Possibility of Hypothetical Stable Micro Black Hole Production at Future 100 Tev Collider

Eur. Phys. J. C (2017) 77:908 https://doi.org/10.1140/epjc/s10052-017-5464-7

Regular Article - Theoretical Physics

Possibility of hypothetical stable micro black hole production at future 100 TeV collider

A. V. Sokolov1,3,2,a,M.S.Pshirkov1,4,5 1 Institute for Nuclear Research of the Russian Academy of Sciences, 117312 Moscow, Russia 2 Physics Department, Lomonosov Moscow State University, 119991 Moscow, Russia 3 Institute of Theoretical and Experimental Physics, B.Cheremushkinskaya 25, 117218 Moscow, Russia 4 Sternberg Astronomical Institute, Lomonosov Moscow State University, Universitetsky prospekt 13, 119234 Moscow, Russia 5 P.N. Lebedev Physical Institute, Pushchino Radio Astronomy Observatory, 142290 Pushchino, Russia

Received: 12 November 2017 / Accepted: 9 December 2017 / Published online: 27 December 2017 © The Author(s) 2017. This article is an open access publication

Abstract We study the phenomenology of TeV-scale black LHC luminosity by a factor of seven and the Future Circu- holes predicted in theories with large extra dimensions, under lar Collider project [2] that can ultimately reach the hadron the further assumption that they are absolutely stable. Our collision energy of 100 TeV. goal is to present an exhaustive analysis of safety of the Before the launch of the LHC the question of its safety proposed 100 TeV collider, as it was done in the case of was examined in detail by the LHC Safety Assessment the LHC. We consider the theories with different number Group (e.g. see review [15]). It was shown that the hypo- of extra dimensions and identify those for which a possible thetical exotic kinds of matter, such as strangelets, mag- accretion to macroscopic size would have timescales shorter netic monopoles, true vacuum bubbles and stable micro black than the lifetime of the Solar system. We calculate the cross holes, cannot be produced at the LHC, for their existence at sections of the black hole production at the proposed 100 the energies below 14 TeV strongly contradicts astrophysi- TeV collider, the fraction of the black holes trapped inside cal observations. In this work we study the case of the future the Earth and the resulting rate of capture inside the Earth 100 TeV collider and limit our research to the hypothetical via an improved method. We study the astrophysical conse- stable micro black holes. The case of 100 TeV collider dif- quences of stable micro black holes existence, in particular its fers significantly from the case of the LHC, because relevant influence on the stability of white dwarfs and neutron stars. astrophysical scenarios are considerably altered at progres- We obtain constraints for the previously unexplored range of sively higher energies. higher-dimensional Planck mass values. Several astrophysi- Microscopic black holes with the masses under 100 TeV cal scenarios of the micro black hole production, which were can naturally appear in the extra-dimensional theories [9, not considered before, are taken into account. Finally, using 10,30]. In these theories the value of the higher-dimensional the astrophysical constraints we consider the implications for Panck mass can be as low as several TeV.Existing constraints future 100 TeV terrestrial experiments. We exclude the pos- on the parameters of the models with extra dimensions come sibility of the charged stable micro black holes production. 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 graviton- < 1 Introduction emission processes: the radii of the extra dimensions RD 37 µm and the Planck mass MD > 3.5 TeV. In this work > Unsolved puzzles of fundamental physics encourage scien- we will conservatively assume MD 3 TeV. The minimum tists to probe interactions at progressively higher energies. black hole mass corresponding to the certain MD can be The Large Hadron Collider has not so far found any hints found from the condition that the entropy of the black hole on ’new physics’, so there are plans to construct even more be large (see [20]). In the case of six dimensions the black = energetic and luminous experiment. In particular, there is the hole mass M 5MD corresponds to the entropy SBH 24 High-Luminosity LHC project [1] that aims to increase the 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 a e-mail: [email protected] 123 908 Page 2 of 9 Eur. Phys. J. C (2017) 77 :908 1 √ 1 the lack of the experimental data concerning the Hawking S = d D x −g · R, (1) radiation as well as due to the theoretical uncertainties in the 8πG D 2 2 2A(y) μ m n field of quantum gravity. However, one should keep in mind ds = e dx dxμ + gmn(y)dy dy , (2) that the black hole evaporation is an inevitable consequence μ of quantum theory. Even though there are theoretical sug- where x are the usual non-compact Minkowski coordinates ¯ = = = gestions [36,37] that the process of Hawking radiation could (here and below we use natural units