Zooming in on fermions and quantum gravity

# Zooming in on fermions and quantum gravity

Astrid Eichhorn CP3-Origins, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany    Stefan Lippoldt Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany    Marc Schiffer Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
###### Abstract

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 derivative coupling between the curvature and fermion fields in the ultraviolet. Its backreaction on the properties 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 indicate the subleading nature of higher-derivative couplings. Moreover our study provides further indications that the critical surface of the Reuter fixed point has a low dimensionality even in the presence of matter.

## I Introduction

In the search for a quantum theory of gravity that is viable in our universe, the existence of fermionic matter must be accounted for. Our strategy to achieve this is based on a quantum field theoretic framework that includes the metric field and fermion fields at the microscopic level. Such a setting requires an ultraviolet completion or extension of the effective field theory framework within which a joint description of gravity and matter is possible up to energies close to the Planck scale. Asymptotic safety Weinberg (1979); Reuter (1998) is the idea that scale-invariance provides a way to extend the dynamics to arbitrarily high momentum scales without running into Landau poles which would indicate a triviality problem. Moreover, scale-invariance is a powerful dynamical principle, that is expected to fix all but a finite number of free parameters in an infinite dimensional space of theories. It can be reached at a fixed point of the Renormalization Group (RG), which can be free (asymptotic freedom) or interacting (asymptotic safety). Compelling indications for the existence of the asymptotically safe Reuter fixed point in four-dimensional gravity have been found, e.g., in Reuter and Saueressig (2002); Lauscher and Reuter (2001); Litim (2004); Codello et al. (2009); Benedetti et al. (2009); Manrique et al. (2011); Becker and Reuter (2014); Demmel et al. (2015); Gies et al. (2016); Denz et al. (2018); Christiansen et al. (2018a); Falls et al. (2018a). For recent reviews and introductions including a discussion of open questions, see Reuter and Saueressig (2012); Ashtekar et al. (2014); Eichhorn (2018a); Percacci (2017); Eichhorn (2018b).

A central part of the interplay of the Standard Model with gravity is the impact of quantum gravity on the microscopic dynamics for fermions as well as the corresponding “backreaction” of fermionic matter on the quantum structure of spacetime. In line with the observation that asymptotically safe quantum gravity could be near-perturbative Eichhorn et al. (2018a, b) or “as Gaussian as it gets” Falls et al. (2013, 2016, 2018b, 2018a), studies of fermion-gravity-systems follow a truncation scheme by canonical power counting. Furthermore, the chiral structure of the fermion sector of the Standard Model is a key guiding principle. Thus, the leading-order terms according to canonical power counting have been explored in the sector of chirally symmetric fermion self interactions Eichhorn and Gies (2011); Meibohm and Pawlowski (2016); Eichhorn and Held (2017), and fermion-scalar interaction sector Eichhorn et al. (2016a); Eichhorn and Held (2017). These are dimension-6 and dimension-8-operators, respectively. Explicitly chiral-symmetry breaking interactions, including a mass term and two dimension-5, nonminimal couplings of fermions to gravity Eichhorn and Lippoldt (2017), have been studied. The effect of quantum gravity on a Yukawa coupling of fermions to scalars has been studied in Zanusso et al. (2010); Vacca and Zanusso (2010); Oda and Yamada (2016); Eichhorn et al. (2016a); Hamada and Yamada (2017); Eichhorn and Held (2017, 2018a). Conversely, the impact of fermionic fluctuations on the Reuter fixed point has been explored in Donà and Percacci (2013); Donà et al. (2014); Meibohm et al. (2016); Eichhorn and Lippoldt (2017); Biemans et al. (2017a); Alkofer and Saueressig (2018); Alkofer (2019). An asymptotically safe fixed point exists in all of these studies, as long as the fermion number is sufficiently small. Moreover, all operators that have been explored follow the pattern that canonical dimensionality is a robust predictor of relevance at the interacting fixed point. Further, they confirm the conjecture that asymptotically safe quantum gravity could preserve global symmetries Eichhorn and Held (2017), at least in the Euclidean regime. Thus, all symmetry-breaking interactions can be set to zero consistently. Additionally, interacting fixed points could, but need not exist for these, as in the case of the Yukawa coupling Zanusso et al. (2010); Vacca and Zanusso (2010); Oda and Yamada (2016); Eichhorn et al. (2016a); Hamada and Yamada (2017); Eichhorn and Held (2017, 2018a). In contrast, the interacting nature of asymptotically safe gravity percolates into the symmetric sector, where interactions can typically not be set to zero consistently Eichhorn (2012); Eichhorn and Held (2017). Hence, their “backreaction” on the asymptotically safe fixed point could be critical. Further, this sector is a potential source of important constraints on the microscopic gravitational parameter space: Strong gravity fluctuations could trigger new divergences in the matter sector, manifesting themselves in complex fixed-point values for matter interactions. The corresponding bound on the gravitational parameter space that separates the allowed, weakly coupled gravity regime from the forbidden strongly-coupled regime, is called the weak-gravity bound Eichhorn et al. (2016a); Christiansen and Eichhorn (2017); Eichhorn and Held (2017).

