The Direct Detection of Boosted Dark Matter at High Energies and PeV events at IceCube

We study the possibility of detecting dark matter directly via a small but energetic component that is allowed within present-day constraints. Drawing closely upon the fact that neutral current neutrino nucleon interactions are indistinguishable from DM-nucleon interactions at low energies, we extend this feature to high energies for a small, non-thermal but highly energetic population of DM particle $\chi$, created via the decay of a significantly more massive and long-lived non-thermal relic $\phi$, which forms the bulk of DM. If $\chi$ interacts with nucleons, its cross-section, like the neutrino-nucleus coherent cross-section, can rise sharply with energy leading to deep inelastic scattering, similar to neutral current neutrino-nucleon interactions at high energies. Thus, its direct detection may be possible via cascades in very large neutrino detectors. As a specific example, we apply this notion to the recently reported three ultra-high energy PeV cascade events clustered around $1-2$ PeV at IceCube (IC). We discuss the features which may help discriminate this scenario from one in which only astrophysical neutrinos constitute the event sample in detectors like IC.


INTRODUCTION
The observation of ultra-high energy (UHE, E ν ≥ 30 TeV) neutrino events at IceCube (IC) [1,2] is one of the most striking of recent experimental results in all of physics. When statistically buttressed by imminent additional observations by IC and other high energy neutrino observatories like ANTARES [3], AUGER [4] and the upcoming KM3NET [5] they promise to open hitherto unprecedented windows of understanding on the highest energy processes in our Universe. When their import is fully understood, these observations will shed light on UHE cosmic and gamma rays and the acceleration mechanisms powering the most energetic of remotely sited astrophysical engines.
Astrophysically generated neutrinos carry the imprint of physics from their sources over distance scales extending to the very edges of the observable universe. This is primarily due to their low interaction cross-sections with the intervening matter, which, while allowing them to traverse these cosmic distances without significant attenuation, also entails the use of very massive detectors here at earth to accumulate statistically meaningful observations. In IC, neutrino detection occurs via weak charge and neutral current (CC and NC respectively) interactions with nucleons in ice, resulting in the deposition of visible energy in the form of Cerenkov radiation. Observed events are categorized into two distinct types: • ν µ CC and a subset of ν τ CC interactions produce tracks of highly energetic charged leptons traversing a significant length of the detector, while • ν e CC, a subset of ν τ CC and NC interactions of all three flavors produce cascades characterized by their collective light deposition in a bulbous signature distributed around the interaction vertex.
Additionally, in spite of the belief that sources do not produce ν τ , the flavour ratios for neutrinos are rendered close to 1 : 1 : 1 at earth due to oscillations over large distance scales. In this situation, cascade events are expected to constitute about 75-80% of the total observed sample [6]. The background to these events is provided by the rapidly falling atmospheric neutrino flux and the muons created in cosmic-ray showers in the atmosphere.

THE IC EVENTS: CHARACTERISTICS AND POSSIBLE ORIGINS
The 988-day IC data reveals 37 events (9 track, 28 cascades) with energies between 30 TeV and 2 PeV, consistent with a diffuse neutrino flux given by in the energy range 60 TeV-3 PeV, where φ ν represents the per-flavor flux. A purely atmospheric/cosmic-ray shower origin of these events is rejected at the 5.7σ level.
We mention three characteristics of the event sample which will be pertinent to our work below: a) the three highest energy events are closely clustered, with energies of 1 PeV, 1.1 PeV and 2.1 PeV, b) there are no events between 400 TeV and 1 PeV, a gap which is presently believed to be statistical, and c) there are no events beyond 2 PeV, although 3 events are expected between 3-10 PeV for an unbroken E −2 spectrum[7].
Since the bulk of DM is known to be non-relativistic, its direct detection has focussed on its low-energy coherent scattering off nuclei, leading to nuclear recoils which have energies of a few keV, making them very challenging to detect over backgrounds. In general, most efforts have focussed on the parameter space spanned by thermal DM masses in the 10-100 GeV range with weak-scale interaction cross-sections with nucleons [28].
While theoretical biases have served as a guide for searches and model-building, in principle, very little is known about the nature and properties of DM. Specifically, the DM mass can span the range 10 −15 -10 15 GeV, and its interaction cross-section with nucleons and annihilation cross-section into SM particles can lie in the range 10 −76 -10 −41 cm 2 .
In the present work, we assume DM to be primarily non-thermal [29], with its bulk comprised of a very massive relic φ (with mass m φ and a lifetime τ φ greater than the age of the Universe) which decays preferentially to another much lighter DM particle χ (as opposed to decaying to SM daughters). This leads to a small but significant population of ultra-high energy relativistic DM particles, non-thermally created in the narrow energy region spanning m φ . In this context we note there is currently some motivation for the existence of small amounts of additional relativistic degrees of freedom N rel from cosmic microwave background anisotropy measurements, which imply N rel = 3.36 ± 0.34. [30]. This is somewhat at variance with the number of effective neutrino species implied by standard cosmology and particle physics, N eff ν = 3.04 [31]. Non-thermal relativistic DM particles could be a possible way to resolve this difference [32].
We further assume that χ interacts with SM particles with cross-sections much smaller than standard weak interactions via the exchange of a heavy gauge boson which connects the SM and DM sectors. At high energies, this may result in deeply inelastic interactions (DIS) of DM with SM particles, and mimic UHE neutrino-nucleon NC interactions in a detector like IC, creating cascades which are indistinguishable from those created by neutrinos. In analogy with the neutrino sector, where the low energy coherent elastic scattering cross-section on atomic nuclei is about 10 −39 cm 2 and is undetectable at present experimental sensitivities, one could envisage the low energy χ-nucleus cross-section to be very small. However, similar to the ν-nucleon interaction, it would rise to detectable levels at ultra-high energies in detectors like IC. Thus, in the same way that the neutrino-nucleus coherent crosssection is an anticipated background to DM detection at low energies [33] because it produces similar nuclear recoiling final states, its deep inelastic counterpart would produce final states which would make it difficult to distinguish from DM signatures at high energies. In what follows, we pursue this idea further by examining its consequences qualitatively and quantitatively when applied to the PeV events at IC, using our current understanding of the neutrino-nucleon deep inelastic interactions [34,35] as a guide.