h c kB 1). The depend on the details of the Planck-scale degrees of freedom, characteristic radius RD of the extra dimensions is connected once the black hole acquires enough mass to enter a semi- with the Planck mass MD as follows: classical regime, the universal Hawking radiation starts. The M2 timescale of evaporation is faster than that of mass acquisi- 4 ( )D−4 · 2ΔA, 2 RD MD e (3) tion, so the black hole cannot grow macroscopically accord- MD ing to any theoretical considerations. Based on the data from 15 astrophysical observations, our work conducts a test of safety, where M4 = 2.4 · 10 TeV, ΔA is a difference in warping independent of the reliance on theoretical results. between the region with the maximum warp factor and the The produced micro black holes, while in general having standard model region. In the case of zero warping and D = 5 7 non-zero charge, can either lose their charge immediately the Planck mass MD ∼ 10 TeV gives the value R5 ∼ 10 km, via the Schwinger mechanism [34] or remain charged. On the that is obviously excluded. In the case of the theories with one hand, there is a similarity between the Hawking radiation the larger number of dimensions RD decreases and the value and the Schwinger mechanism, studied in many works, e.g. of R6 ∼ 5 µm is already smaller than the existing constraint [15,25,35]. This suggests that in the hypothetical case of the on RD (RD < 37 µm). absence of the Hawking radiation the Schwinger discharge The micro black hole accretion generally goes through can be absent as well and the micro black holes would remain three different phases: subnuclear, subatomic and macro- charged. On the other hand, there is also a difference between scopic, that follow each other while the black hole is grow- these effects: the Hawking radiation, unlike the Schwinger ing. In the cases of 1 or 2 extra dimensions the black hole mechanism, is a trans-horizon effect and the conversion of initial capture radius in matter is larger than the nuclear vacuum fluctuations into particles in this case does not occur size. For instance, in the case D = 6 the capture radius −12 over a well-defined space-time domain. If we assume that is REM ∼ 10 cm. As a consequence, the accretion goes it is the horizon physics that, despite all the theoretical evi- through the subatomic phase from the very beginning and dence, forbids the Hawking evaporation, then the Schwinger there is no need to consider the subnuclear phase. In the case mechanism can still operate, hence the neutrality of the black of the bigger number of extra dimensions we conservatively holes. For the sake of robustness we consider the both cases: neglect the subnuclear phase, for it is sufficient to find the the neutral stable micro black holes as well as the charged lower constraint on the black hole accretion time. ones. The micro black hole accretion is the fastest in the theories In our work we refer often to the methods proposed in without warping (ΔA = 0), for the characteristic radius of the study [19] of safety of the LHC in the context of the extra dimensions in this case is maximal and the transition stable micro black holes production. However, we propose to the slow four-dimensional accretion regime occurs later. an improved method of the calculation of the number of Thus, in order to find the lower constraint on the black hole the black holes trapped inside the Earth during the work accretion time, we consider the case ΔA = 0, apart from the of high energy collider. Besides, we examine some astro- theory with D = 5, where the warping is needed to meet the physical mechanisms of micro black hole production, that experimental constraints. In the latter case we set the warp were not considered in [19] and which provide conservative factor value so that R5 be maximal and equal to 37 µm. model-independent constraints. The constraints on the times of the micro black hole accretion inside the Earth in the theories with D = 5 − 11 are presented in Table 1. The capture radius during the phase of macroscopic (Bondi) accretion is called Bondi radius RB, 2 Accretion times while the radius RC denotes the distance of the crossover, where the higher-dimensional gravity force equals the four- This section mainly reviews and structurizes the results of the dimensional one. The accretion times depend on the proper- article [19] about the higher-dimensional black hole accre- ties of the matter inside the Earth. One can parametrize this tion inside the Earth in order to justify the choice of the dependence by the Debye temperature TDeb (that is about gravitational theories we consider in the following sections. 400K for the materials typical for Earth’s composition) and Let us consider the D-dimensional gravitational action numerical O(1) constants χ and λD (4 λ4 < 18, 3 < with a general compact metric gmn(y) and a warp factor A: λD < 6.6 in the case D > 4), which slightly depend on 123 Eur. Phys. J. C (2017) 77 :908 Page 3 of 9 908

