Probing Particle Physics with IceCube
Markus Ahlersa,1, Klaus Helbingb,2, Carlos P´erezde los Herosc,3
1Niels Bohr International Academy & Discovery Centre, Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark 2Department of Physics, University of Wuppertal, D-42119 Wuppertal, Germany 3Department of Physics and Astronomy, Uppsala University, S-75120 Uppsala, Sweden
Abstract The IceCube observatory located at the South experimental setups that utilise natural resources. Not Pole is a cubic-kilometre optical Cherenkov telescope only that – the detector material has to be suitable so that primarily designed for the detection of high-energy astro- these few interactions can be made visible and separated physical neutrinos. IceCube became fully operational in from large atmospheric backgrounds. 2010, after a seven-year construction phase, and reached a Despite these obstacles, there exist a variety of exper- milestone in 2013 by the first observation of cosmic neutri- imental concepts to detect high-energy neutrinos. One nos in the TeV-PeV energy range. This observation does particularly effective method is based on detecting the not only mark an important breakthrough in neutrino radiation of optical Cherenkov light produced by rela- astronomy, but it also provides a new probe of particle tivistic charged particles. This requires the use of optically physics related to neutrino production, mixing, and inter- transparent detector media like water or ice, where the action. In this review we give an overview of the various Cherenkov emission can be read out by optical sensors possibilities how IceCube can address fundamental ques- deployed in the medium. This information then allows tions related to the phenomena of neutrino oscillations to reconstruct the various Cherenkov light patterns pro- and interactions, the origin of dark matter, and the ex- duced in neutrino events and infer the neutrino flavour, istence of exotic relic particles, like monopoles. We will arrival direction, and energy. The most valuable type summarize recent results and highlight future avenues. of events for neutrino astronomy are charged current interactions of muon-neutrinos with matter near the de- Keywords IceCube physics beyond the Standard · tector. These events produce muons that can range into Model neutrino telescopes neutrino oscillation · · · the detector and allow the determination of the initial neutrino interactions sterile neutrinos dark matter · · · muon-neutrino direction within a precision of better than magnetic monopoles one degree. PACS 95.35.+d 14.80.Hv 13.15.+g 14.60.Lm 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. 1 Introduction The principal challenge of any neutrino telescope is the large background of atmospheric muons and neutrinos Not long after the discovery of the neutrino by Cowan produced in cosmic ray interactions in the atmosphere. and Reines in 1956 [1], the idea emerged that it repre- High-energy muons produced in the atmosphere have a sented the ideal astronomical messenger [2]. Neutrinos limited range in ice and bedrock. Nevertheless IceCube, arXiv:1806.05696v1 [astro-ph.HE] 14 Jun 2018 are only weakly interacting with matter and can cross at a depth of 1.5 kilometres, observes about 100 billion at- cosmic distances without being absorbed or scattered. mospheric muon events per year. This large background However, this weak interaction is also a challenge for the can be drastically reduced by only looking for up-going observation of these particles. Early estimates of the ex- events, i.e., events that originate below the horizon. This pected flux of high-energy neutrinos associated with the cut leaves only muons produced by atmospheric neutri- observed flux of extra-galactic cosmic rays indicated that nos at a rate of about 100,000 per year. While these large neutrino observatories require gigaton masses as a neces- backgrounds are an obstacle for neutrino astronomy they sary condition to observe a few neutrino interactions per provide a valuable probe for cosmic ray physics in gen- year [3]. These requirements can only be met by special eral and for neutrino oscillation and interaction studies ae-mail: [email protected] in particular. be-mail: [email protected] In this review we want to highlight IceCube’s po- ce-mail: [email protected] tential as a facility to probe fundamental physics. There 2 exist a variety of methods to test properties of the Stan- Any review has its limitations, both in scope and tim- dard Model (SM) and its possible extensions. The flux ing. We have given priority to present a comprehensive of atmospheric and astrophysical neutrinos observed in view of the activity of IceCube in areas related to the IceCube allows to probe fundamental properties in the topic of this review, rather than concentrating on a few neutrino sector related to the standard neutrino oscil- recent results. We have also chosen at times to include lations (neutrino mass differences, mass ordering, and older results for completeness, or when it was justified flavour mixing) and neutrino-matter interactions. It also as an illustration of the capabilities of the detector on a provides a probe for exotic oscillation effects, e.g., related given topic. The writing of any review develops along to the presence of sterile neutrinos or non-standard neu- its own plot and updated results on some analyses have trino interactions with matter. The ultra-long baselines been made public while this paper was in preparation, associated with the propagation of cosmic neutrinos ob- and could not be included here. This only reflects on the served beyond 10 TeV allow for various tests of feeble lively activity of the field. neutrino oscillation effects that can leave imprints on the Throughout this review we will use natural units, h¯ = oscillation-averaged flavour composition. c = 1, unless otherwise stated. Electromagnetic expres- One of the fundamental questions in cosmology is the sions will be given in the Heaviside-Lorentz system with origin of dark matter that today constitutes one quarter of e = µ = 1, α = e2/4π 1/137 and 1Tesla 195eV2. 0 0 ' ' the total energy density of the Universe. Candidate par- ticles for this form of matter include weakly interacting massive particles (WIMPs) that could have been ther- mally produced in the early Universe. IceCube can probe 2 The IceCube Neutrino Observatory the existence of these particles by the observation of a flux of neutrinos produced in the annihilation or decay of The IceCube Neutrino Observatory [4] consists of an in- WIMPs gravitationally clustered in nearby galaxies, the ice array (simply “IceCube” hereafter) and a surface air halo of the Milky Way, the Sun, or the Earth. In the case shower array, IceTop [5]. IceCube utilises one cubic kilo- of compact objects, like Sun and Earth, neutrinos are the metre of the deep ultra-clear glacial ice at the South Pole only SM particles that can escape the dense environments as its detector medium (see left panel of Fig.1). This vol- to probe the existence of WIMPs. ume is instrumented with 5,160 Digital Optical Modules Neutrino telescopes can also probe exotic particles (DOMs) that register the Cherenkov light emitted by rel- leaving direct or indirect Cherenkov signals during their ativistic charged particles passing through the detector. passage through the detector. One important example The DOMs are distributed on 86 read-out and support are relic magnetic monopoles, topological defects that cables (“strings”) and are deployed between 1.5 km and could have formed during a phase transition in the early 2.5 km below the surface. Most strings follow a triangu- Universe. Light exotic particles associated with exten- lar grid with a width of 125 m, evenly spaced over the sions of the Standard Model can also be produced by volume (see green markers in right panel of Fig.1). the interactions of high-energy neutrinos or cosmic rays. Eight strings are placed in the centre of the array and Collisions of neutrinos and cosmic rays with nucleons in are instrumented with a denser DOM spacing and typical the vicinity of the Cherenkov detector can reach center-of- inter-string separation of 55 m (red markers in right panel mass energies of the order √s 1 TeV (neutrino energy of Fig.1). They are equipped with photomultiplier tubes ' E 1015 eV) or even √s 100 TeV (cosmic ray energy with higher quantum efficiency. These strings, along with ν ' ' E 1020 eV), respectively, only marginally probed by the first layer of the surrounding standard strings, form CR ' collider experiments. the DeepCore low-energy sub-array [6]. Its footprint is The outline of this review is as follows. We will start in depicted by a blue dashed line in Fig.1. While the original sections2 and3 with a description of the IceCube detec- IceCube array has a neutrino energy threshold of about tor, atmospheric backgrounds, standard event reconstruc- 100 GeV,the addition of the denser infill lowers the energy tions, and event selections. In section4 we summarise the threshold to about 10 GeV. The DOMs are operated to phenomenology of three-flavour neutrino oscillation and trigger on single photo-electrons and to digitise in-situ IceCube’s contribution to test the atmospheric neutrino the arrival time of charge (“waveforms”) detected in the mixing. We will cover standard model neutrino inter- photomultiplier. The dark noise rate of the DOMs is about actions in section5 and highlight recent measurements 500 Hz for standard modules and 800 Hz for the high- of the inelastic neutrino-nucleon cross sections with Ice- quantum-efficiency DOMs in the DeepCore sub-array. Cube. We then move on to discuss IceCube’s potential Some results highlighted in this review were derived to probe non-standard neutrino oscillation with atmo- from data collected with the AMANDA array [7], the spheric and astrophysical neutrino fluxes in section6. In predecessor of IceCube built between 1995 and 2001 at section7 we highlight IceCube results on searches for the same site, and in operation until May 2009. AMANDA dark matter and section8 is devoted to magnetic mono- was not only a proof of concept and a hardware test-bed poles while section9 covers other massive exotic particles for the IceCube technology, but a full fledged detector and Big Bang relics. which obtained prime results in the field. 3
Fig. 1 Sketch of the IceCube observatory. The right plot shows the surface footprint of IceCube. The green circles represent the standard IceCube strings, separated by 125 m, and the red ones the more densely instrumented strings with high quantum efficiency photomultiplier tubes. Strings belonging to the DeepCore sub-array are enclosed by the dashed line.
2.1 Neutrino Event Signatures cascades in the detector is rather spherical, see right panel of Fig.2. For cascades or tracks fully contained in the de- As we already highlighted in the introduction, the main tector, the energy resolution is significantly better since event type utilised in high-energy neutrino astronomy are the full energy is deposited in the detector and it is pro- charged current (CC) interactions of muon neutrinos with portional to the detected light. The ability to distinguish nucleons (N), νµ + N µ− + X. These interactions pro- these two light patterns in any energy range is crucial, → duce high-energy muons that lose energy by ionisation, since cascades or tracks can contribute to background or bremsstrahlung, pair production and photo-nuclear inter- signal depending on the analysis performed. actions in the ice [8]. The combined Cherenkov light from the primary muon and secondary relativistic charged The electrons produced in charged current interac- particles leaves a track-like pattern as the muon passes tions of electron neutrinos, ν + N e + X, will con- e → − through the detector. An example is shown in the left tribute to an electromagnetic cascade that overlaps with panel of Fig.2. In this figure, the arrival time of Cherenkov the hadronic cascade X at the vertex. At energies of E ν ' light in individual DOMs is indicated by colour (earlier 6.3 PeV, electron anti-neutrinos can interact resonantly in red and later in blue) and the size of each DOM is with electrons in the ice via a W-resonance (“Glashow” 1 proportional to the total Cherenkov light it detected. . resonance) [10]. The W-boson decays either into hadro- Since the average scattering angle between the incoming nic states with a branching ratio (BR) of 67%, or into ' neutrino and the outgoing muon decreases with energy, leptonic states (BR 11% for each flavour). This type of 0.7 ' Ψν µ 0.7°(Eν/TeV)− [9], an angular resolution be- event can be visible by the appearance of isolated muon → ∼ low 1° can be achieved for neutrinos with energies above tracks starting in the detector or by spectral features in a few TeV, only limited by the detector’s intrinsic angular the event distribution [11]. resolution. This changes at low energies, where muon tracks are short and their angular resolution deteriorates Also the case of charged current interactions of tau rapidly. For neutrino energies of a few tens of GeVs the neutrinos, ντ + N τ + X, is special. Again, the hadro- angular resolution reaches a median of 40°. → ∼ nic cascade X is visible in Cherenkov light. The tau has All deep-inelastic interactions of neutrinos, both neu- a lifetime (at rest) of 0.29 ps and decays to leptons as tral current (NC), να + N να + X and charged current, τ µ + ν + ν (BR 18%) and τ e + ν + ν → − → − µ τ ' − → − e τ να + N `α− + X, create hadronic cascades X that are (BR 18%) or to hadrons (mainly pions and kaons, → ' visible by the Cherenkov emission of secondary charged BR 64%) as τ ν + mesons. With tau energies ' − → τ particles. However, these secondaries can not produce below 100 TeV these charged current events will also con- elongated tracks in the detector due to their rapid scatter- tribute to track and cascade events. However, the delayed ing or decay in the medium. Because of the large separa- decay of taus at higher energies can become visible in Ice- tion of the strings in IceCube and the scattering of light Cube, in particular above around a PeV when the decay in the ice, the Cherenkov light distribution from particle length becomes of the order of 50 m. This allows for a 1Note that in this particular example, also the Cherenkov light variety of characteristic event signatures, depending on emission from the hadronic cascade X is visible in the detector. the tau energy and decay channel [12,13]. 4
Fig. 2 Two examples of events observed with IceCube. The left plot shows a muon track from a νµ interaction crossing the detector. Each coloured dot represents a hit DOM. The size of the dot is proportional to the amount of light detected and the colour code is related to the relative timing of light detection: read denotes earlier hits, blue, later hits. The right plot shows a νe or ντ charged-current (or any flavour neutral-current) interaction inside the detector.
3 Event Selection and Reconstruction directions using sophisticated likelihood-based recon- structions [16,17]. These reconstructions maximise the In this review we present results from analyses which use likelihood function built from the probability of obtain- different techniques tailored to the characteristics of the ing the actual temporal and spatial information in each signals searched for. It is therefore impossible to give a DOM (“hit”) given a set of track parameters (vertex, time, description of a generic analysis strategy which would energy, and direction). For low-energy events, where the cover all aspects of every approach. There are, however, event signature is contained within the volume of the de- certain levels of data treatment and analysis techniques tector, a joint fit of muon track and an hadronic cascade that are common for all analyses in IceCube, and which at the interaction vertex is performed. For those events we cover in this section. 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 3.1 Event Selection each event. This allows, for example, to estimate the prob- ability of each event to be either a track or a cascade. Each Several triggers are active in IceCube in order to prese- analysis will then use complex classification methods lect potentially interesting physics events [14]. They are based on machine-learning techniques to further sepa- based on finding causally connected spatial hit distribu- rate a possible signal from the background. Variables that tions in the array, typically requiring a few neighbour or describe the quality of the reconstructions, the time de- next-to-neighbour DOMs to fire within a predefined time velopment and the spatial distribution of hit DOMs in window. Most of the triggers aim at finding relativistic the detector are usually used in the event selection. 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., monopoles 3.2 Effective Area and Volume catalysing nucleon-decay, which can induce events last- ing up to milliseconds, a dedicated trigger sensitive to After the analysis-dependent event reconstruction and non-relativistic particles with velocities down to β 4 has − selection, the observed event distribution in energy and also been implemented. arrival direction can be compared to the sum of back- When a trigger condition is fulfilled the full detector ground and signal events. For a given neutrino flux, φ , is read out. IceCube triggers at a rate of 2.5 kHz, collect- ν the total number of signal events, µ , expected at the ing about 1 TB/day of raw data. To reduce this amount s detector can be expressed as of data to a more manageable level, a series of software filters are applied to the triggered events: fast reconstruc- Z Z tions [15] are performed on the data and a first event = eff( ) ( ) µs T ∑ dΩ dEν Aνα Eν, Ω φνα Eν, Ω , (1) selection carried out, reducing the data stream to about α 100 GB/day. These reconstructions are based on the posi- eff tion and time of the hits in the detector, but do not include where T is the exposure time and Aνα the detector effec- information about the optical properties of the ice, in or- tive area for neutrino flavour α. The effective area encodes der to speed up the computation. The filtered data is the trigger and analysis efficiencies and depends on the transmitted via satellite to several IceCube institutions in observation angle and neutrino energy. the North for further processing. In practice, the figure of merit of a neutrino telescope Offline processing aims at selecting events according is the effective volume, Veff, the equivalent volume of a to type (tracks or cascades), energy, or specific arrival detector with 100% detection efficiency of neutrino events. 5
This quantity is related to the signal events as cosmic ray Z Z = eff( ) ( ) ( ) Atmosphere µs ∑ dΩ dEνVνα Eν, Ω Tnσ Eν φeνα Eν, Ω , α (2) where φe is the neutrino flux after taking into account atmospheric cosmic muon Earth absorption and regeneration effects, n is the local neutrino target density, and σ the neutrino cross section for the π-θ relevant neutrino signal. The effective volume allows to
express the event number by the local density of events, ~12,700 km i.e., the quantity within [ ]. This definition has the prac- · tical advantage that the effective volume can be simu- lated from a uniform distribution of neutrino events: if IceCube ngen(Eν, Ω) is the number of Monte-Carlo events gener- up-going gen ated over a large geometrical generation volume V by down-going neutrinos with energy injected into the direction Ω, atmospheric Eν neutrino then the effective volume is given by cosmic ray eff ns gen V (Eν, Ω) = V (Eν, Ω) , (3) ν n (E , Ω) gen ν Fig. 3 An illustration of neutrino detection with IceCube located at the South Pole. Cosmic ray interactions in the atmosphere produce where ns is the number of remaining signal events after a large background of high-energy muon tracks (solid blue arrows) all the selection cuts of a given analysis. in IceCube. This background can be reduced by looking for up- going tracks produced by muon neutrinos (dashed blue arrows) that cross the Earth and interact close to the detector. The remaining 3.3 Background Rejection background of up-going tracks produced by atmospheric muon neutrinos can be further reduced by energy cuts. There are two backgrounds in any analysis with a neu- trino telescope: atmospheric muons and atmospheric neu- trinos, both produced in cosmic-ray interactions in the spectra of the signal with respect to the background, and atmosphere. The atmospheric muon background mea- are less sensitive to the absolute normalisation of the lat- sured by IceCube [18] is much more copious than the at- ter. For others, like monopole searches, misreconstructed mospheric neutrino flux, by a factor up to 106 depending atmospheric muons can reduce the sensitivity of the de- on declination. Note that cosmic ray interactions can pro- tector. We will describe in more detail how each analysis duce several coincident forward muons (“muon bundle”) deals with this background when we touch upon specific which are part of the atmospheric muon background. analyses in the rest of this review. Muon bundles can be easily identified as background in The atmospheric neutrino flux constitutes an irre- some cases, but they can also mimic bright single tracks ducible background for any search in IceCube, and sets (like monopoles for example) and are more difficult to the baseline to define a discovery in many analyses. It is separate from the signal in that case. Even if many of the therefore crucial to understand it both quantitatively and IceCube analyses measure the atmospheric muon back- qualitatively. The flux of atmospheric neutrinos is domi- ground from the data, the CORSIKA package [19] is gener- nated by the production and decay of mesons produced ally used to generate samples of atmospheric muons that by cosmic ray interactions with air molecules [20]. The are used to cross-validate certain steps of the analyses. behaviour of the neutrino spectra can be understood from The large background of atmospheric muons can be the competition of meson (m) production and decay in efficiently reduced by using the Earth as a filter, i.e., by the atmosphere: At high energy, where the meson decay selecting up-going track events, at the expense of reducing rate is much smaller than the production rate, the meson the sky coverage of the detector to the Northern Hemi- flux is calorimetric and simply follows the cosmic ray spec- Γ sphere (see Fig.3). Still, due to light scattering in the ice trum, Φm ∝ E− . Below a critical energy em, where the and the emission angle of the Cherenkov cone, a frac- decay rate becomes comparable to the production rate, tion of the down-going atmospheric muon tracks can be the spectrum becomes harder by one power of energy, 1 Γ misreconstructed as up-going through the detector. This Φm ∝ E − . The corresponding neutrino spectra from typically leads to a mismatch between the predicted atmo- the decay of mesons are softer by one power of energy, spheric neutrino rate and the data rate at the final level of Φν ∝ Φm/E due to the energy dependence of the meson many analyses. There are analyses where a certain atmo- decay rate. spheric muon contamination can be tolerated and it does The neutrino flux arising from pion and kaon decay not affect the final result. These are searches that look for is reasonably well understood, with an uncertainty in a difference in the shape of the energy and/or angular the range 10%-20% [20]. Figure4 shows the atmospheric 6
