Bounds on Extra Dimensions from Micro Black Holes in the Context of the Metastable Higgs Vacuum

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, NC 27695-8202, USA

Robert McNees† Loyola University Chicago, Department of Physics, Chicago, IL 60660, USA.

We estimate the rate at which collisions between ultra-high energy cosmic rays can form small black holes in models with extra dimensions. If recent conjectures about false vacuum decay catalyzed by black hole 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 gravity 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 GZK 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 17 18.8 data, the excluded range is approximately 10 eV . E∗ ≤ 10 eV for theories with n = 1 extra 17 18.1 dimension, narrowing to 10 eV . E∗ ≤ 10 eV for n = 6. These bounds rule out regions of parameter space that are inaccessible to collider experiments, small-scale gravity tests, or estimates of Kaluza-Klein processes in neutron stars and supernovae.

I. INTRODUCTION have already led to vacuum decay. The most recent work in the series [9] explicitly conﬁrms that evaporating black holes formed in theories with extra dimensions are capa- In models with extra dimensions, the fundamental scale ble of seeding vacuum decay. The decay of the false vac- of gravity may be lower than the four-dimensional Planck uum is a dramatic consequence that presents an unmis-

scale, MPl. This presents the possibility that high-energy takable (and fatal) observational signature of microscopic collisions between particles, for instance in colliders or black hole production. Thus, its non-observation allows via cosmic rays, may form black holes if a high enough us to place limits on the higher-dimensional fundamental center-of-mass energy is achieved [1–4]. Large extra di- scale, excluding a range of values that are several orders mensions, if discovered, would constitute new physics of magnitude beyond the scales probed by other tests in- and potentially provide an explanation for the relative volving micro black hole production, such as via signs of weakness of gravity in relation to the other fundamental Hawking evaporation in colliders or from nearby cosmic forces. In addition to searches for microscopic black holes ray collisions. formed in particle collisions, experimental constraints on extra dimensions have generally come from searches for Our analysis relies on two main assumptions, both of modifications of the inverse-square force law of gravity which come with some important caveats that we de- at small scales or from signatures of Kaluza-Klein gravi- scribe here. The first is the metastability of the Higgs tons or other exotic particles. Constraints on the higher- vacuum, as implied by recent measurements of the Higgs dimensional fundamental scale depend on the number of boson and top quark masses [10]. This result is based extra dimensions proposed, with collider limits in the TeV on the validity of the Standard Model of particle physics, range and astrophysical limits as high as O(102)-O(103) so any beyond-Standard-Model (BSM) physics may alter arXiv:1809.05089v2 [hep-ph] 18 Jan 2019 TeV [1,5]. the effective potential of the Higgs field in a way that rescues our vacuum from metastability [11]. A high en- We present new limits from black hole creation in the ergy scale of inflation, were it to be confirmed, would context of recent work proposing that Hawking evapora- give evidence that new physics stabilizes the vacuum in tion of black holes can induce the decay of the standard some way, since high-energy-scale inflationary fluctua- model Higgs vacuum [6–9]. These papers argue that the tions would likely have instigated a transition to the true nucleation of a bubble of true vacuum in general precedes vacuum in the early universe [12, 13]. While we fully the final evaporation of the black hole, suggesting that recognize (and, in fact, hope) that vacuum metastability any production of black holes with evaporation times less ends up being ruled out by BSM physics or a better un- than the age of the Universe in our past light cone should derstanding of inflation, we will, for the purpose of this study, rely on the great successes of the Standard Model to justify the apparent metastability of the Higgs vac-