In Tab. 1 we provide an overview over interactions in the fermion sector that have been explored in an asymptotically safe context. The table contains a crucial gap, namely nonminimal, chirally symmetric interactions.

This is the sector that we will begin to tackle in this paper. For an analogous study in the scalar sector see Eichhorn et al. (2018c).

As our key result we find a continuation of the asymptotically safe Reuter fixed point to finite fermion numbers that passes a nontrivial test by remaining robust under a crucial extension of the approximation to the full dynamics. Moreover, we find further indications that the critical hypersurface of the Reuter fixed point has a low dimensionality also in the presence of matter.

This paper is structured as follows: In Sec. II, we provide an overview of the setup, and specify the approximation to the full dynamics that we will explore in the following. In Sec. III we discuss in some detail how to derive the beta functions in our setting. In particular, we discuss the relation of the derivative expansion to the projection at finite momenta. Sec. IV provides an overview of the fixed-point results for , which are representative for the results at small fermion numbers. We discuss tests of the robustness of the fixed point, the impact of the newly included nonminimal derivative interaction on the fixed-point results in a smaller truncation, and the feature of effective universality. Sec. V contains a discussion of structural aspects of the weak-gravity bound for cubic beta functions and highlights that no such bound exists for the nonminimal derivative interaction in the regime of gravitational parameter space where our truncation remains viable. In Sec. VI we extend our investigations to , and discuss the continuation of the Reuter fixed point to larger fermion numbers. In Sec. VII we provide a short summary of our key results and highlight possible routes forward in gravity-matter systems in an outlook. App. A includes a general derivation of the form of the flow equation for the dimensionless effective action. This form can be used to directly derive dimensionless beta functions, in contrast to the usual procedure of only introducing dimensionless quantities after a truncation has been specified.

## Ii Setup

The system we analyze contains a gravitational sector and a matter sector with chiral fermions. We aim at deriving the beta functions in this system, and will employ the well-suited functional Renormalization Group. It is based on the flow equation for the scale-dependent effective action, the Wetterich-equation Wetterich (1993); Ellwanger (1994); Morris (1994),

 ˙Γk[Φ;¯g]=12STr[(Γ(2)k[Φ;¯g]+Rk[¯g])−1˙Rk[¯g]]. (1)

