PHYSICAL REVIEW D 101, 055007 (2020) Anomalous anomalies from virtual black holes Joseph Bramante and Elizabeth Gould The McDonald Institute and Department of Physics, Engineering Physics, and Astronomy, Queen’s University, Kingston, Ontario K7L 2S8, Canada and Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada (Received 15 November 2019; accepted 17 February 2020; published 4 March 2020) We investigate gravitational UV/IR mixing models that predict a breakdown of low energy effective field theory, from loop-level, nonlocal gravitational corrections to particle processes. We determine how the choice of IR cutoff in these theories alters predictions for lepton magnetic moments. Using Brookhaven E821 muon magnetic moment data, we exclude models of UV/IR mixing with an IR cutoff set by a spherical volume enclosing the experiment. On the other hand, an IR cutoff defined by the simply connected spatial volume containing the trajectories of the muons implies a correction to the muon magnetic moment which may have already been observed. DOI: 10.1103/PhysRevD.101.055007 I. INTRODUCTION the dynamics of a region with size L and corresponding infrared cutoff Λ ∼ 1, may break down if the virtual It is a truth universally acknowledged that physicists, IR L lacking Planck density experiments, do not expect to find energy density inside that region implies a black hole quantum gravity in the laboratory. Nevertheless, there are horizon larger than L. This theory also provides a possible – some quantum facets of gravity that arise in regions with resolution of the cosmological constant problem [16 19]. “ sub-Planckian density. In effective field theory terms, this gravitational UV/IR ” Indeed, progress is still being made understanding quan- mixing indicates a correspondence between UV and IR tum mechanics around black holes, where the energy density cutoffs, which applies going from the UV to the IR, and is sub-Planckian. Nevertheless, it has become apparent that from the IR to the UV. new quantum effects should arise as a result of gravitational First, let us assume there is an effective theory with Λ horizons. This spurred the development of black hole ultraviolet cutoff UV. By definition, this effective field ρ ∼ Λ4 radiation, which is too dim to be observed in nature with theory must describe energy densities up to UV. But if present technologies [1–3]. Theoretical analysis of the black the effective field theory is applied to a region so large that Λ4 hole temperature has led to a set of laws governing black the energy density UV implies a black hole, the effective hole thermodynamics, analogous to the known laws of field theory may break down. The size of this region will thermodynamics [4]. The entropy of a black hole’s horizon define an IR cutoff. Approaching from the other direction, as defined by these laws has been conjectured as an upper let us assume an effective field theory with an infrared cutoff – Λ ∼ 1 2 bound on the entropy of any region [5 12] (although it is IR L. The Schwarzschild radius of this theory GM possible to derive an entropy bound without black hole implies an effective field theory validity bound in terms – 2 ¼ 2 ρ 4π 3 ¼ 2 Λ4 4π 3 horizons [13 15]). This implies that a quantum field theory of the UV cutoff, GM G 3 L G UV 3 L <L. describing a region should somehow incorporate the fact This restricts the UV cutoff for the theory to that the entropy of quantum fields in some regions cannot exceed the black hole entropy bound. 3 M2 3 Λ4 ≲ P ¼ M2 Λ2 ; ð1Þ However, there are reasons to think that a quantum UV 8π L2 8π P IR field theory’s character is altered before the black hole entropy bound is saturated. As identified by Cohen, ¼ 1 ℏ ¼ ¼ 1 where G M2 , and we will use units with c .It Kaplan, and Nelson [16], effective field theories describing P follows that the UV scale at which this theory breaks down unless black holepffiffiffiffiffiffiffiffiffiffiffiffiffiffi states are accounted for is approxi- Λ ∼ Λ mately UV MP IR. Published by the American Physical Society under the terms of It is useful to considerpffiffiffiffiffiffiffiffiffiffiffiffiffiffi the physics behind the correction the Creative Commons Attribution 4.0 International license. Λ ∼ Λ Further distribution of this work must maintain attribution to arising at UV MP IR. For the moment neglecting field the author(s) and the published article’s title, journal citation, masses, consider the momenta carried by virtual fields in and DOI. Funded by SCOAP3. loop-level processes for some experiment with an infrared 2470-0010=2020=101(5)=055007(5) 055007-1 Published by the American Physical Society JOSEPH BRAMANTE and ELIZABETH GOULD PHYS. REV. D 101, 055007 (2020) Λ ¼ 1 gravitational UV/IR mixing relation for an infrared cutoff cutoff IR L. If the momenta carriedpffiffiffiffiffiffiffiffiffiffiffiffiffiffi in loop-level