∗ uum as an observational tool for establishing constraints [email protected]; 7 @AstroKatie on higher-dimensional theories. Second, we are assuming † [email protected]; 7 @mcnees 2 that the results of [6–8] hold in a qualitatively similar way knowing more about the mechanisms involved we restrict for theories with more than four spacetime dimensions, our analysis to cosmic rays with energies below the GZK i.e., that black hole evaporation can seed vacuum decay. cut-off. Thus, our calculation probably under-estimates This was explored in [9], where the authors construct an both black hole formation rates and the maximum center- approximate instanton solution for a braneworld black of-momentum energies achieved in these collisions. As a hole in a theory with one extra dimension and then es- result, the excluded range for the fundamental scale of timate that their results extend to regimes where small theories with extra dimensions may extend to even higher black holes are produced in particle collisions. We will values than those we present here. apply these results beyond one extra dimension, though earlier calculations suggest the effect may be somewhat In SectionII we discuss a method for estimating the num- suppressed [7].1 More importantly, the conclusions of [9] ber of collisions that have taken place in our past light cone between UHE cosmic rays, and review the Pierre require the instability scale ΛI of the Higgs vacuum to lie Auger Observatory’s spectrum of these particles. In Sec- below the fundamental scale E∗ of the higher-dimensional theory. Otherwise, the Standard Model calculation of the tionIII we extend this to collisions capable of forming Higgs potential no longer applies. For our analysis, we black holes in higher-dimensional theories, obtain bounds must assume that the instability scale is at the low end of on the fundamental scale of these theories in SectionIV, the range consistent with experimental limits. The most and then discuss these results in SectionV. AppendixA 19 20 considers the various criteria that must be satisfied for likely range calculated by [10] is ΛI ∼ 10 − 10 eV, with some uncertainty around that value. For our anal- a reliable semiclassical analysis of black hole formation, ysis to be fully reliable, we require scales no higher than and AppendixB discusses an analytical result for black 18 hole formation rates that supports the numerical results ΛI ∼ 10 eV for theories with one or two extra dimen- 17 used in the main text. sions and ΛI ∼ 10 eV for theories with up to six. This is an important qualifier on our main results, and will be discussed in more detail at the end of the paper. II. COLLISIONS OF ULTRA-HIGH-ENERGY The structure of our calculation is as follows. Assuming COSMIC RAYS that the Higgs vacuum is metastable and that its decay is catalyzed by black hole evaporation, we take its persis- tence as evidence against black hole evaporation in our At ultra-high energies, cosmic rays are rare enough that past light cone. While this observation can also constrain we expect interactions between them to be exceedingly the production of low mass primordial black holes in the infrequent. But on timescales comparable to the age of early universe, we apply it here to the production of mi- the Universe, even a low rate can lead to an apprecia- croscopic black holes in particle collisions. Specifically, ble number of collisions with center-of-momentum (CM) we consider the formation of black holes in collisions be- energies several orders of magnitude greater than any- tween ultra-high-energy (UHE) cosmic rays, in theories thing that can be achieved in existing accelerators. Let with extra dimensions and a fundamental scale well be- us quickly review the estimate of collisions between UHE low the four-dimensional Planck scale. If the instability cosmic rays with energies above 1020 eV given by Hut and scale for the Higgs vacuum is low enough, this allows us Rees in [17]. to exclude a range of values for the fundamental scale that are not probed by the accelerator and astrophysical Assuming a homogeneous and isotropic distribution, the processes associated with current bounds. For a given density of UHE cosmic rays with energy greater than E value of the fundamental scale, we use the UHE cosmic is proportional to the integrated flux ray spectrum from the Auger experiment (see SectionII 4π Z ∞ and ref. [14]) to make a conservative estimate of the ρ(E) = dE0 J(E0) , (1) number of black holes formed in particle collisions in our c E past light cone. We note that the measured cosmic ray spectrum includes a steep drop-off at high energies. This where J(E) is the differential flux. For constant density is believed to be due to the GZK effect [15, 16], which ρ and cross section σ, the rate of collisions per particle is prevents the highest-energy cosmic rays from traveling ρσc, and the overall rate of collisions per unit volume is unimpeded across cosmological distances. If this is the R = ρ2σc . (2) case, then it is likely that even more energetic particles are plentiful in parts of the cosmos where they are acceler- The total number of collisions in our past light cone is ated by high-energy astrophysical processes. But without given by this rate times the spacetime volume

N = R c3T 4 , (3)

1 The reduced branching ratio of false vacuum decay rate to the where T is the time over which these collisions have oc- Hawking evaporation rate may be offset by the production of curred and our assumptions hold. Hut and Rees calcu- large numbers of black holes. lated the density of cosmic rays above 1020 eV using the 3 differential flux given by Cunningham et. al. in [18]

1.14 × 10−33 1019 eV2.31 J(E) = . (4) m2 · s · sr · eV E Then (1) gives a density of 1.8 × 10−23 m−3. The cross section is taken to be of order the Compton wavelength squared