Table 1 Accretion times of the stable micro black holes inside the sion governed by the radius of extra dimensions RD and the crossover Earth divided into subatomic (capture radius smaller than a = 1Å)and radius RC . The last phase corresponds to the growth from RC to the macroscopic phases. The macroscopic phase is divided into three, divi- size of the black hole with the mass comparable to the mass of the Earth

2 11 T4 f (M5) = 19 − 1.5lnm5, t = 3.1 · 10 · χ year, m D = MD/M0, M0 = 1TeV, T4 = TDeb/400 K, a = 1Å, χ 1, 4 λ4 < 18, 3 <λD < 6.6, ΔA—warp factor

3 Production of gravitationally bound black holes

1 × 107 The cross section of the black hole production in pp collisions at the energy of 100TeV according to the factorization 5 × 106 theorem [13]is:

Time, yr 1 1 √ dx 1 × 106 σ (M > M )= dτ f (x) f (τ/x)σ ( s ), BH min x i j ij τ τ 5 × 105 min (5) 5 10 15 20 Planck mass, TeV where fi (x) are parton distribution functions (we use the M2 set CT14qed [33]), τ = min , M = 5M , τ = Fig. 1 Lower bound on the Earth accretion time in the theories with min y2s min 6 D = 6 dimensions and values of the Planck mass from 3 to 20 TeV x1x2, s = s · τ, y 0.5 − 0.7 is the inelasticity factor (see [14]). The double sum implies the summation over the the material. We see that in the theories with more than six all pairs of partons. The cross section for the collision of two dimensions the accretion times are larger than the lifetime of partons is: the Solar system. In the cases of 5 and 6 dimensions we get √ √ 1/3 5 − 6 √ √ 1 3 τs accretion times t 10 10 year, which are not exceed- σ ( s) = π R2( τs)/4, R( τs) = · , ingly large from the geological point of view, though the case M6 4M6 D = 5 requires a theory with the special choice of warping: (6) R is the Schwarzshild radius. We allow black hole production − . M5 <Δ < − . M5 , 19 1 5ln A 24 1 5ln (4) only for the partonic collisions with the impact parameter M0 M0 b < 0.5R, following the work [18]. The parton distribution where M0 = 1 TeV and the limits are given by the experi- functions should be taken at the scale Q ∼ 1/R, as discussed mental constraint on the value of R5 and the condition that in [20]. The number of the black holes produced at the future ∼ 4 −1 the accretion time is shorter than the Solar system lifetime. collider√ (integrated luminosity L 10 fb , center of mass The results for both 5 and 6 dimensions are similar, and, bear- energy s = 100 TeV, PDF scale Q ∼ 10 TeV) is plotted ing in mind that the theory with D = 5 requires an extreme as a function of M6 in Fig. 2. ﬁne-tuning, we limit our research to the case D = 6. The full Black holes on average will be produced with velocities time of the black hole growth inside the Earth in this case is much larger than the escape velocity, thus only a tiny fraction shown in Fig. 1. of them will be trapped inside the Earth. In order to calculate 123 908 Page 4 of 9 Eur. Phys. J. C (2017) 77 :908

L = –1 8 10 ab –2

6 –3

Log(s) –

Log(N) 4 y = 0.7 2 –5 = 0 y 0.5 –6 –2 4 6 8 10 12 14 4 6 8 10 12 14 16 Planck mass, TeV Planck mass,TeV