The “superfield” is simply a collection of all fields in our system,

 (ΦA)=(hμν(x),ψi(x),¯ψi(x),cμ(x),¯cμ(x)), (2)

where Einsteins summation convention over the “superindex” contains a summation over discrete spacetime, spinor and flavor indices and an integration over the continuous coordinates. Here is a scale-dependent regulator that implements a momentum-shell wise integration of quantum fluctuations and the dot in refers to a derivative with respect to , the RG-“time” with an arbitrary reference scale. The IR-regulator enters the generating functional in the form of a term that is quadratic in the fluctuation fields and renders the Wetterich equation UV and IR finite. Specifically, we choose a Litim-type cutoff Litim (2001) with appropriate factors of the wave-function renormalization for all fields. Next to the gauge-fixing term for the metric fluctuations, it is a second source of breaking of diffeomorphism invariance. It must be set up with respect to an auxiliary metric background , which provides a notion of locality and thereby enables a local form of coarse graining. In the main part of this paper we focus on a flat background,

 ¯gμν=δμν, (3)

while in this section we will keep arbitrary for pedagogical reasons. For introductions and reviews of the method, see, e.g., Berges et al. (2002); Delamotte (2007); Rosten (2012); Braun (2012); specifically for gauge theories and gravity, see, e.g., Pawlowski (2007); Gies (2012); Reuter and Saueressig (2012).

The Wetterich equation provides a tower of coupled differential equations for the scale dependence of all infinitely many couplings in theory space. In practice, this has to be truncated to a (typically) finite-dimensional tower. Let us briefly summarize how we proceed, before providing more details. To construct our truncation, we define a diffeomorphism invariant “seed action”. Next, we expand the terms in this seed action to fifth order in metric fluctuations, defined as

 hμν=¯gμν−gμν. (4)

This corresponds to an expansion of the seed action in vertices. At this point all terms in the seed action, except those arising from the kinetic term for fermions, come with one of the couplings of the seed action. We next take into account that in the presence of a regulator and gauge fixing, the beta functions for those couplings generically differ, when extracted from different terms. Accordingly, we introduce a separate coupling in front of each term in the expanded action. This provides the truncation which we analyze in the following. To close the truncation, the couplings of higher-order vertices are partially identified with those of lower-order ones.

In more detail, these steps take the following form: Our seed action reads

 S=Sgrav+Sgh+Smat. (5)

Classical gravity is described by the Einstein-Hilbert action ,

 SEH=−116π¯GN∫d4x√g(R−2¯λ). (6)

In order to tame the diffeomorphism symmetry of gravity, we choose a gauge-fixing condition ,

 Fμ=(¯gμκ¯Dλ−1+β4¯gκλ¯Dμ)hκλ,β=0. (7)

The gauge choice is incorporated using the gauge fixing action ,

 Sgf=132π¯GNα∫d4x√¯gFμ¯gμνFν,α→0. (8)

To take care of the resulting Faddeev-Popov determinant, we use ghost fields and with the appropriate ghost action ,

 Sgh=∫d4x√¯g¯cμδFμδhαβLcgαβ, (9)

where is the Lie derivative of the full metric in ghost direction,

 Lcgαβ=2¯gρ(α¯Dβ)cρ+cρ¯Dρhαβ+2hρ(α¯Dβ)cρ. (10)

In the following, we choose the Landau gauge, i.e., . By employing a York decomposition of we see that this choice of gauge-fixing parameters leads to contributions from only a transverse-traceless (TT) mode and a trace mode ,

 hμνˆ=hTTμν+14¯gμνhTr, (11)

where the TT-mode satisfies and , while the trace mode is given by . All other modes drop out of the flow equation once it is projected onto monomials with nonvanishing powers of the field. It is important to note that the TT-mode is present in any gauge and to linear order in a gauge invariant quantity. Thus, for external metric fluctuations we exclusively consider the TT-mode. For internal metric fluctuations, also the remaining trace mode is taken into account. We summarize the purely gravitational parts of the action as ,

 Sgrav=SEH+Sgf. (12)

Next we turn to the chiral fermions. Their minimal coupling to gravity is via the kinetic term ,

 Skinmat=Nf∑i=1∫d4x√g¯ψi⧸∇ψi. (13)