2π c2 σ(E) ' ~ , (5) E which at 1020 eV gives 1.5 × 10−52 m2. For their order of magnitude estimate, Hut and Rees use σ ' 10−52 m2. The per-particle rate of collision is then ' 3 × 10−67 s−1, and the rate of collisions per unit volume comes out to FIG. 1. The Auger spectrum of UHE cosmic rays with E > R ' 3 × 10−90 m−3 s−1. Taking the age of the Universe 1018 eV, approximated as a set of power laws [14]. to be about T = 1010 yr, the number of collisions in the past light cone is roughly N' 8 × 105. Hut and Rees give a final estimate of N ≈ 105. a homogeneous and isotropic distribution for cosmic rays at those energies over the full volume of our past light- For our calculations, we will use more recent results for cone. Without knowing more about the origin of UHE the differential flux of UHE cosmic rays in place of (4). cosmic rays, we will conservatively limit all of our cal- The Pierre Auger Observatory is a hybrid cosmic-ray ob- culations to cosmic rays with energies below the break servatory consisting of surface Cerenkov detectors and energy in the Auger spectrum. air-shower observing telescopes, which allows it to collect large samples of cosmic rays across a wide range of ener- The approximation (2) for the rate of collisions per unit gies. Its measurements of the cosmic ray energy spectrum volume treats all particles as if they had roughly the above 1018 eV are well described by a series of power laws same energy, with a constant cross section for collisions. with free breaks between them, or else by broken power We can refine the estimate by dropping these assump- laws with an additional smooth suppression factor at the tions, accounting instead for all collisions above a given highest energies [14, 19–21]. Here we adopt the values CM energy and including the energy dependence of the given in [14], with the differential flux in each range of cross section. Assuming once again a homogeneous and energies taking the form isotropic distribution of UHE cosmic rays, the rate per unit volume of collisions with CM energy greater than E J(E) ∝ E−γ . (6) is given by The flux is shown in Fig.1. Below the ‘ankle’ energy, E = 1018.61±0.01 eV, the spectral index is γ = 16π2 Z Ebreak Z 1 ankle 1 R = dE0dE00 du σ(E ) 3.27±0.02. Above the ankle energy, but below the ‘break’ c CM 19.46±0.03 Emin 0 energy Ebreak = 10 eV, the spectral index flat- 0 00 J(E )J(E ) Θ(ECM − E) , (7) tens to γ2 = 2.59 ± 0.02. Above the break energy the spectral index drops off to γ = 4.3 ± 0.2. The spectrum 3 where u = (1−cos ψ)/2 is related to the angle ψ between between E and E is thought to possibly repre- √ ankle break the momenta of the colliding particles, E = 2 E0E00u sent the transition to a population of extragalactic cos- CM is the CM energy, and the Heaviside theta function re- mic rays, while the steep fall off above E is likely due break stricts the domain of integration to collisions with E > to the GZK effect [15, 16]. CM E. The upper limit in the energy integrals is taken to be

Repeating the estimate of Hut and Rees with the Auger Ebreak, which restricts the calculation to cosmic rays with spectrum yields fewer collisions above 1020 eV – on the energies below the GZK cut-off, while the lower limit is order of a few thousand – because of the steep drop off E2 in the flux above Ebreak that is not accounted for in (4). Emin = , (8) Indeed, the presence of this feature in the spectrum sug- 4 Ebreak gests that the upper limit in (1) should not extend to ar- bitrarily high energies. The GZK effect prevents cosmic rays from traveling cosmological distances with energies 2 greater than Ebreak. So it seems unwarranted to assume drop-off in flux above Ebreak, which may also reflect, for instance, the maximum energies that can be achieved by the sources that accelerate the particles [19]. While its origin does not impact our estimates, we interpret Ebreak as the scale associated with 2 It is possible that the GZK effect is only partly responsible for the the GZK effect. 4

which is the minimum energy of a particle that can par- and collisions with ECM ≥ λ E∗ are considered to form ticipate in a collision with CM energy of at least E. As semiclassical black holes. before, the number of events in our past lightcone is N = R c3T 4. In our calculations, we will take T = 1010 We also require that the black holes be small enough years. This may be a conservative assumption, as the compared with the compactiﬁcation scale L that a ﬂat- production rate for UHE cosmic rays is likely to have space approximation is valid. For a discussion of this been higher toward the early part of that time window, requirement, see AppendixA. closer to the peak of active galactic nucleus (AGN) ac- For the collisions between UHE cosmic rays in the previ- tivity around a redshift of 2. ous section, the cross section (5) was proportional to the In the next section we will employ (7) to estimate the rate square of the Compton wavelength and hence decreased of black hole formation in models where the fundamen- at higher energies. But at CM energies well above E∗ the tal scale of gravity is below the maximum CM energies cross section for black hole formation exhibits the oppo- accessible in collisions between UHE cosmic rays. site behavior. The geometric cross section for black hole formation is [22]-[23]

