An Estimate of Λ in Resummed Quantum Gravity in the Context of Asymptotic Safety †

We show that, by using recently developed exact resummation techniques based on the extension of the methods of Yennie, Frautschi and Suura to Feynman’s formulation of Einstein’s theory, we get quantum field theoretic descriptions for the UV fixed-point behaviors of the dimensionless gravitational and cosmological constants postulated by Weinberg. Connecting our work to the attendant phenomenological asymptotic safety analysis of Planck scale cosmology by Bonanno and Reuter, we predict the value of the cosmological constant Λ. We find the encouraging estimate ρΛ ≡ Λ 8πGN ≃ (2.400 × 10eV ). † Work partly supported by NATO Grant PST.CLG.980342.


Introduction
In Ref. [1], Weinberg suggested that the general theory of relativity may have a non-trivial UV fixed point, with a finite dimensional critical surface in the UV limit, so that it would be asymptotically safe with an S-matrix that depends on only a finite number of observable parameters. In Refs. [2][3][4][5][6], strong evidence has been calculated using Wilsonian [7] field-space exact renormalization group methods to support Weinberg's asymptotic safety hypothesis for the Einstein-Hilbert theory. As we review briefly below, in a parallel but independent development [8][9][10][11][12][13][14][15][16][17], we have shown [18] that the extension of the amplitudebased, exact resummation theory of Ref. [19] to the Einstein-Hilbert theory leads to UV-fixed-point behavior for the dimensionless gravitational and cosmological constants, but with the added bonus that the resummed theory is actually UV finite when expanded in the resummed propagators and vertices's to any finite order in the respective improved loop expansion. We have called the resummed theory resummed quantum gravity. More recently, more evidence for Weinberg's asymptotic safety behavior has been calculated using causal dynamical triangulated lattice methods in Ref. [20] 1 . At this point, there is no known inconsistency between our analysis and those of the Refs. [2][3][4][5][6]20]. The stage is therefore prepared for us to make contact with experiment, as such contact is the ultimate purpose of theoretical physics.
Toward this end, we note that, in Refs. [22,23], it has been argued that the attendant phenomenological asymptotic safety approach in Refs. [2][3][4][5][6] to quantum gravity may indeed provide a realization 2 of the successful inflationary model [25,26] of cosmology without the need of the as yet unseen inflaton scalar field: the attendant UV fixed point solution allows one to develop Planck scale cosmology that joins smoothly onto the standard Friedmann-Walker-Robertson classical descriptions so that then one arrives at a quantum mechanical solution to the horizon, flatness, entropy and scale free spectrum problems. In Ref. [18], we have shown that, in the new resummed theory [8][9][10][11][12][13][14][15][16][17] of quantum gravity, we recover the properties as used in Refs. [22,23] for the UV fixed point of quantum gravity with the added results that we get "first principles" predictions for the fixed point values of the respective dimensionless gravitational and cosmological constants in their analysis. In what follows here, we carry the analysis one step further and arrive at a prediction for the observed cosmological constant Λ in the context of the Planck scale cosmology of Refs. [22,23]. We comment on the reliability of the result as well, as it will be seen already to be relatively close to the observed value [27,28]. While we obviously do not want to overdo the closeness to the experimental value, we do want to argue that this again gives more credibility to the new resummed theory as well as to the methods in Refs. [2][3][4][5][6]20]. More reflections on the attendant implications of the latter credibility in the search for an experimentally testable union of the original ideas of Bohr and Einstein will be taken up elsewhere [29].
The discussion is organized as follows. We start by recapitulating the Planck scale cosmology presented phenomenologically in Refs. [22,23]. This is done in the next section. We then review our results in Ref. [18] for the dimensionless gravitational and cosmological constants at the UV fixed point. In the course of this latter review, which is done in Section 3, we give a new proof of the UV finiteness of the resummed quantum gravity theory for the sake of completeness. In Section 4, we then combine the Planck scale cosmology scenario in Refs. [22,23] with our results to predict the observed value of the cosmological constant Λ. The Appendices contain relevant technical details.