Fig. 2 The number (here and below the logarithmic scale in plots is Fig. 3 The fraction of the neutral black holes produced at the future based on decimal logarithms) of the black holes produced during the 100 TeV collider which are trapped inside the Earth (theories with 6 lifetime of the future 100 TeV collider with the integrated luminosity dimensions) L = 104 fb−1 in the theories with D = 6 dimensions, Planck masses 3 − 15 TeV and for two values of the inelasticity parameter y = 0.5 and y = 0.7 velocity is smaller than the escape velocity vE one should take the latter as a trapping threshold. Note that l/d equals this fraction we need the distribution of not only the lon- vr /v, the ratio of the radial (directed to the center of the Earth) gitudinal momenta of partons, given by the standard parton speed to the total speed of the black hole; vr = (k/M) cos φ, distribution functions, but also of the transverse momenta as where φ is the angle between the direction to the center of well, that is why we use the transverse momentum depen- the Earth and the transverse momentum, −π/2 <φ<π/2, dent parton distribution functions (TMDPDF) gi (x, k) from the velocities are non-relativistic. Then the condition for the the library tmdlib-1.0.7 [22]. According to the TMD factor- black hole to be trapped reads as: ization [31], the cross section of the black hole production v< v ,v ⇒ 2 + 2 (its mass being larger than M, longitudinal momentum less max [ max E ] p k than p and transverse momentum less than k) is given by: 8/3 1/3 < . · −3 M0 M0 φ, 2v2 . max 11 2 10 kMcos M E 2π M6 M 1 σBH(M, p, k) = dα (11) 2π ij 0 Thus, the fraction of the black holes that will be trapped × · ( , ) ( , )σ( , ), dk1dk2dx1dx2 gi x1 k1 g j x2 k2 x1 x2 (suppression factor) is given by: R(p,k,α) (7) π/2 √ / 1 π 3 x x s 2 3 s(M ) = dφ σ( , ) = · 1 2 , 6 2πσ (M ) x1 x2 2 (8) tot 6 4M 4M6 −π/2 6 √ (12) 3σ ( ) M s d BH M6 R(p, k,α) = √ ≤ x1, x2 ≤ 1, |x1 − x2|≤p, × dpdkdM, s 2 dpdkdM D(M6,φ) 2 + 2 + α ≤ . k1 k2 2k1k2 cos k (9) where the region D(M6,φ) is given by the inequality (11) σ ( ) At the present moment we will assume that the black holes and the total cross section tot M6 ÐbytheEq.(5). The are neutral. Neutral black holes will slow down inside the values of the suppression factor are presented in Fig. 3 and Earth due to the accretion and gravitational scattering. This the number of the black holes trapped inside the Earth for the 4 −1 process was in detail studied in [19], where the maximum integrated luminosity L = 10 fb is plotted in Fig. 4. speed for a black hole to be still trapped inside the Earth was It is worth noting that the calculation of the suppression calculated. In our case (D = 6): factor in Ref. [19] was considerably simpliﬁed: the authors assumed that all the black holes which were produced had 8/3 1/3 the same lowest possible mass M and that their transverse v = . · −3 · l M0 M0 , min max 11 2 10 (10) momenta are identical. These assumptions lead to the under- d M6 M stated value of the suppression factor, because the maximal where l is the path length of the black hole inside the Earth, initial momentum for a black hole to be still trapped inside 2/3 d is the diameter of the Earth, M0 = 1 TeV. In case this the Earth is pmax = max [Mvmax, MvE ] ∝ M or M and, 123 Eur. Phys. J. C (2017) 77 :908 Page 5 of 9 908

4 – number A and high energy E hitting a target nucleon. The L = 10 ab 1 2 initial energy of the produced black hole γi M = yxE/A (y

