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3-5-2019

Bounds on extra from micro black holes in the context of the metastable Higgs

Katherine J. Mack North Carolina State University

Robert A. McNees IV Loyola University Chicago, [email protected]

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Recommended Citation Mack, Katherine J. and McNees, Robert A. IV, "Bounds on extra dimensions from micro black holes in the context of the metastable Higgs vacuum" (2019). Physics: Faculty Publications and Other Works. 56. https://ecommons.luc.edu/physics_facpubs/56

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This work is licensed under a Creative Commons Attribution 4.0 License. © American Physical Society, 2020. PHYSICAL REVIEW D 99, 063001 (2019)

Bounds on extra dimensions from micro black holes in the context of the metastable Higgs vacuum

Katherine J. Mack* North Carolina State University, Department of Physics, Raleigh, North Carolina 27695-8202, USA † Robert McNees Loyola University Chicago, Department of Physics, Chicago, Illinois 60660, USA

(Received 31 January 2019; published 5 March 2019)

We estimate the rate at which collisions between ultrahigh- cosmic rays can form small black holes in models with extra dimensions. If recent conjectures about catalyzed by evaporation apply, the lack of vacuum decay events in our past light cone may place new bounds on the black hole formation rate and thus on the fundamental scale of in these models. For theories with fundamental scale E above the Higgs instability scale of the Standard Model, we find a lower bound on E that is within about an order of magnitude of the energy where the cosmic-ray spectrum begins to show suppression from the Greisen-Zatsepin-Kuzmin effect. Otherwise, the abundant formation of semiclassical black holes with short lifetimes would likely initiate vacuum decay. Assuming a Higgs instability scale at the low end of the range compatible with experimental data, the excluded range is approximately 1017 eV ≲ 18.8 17 18.1 E ≤ 10 eV for theories with n ¼ 1 extra , narrowing to 10 eV ≲ E ≤ 10 eV for n ¼ 6. These bounds rule out regions of parameter that are inaccessible to collider experiments, small-scale gravity tests, or estimates of Kaluza-Klein processes in neutron stars and supernovae.

DOI: 10.1103/PhysRevD.99.063001

I. INTRODUCTION in the TeV range and astrophysical limits as high as Oð102Þ–Oð103Þ In models with extra dimensions, the fundamental scale TeV [1,5]. We present new limits from black hole creation in the of gravity may be lower than the four-dimensional context of recent work proposing that Hawking evaporation scale, M . This presents the possibility that high-energy Pl of black holes can induce the decay of the standard model collisions between particles, for instance in colliders or Higgs vacuum [6–9]. These papers argue that the nucle- via cosmic rays, may form black holes if a high enough ation of a bubble of true vacuum in general precedes the center-of-mass energy is achieved [1–4]. Large extra final evaporation of the black hole, suggesting that any dimensions, if discovered, would constitute new physics production of black holes with evaporation less than and potentially provide an explanation for the relative the age of the in our past light cone should have weakness of gravity in relation to the other fundamental already led to vacuum decay. The most recent work in the forces. In addition to searches for microscopic black series [9] explicitly confirms that evaporating black holes holes formed in particle collisions, experimental con- formed in theories with extra dimensions are capable of straints on extra dimensions have generally come from seeding vacuum decay. The decay of the false vacuum is a searches for modifications of the inverse-square force law dramatic consequence that presents an unmistakable (and of gravity at small scales or from signatures of Kaluza- fatal) observational signature of microscopic black hole Klein gravitons or other exotic particles. Constraints on production. Thus, its nonobservation allows us to place the higher-dimensional fundamental scale depend on the limits on the higher-dimensional fundamental scale, number of extra dimensions proposed, with collider limits excluding a range of values that are several orders of magnitude beyond the scales probed by other tests involv- *[email protected]; https://twitter.com/AstroKatie ing production, such as via signs of † [email protected]; https://twitter.com/mcnees Hawking evaporation in colliders or from nearby cosmic- ray collisions. Published by the American Physical Society under the terms of Our analysis relies on two main assumptions, both of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to which come with some important caveats that we describe the author(s) and the published article’s title, journal citation, here. The first is the of the Higgs vacuum, as and DOI. Funded by SCOAP3. implied by recent measurements of the and