2 σBH = O(1) πrH , (11) III. BLACK HOLE FORMATION VIA COSMIC

RAY COLLISION where rH is the horizon radius of a black hole of mass 2 MBH = ECM/c , and an overall factor of order 1 reﬂects various corrections. Collisions at higher energies produce In higher-dimensional theories the fundamental scale of black holes with larger mass, and hence larger horizon gravity may be lower than MPl. We will consider a radius, resulting in a cross section that grows as a positive generic 4 + n-dimensional theory with fundamental scale power of ECM. 2 M∗ = E∗/c related to the Newton’s constant by Assuming the collision forms a Schwarzschild black hole, cn+5 n+1 the horizon radius in 4 + n dimensions is [24] G (4+n) = ~ . (9) N E 2+n ∗ 1 1 2 n+1 3+n n+1 ~ c MBHc 8π Γ( 2 ) If the scale E∗ is low enough, collisions between UHE rH = 3+n . (12) cosmic rays at suﬃciently high CM energy are expected E∗ E∗ n + 2 π 2 to form black holes of mass M ∼ E /c2. BH CM Then, up to the O(1) factor in Eq. (11), the cross section 2 For our estimates of black hole formation via scattering for a collision forming a black hole with MBH = ECM/c to make sense, the black holes should be large enough is that a semiclassical treatment is appropriate. We enforce 2 2 2 n+1 8 Γ( 3+n ) n+1 this by considering only black holes with entropy above (4+n) ~ c ECM 2 σBH = . (13) some minimum value Smin (see AppendixA for a brief E∗ E∗ n + 2 discussion). This implies that the ratio of MBH/M∗ must be greater than Using this cross section in Eq. (7), we can estimate the rate at which black holes are formed by collisions between 1 n+3 ! n+2 2 n+1 UHE cosmic rays in a higher-dimensional theory with n + 2 π n+2 λ = n+3 Smin . (10) fundamental scale E∗. 4π 2 Γ( 2 ) As an example, consider a theory with one extra dimen- 2 We will typically take Smin = 10 , which implies that sion (n = 1). Using Eq. (10), black holes with entropy 2 MBH must be greater than M∗ by a factor that is O(10) SBH ≥ 10 have mass MBH ≥ 8.76 M∗. The number of for n = 1 and increases to O(102) for n = 6. Neglecting such black holes formed in our past light cone by colli- 2 energy loss during the formation process, MBH = ECM/c sions between UHE cosmic rays is approximately

2 √ 2 Z Ebreak Z 1 0 00 √ 3 4 128π ~ c 0 00 E E u 0 00 0 00 N = c T dE dE du J(E )J(E )Θ 2 E E u − 8.76 E∗ . (14) 3 c E∗ Emin 0 E∗

Fixing a fundamental scale allows us to evaluate this ex- producing events above that scale. As an illustrative cal- pression explicitly to determine a number of black hole- culation, we use the Auger results for the diﬀerential ﬂux, 5

T = 1010 yr, and a fundamental scale of 1018.5 eV. The ∼5 and still allow a significant number of black holes to numerical evaluation of this integral gives N' 1.6×1011. form and evaporate over the history of our universe. But Raising the fundamental scale to 1018.8 eV lowers the the majority of those collisions would involve particles number of events to N' 2.6 × 106. with energies above the GZK cut-off, and as explained in the previous section, our assumption of a homogeneous Since we restrict our attention to UHE cosmic rays with 19.46 and isotropic distribution seems questionable for that energy below the GZK cut-off at Ebreak = 10 eV, population of UHE cosmic rays. the maximum possible CM energy in our analysis is 19.76 2 Ebreak = 10 eV. This places an upper limit on the mass of the black hole, so the requirement MBH ≥ λ M∗ implies that our analysis is valid only for E∗ < 2 Ebreak/λ. In this example with n = 1, the largest fundamental scale IV. BOUNDS ON E∗ FROM BLACK HOLE 18.82 we can consider is 10 eV, and N plunges to zero as E∗ FORMATION approaches this value. The drop-off is steep enough that 6 3 the difference between E∗ for N ∼ 10 and N ∼ 10 Now we calculate the number of black holes formed via is much less than the uncertainties in Ebreak and other factors. If vacuum decay triggered by black hole evapo- collisions between UHE cosmic rays for different numbers ration is as likely as the claims of [6–9], then essentially of extra dimensions n, and use this to establish a range of excluded values for the fundamental scale E∗. any value of E∗ up to 2 Ebreak/λ results in too many black holes being formed. As in the previous section, we consider only black holes Note that by raising the maximum cosmic ray energy in with entropy above a minimum value Smin that justifies the use of semiclassical methods. This implies MBH ≥ Eq. (7), E∗ could be greater than 2 Ebreak/λ by a factor of λ M∗, where λ is given by Eq. (10). Then the number of black holes formed is approximately

