On the origin of almost-Gaussian scaling
in asymptotically safe quantum gravity

The primary tool for investigating RG fixed points in quantum gravity is the effective average action $\Gamma_{k}$, a scale-dependent generalization of the ordinary effective action $\Gamma$. By construction, $\Gamma_{k}$ provides a description of the effective dynamics which incorporates the effects of quantum fluctuations with momenta $p^{2}\gtrsim k^{2}$ in its effective vertices. In the context of gravity, $\Gamma_{k}$ depends on a background metric $\bar{g}_{\mu\nu}$ (which may be left unspecified), metric fluctuations $h_{\mu\nu}$ in this background, and ghost fields $\bar{C}_{\mu},C^{\nu}$, arising from gauge-fixing the theory via the Faddeev-Popov construction. Hereby, the background metric $\bar{g}_{\mu\nu}$ and the fluctuations $h_{\mu\nu}$ combine to the full spacetime metric $g_{\mu\nu}$. The presence of a fixed, non-fluctuating background is essential to be able to define the coarse-graining scale $k$. The interaction monomials in $\Gamma_{k}$ are taken to be invariant under (background) diffeomorphisms. Conceptually, the gravitational effective average action may be organized according to

$$\begin{split}\Gamma_{k}[h,\bar{C},C;\bar{g}]=&\,\bar{\Gamma}_{k}[g]+\hat{\Gamma}_{k}[h;\bar{g}]+\Gamma^{\rm ghost}_{k}[h,\bar{C},C;\bar{g}]\,,\end{split}$$

(4)

where $\bar{\Gamma}_{k}[g]$ is the “diagonal part” of $\Gamma_{k}$ which depends on $\bar{g}_{\mu\nu}$ and $h_{\mu\nu}$ in the combination of $g_{\mu\nu}$ only, $\hat{\Gamma}_{k}[h;\bar{g}]$ is the off-diagonal part that also incorporates the gauge-fixing term, and $\Gamma^{\rm ghost}_{k}[h,\bar{C},C;\bar{g}]$ is the ghost-action. In addition, we may supplement the construction by (potentially non-local) “composite operators” $\{{\tilde{\mathcal{O}}}_{n}(k)\}$ coupled to their own sources $\varepsilon_{n}$,

$$\begin{split}\Gamma_{k}[h,\bar{C},C;\bar{g};\varepsilon]=&\,\Gamma_{k}[h,\bar{C},C;\bar{g}]+\sum_{n}\varepsilon_{n}\,{\tilde{\mathcal{O}}}_{n}(k)\,.\end{split}$$

(5)

This extension coincides with the standard definition (4) for $\varepsilon_{n}=0$.

The dependence of $\Gamma_{k}$ on the coarse-graining scale $k$ is governed by the Wetterich equation Wetterich:1992yh ; Morris1994 ; Reuter:1993kw ; Reuter:1996cp ,

$$\frac{\mathrm{d}}{\mathrm{d}t}\Gamma_{k}=\frac{1}{2}\text{STr}\left[\left(\Gamma^{(2)}_{k}+\mathcal{R}_{k}\right)^{-1}\,\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{R}_{k}\right]\,.$$

(6)

Eq. (6) is a functional renormalization group equation (FRGE), in which STr denotes a sum over fluctuation modes (including a sum over all fields that $\Gamma_{k}$ is a functional of) while $\Gamma_{k}^{(2)}$ denotes the second functional derivative of $\Gamma_{k}$ with respect to these fluctuation fields. Moreover, ${\mathcal{R}}_{k}$ is a regulator that equips fluctuations of low momenta with a $k$-dependent mass term, while it vanishes for fluctuations with $p^{2}\gg k^{2}$. Further, we introduced the RG time $t\equiv\ln(k/k_{0})$ for notational convenience.

Retaining the composite operators and expanding (6) to first order in $\varepsilon_{i}$ gives the composite operator equation Pagani:2016pad ; Pagani:2016dof ; Becker:2018quq ; Becker:2019tlf ; Becker:2019fhi ; Houthoff:2020zqy ; Kurov:2020csd ,

$$\frac{\mathrm{d}}{\mathrm{d}t}\tilde{{\mathcal{O}}}_{n}(k)=-\frac{1}{2}\text{STr}\left[\left(\Gamma^{(2)}_{k}+\mathcal{R}_{k}\right)^{-1}\,\tilde{{\mathcal{O}}}_{n}^{(2)}(k)\left(\Gamma^{(2)}_{k}+\mathcal{R}_{k}\right)^{-1}\,\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{R}_{k}\right]\,.$$

(7)

The main virtue of this equation is that it allows to study the scaling dimension of geometric quantities which typically do not appear in $\Gamma_{k}$. A prototypical example of the latter can be the geodesic distance between points studied in Becker:2018quq ; Becker:2019tlf . The use of (7) requires information about the propagators $\left(\Gamma^{(2)}_{k}+\mathcal{R}_{k}\right)^{-1}$ though. This must be provided as an input, e.g., by solving the Wetterich equation appearing at zeroth order in $\varepsilon_{n}$. In the sequel, we will focus on composite operators which could also appear within $\Gamma_{k}[h,\bar{C},C;\bar{g}]$.

