Demystifying the PeV Cascades in IceCube: Less (Energy) is More (Events)
Ranjan Laha, John F. Beacom, Basudeb Dasgupta, Shunsaku Horiuchi, Kohta Murase
DDemystifying the PeV Cascades in IceCube: Less (Energy) is More (Events)
Ranjan Laha,
1, 2
John F. Beacom,
1, 2, 3
Basudeb Dasgupta, Shunsaku Horiuchi, and Kohta Murase Center for Cosmology and AstroParticle Physics (CCAPP), Ohio State University, Columbus, OH 43210 Department of Physics, Ohio State University, Columbus, OH 43210 Department of Astronomy, Ohio State University, Columbus, OH 43210 International Center for Theoretical Physics, 34014 Trieste, Italy Center for Cosmology, Department of Physics and Astronomy, University of California, Irvine, CA 92697 Hubble Fellow, School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540 [email protected], [email protected], [email protected], [email protected], [email protected] (Dated: July 29, 2013)The IceCube neutrino observatory has detected two cascade events with energies near 1 PeV [1, 2].Without invoking new physics, we analyze the source of these neutrinos. We show that atmosphericconventional neutrinos and cosmogenic neutrinos (those produced in the propagation of ultra-high-energy cosmic rays) are strongly disfavored. For atmospheric prompt neutrinos or a diffuse back-ground of neutrinos produced in astrophysical objects, the situation is less clear. We show thatthere are tensions with observed data, but that the details depend on the least-known aspects ofthe IceCube analysis. Very likely, prompt neutrinos are disfavored and astrophysical neutrinos areplausible. We demonstrate that the fastest way to reveal the origin of the observed PeV neutrinosis to search for neutrino cascades in the range below 1 PeV, for which dedicated analyses with highsensitivity have yet to appear, and where many more events could be found.
I. INTRODUCTION
Neutrino astronomy has long promised to reveal the as-trophysical sites of particle acceleration and the nature ofcosmic rays [3–11]. The lack of adequately-sized neutrinodetectors has been a deterrent in turning this dream intoreality. The recent completion of the IceCube detectorhas raised hope of addressing these long-standing prob-lems [12]. Encouraging this hope, an analysis of veryhigh energy neutrino events in the IceCube detector dur-ing 2010–2012, as construction was finishing, found twocandidate neutrino cascade events with energies near 1PeV [1, 2].These are the highest energy neutrinos ever detected –they are 10 times more energetic than typical GeV at-mospheric neutrinos. They signal the entry of neutrinoastronomy into the PeV era, made possible by the hugesize of IceCube. However, these events have led to sev-eral mysteries. Where did they come from? Although weexpect ν µ to be more detectable than ν e due to the longrange of the muons, why are there two cascade events andzero muon track events? Why are the two event energiesso close to each other and to the analysis threshold? Isthe neutrino flux required to explain these events consis-tent with previous limits and with other data?These PeV neutrino events have spurred a flurry ofactivity, due to the importance of the potential first dis-covery of non-atmospheric high-energy neutrinos. As-trophysical neutrinos – those produced inside distantsources – have been considered [13–24]. Cosmogenicneutrinos – those produced in the propagation of ultra-high-energy cosmic rays – have also been considered [25–28]. Other papers have proposed more exotic explana-tions [29–33]. Novel tests of the data or of new physicshave been noted [34–36].We provide a new general analysis of the source of these two events, focusing on the simplest and most straight-forward scenarios, and including many realistic aspectsof neutrino detection in IceCube (for our early results,see Refs. [37, 38]). We assume that both events wereneutrino-induced and that neutrinos have only standardproperties and interactions. We assess which scenariosare compatible with the present data and the implica-tions of this discovery. Importantly, we detail how thesescenarios can be tested by new analyses.The flux of atmospheric conventional neutrinos at PeVenergies is much too low to give rise to these two cascadeevents. Cosmogenic neutrinos are also very unlikely to bethe source, due to the lack of higher-energy events. At-mospheric prompt neutrinos do not appear to be a plau-sible source, but they should not be dismissed lightly. Adiffuse background of neutrinos from astrophysical ob-jects can reasonably explain the observed data, thoughthere are strong constraints on the spectrum. A full as-sessment of these models will require more details aboutthe IceCube search strategies.New analyses optimized for energies near and below1 PeV are urgently needed. The cascade or shower chan-nel for electron neutrinos is especially important, be-cause its atmospheric conventional neutrino backgroundis much lower than for muon neutrinos, as first shownby Beacom and Candia in 2004 [39]. There are greatopportunities to better exploit this detection channel.In Sec. II, we begin with the basic information on thesetwo PeV cascade events and what it suggests, which wesupport with quantitative details in later sections. InSec. III, we test whether various neutrino fluxes can bethe source of these two events. In Sec. IV, we detailhow searches for cascades and tracks in the energy rangebelow 1 PeV will robustly distinguish between varioussources. We conclude in Sec. V, including commentingon preliminary new IceCube events below 1 PeV. a r X i v : . [ a s t r o - ph . H E ] J u l II. WHAT IS KNOWN ABOUT THE EVENTS
These two events were detected as PeV cascades dur-ing the 2010–2012 runs. They were identified in the ex-tremely high energy (EHE) search, which is optimizedfor the detection of EeV = 10 PeV cosmogenic neutri-nos [2]. This search has strong cuts to decisively rejectdetector backgrounds, and these cuts greatly affect theacceptance for signal events, especially in the PeV range,which is the edge of the considered energy range, becauserelatively few cosmogenic events are expected there.Our analysis focuses on the PeV range and below. Thissection introduces the events and their implications. Thereconstructed event energies are 1 . ± .
16 PeV and1 . ± .
17 PeV [2]. This disfavors neutrino interactionsat the Glashow resonance at 6.3 PeV, for which the cas-cade energy should generally be the same; we discussexceptions below. The absence of higher-energy eventsdisfavors cosmogenic neutrinos, as their detection prob-ability is largest in the EeV range.The values of the energies, and especially their prox-imity to each other, are crucial. We assume that thedetected energies are probable values in the distributionof possible values; this is reinforced by there being twosimilar events. The minimal explanation of the two ener-gies is that this distribution is peaked at ∼ ∼ ν e + ¯ ν e [2]. The efficiency for ν µ + ¯ ν µ , which should be more detectable due to the longrange of the muons, is suppressed, because the muons donot deposit their full energy in the detector. The effi-ciency for ν τ + ¯ ν τ is suppressed because of the tau-leptondecay energy carried by neutrinos. This explains the non-observation of muon track and tau-lepton events; futuresearches can be optimized to find them.The most likely scenario is that both cascade eventsarise from charged current (CC) interactions of ν e + ¯ ν e ,for which the detectable cascade energy is nearly the fullneutrino energy. Because of the above suppressions, weneglect the rare cases in which ν µ + ¯ ν µ or ν τ + ¯ ν τ CCevents resemble ν e + ¯ ν e cascades, due to the muon gettinga small fraction of the neutrino energy or the tau leptondecaying quickly. Neutral current (NC) interactions of all E d (cid:92) / d E [ G e V c m - s - s r - ] E [GeV] -10 -9 -8 -7 A t m . C onv . (cid:105) µ A t m . C onv . (cid:105) e Atm. Prompt (cid:105) µ AhlersTakamiE -2 IC40 (cid:105) µ U.L.
