Graviballs and Dark Matter

We investigate the possible existence of graviballs, a system of bound gravitons, and show that two gravitons can be bound together by their gravitational interaction. This idea connects to the gravitational geon studied by Brill and Hartle. However the calculations performed here rely on the formalism and techniques of quantum field theory, specifically on low-energy quantum gravity. We argue that the graviball is a viable dark matter candidate and we compute the associated gravitational lensing.


Introduction
While the origin of the missing mass in galaxy and in galaxy cluster continue to be actively studied, the existence of large amount dark matter (DM) made of yet-to-be-identified particles has emerged as the leading explanation. Under this hypothesis, the near-consensus is that these particles are non-relativistic (cold DM). The main argument against hot DM particles -that their small mass, i.e. relativistic nature, prevents them to make the small initial overdensities necessary for structures to form fast enough -applies even more to massless particles. However, that argument is not systematically applicable, as exemplified by the case of axions as viable cold DM candiates despite of their diminutive mass, or by the massless gluons of Quantum Chromodynamics (QCD), confined in hadronic-size volumes. In this article, we discuss another case where that argument does not apply, and study the possible relation between the graviton and the missing mass problem.
It is interesting to draw a parallel between nucleons and galaxies. The mass of these objects is much larger than the sum of their (known) component masses. In the case of a nucleon, lattice calculations taught us that 99% of its mass is binding energy from the strong interaction, a result in which gluon self-interaction plays a central role. Similarly to gluons, gravitons can self-interact. In Refs. [1], lattice calculations were used to find the influence of gravity's self-interaction on galaxy dynamics, and found that it alleviates the missing-mass problem. However, the lattice calculations of Ref. [1] are based on the Einstein-Hilbert Lagrangian and obtained in the classical limit. Hence, the results are within the classical framework of General Relativity (GR), without explicit link to gravitons (except as a vocabulary convenience for the QCD-GR parallel).
The main goal of this article is to investigate the possible existence of an object made of gravitons bound together by their gravitational interactions. By analogy with QCD's glueballs, we call these objects graviballs. The idea relates to the gravitational geon 1 [2,3], the classical equivalent of the graviball. If graviballs are indeed possible, they would be an evident solution to the missing mass problem. This candidate is more natural than beyondthe-standard-model particles in the sense that no new physics is postulated. Indeed, all the necessary tools for our study have already been developed for other purposes, and our calculations require no additional hypothesis nor free parameters. These tools, those of effective field theory of gravity (or low-energy quantum gravity) [4][5][6] and semi-classical calculations, have proven to give accurate results. Here, the term semi-classical refers to calculations based on the potential energy, extracted from scattering amplitudes, see Sec. 2.
In this article, we start with the simplest possible case of two gravitons with identical energy. We will consider the case of more than two gravitons in a future work. For the simple case considered here, we find that two gravitons can indeed form a bound system of small size. This is not unexpected since objects able to capture massless particles with their gravitational interaction are known: black holes. Evidently, the result depends on the energy of the gravitons and on the system's impact parameter.
We simulated the graviton system using semi-classical calculations similar to those performed in Ref. [7] for the bending of light by the sun, and solved numerically the relativistic equations of motion. In section 2, we present the general framework of our calculations. In section 3, we use the graviton-graviton amplitude given in Refs. [11][12][13] and determine the graviton-graviton potential energy, the central ingredient for our calculations. In section 4, we describe in detail our simulation and present its main results. These results are discussed further in section 5, with an emphasis on the missing mass problem. Throughout this article, we use the signature (+, −, −, −) for the flat metric η µν , and = c = 1.
1 Geons are discussed in appendix A 2 Low-energy quantum gravity and semi-classical calculations Non-renormalizable effective theories, e.g. chiral effective field theory [14], are commonly used. Despite their name, they are renormalizable in a more general sense and can make accurate predictions. The effective Lagrangian is organized in an energy expansion where L j is suppressed by powers of the small energy ratio E/Λ HE compared to L i , i < j. Here, E is the low energy scale characterizing the problem while Λ HE is a high energy scale. At this scale and beyond, the effective theory becomes inapplicable. It is usual to consider that the Standard Model of particle physics corresponds only to the L 0 of the true high energy theory, and physics beyond the Standard Model can be studied with the higherorder Lagrangians L i>0 . Each L i must respect the symmetries of the theory: in the case of gravitation, they must be invariant under general coordinate transformations. For pure gravity 2 , the most general Lagrangian reads [4] where κ 2 = 32πG, G is the Newton constant, Λ is the cosmological constant, g = detg