PLB34430S0370-2693(19)30092-910.1016/j.physletb.2019.02.003PhenomenologySupersymmetric sphaleron configurations as the origin of the perplexing ANITA eventsLuis A.Anchordoquiabcluis.anchordoqui@gmail.comIgnatiosAntoniadisdeaDepartment of Physics and Astronomy, Lehman College, City University of New York, NY 10468, USADepartment of Physics and AstronomyLehman CollegeCity University of New YorkNY10468USAbDepartment of Physics, Graduate Center, City University of New York, NY 10016, USADepartment of PhysicsGraduate CenterCity University of New YorkNY10016USAcDepartment of Astrophysics, American Museum of Natural History, NY 10024, USADepartment of AstrophysicsAmerican Museum of Natural HistoryNY10024USAdLPTHE, Sorbonne Université, CNRS, 4 Place Jussieu, 75005 Paris, FranceLPTHESorbonne UniversitéCNRS4 Place JussieuParis75005FranceeAlbert Einstein Center, Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, SwitzerlandAlbert Einstein CenterInstitute for Theoretical PhysicsUniversity of BernSidlerstrasse 5BernCH-3012SwitzerlandEditor: M. CvetičAbstractThe ANITA experiment has observed two air shower events with energy ∼500PeV emerging from the Earth with exit angles of ∼30∘. We explain ANITA events as arising from neutrino-induced supersymmetric sphaleron transitions. These high-multiplicity configurations could contain a large number of long-lived supersymmetric fermions, which can traverse the Earth and decay in the atmosphere to initiate upward-pointing air showers at large angles above the horizon. We comment on the sensitivity of new generation LHC detectors, designed to searching for displaced decays of beyond standard model long-lived particles, to test our model.KeywordsSupersymmetric sphaleron transitionsThe SU(3)C⊗SU(2)L⊗U(1)Y standard model (SM) of electroweak and strong interactions has recently endured intensive scrutiny at the Large Hadron Collider (LHC) using a dataset corresponding to an integrated luminosity of 63.9fb−1 of 2018 pp collisions at center-of-mass energy s=13TeV, and it has proven once again to be a remarkable structure that is consistent with all experimental results by tuning more or less 19 free parameters. However, the Antarctic Impulsive Transient Antenna (ANITA) experiment, designed to observe ultrahigh-energy cosmic rays and neutrinos from outer space, has detected particles that seemed to be blasting up from Earth instead of zooming down from space, challenging SM explanations [1,2]. As a matter of fact, several beyond standard SM physics models have been proposed to accommodate ANITA observations [3–10], but a convincing explanation is yet to see the light of day. In this Letter, we entertain the possibility that ANITA events originate in a supersymmetric sphaleron transition produced in the scattering of extremely high-energy (Eν≳1010.5GeV) cosmic neutrinos with nucleons inside the Earth. Such a non-perturbative process yield a high-multiplicity final state containing several long-lived supersymmetric fermions, one of which would survive propagation through the Earth crust before decaying into SM particles to initiate an upward-pointing shower in the atmosphere, just below the ANITA balloon.The advantages of our interpretation of ANITA events over previous supersymmetry (SUSY) models [8,9] go in two directions:•The ratio BR(νN→SUSY) of the neutrino–nucleon cross section into SUSY particles over the total νN cross section dominates over the branching ratio of charged current (CC) νN interactions. Furthermore, the particle content of the final state in sphaleron-induced transitions could contain a large multiplicity of SUSY fermions. All of this is in sharp contrast with the production of SUSY pairs in perturbation theory, for which BR(νN→SUSY)≲10−4 [11–15].•The νN scattering process requires a center-of-mass energy s≳245TeV, thus probing Eν≳1010.5GeV. In this energy range a large flux of neutrinos is expected from the decay of cosmic strings [16]. Moreover, in our model all three neutrino flavors would contribute to the ANITA signal. We begin our discussion by highlighting the main characteristics of ANITA events and after that we provide a phenomenological analysis of data.After three balloon flights, the ANITA experiment has detected two perplexing upgoing showers with energies of (600±400)PeV [1] and (560+300−200) PeV [2].11The trigger algorithm used for the second flight was not sensitive to this type of events [1]. The energy estimates are made under the assumption that the showers are initiated close to the event's projected position on the ice. These estimates are lowered significantly if the showers are initiated far above the ice. For example, the energy of the second event is lowered by 30% if the shower is initiated four