Pregeometry and Euclidean Quantum Gravity

Available online at www.sciencedirect.com ScienceDirect

Nuclear Physics B 971 (2021) 115526 www.elsevier.com/locate/nuclphysb

Pregeometry and euclidean quantum gravity

Christof Wetterich

Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany Received 7 May 2021; received in revised form 6 August 2021; accepted 24 August 2021 Available online 26 August 2021 Editor: Stephan Stieberger

Abstract Einstein’s general relativity can emerge from pregeometry, with the metric composed of more fundamental fields. We formulate euclidean pregeometry as a SO(4) - Yang-Mills theory. In addition to the gauge fields we include a vector field in the vector representation of the gauge group. The gauge -and diffeomorphism -invariant kinetic terms for these fields permit a well-defined euclidean functional integral, in contrast to metric gravity with the Einstein-Hilbert action. The propagators of all fields are well behaved at short distances, without tachyonic or ghost modes. The long distance behavior is governed by the composite metric and corresponds to general relativity. In particular, the graviton propagator is free of ghost or tachyonic poles despite the presence of higher order terms in a momentum expansion of the inverse propagator. This pregeometry seems to be a valid candidate for euclidean quantum gravity, without obstructions for analytic continuation to a Minkowski signature of the metric. © 2021 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction

In a modern view, quantum field theories are defined by functional integrals. For many models of particle physics one can employ an euclidean functional integral and continue analytically to Minkowski signature. No well defined functional integral is known for a continuum formulation of quantum gravity based on the metric field. Such a functional integral would require the explicit specification of the microscopic or classical action. For a formulation in terms of the metric any action involving only a finite number of derivatives is problematic, however. The Einstein-Hilbert action with a cosmological constant involves up to two derivatives of the metric. The restriction

E-mail address: [email protected]. https://doi.org/10.1016/j.nuclphysb.2021.115526 0550-3213/© 2021 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. C. Wetterich Nuclear Physics B 971 (2021) 115526 to two derivatives is not compatible with a renormalizable theory. Furthermore, the euclidean action is not bounded from below, such that no corresponding euclidean functional integral can be defined. Adding invariants with up to four derivatives of the metric leads to a renormalizable theory [1–3]. Also the euclidean action can be bounded from below for appropriate signs of the couplings, such that an euclidean functional integral can be defined. The problem arises now from the presence of instabilities in the propagators in flat or weakly curved geometries. In flat space, ghosts or tachyons cannot be avoided for any finite polynomial momentum expansion of the inverse propagator beyond quadratic order in momenta. This implies that the addition of terms with a finite number of derivatives cannot cure the problem. Quantum gravity based on the metric may be a non-perturbatively renormalizable quantum field theory. This is the case if the functional flow of a scale-dependent effective action [4] admits an ultraviolet fixed point. Asymptotically safe quantum gravity [5–10]is defined by choosing this ultraviolet fixed point for a definition of the short distance behavior of quantum gravity. The fixed point behavior permits to extrapolate this short distance behavior to arbitrarily short distances or high momenta. A fixed point corresponds to a scaling solution of an exact functional flow equation [4]. Such scaling solutions have been found for many approximated or “truncated” forms of the scale- dependent effective action [11]. This makes it plausible that an ultraviolet fixed point indeed exists, rendering metric quantum gravity non-perturbatively renormalizable. A functional integral can be defined formally as a solution of the exact flow equation. In practice, this is of little help if the solution gets too complicated. This is presently the case for the functional renormalization group approach. The fixed point involves many invariants, and it is not clear what approximation is needed in order to obtain a well defined propagator for the graviton without ghost or tachyonic instabilities [12]. This explains why so far no proposal for a simple classical action defining the functional integral for quantum gravity has been formulated in this approach. One possibility to avoid these problems is a formulation of the functional integral in terms of fields different from the metric. This approach stays within the setting of quantum field theories. In analogy to quantum chromodynamics (QCD) for the strong interactions in particle physics, the short distance behavior - gluons and quarks in QCD -is described by fields different from the metric. The metric arises as a collective or composite field in an effective low energy theory, in analogy to mesons and baryons in QCD. This type of quantum field theory with an emergent metric may be called “pregeometry” [13]. The most radical approach suggests to use only fermions as fundamental degrees of freedom [13–15]. In spinor gravity a formulation with local Lorentz symmetry and diffeomorphism invariance has indeed been achieved [16–18]. (For first formulations with global Lorentz symmetry see ref. [19,20].) The vierbein and the metric have been constructed as composites of the fermions. While conceptually rather attractive, practical com- putations in this purely fermionic setting have not yet advanced very much, due to the structure of the action containing only invariants with a rather high number of fermions. The present work proposes a formulation of euclidean pregeometry as a SO(4) - Yang-Mills theory. Besides fermions, which we do not discuss explicitly in the present paper, the degrees of freedom of pregeometry are the six gauge fields Aμ of the SO(4) - gauge symmetry, and four additional vector fields e˜μ which belong to the vector representation of SO(4). This model has been discussed earlier [21,22], see also ref. [23–27]for related ideas. A diffeomorphism invariant action is based on the gauge invariant kinetic terms of the two sorts of vector fields Aμ and e˜μ,

2 C. Wetterich Nuclear Physics B 971 (2021) 115526 Z 1 S = e˜ F F mn + U˜ U˜ m e˜ μe˜pρ e˜ νe˜qσ. (1) 8 μνmn ρσ 4 μνm ρσ p q x ˜ m Here Fμνmn is the covariant field strength of the gauge boson Aμmn =−Aμnm and Uμν = m m Dμe˜ν the covariant derivative of the other vector field. With e˜ = det(e˜μ ) > 0we can de- ˜ μ ˜ μ ˜ n = n fine em by the relation em eμ δm. The SO(4) - indices m, n, p, q are raised and lowered mn with δ or δmn, such that their position can be regarded as arbitrary. This action has a single dimensionless coupling Z which is related to the gauge coupling. In the present paper we demonstrate that the euclidean action involving the sum of the invariant kinetic terms for Aμ and e˜μ is bounded from below. The euclidean functional integral is well defined for this type of pregeometry. The short distance propagators show neither ghosts nor tachyons. These stability properties may be expected since only terms with two derivatives of the fields are included in the effective action. The metric is a composite field defined as a SO(4) - invariant bilinear of the vector fields e˜μ. After a suitable rescaling the dimensionless vector fields eμ play the role of generalized vierbeins. In contrast to Cartan’s geometry [28]the covariant derivative of eμ does not vanish, however. This distinguishes the present approach from many formulations where the gauge fields are associated to the spin connection which is a functional of the vierbein [28–39]. In our approach both the vierbein and the gauge fields contain propagating degrees of freedom beyond the ones contained in the metric. They are crucial for the different short distance behavior. A formulation of generalized gravity in Minkowski space that is close to our setting has been developed in refs. [40–45]. (For first functional renormalization group investigations of Cartan geometry see refs. [46–48], and for alternative ideas on diffeomorphism invariant gauge theories cf. ref. [49,50].) The classical field equations have a solution with vanishing gauge fields Aμ and constant vierbeins eμ, leading for the composite metric gμν to a geometry of flat space, gμν = δμν . This solution entails a “spontaneous symmetry breaking” of the SO(4) - gauge symmetry. As a consequence of this “generalized Higgs mechanism” the gauge bosons acquire a mass term and mix with the other vector field e˜μ. Linear combinations of both fields decouple from the effective low energy theory, which can be defined for momenta smaller than the gauge boson mass. Field equations and effective low energy theory should be discussed in terms of the quantum effective action which includes all effects of quantum fluctuations. We will not perform here a computation of the quantum effective action. We rather make the assumption that the quantum fluctuations generate for the effective action all terms allowed by diffeomorphism and gauge symmetries. In particular, this includes a term linear in the derivatives of the gauge fields, and a cosmological constant. There are also additional terms with two derivatives of Aμ. Terms with more than two derivatives will be neglected in the present paper. The effective low energy theory is dominated by the terms with a small number of derivatives. In leading order this turns out to be general relativity for the metric, with an Einstein-Hilbert action and a possible cosmological constant that we assume to be very small, as required by observation. General relativity emerges naturally from the present formulation of pregeometry. This holds provided that the effective theory is stable in the sense that there are no tachyonic instabilities with mass terms of the order of the Planck mass or larger. We will see that this requires for the effective action the presence of invariants beyond those for the classical action (1). Beyond the leading Einstein-Hilbert term the effective low energy theory contains terms with more than two derivatives for the metric. An expansion in the number of derivatives yields at fourth order an effective theory of the type of ref. [1]. One may therefore wonder what happens to the ghost instability in the graviton propagator for this fourth order gravity.

3 C. Wetterich Nuclear Physics B 971 (2021) 115526

In our formulation of pregeometry the graviton fluctuation is a linear combination of fields contained in Aμ and e˜μ. As a result of the diagonalization of the propagator matrix we find that the dependence of the inverse graviton propagator on the squared momentum q2 is given by a function 2 − m G 1 (q2) = (Z + 1)q2 + m2 − M2 grav 8 (2) q2 − (Z − 1)q2 + m2 − M2 2 + 4 m2 − M2 2 . m2

It involves three model parameters, namely m2 which multiplies the kinetic term for the vierbein and determines the effective mass for the gauge bosons, the coupling Z which multiplies the gauge boson kinetic term and equals the inverse squared gauge coupling, and M2 which multiplies the term linear in derivatives and can be associated with the squared Planck mass. Here m2 only sets the overall mass scale, such that only two parameters M2/m2 and Z specify a given model. The absence of ghost or tachyonic instabilities is guaranteed for the parameter range

2 2 0

For this parameter range the graviton propagator is well behaved with a single pole at q2 = 0 and without any instability. The ghost in fourth order gravity is an artefact of the truncated polynomial expansion [12]. The euclidean classical action can be continued analitically to Minkowski signature. The same holds for the graviton propagator (2). So far we have not found any obstruction to analytic continuation. We may therefore consider our model of pregeometry as a proposal for a functional integral for quantum gravity. A recent investigation [51] reveals that the field equations derived from the effective action with Minkowski signature lead to consistent and realistic cosmology if a scalar field is added. This includes an inflationary epoch and dynamical dark energy. Our model of pregeometry is introduced in sect. 2, where we also show that flat space solves the classical field equations. Sect. 3 discusses the generalized Higgs mechanism. It proceeds to a decomposition of the fluctuations around flat space according to different representations of its symmetry. This is the basis for the stability analysis of the classical theory which we address in sect. 4. There we establish that no ghosts or tachyons are present in our model. In sect. 5 we turn to the quantum effective action and the emergence of general relativity as an effective theory for small momenta and curvature. The graviton propagator is discussed in detail in sect. 6. Together with a stable graviton our model of pregeometry also contains a stable massive spin-two particle. Sect. 7 turns to the sector of scalar fields. For suitable values of the couplings multiplying the invariants in the effective action we find that the scalar sector is stable as well. Flat space is not the absolute minimum of the euclidean effective action in the space of all possible field configurations. It may, however, be the minimum in the space of solutions to the field equations. In sect. 8 we add the effects of a cosmological constant. Sect. 9 finally discusses our findings and possible extensions in various directions.

4 C. Wetterich Nuclear Physics B 971 (2021) 115526

2. Pregeometry

Similar to the electroweak or strong interactions our model is a non-abelian local gauge theory or Yang-Mills theory. For our euclidean setting the gauge group is SO(4). The particularity as compared to the gauge theories in particle physics is the presence of an additional vector field m eμ (x) in the vector representation of SO(4). We restrict the space of field configurations in the m m functional integral to e = det(eμ ) > 0, where eμ is considered as a 4 × 4 - matrix. This allows μ for the definition of the inverse matrix em , and the construction of diffeomorphism invariant kinetic terms for both the gauge fields Aμ and the vector fields eμ.

2.1. Diffeomorphism invariant Yang-Mills theory

Our starting point is a SO(4) - gauge theory with gauge ﬁelds

Aμmn =−Aμnm ,n,m= 0, ··· , 3 , (5) and ﬁeld strength

p p Fμνmn = ∂μAνmn − ∂νAμmn + Aμm Aνpn − Aνm Aμpn . (6)

The double-index (m, n) = z labels the six gauge bosons Aμz in the adjoint representation of m SO(4). We further include vector ﬁelds eμ in the vector representation of SO(4). Accordingly, the covariant derivative reads

m m m σ m m n Uμν = Dμeν = ∂μeν − μν eσ + Aμ neν . (7) These vector ﬁelds are restricted to obey

m e = det(eμ )>0 , (8) m where the components eμ are considered as elements of a regular 4 × 4 matrix. This permits μ the definition of inverse vector fields em , with m ν = ν μ n = n eμ em δμ ,em eμ δm . (9) m m While the vector field e˜μ has the dimension mass usual for vector fields, the field eμ = −1 m m e˜μ is dimensionless. We introduce a composite metric gμν as a bilinear in the vector field and note that for e = 0 its inverse gμν exists, = m n μν = μ gμν eμ eν δmn,ggνρ δρ . (10) σ The connection μν in Eq. (7)is the Levi-Civita connection formed with the composite metric gμν . Inserting eq. (10)it can be expressed in terms of the vector field and its derivatives. For m m Dμeν = 0the vector field eμ can play the role of the usual vierbein. We will not impose this constraint here such that the vector field contains additional degrees of freedom. The field strength (6) and the covariant derivative (7) transform as tensors. These tensors can be used as basic building blocks for the construction of invariant kinetic terms for the gauge m fields Aμmn and the vector field eμ . We define a diffeomorphism invariant action 4 S = d xe(LF + LU ), (11)

5 C. Wetterich Nuclear Physics B 971 (2021) 115526 with gauge field kinetic term Z L = F F μνmn , (12) F 8 μνmn and vector field kinetic term m2 L = U U μνm . (13) U 4 μνm m The mass m arises uniquely from the rescaling of e˜μ and should not be regarded as a free μν parameter. World (space-time) indices μ, ν are raised and lowered with g and gμν given by mn eq. (10), while “Lorentz indices” m, n are raised and lowered with δ and δmn. We can convert m μ Lorentz indices to world indices by multiplication with eμ or em , m Uμνρ = Uμν eρm . (14) m For Uμν = 0the covariant derivatives commute with the raising and lowering of world indices or Lorentz indices, but not with the conversion from world to Lorentz indices, m m Dσ Uμνρ = Dσ Uμν eρm + Uμν Uσρm . (15) Because of their analogous role to the vierbein in Cartan’s geometry [28]we will call the vector m fields eμ “generalized vierbeins”, or simply “vierbeins”. Two different connections are present in our setting, the gauge connection Aμmn and the ρ geometric connection μν . The tangent bundle and the gauge bundle are, a priori, not related. For example, one can have the metric corresponding to a sphere and nevertheless vanishing gauge fields, Aμmn = 0. This permits, for example, to have massless fermions (zero eigenvalues of the fermionic kinetic operator) on a sphere [21]. A relation between the connections can be formulated by the identity

