Composite higher derivative operators in $d=2+\epsilon$ dimensions and the spectrum of asymptotically safe gravity

TL;DR

This paper investigates the renormalization and spectrum of higher derivative operators in quantum gravity near two dimensions, confirming the asymptotic safety scenario and identifying relevant operators at the fixed point.

Contribution

It introduces a scheme- and gauge-independent approach to analyze composite operators in quantum gravity and determines their scaling spectrum at the fixed point.

Findings

Existence of an ultraviolet fixed point for Newton's constant.

Identification of a relevant operator near four dimensions.

Other previously known operators are marginal or trivial on-shell.

Abstract

We discuss the renormalization of Einstein-Hilbert's gravity in $d=2+\epsilon$ dimensions. We show that the application of the path-integral approach leads naturally to scheme- and gauge-independent results on-shell, but also gives a natural notion of quantum metric off-shell, which is the natural argument of the effective action, even at the leading order in perturbation theory. The renormalization group of Newton's constant is consistent with the asymptotic safety scenario for quantum gravity in that it has an ultraviolet relevant fixed point. We extend the approach to the analysis of curvature square operators, understood as composites operators, which allows for the determination of the spectrum of scaling operators at the scale invariant fixed point. The analysis suggests that there is one operator that becomes relevant close to $d=4$ dimensions, while other operators previously found in the literature are either marginal or trivial on-shell.

Equations

$$\begin{split}S_{E}[g]&=\int{\rm d}^{d}x\sqrt{g}\Bigl{\{}g_{0}-g_{1}R\Bigr{\}}\,,\end{split}$$

$$\begin{array}{rl}{\delta }_{\xi }{g}_{\mu \nu }& ={\mathcal{L}}_{\xi }{g}_{\mu \nu }={\nabla }_{\mu }{\xi }_{\nu }+{\nabla }_{\nu }{\xi }_{\mu },\end{array}$$

$$\begin{array}{rl}{g}_{\mu \nu }& ={\overline{g}}_{\mu \nu }+h_{\mu \nu }+\frac{\lambda }{2}{h}_{\mu \rho }{\overline{g}}^{\rho \theta }{h}_{\theta \nu }+\mathcal{O}(h^3),\end{array}$$

$$\begin{array}{r}{\delta }_{\xi }{h}_{\mu \nu }={\overline{g}}_{\rho \nu }{\overline{\nabla }}_{\mu }{\xi }^{\rho }+{\overline{g}}_{\rho \mu }{\overline{\nabla }}_{\nu }{\xi }^{\rho }+\mathcal{O}(h),\end{array}$$

$$\begin{array}{rl}{S}_{\mathrm{gf}}& =\frac{1}{2}\int {\mathrm{d}}^{d}x\sqrt{\overline{g}}{\overline{g}}^{\mu \nu }{F}_{\mu }{F}_{\nu },\\ F_{\mu }& ={\overline{\nabla }}^{\rho }{h}_{\rho \mu }-\frac{\beta }{2}{\overline{g}}^{\rho \theta }{\overline{\nabla }}_{\mu }{h}_{\rho \theta },\end{array}$$

$$\begin{array}{rl}{S}_{\mathrm{gh}}& =\int {\mathrm{d}}^{d}x\sqrt{\overline{g}}{\overline{c}}^{\mu }{{\delta }_{\xi }{F}_{\mu }|}_{\xi \to c}.\end{array}$$

$$\begin{array}{rl}Z[j]& =\int \mathrm{D}h{\mathrm{e}}^{-S_E[\overline{g}+h]+S_{\mathrm{gf}}+S_{\mathrm{gh}}+j\cdot h},\end{array}$$

$$\begin{array}{rl}{\mathrm{e}}^{-\Gamma }& =\int \mathrm{D}h{\mathrm{e}}^{-S_E[\overline{g}+h]+S_{\mathrm{gf}}+S_{\mathrm{gh}}+\frac{\delta \Gamma }{\delta ⟨h⟩}\cdot (h-⟨h⟩)}.\end{array}$$

$$\begin{array}{rl}\Gamma & =S_E+\frac{1}{2}\text{mbox}Tr\log {\mathcal{O}}_h^{\beta ,\lambda }-\text{mbox}Tr\log {\mathcal{O}}_{\mathrm{gh}}^{\beta ,\lambda },\end{array}$$

$$\begin{array}{r}{\mathcal{O}}_h^{\beta ,\lambda }={\mathcal{O}}_h^{\lambda }+\delta \beta {\hat{P}}^{\lambda }(\overline{\nabla }),\end{array}$$

$$\begin{array}{r}{\mathcal{O}}_h^{\lambda }=-{\overline{\nabla }}^{2}K+U^{\lambda },\end{array}$$

$$\begin{array}{r}{K}^{\mu \nu }{}_{\rho \theta }={\delta }_{\rho }^{(\mu }{\delta }_{\theta }^{\nu )}-\frac{1}{2}{\overline{g}}^{\mu \nu }{\overline{g}}_{\rho \theta },\end{array}$$

$$\begin{array}{r}{K}^{-1}{}^{\mu \nu }{}_{\rho \theta }={\delta }_{\rho }^{(\mu }{\delta }_{\theta }^{\nu )}-\frac{1}{d-2}{\overline{g}}^{\mu \nu }{\overline{g}}_{\rho \theta },\end{array}$$

