**6. Appendix: Gravitational infrared exponent**

In the text, we use several limits of the gravitational infrared exponent *B*�� *<sup>g</sup>* defined in (6). This appendix contains these evaluations for completeness.

We have to consider

$$-B\_{\mathcal{S}}^{\prime\prime}(p) = \frac{2i\kappa^2 p^4}{16\pi^4} \int \frac{d^4k}{(k^2 - \lambda^2 + i\epsilon)} \frac{1}{(k^2 - 2kp + \Delta + i\epsilon)^2} \tag{24}$$

where <sup>Δ</sup> <sup>=</sup> *<sup>p</sup>*<sup>2</sup> <sup>−</sup> *<sup>m</sup>*2. The integral on the RHS of (24) is given by

$$\begin{split} I &= \int \frac{d^4k}{(k^2 - \lambda^2 + i\epsilon)} \frac{1}{(k^2 - 2kp + \Delta + i\epsilon)^2} \\ &= \frac{-i\pi^2}{p^2} \frac{1}{\mathbf{x}\_+ - \mathbf{x}\_-} \left[ \mathbf{x}\_+ \ln(1 - 1/(\sqrt{2}\mathbf{x}\_+)) - \mathbf{x}\_- \ln(1 - 1/(\sqrt{2}\mathbf{x}\_-)) \right] \end{split}$$

<sup>4</sup> The method of the operator field forces the vacuum energies to follow the same scaling as the non-vacuum excitations.

<sup>5</sup> See also Ref. [39] for an analysis that suggests a value for *ρ*Λ(*t*0) that is qualitatively similar to this experimental result.

with

8 Will-be-set-by-IN-TECH

and use the arguments in Refs. [38] (*teq* is the time of matter-radiation equality) to get the first

*tr*/(360*πM*<sup>2</sup>

)3

*Pl*(1.0362)2(−9.197 <sup>×</sup> <sup>10</sup>−3) 64

where we take the age of the universe to be *<sup>t</sup>*<sup>0</sup> <sup>∼</sup><sup>=</sup> 13.7 <sup>×</sup> 109 yrs. In the latter estimate, the first factor in the square bracket comes from the period from *ttr* to *teq* (radiation dominated) and the second factor comes from the period from *teq* to *t*<sup>0</sup> (matter dominated) 4. This estimate should be compared with the experimental result [29–31]5 *<sup>ρ</sup>*Λ(*t*0)|expt <sup>∼</sup><sup>=</sup> (2.368 <sup>×</sup> <sup>10</sup>−3*eV*(<sup>1</sup> <sup>±</sup>

To sum up, our estimate, while it is definitely encouraging, is not a precision prediction, as possible hitherto unseen degrees of freedom have not been included and *ttr* is not precise, yet.

We thank Profs. L. Alvarez-Gaume and W. Hollik for the support and kind hospitality of the CERN TH Division and the Werner-Heisenberg-Institut, MPI, Munich, respectively, where a

<sup>64</sup> ∑

*Pl*))<sup>2</sup>

*j*

(25)<sup>2</sup> *t* 2 0

1

2*x*+)) − *x*<sup>−</sup> ln(1 − 1/(

(*k*<sup>2</sup> <sup>−</sup> <sup>2</sup>*kp* <sup>+</sup> <sup>Δ</sup> <sup>+</sup> *<sup>i</sup>�*)<sup>2</sup> (24)

√ <sup>2</sup>*x*−))

(−1)*Fnj ρ*2 *j*

(23)

*<sup>g</sup>* defined in (6). This

*Pl*(<sup>1</sup> <sup>+</sup> *<sup>c</sup>*2,*eff <sup>k</sup>*<sup>2</sup>

× *t* 2 *tr t* 2 *eq* × ( *t* 2/3 *eq t* 2/3 0

<sup>∼</sup><sup>=</sup> (2.400 <sup>×</sup> <sup>10</sup>−3*eV*)4.

In the text, we use several limits of the gravitational infrared exponent *B*��

*d*4*k*

(*k*<sup>2</sup> − *<sup>λ</sup>*<sup>2</sup> + *<sup>i</sup>�*)

√

<sup>4</sup> The method of the operator field forces the vacuum energies to follow the same scaling as the

<sup>5</sup> See also Ref. [39] for an analysis that suggests a value for *ρ*Λ(*t*0) that is qualitatively similar to this

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

*x*<sup>+</sup> ln(1 − 1/(

principles estimate, from the method of the operator field,

<sup>∼</sup><sup>=</sup> <sup>−</sup>*M*<sup>2</sup>

*<sup>ρ</sup>*Λ(*t*0) <sup>∼</sup><sup>=</sup> <sup>−</sup>*M*<sup>4</sup>

0.023))4.

**5. Acknowledgments**

part of this work was done.