µν and g µν is the metric. The Riemann curvature tensor R µναβ , the Ricci tensor R µν and the Ricci scalar R involve two derivatives of the metric. In momentum space, these derivatives are associated to the 4-momentum, q, transferred during the interaction of one graviton with another, see Fig. 1, and Eq. (2) is an expansion in powers of q. In this equation the dots denote higher powers of R, R µν and R µναβ . The cosmological constant is generally neglected due to its small experimental value. The Einstein-Hilbert action, corresponds to the lowest order term in Eq. (2) (neglecting Λ), and provides accurate results for small values of the curvature, κ 2 c 1 R 2 R. Eq. (2) shows how general relativity can be treated as any effective field theory. The constants c i are not predicted by the theory and must be extracted by measurements. At a given order in the energy expansion, there is always a finite number of these constants, and the effective field theory can still make predictions if the number of observables is larger than the number of constants to be determined. The gravitational field h µν is defined as a perturbation about the background metricḡ µν g µν =ḡ µν + κh µν .
Its quantization involves in particular a gauge fixing term and ghosts, see Ref. [4].
The Ultra-Violet (UV) divergences due to loop calculations performed with the Einstein-Hilbert action are of higher order in the energy expansion, and can consequently be absorbed in the renormalization of the constants c i , see Ref. [16] and references therein. As an example, consider the UV divergent part of the one-loop effective Lagrangian, including matter loops [16]: where γ is the Euler constant. As expected, this Lagrangian is of higher order in the curvature compared to the Einstein-Hilbert Lagrangian. An inspection of Eqs. (2) and (5) shows that the UV divergence can be absorbed in the renormalization of the c 1 and c 2 constants: where M S indicates that the modified minimal subtraction scheme is used. The loop corrections also contain non-analytic terms of the form ln(−q 2 ) or (q 2 ) −1 which cannot be absorbed in the higher-order terms proportional to q n with n > 0. Consequently, they are separate from the c i and genuine predictions of the low-energy effective theory. One of the advantages of effective field theories is this clear separation of high and low energies. In our case, we have a separation of large and small momentum transfers, corresponding to small and long distances. When studying graviballs, it is useful to remember that at the event horizon of a black hole, the curvature is still sufficiently small for the application of semi-classical calculations. Then, the dynamics of a particle captured by a black hole can be described to good approximation by the Einstein-Hilbert action, neglecting higher orders in Eq. (2). However, when the particle comes too close to the black hole center, the unknown high energy theory is required. The similar limit of our calculations will be discussed at the end of Sec. 4.
Semi-classical calculations are based on the long-distance potential involved in the scattering of two particles. It is given by the Fourier transform of the non-analytic part of the amplitude M( q) [10,15] with E i the energy of particle i participating to the process, see Fig. 1. In Figure 1: Tree-level diagram for the scattering of particles A and B in the t-channel. The P i denote the particle 4-momenta and the Mandelstam variable t is defined by t = q 2 .
Eq. (8), the long-distance limit t = q 2 ≈ − q 2 → 0 is used. The distance between the two particles is denoted r = | r|.
As a first example, we consider the muon-electron interaction µ − e − → µ − e − . At leading order, there is only one Feynman diagram with a photon exchanged in the t channel. The amplitude is where e is the elementary electric charge and the 4-vector s i describes the spin of particle i. The 4-momenta are shown in Fig. 1 and in this example, the electron corresponds to particle A. In the non-relativistic limit, M is dominated by [17] where m is the fermion mass. Since in this limit with m e and m µ the electron and muon masses, respectively, we obtain the repulsive Coulomb potential, Another example is given in Ref. [7], where the authors compute the bending of light (and massless scalar particles) by the sun, see also [8,9]. At leading order, the amplitude reads where M is the mass of the sun and ω the massless particle energy in the sun reference frame. Applying Eq. (8), one finds reminiscent of the Newtonian potential between two masses. Note that here, the energy denominator in Eq. (8) is 4M ω. The 1-loop calculation gives two types of contributions where E = M or E = µ, and µ is an arbitrary mass scale. As explained in [7], the first contribution, whose Fourier transform is proportional to ( ) 0 G 2 r −2 , is of classical nature and corresponds to one of the post-Newtonian corrections to the leading-order potential. The other term is a quantum correction proportional to At large distances, these quantum corrections are negligible compared to the classical contributions. The potential found in Ref. [7] gives a bending angle in agreement with measurements, showing the adequacy of low-energy quantum gravity and semi-classical calculations at large enough distances.
Finally, note that dimensional analysis allows contributions to the potential such as (M ω) 2 G 2 / r. These contributions are clearly problematic for the classical limit, and are absent in [7]. In Ref. [18], the authors showed that in the case of the gravitational interaction between two masses, such contributions can arise from individual Feynman diagrams. However, a cancellation occurs once the diagrams are properly summed.

