ∗ trunc denotes the full β V (34) evaluated at thetruncated fixed point V ∗ trunc , as opposed to β trunc V (36), whichonly contains the beta functions calculated in our truncation. . . . . . . q / k . . . . E (cid:0) q k (cid:1) n = 1 n = 2 , σ = 0 n = 2 , σ = 0 Figure 6 . Integrated deviation in the momentum depen-dence in different truncations as defined in Eq. (49). We in-tegrate from p = k down to p = q . Here E refers to the eval-uation of Eq. (49) for a given truncation at the fixed pointin the particular truncation. The deviation of this expressionfrom zero encodes the accuracy of the projection. to our truncation. Therefore not all differences betweenthe curves are significant.Two aspects of the curves are of particular interestto us. Firstly, momenta p ≈ k are expected to pro-vide the main contribution to the flow. Accordingly, itis desirable for an approximation to capture the momen-tum dependence at p ≈ k accurately. This happens if β V (cid:12)(cid:12) V = V ∗ trunc ≈ p = 0) contribute to the flow, even ifmomenta with p (cid:28) k contribute less. Accordinglyit is desirable to minimize the integrated deviation of β V (cid:12)(cid:12) V = V ∗ trunc from zero. Fig. serves as a summary ofour results for the momentum-dependence of the system:While the momentum-dependence of the flow is well-captured by the n =1 scheme and the n =2 scheme includ-ing σ , the latter highlights the necessity for further ex-tensions of the truncation, i.e., inclusion of higher-orderoperators. Further, the n = 2 scheme including σ cap-tures the momentum-dependence slightly better at lowvalues of p . Nevertheless, the performance of the n =1scheme and the n =2 scheme including σ are comparableat N f = 1.To provide a quantitative characterization, we define E (cid:18) q k (cid:19) = (cid:90) q k d y β V ( y ) | V = V ∗ trunc . (49)For a truncation that captures the full momentum depen-dence, E = 0. As we are mostly interested in capturingthe momentum dependence around p ≈ k correctly, weintegrate β V from a lower boundary q k up to 1, cf. Fig. .We find that the n = 2 projection with σ = 0 performsworst. The n =1 projection matches the flow most closelydown to q k ≈ .
5. In this region, the approximationwith n =2 sample points only yields a small integrateddeviation of order 0 .
04. The total integrated deviation is smallest for the n =2 projection with the inclusion of σ , indicating that it captures the full momentum depen-dence best.As a tentative conclusion of our analysis, we emphasizethat the inclusion of σ leads to a setting which capturesthe momentum-dependence of the full flow with a rea-sonable accuracy. Nevertheless, both Fig. and Fig. suggest that full accuracy requires further extensions ofthe truncation to include higher-order momentum depen-dencies. C. Effective universality
In a theory space where a gauge symmetry is broken,different avatars of this gauge coupling no longer agree. Ifthe symmetry-breaking is a consequence of gauge fixingand the regulator, then modified Slavnov-Taylor iden-tities select a hypersurface in this theory space. Thishypersurface does not coincide with the symmetric the-ory space. It reduces to that of the standard Slavnov-Taylor identities at k = 0, where the regulator vanishes.The initial condition for the flow at k → ∞ should con-tain symmetry-breaking terms such that the symmetry-breaking introduced by the regulator during the flow canbe compensated for by the initial condition, see the peda-gogical introduction in [53]. Accordingly, the fixed pointmight exhibit a difference between distinct avatars of thegauge coupling. Additionally, there is a modified shiftWard-identity, see, e.g., [59] as well as [52, 70, 71] forgauge theories, encoding the difference between correla-tion functions of the background field and the fluctua-tion field, which has been explored, e.g., in [20, 72–75].Effective universality was defined in [20, 21] as a near-agreement of different avatars of the Newton coupling. Inthe case of Yang-Mills theories in four dimensions witha marginal gauge coupling, effective universality holdsin the perturbative regime as a consequence of two-loopuniversality: Up to two loops, the avatars of the gaugecoupling agree exactly. Within perturbation theory, thehigher-loop terms are subleading. Accordingly, effectiveuniversality follows due to the marginal nature of thegauge coupling. In quantum gravity, the case is moresubtle. As the deviation from universality is a conse-quence of quantum effects (the different avatars agreetrivially in a classical theory), the perturbative regime,where quantum effects are small, is expected to exhibiteffective universality.Effective universality is not a feature that a viableasymptotically safe fixed point necessarily must exhibit.Instead, it can be viewed as a criterion that distin-guishes nonperturbative from near-perturbative fixedpoints. Other hints at a near-perturbative nature of theReuter fixed point consist in the near-canonical scalingspectrum of higher-order curvature operators [14, 22–24]which is also exhibited by σ in our system, and thepossibility to uncover the fixed point within one-loopperturbation theory [76, 77]. Further, the Reuter fixed3point might be connected continuously to the perturba-tive asymptotically safe fixed point seen within the ep-silon expansion around d = 2 dimensions [1, 78]. Indica-tions for this have been found, e.g., in [6, 79–81], however,see also [5, 82].Moreover, such a feature appears desirable from a phe-nomenological point of view: According to the findings in[34, 83, 84], a near-perturbative gravitational fixed pointcould induce an asymptotically safe UV completion ofthe Standard Model, which matches onto the perturba-tive RG flow of the Standard Model from the Planck scaleto the IR.To investigate whether the effective universality is re-alized in this system we expand the beta functions forboth avatars of the Newton coupling at the correspondingfixed-point values for λ and λ , cf. Eq. (46). The mecha-nism behind asymptotic safety in gravity is a cancellationof canonical scaling with quantum scaling. The quantumterm must have an antiscreening nature to generate aviable fixed point at G ∗ >
0. For a fixed point whichcan be traced back to the free fixed point as d →
2, thequantum effects are captured qualitatively by the lead-ing term in the expansion in G . Therefore we considerthe two quadratic coefficients A G i and A f ,G i as encod-ing key physics of asymptotic safety. The comparison ofthese coefficients for different avatars of the Newton cou-pling also allows to deduce whether effective universalityis realized.We discover a quantitative agreement of the twoavatars in the n = 1 projection. This holds for the N f -independent part, with A G ψ /A G h ≈ .
96 as well for the N f -dependent part with A f ,G ψ /A f ,G h ≈ .
73. The closeagreement of these coefficients is reflected in the goodagreement of G ∗ h and G ∗ ψ .A measure for the deviation from effective universalityhas been introduced in [20]: ε ( G, µ, λ ) = (cid:12)(cid:12)(cid:12)(cid:12) ∆ β G h − ∆ β G ψ ∆ β G h + ∆ β G ψ (cid:12)(cid:12)(cid:12)(cid:12) G h = G ψ = G , (50)where ∆ β G i = β G i − G i is the anomalous part of β G i .Expressed in this measure, ε ≈ .
01 at the interactingfixed point of the n =1 truncation. This value indicatesan almost exact agreement of the beta functions of bothavatars of the Newton coupling. The systematic errorof our truncation has been estimated in [21] to resultin an uncertainty δε ≈ .
2. Within this estimate for thesystematic error, the fixed point is compatible with exactuniversality according to the measure in Eq. (50). Thisindicates the near-perturbative nature of the interactingfixed point in this truncation, as pointed out in [21].Evaluating this measure of effective universality in the n =2 projection without the inclusion of σ yields a valueof ε ≈ .
2. While the value is larger than that in the n =1 projection, it is still compatible with exact univer-sality within the estimate for the systematic uncertaintyof ε . The discovery of effective universality in this systemaccordingly appears to be quite robust. At a first glance, the presence of σ at the fixed-pointcould look like a potential source of deviation from effec-tive universality. After all, σ couples differently into β G h than it does into β G ψ . Yet, we caution that this conclu-sion might be premature: If the Reuter fixed point fea-tures effective universality, this is a consequence of manystructurally different contributions in the beta functions.At the fixed point, nontrivial cancellations occur whichresult in effective universality. In fact, this is the case inthe truncation explored in [20, 21]: Although λ and λ couple differently into β G h and β G ψ , effective universal-ity is realized. Hence, a priori one cannot infer whetheror not the inclusion of σ will spoil effective universality,as a dynamical adjustment of fixed-point values can leadto effective universality at one point in theory space eventhough the impact of σ on β G h and β G ψ is structurallydifferent. Therefore we now explicitly check this ques-tion. Including σ , we find ε ≈ .