Planck Scale Cosmology
More precisely, we recall the Einstein-Hilbert theory where R is the curvature scalar, g is the determinant of the metric of space-time g µν , Λ is the cosmological constant and κ = √ 8πG N for Newton's constant G N . Using the phenomenological exact renormalization group for the Wilsonian [7] coarse grained effective average action in field space, the authors in Ref. [22,23] have argued that the attendant running Newton constant G N (k) and running cosmological constant Λ(k) approach UV fixed points as k goes to infinity in the deep Euclidean regime in the sense that k 2 G N (k) → g * , Λ(k) → λ * k 2 for k → ∞ in the Euclidean regime.
The contact with cosmology then proceeds as follows. Using a phenomenological connection between the momentum scale k characterizing the coarseness of the Wilsonian graininess of the average effective action and the cosmological time t, the authors in Refs. [22,23] show that the standard cosmological equations admit of the following extension: in a standard notation for the density ρ and scale factor a(t) with the Robertson-Walker metric representation as so that K = 0, 1, −1 correspond respectively to flat, spherical and pseudo-spherical 3spaces for constant time t. Here, the equation of state is taken as where p is the pressure. In Refs. [22,23] the functional relationship between the respective momentum scale k and the cosmological time t is determined phenomenologically via for some positive constant ξ determined from requirements on physically observable predictions.
Using the UV fixed points as discussed above for k 2 G N (k) ≡ g * and Λ(k)/k 2 ≡ λ * obtained from their phenomenological, exact renormalization group (asymptotic safety) analysis, the authors in Refs. [22,23] show that the system in (2) admits, for K = 0, a solution in the Planck regime where 0 ≤ t ≤ t class , with t class a "few" times the Planck time t P l , which joins smoothly onto a solution in the classical regime, t > t class , which coincides with standard Friedmann-Robertson-Walker phenomenology but with the horizon, flatness, scale free Harrison-Zeldovich spectrum, and entropy problems all solved purely by Planck scale quantum physics.
While the dependencies of the fixed-point results g * , λ * on the cut-offs used in the Wilsonian coarse-graining procedure, for example, make the phenomenological nature of the analyses in Refs. [22,23] manifest, we note that the key properties of g * , λ * used for these analyses are that the two UV limits are both positive and that the product g * λ * is cut-off/threshold function independent. Here, we review the predictions in Refs. [18] for these UV limits as implied by resummed quantum gravity theory as presented in [8][9][10][11][12][13][14][15][16][17] and show how to use them to predict the current value of Λ. In view of the lack of familiarity of the resummed quantum gravity theory, we start the next section with a review of its basic principles in the interest of making the discussion self-contained.
For definiteness, let us start with the Lagrangian density for the basic scalar-graviton system which was considered by Feynman in Refs. [30,31]: Here, ϕ(x) can be identified as the physical Higgs field as our representative scalar field for matter, ϕ(x) ,µ ≡ ∂ µ ϕ(x), and g µν (x) = η µν + 2κh µν (x) where we follow Feynman and expand about Minkowski space so that η µν = diag{1, −1, −1, −1}. Following Feynman, we have introduced the notationȳ µν ≡ 1 2 (y µν + y νµ − η µν y ρ ρ ) for any tensor y µν 3 . The bare(renormalized) mass of our otherwise free Higgs field is m o (m) and for the moment we set the small observed [27,28] value of the cosmological constant to zero so that our quantum graviton, h µν , has zero rest mass. We return to the latter point, however, when we discuss phenomenology. Feynman [30,31] has essentially worked out the Feynman rules for (6), including the rule for the famous Feynman-Faddeev-Popov [30,32,33] ghost contribution needed for unitarity with the fixing of the gauge (we use the gauge of Feynman in Ref. [30], ∂ µh νµ = 0), so for this material we refer to Refs. [30,31]. Accordingly, we turn now directly to the quantum loop corrections in the theory in (6).
Referring to Fig. 1, we have shown in Refs. [8][9][10][11][12][13][14][15][16][17] that the large virtual IR effects in the respective loop integrals for the scalar propagator in quantum general relativity can be resummed to the exact result for 3 Our conventions for raising and lowering indices in the second line of (6) are the same as those in Ref. [31]. where the latter form holds for the UV regime, so that (7) falls faster than any power of |k 2 |. An analogous result [8] holds for m=0; we show this in our Appendix 1 for completeness. Here, −iΣ s (k) is the 1PI scalar self-energy function so that i∆ ′ F (k) is the exact scalar propagator. As Σ ′ s starts in O(κ 2 ), we may drop it in calculating one-loop effects. It follows that, when the respective analogs of (7) are used for the elementary particles, one-loop corrections are finite. It can be shown actually that the use of our resummed propagators renders all quantum gravity loops UV finite [8][9][10][11][12][13][14][15][16][17]. We have called this representation of the quantum theory of general relativity resummed quantum gravity (RQG).
