Nonperturbative spectrum of nonlocal gravity

We investigate the nonperturbative degrees of freedom of a class of weakly nonlocal gravitational theories that have been proposed as an ultraviolet completion of general relativity. At the perturbative level, it is known that the degrees of freedom of nonlocal gravity are the same of the Einstein--Hilbert theory around any maximally symmetric spacetime. We prove that, at the nonperturbative level, the degrees of freedom are actually eight in four dimensions, contrary to what one might guess on the basis of the"infinite number of derivatives"present in the action. It is shown that six of these degrees of freedom do not propagate on Minkowski spacetime, but they might play a role at large scales on curved backgrounds. We also propose a criterion to select the form factor almost uniquely.


Introduction
A quantum theory of gravity [1][2][3] should be able to solve, or say something constructive about, some problems left open in general relativity, such as the singularity problem (there exist spacetime points where the laws of physics break down, as in the big bang at the beginning of the Universe or inside black holes), the cosmological constant problem (two thirds of the content of our patch of the cosmos is made of a "dark energy" component not adequately described by general relativity or particle physics), and the mystery surrounding the birth and first stage of development of the Universe (the actual origin of the inflaton is unknown). In recent years, a new perturbative quantum field theory of gravity has rapidly emerged as a promising and accessible framework where the gravitational force consistently obeys the laws of quantum mechanics and all infinities seem to be tamed [4][5][6][7][8][9][10][11][12][13][14][15]. This proposal adapts ordinary techniques of perturbative field theory to an action with nonlocal operators. Fulfilling initial expectations based on naïve power-counting arguments, the theory turns out to be unitary and super-renormalizable or finite [10] at the quantum level thanks to the nonlocal nature of its dynamics. Causality is not violated in the usual eikonal limit [16] and one expects non-offensive microcausality violations at the nonlocality scale. The theory may resolve the big bang [17][18][19][20][21][22][23][24][25] and black-hole singularities [26][27][28][29][30][31], and its cosmological solutions may unravel interesting bottom-up scenarios in the early universe (inflation) and at late times (dark energy).
Surprisingly, these encouraging features are accompanied by a number of appalling gaps of knowledge on basic questions on the classical theory, such as how to find solutions of the dynamics and whether they match the singularity-free geometries found when linearizing the equations of motion. The dynamics is usually solved with approximations or assumptions which do not give access to all admissible solutions [32]. Also, when considering nonlinear interactions (gravity is as nonlinear as it can be!) it becomes unclear how many initial conditions one must specify and how many degrees of freedom (d.o.f.) populate the spectrum of physical particles of the theory. The purpose of this Letter is to pave the way to fill these gaps and infer the total number of nonperturbative d.o.f. in minimal nonlocal gravity [7,9] (super-renormalizable in D = 4 dimensions; finite in D = 5).
We can summarize our findings in three statements. (A) For the types of nonlocality giving rise to a renormalizable theory, the number of field d.o.f. is finite and equal to D(D − 2) = 8 in four dimensions. Two of these d.o.f. correspond to the graviton, they propagate on flat spacetime at short distances, and are a familiar acquaintance in the nonlocal linearized dynamics. The other d.o.f. are a novelty because they emerge only when the fully nonlinear nonlocal dynamics is considered. Since they are not visible in a perturbative treatment on Minkowski spacetime, these d.o.f. are nonperturbative, typical of curved backgrounds, and, hence, may be important in the description of large-scale, long-range physics, such as at astrophysical or cosmological scales. (B) Also the number of initial conditions to specify in order to find classical solutions is finite. This contributes to, or even settles, a seventy-year-old debate about whether nonlocal theories are predictive at all, due to the peculiarities of their problem of initial conditions. The answer is Yes, for the specific nonlocalities considered here. Knowing how to construct dynamical solutions makes a fundamental step in the understanding of the capabilities of the specific theory under examination, both at the classical and the quantum level. (C) The system can be recast as a set of finite-order differential equations, and hence the number of initial conditions is finite, only for two specific nonlocal form factors among those circulating in the literature. This constrains the ambiguity on the choice of form factors, i.e, of nonlocal theory.
