Bounds on Extra Dimensions from Micro Black Holes in the Context of the Metastable Higgs Vacuum
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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 McNeesy 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 y [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 tion III 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 1 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.