2 Z Ebreak Z 1 3 4 16π 0 00 (4 + n) 0 00 N = c T dE dE du σBH (ECM)J(E )J(E )Θ ECM − λ E∗ , (15) c Emin 0

√ 0 00 where ECM = 2 E E u. Since we only consider UHE The excluded range of E∗ for different values of n is cosmic rays with energies below Ebreak, these collisions shown in Fig.3, assuming a Higgs instability scale of 17 2 can only form semiclassical black holes when the fun- ΛI ∼ 10 eV. For SBH ≥ 10 , the upper end of this damental scale satisfies E∗ ≤ 2Ebreak/λ. Because the range is more or less within an order of magnitude of Auger spectrum is defined piecewise for different values the Auger break energy, and safely above our assumed of the energy, Eq. (15) is most easily evaluated numeri- instability scale for the Higgs vacuum. A more conserva- 3 cally. The number of black holes formed as a function of tive condition SBH ≥ 10 would require the formation of E∗ is shown in Fig.2, for 1 ≤ n ≤ 6. larger black holes, which may not be consistent with the assumed instability scale. Fig.4 translates the excluded As in the n = 1 example, N 1 for all values of E∗ 2 range for E∗ (with SBH ≥ 10 ) into a bound on the size of up to the maximum value 2Ebreak/λ that can be probed the extra dimensions (A7) with the typical size of extra using our method. For these values of E∗, collisions dimensions for TeV-scale gravity [25] included for com- between UHE cosmic rays that form rapidly evaporat- parison. The values for E∗ and L are summarized in ing black holes are plentiful within our past light cone. TableI. Thus, avoiding vacuum decay catalyzed by the evaporation of these black holes excludes fundamental scales in The values of N used in Fig.2 were calculated by numer- the range ically evaluating Eq. (15). However, it is easy to show

that once E∗ is larger than about 0.8 Ebreak/λ, both of 19.76 2 Ebreak 10 eV the UHE cosmic rays participating in the collision must ΛI < E∗ ≤ = , (16) λ λ have E > Eankle. In that case, the relevant part of the Auger diﬀerential ﬂux is given by a single power law and where λ is given in Eq. (10). Collisions between cosmic N can be evaluated analytically as a function of E∗ and rays at even higher energies are of course possible, and n. This is described in more detail in AppendixB. would raise the upper end of the range of excluded values for E∗, but they cannot be reliably estimated with the method used here. 6

FIG. 4. Bounds on the size of extra dimensions (blue), with typical values for TeV-scale gravity (red), for different values FIG. 2. Number of collisions N forming a black hole with of the number of extra dimensions, n. The blue shaded region 2 10 entropy SBH ≥ 10 over the past T = 10 yr, as a function of the plot is excluded by our constraints. of the fundamental scale. The colored lines are for different values of n, the number of extra dimensions. N drops off rapidly as E∗ approaches 2Ebreak/λ. 2 n λ (Smin = 10 ) log10(E∗/eV) log10(L/m) 1 8.8 18.8 −7.8 2 16.1 18.6 −16.5 3 23.9 18.4 −19.4 4 31.5 18.3 −20.9 5 38.9 18.2 −21.7 6 46.0 18.1 −22.3

TABLE I. Upper end of the excluded range of values for the fundamental scale E∗, with the associated size L of extra dimensions assuming a toroidal compactiﬁcation.

