The continuum limit of quantum gravity at first order in perturbation theory

The Wilsonian renormalization group (RG) properties of the conformal factor of the metric are profoundly altered by the fact that it has a wrong-sign kinetic term. The result is a novel perturbative continuum limit for quantum gravity, which is however non-perturbative in $\hbar$. The ultraviolet part of the renormalized trajectory lies outside the diffeomorphism invariant subspace, entering this subspace only in the infrared, below a dynamically generated amplitude suppression scale. Interactions are dressed with coefficient functions of the conformal factor, their form being determined by the RG. In the ultraviolet, the coefficient functions are parametrised by an infinite number of underlying couplings. Choosing these couplings appropriately, the coefficient functions trivialise on entering the diffeomorphism invariant subspace. Here, dynamically generated effective diffeomorphism couplings emerge, including Newton's constant. In terms of the Legendre effective action, we establish the continuum limit to first order, characterising the most general form of such coefficient functions so as to verify universality.


Introduction
In this paper we develop further the perturbative continuum limit of quantum gravity begun in refs. [1][2][3][4]. The theory is perturbative in κ ∼ √ G, the natural coupling constant (where G is Newton's coupling), but non-perturbative in . It is the logical consequence of combining the Wilsonian RG (renormalization group) with the action for free gravitons, while respecting the wrong-sign kinetic term that then naturally appears in the conformal sector. Although this renders the partition function meaningless without further reworking [5], the Wilsonian RG remains well defined and provides us with an alternative and actually more powerful route to defining the quantum field theory. As such it then has all the usual desired properties (locality, microcausality, unitarity, gauge invariance etc.) built in. Nevertheless what we are led to is something conceptually different from all other approaches to quantum gravity, and indeed a construction crucially different from all other constructions of quantum field theories. invariance for Λ > aΛ p , where Λ p is a characteristic of the renormalized trajectory and is called the amplitude suppression scale (or amplitude decay scale), and a is a non-universal number. By appropriate choice of the underlying couplings g σ n , diffeomorphism invariance is then recovered at scales Λ, ϕ Λ p where also we recover an expansion in the effective coupling κ ∼ √ G.
The basic structure of this continuum limit is illustrated in fig. 1.1, where we sketch the 'theory space' of effective actions. In order to implement the Wilsonian RG structure one introduces a physical cutoff Λ which sets the scale down to which modes are integrated out and allows us to define the Wilsonian effective action at this scale. Naïvely one thinks of this cutoff as breaking the diffeomorphism invariance. However the Slavnov-Taylor identities get replaced by modified Slavnov-Taylor identities (mST), "Σ = 0", that reduce to the usual ones in the limit that we integrate out all the modes [6,7]. This limit is Λ → 0, which is the limit we need, in order to compute the desired physical observables. Now, because the mST are compatible with the flow equation, if the effective action enters this "diffeomorphism invariant" theory subspace at some (finite) scale Λ, i.e. such that Σ vanishes there, it never leaves this subspace, and physical quantities are then guaranteed to be diffeomorphism invariant.
So far so standard. However, what we find is that the ultraviolet fixed point which supports the continuum limit (at Λ → ∞ and which for us is just the Gaussian fixed point, hence perturbatively describable), is located outside the diffeomorphism invariant subspace, so that interactions constructed from the relevant operators cannot be made to satisfy Σ = 0 there. Instead, by appropriate choice of the associated couplings g σ n , the renormalized trajectory joins the diffeomorphism invariant subspace in the limit as Λ Λ p (and also the conformal mode must have amplitude ϕ Λ p ) where Λ p is a dynamically generated scale determined by the underlying couplings, called the amplitude suppression scale [1]. Equivalently, in the limit in which this new scale Λ p → ∞, we have Σ → 0 and diffeomorphism invariance is recovered. Here we recover Newton's constant as another dynamically generated scale determined by these underlying couplings, and as we'll see also the cosmological constant. At higher orders in perturbation theory other such diffeomorphism invariant effective couplings may potentially arise [8].
Let us emphasise that this structure follows inevitably from imposing the principles of the Wilsonian RG about the Gaussian fixed point, while taking seriously the consequences of the wrong-sign kinetic term in the conformal sector and requiring that in physical amplitudes we recover diffeomorphism invariance [4]. It is therefore well grounded and indeed thus may not seem so different from the usual picture. However all other quantum field theories have Wilsonian RG flows that can be defined within the gauge invariant subspace. For example for (non-Abelian) gauge theories the continuum solution can be chosen to respect the corresponding Σ = 0 identities at all scales, e.g. [7], in fact the gauge invariance can even be manifestly respected through e.g. lattice regularisation [9] or directly in the continuum (e.g. [10][11][12]).
In all other approaches to quantum gravity, it has been assumed that the Wilsonian RG properties defining the continuum limit, and the diffeomorphism gauge invariance, can coexist in the same region of the renormalized trajectory. This tacit assumption for example lies behind intuitive arguments against the existence of an ultraviolet fixed point in quantum gravity, based on black hole entropy considerations [13,14]. We see that these arguments are actually inapplicable in this case. To put it pithily, such tensions in quantum gravity are resolved since a crucial element of quantum gravity is constructed off space-time. This is to be contrasted with classical General Relativity which is a construction of space-time, and with normal quantum field theories which are constructed on space-time.
There are other important properties, which are key to a complete understanding of fig. 1.1, especially the fact the operators are not those of the usual expansion but non-polynomial in ϕ, that infinitely many of these are relevant, that the expansion in terms of these operators actually only makes sense at scales above aΛ p , and that flows in the conformal sector go in the reverse direction (from infrared to ultraviolet). In ref. [4] we highlighted how these novelties lead to differences that need careful treatment. These include differences in limiting procedures, in particular ultraviolet divergences are now absorbed by the underlying couplings, while at low scales outside the diffeomorphism invariant subspace new infrared divergences appear [1]. For these reasons, in this paper we develop further the properties at first order, and provide a tight characterisation of the most general form of the continuum limit at this order as needed for the higher order computations.
The first order continuum limit was formulated in ref. [4] in terms of the Wilsonian effective action and a regularised Quantum Master Equation.
Although this allows for an elegant analysis since the latter effectively leaves BRST invariance unmodified, in sec. 2 we switch to an equivalent [15,16] description in terms of the infrared regulated Legendre effective action [15,17,18] and the mST [6,7]. Although more cumbersome, this then gives us direct access to the one-particle irreducible amplitudes in the physical limit, and leads to useful simplifications at higher orders [1,8,19]. Furthermore it can still be solved in terms of the total free quantum BRST chargeŝ 0 [7] that naturally incorporates a regularised Batalin-Vilkovisky measure operator ∆ [26,27].
In secs. 2 and 3 we review the development of this free BRST algebra and how computations can be couched in minimal gauge invariant basis [7]. Then in sec. 3.1, we choose the first order non-trivial quantum BRST cohomology representative on which to build the continuum limit to first order (in the new quantisation these two are not the same). In order to simplify the higher order computations [8], we choose one that corresponds to expressing diffeomorphism invariance as a Lie derivative, and demonstrate that this differs from the previous choice [4,31] by anŝ 0 -exact piece, such that the regularised measure term ∆ provides a contribution crucial for consistency.