.1 At tree level
The graviton-gravition scattering amplitude has been reported in several articles, see for instance [11]. Accounting for the convention used in [11], where iM is written M, the amplitude reads where s, t and u are the usual Mandelstam variables, s = (P 1 + P 2 ) 2 , t = (P 1 − P 3 ) 2 = q 2 , and u = (P 1 − P 4 ) 2 . The + denotes the helicity of the gravitons, and we use the physical helicity 3 . At tree level, helicity combinations other than (++; ++) are zero. As explained in Sec. 2 the long-distance potential is proportional to the Fourier transform of the amplitude in the limit t → 0. Dividing by s due to the energy factors in Eq. (8) and using the relation lim for massless particles, we obtain the leading-order potential Since s ≥ 0, the potential is always attractive, as expected.

The 1-loop result
The potential given by Eq. (18) can receive two kinds of corrections: from higher order terms in Eq. (2) and from loop calculations in Einstein gravity 4 . We already mentioned that the formers are negligible at sufficiently large distance, and in particular at the graviball event horizon. Power counting indicates that an n-loop amplitude in Einstein gravity is of the same order as L n ∝ R n+1 [6,18]. Here and in the following, we use the notation R n to denote the terms formed by the appropriate contraction of n factors chosen among R, R µν and R µναβ . Power counting being sometimes misleading, the corrections generated by the 1-loop amplitude must be checked explicitly. plitude has been long known [21], the 2-loop calculation has been published 3 Another convention is to consider all particles in the amplitude as outgoing. In this case, particles 1 and 2 have their helicity reversed. 4 In the context of effective field theory, Einstein gravity refers to the Einstein-Hilbert Lagrangian, linear in R. Higher order terms in Eq. (2) are not used for the 1-loop calculation.
only recently [22]. In pure gravity, the 1-loop result for M 1−loop ++;++ is given in Ref. [21] by the involved equations (4.4) and (4.8) 5 . Note also that while the amplitudes M 1−loop +−;++ and M 1−loop −−;++ are not zero at 1-loop, in contrast to the tree-level case, they do not contain non-analytic terms and consequently do not contribute to the long range potential.
It has been shown by Donoghue and Torma [19] that the infrared divergences present in M 1-loop ++;++ cancel at the cross section level against those from Bremsstrahlung, see Fig. 2b. While the 1-loop diagrams with at least one massive external particle contain in general a term in 1/ √ t, whose Fourier transform gives a contribution ∝ 1/r 2 to the potential, no such terms are present in the four-graviton case. Nevertheless, M 1-loop ++;++ contains several non-analytic terms, in particular 6

2F
ln with F given by: where M tree ++;++ is given in Eq. (16), < 0 and vanishingly small, and Γ is the gamma function. The tilde signals the definition of the Mandelstam variables used in [21], since the authors work in the unphysical regime wherẽ s < 0. In this case, the relation between Mandelstam variables ist +ũ =s. In the limit |t| |s| |ũ|, the two terms in Eq. (19) are identical and give the contribution to the amplitude: Using Eqs. (8) and (21), the correction to the leading-order potential is: where γ E is the Euler constant. Restoring momentarily and c, we find The in the denominator suggests that this expression is a calculation artifact. In the following, we will then ignore the two terms given by Eq. (19) as well as the three infrared divergent terms in [21], equation (4.8). This will be discussed in more details in Appendix B.
The remaining terms in equation (4.8) of Ref. [21] are: and They give negligible quantum corrections to the potential where n > 0. Outside the pure gravity sector, the loops can contain other particles like photons. These contributions give also negligible quantum corrections, with at least one power of in the numerator.
All in all, in contrast to the case with at least one massive external particle, the 1-loop amplitude for the graviton-graviton interaction does not yield any classical corrections. The quantum corrections being small, the leading order potential, Eq. (18), is expected to provide accurate results.