2470-0010=2019=99(6)=063001(10) 063001-1 Published by the American Physical Society KATHERINE J. MACK and ROBERT MCNEES PHYS. REV. D 99, 063001 (2019) top masses [10]. This result is based on the validity of the four-dimensional Planck scale. If the instability scale the Standard Model of , so any beyond- for the Higgs vacuum is low enough, this allows us to Standard-Model (BSM) physics may alter the effective exclude a range of values for the fundamental scale that are potential of the Higgs in a way that rescues our not probed by the accelerator and astrophysical processes vacuum from metastability [11]. A high energy scale of associated with current bounds. For a given value of the , were it to be confirmed, would give evidence that fundamental scale, we use the UHE cosmic-ray spectrum new physics stabilizes the vacuum in some way, since high- from the Auger experiment (see Sec. II and Ref. [14])to energy-scale inflationary fluctuations would likely have make a conservative estimate of the number of black holes instigated a transition to the true vacuum in the early formed in particle collisions in our past light cone. We note Universe [12,13]. While we fully recognize (and, in fact, that the measured cosmic-ray spectrum includes a steep hope) that vacuum metastability ends up being ruled out by dropoff at high . This is believed to be due to the BSM physics or a better understanding of inflation, we will, Greisen-Zatsepin-Kuzmin (GZK) effect [15,16], which for the purpose of this study, rely on the great successes of prevents the highest-energy cosmic rays from traveling the Standard Model to justify the apparent metastability of unimpeded across cosmological distances. If this is the the Higgs vacuum as an observational tool for establishing case, then it is likely that even more energetic particles are constraints on higher-dimensional theories. Second, we are plentiful in parts of the cosmos where they are accelerated assuming that the results of [6–8] hold in a qualitatively by high-energy astrophysical processes. But without know- similar way for theories with more than four ing more about the mechanisms involved we restrict our dimensions, i.e., that black hole evaporation can seed analysis to cosmic rays with energies below the GZK vacuum decay. This was explored in [9], where the authors cutoff. Thus, our calculation probably underestimates both construct an approximate solution for a brane- black hole formation rates and the maximum center-of- world black hole in a theory with one extra dimension and momentum energies achieved in these collisions. As a then estimate that their results extend to regimes where result, the excluded range for the fundamental scale of small black holes are produced in particle collisions. We theories with extra dimensions may extend to even higher will apply these results beyond one extra dimension, values than those we present here. though earlier calculations suggest the effect may be In Sec. II we discuss a method for estimating the number somewhat suppressed [7].1 More importantly, the conclu- of collisions that have taken place in our past light cone sions of [9] require the instability scale ΛI of the Higgs between UHE cosmic rays, and we review the Pierre Auger ’ vacuum to lie below the fundamental scale E of the Observatory s spectrum of these particles. In Sec. III we higher-dimensional theory. Otherwise, the Standard Model extend this to collisions capable of forming black holes in calculation of the Higgs potential no longer applies. higher-dimensional theories, obtain bounds on the funda- For our analysis, we must assume that the instability scale mental scale of these theories in Sec. IV, and then discuss is at the low end of the range consistent with experimental these results in Sec. V. Appendix A considers the various limits. The most likely range calculated by [10] is criteria that must be satisfied for a reliable semiclassical 19 20 analysis of black hole formation, and Appendix B discusses ΛI ∼ 10 –10 eV, with some uncertainty around that an analytical result for black hole formation rates that value. For our analysis to be fully reliable, we require 18 supports the numerical results used in the main text. scales no higher than ΛI ∼ 10 eV for theories with one or Λ ∼ 1017 two extra dimensions and I eV for theories with II. COLLISIONS OF ULTRAHIGH-ENERGY up to six. This is an important qualifier on our main results COSMIC RAYS and will be discussed in more detail at the end of the paper. The structure of our calculation is as follows. Assuming At ultrahigh energies, cosmic rays are rare enough that that the Higgs vacuum is metastable and that its decay is we expect interactions between them to be exceedingly catalyzed by black hole evaporation, we take its persistence infrequent. But on timescales comparable to the age of the as evidence against black hole evaporation in our past Universe, even a low rate can to an appreciable number light cone. While this observation can also constrain the of collisions with center-of-momentum (CM) energies production of low mass primordial black holes in the several orders of magnitude greater than anything that early Universe, we apply it here to the production of can be achieved in existing accelerators. Let us quickly microscopic black holes in particle collisions. Specifically, review the estimate of collisions between UHE cosmic rays 20 we consider the formation of black holes in collisions with energies above 10 eV given by Hut and Rees in [17]. between ultrahigh-energy (UHE) cosmic rays, in theories Assuming a homogeneous and isotropic distribution, the with extra dimensions and a fundamental scale well below density of UHE cosmic rays with energy greater than E is proportional to the integrated flux

