**3.** *g*<sup>∗</sup> **and** *λ*<sup>∗</sup> **in resummed quantum gravity**

We start with the prediction for *g*∗, which we already presented in Refs. [9–19]. For the sake of completeness, let us we recapitulate the main steps in the calculation. Referring to Fig. 1, we have shown in Refs. [9–18] 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

$$\dot{\iota} \Delta\_F'(k)|\_{\text{sommed}} = \frac{\dot{\iota} e^{B\_{\text{g}}''(k)}}{(k^2 - m^2 - \Sigma\_{\text{s}}' + i\varepsilon)}\tag{5}$$

where we define for convenience −Σ*s*,0(*k*) = −Σ�

<sup>L</sup>(*x*) = <sup>1</sup>

2*κ*<sup>2</sup>

orders in *κ*2/(4*π*) for all values of *k*2. QED.

in Resummed Quantum Gravity: An Estimate of

with *I*<sup>2</sup> given in Refs. [9–18] by

*j πM*<sup>2</sup> *Pl*

and with *<sup>λ</sup>c*(*j*) = <sup>2</sup>*m*<sup>2</sup>

Refs. [9–18],

where *L*G

mass)

for

*<sup>s</sup>*,0(*k*) = <sup>Δ</sup>−<sup>1</sup>

360*πM*<sup>2</sup> *Pl*

. *nj* is the number of effective degrees of freedom [9–18] of particle

*dxx*3(1 + *x*)−4−*λcx* (12)

*njI*2(*λc*(*j*))

Feynman diagram contribution to Σ*s*(*k*) corresponds to a unique contribution to Σ�

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 (see Refs. [9– 19]) of (1) plus the SM Lagrangian written in diffeomorphism invariant form as explained in

<sup>31</sup> Planck Scale Cosmology and Asymptotic Safety

in Refs. [9–18], the denominator of the graviton propagator becomes [9–18] (*MPl* is the Planck

*<sup>q</sup>*<sup>2</sup> <sup>+</sup> <sup>Σ</sup>*T*(*q*2) + *<sup>i</sup>�* <sup>∼</sup><sup>=</sup> *<sup>q</sup>*<sup>2</sup> <sup>−</sup> *<sup>q</sup>*<sup>4</sup> *<sup>c</sup>*2,*eff*

SM particles j

*j* of mass *mj*. In *c*2,*eff* in (11), we take the SM masses as explained in Refs. [9–19] following Refs. [29–31, 33–35] : for the now presumed three massive neutrinos [33], 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 [34] *me* ∼= 0.51 MeV, *m<sup>μ</sup>* ∼= 0.106 GeV, *m<sup>τ</sup>* ∼= 1.78 GeV, *mu* ∼= 5.1 MeV, *md* ∼= 8.9 MeV, *ms* ∼= 0.17 GeV, *mc* ∼= 1.3 GeV, *mb* ∼= 4.5 GeV and *mt* ∼= 174 GeV and for the massive vector bosons *W*±, *Z* we use the masses *MW* ∼= 80.4 GeV, *MZ* ∼= 91.19 GeV, respectively. We set the Higgs mass at *mH* ∼= 120GeV, in view of the limit from LEP2 [35]. We note that (see the Appendix 1 here and the Appendix 1 in Ref. [9]) when the rest mass of particle *<sup>j</sup>* is zero, such as it is for the photon and the gluon, the value of *mj* turns-out to be <sup>√</sup><sup>2</sup> times the gravitational infrared cut-off mass [29–31], which is *mg* <sup>∼</sup><sup>=</sup> 3.1 <sup>×</sup> <sup>10</sup>−33eV. We also note that from Ref.[36] it also follows that the value of *nj* for the graviton and its attendant

<sup>∼</sup><sup>=</sup> 2.56 <sup>×</sup> 104

0

*c*2,*eff* = ∑

*<sup>I</sup>*2(*λc*) = <sup>∞</sup>

ghost is 42. For *λ<sup>c</sup>* → 0, we have found the approximate representation

*λc*

<sup>−</sup> ln ln <sup>1</sup> *λc*

*GN*(*k*) = *GN*/(1 +

<sup>−</sup> ln ln <sup>1</sup>

*c*2,*eff k*<sup>2</sup> 360*πM*<sup>2</sup> *Pl*

ln <sup>1</sup>

*λc*

*λc*

<sup>−</sup> <sup>11</sup>

<sup>6</sup> . (13)