$$\begin{split}\frac{1}{2}\mbox{Tr}\log{\cal O}^{\beta,\lambda}_{h}&=\frac{1}{2}\int\frac{{\rm d}s}{s}\,\mbox{Tr}\,{\rm e}^{-s{\cal O}^{\beta,\lambda}_{h}}\\ &=\frac{1}{2}\int\frac{{\rm d}s}{s}\,\mbox{Tr}\Bigl{(}{\rm e}^{-s{\cal O}^{\lambda}_{h}}-s\,\delta\beta\,\hat{P}^{\lambda}{\rm e}^{-s{\cal O}^{\lambda}_{h}}\Bigr{)}\,.\end{split}$$

$$\begin{array}{r}\text{mbox}Tr{\mathrm{e}}^{-s{\mathcal{O}}_h^{\lambda }}\sim \frac{1+O(s)}{s^{d/2}},\text{mbox}Tr{\hat{P}}^{\lambda }{\mathrm{e}}^{-s{\mathcal{O}}_h^{\lambda }}\sim \frac{1+O(s)}{s^{d/2+1}}.\end{array}$$

$$\begin{split}&\Gamma_{\rm div}=\frac{\mu^{-\zeta}}{\zeta}\int{\rm d}^{d}x\sqrt{\overline{g}}\Bigl{\{}A\overline{R}+J_{\mu\nu}E[\overline{g}]^{\mu\nu}\Bigr{\}}\,,\\ &E[\overline{g}]^{\mu\nu}=\overline{G}^{\mu\nu}+\frac{g_{0}}{2g_{1}}\overline{g}^{\mu\nu}=\overline{G}^{\mu\nu}+\frac{\Lambda}{2}\overline{g}^{\mu\nu}\,,\end{split}$$

$$\begin{split}A&=\frac{1}{4\pi}\frac{36+3d-d^{2}}{12}\,,\\ J_{\mu\nu}&=\frac{\overline{g}_{\mu\nu}}{4\pi}\Bigl{\{}\frac{d^{2}-d-4}{2(d-2)}\lambda-\delta\beta\left(2+\frac{2\lambda}{d-2}\right)-d-1\Bigr{\}}\,.\end{split}$$

$$\begin{array}{rl}{\beta }_G& =(d-2)G-\frac{36+3d-d^2}{48\pi }{G}^2,\end{array}$$

$$\begin{array}{r}{G}^{\ast }=\frac{48\pi (d-2)}{36+3d-d^2},\end{array}$$

$$\begin{array}{rl}{\Gamma }_{\mathrm{ren}}[\overline{g}]-\frac{{\mu }^{-\zeta }}{\zeta }{J}_{\mu \nu }\frac{\delta {\Gamma }_{\mathrm{ren}}[\overline{g}]}{\delta {\overline{g}}_{\mu \nu }}& =-\frac{1}{G}\int {\mathrm{d}}^{d}x\sqrt{\overline{g}}\overline{R},\end{array}$$

$$\begin{array}{r}{\tilde{g}}_{\mu \nu }={\overline{g}}_{\mu \nu }-\frac{{\mu }^{-\zeta }}{g_1\zeta }{J}_{\mu \nu },\end{array}$$

$$\begin{array}{rl}{\Gamma }_{\mathrm{ren}}[\tilde{g}]& =-\frac{1}{G}\int {\mathrm{d}}^{d}x\sqrt{\tilde{g}}\tilde{R},\end{array}$$

$$\begin{split}S_{\rm hd}[g]&=\int{\rm d}^{d}x\,\sqrt{g}\Bigl{\{}c_{1}R_{\mu\nu\rho\sigma}^{2}+c_{2}R^{2}+c_{3}R_{\mu\nu}^{2}\\ &\qquad\qquad\qquad+c_{4}\Lambda R+c_{5}\Lambda^{2}\Bigr{\}}\end{split}$$

$$\begin{split}S_{\rm hd}[g]&=\int{\rm d}^{d}x\,\sqrt{g}\Bigl{\{}\alpha_{1}R_{\mu\nu\rho\sigma}^{2}+\alpha_{2}R^{2}+I_{2,\mu\nu}E[g]^{\mu\nu}\Bigr{\}}\,,\\ I_{2,\mu\nu}&=\alpha_{3}Rg_{\mu\nu}+\alpha_{4}R_{\mu\nu}+\alpha_{5}\Lambda g_{\mu\nu}\,.\end{split}$$

$$\begin{array}{rl}{S}_{\mathrm{hd}}[g]& =\alpha \cdot {\mathcal{R}}_2[g],\end{array}$$

$$\begin{array}{rl}Z[j;\alpha ]& =\int \mathrm{D}h{\mathrm{e}}^{-S_E[\overline{g}+h]-S_{\mathrm{hd}}[\overline{g}+h]-S_{\mathrm{gf}}-S_{\mathrm{gh}}+j\cdot h},\end{array}$$

$$\begin{array}{r}⟨{\mathcal{R}}_2⟩=-\frac{\partial }{\partial \alpha }Z[j;\alpha {]}_{j={\alpha }_i=0}.\end{array}$$

$$\begin{split}-\alpha\cdot\frac{1}{2}\mbox{Tr}\Bigl{\{}\frac{\delta^{2}{\cal R}_{2}}{\delta h_{\mu\nu}\delta h_{\rho\sigma}}{\cal G}_{h,\mu\nu\rho\sigma}\Bigr{\}}_{\langle h\rangle=0}\,,\end{split}$$

$$\begin{split}\Delta\Gamma=S_{\rm hd}-\alpha\cdot\mbox{Tr}\Bigl{\{}\frac{1}{2}\frac{\delta^{2}{\cal R}_{2}}{\delta h_{\mu\nu}\delta h_{\rho\sigma}}{\cal G}_{h,\mu\nu\rho\sigma}-\frac{GJ_{\mu\nu}}{\zeta}\frac{\delta{\cal R}_{2}}{\delta h_{\mu\nu}}\Bigr{\}}\,,\end{split}$$

$$\begin{array}{r}\text{mbox}Tr{\delta }^2{\mathcal{R}}_2{\mathrm{e}}^{-s{\mathcal{O}}_h^{\lambda }}\sim \frac{1}{s^{d/2+2}}(1+O(s)),\end{array}$$