26. Within our estimatefor the systematic uncertainty of ε , this appears to becompatible with effective universality defined as a near-agreement of the fixed-point values of the avatars, eventhough it appears to be just incompatible with an exactagreement.In summary, both truncations ( σ = 0 and σ (cid:54) = 0),with the n =1 and n =2 projection schemes that we haveexplored indicate the possibility of effective universalityat the Reuter fixed point, hinting at a potentially near-perturbative nature of the fixed point. V. STRUCTURAL ASPECTS OF THEWEAK-GRAVITY BOUND
In this section, we broaden our view away from thefixed point in the above truncation, and instead analyze β σ with all other couplings treated as external parame-ters. Varying these allows us to explore the behavior ofthe system away from the results in one specific trunca-tion. For this section, we assume effective universality,i.e., G ψ = G h = G , motivated by our results in the pre-vious section. Further, we stress that our analysis for thecritical exponent from β σ can be translated to the largersystem in which all couplings are dynamical, if the stabil-ity matrix is approximately block diagonal. Our resultsin the previous section highlight that this is the case, atleast at the fixed point explored there.In [27, 28, 42], the weak-gravity bound for asymptoticsafety was introduced. It is based on the observationthat strong metric fluctuations can lead to the loss ofa predictive fixed point in matter interactions. Specifi-cally, these are such couplings that cannot be set to zeroin the presence of asymptotically safe gravity. Due tothe symmetry-structures in the matter sector, these cou-plings are all canonically irrelevant. For a subset of those,analyzed in truncated flows, strong metric fluctuationslead to a loss of the shifted Gaussian fixed point (sGFP)at a fixed-point collision. The maximum strength of grav-itational fluctuations that is still compatible with a real4shifted Gaussian fixed point leads to a bound on the grav-itational fixed-point values, the weak-gravity bound. Westress that even in the region beyond the weak-gravitybound, the beta functions might allow for other zeros toexist. These additional zeros need not correspond to ac-tual fixed points and could be truncation artifacts. Moreimportantly, these zeroes are typically associated with acritical exponent that deviates rather significantly froma canonical power-counting, invalidating the truncationscheme that is typically used. In particular, the cou-plings in question are all canonically irrelevant in d = 4,but might be relevant at the additional zero of the betafunction. This would imply the existence of an additionalfree parameter, corresponding to a reduced predictivity.In particular, when it comes to matter couplings, thereare no experimental indications that such free parame-ters beyond the couplings of the Standard Model exist innature. Therefore, the region beyond the weak-gravitybound, where the sGFP ceases to exist, is not strictlyexcluded as a viable region for the Reuter fixed point.Yet, the “weak-gravity” region appears preferable bothfrom a phenomenological point of view as well as regard-ing the aspect of controlling the truncation.In [27], a comprehensive analysis of the conditions un-der which a weak-gravity bound exists for quartic mattercouplings was put forward. The corresponding beta func-tions are quadratic in the matter coupling. Here, we willanalyze the case of beta functions that are cubic in thecoupling. The beta function for σ falls into this category.In the previous sections, we have found it convenientto choose a parametrization of the action where G ap-pears in those terms arising from the Einstein-term, inthe minimally coupled interactions and in the nonmin-imal vertex. An alternative parametrization, where G does not appear in the nonminimal vertex, is related bya transformation of the basis in theory space. Specif-ically, the difference between the two parametrizationslies in the redefinition (37). To explore the weak-gravitybound, the modified parametrization is more suitable.This parametrization allows us to test the response of onesector, the nonminimal one, to the strengthening of met-ric fluctuations. In this parametrization, the strengthen-ing can most conveniently be encoded in an increase of G . The same cannot be done in our original parametriza-tion, as an increasing of G simultaneously “dials” thestrength of the nonminimal interaction term. In this al-ternative parametrization, the flow equations for G ψ and σ mod in the n =2 scheme take the form given in Eqs. (40)and (41). In this parametrization, β σ mod is, up to cor-rections from the loop contributions to the anomalousdimensions, cubic in σ mod . In the perturbative approxi-mation, i.e. neglecting the anomalous dimensions comingfrom the scale-derivative of the regulator it becomes ex-actly cubic. This motivates a more general analysis ofthe existence of the shifted Gaussian fixed point in cubicbeta functions.Consider a generic beta function cubic in the coupling − − σ mod − − β σ m o d G = 0 G = 2 G = 4 Figure 7 . We show β σ mod after rescaling σ mod , such that G does not appear in the nonminimal vertex, at λ = 0 . λ = 0 with G = 0 (red, solid), G = 2 (blue, dashed) and G = 4 (green, dotted). γ , i.e., β γ = a + b γ + c γ + d γ , (51)where a, b, c and d are coefficients that are functions ofother couplings of the system. b in general also con-tains a contribution from the canonical dimension of γ .The shifted Gaussian fixed point (sGFP) is defined asthe fixed point that is a continuous deformation of thefree fixed point for a (cid:54) = 0. In the case of gravitationalsystems, it is the effective strength of metric fluctuationsthat leads to this deformation. As the effective strengthof metric fluctuations increases, the critical exponent as-sociated to γ changes. As θ = 0 is associated to a doublezero of a beta function, a change of sign of the criticalexponent of the sGFP is necessarily tied to a fixed-pointcollision. Such collisions can (but need not, see, e.g.,[85, 86] for exceptions) lead to a loss of real fixed points.Up until a possible collision, a canonically irrelevant cou-pling is therefore irrelevant at the sGFP. For d <
0, thebeta function can either feature one or three zeros. Forthe former case, the zero comes with negative slope, i.e.,it corresponds to a fixed point at which the coupling isrelevant. Therefore, in the case of d <
0, the sGFPonly exists if the beta function has three real zeros. Thiscondition is equivalent to demanding that the local min-imum of β γ is negative and the local maximum positive.Expressed in terms of the coefficients, this leads to c ≥ b d, β γ ( γ max ) > , and β γ ( γ min ) < . (52)Given any cubic beta function, this set of conditions canbe checked. Regions in the gravitational parameter spacewhere these conditions are violated do not allow for thesGFP to exist.The case of d > β γ . In the latter case, thecoupling γ is irrelevant at two of the zeros. Hence it is not5 G − . . . . λ − . − . − . − . − . − . − . σ ∗ m o d Figure 8 . Value of the sGFP for the nonminimal coupling σ mod at λ = 0, as a function of the gravitational parameters G and λ . The existence of a real sGFP in the whole plot-range implies the absence of a weak-gravity bound in the sameregion. The red line indicates where one of the anomalousdimensions reaches η = 2. clear from the values of the parameters, which of them isthe sGFP. To establish this requires tracking the sGFP allthe way into the GFP as a →
0. Moreover, as the valuesof the parameters in the beta function are changed, thecase with one zero can turn into the case with three zeros.Overall we conclude that for d > a = 0 to determine whether it existsin a given range of parameter space. Interestingly, in theasymptotic-safety literature, [11] constitutes an exampleof the d > σ mod , withbeta function shown in Fig. . In the case G = 0, the betafunction features a Gaussian fixed point, as expected. Inaccordance with symmetry arguments, this fixed pointis shifted away from zero for G (cid:54) = 0, giving rise to ansGFP. At the sGFP, σ mod is shifted further into irrele-vance. This suggests indeed that canonical power count-ing is a suitable guiding principle to determine whichoperators are relevant or irrelevant at the Reuter fixedpoint.Following our analysis above, exploring whether thesGFP exists everywhere in the gravitational parameterspace is best done by explicitly computing the value ofthe sGFP as a function of the other couplings of the sys-tem. Fig. shows the value of the sGFP as a function ofthe gravitational parameters G and λ at a fixed valueof λ . There is no indication for a weak-gravity boundin the shown region. Fig. only covers a bounded re-gion in theory space. Beyond the red line in Fig. , i.e.,for larger values of G and λ , at least one of the anoma-lous dimensions violates η ≤
2, such that the truncationis not reliable any more. Even though the anomalousdimensions must increase even further in order to flipthe sign of diagrams contributing to the flow, η = 2 is G − . . . . λ θ σ mod < − − . − . − . − . − . − . − . − . θ σ m o d Figure 9 . Values of the critical exponent θ σ mod at the sGFPas a function of the gravitational parameters G and λ . the point where the regulator bound discussed in [37] isviolated. For more negative values of λ , the effectivestrength of gravity fluctuations decreases, cf. the discus-sion in Sec. VI. As the weak-gravity bound is expectedto be induced by strong gravity fluctuations, we do notexpect the sGFP to vanish into the complex plane formore negative values of λ . Fig. shows the values ofthe critical exponent θ σ mod at the sGFP as a function ofthe gravitational parameters. We observe that the crit-ical exponent at the sGFP is almost everywhere shiftedfurther into irrelevance starting from the canonical scal-ing θ σ mod = − G = λ = 0. This signals that thesystem is actually driven away from a fixed-point colli-sion. We conclude that in the region where our trunca-tion is expected to provide robust results, there are noindications for a weak-gravity bound. Accordingly, theinclusion of the nonminimal interaction σ mod does notlead to new constraints on the microscopic gravitationalparameter space. Thus, the asymptotic-safety scenariopasses a nontrivial test: The presence of a nonzero in-teraction, which has been neglected in previous studies,is innocuous in that its inclusion does not impose newconstraints and only results in subleading corrections tofixed-point values from previous studies. VI. ASYMPTOTIC SAFETY FOR MOREFLAVORS
A key question on the asymptotic-safety scenario iswhether it is compatible with arbitrary matter modelsor whether it places restrictions on the matter sector. Ifthis was not the case, there would be a huge “landscape”for asymptotic safety. This would clearly make it muchmore challenging to confront the asymptotic-safety sce-nario with data, as experimental data are only availableat energies where the fixed-point scaling itself is not yetdetectable. If asymptotically safe gravity is only com-patible with a very small set of matter models, then onecan hope for it to be ruled out by particle-physics data.6 N f . . . . . . λ ∗ G ∗ h λ ∗ G ∗ ψ N f − θ θ Re ( θ , ) N f − − η h TT (0) η h TT ( k ) η ψ (0) η ψ ( k ) N f . . . . . λ ∗ G ∗ h λ ∗ G ∗ ψ N f − θ θ Re ( θ , ) N f − − η h TT (0) η h TT ( k ) η ψ (0) η ψ ( k ) Figure 10 . Upper panels: Fixed-point structure as a function of N f in n =1 projection. Lower panels: Fixed-point structurefor the system in n =2 projection with σ = 0 as function of the number of fermions N f . N f . . . G ∗ h λ ∗ G ∗ ψ λ ∗ σ ? N f − θ Re ( θ , ) θ θ N f − − η h TT (0) η h TT ( k ) η ψ (0) η ψ ( k ) Figure 11 . Fixed-point structure for the system with n =2 projection with σ (cid:54) = 0 evaluated Γ ( h ¯ ψψ ) as function of the numberof fermions N f . To establish which models lie in the asymptotically saferegime and which lie in the “swampland”, the interac-tion structure of models also has to be taken into ac-count. Here, we restrict ourselves to the first step in thelandscape/swampland classification, and ask whether abound exists on the number of fermion flavors compati- ble with a fixed point in our truncation. Note that therealization of scale invariance in the UV relies on a del-icate balance of competing effects of quantum fluctua-tions of different fields. Accordingly one might expectthe asymptotically safe region to be a tiny “island”.Further, the gravitational fixed-point values are a key7 N f G ∗ eff , G ∗ eff , G ∗ eff , N f . . . . . . G ∗ eff , G ∗ eff , G ∗ eff , N f G ∗ eff , G ∗ eff , G ∗ eff , Figure 12 . Value of the effective gravitational coupling G ∗ eff , n for different values of n . Left panel: n =1 projection, centralpanel: n =2 with σ = 0 and right panel: n =2 with σ (cid:54) = 0. input in determining whether asymptotically safe grav-ity could provide accurate “retrodictions” of StandardModel couplings [34, 83, 84]. These fixed-point valuesdepend on the number of matter fields, and accordinglyan accurate determination at N f = 22 . N f = 24(SM+ ν ’s) is required. Our study is also a step towardsreducing the systematic error of the gravitational fixed-point values.There are compelling indications for a screening effectof fermions on the running Newton coupling, as it is seenin perturbative studies [87, 88], background studies [29,35, 36] and fluctuation field studies [37], including thepresent one for N f > .
