Global Flows in Quantum Gravity

Global Flows in Quantum Gravity

N. Christiansen Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany    B. Knorr Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany    J. M. Pawlowski Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung mbH, Planckstr. 1, 64291 Darmstadt, Germany    A. Rodigast Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
Abstract

We study four-dimensional quantum gravity using non-perturbative renormalization group methods. We solve the corresponding equations for the fully momentum-dependent propagator, Newton’s coupling and the cosmological constant. For the first time, we obtain a global phase diagram where the non-Gaussian ultraviolet fixed point of asymptotic safety is connected via smooth trajectories to a classical infrared fixed point. The theory is therefore ultraviolet complete and deforms smoothly into classical gravity as the infrared limit is approached.

I Introduction

Understanding the quantization of the gravitational force is an outstanding problem in theoretical physics. Any viable theory of quantum gravity must connect stable infrared (IR) physics with a well-behaved ultraviolet (UV) limit. The asymptotic safety scenario provides a UV-completion in a natural way. It is based on a non-Gaussian UV fixed point, which leads to vanishing -functions in the limit of arbitrarily high energy scales and renders the couplings finite even beyond the Planck scale Weinberg (1979).

The asymptotic safety scenario received growing attention during the past decades and has been studied with different methods. The underlying fixed point structure was found in the non-perturbative continuum approach Donkin and Pawlowski (2012); Christiansen et al. (2014a); Reuter (1998); Litim (2004); Niedermaier and Reuter (2006); Percacci (2007); Codello et al. (2009); Litim (2011); Reuter and Saueressig (2012); Nagy (2012), as well as in lattice simulations Hamber (2009); Ambjorn et al. (2012, 2013). The former is based on the functional renormalization group, in particular on its formulation for the effective action Wetterich (1993). The crucial UV fixed point is confirmed in various approximations, including the coupling to gauge and matter fields Percacci and Perini (2003); Daum et al. (2010); Eichhorn and Gies (2011); Folkerts et al. (2012); Harst and Reuter (2011); Donà et al. (2013), dilaton gravity Henz et al. (2013) and higher derivative calculations Benedetti et al. (2009); Falls et al. (2013); Ohta and Percacci (2014); Benedetti (2013); Demmel et al. (2014). There is also a rich field of phenomenological applications based on asymptotically safe quantum gravity. This includes e.g. implications for the standard model and its extensions Shaposhnikov and Wetterich (2010); Antipin et al. (2013), black hole physics Falls et al. (2012); Falls and Litim (2012); Koch and Saueressig (2014a, b), collider experiments Litim and Plehn (2008); Gerwick et al. (2011) and cosmology Weinberg (2010); Copeland et al. (2013). However, the standard calculations lead to an ill-defined IR limit since the trajectories exhibit a singular behaviour on large length scales.

In Donkin and Pawlowski (2012) an IR fixed point has been found as the endpoint of a singular line. The existence of this fixed point was only seen within a proper distinction of background and dynamical couplings. The singular behaviour in the vicinity of the IR fixed point was attributed to the approximation. The first smooth IR fixed point in asymptotically safe gravity, allowing for a theory that is well defined on all energy scales, was found in Christiansen et al. (2014a). There, however, a non-classical behaviour in the vicinity of the IR fixed point has been computed, therefore leading to modified gravity on very large length scales. Again, the non-classical behaviour may be attributed to the approximation. Further discussions on this issue from different perspectives can be found in Litim and Satz (2012); Nagy et al. (2012); Rechenberger and Saueressig (2012); Contreras and Litim (2014).

In the present work, we construct a qualitatively enhanced approximation within the systematic vertex expansion scheme introduced in Christiansen et al. (2014a). With this enhanced approximation the theory is asymptotically safe in the UV, and exhibits an IR fixed point which describes classical gravity. The corresponding renormalization group (RG) trajectories are globally smooth. They connect the known IR physics with classical gravity on large length scales with a viable theory of Planckian and trans-Planckian gravity. The present scenario encodes that quantum gravity effects set in at about the Planck-scale, and are absent for energy-scales .

This work improves the vertex expansion scheme set-up in Christiansen et al. (2014a) in several aspects. We compute, for the first time, fully momentum-dependent wave-function renormalizations for the graviton and the ghost. Note that the wave function renormalizations are functions of the covariant Laplacian, and hence this takes into account infinitely many terms in an expansion of the effective action in powers of the covariant derivative. Additionally, multi-graviton interactions are constructed from their scaling behaviour, see Fischer and Pawlowski (2009): the dependence on the running RG-scale is deduced from consistent RG-scaling of the vertex. This novel self-consistent vertex construction is of major importance for the transition from UV to IR scaling, and the stability of the IR regime.