We have to consider

**6. Appendix: Gravitational infrared exponent**

− *B*��

*d*4*k*

(*k*<sup>2</sup> − *<sup>λ</sup>*<sup>2</sup> + *<sup>i</sup>�*)

1 *x*<sup>+</sup> − *x*<sup>−</sup>

*I* =

non-vacuum excitations.

experimental result.

<sup>=</sup> <sup>−</sup>*iπ*<sup>2</sup> *p*2

appendix contains these evaluations for completeness.

*<sup>g</sup>* (*p*) = <sup>2</sup>*iκ*<sup>2</sup> *<sup>p</sup>*<sup>4</sup> 16*π*<sup>4</sup>

where <sup>Δ</sup> <sup>=</sup> *<sup>p</sup>*<sup>2</sup> <sup>−</sup> *<sup>m</sup>*2. The integral on the RHS of (24) is given by

$$\propto \pm = \frac{1}{2\sqrt{2}} \left( \bar{\Delta} + \bar{\lambda}^2 \pm ( (\bar{\Delta} + \bar{\lambda}^2)^2 - 4(\bar{\lambda}^2 - i\varepsilon) )^{1/2} \right) \tag{25}$$

for <sup>Δ</sup>¯ <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>m</sup>*2/*p*2, *<sup>λ</sup>*¯ <sup>2</sup> <sup>=</sup> *<sup>λ</sup>*2/*p*2and *�*¯ <sup>=</sup> *�*/*p*2. In this way, we arrive at the results, for *<sup>p</sup>*<sup>2</sup> <sup>&</sup>lt; 0,

$$B\_{\mathcal{S}}^{\prime\prime}(p) = \begin{cases} \frac{\kappa^2 |p^2|}{8\pi^2} \ln\left(\frac{m^2}{m^2 + |p^2|}\right), \quad m \neq 0\\ \frac{\kappa^2 |p^2|}{8\pi^2} \ln\left(\frac{m\_{\mathcal{S}}^2}{m\_{\mathcal{S}}^2 + |p^2|}\right), \quad m = m\_{\mathcal{S}} = \lambda \\\ \frac{2\kappa^2 |p^2|}{8\pi^2} \ln\left(\frac{m\_{\mathcal{S}}^2}{|p^2|}\right), \; m = 0, \; m\_{\mathcal{S}} = \lambda \end{cases} \tag{26}$$

where we have made more explicit the presence of the observed small mass, *mg*, of the graviton. When m=0 and one wants to use dimensional regularization for the IR regime instead of *mg*, we normalize the propagator at a Euclidean point *<sup>k</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>*μ*<sup>2</sup> and use standard factorization arguments [40–44] to take the factorized result for *B*�� *<sup>g</sup>* from (26) as

$$B\_{\mathcal{g}}^{\prime\prime}(p)|\_{\text{factorized}} = \frac{2\pi^2|p^2|}{8\pi^2}\ln\left(\frac{|\mu^2|}{|p^2|}\right), \ m = 0, \ m\_{\mathcal{\mathcal{g}}} = 0. \tag{27}$$

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 � *<sup>d</sup>*4*<sup>k</sup>* with *<sup>k</sup>* = (*k*0, *<sup>k</sup>*1, *<sup>k</sup>*2, *<sup>k</sup>*3) for real *<sup>k</sup><sup>j</sup>* and *<sup>k</sup>*<sup>2</sup> <sup>=</sup> *<sup>k</sup>*0<sup>2</sup> <sup>−</sup> *<sup>k</sup>*1<sup>2</sup> <sup>−</sup> *<sup>k</sup>*2<sup>2</sup> <sup>−</sup> *<sup>k</sup>*3<sup>2</sup> into the integral *<sup>i</sup>* � *<sup>d</sup>*<sup>4</sup>*kE* with *<sup>k</sup>* = (*ik*0, *<sup>k</sup>*1, *<sup>k</sup>*2, *<sup>k</sup>*3) and *<sup>k</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>*k*0<sup>2</sup> <sup>−</sup> *<sup>k</sup>*1<sup>2</sup> <sup>−</sup> *<sup>k</sup>*2<sup>2</sup> <sup>−</sup> *<sup>k</sup>*3<sup>2</sup> ≡ −*k*<sup>2</sup> *<sup>E</sup>* with *kE* the Euclidean 4-vector *kE* = (*k*0, *k*1, *k*2, *k*3) with metric *δμν* = *diag*(1, 1, 1, 1). Thus our results rigorously correspond to <sup>|</sup>*p*2<sup>|</sup> <sup>=</sup> <sup>−</sup>*p*<sup>2</sup> in (26), (27) with *<sup>m</sup>*<sup>2</sup> replaced with *<sup>m</sup>*<sup>2</sup> <sup>−</sup> *<sup>i</sup>�*, with *�* <sup>↓</sup> 0, following Feynman, for *p*<sup>2</sup> < 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.