5, cf. Eq. (B2) and Eq. (B1) inApp. B and Tab. , where A f ,G ψ > N f > . G N f -term in β G ψ and β G h is significantlysmaller than for the background system, where it is A f ,G B /A G B ≈ .
04 for Dirac fermions [36], with the A ’sdefined in analogy to Eq. (46). In contrast, A f ,G ψ /A G ψ ≈ . · − . Accordingly, the Reuter fixed point in the fluc-tuation system changes much slower as N f is increased.Up to values of N f in the range N f ≈ ...
30, only slightchanges are observed in the system. To understand theobserved slight changes, the effect of fermions on λ and λ needs to be taken into account. This is the point wherebackground calculations and fluctuation calculations dif-fer: As a function of increasing N f , the fixed-point valuefor the background cosmological constant becomes in-creasingly negative, see, e.g., [36]. At the same time, thefluctuation quantities λ ∗ , λ ∗ both become more positive.Our results are shown in Fig. for σ = 0 and Fig. .We observe a growth of λ ∗ and λ ∗ . As λ grows, itenhances the effective strength of metric fluctuations inthe system of beta functions at hand. The contributionof gravitons comes with 1 / (1 − λ ) , where , λ ∗ strength-ens the metric contribution to β G ψ/h sufficiently to over-compensate the effect of fermions, such that the fixed-point values in G ψ/h decrease as a function of N f up to N f ≈ −
30. We caution that such threshold effects areregulator dependent, and accordingly it is an interestingopen question how a similar self-stabilization of the grav-itational system could be encoded in beta functions witha different choice of regulator. For a related discussion,see [68].At N f ≈ −
30, we observe hints for the onset of amore strongly coupled regime in our truncation. For in-stance, a subset of the critical exponents deviates furtherfrom the canonical values. This goes hand in hand witha more significant impact of σ , i.e., the fixed-point re-sults with σ start to deviate more significantly from thosewithout. Within the minimally coupled system, the twoprojections, shown in Fig. begin to show slight dif-ferences. In particular, a difference between G ∗ h and G ∗ ψ begins to develop in the n = 2 projection. Moreover, ef-fective universality for G ∗ h and G ∗ ψ appears to be lost inthis regime. That this corresponds to a more strongly-coupled regime can also be inferred by exploring the ef-fective strength of metric fluctuations. It is for instanceencoded in the quantity G eff , n = G (1 − λ ) n . (53)In the present projection, β G ψ and β G h are sensitive to G eff , / . Higher powers of (1 − λ ) can play a role insome diagrams. As λ grows towards the pole at λ =1 /
2, the effective strength of metric fluctuations can groweven if G itself does not grow. We observe that the G eff decrease as a function of N f until about N f ≈
20, whenan increase is observed. For the truncation without σ in n =1 projection the increase only occurs for the higher-order G eff , cf. Fig. .Taken together, this indicates a need to extend thetruncation for a quantitatively reliable determination offixed-point properties in the regime beyond N f ≈ − N f . . . G ∗ e ff ( N f ) G ∗ e ff ( ) G ∗ eff , G ∗ eff , G ∗ eff , n=1n=2, σ = 0 n=2, σ = 0 Background
Figure 13 . We show the value of G ∗ eff , for different trunca-tions as a function of the fermion number. The values for thebackground system are taken from [36]. essary to reliably probe its existence and possible prop-erties. Nevertheless, we highlight that our results couldbe interpreted to hint at the existence of a fixed pointalso at large N f , which could be relevant for asymptoticsafety in matter models [89, 90]. Accordingly, a scenarioin which a crossover from asymptotically safe fixed-pointscaling with gravity to asymptotic safety without grav-ity determines high-energy physics, might potentially berealizable. Such a setting requires further investigation.Finally, we point out that G eff , exhibits similar behav-ior in the background- as in the fluctuation system for N f = 1 ...
10. In this region, G eff , falls for both systems,cf. Fig. , although the underlying mechanisms at thelevel of fixed-point values differ. This could be inter-preted as a hint that at the level of physically relevantcombinations of couplings, the background- and fluctua-tion system could behave similarly. VII. CONCLUSIONS AND OUTLOOKA. Key finding: A robust fermion-gravity fixedpoint for finitely many fermions
In this paper, we have zoomed in on the microscopicdynamics of gravity and fermions. Going beyond pre-vious studies in the literature, we have explored a newdirection in the space of couplings: Nonminimal deriva-tive couplings for fermions are expected to be present atthe Reuter fixed point according to symmetry arguments[27]. Here, we have included the leading-order one of thisfamily of couplings, and confirmed that it cannot be con-sistently set to zero. Accordingly its inclusion constitutesa nontrivial test of asymptotically safe gravity. In partic-ular, we find good indications for a robust continuation ofthe Reuter fixed point from zero to finite fermion number N f ≈ −
30. Specifically, the inclusion of the canonicallyirrelevant, nonminimal coupling σ has very little impacton properties of the system determined in smaller trun- . . . . . . p / k − . . . . q p k q G ∗ ψ (cid:12)(cid:12) N f =25 q p k V ∗ trunc (cid:0) p k (cid:1)(cid:12)(cid:12) N f =25 q p k V (cid:0) p k (cid:1)(cid:12)(cid:12) n =1 N f =25 q p k V (cid:0) p k (cid:1)(cid:12)(cid:12) n =2 N f =25 Figure 14 . Momentum dependence of the graviton-fermionvertex for N f = 25 fermions for the n =1 and the n =2 projec-tion including σ . cations. Most importantly, the introduction of σ addsanother irrelevant direction at the interacting fixed point.Hence, our study provides further evidence for the smallfinite dimensionality of the UV-critical hypersurface ofthe Reuter fixed point also in the presence of matter.Our results further reinforce the scenario that asymptot-ically safe gravity could be near-perturbative, implyingthat the spectrum of higher-order critical exponents fol-lows a near-canonical scaling. Moreover, we find furtherhints for a possible near-perturbative nature of asymp-totic safety by comparing two “avatars” of the Newtoncoupling. These are expected to exhibit a near-equalityof fixed-point values, referred to as effective universality[20, 21], in a near-perturbative regime. Our investiga-tion also includes a detailed analysis of the momentum-dependence of the flow, highlighting the small impact ofthe nonminimal coupling in terms of a suppressed higher-momentum dependence of the graviton-fermion vertex.Beyond N f ≈ −
30, we find indications that fur-ther extensions of the truncation are required. In thisregime, the inclusion of σ as well as changes in the pro-jection scheme have an appreciable impact on the fixed-point properties. Moreover, critical exponents deviatefurther from canonical scaling. To reliably test whetherthe Reuter fixed point can be extended to much higherfermion numbers, extensions of the truncation are re-quired to reach quantitatively robust control of the sys-tem at N f (cid:39) − B. Outlook
For values of N f >
30, the large deviation of the criticalexponents from canonical scaling indicates the need forfurther extensions of the truncation. In principle, manydifferent directions in theory space are available for suchan extension. Accordingly it is highly desirable to obtainan educated guess, which direction is the most likely toprovide an important step towards apparent convergence9 . . . . . . p / k . . . . . . V ∗ h (cid:0) p k (cid:1)(cid:12)(cid:12) N f =1 V ∗ h (cid:0) p k (cid:1)(cid:12)(cid:12) N f =25 V h (cid:0) p k (cid:1)(cid:12)(cid:12) N f =1 V h (cid:0) p k (cid:1)(cid:12)(cid:12) N f =25 Figure 15 . Momentum dependence for the graviton three-point function evaluated at the corresponding fixed point. of the results. We have already observed that the sys-tem is rather stable under the inclusion of the coupling σ . Specifically, it is useful to consider the momentum-dependence of the flow at N f >
