Fixed Points of Quantum Gravity from Dimensional Regularisation

The Einstein-Hilbert action is not renormalisable in four-dimensional spacetime without the inclusion of additional counterterms [1, 2, 3]. Due to the negative mass dimension of Newton’s coupling, each order in a perturbative loop expansion generates ultraviolet (UV) divergences which cannot be subtracted by a renormalisation of the Einstein-Hilbert couplings alone. Within the traditional approach to quantum field theory (QFT), this means that the theory can only be formulated as an effective field theory below the Planck scale [4]. At energies above the Planck scale, predictivity breaks down since an infinite number of couplings needs to be measured by experiments.

One approach towards a predictive theory of quantum gravity beyond the Planck scale is asymptotic safety [5, 6, 7]. It stipulates the existence of a non-trivial fixed point of the renormalisation group flow. If its number of relevant directions is finite, the theory can be predictive even though an infinite number of operators has to be included in the gravitational action. This mechanism is sometimes also referred to as “non-perturbative” renormalisability. Much evidence for the existence of a non-trivial fixed point in quantum gravity has been found using the functional renormalisation group (FRG) [8, 9]. For recent reviews, see [10, 11, 12, 13, 14, 15, 16, 17]. Notably, these non-perturbative studies frequently observe near-perturbative properties [18, 19, 20], such as the Gaussian scaling of eigenvalues [21, 22, 23, 24]. This suggests that this fixed point may also be found within perturbation theory. In fact, dimensional continuation in $d=2+\varepsilon$ at one loop has provided some of the earliest evidence for the existence of a non-trivial fixed point in four-dimensional quantum gravity [25, 26, 27, 28, 29, 30, 31, 32]. However, this procedure faces subtleties due to the non-trivial pole structure of the graviton propagator in $d=2$ [13].

As an alternative to dimensional continuation in $d=2+\varepsilon$, perturbative quantum gravity can also be studied in $d=4$. However, results for fixed points seem to depend non-trivially on the chosen renormalisation scheme. The most commonly used renormalisation scheme in perturbation theory is dimensional regularisation (DR) with minimal subtraction (MS) [33]. Its ability to preserve symmetries, such as gauge invariance or diffeomorphisms, and its straightforward applicability to high order loop calculations [34, 35], have established DR as a central tool to study QFT. However, when DR with MS is applied to quantum gravity in $d=4-2\varepsilon$, a one-loop fixed point is absent.

The absence of a fixed point in DR with MS is attributed to the fact that (scheme dependent) power-law divergences are required to observe the fixed point in perturbation theory. In DR with MS, power-law divergences do not contribute to $\beta$-functions. Nevertheless, by employing renormalisation schemes that track power-law divergences, the non-trivial fixed point can be seen at one loop. Examples include momentum cutoffs, the background covariant operator regularisation, or one-loop approximations of the FRG [36, 37, 38, 39]. However, these regularisations present difficulties by breaking the underlying diffeomorphism symmetry [40, 17] or do not have appropriate generalisations at higher loop orders [41].

In this letter, we aim at studying quantum gravity within a perturbative renormalisation scheme that retains power-law divergences, and can be straightforwardly generalised to higher loop orders. For this, we employ the non-minimal power divergence subtraction (PDS) scheme [7, 42]. Even though this scheme is based on DR, it allows tracking the effects of power-law divergences in perturbative $\beta$-functions. This is done by requiring to subtract all divergences in $d\leq 4$. We exemplify the usage of PDS at the case of one-loop quantum gravity.

In this section, we want to review some basic implications of scheme transformations for the behaviour of $\beta$-functions and their properties. This is followed by a brief discussion of implications for perturbative studies of quantum gravity.

Consider a theory with $\beta$-functions $\beta_{i}$ and their corresponding couplings $g_{i}$. A scheme transformation can be represented by a map from the couplings $g_{i}$ to new couplings $\overline{g}_{i}$,

Since we will focus on perturbative renormalisation schemes below, we choose 1 such that it does not affect the limit of vanishing couplings. Using the chain rule, we can derive how $\beta$-functions transform,

As this expression shows explicitly, $\beta$-functions are not invariant under scheme transformations. However, 2 can be used to show the scheme invariance of various properties of $\beta$-functions. It is readily confirmed that zeros (infinities) of $\beta_{i}$ also imply zeros (infinities) for $\overline{\beta}_{i}$. This is true as long as scheme transformations are viable, i.e. $\text{d}\overline{g}_{i}/\text{d}g_{j}$ is finite. In this sense, the existence of fixed points or divergences is scheme invariant.¹¹1Note that the coordinates of zeros or infinities of $\beta$-functions generally change under scheme transformations. Using 2, it can also be shown that further quantities, such as critical exponents, are invariant. However, these invariance properties generally only hold for exact expressions. When approximations are used, they can be much more delicate.