0 is the inelasticity parameter, x is the fraction of the nucleon centre of mass momentum carried by the incident parton). –2 y = 0.7 Then the condition for the column density δ required to stop – Log(N) 4 the black hole is: –6 y = 0.5 3 δ M 9 A –8 δ>δ ⇒ < 0 · · . min xE 3 M6 (14) 0.27M M6 y –10 0 4 6 8 10 12 14 Planck mass,TeV The column density along the diameter of a white dwarf with 16 2 the mass MWD = 1.2M is δWD = 3.8 · 10 g/cm ,the Fig. 4 The amount of the neutral black holes trapped inside the Earth δ ∼ 20 / 2 during the lifetime of the future 100 TeV collider (integrated luminosity same value for a neutron star is NS 10 g cm .Due L = 104 fb−1) with the suppression factor Eq. (12) taken into account to the inequality (14), a neutron star can stop practically all and for two values of the inelasticity y = 0.5andy = 0.7 the black holes going through it: E is limited from above by more than 108 TeV. For such energies the cosmic ray = setting M Mmin, one gets the underestimated value of this flux is negligible. That is not the case for the white dwarfs, > momentum for the black holes with M Mmin. This means so we have to account for the condition (14) in the further that not all the trapped black holes were taken into account. calculations of the white dwarf constraints. = For example, in our case assumption that M Mmin for all There are several mechanisms that could provide signifi- the black holes produced leads to the decrease in the number cant fluxes of the micro black holes. The most efficient mech- . − . of the trapped black holes by the factor 0 5 0 7 depending anism is the collision of the high energy cosmic rays with on the Planck mass. the surfaces of dense stars. However, considering this mechanism, one has to account for the large magnetic fields of 4 Astrophysical constraints these stars, which presence can lead to the considerable synchrotron energy losses of the cosmic rays. This question was 4.1 General considerations considered by [19] (see Appendix G there). It was shown that the magnetic screening prevents the cosmic rays with If the stable micro black holes could be produced at the collid- the energies more than Emax from reaching the surface, ers, they would be as well naturally produced in the Universe 4 8 2 in the interactions of high energy cosmic rays. This can con- 17 A 10km 10 G Emax 1.8 · 10 eV · , (15) tradict the observed long lifetimes of the dense astrophysical Z 4 R B sin θ objects, in which these black holes can get stuck and accrete. where Z is the charge of the cosmic ray nucleus, R is the First of all, we would like to consider the stopping power star’s radius, and θ is the angle between the momentum of the of different astrophysical objects. The theory of the deceler- cosmic ray and the magnetic axis of the star. The synchrotron ation of the microscopic black holes inside celestial bodies losses are negligible in the case of the ordinary white dwarfs was developed in [19]. It was calculated that the Earth has with the polar magnetic field B ∼ 105 G(E 1020 eV), not sufficient power to stop neither neutral nor charged (with max but they play an important role in the case of neutron stars, for masses more than 7 TeV) relativistic micro black holes (we which the smallest detected magnetic field is B 7 · 107 G consider the black holes produced in the collisions of the high [27]. energy cosmic rays with some slow moving particles), while the Sun can stop the charged relativistic black holes with the masses well in excess of 100 TeV and cannot stop the neu- 4.2 Constraints from white dwarfs tral relativistic black holes. Finally, the general expression for the minimum column density required to stop the neutral The flux of the micro black holes produced by the cosmic micro black hole with the mass M was derived. In our case rays hitting the white dwarf surface is: (D = 6) this required column density is: Emax 3 γ 1/3 φBH = b A(E)J(E)dE δ = . · M6 i M 3, min 0 27 M0 (13) E M0 M6 min (16) 1 1 √ 3 12 2 dx where M = 4.6·10 g/cm , γi is the initial Lorentz factor × dτ f (x) f (τ/x)σ ( s ), 0 x i j ij of the black hole. Consider a cosmic ray nucleus with atomic τmin τ 123 908 Page 6 of 9 Eur. Phys. J. C (2017) 77 :908

10 where b = 1/σNN, σNN = 100 mb Ð total nucleon-nucleon = [ 2 /( 2)]= inelastic cross section, Emin min Mmin A 2m p y 8 5 8 7 · 10 TeV, Emax = 2 · 10 TeV, y = max[0.5, Mmin/ 100 TeV] (most conservative case, corresponding to the 6 100 % p minimum production at the 100 TeV collider), τmin = 2 /( 2 ) ( ) Log(N) 4 10% p Mmin A 2m p y E , J E is the combined energy spectrum of cosmic rays as measured by the Auger Observatory, 2