The effective average action lives on the gravitational theory space ${\mathcal{T}}$. By definition, this linear space comprises all action functionals that can be constructed from the field content and are invariant under background diffeomorphism symmetry. Introducing a basis $\{{\mathcal{O}}_{n}\}$ on ${\mathcal{T}}$, we can expand,³³3Two comments are in order. Firstly, one can obviously use different bases, say, $\{{\mathcal{O}}_{n}\}$ and $\{{\mathcal{O}}^{\prime}_{n}\}$, to expand $\Gamma_{k}$ and $\tilde{\mathcal{O}}_{m}$, respectively. In this work, however, we are only interested in the case ${\mathcal{O}}_{n}={\mathcal{O}}_{n}^{\prime}$. Secondly, the expansions (8) put a non-trivial constraint towards the basis $\{{\mathcal{O}}_{n}\}$, since our definition of $\Gamma_{k}$ includes the gauge-fixing action which also depends on the theory’s couplings. While one might separate the gauge-fixing action from $\Gamma_{k}$ in order to be able to choose the basis $\{{\mathcal{O}}_{n}\}$ more freely, here, we refrain from doing so for the sake of convenience. Moreover, in Sec. 3 below we will employ the physical gauge-fixing condition in which the gauge degrees of freedom do not source the anomalous scaling dimension of $\tilde{O}_{n}$.

$$\Gamma_{k}[h;\bar{g}]=\sum_{n}\,\bar{u}^{n}(k)\,{\mathcal{O}}_{n}\,,\qquad\tilde{{\mathcal{O}}}_{m}(k)=\sum_{n}\bar{Z}_{m}{}^{n}(k)\,{\mathcal{O}}_{n}\,.$$

(8)

Here, $\bar{u}^{n}(k)$ and $\bar{Z}_{m}{}^{n}(k)$ are $k$-dependent, dimensionful couplings. Denoting the mass-dimension of the operators as $[{\mathcal{O}}_{n}]=-d_{n}$, it is convenient to introduce their dimensionless counterparts as

$$u^{n}(k)\equiv\bar{u}^{n}(k)k^{-d_{n}}\,,\qquad{\gamma}_{m}{}^{n}(k)\equiv k^{d_{m}-d_{n}}{\left[\bar{Z}^{-1}{\partial_{t}}\bar{Z}\right]_{m}}^{n}\,.$$

(9)

The couplings $u^{n}$ provide natural coordinates on the theory space, while ${\gamma}_{m}{}^{n}(k)$ encodes the anomalous scaling dimensions of the composite operators Pagani:2016pad .

Plugging the expansion of $\Gamma_{k}$, Eq. (8), into the Wetterich equation (6), matching the coefficients of the basis elements ${\mathcal{O}}_{n}$, converting to dimensionless couplings (9), and solving the resulting system of equations for $\partial_{t}u^{n}(k)$, we arrive at the beta functions governing the dependence of the couplings $u^{n}(k)$ on the coarse graining scale,

$$\partial_{t}u^{n}(k)=\beta^{n}(u)\,.$$

(10)

Analogously, the subsitution of (8) into the COE (7) yields Pagani:2016pad ; Pagani:2016dof ; Becker:2018quq ; Becker:2019tlf ; Becker:2019fhi ; Houthoff:2020zqy ; Kurov:2020csd ,

$$\sum_{m}{\gamma_{n}}^{m}\,k^{d_{m}}{\mathcal{O}}_{m}=-\frac{1}{2}\,k^{d_{n}}\,\text{STr}\left[\left(\Gamma^{(2)}_{k}+\mathcal{R}_{k}\right)^{-1}\,{\mathcal{O}}_{n}^{(2)}\,\left(\Gamma^{(2)}_{k}+\mathcal{R}_{k}\right)^{-1}\,\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{R}_{k}\right]\,,$$

(11)

In this way, $\gamma_{n}{}^{m}$ becomes a function of the couplings $u^{n}(k)$, i.e., $\gamma_{n}{}^{m}(k)\equiv\gamma_{n}{}^{m}(u(k))$. Note that $\gamma_{n}{}^{m}$ also entails an dependence on the beta functions $\beta^{n}(u(k))$, due to the presence of $\mathrm{d}\mathcal{R}_{k}/\mathrm{d}t$ under the trace.

The goal of this work is to understand the predictive power of the Reuter fixed point based on abstract properties of two-point correlation functions. To do so, we compare the stability matrix $B_{*}$ as in eq. (2) with scaling dimensions obtained from a composite-operator equation. Generally, one may interpret the eigenvalues of the matrix $-d_{n}\delta_{n}^{m}+\gamma_{n}{}^{m}$ as the full scaling dimensions of the operators $\{{\mathcal{O}}_{n}\}$. Thus, we have two different methods at hand to evaluate critical exponents – on the one hand, the (negative) eigenvalues of the stability matrix $(B_{*}^{\rm FRGE})^{n}{}_{m}$ obtained from solving the Wetterich equation, and, on the other hand, the eigenvalues of $(B_{*}^{\rm COE})_{n}{}^{m}=-d_{n}\delta_{n}^{m}+\gamma_{n}{}^{m}$ obtained from the composite operator equation.