In sec. 4 we review how the wrong sign kinetic term in the conformal sector profoundly alters RG properties that are central to defining the continuum limit, however framing the discussion now in terms of the Legendre effective action. In particular we recall how this leads to all interactions σ being dressed with a coefficient function f σ Λ (ϕ). This latter is parametrised by the underlying couplings g σ n . At the linearised level only those couplings of non-negative mass dimension must be non-vanishing. Here we work with the most general such coefficient functions that are consistent with the RG properties as determined by the flow equation, and such that the renormalized trajectory enters the diffeomorphism invariant subspace as sketched in fig. 1.1. We do so in order to verify the universality of this continuum limit, here at first order, and later at higher orders [8,19].
We tighten and further develop the arguments from refs. [1,4], that show how the RG properties determine the form of the dressed interactions and their coefficient functions. In doing so, we demonstrate once again that these results follow inevitably from combining the Wilsonian RG and the Gaussian fixed point action for free gravitons, after taking seriously the consequences of the resulting wrong-sign kinetic term in the conformal sector. In particular we give closed expressions for the tadpole corrections appearing in the dressed interactions, prove that there exists a dynamically generated amplitude suppression scale Λ σ that determines the large ϕ behaviour of each coefficient function f σ Λ (ϕ) for all Λ ≥ 0 and prove that f σ Λ (ϕ) itself is determined uniquely by its physical limit. Finally we show that these are given in conjugate momentum space by an entire function f σ (π) whose Taylor expansion coefficients are the underlying couplings g σ n . In sec. 4 and sec. 5 we show that in turn the amplitude suppression scale characterises the asymptotic behaviour of the underlying couplings g σ n at large n. In sec. 5 we define what it means for the coefficient functions to trivialise in the the large Λ σ limit. From ref. [4] we know that the underlying couplings must be chosen so that this trivialisation happens, in order to enter the diffeomorphism invariant subspace at the linearised level. In the simplest case this means that the coefficient function must tend to a constant in this limit; more generally we show that it must tend to a Hermite polynomial of degree α, whose functional form is then fixed.
In sec. 5.1 we show how to derive new solutions for coefficient functions from a given one, and derive formulae for their underlying couplings, either by multiplying the physical coefficient by a power of ϕ or by differentiating with respect to ϕ. These tricks prove useful later.
Then in sec. 5.2 we characterise the most general form of coefficient functions that trivialise in the large Λ σ limit. This is most efficiently expressed in terms of their Fourier transform. In particular we show that f σ (π) must tend to (the α th derivative of) a Dirac δ-function. We make two powerful simplifying assumptions which still leave us with an infinite dimensional function space of solutions flexible enough to encompass the higher order computations. Firstly we specialise to coefficient functions that have definite parity (are even or odd functions). Secondly we insist that at the linearised level the coefficient functions contain only one amplitude suppression scale. 1 Putting all these properties together, allows us to give a complete characterisation of f σ (π) in terms of its large and small π behaviour, its normalisation, and limiting behaviour of key integrals at large Λ σ .
In particular we use this to characterise the approach to the trivialisation limit. In sec. 5.3 we verify all these general properties on a series of instructive examples.
Finally in sec. 6 we construct a very general continuum limit to first order, and verify that its renormalization group trajectory fulfills the properties sketched in fig. 1.1. 1 However in app. A, we also develop their properties when there is a spectrum of amplitude suppression scales.
2 Legendre effective action, mST, and quantum gravity We begin by briefly recalling some key steps from refs. [1,4,7]. This will also serve to set out our choice of notation and formulation for this paper. In ref. [4], we worked with the continuum Wilsonian effective action. Here we will work directly with the renormalized infrared cutoff Legendre effective action Γ, which is also in fact the one-particle irreducible part of the continuum Wilsonian effective action [15]. However it will mean that BRST invariance is no longer expressed as unbroken through the Quantum Master Equation but rather through modified Slavnov-Taylor identities (mST) [6,7], so that we recover (off-shell) nilpotency at the interacting level, only in the limit Λ → 0. The free charges are still nilpotent however, and it is their cohomology that is central to solving for the effective action [7]. In any case the loss of some elegance is outweighed by the advantages: the simplification that comes from not computing also the one-particle reducible parts and especially the fact that the limit then gives us direct access to the physical amplitudes: (2.1) The flow equation for the interacting part thus takes the form [15,17,18] (see also [16,[20][21][22][23]): where the over-dot is ∂ t = −Λ∂ Λ . The BRST invariance is expressed through the mST [6,7]: These equations are both ultraviolet (UV) and infrared (IR) finite thanks to the presence of the UV cutoff function C Λ (p) ≡ C(p 2 /Λ 2 ) which, since it is multiplicative, satisfies C(0) = 1, and its associated IR cutoff C Λ = 1−C Λ , which appears in the IR regulated propagators as AB The cutoff function is chosen so that C(p 2 /Λ 2 ) → 0 sufficiently fast as p 2 /Λ 2 → ∞ to ensure that all momentum integrals are indeed UV regulated (faster than power fall off is necessary and sufficient).
It is also required to be smooth (differentiable to all orders), corresponding to a local Kadanoff blocking. It thus permits for Λ > 0, a quasi-local solution for Γ I , namely one that has a space-time derivative expansion to all orders. We insist on this: it is equivalent to imposing locality on a bare action.
The two equations are compatible: if Σ = 0 at some generic scale Λ, it remains so on further evolution, in particular as Λ → 0. The second term in (2.3) is a quantum modification due to the cutoff Λ > 0. At non-exceptional momenta (i.e. such that no internal particle in a vertex can go on shell) it remains IR finite, and thus vanishes as Λ → 0, thanks to the UV regularisation. We are then left with just the Zinn-Justin equation 1 2 (Γ, Γ) = 0 [24,25], which gives us the standard realisation of quantum BRST invariance through the Slavnov-Taylor identities for the corresponding vertices.
Here Φ and Φ * are the collective notation for the classical fields and antifields (sources of BRST transformations) respectively, while Γ is the "effective average action" [18] part of the infrared cutoff Legendre effective action [7,15]: where R AB is the infrared cutoff expressed in additive form. Γ is expressed in terms of a free part, Γ 0 , which includes the free BRST transformations, plus the interaction part Γ I [Φ, Φ * ]: Note that the free part carries no regularisation. The antibracket in the mST is similarly expressed without regularisation. For arbitrary functionals of the classical (anti)fields, it is given by Notice that in Γ 0 we have chosen left-acting BRST transformations [4] (see also app. A2 of [7]) so that the free BRST transformation is given by the first of the following equations: Here we have taken the opportunity also to define the free Koszul-Tate operator Q − 0 . We will be interested in expanding Γ I perturbatively in its interactions, assuming the existence of an appropriate small parameter : Importantly, in the quantisation established in [1,4], we need however to work non-perturbatively in , so there will be no loop expansion. In the above, is a formal perturbation-order counting parameter, which we set to = 1 at the end. The actual small physical parameter, (where G is Newton's gravitational constant) will properly make its appearance in the theory only in sec. 6, where it arises as a collective effect of all the underlying couplings.