Full text

I Renormalized metric’s action

The asymptotic safety conjecture is based on the premise that Einstein’s gravity, if seen as a quantum field theory of the metric tensor, is ultraviolet complete thanks to the presence of a suitable fixed point of the renormalization group Weinberg:1976xy . The ultraviolet fixed point must be non-Gaussian in order to circumvent the established fact that the Einstein-Hilbert action is not perturbatively renormalizable Goroff:1985th . This was confirmed in a seminal paper using background and Wilsonian renormalization group methods Reuter:1996cp , on which most of the recent literature of the topic is now based.

The combined application of background and Wilsonian methods, however, often comes at the price of having to deal with an effective action that is scheme- and gauge-dependent, that is one of the most pressing proplems with which the proposal has to deal Bonanno:2020bil . This happens because the effective action is off-shell and the gauge-fixed symmetry is nonlinear. In fact, it is not quite clear how to go on-shell given the approximations that are usually involved in splitting the metric over an arbitrary background and fluctuations on it. Recently, new developments in extracting the on-shell physics were obtained in the context of Wilsonian renormalization and its application to quantum gravity Baldazzi:2021ydj ; Baldazzi:2021orb .

A way around the problem would be to address scheme- and gauge-dependence in a setting in which perturbation theory is applicable, e.g., one in which the ultraviolet fixed point is still perturbative. This setting is provided by gravity in $d=2+\epsilon$ dimensions, that exhibits an order-$\epsilon$ ultraviolet fixed point for Newton’s constant motivating the asymptotic safety since its inception Weinberg:1980gg . In fact, this setting was the first testing ground of asymptotic safety Kawai:1989yh . The obvious limitation is that the continuation to $d=4$ requires $\epsilon=2$, which is certainly outside the validity of perturbation theory, which is why it should be regarded as a complementary approach to the functional one Martini:2021lcx .

In a relatively recent paper, Ref. Martini:2021slj , the approach in $d=2+\epsilon$ was reconsidered with an ample discussion on the diffeomorphisms symmetry, its gauge fixing, and the parametric dependence induced by the background splitting. The starting point is a naive dimensional continuation of the Einstein-Hilbert action

[TABLE]

where $g_{1}$ is the inverse of Newton’s constant, modulo a normalization, and $g_{0}$ is a cosmological constant-like contribution. In two dimensions, $g_{1}$ is dimensionless and its inverse is a suitable coupling to construct the perturbative expansion, while $g_{0}$ gives a dimension two deformation that is going to be useful when going on-shell. In general dimension $d$, we have that $\Lambda=g_{0}/g_{1}$ is a dimension two constant (the cosmological constant) and this is going to play an important role in the next section because it could mix with dimensionful composite operators. The reader familiar with Ref. Martini:2021slj can skip to the discussion preceding Eq. (16).

The scalar curvature term of (1) is topological in two dimensions, so there are two distinct ways to implement the diffeomorphisms symmetry at the quantum level given the discontinuity of the number of degrees of freedom of $g_{\mu\nu}$. The discussion of Ref. Martini:2021slj strongly suggests that the simplest realization of the (infinitesimal) diffeomorphisms

[TABLE]

is the one best suited for analytic continuation to general $d$ (though earlier results were often based on an isomorphic one that isolates the conformal mode in a manner close to string theory Kawai:1989yh ). The algebra of the group is isomorphic to the one of Lie-brackets, $\left[\delta_{\xi_{1}},\delta_{\xi_{2}}\right]=\delta_{\left[\xi_{1},\xi_{2}\right]}$.

A path-integral for the action (1) should be constructed integrating over all possible metrics. In order to manifestly maintain covariance, we use the background field method introducing an arbitrary metric $\overline{g}_{\mu\nu}$ and fluctuations $h_{\mu\nu}$

[TABLE]

and integrate the path-integral over the fluctuations. Notice that the above split introduces the constant parameter $\lambda$, which is fundamental for us to test the scheme dependence of the procedure (further orders in the expansion will not play a role in the following).

The manifest symmetry is the one for which the background version of (2) is realized. The symmetry that is gauge-fixed, instead, is the one for which the background is kept fixed, implying that $h_{\mu\nu}$ transforms nonlinearly

[TABLE]

further orders can be computed with some work, but are not needed in what follows. A simple gauge-fixing is the Feynman-deDonder one, which we choose

[TABLE]

that it introduces an additional parameter $\beta$. In Ref. Martini:2021slj the parameter is chosen as $\beta=1+\delta\beta$ with $\delta\beta\ll 1$ for simplicity. The independence on $\delta\beta$ is not enough to fully establish gauge-independence of the final results (it is necessary but not sufficient condition), which is why the $\lambda$ dependence of (3) is most important in the following. From (5) and (4), it is easy to determine the ghosts action

[TABLE]

We have all the ingredients to construct the Euclidean path-integral

[TABLE]

where the measure contains also the integration over the ghosts, and the effective action

[TABLE]

Using standard manipulations, it is straightforward to obtain the one-loop effective action