The most important consequence of identifying new d.o.f. and of knowing how to formulate the initial-condition problem is that now one can, on one hand, construct nontrivial cosmological and black-hole classical solutions previously inaccessible with current methods and, on the other hand, build a robust, rigorous, and systematic set of phenomenological models that can be tested against experiments and observations, a vital task for any candidate theory of natural phenomena. Therefore, the results of the present work may be of interest for the applied mathematician, the quantum-gravity theoretician and phenomenologist, the particle-physics theoretician, the cosmologist, and the astrophysicist.
After a brief overview of the most prominent nonlocal gravities, we show that the number of initial conditions and d.o.f. is finite and we count them explicitly. The logic to follow is simple: (i) we write the nonlocal dynamics with infinitely many derivatives as a system of "master equations" which are second order in spacetime derivatives, hence the Cauchy problem is well defined; (ii) from this reformulation, we extract the number of initial conditions and the d.o.f.. The proof is self-contained and may be skipped by the reader interested only in the physical consequences of the theory, which are discussed above and in the final section.  [5] Tomboulis [6,7] H exp (✷) := αP (✷) −l 2 ✷ l 4 ✷ 2 1 − ω string-related [9,17] Krasnikov [4] 2 Master equations Consider the classical action where G µν is the Einstein tensor and γ(✷) is a nonlocal form factor, an entire function of ✷ having special asymptotic properties [5-7, 10, 12, 13, 33]. It can be parametrized as where H(✷) depends on the dimensionless combination l 2 ✷ and l is a fixed length scale. The four principal theories are shown in Tab. 1. In the first case ("pol": asymptotically polynomial), P (z) is a real positive polynomial of degree n (2n derivatives) with P (0) = 0, Γ is Euler function, and γ E is the Euler-Mascheroni constant. In the second case ("exp") the form factor is asymptotically exponential. Quantum gravity with Kuz'min or Tomboulis form factor is renormalizable [5][6][7]; with the stringy profile H exp (✷) = −l 2 ✷, it is renormalizable if perturbative expansions with the resummed propagator are allowed [11]; with Krasnikov profile H exp (✷) = l 4 ✷ 2 , it is believed to be renormalizable, but the proof is not complete [4]. The role of quadratic curvature operators is to make the theory renormalizable, while the role of the nonlocal form factors is to preserve unitarity. The physics stemming from the action (2.1) is a hard nut to crack. Even before quantizing the theory, two fundamental questions arise. How many degrees of freedom are there? Can one solve the dynamics once a finite number of initial conditions are given? While it is easy to see that linear systems have a finite number of d.o.f. and of initial conditions, the case with nonlinear interactions is highly nontrivial. Directly manipulating the infinitely many derivatives of Eq. (2.2) makes very difficult to answer the above questions.
To put the main result of this Letter in the most direct terms, we trade Eq. (2.2) for kernel functions that obey finite-order differential equations (master equations). Equation whereG is the Green function solving [38] ✷G(y, z) Intuitively, ✷ −1 is expressed in terms ofG, while exp H is replaced by G, everything being integrated three times instead of one. To complete the system, we need to write down the master equations for G, but before doing so we need to specify H. The profile H(✷) can be defined through the integral where α > 0 is real, P (✷) is a generic function of l 2 ✷, and f (ω) is arbitrary. The parameter σ is fictitious and will be commented upon shortly. The kernel G is formally given by G(x, y; σ) = e Hσ(✷x) G(x, y; 0), but this expression is especially difficult to deal with because in nonlocal quantum gravities the propagator is suppressed in the ultraviolet, so that its inverse (the form factor exp H) explodes in Euclidean momentum space in all realistic cases. 1 To avoid this, we consider the inverse F = G −1 , for which the nonlocal operator in is damped at high energies. Once F is found, one can determine G = e 2Hσ F with deconvolution methods [35]. However, Eq. (2.6) still carries all the above-mentioned unknowns related to infinitely many derivatives.