At first order (2.2) and (2.3) becomė where the first equation is the flow equation satisfied by eigenoperators: their RG time derivative is given by the action of the tadpole operator [4]. In the second equation we recognise that we recover the Batalin-Vilkovisky measure operator [26,27]: UV regulated as in refs. [4,7], and we have defined the corresponding full free quantum BRST chargeŝ 0 . Note that ∆ thus generates Λ-dependent tadpole integral corrections to the full free classical BRST transformations. Thanks to compatibility, these corrections are as required in order to find simultaneous solutions of (2.11) and (2.12). Indeed, as shown in [4], theŝ 0 -cohomology can then be defined within the space spanned by the eigenoperators with constant coefficients (a.k.a. couplings).
In this paper, any explicit expression for an action functional should be understood as integrated over four flat Euclidean spacetime dimensions and determined only up to integration by parts. As we will explain shortly, we can in effect work in minimal gauge invariant basis [7] where is the action for free graviton fields H µν , plus the fermionic antifield H * µν source term for the only non-vanishing free linearised BRST transformation in this basis, cf. (2.6), c µ being the (fermionic) ghost fields. Contraction is with the flat metric δ µν , and we write ϕ = 1 2 H µµ . Since raising an index makes no difference we will usually leave all indices as subscripts.
We note in passing that (2.14) is also the action one gets from the Einstein-Hilbert Lagrangian if one expands the metric as Similarly the invariance (2.15) follows from expanding diffeomorphisms (regarding κc µ as the small diffeomorphism).
The only extra (anti)field we will need is the bosonic c * µ , the source for BRST transformations of c µ that will appear at the interacting level. From (2.8) and (2.14), the non-vanishing free Kozsul-Tate differentials are: µν is the linearised Einstein tensor: the linearised curvatures being 2 It is evident that the transformations (2.18) are invariances of the free action (2.14), the former by the linearised Bianchi identity and the latter trivially so.
In order to derive the propagators, which are used in both the flow equation (2.2) and the mST (2.3), we need to introduce gauge fixing. To do this we first extend to the non-minimal basis by adding the bosonic auxiliary field b µ that allows off-shell BRST invariance, andc * µ which sources BRST transformations of the antighostc µ . Then the free effective action is written as [4]: where α is our gauge fixing parameter. Gauge fixing is implemented by a finite quantum canonical transformation [28,29] that takes us to gauge fixed basis Φ * The free action in gauge fixed basis is therefore: It has kinetic operators that can be inverted. The H µν propagator simplifies in "Feynman gauge" α = 2, which as in ref. [4] we set from now on. Splitting H µν into its SO(4) irreducible parts, (thus h µ µ = 0 is traceless), in this gauge the two parts decouple. The propagators we need are where we have written Note that h µν propagates with the right sign, and that the numerator is just the projector onto traceless tensors, while ϕ propagates with wrong sign.
There is a propagator involving b α [4] but it is not needed. Indeed, we will later confirm that the first order interaction Γ 1 can be constructed just from the minimal set. Then in gauge fixed basis, Γ 1 still does not depend on b µ orc * µ and will depend onc µ only through the combination on the right hand side (RHS) of (2.22). By iteration, using the flow equation (2.2), these properties are inherited by all the higher order interactions Γ n>1 . Mapping back to gauge invariant basis using the equations above, we therefore see that Γ I will not depend on b µ ,c * µ orc µ . This means in particular that the full Γ I remains in minimal gauge invariant basis.
Therefore we can most simply express the calculation in this basis [7] as we will do from now on. What this means is that when we compute corrections from the flow equation (2.2) or from the quantum correction part of the mST (2.3), we temporarily make the shift (2.22), which in particular then allows corrections computed using the ghost propagator (2.27), after which we absorb the antighost by shifting back to minimal gauge invariant basis using the inverse of (2.22).
Notice that since the transformation is canonical, it has no effect on the antibracket part of the mST (2.3) which thus can be computed whilst remaining in (minimal) gauge invariant basis.

Free quantum BRST cohomology
Following Henneaux et al [30], we can simplify finding solutions of theŝ 0 -cohomology by splitting the problem up (a.k.a. grading) by antighost, a.k.a. antifield, number. We thus have the weights where the first entry is the ghost number, the second entry the antighost/antifield number, and the final entry the mass dimension. (A full table of weights is given in ref. [4].) Thus all parts ofŝ 0 increase ghost number and mass dimension by one. While ghost number and mass dimension are respected, antighost number is not, but it is chosen so that the free BRST charges have definite antighost number. We anticipated this with our labelling: Q 0 leaves antighost number unchanged, while Q − 0 lowers it by one. Under this grading, the measure operator splits into two parts that lower antighost number by one or two respectively (∆ − simplifies to this in minimal basis [4]): The point of this extra grading is that Γ itself does not have definite antifield number but splits into parts of definite antifield number n: Γ = n=0 Γ n . This means that an (integrated) operator O = n m=0 O m with some maximum antighost number n, that is annihilated byŝ 0 , must satisfy the descent equations: Starting with the top (left-most) equation, these are often easier to analyse than trying to work withŝ 0 O = 0 directly. Grading the square we also have the useful identities [4,7]: 4

Non-trivial free quantum BRST cohomology representatives
As we will review in sec. 6, our choice of non-trivialŝ 0 -cohomology representative,Γ 1 , will lead us to the solution for the first order interactions Γ 1 . (The latter is not simply κΓ 1 as it would be in standard quantisation [4].) In order to get a theory that is consistent with unitarity and causality, we restrictΓ 1 to have a maximum of two space-time derivatives. ThenΓ 1 must be a linear combination of a term involving space-time derivatives and a unique non-derivative piece: This latter is nothing but the O(κ) part of √ g, i.e. what one gets from the expansion (2.17) of a classical cosmological constant term. The derivative part has a unique expression with maximum antighost number two [4,31], up to addition ofŝ 0 -exact pieces. Previously we followed [31] in using the simplest form for thisŝ 0 -cohomology representative, which corresponds to treating c µ as a covariant vector field [4]. At higher orders the formulae will simplify however if we treat c µ as a contravariant vector field since diffeomorphisms can then be expressed through the Lie derivative and thus be independent of the metric. Then the maximum antighost number piece iš where the first bracketed term is half the Lie bracket as required [4], and in the second equality we use (2.15) to express it as the old choice plus a Q 0 -exact piece (the first term on (3.7)'s RHS). Now we could use the second expression and descend via (3.2,3.3), but since we know that the expression is unique up to addition ofŝ 0 -exact pieces, we see immediately that our new choice must bě up to possible furtherŝ 0 -exact terms of lower antighost number. Using also (2.18) and (3.1), where we note that ∆ − trivially annihilates, but ∆ = yields a UV regulated quartically divergent contribution, b being the non-universal number already introduced in refs. [1,4] From (3.6) and the antighost level one part of (3.7), we havě where the previous choice involves the linearised connection Γ . Integrating by parts we get the second expression, and we recognise that inside the brackets we already have the desired Lie derivative form. Combining it with the last term of the free action (2.14), the expression for the Lie derivative of the metric is given exactly, provided that the metric is taken to be exactly (2.17). In other words at the classical level neither (3.5) nor (3.9) receives corrections at higher order in perturbation theory. Finally, from (3.6,3.7) we havě Γ 0 1 | old coincides with the classical three-graviton vertex that one would get from expansion of the Einstein-Hilbert action (2.16) using (2.17), except for a quantum correction, 5 3 2 bΛ 4 ϕ, which is generated by the action of the tadpole operator, the RHS of (2.11), on this triple-graviton vertex [4].