Let us discuss the structure of the beta functions and the stability matrix in more detail. A critical element in their construction is the regulator ${\mathcal{R}}_{k}$ which is conveniently written as⁴⁴4More precisely, in practice one will set ${\mathcal{R}}_{k}(u;z)=\sum_{i}{\mathcal{Z}}^{(i)}(\bar{u})\left[P_{k}^{i}(z)-z^{i}\right]$, where $P_{k}(z)=z+R_{k}(z)$ and the index $i$ runs from 1 to highest power of $\Delta=-\bar{D}^{2}$ appearing in $\Gamma_{k}^{(2)}$. For the sake of simplicity and without loss of generality, we will remain with the simplified form (12).

$${\mathcal{R}}_{k}(\bar{u};z)={\mathcal{Z}}(\bar{u})\,R_{k}(z)\,.$$

(12)

Here, ${\mathcal{Z}}(\bar{u})$ has a status of a wave function renormalization and carries a tensor structure with respect to internal field indices. It depends on $t$ through the dimensionful couplings $\bar{u}$ only. Furthermore, $R_{k}$ is the scalar regulator. At the level of the Wetterich equation this $\bar{u}$-dependence is dictated by the choice of regularization procedure and is a generic feature Codello:2008vh . At the level of the composite operator equation, there is more freedom since there is no direct relation between $\Gamma_{k}^{(2)}$ and ${\mathcal{O}}_{n}$ which would fix such a dependence based on the composite operators under consideration. As a result of the decomposition (12), one has

$$\frac{\mathrm{d}}{\mathrm{d}t}{\mathcal{R}}_{k}(\bar{u};z)=\left(\sum_{n}\frac{\partial{\mathcal{Z}}(\bar{u})}{\partial\bar{u}^{n}}\frac{\partial}{\partial t}\bar{u}^{n}\right)R_{k}(z)+{\mathcal{Z}}(\bar{u})\left(\frac{\partial}{\partial t}R_{k}\right)\,.$$

(13)

Substituting Eqs. (8) and (13) into the Wetterich equation, equating coefficients for the basis element ${\mathcal{O}}_{n}$, and retaining the LHS$=$RHS form yields

$$\partial_{t}u^{n}+d_{n}u^{n}=\omega^{n}(u)+\sum_{m}\tilde{\omega}^{n}{}_{m}(u)\left(\partial_{t}u^{m}+d_{m}u^{m}\right)\equiv\hat{\omega}^{n}(u;\partial_{t}u)\,.$$

(14)

This structure disentangles the classical contributions (LHS) and the contributions of the quantum fluctuations (RHS) to the flow. We may think of the latter as a vector field $\hat{\omega}^{n}(u,\partial_{t}u)$ on the “theory’s configuation space”. The vector $\omega^{n}(u)$ and matrix $\tilde{\omega}^{n}{}_{m}(u)$ depend on the dimensionless couplings $u^{n}$, but not on their derivatives $\partial_{t}u^{n}$. We also observe that $\tilde{\omega}^{n}{}_{m}(u)$ may not depend on all couplings, but only those which appear in the regulator ${\mathcal{R}}_{k}(\bar{u})$. Explicit expressions for $\omega^{n}(u)$ and $\tilde{\omega}^{n}{}_{m}(u)$ can be given in terms of functional traces

$$\begin{split}\omega^{n}(u)=&\left.\frac{1}{2}\,k^{-d_{n}}\,\text{STr}\left[\left(\Gamma^{(2)}_{k}+\mathcal{R}_{k}\right)^{-1}\,{\mathcal{Z}}(\bar{u}^{i})\left(\frac{\partial}{\partial t}R_{k}\right)\right]\right|_{{\mathcal{O}}_{n}}\,,\\ \tilde{\omega}^{n}{}_{m}(u)=&\left.\frac{1}{2}\,k^{d_{m}-d_{n}}\,\text{STr}\left[\left(\Gamma^{(2)}_{k}+\mathcal{R}_{k}\right)^{-1}\,\frac{\partial{\mathcal{Z}}(\bar{u}^{i})}{\partial\bar{u}^{m}}\,R_{k}\right]\right|_{{\mathcal{O}}_{n}}\,.\end{split}$$

(15)

Here, we used the notation $|_{{\mathcal{O}}_{n}}$ to denote the projection of the trace onto the basis element ${\mathcal{O}}_{n}$ and included factors of $k$ in such a way that the LHS is given by dimensionless quantities. The beta functions (10) are then obtained by solving (14) for $\partial_{t}u^{n}$:

$$\beta^{n}(u)=-d_{n}u^{n}+\sum_{m}\left[(\mathds{1}-\tilde{\omega}(u))^{-1}\right]^{n}{}_{m}\,\omega^{m}(u)\,.$$

(16)

The beta functions (16) solve Eq. (14), i.e., they fulfill the identity

$$\beta^{n}(u)=-d_{n}u^{n}+\hat{\omega}^{n}(u,\beta(u))\,.$$

(17)

We can use this identity to organize the stability matrix obtained from Eq. (16). Thus, let us consider the (generalized) stability matrix

$$B^{n}{}_{m}(u)=\frac{\partial}{\partial u^{m}}\beta^{n}(u)$$

(18)

at a generic point along the RG trajectory,

$$\begin{split}B^{n}{}_{m}(u)&=-D^{n}{}_{m}+\frac{\partial}{\partial u^{m}}\hat{\omega}^{n}(u;\beta(u))\\ &=-D^{n}{}_{m}+\frac{\partial}{\partial u^{m}}\hat{\omega}_{0}^{n}(u)+\sum_{l}\left[\left(\frac{\partial}{\partial u^{m}}\tilde{\omega}^{n}{}_{l}(u)\right)\beta^{l}(u)+\tilde{\omega}^{n}{}_{l}(u)B^{l}{}_{m}(u)\right]\,.\end{split}$$

(19)

