Ghost wave-function renormalization in
[1.5ex] Asymptotically Safe Quantum Gravity
[10mm] Kai Groh and Frank Saueressig
[3mm] Institute of Physics, University of Mainz
Staudingerweg 7, D-55099 Mainz, Germany
Motivated by Weinberg’s asymptotic safety scenario, we investigate the gravitational renormalization group flow in the Einstein-Hilbert truncation supplemented by the wave-function renormalization of the ghost fields. The latter induces non-trivial corrections to the -functions for Newton’s constant and the cosmological constant. The resulting ghost-improved phase diagram is investigated in detail. In particular, we find a non-trivial ultraviolet fixed point, in agreement with the asymptotic safety conjecture which also survives in the presence of extra dimensions. In four dimensions the ghost anomalous dimension at the fixed point is , supporting space-time being effectively two-dimensional at short distances.
Constructing a consistent and predictive quantum theory for gravity is one of the prime challenges of theoretical high energy physics today. One proposal in this direction, which recently received a lot of attention, is Weinberg’s asymptotic safety scenario [1, 2, 3], see [4, 5, 6, 7, 8, 9] for reviews. This scenario adopts Wilson’s modern viewpoint on renormalization , assessing that gravity has a fundamental description within the framework of non-perturbatively renormalizable quantum field theories. The key ingredient underlying this idea is a conjectured non-trivial (or non-Gaussian) fixed point of the gravitational renormalization group (RG) flow. For RG trajectories attracted to it at high energies, the fixed point ensures that the dimensionless coupling constants remain finite, so that physical quantities are safe from unphysical UV divergences. These trajectories, also including the one describing our world , span the UV critical surface of the fixed point. The precise position of “our RG trajectory” within this surface is determined by free parameters, which have to be fixed by experiment. For a finite-dimensional this construction is then as predictive as a standard, perturbatively renormalizable, quantum field theory. Elucidating the fixed point structure of the gravitational RG flow and understanding the properties of the corresponding UV critical surfaces is a central aspect of the asymptotic safety program to date.
An important technical tool in this program is the functional renormalization group equation (FRGE) for gravity . Formulated in terms of the Wetterich equation , the FRGE describes the dependence of the effective average action on the coarse graining scale
Here, denotes the second functional derivative of the effective average action with respect to the fields of the theory and STr is a generalized functional trace which includes a minus sign for ghosts and fermions. Furthermore, is a matrix valued IR cutoff, which provides a -dependent mass-term for fluctuations with momenta . The interplay between the full regularized propagator and ensures that the STr receives contributions from a small -interval around only, rendering the trace-contribution finite.111The FRGE captures the RG-dependence of the effective action. The relation between the effective action at a fixed point and the corresponding fundamental action has recently been discussed in .
While (1.1) constitutes a formally exact equation, it comes with the drawback that it cannot be solved exactly. A widely used approximation scheme for obtaining non-perturbative information from the FRGE consists of making an ansatz (truncation) for which retains a finite number of -dependent coupling constants only. Projecting the the resulting RG flow onto the truncation subspace allows to read off the -functions for the running couplings as the coefficients of the interaction monomials retained by the ansatz without resorting to perturbation theory. Employing truncations, an important point is to establish the robustness of the emergent physical picture. Within a given ansatz, its reliability may be tested by studying the dependence of physical quantities on the shape of the unphysical IR regulator . A second, more laborious, route consists in extending the truncation ansatz, thereby showing that all physical results, as, e.g., the fixed points of the RG flow, are robust under the extension.
