Probing particle physics with IceCube
Abstract
The IceCube observatory located at the South Pole is a cubickilometre optical Cherenkov telescope primarily designed for the detection of highenergy astrophysical neutrinos. IceCube became fully operational in 2010, after a sevenyear construction phase, and reached a milestone in 2013 by the first observation of cosmic neutrinos in the TeV–PeV energy range. This observation does not only mark an important breakthrough in neutrino astronomy, but it also provides a new probe of particle physics related to neutrino production, mixing, and interaction. In this review we give an overview of the various possibilities how IceCube can address fundamental questions related to the phenomena of neutrino oscillations and interactions, the origin of dark matter, and the existence of exotic relic particles, like monopoles. We will summarize recent results and highlight future avenues.
1 Introduction
Not long after the discovery of the neutrino by Cowan and Reines [1], the idea emerged that it represented the ideal astronomical messenger [2]. Neutrinos are only weakly interacting with matter and can cross cosmic distances without being absorbed or scattered. However, this weak interaction is also a challenge for the observation of these particles. Early estimates of the expected flux of highenergy neutrinos associated with the observed flux of extragalactic cosmic rays indicated that neutrino observatories require gigaton masses as a necessary condition to observe a few neutrino interactions per year [3]. These requirements can only be met by special experimental setups that utilise natural resources. Not only that – the detector material has to be suitable so that these few interactions can be made visible and separated from large atmospheric backgrounds.
Despite these obstacles, there exist a variety of experimental concepts to detect highenergy neutrinos. One particularly effective method is based on detecting the radiation of optical Cherenkov light produced by relativistic charged particles. This requires the use of optically transparent detector media like water or ice, where the Cherenkov emission can be read out by optical sensors deployed in the medium. This information then allows to reconstruct the various Cherenkov light patterns produced in neutrino events and infer the neutrino flavour, arrival direction, and energy. The most valuable type of events for neutrino astronomy are charged current interactions of muonneutrinos with matter near the detector. These events produce muons that can range into the detector and allow the determination of the initial muonneutrino direction within a precision of better than one degree.
Presently the largest optical Cherenkov telescope is the IceCube Observatory, which uses the deep glacial ice at the geographic South Pole as its detector medium. The principal challenge of any neutrino telescope is the large background of atmospheric muons and neutrinos produced in cosmic ray interactions in the atmosphere. Highenergy muons produced in the atmosphere have a limited range in ice and bedrock. Nevertheless IceCube, at a depth of 1.5 kilometres, observes about 100 billion atmospheric muon events per year. This large background can be drastically reduced by only looking for upgoing events, i.e., events that originate below the horizon. This cut leaves only muons produced by atmospheric neutrinos at a rate of about 100,000 per year. While these large backgrounds are an obstacle for neutrino astronomy they provide a valuable probe for cosmic ray physics in general and for neutrino oscillation and interaction studies in particular.
In this review we want to highlight IceCube’s potential as a facility to probe fundamental physics. There exist a variety of methods to test properties of the Standard Model (SM) and its possible extensions. The flux of atmospheric and astrophysical neutrinos observed in IceCube allows to probe fundamental properties in the neutrino sector related to the standard neutrino oscillations (neutrino mass differences, mass ordering, and flavour mixing) and neutrinomatter interactions. It also provides a probe for exotic oscillation effects, e.g., related to the presence of sterile neutrinos or nonstandard neutrino interactions with matter. The ultralong baselines associated with the propagation of cosmic neutrinos observed beyond 10 TeV allow for various tests of feeble neutrino oscillation effects that can leave imprints on the oscillationaveraged flavour composition.
One of the fundamental questions in cosmology is the origin of dark matter that today constitutes one quarter of the total energy density of the Universe. Candidate particles for this form of matter include weakly interacting massive particles (WIMPs) that could have been thermally produced in the early Universe. IceCube can probe the existence of these particles by the observation of a flux of neutrinos produced in the annihilation or decay of WIMPs gravitationally clustered in nearby galaxies, the halo of the Milky Way, the Sun, or the Earth. In the case of compact objects, like Sun and Earth, neutrinos are the only SM particles that can escape the dense environments to probe the existence of WIMPs.
The outline of this review is as follows. We will start in Sects. 2 and 3 with a description of the IceCube detector, atmospheric backgrounds, standard event reconstructions, and event selections. In Sect. 4 we summarise the phenomenology of threeflavour neutrino oscillation and IceCube’s contribution to test the atmospheric neutrino mixing. We will cover standard model neutrino interactions in Sect. 5 and highlight recent measurements of the inelastic neutrinonucleon cross sections with IceCube. We then move on to discuss IceCube’s potential to probe nonstandard neutrino oscillation with atmospheric and astrophysical neutrino fluxes in Sect. 6. In Sect. 7 we highlight IceCube results on searches for dark matter and Sect. 8 is devoted to magnetic monopoles while Sect. 9 covers other massive exotic particles and Big Bang relics.
Any review has its limitations, both in scope and timing. We have given priority to present a comprehensive view of the activity of IceCube in areas related to the topic of this review, rather than concentrating on a few recent results. We have also chosen at times to include older results for completeness, or when it was justified as an illustration of the capabilities of the detector on a given topic. The writing of any review develops along its own plot and updated results on some analyses have been made public while this paper was in preparation, and could not be included here. This only reflects on the lively activity of the field.
Throughout this review we will use natural units, \(\hbar =c=1\), unless otherwise stated. Electromagnetic expressions will be given in the Heaviside0Lorentz system with \(\epsilon _0=\mu _0=1\), \(\alpha = e^2/4\pi \simeq 1/137\) and \(1\mathrm{Tesla} \simeq 195\mathrm{eV}^2\).
2 The IceCube Neutrino Observatory
Eight strings are placed in the centre of the array and are instrumented with a denser DOM spacing and typical interstring separation of 55 m (red markers in right panel of Fig. 1). They are equipped with photomultiplier tubes with higher quantum efficiency. These strings, along with the first layer of the surrounding standard strings, form the DeepCore lowenergy subarray [6]. Its footprint is depicted by a blue dashed line in Fig. 1. While the original IceCube array has a neutrino energy threshold of about 100 GeV, the addition of the denser infill lowers the energy threshold to about 10 GeV. The DOMs are operated to trigger on single photoelectrons and to digitise insitu the arrival time of charge (“waveforms”) detected in the photomultiplier. The dark noise rate of the DOMs is about 500 Hz for standard modules and 800 Hz for the highquantumefficiency DOMs in the DeepCore subarray.
Some results highlighted in this review were derived from data collected with the AMANDA array [7], the predecessor of IceCube built between 1995 and 2001 at the same site, and in operation until May 2009. AMANDA was not only a proof of concept and a hardware testbed for the IceCube technology, but a full fledged detector which obtained prime results in the field.
2.1 Neutrino event signatures
As we already highlighted in the introduction, the main event type utilised in highenergy neutrino astronomy are charged current (CC) interactions of muon neutrinos with nucleons (N), \(\nu _\mu + N \rightarrow \mu ^ + X\). These interactions produce highenergy muons that lose energy by ionisation, bremsstrahlung, pair production and photonuclear interactions in the ice [8]. The combined Cherenkov light from the primary muon and secondary relativistic charged particles leaves a tracklike pattern as the muon passes through the detector. An example is shown in the left panel of Fig. 2. In this figure, the arrival time of Cherenkov light in individual DOMs is indicated by colour (earlier in red and later in blue) and the size of each DOM is proportional to the total Cherenkov light it detected.^{1} Since the average scattering angle between the incoming neutrino and the outgoing muon decreases with energy, \(\Psi _{\nu \rightarrow \mu }\sim 0.7^{{\circ }}(E_{\nu }/\mathrm{TeV})^{0.7}\) [9], an angular resolution below \(1^{{\circ }}\) can be achieved for neutrinos with energies above a few TeV, only limited by the detector’s intrinsic angular resolution. This changes at low energies, where muon tracks are short and their angular resolution deteriorates rapidly. For neutrino energies of a few tens of GeVs the angular resolution reaches a median of \(\sim 40^{{\circ }}\).
All deepinelastic interactions of neutrinos, both neutral current (NC), \(\nu _{\alpha } + N \rightarrow \nu _{\alpha } + X\) and charged current, \(\nu _{\alpha } + N \rightarrow \ell ^_{\alpha } + X\), create hadronic cascades X that are visible by the Cherenkov emission of secondary charged particles. However, these secondaries can not produce elongated tracks in the detector due to their rapid scattering or decay in the medium. Because of the large separation of the strings in IceCube and the scattering of light in the ice, the Cherenkov light distribution from particle cascades in the detector is rather spherical, see right panel of Fig. 2. For cascades or tracks fully contained in the detector, the energy resolution is significantly better since the full energy is deposited in the detector and it is proportional to the detected light. The ability to distinguish these two light patterns in any energy range is crucial, since cascades or tracks can contribute to background or signal depending on the analysis performed.
The electrons produced in charged current interactions of electron neutrinos, \(\nu _{e} + N \rightarrow e^ + X\), will contribute to an electromagnetic cascade that overlaps with the hadronic cascade X at the vertex. At energies of \(E_\nu \simeq 6.3\) PeV, electron antineutrinos can interact resonantly with electrons in the ice via a Wresonance (“Glashow” resonance) [10]. The Wboson decays either into hadronic states with a branching ratio (BR) of \(\simeq 67\)%, or into leptonic states (\(\mathrm{BR}\simeq 11\)% for each flavour). This type of event can be visible by the appearance of isolated muon tracks starting in the detector or by spectral features in the event distribution [11].
Also the case of charged current interactions of tau neutrinos, \(\nu _{\tau } + N \rightarrow \tau + X\), is special. Again, the hadronic cascade X is visible in Cherenkov light. The tau has a lifetime (at rest) of 0.29 ps and decays to leptons as \(\tau ^\rightarrow \mu ^+\overline{\nu }_\mu +\nu _\tau \) (BR \(\simeq 18\%\)) and \(\tau ^\rightarrow e^+\overline{\nu }_e+\nu _\tau \) (BR \(\simeq 18\%\)) or to hadrons (mainly pions and kaons, BR \(\simeq 64\%\)) as \(\tau ^\rightarrow \nu _\tau +\mathrm{mesons}\). With tau energies below 100 TeV these charged current events will also contribute to track and cascade events. However, the delayed decay of taus at higher energies can become visible in IceCube, in particular above around a PeV when the decay length becomes of the order of 50 m. This allows for a variety of characteristic event signatures, depending on the tau energy and decay channel [12, 13].
3 Event selection and reconstruction
In this review we present results from analyses which use different techniques tailored to the characteristics of the signals searched for. It is therefore impossible to give a description of a generic analysis strategy which would cover all aspects of every approach. There are, however, certain levels of data treatment and analysis techniques that are common for all analyses in IceCube, and which we cover in this section.
3.1 Event selection
Several triggers are active in IceCube in order to preselect potentially interesting physics events [14]. They are based on finding causally connected spatial hit distributions in the array, typically requiring a few neighbour or nexttoneighbour DOMs to fire within a predefined time window. Most of the triggers aim at finding relativistic particles crossing the detector and use time windows of the order of a few microseconds. In order to extend the reach of the detector to exotic particles, e.g., magnetic monopoles catalysing nucleondecay, which can induce events lasting up to milliseconds, a dedicated trigger sensitive to nonrelativistic particles with velocities down to \(\beta ^{4}\) has also been implemented.
When a trigger condition is fulfilled the full detector is read out. IceCube triggers at a rate of 2.5 kHz, collecting about 1 TB/day of raw data. To reduce this amount of data to a more manageable level, a series of software filters are applied to the triggered events: fast reconstructions [15] are performed on the data and a first event selection carried out, reducing the data stream to about 100 GB/day. These reconstructions are based on the position and time of the hits in the detector, but do not include information about the optical properties of the ice, in order to speed up the computation. The filtered data is transmitted via satellite to several IceCube institutions in the North for further processing.
Offline processing aims at selecting events according to type (tracks or cascades), energy, or specific arrival directions using sophisticated likelihoodbased reconstructions [16, 17]. These reconstructions maximise the likelihood function built from the probability of obtaining the actual temporal and spatial information in each DOM (“hit”) given a set of track parameters (vertex, time, energy, and direction). For lowenergy events, where the event signature is contained within the volume of the detector, a joint fit of muon track and an hadronic cascade at the interaction vertex is performed. For those events the total energy can be reconstructed with rather good accuracy, depending on further details of the analysis. Typically, more than one reconstruction is performed for each event. This allows, for example, to estimate the probability of each event to be either a track or a cascade. Each analysis will then use complex classification methods based on machinelearning techniques to further separate a possible signal from the background. Variables that describe the quality of the reconstructions, the time development and the spatial distribution of hit DOMs in the detector are usually used in the event selection.