1. Performing a similarcomparison as for the case N f = 1 in Fig. , we find thatthe n =2 approximation with σ captures the momentum-dependence of the flow more accurately than the approx-imation without σ ( n = 1), cf. Fig. . Therefore, ourextension of the truncation is an important step towardsquantitative precision for N f ≈
25. A further extensionto p terms in the momentum-dependence also appearsindicated by the results, cf. Fig. .Further contributions from the matter sector to β G ψ might be relevant at large N f . In particular, this couldinclude induced, chirally symmetric four-fermions inter-actions [25]. In [27] a suppression of the contribution ofthese induced interactions to other beta functions was ob-served at N f = 1. Yet, an explicit study of the impact ofthese interactions to β G ψ is still outstanding. Moreover,a scaling of this contribution with N f could be possible,and could therefore enhance the corresponding effects atlarge N f .Fig. shows the momentum dependence of the gravi-ton three-point vertex at N f = 1 and N f = 25. For theformer case, it is well approximated by a polynomial up tofirst order in p . This is not the case for N f = 25. There,the momentum dependence shows a clear p dependencewhich is not captured by the n = 2 projection at p = 0and p = k which is the approximation we work in for G h in this paper. Accordingly, at large fermion num-bers, the n =2 projection introduces an error. This mightbe the reason for the deviation of the fixed-point valuesof both avatars of the Newton coupling, visible, e.g., inFig. . Therefore, an extension of the truncation inthe pure-gravity direction is indicated, in order to obtainmore reliable results for the behavior at N f >
30. InFig. we observe that a polynomial up to second or-der in p might capture the full momentum dependenceof the graviton vertex. Therefore the inclusion of R and R µν would be the first step towards a more reliable large- N f behavior. One might interpret our observationas the generation of a sizable R µν R µν and/or R term byfermionic fluctuations. This is particularly intriguing, asthe corresponding change in the effective graviton prop-agator might play a role in generating quantum-gravitycontributions to Standard Model beta functions of theappropriate size to be phenomenologically viable [27].In summary, we aim at reaching quantitative conver-gence of fixed-point values at finite fermion numbers. Acentral motivation is that this is a key input for investiga-tions into the phenomenology of asymptotic safety, bothin particle physics, see, e.g., [19] as well as in cosmology,see, e.g., [91]. Acknowledgements:
We thank J. M. Pawlowski andM. Reichert for discussions. A. E. and M. S. are sup-ported by the DFG under grant no. Ei/1037-1. A. E. isgrateful to the African Institute for Mathematical Sci-ences (AIMS) Ghana for hospitality during the finalstages of this work. S.L. is supported by the DFG Col-laborative Research Centre ”SFB 1225 (ISO-QUANT)”.
Appendix A: Dimensionless form of the Wetterichequation
In this section we explain in detail, how one can castthe flow equation into a completely dimensionless and k -independent form. This leads us to a compact expressioncontaining every beta function of the system and pro-vides a formally exact equation underlying all fixed-pointsearches in truncations. After that, we give an explicitexample on how to extract specific beta functions.
1. Generic case
In order to get the general picture we will, for now,mostly use the DeWitt notation with super indices A ,the super field Φ A ,(Φ A ) = (cid:0) h µν ( x ) , ψ i ( x ) , ¯ ψ i ( x ) , c µ ( x ) , ¯ c µ ( x ) (cid:1) , (A1)and the effective action Γ k without reference to any trun-cation. For example, in this language the vertex expan-sion of the effective action readsΓ k [Φ; ¯ g ] = ∞ (cid:88) n =0 n ! Γ ( n ) k A ...A n [0; ¯ g ]Φ A n . . . Φ A , (A2)where the Γ ( n ) k are the functional derivatives of the effec-tive action with respect to the fields,Γ ( n ) k A ...A n [Φ; ¯ g ] = Γ k [Φ; ¯ g ] ←− δδ Φ A . . . ←− δδ Φ A n . (A3)Note the order of the indices and fields, which is impor-tant to keep in mind for the Grassmann-valued quanti-ties. When performing RG steps, i.e., when changing k ,all n -point functions change. However, some parts of that0we can absorb by a simple rescaling of the field. To cap-ture this, let us define the (Grassmann even) generalizedwave-function renormalization Z k such that the secondfunctional derivative of Γ k agrees with the standard ki-netic term,12 Tr (cid:0) Γ (2) k [0; ¯ g ]ΦΦ T (cid:1) = 12 Tr (cid:0) Z k [¯ g ] T Kin k [¯ g ] Z k [¯ g ]ΦΦ T (cid:1) , (A4)where Kin k is the standard kinetic operator. The wave-function renormalization is a matrix in field space. Thus,for practical purposes we assign a factor of Z k insteadof Z / k to each field. Further, note the remaining k -dependence in Kin k , which is present due to the followingreason: In principle we can bring Γ (2) k into any desiredform, however as Z k is supposed to be a rescaling it mustnot vanish. Therefore we cannot remove, e.g., mass poles,whose location we still need to follow while changing k .One important ingredient for a full RG step is therescaling of the (background) spacetime. This is usu-ally implemented by a rescaling of the coordinates, x µ → k − x µ . Here we will follow a different route:as the diffeomorphism invariant (background) distanceis d s = ¯ g µν d x µ d x ν , we can implemented the rescalingby reparametrizing the background metric ¯ g in terms ofa dimensionless metric ˆ¯ g , ¯ g µν → ¯ g k µν [ˆ¯ g ] = k − ˆ¯ g µν . (A5)In particular this implies that we treat coordinates as di-mensionless, while the metric carries the dimensionality.For example the operator ¯ / ∇ still scales with k , however,the k now does not arise from the dimensionless ¯ ∇ µ , butfrom the dimensionful ¯ γ µ = ¯ g µν ¯ γ ν , { ¯ γ µ , ¯ γ ν } = 2¯ g µν → { k − ˆ¯ γ µ , k − ˆ¯ γ ν } = 2 k − ˆ¯ g µν . (A6)The advantage of this assignment of dimensionality istwofold. Firstly this fits better to the diffeomorphic na-ture of gravity, where one can choose coordinates as onepleases. For instance, for spherical coordinates, it is ob-vious that we do not want the angles to carry any dimen-sion. Yet, if we insist on the radial coordinate to carrya dimension, then the metric tensor would not have ahomogenous dimension, as the purely radial componentwould be dimensionless, while the angular componentsof the metric do carry nonzero dimensionality. The sec-ond advantage is that we can make the rescaling of thespacetime manifest in the effective action, by using k − ˆ¯ g instead of ¯ g as the (background) metric argument of Γ k . In this appendix we use a hat to indicate dimensionless quanti-ties, e.g., ˆ¯ g , whereas dimensionful quantities “wear” no additionalsymbols, in order to avoid confusion of dimensionful quantitieswith the bar of background quantities, e.g., ¯ g . This would be somewhat more complicated if we insistedon rescaling the coordinates.Considering the rescaling of the (background) space-time, we define the (in field space) diagonal operator K k ,such that it accounts for the canonical mass dimensionof the fields. We can read off the canonical scaling with k from the behavior of the kinetic term under a rescalingof the (background) metric,Kin k [ k − ˆ¯ g ] = K − k (cid:100) Kin k [ˆ¯ g ] K − k , K T k = K k , (A7)where (cid:100) Kin k is the dimensionless kinetic operator