For gravity, these ideas have been implemented systematically, so far focusing on the gravitational sector of mostly. In this class the truncation ansatz is spanned by diffeomorphism invariant operators build from the physical metric , e.g., . The most studied case, the so-called Einstein-Hilbert truncation, encompasses a scale-dependent Newton’s constant and cosmological constant , and has been analyzed in a number of works [15, 16, 17, 18, 19, 20, 21]. Subsequently, this ansatz has been refined by including higher-derivative -interactions [22, 23, 24], higher order polynomials in up to within the framework of -gravity [25, 26, 27], non-local operators [28, 26], and lately also the Weyl-squared interactions [29, 30, 31] capturing the characteristic features of higher-derivative gravity. All these computations have identified a NGFP of the gravitational RG flow, providing substantial evidence for the asymptotic safety scenario. Moreover, the - and -results point at the dimension of the associated UV critical surface being finite, possibly even as low as three. Notably, the essential features of this picture already emerge from the structurally significantly simpler flow equations obtained within the conformally reduced gravity framework [32, 33, 34].
In this work we go beyond the gravitational approximation of , including quantum effects from the ghost sector. More specifically, we will augment the Einstein-Hilbert truncation by a non-trivial wave-function renormalization of the ghost fields (for a similar computation in Yang-Mills theory see ). Our main motivation for this ghost-improvement originates from the analogy to QCD where this coupling plays an essential role for the IR physics of theory [36, 37, 38, 39]. While it is clear, that there is also a non-trivial interplay between the ghosts and the gravitational -functions in gravity, explicit computations are not available yet. Here we close this gap, computing the anomalous dimension of the ghost fields and its effect on the running of Newton’s constant and the cosmological constant. The explicit computation is based on a new perturbative heat-kernel technique developed by Anselmi and Benini  which allows the systematic expansion of the flow equation in the presence of background ghost fields in a curved background.222In a companion paper , a similar analysis will be carried out using a spectrally adjusted cutoff and a flat-space projection technique. We thank A. Eichhorn and H. Gies for informing us on their upcoming work. This continues the exploration of quantum gravity effects in the ghost sector .
The rest of the paper is organized as follows. In Section 2 we derive the -functions governing the RG dependence of Newton’s constant, the cosmological constant, and the wave-function renormalization in the ghost sector, using a perturbative heat-kernel technique for non-minimal differential operators. The ghost-improved fixed point structure and phase diagram is analyzed in Section 3 and we comment on our findings in Section 4. Some technical details on the heat-kernel techniques and threshold functions employed in this paper are relegated to Appendix A, while the functions determining the gravitational and ghost anomalous dimensions are defined in Appendix B.
2 The ghost-improved Einstein-Hilbert truncation
The main purpose of this paper is the investigation of the gravitational RG flow, including the quantum effects captured by the wave-function renormalization in the ghost-sector. Our truncation ansatz, which will be called the “ghost-improved Einstein-Hilbert truncation”, encompasses three scale-dependent coupling constants: Newton’s constant , the cosmological constant , and the power-counting marginal multiplying the ghost-kinetic term. In this section, we will derive the non-perturbative -functions capturing the RG dependence of these couplings.
2.1 The truncation ansatz
Our ansatz for the effective average action is of the general form
Besides on the physical metric and the classical ghost fields , it also depends on the corresponding background fields and . They are related by
where and denote the expectation value of the quantum fluctuations around the background, which are not necessarily small. In the gravitational approximation the computations are simplified by setting the background ghost fields to zero. This, however, does not allow to keep track of the ghost-kinetic term, so that we work with a non-trivial ghost background in the following.
The gravitational part is build from the physical metric and taken to be of the Einstein-Hilbert form
where with a fixed reference scale and denotes the wave-function renormalization for the graviton. is supplemented by the gauge-fixing term . Employing the harmonic gauge, the latter reads
The resulting Faddeev-Popov determinant is captured by the ghost term
containing the wave-function renormalization of the ghosts . The gauge-choice (2.4) has the main virtue, that it allows for a straightforward comparison to earlier results  obtained in the Einstein-Hilbert truncation without ghost-improvement, fixing .
Before entering into the explicit computation of the -functions arising from the ansatz (2.1), it is useful to first consider the left-hand-side of the flow equation and identify the interaction monomials whose coefficients encode the running of our coupling constants. Taking the -derivative of our ansatz (2.1) and setting the fluctuation fields to zero afterwards yields
Thus it suffices to extract the Einstein-Hilbert monomials and the ghost kinetic term from the right-hand-side of the flow equation. Comparing the coefficients multiplying these interactions then allows us to read off the desired -functions for and .