Here, we have introduced the matrix of classical scaling dimensions, $D^{n}_{m}\equiv d_{n}\delta^{n}_{m}$, and the vector $\hat{\omega}^{n}_{0}(u)\equiv\hat{\omega}^{n}(u;0)$. While it might be tempting to interpret the RHS of Eq. (19) as the classical scaling, in form of $-D^{n}_{m}$, and its appertaining quantum corrections, in form of $\partial_{m}\hat{\omega}^{n}$, one must take care with such interpretations at this stage, since $\partial_{m}\hat{\omega}^{n}$ depends on the classical scaling dimensions $d_{n}$ and the stability matrix $B$ itself. Solving Eq. (19) for $B$, one obviously has

$$B^{n}{}_{m}(u)=\sum_{l}\left[(\mathds{1}-\tilde{\omega}(u))^{-1}\right]^{n}{}_{l}\,\left(-D^{l}_{m}+\frac{\partial\hat{\omega}_{0}^{l}(u)}{\partial u^{m}}+\sum_{l^{\prime}}\frac{\partial\tilde{\omega}^{l}{}_{l^{\prime}}(u)}{\partial u^{m}}\,\beta^{l^{\prime}}(u)\right)\,.$$

(20)

In order to relate this expression for the stability matrix with the anomalous scaling dimensions obtained within the composite-operator formalism, let us re-arrange Eq. (19) one more time. Having in mind that $\hat{\omega}^{n}=k^{-d_{n}}(\text{RHS of Eq.~{}\eqref{FRGE}}|_{{\mathcal{O}}_{n}})$, it is a straightforward calculation to see that

$$\frac{\partial}{\partial u^{m}}\hat{\omega}^{n}(u;\beta(u))=\gamma_{m}{}^{n}(u)+\tilde{\gamma}_{m}{}^{n}(u)+\sum_{l}\tilde{\omega}^{n}{}_{l}(u)B^{l}{}_{m}(u)\,,$$

(21)

where

$$\gamma_{m}{}^{n}(u)\equiv-\frac{1}{2}\,k^{d_{m}-d_{n}}\,{\rm STr}\left[\left(\Gamma^{(2)}_{k}+\mathcal{R}_{k}\right)^{-1}{{\mathcal{O}}}_{m}^{(2)}\left(\Gamma^{(2)}_{k}+\mathcal{R}_{k}\right)^{-1}\,\left.\frac{\mathrm{d}}{\mathrm{d}t}{\mathcal{R}}_{k}(\bar{u}(k))\right]\right|_{{\mathcal{O}}_{n}}\,,$$

(22)

is the anomalous-dimension matrix of Eq. (11) for the family of operators $\{{\mathcal{O}}_{n}\}$,⁵⁵5Note that Eq. (22) involves the relation ${\mathcal{O}}_{m}^{(2)}=\frac{\partial\Gamma_{k}^{(2)}}{\partial\bar{u}^{m}}$. As often the composite-operator equation (22) is evaluated for the case where $\{{\mathcal{O}}_{n}\}$ does not include gauge-fixing contributions, but where $\Gamma_{k}$ does, we will denote $\gamma$ as $\gamma^{\mathrm{COE+GF}}$ and/or $\gamma^{\mathrm{COE}}$ to distinguish the results in which the gauge-fixing term is included into $\{{\mathcal{O}}_{n}\}$ or not. and the matrix $\tilde{\gamma}_{m}{}^{n}(u)$ contains terms involving the $\bar{u}$-derivative of the regulator ${\mathcal{R}}_{k}$,

$$\begin{split}\tilde{\gamma}_{m}{}^{n}(u)\equiv&-\frac{1}{2}\,k^{d_{m}-d_{n}}\,{\rm STr}\left[\left(\Gamma^{(2)}_{k}+\mathcal{R}_{k}\right)^{-1}\left(\frac{\partial{\mathcal{Z}}(\bar{u})}{\partial\bar{u}^{m}}R_{k}\right)\left(\Gamma^{(2)}_{k}+\mathcal{R}_{k}\right)^{-1}\,\left.\frac{\mathrm{d}}{\mathrm{d}t}{\mathcal{R}}_{k}(\bar{u}(k))\right]\right|_{{\mathcal{O}}_{n}}\\ &+\frac{1}{2}\,k^{d_{m}-d_{n}}\,{\rm STr}\left[\left(\Gamma^{(2)}_{k}+\mathcal{R}_{k}\right)^{-1}\,\left.\frac{\partial}{\partial\bar{u}^{m}}\left(\sum_{l}\frac{\partial{\mathcal{Z}}(\bar{u})}{\partial\bar{u}^{l}}d_{l}\bar{u}^{l}R_{k}+{\mathcal{Z}}(\bar{u})\partial_{t}R_{k}\right)\right]\right|_{{\mathcal{O}}_{n}}\,.\end{split}$$

(23)

In summary, for the stability matrix at the non-Gaussian fixed point (NGFP) we have established the formulae

$$\begin{split}(B_{*})^{n}{}_{m}&=\sum_{l}\left[(\mathds{1}-\tilde{\omega}(u_{*}))^{-1}\right]^{n}{}_{l}\,\left(-D^{l}_{m}+\left.\frac{\partial\hat{\omega}_{0}^{l}(u)}{\partial u^{m}}\right|_{u=u_{*}}\right)\\ &=\sum_{l}\left[(\mathds{1}-\tilde{\omega}(u_{*}))^{-1}\right]^{n}{}_{l}\,\left.\left(-D^{l}_{m}+\frac{\partial\omega^{l}(u)}{\partial u^{m}}+\frac{\partial}{\partial u^{m}}\left(\sum_{l^{\prime}}\tilde{\omega}^{l}{}_{l^{\prime}}d_{l^{\prime}}u^{l^{\prime}}\right)\right)\right|_{u=u_{*}}\\ &=\sum_{l}\left[(\mathds{1}-\tilde{\omega}(u_{*}))^{-1}\right]^{n}{}_{l}\,\left(-D^{l}_{m}+\gamma_{m}{}^{n}(u_{*})+\tilde{\gamma}_{m}{}^{n}(u_{*})\right)\,.\end{split}$$