PeV events: NC scattering of a relativistic dark matter species with ice nuclei
We assume that the DM sector consists of at least two particle species with the following properties: • A co-moving non-relativistic real scalar species φ, with a mass of O(10 PeV), which is unstable but decays with a very large lifetime to χ, and does not have any decay channels to SM particles. We call this species the PeV Dark Matter (PDM), and it comprises the bulk of present-day DM.
• A lighter fermionic DM species (FDM), χ with mass m χ m φ , which we assume is produced in a monochromatic pair when the PDM decays, i.e., φ →χχ, each with energies of m φ /2.
Since φ does not decay to SM particles, constraints relevant here are those based on CMB anisotropies [36], light nuclei abundances during Big-Bang Nucleosynthesis (BBN) [37] and limits from structure formation [38], which only constrain the total relativistic particle density of the universe at the respective epochs, independent of what those particles are [39]. Consistent with these constraints, we assume that the PDM decays with a (rather large) lifetime of τ φ 10 27 s. Additionally, the lighter (and stable) FDM species is assumed to be produced only non-thermally, via the decay of the long-lived PDM. Its contribution to the DM mass density is thus expected to be small.
The FDM flux is composed of galactic and extragalactic components of comparable magnitudes [40]. Thus, the total flux Φ = Φ G + Φ EG , where, Φ G and Φ EG represent the galactic and extra-galactic components of the total FDM flux respectively [25,40]): and, with Here, z represents the red-shift of the source, ρ c = 5.6 × 10 −6 GeV cm −3 denotes the critical density of the universe, and we have used H(z) = Ω Λ + Ω m (1 + z) 3 , and Ω Λ = 0.6825, Ω m = 0.3175, Ω DM = 0.2685 and H 0 = 67.1 km s −1 Mpc −1 from the recent PLANCK data [41]. For the two-body decay φ →χχ The FDM interacts with the nucleus within the Ice-Cube detector via a neutral current interaction mediated by a beyond-SM heavy gauge boson, Z (Fig. 1) that couples to both the χ and quarks and gluons. For both the χχZ and qqZ interactions we assume the interaction vertex to be vector-like, with hitherto undetermined coupling constants g χχZ and g qqZ respectively. The overall normalisation to the cross-section is set by the product of coupling constants G, and is here arbitrarily chosen to be G = 0.05. The real magnitude of G will be determined by comparing event rates to those seen at IC in the succeeding section.
The DIS cross-section for χN → χX is then computed in the lab-frame, with the product G = g χχZ g qqZ as the undetermined parameter, over a broad range of incoming FDM energies, 100 GeV ≤ E in χ ≤ 10 PeV, using tree-level CT10 parton distribution functions [42]. We set the Z mass to be 5 TeV. For Z with mass 2.9 TeV, the couplings g χχZ and g qqZ are largely unconstrained by collider searches [43], therefore being limited only by unitarity.
Since the IC can only measure the deposited energy E dep for neutral current events, it is important to determine the nature of the inelasticity parameter, relating the deposited energy to the incoming particle energy: The DIS differential cross-section with respect to the inelasticity parameter is then expressed as The results for the total cross-section and the mean inelasticity parameter, are shown in Fig. 2.