1 Z The reduced branching ratio of false vacuum decay rate to the 4π ∞ Hawking evaporation rate may be offset by the production of ρðEÞ¼ dE0JðE0Þ; ð1Þ large numbers of black holes. c E

063001-2 BOUNDS ON EXTRA DIMENSIONS FROM MICRO BLACK HOLES … PHYS. REV. D 99, 063001 (2019) where JðEÞ is the differential flux. For constant density ρ and cross section σ, the rate of collisions per particle is ρσc, and the overall rate of collisions per unit volume is

R ¼ ρ2σc: ð2Þ

The total number of collisions in our past light cone is given by this rate times the spacetime volume

N ¼ Rc3T4; ð3Þ where T is the over which these collisions have occurred and our assumptions hold. Hut and Rees calcu- lated the density of cosmic rays above 1020 eV using the FIG. 1. The Auger spectrum of UHE cosmic rays with differential flux given by Cunningham et al. in [18]: E>1018 eV, approximated as a set of power laws [14]. 1 14 10−33 1019 2.31 JðEÞ¼ . × eV : ð Þ γ ¼ 4 3 0 2 2 4 index drops off to 3 . . . The spectrum between m ·s·sr·eV E E E ankle and break is thought to possibly represent the −23 −3 transition to a population of extragalactic cosmic rays, Then (1) gives a density of 1.8 × 10 m . The cross E while the steep falloff above break is likely due to the GZK section is taken to be of order the Compton wavelength effect [15,16]. squared, Repeating the estimate of Hut and Rees with the Auger 1020 — 2πℏc 2 spectrum yields fewer collisions above eV on the σðEÞ ≃ ; ð5Þ order of a few thousand—because of the steep dropoff in E E the flux above break that is not accounted for in (4).Indeed, the presence of this feature in the spectrum suggests which at 1020 eV gives 1.5 × 10−52 m2. For their order of −52 2 that the upper limit in (1) should not extend to arbitrarily magnitude estimate, Hut and Rees use σ ≃ 10 m . The high energies. The GZK effect prevents cosmic rays from −67 −1 per-particle rate of collision is then ≃3 × 10 s , and the traveling cosmological distances with energies greater than rate of collisions per unit volume comes out to E .2 So it seems unwarranted to assume a homogeneous −90 −3 −1 break R ≃ 3 × 10 m s . Taking the and isotropic distribution for cosmic rays at those energies to be about T ¼ 1010 yr, the number of collisions in the over the full volume of our past light cone. Without knowing past light cone is roughly N ≃ 8 × 105. Hut and Rees give more about the origin of UHE cosmic rays, we will a final estimate of N ≈ 105. conservatively limit all of our calculations to cosmic rays For our calculations, we will use more recent results for with energies below the break energy in the Auger spectrum. the differential flux of UHE cosmic rays in place of (4). The approximation (2) for the rate of collisions per unit The Pierre Auger Observatory is a hybrid cosmic-ray volume treats all particles as if they had roughly the same observatory consisting of surface Cerenkov detectors and energy, with a constant cross section for collisions. We can air-shower observing telescopes, which allows it to collect refine the estimate by dropping these assumptions, account- large samples of cosmic rays across a wide range of ing instead for all collisions above a given CM energy and energies. Its measurements of the cosmic-ray energy including the energy dependence of the cross section. spectrum above 1018 eV are well described by a series Assuming once again a homogeneous and isotropic dis- of power laws with free breaks between them, or else by tribution of UHE cosmic rays, the rate per unit volume of broken power laws with an additional smooth suppression collisions with CM energy greater than E is given by factor at the highest energies [14,19–21]. Here we adopt the Z Z 16π2 E 1 values given in [14], with the differential flux in each range R ¼ break dE0dE00 duσðE Þ c CM of energies taking the form Emin 0 JðE0ÞJðE00ÞΘðE − EÞ; ð Þ × CM 7 JðEÞ ∝ E−γ: ð6Þ

