Absence of Black Holes at LHC due to Gravity's Rainbow

In this paper, we investigate the effect of Planckian deformation of quantum gravity on the production of black holes at colliders using the framework of gravity's rainbow. We demonstrate that a black hole remnant exists for Schwarzschild black holes in higher dimensions using gravity's rainbow. The mass of this remnant is found to be greater than the energy scale at which experiments were performed at the LHC. We propose this as a possible explanation for the absence of black holes at the LHC. Furthermore, we demonstrate that it is possible for black holes in six (and higher) dimensions to be produced at energy scales that will be accessible in the near future.


Introduction
Black holes are one of the most important objects in quantum gravity. However, there is little hope of detecting a four dimensional black hole directly in particle accelerators. This is because in order to produce black holes, an energy of the order of the Planck energy (∼ 10 19 GeV) is needed, and this energy is way beyond what can be achieved in the near future. However, if large extra dimensions exist, then there is a hope of observing black holes at colliders, in the near future. This is because the existence of large extra dimensions can lower the effective Planck scale to TeV scales at which experiments can be done [1]. This lowering of Planck scale occurs in Type I and Type II string theories by localizing the standard model particles on a D-brane, while gravity propagates freely in the higher dimensional bulk. Using this model, it was predicted that due to this lowering of effective Planck scale, black holes could be produced at the LHC [2][3][4][5][6]. Furthermore, the production of such black holes would also serve to prove the existence of extra dimensions, and thus provide a strong indication for string theory to be a correct theory describing the natural world (since string theory is critically based on the existence of higher dimensions).
In the experiments performed at the LHC, no black holes have been detected [7,8]. This result has been interpreted to imply the absence of large extra dimensions, at least at the energy scale at which experiments have been performed at the LHC. However, in this paper, we will demonstrate that these results should rather be interpreted as an indication of a suppression of higher dimensional black hole production due to Planckian deformation of quantum gravity. Since large extra dimensions can lower the effective Planck scale to scales at which such experiments are talking place, it becomes very important to consider the Planckian deformation of quantum gravity. We can implement the Planckian deformation of quantum gravity by introducing rainbow functions in the original classical metric using a formalism called gravity's rainbow.
Gravity's rainbow is motivated by doubly special relativity (DSR), which in turn is motivated by the fact that almost all approaches to quantum gravity suggest that standard energy-momentum dispersion relation gets deformed near Planck scale. This deformation of the energy-momentum relation has been predicted from spacetime discreteness [9], spontaneous symmetry breaking of Lorentz invariance in string field theory [10], spacetime foam models [11], spin-network in loop quantum gravity (LQG) [12], non-commutative geometry [13], and Horava-Lifshitz gravity [14,15]. As such a deformation of the dispersion relation is a common prediction of various approaches to quantum gravity, we can expect that this will even hold in any quantum theory of gravity. The modification of the dispersion relation generally takes the form, where E P is the Planck energy, and the functions f (E/E P ) and g(E/E P ) satisfy The modified dispersion relation occurs in DSR because there is a maximum invariant energy scale in addition to the speed of light [16,17]. The most compelling argument for the existence of such a maximum energy scale comes from string theory. This is because it is not possible to probe spacetime below the string length scale. Thus, string theory comes naturally equipped with a minimum length scale, which can be translated into a maximum energy scale [18,19]. DSR can naturally incorporate this maximum energy scale corresponding to string length scale [20,21]. The gravity's rainbow is the generalization of DSR to curved spacetime. This is done by incorporating the functions f (E/E p ) and g(E/E p ) in general curved spacetime metric. So, in gravity's rainbow the structure of spacetime depends on the energy used to probe it [22].
The choice of the rainbow functions f (E/E P ) and g(E/E P ) is important for making predictions. This choice should be phenomenologically motivated. Different aspects of Gravity's Rainbow with various choices of rainbow functions have been studied in [23][24][25][26][27][28][29][30][31][32][33][34][35], Among these choices, the rainbow functions proposed by Amelino-Camelia, et al., are both phenomenologically important and theoretically interesting [36,37], where η is a constant of order unity, and the integer n > 0. This is because the MDR produced by these functions is compatible with some results from non-critical string theory, loop quantum gravity and κ-Minkowski non-commutative spacetime [38]. Furthermore, this MDR was first used to study the possible dispersion of electromagnetic waves from gamma ray bursters [37], and it resolved the ultra high energy gamma rays paradox [39,40]. In fact, it was used for providing an explanation for the 20 TeV gamma rays from the galaxy Markarian 501 [39,41]. Apart from that, it also provides stringent constraints on deformations of special relativity and Lorentz violations [42,43]. A detailed analysis of the phenomenological aspects of these functions has been done in [38]. An outline of the paper is as follows. In section 2, we review the thermodynamics of higher dimensional Schwarzschild black holes, and in section 3, we study their modified thermodynamics using gravity's rainbow with the rainbow functions Eq. (1.3). This is the higher dimensional study of rainbow Schwarzschild black hole which was studied by one of the authors in [44], and reached the conclusion that black holes end in a remnant. In section 4, we discuss this result and compare it with the energy scale of the LHC. In this paper, we use natural units, in which c = 1, = 1, G = 6.708 × 10 −39 GeV −2 and E P = 1/ √ G = 1.221 × 10 19 GeV.