For the construction of the covariant derivative for fermions, we use the spin-base invariance formalism Gies and Lippoldt (2014, 2015); Lippoldt (2015). For our purposes, this is equivalent to using the vierbein formalism with a Lorentz symmetric gauge. Upon expansion in , this minimal interaction between fermions and gravity gives rise to an invariant linear in derivatives. There are several invariants containing terms of third order in derivatives and canonical mass dimension, namely:

 S∇3mat = Nf∑i=1∫d4x√g(¯κR¯ψi⧸∇ψi+¯τ(DμR)¯ψiγμψi +¯ξ¯ψi⧸∇3ψi+¯σRμν(¯ψiγμ∇νψi−(∇ν¯ψi)γμψi)),

where each of the invariants respects the Osterwalder-Schrader positivity of the Euclidean action. Out of these four invariants, the ones corresponding to and do not contribute linearly to an external , as does not contain a transverse traceless part to linear order. In the following, we restrict ourselves to the nonminimal coupling and neglect the term. Thus, the kinetic matter action is complemented with

 SRmat=Nf∑i=1¯σ∫d4x√gRμν(¯ψiγμ∇νψi−(∇ν¯ψi)γμψi). (15)

has all the symmetries of the original action (5) and therefore does not enlarge the theory space.

The nonminimal coupling introduces an invariant of cubic order in derivatives, capturing parts of the higher-derivative structure of the fermion-gravity interaction. Once expanded around a flat background, the interaction with is given by

 SRmat=Nf∑i=1¯σ∫d4x(□hTTμν)¯ψiγμ∂νψi+O(h2), (16)

where is the d’Alambertian in flat Euclidean space. Eq. (16) is the unique invariant consisting of one , , and together with two derivatives acting on the TT-mode and one derivative acting on the . We summarize the matter parts of the action as ,

 Smat=Skinmat+SRmat. (17)

After having specified our complete seed action, we expand the scale-dependent effective action in powers of the fluctuation field,

 Γk[Φ;¯g]=∞∑n=01n!Γ(n)kA1…An[0;¯g]ΦAn…ΦA1, (18)

where refers to functional derivatives with respect to the field ,

 Γ(n)kA1…An[Φ;¯g]=Γk[Φ;¯g]←δδΦA1…←δδΦAn. (19)

Note the order of the indices and fields, which is important to keep in mind for the Grassmann-valued quantities.

By using this vertex form, the flow of 5 individual couplings , , , and as well as the anomalous dimension of two wave-function renormalizations and is disentangled, cf. Tab. 2 and see Sect. III for more details.

Here the barred couplings, e.g., and , refer to dimensionful couplings.

For the gravity-fermion vertex the contributing diagrams are shown in Fig. 1. This highlights the necessity to truncate the tower of vertices, as the flow of each -point vertex depends on the - and -point vertices. We use the seed action in Eq. (5) to parametrize the vertices appearing in the diagrams. When generating, e.g., a graviton three-point vertex or a graviton four-point vertex for the scale-dependent effective action from the seed action by expanding to the appropriate power in , both would depend on the same Newton coupling and the same cosmological constant due to diffeomorphism symmetry. However, the gauge fixing and the regulator break diffeomorphism symmetry. Hence, the effective action is known to satisfy Slavnov-Taylor identities instead, Ellwanger et al. (1996); Reuter (1998); Pawlowski (2007, 2003); Manrique and Reuter (2010); Donkin and Pawlowski (2012). As these identities in general are much more involved, there is no such simple relation between the three- and four-point vertex of the effective action as there is for the seed action. In other words, the breaking of diffeomorphism symmetry leads to an enlargement of theory space in which the couplings parameterizing the vertices are independent. There are different routes towards a truncation of this large theory space. In principle, one could pick some random tensor structure and momentum-dependence in each point function and parameterize this by some coupling. Then, the connection to the diffeomorphism-invariant seed action would be lost completely. Instead, we derive the tensor structures of the vertices from the seed-action, but also take into account that the various couplings are now independent. Specifically, we proceed using the following recipe: The structure of the -point vertex is drawn from the seed action,