This quantum correction turnsΓ 1 | old into a (dimension five) eigenoperator in standard quantisation.
Around the Gaussian fixed point and in dimensionful variables, as in our case, an eigenoperator in standard quantisation is a local solution of the linearised flow equation (2.11) which contains no dimensionful parameters and is polynomial in the fields. Since H µν c µ c * ν is trivially such an eigenoperator (it has no tadpole corrections) andŝ 0 maps the vector space of eigenoperators into itself, cf. below (2.13) [4], we know that (3.7) is an eigenoperator. Indeed the last term of (3.7) is exactly right to balance the action of the (ghost) tadpole operator on 2H µν c µ ∂ α H * αν . Since (3.6) is thus the sum of two eigenoperators of the same dimension, our newΓ 1 is also an eigenoperator (which of course one can also confirm by direct calculation).

Renormalization group properties at the linearised level
The wrong sign ϕ propagator (2.26) reflects the wrong sign kinetic term for ϕ in this gauge, which in turn is a reflection of the instability caused by the unboundedness of the Euclidean Einstein-Hilbert action (see [1,3,4] for further discussion). The Euclidean partition function is then more than usually ill-defined, which the authors of ref. [5] proposed to solve by analytically continuing the ϕ integral along the imaginary axis. However this wrong sign does not invalidate the Wilsonian renormalization group (RG) flow equations, for example (2.2), which provide an alternative and anyway more powerful route to defining a continuum limit (see [1][2][3][4] and e.g. ref. [32] for further discussion). As shown in refs. [1,4], the wrong sign then profoundly alters the RG properties that are central to defining such a continuum limit. (For earlier observations see refs. [33,34].) We review and refine some of those discoveries in this section.
Consider some arbitrary infinitesimal perturbation around the Gaussian fixed point (2.14), whose ϕ-amplitude dependence 6 is given by f Λ (ϕ). Recalling the wrong sign in (2.26), and usinġ C Λ = −Ċ Λ , the linearised flow equation (2.11) implies that this coefficient function must satisfẏ where prime is ∂ ϕ , and is the modulus of the ϕ tadpole integral regularised by the UV cutoff (a > 0 is a dimensionless nonuniversal constant). With the now positive sign on the right hand side of this parabolic equation, the first dramatic conclusion is that the natural direction of RG flow in this sector reverses: solutions are guaranteed to exist only when flowing from the IR towards the UV. This property will play an important rôle here and in later papers [8,19]. Most importantly, the perturbation can be written as a convergent sum over eigenoperators and their couplings only if the coefficient function is square-integrable under the corresponding Sturm-Liouville measure: where the measure is now a growing exponential. We call L − , the (Hilbert) space of such coefficient functions. If f Λ ∈ L − , then it can be written as a (typically infinite) linear combination over the operators: (integer n ≥ 0) with convergence of the sum being in the square-integrable sense under (4.3), under which also the operators are orthonormal: These δ (n) Λ (ϕ) are solutions of (4.1), and are nothing but the tower of non-derivative eigenoperators in the ϕ sector that span L − , the general solution of (4.1) in this space being a linear combination of these eigenoperators with constant coefficients, a.k.a. couplings. The δ Since Ω Λ ∝ , the δ (n) Λ (ϕ) are non-perturbative in . It is for this reason that we must develop the theory whilst remaining non-perturbative in . We mention also that they are also evanescent, i.e.
vanish as Λ → ∞, and have the property that the physical operators, gained by sending Λ → 0, are δ (n) (ϕ), the n th -derivatives of the Dirac delta function.
In the h µν sector and the ghost sector, convergent sums are over eigenoperators that are polynomials in the fields, justifying the usual form of expansion. Altogether, the general eigenoperator can be expressed as [4] δ (n)   The eigenoperator is equal to its physical limit σ δ (n) (ϕ), plus all possible tadpole corrections. Those corrections generated by attaching to σ, terminate eventually (since the monomial will run out of fields), while ϕ-tadpole corrections to δ (n) (ϕ) go on forever but resum to δ In fact we can give (4.6) in closed form. Note that (2.11) implieṡ where Λ AB = C Λ AB is the UV regulated propagator. The solution we need is therefore where ∂ L acts only on the left-hand factor, here σ, and ∂ R acts only the right-hand factor, here δ (n) (ϕ). Thus (factoring out −C Λ for later convenience): The exponential in (4.8) therefore factors into three exponentials. Since δ (n) (ϕ) only depends on ϕ, the third exponential collapses to [1]: where we used (2.26), giving the tadpole integral (4.2) and derivatives ∂ ϕ with respect to the amplitude (i.e. no longer functional), and expressed the result in conjugate momentum π space, after which the integral evaluates to (4.4). Thus the entire eigenoperator can be written as where the term in braces expresses all the tadpole corrections acting purely on σ, and the leftmost term generates ϕ-propagator (2.26) corrections that attach to both σ and δ (n) Λ (ϕ) (each such attachment will increase n → n+1).
A simple example eigenoperator [4] will prove useful later: The second term has the ghost tadpole correction to the top monomial σ = −∂ µ c ν H * µν , that we already derived in (3.7). (To see this immediately, substitute (2.24) into (3.7), integrate by parts, and recall the remark at the end of sec. 3.1.) The continuum limit is described by the renormalized trajectory [32,35], the RG trajectory that shoots out of the (Gaussian) fixed point, parametrised by (marginally) relevant couplings that are finite at physical scales. Close to the fixed point, the linearised approximation is justified. The interaction there is therefore expanded only over the marginal and relevant eigenoperators (4.6) with constant couplings g σ n whose mass-dimensions must all be non-negative. Every monomial σ is therefore associated to an infinite tower of operators, which can be subsumed into where the coefficient function of the top term is given by (at the linearised level) 16) and the tadpole corrections are the same as before (now with f σ Λ differentiated according to the number of times the left-most operator acts on it). In general all the (marginally) relevant couplings [g σ n ] ≥ 0 will be needed [1] and thus at the linearised level For d σ ≥ 5, we are thus including the marginal coupling [g σ nσ ] = 0. The eigenoperators (4.6,4.12) span the complete (Hilbert) space L of interactions whose combined amplitude dependence is square integrable under the Sturm-Liouville measure At the bare level we require that Γ I is inside L, so that expansion over eigenoperators is meaningful.