fitted with a flux model (see [3]). The dependence of the Auger mean atomic number A on the energy E is taken from [3]as 0 well: we interpolated Auger data and averaged over the two 3 4 5 6 7 hadronic interaction models EPOS-LHC and QGS-JetII-04. Planck mass,TeV However in the case of the white dwarf the resulting bounds Fig. 5 The amount of the neutral black holes, stopped by a white dwarf weakly depend on the cosmic ray composition: the black hole with a radius 5600 km during 109 year for the different cosmic ray production at the given energy decreases with increasing A, composition: 100% p, 50% p, 10% p fractions and corresponding to but also the condition (14) becomes less constraining and Auger data on the mean atomic number allows higher energies to be involved. The total number of the black holes, produced on a surface we examine another, less effective mechanisms of the black of a white dwarf of radius R during the time t,isNBH = hole production: 2 4π R Ωt · φBH, where the solid angle Ω = 2π. In order to calculate the whole number of the black holes captured Ð the production on baryons during the lifetime of the Uni- during the lifetime of the white dwarf, we have to limit the verse. We conservatively assume that the cosmic ray flux region of the integration in the Eq. (16) by the condition (14) is constant till the redshift z = 1 (in reality it is sup- and account for a decrease of the column density δ for non- posed to increase with redshift). We also do not take into zero values of the angle of incidence of the cosmic ray α.The account the processes in the early Universe, which are latter is made using the dependence δ(α) (Fig. 1 from [19]). quite model-dependent, considering redshifts z < 1. The The overwhelming majority of the trapped black holes is flux of the black holes can be obtained from Eq. (16), given by the small α, so we conservatively take δ = 0.8δWD substituting parameter b for the following value: (decreasing the value of 0.9δWD by 10% due to the possible systematic error), 1 − cos α<0.01. The calculated number 0 ˆ of the black holes stopped (t = 109 year, R = 5600 km) b = n(z)dt(z) is plotted as a function of M6 in Fig. 5, for the Auger data 1 A(E) and for the 100%, 50%, 10% proton fraction in the 1 cosmic rays. The calculation that uses Auger data on the 3 dz = n0(1 + z) 3 composition sets the lower bound on the number of the black H0(1 + z) ΩM (1 + z) + Ω holes, because cosmic rays with the small atomic number 0 1 2 A (e.g. protons) give the main contribution to the black hole n0(1 + z) dz = = , production, while increasing A (till A 56 of iron) yields the 3 H0 ΩM (1 + z) + Ω number of black holes by orders of magnitude smaller. Thus, 0 considering the mean value of A, we get the conservative (17) estimates of the production. One can see that in the theories −7 −3 with M6 < 7.3 TeV there is at least one black hole that where n0 = 2 · 10 cm is the current baryon density, = · km/s Ω = . is stopped by the white dwarf. One can see also that, due H0 68 Mpc is the Hubble constant, M 0 31, to the stopping condition, this constraint on M6 would not Ω = 0.69 (see [7]). In order to obtain reliable con- 19 improve significantly, even if we knew the precise cosmic ray straints, we set Emax = 5 · 10 eV in Eq. (16), consider- composition. The composition that yields the biggest number ing the cosmic rays with the energies below the GreisenÐ of the black holes trapped (the most optimistic scenario) is ZatsepinÐKuzmin (GZK) limit [21,38]. For these ener- 100% p fraction and it gives only M6 < 7.4TeV. gies the energy loss length for protons is more than 1 Gpc. We have also checked that explicit account for non-uniform distribution of extragalactic baryons [11] 4.3 Constraints from neutron stars very weakly affects our estimate. Eq. (17) gives the value bˆ = 4.6 · 1021 cm−2. Now let us consider a neutron star. As the high energy cosmic Ð The production in binary systems of a neutron star rays cannot reach its surface due to the magnetic screening, and a red giant: cosmic rays hit the giant and produce 123 Eur. Phys. J. C (2017) 77 :908 Page 7 of 9 908