In the case of perturbation theory, issues arise from the fact that 2 mixes different orders of the loop expansion. Therefore, 2 can only be explicitly verified when working non-perturbatively in the coupling. Thus, the invariance of critical exponents or the existence of fixed points may break down in perturbation theory. This does not immediately render perturbation theory useless to study non-trivial fixed points, but it emphasizes the importance of choosing an appropriate renormalisation scheme. In particular, we may expect that a fixed point, even if it exists in the physical theory, may converge in some, but not all perturbative renormalisation schemes. If the radius of convergence of a perturbative $\beta$-function is finite, this can be explicitly seen by the fact that scheme transformations change the radius of convergence.²²2Let us note that questions about the radius of convergence of perturbative $\beta$-functions are difficult to answer in general. On the one hand, one might expect them to be asymptotic, just as observables in QFT are expected to be asymptotic in perturbation theory [43]. On the other hand, there are explicit results for $\beta$-functions in the literature featuring a perturbative expansion with a non-zero radius of convergence [44, 45, 46]. The generality of such statements is further complicated by the possibility of performing scheme transformations.

This point is also relevant for perturbative studies of the fixed point structure in quantum gravity. While a fixed point can be identified at one loop in schemes that retain power-law divergences [36, 37, 38, 39], it does not appear in schemes where these divergences do not contribute to $\beta$-functions, such as DR with MS in $d=4-2\varepsilon$. Nevertheless, as the discussion above shows, this does not disprove the existence of a non-trivial gravitational fixed point. Instead, we must be aware that the convergence of such a fixed point cannot be guaranteed in any perturbative renormalisation scheme. Conversely, if a fixed point can be found in a given renormalisation scheme, it is necessary to observe its convergence towards higher orders before we can conclude that it is a genuine fixed point of the theory.

If power-law divergences are present, we may doubt that schemes like MS which simply ignore their effects on $\beta$-functions always lead to the best perturbative convergence. Instead, non-minimal renormalisation schemes may improve convergence by tracking theory information encoded in power-law divergences. For a perturbative investigation of the fixed point structure in quantum gravity, this suggests that we should employ such non-minimal renormalisation schemes as well. However, applying them to higher loop orders often results in significant challenges. Momentum cutoff regularisations, or loop expansions of the FRG, break diffeomorphism invariance and require the introduction of counterterms to restore the symmetry [40, 17]. Schwinger proper time cutoffs, or more generally the background covariant operator regularisation [36], do not violate symmetries in combination with the background field method [47, 48], but lack a higher loop generalisation [41]. Finally, dimensional continuation in $d=2+\varepsilon$ is plagued by subtleties due to poles of the graviton propagator in $d=2$ [13]. These issues highlight the motivation for focusing on a DR-based renormalisation scheme in $d=4-2\varepsilon$ that is sensitive to power-law divergences. Such a scheme naturally respects symmetries such as diffeomorphism invariance, and can be applied straightforwardly to higher loop orders, thus, enabling perturbative explorations of non-trivial fixed points in quantum gravity.

To understand the behaviour of power-law divergences in more detail, let us start by investigating them in MS with a cutoff regulator. For simplicity, we focus on the renormalisation of Newton’s coupling $G$ in one-loop quantum gravity. Regularising the theory with an UV cutoff $k$, the one-loop renormalisation of $G$ can be given in MS by

with $G_{0}$ the bare Newton coupling, $\mu$ the sliding scale, and $G$ and $\Lambda$ the renormalised Newton coupling and cosmological constant, respectively. The coefficients $B_{i}$ are numbers which can be determined from the structure of one-loop divergences. Note that 3 does not imply any effects for the $\beta$-function from the power-law divergence. Indeed, using $\tfrac{\text{d}}{\text{d}\log\mu}G_{0}=0$, we find

with $g=\mu^{2}G$ and $\lambda=m^{-2}\Lambda$ the dimensionless Newton coupling and cosmological constant, respectively. Only the logarithmic divergence $B_{2}$ has left a mark in this $\beta$-function, while the power-law divergence related to $B_{1}$ is set to zero. This is because power-law divergences in bare couplings do not have an explicit $\mu$-dependence in MS. Thus, they do not leave any imprints on the $\beta$-functions. In this way, MS sets all contributions of power-law divergences in $\beta$-functions to zero, even if the regulator shows an explicit power-law divergence.