) (14)

*<sup>λ</sup><sup>c</sup>* <sup>−</sup> ln ln <sup>1</sup>

*<sup>I</sup>*2(*λc*) <sup>∼</sup><sup>=</sup> ln <sup>1</sup>

We thus identify (we use *GN* for *GN*(0))

<sup>−</sup>*<sup>g</sup>* (*<sup>R</sup>* <sup>−</sup> <sup>2</sup>Λ) <sup>+</sup> <sup>−</sup>*gL*<sup>G</sup>

*SM*(*x*) is the SM Lagrangian written in diffeomorphism invariant form as explained

*<sup>F</sup>* (*k*). This proves that every

*SM*(*x*) (9)

, (10)

*<sup>s</sup>*(*k*) to all

(11)

Fig. 1. Graviton loop contributions to the scalar propagator. *q* is the 4-momentum of the scalar.

for (<sup>Δ</sup> <sup>=</sup> *<sup>k</sup>*<sup>2</sup> <sup>−</sup> *<sup>m</sup>*2)

$$\begin{split}B\_{\mathcal{S}}^{\prime\prime}(k) &= -2i\kappa^2 k^4 \frac{\int d^4\ell}{16\pi^4} \frac{1}{\ell^2 - \lambda^2 + i\epsilon} \\ &\quad \frac{1}{(\ell^2 + 2\ell k + \Delta + i\epsilon)^2} \\ &= \frac{\kappa^2 |k^2|}{8\pi^2} \ln\left(\frac{m^2}{m^2 + |k^2|}\right), \end{split} \tag{6}$$

where the latter form holds for the UV regime, so that (5) the resummed scalar propagator falls faster than any power of <sup>|</sup>*k*2|. An analogous result [9–18] holds for *<sup>m</sup>* <sup>=</sup> 0 (See Appendix 1.). As Σ� *<sup>s</sup>*, the residual self-energy function, starts in <sup>O</sup>(*κ*2), we may drop it in calculating one-loop effects. It follows that, when the respective analogs of *i*Δ� *<sup>F</sup>*(*k*)|resummed are used for the elementary particles, all quantum gravity loop corrections are UV finite [9–18].

We stress that our resummed scalar propagator representation (5) is not limited to the regime where *k*<sup>2</sup> ∼= *m*<sup>2</sup> but is an identity that holds for all *k*<sup>2</sup> – see Refs. [9–18]. This is readily shown as follows. If we invert both sides of (5) and recall the formula for the exact inverse propagator, we get

$$
\Delta\_{\rm F}^{-1}(k) - \Sigma\_{\rm s}(k) = (\Delta\_{\rm F}^{-1}(k) - \Sigma\_{\rm s}'(k))e^{-\mathcal{B}\_{\rm g}^{\prime\prime}(k)}\tag{7}
$$

where the free inverse propagator is Δ−<sup>1</sup> *<sup>F</sup>* (*k*) = Δ(*k*) + *i�* and Σ*s*(*k*) is the exact proper self-energy part. We introduce here the loop expansions

$$\begin{aligned} \Sigma\_{\rm s}(k) &= \sum\_{n=1}^{\infty} \Sigma\_{\rm s,\mathcal{U}}(k) \left( \frac{\kappa^2}{4\pi^2} \right)^n \\ \Sigma\_{\rm s}'(k) &= \sum\_{n=1}^{\infty} \Sigma\_{\rm s,\mathcal{U}}'(k) \left( \frac{\kappa^2}{4\pi^2} \right)^n \end{aligned}$$

and we get, from elementary algebra, the exact relation

$$-\Sigma\_{\rm s,n}(k) = -\sum\_{j=0}^{n} \Sigma\_{\rm s,j}'(k) \left(\frac{-4\pi^2 B\_{\rm g}^{\prime\prime}(k)}{\kappa^2}\right)^{n-j} / (n-j)! \tag{8}$$

where we define for convenience −Σ*s*,0(*k*) = −Σ� *<sup>s</sup>*,0(*k*) = <sup>Δ</sup>−<sup>1</sup> *<sup>F</sup>* (*k*). This proves that every Feynman diagram contribution to Σ*s*(*k*) corresponds to a unique contribution to Σ� *<sup>s</sup>*(*k*) to all orders in *κ*2/(4*π*) for all values of *k*2. QED.