[TABLE]

where ${\cal O}^{\beta,\lambda}_{h}$ and ${\cal O}^{\beta,\lambda}_{\rm gh}$ are the Hessians of $h_{\mu\nu}$ and ghosts, respectively, and they depend parametrically on the gauge and the split thorugh $\beta$ and $\lambda$. At this stage, the effective action $\Gamma$ is naively a functional of the expectation value $\langle h_{\mu\nu}\rangle$ and parametrically depends on $\overline{g}_{\mu\nu}$, i.e., $\Gamma=\Gamma[\overline{g}_{\mu\nu},\langle h_{\mu\nu}\rangle]$. The double dependence is expected to merge in a suitable way according to the Ward identity of a split symmetry induced by all combined transformations preserving (3). This symmetry is not anomalous if we use a regularization prescription that is cutoff independent, which is why we choose dimensional regularization, and this represents a main departure point from the Wilsonian approaches as in Reuter:1996cp . The red herring is that we are tempted to assume that $\Gamma$ is a natural function of a single “quantum” metric, $\Gamma=\Gamma[\langle g_{\mu\nu}\rangle]$, where $\langle g_{\mu\nu}\rangle=\overline{g}_{\mu\nu}+\langle h_{\mu\nu}\rangle+\dots$ and the dots hide corrections that should be computed through external legs of $\langle h_{\mu\nu}\rangle$.

The evaluation of the divergent part of $\Gamma$ can be carried in the limit $\langle h_{\mu\nu}\rangle=0$, which is the most important feature of the background method. In this limit ${\cal O}^{\beta,\lambda}_{h}$ and ${\cal O}^{\beta,\lambda}_{\rm gh}$ are differential operators that depend on covariant derivatives and curvatures of the background. Furthermore, the differential operators are of Laplace-type, except for the insertion caused by $\delta\beta$ that can be expanded because of our choice. We illustrate this considering ${\cal O}^{\beta,\lambda}_{h}$, but a similar discussion applies to ${\cal O}^{\beta,\lambda}_{\rm gh}$. It is expanded

[TABLE]

where

[TABLE]

$K$ and $U^{\lambda}$ are metric-dependent endomorphisms acting on symmetric tensor fields, and $\hat{P}^{\lambda}$ is a differential operator with uncontracted derivatives. It is useful to display $K$ explicitly

[TABLE]

so that it is clear that the Green function of ${\cal O}^{\lambda}_{h}$, defined ${\cal O}^{\lambda}_{h,x}\cdot{\cal G}_{h,xy}=\delta_{xy}$, is proportional to the inverse of (12)

[TABLE]

and contains a kinematical pole $1/(d-2)$ caused by the fact that the Einstein-Hilbert action is topological in two dimensions and the number of degrees of freedom changes discontinuously across $d=2$. In the past a debate on how to deal with this pole has lead to different conclusions Jack:1990ey ; Aida:1996zn , but it will cancel in our physical results.

The computation is most easily performed using standard heat kernel techniques with the Seeley-deWitt expansion, and we continue the illustration using ${\cal O}^{\beta,\lambda}_{h}$. The second term of (9) is expressed as

[TABLE]

Both the trace of the heat kernel, $\mbox{Tr}\exp(-s{\cal O}^{\lambda}_{h})$, and the heat kernel with derivative insertions, $\mbox{Tr}\hat{P}^{\lambda}\exp(-s{\cal O}^{\lambda}_{h})$, admit asymptotic expansions in terms of well-known computable coefficients Groh:2011dw .111A technical point: in Ref. Groh:2011dw the coefficients of the expansion are normalized without the square root of the Van Vleck determinant, as was suggested in Ref. Anselmi:2007eq . For the computations of this paper we have used the more conventional normalization of Refs. Jack:1983sk ; Martini:2018ska . The final results are independent of this choice of course.

Schematically

[TABLE]

In dimensional regularization close to $d=2$, it is important to continue the $d$ in leading power’s dependence to select the terms scaling as $s^{-d/2-1}$ in (14) that correspond to the universal poles (i.e., the ultraviolet logarithmic divergences) Jack:1983sk . A crucial point made in Ref. Martini:2021slj is that the covariant diagrammatics is always convergent if $d<2$ (the real part). For this reason, we introduce the analytic continuation $d=2-\zeta$ with $\zeta$ playing the role of the $\epsilon$-parameter. In the heat kernel approach this changes the asymptotic expansion as $s^{-d/2}\to s^{-1+\zeta/2}$, similarly for the case with one insertion, and gives the pole $1/\zeta$. It is not necessary to continue all other instances of the spacetime dimensionality $d$, instead, including, for example, those coming from contractions of the metric $\overline{g}^{\mu}{}_{\mu}=d$ (which is left as is). The motivation of this prescription is to treat the latter in a way similar to the $N$-dependence of renormalization group functions of $SU(N)$ gauge theories, while simultaneously continuing the integrals for convergence. The idea is to take the limit $d=2+\epsilon=2-\zeta$ at the very end of the computation.

The result of the above procedure in the limit $\langle h_{\mu\nu}\rangle=0$ is the divergent part of the effective action and depends only on the background Martini:2021slj . The most convenient way to write the result is to use the background scalar curvature and the equations of motion of the bare action,

[TABLE]

where $E[\overline{g}]^{\mu\nu}$ are the equations of motion evaluated on the background metric, $\overline{G}_{\mu\nu}=\overline{R}_{\mu\nu}-\frac{1}{2}\overline{R}\,\overline{g}_{\mu\nu}$ is the background’s Einstein tensor, the coefficient $A$ is just a scalar that depends parametrically on $d$, and $J_{\mu\nu}$ depends on gauge- and scheme-parameters

[TABLE]

In essence, the nonzero value for $g_{0}$ in (1) was chosen precisely because it allows to have $\overline{R}\neq 0$ and extract unambiguously the coefficient $A$. Most notably, $J_{\mu\nu}$ contains all the “unwanted” dependencies, including the kinematical pole, suggesting a meaningful $d\to 2$ limit despite the discontinuity of the number of degrees of freedom.