IC40 U.L. EHE search
FIG. 1. Neutrino fluxes as a function of neutrino energy. Theatmospheric conventional ν µ + ¯ ν µ and ν e + ¯ ν e spectra are fromRef. [45, 46]. The atmospheric prompt ν µ + ¯ ν µ spectrum (the ν e + ¯ ν e flux is the same) is the Enberg (std.) model [47]. Ex-ample cosmogenic EHE neutrino fluxes ( ν + ¯ ν for one flavor)are from Refs. [48, 49]. An E − astrophysical neutrino spec-trum for one flavor of ν + ¯ ν , normalized as discussed below, isshown, along with current upper limits from IceCube [43, 46]. flavors of neutrinos also give cascades. The cross sectionis 2.4 times smaller near 1 PeV, though three neutrinoflavors may contribute. The more important point is thatthe average cascade energy in a NC interaction is only ∼ .
25 of the neutrino energy in the PeV range, whichmakes the event much less detectable [2]. It is unlikelythat NC interactions could be the source of these events,especially both of them, because the cascade energies areso close to each other and the analysis threshold.These events are consistent with a steady, isotropicdiffuse source, and we assume this, though other possi-bilities are not excluded. The events were separated tem-porally by 5 months; the search ran for about 2 years. Itis difficult to measure the directions of cascade events, asthe signal regions in the detector are large and sphere-like. No event directions are reported in the IceCube pa-per [2], and preliminary IceCube results from conferencesvary significantly [40, 41]. Future analyses are expectedto have an angular resolution of ∼
10 degrees for cas-cades near 1 PeV (and worse at lower energies) [40]. Forupgoing events that pass through Earth’s core, with azenith angle greater than ∼ ◦ ( ∼
7% of the full sky),there would be especially significant attenuation due tointeractions in Earth [42, 43]. Prompt neutrinos that aresufficiently downgoing will be accompanied by cascadesthat trigger the IceTop surface detector [1, 44]; this wasnot seen, and studies of its efficiency are ongoing.Figure 1 shows some relevant neutrino spectra.
III. WHAT CAN BE THE SOURCE?
In this section, we first discuss our general approachto testing possible spectra, given that much is not yetknown. We then discuss cascade detection in IceCube,followed by detailed discussions of possible sources ofthese events and a summary of remaining issues.
A. Our approach to assessing source spectra
The two PeV events were found in the EHE search,which is not optimized for detection in the PeV energyrange. The cuts required to reject backgrounds reducethe probability of detecting signal events, especially atthese relatively low energies. The effective area plot inRef. [2] shows that the neutrino detection probability fallsvery quickly with decreasing neutrino energy, plummet-ing below ∼ ν e + ¯ ν e flavors [2]. A straightfor-ward calculation – not including the effects of the strongcuts – is about one order of magnitude larger than the ef-fective area of Ref. [2] near 1 PeV, and this point has notbeen noted before. (We can reproduce the effective areafor other IceCube searches, e.g., Ref. [46].) However, asboth events were detected at ∼ E d (cid:92) / d E [ G e V c m - s - s r - ] E [GeV] -10 -9 -8 -7 -6 -5 -4 -3 P r o m p t (cid:105) µ E -2 E -2.5 E -3 FIG. 2. Example neutrino fluxes (for one flavor of ν + ¯ ν )that might produce the PeV events, compared to the atmo-spheric conventional ν µ + ¯ ν µ (upper points) and ν e + ¯ ν e (lowerpoints) fluxes measured by IceCube [50, 51]. The power-lawastrophysical fluxes are normalized so that they do not exceedthe measured data. The atmospheric prompt neutrino flux isonly shown above 1 TeV, following Ref. [47]. then this would conflict with the observation of the twoPeV events. We choose acceptable normalizations inFig. 2 and later estimate the probabilities of detectingtwo events in the PeV range. The normalizations couldbe increased, given the large uncertainties; the power-lawfluxes could be increased by about a factor of 2, and theprompt flux by more. A second tension appears in theslope of a possible source spectrum. If it is too steep,then the spectrum will exceed measurements of atmo-spheric conventional neutrinos at lower energies unlessthe spectrum breaks. If it is not steep enough, then itwill have too many events expected above 1 PeV.For both of these issues, the degree of statistical ten-sion would be calculable in a full analysis, whereas herewe can only estimate it. We consider two energy bins;these were chosen post hoc, but the fact the event ener-gies are so close to each other and the threshold at 1 PeVseems to be a strong clue. The first bin is 1–2 PeV, whicheasily contains both points within energy uncertainties.Detections at lower energies are assumed impossible dueto the threshold. Detections at higher energies are con-sidered with a second bin, 2–10 PeV; for falling spectra,the exact value of the upper limit is not very important.We present our results in terms of detectable energy,which is not always the same as neutrino energy, as ex-plained below. This is closer to what is actually mea-sured, allowing for much better control in separating sig-nals and backgrounds. B. Cascade detection in IceCube
The neutrino-nucleon cross sections σ ( E ν ) near 1 PeVare well known [52–55]. In CC cascade events initiated by ν e + ¯ ν e , the neutrino interacts with a nucleon, leading to ahadronic cascade, and produces an electron or positron,leading to an electromagnetic cascade. The division ofthe neutrino energy E ν depends on the inelasticity y , forwhich (cid:104) y (cid:105) (cid:39) .