(24)

Note that there is no simple identity expressing $\gamma_{m}{}^{n},\tilde{\gamma}_{m}{}^{n}$ in terms of $\omega^{n}$ and $\tilde{\omega}^{n}{}_{m}$, individually. The second line of Eq. (24) should then be seen as a rearrangement of the first line, in terms of the piece given by $\gamma_{m}{}^{n}$, appearing in the COE, and contributions associated with the dependence of the regulator on the couplings $\bar{u}$, which have been collected in $\tilde{\gamma}_{m}{}^{n}$. Only their sum is related to $\hat{\omega}_{0}^{n}{}_{m}$, namely at the fixed point via $\partial\hat{\omega_{0}}^{n}{}/\partial u^{m}(u_{*})=\gamma_{m}{}^{n}(u_{*})+\tilde{\gamma}_{m}{}^{n}(u_{*})$.

The formulae (24) are one of the main results of the present work. We stress that if the employed regulator ${\mathcal{R}}_{k}$ in the Wetterich equation (6) was $\bar{u}$-independent (which in Morris:2022btf is called a non-adaptive cutoff), one has $\tilde{\omega}=0=\tilde{\gamma}$ and the critical exponents of the theory would be given by $(B_{*})^{n}{}_{m}=-D^{n}_{m}+\gamma_{m}{}^{n}(u_{*})$. For this reasoning, one might naively expect this formula to give a generally “good” approximation of the stability matrix,

$$(B_{*})^{n}{}_{m}\approx-D^{n}_{m}+\gamma_{m}{}^{n}(u_{*})\,,$$

(25)

even in the case in which the regulator is coupling-dependent. In particular, one would expect that “almost-Gaussian scaling” of the stability coefficients Falls:2013bv ; Falls:2014tra ; Falls:2017lst ; Falls:2018ylp means that $B_{*}$ is dominated by $-D^{n}_{m}$ with $\gamma_{m}{}^{n}(u_{*})$ providing a small correction. With the examples below in Sec. 3, we will illustrate that this is not the case though. In other words, the matrices $\tilde{\gamma}_{m}{}^{n}(u_{*})$ and especially $\tilde{\omega}^{n}{}_{m}(u_{*})$ are crucial for determining the spectrum of the stability matrix $B_{*}$. This observation is important because in many systems, the critical exponents obtained from $B_{*}$ constitute physical observables. (Whether this is actually the case for gravity is nevertheless up to debate, due to the complexity of constructing observables.) In non-gravitational settings, it is often customary to calculate the stability matrix for fixed values of the anomalous dimension, cf. Hellwig:2015woa , following Eq. (25). This raises the question about the physical relevance of the critical exponents obtained from $(B_{*})^{n}{}_{m}$ versus those obtained from $-D^{n}_{m}+\gamma_{m}{}^{n}(u_{*})$, in case there is no qualitative agreement among the two.

Lastly, we point out that for a fixed point corresponding to an asymptotically free theory, the quantum corrections to the stability matrix vanish ($\gamma=\tilde{\gamma}=\tilde{\omega}=0$). In this case the stability coefficients agree with the mass dimensions of the couplings.

We start by illustrating the general structures derived in the previous section based on the background Einstein-Hilbert truncation Reuter:1996cp ; Reuter:2001ag ; Lauscher:2001ya ; Litim:2003vp ; Gies:2015tca , adapting the derivation given in Reuter:2019byg . Thus, we solve the Wetterich equation (6) and the COE (11) based on the geometric operators

$$\begin{split}{\mathcal{O}}_{1}^{\rm COE}=&\int\mathrm{d}^{d}x\sqrt{g}R\,,\\ {\mathcal{O}}_{1}^{\rm COE+GF}=&\int\mathrm{d}^{d}x\sqrt{g}R+\frac{1}{2\alpha}\int\mathrm{d}^{d}x\sqrt{\bar{g}}\bar{g}^{\mu\nu}F_{\mu}[h;\bar{g}]F_{\nu}[h;\bar{g}]\,.\end{split}$$

(28)

These operators coincide with (26) once the fluctuation fields are set to zero. The gauge-condition is taken of the general form

$$F_{\mu}[h;\bar{g}]=\bar{D}^{\nu}h_{\mu\nu}-\frac{1+\beta}{d}\bar{D}_{\mu}h_{\nu}^{\nu}\,.$$

(29)

This subsection chooses harmonic gauge, $\alpha=1,\beta=(d-2)/2$. The inclusion of the gauge-fixing term ensures that ${\mathcal{O}}^{(2)}_{1}$ depends on the background covariant derivative $\bar{D}_{\mu}$ in the form of the background Laplacian $\Delta=-\bar{g}^{\mu\nu}\bar{D}_{\mu}\bar{D}_{\nu}$ only.