FIG. 3. Excluded range for the fundamental scale for 1 ≤ n ≤ 2 17 6 with SBH ≥ 10 and ΛI ∼ 10 eV, where n is the number supported by recent measurements of the mass of the of extra dimensions, SBH is the entropy of the black hole (set Higgs boson and top quark, at a scale below the funda- to ensure the validity of a semiclassical treatment), and ΛI is mental scale of the higher-dimensional theory. While this the Higgs instability scale. Existing constraints are at scales concept has been used to place limits on the production 2 3 O(10 )-O(10 ) TeV, well below the plot range shown here. of primordial black holes [7,8], ours is the ﬁrst analysis The darker shaded region of the plot is reliably excluded by to place quantitative limits on the fundamental scale of our constraints. The lighter shaded region between 1017.5 eV 18 extra dimensional theories based on this method. and 10 eV, where E∗ is comparable to ΛI , indicates uncertainties surrounding the applicability of our calculation. TableII summarizes current limits on the fundamental scale and/or size of extra dimensions from a range of methods. Comparing with TableI, limits from vacuum V. DISCUSSION decay catalyzed by black hole evaporation exclude ranges of these parameters that are many orders of magnitude beyond what can be probed with current tests. We have presented constraints on the fundamental scale and the size of extra dimensions in higher-dimensional Our analysis is limited by the assumptions underlying theories, based on the non-observation of vacuum decay the conclusions of [6–9]. The most important of these as- catalyzed by microscopic black holes. This scenario is sumptions is the requirement that ΛI < E∗. If the Higgs based on the mechanism outlined in [6–9], in which black instability scale approaches 2 Ebreak/λ then the excluded holes seed vacuum decay before their evaporation is com- range collapses and there is no bound. Likewise, we are plete. It assumes the meta-stability of the Higgs vacuum, not able to draw any conclusions from this analysis con- 7

ray observatories have detected particles with energies as Method Reference n log (E∗/eV) log (L/m) 10 10 high as E = 3 × 1020 eV. The propagation of such par- Grav force [26] 2 12.5 −4.36 ticles on cosmological scales is suppressed by the GZK SN1987A [27] 2 13.4 −6.18 effect, so our method for estimating the number of collisions forming black holes is not applicable. But if colli- 3 12.4 −9.10 sions between particles at these energies occur in regions NS cooling [28] 1 −4.35 where UHE cosmic rays are produced, then CM energies E ∼ 1021 may be achieved. In that case rapidly evapo- 2 −9.81 CM rating black holes may have been formed for fundamental 20 3 −11.6 scales as high as E∗ ∼ 10 eV, potentially inducing vac- 19 4 −12.5 uum decay even if the instability scale is ΛI ∼ 10 eV, which is in the most likely range of values [10]. For larger 5 −13.0 values of the instability scale it seems unlikely that black 6 −13.4 hole formation via UHE cosmic ray collisions could be CMS [29] 2 13.0 used to constrain E∗. 3 12.9 The approach we have taken here comes with important 4 12.8 caveats described above and in the introduction. Never- theless, the possibility of establishing bounds on extra di- 5 12.8 mensions at scales that are not probed by other methods 6 12.7 makes this is a promising direction for continued research. Additionally, our method relies on qualitatively different TABLE II. Current bounds on extra dimensions from: gravi- physics than the tests responsible for the strongest ex- tational force law tests [26]; constraints on the production of isting constraints on extra dimensions. It does not rely Kaluza-Klein gravitons from the supernova 1987A [27]; con- on assumptions about gravitons or other BSM particle straints based on the expectation that Kaluza-Klein gravitons physics, and is therefore an interesting complement to would decay into photons and heat neutron stars [28]; and col- existing methods. lider searches, the most stringent of which currently provided by the CMS collaboration [29]. We provide values for both E∗ We expect that our analysis can be made more robust and L when provided in the cited references. In other cases, with improved inferences about the instability scale from it is possible to deduce the corresponding value via equation collider data, along with a more complete accounting of A7 for the toroidal compactifications considered here. It is of the full range of high-energy particle interactions in our note that the most stringent constraints (SN1987A and NS past light cone. In particular, collisions in regions where cooling) require some assumptions about Kaluza-Klein gravi- UHE cosmic rays are accelerated to energies above the tons. GZK cut-off may achieve higher CM energies than we considered. Such collisions could form black holes for 3 even larger values of the fundamental scale E∗, making cerning scenarios where E∗ is below ΛI . The authors of 17 our analysis relevant for a wider range of values of the [9] quote ΛI ∼ 10 eV as the lowest value consistent with experimental limits on the top quark mass, which is well Higgs instability scale. below our lower bounds on E∗ for 1 ≤ n ≤ 6. They esti- 17 mate that for n = 1 extra dimension, with ΛI ∼ 10 eV 18 and E∗ ∼ 10 eV, vacuum decay would be caused by 20 18.8 black holes with MBH ∼ 10 eV. For E∗ < 10 eV, VI. ACKNOWLEDGMENTS we find that a significant number of black holes with 20 MBH ∼ 10 eV are produced. Even if the instability 18 scale is as high as ΛI ∼ 10 eV, our n = 1 (and pos- RM was supported in part by the National Science sibly n = 2) value for E∗ seems high enough to justify Foundation under Grant No. NSF PHY-1748958 through concerns about vacuum decay. However, the most likely the KITP Scholars program, and by Loyola University range of values for the instability scale ΛI appears to be Chicago through a Summer Research Stipend. KJM and around 1019-1020 eV. In that case the fundamental scale RM acknowledge the hospitality of Caltech and the Burke 20 21 E∗ would have to be greater than 10 -10 eV, which Institute, especially for the workshop “Unifying Tests of is outside the regime that can be probed with this con- General Relativity” (July 2016) during which this project servative calculation. On the other hand, UHE cosmic was conceived. KJM also acknowledges the Australian Research Council and Melbourne University for support during early stages of this project, and RM thanks the Kavli Institute for Theoretical Physics and the KITP 3 Recall that the original motivation for large extra dimensions – a Scholars Program for hospitality and a productive work natural explanation for the apparent weakness of gravity – would environment. We are grateful to Leo Stein and Walter require a fundamental scale well below the Higgs instability scale. Tangarife for helpful comments on an earlier draft of this 8 paper, and to Nima Arkani-Hamed for an important cor- rection that improved the quality of our analysis.