kilometers above the ice [2]. Note that even with the 30% energy reduction, the center-of-mass energy of the collisions initiating these showers is beyond s of the LHC beam.In principle, ANITA events could originate in the atmospheric decay of an upgoing τ-lepton produced through a CC interaction of a ντ inside the Earth [17]. However, the relatively steep arrival angles of these events (27.4∘ and 34.5∘ above the horizon) create a tension with the SM neutrino–nucleon interaction cross section. More concretely, the second event implies a propagating chord distance through the Earth =2R⊕cosθn∼7.2×103km, which corresponds to 1.9×104km water equivalent (w.e.) and a total of 18 SM interaction lengths at Eν∼103PeV [18]. Here, R⊕ is the radius of the Earth and θn the nadir angle of the event. The first event emerged at θn≃62.6∘ implying a chord through the Earth of 5.9×103km, which corresponds to 1.5×104kmw.e. for Earth's density profile [1]. Because the energy deposited in a shower is roughly 80% of the incident neutrino energy, the cosmic neutrino energy range of interest is 200≲Eν/PeV≲1000. Taking the view that the event distribution is maximized at θn=60∘, in our calculations we will consider an average chord distance in traversing the Earth of ∼6×103km.Next, in line with our stated plan, we study the structural properties of our model. In the mid-seventies 't Hooft pointed out that the SM does not strictly conserve baryon and lepton number [19,20]. Rather, non-trivial fluctuations in SU(2) gauge fields generate an energy barrier interpolating between topologically distinct vacua. An index theorem describing the fermion level crossings in the presence of these fluctuations reveals that neither baryon nor lepton number is conserved during the transition, but only the combination B−L. Inclusion of the Higgs field in the calculation modifies the original instanton configuration [21]. An important aspect of this modification (called the “sphaleron”) is that it provides an explicit energy scale Esph∼MW/αW∼9TeV for the height of the barrier, where MW is the mass of the charged vector bosons W± and αW≃1/30. When the energy reach is much lower than Esph the tunneling rate through the barrier is exponentially suppressed Γtunneling∝e−4π/αW∼e−164. However, the sphaleron barrier can be overcome through thermal transitions at high temperatures, providing an important input to any calculation of cosmological baryogenesis [22–24]. Indeed, the rate over the barrier (thermal excitation) contains a Boltzmann factor Γthermal∝T4e−Esph/T, and hence the rate becomes large as the temperature approaches MW.More speculatively, it has been suggested that the topological transition could take place in two particle collisions at very high energy [25–27]. The anomalous electroweak contribution to the partonic process can be written as(1)σˆi(sˆ)=5.3×103e−(4π/αW)FW(ϵ)mb, where the tunneling suppression exponent FW(ϵ) is usually referred to as the “holy-grail function” and ϵ≡sˆ/(4πMW/αW)≃sˆ/30TeV [28–30]. Altogether, it is possible that at or above the sphaleron energy the cross section could be of O(mb) [31].The argument for strong damping of the anomalous cross section for sˆ≳30TeV was convincingly demonstrated in [32,33], in the case that the classical field providing the saddle point interpolation between initial and final scattering states is dominated by spherically symmetric configurations. This O(3) symmetry allows the non-vacuum boundary conditions to be fully included in extremizing the effective action. In [34] it was shown that a sufficient condition for the O(3) dominance is that the interpolating field takes the form of a chain of “lumps” which are well-separated, so that each lump lies well into the exponentially damped region of its nearest neighbors. However, we are not aware of any reason that such lumped interpolating fields should dominate the effective action. It is thus of interest to explore the other extreme, in which non-spherically symmetric contributions dominate the effective action (and let experiment rather than theory [35–37] be the arbiter). Thus far, the searches for instanton-induced processes in LHC data have shown no evidence for excesses of high-multiplicity final states above the predicted background [38–40].Of particular interest here would be an enhancement of the νN cross section over the perturbative SM estimates, say by an order of magnitude, for Eν≳1010.5GeV. To get an estimate of this cross section we first note that for the simple sphaleron configuration s-wave unitarity is violated for sˆ>4πMW/αW∼36TeV [31]. If for sˆ>36TeV we saturate unitarity in each partial wave, then this yields a geometric parton cross section πR2, where R is some average size of the