The flux is shown in Fig. 1. Below the “ankle” energy, 2It is possible that the GZK effect is only partly responsible E ¼ 1018.610.01 γ ¼ 3 27 for the dropoff in flux above E , which may also reflect, for ankle eV, the spectral index is 1 . break 0 02 “ ” instance, the maximum energies that can be achieved by the . . Above the ankle energy, but below the break energy sources that accelerate the particles [19]. While its origin does not E ¼ 1019.460.03 break eV, the spectral index flattens to impact our estimates, we interpret Ebreak as the scale associated γ2 ¼ 2.59 0.02. Above the break energy the spectral with the GZK effect.

063001-3 KATHERINE J. MACK and ROBERT MCNEES PHYS. REV. D 99, 063001 (2019) where u ¼ð1 − cos ψÞ=2 is related to the angle ψpffiffiffiffiffiffiffiffiffiffiffiffiffibetween We also require that the black holes be small enough E ¼ 2 E0E00u L the momenta of the colliding particles, CM is compared with the compactification scale that a flat- the CM energy, and the Heaviside theta function restricts space approximation is valid. For a discussion of this E >E the domain of integration to collisions with CM . The requirement, see Appendix A. E upper limit in the energy integrals is taken to be break, For the collisions between UHE cosmic rays in the which restricts the calculation to cosmic rays with energies previous section, the cross section (5) was proportional to below the GZK cutoff, while the lower limit is the square of the Compton wavelength and hence decreased at higher energies. But at CM energies well above E the E2 E ¼ ; ð Þ cross section for black hole formation exhibits the opposite min 4E 8 behavior. The geometric cross section for black hole break formation is [22–23] which is the minimum energy of a particle that can E σ ¼ Oð1Þπr2 ; ð Þ participate in a collision with CM energy of at least . BH H 11 As before, the number of events in our past light cone 3 4 is N ¼ Rc T . In our calculations, we will take T ¼ where rH is the horizon radius of a black hole of mass 10 M ¼ E =c2 10 years. This may be a conservative assumption, as BH CM , and an overall factor of order 1 reflects the production rate for UHE cosmic rays is likely to have various corrections. Collisions at higher energies produce been higher toward the early part of that time window, black holes with larger mass, and hence larger horizon closer to the peak of (AGN) activity radius, resulting in a cross section that grows as a positive E around a of 2. power of CM. In the next section we will employ (7) to estimate the rate Assuming the collision forms a Schwarzschild black of black hole formation in models where the fundamental hole, the horizon radius in 4 þ n dimensions is [24] scale of gravity is below the maximum CM energies 2 1 3þn 1 ℏc M c nþ1 8π Γð Þ nþ1 accessible in collisions between UHE cosmic rays. BH 2 rH ¼ 3þn : ð12Þ E E n þ 2 π 2 III. BLACK HOLE FORMATION VIA COSMIC-RAY COLLISION Then, up to the Oð1Þ factor in Eq. (11), the cross section for M ¼ E =c2 In higher-dimensional theories the fundamental scale a collision forming a black hole with BH CM is M of gravity may be lower than Pl. We will consider a 2 2 3þn 2 ð4þ Þ ℏc E nþ1 8Γð Þ nþ1 generic 4 þ n-dimensional theory with fundamental scale σ n ¼ CM 2 : ð13Þ 2 BH E E n þ 2 M ¼ E=c related to the Newton’s constant by cnþ5ℏnþ1 Using this cross section in Eq. (7), we can estimate the rate Gð4þnÞ ¼ : ð Þ N 2þn 9 at which black holes are formed by collisions between UHE E cosmic rays in a higher-dimensional theory with funda- If the scale E is low enough, collisions between UHE mental scale E. cosmic rays at sufficiently high CM energy are expected to As an example, consider a theory with one extra M ∼ E =c2 n ¼ 1 form black holes of mass BH CM . dimension ( ). Using Eq. (10), black holes with S ≥ 102 M ≥ 8 76M For our estimates of black hole formation via scattering entropy BH have mass BH . . The number to make sense, the black holes should be large enough that a of such black holes formed in our past light cone by semiclassical treatment is appropriate. We enforce this by collisions between UHE cosmic rays is approximately considering only black holes with entropy above some Z Z pffiffiffiffiffiffiffiffiffiffiffiffiffi S 2 2 E 1 0 00 minimum value min (see Appendix A for a brief dis- 3 4 128π ℏc break 0 00 E E u M =M N ¼ c T dE dE du cussion). This implies that the ratio of BH must be 3c E E 0 E greater than pffiffiffiffiffiffiffiffiffiffiffiffiffimin 0 00 0 00 × JðE ÞJðE ÞΘð2 E E u − 8.76EÞ: ð14Þ nþ3 1 n þ 2 π 2 nþ2 nþ1 λ ¼ ðS Þnþ2: ð Þ 4π nþ3 min 10 Fixing a fundamental scale allows us to evaluate this 2Γð 2 Þ expression explicitly to determine a number of black-hole- S ¼ 102 M We will typically take min , which implies that BH producing events above that scale. As an illustrative calcu- must be greater than M by a factor that is Oð10Þ for n ¼ 1 lation, we use the Auger results for the differential flux, 10 18 5 and increases to Oð102Þ for n ¼ 6. Neglecting energy loss T ¼ 10 yr, and a fundamental scale of 10 . eV. The M ¼ E =c2 N ≃ 1 6 1011 during the formation process, BH CM and colli- numerical evaluation of this integral gives . × . E ≥ λE 1018.8 sions with CM are considered to form semiclassical Raising the fundamental scale to eV lowers the black holes. number of events to N ≃ 2.6 × 106.

