Possibility of hypothetical stable micro black hole production at future 100 TeV collider
Abstract
We study the phenomenology of TeVscale black holes predicted in theories with large extra dimensions, under the further assumption that they are absolutely stable. Our goal is to present an exhaustive analysis of safety of the proposed 100 TeV collider, as it was done in the case of the LHC. We consider the theories with different number of extra dimensions and identify those for which a possible accretion to macroscopic size would have timescales shorter than the lifetime of the Solar system. We calculate the cross sections of the black hole production at the proposed 100 TeV collider, the fraction of the black holes trapped inside the Earth and the resulting rate of capture inside the Earth via an improved method. We study the astrophysical consequences of stable micro black holes existence, in particular its influence on the stability of white dwarfs and neutron stars. We obtain constraints for the previously unexplored range of higherdimensional Planck mass values. Several astrophysical scenarios of the micro black hole production, which were not considered before, are taken into account. Finally, using the astrophysical constraints we consider the implications for future 100 TeV terrestrial experiments. We exclude the possibility of the charged stable micro black holes production.
1 Introduction
Unsolved puzzles of fundamental physics encourage scientists to probe interactions at progressively higher energies. The Large Hadron Collider has not so far found any hints on ’new physics’, so there are plans to construct even more energetic and luminous experiment. In particular, there is the HighLuminosity LHC project [1] that aims to increase the LHC luminosity by a factor of seven and the Future Circular Collider project [2] that can ultimately reach the hadron collision energy of 100 TeV.
Before the launch of the LHC the question of its safety was examined in detail by the LHC Safety Assessment Group (e.g. see review [15]). It was shown that the hypothetical exotic kinds of matter, such as strangelets, magnetic monopoles, true vacuum bubbles and stable micro black holes, cannot be produced at the LHC, for their existence at the energies below 14 TeV strongly contradicts astrophysical observations. In this work we study the case of the future 100 TeV collider and limit our research to the hypothetical stable micro black holes. The case of 100 TeV collider differs significantly from the case of the LHC, because relevant astrophysical scenarios are considerably altered at progressively higher energies.
Microscopic black holes with the masses under 100 TeV can naturally appear in the extradimensional theories [9, 10, 30]. In these theories the value of the higherdimensional Panck mass can be as low as several TeV. Existing constraints on the parameters of the models with extra dimensions come from the direct measurements of Newton’s law of gravitation at small distances [8, 29] and from the LHC searches [4, 24] for the missing transverse energy of jets due to gravitonemission processes: the radii of the extra dimensions \(R_D < 37~\upmu \hbox {m}\) and the Planck mass \(M_D > 3.5~\hbox {TeV}\). In this work we will conservatively assume \(M_D > 3~\hbox {TeV}\). The minimum black hole mass corresponding to the certain \(M_D\) can be found from the condition that the entropy of the black hole be large (see [20]). In the case of six dimensions the black hole mass \(M=5M_D\) corresponds to the entropy \(S_{BH} \simeq 24\) and we will take it as the lower threshold of mass.
In this work we assume that there is no Hawking radiation [23]. This scenario should be taken into account due to the lack of the experimental data concerning the Hawking radiation as well as due to the theoretical uncertainties in the field of quantum gravity. However, one should keep in mind that the black hole evaporation is an inevitable consequence of quantum theory. Even though there are theoretical suggestions [36, 37] that the process of Hawking radiation could depend on the details of the Planckscale degrees of freedom, once the black hole acquires enough mass to enter a semiclassical regime, the universal Hawking radiation starts. The timescale of evaporation is faster than that of mass acquisition, so the black hole cannot grow macroscopically according to any theoretical considerations. Based on the data from astrophysical observations, our work conducts a test of safety, independent of the reliance on theoretical results.
The produced micro black holes, while in general having nonzero charge, can either lose their charge immediately via the Schwinger mechanism [34] or remain charged. On the one hand, there is a similarity between the Hawking radiation and the Schwinger mechanism, studied in many works, e.g. [15, 25, 35]. This suggests that in the hypothetical case of the absence of the Hawking radiation the Schwinger discharge can be absent as well and the micro black holes would remain charged. On the other hand, there is also a difference between these effects: the Hawking radiation, unlike the Schwinger mechanism, is a transhorizon effect and the conversion of vacuum fluctuations into particles in this case does not occur over a welldefined spacetime domain. If we assume that it is the horizon physics that, despite all the theoretical evidence, forbids the Hawking evaporation, then the Schwinger mechanism can still operate, hence the neutrality of the black holes. For the sake of robustness we consider the both cases: the neutral stable micro black holes as well as the charged ones.