 Γ(n)kA1…An=Z12B1ΦA1…Z12BnΦAnS(n)B1…Bn∣∣¯λ→¯λn, (20)

where the replacement only affects pure gravity vertices. Furthermore in Eq. (20) the metric fluctuations of the purely gravitational action are rescaled according to

 Sgrav:(hn)μν→(16π)n2¯GN(¯Gh)n2−1(hn)μν, (21)

whereas the graviton in and is rescaled to

 Sgh: (hn)μν→(16π)n2(¯Gh)n2(hn)μν, (22) Smat: (hn)μν→(16π)n2(¯Gψ)n2(hn)μν. (23)

This rescaling breaks diffeomorphism symmetry and helps 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 in the effective action. They are just a way of arriving at a parameterization of the truncated effective action in the enlarged theory space.
In the following we use the term “avatar”, when a single coupling in the seed action leads to various incarnations in the effective action, e.g., and are avatars of the Newton coupling .

In order to close the flow equation, we identify couplings of higher order -point vertices with the corresponding couplings of the three-point vertex. This was already implicitly done with the rescaling in Eqs. (21), (22) and (23) and with the usage of one single coupling . Similarly, all -point vertices arising from the cosmological-constant part of the seed action are parametrized by one coupling , i.e., and . The relation between and the gravitational mass-parameter that is often used in the literature, reads . In the next section we provide details on how the beta functions are extracted from the sum of the diagrams in Fig. 1.

## Iii How to obtain beta functions

We now discuss in some detail how to derive beta functions. We concentrate on the dimensionless couplings, which are obtained from their dimensionful counterparts by a multiplication with an appropriate power of . Dimensionful couplings are denoted with overbars, e.g., etc., whereas their dimensionless counterparts lack the overbar, e.g., etc.

A key goal of ours is to test the quality of our truncation. Thus, we place a main focus on the momentum-dependence of the flow, i.e., the dependence of the -point vertices on the momenta of the fields. Higher-order momentum-dependencies than those included in the truncation are in general present. This implies that different projection schemes might yield different results when working in truncations. We will discuss these different schemes and their relation to each other in the following.

### iii.1 Fermionic Example

As a concrete example let us consider the fermionic sector. To arrive at beta functions, we have to take several steps. First we define a projector on the gravity-fermion vertex. Its form is motivated by the tensor structure of the considered three-point function,

 Skinmat [g=δ+(16π¯Gψ)12hTT,ψ,¯ψ] (24) =Nf∑i=1∫d4x[¯ψi⧸∂ψi−2π12¯G12ψhTTμν¯ψiγμ∂νψi]+O(h2).

By taking the corresponding functional derivatives of Eq. (24) and evaluating in momentum space, while using the projector onto transverse traceless symmetric tensors , we find that,

 ∫x,y,zei(p1⋅x+p2⋅y+p3⋅z)Skinmat←δδhTTμν(x)←δδψi(y)←δδ¯ψj(z) (25) =(2π)4δ(p1+p2+p3)(−i2π12¯G12ψ)ΠTTμνρσ(p1)γρpσ2δij.

Of the three momenta, only two are independent, the third can be eliminated by momentum conservation. Thus we define the projector on as

 \mathbbmP(3)ijp2,p3μν (x,y,z) =iγρp2σ10π12Nfp2ΠTTρσμν(p1)ei(p2⋅y+p3⋅z)δ(x)δij, (26)

which we evaluate at the symmetric point for the momenta, . The normalization of follows from

 ΠTTμνμσ(p1)pσ2p2ν=53(p22−(p1⋅p2)2p21). (27)

Using we define the projected dimensionful vertex as

 ¯V(p2)= ∫x,y,ztr[\mathbbmP(3)ijp2,p3μν(x,y,z)Γk←δδhμν(x)←δδψi(y)←δδ¯ψj(z)]Φ=0, (28)

where implies the trace over Dirac and flavor indices. This definition is independent of any truncation, while a truncation for can be viewed as choosing a specific point in theory space. For instance, when evaluating for our chosen truncation we find that is equal to . Having defined , we aim at deriving the beta function for the dimensionless counterpart ,