3.2 Effective area and volume
3.3 Background rejection
There are two backgrounds in any analysis with a neutrino telescope: atmospheric muons and atmospheric neutrinos, both produced in cosmicray interactions in the atmosphere. The atmospheric muon background measured by IceCube [18] is much more copious than the atmospheric neutrino flux, by a factor up to \(10^6\) depending on declination. Note that cosmic ray interactions can produce several coincident forward muons (“muon bundle”) which are part of the atmospheric muon background. Muon bundles can be easily identified as background in some cases, but they can also mimic bright single tracks (like magnetic monopoles for example) and are more difficult to separate from the signal in that case. Even if many of the IceCube analyses measure the atmospheric muon background from the data, the CORSIKA package [19] is generally used to generate samples of atmospheric muons that are used to crossvalidate certain steps of the analyses.
The large background of atmospheric muons can be efficiently reduced by using the Earth as a filter, i.e., by selecting upgoing track events, at the expense of reducing the sky coverage of the detector to the Northern Hemisphere (see Fig. 3). Still, due to light scattering in the ice and the emission angle of the Cherenkov cone, a fraction of the downgoing atmospheric muon tracks can be misreconstructed as upgoing through the detector. This typically leads to a mismatch between the predicted atmospheric neutrino rate and the data rate at the final level of many analyses. There are analyses where a certain atmospheric muon contamination can be tolerated and it does not affect the final result. These are searches that look for a difference in the shape of the energy and/or angular spectra of the signal with respect to the background, and are less sensitive to the absolute normalisation of the latter. For others, like searches for magnetic monopoles, misreconstructed atmospheric muons can reduce the sensitivity of the detector. We will describe in more detail how each analysis deals with this background when we touch upon specific analyses in the rest of this review.
The neutrino flux arising from pion and kaon decay is reasonably well understood, with an uncertainty in the range 10–20% [20]. Figure 4 shows the atmospheric neutrino fluxes measured by IceCube. The atmospheric muon neutrino spectrum (\(\nu _\mu +\overline{\nu }_\mu \)) was obtained from one year of IceCube data (April 2008–May 2009) using upgoing muon tracks [23]. The atmospheric electron neutrino spectra (\(\nu _e+\overline{\nu }_e\)) were analysed by looking for contained cascades observed with the lowenergy infill array DeepCore between June 2010 and May 2011 in the energy range from 80 GeV to 6 TeV [21]. This agrees well with a more recent analysis using contained events observed in the full IceCube detector between May 2011 and May 2012 with an extended energy range from 100 GeV to 100 TeV [22]. All measurements agree well with model prediction of “conventional” atmospheric neutrinos produced in pion and kaon decay. IceCube uses the public Monte Carlo software GENIE [27] and the internal software NUGEN (based on [28]) to generate samples of atmospheric neutrinos for its analyses, following the flux described in [29].
Kaons with an energy above 1 TeV are also significantly attenuated before decaying and the “prompt” component, arising mainly from very shortlived charmed mesons (\(D^\pm ,~D^0,~D_s\) and \(\Lambda _c\)) is expected to dominate the spectrum. The prompt atmospheric neutrino flux, however, is much less understood, because of the uncertainty on the cosmic ray composition and relatively poor knowledge of QCD processes at small Bjorkenx [30, 31, 32, 33, 34]. In IceCube analyses the normalisation of the prompt atmospheric neutrino spectrum is usually treated as a nuisance parameter, while the energy distributions follows the model prediction of Ref. [30].
For high enough neutrino energies (\({\mathcal {O}}\)(10) TeV), the possibility exists of rejecting atmospheric neutrinos by selecting starting events, where an outer layer of DOMs acts as a virtual veto region for the neutrino interaction vertex. This technique relies on the fact that atmospheric neutrinos are accompanied by muons produced in the same air shower, that would trigger the veto [35, 36]. The price to pay is a reduced effective volume of the detector for downgoing events and a different sensitivity for upgoing and downgoing events. This approach has been extremely successful, extending the sensitivity of IceCube to the Southern Hemisphere including the Galactic centre. There is not a generic veto region defined for all IceCube analyses, but each analysis finds its optimal definition depending on its physics goal. Events that present more than a predefined number of hits within some time window in the strings included in the definition of the veto volume are rejected. A reduction of the atmospheric muon background by more than 99%, depending on analysis, can be achieved in this way (see for example [26, 36]).
This approach has been also the driver behind one of the most exciting recent results in multimessenger astronomy: the first observation of highenergy astrophysical neutrinos by IceCube. The first evidence of this flux could be identified from a highenergy starting event (HESE) analysis, with only two years of collected data in 2013 [24, 25, 37]. The event sample is dominated by cascade events, with only a rather poor angular resolution of about \(10^{{\circ }}\). The result is consistent with an excess of events above the atmospheric neutrino background observed in upgoing muon tracks from the Northern Hemisphere [26, 38]. Figure 4 summarises the neutrino spectra inferred from these analyses. Based on different methods for reconstruction and energy measurement, their results agree, pointing at extragalactic sources whose flux has equilibrated in the three flavours after propagation over cosmic distances [39] with \(\nu _e:\nu _\mu :\nu _\tau \sim 1:1:1\). While both types of analyses have now reached a significance of more than \(5\sigma \) for an astrophysical neutrino flux, the origin of this neutrino emission remains a mystery (see, e.g., Ref. [40]).
4 Standard neutrino oscillations
The previous mixing and oscillation parameters are derived under the assumption of a constant electron density \(N_e\). If the electron density along the neutrino trajectory is only changing slowly compared to the effective oscillation frequency, the effective mass eigenstates will change adiabatically. Note that the oscillation frequency and oscillation depth in matter exhibits a resonant behaviour [55, 56, 57]. This Mikheyev–Smirnov–Wolfenstein (MSW) resonance can have an effect on continuous neutrino spectra, but also on monochromatic neutrinos passing through matter with slowly changing electron densities, like the radial density gradient of the Sun. Once these matter effect is taken into account, the observed intensity of solar electron neutrinos at different energies compared to theoretical predictions can be used to extract the solar neutrino mixing parameters. In addition to solar neutrino experiments, the KamLAND Collaboration [58] has measured the flux of \(\overline{\nu }_e\) from distant reactors and find that \(\overline{\nu }_e\)’s disappear over distances of about 180 km. This observation allows a precise determination of the solar mass splitting \(\Delta m^2_\odot \) consistent with solar data.
Results of a global analysis [62] of mass splittings, mixing angles, and Dirac phase for normal and inverted mass ordering. We bestfit parameters are shown with \(1\sigma \) uncertainty
Normal ordering  Inverted ordering  

\(\Delta m_{21}^2\) (\(\mathrm{eV}^2\))  \(7.40^{+0.21}_{0.20}\times 10^{5}\)  \(7.40^{+0.21}_{0.20}\times 10^{5}\) 
\(\Delta m_{31}^2\) (\(\mathrm{eV}^2\))  \( 2.494^{+0.033}_{0.031}\times 10^{3}\)  – 
\(\Delta m_{23}^2\) (\(\mathrm{eV}^2\))  –  \(2.465^{+0.032}_{0.031}\times 10^{3}\) 
\(\theta _{12}\) (\({}^{{\circ }}\))  \(33.62^{+0.78}_{0.76}\)  \(33.62^{+0.78}_{0.76}\) 
\(\theta _{23}\) (\({}^{{\circ }}\))  \(47.2^{+1.9}_{3.9}\)  \(48.1^{+1.4}_{1.9}\) 
\(\theta _{13}\) (\({}^{{\circ }}\))  \(8.54^{+0.15}_{0.15}\)  \(8.58^{+0.14}_{0.14}\) 
\(\delta _{\mathrm{CP}}\) (\({}^{{\circ }}\))  \(234^{+43}_{31}\)  \(278^{+26}_{29}\) 
The global fit to neutrino oscillation data is presently incapable to determine the ordering of neutrino mass states. The fit to the data can be carried out under the assumption of normal (\(m_1<m_2<m_3\)) or inverted (\(m_3<m_1<m_2\)) mass ordering. A recent combined analysis [62] of solar, atmospheric, reactor, and accelerator neutrino data gives the values for the mass splittings, mixing angles, and CPviolating Dirac phase for normal or inverted mass ordering shown in Table 1. Note that, presently, the Dirac phase is inconsistent with \(\delta =0\) at the \(3\sigma \) level, independent of mass ordering.
Neutrino oscillation measurements are only sensitive to the relative neutrino mass differences. The absolute neutrino mass scale can be measured by studying the electron spectrum of tritium (\({}^3\hbox {H}\)) \(\beta \)decay. Present upper limits (95% CL) on the (effective) electron antineutrino mass are at the level of \(m_{\overline{\nu }_e}<2\) eV [63, 64]. The KATRIN experiment [65] is expected to reach a sensitivity of \(m_{\overline{\nu }_e}<0.2\) eV. Neutrino masses are also constrained by their effect on the expansion history of the Universe and the formation of largescale structure. Assuming standard cosmology dominated at late times by dark matter and dark energy, the upper limit (95% CL) on the combined neutrino masses is \(\sum _im_i<0.23\) eV [66].
The mechanism that provides neutrinos with their small masses is unknown. The existence of righthanded neutrino fields, \(\nu _{\mathrm{R}}\), would allow to introduce a Dirac mass term of the form \(m_{\mathrm{D}}\overline{\nu }_L\nu _R + h.c.\), after electroweak symmetry breaking. Such states would be “neutral” with respect to the standard model gauge interactions, and therefore sterile [44]. However, the smallness of the neutrino masses would require unnaturally small Yukawa couplings. This can be remedied in seesaw models (see, e.g., Ref. [67]). Being electrically neutral, neutrinos can be Majorana spinors, i.e., spinors that are identical to their chargeconjugate state, \(\psi ^{\mathrm{c}} \equiv {\mathcal {C}}\overline{\psi }^T\), where \({\mathcal {C}}\) is the chargeconjugation matrix. In this case, we can introduce Majorana mass terms of the form \(m_L\overline{\nu _{L}}{\nu }^{\mathrm{c}}_{L}/2 + h.c.\) and the analogous term for \(\nu _R\). In seesaw models the individual size of the mass terms are such that \(m_L\simeq 0\) and \(m_D \ll m_R\). After diagonalization of the neutrino mass matrix, the masses of active neutrinos are then proportional to \(m_i\simeq m_D^2/m_R\). This would explain the smallness of the effective neutrino masses via a heavy sector of particles beyond the Standard Model.
4.1 Atmospheric neutrino oscillations with IceCube
The atmospheric neutrino “beam” that reaches IceCube allows to perform highstatistics studies of neutrino oscillations at higher energies, and therefore is subject to different systematic uncertainties, than those typically available in reactor or acceleratorbased experiments. Atmospheric neutrinos arrive at the detector from all directions, i.e., from travelling more than 12,700 km (vertically upgoing) to about 10 km (vertically downgoing), see Fig. 3. The path length from the production point in the atmosphere to the detector is therefore related to the measured zenith angle \(\theta _{\mathrm{zen}}\). Combined with a measurement of the neutrino energy, this opens the possibility of measuring \(\nu _\mu \) disappearance due to oscillations, exploiting the dependence of the disappearance probability with energy and arrival angle.
Given this relatively narrow energy response of DeepCore compared with the wide range of path lengths, it is possible to perform a search for \(\nu _{\mu }\) disappearance through a measurement of the rate of contained events as a function of arrival direction, even without a precise energy determination. This is the approach taken in Ref. [73]. Events starting in DeepCore were selected by using the rest of the IceCube strings as a veto. A “highenergy” sample of events not contained in DeepCore was used as a reference, since \(\nu _{\mu }\) disappearance due to oscillations at higher energies (\({\mathcal {O}}\)(100) GeV) is not expected. The atmospheric muon background is reduced to a negligible level by removing tracks that enter the DeepCore fiducial volume from outside, and by only considering upgoing events, i.e., events that have crossed the Earth (\(\cos \theta _{\mathrm{zen}} \le 0\)), although a contamination of about 10–15% of \(\nu _e\) events misidentified as tracks remained, as well as \(\nu _{\tau }\) from \(\nu _{\mu }\) oscillations. These two effects were included as background.