We can interpret this as a 'quantisation condition' that is thus both natural and necessary for the Wilsonian RG. However, since we will be solving for Γ I directly in the continuum, our bare cutoff is already sent to infinity. Then this condition is replaced by the requirement that Γ I ∈ L for sufficiently large Λ, where as a consequence we also have f σ Λ ∈ L − . We define the amplitude suppression scale Λ σ ≥ 0 to be the smallest scale such that for all Λ > aΛ σ , the coefficient function is inside L − . The coefficient function exits L − as Λ falls below aΛ σ , either because it develops singularities after which the flow to the IR ceases to exist, or because it decays too slowly at large ϕ.
We need to choose the g σ n so that the flow all the way to Λ → 0 does exist, so that all modes can be integrated over and so that the physical Legendre effective action (2.1) can be defined. Note that we mean by Γ phys the resulting Λ → 0 limit, thus removing the infrared cutoff (lim Λ→0 C Λ = 0).
The results are not yet physical in terms of properly incorporating diffeomorphism invariance. That requires another limit as we will shortly see.
Since the coefficient function thus exits L − by decaying too slowly, we know from (4.3) that asymptotically: for at least one of ϕ → ±∞, with the other side decaying at the same rate or faster, where is a dimensionful constant, and o(· · ·) is a dimensionless term of either sign that grows slower than its argument. (Because of the presence of such undetermined terms, (4.19) only yields A σ up to a dimensionless proportionality constant.) The asymptotic behaviour (4.19) gives us a boundary condition which then fixes the solution of the flow equation (4.1) at large ϕ. Thus we find (at the linearised level) the asymptotic behaviour for any Λ: (on at least one side with the other side being the same rate or faster). From (4.3), our definition of Λ σ is verified: f σ Λ ∈ L − for all Λ > aΛ σ , while f σ Λ / ∈ L − for Λ < aΛ σ (in fact for all such Λ).
Setting Λ = 0 shows that the physical coefficient function f σ phys (ϕ), which following [4] we write simply as f σ (ϕ), is characterised by the decay (on at least one side with the other side being the same rate or faster): From (4.1), this solution can be written in terms of the Fourier transform over π: where f σ is Λ-independent and is thus the Fourier transform of the physical f σ (ϕ). From (4. 16) and (4.11), the couplings are its Taylor expansion coefficients: Since the behaviour (4.22) ensures that the inverse Fourier transform exists for all complex π, f σ is an entire holomorphic function (Paley-Wiener theorem). 8 The asymptotic behaviour (4.22) is reproduced by setting f σ (π) proportional to which also reproduces (4.21). However at this stage it needs to be interpreted with care since it captures only the fastest decaying part, corresponding to the slowest decaying behaviour in ϕ-space.
(See app. A for an example. This corrects part of the characterisation given in ref. [4].) It does however control the large-n behaviour of the couplings: where we Taylor expanded (4.26) and used Stirling's approximation. Indeed from (4.16), (4.3) and (4.5), we see that By its definition, Λ = aΛ σ marks the radius of convergence, and thus we see that g σ n must at large n behave roughly like Ω n aΛσ /n!. Using Stirling's approximation we regain (4.27) (up to sign dependence). This large-n behaviour also verifies that f σ is entire.
As mentioned already below (4.1), flows in the ϕ-sector are guaranteed to exist in the reverse direction, i.e. from the IR towards the UV. In particular, the linearised f σ Λ (ϕ) exists for all Λ ≥ 0 and is unique, once the coefficient function at Λ = 0 is specified, as is also clear from the integral representation (4.24). Given (4.22), this is also clear from the Green's function representation: It is clear that this is the Green's function representation since it satisfies (4.1) by virtue of the fact that the shifted eigenoperator δ (0) Λ (ϕ − ϕ 0 ) does, and returns the boundary condition in the limit 8 Then since f σ is also square integrable, the exponential decay part in (4.24) ensures that the Fourier integral converges for all complex ϕ provided Λ > 0, and thus that f σ Λ>0 (ϕ) is also an entire holomorphic function.
Λ → 0, since in this limit δ Λ (ϕ−ϕ 0 ) about ϕ, we recover (4.16) (and the series converges for Λ > aΛ σ ), and read off a formula for the couplings in terms of the moments of the physical coefficient function [1]: We see therefore that the general form of the solution is given by specifying the physical coefficient function. At this stage it is subject only to the constraints that it satisfy the asymptotic condition (4.22) and be such that its Fourier transform (4.25) has vanishing Taylor expansion coefficients for π n<nσ , equivalently that its moments (4.30) vanish for n < n σ . Indeed the property

Trivialisation in the limit of large amplitude suppression scale
All of the above properties for the linearised solutions are inevitable consequences of respecting the wrong sign kinetic term for the conformal factor ϕ, while insisting that the Wilsonian RG remains meaningful. However this general form must now be married with the first order BRST constraint (2.12). In ref. [4], we proved that this is possible only if the coefficient function trivialises in the sense defined below, 9 and we showed that such trivialisations are possible if we now send Λ σ to infinity. In other words, we can arrange for violations of BRST to be as small as desired by taking sufficiently large Λ σ . In this way, at first order, we get both the continuum limit and diffeomorphism invariance of the renormalized solution.
In the majority of cases the coefficient function has to become ϕ-independent, i.e. we need linearised renormalized trajectories that satisfy: (where we hold Λ, ϕ and A σ fixed and finite) such that also its ϕ-derivatives have a limit, which is thus that they vanish. However if BRST invariance demands a physical vertex of the same dimension but containing an undifferentiated ϕ α factor (α a positive integer), then this would appear as in (4.23), where thus the new monomial σ α has and the ϕ α amplitude dependence must be absorbed by the physical coefficient function.
This will correspond to linearised renormalized trajectories satisfying such that also their ϕ-derivatives have a limit, where H α is the α th Hermite polynomial. This follows because is the unique solution of the flow equation (4.1) with the boundary condition that it just becomes ϕ α at Λ = 0. 10 Notice that the above conditions (5.4,5.5) actually apply also at α = 0, where they just give back (5.1) as a special case. Since we require the ϕ-derivatives to have a limit, by l'Hôpital's rule this limit is given by the ϕ-derivative of the right hand side.
We say that a coefficient function trivialises in the limit of large amplitude suppression scale if it satisfies the limiting condition (5.4) for some α. Since at finite Λ σα (with σ = σ α ), the coefficient functions satisfy (4.21), they are non-trivial, in particular they cannot be polynomial in ϕ.
From (4.27) we see that the couplings g σ n must diverge in the limit Λ σ → ∞. However the vertices are nevertheless well behaved since the coefficient function goes smoothly over to A σ as in (5.1), or more generally to the finite polynomial in (5.4). What is happening is that the Λ = aΛ σ boundary, above which f σ Λ (ϕ) enters L − , is being sent to ever higher scales. In this sense we are taking a limit towards the boundary of this Hilbert space (and thus also L) [3,4].