with di-mensionless couplings (mass poles). The scale derivativeof K k then gives a factor N corresponding to the canon-ical mass dimension, ∂ t K k = N K k = K k N . (A8)With the above definitions we can parametrize the di-mensionful field Φ in terms of the dimensionless field ˆΦ,Φ → Φ k [ ˆΦ; ˆ¯ g ] = Z − k [ k − ˆ¯ g ] K k ˆΦ . (A9)In our case this translates into the following canonicalpowers of k ( K kAB ˆΦ B ) (A10)= k d − (cid:0) ˆ h µν ( x ) , k ˆ ψ ( x ) , k ˆ¯ ψ ( x ) , k ˆ c µ ( x ) , k ˆ¯ c µ ( x ) (cid:1) , and corresponding canonical mass dimension N ( N AB ˆΦ B ) (A11)= (cid:0) d − ˆ h µν ( x ) , d − ˆ ψ ( x ) , d − ˆ¯ ψ ( x ) , d − ˆ c µ ( x ) , d − ˆ¯ c µ ( x ) (cid:1) . Note the canonical scaling of h µν ∼ k d − ˆ h µν . At firstsight this seems different from the standard one of abosonic field ∼ k d − . This “different” factor arises fromthe positioning of the indices in the kinetic term for h µν ,12 Tr (cid:0) Kin k [ k − ˆ¯ g ] hh T (cid:1) ∼ k − d (cid:90) (cid:113) ˆ¯ g h µν ˆ¯∆ h ρσ ˆ¯ g µρ ˆ¯ g νσ . (A12)If we were to use ˜ h µν = ¯ g µρ h ρν instead of h µν ,12 Tr (cid:0) Kin k [ k − ˆ¯ g ]˜ h ˜ h T (cid:1) ∼ k − d (cid:90) (cid:113) ˆ¯ g ˜ h µν ˆ¯∆ ˜ h νµ , (A13)we would find the usual ∼ k d − . One can easily see thatthis choice has no impact on any of the beta functions,as every such factor can be reabsorbed into the wave-function renormalization.Finally let us define the generalized anomalous dimen-sion η k , which captures the anomalous scaling of thefields, i.e., the scaling not coming from K k but from Z k , η k [ˆ¯ g ] = − K − k ˙ Z k [ k − ˆ¯ g ] Z − k [ k − ˆ¯ g ] K k . (A14)1The full scaling of the dimensionful field Φ k then reads K − k Z k [ k − ˆ¯ g ] ∂ t Φ k [ ˆΦ; ˆ¯ g ] = ( η k [ˆ¯ g ] + N ) ˆΦ . (A15)Using the above we can define the dimensionless effectiveaction ˆΓ k ,ˆΓ k [ ˆΦ; ˆ¯ g ] = Γ k [ Z − k [ k − ˆ¯ g ] K k ˆΦ; k − ˆ¯ g ] . (A16)Next we choose a regulator R k of the form R k [¯ g ] = Z T k [¯ g ] K − k ˆ R [ k ¯ g ] K − k Z k [¯ g ] , (A17)where ˆ R [ˆ¯ g ] is a dimensionless and k independent regu-lator shape function. This is in full agreement of thestandard way of writing the shape-function as a functionof a dimensionful Laplacian divided by k . Expressed interms of the dimensionless Laplacian, the shape functionhas no k -dependence.Now we can move on to the calculation of the flow ofthe dimensionless effective action, which explicitly con-tains all beta functions of the considered system, β ˆΓ k (cid:2) ˆΓ k ; ˆ R, ˆ¯ g (cid:3) [ ˆΦ] = ∂ t ˆΓ k [ ˆΦ; ˆ¯ g ] . (A18)We start with the standard flow equation,˙Γ k [Φ; ¯ g ] = 12 STr (cid:2) (Γ (2) k [Φ; ¯ g ] + R k [¯ g ]) − ˙ R k [¯ g ] (cid:3) . (A19)On the right hand side we can insert the functionalderivative of the dimensionless effective action (A16) andour chosen regulator (A17), leading to (cid:0) Γ (2) k (cid:2) Φ k [ ˆΦ; ˆ¯ g ]; ¯ g k [ˆ¯ g ] (cid:3) + R k (cid:2) ¯ g k [ˆ¯ g ] (cid:3)(cid:1) − (A20) = Z − k [ k − ˆ¯ g ] K k (ˆΓ (2) k [ ˆΦ; ˆ¯ g ] + ˆ R k [ˆ¯ g ]) − K k Z − k [ k − ˆ¯ g ] , where ¯ g k and Φ k are given in Eqs. (A5) and (A9). Interms of dimensionless and k -independent quantities, thescale derivative of the regulator can be expressed via K k Z − k [ k − ˆ¯ g ] ˙ R k [ k − ˆ¯ g ] Z − k [ k − ˆ¯ g ] K k (A21)= − (cid:0) η T k [ˆ¯ g ]+ N (cid:1) ˆ R [ˆ¯ g ] − ˆ R [ˆ¯ g ] (cid:0) η k [ˆ¯ g ]+ N (cid:1) − ∂ q ˆ R [ q − ˆ¯ g ] | q =1 . In the last term we treated the k -derivative in terms of anauxiliary dimensionless parameter q . This trick helps usmaking everything explicitly k independent. We use the q -derivative in order to capture the scaling with the met-ric. In practice such a term essentially leads to derivativeswith respect to the (dimensionless) momentum, e.g.,∆[ q − ˆ¯ g ] = q ∆[ˆ¯ g ] . (A22)By comparison with the definition of the dimensionlesseffective action (A16), and applying the chain rule forderivatives, we observe that the scale derivative of ˆΓ k decomposes into the diagrams, coming from the standardflow equation, and the scaling of the fields: ∂ t ˆΓ k [ ˆΦ; ˆ¯ g ] = ˙Γ k (cid:2) Z − k [ k − ˆ¯ g ] K k ˆΦ; k − ˆ¯ g (cid:3) (A23)+ ˆΓ (1) k [ ˆΦ; ˆ¯ g ]( η k [ˆ¯ g ]+ N ) ˆΦ+ ∂ q ˆΓ k [ ˆΦ; q − ˆ¯ g ] | q =1 . Replacing the first line with the standard flow equation(A19), while using the expressions (A20) and (A21), weare led to the flow equation for ˆΓ k , β ˆΓ k [ˆΓ k ; ˆ R, ˆ¯ g ][ ˆΦ] = −
12 STr (cid:104) (ˆΓ (2) k [ ˆΦ; ˆ¯ g ] + ˆ R [ˆ¯ g ]) − (cid:16)(cid:0) η T k [ˆ¯ g ] + N (cid:1) ˆ R [ˆ¯ g ] + ˆ R [ˆ¯ g ] (cid:0) η k [ˆ¯ g ] + N (cid:1) + ∂ q ˆ R [ q − ˆ¯ g ] | q =1 (cid:17)(cid:105) (A24)+ ˆΓ (1) k [ ˆΦ; ˆ¯ g ]( η k [ˆ¯ g ] + N ) ˆΦ + ∂ q ˆΓ k [ ˆΦ; q − ˆ¯ g ] | q =1 . Note that we now have a completely k independent flowequation, with a compact form for the fixed-point equa-tion, β ˆΓ k (cid:2) ˆΓ ∗ k [ ˆ R, ˆ¯ g ]; ˆ R, ˆ¯ g (cid:3) = 0 , (A25)where ˆΓ ∗ k [ ˆ R, ˆ¯ g ] is the fixed point action as a function ofthe chosen regulator shape and the chosen (background)spacetime.Within the vertex expansion scheme a point in theoryspace can be specified by choosing the ˆΓ ( n ) k [0; ˆ¯ g ] for n (cid:54) = 2and choosing the remaining parameters (mass poles) in (cid:100) Kin k [ˆ¯ g ], while ˆΓ (2) k [0; ˆ¯ g ] is then given byˆΓ (2) k [0; ˆ¯ g ] = (cid:100) Kin k [ˆ¯ g ] . (A26) The dimensionless effective action ˆΓ k then reads,ˆΓ k [ ˆΦ; ˆ¯ g ] = ˆΓ k [0; ˆ¯ g ] + ˆΓ (1) k [0; ˆ¯ g ] ˆΦ + Tr (cid:0)(cid:100) Kin k [ˆ¯ g ] ˆΦ ˆΦ T (cid:1) + ∞ (cid:88) n =3 n ! ˆΓ ( n ) k A ...A n [0; ˆ¯ g ] ˆΦ A n . . . ˆΦ A . (A27)The flow β ˆΓ ( n ) k of the vertices ˆΓ ( n ) k [0; ˆ¯ g ] ( n (cid:54) = 2) is calcu-lated by taking n functional derivatives of equation (A24)with respect to ˆΦ and evaluate the expression at ˆΦ = 0,2 β ˆΓ ( n ) k [ˆΓ k ; ˆ R, ˆ¯ g ] A ...A n = Flow ( n ) A ...A n [ˆ¯ g ] + n (cid:88) l =1 ˆΓ ( n ) k A ...A l − BA l +1 ...A n [0; ˆ¯ g ]( η k [ˆ¯ g ] + N ) BA l + ∂ q ˆΓ ( n ) k A ...A n [0; q − ˆ¯ g ] . (A28)where Flow ( n ) is a short hand for the corresponding diagrams,Flow ( n ) A ...A n [ˆ¯ g ] = STr (cid:2) (ˆΓ (2) k [ ˆΦ; ˆ¯ g ]+ ˆ R [ˆ¯ g ]) − (cid:0) − (cid:0) η T k [ˆ¯ g ]+ N (cid:1) ˆ R [ˆ¯ g ] − ˆ R [ˆ¯ g ] (cid:0) η k [ˆ¯ g ]+ N (cid:1) − ∂ q ˆ R [ q − ˆ¯ g ] | q =1 (cid:1)(cid:3) ←− δδ ˆΦ A . . . ←− δδ ˆΦ A n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆΦ=0 . (A29)Furthermore the generalized anomalous dimension needsto be determined self-consistently, by taking two func-tional derivatives of equation (A24) and evaluating atˆΦ = 0, β (cid:100) Kin k [ˆΓ k ; ˆ R, ˆ¯ g ] = Flow (2) [ˆ¯ g ] + ( η T k [ˆ¯ g ] + N ) (cid:100) Kin k [ˆ¯ g ] (A30)+ (cid:100) Kin k [ˆ¯ g ]( η k [ˆ¯ g ] + N ) + ∂ q (cid:100) Kin k [ q − ˆ¯ g ] . However, usually we only consider the flow of somehow projected n -point functions ˆ V ( n ) k,λ ,ˆ V ( n ) k,λ [ˆ¯ g ] = ˆΓ ( n ) k A ...A n [0; ˆ¯ g ] P ( n ) A ...A n λ [ˆ¯ g ] , (A31)where P ( n ) λ projects onto the physically most relevantor technically most feasible structures of ˆΓ ( n ) k . Here weuse the index λ to denote some tunable external param-eter for the projection. In practice this can, e.g., be anexternal momentum scale or numerating various tensorstructures. In order to derive the running of ˆ V ( n ) k,λ , β ˆ V ( n ) k,λ [ˆΓ k ; ˆ R, ˆ¯ g ] = ∂ t ˆ V ( n ) k,λ [ˆ¯ g ] , (A32)we contract equation (A28) with P ( n ) λ , β ˆ V ( n ) k,λ [ˆΓ k ; ˆ R, ˆ¯ g ] = Flow ( n ) A ...A n [ˆ¯ g ] P ( n ) A ...A n λ [ˆ¯ g ] + ∂ q ˆ V ( n ) k,λ [ q − ˆ¯ g ] (A33)+ n (cid:88) l =1 ˆΓ ( n ) k A ...A l − BA l +1 ...A n [0; ˆ¯ g ]( η k [ˆ¯ g ] + N ) BA l P ( n ) A ...A n λ [ˆ¯ g ] − ˆΓ ( n ) k A ...A n [0; ˆ¯ g ] ∂ q P ( n ) A ...A n λ [ q − ˆ¯ g ] . In some cases the terms in the second line of equation(A33) boil down to a simple factor multiplying ˆ V ( n ) k,λ .