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 (see Refs. [9– 19]) of (1) plus the SM Lagrangian written in diffeomorphism invariant form as explained in Refs. [9–18],

$$\mathcal{L}(\mathbf{x}) = \frac{1}{2\mathbf{x}^2}\sqrt{-\mathbf{g}}\left(\mathbf{R} - 2\boldsymbol{\Lambda}\right) + \sqrt{-\mathbf{g}}L\_{SM}^{\mathcal{G}}(\mathbf{x})\tag{9}$$

where *L*G *SM*(*x*) is the SM Lagrangian written in diffeomorphism invariant form as explained in Refs. [9–18], the denominator of the graviton propagator becomes [9–18] (*MPl* is the Planck mass)

$$q^2 + \Sigma^T(q^2) + i\epsilon \cong q^2 - q^4 \frac{c\_{2,eff}}{360\pi M\_{Pl}^2},\tag{10}$$

for

4 Will-be-set-by-IN-TECH

 q ✒

1 <sup>2</sup> − *<sup>λ</sup>*<sup>2</sup> + *<sup>i</sup>�*

> ,

☛ ✡✁

✓✓✂✁ ✟✂✁ ☛✄ ✂

✡ ✄ ✒✒✠ ✄ ✁

❅ ❅❅❘

k ✛

> ✂ ✄

✏✁ ✏✄ ✂ ✠✟

✑ ✑

+ ···

*<sup>F</sup>*(*k*)|resummed are used for

*<sup>g</sup>* (*k*) (7)

/(*n* − *j*)! (8)

(6)

<sup>q</sup> ✲ ☛ ✡✁

scalar.

1.). As Σ�

we get

for (<sup>Δ</sup> <sup>=</sup> *<sup>k</sup>*<sup>2</sup> <sup>−</sup> *<sup>m</sup>*2)

✓✓✂✁ ✟✂✁ ☛✄ ✂

k ✛

k + q

*B*��

one-loop effects. It follows that, when the respective analogs of *i*Δ�

Δ−<sup>1</sup>

self-energy part. We introduce here the loop expansions

and we get, from elementary algebra, the exact relation

− Σ*s*,*n*(*k*) = −

where the free inverse propagator is Δ−<sup>1</sup>

*<sup>g</sup>* (*k*) = <sup>−</sup>2*iκ*2*k*<sup>4</sup>

<sup>=</sup> *<sup>κ</sup>*2|*k*2<sup>|</sup> <sup>8</sup>*π*<sup>2</sup> ln

the elementary particles, all quantum gravity loop corrections are UV finite [9–18].

*<sup>F</sup>* (*k*) <sup>−</sup> <sup>Σ</sup>*s*(*k*)=(Δ−<sup>1</sup>

Σ*s*(*k*) =

*n* ∑ *j*=0 Σ� *s*,*j* (*k*)

Σ� *<sup>s</sup>*(*k*) =

∞ ∑ *n*=1

∞ ∑ *n*=1 Σ� *<sup>s</sup>*,*n*(*k*)

Σ*s*,*n*(*k*)

<sup>−</sup>4*π*2*B*��

*κ*2

✏✁ ✏✄ ✂ ✠✟

✲ ✲ +

(a) (b)

Fig. 1. Graviton loop contributions to the scalar propagator. *q* is the 4-momentum of the

 *d*<sup>4</sup> 16*π*<sup>4</sup>

1 (<sup>2</sup> + 2*k* + Δ + *i�*)<sup>2</sup>

> *m*<sup>2</sup> *<sup>m</sup>*<sup>2</sup> + |*k*2|

*<sup>s</sup>*, the residual self-energy function, starts in <sup>O</sup>(*κ*2), we may drop it in calculating

*<sup>F</sup>* (*k*) − Σ�

 *κ*<sup>2</sup> 4*π*<sup>2</sup>

 *κ*<sup>2</sup> 4*π*<sup>2</sup>

*<sup>g</sup>* (*k*)