Uμνρ = ωμνρ − Aμνρ , (16) with ωμνρ related to the “spin connection” formed from the vector ﬁeld 1 m m m m ωμνρ = eμm ∂ρeν − ∂νeρ + eνm ∂ρeμ − ∂μeρ 2 m m m n + eρm ∂μeν − ∂νeμ = eν eρ ωμmn . (17)

This expresses the Levi-Civita connection in terms of the vierbein by σ m ωμνρ = μρ gσν − eνm∂μeρ . (18) We observe the antisymmetry in the last two indices

ωμρν =−ωμνρ ,Aμρν =−Aμνρ ,Uμρν =−Uμνρ . (19)

For a covariantly conserved vector field, Uμνρ = 0, the gauge field equals the spin connection formed from the vierbein. This is the setting of Cartan’s geometry [28]. One can define the torsion tensor m m Tμνρ = Aρμν − Aνμρ − eμm(∂νeρ − ∂ρeν ). (20) For Minkowski signature and identified gauge and tangent bundles this is the object investigated in refs. [40–45]. The covariant vierbein derivative can be expressed in terms of the torsion as 1 U = (T − T + T ). (21) μνρ 2 μνρ νρμ ρνμ

6 C. Wetterich Nuclear Physics B 971 (2021) 115526

2.2. Boundedness of the action and euclidean functional integral

The action (11)is bounded from below, S 0, as long as m2 0, Z 0. In order to see this, we ﬁrst show that the composite metric has only positive eigenvalues. Since gμν is a real symmetric matrix, it can be diagonalized by an orthogonal transformation O,

μ ν μ m ν n g˜ρσ =Oρ Oσ gμν = Oρ eμ Oσ eν δmn m n =˜eρ e˜σ δmn , (22) m μ m with e˜ρ = Oρ eμ . The diagonal elements are indeed positive (no sum over ρ) m 2 g˜ρρ = (e˜ρ ) > 0 . (23) m

For e>0 one can exclude vanishing eigenvalues g˜ρρ . We next write for z = (m, n) μνmn αβ Fμνmn = Fαz ,FμνmnF = FαzG Fβz , (24) z with α = (μ, ν), β = (ρ, σ)double indices, and real symmetric matrix

Gαβ = gμρ gνσ = Gβα . (25) αβ The expression LF is positive semidefinite if G has only positive eigenvalues. This is indeed the case, as can be seen by diagonalizing Gαβ by an orthogonal transformation. We can choose the required orthogonal matrix as the direct product of the matrices that diagonalize gμν, result- ˜ αα μμ νν ing in G =˜g g˜ > 0. The same argument shows LU 0, with z = m in this case. With both LU and LF positive semidefinite and e>0, the action is indeed positive semidefinite, S ≥ 0. The action vanishes precisely if both LF and LU vanish. The positivity of the classical action (11)for Z>0, m2 > 0 can also be seen from the iden- μνmn pqmn μνm pqm μ ν tities FμνmnF = FpqmnF , UμνmU = UpqmU , where Fpqmn = ep eq Fμνmn. Transmutations of world indices μ, ν to Lorentz indices p, q are achieved by multiplication with the vierbein or inverse vierbein. Since index contractions of Lorentz indices are simply index summations, both LF and LU are squares. With positive semidefinite classical action S an euclidean functional integral can be defined by integrating over unconstrained gauge fields Aμ and vector fields eμ obeying the constraint (8). The constraint e>0is invariant under local SO(4) - gauge transformations and diffeomorphisms. In presence of these two local symmetries the action needs a regularization, either by the usual gauge fixing procedure for a continuous formulation, or perhaps by a lattice formulation that still has to be found. We consider the action (11)as a well defined and simple starting point for an investigation of quantum fluctuations in euclidean gravity. The analytic continuation to a Minkowski signature will be discussed in sect. 6. Configurations with a constant vierbein and vanishing gauge fields include the “flat space configuration” m = m = = eμ δμ ,gμν δμν ,Aμmn 0 . (26) m These configurations yield Uμν = 0 and are therefore minima of the action (11). They are solutions of the corresponding field equations that we discuss in more detail in sect. 5.

7 C. Wetterich Nuclear Physics B 971 (2021) 115526

3. Generalized Higgs mechanism

In order to understand the physical content of our model at short distances we decompose the fluctuations of the vector field and the gauge fields around flat space and zero gauge fields. We work in euclidean flat space. Analytic continuation to Minkowski space can be performed at the end. For sufficiently short distances one can neglect any effects of a given geometry, as m encoded in eμ or gμν , or of a given gauge field configuration Aμmn. For the short distance limit m = m = the flat space configuration eμ δμ , Aμmn 0is always an appropriate starting point for a field-expansion. The decomposition involves different representations of the “euclidean Lorentz group” SO(4) that cannot mix in quadratic order. Mixing is observed, however, between modes belonging to the same representation. This renders the discussion of stability somewhat complex. In the present section we discuss the general structure of the mixing and focus subsequently on the graviton propagator and the scalar propagator in the next section. We will find a stable graviton propagator without ghosts or tachyons. The discussion will be extended in sect. 5 to the quantum effective action. The graviton propagator for this extended setting is discussed in sect. 6, and scalars are addressed in sect. 7. m = m Due to spontaneous symmetry breaking by the non-vanishing value of the vierbein eμ δμ mass terms are generated for the gauge bosons. This “generalized Higgs mechanism” differs from the usual Higgs mechanism since the agent of spontaneous symmetry breaking is a vector field and not a scalar field. The mode decomposition reveals the detailed structure of the generalized Higgs mechanism by focusing on the effects for the propagators of the various fluctuation modes. The mode mixing is a characteristic feature of this generalized Higgs mechanism.

3.1. Fluctuations in ﬂat space

m = m The field equations derived from the action (11) admit as a solution flat space, eμ δμ , gμν = δμν , with vanishing gauge fields Aμmn = 0. This solution will be found to be stable, as may be expected since it corresponds to the minimum of the action. For a more detailed investigation of the stability for various modes we expand in quadratic order in Hμν and Aμmn, 1 e m = δm + H δνm . (27) μ μ 2 μν The standard kinetic term for the transverse gauge fields reads in momentum space Z eL = A (−q) q2δμν − qμqν A mn(q) . (28) F 4 μmn ν x q This dominates the high momentum behavior and ensures stability in this range. For the discus- μν sion of fluctuations world indices are raised and lowered with δ and δμν . In quadratic order the covariant kinetic term LU for the vierbein contains three pieces LU = (1) + (2) + (3) 2 = ρ LU LU LU . The first piece is a kinetic term which reads, with ∂ ∂ ∂ρ , m2 eL(1) =− H (A)∂2H (A)μν (29) U 16 μν x x + (S) 2 ν − ν (S)ρμ + (A) ν (A)ρμ 2Hμν (∂ δρ ∂ ∂ρ)H 4Hμν ∂ ∂ρH .

8 C. Wetterich Nuclear Physics B 971 (2021) 115526

(S) (A) Here Hμν = (Hμν + Hνμ)/2 and Hμν = (Hμν − Hνμ)/2are the symmetric and antisymmetric parts of the fluctuation Hμν. The second piece acts as a mass term for the gauge bosons. Without mixing effects (see below), it would generate an equal mass for all gauge bosons m2 eL(2) = A Aμmn . (30) U 4 μmn Finally, the third piece is a type of source term for the gauge bosons, (3) =− μνρ eLU JμνρA , (31) with m2 J = ∂ H (A) + ∂ H (S) − ∂ H (S) . (32) μνρ 2 μ νρ ρ μν ν μρ m = m The field value eμ δμ breaks the SO(4) - gauge symmetry spontaneously. Similarly to the Higgs mechanism, this spontaneous symmetry breaking generates a mass term for the gauge m bosons. In contrast to the Higgs mechanism the field eμ is a vector, not a scalar. Flat space preserves a global SO(4) -“Lorentz” symmetry of simultaneous SO(4) - gauge rotations and SO(4) - coordinate transformations. It is this residual symmetry that we employ for the mode expansion.

3.2. Decomposition of vierbein ﬂuctuations

For an analysis of the kinetic term (30)for the vierbein we employ the decomposition which orders the scalar ﬂuctuations into a physical and a gauge degree of freedom [52],

H (S) = t + ∂ κ + ∂ κ μν μν μ ν ν μ 1 ∂ ∂ ∂ ∂ + δ − μ ν σ + μ ν u, (33) 3 μν ∂2 ∂2 and (A) = + − Hμν bμν ∂μγν ∂νγμ . (34)

Here tμν , bμν , κμ and γμ are all transversal, μ ν μ ν ∂ tμν = ∂ tμν = ∂ bμν = ∂ bμν = 0 , μ μ μ ∂ κμ = ∂ γμ = 0 ,tμ = 0 . (35) The kinetic piece reads in momentum space m2 eL(1) = 2t (−q)q2tμν(q) + b (−q)q2bμν(q) U 16 μν μν x q + 2 κ (−q)− γ (−q) q4 κμ(q) − γ μ(q) μ μ 2 + σ(−q)q2σ(q) . (36) 3 4 The factor q for κμ −γμ is a consequence of the particular normalization of these ﬁelds. Indeed, 2 we can absorb a factor 4q in a dimensionless normalization of κμ and γμ. All kinetic terms are

9 C. Wetterich Nuclear Physics B 971 (2021) 115526

∼ 2 (1) + then positive and increase q . The kinetic term LU does not involve the combination κμ γμ μ and the scalar u. Similarly, LF does not involve the longitudinal components ∂ Aμmn of the gauge ﬁelds. These ﬁelds correspond to gauge degrees of freedom.

3.3. Decomposition of gauge ﬁeld ﬂuctuations

In flat space we need no longer to distinguish between world and Lorentz indices. Covariant derivatives become partial derivatives. We decompose the linear gauge fields Aμνρ into transversal modes Bμνρ and longitudinal modes Lνρ , μ Aμνρ = Bμνρ + ∂μLνρ ,∂Bμνρ = 0 . (37) Using the projector ∂ ∂ν P ν = δν − μ ,∂μP ν = 0 ,Pν∂ = 0 , (38) μ μ ∂2 μ μ ν we can write = σ = σ Bμνρ Pμ Aσνρ Pμ Bσνρ . (39) The transversal fluctuations can be decomposed as

= 1 στ − Bμνρ ενρ (Pμσ vτ Pμτ vσ ) 4 (40) 1 + (P w − P w ) + D , 3 μν ρ μρ ν μνρ where vμ and wμ are (reducible) four-vectors. The decomposition of the remaining part Dμνρ into irreducible representations is given by 1 D = (∂ E − ∂ E ) + C . (41) μνρ 2 ν μρ ρ μν μνρ The transversal traceless symmetric tensor Eμν obeys μ μν ∂ Eμν = 0 ,δEμν = 0 ,Eμν = Eνμ , (42) and Cμνρ is subject to the constraints μ μνρσ Cμνρ =−Cμρν ,∂Cμνρ = 0 ,ε Cμνρ = 0 , μν μρ ˜ νρ δ Cμνρ = δ Cμνρ = 0 , Pστ Cμνρ = 0 . (43) We have introduced here a further projector 1 P˜ νρ = (∂ ∂νδρ − ∂ ∂νδρ − ∂ ∂ρδν + ∂ ∂ρδν ), (44) στ 2∂2 σ τ τ σ σ τ τ σ which obeys ˜ αβ ˜ νρ = ˜ νρ ˜ νρ = Pστ Pαβ Pστ , Pνρ 3 , 1 ∂τ P˜ νρ = (∂ρδν − ∂νδρ ). (45) στ 2 σ σ This results in the identities

10 C. Wetterich Nuclear Physics B 971 (2021) 115526

Table 1 Physical and gauge modes in the expansion of vierbein and gauge fields in flat space. We indicate tensors (T ), transverse vectors (V ) and scalars S, with the corresponding number of independent modes. In the vector sector the physical modes are actually linear combinations, given by the ones appearing in eqs (65), (66). Setting κμ = lμ = Mμν = 0 we can interpret γμ and bμν as substitute for these linear combinations. Physical modes Gauge modes Type vierbein gauge field vierbein gauge field

tμν Eμν , Cμνρ T ,5 σ v˜, wu˜ S,1 (t) (t) γμ, bμν vμ , wμ κμ Mμν , lμ V ,3

˜ νρ Pστ Dμνρ = (∂σ Eμτ − ∂τ Eμσ ), (46) and

˜ νρ Pστ (∂νEμρ − ∂ρEμν) = (∂σ Eμτ − ∂τ Eμσ ). (47)

We can therefore see Cμνρ as a projection,

˜ στ Cμνρ = Dμνρ − Pνρ Dμσ τ . (48)

Out of the 24 components Aμνρ six degrees of freedom Lνρ =−Lρν are longitudinal degrees of freedom. Thus there are 18 transversal modes Bμνρ. The four modes vμ correspond to μν the totally antisymmetric part A[μνρ], while the four modes wρ account for the trace Aμνρδ . ˜ There remain 10 modes for Dμνρ. The projector P eliminates half of them, and the traceless transversal symmetric tensor Eμν accounts indeed for ﬁve modes. The other ﬁve modes correspond to Cμνρ =−Cμρν , which indeed is subject to 19 constraints. The vectors vμ and wμ can be decomposed into transversal vectors and scalars

= (t) + ˜ = (t) + ˜ vμ vμ ∂μv, wμ wμ ∂μw, μ (t) = μ (t) = ∂ vμ 0 ,∂wμ 0 . (49)

The irreducible representations of the transversal modes Bμνρ are 2 × 5 + 2 × 3 + 2 × 1. In particular, the scalar part of Bμνρ reads

1 1 B(s) = ε τ ∂ v˜ + (δ ∂ w˜ − δ ∂ w)˜ . (50) μνρ 2 μνρ τ 3 μν ρ μρ ν Finally, the six longitudinal modes are two triplets

Lνρ = Mνρ + ∂νlρ − ∂ρlν , (51) with

ν ρ ν ∂ Mνρ = ∂ Mνρ = 0 ,∂lν = 0 . (52) We summarize the various modes in Table 1, with a slight abuse of notation for the physical modes in the vector sector.