Simulation and results
We consider the case of two gravitons with identical energy ω and initial 4-momenta P 1 = (ω, −ω, 0, 0) and P 2 = (ω, ω, 0, 0), where the non-zero component of the 3-momenta is on the x axis. Then, Eq. (18) reads where, for later convenience, and restoring momentarily c, we define M g = ω/c 2 . We first investigate the academic case where M g = 5.9×10 24 kg, equal to the mass of the Earth. This is also useful in the context of more realistic studies with more gravitons, where each graviton with energy M g may be viewed as approximating a large number of gravitons with small energies.
Using the Runge-Kutta method, we solve numerically the relativistic equations of motion, a system of first order (non-linear) differential equations , F x = F cos ϕ, F y = F sin ϕ, and ϕ is the angle with the x axis. We define the impact parameter b, to be the initial separation on the y axis, and we choose the initial separation on the x axis to be significantly larger than b. At large impact parameter, the interaction causes only a deflection of the gravitons, as illustrated in the top panel of Fig. 3. Reducing the value of the impact parameter one reaches the critical value b hor where the gravitons can no longer escape their mutual attraction. The index "hor" stand for horizon, by analogy with the event horizon of a black hole. As mentioned before, in this region the curvature is still small enough for semi-classical calculations to apply. The trajectory obtained in this case is shown in the bottom panel of Fig. 3, where we observe the gravitons spiraling around each other. Videos associated to these trajectories are available in the arxiv ancillary files. With the chosen input energies, we find that b hor ∼ 4 cm. As mentionned in the introduction, this provides another example of a compact system made of massless particles, here within gravity rather than QCD.
The validity of our calculations stops at small r, where the higher orders in Eq. (2) will start to contribute significantly. It is however not possible to determine precisely at which value of r they occur, since it depends on the unknown constants c i . In the case of external particles of mass m 1 and m 2 , the correction to the potential given by the R 2 terms is [23]: where In the case of four external gravitons, we expect corrections similar to those in Eq. (32), since they originate from the graviton propagator. Because the constants c 1 and c 2 are essentially unknown (the limit from laboratory tests Figure 3: Gravitons trajectories represented by the blue and red lines. In the top panel, the initial impact parameter is larger than the critical value b hor and the gravitons scatter off each others. In the bottom panel, b < b hor and the gravitons start to spiral around each others, thereby forming a bound object. Videos associated to these trajectories are available in the arxiv ancillary files. of gravity at short scales is c 1,2 < 10 56 [20]) it is not possible to determine for which value of r these corrections are important. But any reasonable value of the c i constants will give a negligible contribution; in particular for c 1,2 ∼ 1, these corrections start to contribute at r ∼ 10 −35 m.
It is worth noting that we do not expect the conclusion of our study to be changed by the unknown small-distance behavior. Let us consider the situation where higher orders give significant contribution at r ∼ 10 −35 m (c 1,2 ∼ 1), and where the separation between the gravitons has reached this value. If the small-distance potential is strongly attractive, the two gravitons will still be bound together, and the size of the system will be even smaller than expected by semi-classical calculations. If the small-distance potential is weakly attractive or even repulsive, as exemplified by the third term in the r.h.s of Eq. (32), the two gravitons will move away, increasing the distance between them. Doing so, the system will be back in a region where the physics is dictated by the potential Eq. (27). This will happen at a distance much smaller than the event horizon, found to be b hor ∼ 4 cm for our academic example. In this case the size of the system will fluctuate between 10 −35 m and 4 cm.

Discussion
Because we investigated the simplest case of two gravitons of equal 4-momenta, certain aspects of the bound system remain to be investigated, requiring in particular a study with more gravitons. One reason for this is that accelerated gravitons should radiate particles, including gravitons. However, we do not expect this effect to change significantly our conclusion. Indeed, if the particle is emitted inside the event horizon, it will not be able to escape the gravity well 7 . Then, the number of particles will have changed, but not the total amount of energy. If the particle is emitted when the distance between the gravitons is still larger than the event horizon, the graviton energy, and consequently b hor , will be slightly reduced. Another reason to study cases with more than two gravitons is because the large amount of 4-momentum required to form the graviball studied in Sec. 4 may come from a large number of gravitons of small energies rather than two energetic gravitons.
Our results for the two-graviton case readily suggests two candidates for the missing mass problem: 1. Heavy graviballs.