The present work is organized as follows: We introduce our approximation scheme in section II, and the vertex construction in subsection III.1. The flow equations for the fully momentum-dependent propagators are derived in the subsequent subsections. The consistency analysis in subsection III.4 constrains the momentum-independent parts of the vertex functions, and is crucial for the global properties of the phase diagram. The latter is the topic of the first section of the results, subsection IV.1. Moreover, the reliability of all results is tested within a regulator study. To this end we use the optimized regulator as well as a class of exponential ones. The results are found to be stable against variations of the regulator. The properties of the UV regime are discussed in subsection IV.2. In this section we also investigate the stability and reliability of the derivative expansion, at the basis of the full momentum-dependence. In subsection IV.3 the properties of the IR regime are discussed.

Ii Flows in quantum gravity

A quantum field theory is entirely described by a complete set of correlation functions. The generating functional for the 1PI correlators is the effective action , where we have already introduced a fixed background metric and a fluctuation super-field . In the case of gravity this super-field is given by the vector , where is the graviton field and are the corresponding ghost and anti-ghost fields.

In the present work we use the functional renormalization group approach, for reviews on quantum gravity see Litim (2004); Niedermaier and Reuter (2006); Percacci (2007); Codello et al. (2009); Litim (2011); Reuter and Saueressig (2012); Nagy (2012), for general reviews and other applications see e.g. Litim and Pawlowski (1999); Berges et al. (2002); Aoki (2000); Bagnuls and Bervillier (2001); Polonyi (2003); Pawlowski (2007); Gies (2012); Schaefer and Wambach (2008); Rosten (2012); Pawlowski (2011); Scherer et al. (2010); Braun (2012); Metzner et al. (2012); Boettcher et al. (2012); Delamotte (2012) . With the functional renormalization group, the effective action can be determined via a functional differential equation for the scale-dependent effective action , which for quantum gravity is given by Reuter (1998)

(1)

It involves an IR regulator which is implemented on the level of the path integral and carries an IR cutoff-scale . In addition to that, denotes the logarithmic RG scale, , with an arbitrary normalization scale , and the Tr implies an integral over all continuous and a sum over all discrete indices. We will also make use of the notation for any scale-dependent quantity. Moreover, we have introduced the notation

(2)

for the 1PI vertex functions, which are derivatives of the effective action with respect to the fluctuation fields and are the elements of the full super space matrix .

The right hand side of the flow equation (II) depends on the the two-point correlators of the fluctuation field . It is important to note that the fluctuation correlation functions do not agree with the background correlations, i.e.

(3)

for details see Pawlowski (2002); Litim and Pawlowski (2002); Braun et al. (2010); Folkerts et al. (2012); Christiansen et al. (2014a). In other words, one cannot extract dynamical couplings from the flow of the background field effective action at vanishing fluctuation fields, . This directly relates to the fact that the flow equation (II) for the effective action at is not closed within the standard background field approach. More importantly, avoiding unphysical background contributions can be crucial for capturing the correct non-perturbative physics. In conclusion, in the FRG setup, this strongly suggests to start from the exact equation for the inverse fluctuation propagator.

Moreover, the master flow equation (II) leads to an infinite hierarchy of coupled partial integro-differential equations for the scale-dependent vertex functions . More precisely, the equation for the -point vertex function contains vertex functions of order and . This system is usually not exactly solvable. Therefore, one has to employ certain approximation schemes. We assume that the effective action can be expanded in a functional Taylor-series around the fixed background metric . Moreover, we choose the flat Euclidean metric, i.e. the identity , as the expansion point. We will need this expansion up to fourth order in the graviton field. In symbolic notation, the effective action takes the form

(4)

The first and second term are of order and and do not enter the RHS of the flow equations for any .

The above expansion of the effective action in powers of the fluctuating field around a flat Euclidean background also restricts the number of higher derivative operators which can contribute to the vertex functions of -th order. For instance, the most general form of the two-point function derives from an action which includes at most operators. All higher order terms vanish after two functional differentiations and evaluation on a flat background. Terms with higher order than this only contribute to vertex functions of order .

Iii Vertex expansion

In the present work we use the systematic vertex expansion scheme as suggested in (4). The hierarchy of flow equations that has been introduced in the last section has to be truncated at finite order. This means that one can calculate the flow of a vertex function of given order and use an ansatz for and . We will compute the basic quantity of the present approach, the full two-point correlation functions of the fluctuation fields , that is . The corresponding flows rely on the two- but also on the three- and four point functions of the fluctuation fields. Hence we also introduce approximations for and that are consistent with the symmetries of the theory and have the correct RG-scaling. The latter property is essential for the global UVIR flows considered here.

iii.1 Structure of the vertex functions

First of all, we have to specify the tensor structures of the vertices. In the present work, we use the classical tensor structures which arise from functional differentiation of the Einstein-Hilbert action. The gauge-fixed Einstein-Hilbert action, including the ghost part, is given by

(5)

In (III.1), is the Newton constant and is the cosmological constant. The Fadeev-Popov operator is given by

(6)

and the linear gauge fixing conditions reads

(7)

Moreover, in this work we restrict ourselves to Landau gauge, that is . Extended ghost interactions are studied e.g. in Eichhorn et al. (2009); Eichhorn (2013).