The next step in complexity in an oscillation analysis with IceCube is to add the measurement of the neutrino energy, so the quantities L and \(E_\nu \) in Eq. (7) can be calculated separately. This is the approach followed in Ref. [79], where the energy of the neutrinos is obtained by using contained events in DeepCore and the assumption that the resulting muon is minimum ionising. Once the vertex of the neutrino interaction and the muon decay point have been identified, the energy of the muon can be calculated assuming constant energy loss, and it is proportional to the track length. The energy of the hadronic particle cascade at the vertex is obtained by maximising a likelihood function that takes into account the light distribution in adjacent DOMs. The neutrino energy is then the sum of the muon and cascade energies, \(E_{\nu }=E_{\mathrm{cascade}} + E_{\mu }\). The most recent oscillation analysis from IceCube [78] improves on the mentioned techniques in several fronts. It is an allsky analysis and also incorporates some degree of particle identification by reconstructing the events under two hypotheses: a \(\nu _{\mu }\) chargedcurrent interaction which includes a muon track, and a particleshower only hypothesis at the interaction vertex. This latter hypothesis includes \(\nu _{\mathrm{e}}\) and \(\nu _{\tau }\) chargedcurrent interactions, although these two flavours can not be separately identified. The analysis achieves an energy resolution of about 25% (30%) at \(\sim 20~\hbox {GeV}\) for muonlike (cascadelike) events and a median angular resolution of \(10^{{\circ }}\) (\(16^{\circ }\)). Full sensitivity to lower neutrino energies, for example to reach the next oscillation minimum at \(\sim 6~\hbox {GeV}\), can only be achieved with a denser array, like the proposed PINGU lowenergy extension [80].
In order to determine the oscillation parameters, the data is binned into a twodimensional histogram where each bin contains the measured number of events in the corresponding range of reconstructed energy and arrival direction. The expected number of events per bin depend on the mixing angle, \(\theta _{23}\), and the mass splitting, \(\Delta m^2_{32}\), as shown in Fig. 5. This allows to determine the mixing angle \(\theta _{23}\) and the mass splitting \(\Delta m^2_{32}\) as the maximum of the binned likelihood. The fit also includes the likelihood of the track and cascade hypotheses. Systematic uncertainties and the effect of the Earth density profile are included as nuisance parameters. In this analysis, a full threeflavour oscillation scheme is used and the rest of the oscillation parameters are kept fixed to \(\Delta m^2_{21}=7.53\times 10^{5}\hbox {eV}^2\), \(\sin ^2\theta _{12}=3.04\times 10^{1}\), \(\sin ^2\theta _{13}=2.17\times 10^{2}\) and \(\delta _{\mathrm{CP}}=0\). The effect of \(\nu _{\mu }\) disappearance due to oscillations is clearly visible in the left panel of Fig. 7, which shows the number of events as a function of the reconstructed \(L/E_\nu \), compared with the expected event distribution, shown as a dotted magenta histogram, if oscillations were not present. The results of the best fit to the data are shown in the right panel of Fig. 7. The bestfit values are \(\Delta m^2_{32}= 2.31^{+0.11}_{0.13} \times 10^{3}~\hbox {eV}^2\) and \(\sin ^2 2\theta _{23}=0.51^{+0.07}_{0.09}\), assuming a normal mass ordering.
The results of the two analyses mentioned above are compatible within statistics but, more importantly, they agree and are compatible in precision with those from dedicated oscillation experiments.
4.2 Flavour of astrophysical neutrinos
The neutrino oscillation phase in Eq. (7) depends on the ratio \(L/E_\nu \) of distance travelled, L, and neutrino energy, \(E_\nu \). For astrophysical neutrinos we have to consider ultralong oscillation baselines L corresponding to many oscillation periods between source and observer. The initial mixed state of neutrino flavours has to be averaged over \(\Delta L\), corresponding to the size of individual neutrino emission zones or the distribution of sources for diffuse emission. In addition, the observation of neutrinos can only decipher energies within an experimental energy resolution \(\Delta E_\nu \). The oscillation phase in (7) has therefore an absolute uncertainty that is typically much larger than \(\pi \) for astrophysical neutrinos. As a consequence, only the oscillationaveraged flavour ratios can be observed.
Figure 8 shows a visualisation of the observable neutrino flavour. Each location in the triangle corresponds to a unique flavour composition indicated by the three axis. The coloured markers correspond to the oscillationaveraged flavour ratios from the three scenarios (\(x_e=1/3\), \(x_e=0\), and \(x_e=1\)) discussed earlier, where the bestfit oscillation parameters have been used (instead of “tribimaximal” mixing). The blueshaded regions show the relative flavour loglikelihood ratio of a global analysis of IceCube data [81]. The bestfit is indicated as a white cross. IceCube’s observations are consistent with the assumption of standard neutrino oscillations and the production of neutrino in pion decay (full or “muondamped”). Neutrino production by radioactive decay is disfavoured at the \(2\sigma \) level.
5 Standard model interactions
The measurement of neutrino fluxes requires a precise knowledge of the neutrino interaction probability or, equivalently, the cross section with matter. At neutrino energies of less than a few GeV the cross section is dominated by elastic scattering, e.g., \(\nu _x+p\rightarrow \nu _x+p\), and quasielastic scattering, e.g., \(\overline{\nu }_e+p\rightarrow e^++n\). In the energy range of 1–10 GeV, the neutrinonucleon cross section is dominated by processes involving resonances, e.g. \(\nu _e+p\rightarrow e^+\Delta ^{++}\). At even higher energies neutrino scattering with matter proceeds predominantly via deep inelastic scattering (DIS) off nucleons, e.g., \(\nu _\mu +p\rightarrow \mu ^+X\), where X indicates a secondary particle shower. The neutrino cross sections have been measured up to neutrino energies of a few hundreds of GeV. However, the neutrino energies involved in scattering of atmospheric and astrophysical neutrinos off nucleons far exceed this energy scale and we have to rely on theoretical predictions.
We will discuss in the following the expected cross section of highenergy neutrinomatter interactions. In weak interactions with matter the lefthanded neutrino couples via \(Z^0\) and \(W^\pm \) exchange with the constituents of a proton or neutron. Due to the scaledependence of the strong coupling constant, the calculation of this process involves both perturbative and nonperturbative aspects due to hard and soft processes, respectively.
5.1 Deep inelastic scattering
Due to the strength of the QCD coupling at small scales the neutrinonucleon interactions cannot be described in a purely perturbative way. However, since the QCD interaction decreases as the renormalisation scale increases (asymptotic freedom) the constituents of a nucleon may be treated as loosely bound objects within sufficiently small distance and time scales (\(\Lambda _{\mathrm{QCD}}^{1}\)). Hence, in a hard scattering process of a neutrino involving a large momentum transfer to a nucleon the interactions between quarks and gluons may factorise from the subprocess (see Fig. 9). Due to the renormalisation scale dependence of the couplings this factorisation will also depend on the absolute momentum transfer \(Q^2\equiv  \ q^2\).
Figure 9 shows a sketch of a general lepton–nucleon scattering process. A nucleon N with mass M scatters off the lepton \(\ell \) by a tchannel exchange of a boson. The final state consist of a lepton \(\ell '\) and a hadronic state H with centre of mass energy \((P+q)^2=W^2\). This scattering process probes the partons, the constituents of the nucleon with a characteristic size \(M^{1}\) at length scales of the order of \(Q^{1}\). Typically, this probe will be “deep” and “inelastic”, corresponding to \(Q\gg M\) and \(W\gg M\), respectively. The subprocess between lepton and parton takes place on time scales which are short compared to those of QCD interactions and can be factorised from the soft QCD interactions. The intermediate coloured states, corresponding to the scattered parton and the remaining constituents of the nucleus, will then softly interact and hadronise into the final state H.
The kinematics of a lepton–nucleon scattering is conveniently described by the Lorentz scalars \(x=Q^2/(2q\cdot P)\), also called Bjorkenx, and inelasticity \(y=(q\cdot P)/(k\cdot P)\) (see Fig. 9 for definitions). In the kinematic region of deep inelastic scattering (DIS) where \(Q\gg M\) and \(W\gg M\) we also have \(Q^2\simeq 2q\cdot p\) and thus \(x\simeq (q\cdot p)/(q\cdot P)\). The scalars x and y have simple interpretations in particular reference frames. In a reference frame where the nucleon is strongly boosted along the neutrino 3momentum \(\mathbf {k}\) the relative transverse momenta of the partons is negligible. The parton momentum p in the boosted frame is approximately aligned with P and the scalar x expresses the momentum fraction carried by the parton. In the rest frame of the nucleus the quantity y is the fractional energy loss of the lepton, \(y=(EE')/E\), where E and \(E'\) are the lepton’s energy before and after scattering, respectively.
5.2 Charged and neutral current interactions
5.3 Highenergy neutrinomatter cross sections
The expressions for the total charged and neutral current neutrino cross sections are derived from Eqs. (14) and (15) after integrating over Bjorkenx and momentum transfer \(Q^2\) (or equivalently inelasticity y). The evolution of PDFs with respect to factorisation scale \(\mu \) can be calculated by a perturbative QCD expansion and results in the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) equations [84, 85, 86, 87]. The solution of the (leadingorder) DGLAP equations correspond to a resummation of powers \((\alpha _s\ln (Q^2/\mu ^2))^n\) which appear by QCD radiation in the initial state partons. However, these radiative processes will also generate powers \((\alpha _s\ln (1/x))^n\) and the applicability of the DGLAP formalism is limited to moderate values of Bjorkenx (small \(\ln (1/x)\)) and large \(Q^2\) (small \(\alpha _s\)). If these logarithmic contributions from a small x become large, a formalism by Balitsky, Fakin, Kuraev, and Lipatov (BFKL) may be used to resum the \(\alpha _s\ln (1/x)\) terms [88, 89]. This approach applies for moderate values of \(Q^2\), since contributions of \(\alpha _s\ln (Q^2/\mu ^2)\) have to be kept under control.
There are unified forms [90] and other improvements of the linear DGLAP and BFKL evolution for the problematic region of small Bjorkenx and large \(Q^2\). The extrapolated solutions of the linear DGLAP and BFKL equations predict an unlimited rise of the gluon density at very small x. It is expected that, eventually, nonlinear effects like gluon recombination \(g+g\rightarrow g\) dominate the evolution and screen or even saturate the gluon density [91, 92, 93].
5.4 Neutrino cross section measurement with IceCube
The analysis results in a value of \(R= 1.30^{+0.21}_{0.19} (\mathrm{stat}) ^{+0.39}_{0.43} (\mathrm{sys})\). This is compatible with the Standard Model prediction (\(R=1\)) within uncertainties but, most importantly, it is the first measurement of the neutrinonucleon cross section at an energy range (few TeV to about 1 PeV) unexplored so far with accelerator experiments [41]. This is illustrated in Fig. 14 which shows current accelerator measurements (within the yellow shaded area) and the results of the IceCube analysis as the light brown shaded area. The authors of Ref. [100] performed a similar analysis based on six years of highenergy starting event data. Their results are also consistent with perturbative QCD predictions of the neutrinomatter cross section.
5.5 Probe of cosmic ray interactions with IceCube
On a slightly different topic, but still related to the products of cosmic ray interactions in the atmosphere, the high rate of atmospheric muons detected by IceCube can be used to perform studies of hadronic interactions at high energies and high momentum transfers. Muons are created from the decays of pions, kaons and other heavy hadrons. For primary energies above about 1 TeV, muons with a high transverse momentum, \(p_t > rsim 2\) GeV, can be produced alongside the many particles created in the forward direction, the “core” of the shower. This will show up in IceCube as two tracks separated by a few hundred meters: one track for the main muon bundle following the core direction, and another track for the high\(p_t\) muon. The muon lateral distribution in cosmicray interactions depends on the composition of the primary flux and details of the hadronic interactions [101, 102]. If the former is sufficiently well known, the measurement of high\(p_t\) muons can be used to probe hadronic processes involving nuclei and to calibrate existing MonteCarlo codes at energies not accessible with particle accelerators.