Non-trivial contributions from power-law divergences in $\beta$-functions can be obtained using a non-minimal subtraction. An example is given by the Wilson inspired scheme used by Niedermaier at one-loop quantum gravity [38, 39]. In this scheme, we require that renormalised and bare couplings are equal when the sliding scale is set to the cutoff scale,

To ensure 5, the bare coupling must include an additional finite and $\mu$-dependent contribution,

These terms lead to a non-trivial contribution of power-law divergences in $\beta$-functions,

Thus, non-minimal renormalisation schemes are required to retain the effects of power-law divergences in $\beta$-functions, even if we regularise the theory using a cutoff regulator.

Let us now define a non-minimal renormalisation scheme based on DR which leads to non-trivial power-law divergences. Translating 6 to DR, we want to find a renormalisation scheme that fixes the ansatz

The crucial point is a requiring a non-trivial value for $\widetilde{B}_{1}$.³³3The $\mu$-dependent coefficient in front of $\widetilde{B}_{1}$ follows from dimensional analysis. A vanishing $\widetilde{B}_{1}$ corresponds to MS.

We can motivate a subtraction scheme that leads to a non-trivial 8 by analysing how power-law divergences can be identified in DR. Consider the integral

which is quadratically divergent in $d=4$. Setting $d=4-2\varepsilon$, we observe a $\tfrac{1}{\varepsilon}$-pole on the right-hand side of 9. This stems from a logarithmic UV divergence of the integral which is encountered by an expansion to first order in $m^{2}$. In contrast, the quadratic divergence is seen as a $\tfrac{1}{\epsilon}$-pole in $d=2-2\varepsilon$. This is a well-known and general feature of DR [49]: Logarithmic divergences lead to $\tfrac{1}{\varepsilon}$-poles when expanding in $d=4-2\varepsilon$, while power-law divergences lead to $\tfrac{1}{\varepsilon}$-poles when expanding in $d=n-2\varepsilon$, with $n<d$. Power-law divergences are encoded as divergences in lower dimensions.

This fact can be used to define a non-minimal renormalisation scheme in which power-law divergences generate non-trivial contributions to $\beta$-functions. The one that we follow here is given by:

This scheme can be thought of as a minimal extension of MS which captures power-law divergences. Such a scheme was suggested by Weinberg [7], but first applied in the context of nucleon-nucleon interactions in effective field theories under the term power divergence subtraction (PDS) [42]. The requirement of renormalising divergences in $d<4$ only leads to additional finite contributions in bare couplings when $d=4$. As such, it is a valid, non-minimal renormalisation scheme based on DR.

Let us discuss the application of PDS at the example of renormalising Newton’s coupling and the cosmological constant at one-loop. This requires an ansatz of the form,

The ansatz 11 contains one term for each possible UV divergence that can be encountered at a given loop order. The cosmological constant obtains logarithmic, quadratic, and quartic UV divergences. In DR, these are related to divergences in $d=4$, $d=2$, and $d=0$, respectively, generating three terms in the one-loop renormalisation of $\Lambda_{0}$ in 11. Newton’s coupling only receives logarithmic and quadratic UV divergences at one loop. These are related to divergences in $d=4$ and in $d=2$, giving rise to two terms in 11. Each coefficient in 11 can be fixed by computing UV divergences in the corresponding dimensions.

Using the ansatz 11, we find for the running couplings,

where we have used the dimensionless Newton coupling as $g=\mu^{d-2}G$, and the dimensionless cosmological constant as $\lambda=\mu^{-2}\Lambda$. Thus, following PDS we obtain $\beta$-functions that pick up non-trivial contributions from power-law divergences. Below, we will fix the coefficients $A_{i}$ and $B_{i}$ using a one-loop computation and analyse the implications.

Let us make some remarks concerning the generalisation of PDS to higher loop orders. Consider PDS at $L$ loop with power-law divergences of degree $N$. These power-law divergences generate $\tfrac{1}{\varepsilon}$-poles when $d=d_{\text{crit}}-2\varepsilon$. We call $d_{\text{crit}}$ the critical dimension. This critical dimension depends on the degree of divergence, and the loop order. Using dimensional analysis, it can be shown that

Note that the critical dimension shifts at each loop order for power-law divergences. Only logarithmic divergences ($N=0$) are consistently found as divergences in the same dimension, namely at $d=4$.