*<sup>s</sup>*(*k*))*e* −*B*��

*n*

*n*

*n*−*<sup>j</sup>*

*<sup>F</sup>* (*k*) = Δ(*k*) + *i�* and Σ*s*(*k*) is the exact proper

where the latter form holds for the UV regime, so that (5) the resummed scalar propagator falls faster than any power of <sup>|</sup>*k*2|. An analogous result [9–18] holds for *<sup>m</sup>* <sup>=</sup> 0 (See Appendix

We stress that our resummed scalar propagator representation (5) is not limited to the regime where *k*<sup>2</sup> ∼= *m*<sup>2</sup> but is an identity that holds for all *k*<sup>2</sup> – see Refs. [9–18]. This is readily shown as follows. If we invert both sides of (5) and recall the formula for the exact inverse propagator,

$$\begin{split} c\_{2,eff} &= \sum\_{\text{SM particles j}} n\_j I\_2(\lambda\_c(j)) \\ &\cong 2.56 \times 10^4 \end{split} \tag{11}$$

with *I*<sup>2</sup> given in Refs. [9–18] by

$$I\_2(\lambda\_c) = \int\_0^\infty d\mathbf{x} \mathbf{x}^3 (1+\mathbf{x})^{-4-\lambda\_c \ge} \tag{12}$$

and with *<sup>λ</sup>c*(*j*) = <sup>2</sup>*m*<sup>2</sup> *j πM*<sup>2</sup> *Pl* . *nj* is the number of effective degrees of freedom [9–18] of particle *j* of mass *mj*. In *c*2,*eff* in (11), we take the SM masses as explained in Refs. [9–19] following Refs. [29–31, 33–35] : for the now presumed three massive neutrinos [33], 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 [34] *me* ∼= 0.51 MeV, *m<sup>μ</sup>* ∼= 0.106 GeV, *m<sup>τ</sup>* ∼= 1.78 GeV, *mu* ∼= 5.1 MeV, *md* ∼= 8.9 MeV, *ms* ∼= 0.17 GeV, *mc* ∼= 1.3 GeV, *mb* ∼= 4.5 GeV and *mt* ∼= 174 GeV and for the massive vector bosons *W*±, *Z* we use the masses *MW* ∼= 80.4 GeV, *MZ* ∼= 91.19 GeV, respectively. We set the Higgs mass at *mH* ∼= 120GeV, in view of the limit from LEP2 [35]. We note that (see the Appendix 1 here and the Appendix 1 in Ref. [9]) when the rest mass of particle *<sup>j</sup>* is zero, such as it is for the photon and the gluon, the value of *mj* turns-out to be <sup>√</sup><sup>2</sup> times the gravitational infrared cut-off mass [29–31], which is *mg* <sup>∼</sup><sup>=</sup> 3.1 <sup>×</sup> <sup>10</sup>−33eV. We also note that from Ref.[36] it also follows that the value of *nj* for the graviton and its attendant ghost is 42. For *λ<sup>c</sup>* → 0, we have found the approximate representation

$$I\_2(\lambda\_\varepsilon) \cong \ln \frac{1}{\lambda\_\varepsilon} - \ln \ln \frac{1}{\lambda\_\varepsilon} - \frac{\ln \ln \frac{1}{\lambda\_\varepsilon}}{\ln \frac{1}{\lambda\_\varepsilon} - \ln \ln \frac{1}{\lambda\_\varepsilon}} - \frac{11}{6}.\tag{13}$$

We thus identify (we use *GN* for *GN*(0))

$$\mathbf{G}\_{N}(k) = \mathbf{G}\_{N} / \left( 1 + \frac{c\_{2,eff} k^2}{360 \pi M\_{Pl}^2} \right) \tag{14}$$

world – *λ*<sup>∗</sup> would vanish in an exactly supersymmetric theory. Our results for (*g*∗, *λ*∗) agree

<sup>33</sup> Planck Scale Cosmology and Asymptotic Safety

For reference, we note that, if we restrict our resummed quantum gravity calculations above

*g*<sup>∗</sup> = .0533, *λ*<sup>∗</sup> = −.000189 . We see that our results suggest that there is still significant cut-off effects in the results used for *g*∗, *λ*<sup>∗</sup> in Refs. [2, 25], which already seem to include an effective matter contribution when viewed from our resummed quantum gravity perspective, as an artifact of the obvious gauge and cut-off dependences of the results. Indeed, from a purely quantum field theoretic point of