The lateral separation, \(d_t\), of high \(p_t\) muons from the core of the shower is given by \(d_t=p_t H/E_{\mu } \cos \theta _{\mathrm{zen}}\), where H is the interaction height of the primary with a zenith angle \(\theta _{\mathrm{zen}}\). The initial muon energy \(E_{\mu }\) is close to that at ground level due to minimal energy losses in the atmosphere. That is, turning the argument around, the identification of single, laterally separated muons at a given \(d_t\) accompanying a muon bundle in IceCube is a measurement of the transverse momentum of the muon’s parent particle, and a handle into the physics of the primary interaction. Given the depth of IceCube, only muons with an energy above \(\sim 400\) GeV at the surface can reach the depths of the detector. This, along with the interstring separation of 125 m, sets the level for the minimum \(p_t\) accessible in IceCube. However, since the exact interaction height of the primary is unknown and varies with energy, a universal \(p_t\) threshold can not be given. For example, a 1 TeV muon produced at 50 km height and detected at 125 m from the shower core has a transverse momentum \(p_t\) of 2.5 GeV.
Our current understanding of lateral muon production in hadronic interactions shows an exponential behaviour at low \(p_t\), \(\exp (p_t/T)\), typically below 2 GeV, due to soft, nonperturbative interactions, and a powerlaw behaviour at high \(p_t\) values, \((1+p_t/p_0)^{n}\), reflecting the onset of hard processes described by perturbative QCD. The approach traces back to the QCD inspired “modified Hagedorn function” [103, 104]. The parameters T, \(p_0\) and n can be obtained from fits to proton–proton or heavy ion collision data [104, 105].
This is also the behaviour seen by IceCube. Figure 15 shows the muon lateral distribution at high momenta obtained from a selection of events reconstructed with a twotrack hypothesis in the 59string detector [106], along with a fit to a compound exponential plus powerlaw function. Due to the size of the 59string detector and the short live time of the analysis (1 year of data), the statistics for large separations is low and fluctuations in the data appear for track separations beyond 300 m. Still, the presence of an expected hard component at large lateral distances (high \(p_t\)) that can be described by perturbative quantum chromodynamics (a powerlaw behaviour) is clearly visible.
6 Nonstandard neutrino oscillations and interactions
In the previous two sections we have summarised the phenomenology of weak neutrino interactions and standard oscillations based on the mixing between three active neutrino flavour states and the eigenstates of the Hamiltonian (including matter effects). However, the Standard Model of particle physics does not account for neutrino masses and is therefore incomplete. The necessary extensions of the Standard Model that allow for the introduction of neutrino mass terms can also introduce nonstandard oscillation effects that are suppressed in a lowenergy effective theory. This is one motivation to study nonstandard neutrino oscillations. In the following, we will discuss various extensions to the Standard Model that can introduce new neutrino oscillation effects and neutrino interactions. The large energies and very long baselines associated with atmospheric and cosmic neutrinos, respectively, provide a sensitive probe for these effects.
6.1 Effective Hamiltonians
It is convenient to group these contributions into CPTeven terms obeying the relation \(\delta _{\mathfrak {a}} = \overline{\delta }_{\mathfrak {a}}\) and CPTodd terms with \(\delta _{\mathfrak {a}} = \overline{\delta }_{\mathfrak {a}}\). For a CPTsymmetric process we have \(P(\nu _\alpha \rightarrow \nu _\beta ) = P(\overline{\nu }_\beta \rightarrow \overline{\nu }_\alpha )\). In particular, the survival probability between neutrinos and antineutrinos is the same. CPTodd terms break this symmetry. Indeed, we have already encountered such a CPTodd term as the CPviolating matter effect contributing by the effective potential of Eq. (29). The corresponding expansion in terms the effective Hamiltonian of Eq. (32) is given by \({\widetilde{U}} = {\mathbf {1}}\), \(\delta _e= \overline{\delta }_e =\sqrt{2}G_FN_e\), \(\delta _{\mu ,\tau }=0\), and \(n=0\). On the other hand, the free Hamiltonian in Eq. (28) is a CPTeven term with \({\widetilde{U}} = U_{\mathrm{PMNS}}\), \(\delta _i= \overline{\delta }_i = m_i^2/2\), and \(n=1\).
Figure 16 shows the results of an analysis of atmospheric muon neutrino data in the range 100 GeV to 10 TeV taken by AMANDAII in the years 2000 to 2006 [113]. The data was binned into twodimensional histograms in terms of the number of hit optical modules (as a measure of energy) and the zenith angle (\(\cos \theta _{\mathrm{zen}}\)). The predicted effect of nonstandard oscillation parameters can be compared to the data via a profile likelihood method. No evidence of nonstandard neutrino oscillations was found and the statistically allowed region of the \([\Delta \delta , \sin ^22\xi ]\)plane is shown as 90%, 95%, 99% CL. These results are derived under the assumption that \(\eta =\pi /2\) in the unitary mixing matrix. The red dashed line shows the 90% limit of a combined analysis by SuperKamiokande and K2K [112] which is compatible with the AMANDAII bound. The projected IceCube 90% sensitivity after ten years of data taking is given as a yellow dotted line and may improve the limit on \(\Delta \delta \) by one order of magnitude [111].
6.2 Violation of Lorentz invariance
One of the foundations of the Standard Model of particle physics is the principle of Lorentz symmetry: The fundamental laws in nature are thought to be independent of the observer’s inertial frame. However, some extensions of the Standard Model, like string theory or quantum gravity, allow for the spontaneous breaking of Lorentz symmetry, that can lead to Lorentzinvariance violating (LIV) effects in the lowenergy effective theory. There also exist a deep connection between the appearance of LIV effects with the violation of CPTinvariance^{3} in local quantum field theories [115]. Such effects were incorporated in the Standard Model Extension (SME), an effectivefield Lorentzviolating extension of the Standard Model, which includes CPTeven and CPTodd terms [116]. The SME provides a benchmark for experiments to gauge possible Lorentz violating processes in nature, by expressing experimental results in terms of the parameters of the model. The size of LIV effect is expected to be suppressed by Planck scale \(M_{\mathrm{P}} \simeq 10^{19}\) GeV (or Planck length \(\lambda _{\mathrm{P}}\simeq 10^{33}\) cm), consistent with the strong experimental limits on the effect [117, 118].
Oscillations of atmospheric neutrinos with energies above 100 GeV provide a sensitive probe of LIV effects. For instance, LIV in the neutrino sector can lead to small differences in the maximal attainable velocity of neutrino states [120]. Since the “velocity eigenstates” are different from the flavour eigenstates, a new oscillation pattern can arise. The effect can be described in the framework of effective Hamiltonians described in the previous section by CPTeven states with \(n=1\). For the approximate twolevel system with survival probability described by Eq. (35) we can identify effective Hamiltonian parameters as the velocity difference \(\Delta \delta = \Delta c\), together with a new mixing angle \(\xi \) and a phase \(\eta \). Higher order contributions \(n>1\) have been considered for nonrenormalisable LIV effects caused by quantum mechanical fluctuations of the spacetime metric and topology [121]. Both the \(\Delta \delta \propto E\) (\(n=1\)) and the \(\Delta \delta \propto E^{3}\) (\(n=3\)) cases have been examined in the context of violations of the equivalence principle (VEP) [122, 123, 124]. The 90% CL upper limits on the corresponding coefficients \(\Delta \delta /E_\nu ^n~ [\hbox {GeV}{}^{1n}\)] are shown in Table 2.
(From Ref. [113]) 90% CL upper limits on Lorentzinvariance violation (LIV) and quantum decoherence (QD) effects. LIV upper limits are for the case of maximal mixing (\(\sin 2\xi = 1\)), and quantum decoherence upper limits for the case of a 3level system with universal decoherence parameters D (see text for details)
n  LIV (\(\Delta \delta /E_\nu ^n\))  QD (\(D/E_\nu ^n\))  Units 

1  \( 2.8\times 10^{27}\ \)  \( 1.2\times 10^{27}\ \)  – 
2  \( 2.7\times 10^{31}\ \)  \( 1.3\times 10^{31}\ \)  \(\ \mathrm{GeV}^{1}\) 
3  \( 1.9\times 10^{35}\ \)  \( 6.3\times 10^{36}\ \)  \(\ \mathrm{GeV}^{2}\) 
The violation of Lorentz invariance associated with Planckscale physics can also affect neutrino spectra over long baselines. The LIV effects can result in modified dispersion relations, e.g., \(E^2p^2 = m^2  \epsilon E^2\), that introduce nontrivial maximal particle velocities [120]. Whereas at low energies the Lorentz invariance is recovered, \(E^2p^2\simeq m^2\), at high energies we can observe sub or superluminal maximal particle velocities. These can allow otherwise forbidden neutrino decays, in particular, vacuum pair production, \(\nu _\alpha \rightarrow \nu _\alpha +e^++e^\), and vacuum neutrino pair production, \(\nu _\alpha \rightarrow \nu _\alpha +\overline{\nu }_\beta +\nu _\beta \). The secondary neutrinos introduce nontrivial flavour compositions as well as spectral bumps and cutoffs. As discussed in Refs. [126, 127, 128], these small effects can be probed by the IceCube TeV–PeV diffuse flux.
6.3 Nonstandard matter interactions
We have already discussed coherent scattering of neutrinos and antineutrinos in dense matter, that can be accounted for by an effective matter potential in the standard Hamiltonian. Since neutrino oscillations are only sensitive to nonuniversal matter effects, only the unique chargedcurrent interactions of electron neutrinos and antineutrinos with electrons are expected to contribute. However, nonstandard interactions (NSI) can change this picture.
The effect of NSI in atmospheric neutrino oscillations with IceCube data were studied in Refs. [132, 134]. The IceCube analysis of Ref. [132] is based on three years of data collected by the lowenergy extension DeepCore, that was also used in the standard neutrino oscillation analysis discussed in Sect. 4. The data was binned into a twodimensional histogram in terms of reconstructed neutrino energy and zenith angle, \(\cos \theta _{\mathrm{zen}}\), and analysed via a profile likelihood method. The analysis focused on NSI interactions with dquarks assuming \(\epsilon ^{(d)}_{\mu \tau } = \epsilon ^{(d)}_{\mu \tau }\) and \(\epsilon ^{(d)}_{\mu \mu }=\epsilon ^{(d)}_{\tau \tau }\). The constraints for \(\epsilon _{\mu \tau }\) are shown in Fig. 19.
6.4 Neutrino decoherence
The Hamiltonian evolution in Eq. (26) is a characteristic of physical systems isolated from their surroundings. The time evolution of such a quantum system is given by the continuous group of unitarity transformations, \(U_t = \mathrm{e}^{i Ht},\) where t is the time. The hermiticity of the Hamiltonian guarantees the reversibility of the processes, \(U^{1}=U^\dagger \). For open quantum systems, the introduction of dissipative effects lead to modifications of Eq. (26) that account for the irreversible nature of the evolution. The transformations responsible for the time evolution of these systems are defined by the operators of the Lindblad quantum dynamical semigroups [110]. Since this does not admit an inverse, such a family of transformations has the property of acting only forward in time. The monotonic increase of the von Neumann entropy, \(S(\rho ) =  \mathrm{Tr}\,\, (\rho \, \ln \rho )\), implies the hermiticity of the Lindblad operators, \(L_j = L_j^\dagger \) [135]. In addition, the conservation of the average value of the energy can be enforced by taking \([H, L_j] = 0\) [136].
6.5 Neutrino decay
Note, that the decay rates \(\Gamma _i\) are expected to decrease with energy due to relativistic boosting of the mass eigenstate’s lifetime. Therefore, the neutrino flavour composition can experience strong energy dependencies. On the other hand, active neutrino decay into sterile neutrinos can introduce spectral cutoffs due to the energy dependence of the neutrino lifetime. This process is limited by the observation of IceCube’s TeV–PeV neutrino flux and could be responsible for a tentative cutoff [140]. Neutrino decay has also been considered as a possibility to alleviate a mild tension in the bestfit powerlaw spectra between cascade and trackdominated IceCube data [142].
Astrophysical neutrinos propagating over cosmic distances are also susceptible to feeble interactions with cosmic backgrounds. In particular, feeble interactions with the cosmic neutrino background (C\(\nu \)B) that can be enhanced by resonant interactions, e.g., \(\overline{\nu }_\alpha +\nu _\alpha \rightarrow Z'\rightarrow \overline{\nu }_\beta +\nu _\beta \) have been discussed as a source for absorption features [148, 149, 150, 151]. The evolution of the neutrino density matrix is identical to that for neutrino decay with dissipation term as in Eq. (50). However, in this case the interaction rates have a nontrivial dependence on redshift via the density evolution of the C\(\nu \)B.