To solve this issue, we make a crucial observation: any form factor can be written in terms of a kernel F governed by a simple system of renormalization-group-like equations, which determine how F varies in the space of all possible functionals P (✷). This space is parametrized by σ, where σ = 1 corresponds to the end of the flow. The exact form of these master equations depends on the choice of H(✷), which is limited by renormalizability and unitarity.
For the general class f (ω) = exp(−ω) (asymptotically polynomial H pol σ ; includes Kuz'min and Tomboulis form factors), we can finally write the finite-order differential equations: where K is the kernel associated with the operator f [σP (✷)]. The solution of Eq. (2.7a) is formally given by Eq. (2.6). Note that Eq. (2.7c) corresponds, in quantum gravities, to diffusion in the correct direction and its solution K(x, y; σ) = e −σP (✷x) K(x, y; 0) is well defined.
In the much simpler case of the general class f (ω) = 1 − ω (exactly monomial H exp σ ; includes the string-related and Krasnikov exponential form factors), Eqs. (2.7) are replaced by just one master equation: is well defined and coincides with Eq. (2.6). The integro-differential equations (2.7) or (2.8) are the generalizations of the diffusionequation method [34,36] and of the case with an exactly polynomial H(✷) treated in Ref. [37]. Thus, the original, formidable nonlocal problem is reduced to one with a finite number of derivatives where we can count the initial conditions and the field degrees of freedom.
Both turn out to be finite.

Initial conditions for special form factors
Let us consider the system (2.7) with P (✷) = −l 2 ✷. The system given by Eqs. (2.7d), (2.7a), and (∂ σ − l 2 ✷ x )K(x, x ′ ; σ) = 0 is second order in spacetime coordinates and first order in the diffusion parameter σ. In synchronous gauge, the metric in D dimensions simplifies to ds 2 = −dt 2 + h ij (t, x)dx i dx j , where i, j = 1, . . . , D − 1, h ij is the metric of the spatial section, and the covariant d'Alembertian operator on a scalar takes the form Therefore, to solve the second-order master equations we only need to specify the spatial metric and its first time derivative. If we insist in having only second-order (in spacetime coordinates) differential equations with a diffusion-like structure, then there are only two choices for the form factor

Equations of motion
The variation of the nonlocal action with respect to the metric for a generic form factor is doable but complicated for the nonexpert. While variation of curvature terms gives at most two derivatives, the source of infinite derivatives in the EOM is the variation δγ(✷)/δg µν when γ is expressed (as done in most approaches) as a series of ✷ powers. There lies the difficulty in understanding the Cauchy problem in nonlocal quantum gravity. However, thanks to the kernel (instead of series) representation and to the master equations (2.7) or (2.8), we are in a position to count the initial conditions without performing any lengthy calculation. In fact, on one hand we have the action (2.1). Variation of the curvature terms R µν and G µν gives at most two derivatives each, for a total of four. Two of these derivatives are reabsorbed by the inverse ✷ represented byG. The variation of γ only involves form factors of the type ✷ −1 , exp ✷ n , and exp H, and thus they admit kernel representations satisfying Eq. (2.7d) and (2.8) for the case of exp ✷ or the master equations (2.7) for the case of exp H. For a monomial P (✷) = (l 2 ✷) n , these equations are of order 2n. This is one of the main findings of the present work: that there exist suitable kernel representations of the form factors obeying finite-order equations. The derivative order of the system given by the EOM and the master equations is thus 2n + 2.