<sup>2</sup> <sup>&</sup>lt; *<sup>h</sup>*, <sup>R</sup>grav

lim *p*2/*k*<sup>2</sup>→∞

lim *p*2/*k*<sup>2</sup>→0

where *g*¯ is the general background metric, which is the Minkowski space metric *η* here, and

*<sup>k</sup>* , <sup>R</sup>gh

R*<sup>k</sup>* = 0,

for some Z*<sup>k</sup>* [2]. Here, the inner product is that defined in the second paper in Refs. [2] in its Eqs.(2.14,2.15,2.19). The result is that the modes with *p k* have a shift of their vacuum energy by the cut-off operator. There is therefore no disagreement in principle between our gauge invariant and cut-off independent results and the gauge dependent and cut-off dependent

To estimate the value of Λ today, we take the normal-ordered form of Einstein's equation,

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 above in lowest order. Taking the transition time between the Planck regime and the classical

8*πGN*(*ttr*)

*Pl*(*ktr*) <sup>64</sup> ∑ *j*

Friedmann-Robertson-Walker regime at *ttr* ∼ 25*tPl* from Refs. [25], we introduce

*<sup>ρ</sup>*Λ(*ttr*) <sup>≡</sup> <sup>Λ</sup>(*ttr*)

<sup>=</sup> <sup>−</sup>*M*<sup>4</sup>

<sup>R</sup>*<sup>k</sup>* <sup>→</sup> <sup>Z</sup>*kk*2,

*<sup>k</sup> <sup>h</sup>* <sup>&</sup>gt; <sup>+</sup> <sup>&</sup>lt; *<sup>C</sup>*¯, <sup>R</sup>gh

: *<sup>G</sup>μν* : <sup>+</sup><sup>Λ</sup> : *<sup>g</sup>μν* :<sup>=</sup> <sup>−</sup>*κ*<sup>2</sup> : *<sup>T</sup>μν* : . (21)

(−1)*Fnj ρ*2 *j*

*<sup>k</sup> C* > (20)

(22)

*<sup>k</sup>* implement the course graining as they

for *g*∗, *λ*<sup>∗</sup> to the pure gravity theory with no SM matter fields, we get the results

<sup>Δ</sup>*kS*(*h*, *<sup>C</sup>*, *<sup>C</sup>*¯; *<sup>g</sup>*¯) = <sup>1</sup>

*<sup>C</sup>*, *<sup>C</sup>*¯ are the ghost fields and the operators <sup>R</sup>grav

qualitatively with those in Refs. [2, 25].

in Resummed Quantum Gravity: An Estimate of

view, the cut-off action is

satisfy the limits

results in Refs. [2, 25].

**4. An estimate of** Λ

and compute the UV limit *g*<sup>∗</sup> as

$$\text{g}\_{\*} = \lim\_{k^{2} \to \infty} k^{2} G\_{N}(k^{2}) = \frac{360\pi}{\text{c}\_{2,eff}} \cong 0.0442.\tag{15}$$

We stress that this result has no threshold/cut-off or gauge effects in it. a pure property of the known world.

Turning now to *λ*∗, we use Einstein's equation

$$\mathbf{G}\_{\mu\nu} + \Lambda \mathbf{g}\_{\mu\nu} = -\kappa^2 T\_{\mu\nu} \tag{16}$$

in a standard notation where *<sup>G</sup>μν* <sup>=</sup> *<sup>R</sup>μν* <sup>−</sup> <sup>1</sup> <sup>2</sup>*Rgμν*, *Rμν* is the contracted Riemann tensor, and *Tμν* is the energy-momentum tensor. Working with the representation *gμν* = *ημν* + 2*κhμν* for the flat Minkowski metric *ημν* = diag(1, −1, −1, −1) we may isolate Λ in Einstein's equation by evaluating its VEV (vacuum expectation value). For any bosonic quantum field *ϕ* we use the point-splitting definition (here, : : denotes normal ordering)