25 near 1 PeV and varying slightly withenergy [56]. The outgoing lepton has energy (1 − (cid:104) y (cid:105) ) E ν ,with the remainder going to the hadrons, so that the de-tectable total cascade energy is (cid:39) E ν . The cascade leadsto a roughly spherical distribution of hit phototubes overa diameter of a few ×
100 m, though the cascade size isseveral meters. Cascades produced by the NC interac-tions of all flavors are similar, though the hadronic cas-cade energy is just (cid:104) y (cid:105) E ν instead of E ν , so NC cascadescan normally be neglected for all but atmospheric con-ventional neutrinos [39].In the “theorist’s approach” or ideal case, the eventrate spectrum for ν e + ¯ ν e CC cascades is dNdE casc (cid:39) π ρ N A V T (1) × (cid:90) +1 − d (cos θ z ) d Φ dE ν ( E ν ) σ ( E ν ) e − τ ( E ν , cos θ z ) . The number of target nucleons is ρ N A V , where ρ is theice density (in g cm − ), N A the Avogadro number, andthe IceCube volume is V (cid:39) . The observation timeis T = 615 . σ (incm ) and the neutrino intensity spectrum d Φ /dE ν (inGeV − cm − s − sr − ) are evaluated at E ν (cid:39) E casc (inGeV). Neutrino flux attenuation en route to the detector,which depends on energy and zenith angle, is taken intoaccount in the optical depth τ = (cid:96)/λ assuming a constantdensity of 3 g cm − for Earth, where (cid:96) is the path lengthand λ the mean free path. We include NC interactionsvia simple modifications to the above, including a factor1 / (cid:104) y (cid:105) due to the change in the energy differential.The CC cross section varies smoothly with energy, ex-cept near the Glashow resonance at 6.3 PeV, which iscaused by the resonant production of an on-shell W bo-son by ¯ ν e + e − → W − [52, 57]. The W decays promptly,typically depositing most of its energy in the detector.About 10% of the time, the decay to an electron and anantineutrino leads to a range of smaller deposited ener-gies; assuming that there are enough such interactions,the probability for this to happen twice is thus (cid:46)
1% [58].At 6.3 PeV, the ratio of the cross section for ¯ ν e to inter-act with an electron instead of a nucleon is 350 [31, 52].The overall importance of this is reduced by an equal fluxof ν e , half as many electron as nucleon targets, and theopacity of Earth to ¯ ν e at this energy. In the effective areaplot of Ref. [2], the enhancement is thus only a factor of (cid:39)
15 in a bin of width ∆(log E ) = 0 . ν τ + ¯ ν τ can besimilar those those initiated by ν e + ¯ ν e . At ∼ ∼
50 m. (Above ∼ (cid:39) .
3; thefraction of E ν deposited for ν τ + ¯ ν τ events with prompttau-lepton decays is then (cid:39) (cid:104) y (cid:105) + 0 . − (cid:104) y (cid:105) ) (cid:39) . ν τ + ¯ ν τ eventsin our calculations of cascade spectra above 1 PeV forcomparison with present data, but we do in our calcula-tions below of possible future spectra below 1 PeV, whichincreases the rates by somewhat less than a factor of 2.As a more realistic estimate, we calculate the cascadespectra using the effective area from Ref. [2], which leadsto significantly smaller yields, due to the effects of thestrong cuts in this search. In this approach, the eventrate spectrum for ν e + ¯ ν e cascades is dNdE casc = 4 π A eff T × d Φ dE ν ( E ν ) (2)where A eff takes into account all of the factors in Eqn. (1)plus the detailed search cuts.In both approaches, the effect of detector energy reso-lution on the spectrum must be taken into account. Wesmooth the calculated spectra with a Gaussian of width δE/E = 15%, taken to match the uncertainty on the en-ergy of the two events. Future analyses will likely havebetter energy resolution, more like 10% [40]. The effectof energy resolution on the Glashow resonance is espe-cially significant, reducing its height and increasing itswidth while preserving the number of events.Figure 3 shows our results (ideal and realistic) for thesignal and background spectra. The numbers of eventsin each bin for the realistic approach are given in Table I.Energies in IceCube are measured with fractional, notfixed, precision, so log E is a more natural variable than E . The number of bins of fixed width dE = 1 GeV in eachdecade of log E increases ∝ E , so measured event spectrashould then be presented as EdN/dE = dN/d (ln E ) =2 . − dN/d (log E ) instead of dN/dE . Using EdN/dE gives a correct visual representation of the relative de-tection probabilities in different ranges of log E . Further,this makes it much easier to estimate the area, i.e., thetotal number of events. Using EdN/dE and log E to es-timate area means that both the height and width aredimensionless. To get 1 event, the height must be ∼ d (ln E ) = 2 . d (log E ) = ln 2 = 0 . E ca s c ( d N / d E ca s c ) [ c oun t s ] E casc [GeV] -4 -3 -2 -1 E -2 E -2.5 Atm. Conv. Atm. PromptIdeal; Cascades 4 (cid:47) E -3 T a k a m i A h l e r s E ca s c ( d N / d E ca s c ) [ c oun t s ] E casc [GeV] -4 -3 -2 -1 E -2 E -2.5 Atm. Conv. Atm. PromptRealistic; Cascades 4 (cid:47) E -3 T a k a m i A h l e r s FIG. 3.
EdN/dE for neutrino-induced cascade spectra. The left panel is for the ideal case or “theorist’s approach,” and the right is for the realistic case using the effective area from Ref. [2]. These results are for the 615.9 days of exposure that includedthe two PeV events. The power-law fluxes are normalized in Fig. 2. The thin vertical line denotes the boundary between ourtwo bins. The y-axis has a large logarithmic range to show several spectra. The number of events in a region is proportionalto the integrated area, i.e., to the height times the logarithmic energy range, so curves with low heights have very few events.