The standard Einstein-Hilbert truncation amounts to replacing the gravitational coupling and the cosmological constant in (III.1) by running couplings and . The vertex functions are then given by functional derivatives of this effective action. However, this approximation turns out to be inconsistent in the physical IR limit, which will be discussed in detail later. We also mention that the basic Einstein-Hilbert truncation does not disentangle the difference between a wave-function renormalization and a running coupling, since the running of the latter is simply identified with the running of the former. Note that in Yang-Mills theory such an approximation gives a deconfining potential of the order parameter even in the confining regime, see Braun et al. (2010); Marhauser and Pawlowski (2008); Fister and Pawlowski (2013). In this case, it is also the non-trivial momentum-dependence of the correlation functions that plays a crucial rle for capturing the correct non-perturbative physics. Additionally, in the UV limit, , the full momentum-dependence of correlation functions is potentially relevant. In particular, a derivative expansion implies which relates to low energy physics. So far, these momentum-dependencies have not been taken into account.

Consequently, we construct more general vertex functions that take into account the above properties while keeping the classical tensor structures. The construction of such vertex expansions of the scale-dependent effective action was introduced in Fischer and Pawlowski (2009) and applied in the context of Yang-Mills theories. A similar truncation based on these ideas was recently applied in quantum gravity to Codello et al. (2013). One guiding principle in this construction is RG invariance, i.e. invariance of the full effective action under a change of the renormalization scale :

(8)

where should not be confused with the running IR cutoff scale , for a detailed discussion see Pawlowski (2007). In addition to that, we parameterize the vertex functions, i.e. the coefficients in the expansion (4) schematically as

(9)

with a -independent part and a -dependent wave-function renormalization of the attached fields. The above construction implies the correct scaling behaviour for the fields according to

(10)

In the present work we use a uniform wave function renormalization, , for all components of the graviton. Non-uniform -factors will be subject to a forthcoming publication Christiansen et al. (2014b). The transverse-traceless () part of the full propagator is now parameterized as

(11)

with the transverse-traceless projector , and an effective mass term representing the momentum-independent part of the two-point function. Note that the -factors are functions of the covariant Laplacian , and therefore include infinitely many terms in a covariant expansion of the effective action in powers of the Laplace operator. This is the first RG study of quantum gravity taking into account this general momentum dependence of the graviton and the ghost propagator. The fully momentum-dependent -factors lead to a vertex construction with the required RG scaling properties. They also embody a corresponding implicit, non-canonical momentum-dependence of the vertex functions, in line with the co-linear singularity structure of vertex functions. Note that a similar vertex construction is achieved by simply taking further -derivatives of the two-point correlation function (11) with .

Finally, we allow for additional running parameters which govern the (consistent) scale-dependence of the momentum-independent part of the vertex functions. This takes into account scaling properties of the vertex functions that are crucial for the global flows structure, and has not been considered before. The construction of the vertex functions also include appropriate powers of a scale-dependent Newton coupling as prefactors of the vertex functions. These factors, apart from the wave function renormalization factors, encode the correct scale-dependence of the vertices, see Fischer and Pawlowski (2009). Note that in general there are separate coupling constants for each vertex function, but we identify in the present work. Still, this construction goes far beyond the approximations considered so far in RG-gravity. In summary, the vertex functions take the form

(12)

with tensor structures

(13)

that arise from functional differentiation of the classical Einstein-Hilbert action given in (III.1). We left out the momentum arguments on the LHS as well as the functional dependence on the fields. If we leave out the functional argument, it is implicitly understood that the functional is evaluated at vanishing fluctuation fields and on a flat background. The tensor structures carry not only the canonical explicit momentum-dependence of the vertex functions, but also the running parameters . They are defined by

(14)

where is the tensor structure arising from functional differentiation of the integral of the volume form with respect to the metric tensor. The factor of in the definition is introduced such that we recover the classical cosmological constant in the Einstein-Hilbert action. We emphasize again that the consistent RG-scaling of the parameters is of major importance for the transition from the UV regime to the IR regime, as well as for the stability of the IR regime. In this regime the are determined via a self-consistency analysis of the scaling behaviour. Details will be presented in the corresponding results sections.

An example for the above vertex construction is the scalar coefficient of the TT-two point function of the graviton,

(15)

see also (11). The tensor structures for the general two-point function, from which the above results via TT-projection, are given by (13) and arise from functional differentiation of the Einstein-Hilbert action. Equation (15) and (11) entail that is the effective graviton mass,

(16)

Once again, note that this mass term is not the cosmological constant.

iii.2 Flow of the propagator

The flow of in a standard Einstein Hilbert truncation has been calculated in Christiansen et al. (2014a). The corresponding flow equation is obtained by functional differentiation of (II), and its diagrammatic representation is shown in Figure 1.

Figure 1: Diagrammatic representation of the flow of the second order vertex functions. The dressed graviton propagator is represented by a double line, the dressed ghost propagator by a dashed line, while a dressed vertex is denoted by a dot and the regulator insertion by a crossed circle.