2. Example: fermionic sector
Let us be more explicit in the following. As an ex-ample we consider again the fermionic sector from themain text, now in d -dimensional Euclidean space. Thefermionic terms we are following in our truncation canbe written asΓ ψk [ h, ψ, ¯ ψ ; k − δ ] (A34)= k − d N f (cid:80) i =1 (cid:90) (cid:2) k ¯ ψ i Z ψk ( (cid:3) k ) /∂ψ i − π k (cid:0) V k ( (cid:3) k ) h TT µν (cid:1) ¯ ψ i γ µ ∂ ν ψ i (cid:3) , where (cid:3) k = k (cid:3) is the dimensionful version of the di-mensionless (cid:3) = − ∂ . Here we already inserted a flat background, k − ˆ¯ g µν = k − δ µν . The standard kineticterm for chiral fermions isKin ψk [ k − δ ] ij ( x, y ) = k − d /∂δ ( x − y ) δ ij , (A35)where a factor k − d arises from the √ ¯ g and one factor k from the ¯ γ µ in the covariant formulation of the kineticterm, (cid:82) √ ¯ g ¯ ψ ¯ γ µ ¯ ∇ µ ψ . Note that Kin ψk corresponds to the ψ - ¯ ψ sector in field space,Kin k [¯ g ] = Kin hk [¯ g ] 0 0 0 00 0 Kin ψk [¯ g ] 0 00 − Kin ψ T k [¯ g ] 0 0 00 0 0 0 Kin ck [¯ g ]0 0 0 − Kin c T k [¯ g ] 0 . (A36)3The generalized wave function renormalization Z ψk thenreads Z ψk [ k − δ ] ij ( x, y ) = Z ψ k ( (cid:3) k ) δ ( x − y ) δ ij . (A37)As we are dealing with chirally symmetric fermionsthere is no explicit mass term, leading to an actually k -independent dimensionless kinetic operator (cid:100) Kin ψk , (cid:100) Kin ψk [ q − δ ] ij ( x, y ) = q − d /∂δ ( x − y ) δ ij . (A38)Here we can read off the canonical scaling K ψk and thecanonical mass dimension N ψk , K ψk ij ( x, y ) = k d − δ ( x − y ) δ ij , (A39) N ψk ij ( x, y ) = d − δ ( x − y ) δ ij . Thus the fermionic generalized anomalous dimension η ψk [ δ ] is essentially given by the standard fermionicanomalous dimension η ψk ( (cid:3) ), η ψk [ δ ] ij ( x, y ) = − ˙ Z ψk ( (cid:3) k ) Z ψk ( (cid:3) k ) δ ( x − y ) δ ij = η ψk ( (cid:3) ) δ ( x − y ) δ ij . (A40)Putting everything together the dimensionless version ofequation (A34) takes the formˆΓ ψk [ˆ h, ˆ ψ, ˆ¯ ψ ; q − δ ] (A41)= q − d N f (cid:88) i =1 (cid:90) (cid:2) q ˆ¯ ψ i /∂ ˆ ψ i − π q (cid:0) ˆ V k ( ˆ (cid:3) q )ˆ h TT µν (cid:1) ˆ¯ ψ i γ µ ∂ ν ψ i (cid:3) , where ˆ (cid:3) q = q (cid:3) is the dimensionless metric scaled ver-sion of (cid:3) . To self-consistently determine the generalizedanomalous dimension we plug the above into equation(A30),0 = Flow (2) ψ [ δ ] ij ( x, y ) + η ψk ( (cid:3) ) /∂δ ( x − y ) δ ij . (A42)By first multiplying with i /pd γ N f p δ ( x ) e ip · y δ ij , then takingthe Dirac and flavor trace and finally integrating withrespect to x and y , we arrive at η ψk ( p ) = (cid:90) x,y tr (cid:16) i /pd γ N f p e ip · y δ ( x ) δ ij Flow (2) ψ [ δ ] ij ( x, y ) (cid:17) . (A43)This projection can be summarized as P (2) p [ q − δ ] ij ( x, y ) = i q d − /pd γ N f p δ ( x ) e ip · y δ ij . (A44) If we dealt with nonchiral fermions we had (cid:100)
Kin ψk [ q − δ ] ij ( x, y ) = q − d ( q /∂ + ˆ m ψk ) δ ( x − y ) δ ij , where ˆ m ψk would be the dimensionlessfermionic mass. Finally let us move on to the running of ˆ V k . The fullgraviton-fermion three-point function depends on threecoordinates x , y and z , and therefore on three externalmomenta in Fourier space, p , p and p . One of themcan be eliminated due to momentum conservation. Asprojector P (3) p ,p on the desired quantity ˆ V k , we use P (3) p ,p [ q − δ ] ijµν ( x, y, z )= iq d − d − γ ρ p σ π d γ N f ( d +1)( d − p Π TT ρσµν ( p ) e i ( p · y + p · z ) δ ( x ) δ ij , (A45)which we evaluate at the momentum symmetric point,where p = p = − p · p ) = p and p = − p − p . This is the d -dimensional version of Eq. (26). Thenormalization of P (3) p ,p again follows fromΠ TT µνµσ ( p ) p σ p ν = ( d + 1)( d − d − (cid:16) p − ( p · p ) p (cid:17) , (A46)when evaluated at the momentum symmetric point, p = p = − p · p ) = p . Using the general expression (A33)for β ˆ V ( n ) k,λ and inserting the projector (A45) we finallyarrive at the d -dimensional version for β ˆ V k , cf. (34), β ˆ V k ( p ) = Flow (3) ψ ( p )+ ˆ V k ( p ) (cid:0) d − + η ψk ( p )+ η hk ( p ) (cid:1) +2 p ˆ V (cid:48) k ( p ) . (A47) Appendix B: Beta functions for avatars of theNewton coupling from the derivative expansion
Extracting all beta functions and anomalous dimen-sions from the derivative expansion and then equating G ψ = G = G h results in the following two beta functionsfor the avatars of the Newton coupling. We highlight thelast term in each expression, which encodes the effectof fermionic fluctuations. These screen the gravitationalcouplings for σ > − . TT = − λ and P Tr = − λ denote the pole structure of the TT-and trace mode,respectively. β G h =2 G + G π (cid:18) − Tr5 + 175152 P
Tr4 + 9P
Tr2 − TT2 −
15 P
Tr3 P TT2 + 5495 P
Tr4 P TT + 2772280 P Tr3 P TT + 2( −
439 + 932 λ )855 P Tr P TT4 − − λ )4104 P Tr P TT2 + 5(1069 + 1728 λ )1368 P Tr2 P TT + ( − λ )285 P Tr2 P TT3 − − λ )61560 P Tr P TT3 + ( − λ )20520 P Tr2 P TT2 + ( − λ + 30664 λ )5130 P TT3 + ( − λ − λ + 20768 λ )2565 P TT5 + (251791 − λ + 1676088 λ + 2606848 λ )61560 P TT4 − N f σ (cid:19) + O ( G ) , (B1) β G ψ =2 G + G π (cid:18)
38 P
Tr4 − Tr2 P TT2 + 5(71 − λ + 296 λ )216 P TT4 + 4( − σ )81 P TT − −
11 + 28 σ )231 P Tr P TT + 2( −
81 + 180 σ + 80 σ )81 P TT2 + 27 + 112 σ + 252 σ