6 2 in this case is smaller than 5.9 · 10 TeV due to the condi-

1 tion (15); this energy is not sufficient for the production of the black holes with large masses. The magnetic field can 0 be neglected at the poles, however at the high energies the –1 surface area that can be reached there is too small and the Log(N) –2 flux is less than in the case of the mechanisms studied above. In general, one can see that the constraints on M are worse –3 100% p 6 in the case of neutron stars than in the case of white dwarfs. –4 50% p 10% p However, detection of the old neutron stars in the Central 5 10 15 20 Molecular Zone and data on the composition of the cosmic Planck mass,TeV rays can improve the existing constraints from neutron stars Fig. 6 The amount of the neutral black holes, stopped by a neutron and make them the most robust. star with a radius 10 km during 1010 year, the cosmic ray energy is E < 5 · 1019 eV, for the different cosmic ray composition: 100% p, 4.4 Case of the charged black holes 50% p, 10% p Till now we considered the neutral black holes. The charged black holes will be stopped in a white dwarf for the whole black holes which then impinge on the neutron star (in range of energies due to electromagnetic interactions. In the case of neutron stars we consider the neutral black order to get the number of the black holes stopped during holes, so they are not affected by the magnetic field); the lifetime of a white dwarf, we have to do the same cal- bˆ = 1/σ = 1025 cm−2, however t ≡ ‘full coverage NN culations, as in the case of the neutral black holes, however equivalent’ 30 Myr (see [19], Appendix H). ignoring the inequality (14). For 100% proton composition Ð The production on interstellar medium (bˆ = n , where H of the cosmic rays the number of the stopped black holes n ∼ 1021 cm−2 is the average column density of hydro- H exceeds 6.6 · 104 till the Planck mass of 14 TeV. For this gen in the galaxy). and bigger values of the Planck mass the production of black Ð The production on the Central Molecular Zone of our holes at the 100 TeV collider is zero, see Fig. 2. Thus, the the- Galaxy [28], for which bˆ = nL = 6·1022 cm−2. However ories without the mechanism of Schwinger discharge yield no long-lived neutron stars have been observed yet in more than one black hole trapped in a white dwarf during its this zone: maximal ages of the observed ones are t = lifetime, if the fraction of protons in the cosmic rays at ener- 104 − 105 year. gies from the process threshold 5 · 1018Ð5 · 1019 eV exceeds 1.5·10−5, which is actually the case, according to the results The first three mechanisms give comparable fluxes of the of Auger [3] and Telescope Array [6]. black holes. The largest flux is achieved in the second mech- According to the accretion theory, a micro black hole will anism, 6 times higher than in the first one and 30 times higher accrete a neutron star very fast relative to the lifetimes of than in the third. The fourth mechanism produces flux that the known neutron stars (in our case accretion time tacc 2 is 4Ð5 times lower. It is worth noting that the last mecha- 5.3 · (M6/M0) min < 1.5 days). A white dwarf in our case 2 2 4 nism will be the most constraining, if old millisecond pul- will be destroyed in t = 10 ·(M6/M0) year < 4·10 year, sars are detected in the Central Molecular Zone. We use the that is also negligible in comparison with the observed life- first mechanism for our estimates, because the second one times of the white dwarfs. Thus, astrophysical observations is more model-dependent and yields large systematic errors. constrain the theories in question and forbid the production The number of the black holes stopped by the neutron star of the charged stable micro black holes at the 100 TeV col- 10 (t = 10 year, R = 10 km) is plotted as a function of M6 lider. The production of the neutral stable micro black holes in Fig. 6 in case of the different fractions of protons in the is forbidden in the theories with Planck mass smaller than cosmic rays. The maximum M6, that leads to more than one 7.3 TeV. black hole stopped, is 4.1 TeV for 10% proton composition, 5.0 TeV for 50% proton composition and 5.4 TeV in case 4.5 Constraints from astrophysical neutrinos of 100% proton composition. In this calculation we limit the energy of the cosmic rays from above by 5 · 1019 eV, so the One has to mention that robust astrophysical constraints most optimistic case of 100% p composition is quite possible. hypothetically can be obtained from the mechanism of Worse result, with the maximum M6 around 3.5 TeV,is given high energy neutrinos hitting the surface of a neutron star. by the mechanism of the cosmic rays (we take A/Z ∼ 2) hit- The known effect of the decay of high energy neutrino in ting the surface of the neutron star with the minimum mag- the large magnetic field into an electron and a W-boson netic field B = 7 · 107 G. The energy of the cosmic rays [12,16,26,32] does not influence these constraints. Indeed, 123 908 Page 8 of 9 Eur. Phys. J. C (2017) 77 :908