We stress that (7) is not limited to the regime where k 2 ∼ = m 2 but is an identity that holds for all k 2 . This is readily shown as follows. If we invert both sides of (7) we get where the free inverse propagator is ∆ −1 F (k) = ∆(k) + iǫ. We introduce here the loop expansions and we get, from elementary algebra, the exact relation where we define for convenience −Σ s,0 (k) = −Σ ′ s,0 (k) = ∆ −1 F (k) and A s,n is the n-loop contribution to A s . This proves that every Feynman diagram contribution to Σ s (k) corresponds to a unique contribution to Σ ′ s (k) to all orders in κ 2 /(4π) for all values of k 2 . QED.
The key question is whether the terms which we have extracted from the Feynman series were actually in that series. When we take the limit that k 2 → m 2 , the result is known to be valid from the discussion in Ref. [34]. Indeed, one generally has to introduce a regulator for the IR divergence and one shows that the terms which diverge as the regulator vanishes exponentiate in the factor B ′′ g (k). When k 2 = m 2 , the IR divergence is regulated by ∆(k), so that we can use ∆(k) as our IR regulator. We can then isolate that part of the amplitude which diverges when ∆(k) → 0 when the UV divergences are themselves regulated, by n-dimensional methods [35] for example, so that they remain finite in this limit. At this point we stress the following: when we impose a gauge invariant regulator for the UV regime, to any finite order in the loop expansion, all UV divergences are regulated to finite results. If we then resum the IR dominant terms in this the UVregulated theory, that resummation is valid independent of whether or not the theory is UV renormalizable, as the theory is finite order by order in the loop expansion in the UV when the UV regulator is imposed independent of whether or not it is renormalizable. The latter issue arises only if we remove the UV regulator. What we show now is that, after the IR resummation, the UV regulator can be removed and the UV regime remains finite order by order in the loop expansion after the IR resummation.
We call attention as well to the close analogy between our use of IR resummation in the presence of n-dimensional UV regularization to study the UV limit of quantum gravity with the use of exact Wilsonian coarse graining in Refs. [2][3][4][5][6] to arrive at an effective average action for any given scale k which has both an IR cut-off for momentum scales much smaller than k and a UV cut-off for momentum scales much larger than k so that the resulting field-space renormalization group equation is well-defined even for a non-renormalizable theory like quantum gravity. In both cases the UV limit can be studied by taking the UV limit of the resulting non-perturbative solution and in both cases the same result obtains: a non-gaussian UV fixed point is found, as we present below.
To show that (7) holds with B ′′ g (k) given by the expression in (8), we proceed as follows. We represent the respective m-loop contribution as defined above to the proper self-energy contribution to the inverse propagator as where n is the analytically continued dimension of space-time to regulate UV divergences and the function ρ m is symmetric under the interchange of any two of the m virtual graviton n-momenta that are exchanged in (13), by the Bose symmetry obeyed by the spin 2 gravitons and the symmetry of the respective multiple integration volume. Here is the point in the discussion where the power of exact rearrangement techniques such as those in Ref. [19] enters. For the case m = 1, let S ′′ g (k)ρ 0 represent the leading contribution in the the limit k → 0 to ρ 1 . We have where this equation is exact and serves to define β 1 if we specify S ′′ g (k), the soft graviton emission factor, and recall that This can be determined from the Feynman rules for the Feynman [8,30,31] formulation of the scalar-graviton system in (6) or one can also use the off-shell extension of the formulas in Ref. [34]. We get [8] To see this, from Fig. 1, note that the Feynman rules [8,30,31] give us the following result where we have defined from the Feynman rules the respective 3-point(hϕϕ and 4-point(hhϕϕ) vertices using the standard conventions so that p is incoming and p' is outgoing for the scalar particle momenta at the respective vertices. In this way, we see that we may isolate the IR dominant part of iΣ 1 (p) by the separation from which we can see that the first term on the RHS gives, upon insertion into (17), the IR-divergent contribution for the coefficient of the lowest order inverse propagator for the on-shell limit ∆ → 0. The second term does not produce an IR-divergence and the remaining terms vanish faster than ∆ in the on-shell limit so that they do not contribute to the field renormalization factor which we seek to isolate. In this way we get finally which agrees with (14,15,16) with One can see that the result in (16) differs from the corresponding result in QED in eq.(5.13) of Ref. [19] by the replacement of the electron charges e by the gravity charges κpμ, κp ′ν with the corresponding replacement of the photon propagator numerator −iη µν by the graviton propagator numerator i 1 2 (η µν ημν + η µν ημ ν − η µμ η νν ). That the squared modulus of these gravity charges grows quadratically in the deep Euclidean regime is what makes their effect therein in the quantum theory of general relativity fundamentally different from the effect of the QED charges in the deep Euclidean regime of QED, where the latter charges are constants order-by-order in perturbation theory.