063001-4 BOUNDS ON EXTRA DIMENSIONS FROM MICRO BLACK HOLES … PHYS. REV. D 99, 063001 (2019)

Since we restrict our attention to UHE cosmic rays E ¼ 1019.46 with energy below the GZK cutoff at break eV, the maximum possible CM energy in our analysis is 2E ¼ 1019.76 break eV. This places an upper limit on the M ≥ λM mass of the black hole, so the requirement BH E < 2E =λ implies that our analysis is valid only for break .In this example with n ¼ 1, the largest fundamental scale we 18.82 can consider is 10 eV, and N plunges to zero as E approaches this value. The dropoff is steep enough that the 6 3 difference between E for N ∼ 10 and N ∼ 10 is much E less than the uncertainties in break and other factors. If vacuum decay triggered by black hole evaporation is as likely as the claims of [6–9], then essentially any value of E 2E =λ up to break results in too many black holes being formed. Note that by raising the maximum cosmic-ray energy in FIG. 2. Number of collisions N forming a black hole with E 2E =λ ∼5 entropy S ≥ 102 over the past T ¼ 1010 yr, as a function of the Eq. (7), could be greater than break by a factor of BH and still allow a significant number of black holes to form fundamental scale. The colored lines are for different values of n, the number of extra dimensions. N drops off rapidly as E and evaporate over the history of our Universe. But the 2E =λ majority of those collisions would involve particles with approaches break . energies above the GZK cutoff, and as explained in the previous section, our assumption of a homogeneous and isotropic distribution seems questionable for that popula- vacuum decay catalyzed by the evaporation of these black tion of UHE cosmic rays. holes excludes fundamental scales in the range 2E 1019.76 break eV IV. BOUNDS ON E FROM BLACK Λ

063001-5 KATHERINE J. MACK and ROBERT MCNEES PHYS. REV. D 99, 063001 (2019)