3 atmospheric neutrino spectrum is usually treated as a nui- 10− sance parameter, while the energy distributions follows neutrino fluxes the model prediction of Ref. [30]. 4 (per flavor) ] 10− For high enough neutrino energies ( (10) TeV), the 1 O − possibility exists of rejecting atmospheric neutrinos by + sr atmo. νµ ν¯µ
1 5 (before HESE veto) selecting starting events, where an outer layer of DOMs
− 10− s acts as a virtual veto region for the neutrino interaction 2
− vertex. This technique relies on the fact that atmospheric 6 10− neutrinos are accompanied by muons produced in the same air shower, that would trigger the veto [35,36]. The atmo. νe + ν¯e
[GeV cm 7 (before HESE veto) price to pay is a reduced effective volume of the detector
¯ 10 ν −
+ for down-going events and a different sensitivity for up-
ν IC HESE φ going and down-going events. This approach has been 2 8 E 10− extremely successful, extending the sensitivity of IceCube to the Southern Hemisphere including the Galactic cen- 9 IC ν + ν¯ tre. There is not a generic veto region defined for all Ice- 10− µ µ Cube analyses, but each analysis finds its optimal defini- 10 102 103 104 105 106 107 108 tion depending on its physics goal. Events that present more than a predefined number of hits within some time E [GeV] window in the strings included in the definition of the Fig. 4 Summary of neutrino observations with IceCube (per veto volume are rejected. A reduction of the atmospheric flavour). The black and grey data shows IceCube’s measurement of the atmospheric νe + νe [21,22] and νµ + νµ [23] spectra. The green muon background by more than 99%, depending on anal- data show the inferred bin-wise spectrum of the four-year high- ysis, can be achieved in this way (see for example [36, energy starting event (HESE) analysis [24,25]. The green line and 26]). green-shaded area indicate the best-fit and 1σ uncertainty range of a power-law fit to the HESE data. Note that the HESE analysis ve- This approach has been also the driver behind one toes atmospheric neutrinos, and the true background level is much of the most exciting recent results in multi-messenger lower as indicated in the plot. The green line and green-shaded area astronomy: the first observation of high-energy astro- indicate the best-fit and 1σ uncertainty range of a power-law fit of physical neutrinos by IceCube. The first evidence of this the up-going muon neutrino analysis [26]. flux could be identified from a high-energy starting event (HESE) analysis, with only two years of collected data in neutrino fluxes measured by IceCube. The atmospheric 2013 [37,24,25]. The event sample is dominated by cas- cade events, with only a rather poor angular resolution muon neutrino spectrum (νµ + νµ) was obtained from one year of IceCube data (April 2008 to May 2009) us- of about 10°. The result is consistent with an excess of ing up-going muon tracks [23]. The atmospheric electron events above the atmospheric neutrino background ob- served in up-going muon tracks from the Northern Hemi- neutrino spectra (νµ + νµ) were analysed by looking for contained cascades observed with the low-energy infill sphere [38,26]. Figure4 summarises the neutrino spectra array DeepCore between June 2010 and May 2011 in the inferred from these analyses. Based on different methods energy range from 80 GeV to 6 TeV [21]. This agrees well for reconstruction and energy measurement, their results with a more recent analysis using contained events ob- agree, pointing at extra-galactic sources whose flux has served in the full IceCube detector between May 2011 and equilibrated in the three flavours after propagation over cosmic distances [39] with ν : ν : ν 1 : 1 : 1. While May 2012 with an extended energy range from 100 GeV e µ τ ∼ to 100 TeV [22]. All measurements agree well with model both types of analyses have now reached a significance prediction of “conventional” atmospheric neutrinos pro- of more than 5σ for an astrophysical neutrino flux, the duced in pion and kaon decay. IceCube uses the public origin of this neutrino emission remains a mystery (see, Monte Carlo software GENIE [27] and the internal soft- e.g., Ref. [40]). ware NUGEN (based on [28]) to generate samples of at- mospheric neutrinos for its analyses, following the flux described in [29]. 4 Standard Neutrino Oscillations Kaons with an energy above 1 TeV are also signifi- cantly attenuated before decaying and the “prompt” com- Over the past decades, experimental evidence for neu- ponent, arising mainly from very short-lived charmed trino flavour oscillations has been accumulating in solar 0 mesons (D±, D , Ds and Λc) is expected to dominate the (νe), atmospheric (νe,µ & νe,µ), reactor (νe), and accelerator spectrum. The prompt atmospheric neutrino flux, how- (νµ & νµ) neutrino data (for a review see [41]). These oscil- ever, is much less understood, because of the uncertainty lation patterns can be convincingly interpreted as a non- on the cosmic ray composition and relatively poor knowl- trivial mixing of neutrino flavour and mass states with a edge of QCD processes at small Bjorken-x [30,31,32,33, small solar and large atmospheric mass splitting. Neutrinos 34]. In IceCube analyses the normalisation of the prompt να with flavour α = e, µ, τ refer to those neutrinos that 7 couple to leptons `α in weak interactions. Flavour oscilla- The first compelling evidence for the phenomenon tions are based on the effect that these flavour states are a of atmospheric neutrino oscillations was observed with non-trivial superposition of neutrino mass eigenstates νj Super-Kamiokande (SK) [45]. The simplest and most di- (j = 1, 2, 3) expressed as rect interpretation of the atmospheric data [46] is oscilla- tions of muon neutrinos, most likely converting into tau να = U∗ ν , (4) | i ∑ αj| ji neutrinos. The survival probability of νµ can be approxi- j mated as an effective two-level system with where the Uαj’s are elements of the unitary neutrino mass- 2 2 to-flavour mixing matrix, the so-called Pontecorvo-Maki- Pν ν = 1 sin 2θatm sin ∆atm (8) µ→ µ − Nagakawa-Sakata (PMNS) matrix [42,43,44]. In general, the mixing matrix U has nine degrees of freedom, which The angular distribution of contained events in SK shows can be reduced to six by absorbing three global phases that for E 1 GeV, the deficit comes mainly from ν ∼ into the flavour states να. The neutrino mixing matrix L 102 104 km. The corresponding oscillation phase atm ∼ − U is then conveniently parametrised [41] by three Euler must be nearly maximal, ∆atm 1, which requires a mass 2 4 ∼2 2 rotations θ12, θ23, and θ13, and three CP-violating phases splitting ∆m 10 10 eV . Moreover, assum- atm ∼ − − − δ, α1 and α2, ing that all up-going νµ’s which would yield multi-GeV events oscillate into a different flavour while none of iδ 1 0 0 c13 0 s13e− c12 s12 0 the down-going ones do, the observed up-down asym- U = 0 c s 0 1 0 s c 0 23 23 − 12 12 metry leads to a mixing angle very close to maximal, 0 s c s eiδ 0 c 0 0 1 2 − 23 23 − 13 13 sin 2θatm > 0.92 at 90%C.L. These results were later con- diag(eiα1/2, eiα2/2, 1) . (5) firmed by the KEK-to-Kamioka (K2K) [47] and the Main · Injector Neutrino Oscillation Search (MINOS) [48] experi- Here, we have made use of the abbreviations sin θij = sij ments, which observed the disappearance of accelerator and cos θij = cij. The phases α1/2 are called Majorana νµ’s at a distance of 250 km and 735 km, respectively, as a phases, since they have physical consequences only if distortion of the measured energy spectrum. the neutrinos are Majorana spinors, i.e., their own anti- Furthermore, solar neutrino data collected by SK [49], particles. Note, that the phase δ (Dirac phase) appears the Sudbury Neutrino Observatory (SNO) [50] and Borex- only in combination with non-vanishing mixing sin θ13. ino [51] show that solar νe’s produced in nuclear pro- Neutrino oscillations can be derived from plane-wave cesses convert to νµ or ντ. For the interpretation of solar solutions of the Hamiltonian, that coincide with mass neutrino data it is crucial to account for matter effects eigenstates in vacuum, exp( i(ET pL)). To leading or- − − that can have a drastic effect on the neutrino flavour der in m/E, the neutrino momentum is p E m2/(2E) ' − evolution. The coherent scattering of electron neutrinos and a wave packet will travel a distance L T. There- off background electrons with a density Ne introduces ' 2 fore, the leading order phase of the neutrino at distance L a unique potential term Vmat = √2GF Ne, where GF is from its origin is exp( im2L/(2E)). From this expression − the Fermi constant [52]. In the effective two-level system we see that the effect of neutrino oscillations depend on for the survival of electron neutrinos, the effective matter the difference of neutrino masses, ∆m2 m2 m2. After 2 ij i j oscillation parameters (∆meff & θeff) relate to the vacuum ≡ − 2 traveling a distance L an initial state να becomes a su- values (∆m & θ ) as perposition of all flavours, with probability of transition 2 1 to flavour β given by Pνα νβ = νβ να . This can be " # → |h | i| 2 2 2 expressed in terms of the PMNS matrix elements as [41] ∆meff Ne 2 2 2 = 1 cos 2θ + sin 2θ , (9) ∆m − Nres 2 Pνα νβ = δαβ 4 ∑ (Uα∗i Uβi Uαj Uβ∗j) sin ∆ij 1 → − < tan 2θ Ne − i>j eff = 1 , (10) tan 2θ − Nres + 2 ∑ (Uα∗i Uβi Uαj Uβ∗j) sin 2∆ij , (6) i>j = where the resonance density is given by where the oscillation phase ∆ij can be parametrised as ∆m2 cos 2θ 2 2 1 Nres = . (11) ∆mij L ∆mij L Eν − √22EGF ∆ij = 1.27 2 . (7) 4Eν ' eV km GeV The effective oscillation parameters in the case of electron Note, that the third term in Eq. (6) comprises CP-violating anti-neutrinos are the same as (9) and (10) after replacing effects, i.e., this term can change sign for the process Ne Ne. Pνα νβ , corresponding to the exchange U U∗ in Eq. (6). → − → ↔ For the standard parametrisation (5) the single CP-violating 2All neutrino flavours take part in coherent scattering via neutral contribution can be identified as the Dirac phase δ; oscil- current interactions, but this corresponds to a flavour-universal lation experiments are not sensitive to Majorana phases. potential term, that has no effect on oscillations. 8
normal ordering inverted ordering 2 2 +0.21 5 +0.21 5 ∆m21 [eV ] 7.40 0.20 10− 7.40 0.20 10− 2 2 −+0.033× 3 − × ∆m31 [eV ] 2.494 0.031 10− — 2 2 − × +0.032 3 ∆m23 [eV ] — 2.465 0.031 10− − × +0.78 +0.78 θ12 [°] 33.62 0.76 33.62 0.76 +−1.9 +−1.4 θ23 [°] 47.2 3.9 48.1 1.9 +−0.15 +−0.14 θ13 [°] 8.54 0.15 8.58 0.14 −+43 −+26 δCP [°] 234 31 278 29 − − Table 1 Results of a global analysis [58] of mass splittings, mixing angles, and Dirac phase for normal and inverted mass ordering. We best-fit parameters are shown with 1σ uncertainty.
Fig. 5 The survival probability of muon neutrinos (averaged over νµ and νµ) as a function of zenith angle and energy. Figure from The previous mixing and oscillation parameters are Ref. [64] derived under the assumption of a constant electron den- sity Ne. If the electron density along the neutrino tra- jectory is only changing slowly compared to the effec- Neutrino oscillation measurements are only sensitive tive oscillation frequency, the effective mass eigenstates to the relative neutrino mass differences. The absolute will change adiabatically. Note that the oscillation fre- neutrino mass scale can be measured by studying the quency and oscillation depth in matter exhibits a resonant electron spectrum of tritium (3H) β-decay. Present upper behaviour [53,54,52]. This Mikheyev-Smirnov-Wolfenstein limits (95% C.L.) on the (effective) electron anti-neutrino
(MSW) resonance can have an effect on continuous neu- mass are at the level of mνe < 2 eV [59,60]. The KATRIN trino spectra, but also on monochromatic neutrinos pass- experiment [61] is expected to reach a sensitivity of mνe < ing through matter with slowly changing electron den- 0.2 eV. Neutrino masses are also constrained by their sities, like the radial density gradient of the Sun. Once effect on the expansion history of the Universe and the these matter effect is taken into account, the observed formation of large-scale structure. Assuming standard intensity of solar electron neutrinos at different energies cosmology dominated at late times by dark matter and compared to theoretical predictions can be used to ex- dark energy, the upper limit (95% C.L.) on the combined tract the solar mixing parameters. In addition to solar neutrino masses is ∑i mi < 0.23 eV [62]. neutrino experiments, the KamLAND Collaboration [55] The mechanism that provides neutrinos with their has measured the flux of νe from distant reactors and find small masses is unknown. The existence of right-handed that νe’s disappear over distances of about 180 km. This neutrino fields, νR, would allow to introduce a Dirac mass observation allows a precise determination of the solar term of the form mDνLνR + h.c., after electroweak sym- mass splitting ∆m2 consistent with solar data. metry breaking. Such states would be “neutral” with The results obtained by short-baseline reactor neu- respect to the standard model gauge interactions, and trino experiments show that the remaining mixing angle therefore sterile [44]. However, the smallness of the neu- θ13 is small. This allows to identify the mixing angle θ12 trino masses would require unnaturally small Yukawa as the solar mixing angle θ and θ23 as the atmospheric couplings. This can be remedied in seesaw models (see, mixing angle θatm. Correspondingly, the mass splitting e.g., Ref. [63]). Being electrically neutral, neutrinos can 2 2 2 2 be Majorana spinors, i.e., spinors that are identical to can be identified as ∆m ∆m21 and ∆matm ∆m32 ' ' | | ' T ∆m2 . However, observations by the reactor neutrino their charge-conjugate state, ψc ψ , where is the | 31| ≡ C C experiments Daya-Bay [56] and RENO [57] show that the charge-conjugation matrix. In this case, we can introduce c small reactor neutrino mixing angle θ13 is larger than zero. Majorana mass terms of the form mLνLνL/2 + h.c. and As pointed out earlier, this is important for the observa- the analogous term for νR. In seesaw models the individ- tion of CP-violating effects parametrised by the Dirac ual size of the mass terms are such that m 0 and L ' phase δ in the PMNS matrix (5). m m . After diagonalization of the neutrino mass D R The global fit to neutrino oscillation data is presently matrix, the masses of active neutrinos are then propor- incapable to determine the ordering of neutrino mass tional to m m2 /m . This would explain the smallness i ' D R states. The fit to the data can be carried out under the of the effective neutrino masses via a heavy sector of assumption of normal (m1 < m2 < m3) or inverted (m3 < particles beyond the Standard Model. m1 < m2) mass ordering. A recent combined analysis [58] of solar, atmospheric, reactor, and accelerator neutrino data gives the values for the mass splittings, mixing an- 4.1 Atmospheric Neutrino Oscillations with IceCube gles, and CP-violating Dirac phase for normal or inverted mass ordering shown in Table1. Note that, presently, the The atmospheric neutrino “beam” that reaches IceCube Dirac phase is inconsistent with δ = 0 at the 3σ level, allows to perform high-statistics studies of neutrino os- independent of mass ordering. cillations at higher energies, and therefore is subject to 9
× -6 low-energy sample 10 7 MINOS, 2012 90% best fit ANTARES 6 Super-K, 2012, 90% ANTARES, 68% rate (Hz) 6 best fit IceCube ANTARES, 90% IceCube-79, 68% 5 best fit MINOS IceCube-79, 90% )
2 5 eV
4 -3 4 | (10 2
3 m ∆ | 3 2
2 1
-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.4 0.5 0.6 0.7 0.8 0.9 1 2 θ cos(reconstructed zenith angle) sin (2 23) Fig. 6 Left Panel: Angular distribution of contained events in DeepCore (i.e., with energies between approximately 10 GeV and 60 GeV), compared with the expectation from the non-oscillation scenario (red area) and with oscillations (grey area) assuming current best-fit values 2 3 2 2 of ∆m32 = 2.39 10− eV and sin (2θ23) = 0.995, from [65]. Systematic uncertainties are split into the normalisation contribution (dashed areas)| and| the shape× contribution (filled areas) for each assumption shown. Right Panel: Significance contours at 68% and 90% C.L. for the best-fit values of the IceCube analysis (red curves), compared with results of the ANTARES [66], MINOS [67] and Super-Kamiokande [68] experiments. Figures reprinted with permission from Ref. [69] (Copyright 2013 APS)
different systematic uncertainties, than those typically a reference, since νµ disappearance due to oscillations available in reactor- or accelerator-based experiments. At- at higher energies ( (100) GeV) is not expected. The O mospheric neutrinos arrive at the detector from all direc- atmospheric muon background is reduced to a negligi- tions, i.e., from travelling more than 12,700 km (vertically ble level by removing tracks that enter the DeepCore up-going) to about 10 km (vertically down-going), see fiducial volume from outside, and by only considering Fig.3. The path length from the production point in the up-going events, i.e., events that have crossed the Earth atmosphere to the detector is therefore related to the mea- (cos θ 0), although a contamination of about 10%- zen ≤ sured zenith angle θzen. Combined with a measurement 15% of νe events misidentified as tracks remained, as well of the neutrino energy, this opens the possibility of mea- as ντ from νµ oscillations. These two effects were included suring νµ disappearance due to oscillations, exploiting as background. the dependence of the disappearance probability with After all analysis cuts, a high-purity sample of 719 energy and arrival angle. events contained in DeepCore were detected in a year. Although the three neutrino flavours play a role in The left panel of Fig.6 shows the angular distribution the oscillation process, a two-flavour approximation as of the remaining events compared with the expected in Eq. (8) is usually accurate to the percent level with event rate without oscillations (red-shaded area) and ∆ ∆ and θ θ . The survival probability of with oscillations using current world-average values for atm ' 23 atm ' 23 muon neutrinos as a function of path length through sin2 θ and ∆m2 [65] (grey-shaded area). A statisti- 23 | 32| the Earth and neutrino energy is shown in Fig.5. It can cally significant deficit of events with respect to the non- be seen that, for the largest distance travelled by atmo- oscillation scenario can be seen near the vertical direction spheric neutrinos (the diameter of the Earth), Eq. (8) ( 0.6 < cos θ < 1.0), while no discrepancy was ob- − zen − shows a maximum νµ disappearance at about 25 GeV. served in the reference high-energy sample (see Fig. 2 This is precisely within the energy range of contained in [69]). The discrepancy between the data and the non- events in DeepCore. Simulations show that the neutrino oscillation case can be used to fit the oscillation parame- energy response of DeepCore spans from about 6 GeV to ters, without assuming any a priori value for them. The about 60 GeV, peaking at 30 GeV. Neutrinos with higher right panel of Fig.6 shows the result of that fit, with 68% energies will produce muon tracks that are no longer (1σ) and 90% contours around the best-fit values found: 2 2 +0.6 3 2 contained in the DeepCore volume. sin (2θ23) = 1 and ∆m32 = 2.3 0.5 10− eV . | | − × Given this relatively narrow energy response of Deep- The next step in complexity in an oscillation analysis Core compared with the wide range of path lengths, it is with IceCube is to add the measurement of the neutrino possible to perform a search for νµ disappearance through energy, so the quantities L and Eν in equation (7) can be a measurement of the rate of contained events as a func- calculated separately. This is the approach followed in tion of arrival direction, even without a precise energy Ref. [75], where the energy of the neutrinos is obtained determination. This is the approach taken in Ref. [69]. by using contained events in DeepCore and the assump- Events starting in DeepCore were selected by using the tion that the resulting muon is minimum ionising. Once rest of the IceCube strings as a veto. A “high-energy” the vertex of the neutrino interaction and the muon de- sample of events not contained in DeepCore was used as cay point have been identified, the energy of the muon 10
2 Cascade-like Track-like FC 68% 2
χ 1
2000 ∆ no osc. νµ CC µatm ν CC σ uncor 0 e ν +µatm IC2017 [NO] (this work) SK IV 2015 [NO] 1500 3.4 MINOS w/atm [NO] NOνA 2017 [NO] ντ CC data 3.2 T2K 2017 [NO] ν NC ) 2
1000 V 3.0 e 3 −
0 2.8 1 Number of Events 500 ( FC 68% |
2 2.6 2 3