Actually, from (4.27), we can keep the couplings g σ n perturbative in this limit if we choose A σ to vanish fast enough with Λ σ . For example if we set A σ = a σ e −Λσ/µ for fixed a σ and µ, then for any finite n, the couplings g σ n → 0 as Λ σ → ∞. Although this means that the coefficient function, and thus the vertex itself, vanishes in the limit, this does not stop us from computing perturbative corrections in the usual way [4], as reviewed in sec. 6. We can also choose A σ to vanish fast enough to ensure that couplings remain uniformly perturbative (as opposed to pointwise in n as in the above example). From (4.27) one sees that for large Λ σ , the couplings first grow with n and then decay once the n −n/2 factor dominates. Thus we can estimate the maximum size coupling by differentiating with respect to n and finding the stationary point. We find which implies that we can keep the couplings uniformly perturbative if we set A σ to vanish faster than Λ −1 σ e −Λ 2 σ /4 . The above result already suggests that it is the large-n g σ n couplings that should be important in the limit of large amplitude suppression scale. We will see this more dramatically from a different point of view in ref. [8].

Relations
In this subsection, we pause the main development to explore two rather natural ways for generating new solutions.
On the one hand, we can convert any solution to (5.1), into one satisfying (5.4), by multiplying the physical coefficient function by ϕ α and using the fact that the flow to all Λ > 0 then exists and is unique. Recalling that we defined o(· · ·) to be dimensionless, we thus identify from (4.22): where Λ σα is the amplitude suppression scale, and A σα the dimensionful constant, in the asymptotic behaviour of the physical coefficient function associated to the new monomial σ α . Using (4.24) at Λ = 0, and integration by parts, we see that the new physical coefficient function is given by setting: We confirm that f σα (π) thus satisfies the same general formula (4.25), with i.e. defined as in (4.17), since unless d σα < 5 in which case n σα = 0. Reading off the couplings from (4.25) and (5.8), we have g σα n = (−) α (n + 1)(n + 2) · · · (n + α) g σ n+α = (−) α (n + α)! n! g σ n+α . (5.11) Using this, (4.27) and (5.7), we confirm that in terms of the appropriate σ α -labelled quantities, these couplings have the expected limiting behaviour at large n.
On the other hand, thanks to the recurrence relation H α (x) = αH α−1 (x), one easily verifies that taking the ϕ-derivative of (5.4) just maps it to (α times) the (α−1) th case, as it must since the derivative is still a solution of the flow equation (4.1) and the result is determined by the physical (Λ → 0) limit, in this case αA σ ϕ α−1 . Of course this does not mean in general that , since there are infinitely many solutions with these limits. Indeed while f σα satisfies (5.9) for each α in general, the coefficient function defined by is more restricted. From (4.24) and (4.25) we see that it has couplings with the lowest n in the sum thus being where in the last step we use (5.3) for the (α−1) th case. Thus for d σ α−1 ≤ 5, f σ α−1 has no g σ α−1 0 coupling in contrast to the general case for f σ α−1 viz. (5.9).

Simplifications and general form
In order to check the universal nature of the final result, we want to work with very general solutions for linearised coefficient functions satisfying the required constraints (5.1,5.4). These not only determine the form of the interactions at the linearised level, but then contribute at the non-linear level through higher order contributions in the perturbative expansion (2.9). As will become clear [19], the most powerful way to handle these higher order contributions is to express the solutions in conjugate momentum space. Thus we use the fact that the linearised coefficient functions are given by (4.24) via a Λ-independent f σ (π) which, from (4.25) and the discussion below it, we know can be written as an entire function times a π nσ factor. The constraint (5.1) is equivalent to understood in the usual distributional sense (see also below) while more generally from (5.4): as we see immediately from (5.15) and (5.8), and which includes (5.15) as the special case α = 0.
(From here on for notational simplicity, we use (5.7) to write Λ σα = Λ σ .) These constraints evidently still leave us with a huge (infinite dimensional) function space of renormalized trajectories. We now make two further restrictions that do not result in any significant loss of generality but greatly strengthen and streamline the analysis.
Firstly, we insist that the coefficient functions are of definite parity, i.e. even or odd functions of ϕ. Thus those satisfying (5.1) will be even parity, and those satisfying (5.4) will be even or odd, depending on whether α is even or odd respectively. This also implies the same of f σα (π) in (5.15,5.16), and enforces that the asymptotic estimates (4.21,4.22) apply for both limits ϕ → ±∞.
We see from either (4.16) or (4.25), that the couplings g σα n will be indexed by an integer of the same parity, and in particular the minimum index (5.9) required in order that the coefficient function represents a linearised renormalized trajectory, actually has this parity, so now n σα is the smallest index of the same parity as α such that n σα ≥ max(0, d σα − 5) . (5.17) Secondly we insist that such linearised solutions contain only one amplitude suppression scale, so that (4.26) now genuinely captures their large π behaviour. 11 Then for cases satisfying (5. 15) we have that where n σ is even i.e. satisfies (5.17) for α = 0,π = Λ σ π is dimensionless, andf σ is a dimensionless entire function which from (4.26) takes the form f σ (π 2 ) = e −π 2 /4+o(π 2 ) , (5.19) at largeπ. Likewise for general α, whereπ has the same definition, andf σα is also a dimensionless entire function satisfying (5.19).
Note that the Λ α+1 σ factor is fixed by dimensions, e.g. using (5.16). Together with A σ , these factors appear in the same form as (5.18) if we use the identifications (5.7).
Note that the parity is carried by ∂ ᾱ π , and thusn σα is even. If α is even and n σα = 0 we do not require a separateπ power, likewise if α is odd and n σα = 1 since the ∂π differentials will generate a Taylor expansion with only odd powers ofπ. However if n σα from (5.17), is larger than these absolute minima, then the Taylor expansion of the term in square brackets must be such that all powersπ n>α are missing up to the point where we are left with an overall factor ofπ nσ α after differentiation by ∂ ᾱ π . Without loss of generality we capture this by factoring out this power, leaving behind a function that is still entire. Thus we see that n σα = 0 if n σα = ε , otherwisen σα = n σα + α , where we define ε = 0 or 1 according to whether the coefficient function is even or odd.
Constraint (5.16) is then satisfied (on finite smooth functions) provided that (for n ≥ 0) (or we get these constraints directly from the physical limit A σ ϕ α , by Taylor expanding the Fourier representation (4.24) in ϕ), and from (5.20) these are in turn satisfied iff σα is normalised as and provided that for any integer p > 0, we have (These integrals converge for largeπ by virtue of (5.19).) At first order in the perturbation theory (2.9),f σα can be chosen to be a finite function and independent of Λ σ , and thus (5.24) follows trivially. At second order in perturbation theory, we will find that we need linearised coefficient functions for whichf σα depends on Λ σ . In the majority of cases we can choose it to tend to a finite function as Λ σ → ∞, but exceptionally it will prove useful to allow it to contain terms with coefficients that diverge logarithmically with Λ σ . Clearly this mild divergence is well within the bounds implied by (5.24).
We see that the difference between the left and right hand sides of (5.4) is bounded by a term of order 1/Λ 2 σ . Furthermore this is true for every relation obtained by differentiating with respect to ϕ on both sides until the RHS vanishes. At this point successive differentials will bring down further powers of 1/Λ 2 σ from (5.28) via ϕ 2 /Λ 2 σ . Thus we have for large Λ σ : for p > α , (5.29) which since this applies for p = 0, refines the earlier characterisation (5.1,5.4), and where again one should understand that the RHS is corrected by a factor of ln(Λ σ ) in some cases at second order.