In the first step, we project the Wetterich equation onto the subspace spanned by $\bar{{\mathcal{O}}}_{1}[\bar{g}]$. Thus, all vectors and matrices associated with this example are one-dimensional. Including the hamonic gauge-fixing condition, the corresponding ansatz for the effective average action reads

$$\bar{\Gamma}_{k}[g]+\hat{\Gamma}_{k}[h;\bar{g}]\simeq\bar{u}^{1}\,{\mathcal{O}}_{1}^{\rm COE+GF}\,,\quad\text{with}\quad u^{1}\equiv k^{2-d}\,\bar{u}^{1}\,.$$

(30)

The coupling $\bar{u}^{1}$ is related to Newton’s coupling $G_{k}$ by $\bar{u}^{1}=-\frac{1}{16\pi G_{k}}$ and we have $[\bar{u}^{1}]\equiv D_{u}=d-2>0$. Following the computation Reuter:2019byg , the LHS$=$RHS form of the projected Wetterich equation, introduced in (14), is

$$\partial_{t}u^{1}+D_{u}\,u^{1}=\omega+\tilde{\omega}(u^{1})\,\left(\partial_{t}u^{1}+D_{u}\,u^{1}\right)\,.$$

(31)

The vector $\omega(u)$ and matrix $\tilde{\omega}(u)$ given by $\omega=B_{1}$ and $\tilde{\omega}(u^{1})=B_{2}/u^{1}$, with the constants

$$\begin{split}B_{1}=&\frac{1}{(4\pi)^{d/2}}\,\frac{1}{12}\left(d(d-3)\Phi^{1}_{d/2-1}-6(d^{2}-d+4)\Phi^{2}_{d/2}\right)\,,\\ B_{2}=&\frac{1}{(4\pi)^{d/2}}\,\frac{1}{24}\,\left(d(d+1)\tilde{\Phi}^{1}_{d/2-1}-6d(d-1)\tilde{\Phi}^{2}_{d/2}\right)\,.\end{split}$$

(32)

The dimensionless threshold functions $\Phi^{p}_{n}(w)$ and $\tilde{\Phi}^{p}_{n}(w)$ introduced in Reuter:1996cp are evaluated at vanishing argument and carry the dependence of the result on the choice of scalar regulator. For the Litim-type profile (27) they evaluate to

$$\Phi^{p}_{n}=\frac{1}{\Gamma(n+1)}\,,\qquad\tilde{\Phi}^{p}_{n}=\frac{1}{\Gamma(n+2)}\,.$$

(33)

Notably, the present case is special in the sense that $\omega$ is independent of the coupling $u^{1}$.

Solving (31) for $\partial_{t}u^{1}$ gives the beta function

$$\beta_{u}(u^{1})=-D_{u}\,u^{1}+\frac{\omega}{1-\tilde{\omega}(u^{1})}\,.$$

(34)

This beta function admits a single NGFP characterized by

$$u^{1}_{*}=B_{2}+\frac{B_{1}}{D_{u}}\,,\qquad B_{*}^{\rm FRGE}=\,-D_{u}+D_{u}^{2}\,\frac{B_{2}}{B_{1}}\,.$$

(35)

Evaluating these expressions for $d=4$ yields

$$u^{1}_{*}\simeq-0.012\,,\qquad B_{*}^{\rm FRGE}=-2.09\,.$$

(36)

Thus, we encounter a NGFP situated at a positive Newton’s coupling and a UV-attractive eigendirection.

In the next step, we compare this result to the one found in the composite operator approximation (25). Thus, we seek to find the anomalous scaling dimension of the operator $\bar{{\mathcal{O}}}_{1}[\bar{g}]$. Without the inclusion of the gauge-fixing term in (28), this computation has been carried out in Houthoff:2020zqy , yielding

$$\begin{split}\gamma^{\rm COE}(u^{1}_{*})=-\frac{1}{(4\pi)^{d/2}}&\,\frac{1}{u^{1}_{*}}\,\Bigg{[}\frac{1}{24}\left(d^{3}+9d^{2}-22d-24\right)\,\left(\Phi^{2}_{d/2}-\frac{1}{2}(2-d)\tilde{\Phi}^{2}_{d/2}\right)\\ &-\frac{1}{2}\left(d^{3}-3d^{2}+4d-8\right)\left(\Phi^{3}_{d/2+1}-\frac{1}{2}(2-d)\tilde{\Phi}^{3}_{d/2+1}\right)\Bigg{]}\,.\end{split}$$

(37)

In comparison to the “exact definition” of $\gamma$, Eq. (22), this result neglects the contribution of the gauge-fixing terms in $\frac{\partial\Gamma_{k}^{(2)}}{\partial\bar{u}^{m}}$. Adding the gauge-fixing term (implementing harmonic gauge) changes this result to

$$\begin{split}\gamma^{\rm COE+GF}(u^{1}_{*})=-\frac{1}{(4\pi)^{d/2}}&\,\frac{1}{u^{1}_{*}}\,\Bigg{[}\frac{1}{24}d\,(d^{2}+11d-14)\,\left(\Phi^{2}_{d/2}-\frac{1}{2}(2-d)\tilde{\Phi}^{2}_{d/2}\right)\\ &-\frac{1}{2}d^{2}(d-1)\left(\Phi^{3}_{d/2+1}-\frac{1}{2}(2-d)\tilde{\Phi}^{3}_{d/2+1}\right)\Bigg{]}\,.\end{split}$$

(38)

Using these results in (25) and specializing to $d=4$ then yields

$$B_{*}^{\rm COE}=-2+\gamma^{\rm COE}(u^{1}_{*})=-1.91\,,\qquad B_{*}^{\rm COE+GF}=-2+\gamma^{\rm COE+GF}(u^{1}_{*})=-1.94\,.$$

(39)

Hence, we again encounter a UV-attractive eigendirection. In both computations $B_{*}$ is governed by the classical part $-D_{u}$ and receives a subleading correction from the quantum corrections. We also observe that there is a reasonable qualitative agreement between the exact projection and its approximation by the composite operator contribution. Subsequently, we will observe how this approximation becomes increasingly worse when going to increasing the order of the truncation.