$$\begin{aligned} \varphi(0)\varrho(0) &= \lim\_{\epsilon \to 0} \varrho(\epsilon)\varrho(0) \\ &= \lim\_{\epsilon \to 0} T(\varrho(\epsilon)\varrho(0)) \\ &= \lim\_{\epsilon \to 0} \{ : \left( \varrho(\epsilon)\varrho(0) \right) : + < 0 | T(\varrho(\epsilon)\varrho(0)) | 0 > \} \end{aligned} \tag{17}$$

where the limit is taken with time-like *�* <sup>≡</sup> (*�*,�0) <sup>→</sup> (0, 0, 0, 0) <sup>≡</sup> 0 respectively. A scalar then makes the contribution [9–18] to Λ given by3

$$\begin{split} \Lambda\_{s} &= -8\pi G\_{N} \frac{\int d^{4}k}{2(2\pi)^{4}} \frac{(2k\_{0}^{2})e^{-\lambda\_{\ell}(k^{2}/(2m^{2}))\ln(k^{2}/m^{2}+1)}}{k^{2}+m^{2}} \\ &\cong -8\pi G\_{N} [\frac{1}{G\_{N}^{2}64\rho^{2}}], \end{split} \tag{18}$$

where *ρ* = ln <sup>2</sup> *<sup>λ</sup><sup>c</sup>* and we have used the calculus of Refs. [9–18]. We note that the standard equal-time (anti-)commutation relations algebra realizations then show that a Dirac fermion contributes −4 times Λ*<sup>s</sup>* to Λ. The deep UV limit of Λ then becomes

$$\begin{aligned} \Lambda(k) & \underset{k^2 \to \infty}{\longrightarrow} k^2 \lambda\_\*, \\ \lambda\_\* &= -\frac{c\_{2\varepsilon ff}}{2880} \sum\_j (-1)^{F\_j} n\_j / \rho\_j^2 \\ & \cong 0.0817 \end{aligned} \tag{19}$$

where *Fj* is the fermion number of *j* and *ρ<sup>j</sup>* = *ρ*(*λc*(*mj*)). We see again that *λ*<sup>∗</sup> is free of threshold/cut-off effects and of gauge artifacts and is a pure prediction of our known

<sup>3</sup> We note the use here in the integrand of 2*k*<sup>2</sup> <sup>0</sup> rather than the 2( �*k*<sup>2</sup> + *m*2) in Ref. [19], to be consistent with *ω* = −1 [37] for the vacuum stress-energy tensor.

6 Will-be-set-by-IN-TECH

*<sup>k</sup>*<sup>2</sup>→<sup>∞</sup> *<sup>k</sup>*<sup>2</sup>*GN*(*k*2) = <sup>360</sup>*<sup>π</sup>*

We stress that this result has no threshold/cut-off or gauge effects in it. a pure property of the

*Tμν* is the energy-momentum tensor. Working with the representation *gμν* = *ημν* + 2*κhμν* for the flat Minkowski metric *ημν* = diag(1, −1, −1, −1) we may isolate Λ in Einstein's equation by evaluating its VEV (vacuum expectation value). For any bosonic quantum field *ϕ* we use

where the limit is taken with time-like *�* <sup>≡</sup> (*�*,�0) <sup>→</sup> (0, 0, 0, 0) <sup>≡</sup> 0 respectively. A scalar then

equal-time (anti-)commutation relations algebra realizations then show that a Dirac fermion

<sup>2880</sup> ∑ *j*

where *Fj* is the fermion number of *j* and *ρ<sup>j</sup>* = *ρ*(*λc*(*mj*)). We see again that *λ*<sup>∗</sup> is free of threshold/cut-off effects and of gauge artifacts and is a pure prediction of our known

<sup>0</sup> rather than the 2(

(2*k*<sup>2</sup>

*c*2,*eff*

{: (*ϕ*(*�*)*ϕ*(0)) : + < 0|*T*(*ϕ*(*�*)*ϕ*(0))|0 >}

*<sup>λ</sup><sup>c</sup>* and we have used the calculus of Refs. [9–18]. We note that the standard

(−1)*Fj*

*nj*/*ρ*<sup>2</sup> *j*

<sup>0</sup>)*e*−*λ<sup>c</sup>* (*k*2/(2*m*<sup>2</sup>))ln(*k*2/*m*<sup>2</sup>+1) *k*<sup>2</sup> + *m*<sup>2</sup>