The immediate conclusion of (16) is that on-shell all the unwanted parameters and the kinematical pole in $J_{\mu\nu}$ decouple on-shell. The remaining coefficient $A$ is naturally subtracted by renormalizing the coupling $g_{1}$, so it is used to obtain the beta function of the dimensionless Newton’s constant $G=\mu^{-\zeta}/g_{1}$ in the limits $d=2+\epsilon=2-\zeta$, which is

[TABLE]

that confirms a result of Ref. Falls:2017cze obtained with a cutoff prescription (see also Ref. Bastianelli:2022pqq ). Notice that we have now moved to the renormalized coupling $G$, expressed in units of the renormalization group scale $\mu$. The solution of $\beta_{G}=0$ give a $G^{*}\sim O(d-2)$ ultraviolet fixed point. Extrapolation to finite values of $\epsilon$ gives $G^{*}$ as a function of $d$

[TABLE]

which is valid up to $d\approx 7.7$, so “safely” above $4$ (although further corrections should be computed).

One observation, that is going to be very important in the following, comes from realizing that (16) is telling us what is the wavefunction renormalization of the metric. Subtracting the counterterms, we have that (16) can be rewritten in terms of the renormalized coupling

[TABLE]

to the leading order in $\hbar$ (which is implicit in our notation). This shows that the renormalized action $\Gamma_{\rm ren}$ is actually a natural function of a renormalized metric that is not $\overline{g}_{\mu\nu}$, even in the limit $\langle h_{\mu\nu}\rangle=0$. To the leading order in perturbation theory the renormalized metric is

[TABLE]

where the expression for $J_{\mu\nu}$ is given in (17) and the result depends on the unphysical parameters because the metric itself is not straightforwardly an observable. Using the renormalized metric

[TABLE]

which, again, is valid to the leading order in perturbation theory. This is the first important result of this paper: a finite renormalized action that depends only on the finite renormalized coupling, denoted $G$, and the renormalized metric $\tilde{g}_{\mu\nu}$. The use of (21) and the second term of the lhs of (20) is fundamental to have a completely finite and manifestly gauge- and scheme-independent renormalized result, in a manner similar to other nonlinear theories Osborn:1987au , which corrects the naive assumption discussed after Eq. (9). This will become particularly important in the next application that involves the deformation of the original bare action through the inclusion of composite operators that require further renormalization.

II Higher derivative operators

A suitable ultraviolet fixed point for asymptotic safety must satisfy few important properties. The first and possibily most important of these is that the dimensionality of the (ultraviolet) critical surface of the fixed point in the space of couplings is finite, since this dimensionality is linked to the number of uv relevant operators and hence of undetermined parameter of the theory Weinberg:1976xy . The critical surface can be obtained by studying the linearized renormalization group flow close to the uv fixed point.

In Wilsonian approaches the number of uv relevant operators is generally obtained by truncating the effective action with a controlled number of operators, which may or may not depend on the background. So the flow is projected onto the truncation and the number of relevant operators can be found. This operation, however, shares some limitation with the rest of the framework, even neglecting the cases in which the background specializes the projection. In fact, the renormalization group flow that is generally projected is the one of the off-shell action, which, as we discussed before, is gauge- and parameter-dependent. Furthermore, the spectrum of operators may not be entirely physical given that some scaling operators can be constructed from traces and contractions of the equations of motion.222Having an off-shell functional effective action is not intrinsically bad, but drawing physical conclusions on the number of (ir)relevant operators is potentially dangerous. For an overview of the essential scheme in perturbation theory and its application to composite operators see Ref. Anselmi:2012aq .

For example, the current understanding of the spectrum is that a linear combination of square curvature operators is a relevant operator in $d=4$, while two more combinations are irrelevant Codello:2007bd . A similar story might happen for cubic curvature operators Kluth:2020bdv ; Kluth:2022vnq . In this section we want to show, within the limitations of perturbation theory, that there is one and only one additional operator that is uv relevant and quadratic in the curvatures.

To begin with, we consider a basis for the bare higher derivative operators, which we could parametrize as

[TABLE]

where $R_{\mu\nu}^{2}=R_{\mu\nu}R^{\mu\nu}$ and $R_{\mu\nu\rho\sigma}^{2}=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ and the last two terms are added as potential mixing on the basis of dimensional analysis (recall that $\Lambda=g_{0}/g_{1}$ is dimension two). However, the above parametrization in terms of the couplings $c_{i}$ is not convenient when going on-shell, and we do need to go on-shell later on. For this reason, we introduce the linearly related couplings $\alpha_{i}$ through

[TABLE]

There is some freedom in the way in which the tensor $I_{2,\mu\nu}$ is parametrized, as itself can contain the equations of motion, however the important points are that $I_{2,\mu\nu}$ is dimension two, that all the terms of (23) appear, and, finally, that on-shell the tensor $I_{2,\mu\nu}$ decouples. For convenience, we introduce the shorthand notation

[TABLE]

where $\alpha$ and ${\cal R}_{2}[g]$ are interpreted as vectors in the five-dimensional spaces of couplings and dimension four operators, respectively. The explicit basis can be written down easily inserting $I_{2,\mu\nu}$ and $E[g]^{\mu\nu}=G^{\mu\nu}+\Lambda/2g^{\mu\nu}$ in $S_{\rm hd}[g]$.