classical configuration. As a fiducial value we take the core size of the Manton–Klinkhamer sphaleron, R∼10−2fm. In this simplistic model, the νN cross section is found to be(2)σνNblack disk(Eν)=πR2∫xmin1∑partonsf(x)dx, where xmin=sˆmin/s=(36)2/2mNEν≃0.065, where mN is the mass of an isoscalar nucleon, N≡(n+p)/2, in the renormalization group-improved parton model. In the region 0.065<x<3(0.065) the parton distribution function for the up and down quarks is well approximated by f≃0.5/x, so the expression for the cross section becomes(3)σνNblack disk(Eν)≃πR2(0.5)(ln3)(2/2)≃1.5×10−30cm2, where the last factor of 2/2 takes into account the (mostly) 2 contributing quarks (u,d) in this range of x, and the condition that only the left-handed ones contribute to the scattering. This is about 80 times the SM cross section. Of course this calculation is very approximate and the cross section can easily be smaller by a factor of 10 (e.g., if R is 1/3 of the fiducial value used). The sphaleron production cross section derived “professionally” [41] is consistent with our back-of-the-envelope estimate, and shows an enhancement of the νN cross section over the perturbative SM estimates by about an order of magnitude in the energy range Eν≳1010.5GeV. Previous estimates pointed to even larger cross section enhancements above perturbative SM prediction [42,43]. In our calculations we will adopt the estimate of [41].A point worth noting at this juncture is that the energy for the height of the barrier in SUSY models is also about 10 TeV [44], and consequently the expected production rate of supersymmetric sphaleron configurations is comparable to the SM one [45]. Most importantly, the decay BR increases if the final state contains a large number of SUSY fermions [45]. To develop our program in the simplest way, we will work within a construct with gauge mediated SUSY breaking, in which the gravitino ψ3/2 is the lightest supersymmetric particle (LSP) and the next-to-lightest supersymmetric (NLSP) is a long-lived bino B˜ [46]. Note that for MB˜∼700GeV [47], NLSPs could be copiously produced through instanton-induced processes at sˆ≳50TeV (see Fig. 3 in [45]), and could propagate inside the Earth without suffering catastrophic energy losses from electromagnetic interactions. The bino decays into a gravitino and a gauge boson (i.e., photon or Z-boson) with Planck-suppressed partial widths,(4)Γ(B˜→ψ3/2γ)=cos2θW48πMPl2MB˜5m3/22(1−x3/22)3(1+3x3/22),(5)Γ(B˜→ψ3/2Z)=sin2θWβB˜→ψ3/2Z48πMPl2MB˜5m3/22[(1−x3/22)2(1+3x3/22)−xZ2×{3+x3/23(−12+x3/2)+xZ4−xZ2(3−x3/22)}], where MPl∼1019GeV, sin2θW≈0.23, x3/2≡m3/2/MB˜, and xZ≡MZ/MB˜, and where(6)βB˜→ψ3/2Z≡[1−2(x3/22+xZ2)+(x3/22−xZ2)2]1/2, for MB˜>m3/2+MZ, and βB˜→ψ3/2Z=0 otherwise [48]. For MB˜>m3/2+MZ, the total decay width is well approximated by(7)τB˜−1≃Γ(B˜→ψ3/2γ)+Γ(B˜→ψ3/2Z), and the NLSP lifetime is estimated to be(8)τB˜∼5×1014m3/22MB˜5s, when masses are given in GeV [49].Before proceeding, we pause to discuss existing limits from searches of long-lived neutral particles at the Tevatron and at the LHC. The CDF Collaboration searched for long-lived particles which decay to Z-bosons by looking for Z→e+e− decays with displaced vertices and excluded proper decay lengths cτ<20cm for masses <110GeV [50]. Searches by D0 Collaboration exclude long-lived neutral particles of comparable lifetimes and masses [51,52]. The CMS Collaboration has searched for long-lived neutralinos decaying into a photon and an invisible particle, excluding cτ<50cm for masses <220GeV [53]. The ATLAS Collaboration searched for high-mass long-lived particles that decay within the inner detector to give displaced dilepton vertices excluding cτ<100cm [54]. ATLAS has also searched for very low mass (<10GeV) long-lived particles by considering pairs of highly collimated leptons [55], with sensitivity to cτ≲20cm. The most restrictive constraints on the lifetime of a long-lived particle come from a search by the ATLAS Collaboration for final states with displaced dimuon vertices in collisions at s=13TeV [56]. Proper decay lengths cτ<14m are excluded for SUSY models in which the lightest neutralino is the NLSP, with a relatively long lifetime due to its weak coupling to the LSP-gravitino. The lifetime limits are determined for very light gravitino mass and a neutralino mass of 700 GeV. Altogether, we can remain consistent with LHC bounds requiring τB˜∼44ns for MB˜∼700GeV. Substituting the bino lifetime in (8) we obtain m3/2∼122keV.SUSY models with a gravitino LSP are also constrained by a variety of cosmological observations. Of relevance to our analysis: (i) if τB˜∼44ns, NLSP decay does not perturb light element abundances which are synthesized during Big Bang nucleosynthesis [57,58]; (ii) if