Indeed, proceeding recursively, we write where here the notation indicates that the residual β m does not contain the leading infrared contribution for k m that is given by the first term on the RHS of (22) 4 . We iterate (22) to get The symmetry of ρ m implies that the quantity in curly brackets is also symmetric in the interchange of k m−1 and k m . We indicate this explicitly with the notation Repeated application of (22) and use of the symmetry of ρ m leads us finally to the exact result where the case m=1 has already been considered in (14) with ρ 0 ≡ β 0 . Here, we defined as well β (i) i ≡ β i . We can use the symmetry of the residuals β i to re-write ρ m as so that we finally obtain, upon substitution into (13), and the identification we introduce the result (27) into (9) via (10) to get In this way, our resummed exact result for the complete scalar propagator in quantum general relativity is seen to be [8,[10][11][12] i∆ We have introduced the shorthand "rsm" for "resummed" in the last line of (31) for later convenience.
This result (32) becomes identical to (7) when we take the limit n → 4 in it. In taking this limit, we note that B ′′ g (k) is UV finite so that the limit exists without further ado. As the IR limit of the coupling of the graviton to a particle is well-known [34] to independent of its spin, the entirely analogous result to (32) holds for the propagators of all particles [8,[10][11][12] with corresponding exponent B ′′ g (k) and the attendant IR-improved proper self-energy function. We note that in Σ ′ s (p) the limit n → 4 can be taken if we represent it by its IR-improved propagator expansion in which, to any finite order in the loop expansion, the usual free Feynman propagator is replaced by its resummed version with the attendant IR-improved proper self-energy function, Σ ′ s (p) or its graviton analog, set to zero on at least one internal line (per loop): for the scalar case this reads with a corresponding result for the graviton case. Standard resummation algebra then can be used to remove any double counting effects to any finite order in the loop expansion, as B ′′ g (k) is a UV finite one-loop effect. Let us now see how one proves this last remark.
To this end, let Γ ℓ,m (k 1 , . . . , k ℓ ; k ′ 1 , . . . , k ′ m ) be the 1PI ℓ-graviton, m-scalar proper vertex function, where we suppress all Lorentz indices without loss of content. We follow Ref. [36] and write Γ ℓ,m (k 1 , . . . , k ℓ ; k ′ 1 , . . . , k ′ m ) in terms of its skeleton expansion in which, to any finite order in the respective loop expansion, each graph G is mapped into a unique skeleton S in which all corrections to propagators and interaction vertices are removed. We then have the identification following the recipe in Ref. [36] so that here one uses the complete propagators, ∆ ′ F , D ′ F , for the scalar and the graviton on the lines of the skeleton and one uses the complete interaction vertex foundations {Γ j } at each respective vertex in the skeleton to produce the exact, complete result for Γ ℓ,m (k 1 , . . . , k ℓ ; k ′ 1 , . . . , k ′ m ). In this representation, it is immediate how to obtain the attendant N-th loop result accurate up to and including the N-th loop for Γ ℓ,m (k 1 , . . . , k ℓ ; k ′ 1 , . . . , k ′ m ): one expands the propagators and complete interaction vertices to the appropriate order, ≤ N and retains all terms with ≤ N loops in the sum on the RHS of (34). In the case of the exact scalar propagator, for example, we expand it as usual in each term in (34), and we stop at the term with N-factors of (−iΣ s (k)) each one of which we evaluate only to one loop order in this last term, with the attendant higher loop evaluations in the terms with less than N factors by the standard methodology. Inserting this result into (34) with the analogous ones for the graviton propagator and the interaction vertices we isolate the result accurate up to and including the N-th loop by dropping all contributions that involve more than N-loops. This is the standard Feynman diagrammatic practice.