Appendix A: Black Hole Formation and Semiclassical Methods

Our analysis is based on the formation of black holes in collisions between UHE cosmic rays, which rapidly evaporate via Hawking radiation. Thus, we must consider three questions. First, under what conditions can we say that a collision has formed a black hole? Second, when can those black holes be described semiclassically? And third, since the extra dimensions of spacetime are assumed to be small and compact, when can the black FIG. 5. The ratio λ = MBH/M∗ for a black hole with entropy holes be described using results that assume an asymp- Smin. totically flat space time? A basic criteria for saying that a black hole has formed is The first two factors give a number greater than 1 for that the decay time should be very long compared to the 1 ≤ n ≤ 9, and of O(1) out to n ∼ 35. So the condition time scale associated with the formation process. For SBH 1 is essentially the same as the previous condition, black holes formed via collision, we take that to mean τD τC, in all cases of interest. that the decay time should be much longer than the time For a black hole of mass M = 10 M , the entropy needed for the particles to cross a region of linear size r . BH ∗ H ranges from S ' 120 when n = 1, down to S ' 20 for In 4 + n dimensions the decay time is of order BH BH n = 6. Since SBH decreases with n for fixed MBH/M∗, we n+3 will always set a minimum entropy S that is sufficient M n+1 min ~ BH to justify semiclassical calculations, and then restrict our τD ∼ 2 , (A1) M∗c M∗ attention to black holes with entropy at or above this cut-off. Using Eq. (A5), this fixes the minimum value λ while the crossing time τC ' rH/c is of the ratio MBH/M∗ for a semiclassical black hole in our 1 1 1 8 Γ( 3+n ) n+1 M n+1 analysis as ~ √ 2 BH τC = 2 . (A2) M∗c π n + 2 M∗ 1 n + 2 w n+2 n+1 n+2 n+2 λ = (Smin) . (A6) In all cases of interest, the n-dependent factors in τC are 4π 4 O(1), so the condition τ τ is equivalent to D C This is shown for 1 ≤ n ≤ 6 in Fig.5. Black holes with n+2 M n+1 MBH ≥ λ M∗ have SBH ≥ Smin. BH 1 . (A3) M∗ For the higher dimensional theories considered in this The power on the left-hand side of this inequality is al- paper we assume that n dimensions are compactified with length scale L. The formulas above assume that ways greater than 1, so black holes with MBH M∗ satisfy τ τ . spacetime is asymptotically flat, but we may regard D C them as approximately true when the black hole radius To justify a semiclassical treatment, the entropy of the (12) is much smaller than the compactification scale: black hole should satisfy SBH 1. In any dimension rH L. In the case of toroidal extra dimensions, the the entropy is given by one-quarter of the horizon area in four-dimensional Planck mass is related to the compact- units of the fundamental length scale. For a non-rotating ification scale of the higher dimensional theory by black hole this is: n 2 n 2+n c 3 2+n 3 M = (2πL) (M∗) . (A7) AHc ω2+n rH c Pl n SBH = (4+n) = (4+n) , (A4) ~ 4 ~ GN 4 ~ GN Ignoring factors of O(1), the condition r L becomes 3+n H 2 3+n where ω2+n = 2π /Γ( 2 ) is the area of a unit 2 + n- 1 2 sphere. Using Eq. (9) and Eq. (12), the entropy can be n+1 n MBH MPl expressed as (A8) M∗ M∗ n+2 1 n+2 4π n+1 4 n+1 M n+1 BH Thus, M /M should be large enough to justify a semi- SBH = . (A5) BH ∗ n + 2 ωn+2 M∗ classical calculation, but not so large that the black hole 9