6.6 Sterile neutrinos
Many extensions of the Standard Model that relate to the appearance of neutrino mass terms predict the existence of sterile neutrinos. As discussed earlier, the righthanded neutrino field, that can provide a Dirac mass term \(m_{\mathrm{D}}\overline{\nu }_L\nu _R+h.c.\), does not interact via weak interactions and is therefore sterile. However, in the minimal type I seesaw models (see, e.g., Ref. [67]) these sterile neutrinos have a large Majorana mass term, \(M\overline{\nu }_{R}{\mathcal {C}}\overline{\nu }_{R}^T/2 + h.c.\), with \(m_{\mathrm{D}}\ll M\), that give rise to a large effective neutrino mass after diagonalising the mass matrix. These massive sterile states are practically unobservable in lowenergy oscillation experiments. On the other hand, for values of \(M\ll m_{\mathrm{D}}\) (“pseudoDirac” case), the active and sterile state mass states are degenerate after diagonalisation, leading to maximal mixing between the left (active) and right (sterile) states.
The minimal sterile neutrino model is a “3+1” model where, in addition to the three standard weaklyinteracting neutrino flavours, one additional heavier sterile neutrino state is added. Such a simple extension of the neutrino sector has been advocated to explain certain tensions between experimental results from accelerator [159, 160, 161], reactor [162] and radiochemical [163] experiments, and the predictions from the standard three active flavour scenario. In the most general case, the introduction of one sterile neutrino adds six new parameters to the neutrino oscillation phenomenology [156]: three mixing angles \(\theta _{\mathrm{14}}\), \(\theta _{\mathrm{24}}\), \(\theta _{\mathrm{34}}\), two CPviolating phases, \(\delta _{\mathrm{14}}\) and \(\delta _{\mathrm{34}}\), and one mass difference, \(\Delta m^2_{\mathrm{41}}\), where the indexes ‘1–3’ stand for the known neutrino mass states and ‘4’ for the sterile state.
Although sterile neutrinos can not be detected directly, their existence can leave an imprint on the oscillation pattern of active neutrinos. The sterile neutrino modifies the oscillation pattern of the standard neutrinos since these can now undergo vacuum oscillations into the new state, with a probability that is proportional to the new mixing angles. The period of these oscillations can be small, smaller than the directional resolution of IceCube, and the net effect is then to distort the overall \(\nu _{\mu }\) flux normalisation with respect to the threeflavour case. An additional effect arises from the different interactions of flavours with matter when traversing the Earth [164, 165]. The new possibility to oscillate to a state that does not interact results in energy and angular dependent oscillation amplitudes that depend on the mixing angles, but also on the new \(\Delta m^2_{\mathrm{41}}\). More precisely, the comparison between the oscillation pattern in the \(\mathrm{(energy, zenith)}\) phase space predicted by the sterile “3+1” model and the pattern seen in experimental data is what IceCube exploits to set limits on the sterile mixing parameters. As for the standard oscillation case described in Sect. 4, searches for sterile neutrinos in IceCube are based on the detection of the disappearance of muon neutrinos and are more sensitive to \(\theta _{\mathrm{24}}\). Therefore the choice of a simplified minimal mixing scenario where only \(\theta _{\mathrm{24}}\) is not zero is justified, and is indeed the approach followed in the analyses described below. Nonzero values of the weakly constrained \(\theta _{\mathrm{34}}\) within current limits would not significantly affect the results presented here [166].
There have been several searches in the past at subTeV energies with neutrinos from the Sun, reactors, and accelerator setups (see, e.g., Refs. [156, 167, 168, 169] and references therein). The large flux of atmospheric neutrinos that reach IceCube and the wide range of energy response of the detector allow to search for signatures of anomalous \(\nu _{\mu }+{\overline{\nu }_{\mu }}\) disappearance caused by oscillations to an sterile neutrino, \(\nu _{\mathrm{s}}\), at TeV energies, an energy not probed before. Furthermore, the DeepCore detector can extend the search down to about 6 GeV, improving previous limits from smaller detectors.
The analysis techniques are very similar to the search for standard oscillations and are based on comparing the equivalent oscillogram from Fig. 5 for oscillations with an sterile component, to the measured data. Although the lowenergy tracks provide a quite short lever arm to reconstruct their direction with precision, and the angular resolution of the analysis varies between \(6^{{\circ }}\) and \(12^{{\circ }}\), depending on energy. This is enough, though, to perform the analysis (see Fig. 5 in Ref. [129]) since the differences in the oscillation pattern in the presence of a sterile neutrino can still be distinguished from the nonsterile case with such angular resolution. At higher energies the analyses necessarily follow a slightly different approach since the tracks originate outside the detector and the energy deposited in the hadronic shower at the interaction vertex is not accessible. Such analyses use measured muon energies instead of neutrino energies, like the one presented in Ref. [158]. The muon energy is reconstructed from the light emission profile from stochastic energy losses of the muon along its trajectory [17], achieving an energy resolution of \(\sigma _{\mathrm{log_{10}}}(E_{\mu }/\mathrm{GeV})\sim 0.5\). Since at the energies of this analysis the muon track can be well reconstructed, the angular resolution reaches values between \(0.2^{{\circ }}\) and \(0.8^{{\circ }}\), depending on incoming angle.
As in the previous cases, the presence of sterile neutrinos can also have an effect on neutrino spectra from astrophysical sources. In the case of a pseudoDirac scenario, where the mass splitting between active and sterile neutrinos is very small, there exist the possibility that the corresponding neutrino oscillation effects are still visible, i.e., \(L\Delta m^2/E\simeq 1\) [141, 170].
7 Indirect dark matter detection
There is a large corpus of evidence that supports the existence of a nonbaryonic, nonluminous component of matter in the cosmos. A way to understand the rotation curves of galaxies, the peculiar velocities of galaxies in clusters, and the formation of first galaxies growing out of small density perturbations imprinted in the cosmic microwave background, is to introduce a “dark matter” component in the energy budget of the Universe [171, 172]. Attractive candidates for dark matter consists of stable relic particles whose present density is determined by the thermal history of the early universe [173, 174, 175, 176]. The present abundance of dark matter can be naturally explained by physics beyond the Standard Model providing stable, weaklyinteracting massive particle (WIMP) in the few GeV–TeV mass range. For thermally produced WIMPs, the upper mass limit arises from theoretical arguments in order to preserve unitarity [177], although higher masses can be accommodated in models where the dark matter candidates are not produced thermally [178].
There is a vast ongoing experimental effort to try to identify the nature of dark matter through different strategies: production at colliders [179] or through the detection of nuclear recoils in a selected target in “direct detection” experiments [180]. A complementary, “indirect”, approach is based on searching for the products of the annihilation of dark matter particles gravitationally trapped in the halo of galaxies or accumulated in heavy celestial objects like the Sun or Earth [181, 182, 183, 184, 185, 186, 187, 188]. In this latter case, neutrinos are the only possible messengers, since other particles produced in the annihilations will be absorbed. These search techniques are competitive since they can set limits on the same physical quantities (the dark matternucleon cross section for example). But they are also complementary since they are subject to different backgrounds (the gammaray sky is very different from the proton or neutrino sky), different astrophysical inputs (dark matter density and velocity distribution) and different systematics (nucleon and nuclear form factors of different targets). Additionally the authors in [189] have used arguments based on the observed heat flow of the Earth to constrain in a rather modelindependent way the dark matter spinindependent cross section: the energy deposition from the annihilation of dark matter should not produce heat that exceeds experimental measurements. Such argument provides competitive limits in the case of strongly interacting dark matter.
Dark matter searches from our own Galaxy, nearby galaxies or galaxy clusters present some distinct features with respect to searches from the Sun or Earth which are advantageous. Firstly, capture is not an issue since the presence of dark matter overdensities has been an essential part in the process of galaxy formation. What can be measured then is the velocityaveraged WIMP selfannihilation cross section, \(\langle \sigma _{\mathrm A} v \rangle \). Secondly, the products of the annihilations are not necessarily absorbed at the production site, and other indirect signatures (photons, antiprotons, etc.) can also be searched for in \(\gamma \)ray and cosmicray observatories. These multiwavelength and/or multimessenger searches can increase the sensitivity of dark matter searches. Neutrinos remain, however, an attractive signature since they do not suffer from uncertainties in their propagation (as charged particles do) and no background or foreground from astrophysical objects is present (as in the case of \(\gamma \)rays). Note, that some of the sources are extended (the Galactic halo for example) and pointsource analysis techniques have to be modified. On the other hand we expect that the flux of secondaries from these distant objects is much lower than that predicted from WIMP annihilations in the Sun and, furthermore, there are new systematics effecting the calculations. For example, the assumed shape of the dark matter halo profile effects significantly the interpretation of the results since the annihilation rate depends on the square of the dark matter number density. We will discuss these issues in Sect. 7.4.
7.1 Neutrinos from WIMP annihilation and decay
In the absence of a signal, the 90% confidence level limit on the number of signal events, \(\mu _{\mathrm{s}}^{\mathrm{90}}\), can then be directly translated into a limit on either the velocityaveraged annihilation cross section \(\langle \sigma _Av\rangle \) or the dark matter lifetime \(\Gamma _X\). The interplay between the total number of observed events in a given data sample, \(n_{\mathrm{obs}}\), and the estimated number of background events, \(n_{\mathrm{bg}}\), is the basis to perform a simple eventcounting statistical analysis to constrain \(\mu _{\mathrm{s}}\), i.e., to constrain a given model. This was the approach followed in early IceCube publications, e.g., Refs. [195, 196].
There are systematic uncertainties in the translation of the number of detected events into capture cross section values due to uncertainties in the element composition of the Sun [198], the effect of planets on the capture of WIMPS from the halo [199], astrophysical uncertainties [200, 201] and the uncertainty on the values of the nuclear form factors needed in the rather complex capture calculations [190, 202, 203]. These effects can be of relevance when comparing results from different search techniques [204].
The experimental effort to detect neutrinos as a signature of dark matter annihilations in celestial bodies came of age in the mid 90’s with underground detectors like MACRO, Baksan, Kamiokande and SuperKamiokande. These detectors provided the first limits on the flux of neutrinos from dark matter annihilations in the Earth or the Sun [205, 206, 207, 208]. Baksan and SuperKamiokande continue to be competitive in the field today [209, 210, 211]. Baikal [212] and AMANDA were the first largescale neutrino detectors with an open geometry to perform dark matter searches in the late 90’s, soon followed by ANTARES [213]. Early results of these experiments can be found in Refs. [195, 212, 214, 215, 216, 217] (Fig. 23).
7.2 Dark matter signals from the Sun
Traditionally, solar WIMP searches with IceCube have used the muon channel since it gives better pointing and, in the end, dark matter searches from the Sun are really pointsource searches. The first analyses used the Earth as a filter of atmospheric muons and “looked” at the Sun only in the austral winter, when the Sun is below the horizon at the South Pole [223, 234, 235]. With the completion of IceCube79 and DeepCore, it was possible to define effective veto regions to efficiently reject incoming atmospheric muons from above [236]. Since then the IceCube solar WIMP searches cover also the austral summer, doubling the exposure of the detector per calendar year. DeepCore has also allowed to extend the search for neutrinos from WIMPs with masses as low as \(20 \hbox { GeV/c}^2\), whereas past IceCube searches have only been sensitive above \(50~\hbox {GeV/c}^2\). Additionally, allflavour analyses are being developed [237], since the addition of \(\nu _e\) and \(\nu _{\tau }\) events triples the expected signal. Improved lowenergy reconstruction techniques allow to reconstruct electron and tau neutrino interactions with sufficiently good angular resolution to be useful in solar and Earth dark matter searches.
There is an additional irreducible background in indirect dark matter searches from the Sun which originates from cosmic ray interactions in the solar atmosphere producing neutrinos. These neutrinos constitute what is called the solar atmospheric neutrino flux, and provide a sensitivity floor for dark matter searches with neutrino telescopes [239, 240, 241, 242, 243]. Predictions of the level of this flux are at the order of one event per year, with an energy distribution that, in principle, can be different from the flux from neutrinos from dark matter annihilations. However, due to the predicted level of this flux (about one order of magnitude below the present sensitivity of \(\hbox {km}^3\) neutrino detectors), it has not been taken into account as an additional background in the results shown below.