11 C. Wetterich Nuclear Physics B 971 (2021) 115526

3.4. Generalized Higgs mechanism

In contrast to spontaneous symmetry breaking by a scalar field the expectation value of a vector field leads to a mixing of the gauge field fluctuations and the vierbein fluctuations. This (3) particular feature of the generalized Higgs mechanism is due to the “source term” LU . It is our aim to separate the various physical modes. For this purpose we have to diagonalize the inverse propagator which corresponds to the second functional derivative of the action. In other words, we have to find the field combinations for which the quadratic expansion of the action in the fluctuations becomes diagonal. We end this section by writing the quadratic expansion of the action in terms of the various fields appearing in our decomposition. In the next section we will proceed to the diagonalization. (3) The source term LU mixes the gauge fields Aμνρ and the vierbein fluctuations Hμν. In quadratic order one finds from eqs. (31), (32) m2 eL(3) =− Aμνρ ∂ H (A) + ∂ H (S) − ∂ H (S) U 4 μ νρ ρ μν ν μρ x x m2 = Lνρ ∂2H (A) + ∂ ∂μH (S) − ∂ ∂μH (S) 4 νρ ρ μν ν μρ (53) x m2 +2∂ BμνρH (S) = (Y + Y ). ρ μν 4 l t x

The longitudinal part Yl concerns the mixing effects for the longitudinal gauge bosons. By use of the decomposition (34), (51), this results in

νρ 2 μ 4 Yl = M ∂ bνρ + 2l ∂ (κμ − γμ). (54)

The transversal part Yt accounts for the mixing of the transversal gauge bosons. For its evalua- μνρ tion we decompose the tensor ∂ρB into its traceless symmetric, antisymmetric, and (modiﬁed) trace parts 1 ∂ Bμνρ = B˜ (S)μν + B˜ (A)μν + P μνb,˜ (55) ρ 3 with

˜ (S)μν ˜ (S)νμ ˜ (S)μν ˜ (A)μν ˜ (A)νμ B = B , B δμν = 0, B =−B , ˜ (S)μν ˜ (A)μν ∂μB + ∂μB = 0 . (56) Comparing with the expansion (40)we identify 1 1 B˜ (S)μν =− ∂2Eμν , B˜ (A)μν = εμνρτ ∂ v(t) , 2 2 ρ τ b˜ = ∂2w.˜ (57)

The components Cμνρ do not appear due to the identity ρ ∂ Cμνρ = 0 , (58) which follows from eqs. (45)(46).

12 C. Wetterich Nuclear Physics B 971 (2021) 115526

(S) Inserting also the expansion (34)for Hμν , the transversal part of eq. (53) reads 2 2 Y = 2B˜ (S)μν + P μνb˜ H (S) =−Eμν∂2t + w∂˜ 2σ. (59) t 3 μν μν 3 (t) (t) We observe that the twelve transversal gauge ﬁeld ﬂuctuations v , w , v˜, Cμνρ do not appear (3) (2) in LU . They do not take part in the mixing and simply acquire a mass from LU . Also the four gauge modes κμ + γμ and u are absent. For a discussion of the inverse propagator we also have to decompose the gauge boson mass (2) term LU , m2 m2 L(2) = AμνρA = (BμνρB − Lνρ ∂2L ) U 4 μνρ 4 μνρ νρ 2 =m −1 νρ 2 + μνρ + (t)μ (t) − 3 ˜ 2 ˜ E ∂ Eνρ C Cμνρ v vμ v∂ v 4 2 2 4 2 + w(t)μw(t) − w∂˜ 2w˜ − Mνρ ∂2M + 2lμ∂4l . (60) 9 μ 3 νρ μ (1) The corresponding expression for LU is given by eq. (36) 2 (1) =m −1 μν 2 − 1 μν 2 LU t ∂ tμν b ∂ bμν 4 2 4 (61) 1 1 + (κμ − γ μ)∂4(κ − γ ) − σ∂2σ . 2 μ μ 6 3.5. Mode mixing

Taking things together, LU involves a couple of independent pieces 2 m L = L + L ˜ + L + L + L . (62) U 4 tE σ w bM κγl m They can be diagonalized separately. For the transversal traceless symmetric tensors t and E one has 1 L =− (tμν + Eμν)∂2(t + E ), (63) tE 2 μν μν while in the scalar sector σ and w˜ are connected 1 2 L ˜ =− (σ − 2w)∂˜ (σ − 2w)˜ . (64) σ w 6 The sector for the longitudinal gauge bosons involves 1 L =− (bμν − 2Mμν)∂2(b − 2M ), (65) bM 4 μν μν and 1 L = (κμ − γ μ + 2lμ)∂4(κ − γ + 2l ). (66) κγl 2 μ μ μ The remaining part Lm involves only mass terms for the gauge bosons and no mixing

3 4 L = CμνρC + v(t)μv(t) − v∂˜ 2v˜ + w(t)μw(t) . (67) m μνρ μ 2 9 μ

13 C. Wetterich Nuclear Physics B 971 (2021) 115526

The kinetic term for the gauge bosons (28) contributes only for the transversal gauge bosons, Z Z L =− Aμνρ∂2P σ A =− Bμνρ∂2B . (68) F 4 μ σνρ 4 μνρ The decomposition is similar to the transversal part of eq. (60), replacing m2 by −Z∂2, Z 1 L = Eνρ ∂4E − Cμνρ∂2C − v(t)μ∂2v(t) F 4 2 νρ μνρ μ (69) 3 4 2 + v∂˜ 4v˜ − w(t)μ∂2w(t) + w∂˜ 4w˜ . 2 9 μ 3

At this point we have expressed both LU and LF in terms of the fields of our decomposition. The transition to momentum space is straightforward. The pieces ∼ q4 are artefacts of the particular normalization of fields in our decomposition. They can be absorbed by a momentum dependent rescaling which makes all fields in the decomposition of the vierbein dimensionless, and gives all fields in the expansion of the gauge fields the canonical dimension of mass. The high momentum behavior for all modes is then ∼ q2, as appropriate for an action that contains only two derivatives. The different blocks can now be diagonalized separately.

4. Stability of classical theory

In this section we diagonalize the propagator. The question of stability of the classical theory can be read off directly from the form of the propagators. Flat space is stable if there are neither ghosts nor tachyons and all modes are described by massive or massless particles. Sta- bility of flat space extends to stability of the high-momentum modes for arbitrary “background configurations” of the gauge fields and vierbein.

4.1. High momentum limit

Before proceeding to the diagonalization of the inverse propagator it is instructive to discuss 2 →∞ (2) (3) the limit q . In this limit LU and LU can be neglected. The infinite momentum limit becomes a good approximation for q2 m2. In consequence, the leading short distance behavior + (1) is well described by an action based on LF LU , evaluated for the flat space solution. This is a free theory with inverse propagators for the physical fluctuations proportional to q2. No instability occurs. For this free theory the analytic continuation to Minkowski space is straightforward. We therefore consider our pregeometry based on a Yang-Mills theory as a candidate for an ultraviolet completion of quantum gravity. Beyond the leading approximation the action (11)also entails interactions. First, there are the gauge interactions mediated by the gauge field Aμ. The effective gauge coupling g, with α = g2/(4π), is given by 4π α = . (70) Z It vanishes in the limit Z →∞. For large Z a perturbative treatment of the gauge interactions becomes possible. In contrast, the vierbein-mediated “gravitational interactions” cannot be switched off. We can use scaling fields [22]

14 C. Wetterich Nuclear Physics B 971 (2021) 115526

m m 2 μν −2 μν e˜μ = keμ , g˜μν = k gμν , g˜ = k g , (71) with k some “renormalization scale”. As a result one ﬁnds 1 1 1 S = e˜ F F mn + U˜ U˜ m g˜μρ g˜νσ , (72) 16π 2α μνmn ρσ γ μνm ρσ x with

˜ m m m Uμν = Dμe˜ν = kUμν , (73) and dimensionless coupling k2 γ = . (74) 4πm2 m σ A multiplicative rescaling of e˜μ does not change the Levi-Civita connection μν in the co- ˜ m variant derivative (7). It only results in a multiplicative rescaling of Uμν . As a result, γ is not m a free parameter -it can be set to any arbitrary value by a rescaling of e˜μ . In contrast to the gauge interactions, the gravitational interactions do not switch off in the limit γ → 0. Using the freedom of ﬁeld-rescaling the only remaining free parameter for the classical theory is the gauge coupling α.

4.2. Diagonalization of the propagator

Let us next consider the full classical action based on LF + LU . With an appropriate normalization, the inverse propagator for the gauge boson ﬂuctuations C, v(t), w(t) and v˜ is the standard one for massive particles in momentum space

2 − m P = G 1 = q2 + . (75) Z √ This is the inverse propagator of a particle with mass m/ Z. No instability occurs in this sector. For the traceless transversal tensors in the t − E - sector the inverse propagator matrix takes the form 2 2 2 2 − q Zq + m ,m P = G 1 = . (76) 4 m2 ,m2 The only zero eigenvalue of P occurs for q2 = 0. The eigenvalues of P are given by

2 q 2 2 λ± = Zq + 2m ± Z2q4 + 4m4 . (77) 8 These eigenvalues have dimension mass4. We can switch to the more usual normalization for bosonic fields for which the boson fields have dimension of mass and the inverse propagator has dimension mass squared. This can be achieved by a momentum dependent renormalization of the q m fields, E = E , q = |q2|, t = t , resulting in the renormalized inverse propagator Rμν 2 μν Rμν 2 μν matrix Zq2 + m2 ,mq P = . (78) R mq , q2

15 C. Wetterich Nuclear Physics B 971 (2021) 115526

The q-dependence in the renormalization of E corresponds to the standard normalization of gauge fields A, noting A ∼ qE, while the normalization of t provides a canonical mass dimension to this field. Poles in the propagator correspond to vanishing eigenvalues of PR. The condition for the existence of propagator poles reads 4 det PR = Zq = 0 . (79) We conclude that the only possible poles of the propagator occur for q2 = 0. 2 For the renormalized fields the eigenvalues of PR for general values of q are given by 1 2 2 λ± = (Z + 1)q + m 2 (80) ± (Z − 1)2q4 + 2(Z + 1)q2m2 + m4 . They correspond to the inverse propagators after diagonalization and describe the propagation of independent modes. For the high momentum behavior, q2 m2, this yields ⎧ ⎪ m2 ⎨⎪Z q2 + for Z>1 = Z − 1 λ+ ⎪ (81) ⎪ m2 ⎩q2 + for Z<1 , 1 − Z and ⎧ ⎪ m2 ⎨⎪q2 − for Z>1 = Z − 1 λ− ⎪ (82) ⎪ m2 ⎩Z q2 − for Z<1 . 1 − Z The apparent zeros in these approximate expression cannot correspond to poles in the propagators since eq. (79) tells us that all possible poles occur for q2 = 0. Near the pole at q2 = 0 one finds 4 Zq 2 2 λ− = ,λ+ = m + (Z + 1)q . (83) m2 Again, the apparent zero in λ+ is a “fake pole”, since the only possible poles for q2 = 0are associated to λ−. The function λ+(q2) has no zero. No propagating particle is associated to this mode. The zero of the inverse propagator at q2 = 0 corresponds to a double pole in the propagator. A propagator 1/q4 leads to a “secular instability” after analytic continuation to Minkowski space. It does not pose any problem for a well defined euclidean functional integral. A double pole can be considered as the boundary between stability and instability. The stability for low q2 should be discussed within the quantum effective action once quantum fluctuation effects are included. We will see below that additional terms in the effective action modify the behavior for q2 → 0. For the other limit at large q2 one may assume that the short distance behavior of the effective action is close to the classical action. The short distance fluctuations in the t - E - sector are stable. We next turn to the remaining part in the sector of physical scalar fluctuations. In the w˜ − σ - sector the inverse propagator matrix

16 C. Wetterich Nuclear Physics B 971 (2021) 115526 ⎛ ⎞ m2 2 + 2 − 2 ⎜ Zq m , ⎟ = q ⎜ 2 ⎟ P ⎝ m2 m2 ⎠ (84) 3 − , 2 4 has the same structure as eq. (76), up to an overall factor 4/3 and a rescaling of σ by a factor (−2). The poles in the propagators are therefore the same as in the sector of transverse traceless tensors (t, E). Together with eq. (75)for the scalar v˜ we observe in the scalar sector one massive particle and one massless particle with a double pole. The scalar u˜ is a gauge degree of freedom of diffeomorphisms and does not appear in the quadratic action. The sector of transversal vectors contains several independent blocks. The vectors v(t) and w(t) from the expansion of the transversal gauge field fluctuations correspond to massive particles with propagators given by eq. (75). A third physical vector fluctuation is given by the combination 1 s = (κ − γ ) + l . (85) μ 2 μ μ μ 2 4 Its inverse propagator P = m q obtains from Lκγl in LU as given by eq. (66). A canonical normalization of this vector field absorbs in P a factor m2q2. We end with a massless vector 2 particle with normalized inverse propagator PR = q . The transverse vector fluctuations contain also two gauge degrees of freedom, one from diffeomorphisms and the other from the SO(4) - gauge symmetry. We may take them as κμ and lμ at fixed sμ. Only the physical mode sμ appears in the quadratic action. The fluctuations C describe one more massive particle with inverse propagator given by eq. (75). What remains is the sector of antisymmetric tensor fluctuations, consisting of the longitudinal gauge bosons M and the antisymmetric vierbein fluctuations b. For the longitudinal gauge bosons there is no contribution from LF . The inverse propagator matrix in the M − b - sector obtains from LbM in eq. (65) q2m2 4 −2 P = . (86) 8 −21 After proper renormalization of the fields this becomes m2 −mq PR = . (87) −mq q2

One eigenvalue λ+ of this matrix corresponds to the physical ﬂuctuation 1 r = b − M . (88) μν 2 μν μν Only this physical mode appears in LU through LbM in eq. (65). It corresponds to a massive particle with inverse propagator

2 2 λ+ = q + m . (89) 2 The other eigenvalue λ− is zero for all q , corresponding to det P = 0or det PR = 0. This is the gauge mode of local SO(4) - gauge transformations. Indeed, these gauge transformations, = m = m applied to a “vacuum state” Aμmn 0, eμ δμ , do not only shift the longitudinal compo- m (A) nents of Aμmn, but also rotate eμ , contributing inﬁnitesimally to Hμν . For this reason the gauge

17 C. Wetterich Nuclear Physics B 971 (2021) 115526 modes can be taken as linear combinations of Lμν and bμν . We use the freedom in the precise parametrization of the gauge mode to take the fluctuation Lμν at fixed physical fluctuation sμν. In total we observe 9 massless physical modes (t − E, w˜ − σ , s), 15 massive physical modes (v(t), w(t), v,˜ C, r) and 10 gauge degrees of freedom (κ, u, l, M). We recall that the gauge fluctuations have to be taken at fixed physical modes. One linear combination in the t − E-sector as well as in the w˜ − σ -sector (6 modes) corresponds to physical modes that do not describe a propagating particle. Similar to Einstein gravity one expects that due to non-local projectors not all of the modes (t − E, w˜ − σ, s, v(t), w(t), v,˜ C, r) are propagating particles [52]. This reduces the number of modes corresponding to physical propagating particles below 24.