In our work we refer often to the methods proposed in the study [19] of safety of the LHC in the context of the stable micro black holes production. However, we propose an improved method of the calculation of the number of the black holes trapped inside the Earth during the work of high energy collider. Besides, we examine some astrophysical mechanisms of micro black hole production, that were not considered in [19] and which provide conservative modelindependent constraints.
2 Accretion times
This section mainly reviews and structurizes the results of the article [19] about the higherdimensional black hole accretion inside the Earth in order to justify the choice of the gravitational theories we consider in the following sections.
Accretion times of the stable micro black holes inside the Earth divided into subatomic (capture radius smaller than \(a=1~\AA \)) and macroscopic phases. The macroscopic phase is divided into three, division governed by the radius of extra dimensions \(R_D\) and the crossover radius \(R_C\). The last phase corresponds to the growth from \(R_C\) to the size of the black hole with the mass comparable to the mass of the Earth

3 Production of gravitationally bound black holes
4 Astrophysical constraints
4.1 General considerations
4.2 Constraints from white dwarfs
4.3 Constraints from neutron stars
 the production on baryons during the lifetime of the Universe. We conservatively assume that the cosmic ray flux is constant till the redshift \(z=1\) (in reality it is supposed to increase with redshift). We also do not take into account the processes in the early Universe, which are quite modeldependent, considering redshifts \(z<1\). The flux of the black holes can be obtained from Eq. (16), substituting parameter b for the following value:where \(n_0 = 2\cdot 10^{7}~\hbox {cm}^{3}\) is the current baryon density, \(H_0 = 68 \cdot \frac{\text {km}/\text {s}}{\text {Mpc}}\) is the Hubble constant, \(\varOmega _M = 0.31\), \(\varOmega _\Lambda = 0.69\) (see [7]). In order to obtain reliable constraints, we set \(E_{max} = 5\cdot 10^{19}~\hbox {eV}\) in Eq. (16), considering the cosmic rays with the energies below the Greisen–Zatsepin–Kuzmin (GZK) limit [21, 38]. For these energies the energy loss length for protons is more than 1 Gpc. We have also checked that explicit account for nonuniform distribution of extragalactic baryons [11] very weakly affects our estimate. Eq. (17) gives the value \({\hat{b}}=4.6\cdot 10^{21}~\text {cm}^{2}\).$$\begin{aligned} \begin{aligned} {\hat{b}}&=\int \limits _1^0 n(z) dt(z) \\&=\int \limits _0^1 n_0 (1+z)^3 \frac{dz}{H_0 (1+z) \sqrt{\varOmega _M(1+z)^3 + \varOmega _\Lambda }}\\&=\int \limits _0^1 \frac{n_0 (1+z)^2 dz}{H_0 \sqrt{\varOmega _M(1+z)^3 + \varOmega _\Lambda }}, \end{aligned}\nonumber \\ \end{aligned}$$(17)

The production in binary systems of a neutron star and a red giant: cosmic rays hit the giant and produce black holes which then impinge on the neutron star (in the case of neutron stars we consider the neutral black holes, so they are not affected by the magnetic field); \({\hat{b}}=1/\sigma _{NN} = 10^{25}~\text {cm}^{2}\), however \(t\equiv ~\) ‘full coverage equivalent’ \(\lesssim 30~\hbox {Myr}\) (see [19], Appendix H).

The production on interstellar medium (\({\hat{b}}=n_H\), where \(n_H \sim 10^{21}~\hbox {cm}^{2}\) is the average column density of hydrogen in the galaxy).

The production on the Central Molecular Zone of our Galaxy [28], for which \({\hat{b}}=nL=6\cdot 10^{22}~\text {cm}^{2}\). However no longlived neutron stars have been observed yet in this zone: maximal ages of the observed ones are \(t=10^4  10^5~\hbox {year}\).