particle Λ ∼ Λ determined by the spherical region circumscribing particle exchanges is greater than UV MP IR, the theory must contain virtual states with densities that exceed black hole trajectories at precision experiments. On the other hand, densities. Such extremely dense virtual states can be for an IR cutoff determined by the simply connected constructed using sufficiently high-order loop processes, volume containing particle trajectories, we find that the anomalous measurement of the muon anomalous magnetic by linking together a verypffiffiffiffiffiffiffiffiffiffiffiffiffiffi large number of virtual fields Λ ∼ Λ moment [34,35], indicate that the effect of virtual black hole with momentum UV MP IR. The fact that these states would arise at extremely high order in a perturbation theory states may have already been observed. does not necessarily mean they can be neglected. In fact, II. INFRARED CUTOFFS AND NONLOCAL accurately computing virtual gravitational corrections asso- GRAVITATIONAL CORRECTIONS ciated with black holes would seem to require a detailed understanding of how quantum fields assemble to form a In the absence of an explicit theory treating virtual black black hole, which is a topic of active research [20–27]. holes, we will begin our treatment with a general gravi- In the absence of a settled theory for nonlocal black tational UV/IR mixing relation, Λ hole field dynamics, we can attempt an estimate of UV. Reference [16] pointed out that the gravitational ultraviolet/ 2 ¼ 2 Λ4 ≃ ð Þ GM G UVV R; 4 infrared (UV/IR) mixing given by Eq. (1) may have observ- able consequences at particle experiments. Corrections to the where R is the largest radius required to enclose some electron magnetic moment were considered; the electron region of interest, and V is the volume containing fields of magnetic moment is arguably the most precisely measured interest. The choice of R on the right-hand side of this parameter in the Standard Model [28–31], and is sensitive to equation can be motivated by the hoop conjecture [36,37], corrections from new UV states [32]. For the electron which states that a black hole will form if a ring of size anomalous magnetic moment ae, which is related to the 4πGM can be spun around some region of interest with ge electron ge − 2 factor by ae ¼ 2 − 1, the leading order mass M. We expect our black hole formation threshold to UV and IR contributions from a new state coupled to the be similarly defined. There are a number of ways we may electron are define the infrared cutoff for a theory which attempts to include nonlocal gravitational effects. α 2 1 2 δ ≃ me þ ð Þ ae 2π Λ ; 2 A. Spherical infrared cutoff UV meL A simple IR cutoff ansatz is to take V as the spherical α where is the fine structure constant in the Standard Model volume circumscribing the region within which some and me is the mass of the electron. In [16], the gravitational measured particles are localized. In this case, V ¼ 4π R3, Λ2 3 UV/IR relation L ¼ UV was substituted into the second term as was assumed in Eq. (1). The UV cutoff obtained by MP Λ in Eq. (2),and UV wasvariedtoobtaintheminimum assuming a spherical volume for our IR cutoff region is correction to ae, 3 M2 1=4 Λsph ¼ P : ð5Þ α 2=3 UV 8π 2 δ j ¼ me ≃ 0 14 10−17 ð Þ R ae min . × ; 3 2π MP We will shortly see that a spherical cutoff can be ruled out which is 5 orders of magnitude smaller than the present using precise measurements of the muon’s magnetic moment. uncertainty on the electron ae measurement, which is Æ0.28 × 10−12ð1σÞ [28]. It has also been considered in [33] whether UV/IR effects could lead to an unexpected B. Simply connected volume infrared cutoff running of α, when comparing measurements of the electron But perhaps using a spherical volume to set the IR cutoff and muon magnetic moments. is too simple. We may expect the virtual black hole effect On the other hand, Eq. (3) only gives the minimum we are looking for to depend on fluctuations of virtual possible correctionpffiffiffiffiffiffiffiffiffiffiffiffiffiffi from the gravitational UV/IR mixing fields in the space around the particles being measured. In Λ ∼ Λ relation UV MP IR. There are other plausible values most precision experiments, the particles will be confined for the IR cutoff in the gravitational UV/IR relation. Indeed, to a region (usually a ring or beam), but the virtual states are one might expect the IR cutoff to be set by an experimental presumably not confined to that region. However, if a path length scale. In this paper we investigate a number of integral formulation applies to corrections associated with Λ choices for IR, based on some field theoretic arguments black hole formation thresholds, we should expect these and the geometry of particles contained in precision experi- virtual black hole forming contributions to be suppressed ments.