Schwarzschild Black Holes in Higher Dimensions
In this section, we will review the Schwarzschild black holes in higher dimensions. This will be used to motivate a similar analysis based on gravity's rainbow, in the next section. The metric of Schwarzschild black holes in d dimensions takes the form [45,46] where the mass parameter µ is given by and Ω d−2 is the volume of the (d − 2) unit sphere The horizon radius r h is evaluated by solving (1 − µ/r d−3 h ) = 0 leading to (2.5) The Hawking temperature can be calculated via the relation [47] This relation applies to any spherically symmetric black hole with a metric of the form From the Schwarzschild metric in Eq.
and when we substitute the value of r h from Eq. (2.5) we get [48] (2.9) Since d ≥ 4, the temperature goes to infinity as M → 0. Figure 1 is a plot of this equation for d = 4, d = 6, and d = 10, with the generic values n = 4, η = 1, and M P = 1; different values lead to the same qualitative behavior. The black hole entropy can be calculated from the first law of black hole thermodynamics dM = T dS leading to (2.10) which goes to zero as M → 0. The specific heat capacity is calculated from the relation By differentiating the temperature from Eq. (2.9) with respect to M we get (2.12) The emission rate (the energy radiated per unit time) can be calculated from the temperature using the Stefan-Boltzmann law assuming the energy loss is dominated by photons. In m-dimensional brane the emission rate of a black body with temperature T and surface area A m is given by [5] where σ m is the Stefan-Boltzmann constant in m dimensions. Since black holes are radiating mainly on the brane [5], so using m = 4 as in [48], and since A ∝ M (2.14) The exact form can be found in [5,48]. From the relations Eq. (2.9), (2.10), (2.12), and (2.14), we see that when the black hole evaporates and its mass goes to zero, the temperature and emission rate go to infinity, while the entropy and heat capacity vanish. This means that the black hole reaches a stage of catastrophic evaporation as the black hole mass approaches zero, and this definitely needs a resolution. This problem has been tackled in [49], and it has been resolved by considering the generalized uncertainty principle [18] instead of the standard uncertainty principle, and in this picture, black holes end at a remnant that does not exchange hawking radiation with the surroundings. Similar conclusion was obtained by one of the authors in [44], in which it was studied the thermodynamics of Schwarzschild black holes in the context of gravity's rainbow, and it was found that the rainbow black hole ends at a remnant at which the specific heat vanishes and hence the catastrophic behavior is again resolved but this time in the context of gravity's rainbow. In the next section, we shall extend this study into extra dimensions to investigate the phenomenological implications on the productions of black holes at TeV scales.

Schwarzschild Black Holes in Gravity's Rainbow
In this section, we will analyze the Schwarzschild black hole in higher dimensions using gravity's rainbow. The four dimensional Schwarzschild black hole has been analyzed in gravity's rainbow [44], and it was found that a remnant forms. In this section, we extend this analysis into higher dimensional Schwarzschild black holes. In gravity's rainbow, the geometry of spacetime depends on the energy E of the particle used to probe it, and so, the rainbow modified metric can be written as [22] g(E) = η ab e a (E) ⊗ e b (E). (3.1) The energy dependence of the frame fields can be written as where the tilde quantities refer to the energy independent frame fields. So, we can write the modified Schwarzschild metric as [22,50] where f (E) and g(E) are the rainbow functions used in the MDR given in Eq. (1.1). Thus, the modified temperature can be calculated from Eq. (2.6) with the change A(r) → A(r)/f (E) 2 and B(r) → B(r)g(E) 2 leading to where we used the rainbow functions from Eq. (1.3). According to [48,49,51,52], the uncertainty principle ∆p ≥ 1/∆x can be translated to a lower bound on the energy E ≥ 1/∆x of a particle emitted in Hawking radiation, and the value of the uncertainty in position can be taken to be the event horizon radius. Hence, The temperature becomes where we used E P = M P in natural units. From Eq. (3.6), it is clear that the temperature goes to zero at r h = η 1 n /M P , and below this value the temperature has no physical meaning. This minimum horizon radius corresponds to the minimum mass This implies that the black hole ends in a remnant. Figure 2 is a plot of Eq. (3.6) for d = 4, d = 6, and d = 10. The entropy can be calculated from the first law of black hole thermodynamics using the modified temperature from Eq. (3.6) This integral cannot be evaluated exactly for general n and d, but taking as an example d = 4 and n = 4 we get which is the same as the expression derived in [44]. Taking as another example d = 5 and n = 2 we get The heat capacity can be calculated from Eq. (2.11) with the modified temperature in Eq. (3.6), and we get (3.11) Figures 3 and 4 are plots of the heat capacity for d = 4 and d = 10 respectively. We see that the modified heat capacity diverges at a value where the temperature is maximum, then goes to zero at the minimum mass given by Eq. (3.7). The zero value of the heat capacity means the black hole cannot exchange heat with the surrounding space, and hence predicting the existence of a remnant. The emission rate is proportional to T 4 , which means that from the modified temperature in Eq. (3.6), the modified emission rate is which also goes to zero at r h = η 1 n /M P . From the calculations in this section, we conclude that in gravity's rainbow black holes reach a remnant near the Planck scale. In the next section, we investigate whether black hole remnants can be detected in the LHC.