m 2.4 ∆ 0 | 1.2 2.2 1.0 2.0 90% CL contours 0.8 0.4 0.5 0.6 0 1 2 0 1 2 3 0 1 2 3 2 ∆χ2 Ratio to Best Fit sin (θ ) log10(Lreco/Ereco [km/GeV]) 23 Fig. 7 Left Panel: Event count as a function of reconstructed L/E. The expectation with no–oscillations is shown by the dashed line, while the best fit to the data (dots) is shown as a the full line. The hatched histograms show the predicted counts given the best-fit values for each uncor component. σν+µatm represents the uncertainty due to finite Monte Carlo statistics and the data-driven atmospheric muon background 2 2 estimate. The bottom panel shows the ratio of the data to the best fit hypothesis. Right Panel: 90% confidence contours in the sin θ23–∆m32 plane compared with results of Super-Kamiokande [70], T2K [71], MINOS [72] and NOvA [73]. A normal mass ordering is assumed. Figures from Ref. [74] can be calculated assuming constant energy loss, and oscillation parameters are kept fixed to ∆m2 = 7.53 21 × it is proportional to the track length. The energy of the 10 5eV2, sin2 θ = 3.04 10 1, sin2 θ = 2.17 10 2 − 12 × − 13 × − hadronic particle cascade at the vertex is obtained by and δCP = 0. The effect of νµ disappearance due to oscil- maximising a likelihood function that takes into account lations is clearly visible in the left panel of Fig.7, which the light distribution in adjacent DOMs. The neutrino shows the number of events as a function of the recon- energy is then the sum of the muon and cascade en- structed L/Eν, compared with the expected event distri- ergies, Eν = Ecascade + Eµ. The most recent oscillation bution, shown as a dotted magenta histogram, if oscilla- analysis from IceCube [74] improves on the mentioned tions were not present. The results of the best fit to the techniques in several fronts. It is an all-sky analysis and data are shown in the right panel of Fig.7. The best-fit also incorporates some degree of particle identification values are ∆m2 = 2.31+0.11 10 3 eV2 and sin2 2θ = 32 0.13 × − 23 by reconstructing the events under two hypotheses: a ν +0.07 − µ 0.51 0.09, assuming a normal mass ordering. charged-current interaction which includes a muon track, The− results of the two analyses mentioned above are and a particle-shower only hypothesis at the interaction compatible within statistics but, more importantly, they vertex. This latter hypothesis includes νe and ντ charged- agree and are compatible in precision with those from current interactions, although these two flavours can not dedicated oscillation experiments. be separately identified. The analysis achieves an energy resolution of about 25% (30%) at 20 GeV for muon-like ∼ (cascade-like) events and a median angular resolution 4.2 Flavour of Astrophysical Neutrinos of 10° (16°). Full sensitivity to lower neutrino energies, for example to reach the next oscillation minimum at The neutrino oscillation phase in equation (7) depends on 6 GeV, can only be achieved with a denser array, like the ratio L/Eν of distance travelled, L, and neutrino en- ∼ the proposed PINGU low–energy extension [76]. ergy, Eν. For astrophysical neutrinos we have to consider ultra-long oscillation baselines L corresponding to many In order to determine the oscillation parameters, the oscillation periods between source and observer. The ini- data is binned into a two-dimensional histogram where tial mixed state of neutrino flavours has to be averaged each bin contains the measured number of events in the over ∆L, corresponding to the size of individual neutrino corresponding range of reconstructed energy and arrival emission zones or the distribution of sources for diffuse direction. The expected number of events per bin depend emission. In addition, the observation of neutrinos can 2 on the mixing angle, θ23, and the mass splitting, ∆m32, only decipher energies within an experimental energy as shown in Fig.5. This allows to determine the mixing resolution ∆Eν. The oscillation phase in (7) has therefore 2 angle θ23 and the mass splitting ∆m32 as the maximum an absolute uncertainty that is typically much larger than of the binned likelihood. The fit also includes the like- π for astrophysical neutrinos. As a consequence, only the lihood of the track and cascade hypotheses. Systematic oscillation-averaged flavour ratios can be observed. uncertainties and the effect of the Earth density profile are The flavour-averaged survival and transition proba- included as nuisance parameters. In this analysis, a full bility of neutrino oscillations in vacuum, can be derived three-flavour oscillation scheme is used and the rest of the from Eq. (6) by replacing sin2 ∆ 1/2 and sin 2∆ 0. ij → ij → 11
to a unique flavour composition indicated by the three νe : νµ : ντ at source 0 .0 20 0 :1 :0 0 axis. The coloured markers correspond to the oscillation- 1 :2 :0 1.00 0 18 averaged flavour ratios from the three scenarios (xe = 1 :0 :0 .1 7 1/3, xe = 0, and xe = 1) discussed earlier, where the 0.83 16 0 best-fit oscillation parameters have been used (instead .3 3 14 of “tri-bi-maximal” mixing). The blue-shaded regions % 0.67 0 95 . show the relative flavour log-likelihood ratio of a global ντ 5 8 % νµ 12
0 6 L n
0.50 l analysis of IceCube data [77]. The best-fit is indicated 0 10 .6 ∆ 2 as a white cross. IceCube’s observations are consistent 7 0.33 8 − with the assumption of standard neutrino oscillations and 0 .8 95 % 3 6 the production of neutrino in pion decay (full or “muon- 0.17 1 damped”). Neutrino production by radioactive decay is .0 4 0 disfavoured at the 2σ level. 0.00 0 7 3 0 7 3 0 2 .0 .1 .3 .5 .6 .8 .0 0 0 0 0 0 0 1 0 νe 5 Standard Model Interactions
Fig. 8 Observed flavour composition of astrophysical neutrino with The measurement of neutrino fluxes requires a precise IceCube [77]. The best-fit flavour ratio is indicated by a white “ ”, knowledge of the neutrino interaction probability or, equi- with 68% and 96% confidence levels indicated by white lines. The× expected oscillation-averaged composition is indicated for three valently, the cross section with matter. At neutrino ener- different initial compositions, corresponding to standard pion decay gies of less than a few GeV the cross section is domi- (1 : 2 : 0), muon-damped pion decay (0 : 1 : 0), and neutron decay nated by elastic scattering, e.g., ν + p ν + p, and x → x (1 : 0 : 0). The white “+” indicate the best-fit from a previous quasi-elastic scattering, e.g., ν + p e+ + n. In the analysis [39]. (From Ref. [77]) e → energy range of 1-10 GeV, the neutrino-nucleon cross section is dominated by processes involving resonances, ++ The resulting expression can be expressed as e.g. νe + p e− + ∆ . At even higher energies neu- → trino scattering with matter proceeds predominantly via 2 2 Pνα ν Uαi Uβi . (12) deep inelastic scattering (DIS) off nucleons, e.g., ν + p → β ∑ µ ' i | | | | → µ− + X, where X indicates a secondary particle shower. To a good approximation, neutrinos are produced in as- The neutrino cross sections have been measured up to neutrino energies of a few hundreds of GeV. However, the trophysical environments as a mixed state involving νe, neutrino energies involved in scattering of atmospheric νe, νµ, and νµ. Due to the similarity of neutrino and anti- neutrino signals in Cherenkov telescopes we consider and astrophysical neutrinos off nucleons far exceed this in the following only flavour ratios of the sum of neu- energy scale and we have to rely on theoretical predic- tions. trino and anti-neutrino fluxes φν+ν with flavour ratios We will discuss in the following the expected cross Ne : Nµ : Nτ. Note, that the mixing angles shown in Ta- ble1 are very close to the values for “tri-bi-maximal” mix- section of high-energy neutrino-matter interactions. In ing [78] corresponding to sin2 θ 1/3, sin2 θ 1/2 weak interactions with matter the left-handed neutrino 12 ∼ 23 ∼ 0 and sin2 θ 0. If we use this approximation then the couples via Z and W± exchange with the constituents of 13 ∼ oscillation-averaged spectrum will be close to a flavour a proton or neutron. Due to the scale-dependence of the ratio strong coupling constant, the calculation of this process involves both perturbative and non-perturbative aspects 2 7 x 7 x due to hard and soft processes, respectively. N : N : N + x : e : e , (13) e µ τ ' 3 e 6 − 2 6 − 2 where xe = Ne/Ntot is the electron neutrino fraction on 5.1 Deep Inelastic Scattering production. For instance, pion decays π+ µ+ + ν → µ followed by muon decay µ+ e+ + ν + ν produces The gauge coupling of quantum chromodynamics (QCD) → e µ an initial electron fraction of xe = 1/3. The resulting increases as the renormalisation scale µ decreases, a be- flavour ratio is then close to 1 : 1 : 1. It is also feasible haviour which leads to the confinement of quarks and that the muon from pion decay loses energy as a result of gluons at distances smaller that the characteristic size 1 1 synchrotron radiation in strong magnetic fields (“muon- Λ− (200MeV) 1 fm. In nature (except in high QCD ' − ' damped” scenario) resulting in x 0 and a flavour ratio temperature environments (T Λ ) as in the early e ' QCD of 4 : 7 : 7. Radioactive decay, on the other hand, will universe) the only manifestations of coloured representa- produce an initial electron neutrino fraction x 1 and a tions are composite gauge singlets such as mesons and e ' flavour ratio 5 : 2 : 2. baryons. These bound states consist of valence quarks, Figure8 shows a visualisation of the observable neu- which determine the overall spin, isospin, and flavour trino flavour. Each location in the triangle corresponds of the hadron, and a sea of gluons and quark-anti-quark 12
′ s (P + k)2 t q2 Q2 ℓ ℓ ≡ ≡ ≡ − ′ M2 P2 W2 (P + q)2 k k ≡ ≡ q Q2 q P x y · p ≡ 2 q P ≡ k P · · (Bjorken x)(inelasticity) − P } Q MW M {z H (deep)(inelastic) N | Fig. 9 The kinematics of deep inelastic scattering. pairs, which results from QCD radiation and pair-creation. These constituents of baryons and mesons are also called “partons”. Due to the strength of the QCD coupling at small scales the neutrino-nucleon interactions cannot be de- Fig. 10 The kinematic plane investigated by various collider and scribed in a purely perturbative way. However, since the fixed target experiments in terms of Bjorken-x and momentum transfer Q2. Figure from Ref. [41]. QCD interaction decreases as the renormalisation scale increases (asymptotic freedom) the constituents of a nu- cleon may be treated as loosely bound objects within the momentum fraction carried by the parton. In the rest 1 sufficiently small distance and time scales (ΛQCD− ). Hence, frame of the nucleus the quantity y is the fractional energy in a hard scattering process of a neutrino involving a large loss of the lepton, y = (E E )/E, where E and E are the − 0 0 momentum transfer to a nucleon the interactions between lepton’s energy before and after scattering, respectively. quarks and gluons may factorise from the sub-process From the previous discussion we obtain the follow- (see Fig.9). Due to the renormalisation scale dependence ing recipe for the calculation of the total (anti-)neutrino- of the couplings this factorisation will also depend on the nucleon cross section σ(ν(ν)N). The differential lepton- absolute momentum transfer Q2 q2. parton cross section may be calculated using a perturba- ≡ − Figure9 shows a sketch of a general lepton-nucleon tive expansion in the weak coupling. The relative contri- scattering process. A nucleon N with mass M scatters bution of this partonic sub-process with Bjorken-x and 2 off the lepton ` by a t-channel exchange of a boson. The momentum transfer Q in the nucleon N is described final state consist of a lepton `0 and a hadronic state H by structure functions, which depend on the particular 2 with centre of mass energy (P + q)2 = W2. This scatter- parton distribution functions (PDFs) of quarks ( fq(x, Q )) 2 ing process probes the partons, the constituents of the and gluons ( fg(x, Q )) . These functions must be mea- 1 nucleon with a characteristic size M− at length scales sured in fixed target and accelerator experiments, that 1 2 of the order of Q− . Typically, this probe will be “deep” only access a limited kinematic region in x and Q . Fig- 2 and “inelastic”, corresponding to Q M and W M, ure 10 shows the regions in the kinematical x-Q -plane respectively. The sub-process between lepton and parton which have been covered in electron-proton (HERA), anti- takes place on time scales which are short compared to proton-proton (Tevatron), and proton-proton (LHC) colli- those of QCD interactions and can be factorised from the sions as well as in fixed target experiments with neutrino, soft QCD interactions. The intermediate coloured states, electron, and muon beams (see, e.g., Ref. [79] and refer- corresponding to the scattered parton and the remaining ences therein). constituents of the nucleus, will then softly interact and hadronise into the final state H. 5.2 Charged and Neutral Current Interactions The kinematics of a lepton-nucleon scattering is conve- niently described by the Lorentz scalars x = Q2/(2q P), · The parton level charged current interactions of neutrinos also called Bjorken-x, and inelasticity y = (q P)/(k P) · · with nucleons are shown as the top two diagrams (a) and (see Fig.9 for definitions). In the kinematic region of deep (b) of Fig. 11. The leading-order contribution is given by inelastic scattering (DIS) where Q M and W M we 2 2 also have Q 2q p and thus x (q p)/(q P). The 2 2 2 ! ' · ' · · d σCC GF mW scalars x and y have simple interpretations in particular = dQ2dx π Q2 + m2 reference frames. In a reference frame where the nucleon W is strongly boosted along the neutrino 3-momentum k q(x, Q2) + q(x, Q2)(1 y2) , (14) the relative transverse momenta of the partons is negli- · − 5 2 gible. The parton momentum p in the boosted frame is where G 1.17 10 GeV− is the Fermi coupling F ' × − approximately aligned with P and the scalar x expresses constant. The effective parton distribution functions are 13
ν ℓ ν ℓ ν (I =ν 1/2)ν and right-handedν (I = 0) quarks are given 3 ± 3 by
0 1 2 0 1 1 W ± W ± Z L = sin2 θZ , L = + sin2 θ , (18) u 2 − 3 W d − 2 3 W 2 1 R = sin2 θ , R = sin2 θ . (19) u − 3 W d 3 W u d d u u/d u/d u/d u/d (a) (b) (c) As in the case of charged(d) current interactions, the relation ν ℓ ν ℓ ν ν ν ν of neutron structure function fq are given by the exchange u d and u d and for an iso-scalar target one takes ↔ ↔ f ( f + f )/2 and f ( f + f )/2. u/d → u d u/d → u d 0 0 W ± W ± Z Z
5.3 High-Energy Neutrino-Matter Cross Sections u d d u u/d u/d u/d u/d The expressions for the total charged and neutral current (a) (b) (c) (d) neutrino cross sections are derived from Eqs. (14) and (15) Fig. 11 The parton level W (a/b) and Z (c/d) boson exchange after integrating over Bjorken-x and momentum trans- between neutrinos and light quarks. fer Q2 (or equivalently inelasticity y). The evolution of PDFs with respect to factorisation scale µ can be calcu- 2 2 lated by a perturbative QCD expansion and results in the q(x, Q ) = fd + fs + fb and q(x, Q ) = fu + fc + ft. For Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equa- antineutrino scattering we simply have to replace all fq tions [80,81,82,83]. The solution of the (leading-order) by fq. These structure functions fq are determined experi- mentally via deep inelastic lepton-nucleon scattering or DGLAP equations correspond to a re-summation of pow- 2 2 n hard scattering processes involving nucleons. The corre- ers (αs ln(Q /µ )) which appear by QCD radiation in sponding relation of neutron structure function are given the initial state partons. However, these radiative pro- n by the exchange u d and u d due to approximate cesses will also generate powers (αs ln(1/x)) and the ↔ ↔ isospin symmetry. In neutrino scattering with matter one applicability of the DGLAP formalism is limited to mod- 2 usually makes the approximation of an equal mix be- erate values of Bjorken-x (small ln(1/x)) and large Q tween protons and neutrons. Hence, for an iso-scalar tar- (small αs). If these logarithmic contributions from a small get, i.e., averaging over isospin, f ( f + f )/2 and x become large, a formalism by Balitsky, Fakin, Kuraev, u/d → u d f ( f + f )/2. and Lipatov (BFKL) may be used to re-sum the αs ln(1/x) u/d → u d Analogously, the parton level neutral current (NC) terms [84,85]. This approach applies for moderate values 2 2 2 interactions of the neutrino with nucleons are shown of Q , since contributions of αs ln(Q /µ ) have to be kept in the bottom two diagrams (c) and (d) of Fig. 11. The under control. leading-order double differential neutral current cross There are unified forms [86] and other improvements section can be expressed as of the linear DGLAP and BFKL evolution for the problem- atic region of small Bjorken-x and large Q2. The extrapo- !2 lated solutions of the linear DGLAP and BFKL equations d2σ G2 m2 NC = F Z predict an unlimited rise of the gluon density at very dQ2dx π 2 2 Q + mZ small x. It is expected that, eventually, non-linear effects 0 2 0 2 2 like gluon recombination g + g g dominate the evolu- q (x, Q ) + q (x, Q )(1 y ) . (15) → · − tion and screen or even saturate the gluon density [87,88, Here, the structure functions are given by 89]. Note, that neutrino-nucleon scattering in charged (14) 0 2 2 q = ( fu + fc + ft)Lu + ( fu + fc + ft)Ru , and neutral (15) current interactions via t-channel ex- 2 2 change of W and Z bosons, respectively, probe the parton + ( f + fs + f )L + ( f + f + f )R , (16) d b d d s b d content of the nucleus effectively up to momentum trans- fers of Q2 M2 (see Fig. 11). The present range of ' Z/W 0 2 2 Bjorken-x probed by experiments only extends down to q = ( fu + fc + ft)Ru + ( fu + fc + ft)Lu , x 10 4 at this Q-range, and it is limited to 10 6 for arbi- + ( f + f + f )R2 + ( f + f + f )L2 . (17) ' − − d s b d d s b d trary Q values. On the other hand, the Bjorken-x probed The weak couplings after electro-weak symmetry break- by neutrino interactions is, roughly, 2 ing depend on the combination I3 q sin θW, where I3 is − 2 1 the weak isospin, q the electric charge, and θW the Wein- MZ/W 4 Eν − x 10− . (20) berg angle. More explicitly, the couplings for left-handed ' s m2 ' 100PeV − N 11
0.4 Ν CC xsec $HERAPDF1.5% 0.4 Ν NC xsec $HERAPDF1.5% Ν CC xsec $ANIS% Ν NC xsec $ANIS% 0.2 Ν CC xsec $GENIE% 0.2
0.0 0.0 r to HERAPDF1.5 r to HERAPDF1.5 # # !0.2 !0.2
!0.4 !0.4 rel. dev. w rel. dev. w 100 4 6 8 10 12 100 4 6 8 10 12 10 10 10 10 10 11 10 10 10 10 10 EΝ !GeV" EΝ !GeV"
Ν Ν 0.4 CC xsec $HERAPDF1.5% 14 FIG.0.4 12: TheNC relative xsec $HERAPDF1.5 deviation% of the ANIS [57] and GENIE [59] cross-sections from the HERAPDF1.5 central member. Ν CC xsec $ANIS% Ν NC xsec $ANIS% 0.2 Ν CC xsec $GENIE% 0.2 " " Ν CC xsec #HERAPDF1.5$ Ν NC xsec #HERAPDF1.5$
5 pb 5 pb Ν Ν ! 0.0 ! 100.0 CC xsec #HERAPDF1.5$ 10 NC xsec #HERAPDF1.5$ r to HERAPDF1.5 r to HERAPDF1.5 # # !0.2 !0.2 1000 1000 !0.4 !0.4 rel. dev. w rel. dev. w CC cross section 100 4 6 8 10 12 10 100 4 6 8 10 12 NC cross section 10 Ν 10 10 10 10 10 10 10 10 10 10 Ν EΝ !GeV" EΝ !GeV" and and Ν Ν 0.1 0.1 FIG. 12: The relative deviation of the ANIS [57] and GENIE [59] cross-sections100 10 from4 the106 HERAPDF1.5108 10 central10 member.1012 100 104 106 108 1010 1012 EΝ !GeV" EΝ !GeV" " " Ν CC xsec #HERAPDF1.5$ Ν NC xsec #HERAPDF1.5$
5 pb 5 pb Ν FIG. 13: NeutrinoΝ and anti-neutrino cross-sections on isoscalar targets for CC and NC scattering according to HERAPDF1.5. ! ! 10 CC xsec #HERAPDF1.5$ 10 NC xsec #HERAPDF1.5$
1000 using1000 HERAPDF1.5 at NLO are shown in Fig. 13. The general trend of the uncertainties can be understood by noting that as one moves to higher neutrino energy one also movesFig. to 13 lowerTransmissionx where probabilitythe PDF uncertainties assuming the are Standard increasing. Model 2 < < 1 5 2 The PDF uncertainties are smallest at 10− x 10− , correspondingneutrino-nucleon to s cross10 sectionGeV for. Movingneutrinos to crossing smaller the neutrino Earth, as CC cross section 10 NC cross section 10 ∼ Ν energiesΝ brings us into the high x region where∼ ∼ PDF uncertaintiesa function of increase energy and again. zenith angle.This e Neutral-currentffect is greate interactionsrforthe HERAPDF1.5 because the HERA data have less statisticsare at included. high x than The horizontal the fixedwhite target line data shows which the are location included of the and and Ν inΝ CT10; however these data have further uncertainties thatcore-mantle are not fully boundary. accounted Figure fromfor in [93 CT10,]. e.g. heavy target 0.1 0.1 100 104 106 108 1010 1012 corrections,100 deuterium104 corrections106 108 and assumptions1010 1012 regarding higher twist effects. When the high x region becomes important the neutrino and antineutrino cross-sections are different because the valence contribution to xF3 is now EΝ !GeV" EΝ !GeV" 5.4 Neutrino Cross Section Measurement with IceCube2 significant. This is seen in Fig. 13, as is the onset of the linear dependence of the cross-sections for s
0.9 neutrinos. Deviations from the Standard Model cross sec- Neutrino tion σSM were fitted by the ratio R = σνN/σSM. The anal- 0.8 Antineutrino ysis assumes priors on the atmospheric and astrophysical 0.7 Weighted combination neutrino flux based on the baseline models in Refs. [94, This result 0.6
30,24]. In practice, the likelihood maximisation uses the /GeV) 2 0.5
product of the flux and the cross section, keeping the ob- cm served number of events as a fixed quantity. Thus, trials -38 0.4
with higher cross sections must assume lower fluxes (or (10
ν 0.3 vice-versa) in order to preserve the total number of events. /E ν The procedure is thus sensitive to neutrino absorption in σ 0.2 Accelerator Data the Earth alone, and not to the total number of observed 0.1 events. Since the astrophysical flux is still not known to a 0.0 high precision, the uncertainties in the normalisation and 1.5 2.5 3.5 4.5 5.5 6.5 spectral index were included as nuisance parameters in log10(Eν [GeV]) the analysis. Other systematics considered are the Earth Fig. 14 Charged-current neutrino cross section as a function of density and core radius as obtained from the Preliminary energy. Shown are results from previous accelerator measurements Earth Model [95], the effects of temperature variations in (yellow shaded area, from [41]), compared with the result from the atmosphere, which impact the neutrino flux during IceCube for the combined (ν + ν) + N charged-current cross section. the year, and detector systematics. The blue and green lines represent the Standard Model expectation +0.21 for ν and ν respectively. The dotted red line represents the flux– The analysis results in a value of R = 1.30 0.19(stat) weighted average of the two cross sections, which is to be compared − +0.39 with the IceCube result, the black line. The light brown shaded 0.43(sys). This is compatible with the Standard Model prediction− (R = 1) within uncertainties but, most impor- area indicates the uncertainty on the IceCube measurement. Figure from [93]. tantly, it is the first measurement of the neutrino-nucleon cross section at an energy range (few TeV to about 1 PeV) unexplored so far with accelerator experiments [41]. This where H is the interaction height of the primary with is illustrated in Fig. 14 which shows current accelerator a zenith angle θzen. The initial muon energy Eµ is close measurements (within the yellow shaded area) and the to that at ground level due to minimal energy losses in results of the IceCube analysis as the light brown shaded the atmosphere. That is, turning the argument around, area. The authors of Ref. [96] performed a similar analy- the identification of single, laterally separated muons sis based on six years of high-energy starting event data. at a given dt accompanying a muon bundle in IceCube Their results are also consistent with perturbative QCD is a measurement of the transverse momentum of the predictions of the neutrino-matter cross section. 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 400 GeV at the surface 5.5 Probe of Cosmic Ray Interactions with IceCube ∼ can reach the depths of the detector. This, along with On a slightly different topic, but still related to the prod- the inter-string separation of 125 m, sets the level for the ucts of cosmic ray interactions in the atmosphere, the minimum pt accessible in IceCube. However, since the high rate of atmospheric muons detected by IceCube can exact interaction height of the primary is unknown and be used to perform studies of hadronic interactions at varies with energy, a universal pt threshold can not be high energies and high momentum transfers. Muons are given. For example, a 1 TeV muon produced at 50 km created from the decays of pions, kaons and other heavy height and detected at 125 m from the shower core has a hadrons. For primary energies above about 1 TeV, muons transverse momentum pt of 2.5 GeV. with a high transverse momentum, pt & 2 GeV, can be Our current understanding of lateral muon produc- produced alongside the many particles created in the for- tion in hadronic interactions shows an exponential be- ward direction, the “core” of the shower. This will show haviour at low pt, exp( pt/T), typically below 2 GeV, − up in IceCube as two tracks separated by a few hundred due to soft, non-perturbative interactions, and a power- n meters: one track for the main muon bundle following law behaviour at high pt values, (1 + pt/p0)− , reflect- the core direction, and another track for the high-pt muon. ing the onset of hard processes described by perturbative The muon lateral distribution in cosmic-ray interactions QCD. The approach traces back to the QCD inspired depends on the composition of the primary flux and de- ”modified Hagedorn function” [99,100]. The parameters tails of the hadronic interactions [97,98]. If the former T, p0 and n can be obtained from fits to proton-proton or is sufficiently well known, the measurement of high-pt heavy ion collision data [100,101]. muons can be used to probe hadronic processes involv- This is also the behaviour seen by IceCube. Fig. 15 ing nuclei and to calibrate existing Monte-Carlo codes at shows the muon lateral distribution at high momenta energies not accessible with particle accelerators. obtained from a selection of events reconstructed with a The lateral separation, dt, of high pt muons from the two-track hypothesis in the 59-string detector [102], along core of the shower is given by dt = pt H/Eµ cos θzen, with a fit to a compound exponential plus power-law 16