The conclusion is that the EOM can be expressed in terms of the kernels K and G living in the space of form factors and the Green functionG(x, y). These kernels are nonlocal because they depend on two spacetime points, but they are determined by a set of equations independent of the actual EOM. This set may be more or less difficult to solve, but it is well defined. Therefore, the EOM coming from Eq. (2.3) are nonlocal integral equations but finite differential equations: for P (✷) = O(✷ n ), they are of derivative order 2n + 2 and we must specify 2n + 2 initial conditions. This result agrees with the counting in a nonlocal scalar field theory [36,39], and with the diffusion method in Lagrangian formalism [37]. Deser-Woodard nonlocal gravity [38] admits a well defined counting, too, although the diffusion method is not needed due to the different nature of the form factor therein.
Up to this point, we concentrated on a class of super-renormalizable theories, but the main result is insensitive to the introduction of other special operators that make the theory finite in even dimension. Here by special we mean nonlocal "killer operators" [10] at least cubic in the curvature, namely, Rγ k,1 (✷)R 2 , R 2 γ k,2 (✷)R 2 , . . ., R D/2 γ k,··· (✷)R 2 , R ✷ R D 2 −2 γ k,··· (✷)R 2 , where γ k,··· may all differ from one another and, again, R is the generalized curvature. These operators increase the order of the EOM. In D = 4, killer operators are cubic or quartic in the Riemann tensor and the order of the EOM is the same, namely, 2n + 2. In particular, for n = 1 four-dimensional finite nonlocal gravity only needs four initial condition at the nonperturbative level.

Degrees of freedom
Having counted the number of initial conditions, we also comment on the number of field degrees of freedom, i.e., the number of independent components of the tensor fields populating the theory. We introduce two auxiliary fields, a rank-2 symmetric tensor φ µν and a scalar field ψ, to infer the exact number of d.o.f.. Let us consider the Lagrangian The EOM for the tensor φ µν and the scalar ψ are ψ is just the trace of φ µν and is not an independent degree of freedom [37]. Eliminating the auxiliary fields from the action, we end up with the original action (2.1). Notice that Eq. (5.2) implies the transverse condition ∇ µ φ µν = 0 on-shell, as a consequence of Bianchi identity. The EOM for the metric are more involved, but not overly so, and agree with the case without auxiliary fields [37]. It turns out that we only deal with second-order differential equations [37], up to γ factors that, as we have shown above, can be dealt with the diffusion-equation method without increasing the derivative order. On the other hand, by a simple count of the independent components of the fields, we find that the d.o.f. are: We here expand on the nature of these particle d.o.f. in Minkowski spacetime, following closely [41]. First we focus on the theory in D = 4 dimensions.
In order to distinguish the d.o.f. of the theory (2.1), it is useful to separate the massive from the massless fields. We consider the action (2.1) at the quadratic order in the perturbation h µν , the latter being defined by g µν = η µν + 2κh µν : where the projectors P (2) and P (0) are defined as We now proceed as in local quadratic gravity [41] and, in Eq. (5.3), we first replace the graviton h µν with h µν + Σ µν and, after some intermediate field redefinitions, the tensor Σ µν with where Ψ µν is a symmetric rank-2 tensor and χ is a scalar. The outcome at the second order in the perturbations is the following Lagrangian: where ✷ = ∂ µ ∂ µ and we introduced Einstein's linearized Lagrangian (for a generic field Z µν ) Selecting the gauge-independent terms of the action, we get (5.7) Notice that the local limit of the form factors in (5.7) is which reduces to Stelle gravity (mass terms for χ and ψ µν ) only in the theory with stringrelated form factor, where H ∝ ✷. In any other case, the local limit of the theory is not Stelle gravity. The spin structure of the fields is not affected by the form factors, which are entire functions, and is the same as in Stelle gravity. Therefore, we do not need to repeat the discussion in [40][41][42]. However, nonlocality radically changes the propagation of these fields and, ultimately, the physical content of the theory. Thus, on one hand the theory (2.1) describes a massless graviton field, a spin-two field 2 with kinetic term with the wrong sign (analogous to Stelle's Pauli-Fierz massive ghost field), and a scalar field. On the other hand, the gauge-invariant terms of the propagators for χ and Ψ µν are both proportional to the inverse of (5.8), γ e −H , which has no poles by definition of H. The conclusion is that the spin-2 ghost and the scalar present in Stelle local theory [40] do not propagate at the perturbative level in nonlocal gravity.