Black Hole Production at Colliders
In the last section, we found that in gravity's rainbow, black holes end up in a remnant with the mass in Eq. (3.7), which we reproduce here for convenience, (4.1) From this minimum mass, we can calculate the minimum energy needed to form black holes in a collider, such as the LHC. In the ADD model [1], the reduced Planck constant M P in extra dimensions is related to the 4D Planck mass M P (4) ∼ 10 19 GeV via where R is the size of the compactified extra dimensions. Fixing M P at around the electroweak scale ∼TeV, and using Eq. (4.2), we obtain d = 5, 6, ..., 10 → R ∼ 10 9 km, 0.5mm, ..., 0.1MeV −1 [53]. Thus, d = 5 is clearly ruled out, but not d ≥ 6. When we use the latest experimental limits on M P from Ref. [7], and assume that the rainbow parameter η = 1, we obtain the results given in Table 1. We see that in d = 6, black holes can form only at energies not less than 9.5 TeV. This energy scale is larger than the energy scale of the LHC, which means that they cannot be detected in the LHC. Previous work based on theories with large extra dimensions predicted the possibility of forming black holes at energy scales of a few TeVs [3][4][5]48], which has not been experimentally observed at the Compact Muon Solenoid (CMS) detector in LHC where experiments are excluding semiclassical and quantum black holes with masses below 3.8 to 5.3 TeV [7,8].
By considering our proposed approach of studying black holes in the context of gravity's rainbow, we may justify why higher energy scales are needed to form black holes. Furthermore, this energy scale will be accessible in the near future. Thus, the prediction made in this paper is that in future colliders black holes can be formed at 9.5 TeV, if six large dimensions exist in nature. As the ten dimensional case is interesting from a string theory point of view, we also state here that it would take a minimum energy of 11.9 TeV to form black holes in future colliders, if ten large dimensions exist in nature. It would also be an indication of gravity's rainbow to be a correct effective description of such a theory.

Conclusions
In this paper, we have analyzed higher dimensional Schwarzschild black holes in gravity's rainbow. It was expected that black holes will be detected at LHC if large extra dimensions existed. This was because the existence of extra dimensions would lower the effective Planck mass to TeV scale (i.e LHC energy scale). The absence of any black hole at LHC could thus be interpreted as the absence of large extra dimensions, at least at the energy scale of the LHC. However, we argued that black holes were not detected due to Planckian deformation of quantum gravity, which was not taken into account. As the effective Planck scale was reduced due to the existence of large extra dimensions, it is important that these effects are taken into account. When we did that using gravity's rainbow, we found that the energy needed to form black holes is larger than the energy scale of the LHC, but is within reach of the next particle colliders. It may be noted that such a suppression was predicted in the framework of generalized uncertainty principle in [48,54]. The fact that the generalized uncertainty principle can lead to a deformed dispersion relation suggests that this might be a general feature of theories with modified dispersion relation. It would be interesting to analyze this relation in more details. Apart from this phenomenological result, it was demonstrated that a black hole remnant will form for higher dimensional Schwarzschild black holes. Such a remnant forms for a four dimensional Schwarzschild black hole [44]. In fact, recently it was demonstrated that a remnant also forms for black rings [55]. These are strong indications that a remnant might form for all black objects, in gravity's rainbow. It will be appropriate to extend investigate dark matter, cosmological constant, etc in the context of gravity's rainbow. We hope to report on these in the future.