m3/2∼122keV, the relic density of gravitinos can be accommodated to match observations with choice of parameters [59,60].It takes a proper time of order 4.5MW−1 until the sphaleron radiation shows free-field behavior [61]. For neutrino-induced sphaleron transitions, this radiation will be emitted in a cone with half-opening angle δϕ∼O(1/γ), where γ is the Lorentz factor. Taking fiducial values Eν∼1010.5GeV and sˆ∼50TeV, one can have an order of magnitude estimate γ∼6×105. All in all, the bino decay length in the lab frame is γcτ∼8×103km. This means that for emerging angles θn∼60∘, a long-lived bino could survive the trip through the Earth. Note also that the boosted bino would have an energy EB˜∼420PeV, and after decay roughly half of its energy will be deposited in the air shower. These order of magnitude estimates are in good agreement with the energy and opening angle distributions shown in Fig. 4 of [41].Given an isotropic ν+ν¯ flux, the number of binos that emerge from the Earth is proportional to an “effective solid angle” Ωeff≡∫dθndϕcosθnP(θn,ϕ,X), where P(θn,ϕ;X) is the probability for a neutrino with incident nadir angle θn and azimuthal angle ϕ to emerge as a detectable B˜ [62,63]. P(θ,ϕ,X) is a rather complicated function of various unkown (model dependent) parameters X. However, we can provide a rough estimate of the event rates if we adopt the exposure calculations of [8], which suggest a total ANITA exposure for sub-EeV emergent cosmic rays of 2.7km2sryr, for the two flights together. It is noteworthy that this exposure is orders of magnitude larger than the exposure for τ-neutrinos reported by the ANITA Collaboration [64]. This is because τ-neutrinos which do not arrive at very large nadir angles are mostly blocked by the Earth. Observation of 2 events at ANITA would require an integrated neutrino flux Φν(Eν>1010.5GeV)∼10−17.7(cm2ssr)−1. Interestingly, at Eν∼1010.5GeV, the ANITA experiment sets the most restrictive upper limit on the energy weighted cosmic neutrino flux; namely, EνΦν(Eν)≲10−17.5(cm2ssr)−1 at 90% CL [65,66]. Note that neutrino-induced sphaleron transitions with non-negligible (missing) energy carried away by long-lived SUSY fermions would relax limits on the neutrino flux at extreme energies. We end with two comments on the neutrino flux. On the one hand, the required flux level to accommodate ANITA events may be exceptionally high by astronomical standards [67]. On the other hand, for some model parameters, such a flux of extremely high-energy (Eν≳1010.5GeV) neutrinos is consistent with predictions from decay of cosmic strings [16]. The decay of cosmic strings also produces extremely high-energy photons and electrons that interact with the cosmic microwave background and extra galactic background light, producing an electromagnetic cascade, whose energy density is constrained by measurements of the diffuse γ-ray background [68]. A point worth noting at this juncture is that the fluxes of γ-rays and neutrinos expected from the decay of cosmic strings are consistent with existing observations [69]. Moreover, experiments are being designed to search for the neutrino signals of cosmic strings; e.g., the Lunar Orbital Radio Detector (LORD) that will fly aboard the Luna-Resurs Orbiter space mission [70].In summary, we have provided an interpretation of ANITA events in terms of neutrino-induced supersymmetric sphaleron transitions. These high-multiplicity B+L violating transitions may contain a large number of long-lived SUSY fermions, which can traverse the Earth and decay in the atmosphere to initiate an upward-pointing shower just below the ANITA balloon. As a proof of concept, we have framed our discussion in the context of a gauge-mediated breaking scheme, but this model spans only a small region of the SUSY parameter space that can accommodate ANITA events. Indeed, our interpretation of these perplexing events can be encapsulated in the product of three factors:•the differential flux of incident neutrinos,•the ratio of the νN cross section into SUSY particles over the total νN cross section,•the lifetime of the SUSY fermion. Note these three factors are actually generic to a broad class of models in which the messenger of ANITA events does not live inside the Earth neither originate at cosmological distances. New generation LHC experiments dedicated to searching for long-lived particles (such as the ForwArd Search ExpeRiment (FASER) [71,72], the MAssive Timing Hodoscope for Ultra Stable neutraL pArticles (MATHUSLA) [73,74], and the Compact Detector for Exotics at LHCb (CODEX-b) [75]) will provide an important test both of the last two factors and of the ideas discussed in this Letter. 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