C. Atmospheric conventional fluxes: very unlikely
Because atmospheric conventional neutrinos definitelyexist, it is important to ask if they could produce theseevents. We show the ν µ + ¯ ν µ and ν e + ¯ ν e fluxes fromRef. [45, 46] in Fig. 1. The ν τ + ¯ ν τ flux is much smaller,because both direct production and neutrino oscillationsat these energies are suppressed, and it is not shown.In the muon track channel, the atmospheric conven-tional ν µ + ¯ ν µ flux is a significant background to new TABLE I. Expected numbers of cascade events in the twoenergy bins, obtained by integrating the curves in the rightpanel (the realistic approach using the effective area) of Fig. 3.These numbers are typically a factor of ∼ Possible Source N(1 − −
10 PeV)Atm. Conv. [45, 46] 0.0004 0.0003Cosmogenic–Takami [48] 0.01 0.2Cosmogenic–Ahlers [49] 0.002 0.06Atm. Prompt [47] 0.02 0.03Astrophysical E − E − . E − ν e +¯ ν e are significantly less, which means that new signalscan emerge at lower energies. To see this, it is necessaryto plot predicted event spectra in terms of detectable cas-cade energy instead of neutrino energy. For ν e + ¯ ν e CCevents, these are the same. For NC ν µ + ¯ ν µ events, whichhave a small energy deposition, it is a big difference. Go-ing from Fig. 1 to the left panel of Fig. 3, the importanceof atmospheric conventional neutrinos relative to othersources (e.g., the E − spectrum) is greatly reduced. Thisis what makes cascade searches so powerful [39].The complete (CC + NC) ν e + ¯ ν e cascade spectrumfrom atmospheric conventional neutrinos is shown inFig. 3, with the integrated numbers of events for the real-istic case given in Table I. If we also include muon tracks(see below), the total number of events above 1 PeV in-creases to 0.008, which is consistent within uncertaintieswith the 0.012 of Ref. [2]. As these expected numbersare negligible, it is very unlikely that they can yield thePeV events.Most downgoing atmospheric muons are easily identi-fied as such. In some rare cases, including muon bundles,these initiate events that look like neutrino-induced cas-cades. The expected number of such events is 0.04 [2],larger than the background from neutrinos. All together,these conventional backgrounds have a ∼ − probabil-ity of producing at least two observed events. Thesebackgrounds can be studied further at lower energies,where they are larger. D. Cosmogenic neutrinos: very unlikely
Cosmogenic neutrinos [63–72] have been invoked as thesource of the PeV events, in part because the EHE searchwas designed to detect them, albeit at much higher en-ergies. Example spectra [48, 49] are shown in Fig. 1.The ν e + ¯ ν e cascade spectra are shown in Fig. 3 andthe numbers of events are given in Table I. Two problemsare obvious. First, the expected numbers of events arevery small because the spectrum normalization is low.Second, the predicted distribution of events emphasizeshigh, not low, energies.The probability of having two or more ν e + ¯ ν e cascadeevents detected in the first bin is ∼ − for the modelof Ref. [48] and ∼ − for the model of Ref. [49]. Thereshould also be a penalty factor to not have events inthe second bin, but this is modest because the expectednumbers of events are small. For these models, there arecomparable numbers of muon track and tau-lepton eventsthat pass the search criteria, and their sum is comparableto the number of ν e + ¯ ν e cascades in each bin. Includingthese would increase the Poisson probability of detectingtwo or more events by a factor of ∼ = 4.In addition, there is a third problem, that the expectednumber of all events – cascades, muon tracks, and tau lep-tons – at EHE energies is large enough that some eventsmight have been seen, but none were [2, 43]. The nor-malizations of these representative models are based onmeasured gamma-ray and cosmic-ray data [48, 49]. If wearbitrarily increased the normalization to increase theyields in the PeV range, that would cause an unaccept-able increase in the expected number of events in theEeV range. Cosmogenic neutrinos are thus very unlikelyto be the source of the PeV events. If they are, IceCubeshould quickly discover new events at higher energies. E. Atmospheric prompt neutrinos: disfavored
Collisions of cosmic rays with atmospheric nuclei pro-duce many unstable hadrons; these are dominantly pi-ons, with a small fraction of kaons, and a very smallfraction of mesons and baryons with heavy quarks suchas charm [73]. The decays of many of these hadronsproduce atmospheric neutrinos and muons. Where theenergy losses of these hadrons due to hadronic scatteringbefore decay can be neglected, their spectrum and thatof their daughter neutrinos follows the spectrum of thecosmic rays; otherwise, those spectra fall more steeply.At the lowest energies, neutrinos from pions dominate.As the energy increases, pions have increasing losses andthen neutrinos from kaons dominate. Together, these arethe atmospheric conventional neutrinos. As the energyincreases further, kaons have increasing losses and thenneutrinos from the decays of heavy-quark states domi-nate. For these states, the decays are quite rapid, sothe effects of hadron energy losses in the atmosphere aremuch less. These are the atmospheric prompt neutrinos. The conventional neutrinos have a strong zenith-angledependence, due to the varying depth of atmosphere, butprompt neutrinos are closer to isotropic [47].Atmospheric neutrinos have been detected with ener-gies up to a few hundred TeV [51]. The spectra are con-sistent with atmospheric conventional neutrinos, with noprompt component identified yet. Precise prediction ofthe atmospheric prompt fluxes is difficult because of un-certainties in the hadronic physics and the nuclear com-position of the cosmic rays [47, 74–83].One generic prediction is that the prompt componentwill begin to dominate the conventional component atsome high energy, due to its harder spectrum. Anothergeneric prediction is that the ν e + ¯ ν e flux is the same asthe ν µ + ¯ ν µ flux for the prompt component; it is sup-pressed for the conventional component because pionsand kaons decay primarily to muons, which are stoppedin Earth before they decay. This means that the prompt ν e + ¯ ν e component should emerge from the conventionalcomponent at lower energies than the prompt ν µ + ¯ ν µ component, which gives an advantage to cascade searchesover track searches, despite the long range of muons, asemphasized in Ref. [39].We adopt the Enberg (std.) model [47] for the atmo-spheric prompt neutrino flux; the components are shownin Fig. 1. This calculation is based on the dipole formal-ism in a perturbative QCD framework, which providesa way to treat gluon saturation effects at low x , and itassumes that the cosmic rays are protons.There is uncertainty in the hadronic interactions, dueto the extrapolation of the gluon distribution function tolow x , and more experimental data from the LHC areneeded [84–86]. Although other perturbative QCD mod-els may give similar results, e.g., the flux in Ref. [80] isabout a factor of 2 below that of Ref. [47], phenomenolog-ical non-perturbative QCD approaches typically predicthigher fluxes by a factor of ∼ −
10 [74, 87, 88]. Themost extreme models are already ruled out or disfavoredby neutrino data [88–90].For the atmospheric prompt fluxes, the ν e + ¯ ν e cas-cade spectra are shown in Fig. 3 and the numbers ofevents are given in Table I. The slope is reasonable, inthat energies near the threshold at 1 PeV are favored.The expected number of atmospheric prompt events is ∼ .
02 in each of the two bins (including muon tracksand tau leptons would increase these by ∼ ∼ − . An additional problem is that thecosmic ray spectrum steepens at the knee, reducing theprompt neutrino flux [91].However, the normalization of the prompt flux couldeasily be larger, given the substantial hadronic uncertain-ties, without conflicting with the neutrino measurements(which have large uncertainties) shown in Fig. 2. Accord-ing to Refs. [2, 90], the normalization could be about 4times larger; that would improve the probability by afactor of ∼ = 16, but it would still be very small.The atmospheric prompt neutrino flux near 1 PeVwould be even smaller if cosmic rays at higher energies arenuclei, as argued in, e.g, Refs. [91, 92], instead of protons,as assumed here. The neutrino number flux per logarith-mic energy bin depends on the same for the cosmic rays,which falls as Ed Φ /dE ∼ E − γ , where γ (cid:39) .
7. If cosmicrays are protons, this spectrum is used directly. If cos-mic rays are nuclei of mass number A , then the nucleonspectrum must be derived first. To give the same rangeof nucleon energy, cosmic ray nuclei must have energies A times larger, which gives a suppression A − γ . Takinginto account the greater multiplicity of nucleons, the netsuppression of the neutrino flux is A − γ (cid:39) A − . . There-fore, if the initiating cosmic rays are dominantly nuclei,then it is even more unlikely that prompt neutrinos canexplain the two observed events.We emphasize that the atmospheric prompt neutrinohypothesis for the observed events, although disfavored,would not require the first discovery of high-energy astro-physical neutrinos. The prompt neutrino flux has neverbeen experimentally identified, and the theoretical un-certainties are quite large, so a very high standard mustbe met to reject this hypothesis. On the other hand, if itwere confirmed to be the source of the events, that wouldprovide important and constraining information aboutboth low- x QCD and the composition of the cosmic rays.IceCube can test the normalization of the prompt fluxusing both neutrinos and muons [39, 91, 93–95]. TheIceTop detector can reject downgoing prompt neutrinosby detecting accompanying cascades [1, 44].