Examples
For example if there is no o(π 2 ) correction in (5.19), then (5.23) fixes the normalisation of the dimensionless entire function so that 12 In the case (5.18), solutions to (5.1) that keep all possible couplings, so n σ = 0, take the form Using (5.30) with α = 0 to generate an explicit example, we have: which just gives us our previously well-worked specimen [1,4]: √ π n!4 n A σ Λ 2n+1 σ (5.33) (n = 0, 1, · · · ), where the first expression follows from performing the integral in (4.24), the second is its Λ → 0 limit, and the couplings follow from the Taylor expansion relation (4.25). Similarly from (5.20) and (5.21) with α = 1. The explicit example (5.30) again gives (5.32), and thus (n = 0, 1, · · · ), in agreement with (5.11) and (5.33). For α = 2 and n σ 2 = 0 one gets from (5.30), which gives the physical coefficient function and couplings: n!4 n A σ Λ 2n+3 σ (n = 0, 1, · · · ) . (5.37) , is in agreement with (5.4). Differentiating (5.35) with respect to ϕ: gives an alternative example solution for (5.1): as is clear from the large amplitude suppression scale limit. However this solution hasǧ σ 0 = 0 as is immediately clear from integrating (5.38) and using (4.30). In sec. 5.1 we showed that n σ = 1 -or rather n σ = 2 since it is even, cf. (5.17). Indeed differentiating (4.24), and using (5.34) and (5.32), we see that the correspondingf σ (π) takes the form (5.18): 6 Continuum limit at first order in perturbation theory We will treat the first order cosmological constant term, associated to (3.4), at the end of this section. The remaining parts ofΓ 1 that we computed in (3.5), (3.9) and (3.10) will provide us with the top monomials σ that we need to construct the derivative part. In order to be supported on the renormalized trajectory, such that Γ 1 is constructed, these σ need to be 'dressed' with coefficient functions f σ Λ (ϕ) as in (4.15). In the most general case we should give each top term its own coefficient function. This would provide the most complete test of universality of the continuum limit, however at the expense of carrying around a lot more terms and labels. At sufficiently high order of perturbation theory in the expansion (2.9), we expect to have to do this because these Γ 1 couplings will then run independently [4]. In fact we will show in ref. [19] that as a consequence of specialising to coefficient functions of definite parity, the Γ 1 couplings do not run at second order but they can be expected to run at third order.
Here it is not necessary to treat the general case, since we will see that the passage to universality is very generic such that it is clear that this will continue to work when we give each top monomial in Γ 1 its own coefficient function. We thus find that for our purposes just two coefficient functions are sufficient for constructing Γ 1 , the first of which we label as f 1 Λ (ϕ), setting the superscript to σ = 1 i.e. the perturbation level index, and in the second case choose the label σ = 1 1 as in (5.4) to indicate that f 1 1 Λ (ϕ) absorbs a factor of ϕ. Thus f 1 Λ (ϕ) is even, while f 1 1 Λ (ϕ) is odd. Although in principle every vertex can have its own amplitude suppression scale Λ σ , we will find that we can choose them all to be equal. To make clear that it is independent of σ, we set this common amplitude suppression scale to Λ σ = Λ p (borrowing the notation already used in [1]). Now sinceΓ 1 is a dimension d 1 = 5 operator, we have from (4.20) that the dimensionful coefficient [A 1 ] = −1. As the remaining factor in frontΓ 1 , after taking the limit Λ p → ∞, we recognise that it is actually A 1 = κ, where the latter was defined in (2.10), i.e. we have where whenever we now write the limit of large amplitude suppression scale, we mean also the more refined regularity properties (5.29), in particular in these cases the limits are reached at least as fast as 1/Λ 2 p . We see that Newton's constant therefore arises only as a kind of collective effect of all the renormalizable couplings {g 1 2n , g 1 1 2n+1 }, these latter being responsible for forming the continuum limit. Indeed A 1 = κ is not an underlying coupling in its own right but rather appears as the overall proportionality constant when the couplings are expressed in terms of Λ p , through (4.27).
Examples of such coefficient functions were given in [4] and appear in equations (5.33) and (5.35). We stress however that we are working here with very general solutions for these coefficient functions. From (5.17) and (4.16), we have that in general all eigenoperators will be involved: where these sums converge (in the square integrable sense) for Λ > aΛ p . From (4.14), and (5.3): Thus all these couplings are relevant, with the exception of g 1 0 which is marginal. Up to second order it does not run [19] and thus behaves as though it is exactly marginal, parametrising a line of fixed points. From (3.5), we thus set at antighost level two: Since f 1 Λ is taken to satisfy (4.1) and there is no other opportunity to attach tadpoles to (6.4), Γ 2 1 already satisfies the linearised flow equation (2.11), and thus appears correctly as a sum over eigenoperators. Evidently at this antighost level, the linearised mST (2.12) is satisfied in the limit (by (5.29) at least as fast as 1/Λ 2 p ) since: and Γ 2 1 → κΓ 2 1 then coincides with a legitimate choice in the usual perturbative quantisation. As discussed above sec. 5.1, if we keep κ fixed in the large amplitude suppression scale limit, all the couplings {g 1 2n , g 1 1 2n+1 } diverge. As we noted however, we can stay perturbative by requiring instead that κ vanish fast enough. Although this makes the vertex vanish, we can still extract the same results by phrasing the limit more carefully as Γ 2 1 /κ →Γ 2 1 . From here on we will take this phrasing as tacitly understood. 13 In (3.9) we need to substitute (2.24) into the last term to isolate the factor of ϕ, and thus the dressed antighost-level-one piece appears as This time the result does not yet satisfy the linearised flow equation (2.11), unlike with the previous choice in ref. [4], because it requires the tadpole correction in (3.7) or rather as formulated for the new quantisation in (4.13). 14 In other words the sum over eigenoperators is actually Γ 1 1 +2bΛ 4 f 1 1 Λ (ϕ). Since ∆ − Γ 2 1 trivially vanishes, the descendant equation (3.2) that relates Γ 2 1 to Γ 1 1 reads: where we used (2.18) and note from (2.15) that It is clear from (6.1) that as required Q − 0 Γ 2 1 + Q 0 Γ 1 1 → 0 (at least as fast as 1/Λ 2 p ). At the expense of some generality, we could eliminate the first term on the RHS of (6.7) by setting By (5.14) this would also eliminate δ (0) Λ (ϕ), i.e. set g 1 0 = 0. We would still be left with the Λ p < ∞ violations on the second line however.
Finally, extracting the undifferentiated ϕ pieces from (3.10) by using (2.24), we have as in [4] that the first order graviton interaction is made up of twelve top terms and one tadpole contribution: 13 This is in conformity with the reasonable assumption that the expansion in κ is only asymptotic [4]. Then strictly speaking the expansion only anyway makes sense in the κ → 0 limit, i.e. as Taylor expansion coefficients in κ . 14 Tadpole contributions from the first term in (6.6) all vanish, either because the tadpole integral is odd in momentum or because hαα = 0.
except that the tadpole contribution now appears with coefficient 7 2 = 2+ 3 2 . The final descendant equation (3.2) is satisfied in the limit: Note that although these measure terms give contributions proportional to some positive power of Λ, thanks to UV regularisation by C, for example it does not alter the speed at which they vanish in the limit of large Λ p (as can be verified here by integration by parts).