2.2 Quadratic forms and the inverse Hessian
Upon specifying our truncation ansatz, we proceed by computing the second variation of with respect to the fluctuations (2.2). To facilitate the subsequent steps, it is convenient to decompose the metric fluctuations into their traceless and trace part
The forms quadratic in the fluctuations can be simplified further by first identifying and specifying a particular class of backgrounds. In our case, this class has to be general enough to distinguish the interaction monomials in the Einstein-Hilbert term and the ghost-kinetic term. This can be accomplished by choosing the background metric as the one of a -dimensional sphere, implying
Moreover, the background ghost field can be taken transversal
which suffices to keep track of the ghost kinetic term. In the sequel, we will resort to this choice of background to simplify all expressions. For later reference, we also introduce the unit on the space of traceless symmetric tensors (2T) and vectors (1)
Given these preliminaries, we now expand the ansatz (2.1) around the background (2.9) and (2.10), retaining the pieces quadratic in the fluctuations only. For the result has already been given in 
and denoting the covariant Laplacian constructed from the background metric. Owed to the non-trivial background ghost field, the analogous computation for is slightly more involved. Thus we first give the intermediate results obtained from expanding (2.6) contracted with a dummy vector in
Here the bar indicates that we expand around . Remarkably, the last line is independent of the choice of background or gauge. Based on these results, we obtain the quadratic form in the ghost sector
Here the two lines are equivalent up to surface terms and define the Grassmann-valued operators , and their adjoints, respectively. Their explicit expressions read
where the covariant derivatives to the left of the ghost fields act on or only and denotes symmetrization with unit strength. The last lines in the -expressions ensure that the operators are traceless in the 2T-indices . The quadratic forms (2.12) and (2.15) provide the key ingredients for constructing the IR cutoff and the inverse Hessian in the sequel.
The construction of the IR-cutoff follows the prescription in . I.e., at the level of the path integral, the gauge-fixed action is supplemented by an IR regulator of the form
The matrix is designed in such a way, that it adds a -dependent mass-term to the Laplacians appearing in the kinetic terms,
where is the scalar part of the IR regulator and a dimensionless shape-function interpolating monotonically between and . Comparing (2.18) to the quadratic forms (2.12) and (2.15), this prescription fixes
Observe that inherits a non-trivial -dependence via the wave-function renormalization factors. Notably, it is this feature which is partially responsible for the non-perturbative character of the computation.
The next step consists in constructing the Hessian and the inverse of . Here we follow the conventions [37, 35], using a skew-symmetric metric to couple anti-commuting fields (see Appendix A of  for details).333One can explicitly verify that the resulting FRGE is equivalent to . In terms of the multiplets
the Hessian is given by the following matrix
assumes block form with entries
The inverse of (2.23) is then found via the general inversion formula for -block matrices
Expanding the inverse up to second order in the background-ghost fields, and taking into account the minus sign originating from the ghost sector of the supertrace, the right-hand-side of the flow equation becomes
with , , and terms not contributing to the truncation indicated by the dots. As already anticipated in the last line, the full trace decomposes into operator traces on the space of traceless symmetric tensors (2T), scalars (0) and vectors (1). Substituting the explicit expressions for the block matrices, the traces , which by definition, do not include and and are thus independent of the background ghost fields, become
They give rise to the -functions for Newton’s constant and the cosmological constant. The -function for is captured by the terms of second order in the background-ghost fields. Neglecting the curvature terms and making use of the cyclicity of the trace, these are found as
Notably, the insertions including the background ghosts, like, e.g., the and in , always appear in pairs. Each part thereby gives exactly the same contribution to the , consistent with the symmetry factors of the underlying Feynman-diagrams. Evaluating both parts individually provides a highly non-trivial crosscheck when computing the coefficients of the ghost-kinetic term in the next subsection.