8 10 sary extensions of the Standard Model that allow for the Fit Parameters A: 25.13 ± 0.94 introduction of neutrino mass terms can also introduce B: -0.05 ± 0.01 7 non-standard oscillation effects that are suppressed in 10 C: 9.44 ± 1.40 n: -17.56 ± 5.15 a low-energy effective theory. This is one motivation to study non-standard neutrino oscillations. In the follow-
10 6 ing, we will discuss various extensions to the Standard Model that can introduce new neutrino oscillation effects and neutrino interactions. The large energies and very
Number in 335 days 335 in Number 10 5 long baselines associated with atmospheric and cosmic neutrinos, respectively, provide a sensitive probe for these effects. 10 4 For the following discussion it is convenient to intro- 150 200 250 300 350 400 duce neutrino oscillations via the evolution of the density Separation between Bundle and LS Muon [m] operator ρ of a mixed state. For a given Hamiltonian H, the time evolution of the neutrino density operator is Fig. 15 The lateral muon distribution measured by IceCube, normalised to sea level, along with the best fit parameters to governed by the Liouville equation, a combined exponential and power law function of the form C n ρ˙ = i[H , ρ] . (26) exp(A + Bx) + 10 (1 + x/400) . The exponential part of the fit is − SM plotted as a dotted red line and the power law is shown as a dashed blue line. Figure reprinted with permission from [102] (Copyright The solution to the Liouville equation allows to describe 2013 APS). neutrino oscillation effects in a basis-independent way. For instance, a neutrino that is produced at times t = 0 in the flavour state να can be represented by ρ(0) = Πα, function. Due to the size of the 59-string detector and the where Πα = να να is the projection operator onto the short live time of the analysis (1 year of data), the statis- | ih | flavour state να . The transition probability between two tics for large separations is low and fluctuations in the | i flavour states να and νβ can then be recovered as the data appear for track separations beyond 300 m. Still, the expectation value of the projector Πβ, which is given by presence of an expected hard component at large lateral trace distances (high pt) that can be described by perturbative quantum chromodynamics (a power-law behaviour) is Pα β(t) = Tr(ρ(t)Πβ) . (27) → clearly visible. For standard neutrino oscillations the Hamiltonian is Significant discrepancies also exist between interac- composed of two terms, H = H + V , describing tion models on the expected p distribution of muons [103]. SM 0 mat t the free evolution in vacuum and coherent matter effects, The high-p muon yield per collision depends, both, on t respectively. The free Hamiltonian in vacuum can be writ- the primary composition (protons typically producing ten as higher rates of high-pt muons than iron interactions) and the hadronic interaction model. Future IceCube analyses m2 H i (Π + Π ) , (28) with the larger, completed, 86-string detector using Ice- 0 ∑ i i ' i 2Eν Cube/IceTop coincident events can extend the range of the cos θzen distribution as well as provide an updated where the sum runs over the available flavours and we have introduced the projection operators Π ν ν comparison of the muon pt distribution with predictions i ≡ | iih i| and Π ν ν onto neutrino and anti-neutrino mass from existing hadronic models with higher statistics than i ≡ | iih i| that shown in Fig. 15. There is definitely complementary eigenstates. Matter effects introduced earlier can be cast information from neutrino telescopes to be added to the into the form efforts in understanding hadronic interactions using air Vmat = √2GF Ne(Πe Πe) , (29) shower arrays and heavy-ion and hadronic accelerator − experiments [104,105]. where Π ν ν and Π ν ν are the projections e ≡ | eih e| e ≡ | eih e| onto electron neutrino and anti-neutrinos states, respec- tively, and Ne is the number density of electrons in the 6 Non-Standard Neutrino Oscillations and background. Interactions In the following, we will discuss non-standard contri- butions to the Liouville equation that we assume to take In the previous two sections we have summarised the on the form phenomenology of weak neutrino interactions and stan- dard oscillations based on the mixing between three ac- ρ˙ = i[HSM, ρ] i [Hn, ρ] + n[ρ] . (30) − − ∑ ∑ D tive neutrino flavour states and the eigenstates of the n n Hamiltonian (including matter effects). However, the The second term on the right hand side of Eq. (30) ac- Standard Model of particle physics does not account for counts for additional terms in the Hamiltonian, that af- neutrino masses and is therefore incomplete. The neces- fect the phenomenology of neutrino oscillations. These 17 terms are therefore parametrised by a sequence of Her- !24 † mitian operators, Hn = Hn, that preserve the unitarity of the density operator. The third term corresponds to non-unitary effects, related to dissipation or decoherence, !25 that can not only affect the neutrino flavour composition, but also their spectra. In particular, we will focus in the following on cases where the sequence of terms can Dn #∆ be expressed in terms of a set of Lindblad operators Lj as 10 !26 log 1 † † n[ρ] = ∑ [Lj, ρ Lj ] + [Lj ρ, Lj ] . (31) D 2 j !27 These Lindblad operators act on the Hilbert space of the open quantum system, , and satisfy ∑ L† L ( ), H j j j ∈ B H where ( ) indicates the space of bounded operators act- B H !28 ing on [106]. We will discuss in the following various H 0.0 0.2 0.4 0.6 0.8 1.0 instances of these Lindblad operators for neutrino deco- 2 herence, neutrino decay, or hidden neutrino interactions sin 2Ξ with cosmic backgrounds. Fig. 16 Allowed regions at 90%, 95%, and 99% confidence level (from darkest to lightest) for LIV-induced oscillation effects with n = 1. Note we plot sin2 2ξ to enhance the region of interest. Also 6.1 Effective Hamiltonians shown are the Super-Kamiokande + K2K 90% contour [108] (dashed line), and the projected IceCube 10-year 90% sensitivity [107] (dot- ted line). Reprinted with permission from [109]) (Copyright 2013 Non-trivial mixing terms of the effective Hamiltonian can APS) be generated in various ways, including non-standard . interactions with matter and Standard Model extensions that violate the equivalence principle, Lorentz invariance, be treated effectively as a two-level system, analogous or CPT symmetry (see Ref. [107] for references). We will to the case of standard atmospheric neutrino oscillations. first study oscillations that are induced by additional The most general form of the unitary matrix Ue is then terms to the Hamiltonian that can be parametrised in the form [107] cos ξ eiη sin ξ Ue = iη . (34) n e− sin ξ cos ξ Heff = (Eν) ∑(δaΠa + δaΠa) . (32) − a The νµ survival probability can then be expressed as [107] Here we introduced the projectors Πa = νa νa and | ih | Πa = νa νa onto a new set of states νa that are re- | ih | | i 2 lated to the usual flavor states by a new unitary mixing 2 2 ∆matmL Pνµ νµ = 1 sin 2θeff sin R , (35) matrix → − 4Eν where we have introduced the scaling parameter να = ∑ Ueα∗a νa . (33) | i a | i ∆m2 The expansion parameters δa have mass dimensions 1 n = eff = + 2 + ( θ ξ − 2 1 R 2R cos 2 atm cos 2 with integer n. R ∆matm It is convenient to group these contributions into CPT- 1/2 + sin 2θatm sin 2ξ cos η) , (36) even terms obeying the relation δa = δa and CPT-odd terms with δa = δa. For a CPT-symmetric process we and − have P(να νβ) = P(νβ να). In particular, the sur- n → → ∆δE 4Eν vival probability between neutrinos and anti-neutrinos R ν , (37) ≡ 2 2 is the same. CPT-odd terms break this symmetry. Indeed, ∆matm we have already encountered such a CPT-odd term as with ∆δ = δ δ . The effective rotation angle θ is 3 − 2 eff the CP-violating matter effect contributing by the effec- given by tive potential of Eq. (29). The corresponding expansion in terms the effective Hamiltonian of Eq. (32) is given by 2 1 2 sin 2θeff = sin 2θatm Ue = 1, δ = δ = √2G N , δµ,τ = 0, and n = 0. On the 2 e − e F e other hand, the free Hamiltonian in Eq. (28) is a CPT-even R2 2 + R sin 2ξ + 2R sin 2θatm sin 2ξ cos η . (38) term with Ue = U , δ = δ = m2/2, and n = 1. PMNS i i i − The contribution of effective Hamiltonians can be Note that the survival probability in Eq. (35) reduces to tested by the oscillation of atmospheric muon neutri- the familiar expression of atmospheric oscillation in the nos [107]. For simplicity, we will assume that this can limit R 0. → 18
0 n n 100 n LIV (∆δ/Eν ) QD (D/Eν ) Units pion decay 1 2.8 10 27 1.2 10 27 – muon-damped − − × 31 × 31 1 20 neutron decay 2 2.7 10− 1.3 10− GeV− 80 × 35 × 36 2 3 1.9 10 6.3 10 GeV− × − × − n 40 µ Table 2 (From Ref. [109]) 90% CL upper limits on Lorentz- fraction [%] 60 invariance violation (LIV) and quantum decoherence (QD) effects. LIV upper limits are for the case of maximal mixing (sin 2ξ = 1), 60 and quantum decoherence upper limits for the case of a 3-level sys- fraction [%] 40 t tem with universal decoherence parameters D (see text for details). n
80 20 a non-trivial set of unitarity conditions: 100 0 1 0 20 40 60 80 100 Peνα να , (40) ne fraction [%] → ≥ 3 1 Fig. 17 The range of oscillation-averaged neutrino flavour ratios Peνα νβ (α = β) , (41) from astrophysical sources. The filled markers show the initial → ≤ 2 6 flavor composition from pion decay (1:2:0), muon-damped pion 1 Peν ν + Peν ν (α = β) . (42) decay (0:1:0), and neutron decay (1:0:0). The open markers show α→ β ≤ 24 α→ α 6 the expected observed flavour ratio assuming the best-fit oscillation parameters under the assumption of normal ordering of neutrino The resulting oscillation-averaged flavour composition masses (see Table1). The coloured contours show the range of from these unitarity bounds are shown as the contours in oscillation-averaged flavor ratios based on the unitarity of the neu- Fig. 17. We assume three initial flavour composition from trino mixing matrix. pion decay (1:2:0), muon-damped pion decay (0:1:0), and neutron decay (1:0:0). Figure 16 shows the results of an analysis of atmo- spheric muon neutrino data in the range 100 GeV to 10 TeV taken by AMANDA-II in the years 2000 to 2006 [109]. 6.2 Violation of Lorentz Invariance The data was binned into two-dimensional histograms in terms of the number of hit optical modules (as a measure One of the foundations of the Standard Model of particle of energy) and the zenith angle (cos θzen). The predicted physics is the principle of Lorentz symmetry: The funda- effect of non-standard oscillation parameters can be com- mental laws in nature are thought to be independent of pared to the data via a profile likelihood method. No ev- the observer’s inertial frame. However, some extensions idence of non-standard neutrino oscillations was found of the Standard Model, like string theory or quantum and the statistically allowed region of the [∆δ, sin2 2ξ]- gravity, allow for the spontaneous breaking of Lorentz plane is shown as 90%, 95%, 99% C.L. These results are symmetry, that can lead to Lorentz-invariance violating derived under the assumption that η = π/2 in the uni- (LIV) effects in the low-energy effective theory. There also tary mixing matrix. The red dashed line shows the 90% exist a deep connection between the appearance of LIV 3 limit of a combined analysis by Super-Kamiokande and effects with the violation of CPT-invariance in local quan- K2K [108] which is compatible with the AMANDA-II tum field theories [111]. Such effects were incorporated bound. The projected IceCube 90% sensitivity after ten in the Standard Model Extension (SME), an effective-field years of data taking is given as a yellow dotted line and Lorentz-violating extension of the Standard Model, which may improve the limit on ∆δ by one order of magni- includes CPT-even and CPT-odd terms [112]. The SME tude [107]. provides a benchmark for experiments to gauge possi- Effective Hamiltonians with positive energy depen- ble Lorentz violating processes in nature, by express- dence n > 0 can dominate the standard Hamiltonian H0 ing experimental results in terms of the parameters of at sufficiently high energy. For astrophysical neutrinos the model. The size of LIV effect is expected to be sup- pressed by Planck scale M 1019 GeV (or Planck length this can lead to the situation that the oscillation-averaged P ' λ 10 33 cm), consistent with the strong experimental flavour transition probability is completely dominated P ' − by the new unitary transition matrix leading to limits on the effect [113,114]. Oscillations of atmospheric neutrinos with energies 2 2 above 100 GeV provide a sensitive probe of LIV effects. Peνα νβ ∑ Ueαa Ueβa . (39) → ' a | | | | For instance, LIV in the neutrino sector can lead to small differences in the maximal attainable velocity of neutrino This effect can therefore be tested by looking for statisti- states [116]. Since the “velocity eigenstates” are different cally significant deviations from the predictions of stan- from the flavour eigenstates, a new oscillation pattern dard oscillations in Eq. (12). However, the transition prob- abilities described by Eq. (39) are constrained by lepton 3The invariance of physical laws under simultaneous charge conju- unitary. This has been studied in Ref. [110], who derived gation (C), parity reflection (P), and time reversal (T). 19
Fig. 18 Exclusion region in the LIV parameter space discussed in the text. The parameter ρ is related to the strength of the Lorentz invariance violation. The parameter cos θ is a combination of coefficients that define Lorentz invariance violation in the effective Hamiltonian of Standard Model Extension [112]. Each plot shows results for the dimension d of the operator, from 3 to 8, from left to right, and top to bottom. The red (blue) regions are excluded at 90% (99%) C.L. Figure from [115] can arise. The effect can be described in the framework of A more recent search for isotropic LIV effects with the effective Hamiltonians described in the previous section complete IceCube detector was the subject of the anal- by CPT-even states with n = 1. For the approximate ysis presented in [115], based on data collected during two-level system with survival probability described by the period from May 2010 to May 2012. The LIV effects Eq. (35) we can identify effective Hamiltonian parameters were parametrised by two parameters ρd and θd, which as the velocity difference ∆δ = ∆c, together with a new are related to the strength of the LIV effect and the a mixing angle ξ and a phase η. Higher order contributions combination of other LIV parameters of the SME [112], n > 1 have been considered for non-renormalisable LIV respectively. These parameters accompany the expansion effects caused by quantum mechanical fluctuations of of the Hamiltonian in powers of energy, and the sub- the space-time metric and topology [117]. Both the ∆δ ∝ script d refers to the power of the corresponding operator E (n = 1) and the ∆δ ∝ E3 (n = 3) cases have been in the Hamiltonian. Most experiments are sensitive to examined in the context of violations of the equivalence effects of dimension d = 3 and d = 4, but the energy principle (VEP) [118,119,120]. The 90% C.L. upper limits reach of IceCube makes it possible to extend the search n 1 n on the corresponding coefficients ∆δ/Eν [GeV − ] are to LIV effects induced by operators of dimension up to shown in Tab.2. d = 8. The analysis used binned atmospheric neutrino data into a horizontal (cos θ > 0.6) and a vertical Violations of Lorentz invariance can also manifest zen − (cos θ < 0.6) sample which allowed to study the LIV themselves by the dependence of neutrino oscillations on zen − a preferred orientation of the neutrino arrival direction. effect by the energy-dependent ratio of the samples using This effect can also be described by effective Hamiltoni- a likelihood analysis. No LIV effects were identified, the ans similar to Eq. (32) where the expansion parameters best fit values of all the ρd parameters being compatible not only depend on energy but also on direction. The con- with zero. This allowed to place limits in the [ρd-cos θd] tribution of these terms were studied in the analysis [121] parameter space. The results are shown in Fig. 18 for 3 d 8. using data collected by the 40-string configuration of Ice- ≤ ≤ Cube between April 2008 and May 2009. Due to IceCube’s The violation of Lorentz invariance associated with unique position at the geographic South Pole, the field Planck-scale physics can also affect neutrino spectra over of view of the observatory is constant in time. The at- long baselines. The LIV effects can result in modified dis- mospheric neutrino data extracted from this period was persion relations, e.g., E2 p2 = m2 eE2, that introduce − − binned in terms of sidereal time. If neutrino oscillations non-trivial maximal particle velocities [116]. Whereas at depend on the relative neutrino momenta in the cosmic low energies the Lorentz invariance is recovered, E2 − rest frame, then the muon neutrino data in this reference p2 m2, at high energies we can observe sub- or su- ' frame is expected to show oscillation patterns. The non- perluminal maximal particle velocities. These can allow observation of this effect allowed to constrain these LIV otherwise forbidden neutrino decays, in particular, vac- parameters. uum pair production, ν ν + e+ + e , and vacuum α → α − 20
14 90% CL 0.1 IC 79 , 90 C.L. 90% CI IC 40 , 90 C.L. 12 90% CL Super-K SK , 90 C.L. 90% CI Salvado et al 10 H
L 8 L ∆
2 6 − 4 Ε 0 2
0 0.010 0.005 0.000 0.005 0.010 ²µτ
Fig. 19 Confidence limits on the NSI parameter eµτ using the event b.f. , IC 79 selection from [75,125] shown as solid vertical red lines. Similarly, b.f. , IC 40 b.f. , SK dashed vertical red lines show the 90% credibility interval using a 0.1 flat prior on eµτ and where we have profiled over the nuisance pa- 0.01 0 0.01 rameters. The light blue vertical lines show the Super-Kamiokande ΕΜΤ 90% confidence limit [126]. The light green lines show the 90% credibility region from [127]. Finally, the horizontal dash-dot line in- Fig. 20 Allowed region in the eµτ, e0 plane at 90% C.L. obtained dicates the value of 2∆LLH that corresponds to a 90% confidence − from the combined analysis of low and high energy samples of interval according to Wilks’ theorem. Figure from [128]. data (IceCube-79 and DeepCore respectively), shown by red solid curve. The black dashed curve shows the allowed region from Super-Kamiokande experiment, taken from [126], and the green neutrino pair production, ν ν + ν + ν . The sec- dotted curve is for IceCube-40. The red cross, green + and black α → α β β ondary neutrinos introduce non-trivial flavour composi- star signs show the best-fit values of NSI parameters from IceCube- tions as well as spectral bumps and cutoffs. As discussed 79, IceCube-40 and Super-Kamiokande experiments, respectively. Figure from [130]. in Refs. [122,123,124], these small effects can be probed by the IceCube TeV-PeV diffuse flux. must have Np = Ne. The average neutron abundance according to Earth density models is N 1.051 [95]. h ni ' 6.3 Non-Standard Matter Interactions Therefore, we have
We have already discussed coherent scattering of neu- N 3.051N , N 3.102N . (45) h ui ' e h di ' e trinos and anti-neutrinos in dense matter, that can be accounted for by an effective matter potential in the stan- In the full three-flavour neutrino system, the hermiticity dard Hamiltonian. Since neutrino oscillations are only condition leaves nine degrees of freedom for the eαβ coef- sensitive to non-universal matter effects, only the unique ficients, that reduce to eight after absorption of a global charged-current interactions of electron neutrinos and phase. anti-neutrinos with electrons are expected to contribute. The case of atmospheric muon neutrino oscillations However, non-standard interactions (NSI) can change can again be approximated by oscillations in the νµ-ντ- this picture. system. The oscillations induced by NSI interactions has A convenient theoretical framework to study NSI con- three degrees of freedom that can be parametrized by e e = e and e = e exp(iα). We can make use tributions is given by the Hamiltonian, ττ − µµ 0 µτ of the fact that the NSI Hamiltonian can be recast as an √ HNSI = 2GF Ne ∑ eαβ νβ να . (43) effective Hamiltonian as in Eq. (32). The NSI Hamiltonian α,β | ih | in the two level system can be diagonalised by η = α and tan 2ξ = 2e/e . After removing a global phase, the The dimensionless coefficients e measure the NSI con- 0 αβ effective splitting is ∆δ = √2G N e √1 + 4e2/e 2 with tribution relative to the strength of the standard matter F e 0 0 σ = 1. potential. Hermiticity of the Hamiltonian requires that − The effect of NSI in atmospheric neutrino oscillations e = e . Non-standard interactions with nucleons are αβ ∗βα with IceCube data were studied in Refs. [128] and [130]. not necessarily universal for electrons and valence quarks The IceCube analysis of Ref. [128] is based on three years in matter [129]. Therefore, the parameters eαβ can be con- of data collected by the low-energy extension DeepCore, sidered as effective parameters of the form that was also used in the standard neutrino oscillation analysis discussed in Section4. The data was binned into (e) Nu (u) Nd (d) eαβ = eαβ + h i eαβ + h i eαβ , (44) a two-dimensional histogram in terms of reconstructed Ne Ne neutrino energy and zenith angle, cos θzen, and analysed (u/d) via a profile likelihood method. The analysis focused on where eαβ are the couplings to up- and down-type ( ) ( ) valence quarks in matter. In charge-neutral matter we NSI interactions with d-quarks assuming e d = e d µτ | µτ | 21