Since we have the n-dimensional regulation of the UV divergences, the result we obtain this way is UV finite.
To improve it we substitute the resummed representation for the propagators, which we denote as we have above so that we have To obtain the IR-improved result correct up to an including the N-th IR-improved loop, we repeat the same steps as we did for the un-improved case: for example, we expand the scalar propagator as where we now stop the expansion at the term with N-factors of (−iΣ ′ s (p)) in which each factor is only computed to one-loop order. We then introduce this IR-improved N-loop result for the scalar propagator and the analogous results for the graviton propagator and the interaction vertices's accurate as well to N loops in the IR-improved loops into the the RHS of (36) and drop all terms with more than N IR-improved loops. The result is now UV finite because the exponential factor in the respective propagators render the integration in deep UV finite for any finite order in the interaction strength κ because these exponential factors fall faster than any of the finite powers of the loop momenta that occur at finite orders in κ as given by the Feynman rules that follow from Refs. [30,31] for (6).
Finally, we observe that (12) can be inverted to give as well the identity This allows us to employ the same result (36) in calculating the IR-improved self-energy so that it too is now UV finite with our IR-improved resummation prescription. It follows that, to any finite order in the IR-improved loop expansion, all Γ ℓ,m (k 1 , . . . , k ℓ ; k ′ 1 , . . . , k ′ m ) are UV finite. QED As we have indicated above [8] and as Weinberg has shown in Ref. [34], the IR limit of the coupling of the graviton to a particle is independent of its spin, so that we get the same exponential behavior in the resummed propagator for all particles in the Standard Model. Indeed, when we use our resummed propagator results, as extended to all the particles in the SM Lagrangian and to the graviton itself, working now with the complete theory where L G SM (x) is SM Lagrangian written in diffeomorphism invariant form as explained in Refs. [8,10], we show in the Refs. [8][9][10][11][12][13][14][15][16][17] that the denominator for the propagation of transverse-traceless modes of the graviton becomes (M P l is the Planck mass) where we have defined with I 2 defined [8][9][10][11][12][13][14][15][16][17] by and with λ c (j) = 2m 2 j πM 2 P l and [8][9][10][11][12][13][14][15][16][17] n j equal to the number of effective degrees of particle j. For completeness, we repeat the derivation of (40) in our Appendix 2, using results from Appendix 3. In arriving at the numerical value in (41), we take the SM masses as follows: for the now presumed three massive neutrinos [37,38], we estimate a mass at ∼ 3 eV; for the remaining members of the known three generations of Dirac fermions {e, µ, τ, u, d, s, c, b, t}, we use [39][40][41] 19 GeV, respectively. We set the Higgs mass at m H ∼ = 120GeV, in view of the limit from LEP2 [42,43]. We note that (see the Appendix 1) when the rest mass of particle j is zero, such as it is for the photon and the gluon, the value of m j turns-out to be √ 2 times the gravitational infrared cut-off mass [27,28], which is m g ∼ = 3.1 × 10 −33 eV. We further note that, from the exact one-loop analysis of Ref. [44], it also follows (see Appendix 2) that the value of n j for the graviton and its attendant ghost is 42. For λ c → 0, we have found the approximate representation (see Appendix 3) These results allow us to identify (we use G N for G N (0)) and to compute the UV limit g * as We stress that this result has no threshold/cut-off effects in it. It is a pure property of the known world.
For reference, the UV fixed-point calculated here, (g * , λ * ) ∼ = (0.0442, 0.0817), can be compared with the estimates in Refs. [22,23], which give (g * , λ * ) ≈ (0.27, 0.36). In making this comparison, one must keep in mind that the analysis in Refs. [22,23] did not include the specific SM matter action and that there is definitely cut-off function sensitivity to the results in the latter analyses. What is important is that the qualitative results that g * and λ * are both positive and are less than 1 in size are true of our results as well.

An Estimate of Λ
To see that the results here, taken together with those in Refs. [22,23], allow us to estimate the value of Λ today, we take the normal-ordered form of Einstein's equation : G µν : +Λ : g µν := −κ 2 : T µν : .