42 P
Tr3 − − σ + 2688 σ )5544 P Tr P TT2 − σ + 2688 σ )1848 P Tr2 P TT + 4536 + 9045 σ + 7168 σ Tr − σ + 100072 σ Tr2 + 5( −
17 + 24 λ − σ + 80 λ ( − σ ))162 P TT3 − N f
105 + 1472 σ (cid:19) + O ( G ) , (B2) β σ =2 σ + Gπ (cid:18) − − σ + 6 σ )24 P TT2 P Tr2 + 3(27 + 140 σ + 300 σ )256 P Tr4 − σ (2268 − σ + 896 σ )12096 P Tr − − σ + 17370 σ + 8960 σ TT + 1386 − σ + 10010 σ + 10080 σ TT P Tr + 5082 + 4125 σ + 29260 σ + 20160 σ Tr P TT2 − σ + 134428 σ + 48384 σ Tr3 + 20979 + 128628 σ + 243065 σ + 236544 σ Tr2 + 5(51 − σ + λ ( −
111 + 124 σ ))432 P TT4 − σ + 306590 σ + 53760 σ − λ (49 − σ + 96 σ )54432 P TT2 + 6237 + 11055 σ − σ + 20160 σ Tr2 P TT − − σ − σ + 15 λ ( − − σ + 740 σ )13608 P TT3 (cid:19) + O ( G ) . (B3)In the following we list the general beta function for the fermion-gravity avatar of the Newton coupling G ψ and thenonminimal coupling σ , obtained via a derivative expansion: β G ψ =(2 + η h TT + 2 η ψ ) G ψ + G ψ π (cid:18) − − σ + η ψ ( − σ ))81 P TT + 154( −
891 + 5040 σ + 1280 σ ) + η h TT ( − σ + 17920 σ )99792 P TT2 + − σ + 66556 σ ) η h Tr (62469 + 623172 σ + 552104 σ )221760+ η ψ (4158 + 9405 σ + 3304 σ ) − σ + 3416 σ )83160 P Tr (cid:19) (cid:112) G ψ G h π (cid:18) − σ + η ψ ( −
33 + 56 σ ))2772 P Tr P TT + η ψ (81 + 224 σ + 168 σ ) − σ + 252 σ )2016 P Tr2 + − −
121 + 231 σ + 504 σ ) + 3 η h TT ( − σ + 6720 σ )108108 (cid:0) P Tr P TT2 + 3P
Tr2 P TT (cid:1) − − σ + 252 σ ) + η h Tr (2145 + 6734 σ + 13680 σ )30030 P Tr3 (cid:19) , (B4) β σ =2 σ + (cid:112) G h G ψ π (cid:18) − − σ + 6 σ )24 P Tr2 P TT2 + 3(27 + 140 σ + 300 σ )256 P Tr4 + 5(51 − σ + λ ( −
111 + 124 σ ))432 P TT4 + 5 λ η h TT ( − − σ + 68 σ )27216 P TT3 − η ψ (1848 − σ + 8008 σ + 6720 σ ) − − σ + 10010 σ + 10080 σ )133056 P Tr P TT − − σ − σ + 20160 σ ) + η h Tr (59202 + 368225 σ + 299208 σ + 482840 σ )1729728 P Tr2 P TT + η h TT (10692 − σ − σ ) + 88( −
693 + 4845 σ + 10640 σ ) + 660 λ (749 + 2730 σ − σ )598752 P TT3 − − η ψ (63 − σ + 196 σ ) + 4(189 − σ + 980 σ ) + 75 λ ( η ψ (49 − σ + 48 σ ) − − σ + 96 σ ))54432 P TT2 − η h Tr (521235 + 471900 σ − σ − σ )23063040 P Tr3 − σ ( η ψ (81 + 280 σ + 252 σ ) − σ + 378 σ ))6048 P Tr2 + 567 η ψ (1 + 2 σ ) − σ + 34636 σ + 48384 σ )32256 P Tr3 (cid:19) + G ψ π (cid:18) η ψ (609 − σ + 4360 σ + 1792 σ ) − − σ + 17370 σ + 8960 σ )36288 P TT − σ + 100890 σ + 17920 σ ) − η h TT (1386 + 13728 σ + 143407 σ + 17920 σ )598752 P TT2 − η h Tr (189189 + 766755 σ + 971685 σ + 768320 σ )443520 P Tr2 − η ψ (5 + 32 σ + 44 σ ) − σ + 79155 σ + 76832 σ )16128 P Tr2 − σ + 44 σ )64 P Tr3 (cid:19) . (B5) [1] S. Weinberg, General Relativity: An Einstein centenarysurvey, Eds. Hawking, S.W., Israel, W; Cambridge Uni-versity Press , 790 (1979).[2] M. Reuter, Phys. Rev. D57 , 971 (1998), arXiv:hep-th/9605030.[3] M. Reuter and F. Saueressig, Phys. Rev.
D65 , 065016(2002), arXiv:hep-th/0110054 [hep-th].[4] O. Lauscher and M. Reuter, Phys. Rev.
D65 , 025013(2001), arXiv:hep-th/0108040.[5] D. F. Litim, Phys.Rev.Lett. , 201301 (2004),arXiv:hep-th/0312114 [hep-th].[6] A. Codello, R. Percacci, and C. Rahmede, Annals Phys. , 414 (2009), arXiv:0805.2909 [hep-th].[7] D. Benedetti, P. F. Machado, and F. Saueressig, Mod. Phys. Lett. A24 , 2233 (2009), arXiv:0901.2984 [hep-th].[8] E. Manrique, S. Rechenberger, and F. Saueressig,Phys.Rev.Lett. , 251302 (2011), arXiv:1102.5012[hep-th].[9] D. Becker and M. Reuter, Annals Phys. , 225 (2014),arXiv:1404.4537 [hep-th].[10] M. Demmel, F. Saueressig, and O. Zanusso, JHEP ,113 (2015), arXiv:1504.07656 [hep-th].[11] H. Gies, B. Knorr, S. Lippoldt, and F. Saueressig, Phys.Rev. Lett. , 211302 (2016), arXiv:1601.01800 [hep-th].[12] T. Denz, J. M. Pawlowski, and M. Reichert, Eur. Phys.J. C78 , 336 (2018), arXiv:1612.07315 [hep-th].[13] N. Christiansen, K. Falls, J. M. Pawlowski, and M. Re- ichert, Phys. Rev. D97 , 046007 (2018), arXiv:1711.09259[hep-th].[14] K. G. Falls, D. F. Litim, and J. Schr¨oder, (2018),arXiv:1810.08550 [gr-qc].[15] M. Reuter and F. Saueressig, New J. Phys. , 055022(2012), arXiv:1202.2274 [hep-th].[16] A. Ashtekar, M. Reuter, and C. Rovelli, (2014),arXiv:1408.4336 [gr-qc].[17] A. Eichhorn, Black Holes, Gravitational Waves andSpacetime Singularities Rome, Italy, May 9-12, 2017 ,Found. Phys. , 1407 (2018), arXiv:1709.03696 [gr-qc].[18] R. Percacci, An Introduction to Covariant QuantumGravity and Asymptotic Safety , 100 Years of General Rel-ativity, Vol. 3 (World Scientific, 2017).[19] A. Eichhorn, (2018), arXiv:1810.07615 [hep-th].[20] A. Eichhorn, P. Labus, J. M. Pawlowski, and M. Re-ichert, SciPost Phys. , 031 (2018), arXiv:1804.00012[hep-th].[21] A. Eichhorn, S. Lippoldt, J. M. Pawlowski, M. Reichert,and M. Schiffer, (2018), arXiv:1810.02828 [hep-th].[22] K. Falls, D. Litim, K. Nikolakopoulos, and C. Rahmede,(2013), arXiv:1301.4191 [hep-th].[23] K. Falls, D. F. Litim, K. Nikolakopoulos, andC. Rahmede, Phys. Rev. D93 , 104022 (2016),arXiv:1410.4815 [hep-th].[24] K. Falls, C. R. King, D. F. Litim, K. Nikolakopou-los, and C. Rahmede, Phys. Rev.
D97 , 086006 (2018),arXiv:1801.00162 [hep-th].[25] A. Eichhorn and H. Gies, New J. Phys. , 125012 (2011),arXiv:1104.5366 [hep-th].[26] J. Meibohm and J. M. Pawlowski, Eur. Phys. J. C76 ,285 (2016), arXiv:1601.04597 [hep-th].[27] A. Eichhorn and A. Held, Phys. Rev.