4.3. Short distance completion for euclidean gravity

Taking things together, for the action LU + LF the propagators of the physical ﬂuctuations around euclidean ﬂat space (excluding the gauge modes) have the following properties:

(i) In the complex q2-plane all poles occur on the negative real axis, including q2 = 0. 2 =− 2 (ii) For the massive modes with location of the poles at qp mi the residuum is always positive, as required for standard stable massive particles. (iii) For real q2 0the inverse propagator is positive for all modes, increasing ∼ q2 for q2 → ∞. (iv) At q2 = 0 there are degenerate poles, with eigenvalues of the inverse propagator ∼ q4. These massless modes are linear combinations of gauge boson - and vierbein-ﬂuctuations. The behavior ∼ q4 can be considered as the boundary of the region of stability.

We can consider this model of SO(4) - pregeometry as a valid candidate for the ultraviolet completion of euclidean gravity.

4.4. Analytic continuation

For the momentum dependence of the propagators discussed above there is no obstruction for analytic continuation to a Minkowski signature. Investigating the propagator for a Minkowski 2 | 2| + 2 signature in the complex q0 - plane one finds besides the poles branch cuts for q0 > qc q , 2 with positive non zero |qc| to be discussed in more detail in sect. 6. Such a setting is rather promising for a consistent description of quantum gravity. The general analytic continuation of our gauge theory of pregeometry has to map the compact gauge group SO(4) to the non-compact Lorentz group SO(1, 3). This analytic continuation m can be achieved by a complex continuation of the vierbein eμ and the gauge fields Aμmn by mapping 0 → 0 = iϕ 0 eμ eμ e eμ , → A = −iϕ Aμk0 μk0 e Aμk0 , (90) = k = k = while fields carrying only “Lorentz indices” k 1, 2, 3 remain invariant, eμ eμ , Aμkl = 0 = 0 = Aμkl. For ϕ π/2 one reaches Minkowski signature of the metric. With eμ ieμ , Aμm0 −iAμm0, the analytically continued metric reads = m n = m n gμν eμ eν δmn eμ eν ηmn . (91)

18 C. Wetterich Nuclear Physics B 971 (2021) 115526

We observe the appearance of the SO(1, 3) invariant tensor ηmn = diag(−1, 1, 1, 1). Corre- spondingly, the SO(4) gauge symmetry becomes SO(1, 3) after analytic continuation. This is seen most easily if we define a Minkowski-vierbein and Minkowski gauge fields by the convention (M)0 =− 0 (M) = eμ ieμ ,Aμk0 iAμk0 . (92) (M)m (M) The fields eμ and Aμmn are real. After analytic continuation with ϕ = π/2they equal the (M)m m (M) original euclidean fields, eμ = eμ , Aμmn = Aμmn. The analytic continuation of the metric (91) remains, however, being unaffected by the convention chosen for the vierbein and gauge fields. The analytic continuation of the gauge group follows if one continues the parameters εmn of infinitesimal SO(4)-transformations by the same rules. More details for the analytic continuation can be found in ref. [53]. We emphasize that this procedure of analytic continuation is done at fixed coordinates xμ. It only concerns the Lorentz-indices m of the fields, not the world indices μ. This version of analytic continuation in the vierbein and metric has been proposed previously [54]. As long as only vierbein and metric are concerned it is equivalent to analytic continuation in the time coordinate. 2 μν In particular, for the squared momentum, q = g qμqν , or corresponding expressions with covariant derivatives, the analytic continuation of the metric can be described equivalently with a fixed metric and analytic continuation in the zero component of momentum q0. Propagators depending on q2 do not change. Analytic continuation in flat space only changes the euclidean 2 = 2 + 2 expression q q0 q . For theories with gauge fields carrying Lorentz indices the analytic continuation of fields is mandatory, however. If one would only continue the time coordinate, the gauge group would remain SO(4), resulting in a mismatch with the SO(1, 3)-symmetry of Minkowski space. The analytic continuation of fields with Lorentz-indices has the additional advantage that it is well (M)m defined for an arbitrary geometry. There is a clear rule how the metric transforms, using e μ = m ρ eμ and continuing δmn → ηmn, while coordinates remain fixed. Furthermore, fields as Uμν = m ρ Uμν em remain invariant under analytic continuation since the phase factors cancel out. In (M)m (M) short, by use of the “Minkowski convention” for fields e μ , Aμmn , the effects of analytic m continuation in the fields can equivalently be described by fixed fields eμ , Aμmn and analytic continuation in the invariants δmn → ηmn, εmnpq → iεmnpq . (The change in the ε-tensor accounts for the appearance of a factor i in the exponent e−S in the functional integral, due to e = ie.)

5. Effective action and emergence of general relativity

The action (11)may be considered as the microphysical or classical action which is used to define a functional integral. Quantum fluctuations will lead to a macroscopic or effective action. The quantum effective action is the generating functional of the one-particle-irreducible Green’s functions. It includes all fluctuation effects. The field equations obtained by variation of the effective action are exact. They replace the field equations obtained from the classical action. The second functional derivative of the effective action is the inverse propagator in an arbitrary background of fields. With a proper definition (and in the absence of anomalies) the effective action has the same symmetries as the classical action. In general, this invariant functional of the fields contains infinitely many invariants. The arguments are now macroscopic fields, corresponding to expectation values of microscopic fields in the presence of arbitrary sources.

19 C. Wetterich Nuclear Physics B 971 (2021) 115526

We do not attempt to compute the effective action in this paper. We rather make an ansatz for its form which is consistent with the symmetries. In practice, we replace S by and add further invariants. The parameters m2 and Z are replaced by renormalized parameters which include the effects of the quantum ﬂuctuations. For the additional invariants we investigate a derivative expansion up to second order in the derivatives. It may be possible to include some of the additional invariants into the classical action. We will leave it open here if they are also part of the classical action or if they are generated only by ﬂuctuation effects.

5.1. New invariants

At low momenta a derivative expansion of the effective action is often a good guide. We will next add to the action (11) further invariants with up to two derivatives, 4 = d xe(LF + LU + U + LR + LG). (93)

The term without derivatives is a cosmological constant U. We will first neglect it here for simplicity, such that flat space remains a solution of the field equations. This may be motivated by the observation that in the present universe flat space is a very good approximation. For early cosmology this may no longer hold. We discuss the effects of U = 0in sect. 8. The particular field content with a vierbein permits an invariant that is linear in the fields

M2 L =− F μν . (94) R 2 μν Indeed, contraction of the ﬁeld strength with the inverse vierbein yields a scalar which is invariant under SO(4) - gauge transformations,

μν mμ nν F = Fμν = Fμνmne e . (95)

The presence of LR in the effective action is a crucial ingredient for the emergence of general relativity. Further invariants with two derivatives can be constructed by use of the contractions F and

ρ m nρ Fμν = Fμρν = Fμρmneν e , (96) as well as

μ μν Uμ ρ = g Uμνρ , (97) namely

B C n2 L = F F μν + F 2 + U μ U νρ . (98) G 2 μν 2 2 μ ρ ν The invariants ∼ C and ∼ n2 will play a role for the stability of scalar excitations discussed in sect. 7. To keep formulas short we set B = 0. There exist further invariants with two derivatives partly violating parity or time reversal. They are omitted here. The “observable” parameters of the model are the dimensionsless couplings or ratios as Z, B, C, M2/m2 and n2/m2.

20 C. Wetterich Nuclear Physics B 971 (2021) 115526

5.2. Field equations

The field equations obtain by variation of the effective action (93) with respect to Aμmn and m eμ . For a given effective action they are exact, including all effects of quantum fluctuations. Approximations occur only because the effective action is not known precisely. For a model that is thought to describe quantum gravity these field equations have to lead, after analytic continuation to Minkowski signature, to the gravitational field equations that we use on Earth or for the description of black holes. The field equation for the gauge fields is given by μνmn mμ nν mν nμ μmn ZDνF + 2C∂νF(e e − e e ) = J , (99) with “source” or “current” J μmn = m2U μmn 2 mμn nμm ρm nμ ρn mμ + (2CF − M )(U − U + Uρ e − Uρ e ) 2 ρm nμ ρn mμ − n (Uρ e − Uρ e ). (100) μνmn From the identity DμDνF = 0 one infers for C = 0 that solutions of the field equations need a covariantly conserved current, μmn DμJ = 0 . (101) The field equation for the vierbein can be split into symmetric and antisymmetric parts. The one for the symmetric part reads M2 (F + F − Fg ) = T (U) + T (F ) , (102) 2 μν νμ μν μν μν where the r.h.s. involves the symmetric tensors m2 T (U) = (D U ρ + D U ρ + U τρU (103) μν 2 ρ μν ρ νμ μ ντρ 2 1 στρ n τρ ρ − U Uστρgμν) + 2DρUτ gμν − DμUρ ν 2 2 ρ τρ ρ τσ − DνUρ μ + (Uμνρ + Uνμρ)Uτ − Uρ σ Uτ gμν , and (F ) =Z ρmn − 1 σρmn Tμν (Fμ Fνρmn F Fσρmngμν) 2 4 (104) 1 + C F(F + F ) − F 2g . μν νμ 2 μν (F ) We observe that Tμν is traceless μν (F ) = g Tμν 0 . (105) The antisymmetric part yields m2D U ρ + (M2 − 2CF )(F − F ) (106) ρ μν μν νμ 2 τρ ρ ρ − n (Uμνρ − Uνμρ)Uτ − DμUρ ν + DνUρ μ = 0 . We emphasize that these field equations hold both for euclidean and Minkowski signature.

21 C. Wetterich Nuclear Physics B 971 (2021) 115526

5.3. Simple solutions

An interesting class of solutions is characterized by

m Uμνρ = 0 ,Dμeν = 0 . (107)

In this case one has Fμνρσ = Rμνρσ , with Rμνρσ the curvature tensor constructed from the metric gμν . The ﬁeld equations (99) and (102) become μνρσ σ μρ ρ μσ ZDνR + 2C(∂ Fg − ∂ Fg ) = 0 , 1 M2(R − Rg ) = T (F ) , μν 2 μν μν Z 1 T (F ) = (R ρστR − RσρτηR g ) μν 2 μ νρσ τ 4 στρη μν 1 + CR(2R − Rg ), (108) μν 2 μν ρ μ with Rμν = Rμρν , R = Rμ . Eq. (106)is obeyed identically. The second equation (108)is Einstein’s equation with an effective energy momentum tensor reﬂecting the effects of higher curvature invariants. Taking the trace,

R = 0 , (109) the Bianchi identity implies ν (F ) = D Tμν 0 , (110) which reads for R = 0

στρη ν στρ στρ ν ∂μ(R Rστρη) = 4(D Rμ )Rνστρ + 4Rμ D Rνστρ . (111)

A particular solution is ﬂat space, Rμνρσ = 0. For momenta and curvature much smaller than m2 eq. (107) becomes a valid approximation, with corrections suppressed by factors R/m2 or D2/m2. (We assume implicitly that n2 is not 2 (F ) much larger in size than m .) Furthermore, Tμν can be neglected. One ends with Einstein’s equation in vacuum. The coupling to elementary particles, that we do not discuss in the present paper, adds an energy momentum tensor as for general relativity.

5.4. Emergent general relativity

For a discussion of the low energy effective theory the invariant LR will become dominant. For U = 0this is the leading term in an expansion in derivatives. We will next construct an effective action that is valid for momenta and curvature much smaller than m2. This will reveal the emergence of general relativity with the Einstein-Hilbert action dominating the long distance physics. For the construction of an effective theory at low momenta q2 m2 we write

Aμνρ = ωμνρ − Uμνρ , (112) m and solve the ﬁeld equation for the tensor Uμνρ as a functional of the vierbein eμ which is kept ﬁxed. If the invariant LU dominates, the solution is simply Uμνρ = 0. The vierbein is then

22 C. Wetterich Nuclear Physics B 971 (2021) 115526

m covariantly conserved, Dμeν = 0, and the gauge ﬁelds cease to be independent degrees of 2 freedom, given by Aμmn = ωμmn. This does not change if we add the term ∼ n from LG. For the ﬁeld strength one has [21]

Fμνρσ = Rμνρσ − Vμνρσ . (113) The curvature tensor involves two derivatives of the vierbein. The square of the tensor σ σ m m Vμνρ =em DμUνρ − DνUμρ (114) σ σ στ στ =DμUνρ − DνUμρ − Uμ Uνρτ + Uν Uμρτ provides for a kinetic term for Uμνρ through LF and LG. The invariants LF + LG also involve σ m terms linear in Vμνρ and therefore linear in Uμν . For the demonstration of emergent general relativity we omit for a moment the invariant LG, cf. ref. [51]for a more complete discussion. m In the presence of LF the term linear in Uμν implies that Uμνρ = 0is no longer an exact solution. This linear term arises from the second term in the expression Z L = R Rμνρσ − 2R V μνρσ + V V μνρσ . (115) F 8 μνρσ μνρσ μνρσ After partial integration we can replace μνρσ m μνρ 2Rμνρσ V →−4Uνρ DμR m , (116) such that the term linear Uμνρ involves covariant derivatives of the curvature tensor. Similarly, a linear term arises from LR, M2 L =− R − V μν , (117) R 2 μν with R the curvature scalar. The low momentum effective theory becomes valid if the derivatives of the curvature tensor 3 are small as compared to m . In this case the quadratic term for Uμνρ is dominated by LU and one finds for small M2/m2 the approximate solution Z U =− Dσ R . (118) μνρ m2 σμνρ 2 2 Insertion into LF and LU yields a term ∼ (Z /m )(DRDR) with six derivatives that we may neglect for the low momentum effective theory. For M2 of the same order as m2 the result changes quantitatively, but not qualitatively. As a result, we can use Uμνρ = 0 and find the effective action for low momenta M2 Z C S = e − R + R Rμνρσ + R2 . (119) 2 8 μνρσ 2 x √ Here we have restituted the contribution from LG. With e = g, g = det(gμν), the low energy effective action only involves the metric. General relativity emerges in this limit. Indeed, the first term in eq. (119) constitutes the Einstein-Hilbert action and we can associate M with the (reduced) Planck mass. The second term involves the squared Riemann tensor and the squared curvature scalar. Since it contains four derivatives of the metric its influence at momenta q2 M2/Z will be suppressed. We conclude that for most practical purposes, except perhaps for very early cosmology or the vicinity of the singularity in black holes, our model predicts precisely the field equations for general relativity. Since general relativity is compatible with all present tests of the gravitational interactions, the same holds for our model of pregeometry.