In the present work, we compute the fully momentum-dependent two-point functions with the RG-consistent ansatz (12) for the three- and the four-point function. With Figure 1 and (12) the flows of the two-point functions depend on

(17)

Here we have already dropped the subscript , and if not stated otherwise, all generalized couplings are scale-dependent.

The two-point function contains all tensor structures in the York transverse-traceless decomposition. The transverse-traceless part is not constrained by Slavnov-Taylor identities and is expected to carry the essential properties of the graviton. In this work we identify all wave function renormalizations of the graviton modes with that of the transverse-traceless mode. Hence, from now on, all expressions for the two-point functions relate to the TT-part. A study with all independent tensor structures will be presented elsewhere Christiansen et al. (2014b).

In this work, we use a regulator of the form with , where denotes the tensor structure of the corresponding two-point function evaluated at vanishing mass, and is a dimensionless shape function. Then, the flow of the inverse propagator reads

(18)

where

(19)

In the above equation, are scalar functions that arise from the contraction of the diagrams and a subsequent projection onto the TT-structure, and . This structure follows from our vertex construction discussed above. Explicit expressions for the flow equations are given in Appendix A. For the ghost sector, we apply the same strategy and arrive at the much simpler equation

(20)

Note that with the RG consistent vertex ansatz (12) and the structure of the flow equations for , one can infer that the wave-function renormalization does never enter a flow equation alone, but always in the combination . This motivates the definition of the anomalous dimensions of the graviton,

(21)
and of the ghosts,
(22)

We note that the RHS of the flow of the does depend on and the three- and four-point functions, see (17). In the standard Einstein-Hilbert setup, a scale-dependent gravitational coupling is constructed from the graviton wave-function renormalization, and the momentum-independent parts of the vertex functions are all identified with the cosmological constant, . This procedure closes the equations (III.2) and (20) and was used at least in parts in all FRG gravity calculations so far.

In the present work this identification, which spoils the scaling properties of the correlation functions, is avoided. For the gravitational coupling constant , we use the relation of the present framework at flat backgrounds to that with geometrical effective actions, see Donkin and Pawlowski (2012). This is discussed in more detail below. The couplings are constrained within a self-consistency analysis, see subsection III.4.

Finally we introduce dimensionless, scale-dependent couplings, to wit

(23)
(24)

with and . It is left to project the functional flow onto individual flow equations for all running couplings.

iii.2.1 The running mass

The flow equation for the mass is obtained from the flow of the inverse propagator, evaluated at the pole of the propagator, i.e. . Taking the -derivative of the on-shell two-point function, , yields

(25)

Solving for the running of the mass parameter, we get

(26)

One of the goals of this work is to evaluate the phase diagram of quantum gravity and its fixed point structure. For this reason, it is convenient to derive - functions for the dimensionless parameters. Then, the above equation translates into

(27)

The explicit form of the -function is given in Appendix A. It is clear from (19) that the -function for the running mass shows a functional dependence on the anomalous dimensions. It turns out that the final result for the -function of and the momentum- dependent equation for the anomalous dimension does not depend on the projection point, see Appendix B. The finite difference at is just the most convenient choice.

iii.2.2 Integral equations for the anomalous dimensions

Starting from the general equation (III.2), using the above definition of the anomalous dimension and inserting (26), we obtain an integral equation for which reads

(28)

Note that all isolated - factors drop out, see (III.2). The same procedure can be applied for the ghost sector. Since there is no ghost mass, we trivially arrive from (20) at

(29)

The explicit form of eqs. (28) and (29) is given in Appendix A. The full expressions of are derived and solved numerically, with the help of Form Vermaseren (2000), xTensor Martingarcia et al. (2007), and Eigen Guennebaud et al. (2013).

iii.3 The running gravitational coupling

As already mentioned, we use geometrical flow equations for in order to close the system of differential equations. This approach allows an inherently diffeomorphism- invariant construction of flows in quantum gravity, see Branchina et al. (2003); Pawlowski (2003). It has been applied to the phase structure of quantum gravity in Donkin and Pawlowski (2012), where evolution equations for the dynamical coupling , and for the background coupling are derived. In the present work we utilize the fact that the geometrical approach is directly related to the present approach in a flat background. In particular, the dynamical and background couplings in both approaches agree.

Moreover, with the fully momentum-dependent anomalous dimensions computed in the present work, we are able to directly incorporate effects of arbitrarily high powers of derivatives in the equations for , in Donkin and Pawlowski (2012). The anomalous dimensions enter the geometric flow equations in very much the same way as in (III.2). However, the wave-function renormalizations in Donkin and Pawlowski (2012) are momentum-independent and can be pulled outside the integrals. This is not the case in our set-up. Entering the equations with the momentum-dependent calculated via (28) leads to a modification on the level of the threshold functions . These modified threshold functions are given in Appendix A.