V. DISCUSSION We have presented constraints on the fundamental scale and the size of extra dimensions in higher-dimensional theories, based on the nonobservation of vacuum decay catalyzed by microscopic black holes. This scenario is based on the mechanism outlined in [6–9], in which black holes seed vacuum decay before their evaporation is com- plete. It assumes the metastability of the Higgs vacuum, supported by recent measurements of the mass of the Higgs boson and , at a scale below the funda- mental scale of the higher-dimensional theory. While this concept has been used to place limits on the production of primordial black holes [7,8], ours is the first analysis to FIG. 4. Bounds on the size of extra dimensions (blue), with place quantitative limits on the fundamental scale of extra- typical values for TeV-scale gravity (red), for different values of dimensional theories based on this method. the number of extra dimensions, n. The blue shaded region of the Table II summarizes current limits on the fundamental plot is excluded by our constraints. scale and/or size of extra dimensions from a range of methods. Comparing with Table I, limits from vacuum decay catalyzed by black hole evaporation exclude ranges for E, but they cannot be reliably estimated with the method used here. of these parameters that are many orders of magnitude beyond what can be probed with current tests. The excluded range of E for different values of n is Our analysis is limited by the assumptions underlying shown in Fig. 3, assuming a Higgs instability scale of – Λ ∼ 1017 S ≥ 102 the conclusions of [6 9]. The most important of these I eV. For BH , the upper end of this range is assumptions is the requirement that ΛI

063001-6 BOUNDS ON EXTRA DIMENSIONS FROM MICRO BLACK HOLES … PHYS. REV. D 99, 063001 (2019) range collapses and there is no bound. Likewise, we are not cutoff may achieve higher CM energies than we con- able to draw any conclusions from this analysis concerning sidered. Such collisions could form black holes for even 3 scenarios where E is below ΛI. The authors of [9] quote larger values of the fundamental scale E, making our 17 ΛI ∼ 10 eV as the lowest value consistent with exper- analysis relevant for a wider range of values of the Higgs imental limits on the top quark mass, which is well instability scale. below our lower bounds on E for 1 ≤ n ≤ 6. They estimate 17 that for n ¼ 1 extra dimension, with ΛI ∼ 10 eV and 18 ACKNOWLEDGMENTS E ∼ 10 eV, vacuum decay would be caused by black M ∼ 1020 E < 1018.8 R. M. was supported in part by the National Science holes with BH eV. For eV, we find M ∼ Foundation under Grant No. NSF PHY-1748958 through that a significant number of black holes with BH 1020 eV are produced. Even if the instability scale is as the KITP Scholars program, and by Loyola University 18 Chicago through a Summer Research Stipend. K. J. M. and high as ΛI ∼ 10 eV, our n ¼ 1 (and possibly n ¼ 2) R. M. acknowledge the hospitality of Caltech and the Burke value for E seems high enough to justify concerns about Institute, especially for the workshop “Unifying Tests of vacuum decay. However, the most likely range of values for 19 20 ” during which this project was con- the instability scale ΛI appears to be around 10 –10 eV. ceived. K. J. M. also acknowledges the Australian Research In that case the fundamental scale E would have to be Council and Melbourne University for support during early greater than 1020–1021 eV, which is outside the regime that stages of this project, and R. M. thanks the Kavli Institute can be probed with this conservative calculation. On the for and the KITP Scholars Program for other hand, UHE cosmic-ray observatories have detected E ¼ 3 1020 hospitality and a productive work environment. We are particles with energies as high as × eV. The grateful to Leo Stein and Walter Tangarife for helpful propagation of such particles on cosmological scales is comments on an earlier draft of this paper, and to Nima suppressed by the GZK effect, so our method for estimating Arkani-Hamed for an important correction that improved the number of collisions forming black holes is not the quality of our analysis. applicable. But if collisions between particles at these energies occur in regions where UHE cosmic rays are E ∼ 1021 produced, then CM energies CM may be achieved. APPENDIX A: BLACK HOLE FORMATION In that case rapidly evaporating black holes may have been AND SEMICLASSICAL METHODS 20 formed for fundamental scales as high as E ∼ 10 eV, potentially inducing vacuum decay even if the instability Our analysis is based on the formation of black holes in 19 collisions between UHE cosmic rays, which rapidly evapo- scale is ΛI ∼ 10 eV, which is in the most likely range of rate via . Thus, we must consider three values [10]. For larger values of the instability scale it questions. First, under what conditions can we say that a seems unlikely that black hole formation via UHE cosmic- collision has formed a black hole? Second, when can those ray collisions could be used to constrain E. black holes be described semiclassically? And third, since The approach we have taken here comes with impor- the extra dimensions of spacetime are assumed to be small tant caveats described above and in the Introduction. and compact, when can the black holes be described using Nevertheless, the possibility of establishing bounds on extra dimensions at scales that are not probed by other results that assume an asymptotically flat spacetime? methods makes this is a promising direction for continued A basic criterion for saying that a black hole has formed research. Additionally, our method relies on qualitatively is that the decay time should be very long compared to the different physics than the tests responsible for the strongest timescale associated with the formation process. For black existing constraints on extra dimensions. It does not rely holes formed via collision, we take that to mean that the decay time should be much longer than the time needed for on assumptions about gravitons or other BSM particle r 4 þ n physics and is therefore an interesting complement to the particles to cross a region of linear size H.In existing methods. dimensions the decay time is of order