F. Astrophysical neutrinos: plausible
Neutrinos are inevitably produced by cosmic-ray in-teractions with matter and radiation in astrophysi-cal sources. Many sources that may have large neu-trino fluxes have been proposed, e.g., jets [96–100] andcores [101, 102] of active galactic nuclei, the prompt [103–105] and afterglow [106–108] phases of gamma-ray bursts,newly-born neutron stars [109], early supernovae [110,111], starburst galaxies [112, 113], and large-scale struc-tures and galaxy clusters [114–116]. There is a wide vari-ety of models, each with some parameters, so roughlymeasuring a flux and spectrum may not identify thesource.To survey possible astrophysical diffuse sources, weconsider power-law neutrino spectra, d Φ /dE ∝ E − s . Weassume flavor ratios of ν e : ν µ : ν τ = 1 : 1 : 1 for neutri-nos and antineutrinos, and equal fluxes of each. (Testingflavor ratios will be important [35, 60, 117].) Becauseour focus is a narrow range near 1 PeV, more generalspectra may be fairly characterized by power laws, andwe define three cases: s = 2, s = 2 . s = 3. Theobservation of two events near threshold at 1 PeV andnone at higher energies strongly favors neutrino spectrathat lead to adequately falling cascade spectra EdN/dE beyond 1 PeV. Below, we discuss spectra that are moregeneral than these unbroken power laws. We define the flux normalizations by using the largestpower-law fluxes that do not exceed the measured atmo-spheric neutrino data at any energy, as shown in Fig. 2.For s = 2, the flux normalization for ν + ¯ ν in one flavor is E d Φ /dE (cid:39) . × − GeV cm − s − sr − . This is con-sistent with upper bounds from IceCube [43, 46, 118], andis smaller than the upper range of the Waxman-Bahcallbound, E d Φ /dE (cid:39) (0 . − . × − GeV cm − s − sr − [119]. For s = 2 .
5, the normalization (at 1 PeV) is E d Φ /dE (cid:39) . × − GeV cm − s − sr − . For s = 3,the normalization (at 1 PeV) is E d Φ /dE (cid:39) . × − GeV cm − s − sr − . These latter two are comparable toor smaller than the nucleus-survival bound [72].The ν e + ¯ ν e cascade spectra are shown in Fig. 3 and thenumbers of events are given in Table I. In the results foran ideal detector, both the slopes and normalizations ofthe cascade spectra are favorable, in that the cascadespectra peak near threshold at 1 PeV and reasonablenumbers of events are expected. However, in the cal-culation using the effective area from Ref. [2], the effectof the cuts on the efficiency near 1 PeV is very significant,driving down the total number of events and suppressingthe importance of the first bin. This makes the secondbin, and the Glashow resonance there, much more im-portant; for the power-law spectra, there are comparablenumbers of events in the continuum and in the excess dueto the resonance. Beyond 10 PeV, the detector efficiencyapproaches the ideal case and, for all but the cosmogenicmodels, the cascade spectra are falling and the expectednumbers of events are small.For the different s values in the realistic case, the totalnumbers of expected events might be reasonable, espe-cially if some things are taken into account. The normal-izations for the spectra chosen in Fig. 2 could plausiblybe increased by a factor of 2. Comparable numbers of ν µ + ¯ ν µ and ν τ + ¯ ν τ CC events should be included tomatch the IceCube search criteria, and their sum is com-parable to the number of ν e + ¯ ν e cascades in each bin.In the E − case, ν µ + ¯ ν µ and ν τ + ¯ ν τ NC events in thesecond bin could contribute ∼ . f changesthe probability of getting two or more events by ∼ f .The distribution of events is a larger problem: insteadof favoring the lowest energies, near threshold, these cas-cade spectra favor higher energies in all cases.The astrophysical models considered here are not inobvious agreement with observations, but this dependson the details of the efficiency near threshold, so wemust withhold judgment until there are results from newsearches. It is plausible that astrophysical scenarios couldexplain the observed events. Taking the large uncertain-ties into account, spectra less steep than E − seem tobe disfavored by the spectrum shape, and spectra moresteep than E − seem to be strongly disfavored by thespectrum normalization. The most important thing is toimprove the efficiency at energies below 1 PeV, where thenumber of events might be much larger. G. What conclusions can we draw now?
None of the sources above immediately fits the keyobserved properties of the data: two cascade events, veryclose in energy to each other and the analysis threshold,no cascades at higher energies, and no other types ofevents. How can this be? We focus on steady diffusefluxes here and then mention other possibilities below.One possibility is improbable fluctuations. These twoevents might be caused by astrophysical neutrino signals,and what was seen was a lucky fluctuation. Reconcilingwhat was and was not seen may be challenging. Or thesetwo events might be caused by atmospheric neutrino ormuon backgrounds, and what was seen was an unluckyfluctuation. With the expected rates, this is very un-likely; further study is needed to be sure there are nosurprises with muon-induced backgrounds.Another possibility, which we think is unlikely, is thatthe effective area or the relation between the number ofdetected photoelectrons and cascade energy is not com-pletely understood. The search strategy was optimizedfor cosmogenic neutrinos in the EeV range, and perhapsthere are subtleties near 1 PeV, the edge of their range [2].The IceCube Collaboration takes great care in their anal-yses and papers, but the possibility of some revisions be-ing needed must be considered because of the seemingparadox of detecting two events near threshold, wherethe efficiency is only ∼ E − spectrum in the three energy ranges separately, setby Fig. 2, the observation (and hence expectation) of twoevents in the first bin, and the observation of zero eventsin the second bin, respectively. (We always quote neu-trino fluxes for one flavor of ν + ¯ ν , assuming equal flavorratios, whereas some authors quote the sum of all threeflavors.) These results suggest a break in the spectrumat several hundred TeV and another break or cutoff atabout 2 PeV. For a different spectrum shape or choiceof bins, these constraints would change. Still, the nomi-nal conflicts between fluxes in different energy ranges arestartling, and indicate tensions that need to be resolved. E d (cid:92) / d E [ G e V c m - s - s r - ] E [GeV] -9 -8 -7 -6 E -2 E -2.5 E -3 Real RealIdeal Ideal