The first three mechanisms give comparable fluxes of the black holes. The largest flux is achieved in the second mechanism, 6 times higher than in the first one and 30 times higher than in the third. The fourth mechanism produces flux that is 4–5 times lower. It is worth noting that the last mechanism will be the most constraining, if old millisecond pulsars are detected in the Central Molecular Zone. We use the first mechanism for our estimates, because the second one is more modeldependent and yields large systematic errors. The number of the black holes stopped by the neutron star (\(t = 10^{10}~\hbox {year}\), \(R = 10~\hbox {km}\)) is plotted as a function of \(M_6\) in Fig. 6 in case of the different fractions of protons in the cosmic rays. The maximum \(M_6\), that leads to more than one black hole stopped, is 4.1 TeV for \(10\%\) proton composition, \(5.0~\hbox {TeV}\) for \(50\%\) proton composition and \(5.4~\hbox {TeV}\) in case of \(100\%\) proton composition. In this calculation we limit the energy of the cosmic rays from above by \(5\cdot 10^{19}~\hbox {eV}\), so the most optimistic case of \(100\%~\)p composition is quite possible. Worse result, with the maximum \(M_6\) around 3.5 TeV, is given by the mechanism of the cosmic rays (we take \(A/Z \sim 2\)) hitting the surface of the neutron star with the minimum magnetic field \(B = 7\cdot 10^7~\hbox {G}\). The energy of the cosmic rays in this case is smaller than \(5.9\cdot 10^6~\hbox {TeV}\) due to the condition (15); this energy is not sufficient for the production of the black holes with large masses. The magnetic field can be neglected at the poles, however at the high energies the surface area that can be reached there is too small and the flux is less than in the case of the mechanisms studied above. In general, one can see that the constraints on \(M_6\) are worse in the case of neutron stars than in the case of white dwarfs. However, detection of the old neutron stars in the Central Molecular Zone and data on the composition of the cosmic rays can improve the existing constraints from neutron stars and make them the most robust.
4.4 Case of the charged black holes
Till now we considered the neutral black holes. The charged black holes will be stopped in a white dwarf for the whole range of energies due to electromagnetic interactions. In order to get the number of the black holes stopped during the lifetime of a white dwarf, we have to do the same calculations, as in the case of the neutral black holes, however ignoring the inequality (14). For \(100\%\) proton composition of the cosmic rays the number of the stopped black holes exceeds \(6.6\cdot 10^4\) till the Planck mass of \(14~\hbox {TeV}\). For this and bigger values of the Planck mass the production of black holes at the 100 TeV collider is zero, see Fig. 2. Thus, the theories without the mechanism of Schwinger discharge yield more than one black hole trapped in a white dwarf during its lifetime, if the fraction of protons in the cosmic rays at energies from the process threshold \(5\cdot 10^{18}\)–\(5\cdot 10^{19}~\hbox {eV}\) exceeds \(1.5\cdot 10^{5}\), which is actually the case, according to the results of Auger [3] and Telescope Array [6].
4.5 Constraints from astrophysical neutrinos
5 Conclusion
In this article we have studied the phenomenology of the models with extra dimensions in absence of the Hawking radiation in order to conduct an independent observationsbased check of safety of the proposed 100 TeV collider. The models with more than 6 dimensions always yield Earth’s accretion times larger than the lifetime of the Solar system. A theory with five dimensions could be consistent with the existing experimental constraints on the size of extra dimensions and yield accretion times smaller than the lifetime of the Solar system only with a finetuning of the warpfactor, given by the inequalities (4). The calculation of the number of the micro black holes that would have been produced in the future 100 TeV collider with the integrated luminosity \(L = 10~\hbox {ab}^{1}\) and the astrophysical constraints from the observational data on the lifetime of white dwarfs and cosmic ray composition suggest that it is possible to exclude the production of the charged stable micro black holes already. As it is shown in Fig. 8, the case of the neutral black holes, while broadly addressed for most D and mass values, leaves some loophole, which can be closed with further cosmic ray data (e.g. on the neutrino spectrum and cosmic ray composition) or astrophysical observations.
Notes
Acknowledgements
The authors would like to thank P. Satunin, S. Troitsky, I. Tkachev and the anonymous referee for numerous valuable discussions and comments on the manuscript. The question of the safety of a future collider, in its relation to cosmicray studies, was asked by J.R. Cudell at the Solvay meeting “Facing the scalar sector”. The work of the authors was supported by the Russian Science Foundation Grant 141201340. This research has made use of NASA’s Astrophysics Data System.
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