(d) (d) and eµµ = eττ . The constraints for eµτ are shown in In atmospheric neutrino data we are interested in Fig. 19. the survival probability of muon neutrinos (and anti- A two-dimensional analysis on the parameters e neutrinos). Typically, it is assumed that the 8 8 matrix µτ × and e was carried out by the authors of Ref. [130]. The D takes on a diagonal form, i.e., D δ D , with posi- 0 ij ij ' ij i analysis was performed on the atmospheric neutrino mea- tive diagonal elements. The three-level system including surement done with one year of IceCube-40 data [23] and decoherence and oscillations can be further simplified as- with two years of public DeepCore data, which was ini- suming universal decoherence parameters Di = D. The tially used for the first oscillation analysis by IceCube [69]. muon neutrino survival probability can then be expressed A combined confidence level region of allowed values as [133], for eµτ and e0 is obtained from this analysis, see Fig. 20. Although the analysis is only based on the arrival di- 1 2 DL DL 2 2 Pν ν = + e− 4e− UµiUµj sin ∆ij . µ→ µ ∑ rection distribution of muon-neutrinos and not their en- 3 3 − i The monotonic increase of the von Neumann entropy, Only the parameters D3 and D8 contribute in this case. (ρ) = (ρ ρ) S Tr ln , implies the hermiticity of the Lind- Again, for D3L ∞ and D8L ∞ we recover a flavour − † → → blad operators, Lj = Lj [131]. In addition, the conserva- ratio 1 : 1 : 1, as expected for full decoherence. tion of the average value of the energy can be enforced by taking [H, Lj] = 0 [132]. In the three-flavour basis it is convenient to expand 6.5 Neutrino Decay the density operator ρ, the Hamiltonian H and the Lind- blad operators L in terms of Hermitian 3 3 matrices i × The flavour ratio of neutrinos from astrophysical sources Fµ (µ = 0, ... , 8), given by, F0 = 13 3/√6 and Fi = λi/2, × can be significantly altered if neutrinos can decay en route where λi (i = 1, ... , 8) are the Gell-Mann matrices [133]. to Earth [134,135,136,137,138]. Two-body decays of the This set of basis matrices satisfy the orthogonality con- form ν ν + X with a massless light scalar X (e.g., a † i → j dition Tr(Fµ Fν) = δµν/2. The Liouville equation (30) can Majoron [139,140,141]) are only weakly limited by solar now be expressed in terms of the expansion coefficients 4 neutrino data [142] with τi & 10− s(mi/eV). In the fol- ρi and hi of the density operator and Hamiltonian, respec- lowing, we will assume that the decay of neutrino mass tively, as the matrix equation eigenstates can be written as a dissipation term in Eq. (30) ρ˙k = ∑ hiρj fijk ∑ Dkiρi . (46) of the form i,j − i 1 The coefficients fijk are the SU(3) structure constants de- [ρ] = Γ (E) Π , ρ(E) Ddec − 2 ∑ i i fined by [Fi, Fj] = i ∑ fijk Fk [133]. The decoherence ef- i k Z fects are cast into a matrix + ∑ dE0γi j(E0, E)ΠjTr(ρ(E0)Πi0) . (50) 1 (n) (n) ij → Dij = ∑ lm fimllk fjkl , (47) 2 k,l,m,n The first term describes the disappearance of a neutrino ( ) where l n are the expansion coefficient of the Lindblad mass eigenstate ν with total decay width Γ (E ). The i | ii i i operator Ln. second term describes the appearance of neutrino mass 22 eigenstates from the transition ν and ν with differ- Astrophysical neutrinos propagating over cosmic dis- | ii | ji ential production rate γi j(Ei, Ej). Note, that the corre- tances are also susceptible to feeble interactions with cos- sponding set of Lindblad→ operators representing the ex- mic backgrounds. In particular, feeble interactions with pression in Eq. (50) can be written as the cosmic neutrino background (CνB) that can be en- hanced by resonant interactions, e.g., ν + ν Z q α α → 0 → Lij = γi j(Ei, Ej) νj(Ej) νi(Ei) , (51) νβ + νβ have been discussed as a source for absorption → | ih | features [144,145,146,147]. The evolution of the neutrino where the index i runs over active neutrino states and density matrix is identical to that for neutrino decay with the index j over active and sterile states. In contrast to dissipation term as in Eq. (50). However, in this case the quantum decoherence discussed in the previous section, interaction rates have a non-trivial dependence on red- this set of Lindblad operators are non-Hermitian and do shift via the density evolution of the CνB. not commute with the Hamiltonian. As a further approximation, we will consider neu- trino propagation over cosmological distances, i.e., all 6.6 Sterile Neutrinos oscillation terms from neutrino mass differences are av- eraged. As a representative example, we consider transi- It is conceivable that the three neutrino flavour states, tions that leave the neutrino energy constant, i.e., γi j νe, νµ, and ντ, are augmented by additional states. The → ' Bri jΓiδ(Ei Ej), and normal ordering of neutrino masses number of light “active” neutrinos, i.e., neutrinos that → − with transitions 3 2, 3 1, and 2 1. In the mass take part in the Standard Model weak interactions is lim- → → → eigenbasis the evolution of the density matrix has then ited to the known flavour states from the observed decay the form width of the Z-boson. However, it is feasible that there are 2 light “sterile” neutrino states with no Standard Model in- i∆mij 1 teractions. These sterile states would imply the existence ρ˙ij = ρij (Γi + Γj)ρij + δij ∑ ΓkBrk iρkk . (52) 2Eν − 2 k → of additional neutrino mass eigenstates that could impact the three-flavour neutrino oscillation phenomenology, In the oscillation-averaged solution only the diagonal due to an extended PMNS mixing matrix Ue, elements ρii have non-zero contributions. We can then express the neutrino survival/appearance probability as 3+n να = ∑ Ueα∗j νj . (56) 2 | i j=1 | i Pα β = Mji UβjUαi , (53) → ∑ i,j The term “sterile” referring to neutrinos was introduced by Pontecorvo already in 1968 when discussing the, at where the elements of transition matrix, Mji, can be writ- ten in the form [143] that time hypothetical, possibility of vacuum neutrino oscillations [44]. (c) Many extensions of the Standard Model that relate to Mji = ∑ Mji . (54) c the appearance of neutrino mass terms predict the exis- The sum in the previous equation runs over all produc- tence of sterile neutrinos. As discussed earlier, the right- handed neutrino field, that can provide a Dirac mass term tion chains c1 ... cnc with c1 = i and cnc = j and → → mDνLνR + h.c., does not interact via weak interactions and is therefore sterile. However, in the minimal type nc 1 ! nc nc (c) − LΓ 1 I seesaw models (see, e.g., Ref. [63]) these sterile neutri- M = Γ e− ck . (55) ji ∏ cl+1cl ∑ ∏ Γ Γ nos have a large Majorana mass term, Mν νT /2 + h.c., l=1 k=1 p=1(=k) cp − ck RC R 6 with m M, that give rise to a large effective neutrino D In this particular scenario, the final neutrino flavour ratio mass after diagonalising the mass matrix. These massive for LΓ 1 will thus correspond to the composition of sterile states are practically unobservable in low-energy i the lowest mass eigenstate. oscillation experiments. On the other hand, for values Note, that the decay rates Γ are expected to decrease of M m (“pseudo-Dirac” case), the active and ster- i D with energy due to relativistic boosting of the mass eigen- ile state mass states are degenerate after diagonalisation, state’s lifetime. Therefore, the neutrino flavour composi- leading to maximal mixing between the left (active) and tion can experience strong energy dependencies. On the right (sterile) states. other hand, active neutrino decay into sterile neutrinos The minimal sterile neutrino model is a “3+1” model can introduce spectral cutoffs due to the energy depen- where, in addition to the three standard weakly-inter- dence of the neutrino lifetime. This process is limited by acting neutrino flavours, one additional heavier sterile the observation of IceCube’s TeV-PeV neutrino flux and neutrino state is added. Such a simple extension of the could be responsible for a tentative cutoff [136]. Neutrino neutrino sector has been advocated to explain certain ten- decay has also been considered as a possibility to alleviate sions between experimental results from accelerator [155, a mild tension in the best-fit power-law spectra between 156,157], reactor [158] and radio-chemical [159] experi- cascade- and track-dominated IceCube data [138]. ments, and the predictions from the standard three active 23 8 L 99% C.L. 6 4 90% C.L. 2∆ ln 2 − 0 0.30 C.L. 90% C.L. 99% SK, NO (2015), 90 % C.L. SK, NO (2015), 99 % C.L. 0.25 IceCube, NO (2016), 90 % C.L. IceCube, NO (2016), 99 % C.L. IceCube, IO (2016), 90 % C.L. 24 θ IceCube, IO (2016), 99 % C.L. 2 0.20 cos · 34 θ 2 0.15 = sin 2 | 4 0.10 τ U | 0.05 0.00 3 2 1 10− 10− 10− 0 2 4 6 8 2 2 2∆ ln Uµ4 = sin θ24 − L | | 2 2 Fig. 22 The exclusion contours on the (sin θ24, Uτ4 ) plane at 90% and 99% C.L. (dark and light blue solid lines,| respectively).| The Fig. 21 The 90% C.L. exclusion contour on the (sin2 2θ , ∆m2 ) 24 41 dash lines show results from the Super-Kamiokande [148]. Figure plane (orange solid line). The green and yellow bands contain reprinted with permission from [125] (Copyright 2017 APS). 68% and 95%, respectively, of the 90% contours in simulated pseudo-experiments. The contours and bands are overlaid on 90% C.L. exclusions from the Super-Kamiokande, MINOS, Mini- BooNE and CDHS experiments [148,149,150,151], and the 99% disappearance of muon neutrinos and are more sensi- C.L. allowed region from global fits to appearance experiments in- 2 tive to θ24. Therefore the choice of a simplified minimal cluding MiniBooNE and LSND, assuming Ue4 =0.0023 [152] and 2 | | mixing scenario where only θ24 is not zero is justified, Ue4 =0.0027 [153], respectively. Figure reprinted with permission from| | [154] (Copyright 2016 APS). and is indeed the approach followed in the analyses de- scribed below. Nonzero values of the weakly constrained θ34 within current limits would not significantly affect flavour scenario. In the most general case, the introduc- the results presented here [162]. tion of one sterile neutrino adds six new parameters to There have been several searches in the past at sub- the neutrino oscillation phenomenology [152]: three mix- TeV energies with neutrinos from the Sun, reactors, and ing angles θ14, θ24, θ34, two CP-violating phases, δ14 and accelerator setups (see, e.g., Refs. [152,163,164,165] and 2 δ34, and one mass difference, ∆m41, where the indexes references therein). The large flux of atmospheric neu- ’1-3’ stand for the known neutrino mass states and ’4’ for trinos that reach IceCube and the wide range of energy the sterile state. response of the detector allow to search for signatures Although sterile neutrinos can not be detected directly, of anomalous νµ + νµ disappearance caused by oscilla- their existence can leave an imprint on the oscillation pat- tions to an sterile neutrino, νs, at TeV energies, an energy tern of active neutrinos. The sterile neutrino modifies not probed before. Furthermore, the DeepCore detector the oscillation pattern of the standard neutrinos since can extend the search down to about 6 GeV, improving these can now undergo vacuum oscillations into the new previous limits from smaller detectors. state, with a probability that is proportional to the new The analysis techniques are very similar to the search mixing angles. The period of these oscillations can be for standard oscillations and are based on comparing the small, smaller than the directional resolution of IceCube, equivalent oscillogram from Fig.5 for oscillations with and the net effect is then to distort the overall νµ flux an sterile component, to the measured data. Although normalisation with respect to the three-flavour case. An the low-energy tracks provide a quite short lever arm to additional effect arises from the different interactions of reconstruct their direction with precision, and the angu- flavours with matter when traversing the Earth [160,161]. lar resolution of the analysis varies between 6° and 12°, The new possibility to oscillate to a state that does not in- depending on energy. This is enough, though, to perform teract results in energy and angular dependent oscillation the analysis (see Fig. 5 in Ref. [125]) since the differences amplitudes that depend on the mixing angles, but also on in the oscillation pattern in the presence of a sterile neu- 2 the new ∆m41. More precisely, the comparison between trino can still be distinguished from the non-sterile case the oscillation pattern in the (energy,zenith) phase space with such angular resolution. At higher energies the anal- predicted by the sterile “3+1” model and the pattern seen yses necessarily follow a slightly different approach since in experimental data is what IceCube exploits to set lim- the tracks originate outside the detector and the energy its on the sterile mixing parameters. As for the standard deposited in the hadronic shower at the interaction vertex oscillation case described in section4, searches for sterile is not accessible. Such analyses use measured muon ener- neutrinos in IceCube are based on the detection of the gies instead of neutrino energies, like the one presented 24 in Ref. [154]. The muon energy is reconstructed from the clusters, and the formation of first galaxies growing out of light emission profile from stochastic energy losses of small density perturbations imprinted in the cosmic mi- the muon along its trajectory [17], achieving an energy crowave background, is to introduce a “dark matter” com- resolution of σ (E /GeV) 0.5. Since at the energies ponent in the energy budget of the Universe [167,168]. log10 µ ∼ of this analysis the muon track can be well reconstructed, Attractive candidates for dark matter consists of stable the angular resolution reaches values between 0.2° and relic particles whose present density is determined by the 0.8°, depending on incoming angle. thermal history of the early universe [169,170,171,172]. As with searches for standard oscillations, the aim The present abundance of dark matter can be naturally is to reach a pure sample of atmospheric neutrinos to explained by physics beyond the Standard Model provid- be able to compare the measured number of events as a ing stable, weakly-interacting massive particle (WIMP) function of angle and energy with the model prediction. in the few GeV-TeV mass range. For thermally produced None of the analyses detected a deviation from the ex- WIMPs, the upper mass limit arises from theoretical ar- pected standard oscillation scenario and, therefore, limits guments in order to preserve unitarity [173], although on the oscillation parameters which depend on the addi- higher masses can be accommodated in models where the tional neutrino can be set. The low-energy analysis uses dark matter candidates are not produced thermally [174]. a 8 8 binned grid in the (energy, cos θ ) plane while, There is a vast ongoing experimental effort to try to × zen due to its higher energy resolution, the high-energy anal- identify the nature of dark matter through different strate- ysis uses a 10 20 binned grid. A log-likelihood approach gies: production at colliders [175] or through the detec- × is used to find the best fit to the data given the model tion of nuclear recoils in a selected target in “direct de- parameters. Confidence levels are calculated from the tection” experiments [176]. A complementary, “indirect”, difference between the profile log-likelihood and the log- approach is based on searching for the products of the an- likelihood at the best fit point. The results are illustrated nihilation of dark matter particles gravitationally trapped in Figs. 21 and 22. Figure 21 shows the result of the high- in the halo of galaxies or accumulated in heavy celestial 2 2 energy analysis expressed in the (∆m14, sin 2θ24) plane. objects like the Sun or Earth [177,178,179,180,181,182, The figure shows the 90% confidence level contour (red 183,184]. In this latter case, neutrinos are the only pos- line) compared with 90% exclusions from previous dis- sible messengers, since other particles produced in the appearance searches. The exclusion is compatible with annihilations will be absorbed. These search techniques the sensitivity (green and yellow areas) calculated un- are competitive since they can set limits on the same phys- der the assumption of the no-sterile neutrino hypothesis. ical quantities (the dark matter-nucleon cross section for The result therefore disfavours much of the parameter example). But they are also complementary since they are space of the “3+1” model. On the other hand, Fig. 22 subject to different backgrounds (the gamma-ray sky is shows the results of an analysis using low-energy events very different from the proton or neutrino sky), different (6 GeV to 60 GeV) contained in DeepCore. The results astrophysical inputs (dark matter density and velocity are shown in the ( U 2, U 2) plane, where U is distribution) and different systematics (nucleon and nu- | τ4| | µ4| | τ4| defined as sin2 θ cos2 θ and U is just sin2 θ . The clear form factors of different targets). 34 34 | µ4| 24 best-fit value of these elements of the mixing matrix are The strength of the expected neutrino flux emitted U 2 = 0.0 and U 2 = 0.08, while the 90% confidence from a celestial object depends, among other factors, on | µ4| | τ4| level limits on their values are still compatible with 0, the capture rate of WIMPs, which is proportional to the U 2 0.11 and U 2 0.15. Combinations of the WIMP-nucleon cross section, and the annihilation rate, | µ4| ≤ | τ4| ≤ parameters shown in the axes above the contours are dis- which is proportional to the velocity-averaged WIMP- WIMP annihilation cross section, σ v . The evolution favoured. The figure includes a recent exclusion contour h A i from Super-Kamiokande [148] as a comparison. of the WIMP number density nDM in compact celestial As in the previous cases, the presence of sterile neu- objects follows the balance equation trinos can also have an effect on neutrino spectra from 2 n˙ DM = QC 2 σAv (nDM/2) , (57) astrophysical sources. In the case of a pseudo-Dirac sce- − h i nario, where the mass splitting between active and sterile where QC is the capture rate per unit volume and the neutrinos is very small, there exist the possibility that the numerical factors account for the fact that annihilations corresponding neutrino oscillation effects are still visible, remove two WIMPs per interaction but there are only i.e., L∆m2/E 1 [166,137]. 1/2 of the WIMPS available to form pairs. If the capture ' rate remains constant over a long time, the WIMP density reaches an equilibrium solution 7 Indirect Dark Matter Detection s QC nDM,eq = . (58) σAv There is a large corpus of evidence that supports the ex- h i istence of a non-baryonic, non-luminous component of The WIMP capture rate from interaction with bary- matter in the cosmos. A way to understand the rotation onic matter can have spin-dependent, σSD, and spin- curves of galaxies, the peculiar velocities of galaxies in independent, σSI, contributions [185]. Since the Sun is 25 primarily a proton target (75% of H and 24% of He in with a spectral production rate per unit volume given by mass) [186] the capture of WIMPs from the halo can be 1 σ v dN considered to be driven mainly via the spin-dependent ann(r) = 2 (r) A να Qνα ρDM h 2 i , (59) scattering. Heavier elements constitute less than 2% of the 2 mX dEν mass of the Sun, but can still play a role when considering where ρDM = m nDM is the WIMP mass density at a spin-independent capture, since the spin-independent X given position r. The energy distribution dNν /dEν of cross section is proportional to the square of the atomic α neutrinos is normalised to the total number of neutrinos mass number. Note that these heavy elements can also of flavour να expected from the annihilation. The factor take part in the spin-dependent capture process if WIMPs 1/2 compensates for the symmetry of WIMP combina- present momentum-dependent interactions with normal tions. The differential flux observed from the solid angle matter [187]. The situation for the Earth is rather different, Ω can then be expressed as the line-of-sight integral from since the most abundant isotopes of the main components Earth’s location4 r , of the Earth inner core, mantle and crust, 56Fe, 28Si and ⊕ 16 O[188], are spin 0 nuclei. Searches for dark matter ac- ∞ Z σAv dNν cumulated in the Earth are then more sensitive to the σSI 2 α Fνβ (Eν) = ∑ Pα β dlρDM(r(l, Ω)) h 2i , component of the WIMP-nucleon cross section. Another α → 8πm dEν 0 X difference with respect to solar searches is that equilib- (60) rium between the capture and annihilation rates can not be assumed, and to be able to extract a limit on σSI, an where r(l, Ω) r + ln(Ω) and n(Ω) is a unit vector in ≡ ⊕ b b assumption on the value of the WIMP self-annihilation the direction Ω. The factor Pα β(Eν) accounts for flavour → cross section must be made. oscillations5. The factor in parenthesis [ ] encodes the · Dark matter searches from our own Galaxy, nearby cosmological and astrophysical dependence on the emis- galaxies or galaxy clusters present some distinct features sion from the dark matter density distribution. This is with respect to searches from the Sun or Earth which sometimes called J-factor [189]. are advantageous. Firstly, capture is not an issue since The decay of dark matter can be treated analogously. the presence of dark matter over-densities has been an Here, the spectral production rate is given by essential part in the process of galaxy formation. What can be measured then is the velocity-averaged WIMP dec ΓX dNνα Qνα (r) = ρDM(r) , (61) self-annihilation cross section, σ v . Secondly, the prod- mX dEν h A i ucts of the annihilations are not necessarily absorbed at where ΓX denotes the dark matter lifetime. As before, the the production site, and other indirect signatures (pho- distribution dNνα /dEν of neutrinos is normalized to the tons, anti-protons, etc.) can also be searched for in γ-ray total number of neutrinos of flavour να and it depends and cosmic-ray observatories. These multi-wavelength implicitly on the branching fraction of the decay. The and/or multi-messenger searches can increase the sen- differential flux observed from the solid angle Ω can then sitivity of dark matter searches. Neutrinos remain, how- be expressed as the line-of-sight integral ever, an attractive signature since they do not suffer from uncertainties in their propagation (as charged particles ∞ Z Γ d do) and no background or foreground from astrophysical X Nνα Fνβ (Eν) = ∑ Pα β dlρDM(r(l, Ω)) . α → 4πmX dEν objects is present (as in the case of γ-rays). Note, that 0 some of the sources are extended (the Galactic halo for (62) example) and point-source analysis techniques have to be modified. On the other hand we expect that the flux Again, the factor in parenthesis [ ] encodes the cosmo- · of secondaries from these distant objects is much lower logical and astrophysical dependence and is sometimes than that predicted from WIMP annihilations in the Sun called D-factor. and, furthermore, there are new systematics effecting the IceCube analyses are performed in the most model- calculations. For example, the assumed shape of the dark independent way possible, but the allowed parameter matter halo profile effects significantly the interpretation space of the underlying theoretical models force some of the results since the annihilation rate depends on the assumptions to be made. Since the exact branching ratios square of the dark matter number density. We will discuss of WIMP annihilation into different channels is model- these issues in section 7.4. dependent, two annihilation channels which give ex- treme neutrino emission spectra dN/dE are chosen, and 4Note that in a galactocentric coordinate system considered for 7.1 Neutrinos from WIMP Annihilation and Decay WIMP annihilations in the Galactic halo, the Earth’s position is r r (0, 8.5kpc, 0). 