23 C. Wetterich Nuclear Physics B 971 (2021) 115526

5.5. Stability for low momenta

The issue of stability arises on two different levels. The first concerns the properties of the classical action. This topic is related to the question if the euclidean functional integral is well defined. It typically concerns the microscopic or high momentum physics. We have discussed this issue in sect. 4. The second topic concerns the stability of solutions of the field equations. Since the relevant field equations obtain by variation of the effective action this part of the stability discussion concerns the effective action. In particular, for the stability at momenta below the Planck mass the term LR will play an important role. We therefore extend the stability discussion of sect. 4 to an effective action based on LR + LU + LF + LG. In contrast to LF and LU the invariant LR is not a square and can therefore take negative 2 2 values. Since LR is linear in q, it cannot modify the leading behavior ∼ q for large q . In the infrared limit of small q2 it can play an important role, however. One expects that it dominates the behavior near massless propagator poles at q2 = 0. We will also consider the possibility that 2 C or n in LG take negative values. As a part of the classical action this would destroy the boundedness if m˜ 2 or Z˜ are negative,

m˜ 2 = m2 + 3n2 , Z˜ = Z + 12C. (120) All couplings in the effective action become running couplings. While for the short distance effective action for large q2 one would suggest m˜ 2 > 0, Z>˜ 0in order to make a simple associ- ation with the classical action, the stability discussion concerns typically the range q2 ≈ M2, or q2 M2. In this range m˜ 2 and Z˜ can be negative. We find that such negative values are required for stability in the scalar sector. For a stability analysis we consider here small fluctuations around flat space. As an important requirement for a realistic theory there should be no tachyonic modes. Otherwise small deviations from flat space would grow exponentially, in contradiction with the observation that cosmology has approached at present flat space with high accuracy. This requirement is based on analytic continuation and realistic cosmology. For euclidean signature a tachyonic mode implies metastability of flat space since the euclidean action for some of the small fluctuations is smaller than the euclidean action for flat space.

5.6. Expansion of LR on ﬂat space

The term LR dominates the small momentum behavior. Its contribution is therefore a central ingredient for the analysis of stability for the effective action. We need to expand

M2 eL =− eemμenνF (121) R 2 μνmn m = m = in quadratic order in H and A around eμ δμ , Aμmn 0. Since Fμνmn is a least linear in Aμmn according to eq. (6), we need the vierbein in linear order in H ,

eemμenν = δmμδnν (122) 1 + H δmμδnνδρσ − δmμδnρ δνσ − δmρ δnνδμσ . 2 ρσ μν μν The linear term ∂μAν − ∂νAμ is a total derivative and therefore vanishes.

24 C. Wetterich Nuclear Physics B 971 (2021) 115526

= (1) + (2) In quadratic order one finds two contributions, LR LR LR , M2 eL(1) =− H ∂ A μν − ∂ A μν δρσ (123) R 4 ρσ μ ν ν μ μρ μρ νσ ρν ρν μσ − ∂μAν − ∂νAμ δ − ∂μAν − ∂νAμ δ , and M2 eL(2) =− A μρ A ν − A μρ A ν . (124) R 2 μ νρ ν μρ The first term involves the term linear in Aμ in F and the linear term (122), while the second corresponds to the term quadratic in Aμ in F . For the first term we observe that only the transversal gauge bosons contribute to Fμνmn in linear order, M2 eL(1) = H δρσ ∂ B νμ − ∂ Bσρμ − ∂σ B μρ R 2 ρσ μ ν μ μ M2 2 1 ∂ρ∂σ = H δρσ + b˜ − B˜ (S)σρ − B˜ (A)σρ 2 ρσ 3 3 ∂2 2 − ∂σ w(t)ρ − ∂σ ∂ρw˜ . (125) 3 The decomposition yields M2 2 1 eL(1) = w∂˜ 2σ + Eρσ ∂2t (126) R 2 3 2 σρ 1 2 + εμνρσ v(t)∂ b + w(t)μ∂2(κ − γ ) . 2 μ ν ρσ 3 μ μ This contributes to the mixing between transversal gauge bosons and vierbein fluctuations, similar to eq. (59). As it should be, the gauge modes κμ + γμ and u do not appear. The second term decomposes as M2 1 eL(2) = Eμν∂2E + C Cνρμ R 2 4 μν μνρ + 2 (t)μ (t) − 2 ˜ 2 ˜ + 1 (t)μ (t) − 3 ˜ 2 ˜ w wμ w∂ w v vμ v∂ v 9 3 2 2 4 + w(t)μ∂2l − εμνρσ v(t)∂ M . (127) 3 μ μ ν ρσ

The different contributions of eLR can be listed as follows M2 eL = L + L ˜ + L + L + L + L˜ . (128) R 4 tE σ w vM wL C v In the sector of transversal traceless tensors we ﬁnd a further contribution to the t - E mixing and the term quadratic in E, 1 L = Eμν∂2t + Eμν∂2E . (129) tE μν 2 μν

25 C. Wetterich Nuclear Physics B 971 (2021) 115526

It has the opposite sign as compared to the contributions from eq. (63). In the scalar sector a term

4 2 2 L ˜ = (w∂˜ σ −˜w∂ w)˜ , (130) σ w 3 adds to eqs. (63), (64) further contributions. The term = μνρσ (t) − + (t)μ (t) LvM ε vμ ∂ν(bρσ 2Mρσ ) v vμ (131) can be combined with LbM in eq. (65), involving the same physical combination bμν − 2Mμν = (t) 2rμν . It induces a new momentum-dependent mixing between rμν and vμ . Similarly 4 4 L = w(t)μ∂2(κ − γ + 2l ) + w(t)μw(t)μ (132) wl 3 μ μ μ 9 μ (t) combines with Lκγl in eq. (66). It mixes the physical transversal vector ﬂuctuations sμ and wμ . The remaining parts,

νρμ 2 LC = 2CμνρC ,Lv˜ =−3v∂˜ v,˜ (133) add mass terms to the gauge boson ﬂuctuations C and v˜.

5.7. Expansion of LG on ﬂat space

Contributions from LG will be essential in order to avoid tachyons in the scalar sector. The expansion of eLG reads 2 ∗ n ∗ ∗ eL =2Cw˜ ∂4w˜ − (σ − 2w )∂2(σ − 2w) G 8 2 2 n ∗ 2n ∗ + γ ∂4γ μ + w(t) w(t)μ (134) 8 μ 9 μ 2 n ∗ ∗ − w(t) ∂2γ μ + γ ∂2w(t)μ . 6 μ μ Thus LG does not contribute to the transverse traceless tensor modes. It yields additional terms in the scalar and vector sector, contributing to the blocks Lσ w˜ and Lwl. Adding the contributions of LR + LG to the ones from LU + LF the stability discussion can again be performed separately for different blocks. We need to compute the propagators from a diagonalization of the individual blocks. Stability depends on the behavior of these propagators. In the next section we will compute the propagator in the transverse traceless tensor sector which describes the massless graviton and a massive spin-two particle. In the following sect. 7 we turn to the scalar sector. There we also discuss brieﬂy the other sectors.

6. Graviton propagator

We are now in a position to discuss the propagator of the graviton. In the low momentum limit we expect an inverse propagator ∼ q2 according to the emergence of general relativity discussed in the preceding section. Also for high q2 one has a propagator ∼ q2 according to the ﬁndings of sect. 4. This momentum region is not affected by the addition of LR, which is a subleading correction for q2 M2. One expects some smooth interpolation between the two limits. In the view of later analytic continuation we discuss the graviton propagator for arbitrarily complex q2.

26 C. Wetterich Nuclear Physics B 971 (2021) 115526

6.1. Inverse propagator for transversal traceless tensors

Our model has two ﬁelds transforming as transverse traceless tensors, namely tμν and Eμν . They mix, and the graviton has to be identiﬁed with one of the eigenmodes of the inverse propagator matrix. In the presence of LR the renormalized inverse propagator matrix (78)in the t − E - sector is replaced by ⎛ ⎞ m2 − M2 ⎜ 2 + 2 − 2 ⎟ ⎜ Zq m M , q ⎟ (tE) ⎜ m ⎟ P = ⎜ ⎟ . (135) R ⎝ m2 − M2 ⎠ q, q2 m

This matrix is the central quantity for the discussion of the graviton propagator. In addition to the massless graviton it will describe a massive particle with mass ∼ M. For M → 0 both excitations are massless, as found in sect. 4. The poles of the propagators or zero eigenvalues of PR occur for

q2 = 0 ,q2 =−μ2 , (136) with

m2 M2 μ2 = y(1 − y) , y = . (137) Z m2

This is seen easily from the condition det(PR) = 0. No other poles of the propagator are possible. For M2 → 0the second pole moves to zero, M2 M2 μ2 = 1 − . (138) Z m2 A stable particle requires μ2 0, or

0 y 1 , 0 M2 m2 . (139)

Otherwise one encounters a tachyonic instability. The constraint (139)is a ﬁrst condition for stable gravity. The inverse particle propagators are given by the eigenvalues of PR, 1 2 2 2 λ± = (Z + 1)q + m − M (140) 2 q2 ± (Z − 1)q2 + m2 − M2 2 + 4 (m2 − M2)2 . m2

All properties of the propagation of transverse traceless tensor modes in ﬂat space are encoded in these two eigenvalues. For suitable ranges of parameters we will ﬁnd that the eigenvector to λ− describes the massless graviton, while the eigenvector to λ+ accounts for a stable massive particle. At long distances only the graviton will matter.

27 C. Wetterich Nuclear Physics B 971 (2021) 115526

6.2. Graviton

Indeed, for q2 → 0the eigenvalue λ−(q2) is the inverse propagator for the massless graviton, corresponding to a Taylor expansion

M2q2 M2 q4 λ− = + Z − + ... (141) m2 m2 m2 A factor 4/m2 may be absorbed by a normalization which makes the graviton fluctuation dimensionless. The relation between the expansion (141) and the low-momentum effective action (119) involves a wave function renormalization. We can write M2q2 + Zq4 + ... λ− = (142) m2 + q2 2 2 −1 and absorb the factor (m + q ) into the normalization. Indeed, the tensor fluctuation tμν in the effective low-momentum theory equals the eigenfunction of λ− only in lowest order in an 2 expansion in q . For the effective low-momentum theory one employs Uμνρ = 0or Eμν =−tμν , while the eigenfunction of λ− obeys this relation only in lowest order in q2. The use of the low- energy effective action away from solutions of the field equations (“off shell”) has limitations. For the other eigenvalue one has

2 2 2 λ+(q = 0) = m − M . (143) For M2 > 0 and M2

2 2 1 At m 2 2 − =− − + − + λ mt Z 1 2 (q μ ). (146) 2 mt 2 Indeed, for Z

The coefﬁcient At is given by 1 A = (1 − y) Z + 1 − Z + y . (148) t Z Here we have always assumed the range 0 y 1 required by stability according to eq. (139). With the conditions (139) and (145)the eigenvalue λ−(q2) therefore has a single zero, located at q2 = 0. There is no tachyonic or ghost pole in the graviton propagator.

28 C. Wetterich Nuclear Physics B 971 (2021) 115526

For large |q2| one obtains q2 for Z>1 − = lim λ 2 q2→∞ Zq for Z<1 , Zq2 for Z>1 − = lim λ 2 (149) q2→−∞ q for Z<1 .

2 2 2 For ZZc the graviton propagator λ− (q ) has a second pole at q =−μ , with a behavior for q2 near −μ2 given by 2 2 2 λ−(q ) = Bt (q + μ ), (150) where 1 A m2 Zy B = Z + 1 + t = . (151) t 2 − − 2 mt y Z(1 y) 2 2 + 2 For Z>Zc one has mt < 0 and Bt < 0. The negative prefactor of q μ corresponds to a negative residuum at the pole at q2 =−μ2. This indicates a ghost instability. A stable theory with a well behaved graviton propagator for all momenta therefore requires the “stability condition” (145), Z

2 4 Ggrav(q ) = . (152) m2λ−(q2) This is the graviton propagator (2) mentioned in the introduction. For q2 near zero it reads, cf. eq. (141), 4 q2 −1 G (q2) = 1 + (Z − y) . (153) grav M2q2 M2 While the truncated propagator has a ghost pole at q2 =−M2/(Z − y),this pole is not present in the propagator (152). It is therefore an artefact of the truncation [12].

6.3. Massive spin-two particle

2 In the stable range for Z 0 one infers Bt > 0. The eigenmode λ+(q ) corresponds to a stable massive spin- −1 two particle, with mass μ and positive residuum at the pole Bt > 0. This resembles to similar ﬁndings in refs. [40–45]. In the stable range both eigenvalues of the inverse propagator matrix (135) correspond to stable modes. There is no instability in this sector. At q2 = 0 one ﬁnds a positive value of λ+,

29 C. Wetterich Nuclear Physics B 971 (2021) 115526

2 2 2 λ+(q = 0) = m − M . (155) For the boundary case M2 = m2 one observes a second stable massless particle, with

2 2 λ+ = Zq ,λ− = q . (156)

In this limit Zc diverges. A future exploration of the limit with two massless tensor ﬁelds is conceptually rather interesting, but not the focus of this paper. 2 The condition of stability Zm2(1 − y)2 = M2 , (157) y where we recall eq. (149). Except for the limiting case y → 1the heavy mass is at least of the order M, such that the massive spin-two particle induces only tiny corrections for low energy phenomenology. If for a given m2 and Z one starts at M2 = 0 and switches on M2 continuously, the double pole at q2 = 0for M2 = 0ﬁrst turns to a ghost instability since the condition (145) will be violated for small enough y. Once y reaches the critical value saturating eq. (145)the pole jumps from λ− to λ+ and the sign of the residuum changes correspondingly. This explains why the massive spin-two particle cannot be found as a small deformation in the vicinity of a massless graviton. The massive particle pole is outside the range of validity of a polynomial expansion in q2 for the inverse propagator.