Furthermore, the six nonpropagating d.o.f. Ψ µν and χ are exactly the same of φ µν up to a change of basis. We can safely conclude that the action (5.1) describes exactly the same d.o.f. of Stelle's theory, that are now harmless thanks to the nonlocality of the action. Finally, it was recently proved [43] that Minkowski spacetime is stable not only at higher perturbative order, but also to all orders in the graviton perturbation. Therefore, the field φ µν , or the fields χ and Ψ µν , never propagate at any arbitrarily high perturbative order.
In D dimensions, the number of independent components or the fields changes, but the spin decomposition into a graviton, a spin-2 ghost and a scalar is the same. This was showed in [44] for higher-derivative local theories, but it is true also for nonlocal theories, since the form factors only affect the propagation of these modes, not their spin. The nonpropagation of the spin-2 and the scalar d.o.f. can be checked by coupling the graviton to the most general energy tensor Θ µν and computing the transition amplitude in momentum space, namely, where O −1 µν,ρσ is the propagator for the theory (2.1). The result is Again, the graviton is the only one propagating in any dimension, while for the massive spin-2 field and the scalar we do not have any pole, since [1 − exp(−H)]/k 2 is entire.

Applications and conclusions
Some years ago, the diffusion-equation method revealed that a string-motivated nonlocal scalar field theory with exponential operators has a finite number of initial conditions and nonperturbative degrees of freedom [36]. In this Letter, we converted the infinite number of derivatives of renormalizable nonlocal gravities with highly nontrivial nonlocal operators into integral kernels living in the space of all possible form factors and that do not carry pathological nonlocality. The main action (2.3) must be placed side by side with the finiteorder differential equations given by the system (2.7) or (2.8). The system of EOM for the metric and the kernels is of finite order 2n + 2 for the metric, for a form factor (2.2) with H given by H pol with P (✷) ∝ ✷ n . For the special form factor with n = 1, where kernels satisfy second-order differential equation, the EOM for the metric are forth order and the number of nonperturbative degrees of freedom is eight in four dimensions. Two of these d.o.f. are perturbative and well known in the literature and, thanks to asymptotic freedom, they describe the theory completely at small scales on a Minkowski background [9], i.e., at the scales of the local inertial frame of the observer where the background is approximately flat. The new degrees of freedom found in this paper are nonperturbative and should play a capital role at large scales and on curved backgrounds, in high-curvature regimes where interactions are important. By large scales, one means scales where tidal forces become important. The diffusion method proposed here reaches those solutions that cannot be found with the available methods applied to linearized EOM. This is the reason why the new d.o.f. are not seen when interactions are ignored. This is the most direct and important consequence of our findings for the physics of nonlocal gravity. These results repair the old and bad reputation of nonlocal theories for having an illdefined Cauchy problem. Armed with the generalized diffusion method, one can construct unambiguous analytic solutions of cosmological and astrophysical backgrounds, previously inaccessible via the typical Ansatz ✷R = λR in the literature. From these new, fully nonperturbative solutions, one will be able to extract conclusions on the stability or absence of big-bang and black-hole singularities, find predictions on the cosmic evolution induced by nonlocal gravity, and check the theory at the nonperturbative level against present and near-future observations, such as those of Planck [45,46] and LIGO [47,48], according to a varied battery of phenomenological tests that has already been effective for string cosmology and other quantum gravities [3,49]. Supported by these, other recent, and upcoming data, the phenomenology and testing of nonlocal quantum gravity is about to bloom.