4 10 – WD L = 10 ab 1 2 constraint 8 0

6 –2 y = 0.7

Log(N) – 4 Log(N) 4

–6 y = 0.5 2 –8 0 –10 5 10 15 20 4 6 8 10 12 14 Planck mass,TeV Planck mass,TeV

Fig. 7 Upper bound on the amount of the black holes, produced due Fig. 8 The constraint from white dwarfs on the number of the neutral to hypothetical high energy neutrino collisions with a neutron star with black holes, trapped inside the Earth a radius 10 km during 10 million years and stopped by it

experimentally. Now the highest energy of the detected astro- in the article [16] it was found that above the process thresh- 4 16 physical neutrino does not exceed 10 TeV [5]. old Eth ∼ 2.2 · 10 · (Bcr /B) eV an asymptotic neutrino 2 16 absorption length is l ∼ 1.1 · (Bcr /B) · (10 eV/E) m, where E is the neutrino energy, B is the magnetic field and 13 Bcr = 4.4·10 G. Thus the effect of neutrino decay becomes 5 Conclusion significant only for the neutron stars with the magnetic fields about 1012 G and higher. Observation of the neutron stars In this article we have studied the phenomenology of the with the small magnetic fields, in particular observation of models with extra dimensions in absence of the Hawking a neutron star with the magnetic field B ∼ 7 · 107 G[27], radiation in order to conduct an independent observations- suggests that this effect can be neglected in our study. We based check of safety of the proposed 100 TeV collider. The constrain the number of the black holes trapped inside a neu- models with more than 6 dimensions always yield Earth’s tron star with the small magnetic field, using the upper limit accretion times larger than the lifetime of the Solar system. on the flux of single-flavour high energy neutrinos [3]: A theory with five dimensions could be consistent with the existing experimental constraints on the size of extra dimen- − − − − − N(E)<6.4 · 10 9 · E 2 GeV cm 2 s 1 sr 1, (18) sions and yield accretion times smaller than the lifetime of the Solar system only with a fine-tuning of the warp-factor, valid for the energies E = 0.1 − 25 EeV. The resulting given by the inequalities (4). The calculation of the num- bounds on the number of the black holes trapped per ten ber of the micro black holes that would have been produced million years are presented in Fig. 7. The calculation was in the future 100 TeV collider with the integrated luminos- − made using a formula analogous to the Eq. (16): ity L = 10 ab 1 and the astrophysical constraints from the observational data on the lifetime of white dwarfs and cosmic Emax ray composition suggest that it is possible to exclude the pro- NBH = 2π St 3N(E)dE duction of the charged stable micro black holes already. As it is shown in Fig. 8, the case of the neutral black holes, while Emin (19) broadly addressed for most D and mass values, leaves some 1 √ σ ( s) loophole, which can be closed with further cosmic ray data × dx · f (x) · √ , i σ ( ) + σ ( ) (e.g. on the neutrino spectrum and cosmic ray composition) i s tot E xmin or astrophysical observations.

= = 2 /( 2) where s 2m p Ex, Emin Mmin 2m p y Ð the threshold Acknowledgements The authors would like to thank P. Satunin, of black hole production, xmin = Emin/E, Emax = 25 EeV, S. Troitsky, I. Tkachev and the anonymous referee for numerous valu- able discussions and comments on the manuscript. The question of the σtot is the full neutrino-nucleon inelastic cross section, see = . safety of a future collider, in its relation to cosmic-ray studies, was [17]. Inelasticity was conservatively taken to be y 0 5. asked by J.-R. Cudell at the Solvay meeting “Facing the scalar sec- The number of neutrino flavours was accounted for by the tor”. The work of the authors was supported by the Russian Science multiplier 3 in the integrand. Foundation Grant 14-12-01340. 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