5⊕ ' ' − The annihilation of dark matter into Standard Model par- Strictly speaking, the flavour transition probabilities depend on the distance l and should appear under the line-of-sight integral. ticles can be probed by γ-ray, cosmic ray, and neutrino However, in most situations discussed in the following it can be emission. We focus in the following on neutrino emission treated as a constant. 26 the analysis is optimised for each of them separately. An- nihilation into bb is taken as a representative case for a soft neutrino energy spectrum. This is in part due to the b- quark interaction with the medium (for dense objects like the Sun) and the hadronisation of the final state quarks leading to neutrinos from meson decays. In contrast, the + + annihilation into W W− or τ τ− follows a hard spec- trum. Assuming a 100% branching ratio to each of these channels brackets the expected neutrino spectrum from any realistic model with branching to more channels. The WIMP mass is left free, and independent searches are per- formed for a few benchmark masses. The total number of signal events, µs, expected at the detector from a given Fig. 23 IceCube limits on the spin-dependent WIMP-proton scat- model is then given by, SD tering cross section, σχ p as a function of WIMP mass, compared Z Z to results from other neutrino− detectors and direct detection experi- = eff( ) ( ) µs Tlive ∑ dΩ dEν Aνα Ω, Eν Fνα Eν , (63) ments [211,203,212,213]. The IceCube limits have been scaled up to α the upper edge of the 1σ systematic uncertainty band. The coloured points correspond to models from a scan of the pMSSM. The model where is the exposure time, eff(Ω, ) the detector Tlive Aνα Eν points are shown colour-coded according to the “hardness” of the effective area for neutrino flavour α, that depends on the resultant neutrino spectrum. Red points correspond to models that + detector’s response with respect to observation angle and annihilate predominantly into harder channels (such as τ τ−) and neutrino energy. can hence be excluded by the IceCube red line, while blue points correspond to models that favour annihilations into softer channels In the absence of a signal, the 90% confidence level 90 (such as bb) and are probed by the blue lines. Similar coding applies limit on the number of signal events, µs , can then be for intermediate colours. Figure from [214]. directly translated into a limit on either the velocity- averaged annihilation cross section σ v or the dark h A i matter lifetime ΓX. The interplay between the total num- can be of relevance when comparing results from differ- ber of observed events in a given data sample, nobs, and ent search techniques [197]. the estimated number of background events, nbg, is the The experimental effort to detect neutrinos as a signa- basis to perform a simple event-counting statistical anal- ture of dark matter annihilations in celestial bodies came ysis to constrain µs, i.e., to constrain a given model. This of age in the mid 90’s with underground detectors like was the approach followed in early IceCube publications, MACRO, Baksan, Kamiokande and Super-Kamiokande. e.g., Refs. [190,191]. These detectors provided the first limits on the flux of neu- In order to improve the power of the statistical test, trinos from dark matter annihilations in the Earth or the a distribution-shape analysis can be used instead. The Sun [198,199,200,201]. Baksan and Super-Kamiokande angular distribution of the expected signal events is ex- continue to be competitive in the field today [202,203, pected to peak towards the source, and is very different 204]. Baikal [205] and AMANDA were the first large- from the flatter atmospheric neutrino background. If Ω is scale neutrino detectors with an open geometry to per- the solid angle of a reconstructed track position, we can form dark matter searches in the late 90’s, soon followed weight the event by signal and background probability by ANTARES [206]. Early results of these experiments distributions S(Ω) and B(Ω), respectively. The likelihood can be found in Refs. [205,207,208,209,190,210]. that the data sample contains µs signal events out of nobs observed events is then defined as nobs µs µs 7.2 Dark Matter Signals from the Sun (µs) = ∏ S(Ωi) + 1 B(Ωi) . (64) L nobs − nobs i=1 The interplay between the dark matter capture and anni- From this likelihood, one can determine confidence inter- hilation determines the number of WIMPs accumulated vals for µs by using the likelihood-ratio test statistic as in the Sun. Losses through evaporation due to WIMP- proposed in [192]. This method produces stronger limits nucleus scattering have been shown to be negligible for than just event counting, and it is less prone to unsimu- WIMP masses above a few GeV [178,215,216] and can lated background contamination in the final data sample. therefore be neglected in IceCube analyses. Given the age There are systematic uncertainties in the translation of of the Sun (4.5 Gyr), the estimated dark matter density the number of detected events into capture cross section (ρ 0.4 GeV/cm3) and a weak-scale interaction be- local ∼ values due to uncertainties in the element composition tween dark matter and baryons, many models predict of the Sun [193], the effect of planets on the capture of that dark matter capture and annihilation in the Sun have WIMPS from the halo [194] and the uncertainty on the reached equilibrium with density following Eq. 58. In values of the nuclear form factors needed in the rather this case, the J-factor in Eq. (60) becomes proportional complex capture calculations [185,195,196]. These effects to 1/ σ v and total neutrino emission only depends h A i 27 •34 ) 2 10 ∆ allowed mγ (1), q(1) mer, doubling the exposure of the detector per calendar IceCube•22 LKP γ(1) (2007) 10•35 year. DeepCore has also allowed to extend the search for 2 •36 10 CDMS (2008) neutrinos from WIMPs with masses as low as 20 GeV/c , COUPP (2008) whereas past IceCube searches have only been sensitive 10•37 KIMS (2007) above 50 GeV/c2. Additionally, all-flavour analyses are ∆ (1) = 0.01 10•38 q being developed [231], since the addition of νe and ντ 10•39 events triples the expected signal. Improved low-energy •40 ∆ (1) = 0.1 10 q reconstruction techniques allow to reconstruct electron LKP • proton SD cross•section (cm and tau neutrino interactions with sufficiently good angu- 10•41 ∆ (1) = 0.5 q lar resolution to be useful in solar and Earth dark matter 10•42 searches. •43 10 Ω 2 Signal and background differ, though, not only in their 0.05 < CDMh < 0.20 Ω 2 σ 0.1037 < CDMh < 0.1161 WMAP 1 10•44 angular distribution, but also in their energy spectra. This 102 103 104 LKP mass (GeV) information can be encapsulated in a likelihood function that includes a number-counting (normalisation) term Fig. 24 IceCube limits on the spin-dependent LKP-proton scatter- SD and an angular and energy (spectral) terms, both for the ing cross section, σχ p as a function of LKP mass, compared with limits from direct detection− experiments [221,222,223]. The theoret- signal and background p.d.f.’s. This is the approach taken ically allowed phase space is indicated by the green shaded [224]. in the latest IceCube solar WIMP analyses [232,214]. All- The region below mLKP = 300 GeV is excluded by collider ex- flavour analyses also benefit from this technique since periments [225], and the upper bound on mLKP arises from ar- the better energy resolution of cascade events benefits guments to avoid over-closure in the early universe [226]. The lighter blue region is allowed when considering a wide range from the use of energy information in the likelihood. The 2 of dark matter relic density 0.05 < ΩCDMh < 0.20, and the general form of such an extended likelihood is, darker blue region corresponds to the 1σ WMAP relic density, nobs µ < 2 < µ e tot 0.1037 ΩCDMh 0.1161 [227]. Figure reprinted with permission (µ µ ) = tot − (µ ) (65) from [217](Copyright 2010 APS). ext s, tot s , L nobs! L where the second term is analogous to Eq. (64) and the prefactor is the Poisson number likelihood for observing on the capture Q related to the WIMP-nucleon scatter- C n events given a total of µ predicted signal and back- ing cross section. Under the assumption that the cap- obs tot ground events. The signal and background probability ture rate is fully dominated either by the spin-dependent distribution functions, S and B, introduced in Eq. (64) or spin-independent scattering, conservative limits can now include the event’s reconstructed solid angle Ω and be extracted on either the spin-dependent, σ , or spin- i SD energy E . For instance, the signal probability distribution independent, σ , WIMP-proton scattering cross section. i SI is given by The number of events in the detector observed in a given live time can therefore be translated into a limit on a Z Z S(E , Ω , t ) dΩ0 dE0 Q(E , Ω E0, Ω0) physical quantity that can be used to compare with other i i i ≡ i i| 2 experiments or to test predictions of a specific particle d PS physics model. Different dark matter scenarios can be (E0, Ω0, ti, ξ) , (66) · dE0 dΩ0 probed through the predicted neutrino flux in Eq. (63). where d2P /dE dΩ is the expected signal distribution of Indeed, IceCube has set limits to the muon flux from an- S incident neutrino energies and angles at the time of the nihilations of the lightest Kaluza-Klein mode arising in event’s arrival6, t . For the signal, this term is a prediction models of universal extra dimensions, as well as to its i of the model under consideration and it depends on the scattering cross section with protons [217] (see Fig. 24). free parameters of the model, denoted by the vector ξ. An Other, non-standard, scenarios like strongly interacting equation analogous to Eq. (66) holds for the background dark matter or self-interacting dark matter can also be distribution B(E, Ω). For the background, d2P /dE dΩ tested since the IceCube event selections are quite generic B will typically depend only on the energy and angular and model-independent [218,219,220]. spectra of the atmospheric neutrino flux, and it is inde- Traditionally, solar WIMP searches with IceCube have pendent of arrival time ti and ξ. used the muon channel since it gives better pointing and, In general, the reconstructed neutrino energy E and in the end, dark matter searches from the Sun are really solid angle Ω are different from the true energy E0 and point-source searches. The first analyses used the Earth true arrival direction Ω0. The relation between these quan- as a filter of atmospheric muons and “looked” at the Sun tities on a statistical basis must be obtained from simula- only in the austral winter, when the Sun is below the tions, and it is specific to the annihilation channel under horizon at the South Pole [217,228,229]. With the com- pletion of IceCube-79 and DeepCore, it was possible to 6The dependence on arrival time is relevant for indirect dark matter define effective veto regions to efficiently reject incom- searches from the Sun, which can be effectively parametrised by the angular distance ψ between reconstructed arrival direction and ing atmospheric muons from above [230]. Since then the the known position of the Sun at the arrival time (corrected for the IceCube solar WIMP searches cover also the austral sum- light-travel time of about 8 min). 28 31 IceCube Collaboration 2016 ) 10 − perimental limits. Since large-volume neutrino telescopes 2 ¯ + 32 bb, counting τ τ −, counting are high-energy neutrino detectors, the limits for annihi- (cm 10 − ¯ + ,p bb, full τ τ −, full 33 L L lation channels leading to a soft neutrino spectrum can SD 10 − + σ W W −, counting PRL soft be more than two orders of magnitude less restrictive 34 + 10 − W W −, full PRL hard L than those resulting in harder spectra. Even in the lat- 35 10 − ter case the limits can decrease rapidly for low WIMP 36 10 − masses. Direct search experiments do not reach σSD val- 37 38 2 10 − ues much below 10− cm at their best point, worsening 38 10 − rapidly away from it, while, together, the limits from 39 IceCube and Super-Kamiokande cover the WIMP mass 10 − 40 range between from a few GeV to 100 TeV and reach cross 10 − 40 2 IC79 section values of about 10− cm . In the case of the σSD Dark matter-proton cross-section 41 10 − 10 1 10 2 10 3 10 4 cross section, direct-search experiments have the advan- Dark matter mass mχ (GeV) tage of dedicated spinless targets, and the limits from Fig. 25 Improvement on the dark-matter proton cross section due neutrino telescopes lie about three orders of magnitude to the use of an event-level likelihood analysis (full lines) compared above the best limit from LUX at a WIMP mass of about to a traditional cut-and-count analysis (dashed lines) and an analy- 50 GeV [236]. sis based on the difference in shape of the space-angle distribution for signal and background (dotted lines tagged ’PRL’ and originally The improvement due to using a full event-based like- + + from [230]). 100% annihilation to bb, W W− and τ τ− is assumed lihood in comparison to just an angular shape analysis is in each case shown. Figure from [232]. illustrated in Fig. 25, taken from [232]. The figure shows the limit on the spin-dependent WIMP-proton cross sec- tion as a function of WIMP mass obtained with two anal- study. The true energy E can be related to the number of yses performed on the same data set taken with IceCube- hit DOMs or can be estimated from more elaborate en- 79. Shown are the limits using an event-count likelihood ergy reconstructions. The quantity Q(Ei, Ωi E0, Ω0) is the (dashed lines), the limits obtained using an analysis based | probability density (in effective units of inverse steradian on the difference in shape of the space-angle distribution and proxy energy) for reconstructing Ei and Ωi for the ith for signal and background (tagged ’PRL’ and originally event when the true values are E0 and Ω0, respectively. from [230]) and the limits obtained using a full likelihood Note that systematic uncertainties on the signal and/or like in Eq. (66). Including the event-level energy infor- background prediction or on the angular or energy resolu- mation has the most impact at high WIMP mass, due to tions can be easily incorporated in a likelihood approach the relatively good energy resolution of IceCube at high as nuisance parameters by marginalising over them. The neutrino energies. Note that the full likelihood analysis only knowledge needed is the functional form of the nui- in [232] used a rather simple energy proxy based on the sance parameters. P is a function of energy and angle, number of hit DOMs. Better energy reconstruction algo- which in a simplified approach can be decomposed in rithms being developed within IceCube, particularly at an angle-dependent part (the PSF of the detector) and low energies, will further improve the performance of an energy-dependent part (the energy dispersion of the this method [76]. detector). More generally, the angular response of the There is an additional step in complexity when us- detector can depend on energy, and then this decompo- ing neutrino telescope results to probe specific WIMP sition is not valid. For point-source searches, due to the models, that avoids the need to simulate specific anni- restricted angular region in the sky considered, the PSF hilation benchmark channels [237,238,235]. For a given and energy dispersion can be taken to be uncorrelated. model, this approach takes into account the full neutrino Figure 23 shows the limits on the spin-dependent spectrum from WIMP annihilations including all allowed WIMP-proton cross section using three years of IceCube annihilation channels, i.e., the full d2P/dE dφ(E, φ, ξ) is data, including both angular and energy terms in the like- calculated. The expected number of signal events is then lihood [214]. The plot shows the latest IceCube results obtained through Eq. (63) and, through the use of the as full lines for three benchmark annihilation channels likelihood function in Eq. (65), the model under consid- (in different colours), compared with the results of Super- eration can be accepted or rejected at a given confidence Kamiokande [203] and ANTARES [211], as well as results level. To explore a wide parameter space, which is typi- from the PICO direct detection experiment [212,213]. The cal for supersymmetric extension of the Standard Model, dots correspond to a parameter scan of the phenomeno- each allowed combination of the free parameters of the logical Minimal Supersymmetric extension of the Stan- model needs to be tested, and the procedure becomes dard Model (pMSSM) where the color code represents computationally demanding. Even ad-hoc models like the leading annihilation channel: channels producing a the constrained MSSM (cMSSM) [239] with just 7 param- soft neutrino spectra are marked as blue, and are probed eters pose a computational challenge. But it is a power- by the soft experimental limits, while harder neutrino ful way of assigning a statistically meaningful weight spectra are marked in red and are probed by the hard ex- to different areas of the model parameter space. The au- 29 Fig. 26 Spin-dependent WIMP-proton scattering cross section, Fig. 27 Lightest squark mass as a function of gluino mass for SD σχ p as a function of WIMP mass for points derived from explo- points derived from explorations of the MSSM-25 parameter rations− of the MSSM-25 parameter space. The corresponding space. The points to the bottom left of the magenta line are 90% CL limits from SIMPLE [233] and COUPP [234] direct excluded by ATLAS at 95% CL, based on searches for direct detection experiment are shown as magenta and yellow lines production of coloured sparticles and their decay into jets and respectively. Colour coding indicates predicted IceCube-86 missing transverse energy. Colour coding indicates predicted model exclusion levels. The areas of cyan and blue points show IceCube-86 model exclusion levels. Figure from [235] (Copy- that IceCube-86 has the ability to exclude models beyond the right SISSA Medialab Srl. Reproduced by permission of IOP reach of current direct detection experiments. Figure from [235] Publishing. All rights reserved). (Copyright SISSA Medialab Srl. Reproduced by permission of IOP Publishing. All rights reserved). thors in [237] and [235] have performed sensitivity stud- Earth can annihilate, giving rise to a flux of neutrinos. ies of IceCube and IceCube-DeepCore under different A signal from dark matter annihilations in the centre of model assumptions. Figure 26 shows the results of the the Earth will produce a unique signature in IceCube as model scan in [235]. The figure shows the spin-dependent vertically up-going muons, where each string can act as neutralino-proton cross section versus neutralino mass. a more or less independent detector. Searches from the Each point in the plot represents an allowed combina- vertical direction pose, however, some challenges in Ice- tion of the cMSSM parameters. The colour code indi- Cube due to its geometry. While in any other point source cates which models can be probed by IceCube and dis- search an off-source region at the same declination of the favoured at the 1σ (green), 2σ (light blue) and 5σ (dark source can be defined to measure the background, this blue) level. Figure 27 illustrates the complementarity be- is not possible for the vertical direction, and one needs tween accelerator and neutrino telescope searches for to rely on accurate simulations of the background com- supersymmetry. The plot shows the gluino-squark mass ponents (atmospheric neutrinos and muons). Through parameter space with colour-coded exclusion levels by the use of advanced classification methods to separate IceCube (same colour coding as in Fig. 26). The brown signal and background, the amount of misreconstructed line shows the 95% CL exclusion region from searches atmospheric muons can be reduced to a negligible level, for coloured sparticles in jets+missing transverse energy and a likelihood shape analysis can be performed using 1 with, at the time, 4.71 fb− of data at centre-of-mass ener- Eq. (64). The results of such an analysis using 327 days gies √s 7 TeV from ATLAS [240]. As can be seen, there of live time with the IceCube-79 configuration [241] are ' is a wealth of models not excluded by ATLAS that are shown in Fig. 28. The shape of the IceCube limits reflects under the reach of IceCube (blue dots). Note that more re- the resonant capture of WIMPs of certain masses that cent results from ATLAS and CMS (see, e.g., the summary nearly match the mass of the main isotopes of the Earth on the experimental status of Supersymmetry in [41]) do (the peaks show the Iron, Silicon and Oxygen resonances not change the picture of complementarity between the at 53 GeV/c2, 26 GeV/c2 and 15 GeV/c2, respectively). A parameter space reach of IceCube and the LHC. self-annihilation cross section, σ v , of 3 10 26 cm3 s 1 h A i × − − has been assumed, a typical value for a particle species to be a thermal relic. The lack of a signal can also be used 7.3 Dark Matter Signals from the Earth to set limits on the dependence of σSI on the annihila- tion cross section, as illustrated in Fig. 29. Values above The rationale for searching for dark matter accumulated the full lines are disfavoured at the 90% confidence level in the Earth follows a similar line than that of the Sun: since the combination of capture and annihilation would WIMPs gravitationally accumulated in the centre of the have produced a detectable muon flux in IceCube. 