In general, the reconstructed neutrino energy E and solid angle \(\Omega \) are different from the true energy \(E'\) and true arrival direction \(\Omega '\). The relation between these quantities on a statistical basis must be obtained from simulations, and it is specific to the annihilation channel under study. The true energy E can be related to the number of hit DOMs or can be estimated from more elaborate energy reconstructions. The quantity \(Q(E_i, \Omega _i  E', \Omega ')\) is the probability density (in effective units of inverse steradian and proxy energy) for reconstructing \(E_i\) and \(\Omega _i\) for the ith event when the true values are \(E'\) and \(\Omega '\), respectively.
Note that systematic uncertainties on the signal and/or background prediction or on the angular or energy resolutions can be easily incorporated in a likelihood approach as nuisance parameters by marginalising over them. The only knowledge needed is the functional form of the nuisance parameters. P is a function of energy and angle, which in a simplified approach can be decomposed in an angledependent part (the PSF of the detector) and an energydependent part (the energy dispersion of the detector). More generally, the angular response of the detector can depend on energy, and then this decomposition is not valid. For pointsource searches, due to the restricted angular region in the sky considered, the PSF and energy dispersion can be taken to be uncorrelated.
The improvement due to using a full eventbased likelihood in comparison to just an angular shape analysis is illustrated in Fig. 25, taken from [238]. The figure shows the limit on the spindependent WIMPproton cross section as a function of WIMP mass obtained with two analyses performed on the same data set taken with IceCube79. Shown are the limits using an eventcount likelihood (dashed lines), the limits obtained using an analysis based on the difference in shape of the spaceangle distribution for signal and background (tagged ‘PRL’ and originally from [236]) and the limits obtained using a full likelihood like in Eq. (66). Including the eventlevel energy information has the most impact at high WIMP mass, due to the relatively good energy resolution of IceCube at high neutrino energies. Note that the full likelihood analysis in [238] used a rather simple energy proxy based on the number of hit DOMs. Better energy reconstruction algorithms being developed within IceCube, particularly at low energies, will further improve the performance of this method [80].
7.3 Dark matter signals from the Earth
7.4 Dark matter signals from galaxies and galaxy clusters
The Milky Way centre and halo, as well as nearby dwarf galaxies and galaxy clusters provide natural largescale regions of increased dark matter density. Since dark matter played a significant role in the formation of such structures from primordial density fluctuations, the issue of capture is not relevant, and what neutrino telescopes can prove when considering such objects is the WIMP selfannihilation cross section, \(\langle \sigma _{\mathrm A} v \rangle \). In order to predict the rate of annihilation of dark matter particles in galactic halos, the precise size and shape of the halo needs to be known. There is still some controversy on how dark halos evolve and which shape they have. There are different numerical simulations, observational fits, and parametrisations of the dark matter density around visible galaxies, including the Navarro–Frenk–White (NFW) profile [256], the Kravtsov profile [257], the Moore profile [258], and the Burkert [259]. The common feature of these profiles is a denser spherically symmetric region of dark matter in the centre of the galaxy, with decreasing density as the radial distance to the centre increases. Where they diverge is in the predicted shape of the central region. Profiles obtained from Nbody simulations of galaxy formation and evolution tend to predict a steep powerlaw type behaviour of the dark matter component in the central region, while profiles based on observational data (stellar velocity fields) tend to favour a constant dark matter density near the core. This is the corecusp problem [260], and it is an unresolved issue which affects the signal prediction from dark matter annihilations in neutrino telescopes. Note that the shape of the dark halo can depend on local characteristics of any given galaxy, like the size of the galaxy [261] or on its evolution history [262, 263].
The shape of the dark matter halo is important because the expected annihilation signal depends on the lineofsight integral from the observation point (the Earth) to the source, and involves an integration over the square of the dark matter density. This is included in the Jfactor of Eq. (60), which is galaxydependent, and absorbs all the assumptions on the shape of the specific halo being considered. In the case of our Galaxy, the expected signal from the Galactic Centre assuming one halo parametrisation or another can differ by as much as four orders of magnitude depending on the halo model used (see, e.g., Fig. 2 in Ref. [271]).
The highenergy diffuse astrophysical neutrino flux discovered by IceCube opens a new possibility of probing the galactic dark matter distribution through neutrinodark matter interactions [272, 273, 274]. Indeed dark matter couplings to standard model particles are commonly assumed to exist, and this is the basis of direct detection experiments. Such a coupling can be extended to neutrinos if one assumes the existence of a mediator \(\phi \), which can be either bosonic of fermionic in nature, which couples to dark matter with a coupling g. For simplicity one can assume that the strength of the \(\nu \phi \) coupling is also given by g. Under these assumptions, dark matterneutrino interactions could distort the isotropy of the astrophysical neutrino flux, resulting in an attenuation of the flux towards the Galactic Centre, where the density of dark matter is higher. Therefore, an analysis of the isotropy of the highenergy astrophysical neutrino data of IceCube can be translated into a limit on the strength of the neutrinodark matter coupling, g. Such an analysis has been performed by the authors in [272], where the mass of the dark matter candidate, the mass of the mediator and the strength of the coupling are left as free parameters in a likelihood calculation which aims at evaluating the suppression of astrophysical neutrino events from the direction of the Galactic Centre. The results are shown in Fig. 31. The left panel shows contours of the maximum allowed value of the coupling of fermionic dark matter coupled to neutrinos through a vector mediator, while the right panel shows the case of a scalar dark matter coupled through a fermionic mediator. Interestingly IceCube is sensitive to a region of parameter space complementary to results derived from cosmological arguments alone [275, 276, 277], as indicated by the magenta line. This line delimits the region where limits from analyses using largescale structure data become more restrictive than the IceCube limits shown in the plot.
7.5 TeV–PeV dark matter decay
The origin of the TeV–PeV diffuse flux observed with IceCube is so far unknown. Whereas most models assume an astrophysical origin of the emission, it is also feasible that the emission is produced via the decay of dark matter, as first proposed in Ref. [278]. Various studies have argued that heavy dark matter decay can be responsible for various tentative spectral features in the inferred neutrino spectrum of the HESE analysis [279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290] and also its low energy extension [291, 292, 293]. Some authors have also discussed the necessary condition for PeV dark matter production in the early Universe, e.g., via a secluded dark matter sector [294], resonantlyenhanced freezeout [295, 296], or freezein [297, 298, 299, 300].
If dark matter decay is responsible for the highenergy neutrino emission, the arrival directions of TeV–PeV neutrino events observed with IceCube should correlate with the lineofsight integral of the dark matter distribution (“Dfactor”). The contribution of neutrinos from dark matter decay in the Galactic halo can be similar to the isotropic extragalactic contribution. Half of the neutrino events from dark matter decay in the halo are predicted to fall within \(60^{{\circ }}\) around the Galactic Centre. This introduces a weak large scale anisotropy that can be tested against the observed event distribution [305, 306, 307]. The neutrino emission from galaxies and galaxy clusters could also be identified as (extended) pointsource emission in future IceCube searches [308].
It can be expected that the secondary emission from decaying dark matter scenarios will also include other standard model particles that can be constrained by multimessenger observations. In particle the production of PeV \(\gamma \)rays, that have a pair production length of \({\mathcal {O}}(10)\) kpc in the CMB, would be a smoking gun for a Galactic contributions [308]. However, also the the secondary GeV–TeV emission from electromagnetic cascades initiated by PeV \(\gamma \)rays, electrons, and positrons can constrain the heavy dark matter decay scenario [306, 308, 309, 310, 311].
8 Magnetic monopoles
Maxwell’s equations of classical electrodynamics appear to be asymmetric due to the absence of magnetic charges. However, this is merely by choice. We can always redefine electric and magnetic fields by a suitable duality transformation, \(\mathbf{E}' + i\mathbf{B}' = e^{i\phi }(\mathbf{E} + i\mathbf{B})\), such that Maxwell’s equations in terms of the new fields are completely symmetric. However, this duality transformation requires that for every particle the ratio between magnetic and electric charges are the same. If this is not the case, then it is necessary to include a source term for magnetic charges \(q_m\) (i.e., magnetic monopoles), that create a magnetic field of the form \(\mathbf{B} = q_m\mathbf{e}_r/4\pi r^2\).
In solids, structures have been found which resemble poles. These are sometimes mistakenly called magnetic monopoles, although the poles can only occur in pairs and do not exist as free particles. For distinction, recently the term magnetricity has been coined for the field that exhibits theses poles. Fundamental magnetic monopoles have not been observed so far.
Although, as Dirac showed, magnetic monopoles can be consistently described in quantum theory, they do not appear automatically in that framework. As was first found independently by ’t Hooft [315] and Polyakov [316], this is different in Grand Unified Theories (GUT) which embed the Standard Model interactions into a larger gauge group. These theories are motivated by the observation that the scaledependent Standard Model gauge couplings seem to unify at very high energies. Generally, a ’t Hooft–Polyakov monopole can arise via spontaneously breaking of the GUT group via the Higgs mechanism. The stability of the monopole is due to the Higgs field configuration which cannot smoothly be transformed to a spatially uniform vacuum configuration.
The unification scale is related to the size of the monopole as \(r_{\mathrm{M}}\simeq \Lambda _{\mathrm{GUT}}^{1}\simeq 10^{29}\) cm. Larger radii correspond to different energy scales reflecting various transitions in the Standard Model, in particular, the electroweak transition scale \(M_Z^{1}\simeq 10^{16}\) cm and the confinement scale \(\Lambda _{\mathrm{QCD}}^{1}\simeq 10^{13}\) cm (see Fig. 33). The presence of virtual particles within these “shells” influences the monopole’s interaction with matter. For instance, within the monopole core, the GUT gauge symmetry is restored, and can mediate baryonnumber violating processes. At large distances, only electromagnetic interactions are visible by the magnetic monopole field.
8.1 Cosmological bounds
An elegant solution to this problem is an inflationary universe, i.e., a universe that underwent an exponential expansion of the scale factor, diluting any initial monopole abundance to an (almost) unobservable level. This inflationary mechanism is a very powerful idea since it simultaneously explains why our Universe has been extremely flat at early times (flatness problem), e.g., \(\Omega 1\simeq 10^{16}\) at the epoch of big bang nucleosynthesis, and why the Universe appears to be so homogeneous over causally disconnected distances (Horizon problem), e.g., temperature fluctuations in the CMB of only \(10^{5}\).
8.2 Parker bound
Magnetic monopoles are accelerated in magnetic fields – analogously to charged particle acceleration in electric fields. Therefore, relic monopoles that are initially nonrelativistic are expected to gain energy while they travel along galactic and intergalactic magnetic fields. The requirement that monopoles have to be rare not to shortcircuit these magnetic fields gives the socalled Parker bound [324]. The galactic magnetic field with a strength of a few \(\mu \)G can be generated by a dynamo action on a time scale that is comparable to the Milky Way’s rotation period, \(\tau \simeq 10^8\) yr. A monopole with magnetic charge \(q_m\) will gain an energy of \(\Delta E_{\mathrm{kin}} = \Delta \ell B q_m\) after it travels a distance \(\Delta \ell \) along magnetic field lines. The power density of the galactic dynamo \(\sim B^2/\tau \) should be larger than the energy drained by the magnetic monopoles.
8.3 Nucleon decay catalysis
8.4 Monopole searches with IceCube
 (i)

Direct Cherenkov light is produced at highly relativistic velocities above \( 0.76\, c\) as with any other charged Standard Model particle. Due to the high relative Dirac charge, as shown in Eq. (72), several thousand times more light is radiated with a monopole than from a minimum ionising singly electrically charged particle like a muon [336].
 (ii)

Indirect Cherenkov light from secondary knockoff \(\delta \)electrons is relevant at mildly relativistic velocities \((\simeq 0.5\, c \text { to } 0.76\, c)\). The highenergy \(\delta \)electrons in turn can have velocities above the Cherenkov threshold themselves. The energy transfer of the monopole to the \(\delta \)electrons can be inferred from the differential cross section calculated by Kasama, Yang and Goldhaber (KYG) [337, 338].
 (iii)

Luminescence light from excitation of the ice dominates at low relativistic velocities \((\simeq 0.1\, c \text { to } 0.5\, c)\). The observables of luminescence, such as the wavelength spectrum and decay times, are dependent on the properties of the medium, in particular, temperature and purity. The signature is relatively dim in comparison to muon signatures. Pending further laboratory measurements in ice and water [339], the efficiency of luminescence photon production per deposited energy is in the range of \(\mathrm{d}N_{\gamma }/\mathrm{d}E=0.2\,\gamma /\mathrm {MeV}\) and \(2.4\,\gamma /\mathrm {MeV}\) [340, 341]. Even for the lower plausible light yield, luminescence is a viable signature due to the high excitation of the medium induced by a monopole [329].