Determining the normalization to the interaction strength for χN → χX
The energy at which the χ flux should peak is determined by requiring that the event rates peak at around 1.1 PeV; in turn, this requires that the flux peak at around energies of The total number of events in a given IC bin increases proportionally with the incident flux and the interaction rate of the incident particles with the ice nuclei relevant to the corresponding bin energies. Since, in addition, the (2) and (3)] and dσ/dy ∝ G 2 [Eq. (6)], the ratio G 2 /τ φ of the undetermined parameters G and τ φ can be ascertained by normalising the number of events predicted due to the FDM flux at deposited energies E dep 1 PeV against those seen at the IC. We find that for a reasonable decay lifetime of τ φ = 10 24 s, we need to set G 2 = 0.45 to obtain the 3 PeV+ events from the FDM flux seen over the 988-day IC runtime. The corresponding nature of the FDM extragalactic flux is shown in Fig. 3. The bigger the value of τ φ , the larger would G need to be, to match the IC PeV+ event rate, with the upper bound to the coupling constant and, by consequence, the upper bound to τ φ being set by unitarity limits on G.

Sub-PeV Events: Neutrinos from extra-galactic sources
While the events corresponding to deposited energies E dep 1 PeV are accounted for by the FDM flux, the sub-PeV events up to 400 TeV are consistent with a power-law flux of incident particles, and are, likely, representative of a diffuse flux of neutrinos from extragalactic sources. The term "best-fit" has limited validity at this point in time since given the limited statistics, it is at present unclear if the flux is truly diffuse and extra-galactic, or a superposition of individual extended sources or a combination of these alternatives [44]. Indeed, using only the sub-PeV events to determine the best-fit E −α spectrum, we find that the IC observation is closely matched by a more steeply falling astrophysical flux spectrum than that in Eq. (1), i.e., the best-fit is instead given by (Fig. 3) [45] Consequently, the astrophysical flux drops to below the single-event threshold at energies higher than 400 TeV, rendering it naturally consistent with the lack of events at subsequent energies up to the PeV (see Fig. 4). The FDM flux itself does not contribute appreciably to the sub-PeV event-rate. Predicted and observed total event rates at the Ice-Cube. The gray shaded region represents energies at which we expect events predominantly from the DM sector. The green line shows event-rate predictions from our best fit flux to the sub-PeV event-rates observed at IC, with the flux given by Eq. (7). The event rates predicted due to the IC best-fit E −2 flux (gray dashed line) and the observed data (red diamonds) are shown. The estimated atmospheric background event-rate for each bin is shown as a yellow shaded bar.

DISCUSSION AND CONCLUSIONS
Given present-day constraints on DM, it is possible that it may not be WIMP-like and thermal in nature. In the scenario proposed in this paper, we have focussed on the possible direct detection of high energy DM particles. Such particles cannot form the bulk of DM, which must be non-relativistic, but may be a small but detectable population at and around a specific high energy, created by the decay of another significantly more massive nonthermal DM relic. Such a species may also help mitigate the currently existing tension between the number of effective neutrino species predicted by standard cosmology and particle physics and by CMB measurements. If the lighter DM particle interacts with nucleons, its cross-section at high energies may be detectable as neutrino-like cascades in a massive detector like IC. Using the neutrino-nucleon NC deep inelastic cross-section as a guiding analogy, we have applied this to the cluster of three ∼ PeV events seen at IC.
Thus, this cluster of three events has a different origin from the remainder of the IC event sample, which we assume to be the astrophysical extra-galactic neutrinos. It results in a softer astrophysical spectral best-fit than the one which includes the full-event sample. In this picture, the gap currently seen in the data between 400 TeV-1 PeV is physical, and the result of two distinct spectra with differing origins. Also, the PeV events should continue to cluster in the 1-3 PeV region, with a galactic bias [25] due to the fact that about half of the DM induced PeV flux contribution is expected to be galactic. We note that at present 2 of the 3 events appear to come from the direction of the galaxy. This scenario also provides a natural explanation for the lack of events beyond 3 PeV. Other recent proposals which also account for the cut-off at PeV energies are discussed in [23,24,[47][48][49][50][51].
A feature which would distinguish DM-induced PeV events from neutrino-induced ones is the lack of muon tracks in the 1-3 PeV region. In the standard astrophysical scenario, IC expects to see both down-going and upgoing muons (with the latter being somewhat suppressed) in this range, and they would comprise about 20% of the total events [52]. Additionally, for DM events in the 1 − 3 PeV range, some extra-galactic contribution of cascades could come from the Northern hemisphere, because the low DM-matter cross-section does not cause their flux to attenuate significantly in the earth at PeV energies, unlike neutrinos.These predictions (which also separate the present scenario from other DM induced indirect detection proposals [23,24]) and the persistence of a gap-like feature can be tested as IC accumulates more statistics.
The authors would like to thank Nathan Whitehorn for his patient answering of many questions on the IC data and Arindam Chatterjee for useful discussions related to this work. RG thanks Alejandro Ibarra for very useful discussions. RG also acknowledges support from Fermilab via an Intensity Frontier Fellowship. RG and AG acknowledge support from a XII Plan DAE Neutrino Physics and Astrophysics Grant. AG is also deeply appreciative of help from Mehedi Masud and Titas Chanda related to some of the relevant computational work. This work was supported in part by the US Department of Energy contracts DE-FG02-04ER41298 and DE-FG02-13ER41976 for AB.