The coherent state representation of the thermal density matrix then gives the Einstein equation in the form of thermally averaged quantities with Λ given by our result in (48) summed over the degrees of freedom as specified above in lowest order. In Ref. [23], it is argued that the Planck scale cosmology description of inflation needs the transition time between the Planck regime and the classical Friedmann-Robertson-Walker regime at t tr ∼ 25t P l . We thus introduce and use the arguments in Refs. [46] (t eq is the time of matter-radiation equality) to get the first principles estimate, from the method of the operator field, where we take the age of the universe to be t 0 ∼ = 13.7 × 10 9 yrs. In the latter estimate, the first factor in the second line comes from the period from t tr to t eq which is radiation dominated and the second factor comes from the period from t eq to t 0 which is matter dominated 6 . This estimate should be compared with the experimental result [28] 7 ρ Λ (t 0 )| expt ∼ = (2.368 × 10 −3 eV (1 ± 0.023)) 4 .
To sum up, in addition to our having put the Planck scale cosmology [22,23] on a more rigorous basis, we believe our estimate of ρ Λ (t 0 ) represents some amount of progress in the long effort to understand its observed value in quantum field theory. Evidently, the estimate is not a precision prediction, as hitherto unseen degrees of freedom may exist and they have not been included. Moreover, the value of t tr cannot be taken as precise, yet. We do look forward, however, to additional possible checks from experiment, to which we return elsewhere [29].

Appendix 1: Evaluation of Gravitational Infrared Exponent
In the text, we use several limits of the gravitational infrared exponent B ′′ g defined in (28). Here, we present these evaluations for completeness.
We have to consider where ∆ = p 2 − m 2 . The integral on the RHS of (53) is given by for∆ = 1 − m 2 /p 2 ,λ 2 = λ 2 /p 2 andǭ = ǫ/p 2 . In this way, we arrive at the results, for p 2 < 0, where we have made more explicit the presence of the observed small mass, m g , of the graviton. When m=0 and one wants to use dimensional regularization for the IR regime instead of m g , we normalize the propagator at a Euclidean point k 2 = −µ 2 and use standard factorization arguments [48][49][50][51][52] to take the factorized result for B ′′ g from (55) as In physical applications, such mass singularities are absorbed by the definition of the initial state "parton" densities and/or are canceled by the KLN theorem in the final state; we do not exponentiate them in the exactly massless case. We stress that the standard analytic properties of the 1PI 2pt functions obtain here, as we use standard Feynman rules. Wick rotation changes the Minkowski space Feynman loop integral d 4 k with k = (k 0 , k 1 , k 2 , k 3 ) for real k j and 1, 1, 1). Thus our results rigorously correspond to |p 2 | = −p 2 in (55), (56) with m 2 replaced with m 2 − iǫ, with ǫ ↓ 0, following Feynman, for p 2 < 0; by Wick rotation this is the regime relevant to the UV behavior of the Feynman loop integral. Standard complex variables theory then uniquely specifies our exponent for any value of p 2 .

Appendix 2: Graviton Inverse Propagator
To obtain the result in (40) we first consider [8] the diagrams in Figs. 2 and 3. These graphs have a superficial degree of divergence in the UV of +4 and are a test of our methods because, in the usual treatment of the theory, they generate a UV divergence in the respective 1PI 2-point function for the coefficient of q 4 which can not be removed by the standard field and mass renormalizations.
For example, consider the graph in Fig. 3a. When we use our resummed propagators, we get (here, k → (ik 0 , k) by Wick rotation, and we work in the transverse-traceless space) We see explicitly that the exponential damping in the deep Euclidean regime has rendered so that we need to evaluate the transverse, traceless self-energy function Σ T (q 2 ) that follows from eq.(57) for Fig. 3a and its analogs for Figs. 3b and 2 by the standard methods. Here, we work in the expectation that, in consequence to the newly UV finite calculated quantum loop effects in Figs. 2 and 3, the Fourier transform of the graviton propagator that enters Newton's law, our ultimate goal here, will receive support from from |q| 2 << M 2 P l . We will therefore work in the limit that q 2 /M 2 P l is relatively small, .1, for example 8 . This will allow us to see the dominant effects of our new finite quantum loop effects. In other words, we will work to ∼ 10% (leading-log) accuracy in what follows. See Appendix 2 for more discussion on this point.