FIG. 4. Example neutrino fluxes, as in Fig. 2, for one flavorof ν + ¯ ν , assuming equal flavor ratios. In the 1–2 PeV and2–10 PeV bins, we show our estimates of the flux normaliza-tion required to match the observations of two events and zeroevents, respectively, for an E − spectrum in each bin sepa-rately. We show the 68% confidence-level uncertainty rangefor the first bin and the 90% confidence-level upper limit forthe second [123]. The “Real” case uses the right panel ofFig. 3 (based on Ref. [2]), while the “Ideal” case uses the left. The dominant uncertainties are those shown in Fig. 4.We fix the power-law normalizations in Fig. 2 by de-manding that they not exceed the measured points. Thisleaves no room for the expected atmospheric conventionalneutrinos, but the uncertainties are large, probably evenlarger than the quoted factors of a few up or down. ThePoisson uncertainties on the fluxes in our two bins aresignificant. Our calculations of the expected numbers ofevents are reasonably precise, though we make approxi-mations throughout at the level of a few tens of percent.These include the form of the event rate equations, ap-proximating the dσ/dy distributions and Earth attenua-tion, and neglecting the small numbers of expected eventsbelow 1 PeV and above 10 PeV.If the true spectrum is not peaked, then the most likelyscenario is that there should be an excess in the low-energy muon neutrino data (now seen in Ref. [90]), thatthe observation of the two PeV events was a fortunateupward fluctuation, and that there should be a cutoff atabout 2 PeV. In this case, our results show that the pre-ferred power-law spectrum is around E − . The strongconstraint on an astrophysical neutrino flux shown inFig. 1, E d Φ /dE < . × − GeV cm − s − sr − [43],would apply to an E − spectrum that held over the fullenergy range shown there. See also the preliminary dif-ferential constraints shown in Ref. [1]. IV. FUTURE NEUTRINO OBSERVATIONS
As we show above, the source of the two cascade eventsin IceCube remains unknown, though some possibilitiescan already be excluded. With such a small sample andsuch large uncertainties, it is not yet possible to makevery precise statements. We now show that analyses ofexisting cascade data at lower energies have great poten-tial to quickly reveal the source of these events. Searchesfor muon tracks in IceCube are quite mature, with atmo-spheric neutrino events measured up to a few hundredTeV [51]. To measure the smaller fluxes at higher ener-gies, greater exposure is needed, which will simply taketime. In contrast, searches for cascades with measuredatmospheric neutrino events are relatively recent and thespectra only go up to 10 TeV [50].A comprehensive exploration below 1 PeV, where theremight be many more events, is needed in both the trackand cascade channels. In the following, we first reviewmuon track detection in IceCube. Cascade detection isdiscussed in detail above. Here, one important differenceis that ν τ + ¯ ν τ CC events are now included as cascadesfor the astrophysical scenarios (but not for atmosphericprompt neutrinos, which have a small ν τ + ¯ ν τ flux) be-cause the tau-lepton track length below 1 PeV is short.We show how our new results on the predicted spectracan differentiate between possible scenarios.The following is for the ideal case or “theorist’s ap-proach,” because the detailed properties of IceCube forfuture searches are not yet known, as new strategies toisolate signals from backgrounds will be developed. Thetrue efficiency will be somewhat less, e.g., due to cuts toreject backgrounds and because outward-directed signalevents near the surface will not deposit enough energy.In addition, the spectrum shapes will suffer some smear-ing due to energy resolution. The most important pointof realism that we do include is that we plot our resultsin terms of measurable energy, not neutrino energy, asthis gives better separation of signals and backgrounds. A. Muon tracks in IceCube
Muons are produced by the CC interactions of ν µ +¯ ν µ with nucleons [52–55]. The initial muon energy is E µ (cid:39) (1 − (cid:104) y (cid:105) ) E ν (cid:39) . E ν for E ν ∼ ∼
15 km and varies logarithmi-cally with energy. Those produced inside IceCube arecontained-vertex muons, whereas those produced outsideare through-going muons. For contained-vertex muons,the hadronic energy will be deposited in the detector,while it is lost for through-going muons.We present our results in terms of the energy of themuon as it first appears in the detector, due to beingcreated there or when it first enters. This is measurableand provides the most information about the neutrino spectrum [13, 60, 124]. The average muon energy lossrate is − dE µ /dx = α + βE µ [125, 126]. In the TeVrange and above, the radiative term ( βE µ ) dominatesthe ionization ( α ) term. We take α (cid:39) × − GeVcm g − (its low-energy value) and β (cid:39) × − cm g − (near 1 PeV). The muon energy can be measured bythe fluctuations in its radiative losses, and a precision ofa factor of 2 is expected [40]. The present EHE searchsimply measures the number of detected photoelectronsproduced by an event, which utilizes less information.The complete measurable muon spectrum is (cid:18) dNdE µ (cid:19) tracks = (cid:18) dNdE µ (cid:19) cont + (cid:18) dNdE µ (cid:19) thru , (3)where the same value of E µ comes from different rangesof neutrino energy in the two cases. For simplicity, weadd these event classes, though they should be separa-ble. In the following, through-going events are about 3times more numerous than contained-vertex events for an E − spectrum, and about 1.5 times more so for an E − spectrum. We consider only upgoing neutrino-inducedmuons, to avoid the large backgrounds from downgoingatmospheric muons. In principle, it should be possible toinclude some downgoing contained-vertex events [44].The muon spectrum from contained-vertex events [73,124] is similar to that for electron cascades and is (cid:18) dNdE µ (cid:19) cont (cid:39) π ρ N A V T (4) × (cid:90) − d (cos θ z ) d Φ dE ν ( E ν ) σ ( E ν ) e − τ ( E ν , cos θ z ) . Here we assume E µ (cid:39) E ν because the hadronic cascadewill contribute to the energy deposited.The muon spectrum from through-going events [73,124], taking into account the increase in the effective vol-ume of the detector due to the long muon range, is (cid:18) dNdE µ (cid:19) thru (cid:39) π ρ N A A T (5) × (cid:90) − d (cos θ z ) 1 ρ ( α + βE µ ) × (cid:90) E high E µ dE i d Φ dE i ( E i ) σ ( E i ) e − τ ( E i , cos θ z ) , where E i is the initial neutrino energy and E high its max-imum value, which depends on the distance to the hori-zon at that zenith angle; for upgoing events, E high iseffectively infinite. Instead of the detector volume, thedetector area A (cid:39) and a term reflecting the muonrange appear. We neglect the large fluctuations in themuon energy-loss rate [125, 126]. This and the preced-ing event rate equations also neglect the integration over dσ/dy , which can affect the results by a few tens of per-cent, which is within our uncertainties.0 E µ ( d N / d E µ ) [ c oun t s ] E µ [GeV] -1 E - E - . E - A t m . P r o m p t Ideal; Tracks up-goingAtm. Conv. E ca s c ( d N / d E ca s c ) [ c oun t s ] E casc [GeV] -1 E - E - . Atm. Conv. A t m . P r o m p t Ideal; Cascades 4 (cid:47) E - FIG. 5. Predictions for measurable spectra in two years of the full IceCube for various neutrino spectra considered above.(
Left Panel ) EdN/dE for neutrino-induced muons (upgoing only), where the muon energy is measured as it first appears inthe detector, whether as a contained-vertex or through-going event. (
Right Panel ) EdN/dE for neutrino-induced cascades(all directions), where the cascade energy is measured as deposited in the detector, whether as a CC or NC event. As above,the number of events in a region is proportional to the integrated area, i.e., to the height times the logarithmic energy range.