In renormalisable theories, the critical dimension approaches $d_{\text{crit}}\to 4$ at high loop orders. This is because such theories have an upper bound on $N$. In non-renormalisable theories, this is in general not the case. Considering quantum gravity, the strongest divergences depend on the loop order — they are power-law divergences of order $N=2L+2$. Using this in 13, we find

Thus, the strongest divergences in quantum gravity approach $d_{\text{crit}}^{\text{QG}}\to 2$ at infinite loop order.

A technical complication arising at higher loop orders is the renormalisation of subdivergences. An elegant way to deal with them is the incomplete $R$-operation $R^{\prime}$ [50], which is also necessary for an efficient utilisation of the background field method at higher loop orders [1, 3]. To adapt the $R^{\prime}$ operation for PDS, only minor modifications have to be considered that we review now.

Considering a graph $G$ and all of its subgraphs $G_{i}$, the $R^{\prime}$-operation is defined recursively by

The sum runs over all possible combinations of subgraphs $G_{i}$ of $G$, and $G\,\backslash\left\{G_{1},\dots,G_{m}\right\}$ denotes the graph $G$ without the subgraphs $G_{i}$. In MS, the operator $\mathcal{K}$ gives the divergence of its argument in $d=4-2\varepsilon$,

with div defined such that

i.e. it extracts the divergent terms from a Laurent series in $\varepsilon$. To employ PDS, the operator $\mathcal{K}$ needs to be modified. It does not only need to return the divergence in $d=4$, but divergences in all critical dimensions $d_{\text{crit}}$. For that purpose, we define,

where $L$ is the loop order of the diagram $G$, and $N_{\text{max}}$ the strongest divergence that can be generated by $G$. This allows for $\mathcal{K}$ to pick up all relevant UV divergences in $d\leq 4$. In this way, PDS can be combined with the $R^{\prime}$-operation to subtract all subdivergences at higher loop orders. With this in mind, PDS can be applied straightforwardly to higher loop orders, in particular also in combination with the background field method in quantum gravity.

In this section, we apply PDS to quantum gravity at one loop. The gravitational action that we consider is given by

Using a harmonic gauge fixing in the background field method [51], we compute one loop divergences from the one-loop effective action $\Gamma^{(1)}$. This computation relies on the graviton and ghost propagators. As described in App.​ A, we use a non-trivial procedure to dimensionally continue the graviton propagator away from $d=4$. This modification allows us to avoid a singular graviton propagator in $d=2$. Following App.​ A, we have

where $\mathcal{H}$ is the dimensionally continued version of the graviton Hessian fulfilling

For an explicit definition of $\mathcal{H}$, see 31.

Following PDS and the structure of one-loop UV divergences encoded in the ansatz 11, we compute the divergences of 20 in $d=4$, $d=2$, and $d=0$. The one-loop divergences in $d=4-2\varepsilon$ are given by

with $E=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^{2}$ the topological Euler density. These are the (scheme independent) logarithmic divergences.

The quadratic divergences computed in $d=2-2\varepsilon$ take the form,

Contrary to the logarithmic divergences, these are scheme dependent. In particular, specific choices of our scheme, such as the dimensional continuation of the propagator described in App.​ A affect the result in 23.

In $d=-2\varepsilon$ we do not find any divergences,

This can be understood by writing the traces in 20 in terms of heat kernels. Denoting the Seeley-DeWitt coefficients that enter the traces by $A_{i}$, the divergences in $d=0$ are captured by $A_{0}$, see also 41. The trace of $A_{0}$, which enters the one-loop effective action 20, is given by $\text{Tr}\left\{A_{0}\right\}=\frac{d(d+1)}{2}$ for the graviton, and by $\text{Tr}\left\{A_{0}\right\}=d$ for the ghost. Note that both are proportional to a factor of $d$. It is this factor that cancels the UV divergence encountered in $d=0$. While the UV divergence results in a pole of the form of $\tfrac{1}{d}$ from a Schwinger integral 41, the trace of $A_{0}$ cancels the factor of $d$ in the denominator, leading to a finite result.