 V(p2k2)=kZ12h(p2)Zψ(p2)¯V(p2). (29)

Note that carries a non-trivial dimension, as the gravity-fermion vertex contains an additional momentum . The scale derivative of reads

 βV(p2k2)= V(p2k2)(1+ηψ(p2)+12ηh(p2))+2p2k2V′(p2k2) +k˙¯V(p2), (30)

where one has to take into account the scaling of the momentum, . Here and are the anomalous dimensions,

 ηh(p2)=−˙Zh(p2)Zh(p2),ηψ(p2)=−˙Zψ(p2)Zψ(p2). (31)

We can read off by replacing with in equation (28),

 ˙¯V(p2)=1kFlow(3)ψ(p2), (32)

where is a short hand for the contributing diagrams in Fig. 1,

 Flow(3)ψ(p2)= (33)

Here we made use of Eq. (1).

By inserting the expression for given in Eq. (32) into Eq. (30) for we finally arrive at the beta function for ,

 βV(p2k2)= Flow(3)ψ(p2) (34) +V(p2k2)(1+ηψ(p2)+12ηh(p2))+2p2k2V′(p2k2).

This equation will take center stage in our analysis of the momentum-dependence of the flow and tests of robustness of the truncation. We highlight that in general the right-hand-side of the flow equation generates terms beyond the chosen truncation. In Eq. (34) the consequence is, that our truncation does not capture the full momentum-dependence that is generated. Accordingly, the fixed-point equation cannot be satisfied for all momenta, but instead only at selected points. We will extensively test how large the deviations of from zero are in order to judge the quality of different truncations.

### iii.2 Projection schemes

We perform our analysis in several different projection schemes, as a comparison between the fixed-point structure of the different truncations provides indications for or against the robustness of the fixed point. We now motivate the use and explain the details of these three projection schemes.

Using Eq. (34), the momentum-dependent fixed-point vertex could be found by demanding and solving Eq. (34). In practice, we choose an ansatz for , which is part of choosing a truncation. At a point in theory space defined by , Eq. (34) holds, but indicates that terms not yet captured by are generated. These are present in Eq. (34), so that we need to truncate the beta function in order to close the system. For example, in our setup, we restrict to a polynomial up to first order in , i.e.,

 Vtrunc(p2k2)=√Gψ−2σmodp2k2, (35)

and

 βtruncV(p2k2)=12√GψβGψ−2βσmodp2k2. (36)

Here we introduced a modified version of the coupling ,

 σmod=√Gψσ, (37)

where and are the dimensionless counterparts of and ,

 Gψ=k2¯Gψ,σ=k2¯σ. (38)

However, this specific ansatz does not satisfy Eq. (34) for all values of . Accordingly, the right-hand side of Eq. (34) differs from Eq. (36). This is simply an example for the general fact that, plugging a truncation into the right-hand-side of the Wetterich equation, terms beyond the truncation are generated and therefore the truncation is not closed.

As is not equal to for all momenta, we can choose selected points in the interval for which we demand that is exactly equal to at these points, see, e.g., Eqs. (40) and (41). However, we can also choose superpositions of more values for , see, e.g., Eq. (42) for . Even though this superposition might lead to being not exactly equal to at any point, it can still lead to an overall better description of the full momentum dependence, by being almost equal in a larger region. The values of the coefficients in the ansatz, i.e., and , depend on this choice.