2.3 The ghost-improved -functions
We are now in the position to construct the -functions for , and . In this context, it is useful to introduce the anomalous dimensions of Newton’s constant and the ghost wave-function renormalization
together with the dimensionless couplings
As explained in Section 2.1, the -functions for Newton’s and the cosmological constant are encoded at zeroth order in the background-ghost fields. The desired interaction monomials are generated by the traces , eq. (2.28), which are evaluated straightforwardly by applying the early-time heat-kernel expansion detailed in Appendix A.1 followed by inserting the identity (A.15). Equating the resulting coefficients with (2.7) provides the flow equations for and
In the limit , this result agrees with earlier computations [12, 15]. The terms proportional to are novel and capture the backreaction of the quantum effects in the ghost sector on the running of the gravitational coupling constants.
The final step is the computation of . This requires extracting the background-ghost kinetic term from (2.29). Unfortunately, the differential operators entering into these traces are not minimal, so that the early-time heat-kernel expansion is no longer applicable and a more sophisticated method is needed. Here we follow Anselmi and Benini , which allows to compute operator traces of Laplace-type operators which include arbitrary (non-Laplace type) “perturbations” by considering the heat-kernel expansion at non-coincident points. Adapting these techniques to the case where the perturbation is build from the background-ghost fields, this method becomes applicable to . (See Appendix A.2 for more technical details.) Retaining the ghost-kinetic term only, a tedious but straightforward computation yields
Here the -functionals are defined in (A.4) and
Here the and -terms originating from drop out, due to the cancellation of the corresponding coefficients.
and the anomalous dimensions
The explicit form of the functions and is given in Appendix B. The eqs. (2.38) and (2.39) are the desired -functions governing the RG dependence of , and and constitute the central result of this section.
Some comments are now in order. The inclusion of the wave-function renormalization for the ghosts gives non-trivial contributions to the -functions for . These encompass the terms proportional to in and the qualitatively new -terms in . The results obtained within the standard Einstein-Hilbert truncation [12, 7] are recovered by setting . Comparing powers of , the leading contributions from the ghost sector are suppressed by one power of , relative to the leading Einstein-Hilbert terms. Thus in the classical regime, , the ghost-improvement may be neglected. In the quantum regime close to the NGFP where , however, we expect that these corrections become important. Investigating the influence of these new terms on the gravitational RG flow is then the subject of the next section.
3 Properties of the flow equation
After deriving the -functions (2.37) and (2.39), we now proceed by studying their properties, mostly resorting to numerical methods. In this context, it is useful to observe that enters into via only and is in turn completely determined by . Thus, substituting the explicit formula for into , the running of decouples and allows to analyze the gravitational RG flow in the two-dimensional -subsystem. Once a RG trajectory for is found, it can be plugged back into to obtain the running of the ghost anomalous dimension. We will now exploit this decoupling and first discuss the fixed point structure of the ghost-improved Einstein-Hilbert truncation for in Section 3.1, before focusing on the phase portrait and the fixed point structure including extra-dimensions in Sections 3.2 and 3.3, respectively.
3.1 Fixed points of the four-dimensional -functions
As highlighted in the introduction, the crucial ingredient of the asymptotic safety scenario is the fixed point structure of the gravitational -functions. Thus, we start our investigation by looking for fixed points where simultaneously. In the vicinity of such a point, the linearized -functions are given by , where , . The stability coefficients , defined as minus the eigenvalues of , provide an important characteristic of the fixed point. In particular, eigendirections with a positive (negative) real part are UV-attractive (UV-repulsive) for trajectories close to the fixed point. For the remainder of this subsection, we will set .
Inspection of (2.37) immediately gives the Gaussian fixed point (GFP)
This fixed point corresponds to the free theory, and constitutes a saddle point in the --plane. It has one attractive and one repulsive eigendirection with stability coefficients given by the canonical mass dimensions of and , respectively.
The numerical analysis of the ghost-improved -functions also reveals a unique NGFP. Its position and properties are shown in the first two lines of Table 1.
|EH + ghost||opt|
|EH + ghost||exp ()|
It is situated at