In this section and from now on we restrict our attention to the gauge choice $\beta=1$ (i.e., $\delta\beta=0$) in (5), which will leave us with only $\lambda$ in (3) to test parametric dependencies. An important point: notice that our ability to write down the complete basis (24) hinges on the fact that we are working with an arbitrary dimensionality $d$, while for specific dimensions the basis might collapse in two or less terms, suggesting the fact that some operators are evanescent (e.g., in $d=2$ we have that both $R_{\mu\nu}$ and $R_{\mu\nu\rho\theta}$ can be expressed in terms of $R$ and the metric, while in $d=4$ one combination is the topological Euler term). We return on this point when presenting the spectrum of scaling operators for the special cases $d=2,3$ in Sect. III.

The bare couplings $\alpha_{i}$ can act as sources for the (integrated) higher derivative operators.333It would be possible to source the local composite operators replacing $\alpha_{i}\to\alpha_{i}(x)$, which would require an analysis more akin to the local renormalization group Shore:1986hk . This would be more difficult, but it could have implications for future applications of asymptotic safety, especially for studying local observables. We hope that this is addressed in future research. If we include $S_{\rm hd}$ in (7), we obtain a functional,

[TABLE]

that depends on the new sources and derivatives wrt $\alpha$ realize the expectation values of the operators in (24)

[TABLE]

Since the perturbative expansion is based on Newton’s constant, the additional higher derivative operators can be treated as composites and it is sufficient to determine their linear renormalization. This contribution can be easily deduced by including the Hessian $\alpha\cdot\delta^{2}{\cal R}_{2}$ of the new contributions (24) in Eq. (9) through the replacement ${\cal O}_{h}^{\lambda}\to{\cal O}_{h}^{\lambda}+\alpha\cdot\delta^{2}{\cal R}_{2}$ (recall that we have now set $\beta=1$). Since we only need the linear order in the sources $\alpha_{i}$ for (27), we deduce the first contribution

[TABLE]

where ${\cal G}_{h}$ is the Green function of ${\cal O}^{\lambda}_{h}$.

It is less clear, however, what is the correct argument of the lhs of Eq. (27). The important point is to recall the discussion that has lead to (22) and motivates the redefinition of the metric to $\tilde{g}_{\mu\nu}$ in (21). Combining the redefinition with (28), we have

[TABLE]

to be evaluated at $\langle h\rangle=0$, which is a functional of $\tilde{g}_{\mu\nu}$ and must be made finite by subtraction and renormalization of $\alpha_{i}$ to obtain the contribution $\Delta\Gamma_{\rm ren}$ to (22). Similar previous applications of the composite operators, e.g., Refs. Becker:2019fhi and Houthoff:2020zqy , overlook the second term in parentheses because it is not clear there what the renormalized metric is, but we find that the metric is essential to eliminate the parameter $\lambda$. To reiterate: if the second term of (29) is not included the results depend on the scheme, even on-shell, and thus their physical meaning is unclear.

The difficult part of the computation of (29) is the trace involving the Green function. The second functional derivative $\delta^{2}{\cal R}_{2}$ is a differential operator with up to four covariant derivatives of the background metric that acts on the Green function ${\cal G}_{h}$. One can follow the same strategy as the covariant insertion that appeared in (14) and express ${\cal G}_{h}=\int_{0}^{\infty}\exp(-s{\cal O}^{\lambda}_{h})$. Thus the contribution reduces again to an insertion of a differential operator, $\delta^{2}{\cal R}_{2}$, acting on the heat kernel. The result admits an asymptotic expansion that schematically is

[TABLE]

not unlike (15), the coefficients of which are known Groh:2011dw ; Jack:1983sk . The discussion on how to regularize the above functional trace applies without change, so we continue $d$ in the leading power and extract the $1/\zeta$ poles.

As expected, the computation of (29) reveals a result that is expressed in a “redundant” basis, because, besides the dimension-two background curvatures, we have also introduced the dimensionful parameter $g_{0}$ from the original bare action (1). The rhs of (29) is thus expressed in a basis that includes contractions of the equations of motion $E[g]^{\mu\nu}$, which should be eliminated going on-shell, $E[g]^{\mu\nu}={G}_{\mu\nu}+\frac{g_{0}}{2g_{1}}{g}_{\mu\nu}=0$ (dropping the bar notation for ease of notation, this is our “quantum” equation of motion, that would receive corrections beyond one loop).

To make explicit the role of the equations of motion, we write the counterterm action in the limit $\langle h_{\mu\nu}\rangle=0$ in analogy with Eqs. (16) and (24) as

[TABLE]

where $B_{1}$ and $B_{2}$ are constants linear in $\alpha_{i}/g_{1}$, while $J_{2,\mu\nu}$ is a new source that differs from $J_{\mu\nu}$ of the previous section because it is dimension two, for example. The most important points, that we checked explicitly in our computation, are that the renormalization $J_{2,\mu\nu}$ depends on the parameter $\lambda$ because it is an off-shell quantity, instead the constants $B_{1}$ and $B_{2}$ are independent on the parametrization, implying that we can safely go on-shell even when renormalizing the composite operators.