Note that 24 is not affected by altering the dimensional continuation of the Hessian $\mathcal{H}$. Since $A_{0}$ is given by the identity element of the field space, it is independent of any endomorphisms included in $\mathcal{H}$. Moreover, prefactors in front of the differential operator in $\mathcal{H}$ drop out due to the logarithm in 20. Unless we define a dimensional continuation of the propagator which completely alters the form of the differential operator, such as making it non-minimal, the result 24 seems to be hard-wired in PDS. In this context, it is interesting to note that similar conclusions about the absence of quartic divergences have been made in [52, 53].

We now use the results 22, 23 and 24 to determine the coefficients in our ansatz for the bare couplings 11. Since the Ricci tensor vanishes on-shell, and $E$ is a topological invariant, we will discard the divergences related to quadratic curvature invariants here. They could be taken into account, for example, using field redefinitions [54, 55]. However, for brevity, we will simply neglect these contributions instead.⁴⁴4Following from the vanishing of quartic divergences in 24, one can show that the minimal essential scheme of [55] simply corresponds to keeping $\lambda=0$ in 26. In particular, the divergences related to quadratic curvatures do not affect the $\beta$-function for Newton’s coupling. We find

This gives rise to the $\beta$-functions

It is worth noting that $\beta_{\lambda}=0$ for $\lambda=0$, independently of the value of Newton’s coupling $g$. This originates from the vanishing of quartic divergences, see 24. If the quartic divergences were non-trivial, there would be an additional contribution to $\beta_{\lambda}$ of the form of $ag$, with $a$ a number. In that case, the running of Newton’s coupling would immediately imply a running for the cosmological constant, and it would be impossible to keep $\lambda=0$ fixed for any value of $g$.

The $\beta$-functions 26 feature a total of three fixed points shown in Tab.​ 1. This includes the Gaussian fixed point $\textbf{FP}_{\text{Gauss}}$ located at vanishing couplings, a non-trivial UV fixed point $\textbf{FP}_{\text{UV}}$, and an unphysical fixed point $\textbf{FP}_{\text{unphys}}$ located at negative Newton coupling. The fixed point $\textbf{FP}_{\text{UV}}$ features a vanishing cosmological constant, and a positive Newton coupling of $g\approx 0.0968$. The fact that $\lambda=0$ is closely related to the vanishing of the quartic divergence in 24. Due to that, $\lambda=0$ is a fixed point of $\beta_{\lambda}$ and the fixed point value of $g$ only has to lead to a vanishing of $\beta_{g}$.

Both critical exponents of $\textbf{FP}_{\text{UV}}$ are positive, meaning that the fixed point has two relevant directions. They are also real, with the cosmological constant giving a critical exponent of $\theta_{0}\approx 2.839$, while the critical exponent related to Newton’s coupling is given by $\theta_{1}=2$. Note that $\theta_{1}$ is fixed by the fact that we perform a one-loop computation and observe a vanishing cosmological constant at the fixed point. This dictates $\theta_{1}=2$. Compared to the literature, both values for the critical exponents fall within expected results from non-perturbative computations. However, note that these critical exponents are usually seen as complex conjugate pairs. For a discussion on this topic, see [21]. A disentanglement into two real exponents is sometimes observed in cases where the cosmological constant decouples in some way [56, 57, 55, 58]. In our case, a similar effect takes place in the form of a vanishing quartic divergence for the cosmological constant in 24.

In Fig.​ 1 we show the phase diagram of the $\beta$-functions in 26 in the physical region $g>0$. Note that we have translated the couplings $\{\lambda,g\}$ into the set $\{\lambda g,g\}$. As discussed above, the non-trivial fixed point $\textbf{FP}_{\text{UV}}$ has two relevant directions. Thus, all trajectories in its vicinity end up in $\textbf{FP}_{\text{UV}}$ in the UV. Trajectories flowing out of $\textbf{FP}_{\text{UV}}$ can end up in two different regions. In case they flow towards negative cosmological constant, they show some intermediate scaling, but end up at $\lambda g=g=0$ in the IR. This does not imply that the cosmological constant vanishes in the IR for these trajectories. In fact, the only trajectory connecting $\textbf{FP}_{\text{UV}}$ to a vanishing cosmological constant in the IR is the separatrix on which the cosmological constant vanishes all along. All other trajectories flowing to $\lambda g=0$ in the IR actually have a diverging (negative) cosmological constant in the IR. Trajectories flowing out of $\textbf{FP}_{\text{UV}}$ towards positive cosmological constants lead to $\lambda g$ growing towards large non-perturbative values in the IR.