First let us dispense with the contributions from Figs. 2b and Fig. 3b. These are independent of q 2 so that we use a mass counter-term to remove them and set the graviton mass to 0. Following the suggestion of Feynman in Ref. [31], we will change this to a small non-zero value below to take into account the recently established small value of the cosmological constant [27,28]. See also the discussion in Ref. [53][54][55][56] where it is shown that the quantum fluctuations in the exact de Sitter metric implied by the non-zero cosmological constant correspond in general to a mass for the graviton. Here, as we expand about a flat background, we take this effect into account as a small infrared regulator for the graviton. The deviations from flat space in the deep Euclidean region that we study due to the observed value of the cosmological constant are at the level of e 10 −61 − 1! This is safely well beyond the accuracy of our methods.
Returning to Fig. 3a, when we project onto the transverse, traceless space, that is to say, the graviton helicity space {e µν (±2) = ǫ µ ± ǫ ν ± , where ǫ ν ± = ±(x ± iŷ)/ √ 2 when x,ŷ are purely space-like and( x, ŷ, q/| q|) form a right-handed coordinate basis}, we get (see the Appendix 3) the result where λ c = 2m 2 πM 2 P l ,d = α(1 − α) q 2 /m 2 so that we have made the substitution x = k 2 and imposed the mass counter-term as we noted. We have taken for definiteness q = (0, q). We also use q = | q| when there is no chance for confusion. We are evaluating (59) in the deep UV where m 2 /q 2 << 1 and where q 2 /M 2 P l 0.1 -see footnote 8. Accordingly, we get where Using the usual field renormalization, we see that Fig. 3a makes the contribution to the transverse traceless graviton proper self-energy function.
Turning now to Figs. 2, the pure gravity loops, we use a contact between our work and that of Refs. [44]. In Refs. [44], the entire set of one-loop divergences has been computed for the theory in (6). The basic observation is the following. As we work only to the leading logarithmic accuracy in ln λ c , it is sufficient to identify the correspondence between the divergences as calculated in the n-dimensional regularization scheme in Ref. [44] and as they would occur when λ c → 0. This we do by comparing our result for (59) when q 2 → 0 with the corresponding result in Ref. [44] for the same theory. In this way we see that we have the correspondence This allows us to read-off the leading log result for the pure gravity loops directly from the results in Ref. [44]. Since − ln λ c = ln M P l 2 − ln m 2 − ln 2 π , we see that our exponentiated propagators have cut-off our UV divergences at the scale ∼ M P l and the correspondence in (63) shows the usual relation between the effective UV cut-off scale and the pole in (2 − n/2) in dimensional regularization. Note as well that, if the small cosmological constant [27,28] is set to zero 9 , the graviton is then exactly massless and we normalize its propagator at a Euclidean point p 2 = −µ 2 as is standard for massless non-Abelian gauge theories for example. It follows that for the graviton case and for all other cases where m = 0, as we explain in Appendix 1 (see eq.(56), the mass m in (63) is replaced with m = µ -there is no zero mass divergence in the case that the mass of the respective particle is zero. The UV correspondence is the same in both the m = 0 and m = 0 cases.
Specifically, the result in Ref. [44], when interpreted as we have just explained, is that the pure gravity loops give a factor of 42 times the scalar loops for the coefficient a 2 above when we work in the regime where |q 2 | is relatively small compared to M 2 P l . Here, we again take into account the recent evidence for a non-zero cosmological constant [27,28], which can be seen to provide the small non-zero rest mass for the graviton, m g ∼ = 3.1 × 10 −33 eV, which serves as an IR regulator for the graviton. This is the value of rest mass in λ c which should be used for pure gravitational loops -see footnote 9 for more discussion on this point relevant to Refs. [57,58]. See the Appendix 1 for the derivation of the corresponding infrared exponents.
We note that, for λ c = 0, the constant c 2 is infinite and, as we have already imposed both the mass and field renormalization counter-terms, there would be no physical parameter into which that infinity could be absorbed: this is just another manifestation that QGR, without our resummation, is a non-renormalizable theory.