We expect that our analysis can be made more robust nþ3 ℏ M nþ1 with improved inferences about the instability scale from τ ∼ BH ; ð Þ D 2 A1 collider data, along with a more complete accounting of the Mc M full range of high-energy particle interactions in our past τ ≃ r =c light cone. In particular, collisions in regions where UHE while the crossing time C H is cosmic rays are accelerated to energies above the GZK 3þn 1 1 ℏ 1 8Γð Þ nþ1 M nþ1 τ ¼ pffiffiffi 2 BH : ð Þ C M c2 π n þ 2 M A2 3Recall that the original motivation for large extra dimensions—a natural explanation for the apparent weakness n τ of gravity—would require a fundamental scale well below the In all cases of interest, the -dependent factors in C are Higgs instability scale. Oð1Þ, so the condition τD ≫ τC is equivalent to

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nþ2 M nþ1 BH ≫ 1: ðA3Þ M

The power on the left-hand side of this inequality is M ≫ M always greater than 1, so black holes with BH satisfy τD ≫ τC. To justify a semiclassical treatment, the entropy of the S ≫ 1 black hole should satisfy BH . In any dimension the entropy is given by one-quarter of the horizon area in units of the fundamental length scale. For a nonrotating black hole this is

A c3 ω r2þnc3 S ¼ H ¼ 2þn H ; ð Þ BH A4 λ ¼ M =M 4ℏGð4þnÞ 4ℏGð4þnÞ FIG. 5. The ratio BH for a black hole with N N entropy Smin. 3þn 3þn where ω2þn ¼ 2π 2 =Γð Þ is the area of a unit 2 þ n- 2 1 2 M nþ1 M n sphere. Using Eqs. (9) and (12), the entropy can be BH Pl ≪ ðA8Þ expressed as M M

nþ2 1 nþ2 4π nþ1 4 nþ1 M nþ1 Thus, M =M should be large enough to justify a semi- S ¼ BH : ð Þ BH BH A5 n þ 2 ωnþ2 M classical calculation, but not so large that the black hole begins to notice the extent L of the extra dimensions. In the The first two factors give a number greater than 1 for text we consider theories with a fundamental scale as large 19 1 ≤ n ≤ 9, and of Oð1Þ out to n ∼ 35. So the condition as E ∼ 10 eV, and limit ourselves to UHE cosmic-ray S ≫ 1 E ∼ 1020 BH is essentially the same as the previous condition, collisions with CM energy no greater than CM eV. τD ≫ τC, in all cases of interest. In that case, for collisions forming black holes with entropy M ¼ 10M S r =L For a black hole of mass BH , the entropy greater than min, the ratio H always satisfies S ≃ 120 n ¼ 1 S ≃ 20 ranges from BH when down to BH for n ¼ 6 S n M =M 2 1 . Since BH decreases with for fixed BH ,we rH 17 −1 ð16πÞ n −1 ≤ 2πð1 79 10 Þ n ðS Þ n: ð Þ S . × 2 min A9 will always set a minimum entropy min that is sufficient to L ðn þ 2Þ ωnþ2 justify semiclassical calculations, and then restrict our attention to black holes with entropy at or above this S ¼ 102 10−18 n ¼ 1 For min , this is of order for , and of λ −3 cutoff. Using Eq. (A5), this fixes the minimum value of order 10 for n ¼ 6. In these cases, the black holes we M =M the ratio BH for a semiclassical black hole in our consider are all much smaller than the size of the extra analysis as dimensions and the physics should be well described by