6.4. Analyticity

Analytic continuation can be discussed on different levels. On an overall level one has to specify how all fields are analytically continued. This includes the gauge fields, such that the gauge group SO(4) is analytically continued to the non-compact gauge group SO(1, 3). We will describe this procedure in a separate paper. In the present paper we discuss analytic continuation on the level of propagators in flat space. This amounts to a discussion of the behavior of propagators 2 in the plane of complex q or complex q0. Analytic continuation interpolates from the euclidean 2 = μ = 2 + 2 2 =− 2 + 2 relation q q qμ q0 q to the Minkowski relation q q0 q . It requires that there exists a continuous path between the two limits in the complex plane that is not obstructed by non-analyticities as poles or branch cuts. For an overall picture of the q2-dependence of the two eigenvalues λ±(q2) we further note 2 2 that at a critical value qc < 0 both eigenvalue can coincide at branch points qc , 2 = 2 λ+(qc ) λ−(qc ). (158) This occurs when the square root in eq. (140) vanishes, determining

2 q ± 1 − y c =− Z + 1 − 2y ∓ 2 1 − y Z − y . (159) m2 (1 − Z)2 The condition for the existence of real intersection points is Z y. For Z = y, which belongs to 2 2 2 =− 2 the stable region, λ+(q ) and λ−(q ) touch each other at qc m . For Z>yone has a ﬁnite 2 2 2 region qc− yboth qc− and qc+ are ﬁnite.) We

30 C. Wetterich Nuclear Physics B 971 (2021) 115526 observe that Z>yis compatible with Z0. In the complex q2-plane the graviton propagator has a branch cut for real negative q2, extend- 2 2 2 = ing from qc− to qc+ . Except for the pole at q 0 and this cut, the graviton propagator is analytic, decaying ∼|q|−2 for large |q|. Nothing obstructs analytic continuation from euclidean space to = Minkowski space. We can therefore extend our analysis to a Minkowski metric, gμν ημνand 2 =− 2 + 2 =± 2 q q0 q . In the complex q0 - plane the graviton propagator has two poles at q0 q . 2 =| 2 | + 2 | 2 | + 2 It inherits the cuts on the real q0 - axis, extending from q0 qc− q to qc+ q . The pre- scription for the usual inﬁnitesimal iε -terms is dictated by analytic continuation. This graviton propagator is well behaved in Minkowski space, without any instability. For the particular case Z = y one has the particularly simple behavior 2 M 2 2 2 2 λ− = q ,λ+ = q + m − M . (163) m2 2 =− 2 = =− 2 For qc m the two eigenvalues are equal, λ− λ+ M . Finally, for Z

7. Stability of ﬂat space

In the presence of LR flat space cannot be the minimum of the euclidean effective action. For flat space one has = 0. The curvature scalar R can take positive and negative values, however. Since the term LR is dominant for small momenta, positive R and therefore negative LR lead to < 0. This issue is well known in Einstein gravity. In Einstein gravity it leads to an unbounded Einstein-Hilbert action such that the euclidean functional integral is not well defined. For our model of pregeometry the functional integral based on the classical action is well defined. This extends to the high momentum behavior of the effective action, and therefore the behavior for

31 C. Wetterich Nuclear Physics B 971 (2021) 115526 large |R|, which are well behaved. The presence of LR in the effective action indicates, however, that ﬂat space is a saddle point of the euclidean effective action, but not a minimum. The fact that ﬂat space is not a minimum of the euclidean effective action does not imply instability of Minkowski space. A good example is again the Einstein-Hilbert action. Despite possible negative values of the effective action Minkowski space is stable due to the positive energy theorem. What is required for stability is the absence of tachyonic modes. This issue will be discussed next.

7.1. Scalar propagators

In the scalar sector one has three physical scalar modes, namely σ , w˜ , v˜. The scalar u is a gauge mode. The scalar v˜ does not mix with the other two physical scalars. It describes a massive particle with a stable propagator 2 2 2 −1 Gv˜ ∼ (Zq + m + 2M ) , (164) as obtained from the action 2 3q 2 2 2 S˜ = v(−q) (Zq + m + 2M )v(q) . (165) v 8 q 2 2 The squared mass m + 2M is increased by LR. In the σ −˜w sector one finds the inverse propagator matrix ⎛ ⎞ ˜ 1 ⎜Zq4 + (m˜ 2 + 2M2)q2 − m˜ 2 + M2 q2 ⎟ (σ w)˜ 2 = 1 ⎜ 2 ⎟ P (q ) ⎝ ⎠ . (166) 3 1 m˜ 2 − m˜ 2 + M2 q2 q2 2 4 ˜ 2 2 The addition of the contribution LG changes Z → Z, m →˜m according to eq. (120). The zero eigenvalues of P (σ w)˜ occur for 2M2 M2 q2 = 0 ,q2 = ν2 = (1 + 2y)˜ , y˜ = . (167) Z˜ m˜ 2 If both Z˜ and y˜ are positive, the second pole of the scalar propagator is located at positive q2, 2 2 in contrast to q =−μ for the graviton sector. In the absence of LG this leads to a tachyonic instability, since in this case Z˜ = Z, y˜ = y, and both Z and y have to be positive for a stable graviton. We conclude that stability of flat space requires either Z˜ or m˜ 2 or both to be negative, such that either C or n2 need to be negative. Stability of flat space requires 1 Z<˜ 0 , y>˜ − , (168) 2 or 1 Z>˜ 0 , y<˜ − . (169) 2 We will choose the first condition (168). The discussion of stability in the scalar sector is somewhat complex since different regions in the space of couplings Z˜ , m˜ 2 have to be distinguished. We first start with m˜ 2 < 0 and turn later to positive m˜ 2 > 0. Restoring the normalization to scalar fields with dimension mass the corresponding renormalized propagator matrix becomes for m˜ 2 < 0

32 C. Wetterich Nuclear Physics B 971 (2021) 115526 ⎛ ⎞ q Zq˜ 2 +˜m2 + 2M2 , −(m˜ 2 + 2M2) ˜ 1 ⎜ |˜| ⎟ (σ w) 2 = ⎝ m ⎠ PR (q ) q . (170) 3 −(m˜ 2 + 2M2) , −q2 |˜m| Metastability is expected to become visible in the behavior of one of the eigenfunctions for q2 → 0. According to the effective low energy theory being general relativity, the physical scalar ﬂuctuation in the metric should have the “wrong” sign of the inverse propagator. 2 For general q the two eigenvalues of PR are given by 2 1 ˜ 2 2 λ±(q) = (Z − 1)q +˜m (1 + 2y)˜ (171) 6 2 ± (Z˜ + 1)q2 +˜m2(1 + 2y)˜ − 4(1 + 2y)˜ 2m˜ 2q2 .

For q2 near zero one obtains for m˜ 2 < 0 ˜ 2 2y 2 λ+(q ) = q + ... . (172) 3

For m˜ 2 < 0, y<˜ 0the coefﬁcient of q2 is negative, in contrast to the opposite sign of λ− = yq2 for the graviton sector. This is the expected behavior for the Einstein-Hilbert action. Around q2 = 0the second eigenvalue is negative 2 1 2 ˜ 2 λ−(q ) = m˜ (1 + 2y)˜ + (Z − 1 − 2y)q˜ + ... . (173) 3 At the other pole of the propagator one ﬁnds for 1 + 2y>˜ 2y/˜ Z˜ that it corresponds again to λ+

2 2 λ+(q = ν ) = 0 , ˜ 2 ˜ 2 2 m 2y λ−(q = ν ) = (1 + 2y)˜ 1 + 2y˜ − . (174) 3 Z˜ ˜ On the other hand, for 1 + 2y<˜ 2y/˜ Z the role of λ+ and λ− is exchanged. For m˜ 2 < 0, 1 + 2y>˜ ˜ 2y/˜ Z both poles correspond to zeros of λ+, while λ− remains negative for all q2, without any ˜ zero. For m˜ 2 < 0, 1 + 2y<˜ 2y/˜ Z the pole at q2 = 0 belongs to λ+, while the pole at q2 = ν2 ˜ occurs for λ−. Expanding the eigenvalues in q2 − ν2 one obtains for 1 + 2y>˜ 2y/˜ Z ∂λ+ 2y˜ 2y˜ −1 =− 1 + 2y˜ − > 0 , (175) 2 ˜ ∂q q2=ν2 3 Z ∂λ− 1 2y˜ 2y˜ −1 = (Z˜ − 1) + 1 + 2y˜ − < 0 . 2 ˜ ∂q q2=ν2 3 3 Z The residuum at the pole at q2 = ν2 is therefore positive. This corresponds to a stable particle ˜ with mass ν. For 1 + 2y<˜ 2y/˜ Z the roles of λ+ and λ− are interchanged. The residuum at the pole at q2 = ν2 is now negative, indicating a “ghost-excitation” with a negative kinetic term. We can perform a similar discussion for positive m˜ 2 > 0. The element in the lower right corner of the matrix (170)is now +q2 instead of −q2. Correspondingly, the two eigenvalues read

33 C. Wetterich Nuclear Physics B 971 (2021) 115526 2 1 ˜ 2 2 λ±(q ) = (Z + 1)q +˜m (1 + 2y)˜ (176) 6 ± (Z˜ − 1)q2 +˜m2(1 + 2y)˜ 2 + 4(1 + 2y)˜ 2 m˜ 2q2 .

The pole at q2 = 0 corresponds now to λ−, with expansion ˜ 2 2y 2 λ−(q ) =− q + ... . (177) 3 With y>˜ 0the coefficient of q2 is again negative. For 1 + 2y˜ + 2y/˜ Z>˜ 0the second pole at q2 = ν2 occurs for λ− as well, while ˜ 2 2 1 2y λ+(q = ν ) = (1 + 2y)˜ 1 + 2y˜ + > 0 . (178) 2 Z˜ −1 2 In this case λ+ is not related to a propagating degree of freedom. The function λ+ (q ) does not have any pole in the complex q2-plane. With ∂λ− 2y˜ 2y˜ −1 q2 = ν2 = 1 + 2y˜ + , ∂q2 3 Z˜ ˜ ∂λ+ Z + 1 2y˜ 2y˜ −1 q2 = ν2 = − 1 + 2y˜ + , (179) ∂q2 3 3 Z˜ we find at q2 = ν2 a positive value for ∂λ−/∂q2, indicating a stable particle. For 1 + 2y˜ + ˜ 2y/˜ Z<0the roles of λ+ and λ− are interchanged. The pole at q2 = ν2 occurs now for λ+, with ∂λ+/∂q2 < 0. The pole at q2 = ν2 corresponds to a ghost. In summary, for Z<˜ 0, y>˜ −1/2 there is no tachyonic instability. A stable massive particle can occur for both positive or negative values of m˜ 2 and y˜, provided 2|˜y| 1 + 2y˜ + > 0 . (180) Z˜ If the condition (180)is violated, the massive pole corresponds to a ghost. For the pole at q2 = 0 the coefficient is always negative, indicating for euclidean signature a saddle point, in complete analogy to the Einstein-Hilbert action for general relativity. This reflects the observation that the m = m = configuration with eμ δμ , Aμmn 0 cannot be the minimum of the euclidean action. The term LR linear in F is not positive definite and can take negative values. The flat space configuration with vanishing gauge fields is therefore a saddle point.

7.2. Metastability of ﬂat euclidean space

The euclidean action based on LU +LF involves two squares of real tensors. Since we restrict the vierbein to values for which the determinant e is positive, e 0, an effective action containing only the terms LU + LF is positive definite, 0. The minimum occurs for = 0. This is realized by flat space and zero gauge fields. Adding the invariant LR, the flat space solution with vanishing gauge fields becomes a saddle point. As argued before, this follows from the simple μν 2 fact that F = Fμν can take positive and negative values. For every M = 0 there are therefore field configurations for which the action becomes negative. Flat space remains a solution, but is no longer a minimum of the euclidean action. It is turned to a saddle point. For M2 > 0the graviton sector is stable, i.e. the action increases to positive values > 0for nonzero inhomogeneous

34 C. Wetterich Nuclear Physics B 971 (2021) 115526 small tμν or Eμν . In contrast, the scalar direction has partially the opposite property. For small |q2| the action can decrease for nonzero small values of the scalar fields σ and v˜. For larger momentum also the term ∼ LG plays an important role. For the graviton sector it only replaces Z by Z + B. Thus the graviton sector is unaffected for our choice B = 0. In ˜ ˜ contrast, the scalar sector depends strongly on Z. Omitting LG (taking Z = Z, y˜ = y) one would find a tachyonic instability in the scalar sector. This is avoided for Z<˜ 0. On the other hand, Z<˜ 0 implies in the derivative expansion considered here an unbounded euclidean action. This is apparent if we decompose the squared field strength as Z F F μνρσ = αF 2 + βF˜ F˜ μν + γW W μνρσ , (181) 8 μνρσ μν μνρσ with 1 F˜ = F − Fg , F˜ μνgμν = 0 , (182) μν μν 4 μν and 1 W = F − (g F + g F − g F − g F ) μνρσ μνρσ 2 μρ νσ νσ μρ μσ νρ νρ μσ 1 + F(g g − g g ), (183) 6 μρ νσ μσ νρ where

νσ μρ Wμνρσ g = Wμνρσ g = 0 . (184) The coefficients are given by Z Z Z α = ,β= ,γ= . (185) 48 4 8 One finds for the sum Z Z + 2B L +L + L = W W μνρσ + F˜ F˜ μν F R G 8 μνρσ 4 μν 1 M2 + 2Z˜ − (Z + 2B) F 2 − F. (186) 48 2 Here we generalize for B = 0 Z˜ = Z + 4B + 12C, (187) such that our discussion of the scalar sector remains valid for B = 0. For Z<˜ 0 and Z + 2B>0 the coefficient of the term ∼ F 2 is negative, implying an unbounded euclidean action.