Using the universality of the coupling of the graviton when the momentum transfer scale is relatively small compared to M P l , we can extend the result for the scalar field above to the remaining known particles in the Standard Model by counting the number of physical degrees of freedom for each such particle and replacing the mass of the scalar with the respective mass of that particle. For a massive fermion we get a factor of 4 relative to the scalar result with the appropriate change in the mass parameter from m to m f , the mass of that fermion, for a massive vector, we get a factor of 3 relative to the scalar result, with the corresponding change in the mass from m to m V , the mass of that vector, etc. In this way, we arrive at the result that the denominator of the graviton 9 For the reader unfamiliar with Feynman's original observation [31] that, in his approach to QGR, one of the main effects of the cosmological constant is to give the quantum graviton field h µν a mass, we recall Einstein's equation R µν − 1 2 g µν R + Λg µν = −κ 2 T µν , with R µν and T µν the respective Ricci and energy-momentum tensors. For g µν = η µν + 2κh µν , we get R µν = κr µν + O(κ 2 ), with r µν = h µν − ∂ α ∂ µ h α ν − ∂ α ∂ ν h α µ + ∂ µ ∂ ν h α α so that, absorbing the Λη µν term into the normal ordering constantη µν term in T µν , we get the result r µν − 1 2 η µν r α α + 2Λh µν = κT ′ µν where here T ′ µν is now the normal ordered energy-momentum tensor, including the contribution from the graviton itself. This result shows that the field h µν , as already noted by Feynman [31], now has mass-squared 2Λ working to leading order in Λ. We treat this as an IR regulator mass for a massless spin 2 field in Minkowski space over the Planck scale distances with which we work. Indeed, the non-zero value of Λ means the background metric should be of de Sitter type and this avoids the problems noted in Refs. [57,58] associated with a graviton mass different from zero in Minkowski space, as we explained further in the text above.
propagator becomes, in the Standard Model, where we have defined with I 2 defined above and with λ c (j) = 2m 2 j πM 2 P l and [11] n j equal to the number of effective degrees of particle j as already illustrated. The values for Standard Model masses used in arriving at the numerical value for c 2,ef f in (65) are explained in the text. We also note that (see Appendix 3) for λ c → 0, we have found the approximate representation The results (64), (65) and (66) have been used in the text.

Appendix 3: Evaluation of Gravitationally Regulated Loop Integrals
In this section we present the derivation of the representations which we have used in the text in evaluating the gravitationally regulated loop integrals in Figs. 2,3.
Considering the integrals in Fig. 3 to show the methods, we need the result for In the limit that |q 2 | << M 2 P l , standard symmetric integration methods give us, for the transverse parts, where we have k 4 e λc(k 2 /m 2 ) ln(m 2 /(m 2 +k 2 )) [k 2 + m 2 + |q 2 |α(1 − α)] 2 (69) and where we used the symmetrization, valid under the respective integral sign, and λ c = 2m 2 /(πM 2 P l ). The integral I 0 , with the use of the mass counter-term, then leads us to evaluate the difference, where we define hered = |q 2 |α(1 − α)/m 2 . It is seen that the dominant part of the integrals comes from the regime where x ∼ 1/(ρλ c ) with ρ = − ln λ c , so that we may finally write where we have defined I 1 (λ c ) = The result (72) has been used in the text.
The integral I 1 is a field renormalization constant so, in the usual renormalization program, we do not need it for most of the applications. Here, we will discuss it as well for completeness. We get where, as above, we use Thus, we get Finally, let us show why we can neglect the termsd that were in the denominators of I j , j = 1, 2. It is enough to look into the differences where we note that the integral I 1 is absorbed by the standard field renormalization where here for convenience we do this at |q 2 | = 0 when we neglectd in the denominator of I 1 or at the zero of the respective graviton propagator away from the origin otherwise. From this perspective, the main integral to examine to illustrate the level of our approximation becomes where we have defined σ = λ cd . The approximation, valid for small values of σ, 1 − e −α 2 σ = 2e −α 2 σ/2 sinh(α 2 σ/2) ∼ = α 2 σe −α 2 σ/2 (78) then allows us to get ∆I 2 ∼ = 4σ ∂ 2 ∂σ 2 ∞ 0 dr e −rρ 1 − λ c + σ/2 r + λ c + σ/2 ∼ = 2 + ρσ + 2ρσ(1 + 1 4 ρσ)e ρσ/2 (C + ln(ρσ/2) + ∞ n=1 (−1) n (ρσ/2) n n n! ) which shows that this difference is indeed non-leading log. The analogous analysis holds for ∆I 1 as well.