Let us now compare two popular choices, namely the derivative expansion about , and a projection at various values for . Working within a derivative expansion about , one extracts the flow of the -th coefficient of the polynomial by the -th derivative of Eq. (34), evaluated at . Specifically, for the chosen ansatz in Eq. (35) together with Eq. (36), this yields:

 βDEGψ=2√GψβV(0),βDEσmod=−12β′V(0). (39)

This expansion ensures that and its derivative are equal to and its derivative at . However, the derivative expansion to this order does not satisfy this equality away from . This simply means that higher-order terms in the derivative expansion around are generated by the flow. By the evaluation at a single point in , this scheme is very sensitive to local fluctuations at , which might cause deviations for larger momenta.

Alternatively, we can choose finite momenta, e.g., , to extract one of the beta functions. Equating and at and , and solving for the beta functions yields

 β(0,1)Gψ= (40) β(0,1)σmod= βV(12)−βV(1). (41)

In this scheme, the beta functions and by construction are equal at and . Thus, it provides an interpolation between these momenta, while the derivative expansion provides an extrapolation from onwards. The same projection schemes can analogously be applied to other -point functions, including the anomalous dimensions. We will refer to the projection at different values for the momentum by -sample-point projection in the following. More specifically, starting from Eq. (40), the beta functions for and take the following form

 βGψ= 2√Gψ[2C(k22)Vtrunc(12)−C(k2)Vtrunc(1) +2Flow(3)ψ(k22)−Flow(3)ψ(k2)], (42) βσ= 2σ+12√Gψ(Vtrunc(1)(−C1(k2)+βGψ2Gψ) −Flow(3)ψ(k2)), (43)

with

 C(p2) = 1+12ηh(p2)+ηψ(p2). (44)

In practice, the ingredients to evaluate the beta functions are the following: is given by the sum of diagrams in Fig. 1 which uses xAct Brizuela et al. (2009, 2009); Martín-García (2008); Martín-García et al. (2007, 2008) as well as the FORM-tracer Cyrol et al. (2017), is given by Eq. (36) and the anomalous dimensions are extracted from a projection of the corresponding two-point functions at , as in Christiansen et al. (2016, 2015); Meibohm et al. (2016); Denz et al. (2018).

We now provide our motivations for using projections with and sampling points. The derivative expansion at for the gravity-matter avatars of the Newton coupling does not capture all properties of the flow in a quantitatively reliable way cf. the discussion in Eichhorn et al. (2018a). In particular, a derivative expansion of the Einstein-Hilbert truncation at , together with a momentum-independent anomalous dimension for the graviton results in a slightly screening property of gravity fluctuations on the Newton coupling. We expect that at higher orders in the truncation, the derivative expansion becomes quantitatively reliable, but in our truncation projections at finite momenta are preferable.

Instead, an expansion at finite momenta is expected to be more stable in small truncations. This is easiest to appreciate when thinking of the flow equation in terms of a vertex expansion: The -point functions that enter the flow depend on momenta. Of these, one becomes the loop momentum in the flow equation. Due to the properties of the regulator, the momentum integral over the loop momentum is peaked at . Accordingly, the flow depends on the vertex at a finite momentum, not vanishing momentum, cf. Fig. 2.

Accordingly, a good approximation of the full flow might require higher orders in the derivative expansion around than in projection schemes at finite momentum. For technical simplicity, a symmetric point where the magnitudes of all momenta at the vertex are chosen to be the same nonzero value is preferable, although the example in Fig. 2 showcases that a non-symmetric point is likely to most accurately capture the momentum-dependence of the vertex as it is relevant for the feedback into the flow equation.

We point out that for this type of projections, a one-to-one mapping between the couplings extracted in this way and the couplings of the action written in a derivative expansion in terms of curvature invariants, as it is usually done, becomes more involved. For the derivative expansion about , this mapping is one-to-one. Specifically, projecting onto a term at finite yields a different result than projection at vanishing . This remains the case even in untruncated theory space, where the couplings in a derivative expansion around zero momentum and the couplings in a projection at finite momenta satisfy a nontrivial mapping onto each other. In an untruncated theory space, such a difference in the choice of basis does not matter for the universal properties of the fixed point. In truncations, such choices can make a difference, as some expansions are better suited to capturing the flow already in small truncations. One might tentatively interpret the results in the Einstein-Hilbert truncation and small extensions thereof Christiansen et al. (2016, 2015); Meibohm et al. (2016); Denz et al. (2018); Eichhorn et al. (2018a); Christiansen et al. (2018b); Eichhorn et al. (2018b) as implying that projections with sampling points are preferred over the derivative expansion about vanishing momentum.