1 formulas that assume an asymptotically flat spacetime. n þ 2 wnþ2 nþ2 nþ1 λ ¼ ðS Þnþ2: ð Þ A quick calculation shows that the size of extra dimensions min A6 4π 4 is also much larger than the fundamental length scale L ≫ l, so that corrections may safely be This is shown for 1 ≤ n ≤ 6 in Fig. 5. Black holes with neglected. M ≥ λM have S ≥ S . S ≥ 102 BH BH min Thus, for black holes with entropy BH , the For the higher-dimensional theories considered in this process of formation via collision and subsequent evapo- paper we assume that n dimensions are compactified with ration should be well described using semiclassical meth- L length scale . The formulas above assume that spacetime ods and asymptotically flat-space results for 1 ≤ n ≤ 6. The is asymptotically flat, but we may regard them as approx- case n ¼ 7 is borderline, with the conditions described imately true when the black hole radius (12) is much above and the assumptions outlined elsewhere in the paper r ≪ L smaller than the compactification scale: H . In the case beginning to break down. of toroidal extra dimensions, the four-dimensional Planck mass is related to the compactification scale of the higher dimensional theory by APPENDIX B: ANALYTIC EXPRESSION FOR N n 2 n 2þn c Since we consider cosmic rays with energies below the M ¼ð2πLÞ ðMÞ : ðA7Þ E Pl ℏn GZK cutoff at break, the minimum energy of a that can participate in a collision with CM energy above Ignoring factors of Oð1Þ, the condition rH ≪ L becomes λE is

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2 ðλE Þ where γ2 ¼ 2.59 is the spectral index given by Auger for E ¼ : ð Þ min 4E B1 cosmic rays with energies between E and E . Notice break ankle break that N grows with the minimum entropy for semiclassical ðS Þ2 S If we express the fundamental scale as a fraction of the calculations as min . This is due to the fact that as min E ¼ð1 − χÞ2E =λ maximum value that we can probe, break goes up, the fundamental scales we probe go down like 1=λ, with 0 ≤ χ < 1, then both cosmic rays must have energy resulting in a larger cross section (13). E greater than ankle when The full expression obtained from evaluating the integral sffiffiffiffiffiffiffiffiffiffiffi in Eq. (B3) is not especially illuminating, but was used to E verify the numerical results presented in Sec. IV.Forχ ≪ 1, 1 − χ > ankle ¼ 0 38: ð Þ E . B2 the regime where E is extremely close to the maximum break value for which we can estimate black hole formation rates, N is well approximated by In this regime the differential flux in Eq. (15) is described by a single power law, and the integral can be evaluated 4ðn þ 2Þ2 analytically. N ≃ 2 38 105ðS Þ2 χ3 ð Þ E . × min 3 × B4 Expressing the particle energies in units of break, the S ≥ S number of black holes with BH min formed over the past T ¼ 1010 yr, in a theory with fundamental scale E ¼ð1 − χÞ2E =λ,is break 4n þ 5 2ðnþ2Þ 1 þ γ2χ þ χ : ðB5Þ 1 nþ1 2ðn þ 1Þ N ¼ 2 38 105ðn þ 2Þ2ðS Þ2 . × min 1 − χ Z S ¼ 102 E ¼ 1018.8 1 For min and eV (corresponding to 0 00 1 0 00 1 −γ 0 00 2 × de de duunþ1ðe e Þnþ1 2 Θðe e u − ð1 − χÞ Þ χ ¼ 0.042) this approximation gives N ¼ 2.55 × 106, 2 ð1−χÞ which is within about 2% of the result obtained directly ðB3Þ from Eq. (B3).

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