With these ingredients, the general structure of the -function for the dynamical gravitational coupling is given by

(30)

and the one for the background coupling takes the same form with being replaced by and an individual loop contribution . The functionals and are given in Appendix A. Note that the flow equation of the background coupling depends on the dynamical coupling via the anomalous dimensions, while the converse does not hold.

iii.4 The couplings

In subsection III.1 we have introduced an approximation which takes into account scale-dependent couplings for the momentum-independent part of each vertex function. For the second order, we have identified as the graviton mass , see (16). In the present section we discuss the vertices with .

The Einstein-Hilbert truncation, which identifies all with the cosmological constant , is ill-defined in the limit . As we will see in section IV, this limit is approached by physical RG trajectories in the deep IR. This regime is crucial to understand the global phase structure of Euclidean quantum gravity: the couplings play a distinguished role, as the related singularities arise from the momentum-independent parts of the vertex functions. In order to cure the inconsistencies of the Einstein-Hilbert truncation, we deduce the singularity structure of the couplings with . The full details are given in Appendix C. Essentially, the idea is to expand the right-hand sides of the flow equations for the -point functions in powers of and taking into account the singularities of highest order. Thus, for we use the ansatz

(31)

for . We proceed by inserting this ansatz in the flow equations for and analyze the generic loop integrals to leading order in the singularities that arise in the limits under consideration. Consistent scaling of both sides of the flow equations for arbitrary leads to the relations

(32)

for and

(33)

The parameter obeys the bound

(34)

The value of the parameters and cannot be obtained from the divergence analysis alone. They are dynamically determined by the flow of the three- and four-point function. This highlights again that the standard Einstein-Hilbert approximation with is inconsistent in the IR, and the non-existence of the IR fixed point cannot be inferred from such an approximation. It is also important to stress that the qualitative features of the phase diagram do not depend on the specific choice of and , see subsection IV.1 and Appendix D. In turn, the quantitative behaviour does only mildly depend on variations of these two parameters. Their flows will be studied in a forthcoming publication Christiansen et al. (2014b).

Still, we can estimate based on the saturation of the inequality (33). Moreover, the constant parts of the vertex functions are parametrically suppressed far away from the singular regime. This entails that there it is viable to identify as done in all other approximations used in the literature. From these conditions one obtains . More details are given in Appendix D. In Appendix E it is shown that

(35)

is consistent with all constraints, where parametrizes the deviation from the Einstein-Hilbert approximation. The latter is modeled by

(36)

with a parameter to be tuned to match the aforementioned conditions.

iii.5 The cosmological constant

It is left to discuss the role of the cosmological constant in the present construction. Written on the right hand side of the field equations, it can be interpreted as an additional source for gravity. In the classical limit, the quantum equations

(37)

with an external source , reduce to the classical equations of motion. Hence, it is natural to define the cosmological constant from the one-point function, i.e. we identify as the vacuum energy. More precisely, with the vertex construction (12), the one-point function takes the form

(38)

Note that the one-point function does not enter the flow of higher order vertex functions. Consequently, the cosmological constant decouples from the -functions for Newtons constant, the effective mass and the set of integral equations for the anomalous dimensions. On the other hand, these quantities obviously determine the running of the cosmological constant, i. e. the - function for the dimensionless cosmological constant is of the form

(39)

The explicit form of this flow equation is given in Appendix A.

iii.6 Regulators and stability

In order to test the quality of our truncation, we will use several regulators and vary the parameters and introduced before. As regulators, on the one hand we use the class of exponential regulators given by

(40)

where is the dimensionless squared momentum. In our analysis, we scanned the parameter range . On the other hand, the Litim regulator, Litim (2000), is used,

(41)

where is the Heaviside step function. Note that this regulator is optimized within the leading order derivative expansion but not beyond, see Litim (2000); Pawlowski (2007). Also, with the semi-optimized regulator the divergence analysis for the is slightly different from the one performed in Appendix C, but leads to similar results.

We also have scanned different values for the parameters and in (36), and we have restricted our investigation to the case of equality in (33). It turns out that the results do not depend on the specific choice of . Note that the parameter is bounded from above as otherwise the parametric suppression of the -contribution away from the singularity is lifted and the UV regime is changed. In Table 7 in Appendix D, a table is given where the change of the UV fixed point values under a change of and can be ascertained.

Iv Results

In this section we present our results. First, the global phase diagram is discussed. Subsequently, its UV and IR properties will be examined in more detail. In doing so, we will also make contact with older results. If not stated otherwise, all results and pictures are obtained with the specific choice of the exponential regulator .

iv.1 The phase diagram

The phase diagram for the dynamical couplings is depicted in Figure 2.

Figure 2: Fixed points and global phase diagram in the -plane. Arrows point from the IR to the UV, red solid lines mark separatrices while dots indicate fixed points. The black dashed line is a specific trajectory that connects the UV fixed point with the non-trivial IR fixed point, which is analyzed further in the text. In analogy to Christiansen et al. (2014a), region Ia corresponds to trajectories leading to the massless IR fixed point, whereas region Ib leads to the massive IR fixed point. Region II is not connected to the UV fixed point, and thus physically irrelevant.
Figure 3: The momentum-dependence of the anomalous dimensions of the graviton (left) and the ghost field (right) for different regulators at the respective UV fixed points. Only a weak dependence on the parameter is observed. The difference between optimized and exponential regulators is due to the fact that the modes are not integrated out at the same scale. From the fact that the quadratic external momentum terms cancel in the flow of the graviton Christiansen et al. (2014a), goes to zero in the limit of large external momenta. The same is not true for the ghosts, where the anomalous dimension goes to a constant.