Light graviballs.
In the first case, the situation is similar to that of black holes. Primordial black holes with masses in the range of asteroid masses are still candidates to explain the missing mass problem, while lighter or heavier black holes have been ruled out [24]. Each black hole, or graviball, generates a large gravitational interaction, and the amount of missing mass requires only a small quantity of these objects. As an illustration, we compute the effect of the (academic) graviball studied in Sec. 4, on the light of a distant galaxy. Here, the photon interacts coherently with the whole 2-graviton system of total 4-momentum P = (2M g , 0, 0, 0), i.e. of genuine mass 2M g . One of the main results of Ref. [7] being that, except for negligible quantum corrections, the gravitational interaction is spin independent, we can use the expression obtained by the authors for the bending of light by a massive object: where the constant c has been restored for better readability. For distance of closest approach of the light ray e.g. d = 10 3 m, the first term of this equation gives an angle of 7.2 arcsec, which is approximately 4 times the value obtained for the sun (measured at the sun surface d 6 × 10 8 m). The second term gives a negligible contribution of 1.8 × 10 −4 arcsec. Two distinct scales appear; the size of the graviball, here a few centimeters, and the maximal distance to the graviball where the effects on light are measurable, around 10 4 m.
In case 2, the graviball is made of two low energy gravitons. The event horizon b hor can be estimated by a formula similar to the Schwarzchild ra- Eq. (35) gives the correct order of magnitude as we can check with M = M g 6 × 10 24 kg yielding b hor 3 cm, in agreement with the value found in Sec. 4. As an example of light graviball, we take M g ∼ 10 −6 kg. In this case b hor ∼ 10 −32 m, and the semi-classical calculations are still valid (for reasonable values of the c i constants, see the discussion at the end of Sec. 4). This example shows that graviball can be light enough to escape direct detection, and a large amount of light graviballs, about 10 50 , could explain the missing mass problem. For a uniform distribution of light graviballs in the galaxy, this number corresponds approximately to a density of 10 −15 m −3 . The size of a light graviball being necessarily small, it could be interpreted, e.g., as a light particle. In fact, Wheeler had speculated on the possible relation between geons and elementary particles [25].
A detailed phenomenological analysis will be provided in a future work. It will include a N-body simulation, taking into account the energy lost by a graviton due to its acceleration, or the annihilation of a graviton pair into other particles. One goal of this study will be to determine, once these effects are included, the stability, mass and size of graviballs. They can then be compared to known constraints on dark matter. Finally, the detectability of this object will be discussed.

Conclusion
We have used the formalism low-energy quantum gravity to study the possible existence of graviballs in the simplest case of two graviton constituents. We showed that the two gravitons can form a bound system due to their gravitational interaction. While the more realistic case including more than two gravitons needs to be studied, the fact that a 2-graviton bound state is possible open the prospect of graviballs being a solution to the missing mass problem. In that context, we computed the gravitational lensing caused by graviballs in order to assess their detectability.
of Sec. 4. This could help in the debate on the curvature singularity for gravitational geons. In some articles, e.g. [3], it is said that the curvature has no singularity. However, in Ref. [26] the authors claim that space-time cannot be taken as singularity-free. Our calculations agree with the latter statement. Another advantage is that we do not employ the time averaging used for gravitational geons. In our calculation, the potential energy is computed at each step of the evolution, allowing a realistic dynamical study of the graviball.

B Comments on the 1/ term
We discuss here the 1/ term in the potential, Eq. (23). This term, stemming from the 1-loop terms, Eq. (19) is unwanted because the classical limit, → 0, gives an unreasonable large contribution to the amplitude. As mentioned in Sec. 2, the presence of such terms could indicate missing Feynman diagrams.
However, another explanation is possible. It is interesting to note that, in the long-distance limit |t| |s| |ũ|, two of the infrared divergent terms in Eq. (4.8) of Ref. [21], are the same as those of Eq. (19), except for the 1/ replaced by ln(−t). In particular, the arguments of the logarithms are not dimensionless. This is generally an indication that these terms should be renormalized or cancelled. We already know that the (non-physical) infrared terms cancels at the cross section level against those from Bremsstrahlung [19]. The infrared and the 1/ terms having the same structure, it is reasonable to expect that the latter will also not contribute to the physical potential.