Note that the existence of $\textbf{FP}_{\text{UV}}$ implies a cancellation between the one-loop and the tree-level contributions to $\beta_{g}$. As such, care must be taken when interpreting its physical significance. However, the fact that its properties resemble non-perturbative results indicate that it might be related to a true physical fixed point of quantum gravity. Moreover, the smallness of Newton’s coupling suggests that the fixed point might be perturbative enough to converge once higher order loop corrections are taken into account. However, it should be noted that this argument is only valid if the coefficients of higher order loop corrections in the $\beta$-functions do not grow too rapidly. With this in mind, a computation of higher loop coefficients with PDS would be of great interest.

Finally, let us point out that $\textbf{FP}_{\text{UV}}$ is induced entirely by the UV divergence related to $B_{1}$ in 11. This is due to the fact that the cosmological constant vanishes. As a consequence, the UV behaviour of $\textbf{FP}_{\text{UV}}$ is solely determined by divergences encountered in $d=2$.

In this letter, we have employed dimensional regularisation without minimal subtraction, such that we can account for non-trivial power-law divergences. Such divergences arise naturally in theories with dimensionful couplings, in particular, perturbatively non-renormalisable theories. We have argued that retaining power-law divergences can be beneficial to capture key properties of such theories. For any $d$-dimensional theory, this can be done by subtracting all divergences in dimensions $\leq d$. This scheme is called PDS [7, 42] and preserves underlying global or gauge symmetries, as well as diffeomorphisms. Moreover, it is conceivable that this procedure improves the convergence of perturbative expansions.

We have applied our setup to four-dimensional quantum gravity at one loop. The resulting $\beta$-functions in 26 give rise to a non-trivial fixed point suitable for the asymptotic safety scenario, see Tab.​ 1. Moreover, its properties are consistent with results obtained from cutoff regularisations [9, 37, 38, 39, 16, 17, 11]. While such schemes usually face difficulties at higher loop orders [41] or break underlying symmetries of the theory [40, 17], our setup is free from such subtleties. Furthermore, it is noteworthy that our critical exponents are real rather than complex conjugate pairs [21].

An interesting feature of our results is the impact played by UV divergences in $d=2$. Using 14, we have found that the strongest divergences in quantum gravity approach $d\to 2$ in the infinite loop limit. In this sense, the strongest UV divergences are associated with two dimensional spacetime. Moreover, in the one-loop computation we have found quartic divergences to be absent. As a consequence, the cosmological constant vanishes at the fixed point and the fixed point is entirely determined by divergences in $d=2$. These observations could be interpreted as ramifications that quantum gravity becomes two-dimensional in the UV. This conjecture has been observed across several different approaches to quantum gravity [59].

Our results for one-loop quantum gravity ask for extensions to higher loop orders, particularly to assess the convergence of the non-trivial fixed point. The smallness of Newton’s coupling suggests that this might be possible. A first step towards this is the computation of quantum gravity at two loop. With a recent computation suggesting that the effects of the two-loop counterterms may be only marginal [20], we could expect convergence of our results as well. The two-loop computation would also give rise to a non-trivial critical exponent for Newton’s coupling for the first time. This could be compared to non-perturbative results and provide another benchmark test for the use of PDS in quantum gravity.

More generally, we note that applications of PDS have been focussed on effective field theories, in particular, nuclear interactions [60, 61, 62, 63]. However, as we have shown here, the effects of power-law divergences may also be relevant for other non-renormalisable field theories. Apart from pure quantum gravity, examples include the non-linear sigma model [64, 65, 66, 67], four-fermi theories [68, 69, 70, 71, 46], non-abelian gauge theories in $d>4$ [72], or gravity-matter systems [40, 73, 15]. In the context of gravity, the perturbative approach presented here could also be useful to study the scattering of gravitons within asymptotic safety [74, 75, 10].

On a different tack, it would also be valuable to explore the implications of other non-minimal renormalisation schemes. Since PDS is just one out of many schemes of dimensional regularisation that retains the effects of power-law divergences, we could consider modifications to enhance perturbative convergence. This could be achieved using independent optimisation criteria, such as the principle of minimal sensitivity [76]. Better understood theories, such as four-fermi theories could act as valuable toy models to test such ideas.

I would like to thank Gabriel Assant, Kevin Falls, Daniel Litim, and Peter Millington for helpful discussions and Daniel Litim and Peter Millington for comments on earlier versions of this manuscript. This work was supported by a United Kingdom Research and Innovation (UKRI) Future Leaders Fellowship [Grant No. MR/V021974/2].

No new data were created or analysed in this study.