We find an attractive UV fixed point with coordinates

(42)

and complex critical exponents . This provides further non-trivial evidence for the asymptotically safe UV structure of quantum gravity. We also find the built-in repulsive Gaussian fixed point at , and a massive IR fixed point at . The most striking feature of the present phase diagram is the confirmation of the attractive massless IR fixed point

(43)

which was already found in Christiansen et al. (2014a), where it corresponds to a de-Sitter fixed point with . This fixed point implies the global existence of trajectories connecting the UV fixed point with a finite IR fixed point. The present result is a clear confirmation that this IR fixed point is not a truncation artifact, but rather a physical property of the theory.

Importantly, it turns out to be an IR fixed point describing classical gravity. Physical initial conditions lead to globally defined trajectories that connect the non-trivial UV fixed point with the physical IR fixed point .

Note also that all UV-complete trajectories are also IR-complete, and end in either the massive or massless IR fixed point. In addition to this structure, there is a repulsive fixed point at . This fixed point was also found in Donkin and Pawlowski (2012). All essential features do not depend on the choice of the regulator , and there are only minor quantitative changes induced by variations of the latter. The variation of the UV fixed point values under a variation of the vertex model parameters , (36) is given in Table 7 in Appendix C.

iv.2 Uv regime

2 3 4 5 6 opt
-0.637 -0.641 -0.645 -0.649 -0.651 -0.489
0.621 0.622 0.614 0.606 0.600 0.831
0.574 0.573 0.567 0.559 0.553 0.763
0.319 0.316 0.316 0.318 0.319 0.248
EVs -1.284 -1.284 -1.268 -1.255 -1.244 -1.876
3.247i 3.076i 3.009i 2.986i 2.974i 2.971i
-2 -2 -2 -2 -2 -2
-1.358 -1.360 -1.360 -1.358 -1.356 -1.370
Table 1: UV fixed point values and eigenvalues for different regulator parameters , and the optimized regulator, with parameter values and .
here Litim (2004) Christiansen et al. (2014a) Donkin and Pawlowski (2012) Manrique et al. (2011) Codello et al. (2013) Table 6
0.763 1.178 2.03 0.966 1.055 1.617 1.684
0.248 0.250 0.22 0.132 0.222 -0.062 -0.035
0.189 0.295 0.45 0.128 0.234 -0.100 -0.059
Table 2: Comparison of the UV fixed point coordinates with earlier results for the optimized cutoff. Parameter values are and . Methods of the references (in order): background approximation Litim (2004), bi-local projection Christiansen et al. (2014a), geometric approach Donkin and Pawlowski (2012), bi-metric approach Manrique et al. (2011). The mixed approach is applied in Codello et al. (2013) and is also discussed in the present paper in the last paragraph of this subsection, Table 6.

Let us further investigate the properties of the UV fixed point. First of all, the existence of the fixed point does not depend on the specific choice of the regulator. Moreover, it is attractive in all four directions investigated here. Furthermore, even though the critical exponents of the dynamical quantities are complex, the ones of the physical background couplings are real. This was also found in Christiansen et al. (2014a) and Codello et al. (2013). Notice that the eigenvalue corresponding to is exactly -2, which can be immediately inferred from the specific structure of the background coupling flow equation. Also, the eigenvalue corresponding to is inherently real, as its flow equation is a polynomial of order one in the cosmological constant. All these points are summarized in Table 1.

The connection to earlier results is drawn in Table 2. The present results support the qualitative reliability of the Einstein-Hilbert type approximations in the UV regime.

The couplings as functions of the RG scale along one selected trajectory (marked as a dashed black line in the phase diagram) are shown in Figure 5. One can see how the couplings tend to their finite fixed point values in the UV. The IR regime will be discussed below.

A further quantity of interest is the anomalous dimension. The momentum-dependence of both graviton and ghost anomalous dimension is given in Figure 3 for all used regulators at their respective UV fixed point. As one can see, only quantitative differences occur. The graviton anomalous dimension is of the order of , whereas the ghost anomalous dimension is of the order of . The difference between exponential and optimized regulators is due to the fact that the regulators integrate out modes at different scales. Consequently, the effective cut-off scale is regulator-dependent. A formal discussion of scale-optimization can be found in Pawlowski (2007), and is applied in the context of finite temperature Yang-Mills theory in Fister and Pawlowski (2011). For instance, for the exponential cutoff with , we find that if one rescales

(44)

the momentum-dependence of the anomalous dimension with a optimized regulator matches the one obtained with an exponential regulator, see Figure 4.