7.3. Stability of Minkowski space

As we have discussed above, no tachyonic or ghost instability occurs in the graviton and scalar sectors for the parameter range y m2 > 0 ,y>0 , 0 ˜ − , Z<˜ − . (188) 2 1 + 2y˜

35 C. Wetterich Nuclear Physics B 971 (2021) 115526

We will see below that these conditions ensure also the stability of the other fluctuation modes. The status of ghost instabilities is not completely understood. If one only wants to avoid tachyonic instabilities the conditions on Z and Z˜ get relaxed to Z>0, Z<˜ 0. With parameters in the range (188)this suggests stability of Minkowski space. This property of the field equations becomes more manifest if we add in our setting a further scalar field φ with effective action 2 2 φ = eδ(φ − F) , (189) x with 1 δ =− 2Z˜ − (Z + B) > 0 . (190) 48 The field equations for the extended system are equivalent to the original system if we discard solutions with φ = 0. Indeed, inserting the solution φ2 = F for the field equation for φ into the 2 effective action the contribution φ vanishes. With our choice of δ the coefficient ∼ F of the combination LF + LG + Lφ vanishes. We end in the scalar sector with the expression 2 M L = e − + 2δφ2 F + δφ4 . (191) φ 2 x Replacing F by R the resulting theory of general relativity coupled to a scalar field is well known to be stable. A Weyl scaling to the Einstein frame induces a kinetic term for φ. The scalar field φ is stable, with effective potential in the Einstein frame 4 4 2 2 −2 VE = δM φ (M + 4δφ ) . (192) Here M is the observable Planck mass. The situation is analogous to general relativity with higher derivative invariants. Stability of Minkowski space requires the coefficient of the terms ∼ R2 in the effective action to be negative, similar to the term ∼−δF 2 for pregeometry. In the version with an additional scalar field φ the scalar sector of our model of pregeometry resembles closely general relativity with an additional stable scalar field. The action for the euclidean metric of Einstein gravity exhibits also a saddle point, with scalar fluctuations around flat space leading to a lowering of the euclidean action. At least for the analytically continued model in Minkowski space this is actually not a sign of instability. Due to the positive energy condition Minkowski space is stable in Einstein gravity. The scalar mode is actually not a propagating mode. Combined with the necessary projectors the scalar propagator turns out to be given by (q 2)−1 instead of (q2)−1 [52]. There is therefore no pole for q 2 = 0. The scalar propagator accounts for Newton’s potential, rather than being a physical propagating particle. We expect a similar behavior for our model, at least for the region of momenta q2 M2. The pole at q2 = 0in the stability analysis above is expected to correspond to the physical scalar in the metric for Einstein gravity. The other pole at q2 = ν2 can be associated to the stable scalar field φ.

7.4. Classical action and effective action

The metastability of euclidean ﬂat space and the stability of Minkowski space points to an important difference between the classical action and the effective action. For the classical action

36 C. Wetterich Nuclear Physics B 971 (2021) 115526 all field configurations are included in the functional integral. A lack of boundedness from below obstructs a well defined functional integral. For the effective action we are concerned with the stability of solutions of the field equations derived from it. Not all arbitrary field configurations are reached by such solutions. Einstein’s equations imply constraints that imply that the scalar part in the metric cannot be chosen arbitrarily. It is a generalization of Newton’s potential which is determined as a fixed functional of the other fluctuations. For Minkowski signature this is the basic reason why the apparent metastability of flat space does not induce any instability of the solutions of field equations. One concludes that on the level of the effective action not every instability in the space of all arbitrary field configuration leads to an instability of the field configurations that can be reached by solutions of field equations. It seems rather likely that these features also play a role for the effective action in a euclidean setting. The field configurations that can be reached by solutions of the field equations include the presence of conserved sources and arbitrary boundary conditions. Nevertheless, the metastability for arbitrary configurations in the scalar sector may not lead to metastable solutions of field equations. We conclude that unboundedness of the euclidean effective action points to a possible problem, but is not yet per se and indication for an unacceptable instability. If one formulates the classical or microscopic action for the definition of the functional integral at distances much smaller than m−1, |˜m|−1, there is not a very direct relation to the quantum effective action for momenta q2 of the order of m2, m˜ 2 or M2. By virtue of universality a large family of classical actions can lead to the same effective action in this momentum range. Even if we associate the effective action at some very short distance scale l0 with the classical action, 2 ≈ −2 2 ≈ 2 the effective couplings will change their values between q l0 and q M . This flow of 2 2 couplings can be a very substantial effect if M l0 1. The stability discussion in the sector of scalar fluctuations involves the couplings at q2 ≈ M2. A tachyonic instability can be avoided if the running coupling Z˜ q2 ≈ M2 is negative. This is enough for a stable particle pole at q2 = ν2 ≈ O(M2), ν2 < 0. So far we have not discussed a possible running of the couplings. Constant couplings may be a valid approximation for the momentum range which is relevant for the stability discussion. In general, however, the dimensionless couplings will be functions of q2/k2, where k is some suitable renormalization scale. In a covariant setting q2 stands for squared covariant derivatives, such that the flow of couplings is associated with an effective field dependence. It is well conceivable that couplings Z˜ and m˜ 2 flow for q2 →∞to positive values. This may well lead to a bounded euclidean effective action, in correspondence to the bounded classical action. The boundedness would become visible for q2 m2, |˜m|2, M2 or very large field values. Nevertheless, flat space would not be the minimum of the euclidean effective action but rather be a saddle point.

7.5. Stability of other modes

So far we have found a region (188)in parameter space for which the traceless transversal tensor and scalar fluctuation modes are stable. We will conclude this section by establishing that for this parameter range also all other fluctuation modes are stable. For simplicity we concentrate on B = 0. We also omit the gauge modes, setting lμ = 0, κμ = 0, Mμν = 0, u = 0. We begin with the gauge boson fluctuation Cμνρ. In quadratic order in an expansion around flat space there are no contributions from LG and the inverse propagator is proportional Z (1 + 2y)m2 P = q2 + μ2 ,μ2 = . (193) C 2 C C Z

37 C. Wetterich Nuclear Physics B 971 (2021) 115526

This fluctuation has the same mass as the scalar fluctuation v˜ and is therefore stable. (t) The transverse vectors γμ and wμ are mixed. The inverse propagator matrix, ⎛ ⎞ 3 2 + 2 + 2 + 2 2 2 2 ⎜Zq 4m 2n M q n M ⎟ = 2 ⎜ 2 ⎟ Pwγ ⎝ 3 9 ⎠ , (194) 9 q2n2M2 q4(m2 + n2) 2 8 implies for this sector a massless fluctuation with pole at q2 = 0, and a massive excitation with 2 =− 2 pole at q μwγ , 4m4 + 6m2n2 + M2(m2 − 3n2 − 2M2) μ2 = . (195) wγ Z(m2 + n2) 2 2 2 2 2 For a whole range of n /m and M /m the massive particle is stable, μwγ > 0. (t) Finally, for the sector of the fluctuations v μ and bμν we employ ˜μ μνρσ ˜μ b = ε ∂νbρσ ,∂μb = 0 . (196) (t) ˜ The transverse vectors v μ and bμ mix, with propagator matrix ⎛ ⎞ M2 ⎜Zq2 + m2 + M2 ⎟ 1 ⎜ 2 ⎟ Pvb = ⎝ ⎠ . (197) 2 M2 m2 2 2 In this sector one finds again a stable massive particle with positive mass term 1 + y − 1 y2 m2 μ2 = 2 . (198) bv Z We conclude that no tachyonic fluctuations around flat space are present. For U = 0 Minkowski space is a solution of the analytically continued model with Minkowski signature. The findings in the present section imply that the linearized field equations in the vicinity of Minkowski space have no unstable solution. In this respect our model of pregeometry shares the stability property of general relativity.

8. Cosmological constant

The issue of metastability of the ﬂat space solution concerns partly the physics at low momenta and geometries with large volumes. If we add a cosmological constant U in eq. (93), it plays a central role for the behavior for q2 → 0. The cosmological constant U does not vanish for q2 → 0, in contrast to all other terms that involve derivatives. In this section we include in the effective action a small positive cosmological constant U>0. We will see its profound implications for the effective action for euclidean gravity.

8.1. Spheres with vanishing covariant derivative of vierbein

Let us consider a particular family of ﬁeld conﬁgurations for which the vierbein is covariantly conserved, while the metric describes a sphere with radius L,

38 C. Wetterich Nuclear Physics B 971 (2021) 115526

m Uμν = 0 ,Fμνρσ = Rμνρσ , 1 R = (g g − g g ), μνρσ L2 μρ νσ νρ μσ 3 12 F = R = g ,F= R = . (199) μν μν L2 μν L2 The effective action (93)for this configuration is given for B = 0by 6M2 3Z + 72C (L) =cL4 U − + L2 L4 (200) =c(UL4 − 6M2L2 + 6Z˜ − 3Z) . Here we have employed e = cL4 . (201) x The minimum of (L) obeys 2 − 2 = UL0 3M 0 , (202) or 3M2 L2 = . (203) 0 U This corresponds to the solution of the Einstein equation 1 4U M2(R − Rg ) =−Ug ,R= . (204) μν 2 μν μν M2 At the minimum, the effective action is negative, 9M4 =−c(UL4 − 3Z) =−c − 6Z˜ + 3Z . (205) 0 0 U 4 For a small ratio U/M 1the effective action 0 takes large negative values, and the contributions ∼ Z,˜ Z can be neglected. Nevertheless, the function (L) remains bounded from below even for very small nonzero U/M4. For U>0 flat space is no longer a solution of the field equations. Instead, we find a new solution of the field equations corresponding to the sphere (199) with L2 given by eq. (203).

8.2. Minimum of euclidean effective action

It is an interesting question if this solution corresponds to the minimum of the euclidean action (93)on the space of possible solutions of the field equations. For this question we have to analyze fluctuations around this solution. We present several arguments suggesting that the solution (199), (203) could indeed be the minimum of the euclidean effective action on the space of possible solutions of field equations. We start by discussing the subspace of fluctuations for which the vierbein is covariantly con- m served, Uμν = 0. On this subspace the effective action becomes

39 C. Wetterich Nuclear Physics B 971 (2021) 115526 M2 = e U − R +˜αR2 + βR˜ R˜μν + γC Cμνρσ , 2 μν μνρσ x Z C 1 α˜ = + = (2Z˜ − Z) , (206) 48 2 48 1 with R˜ = R − Rg and C the Weyl tensor. The metric is the only relevant degree μν μν 2 μν μνρσ of freedom, such that the situation is precisely the same as for general relativity with effective action√ (206). We assume a form of LG for which the coefficients β and γ remain positive. With e = g eq. (206)is a particular form of metric gravity. We may discuss the question of the minimum of for arbitrary couplings U>0, M2 > 0, β, γ>0. ˜ Consider first the limit β →∞, γ →∞. The minimum of has to obey in this limit Rμν = 0, Cμνρσ = 0. Solutions of the field equations with these properties are maximally symmetric spaces. On this subspace the spheres have lower action as compared to the ones with R<0. We have already found the minimum for the space of spheres. We conclude that for β →∞, γ →∞ the minimum of the effective action on the space of solutions of the field equations is indeed given by a sphere with radius determined by eq. (203). We can consider an alternative limit with finite β and γ , taking now U, M2 and α˜ to zero. The effective action is positive semidefinite and its minimum occurs for maximally symmetric spaces. For spheres there is a flat direction corresponding to L. Allowing for α˜ = 0, the flat direction remains present, cf. eq. (200). Adding M2 and U the degeneracy is lifted, and (L) 4 develops a rather deep minimum for small U/M . The large negative value of 0 is due to the large volume of the sphere. For U/M4 1the geometry of space is almost flat, as observed for the almost de Sitter geometry of our present Universe for Minkowski signature. In the limit U/M4 → 0the geometry approaches flat space. In this limit the graviton fluctuations around the configuration (199), (203) remain stable. For an expansion around a sphere the graviton fluctuations still correspond to massless excitations with a positive kinetic term. This differs from graviton fluctuations around flat space which develop a tachyonic instability for U>0. Besides the graviton, the only physical fluctuation that remains is the scalar fluctuation of the metric. For flat space the effective action contains a term linear in the scalar fluctuation, indicating a different minimum with 0 < 0. The configuration (199), (203) corresponds to this required new extremum for the scalar fluctuation. In view of these arguments it seems rather likely that the sphere could be on acceptable minimum for the effective action (206)for metric gravity within the space of configurations that can be reached by solutions of field equations. For a pure metric theory the propagators derived from this polynomial effective action show tachyonic or ghost instabilities for q2 around M2. This is of no worry in our case since we have already found stability in this momentum region for the full effective action (93). Relaxing the restriction on configurations with vanishing covariant derivative of the vierbein, m m m it seems useful to consider eμ and Uμν as variables, instead of eμ and Aμmn. The term LU m 2 2 provides for a positive quadratic term for Uμν . For n > −3m this extends to LG. Expanding m m around the sphere there is no term linear in Uμν . The only terms linear in Uμν are from LF via eq. (116), and similarly from LG. They vanish for a covariantly constant curvature tensor. m In principle, mixed terms linear in Uμν and linear in fluctuations of the vierbein could turn the extremum characterizing the sphere to a saddle point. This seems not very likely in the low momentum range, given that the vierbein fluctuations have to generate covariant derivatives of the curvature tensor.

40 C. Wetterich Nuclear Physics B 971 (2021) 115526

8.3. Realistic euclidean quantum gravity

All these arguments do not constitute a proof that the sphere (199), (203)is the minimum of the euclidean effective action (93)on the space of possible solutions of the field equations. If this sphere is the minimum for the possible solutions of field equations, our formulation of pregeometry leads to a rather satisfactory state of euclidean quantum gravity. The minimum corresponds for tiny U/M4 almost to flat space. Its analytic continuation to Minkowski signature would correspond to de Sitter space with a tiny cosmological constant, close to the present Universe for U/M4 ≈ 10−120. With U/M4 = 10−120, the overall view of the ground state of euclidean pregeometry is 120 a very deep minimum with 0 ≈−10 , assumed by a sphere with huge radius in units 60 of the Planck mass, L0M ≈ 10 . The four dimensional volume of the sphere is extremely 4 4 ≈ 240 large, L0M 10 , corresponding to the size of our observable Universe in Planck units, M4/H 4 ≈ 10240. Field configurations with even larger volume L4 will lead to a huge positive ≈ 4 4 ≈ 120 4 action, due to the positive term x eU (L/L0) (M /U) 10 (L/L0) , which overwhelms 4 4 all other contributions to . Fluctuations with effective volume L substantially smaller than L0 change the effective action only by a small relative amount /0, suppressed by a volume 4 4 factor L /L0.