30 + + + IC86 I Earth limit : mχ =1TeV, χχ W W− IC86 I Earth : χχ W W− or τ τ− -38 − → -37 − → 10 + 10 IC86 I Earth limit : m =50GeV, χχ τ τ− SuperK − χ → + + IC79 Sun : χχ W W− or τ τ− -38 → -39 10 SuperCDMS LT 10 − LUX 2013 -39 10 -40 10 -40 IC86 I Earth limit calculated 10 T − − 26 3 1 -41 h assuming σAv = 3 10− (cm s− ) 10 e × r m ) ® ) -41 a 2 2 10 l c m r m IC limit -42 o c c 10 s ( s ( on σ v − A I -42 I S 10 s S e σ c ® σ t i -43 o -43 10 n 10 IC limit on σAv LUX limit mχ =1TeV -44 10-44 10 ® -45 10 -45 10 LUX limit mχ =50GeV -46 10 1 2 3 4 10 10 10 10 -46 10 10-30 10-29 10-28 10-27 10-26 10-25 10-24 10-23 10-22 10-21 10-20 mχ (GeV) 3 1 σAv (cm s− ) SI ® SI Fig. 28 Upper limits at 90% confidence level on σWIMP+N as a func- Fig. 29 Upper limits at 90% confidence level on σX+N as a tion of the WIMP-mass obtained from a dedicated Earth search with function of the velocity-averaged annihilation cross section + IceCube-79 [241], assuming a WIMP annihilation cross section of σAv for 50 GeV WIMPs annihilating into τ τ− and for 1 TeV 26 3 1 h i + σAv = 3 10− cm s− and a local dark matter density of 0.3 WIMPs annihilating into W W−. Limits from LUX [236] are h i 3 × GeV cm− . Results from Super-Kamiokande [242], SuperCDMS [243], shown as dashed lines for comparison. The red vertical line LUX [236] and an IceCube-79 solar search [230] are shown for com- indicates the thermal annihilation cross section for a particle parison. Figure from [241] species to constitute the dark matter. Also indicated are Ice- Cube limits on the annihilation cross section for the respective masses and annihilation channels [244]. Figure from [241] 7.4 Dark Matter Signals from Galaxies and Galaxy matter density near the core. This is the core-cusp prob- Clusters lem [249], and it is an unresolved issue which affects the signal prediction from dark matter annihilations in neu- trino telescopes. Note that the shape of the dark halo can The Milky Way centre and halo, as well as nearby dwarf depend on local characteristics of any given galaxy, like galaxies and galaxy clusters provide natural large-scale the size of the galaxy [250] or on its evolution history [251, regions of increased dark matter density. Since dark mat- 252] ter played a significant role in the formation of such struc- The shape of the dark matter halo is important be- tures from primordial density fluctuations, the issue of cause the expected annihilation signal depends on the capture is not relevant, and what neutrino telescopes can line-of-sight integral from the observation point (the Earth) prove when considering such objects is the WIMP self- to the source, and involves an integration over the square annihilation cross section, σ v . In order to predict the h A i of the dark matter density. This is included in the J-factor rate of annihilation of dark matter particles in galactic of Eq. (60), which is galaxy-dependent, and absorbs all halos, the precise size and shape of the halo needs to be the assumptions on the shape of the specific halo being known. There is still some controversy on how dark halos considered. In the case of our Galaxy, the expected signal evolve and which shape they have. There are different from the Galactic Centre assuming one halo parametri- numerical simulations, observational fits, and parametri- sation or another can differ by as much as four orders of sations of the dark matter density around visible galaxies, magnitude depending on the halo model used (see, e.g., including the Navarro-Frenk-White (NFW) profile [245], Fig. 2 in Ref. [260]). the Kravtsov profile [246], the Moore profile [247], and the Burkert [248]. The common feature of these profiles As before, a few benchmark annihilation channels can is a denser spherically symmetric region of dark matter be chosen to bracket the model expectations, and then in the centre of the galaxy, with decreasing density as dNν/dE represents the neutrino flux assuming 100% an- the radial distance to the centre increases. Where they nihilation to each of the benchmark scenarios. The anal- diverge is in the predicted shape of the central region. yses use the same likelihood approach as described by Profiles obtained from N-body simulations of galaxy for- Eq. (64). The signal hypothesis (excess of events at small mation and evolution tend to predict a steep power-law angular distances ψ to the Galactic Centre) can then be type behaviour of the dark matter component in the cen- tested against the background-only hypothesis (an event tral region, while profiles based on observational data distribution isotropic in the sky). There is, however, an (stellar velocity fields) tend to favour a constant dark additional effect to take into account when dealing with 31 IC79 Halo, Multipole IC59 Dwarfs mapped onto an expansion in spherical harmonics on the IC86 Halo, Cascades ANTARES (2007-15) sphere, IC79 GC Fermi+MAGIC Seg1 95% IC86 GC (2012-14) H.E.S.S. GC 95% (Ein.) 10 20 m=` − m m f (Ω) = ∑ ∑ a` Y` (Ω) , (69) 21 ` m= ` 10− − m ] 22 where the expansion coefficients a are given by the sum 1 10− ` − s over events with reconstructed arrival directions Ω , 3 i 23 10− [cm nobs i m m v a = (Y )∗(Ω ) . (70) A i 24 NFW profile ` ∑ ` σ 10− h + i=1 τ τ −-channel 25 10− Note that this expansion depends on the coordinate sys- Natural scale tem. In particular, in the equatorial coordinate system 26 10− where the event distribution has strong dependence on 101 102 103 104 105 declination angle (equivalent to zenith angle) from the WIMP Mass [GeV] detector acceptance, the expansion is dominated by the Fig. 30 Comparison of upper limits on σAv versus WIMP mass, m = 0 coefficients. It is possible to design a test statistic + h i for the annihilation channel χχ τ τ−. IceCube results obtained of the remaining m = 0 components to separate signal with different detector configurations→ [253,254,244,255] are com- 6 from background (see Eqs. (8) and (9) in [253]). pared to ANTARES [256] and the latest upper limits from γ-ray searches from H.E.S.S. [257] and from a combination of Fermi-LAT Results from the searches performed by IceCube with and MAGIC results [258]. Figure from [259]. different techniques on the Galactic Centre, halo and dwarf spheroid galaxies [253,254,244,255] are shown in Fig. 30, compared with other experiments and theory in- extended sources, like the Galactic halo. Since the signal terpretations. All sources considered showed results com- is allowed to come from anywhere in the halo, the back- patible with the background-only hypothesis yielding ground distribution, which is usually taken from data limits on the velocity-averaged annihilation cross section scrambled over azimuth angle, B = De, is necessarily con- 23 3 1 at the level of few 10− cm s− . Recent results from taminated by a potential signal. Thus, the background γ-ray telescopes on dwarf spheroids currently provide distribution B depends indirectly on the number of sig- the best limits on σ v , due to their accurate pointing h A i nal events µS and needs to be corrected. The effective and lack of foreground or background from these kind of background distribution can be written as sources. The high-energy diffuse astrophysical neutrino flux µs µs B = De Se / 1 , (67) discovered by IceCube opens a new possibility of probing − nobs − nobs the galactic dark matter distribution through neutrino- dark matter interactions [262,263,261]. Indeed dark mat- where Se and De are the probability distribution functions ter couplings to standard model particles are commonly of the azimuth-scrambled arrival directions of signal sim- assumed to exist, and this is the basis of direct detec- ulation and data events respectively. tion experiments. Such a coupling can be extended to There is another, complementary, analysis approach neutrinos if one assumes the existence of a mediator φ, for extended sources that naturally incorporates the dif- which can be either bosonic of fermionic in nature, which fuse character of the signal over a large region in the sky. couples to dark matter with a coupling g. For simplicity It is based on a multipole expansion of the sky map of one can assume that the strength of the ν φ coupling is − event arrival directions. Dark matter annihilations in the also given by g. Under these assumptions, dark matter- halo would produce a diffuse flux of neutrinos with a neutrino interactions could distort the isotropy of the characteristic large scale structure following the shape of astrophysical neutrino flux, resulting in an attenuation the halo, while the atmospheric neutrino background of the flux towards the Galactic Centre, where the den- presents small anisotropies at smaller scales. The sky sity of dark matter is higher. Therefore, an analysis of the map of reconstructed event arrival directions can be con- isotropy of the high-energy astrophysical neutrino data structed as of IceCube can be translated into a limit on the strength of the neutrino-dark matter coupling, g. Such an analy- nobs D D sis has been performed by the authors in [261], where f (θ, φ) = ∑ δ (cos(θ) cos(θi))δ (φ φi) , (68) i=1 − − the mass of the dark matter candidate, the mass of the mediator and the strength of the coupling are left as free where (θi, φi) are the reconstructed coordinates (declina- parameters in a likelihood calculation which aims at eval- tion and right ascension respectively) of event i, nobs is uating the suppression of astrophysical neutrino events the total number of events in the data sample and δD from the direction of the Galactic Centre. The results are is the Dirac-delta-distribution. Such distribution can be shown in Fig. 31. The left panel shows contours of the 32 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 -1.5 -1.5 -2 -2 -2.5 -2.5 Fig. 31 Contours of the maximum allowed value of the coupling of dark matter to neutrinos, gmax, as a function of the dark matter mass, mχ, and the mediator mass mφ. Left Panel: fermionic dark matter coupled through a vector mediator. Right Panel: scalar dark matter coupled through a fermionic mediator. The magenta line in both plots delimits the region where cosmological observations from large scale structure become more restrictive than the IceCube limits. The plots have been obtained by marginalising over the atmospheric and astrophysical fluxes, allowing the astrophysical spectral index to vary between 2 and 3. The diamond and the stars refer to specific models studied in [261]. Reprinted with permission from [261]. Copyright 2017 by the American Physical Society. maximum allowed value of the coupling of fermionic fall within 60° around the Galactic Centre. This introduces dark matter coupled to neutrinos through a vector me- a weak large scale anisotropy that can be tested against diator, while the right panel shows the case of a scalar the observed event distribution [289,290,291]. The neu- dark matter coupled through a fermionic mediator. In- trino emission from galaxies and galaxy clusters could terestingly IceCube is sensitive to a region of parameter also be identified as (extended) point-source emission in space complementary to results derived from cosmolog- future IceCube searches [292]. ical arguments alone [264,265,266], as indicated by the It can be expected that the secondary emission from magenta line. This line delimits the region where limits decaying dark matter scenarios will also include other from analyses using large-scale structure data become standard model particles that can be constrained by multi- more restrictive than the IceCube limits shown in the messenger observations. In particle the production of PeV plot. γ-rays, that have a pair production length of (10) kpc O in the CMB, would be a smoking gun for a Galactic contri- butions [292]. However, also the the secondary GeV-TeV 7.5 TeV-PeV Dark Matter Decay emission from electro-magnetic cascades initiated by PeV γ-rays, electrons, and positrons can constrain the heavy The origin of the TeV-PeV diffuse flux observed with Ice- dark matter decay scenario [292,290,293,294,295]. Cube is so far unknown. Whereas most models assume From an experimental point of view, the search for a an astrophysical origin of the emission, it is also feasi- neutrino signal from heavy dark matter decay follows ble that the emission is produced via the decay of dark closely the strategies used for the search for annihila- matter, as first proposed in Ref. [267]. Various studies tion signatures. A given analysis can be used for both have argued that heavy dark matter decay can be re- constraining dark matter lifetime and annihilation, as sponsible for various tentative spectral features in the can be inferred from the similarities between Eqs. (60) inferred neutrino spectrum of the HESE analysis [268, and (62). A feature of searches for dark matter decay in 269,270,271,272,273,274] and also its low energy exten- the > 100 TeV mass region is that the recently discovered sion [275,276,277]. Some authors have also discussed the astrophysical neutrino flux becomes a background to the necessary condition for PeV dark matter production in search, and its current uncertainties on normalisation the early Universe, e.g., via a secluded dark matter sec- and the presence, or not, of a cut-off effect the interpre- tor [278], resonantly-enhanced freeze-out [279,280], or tation of any search for an additional component due freeze-in [281,282,283,284]. to dark matter decay. IceCube has performed a search If dark matter decay is responsible for the high-energy for signatures of heavy dark matter decay by assuming neutrino emission, the arrival directions of TeV-PeV neu- that the background is a combination of atmospheric and trino events observed with IceCube should correlate with astrophysical neutrinos, where the normalisation and en- the line-of-sight integral of the dark matter distribution ergy dependence of the astrophysical flux is allowed to (“D-factor”). The contribution of neutrinos from dark flow freely [288]. This allows for deviations from a strict matter decay in the Galactic halo can be similar to the power law which could be interpreted as an additional isotropic extra-galactic contribution. Half of the neutrino contribution from dark matter decays. The signal consists events from dark matter decay in the halo are predicted to of both a galactic component and a diffuse component 33 1 ¯ 10 b + b Z + ⌫ ⌫ +¯⌫ + 29 µ + µ 2 + 10 10 W + W b + ¯b + 3 H + ⌫ µ + µ 28 1 10 10 + 4 ⌧ + ⌧ 10 1027 GeV / 5 10 26 dE dN 10 6 10 IceCube (this work) Dark matter lifetime / sec 7 m = 2 PeV 1025 HAWC (dSph, 2018) 10 DM HAWC (GC, 2018) Fermi/LAT (2012) 8 10 1024 3 4 5 6 7 8 9 3 4 5 6 10 10 10 10 10 10 10 10 10 10 10 Dark matter mass / GeV E⌫ / GeV Fig. 32 Left Panel: Neutrino yield per decay as a function of neutrino energy (flavour-averaged) for an assumed dark matter mass of 2 PeV. Right Panel: IceCube lower limits on dark matter lifetime versus dark matter mass assuming a 100% branching ratio decay to bb (full line), + µ µ− (dashed line), compared with limits from HAWC, both for Dwarf Spheroidal Galaxies [285] and the Galactic Centre/Halo [286], and Fermi/LAT [287]. Figures from [288]. from the contribution of distant galaxies. Typical neutrino His argument was based on the observation that the spectra from the decay of a 2 PeV dark matter candidate, magnetic potential of the static magnetic monopole field, assuming 100% decay to each channel, are shown in the B = rot A, has to be singular along a semi-infinite line. left panel of Fig. 32, featuring the characteristic peak at The wave function of a particle with charge qe encircling the dark matter mass from those decays where the neu- this line will pick up a phase that is equal to ∆φ = q q . ± e m trino takes most of the initial available energy. Recent To be unobservable, this phase should be a multiple of results from IceCube on the dark matter lifetime as a 2π independent of the type of monopole or electrically function of the dark matter mass are shown in the right charged particle. This leads to the condition8 panel of Fig. 32, compared with results from HAWC and q q = 2πn (n Z) . (71) Fermi/LAT. Values below the lines are disfavoured at m e ∈ 90% confidence level, with IceCube limits showing the At the time of Dirac’s seminal work, the smallest electric strongest constraint for masses above about 10 TeV. charge unit was considered to be that of the electron, qe = e, and hence the smallest possible (Dirac) magnetic charge would be 8 Magnetic Monopoles g = 2π/e = e/2α 68.5e . (72) D ' Maxwell’s equations of classical electrodynamics appear This elementary monopole charge derives from basic con- to be asymmetric due to the absence of magnetic charges. siderations still valid in modern physics. In the Standard However, this is merely by choice. We can always redefine Model the smallest charge is provided by the down-type electric and magnetic fields by a suitable duality trans- quarks, increasing the minimal magnetic charge to 3g . formation, E + iB = eiφ(E + iB), such that Maxwell’s D 0 0 Note that the large magnetic charge of monopoles pre- equations in terms of the new fields are completely sym- dicted by the Dirac quantisation condition (72) leads metric. However, this duality transformation requires to distinct electromagnetic features as monopoles pass that for every particle the ratio between magnetic and through matter. We will return to this point shortly. electric charges are the same. If this is not the case, then it In solids, structures have been found which resemble is necessary to include a source term for magnetic charges poles. These are sometimes mistakenly called magnetic q (i.e., magnetic monopoles), that create a magnetic field m monopoles, although the poles can only occur in pairs of the form B = q e /4πr2. m r and do not exist as free particles. For distinction, recently P. Dirac was the first to speculate about the existence the term magnetricity has been coined for the field that of magnetic monopoles, guided7 by “new mathematical exhibits theses poles. Fundamental magnetic monopoles features” appearing in the quantum-mechanical descrip- have not been observed so far. tion of electrodynamics [297]. He showed that the pres- Although, as Dirac showed, magnetic monopoles can ence of a magnetic monopole with a minimum charge q m be consistently described in quantum theory, they do would explain why the electric charge is always quan- tised, i.e., in integer multiples of an elementary charge. 8Note, that in a Gaussian system, where the unit of charge is defined 2 via e = hc¯ α, this quantisation condition is expressed as qmqe = 7Previously, he had successfully used this method to propose the nhc¯ /2. However, to avoid confusion, we use the Heaviside-Lorentz existence of electrons with positive charge [296], i.e., positrons. system with the conventional definition e2 = 4πα (in natural units). 34 broken to U(1), there are no topologically nontrivial con- figurations of the Higgs field, and hence no topologically stable monopole solutions exist. However, there exist specific modifications with a non-minimal coupling of W/Z QCD e+e- magnetic core bosons screening screening field the Higgs field that would allow electroweak monopoles with TeV-scale masses, e.g., Refs. [304,305]. 10-29 cm 10-16 cm 10-13 cm 10-11 cm 8.1 Cosmological Bounds Fig. 33 Structure of a GUT monopole with virtual gauge bosons While inaccessible to collider experiments, very heavy populating the outer spheres (after Ref. [298]). GUT monopoles could have been produced in the early universe if the temperature exceeds T Λ . In this cr ∼ GUT case, the expected monopole density would be roughly not appear automatically in that framework. As was first one per correlated volume, which corresponds to the found independently by ’t Hooft [299] and Polyakov [300], horizon size in a second-order phase transition [306,307]. this is different in Grand Unified Theories (GUT) which This gives a naive monopole energy density (relative to embed the Standard Model interactions into a larger the critical density) of gauge group. These theories are motivated by the obser- vation that the scale-dependent Standard Model gauge Tcr 3 Mm Ω h2 1020 . (74) couplings seem to unify at very high energies. Generally, GUT ' 1016 GeV 1017 GeV a ’t Hooft–Polyakov monopole can arise via spontaneously This prediction is in clear conflict with the observed spa- breaking of the GUT group via the Higgs mechanism. tial flatness of the Universe (Ω 1) and known as the tot ' The stability of the monopole is due to the Higgs field “monopole problem”. configuration which cannot smoothly be transformed to An elegant solution to this problem is an inflation- a spatially uniform vacuum configuration. ary universe, i.e., a universe that underwent an expo- An early candidate for a GUT theory studied in this nential expansion of the scale factor, diluting any ini- context has been the Georgi–Glashow model [301], where tial monopole abundance to an (almost) unobservable matter is unified in SU(5) representations, that sponta- level. This inflationary mechanism is a very powerful neously break to the Standard Model gauge representa- idea since it simultaneously explains why our Universe tions below the unification scale Λ 1015 GeV. This GUT ' has been extremely flat at early times (flatness problem), is related to the mass of the monopole as e.g., Ω 1 10 16 at the epoch of big bang nucleosynthe- − ' − sis, and why the Universe appears to be so homogenous ΛGUT 16 17 Mm 10 − GeV , (73) over causally disconnected distances (Horizon problem), ' αGUT ' 5 e.g., temperature fluctuations in the CMB of only 10− . 