It is natural at this stage to subtract the divergences $B_{i}$ by renormalizing the couplings $\alpha_{i}$ for $i=1,2$ in (24), and to subtract the divergence $J_{2,\mu\nu}$ by renormalizing $I_{2,\mu\nu}$, i.e., the couplings $\alpha_{i}$ for $i=3,4,5$, also in (24). Upon subtraction the renormalized couplings, which by abuse of notation we still denote with $\alpha=\{\alpha_{i}\}$, obey renormalization group equations linear in themselves, $\beta_{i}=\mu\partial_{\mu}\alpha_{i}=\gamma_{ij}\alpha_{j}=(\gamma\cdot\alpha)_{i}$, but the operators are subject to mixing through a $5\times 5$ matrix. The Callan-Symanzik equation for the renormalized expectation values $\langle{\cal R}_{2}\rangle_{\rm ren}$ is

[TABLE]

At the ultraviolet fixed point the critical properties of the spectrum come from the diagonalization of the matrix $\gamma$. We use the standard convention of expressing the scaling exponents from the analysis of the dimensionless couplings (in units of the renormalization group scale), ${\tilde{\alpha}}_{i}=\mu^{4-d}\alpha_{i}$, with beta functions ${\tilde{\beta}}_{i}={\tilde{\gamma}}_{ij}{\tilde{\alpha}}_{j}$.

The constants $B_{1}$ and $B_{2}$ are naturally interpreted as the renormalizations of $\alpha_{1}$ and $\alpha_{2}$, as we have done with Newton’s constant in the previous section, and they do not depend on the parametrization $\lambda$. The renormalized couplings $\{{\alpha}_{3},{\alpha}_{4},{\alpha}_{5}\}$, instead, have trivial beta functions, ${\tilde{\beta}}_{i}=(4-d){\tilde{\alpha}}_{i}$, corresponding to the fact that we eliminated three operators in the counterterm Lagrangian using the equations of motion. The corresponding scaling exponents are thus precisely what we expect from traces and contractions of the equations of motion, so we disregard them. The interesting beta functions are given by

[TABLE]

Even though the complete $5\times 5$ mixing matrix $\tilde{\gamma}$ is not block diagonal (there is mixing with the equations of motion), we have directly checked that its remaining two eigenvalues are not affected by the mixing of $\tilde{\alpha}_{1}$ and $\tilde{\alpha}_{2}$ with the redundant sector: we can thus extract the scaling exponents from the $2\times 2$ minor matrix $\hat{\gamma}$

[TABLE]

The minor is equivalent not only to remove rows and columns of the redundant couplings, but also to set the redundant couplings to zero from the very beginning, as would be required by a truly essential scheme. Substituting the value $\zeta=2-d$, the components of $\hat{\gamma}$ are

[TABLE]

with polynomials

[TABLE]

and the result has to be evaluated at the fixed point $G=G^{*}$ given in (19) to deduce the scaling properties in the uv.

The explicit expression of the eigenvalues and of the eigenvectors of $\hat{\gamma}$ for a general $d$ is not particularly illuminating, so we plot critical exponents as functions of $d$ in Fig. 1. It is interesting to notice that, in the physical case $d=4$, one of the exponents vanishes exactly because of the appearance of the Euler density which, on-shell, coincides with the squared Riemann tensor. The eigensystem of $\hat{\gamma}$ in $d=4$ reduces to the following combination of eigenvalues and eigendirections

[TABLE]

We deduce some important facts from the analysis of the critical properties. According to our perturbative analysis, asymptotic safety in $d=4$ dimensions is expected to have only one additional uv relevant direction taking the form of the $R^{2}$ operator, and an exactly marginal one which is the topological term. Most importantly, other operators do not contribute nontrivially to the spectrum, but rather scale trivially as one would expect from the equations of motion. In other words, asymptotic safety begins to be predictive (i.e., there are uv irrelevant directions) beyond the quadratic order in the curvatures.

III The special cases $\bm{d=2}$ and $\bm{3}$

It important to discuss the other integer values $d=2$ and $3$, because they are special limits that need to be treated with care. In fact, in these cases we have less than three independent contractions of the Riemann tensor with itself, so the basis given in (24) is redundant and we need some further manipulations to extract the actual critical exponents. In $d=2$ one has

[TABLE]

implying that the squares of both Riemann and Ricci tensors are proportional to $R^{2}$. Therefore, if we wish to extract the correct result from our computations, we need to take into account the fact that the essential couplings $\{\tilde{\alpha}_{1},\tilde{\alpha}_{2}\}$ are not independent. Since the equations of motion are themselves singular in $d=2$, the best strategy is to consider the general $d$-dependent result given in the previous section, and properly project to a linear combination of beta functions. For the projection, we are interested in the coupling $\tilde{\alpha}_{2}$, but must also take into account that the square of the Riemann tensor is proportional to $R^{2}$ in the limit. Consequently we have

[TABLE]

where on the rhs there are the general $d$-dependent beta function computed before. The complete mixing matrix $\gamma$ becomes $3\times 3$ and, as one might expect, in this limit the scaling of $\tilde{\alpha}_{2}$ becomes the classical one: $\theta=2$. The same remains true for $\tilde{\alpha}_{4}$ and $\tilde{\alpha}_{5}$ since we use the corresponding operators go on-shell. One might be surprised that the scaling of $\tilde{\alpha}_{4}$ does not vanish, as would be suggested by the fact that its conjugate operator becomes the topological $2$-dimensional Euler term. However, the equations of motions are discontinuous in this limit and imposing them for a generic $d$ hides the identification with the Euler term.

In $d=3$ there is one relation among the curvatures

[TABLE]

which implies

[TABLE]

Following the same strategy as the previous case, we can forget about the first coupling, $\tilde{\alpha}_{1}=0$, and the beta function for $\tilde{\alpha}_{2}$ becomes

[TABLE]

which also depends on the general $d$-dependent beta functions on the rhs. At the uv fixed point, we find the nontrivial critical exponent $\theta=-\frac{49}{108}$, while the rest of the spectrum has canonical scaling (in $d=3$ the canonical scaling of higher derivative operators is $-1$).