B. Predicted spectra below 1 PeV
Figure 5 shows our predicted track and cascade spec-tra for two years of the full IceCube; the numbers ofevents are given in Table II. It is likely that much ofthis exposure time can be obtained from existing datawith new analyses targeted to this energy range. All in-put neutrino fluxes are normalized as in previous figures.To avoid over-extrapolating the power-law astrophysicalfluxes and to focus on the energy range with the bestratio of signal to background, we show results only downto 0.1 PeV, though IceCube should go to lower energies.The left panel shows that analyses with muon tracksare limited by the large atmospheric conventional back-ground, so that the astrophysical signals will only emergeabove a few hundred TeV, especially once the smearingeffects of energy resolution are taken into account. Evenif just contained-vertex muons are selected, the back-ground due to atmospheric conventional ν µ + ¯ ν µ will bedominant until high energies, where the statistics are low.There is now some excess at the highest energies in theIceCube neutrino-induced muon data [90]. However, it isdifficult to judge the significance when the results havebeen processed by unfolding to estimate the spectrum interms of neutrino energy, which mixes different rangesof measurable muon energy and gives strongly correlateduncertainties. When spectra are shown in terms of muonenergy, there is better separation of signal and back-ground and then even a small number of signal events at high energy can be quite significant [124].The right panel shows that the prospects for cas-cades are extremely promising, because the atmosphericconventional background is strongly suppressed, as firstshown in Ref. [39]. The difference in cascade rates at 1PeV seen between the left panel of Fig. 3 and the rightpanel of Fig. 5 is due to the latter including ν τ + ¯ ν τ events (factor of 2), the slightly different exposure times,and the former including energy resolution smearing.Even if the efficiency is reduced from that shown inFig. 5, it should still be possible to detect potentiallylarge numbers of cascade events with minimal back-grounds. This could quickly discover an astrophysicalflux. The atmospheric conventional neutrinos and eventhe atmospheric prompt neutrinos are negligible back- TABLE II. Expected numbers of track and cascade events(ideal case or “theorist’s approach”), obtained by integratingthe curves in each panel of Fig. 5 over the range 0.1–1 PeV.
Possible Source N track N casc Atm. Conv. [45, 46] 11 1Atm. Prompt [47] 3 4Astrophysical E −
11 19Astrophysical E − .
10 20Astrophysical E − E casc (cid:39) E ν for the dominant CC events and good en-ergy resolution for cascades. The normalizations of thesespectra are the largest values that do not conflict with themeasured atmospheric neutrino data shown in Fig. 2. Ifthe normalizations were instead set by the requirement ofproducing the two PeV events, then the curves in Fig. 5would cross near 1 PeV and the differences between themwould be much larger below 1 PeV.Even though there are essentially no neutrino-inducedbackgrounds for cascade signals, there may be back-grounds induced by downgoing atmospheric muons [2].The cascade analysis that measured the conventional at-mospheric neutrino spectrum up to 10 TeV, as shown inFig. 2, used the small inner DeepCore detector as theactive volume and the rest of IceCube as a veto [50]. Itshould be possible to extend this idea as a function ofenergy, effecting a series of nested inner detectors andouter veto layers, with larger inner volumes than Deep-Core probing the smaller fluxes at higher energies. V. CONCLUSIONSA. Summary and Outlook
The observation of two cascade events near 1 PeV [1,2] is a remarkable achievement that follows more thantwo decades of pioneering work by the AMANDA andIceCube Collaborations [127–131]. It is very likely thatthese are neutrino-induced events, possibly the first high-energy astrophysical neutrinos ever observed, opening anew era. A high burden of proof will be needed to rejectall hypotheses based on a terrestrial origin and to acceptany based on an astrophysical origin.We provide a comprehensive general study of thesePeV events and their possible origin as a diffuse flux [37,38]. We apply physical insights to characterize the natureof the events and to define the framework for analyzingpossible source spectra. We systematically analyze sev-eral possible neutrino sources and backgrounds and drawconclusions about whether they can explain the observedevents in light of realistic detector modeling and otherconstraints. We show how IceCube can most quickly un-cover the nature of these events with searches at lowerenergies, for which we make detailed predictions.The search efficiency near the analysis threshold at 1PeV is ∼ E − and E − are plausible, with E − (with a cutoff atabout 2 PeV) being the most likely. There are tensionsregarding the normalization and slope of such models,but these are subject to the above uncertainties.The most important thing for IceCube to do is to im-prove the efficiency of searches at and below 1 PeV. Weshow in detail, including in Fig. 5, how such searchescan differentiate between possible scenarios for the ob-served PeV events. Even in the absence of one or bothof these events, there is tremendous discovery potentialfor cascade searches in this energy range. The detectionof cascade events has long been recognized as important,as a probe of ν e + ¯ ν e and because the good fidelity be-tween cascade and neutrino energy allows reconstructionof the neutrino spectrum [132]. As first shown by Bea-com and Candia [39] in 2004, there is a strong suppressionof the atmospheric conventional neutrino background forthe cascade channel relative to the muon track channel,giving improved sensitivity at lower energies.Our results on cascades go well beyond those inRef. [39] and will be generally useful for future searches.In addition to adopting updated fluxes, we provide a de-tailed discussion of the effects of many realistic IceCubedetector properties. We show how to best display andinterpret cascade spectra over a wide energy range, in-cluding near the Glashow resonance, where energy reso-lution effects must be included. We compare cascade andtrack spectra, with both presented in terms of detectableenergy instead of neutrino energy.Many of our considerations would easily carry over forpoint sources or collections thereof. For the same twoPeV events detected, which sets the total flux required,point sources would be easier to separate from the con-ventional atmospheric neutrino background because therelevant solid angle would be smaller than the full sky.Whatever the origin of these two events, their detec-tion is an important milestone in advancing our knowl-edge of the high-energy Universe, and we congratulatethe IceCube Collaboration on this success. Now that theconstruction of the IceCube detector is complete, neutri-nos will be detected at a faster rate, and great progressis expected soon, which we eagerly await.2 B. Impact of new results