One of the most prominent results in the context of asymptotically safe quantum gravity is the “almost-Gaussian” scaling observed when considering polynomial $f(R)$-truncations at sufficiently large order of the polynomials Falls:2013bv ; Falls:2014tra . This statement paraphrases that the spectrum of critical exponents $\{\theta_{I}^{\mathrm{FRGE}}=-\lambda_{I}^{\mathrm{FRGE}}\}$ of $B^{\mathrm{FRGE}}_{*}$ converges to the Gaussian scaling dimensions $\{d_{I}\}$. Hereby, the corrections to the first three classical scaling dimensions, $d_{0}=4$, $d_{1}=2$, and $d_{2}=0$ are such that the first two directions remain relevant, whereas the quantum corrections shift the third direction from a marginal to a relevant direction. All other directions remain irrelevant. For operators ${\mathcal{O}}_{n}$ with sufficiently large value of $n$ this is ensured by the aforementioned convergence pattern. Therewith, the polynomial $f(R)$-truncation exhibits a three-dimensional UV-critical hypersurface.

In this section, we carry out a first reconnoitring study comparing the stability matrices at the fixed point obtained from the standard FRGE procedure for the $f(R)$-truncation and the corresponding composite operator results. The FRGE computation is based on Eq. (24), i.e.,

$$\begin{split}(B^{\mathrm{FRGE}}_{*})^{n}{}_{m}&=\sum_{l}\left[(\mathds{1}-\tilde{\omega}(u_{*}))^{-1}\right]^{n}{}_{l}\,\left.\left(-D^{l}_{m}+\frac{\partial\omega^{l}(u)}{\partial u^{m}}+\frac{\partial}{\partial u^{m}}\left(\sum_{l^{\prime}}\tilde{\omega}^{l}{}_{l^{\prime}}d_{l^{\prime}}u^{l^{\prime}}\right)\right)\right|_{u=u_{*}}\\ &=\sum_{l}\left[(\mathds{1}-\tilde{\omega}(u_{*}))^{-1}\right]^{n}{}_{l}\,\left(-D^{l}_{m}+\gamma_{m}{}^{n}(u_{*})+\tilde{\gamma}_{m}{}^{n}(u_{*})\right)\,.\end{split}$$

(40)

The corresponding results for the scaling corrections emerging from the COE are obtained from Eq. (25), i.e.,

$$(B^{\mathrm{COE}}_{*})^{n}{}_{m}=-D^{n}_{m}+\gamma_{m}{}^{n}(u_{*})\,.$$

(41)

This comparison thus boils down to the question whether we may neglect the contributions of the matrices $\tilde{\omega}^{n}{}_{m}(u_{*})$ and $\tilde{\gamma}_{m}{}^{n}(u_{*})$. If we do so for the contribution of $\tilde{\gamma}_{m}{}^{n}(u_{*})$, we may think of the matrix $\tilde{\omega}^{n}{}_{m}(u_{*})$ as translating between the two approximations.

We start by analyzing Eq. (40). For functional $f(R)$-truncations the ingredients in Eq. (4) take the form

$$\begin{split}\bar{\Gamma}_{k}[g]&=\int\mathrm{d}^{d}x\,\sqrt{g}f_{k}(R)=\int\mathrm{d}^{d}x\,\sqrt{g}k^{d}{\mathcal{F}}_{k}(\rho)\,,\\ \hat{\Gamma}[h;\bar{g}]&=\frac{1}{2\alpha}\int\mathrm{d}^{d}x\sqrt{\bar{g}}\,\bar{g}^{\mu\nu}F_{\mu}[h;\bar{g}]F_{\nu}[h;\bar{g}]\,.\end{split}$$

(42)

Here $f_{k}(R)$ is a generic, $k$-dependent function of the scalar curvature $R$ with dimensionless counterparts, ${\mathcal{F}}_{k}(\rho)=k^{-d}f_{k}(R)$ and $\rho=k^{-2}R$. Supplementing (42) by the corresponding ghost action and substituting the result into the Wetterich equation results in a partial differential equation governing the functional form of ${\mathcal{F}}_{k}(\rho)$. There are several incarnations of this equation which differ by the treatment of the discrete modes appearing on spherical backgrounds Morris:2022btf .

The technical construction of the flow equation for $f(R)$-truncations implements the York decomposition York ; York2 of the fluctuation fields

$$h_{\mu\nu}=h_{\mu\nu}^{\mathrm{TT}}+\bar{D}_{\mu}\xi_{\nu}+\bar{D}_{\nu}\xi_{\mu}+\bar{D}_{\mu}\bar{D}_{\nu}\sigma-\frac{1}{d}\bar{g}_{\mu\nu}\bar{D}^{2}\sigma+\frac{1}{d}\bar{g}_{\mu\nu}h\,.$$

(43)

Here $h_{\mu\nu}^{\mathrm{TT}}$ is the transverse-traceless (TT) part, $\hat{\xi}_{\mu}$ is the transverse vector field of the longitudinal part of $h_{\mu\nu}$, $\sigma$ is the corresponding longitudinal scalar, and $h$ is the trace part. The component fields are subject to the constraints

$$\bar{D}^{\mu}h_{\mu\nu}^{\mathrm{TT}}=0\,,\quad\bar{g}^{\mu\nu}h_{\mu\nu}^{\mathrm{TT}}=0\,,\quad\bar{D}_{\mu}\xi^{\mu}=0\,,\quad\bar{g}^{\mu\nu}h_{\mu\nu}^{\mathrm{TT}}=h\,.$$

(44)

The fields are re-scaled to be orthonormal on Einstein spaces,

$$\hat{\xi}_{\mu}=(2(\Delta-\bar{R}/d))^{1/2}\xi_{\mu}\,,\quad\hat{\sigma}=\left(\frac{d-1}{d}\Delta\left(\Delta-\bar{R}/(d-1)\right)\right)^{1/2}\sigma\,,\quad\hat{h}=\frac{1}{\sqrt{d}}h\,,$$

(45)

which also accounts for the Jacobians arising from the decomposition (43).