Figure 4: Comparison of the momentum dependence of the graviton and the ghost anomalous dimension with the exponential regulator with on the one hand, and the optimized regulator with the cutoff-rescaling (44) on the other.

In general, we observed that the (TT-part of the) graviton anomalous dimension is positive, however there are indications that this does not remain so when the other degrees of freedom of the graviton receive an individual anomalous dimension Christiansen et al. (2014b). On the other hand, the ghost anomalous dimension is strictly negative, as was already found in Eichhorn and Gies (2010) and Groh and Saueressig (2010). We also note that the anomalous dimensions are not the leading contribution to the flow. This means that by setting on the RHS of the flow (19), one captures all qualitative properties dicussed here. Hence, the anomalous dimensions only constitute correction effects while the leading term on the RHS of the flow equations is the one proportional to . In the ghost sector this pattern is even more pronounced, and dynamical ghost effects on the phase diagram and the running couplings are very small.

Derivative expansion:

We close this section with a discussion of the stability of the (covariant) derivative expansion which is the standard approximation scheme used so far. The first calculation of the graviton anomalous dimension has been presented in Christiansen et al. (2014a) within the FRG. There, the flow is projected at . In the work Codello et al. (2013) a derivative expansion around is performed. The full results in the present study show a strong momentum dependence of the correlation functions as well as their flows in the cut-off regime with . Such a strong momentum dependence of the flows either requires higher orders in the derivatives or a non-local expansion that works-in the information of momenta close to zero and those close to , see Fister and Pawlowski (2011); Christiansen et al. (2014a); Schnoerr et al. (2013).

Note that there is a strong cut-off dependence of the graviton anomalous dimension at the UV fixed point for small momenta , see Figure 3. The occurance of this regime is presumably related to the mass-scale set by the fixed point value of . Here we investigate its impact on the value of in the leading order of the derivative expansion. We also consider a variation of the expansion point. We also use the present results with full momentum-dependence in order to investigate the reliability of the derivative expansion. There, the computation of the anomalous dimension requires

(45)

e.g. at vanishing momentum, . On the other hand, the momentum derivative of gives the relation

(46)

Inserting (46) in (45) leads to

(47)

In the lowest order derivative expansion, that is , the anomalous dimension is given by (minus) (47) evaluated at . Moreover, the lowest order implies and we simply arrive at

(48)

For the regulators used in the present work this leads to anomalous dimensions listed in Table 3.

2 3 4 5 6 opt
0.46 0.48 0.51 0.54 0.56 0.51
Table 3: Anomalous dimension in the lowest order derivative expansion derived from the full flow.
2 3 4 5 6 opt
0.44 0.50 0.58 0.66 0.74 0.61
Table 4: Full anomalous dimension in the lowest order derivative expansion derived from the full flow.

However, the full lowest order derivative expansion takes into account the -terms on the right hand side. At vanishing momentum there is the relation

(49)

This is easily derived from

(50)

where both and are in the scaling regime. The latter condition implies that and are independent of . Then we conclude that at we have

(51)

for all in the scaling regime and we are led to (49). Hence, in the scaling regime (with ) the full anomalous dimension in the derivative expansion at is given by

(52)

leading to Table 4. These results seem to be much more stable then the approximation (48).

0 0.25 0.5 0.75 1 1.15
0.57 0.50 0.39 0.25 0.074 -0.016
Table 5: Anomalous dimension in the standard derivative expansion with optimized regulator in an expansion around with .

To complete the present reliability analysis of Taylor expansions in momenta , we also investigate expansions about a general expansion point . We present results for the optimized regulator and evaluate the anomalous dimension for . The conclusions of this study do not depend on the choice of these parameters. As one can see, the anomalous dimension of the graviton in a derivative expansion strongly depends on the specification parameter . This relates to the fact that such an expansion only works well if the full flow of the propagator shows a mild momentum dependence. This is not the case for the flow of the graviton two-point function, see Christiansen et al. (2014a). In general, even the sign of the anomalous dimension depends on the specification parameter. We conclude that a derivative expansion in quantum gravity with has to be used with great caution.

Effect of identifications of couplings in the Uv:

As already mentioned, the present approximation is the first work that employs individual running couplings for the momentum-independent part of each vertex function. In particular, the graviton mass term should not be identified with the cosmological constant. Still, we have shown, that the full expansion with momentum-dependent wave function renormalizations and a mass term for the fluctuating graviton provides UV fixed point results in qualitative agreement with that of the standard background field approach, if we identify the mass a posteriori with (minus 1/2 of) the cosmological constant. Within such an identification we have a deSitter fixed point.