As this paper was being completed (for early results,see Refs. [37, 38]), IceCube announced the detection ofnew events [133]. These preliminary data shed light onthe PeV events and seem to strengthen the case that theirorigin is astrophysical. There are no serious disagree-ments with our results and many of our assumptions arenow confirmed. Here we summarize their most importantnew results and our interpretation of them.The basic aspects of the data fit within the frameworkwe consider. The events are consistent with being uni-form in the volume, so are likely not due to backgroundsinduced by downgoing atmospheric muons. No remarksare made about the arrival times of the events, so pre-sumably they are consistent with being from a steadysource. The distribution of arrival directions is consistentwith isotropy subject to expected attenuation in Earth,so consistent with a diffuse source.New search criteria improved the efficiency at 1 PeVby a factor of 3 (i.e., part of the possible factor of 5noted above); the improvements at nearby energies varywith energy. No new events were found near 1 PeV, or athigher energies, which indicates that our choice of bins inthe PeV range was reasonable and that the former obser-vation of two events must have been a lucky fluctuation.This follows from Fig. 3 and the surrounding discussion,and acts to reduce the tensions shown in Fig. 4.The new criteria also provided some efficiency at en-ergies well below the previous threshold at 1 PeV. Thereare 19 new cascade events between 0.03 and 0.3 PeV.Only six of these are above 0.1 PeV, where the atmo-spheric neutrino backgrounds are minimal. The numberabove 0.1 PeV is reasonable for the E − spectrum above.The lack of events above 0.3 PeV supports the detectionof the two PeV events being a lucky fluctuation.The new events also include 7 contained-vertex muonevents, all between 0.03 and 0.3 PeV; all but one arebelow 0.1 PeV. The new search criteria still suppress ν µ + ¯ ν µ detection relative to ν e + ¯ ν e detection, a smallfraction of track to cascade events is expected. The leftpanel of our Fig. 5 shows that atmospheric conventionalneutrinos dominate in this energy range, including forcontained-vertex muon events. For equal flavor ratios,cascade events are much more likely to be signals thanare track events, so these events should not be mixed.It is stated that an E − spectrum is a reasonable fit,provided there is a spectrum cutoff in the PeV range,as we independently show. The right panel of our Fig. 5shows that the ratios of numbers of events near 0.1 PeV tothose near 1 PeV are quite distinct for different power-lawspectra, so this will be a powerful test of the spectrum.More information is needed on the consistency of an E − spectrum with lower-energy neutrino data of all flavors.The difficulties we point out in Fig. 4 would be somewhatalleviated if analyses of that data show some excessesnear a few hundred TeV, as is now reported. C. Astrophysical implications
Many models of astrophysical neutrino sources havebeen proposed. There are two key requirements for vi-able scenarios to explain the IceCube results. First,the cosmic-ray energy injection rate and meson pro-duction efficiency must be sufficient to give a neutrinoflux of at least E d Φ /dE ∼ − GeV cm − s − sr − near 1 PeV. Second, since protons with energy ε p ata typical redshift z ∼ E ν ∼ ε p /
100 PeV), sources should be able toaccelerate protons to energies close to the iron/secondknee [115]. In addition, a break at high energies seemsto be required, and the spectrum may even be peaked.Proton-photon ( pγ ) interactions are dominant for PeVneutrino production in most models of active galactic nu-clei (AGN) and gamma-ray bursts (GRBs) [134]. Protonstypically interact near threshold with photons of energy ε γ , so ε p ε γ ∼ .
16 GeV Γ , where Γ is the Lorentz factor.Then π − production is suppressed and fewer antineutri-nos are produced. In addition, flavor ratios are affectedby muon cooling in magnetized sources [117]. In the pγ case, the neutrino spectrum is hard at low energies andtypically has a peak depending on source properties.In AGN jet models [96–100], the neutrino spectrumpeaks at ∼ − ∼
10 and theobserved photon spectra of luminous blazars peak at ε γ ∼ . −
10 eV. The spectrum is expected to be rising atenergies above the PeV range, as for cosmogenic models,which are disfavored. In AGN core models [101, 102],where neutrinos are produced not far from accretiondisks, a peak in the PeV range is possible, though op-timistic cases have been ruled out.In GRB prompt emission models [103–105], PeV neu-trinos are expected because ε γ ∼ ∼ −
100 PeV [135, 136]. Althoughthis spectrum shape may be appealing, stacking searchesby IceCube set limits of E d Φ /dE (cid:46) . × − GeVcm − s − sr − [14, 137–139], well below the requiredflux. However, many transients like low-luminosity GRBsare missed; some predictions are ∼
10 times largerthan this limit and have a peak or break in the PeVrange [13, 140, 141]. Although neutrinos can be pro-duced in GRB afterglows [106–108], their typical energyis much higher than 1 PeV, as in the AGN jet model, soexplaining the IceCube PeV events is difficult.Proton-proton ( pp ) interactions are dominant for PeVneutrino production in starburst galaxies and large-scalestructures. Many pions of all types are produced in eachscattering, and the neutrino spectrum basically followsthe proton spectrum [142], with equal ratios of neutrinosand antineutrinos and of flavors after mixing.Starburst galaxies contain many massive stars, whichlead to supernovae that may produce cosmic rays. Mostof the cosmic ray power would be lost to neutrinos andgamma rays due to interactions in the high-column-density material, and detections of gamma rays from3nearby galaxies [143] supports this idea. The predictedflux is E d Φ /dE ∼ (0 . − × − GeV cm − s − sr − ,with a possible cutoff [112, 113], though it is uncertain if ∼
100 PeV protons (rather than heavy nuclei) are pro-duced in these galaxies.Large-scale structures (especially galaxy clusters) aregigantic reservoirs of cosmic rays that may be acceleratedat structure formation shocks and supplied by containedAGN [114]. PeV neutrinos are produced via pp interac-tions with the intracluster medium. The expected flux is E d Φ /dE ∼ (0 . − × − GeV cm − s − sr − , and abreak due to the diffusive escape or maximum energy ofcosmic rays has been predicted [115].The possible connection with extragalactic cosmic raysis intriguing, because a neutrino flux of E d Φ /dE ∼ − GeV cm − s − sr − is comparable to the Waxman-Bahcall bound [119] derived from the ultra-high-energycosmic ray flux. However, PeV neutrinos correspond toprotons at lower energies, near 100 PeV, and higher-energy neutrinos have not been detected, despite the in-creasing effective area. If ultra-high-energy cosmic raysare heavy nuclei, as suggested by Auger, then the neu-trino flux from their sources is much lower than the Waxman-Bahcall bound [72].To conclude our discussion of astrophysical neutrinofluxes, there is so far no obvious source that explains allaspects of the IceCube data. Many models (e.g., GRBprompt, starburst galaxies, and large-scale structures)seem compatible with the data, though some models(e.g., AGN jets and GRB afterglow) are already disfa-vored. Interestingly, the neutrino flux sensitivity is ap-proaching that needed to probe the sources of the ultra-high-energy cosmic rays. More experimental data andtheoretical studies are needed to unravel the mysteries ofthe high- and ultra-high-energy universe. Acknowledgments:
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