Owed to the higher-derivative terms generated by $\bar{\Gamma}_{k}$ it turns out to be convenient to restrict (42) to Landau-type gauges, which fix $\beta=0$ and take the limit $\alpha\rightarrow 0$ once all ingredients in the traces are computed. The ghost-sector arising from this “physical gauge” follows Machado:2007ea . This choice leads to a number of simplifications. Firstly, the matrix $(\Gamma_{k}^{(2)}+{\mathcal{R}}_{k})$ becomes diagonal in field space. Secondly, the contributions of the unphysical transverse vector field $\xi_{\mu}$ and the unphysical scalar field $\sigma$ become independent of the function $f_{k}(R)$. This is readily deduced by analyzing the $\alpha$-dependence of the decomposed propagator (see Table 1 in Machado:2007ea and Eqs. (97)-(99) of Codello:2008vh ). Conversely, the matrix elements for the transverse traceless and $\hat{h}$-fluctuations appearing in $(\Gamma_{k}^{(2)}+{\mathcal{R}}_{k})$ are solely governed by $\bar{\Gamma}_{k}[g]$. Specifying to the Litim-type regulator, they read Codello:2008vh

$$\begin{split}(\Gamma_{k,\mathrm{TT}}^{(2)}+{\mathcal{R}}_{k,\mathrm{TT}})=&-\frac{1}{2}\Bigg{[}\left(k^{2}-\frac{2(d-2)}{d(d-1)}\bar{R}\right)f^{\prime}_{k}+f_{k}\Bigg{]}\,\mathds{1}_{\mathrm{TT}}\,,\\ (\Gamma_{k,\hat{h}\hat{h}}^{(2)}+{\mathcal{R}}_{k,\hat{h}\hat{h}})=&\frac{d-2}{4}\Bigg{[}\frac{4(d-1)^{2}}{d(d-2)}f^{\prime\prime}_{k}\left(k^{2}-\frac{\bar{R}}{d-1}\right)^{2}\\ &\qquad\qquad+\frac{2(d-1)}{d}f^{\prime}_{k}\left(k^{2}-\frac{\bar{R}}{d-1}\right)-\frac{2\bar{R}}{d}f^{\prime}_{k}+f_{k}\Bigg{]}\,.\end{split}$$

(46)

Here, $\mathds{1}_{\mathrm{TT}}$ is the identity on the space of transverse-traceless fields, $f^{\prime}_{k}(R)=\partial_{R}f_{k}(R)$, and we adopt an identical notation for the dimensionless functions ${\mathcal{F}}_{k}^{\prime}(\rho)=\partial_{\rho}{\mathcal{F}}_{k}(\rho)$. The explicit form of corresponding matrix element of the regulator ${\mathcal{R}}_{k}$ arising from implementing the adaptive cutoff $\Delta\mapsto\Delta+R_{k}$ is

$$\begin{split}{\mathcal{R}}_{k,\mathrm{TT}}=&-\frac{1}{2}\,f^{\prime}_{k}\,R_{k}\,\mathds{1}_{\mathrm{TT}}\,,\\ {\mathcal{R}}_{k,\hat{h}\hat{h}}=&\frac{d-1}{2d}\Big{[}(d-2)f^{\prime}_{k}+2(d-1)f^{\prime\prime}_{k}(2\Delta+R_{k})-4f^{\prime\prime}_{k}\bar{R}\Big{]}R_{k}\,.\end{split}$$

(47)

The Wetterich equation takes the schematic form

$$\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}\bar{\Gamma}_{k}=&\,\frac{1}{2}\operatorname{Tr}_{\mathrm{TT}}\left[\left(\bar{\Gamma}_{k,\mathrm{TT}}^{(2)}+{\mathcal{R}}_{k,\mathrm{TT}}\right)^{-1}\frac{\mathrm{d}}{\mathrm{d}t}{\mathcal{R}}_{k,\mathrm{TT}}\right]+\frac{1}{2}\operatorname{Tr}_{\mathrm{S}}\left[\left(\bar{\Gamma}_{k,{\hat{h}\hat{h}}}^{(2)}+{\mathcal{R}}_{k,{\hat{h}\hat{h}}}\right)^{-1}\frac{\mathrm{d}}{\mathrm{d}t}{\mathcal{R}}_{k,{\hat{h}\hat{h}}}\right]\\ &+\bar{\Gamma}_{k}\text{-independent terms}\,,\end{split}$$

(48)

where the $\bar{\Gamma}_{k}$-independent terms are due to the Fadeev-Popov ghosts and the contributions of the $\xi_{\mu}$ and $\sigma$ modes. Owed to the presence of the higher-derivative terms, the regulator does not obey the simple factorization (12). Nevertheless, one is able to separate the trace contributions into parts associated with the regulator and the couplings appearing within (47). The result of this computation is rather lengthy and given as Eq. (78) of Appendix B.