 (iv)

At velocities well below 0.1 c luminescence is expected to fall off (see Fig. 35). The catalysis of nucleon decays is a plausible scenario for GUT monopoles (see Sect. 8.3) and may be observed if its mean free path is small compared to the detector size. The Cherenkov light from secondaries emitted in nucleon decays along the monopole trajectory can lead to a characteristic slow moving event pattern across the detector [335].
For each of these speed ranges, searches for magnetic monopoles at the IceCube experiment are either in progress (luminescence) or have already set the world’s best upper limits on the flux of magnetic monopoles over a wide range of velocities. Examples of magnetic monopole passing through the detector at different velocities are shown in Fig. 36.
 (i)

Relativistic monopoles are selected based on their brightness, arrival direction, and velocity [336]. The high energy astrophysical neutrino flux is an important background to this signal. At similar brightness, monopoles show less stochastic energy loss than Standard Model particles leading to a smoother light yield distribution along the track.
 (ii)

Due to its lower rest mass a Standard Model particle with a velocity below the speed of light in vacuum, c, would not be able to traverse the whole detector. However, the discrimination power of the reconstruction of the velocity is insufficient for the suppression of the vast air shower backgrounds against the identification of mildly relativistic monopoles. Instead variables describing the topology, smoothness, and time distribution of the events are processed in a Boosted Decision Tree (BDT) machine learning [336].
 (iii)

Searching for mildly relativistic monopoles using luminescence light can be performed using analysis techniques that combine the nonrelativistic reconstructed particle velocity and the continuous but dim light production of a throughgoing track in the detector [329, 334].
 (iv)

Catalysed nucleon decay, like \(p + M \rightarrow e^{+} + \pi ^{0} + M\), transfers almost all of the proton’s rest mass to the energy of electromagnetic and hadronic cascades. Because of the high light yield this channel is typically used as a benchmark in analyses. Due to their low speed, the duration of the event is in the order of 10 ms. As obvious from the right event display in Fig. 36 at such timescales random noise pulses are a significant contribution. Various effects contribute to subtle temporal correlations on long time scales of this noise [4, 348] complicating an adequate description in simulation. Instead, a background model is established from reshuffling experimental data. For signal identification, timeisolated local coincidences in neighbouring DOMs are searched for along a monopole trajectory hypothesis consistent with a straight particle track of constant speed. A Kalman filter is used to separated noise from monopole signals and a combination of observables are fed to a BDT to further improve the signal purity [335].
Figure 37 shows a compilation of current flux upper limits of relic monopoles from various experiments. Only in the past decade astrophysical experiments have been able to improve upon the original Parker bound which is shown for comparison. These recent experiments have employed large scale detectors for cosmic rays to achieve the highest sensitivities in the whole \(\beta \) range in which GUT magnetic monopoles are expected. Typically it is assumed that the flux at the respective detector site is isotropic implying sufficient kinetic energy in order to cross the Earth or the overburden above the detector due to their large rest masses. This assumption is justified for monopoles of masses in excess of \(10^{10}\) GeV. The monopole flux limits commonly assume a single Dirac magnetic charge \(q_m = g_D\) (see Eq. (72)) with no additional electric charge. In most detectors and velocity ranges, the detection efficiency for larger magnetic charges or for electrically charged monopoles (dyons) is expected to increase.
Operational until 2000, MACRO searched for magnetic monopoles using three types of subdetectors – liquid scintillation counters, limited streamer tubes and nuclear track detectors. No monopole was found, with an upper flux limit at the 90% confidence level of \(1.4 \times 10^{16} \ \mathrm{cm}^{2} \ \mathrm{s}^{1} \ \mathrm{sr}^{1}\) for monopoles with velocity between \(4 \times 10^{5} \ c\) to c and magnetic charge with \(n \ge 1\) [349, 350]. Under the assumption that monopoles are gravitationally accumulated in the centre of the Sun, SuperKamiokande [351] could impose stringent limits for nonrelativistic velocities in an indirect search for neutrinos from the direction of the Sun. Baikal [340] has investigated the direct Cherenkov light from relativistic monopoles. The analysis by ANTARES includes also the mildly relativistic regime employing similar techniques like IceCube [344]. The reduced scattering in water at the ANTARES site compared to ice leads to a better velocity reconstruction which helps with the Standard Model background suppression. This partly compensates the higher noise level in the detector.
Intermediate and low mass monopoles may acquire highly relativistic velocities in intergalactic magnetic fields reaching Lorentz factors of \(\gamma \simeq 10^{10}\) for the example of a PeVmass [318]. Ultrarelativistic particles with magnetic charge (or large electric charge) dramatically loose energy in their passage through matter, initiating a large number of bright showers along the track. At high Lorentz boost factors the photonuclear effect is the dominant energy loss mechanism generating hadronic showers. While these showers are continuously produced, they may overlap with each other. In the atmosphere of the Earth this leads to a builtup such that the energy deposit increases with slant depth. The Auger experiment has used this feature to distinguish monopoles from the background of electrically singly charged ultrahighenergy cosmic rays like protons [347]. The RICE [345] and ANITA [346] experiments have searched for such multiple subshower signatures in the Antarctic ice sheet with the RadioCherenkov technique, the discriminant against conventional cosmic rays here being primarily the rapid succession of several radio pulses received from each subshower.
The extrapolation of IceCube’s limit towards highly relativistic velocities by a constant line in Fig. 37 is a very conservative approach. It not only neglects the increase in signal detection efficiency with more energy deposited, but it also ignores the onset of the photonuclear effect and pair production. These effects would produce showers with light emission orders of magnitude brighter than from the Cherenkov effect considered here, hence visible also from far outside the instrumented detector volume.
While these flux limits reflect the cosmic density, at the electroweak scale monopoles may be created in accelerator collisions, which is studied at the MoEDAL experiment. Also cosmic ray collisions or high energy neutrino interactions in the Earth may produce monopoles. This might be an additional detection opportunity, also for IceCube.
9 Other exotic signals
The detection principle of Cherenkov telescopes is very general in the sense that it applies to any flux of particles that can penetrate the detector shielding and produce light signals inside the detector volume. We have already covered the possibility to observe relic magnetic monopoles in the previous section. In this section we will discuss the detection potential for other exotic candidates like Qballs and strangelets. We will also address the possibility that longlived charged particles produced in cosmic ray or neutrino interactions may be discovered via their Cherenkov emission. All these particles have in common that their passage through the IceCube detector produces observable features that can be extracted from backgrounds.
9.1 Qballs
There exist nontopological solitonic solutions of a field theory, socalled Qballs [352]. Whereas the stability of topological solitons, e.g., monopoles is guaranteed by the conservation of a topological charge (winding number) associated with a degeneracy of the vacuum state, a Qball can be stable by the conservation of a charge associated with a global symmetry of the theory. This can happen if its energy configuration is lower than the corresponding multiparticle Fock state. For a single complex scalar field \(\phi \) carrying the charge Q, this implies that there exists a nontrivial field value \(\phi _0>0\) where the scalar potential \(U(\phi ^\dagger \phi )\) obeys the condition \(U(\phi _0^\dagger \phi _0) < m_\phi ^2\phi _0^\dagger \phi _0\), where \(m_\phi \) is the mass of the scalar field (for a review see Ref. [353]).
If baryogenesis proceeded via the AffleckDine mechanism [357],^{9} stable Qballs with \(10^{12}\lesssim Q_B\lesssim 10^{30}\) could have been formed copiously as a dark matter contribution by the fragmentation of the AffleckDine condensate [358]. It is an appealing property of this scenario that the baryon and darkmatter abundance, \(\Omega _{B}\) and \(\Omega _{\mathrm{DM}}\), respectively, are related. For \(Q_B\simeq 10^{26}\) one obtains \(\Omega _{\mathrm{DM}} \simeq 10\Omega _{\mathrm{B}}\) close to the observed values.
The global charge Q associated with the Qball can be the same as baryon number (B) or lepton number (L) or some combination of them if these symmetries are connected to a global \(\mathrm{U}(1)\) symmetry. Since the global symmetry is spontaneously broken in the interior of the Qball with \(\phi \ne 0\), the soliton could catalyse nucleon decay traversing the detector volume. This is analogous to the case of monopolecatalysed nucleon decay. Even if the charge is not related to B or L, it is possible that the vacuum state associated with the Qball interior catalyses nucleon decay. This can happen if the scalar potential is very flat such that \(U(\phi _0^\dagger \phi _0) \ll m_\phi ^2\phi _0^\dagger \phi _0\). Interactions that lead to nucleon decay induced by new physics at an ultraviolet scale \(\Lambda \), for instance, in grand unified theories are typically suppressed by powers of \(\langle \phi \rangle /\Lambda \) and can become large in the Qball environment [359].
9.2 Strange quark matter
Using energy and symmetry arguments, it has been speculated that strange quark matter (SQM), a hypothetical form of matter with roughly equal numbers of up (u), down (d), and strange (s) quarks, could be the true ground state of QCD [362, 363, 364]. For a plasma of quarks in thermodynamical equilibrium it might be energetically preferable to condense into a phase containing strange quarks instead of ordinary matter with neutrons (udd) and protons (uud).
An approximate thermodynamical calculation with massless quarks and neglecting strong interactions shows that the average kinetic energy per quark in ordinary bulk matter could be reduced in bulk SQM by a factor of about 0.89 [363]. Therefore, it is feasible that the extra “penalty” paid by the presence of more massive strange quarks is overcompensated by the reduction in energy density. However, the metastable state of protons, neutrons, and composite nuclei would be very longlived, since conversion to the SQM ground state would proceed via weak interactions.
Lumbs of SQM, socalled strangelets, can have a large atomic mass number A and charge Z. Classical strangelets have a quark charge \(Z\sim 0.1A\) for low mass numbers (\(A\ll 700\)). For total quark charges exceeding \(Z\sim \alpha ^{1}\sim 137\) strong field QED corrections lead to screening and \(Z\sim 8A^{1/3}\) (\(A\gg 700\)) [364]. It has also been speculated that colour and flavour symmetries at high baryon densities might be broken simultaneously by the condensation of quark Cooper pairs [365]. In this scenario, the “colourflavourlocked” strangelets have charges of \(Z\sim 0.3A^{2/3}\) [366].
Stable strangelets can absorb ordinary matter in exothermic reactions involving \(u\rightarrow s+e+\bar{\nu }_e\) or \(u+d\rightarrow s+u\) [364]. If the strangelet carries a positive charge the Coulomb barrier will usually prevent a strangelet–nucleus system from collapse. However, neutronrich environments like neutron stars are not protected by this mechanism. In fact, if strange quark matter is stable then all compact stars like white dwarfs or neutron stars are likely to consist of it. Even the capture of a single strangelet would be sufficient to convert a neutronrich environment very rapidly.
Slowly moving strangelets – so called nuclearites [367] – lose their energy in matter due to atomic collisions. The excessive energy released will heat the medium and create thermal shocks. The hot expanding plasma will emit Planck radiation from its surface over a wide range of frequencies. IceCube is sensitive to optical photons energies of about \(2\div 4\) eV (300–600 nm). At nuclearite velocities expected for cold dark matter the fraction of total energy loss emitted in optical photons is of order \(10^{5}c\). Since this leads to signatures similar to slow magnetic monopoles or QBalls the detection efficiency is comparable again. Nuclearites with masses in excess of \(10^{14}\) GeV and typical velocities of order of \(10^{3}c\) reach underground detectors. Correspondingly, the sensitivity to the flux of nuclearites is roughly of the same order as that to slow monopole fluxes catalysing nucleon decays. Results for the MACRO experiments [368] and ongoing studies for the ANTARES detector [369] addressing fluxes of order \(10^{16}\)–\(10^{15}~\mathrm{cm}^{2} \ \mathrm{sr}^{1} \ \mathrm{s}^{1}\) underline the high potential for a corresponding reinterpretation of IceCube analyses.
9.3 Longlived charged massive particles
Many extensions of the Standard Model predict the existence of longlived charged massive particles (CHAMPs). These particles occur naturally in scenarios where the decay of charged particles is limited by (approximate) discrete symmetries and involves final states that have only very weak couplings. Analogously to muons, these CHAMPs have a reduced electromagnetic energyloss in matter due to the suppression of bremsstrahlung by the rest mass. Still, they may be detected by their Cherenkov emission and, due to the long range, even with an enlarged effective detection area.