9. Conclusions and discussion

In this paper we propose an euclidean functional integral for a model of pregeometry that contains quantum gravity. The classical action is bounded from below and all two-point corre- lation functions or propagators have a simple short distance behavior without any instabilities. The inverse propagators in flat space are for all physical excitations proportional to the squared momentum q2 in the limit q2 →∞. Analytic continuation to Minkowski signature does not en- counter any problems, and there are neither ghost nor tachyon instabilities in the high momentum region. The functional integral still needs a regularisation. Since our model of pregeometry is a Yang-Mills gauge theory, the situation is in many respects similar to other gauge theories. The particularity is the diffeomorphism invariance of the classical action that should be preserved by the regularization. Dimensional regularization may be one of the possibilities, needing as all other continuum regularizations the introduction of some type of gauge fixing. It will be interesting to find out if a suitable lattice regularization exists, with Aμ and eμ replaced by variables on the links of a lattice. This may require lattice diffeomorphism invariance [17,55]for the discrete action, which should extend to the usual diffeomorphism symmetry in the continuum limit. We have discussed at length the properties of the quantum effective action and the associated issues of stability of solutions of the field equations. For low momenta it seems reasonable to expect that the invariants consistent with the symmetries and involving a low number of derivatives are present in the quantum effective action. For a realistic theory it is necessary that the term linear in the derivatives of the gauge field is generated by the fluctuations, with a coefficient M2 > 0. Furthermore, the couplings in a derivative approximation to the effective action have to be in a range for which all fluctuations around flat space are stable for a vanishing covariant derivative of eμ, Uμνρ = 0. In this case general relativity with the Einstein-Hilbert action for the composite metric emerges as the effective low momentum theory. By analytic continuation our model of pregeometry can then describe quantum gravity.

41 C. Wetterich Nuclear Physics B 971 (2021) 115526

It is straightforward to add fermions in our model. The corresponding Grassmann variables ψ are scalars with respect to general coordinate transformations and transform as Weyl or Dirac spinors with respect to the SO(4) - gauge group. The covariant derivative contains the gauge field in the usual way. The gauge invariant kinetic term for the fermions reads m μ † 0 Sψ = i eψγ em Dμψ, ψ = ψ γ , x 1 D ψ = ∂ ψ − A mnψ, (207) μ μ 2 μmn with γ m the euclidean Dirac matrices and mn the SO(4) - generators in the spinor representation, 1 mn =− γ m,γn , γ m,γn = 2δmn . (208) 4 μ With eem being a polynomial in the vierbein, μ μνρσ n p q eem ∼ ε εmnpq eν eρ eσ , (209) this action does actually not involve the inverse vierbein. Weyl spinors obtain by suitable projec- tions, ψ± = (1 ± γ 5)/2 ψ, γ 5 = ηγ 0γ 1γ 2γ 3. It is possible that the vierbein arises from an even more fundamental pregeometry as a composite fermion bilinear m m eμ = iψγ Dμψ. (210) Insertion of this expression into eqs. (207), (209) yields an invariant term involving eight powers of the fermion fields ψ. This is a gauge theory based only on fermions and gauge fields. Fi- nally, the gauge fields may also be expressed as composites of fermions similar to ref. [56]. This results in a type of spinor gravity with local “Lorentz” symmetry SO(4) and diffeomorphism invariance [17,18]. While conceptually rather interesting, it is clear that suitable technologies have to be developed for dealing with actions that involve only multi-fermion invariants. In the direction of a more realistic model for particle physics coupled to gravity, one needs to include additional gauge symmetries and corresponding gauge fields, as for the standard model gauge symmetry SU(3) × SU(2) × U(1) or some grand unified extension. Furthermore, scalar fields are needed for a description of spontaneous symmetry breaking. Interesting cosmologies can be obtained if the parameters Z, Z˜ , m2, m˜ 2, M2 become functions of a scalar singlet field χ, similar to variable gravity [57]. A first investigation [51] reveals that rather realistic cosmology with inflation and dynamical dark energy can be obtained for a suitable shape of the coupling functions. The central remaining issue with which we have not addressed in this paper concerns the renormalizability of our model of pregeometry. This involves an investigation of the dependence of the couplings Z, C, m2, n2, M2 and U on some renormalization scale k. More generally, beyond the couplings mentioned explicitly, one needs to understand the dependence of the whole scale-dependent effective action k on k. This question is a typical issue for functional renormalization which can be addressed by suitable truncations of the exact flow equation for the effective average action [4]. The scale-dependent effective action should admit a fixed point or scaling solution for k →∞, which remains sufficiently simple and close to the proposed classical action. Only in this case our model of pregeometry can be considered as a fully satisfactory description of emerging quantum gravity.

42 C. Wetterich Nuclear Physics B 971 (2021) 115526

Declaration of competing interest

The authors declare that they have no known competing ﬁnancial interests or personal rela- tionships that could have appeared to inﬂuence the work reported in this paper.

References

[1] K.S. Stelle, Renormalization of higher-derivative quantum gravity, Phys. Rev. D 16 (1977) 953. [2] E.S. Fradkin, A.A. Tseytlin, Renormalizable asymptotically free quantum theory of gravity, Nucl. Phys. B 201 (1982) 469. [3] I.G. Avramidy, A.O. Barvinsky, Asymptotic freedom in higher-derivative quantum gravity, Phys. Lett. B 159 (1985) 269. [4] Christof Wetterich, Exact evolution equation for the effective potential, Phys. Lett. B 301 (1993) 90–94, arXiv : 1710 .05815 [hep -th]. [5] S. Weinberg, Ultraviolet divergences in quantum theories of gravitation, in: General Relativity: an Einstein Cente- nary Survey, Cambridge University Press, 1980, p. 790. [6] M. Reuter, Nonperturbative evolution equation for quantum gravity, Phys. Rev. D 57 (1998) 971, arXiv :hep -th / 9605030 [hep -th]. [7] D. Dou, R. Percacci, The running gravitational couplings, Class. Quantum Gravity 15 (1998) 3449, arXiv :hep -th / 9707239 [hep -th]. [8] W. Souma, Non-trivial ultraviolet fixed point in quantum gravity, Prog. Theor. Phys. 102 (1999) 181. [9] M. Reuter, F. Saueressig, Renormalization group flow of quantum gravity in the Einstein-Hilbert truncation, Phys. Rev. D 65 (2002) 065016. [10] O. Lauscher, M. Reuter, Ultraviolet fixed point and generalized flow equation of quantum gravity, Phys. Rev. D 65 (2001) 025013, arXiv :hep -th /0108040 [hep -th]. [11] Alfio Bonanno, Astrid Eichhorn, Holger Gies, Jan M. Pawlowski, Roberto Percacci, Martin Reuter, Frank Sauer- essig, Gian Paolo Vacca, Critical reflections on asymptotically safe gravity, Front. Phys. 8 (2020) 269, arXiv : 2004 .06810 [gr-qc]. [12] A. Platania, C. Wetterich, Non-perturbative unitarity and fictitious ghosts in quantum gravity, arXiv :2009 .06637 [hep -th], 2020. [13] K. Akama, An attempt at pregeometry: – gravity with composite metric, Prog. Theor. Phys. 60 (1978) 1900. [14] D. Amati, G. Veneziano, A unified gauge and gravity theory with only matter fields, Nucl. Phys. B 204 (1982) 451. [15] G. Denardo, E. Spallucci, Curvature and torsion from matter, Class. Quantum Gravity 4 (1987) 89. [16] C. Wetterich, Spontaneous symmetry breaking origin for the difference between time and space, Phys. Rev. Lett. 94 (2005) 011602, arXiv :hep -th /0405223 [hep -th]. [17] C. Wetterich, Spinor gravity and diffeomorphism invariance on the lattice, Lect. Notes Phys. 67 (2013), arXiv : 1201 .2871 [gr-qc]. [18] C. Wetterich, Geometry and symmetries in lattice spinor gravity, Ann. Phys. 327 (2012) 2184, arXiv :1201 .6505 [hep -th]. [19] C. Wetterich, Gravity from spinors, Phys. Rev. D 70 (2004) 105004, arXiv :hep -th /0307145 [hep -th]. [20] A. Hebecker, C. Wetterich, Spinor gravity, Phys. Lett. B 574 (2003) 269, arXiv :hep -th /0307109 [hep -th]. [21] C. Wetterich, Dimensional reduction of fermions in generalized gravity, Nucl. Phys. B 242 (1984) 473. [22] C. Wetterich, Fundamental scale invariance, arXiv :2007 .08805 [hep -th], 2020. [23] Roberto Percacci, Spontaneous soldering, Phys. Lett. B 144 (1984) 37–40. [24] Dmitri Diakonov, Towards lattice-regularized quantum gravity, arXiv :1109 .0091 [hep -th], 2011. [25] Alexey A. Vladimirov, Dmitri Diakonov, Phase transitions in spinor quantum gravity on a lattice, Phys. Rev. D 86 (2012) 104019, arXiv :1208 .1254 [hep -th]. [26] Shinya Matsuzaki, Shota Miyawaki, Kin-ya Oda, Masatoshi Yamada, Dynamically emergent gravity from hidden local Lorentz symmetry, Phys. Lett. B 813 (2021) 135975, arXiv :2003 .07126 [hep -th]. [27] Kirill Krasnov, Roberto Percacci, Gravity and unification: a review, Class. Quantum Gravity 35 (2018) 143001, arXiv :1712 .03061 [hep -th]. [28] E. Cartan, Sur une généralisation de la notion de courbure de Riemann et les espaces á torsion, C. R. Acad. Sci. 174 (1922) 593. [29] R. Utiyama, Invariant theoretical interpretation of interaction, Phys. Rev. 101 (1956) 1597. [30] T.W.B. Kibble, Lorentz invariance and the gravitational field, J. Math. Phys. 2 (1961) 212.

43 C. Wetterich Nuclear Physics B 971 (2021) 115526

[31] D.W. Sciama, The physical structure of general relativity, Rev. Mod. Phys. 36 (1964) 463. [32] F.W. Hehl, P.v.d. Heyde, G.D. Kerlick, J.M. Nester, General relativity with spin and torsion: foundations and prospects, Rev. Mod. Phys. 48 (1976) 393. [33] I.L. Shapiro, Physical aspects of the space–time torsion, Phys. Rep. 357 (2002) 113. [34] F.W. Hehl, J.D. McCrea, E.W. Mielke, Y. Ne’eman, Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance, Phys. Rep. 258 (1995) 1. [35] L. Heisenberg, A systematic approach to generalisations of general relativity and their cosmological implications, Phys. Rep. 796 (2019) 1, arXiv :1807 .01725 [gr-qc]. [36] R. Percacci, Towards metric-affine quantum gravity, arXiv :2003 .09486 [gr-qc], 2020. [37] R. Percacci, E. Sezgin, New class of ghost- and tachyon-free metric affine gravities, Phys. Rev. D 101 (2020) 084040, arXiv :1912 .01023 [hep -th]. [38] M. Shaposhnikov, A. Shkerin, I. Timiryasov, S. Zell, Einstein-Cartan gravity, matter, and scale-invariant generalization, arXiv :2007 .16158 [hep -th], 2020. [39] S. Matsuzaki, S. Miyawaki, K.y. Oda, M. Yamada, Dynamically emergent gravity from hidden local Lorentz symmetry, arXiv :2003 .07126 [hep -th], 2020. [40] Kenji Hayashi, Takeshi Shirafuji, Gravity from Poincare gauge theory of the fundamental particles. 1. Linear and quadratic Lagrangians, Prog. Theor. Phys. 64 (1980) 866; Erratum: Prog. Theor. Phys. 65 (1981) 2079. [41] Kenji Hayashi, Takeshi Shirafuji, Gravity from Poincare gauge theory of the fundamental particles. 3. Weak field approximation, Prog. Theor. Phys. 64 (1980) 1435; Erratum: Prog. Theor. Phys. 66 (1981) 741. [42] Kenji Hayashi, Takeshi Shirafuji, Gravity from Poincare gauge theory of the fundamental particles. 4. Mass and energy of particle spectrum, Prog. Theor. Phys. 64 (1980) 2222. [43] E. Sezgin, P. van Nieuwenhuizen, New ghost-free gravity Lagrangians with propagating torsion, Phys. Rev. D 21 (1980) 3269–3280. [44] V.P. Nair, S. Randjbar-Daemi, V. Rubakov, Massive spin-2 fields of geometric origin in curved spacetimes, Phys. Rev. D 80 (2009) 104031, arXiv :0811 .3781 [hep -th]. [45] V. Nikiforova, S. Randjbar-Daemi, V. Rubakov, Infrared modified gravity with dynamical torsion, Phys. Rev. D 80 (2009) 124050, arXiv :0905 .3732 [hep -th]. [46] J.-E. Daum, M. Reuter, Einstein–Cartan gravity, asymptotic safety, and the running Immirzi parameter, Ann. Phys. 334 (2013) 351, arXiv :1301 .5135 [hep -th]. [47] U. Harst, M. Reuter, A new functional flow equation for Einstein–Cartan quantum gravity, Ann. Phys. 354 (2015) 637, arXiv :1410 .7003 [hep -th]. [48] J.-E. Daum, M. Reuter, Renormalization group flow of the Holst action, Phys. Lett. B 710 (2012) 215, arXiv : 1012 .4280 [hep -th]. [49] K. Krasnov, Gravity as a diffeomorphism-invariant gauge theory, Phys. Rev. D 84 (2011) 024034, arXiv :1101 .4788 [hep -th]. [50] K. Krasnov, Pure connection action principle for general relativity, Phys. Rev. Lett. 106 (2011) 251103, arXiv : 1103 .4498 [gr-qc]. [51] C. Wetterich, Cosmology from pregeometry, arXiv :2104 .14013 [gr-qc], 2021. [52] C. Wetterich, Quantum correlations for the metric, Phys. Rev. D 95 (2017) 123525, arXiv :1603 .06504 [gr-qc]. [53] C. Wetterich, Pregeometry and spontaneous time-space asymmetry, arXiv :2101 .11519 [gr-qc], 2021. [54] C. Wetterich, Spinors in euclidean field theory, complex structures and discrete symmetries, Nucl. Phys. B 852 (2011) 174–234, arXiv :1002 .3556 [hep -th]. [55] C. Wetterich, Lattice diffeomorphism invariance, Phys. Rev. D 85 (2012) 104017, arXiv :1110 .1539 [hep -lat]. [56] C. Wetterich, Scalar lattice gauge theory, Nucl. Phys. B 876 (2013) 147–186, arXiv :1212 .3507 [hep -lat]. [57] C. Wetterich, Variable gravity universe, Phys. Rev. D 89 (2014) 024005, arXiv :1308 .1019 [astro -ph .CO].