For the fermion-gravity vertex, we consider the following three approximations

• projection: We set in (Eq. (35)) and in (Eq. (36)) and project onto using the projection point . In analogous systems, such a projection has been called bilocal, Christiansen et al. (2018b); Eichhorn et al. (2018a, b).

• with projection: We set in (Eq. (35)) and in Eq. (36) and project onto using the two projection points and , as in Eq. (42).

• with projection: This projection contains an extension of the truncation by , and uses Eq. (42) and Eq. (43), based on the full expressions for (Eq. (35)) and (Eq. (36)).

For completeness let us explain how we extract the remaining gravitational couplings, , , , and the wave-function renormalizations, , . An analogous parameterization to Eqs. (28) and (35) holds for the two-fermion, as well as the two- and three-graviton vertices. Thus, in all approximation schemes under consideration, we use a projection with sample-points of and at and for these couplings, i.e., we define them as follows:

 Zh(p2)= NZh(p2)∫x,y[Γk[Φ;¯g=δ]←δδhμν(x)←δδhρσ(y)]Φ=0δ(x)eipy∫x′,y′δ(x′)eipy′[SEH[g=δ+hTT]←δδhTTμν(x′)←δδhTTρσ(y′)]hTT=0, Zψ(p2)= ¯λ2= (45) ¯λ3= N¯λ3∫x,y,z[Γk[Φ;¯g=δ]←δδhμν(x)←δδhρσ(y)←δδhκλ(z)]Φ=0δ(x)∫x′,y′,z′δ(x′)[SEH[g=δ+hTT]←δδhTTμν(x′)←δδhTTρσ(y′)←δδhTTκλ(z′)]hTT=0, ¯Gh= N¯Gh∫x,y,z[Γk[Φ;¯g=δ]←δδhμν(x)←δδhρσ(y)←δδhκλ(z)]Φ=0δ(x)(eip2y+ip3z−1) ×∫x′,y′,z′δ(x′)eip2y′+ip3z′[SEH[g=δ+hTT]←δδhTTμν(x′)←δδhTTρσ(y′)←δδhTTκλ(z′)]\scalebox0.85$hTT=0¯λ=0p2=k2$,

where the momenta and are evaluated at the symmetric point for three momenta with and the normalizations , , , and are defined such, that when we plug the from Eqs. (20), (21), (22) and (23) into Eq. (45) we get the corresponding coupling.

This projection is equivalent to the bilocal evaluation of the pure-graviton vertices, as employed, e.g., in Christiansen et al. (2016, 2015); Meibohm et al. (2016); Denz et al. (2018); Reichert (2018).

## Iv Asymptotic safety for one flavor

Phenomenologically, fermion-gravity systems with are of most interest, as this is the number of Dirac fermions in the Standard Model, extended by three right-handed neutrinos. There are indications Donà and Percacci (2013); Donà et al. (2014); Meibohm et al. (2016); Eichhorn and Lippoldt (2017); Alkofer and Saueressig (2018); Alkofer (2019) that such a fermion-gravity system with features an asymptotically safe fixed point that is continuously connected to the pure-gravity one. We explore this hypothesis further, and therefore start by exploring a small deformation of the pure-gravity universality class by fermions.

In this section, we aim at answering three key questions:

1. Is there a fixed point in the fermion-gravity system that is robust under extensions of the truncation and changes of the projection scheme?

2. Is the nonminimal coupling nonzero at the fixed point, and how large is its “backreaction” onto the minimally coupled system?

3. Do the avatars of the Newton coupling exhibit effective universality at this fixed point?