In the following we will consider SUSY breaking scenarios where the righthanded stau is the nexttolightest SUSY particle (NLSP). However, most of the arguments apply equally well to other scenarios of CHAMPs with a decay length larger than other experimental scales (see, e.g., Ref. [370]). If \({\mathcal {R}}\)parity is conserved the stau NLSP can only decay into final states containing the LSP. Depending on the mass and coupling of this particle the lifetime of the stau NLSP can be very long and, in some cases, it can be considered as practically stable on experimental timescales.
In the case of a neutralino LSP, the stau NLSP can be very longlived if its mass is nearly degenerated with that of the neutralino [371]. However, there are also superweakly interacting candidates for the LSP in extensions of the MSSM, which provide the long lifetime of the NLSP more naturally. Possible scenarios include SUSY extensions of gravity with a gravitino \(\widetilde{G}\) LSP, SUSY versions of the PecceiQuinn axion and the corresponding axino LSP [372], and the MSSM with righthanded chiral neutrinos and a righthanded sneutrino LSP.
Staus produced in SUSY interactions of EHE neutrinos could traverse Cherenkov telescopes at the level of a few per year, assuming that extragalactic neutrino fluxes are close to the existing bounds [373, 374, 375, 376, 377, 378] or prompt atmospheric neutrino fluxes close to upper theoretical estimates [377]. The leadingorder SUSY contribution consists of chargino \(\chi \) or neutralino \(\chi ^0\) exchange between neutrinos and quarks, analogous to the partonlevel SM contributions shown in Fig. 11. The reactions produce sleptons and squarks, which promptly decay into lighter stau NLSPs.
Cosmic ray interactions in the atmosphere could also produce a longlived stau signal in neutrino telescopes [379] (see also Ref. [377]). At energies above \(10^4\) GeV the decay length of weakly decaying nucleons, charged pions and kaons is much larger than their interaction length in the air. Therefore, as they propagate in the atmosphere the probability that a longlived hadron (h) collides with a nucleon to produce SUSY particles is just \(\sigma _h^{\mathrm{SUSY}} / \sigma _h^{\mathrm{SM}}\), where \(\sigma _{h}^{\mathrm{SUSY}}\) is the cross section to produce the SUSY particles X (gluinos or squarks) and \(\sigma _h^{\mathrm{SM}}\) the Standard Model cross section of the hadron with nuclei in the atmosphere (which is also the total cross section to a good approximation). However, since \(\sigma _{h}^{\mathrm{SM}}\) is above 100 mb, it is apparent that this probability will be very small and that it would be much larger for a neutrino propagating in matter.
The energy of charged particle tracks observed in neutrino detectors is determined by measuring their specific energy loss, \(\mathrm{d}E/\mathrm{d}x\). For muons with energies above an energy of about 500 GeV, the energy loss rises linearly with energy. However, since the energy loss depends on the particle Lorentz boost, a highenergy stau is practically indistinguishable from a muon with reduced energy, \(E_\mu /E_{{\widetilde{\tau }}} \sim m_\mu /m_{{\widetilde{\tau }}}\). A smokinggun signal for pairproduced stau NLSPs are parallel tracks in the Cherenkov detector [373].
The detection efficiency of stau pairs, i.e., coincident parallel tracks, depends on the energies and directions of the staus, as in the case of single events, but also on their separation. A large fraction of staus reaching the detector will be accompanied by their stau partner from the same interaction. However, not all of the stau pairs might be seen as separable tracks in a Cherenkov telescope if they emerge from interactions too close to or also too far from the detector. The opening angle \(\theta _{\widetilde{\tau }\widetilde{\tau }}\) between staus can be estimated by the initial opening angle between the SUSY particles in deep inelastic scattering. The separation of stau tracks in the detector is then given as \(x\simeq 2\,\Delta \ell \,\tan (\theta _{\widetilde{\tau }\widetilde{\tau }}/2)\), where \(\Delta \ell \) is the distance of the interaction to the detector center.
Double tracks with low track separation are difficult to distinguish from single muons copiously produced either directly in air showers or from atmospheric neutrinos. A required minimum track separation in IceCube of 150 m was found to be necessary to suppress these muons, due to the geometry of the detector [380]. A possible background to the stau pair signal produced in neutrino interactions consists of parallel muon pair events from random coincidences produced by upgoing atmospheric neutrinos. However, this is bounded from above by the number of muons arriving within a coincidence timewindow requiring them to be almost parallel and is several orders of magnitude below the stau pair event rate with \(N_{\mu +\mu } \lesssim {\mathcal {O}}(10^{12})N_\mu \) (Ref. [381]). Muon pairs from charged current muonneutrino interactions involving final state hadrons that promptly decay into a second muon are expected to be more likely. This has been estimated in Ref. [376] for the production and decay of charmed hadrons.
The rate of stau pairs from neutrino production is largely uncertain and depends on the SUSY mass spectrum. If the SUSY mass spectrum close to observational bounds (see, e.g., Ref. [41]), the rate might reach a few events per decade in cubickilometer Cherenkov telescopes [373, 375, 376, 377].
Scenarios where \({\mathcal {R}}\)parity is not conserved are also possible. Although in these cases the new supersymmetric particles are not longlived enough to leave a track in the detector, resonance production in neutrinonucleon interactions would produce a detectable cascadetype signal in IceCube. The absence of prominent resonance features in the IceCube HESE spectrum has been used to set limits on the strength of Rparity violation as a function of squark mass [382]. Due to the high energies of the neutrinos observed by IceCube the limits on squark masses above 1 TeV even go beyond the reach of current accelerator experiments.
9.4 Fractional electric charges
The Standard Model intrinsically does not constrain the elementary charge but observationally it appears as a physical constant, i.e., all observed coloursinglet particles have integer multiples of elementary charge. As outlined in Sect. 8, magnetic monopoles would provide a mechanism to explain electrical charge quantisation. In GUT theories the resulting quantisation is driven by the minimum possible magnetic charge rather than directly. Free fractionally charged states hence are often predicted in multiples of 1 / 2 (e.g., in the PatiSalam model [383]) or 1 / 3, but other and smaller fractions are possible. Beyond this, fractional charges may exist in composite objects with large (\(10^{12}\) GeV) confinement scales, probably also contributing to dark matter [384].
Experimentally, fractionally charged particles in cosmic rays can be observed through their anomalously low energy loss and lower light emission. Cherenkov emission scales as the square of the particle’s electric charge e, so the amount of Cherenkov light emitted by a fractionally charged particle with charge \(\xi e\) will be reduced by a factor \(\xi ^2\) with respect to a minimum ionizing muon. The same quadratic factor enters into the ionization and pair production energy loss factors. Discarding one event from the signal region, the lowest upper limits on the flux of such particles have been reached by a study of the MACRO experiment [385] assuming a simple chargesquared scaling of the energy loss. The search for fractionally charged particles in IceCube can be based on searches for anomalously dim (compared to minimum ionizing muons) throughgoing particles. The combination of the high energy needed to traverse the detector and low light emission could constitute the signal of fractionally charged massive particles. Due to its size, it is expected that IceCube can reach a competitive sensitivity in such searches.
10 Summary
IceCube opened a new window to study the nonthermal universe in 2013 through the discovery of a highenergy neutrino flux of astrophysical origin, and a first identification of a highenergy neutrino point source may be possible through joint multimessenger observations. While these novel results can be taken as the beginning of neutrino astronomy, the potential exists to use IceCube to probe physics topics beyond astrophysics in particle physics, not least due to its sheer size.
The observation of secondary particles produced in interactions of neutrinos or cosmic rays with matter provides a probe of Standard Model interactions at energies only marginally covered or inaccessible by particle accelerator experiments. The continuous flux of atmospheric neutrinos allows to study standard neutrino flavour oscillations at a precision that is compatible with those of dedicated oscillation experiments. Moreover, the very long oscillation baselines (thousands of kilometres for atmospheric neutrinos or gigaparsec in astrophysical neutrinos) and the very high neutrino energies (up to PeV) can probe feeble deviations from the standard threeflavour oscillation scenario, that are otherwise undetectable. IceCube has indeed provided strict limits on the allowed parameter space for an additional light “sterile” neutrino state with no Standard Model interactions, reducing considerably the range of allowed values of the new oscillation variables \(\sin ^2 2\theta _{24}\) and \(\Delta m^2_{41}\). Similar analyses can set limits on the degree of Lorentz invariance violation, an effect that can be factorised in terms of operators proportional to powers of the neutrino energy. The energy reach of IceCube allows to probe higher dimension operators than previous experiments.
Neutrinos are also valuable indirect messengers of dark matter annihilation and decay in the Earth, Sun, Milky Way, local galaxies, or galaxy clusters. In general, neutrino emission does not suffer from large astrophysical fore and backgrounds like electromagnetic emission and does not suffer from deflections in magnetic fields like cosmic rays. Neither indirect dark matter detection with neutrinos shares the same systematic uncertainties from astrophysical or particle physics inputs with other search techniques. In this way, indirect limits on dark matter properties from neutrino observations are complementary to indirect searches with other messengers or direct searches with accelerator or scattering experiments. Furthermore, neutrinos can be visible from very distant dark matter sources like galaxy clusters and also probe the interior of compact sources like our Sun. Besides probing dark matter capture or self annihilation cross sections, IceCube has current world leading limits on the dark matter lifetime, extending the range of the dark matter masses probed up to two orders of magnitude with respect to results from Cherenkov telescopes.
Neutrino telescopes are also sensitive to a variety of exotic signatures produced by rare particles, like Big Bang relics, passing through the detector and emitting direct or indirect Cherenkov light, as well as luminescence. Probably the most interesting signal consists of relic monopoles that could be deciphered from atmospheric and astrophysical backgrounds as extremely bright tracks and/or anomalously slow particles. Other heavy exotic particles that could be visible in this are relic Qballs and strangelets. Cherenkov emission of longlived supersymmetric particles or fractionally charged particles can also be considered. These are just a few examples of the many possibilities how neutrino observatories can be uses as multipurpose particle detectors.
In this review we have summarised the many possibilities how the IceCube Observatory can probe fundamental questions of particle physics. Proposed future extensions of IceCube will enhance the sensitivity of these searches [386]. A lowenergy infill, such as PINGU [387], would provide highly competitive measurements of the atmospheric neutrino oscillation parameters, the neutrino mass ordering, or the rate of tau neutrino appearance. It would also be more sensitive to indirect signals of lowmass dark matter. On the other hand, highenergy extensions would allow to study the astrophysical flux of neutrinos with better precision and over a wider energy range. This would reduce systematic uncertainties regarding neutrino spectra and flavour composition and help to establish astrophysical neutrinos as a probe of neutrino interactions and oscillations over ultralong baselines.
Footnotes
 1.
Note that in this particular example, also the Cherenkov light emission from the hadronic cascade X is visible in the detector.
 2.
All neutrino flavours take part in coherent scattering via neutral current interactions, but this corresponds to a flavouruniversal potential term, that has no effect on oscillations.
 3.
The invariance of physical laws under simultaneous charge conjugation (C), parity reflection (P), and time reversal (T).
 4.
Note that in a galactocentric coordinate system considered for WIMP annihilations in the Galactic halo, the Earth’s position is \(\mathbf{r}_{\oplus } \simeq \mathbf{r}_{\odot } \simeq (0,8.5\mathrm{kpc},0)\).
 5.
Strictly speaking, the flavour transition probabilities depend on the distance l and should appear under the lineofsight integral. However, in most situations discussed in the following it can be treated as a constant.
 6.
The dependence on arrival time is relevant for indirect dark matter searches from the Sun, which can be effectively parametrised by the angular distance \(\psi \) between reconstructed arrival direction and the known position of the Sun at the arrival time (corrected for the lighttravel time of about 8 min).
 7.
Previously, he had successfully used this method to propose the existence of electrons with positive charge [312], i.e., positrons.
 8.
Note, that in a Gaussian system, where the unit of charge is defined via \(e^2 =\hbar c \alpha \), this quantisation condition is expressed as \(q_m q_e = n\hbar c/2\). However, to avoid confusion, we use the Heaviside–Lorentz system with the conventional definition \(e^2 =4\pi \alpha \) (in natural units).
 9.
In this scenario, a combination of MSSM scalar fields with nonzero baryon number B develops a large expectation value at the end of inflation and decays.
Notes
Acknowledgements
We would like to thank our colleagues in the IceCube collaboration for discussions and support. MA acknowledges support by Danmarks Grundforskningsfond (project no. 1041811001) and by VILLUM FONDEN (project no. 18994).
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