begins to notice the extent L of the extra dimensions. E∗ = (1 − χ) 2Ebreak/λ, is In the text we consider theories with fundamental scale 19 2(n+2) as large as E∗ ∼ 10 eV, and limit ourselves to UHE n+1 5 2 2 1 cosmic ray collisions with CM energy no greater than N = 2.38 × 10 (n + 2) (Smin) 20 1 − χ ECM ∼ 10 eV. In that case, for collisions forming black Z 1 holes with entropy greater than Smin, the ratio rH/L al- 1 1 0 00 0 00 −γ2 0 00 2 ways satisﬁes de de du u n+1 (e e ) n+1 Θ e e u − (1 − χ) (1−χ)2

1 (B3) 1 2 n rH − (16π) − 1 ≤ 2π 1.79 × 1017 n (S ) n . 2 min where γ2 = 2.59 is the spectral index given by Auger L (n + 2) ωn+2 (A9) for cosmic rays with energies between Eankle and Ebreak. Notice that N grows with the minimum entropy for semiclassical calculations as (S )2. This is due to the fact For S = 102, this is of order 10−18 for n = 1, and of min min that as S goes up, the fundamental scales we probe go order 10−3 for n = 6. In these cases, the black holes we min down like 1/λ, resulting in a larger cross-section (13). consider are all much smaller than the size of the extra dimensions and the physics should be well-described by The full expression obtained from evaluating the integral formulas that assume an asymptotically ﬂat spacetime. in Eq. (B3) is not especially illuminating, but was used A quick calculation shows that the size of extra dimen- to verify the numerical results presented in sectionIV. sions is also much larger than the fundamental length For χ 1, the regime where E∗ is extremely close to scale L `∗, so that quantum gravity corrections may the maximum value for which we can estimate black hole safely be neglected. formation rates, N is well approximated by 2 Thus, for black holes with entropy SBH ≥ 10 , the pro- 2 5 2 4(n + 2) 3 cess of formation via collision and subsequent evapora- N' 2.38 × 10 (Smin) χ × tion should be well described using semiclassical meth- 3 ods and asymptotically ﬂat-space results for 1 ≤ n ≤ 6. 4n + 5 1 + γ2 χ + χ . (B4) The case n = 7 is borderline, with the conditions de- 2(n + 1) scribed above and the assumptions outlined elsewhere in 2 18.8 the paper beginning to break down. For Smin = 10 and E∗ = 10 eV (corresponding to χ = 0.042) this approximation gives N = 2.55 × 106, which is within about 2% of the result obtained directly from Eq. (B3). Appendix B: Analytic expression for N

Since we consider cosmic rays with energies below the

GZK cut-oﬀ at Ebreak, the minimum energy of a cosmic ray that can participate in a collision with CM energy above λE∗ is

2 (λE∗) Emin = . (B1) 4 Ebreak

If we express the fundamental scale as a fraction of the maximum value that we can probe, E∗ = (1−χ) 2Ebreak/λ with 0 ≤ χ < 1, then both cosmic rays must have energy greater than Eankle when

r E 1 − χ > ankle = 0.38 . (B2) Ebreak

In this regime the diﬀerential ﬂux in Eq. (15) is described by a single power law, and the integral can be evaluated analytically.

Expressing the particle energies in units of Ebreak, the number of black holes with SBH ≥ Smin formed over the past T = 1010 yr, in a theory with fundamental scale 10

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