*N*64*ρ*<sup>2</sup> ], (18)

�*k*<sup>2</sup> + *m*2) in Ref. [19], to be consistent with

∼= 0.0442. (15)

(17)

(19)

*<sup>G</sup>μν* <sup>+</sup> <sup>Λ</sup>*gμν* <sup>=</sup> <sup>−</sup>*κ*2*Tμν* (16)

<sup>2</sup>*Rgμν*, *Rμν* is the contracted Riemann tensor, and

*g*<sup>∗</sup> = lim

the point-splitting definition (here, : : denotes normal ordering)

*�*→0

= lim *�*→0

= lim *�*→0

Λ*<sup>s</sup>* = −8*πGN*

<sup>∼</sup><sup>=</sup> <sup>−</sup>8*πGN*[ <sup>1</sup>

contributes −4 times Λ*<sup>s</sup>* to Λ. The deep UV limit of Λ then becomes

Λ(*k*) −→

*<sup>k</sup>*<sup>2</sup>→<sup>∞</sup> *<sup>k</sup>*2*λ*∗,

*<sup>λ</sup>*<sup>∗</sup> <sup>=</sup> <sup>−</sup>*c*2,*eff*

∼= 0.0817

*ϕ*(*�*)*ϕ*(0)

 *d*4*k* 2(2*π*)<sup>4</sup>

*G*2

*T*(*ϕ*(*�*)*ϕ*(0))

*ϕ*(0)*ϕ*(0) = lim

and compute the UV limit *g*<sup>∗</sup> as

Turning now to *λ*∗, we use Einstein's equation

in a standard notation where *<sup>G</sup>μν* <sup>=</sup> *<sup>R</sup>μν* <sup>−</sup> <sup>1</sup>

makes the contribution [9–18] to Λ given by3

<sup>3</sup> We note the use here in the integrand of 2*k*<sup>2</sup>

*ω* = −1 [37] for the vacuum stress-energy tensor.

known world.

where *ρ* = ln <sup>2</sup>

world – *λ*<sup>∗</sup> would vanish in an exactly supersymmetric theory. Our results for (*g*∗, *λ*∗) agree qualitatively with those in Refs. [2, 25].

For reference, we note that, if we restrict our resummed quantum gravity calculations above for *g*∗, *λ*<sup>∗</sup> to the pure gravity theory with no SM matter fields, we get the results

$$g\_\* = .0533, \ \lambda\_\* = -.000189$$

. We see that our results suggest that there is still significant cut-off effects in the results used for *g*∗, *λ*<sup>∗</sup> in Refs. [2, 25], which already seem to include an effective matter contribution when viewed from our resummed quantum gravity perspective, as an artifact of the obvious gauge and cut-off dependences of the results. Indeed, from a purely quantum field theoretic point of view, the cut-off action is

$$
\Delta\_k \mathcal{S}(h, \mathbb{C}, \bar{\mathbb{C}}; \bar{\mathbb{g}}) = \frac{1}{2} < h, \mathcal{R}\_k^{\text{grav}} h > + < \bar{\mathbb{C}}, \mathcal{R}\_k^{\text{gh}} \mathbb{C} > \tag{20}
$$

where *g*¯ is the general background metric, which is the Minkowski space metric *η* here, and *<sup>C</sup>*, *<sup>C</sup>*¯ are the ghost fields and the operators <sup>R</sup>grav *<sup>k</sup>* , <sup>R</sup>gh *<sup>k</sup>* implement the course graining as they satisfy the limits

$$\lim\_{p^2/k^2 \to \infty} \mathcal{R}\_k = 0,$$

$$\lim\_{p^2/k^2 \to 0} \mathcal{R}\_k \to \mathfrak{Z}\_k k^2.$$

for some Z*<sup>k</sup>* [2]. Here, the inner product is that defined in the second paper in Refs. [2] in its Eqs.(2.14,2.15,2.19). The result is that the modes with *p k* have a shift of their vacuum energy by the cut-off operator. There is therefore no disagreement in principle between our gauge invariant and cut-off independent results and the gauge dependent and cut-off dependent results in Refs. [2, 25].
