Origin of an IR mass scale

It is interesting to discuss possible mechanisms for the generation of nonlocal terms relevant for the IR dynamics and associated to a mass scale. As a starting point, we recall that perturbative loop corrections due to massive matter fields cannot be responsible for producing cosmologically relevant nonlocal structures, as shown in [87]. Let us review how the explanation goes.

Perturbative loop corrections do not affect the IR regime

In gravity the one-loop corrections induced by matter fields can produce nonlocal form factors in the quantum effective action, associated to terms quadratic in the curvature [50,51,88,89,90,91] (see also [49,52,54] for reviews). The resulting QEA can be expressed using form factors as

Γone−^loop = where “GB” denotes a similar nonlocal term that reduces to the topological Gauss-Bonnet term when its form factor is set to one. Consider the contribution to the form factor from a particle of mass M. When the particle is very massive compared to the energies involved (whose scale is set by the Hubble parameterH(t)in a cosmo-logical setting), then the particle decouples and its contribution to the form factor is local and suppressed by a factor O(2/M²) 1. In this case only local terms are produced and the situation, just like we have seen in Section1.2.3about QED when 2/m²_e 1. To get nonlocal operators we should look at the case where the particle of mass M is light with respect to the energy scale involved. In that situation, the form factork_Rhas the form [90,91,92,93] and similarly for k_W. In [94] it was observed that the logarithmic terms and the term(M²/2)have little effect on the cosmological evolution in the present epoch, so one might hope that the leading term is actually given by the term proportional to

M⁴/2², which is the same operator appearing in eq. (1.3.45) for the RR model. Com-paring eq. (1.3.60) with eq. (1.3.45) we see that they match only ifm =O(M²/MP), where M_P is the reduced Planck mass. The expansion in eq. (1.3.60) can only be valid today if M ^H0, and therefore we see that loop corrections from light par-ticles could only generate a nonlocal term m²R2⁻²R with m = O(M²/MP) H₀(H₀/M_P). However, we will see in Chapter 2 that mhas to be of order H₀ to reproduce the dark energy content today and certainly cannot be suppressed with respect to H0 by the very small factor H0/MP. Hence, loop corrections from light particles cannot provide the required value ofmwhile, on the other hand, we have already seen that heavy particles give only local contributions to the form factors, suppressed byO(H₀²/M²)^1.

Dynamical mass generation

In this subsection we will discuss indications, from various non-perturbative tech-niques, in favor of the possibility of a dynamical mass generation in the IR limit of four-dimensional quantum gravity, in particular in relation to the conformal mode.

We follow the presentation in [31].

Functional renormalization group equations. Renormalization group (RG) equa-tions, such as the Polchinski equation [95] and the Wetterich equation [96] are exact equations transform the problem of performing a functional integration for comput-ing the path integral of the QEA into a functional differential equation. In practice, just as the evaluation of the functional integral for an interacting theory requires ap-proximations or numerical methods, one is usually required to truncate the space of action functionals allowed in QEA, to have some hope to solve the functional RG equation. This is the same RG idea used in statistical physics for studying critical phenomena, when one restricts the attention to some couplings among the infinite number of them that is in principle allowed by the symmetries of the microscopic interaction. Therefore the results obtained from functional RG equations can depend on the truncation adopted and require some care in their interpretation. For gravity, functional RG techniques have been developed particularly in connection with the asymptotic safety program, i.e. the search for a non-trivial UV fixed point (see [97]

for review). Only more recently, these tools are being applied to the study of the IR behavior of gravity, and a number of functional RG studies have found indica-tions of strong quantum gravity effect in the IR [98,99, 100,101]. Dynamical mass generation could be compatible with the findings in [102], where, in a truncation of the theory including only the Einstein-Hilbert term and the cosmological constant, it was found that, evolving the RG flow toward the IR, for some trajectories the run-ning of Newton’s constant hits a singularity at a finite momentum scale. But, as we said, the the singularity could also be an artifact of the truncation. Another inter-esting result in [98] suggests that the cosmological constant could be screened by strong IR effects due to the quantum fluctuations of the trasverse-traceless modes, while the conformal mode fluctuations could generate a new mass scale. This is compatible with the phenomelogical RT and RR models, as they exhibit a mass term for the conformal mode.

Lattice gravity. A possible non-perturbative tool is provided by lattice gravity, based either on a simplicial decomposition of the space-time manifold in Euclidean space (see [103] for review), or on causal dynamical triangulations (CDT) (see [104,105]

for reviews).

Using numerical simulation of CDT it is possible to have some informations on the two-point function of the conformal mode, and ref. [106] showed that the nu-merical results provide evidence for a massive conformal mode, corresponding to a linearized nonlocal quantum effective action of the form in eq. (1.3.33), whose co-variantizations can give the RR or RT models. This numerical result was not studied in the continuum limit, but it is still an indication that quantum effects could gener-ate a mass for the conformal mode of the metric.

Naturalness of the mass scale in nonlocal gravity models.

Let us rewrite the quantum effective action (1.3.45) of the RR model as ΓRR = ^M

2P

2 Z

d⁴xp

−^g

R−¹ 6m²R 1

2²R

=

Z

d⁴xp

−^g M²_P

2 R−^RΛ₂^4RR₂ ^R

, (1.3.61)

whereΛRR = (1/12)m²M²_P. In this form, it is clear thatΛRR should be taken as the fundamental scale generated dynamically, corresponding to a dimensionless form factork_R(2) = Λ⁴RR/2² inRk_R(2)R, while the parametermis just a derived quan-tity introduced for convenience.ΛRRis a scale generated generated dynamically and its value cannot be predicted, similarly to what happens for QCD where the value ofΛQCD can only be obtained by comparison with observations. As we already an-ticipated, we needm= O(H₀)in order to have a viable dark energy content in the present epoch. Therefore,

ΛRR=O(H₀m_Pl)^1/2 =O(meV). (1.3.62) A similar analysis can be carried out for the RT model to find a scaleΛRTof the same order as in the RR case. The appearance of a meV scale is not particularly surprising for QFT and is certainly much more natural than what would be needed to explain dark enrgy by introducing some particle of massm. Indeed, in the latter case,mis the fundamental scale with a very small valuem∼ ^H0 ∼^10−33eV.

Chapter 2

Nonlocal gravity: cosmological implications

2.1 Cosmology of the RT model