D96 , 086025(2017), arXiv:1705.02342 [gr-qc].[28] A. Eichhorn, A. Held, and J. M. Pawlowski, Phys. Rev.
D94 , 104027 (2016), arXiv:1604.02041 [hep-th].[29] A. Eichhorn and S. Lippoldt, Phys. Lett.
B767 , 142(2017), arXiv:1611.05878 [gr-qc].[30] O. Zanusso, L. Zambelli, G. P. Vacca, and R. Percacci,Phys. Lett.
B689 , 90 (2010), arXiv:0904.0938 [hep-th].[31] G. P. Vacca and O. Zanusso, Phys. Rev. Lett. ,231601 (2010), arXiv:1009.1735 [hep-th].[32] K.-y. Oda and M. Yamada, Class. Quant. Grav. ,125011 (2016), arXiv:1510.03734 [hep-th].[33] Y. Hamada and M. Yamada, JHEP , 070 (2017),arXiv:1703.09033 [hep-th].[34] A. Eichhorn and A. Held, Phys. Lett. B777 , 217 (2018),arXiv:1707.01107 [hep-th].[35] P. Don`a and R. Percacci, Phys. Rev.
D87 , 045002 (2013),arXiv:1209.3649 [hep-th].[36] P. Don`a, A. Eichhorn, and R. Percacci, Phys.Rev.
D89 ,084035 (2014), arXiv:1311.2898 [hep-th].[37] J. Meibohm, J. M. Pawlowski, and M. Reichert, Phys.Rev.
D93 , 084035 (2016), arXiv:1510.07018 [hep-th].[38] J. Biemans, A. Platania, and F. Saueressig, JHEP ,093 (2017), arXiv:1702.06539 [hep-th].[39] N. Alkofer and F. Saueressig, Annals Phys. , 173(2018), arXiv:1802.00498 [hep-th].[40] N. Alkofer, Phys. Lett. B789 , 480 (2019),arXiv:1809.06162 [hep-th].[41] A. Eichhorn, Phys. Rev.
D86 , 105021 (2012),arXiv:1204.0965 [gr-qc].[42] N. Christiansen and A. Eichhorn, Phys. Lett.
B770 , 154(2017), arXiv:1702.07724 [hep-th]. [43] A. Eichhorn, S. Lippoldt, and V. Skrinjar, Phys. Rev.
D97 , 026002 (2018), arXiv:1710.03005 [hep-th].[44] C. Wetterich, Phys. Lett.
B301 , 90 (1993),arXiv:1710.05815 [hep-th].[45] U. Ellwanger,
Proceedings, Workshop on Quantum fieldtheoretical aspects of high energy physics: Bad Franken-hausen, Germany, September 20-24, 1993 , Z. Phys.
C62 ,503 (1994), arXiv:hep-ph/9308260 [hep-ph].[46] T. R. Morris, Int. J. Mod. Phys.
A09 , 2411 (1994),arXiv:hep-ph/9308265.[47] D. F. Litim, Phys.Rev.
D64 , 105007 (2001), arXiv:hep-th/0103195 [hep-th].[48] J. Berges, N. Tetradis, and C. Wetterich, Phys. Rept. , 223 (2002), arXiv:hep-ph/0005122.[49] B. Delamotte, (2007), arXiv:cond-mat/0702365 [COND-MAT].[50] O. J. Rosten, Phys. Rept. , 177 (2012),arXiv:1003.1366 [hep-th].[51] J. Braun, J.Phys.
G39 , 033001 (2012), arXiv:1108.4449[hep-ph].[52] J. M. Pawlowski, Annals Phys. , 2831 (2007),arXiv:hep-th/0512261 [hep-th].[53] H. Gies, Lect.Notes Phys. , 287 (2012), arXiv:hep-ph/0611146 [hep-ph].[54] H. Gies and S. Lippoldt, Phys.Rev.
D89 , 064040 (2014),arXiv:1310.2509 [hep-th].[55] H. Gies and S. Lippoldt, Phys. Lett.
B743 , 415 (2015),arXiv:1502.00918 [hep-th].[56] S. Lippoldt, Phys. Rev.
D91 , 104006 (2015),arXiv:1502.05607 [hep-th].[57] U. Ellwanger, M. Hirsch, and A. Weber, Z.Phys.
C69 ,687 (1996), arXiv:hep-th/9506019 [hep-th].[58] J. M. Pawlowski, (2003), arXiv:hep-th/0310018 [hep-th].[59] E. Manrique and M. Reuter, Annals Phys. , 785(2010), arXiv:0907.2617 [gr-qc].[60] I. Donkin and J. M. Pawlowski, (2012), arXiv:1203.4207[hep-th].[61] D. Brizuela, J. M. Martin-Garcia, and G. A. Mena Maru-gan, Gen. Rel. Grav. , 2415 (2009), arXiv:0807.0824[gr-qc].[62] J. M. Mart´ın-Garc´ıa, Computer Physics Communications , 597 (2008), arXiv:0803.0862 [cs.SC].[63] J. M. Mart´ın-Garc´ıa, R. Portugal, and L. R. U. Manssur,Computer Physics Communications , 640 (2007),arXiv:0704.1756 [cs.SC].[64] J. M. Mart´ın-Garc´ıa, D. Yllanes, and R. Portugal,Computer Physics Communications , 586 (2008),arXiv:0802.1274 [cs.SC].[65] A. K. Cyrol, M. Mitter, and N. Strodthoff, Comput.Phys. Commun. , 346 (2017), arXiv:1610.09331 [hep-ph].[66] N. Christiansen, B. Knorr, J. M. Pawlowski, andA. Rodigast, Phys. Rev. D93 , 044036 (2016),arXiv:1403.1232 [hep-th].[67] N. Christiansen, B. Knorr, J. Meibohm, J. M. Pawlowski,and M. Reichert, Phys. Rev.
D92 , 121501 (2015),arXiv:1506.07016 [hep-th].[68] N. Christiansen, D. F. Litim, J. M. Pawlowski,and M. Reichert, Phys. Rev.
D97 , 106012 (2018),arXiv:1710.04669 [hep-th].[69] M. Reichert,
Towards a UV-complete Standard Model:From baryogenesis to asymptotic safety , Ph.D. thesis, U.Heidelberg, ITP (2018).[70] M. Reuter and C. Wetterich, Phys.Rev.
D56 , 7893 (1997), arXiv:hep-th/9708051 [hep-th].[71] D. F. Litim and J. M. Pawlowski, JHEP , 049(2002), arXiv:hep-th/0203005 [hep-th].[72] T. R. Morris, JHEP , 160 (2016), arXiv:1610.03081[hep-th].[73] R. Percacci and G. P. Vacca, Eur. Phys. J. C77 , 52(2017), arXiv:1611.07005 [hep-th].[74] P. Labus, T. R. Morris, and Z. H. Slade, Phys. Rev.
D94 , 024007 (2016), arXiv:1603.04772 [hep-th].[75] N. Ohta, PTEP , 033E02 (2017), arXiv:1701.01506[hep-th].[76] M. R. Niedermaier, Phys. Rev. Lett. , 101303 (2009).[77] M. Niedermaier, Nucl. Phys.
B833 , 226 (2010).[78] H. Kawai, Y. Kitazawa, and M. Ninomiya, Nucl. Phys.
B393 , 280 (1993), arXiv:hep-th/9206081 [hep-th].[79] A. Nink and M. Reuter, JHEP , 062 (2013),arXiv:1208.0031 [hep-th].[80] A. Nink and M. Reuter, JHEP , 167 (2016),arXiv:1512.06805 [hep-th].[81] K. Falls, (2015), arXiv:1503.06233 [hep-th].[82] J. Biemans, A. Platania, and F. Saueressig, Phys. Rev. D95 , 086013 (2017), arXiv:1609.04813 [hep-th]. [83] A. Eichhorn and F. Versteegen, JHEP , 030 (2018),arXiv:1709.07252 [hep-th].[84] A. Eichhorn and A. Held, Phys. Rev. Lett. , 151302(2018), arXiv:1803.04027 [hep-th].[85] A. Eichhorn, D. Mesterh´azy, and M. M. Scherer,Phys. Rev. E88 , 042141 (2013), arXiv:1306.2952 [cond-mat.stat-mech].[86] A. Eichhorn, T. Helfer, D. Mesterh´azy, and M. M.Scherer, Eur. Phys. J.
C76 , 88 (2016), arXiv:1510.04807[cond-mat.stat-mech].[87] D. N. Kabat, Nucl. Phys.
B453 , 281 (1995), arXiv:hep-th/9503016 [hep-th].[88] F. Larsen and F. Wilczek, Nucl. Phys.
B458 , 249 (1996),arXiv:hep-th/9506066 [hep-th].[89] D. F. Litim and F. Sannino, JHEP , 178 (2014),arXiv:1406.2337 [hep-th].[90] E. Molinaro, F. Sannino, and Z. W. Wang, Phys. Rev. D98 , 115007 (2018), arXiv:1807.03669 [hep-ph].[91] A. Bonanno and F. Saueressig, Comptes RendusPhysique18