For completeness, we also have investigated a mixed approach: We use a flat anomalous dimension in a derivative expansion about vanishing momentum, or a momentum-dependent one, , for the fluctuating graviton. In turn, the flows of the graviton mass and the Newton coupling are extracted from the flow of the cosmological constant and the Newton coupling in the background field approximation. This can be interpreted as an intermediate step towards the full approximation studied here. Interestingly this leads to a very small and negative fixed point value for the cosmological constant, see also Codello et al. (2013) for such a mixed expansion with a flat anomalous dimension. Our fixed point results for the case with a flat anomalous dimension are given in Table 6. They are in qualitative agreement with the results of Codello et al. (2013). Notably, the results in the mixed approach deviate from both, the background field results and that of the full approximation introduced in the present work. We have also checked that this originates in the identification of the mass term with the cosmological constant, the given alternative choices for the flow of the Newton constant do not alter this result.

2 3 4 5 6 opt
1.68 1.72 1.75 1.77 1.80 1.68
-0.064 -0.076 -0.088 -0.100 -0.110 -0.035
0.81 0.93 1.03 1.14 1.24 0.86
-1.08 -1.05 -1.04 -1.03 -1.01 -0.75
Table 6: Fixed point values and and anomalous dimension at this fixed point in the mixed approach.

iv.3 Ir regime

Figure 5: The running couplings as functions of the renormalization scale in units of the Planck mass. In the UV the couplings tend to their finite fixed point values. As one follows the trajectories down to the IR, one can see the scaling behaviour in the vicinity of the IR fixed point.

The non-trivial IR fixed point is located at . The most important feature is that it is a classical one, i.e. the essential couplings scale classically and all quantum contributions vanish: The gravitational couplings and the cosmological constant scale as

(53)
and the anomalous dimensions vanish,
(54)

see Appendix F. This leads to flow trajectories that connect the asymptotically safe UV regime for and short distances, with a classical IR regime for and large distances. Since approaches a finite value in the limit , the dimensionful mass vanishes in the deep IR. Accordingly, this fixed point corresponds to a massless theory on large distances, which is consistent with gravity as a force with infinite range. Moreover, the scaling of Newton’s coupling (53) allows us to identify a scale in the following way. As (or ) scales classically, the coefficient of proportionality, say , is nothing else than the Newton constant, because

(55)

Thus, scales are measured in units of the Planck mass, . The physical trajectory is then fixed by measuring the relevant couplings, that is the (background) Newton constant and cosmological constant, at a given scale. In Figure 5 one can see both, the classical scaling in the IR as well as the vanishing of the -functions in the vicinity of the UV fixed point for large . The classical scaling regime extends roughly up to one order of magnitude below the Planck scale. This implies the absence of quantum gravity effects for energies , as it is expected in a theory without a large volume compactification of extra-dimensions. Note also that the difference between the two couplings and is hardly visible, which justifies to some extent the background approximation for the Newton constant.

V Summary and outlook

We have presented a quantum gravity calculation that shows a classical regime on large distances, and asymptotically safe physics in the non-perturbative UV limit. The behaviour at large distances relates to an attractive IR fixed point with classical scaling behaviour. This implies that the dimensionful Newton constant and the cosmological constant do not depend on the energy scale (inverse length scale) for large distances. Hence, for the first time, this includes the domain of classical gravity, that has been tested experimentally, in the renormalization group approach to quantum gravity. The classical gravity regime in the vicinity of the IR fixed point is connected to a non-perturbative UV fixed point, which ensures the finiteness of scattering amplitudes at arbitrary high energies. The small dependence of the results on the regulator indicates stability of the present truncation. Technically, this work introduces a novel approximation scheme in RG-gravity calculations. This scheme has two essential features: First, we work in a vertex expansion with fully momentum-dependent wave-function renormalizations for the graviton and for the ghost field. Second, the higher order correlation functions are parametrized by additional couplings for their momentum-dependent and momentum-independent parts. The latter become important in the IR and their properties are determined by a self-consistency scaling analysis. In particular, we show that the momentum-independent part of the two point function cannot be identified with the cosmological constant at large distances.

In summary, this work provides further evidence for the asymptotic safety scenario in quantum gravity. In addition it substantiates the physics at the IR fixed point found in Donkin and Pawlowski (2012); Christiansen et al. (2014a). By now, the approximation is quantitative enough to produce classical scaling for the couplings and for large length scales, in accordance with experimental observations. The present approximation is readily extended to include higher correlation functions which will be reported on in future work Christiansen et al. (2014b).

Acknowledgments

We thank Tobias Henz, Daniel F. Litim and Christof Wetterich for discussions and collaboration on related subjects. We thank Alessandro Codello for providing us with an updated version of Codello et al. (2013) prior to substitution on arXiv. JMP thanks the Yukawa Institute for Theoretical Physics, Kyoto University, where this work was completed during the YITP-T-13-05 on ’New Frontiers in QCD’. This work is supported by Helmholtz Alliance HA216/EMMI and by ERC-AdG-290623.

Appendix A Flow equations

The modified threshold functions introduced in subsection III.3 are given by:

(56)

With these threshold functions, the geometric flow equations in Donkin and Pawlowski (2012), improved by momentum-dependent anomalous dimensions, are given by

(57)
(58)

The flows of the two-point functions are given by

(59)
(60)

Here, we have introduced the short cuts and . The functions and read

(61)

and