The analysis of both special cases confirms that, in the range $2<d<4$, it is the operator $R^{2}$ that naturally determines the dimensionality of the uv critical surface in the space of couplings. In particular, in agreement with the general plot, in $d=3$ the critical surface is two-dimensional while in $d=4$ it is three-dimensional. In the analytic continuation, the dimensionality of the critical surface changes somewhere above $d=3$. The fact that the $R^{2}$ operator is a uv relevant deformation of the critical surface is not new in the literature of asymptotic safety, however, our analysis clearly shows that the other directions are not part of an essential scheme that goes on-shell and further (ir)relevant deformations should be looked for in the spectrum of operators with more than four derivatives.

IV Conclusions

In this paper we have pushed forward the program initiated in Ref. (Martini:2021slj, , Sect. IV) of using dimensional regularization close to two dimensions to discuss the asymptotically safe fixed point of gravity above $d=2$. The technical environment is the one of a background field computation, performed covariantly thanks to heat kernel methods, that reveals the existence of a uv completion of standard Einstein-Hilbert gravity. The completion is based on an ultraviolet fixed point for Newton’s constant $G$ that exists, formally, as an expansion $G^{*}\sim(d-2)$. We have given indication on what the $d$-dependence of the fixed point is even outside the regime $d-2\ll 1$, thanks to the fact that, at least at the leading order in perturbation theory, $d$ can be continued in a way similar to the parameter $N$ of $SU(N)$ grauge theories. At face value the results suggest that in $d=4$ the fixed point exists and is scheme and gauge-independent, as confirmed independently by Ref. Falls:2017cze .

We are aware of the limitations of perturbation theory when applying the results to the physically interesting limit $d=4$, which is why we recommend our approach as one that integrates and complements the results coming from the use of Wilsonian methods Reuter:1996cp .

There are two main results of our analysis. The first is that we can deduce that a renormalization of the metric is necessary to interpret the renormalized effective action as a proper functional of the metric that is also independent of scheme and gauge. The renormalization is somewhat nontrivial and plays an important role for consistency of further applications, such as the computation of the rg equations of expectation values of composite operators. We informally refer to the renormalized metric as “the true quantum metric” of quantum Einstein’s gravity. This renormalization is a test bed for future directions, in particular the much needed extension to two loops of our present analysis, because the nonlinear renormalization of the metric is expected to play a crucial role when subtracting higher order dimensional poles (in a way similar to other nonlinear theories, such as nonlinear sigma models Howe:1986vm ; Osborn:1987au ).

The first result fully clarifies the counterterms structure of Ref. Martini:2021slj and consolidates the importance of going on-shell to prove the full scheme- and gauge-independence of the results. As far as we know, this is largely ignored in the Wilsonian-based literature of asymptotically safe gravity, because an off-shell effective action is deemed satisfactory for physical applications, however such Wilsonian action is manifestly gauge-dependent, making its careless application potentially suspicious. We would like to see this problem addressed in the Wilsonian framework, and we believe that obtaining an equivalent notion of true quantum metric is key.

The second main result of this paper is the analysis of higher derivative operators, quadratic in the curvatures, on the premise of the first result. The analysis is motivated by the need of comparing our approach with the Wilsonian ones, where the spectrum of (ir)relevant operators is generally provided, although often in some approximation, to establish the finiteness of the uv critical surface of the fixed point.

Several facts become clear through the analysis, but the most important ones are that the use of the true quantum metric is necessary to fully subtract the divergences and write everything in terms of renormalized quantities. Furthermore, the spectrum of quadratic operators needs to go on-shell too, as with the rest of the effective action. We find that there is only one physically interesting scaling operator, while off-shell approaches would find two or more, but scheme- and gauge-dependent. The scaling operator corresponds to the square of the Ricci scalar, which is not surprising if the geometrical properties of the square tensors are kept in mind.

One interesting aspect of the analysis is that the critical exponent is computed as a $d$-dependent function, using essentially the same method of continuing the dimensionality as done in gauge theories. If we trust the continuation from $d=2$ to $d=4$ of the result, which is of course an extrapolation of perturbation theory, we see that the critical exponent starts irrelevant in $d=2$ (the coupling has negative canonical dimension), but becomes relevant before $d=4$. This fact confirms the finding of the Wilsonian approaches and, we believe, increase their intrinsic value, as we now expect that the corresponding relevant $R^{2}$ deformation of the critical effective action is truly physical on-shell. All these considerations, in our eyes, give an added value to the past studies of $R+R^{2}$ truncations of the renormalization group.

Our work has two natural directions at this stage. On the one hand, it remains to be proven that our renormalization procedure continues to work beyond one loop (and especially at two loops, where some of its features would be really tested). In order for the procedure to work, we must be able to treat the dimensionality that is continued for dimensional regularization and the $d=g^{\mu}{}_{\mu}$ of spacetime independently in a consistent way. The composite operators are a sort of first test of this fact, given the nontrivial cancellations that occurred here. On the other hand, it would be interesting to expand our approach to cubic curvature operators, because recent work has pushed forward the idea that among all cubic operators there is a uv relevant direction in $d=4$ Kluth:2020bdv , so it would be promising to see this feature confirmed by our approach.

Acknowlegments.

We are grateful to D. Anselmi, K. Falls, H. Gies, D. Litim and R. Percacci for useful discussions.