In the opposing limits there is no sense in which a non-trivial diffeomorphism invariance holds because the dependence on the conformal factor forbids it [4]. For example if ϕ Λ p , Λ, the coefficient functions are no longer given approximately by κ and κϕ, but rather take the exponentially decaying form (4.21).
In dimensionful variables, if Λ is much larger than the other scales Λ p , ϕ, the situation is a little obscured but it is still the case that there is no sense in which a non-trivial diffeomorphism invariance is recovered. The coefficient functions are again dominated by the lowest terms in the expansion (6.2). Using (4.4) we have in the current case The leading terms, and only the leading terms, have the correct ϕ dependence to allow BRST invariance to be recovered, however with g 1 0 = 0 they have the wrong ratio. (They should have equal coefficients, but this is impossible at diverging Λ since g 1 0 and g 1 1 1 must be fixed and finite.) By setting g 1 2 = g 1 1 1 , and g 1 0 = 0, (only) the leading terms have both the correct ϕ dependence and the correct ratio, as in fact would result from the identification (6.9), cf. (5.13), although with an effective κ that then vanishes as κ eff ∼ 1/Λ 3 . Meanwhile the measure terms in (6.11) provide divergent obstructions to satisfyingŝ 0 Γ 1 = 0, if g 1 0 = 0. Thus (6.14) tells us that (dropping total derivative terms), and ∆ − Γ 1 1 provides also such a term but with coefficient − 9 2 and also a g 1 0 Λ(c α ∂ α ϕ + ∂ α c β h αβ )ϕ piece arising from the contribution containing ∆ − (H * µν f 1 Λ ). Setting g 1 0 = 0 removes these divergences but leaves us with subleading terms that violate BRST invariance, as is also true of the subleading terms in (6.16)  This means in particular in the limit Λ p → ∞ and the physical limit (Λ → 0), we recover diffeomorphism invariance precisely in terms of satisfying the standard Slavnov-Taylor (Zinn-Justin) identities, namely at first order (Q 0 + Q − 0 )Γ 1 = (Γ 0 , Γ 1 ) = 0, where we used (2.8), (2.12) and noted that from (2.13), ∆ → 0 as Λ → 0.
Finally, we remark that including a cosmological constant is straightforward at first order. We need to dress (3.4) with its own coefficient function. Since we must absorb the factor of ϕ, the monomial σ = 1 is simply the unit operator, whilst we must choose an odd coefficient function f cc Λ (ϕ) with the trivialisation f cc Λ (ϕ) → λ , as Λ p → ∞ , (6.19) where κ 2 λ/4 is the standard cosmological constant. At this order we do not need a whole separate odd coefficient function and can by (6.1) just set f cc Λ = λf 1 1 Λ /κ. The linearised mST (2.12) is satisfied in the limit because Q 0 f cc Λ (ϕ) = ∂ ·c f cc Λ (ϕ) → 0 at least as fast as 1/Λ 2 p , as follows by integration by parts and using (5.29), or directly by the observation (6.12). Indeed these properties were already used in proving the invariance (in the limit) of the last term in (6.10).

Acknowledgments
AM acknowledges support via an STFC PhD studentship. TRM thanks David Turton for discussions, and acknowledges support from STFC through Consolidated Grant ST/P000711/1.

A Further examples of coefficient functions A.1 Examples with multiple amplitude suppression scales
Here we develop some of the properties of linearised coefficient functions that are constructed from a spectrum of amplitude suppression scales γ k Λ σ . For example for symmetric coefficient functions satisfying (5.1), we can take [4] f σ (π) = A σ N k=0 a k f(π, γ k Λ σ ) , (A.1) where N ≥ nσ 2 will allow us to ensure that couplings g σ 2n<nσ vanish, and we define the function f(π,Λ) = √ πΛ e −π 2Λ2 /4 , (A.2) which is just the simplest choice (5.32), where for convenience we have absorbed the factor of 2πΛ σ from (5.31). The dimensionless parameters γ k > 0 are chosen unequal, and without loss of generality we order them and set the greatest to unity: 0 < γ N < γ N −1 < · · · < γ 0 = 1 , where f Λ (ϕ, γ k Λ σ ) is just (5.33) with Λ σ rescaled by γ k .
From the definition of the amplitude suppression scale, see above (2.1), we see that f σ Λ has overall amplitude suppression scale Λ σ , corresponding to the maximum one γ 0 Λ σ = Λ σ . We verify that it also characterises the exponential decay of the physical coefficient function: setting Λ = 0, where the last equation holds at large ϕ. Thus we satisfy constraint (4.22), but we have here an example where the asymptotic behaviour is fixed by A σ only up to an undetermined dimensionless proportionality constant, cf. comments below (4.20). Importantly note that the large π behaviour in (A.1) is however set by the smallest amplitude suppression scale: f σ (π) ∼ √ π a N γ N A σ Λ σ e −π 2 γ 2 N Λ 2 σ /4 , (A. 8) and thus (4.26) does not hold, hence the comments below it. The couplings in (4.25) are given by and satisfy the constraint that they vanish for 2n < n σ , thanks to (A.5). The last equation holds at large n, which thus verifies that (4.27) nevertheless holds, although again we see the presence of an undetermined proportionality. Finally, since f Λ (ϕ, γ k Λ σ ) → 1 as Λ σ → ∞, we have from (A.4) that (5.1) is satisfied, while since Λ 2 + a 2 γ 2 k Λ 2 σ sets the scale for ϕ-variation in the components, we see that the limit (5.1) is reached at least as fast as O(1/γ N Λ σ ) and more generally (5.29) is satisfied. Notice however that it is the smallest amplitude suppression scale that controls the corrections here.
Since (A.4,A.5) provides nσ 2 +1 linearly independent conditions on N +1 ≥ nσ 2 +1 coefficients a k , they can always be satisfied. By choosing N > nσ 2 large enough, we can go on to fix the numerical coefficient of finitely many of any of the surviving g σ 2n (with n finite) to any value we wish, including forcing them also to vanish. We also have the freedom to alter couplings through changing the 0 < γ k>0 < 1 provided they remain unequal. We see that the limit (5.1) is independent of the value of any finite set of finite-n couplings or indeed of any finite number of relations between these couplings [8]. Therefore, apart from confirming that we can ensure that g σ 2n<nσ = 0, the universal information on the couplings is that captured in the large n asymptotic estimate (4.27), which indeed holds for any linearised solution.

A.2 Other examples with only one amplitude suppression scale
exp − a 2 ϕ 2 + γ 2 Λ 2 /4 Λ 2 + a 2Λ2 cos a 2 γΛϕ Λ 2 + a 2Λ2 , (A. 13) which clearly again has the right limiting properties to satisfy (5.1) and (5.29). These functions have the same amplitude suppression scaleΛ irrespective of the choice of γ. Further examples can be generated by exchanging cosh with cos in the above, or for odd functions, replacing these with sinh and sine.