2 where αGUT = g /4π is the gauge coupling constant at The requirement that the contribution of monopoles in the unification scale. If the original unified group un- Eq. (74) does not exceed today’s dark matter abundance, dergoes secondary symmetry breaking at lower energies ΩGUT . Ωm, results in the overclosure bound on the also monopoles with lower masses may be generated. integrated isotropic flux Depending on details of the GUT model, the monopole 15 2 17 7 17 10− Ωmh v 10 GeV mass can range from 10 GeV to 10 GeV [298,302]. FGUT . . (75) cm2 sr s 0.13 10 3c M The unification scale is related to the size of the mono- − m 1 29 pole as r Λ− 10 cm. Larger radii correspond If monopoles cluster in our local galaxy this bound can be M ' GUT ' − to different energy scales reflecting various transitions in relaxed by several orders of magnitude. Taking the solar 24 3 the Standard Model, in particular, the electroweak tran- halo density ρhalo 10− g cm− we obtain the limit 1 16 ' sition scale MZ− 10− cm and the confinement scale 11 17 1 13 ' 10− v 10 GeV Λ− 10− cm (see Fig. 33). The presence of virtual FGUT . . (76) QCD ' cm2 sr s 10 3c M particles within these “shells” influences the monopole’s − m interaction with matter. For instance, within the mono- pole core, the GUT gauge symmetry is restored, and can 8.2 Parker Bound mediate baryon-number violating processes. At large dis- tances, only electromagnetic interactions are visible by Magnetic monopoles are accelerated in magnetic fields the magnetic monopole field. – analogously to charged particle acceleration in electric Although the greatest interest has been in the super- fields. Therefore, relic monopoles that are initially non- massive monopoles that are motivated by GUTs, with the relativistic are expected to gain energy while they travel advent of the LHC and the MoEDAL experiment [303] the along galactic and intergalactic magnetic fields. The re- possibility of lighter monopoles lately received renewed quirement that monopoles have to be rare not to short- attention. In the electroweak theory, with SU(2) U(1) circuit these magnetic fields gives the so-called Parker × 35 Direct Cherenkov light Luminescence [Qui83] Indirect Cherenkov light (KYG) Proton decay λcat =1 cm Luminescence [Bai08] GUT + Proton decay λcat =10 m e 7 monopole 10 106 5 u ] 10 1 u d − m 104 c [ d d x 3 d 10 / Fig. 34 Illustration of a proton decay into a positron and a neutral γ N 2 pion catalysed by a GUT monopole (after Ref. [312]). d 10 Muon direct Cherenkov 101 light 0 10 4 3 2 bound [308]. The galactic magnetic field with a strength 10− 10− 10− 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 of a few µG can be generated by a dynamo action on a v/c v/c time scale that is comparable to the Milky Way’s rotation Fig. 35 Light yield for the different radiation mechanisms of mag- period, τ 108 yr. A monopole with magnetic charge q ' m netic monopoles [313]. For comparison the direct Cherenkov light will gain an energy of ∆Ekin = ∆`Bqm after it travels a emitted by a minimum ionising muon is shown. distance ∆` along magnetic field lines. The power density of the galactic dynamo B2/τ should be larger than the ∼ 8.3 Nucleon Decay Catalysis energy drained by the magnetic monopoles. During its acceleration the monopole encounters dif- The central core of a GUT monopole (see Fig. 33) contains ferent magnetic field orientations coherent over a length the fields of the superheavy gauge bosons that medi- scale λc which are much smaller than the typical size r of ate baryon number violation, so one might expect that the magnetic halo. If the monopoles stay non-relativistic, baryon number conservation could be violated in baryon- i.e., M q Bλ , the energy gain is always large com- m m c monopole scattering and the possibility that a proton pared to the kinetic energy and the particle will be accel- or a neutron in contact with a GUT monopole can de- erated. The processes can be estimated by a random walk cay. This feature was pointed out by Callan [314] and with N r/λ encounters with coherent field regions. ' c Rubakov [315]. The cross sections for the catalysis pro- For weak magnetic fields, i.e., Mm qmBλc this process cesses such as p + M e+ + π0 + M (Fig. 34) are essen- → loses its efficiency since the monopole does not follow tially geometric: the potential drop along the field lines. A careful analysis ( σ0 for our own galaxy gives the following bound for masses β for β β0 , 17 σcat = ≥ (78) Mm . 10 GeV [309] σ0 F(β) for β < β . β · 0 β γ 15 7 1/2 The correction F(β) = takes into account an ad- 10 B 3 10 yr r/λ β0 F − × c . GUT . cm2 sr s 3 µG τ 100 ditional angular momentum of the monopole-nucleus system. Both parameters γ and β depend on the nu- (77) 0 cleus [316]. Current estimates for the catalysis cross sec- 27 2 21 2 tions are of the order of 10− cm to 10− cm [317]. 17 Very massive monopoles Mm & 10 GeV will not be sig- nificantly deflected by the galactic magnetic field, since the acceleration across the galaxy does not change much 8.4 Monopole Searches with IceCube of the initial virial velocity v 10 3c. The energy drain ∼ − of the field by these monopoles depends thus on the ini- The experimental search for magnetic monopoles has a tial monopole trajectory with respect to the field lines. long history. Searches were pursued at accelerators, in cos- To first order, the effect from the motion of these heavy mic rays, and for bound monopoles in matter. Detection monopoles and their anti-monopoles across the galaxy methods include induction in SQUIDs, the observation will cancel. However, the effect will be visible at second of excessive energy loss of the monopole compared to order which introduces a mass dependence ∆Ekin ∝ Mm. Standard Model particles and particles describing non- 17 The Parker bound beyond Mm & 10 GeV is hence helical paths in a uniform magnetic field, or other un- 17 weaker than (77) by a factor (Mm/10 GeV). Apply- usual trajectories like non-relativistic velocities combined ing Parker’s arguments to the seed magnetic fields of with a high stopping power and long ranges. In indirect galaxies or galaxy clusters leads to tighter bounds [310, searches at accelerators virtual monopole processes are 311]. As these bounds are less secure, and for consistency assumed to influence the production rates of final states. with other literature on the subject, we compare the ex- In a direct search, evidence of the passage of a monopole perimental results with the original Parker bound (see through material is sought. Here we primarily address Fig. 37). direct searches for primordial monopoles with IceCube. 36 Fig. 36 Simulated event topologies for magnetic monopole of different velocities (see Fig.2 for graphical event representation). Left Panel: monopole traveling at 0.982c from the bottom to the top of the detector [318]. The total time of the event is 5000 ns. Less light has been detected in a horizontal plane [4] roughly at the half height of the detector due to dust in the ice. Middle Panel: monopole at 0.3c moving from the top of the detector to the bottom emitting luminescence light. The simulated track of the particle is indicated in red. Only a 3 few noise hits contribute throughout the detector due to the short time frame of the event [318]. Right Panel: Monopole with β = 10− catalysing nucleon decay with λcat = 1 cm with superimposed background noise [319]. The black line represents the simulated monopole track. Different light production mechanisms induced by nopole trajectory can lead to a characteristic slow moving monopoles dominate depending on their velocity (see event pattern across the detector [319]. Fig. 35): For each of these speed ranges, searches for magnetic (i) Direct Cherenkov light is produced at highly relativis- monopoles at the IceCube experiment are either in pro- tic velocities above 0.76 c as with any other charged Stan- gress (luminescence) or have already set the world’s best dard Model particle. Due to the high relative Dirac charge, upper limits on the flux of magnetic monopoles over a as shown in Eq. (72), several thousand times more light is wide range of velocities. Examples of magnetic monopole radiated with a monopole than from a minimum ionising passing through the detector at different velocities are singly electrically charged particle like a muon [320]. shown in Fig. 36. These unique signatures from monopoles lead to the (ii) Indirect Cherenkov light from secondary knock-off following general analysis strategies in the different ve- δ-electrons is relevant at mildly relativistic velocities ( ' locity regions: 0.5 c to 0.76 c). The high-energy δ-electrons in turn can (i) Relativistic monopoles are selected based on their have velocities above the Cherenkov threshold them- brightness, arrival direction, and velocity [320]. The high selves. The energy transfer of the monopole to the δ- energy astrophysical neutrino flux is an important back- electrons can be inferred from the differential cross sec- ground to this signal. At similar brightness, monopoles tion calculated by Kasama, Yang and Goldhaber (KYG) [321, show less stochastic energy loss than Standard Model 322]. particles leading to a smoother light yield distribution (iii) Luminescence light from excitation of the ice domi- along the track. nates at low relativistic velocities ( 0.1 c to 0.5 c). The ' (ii) Due to its lower rest mass a Standard Model particle observables of luminescence, such as the wavelength with a velocity below the speed of light in vacuum, c, spectrum and decay times, are dependent on the proper- would not be able to traverse the whole detector. How- ties of the medium, in particular, temperature and purity. ever, the discrimination power of the reconstruction of the The signature is relatively dim in comparison to muon velocity is insufficient for the suppression of the vast air signatures. Pending further laboratory measurements in shower backgrounds against the identification of mildly ice and water [323], the efficiency of luminescence pho- relativistic monopoles. Instead variables describing the ton production per deposited energy is in the range of topology, smoothness, and time distribution of the events dNγ/dE = 0.2 γ/MeV and 2.4 γ/MeV [324,325]. Even are processed in a Boosted Decision Tree (BDT) machine for the lower plausible light yield, luminescence is a vi- learning [320]. able signature due to the high excitation of the medium (iii) Searching for mildly relativistic monopoles using induced by a monopole [313]. luminescence light can be performed using analysis tech- (iv) At velocities well below 0.1 c luminescence is ex- niques that combine the non-relativistic reconstructed pected to fall off (see Fig. 35). The catalysis of nucleon particle velocity and the continuous but dim light pro- decays is a plausible scenario for GUT monopoles (see duction of a through-going track in the detector [313, Sec. 8.3) and may be observed if its mean free path is 318]. small compared to the detector size. The Cherenkov light (iv) Catalysed nucleon decay, like p + M e+ + π0 + M, → from secondaries emitted in nucleon decays along the mo- transfers almost all of the proton’s rest mass to the energy 37 IceCube 86 DC MACRO up-going IceCube 22 RICE 2008 IceCube 59 ANTARES 2008 IceCube 40 ANITA 2 BAIKAL 1998-2003 ANTARES 2015 Sens. IceCube 86 Auger 2015 -15 10 Parker Bound ] -16 1 10 − r s -17 1 10 − s λcat =1mm -18 2 10 − m c -19 [ 10 0 9 Φ 10-20 Cherenkov Threshold -21 10 3 2 10− 10− 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.995 9 11 13 v/c v/c log10γ Fig. 37 Upper limits on the flux of magnetic monopoles as a function of the velocity of the monopole at the detector [313]. Shown are limits from IceCube [319,326,320], Baikal [324], ANTARES [327,328], RICE [329], ANITA [330] and Auger [331]. of electromagnetic and hadronic cascades. Because of the Operational until 2000, MACRO searched for mag- high light yield this channel is typically used as a bench- netic monopoles using three types of subdetectors – liq- mark in analyses. Due to their low speed, the duration of uid scintillation counters, limited streamer tubes and the event is in the order of 10 milliseconds. As obvious nuclear track detectors. No monopole was found, with from the right event display in Fig. 36 at such timescales an upper flux limit at the 90% confidence level of 1.4 16 2 1 1 × random noise pulses are a significant contribution. Vari- 10− cm− s− sr− for monopoles with velocity between ous effects contribute to subtle temporal correlations on 4 10 5 c to c and magnetic charge with n 1 [333, × − ≥ long time scales of this noise [332,4] complicating an ad- 334]. Under the assumption that monopoles are gravi- equate description in simulation. Instead, a background tationally accumulated in the centre of the Sun, Super- model is established from reshuffling experimental data. Kamiokande [335] could impose stringent limits for non- For signal identification, time-isolated local coincidences relativistic velocities in an indirect search for neutrinos in neighbouring DOMs are searched for along a mono- from the direction of the Sun. Baikal [324] has investi- pole trajectory hypothesis consistent with a straight par- gated the direct Cherenkov light from relativistic mo- ticle track of constant speed. A Kalman filter is used to nopoles. The analysis by ANTARES includes also the separated noise from monopole signals and a combina- mildly relativistic regime employing similar techniques tion of observables are fed to a BDT to further improve like IceCube [328]. The reduced scattering in water at the the signal purity [319]. ANTARES site compared to ice leads to a better veloc- ity reconstruction which helps with the Standard Model Figure 37 shows a compilation of current flux upper background suppression. This partly compensates the limits of relic monopoles from various experiments. Only higher noise level in the detector. in the past decade astrophysical experiments have been able to improve upon the original Parker bound which Intermediate and low mass monopoles may acquire is shown for comparison. These recent experiments have highly relativistic velocities in intergalactic magnetic fields employed large scale detectors for cosmic rays to achieve reaching Lorentz factors of γ 1010 for the example of a ' the highest sensitivities in the whole β range in which PeV-mass [302]. Ultra-relativistic particles with magnetic GUT magnetic monopoles are expected. Typically it is charge (or large electric charge) dramatically loose energy assumed that the flux at the respective detector site is in their passage through matter, initiating a large num- isotropic implying sufficient kinetic energy in order to ber of bright showers along the track. At high Lorentz cross the Earth or the overburden above the detector due boost factors the photo-nuclear effect is the dominant en- to their large rest masses. This assumption is justified for ergy loss mechanism generating hadronic showers. While monopoles of masses in excess of 1010 GeV. The mono- these showers are continuously produced, they may over- pole flux limits commonly assume a single Dirac mag- lap with each other. In the atmosphere of the Earth this netic charge qm = gD (see Eq. (72)) with no additional leads to a built-up such that the energy deposit increases electric charge. In most detectors and velocity ranges, with slant depth. The Auger experiment has used this the detection efficiency for larger magnetic charges or feature to distinguish monopoles from the background of for electrically charged monopoles (dyons) is expected to electrically singly charged ultrahigh-energy cosmic rays increase. like protons [331]. The RICE [329] and ANITA [330] exper- 38 iments have searched for such multiple sub-shower signa- The appearance of these flat scalar potentials is generic tures in the Antarctic ice sheet with the Radio-Cherenkov in supersymmetric (SUSY) field theories, predicting scalar technique, the discriminant against conventional cosmic partners for fermions and gauge bosons, that may carry rays here being primarily the rapid succession of several global charges [338]. Supersymmetry has to be broken at radio pulses received from each sub-shower. low energies to provide mass terms for the SUSY partners The extrapolation of IceCube’s limit towards highly (for a review see, e.g., Ref. [339] and references therein). relativistic velocities by a constant line in Fig. 37 is a very Since a naive spontaneous SUSY breaking by renormal- conservative approach. It not only neglects the increase isable terms in the visible sector predicts SUSY masses in signal detection efficiency with more energy deposited, that are in general too light (below TeV), it is typically as- but it also ignores the onset of the photo-nuclear effect sumed that the breaking occurs in a hidden sector at some and pair production. These effects would produce show- unobservable high-energy scale. The mediation of SUSY ers with light emission orders of magnitude brighter than breaking terms to the visible sector, e.g., by gauge interac- from the Cherenkov effect considered here, hence visible tions or supergravity generates then soft (renormalisable) also from far outside the instrumented detector volume. SUSY breaking masses for SUSY partners of the Standard While these flux limits reflect the cosmic density, at Model matter and gauge bosons. Generically, SUSY ex- the electroweak scale monopoles may be created in accel- tensions of the Standard Model predict flat directions for erator collisions, which is studied at the MoEDAL experi- some combination of scalar fields. For instance, gauge- ment. Also cosmic ray collisions or high energy neutrino mediated SUSY breaking is expected to introduce a scalar 4 † 2 2 interactions in the Earth may produce monopoles. This potential with V m (log(φ φ/M )) (for φ Mm) ∼ F m | | might be an additional detection opportunity, also for with mass scale m and messenger mass Mm m. In this IceCube. case, the mass of a Q-ball with Q 1 can be estimated as [340] 4π√2 m mQ3/4 . (79) 9 Other Exotic Signals Q ' 3 Naturally, the effective mass scale m is expected to lie The detection principle of Cherenkov telescopes is very within the range 100 GeV < m < 100 TeV. general in the sense that it applies to any flux of parti- If baryogenesis proceeded via the Affleck-Dine mech- cles that can penetrate the detector shielding and pro- anism [341] 9, stable Q-balls with 1012 Q 1030 could duce light signals inside the detector volume. We have . B . have been formed copiously as a dark matter contribu- already covered the possibility to observe relic magnetic tion by the fragmentation of the Affleck-Dine conden- monopoles in the previous section. In this section we sate [342]. It is an appealing property of this scenario will discuss the detection potential for other exotic candi- that the baryon and dark-matter abundance, Ω and dates like Q-balls and strangelets. We will also address B Ω , respectively, are related. For Q 1026 one ob- the possibility that long-lived charged particles produced DM B ' tains Ω 10Ω close to the observed values. in cosmic ray or neutrino interactions may be discovered DM ' B Analogously to monopole densities, Q-ball densities via their Cherenkov emission. All these particles have in are also limited by the dark matter abundance. The max- common that their passage through the IceCube detector imal contribution of Q-balls to the observed dark mat- produces observable features that can be extracted from ter Ω h2 Ω h2 results in a bound on the integrated backgrounds. Q . m isotropic flux since 4πFQ . vMΩmρcr/mQ and 3/4 5 10 22 v 1 TeV 1026 − . (80) 9.1 Q-Balls FQ . × 2 3 cm sr s 10− c msoft QB There exist non-topological solitonic solutions of a field Here the mass term (79) from gauge-mediated SUSY theory, so-called Q-balls [336]. Whereas the stability of breaking is used. If we assume that dark matter Q-balls cluster in our galaxy with ρ 10 24 g cm 3 we obtain topological solitons, e.g., monopoles is guaranteed by halo ' − − the conservation of a topological charge (winding num- the limit ber) associated with a degeneracy of the vacuum state, 18 26 3/4 5 10− v 1 TeV 10 a Q-ball can be stable by the conservation of a charge FQ . × 2 3 , (81) cm sr s 10 c msoft QB associated with a global symmetry of the theory. This − can happen if its energy configuration is lower than the corresponding to a few events per year and square kilo- corresponding multi-particle Fock state. For a single com- meter in a 2π sky coverage. plex scalar field φ carrying the charge Q, this implies that The global charge Q associated with the Q-ball can there exists a non-trivial field value φ0 > 0 where the be the same as baryon number (B) or lepton number (L) † | | † scalar potential U(φ φ) obeys the condition U(φ0 φ0) < 9 2 † In this scenario, a combination of MSSM scalar fields with non- mφφ0 φ0, where mφ is the mass of the scalar field (for a zero baryon number B develops a large expectation value at the review see Ref. [337]). end of inflation and decays. 39 or some combination of them if these symmetries are composite nuclei would be very long-lived, since conver- connected to a global U(1) symmetry. Since the global sion to the SQM ground state would proceed via weak symmetry is spontaneously broken in the interior of the interactions. Q-ball with φ = 0, the soliton could catalyse nucleon Lumbs of SQM, so-called strangelets, can have a large 6 decay traversing the detector volume. This is analogous atomic mass number A and charge Z. Classical strange- to the case of monopole-catalysed nucleon decay. Even if lets have a quark charge Z 0.1A for low mass num- ∼ the charge is not related to B or L, it is possible that the bers (A 700). For total quark charges exceeding Z ∼ vacuum state associated with the Q-ball interior catalyses α 1 137 strong field QED corrections lead to screening − ∼ nucleon decay. This can happen if the scalar potential and Z 8A1/3 (A 700) [348]. It has also been specu- ∼ is very flat such that U(φ†φ ) m2 φ†φ . Interactions lated that colour and flavour symmetries at high baryon 0 0 φ 0 0 that lead to nucleon decay induced by new physics at an densities might be broken simultaneously by the con- ultra-violet scale Λ, for instance, in grand unified theories densation of quark Cooper pairs [349]. In this scenario, are typically suppressed by powers of φ /Λ and can the “colour-flavour-locked” strangelets have charges of h i become large in the Q-ball environment [343]. Z 0.3A2/3 [350]. ∼ The cross section of the catalyzed nucleon decay via Stable strangelets can absorb ordinary matter in exo- Q-balls with baryon number Q passing through matter thermic reactions involving u s + e + ν¯ or u + d B → e → can be approximated by its size. In gauge-mediated SUSY s + u [348]. If the strangelet carries a positive charge breaking, it can be estimated as [340] the Coulomb barrier will usually prevent a strangelet– nucleus system from collapse. However, neutron-rich 1/2 2 20 QB 1 TeV 2 environments like neutron stars are not protected by this σcat 10− 26 cm . (82) ' 10 msoft mechanism. In fact, if strange quark matter is stable then all compact stars like white dwarfs or neutron stars are The experimental signature of this process would thus be likely to consist of it. Even the capture of a single strange- similarly spectacular as in the case of nucleon-decay catal- let would be sufficient to convert a neutron-rich environ- ysed by monopoles. Hence, the upper limits discussed ment very rapidly. for slow monopoles can be reinterpreted in terms of elec- Slowly moving strangelets – so called nuclearities [351] trically neutral Q-balls also for neutrino telescopes. An – lose their energy in matter due to atomic collisions. The example of such a recalculation is available for the Baikal excessive energy released will heat the medium and cre- experiment in [344,345] leading to a flux upper limit of ate thermal shocks. The hot expanding plasma will emit F = 3.9 10 16 cm 2sr 1s 1 at β = 10 3 for an as- × − − − − − Planck radiation from its surface over a wide range of fre- sumed cross section of σ = 1.9 10 22 cm2. Since the cat × − quencies. IceCube is sensitive to optical photons energies detection efficiency for Q-balls is comparable to that for of about 2 4 eV (300 nm to 600 nm). At nuclearite ve- ÷ slow monopoles catalysing nucleon decay with the same locities expected for cold dark matter the fraction of total cross section it is to be expected that a reinterpretation 5 energy loss emitted in optical photons is of order 10− c. of the above mentioned IceCube slow monopole limits Since this leads to signatures similar to slow magnetic mo- would also present an improvement of about two orders nopoles or Q-Balls the detection efficiency is comparable of magnitude with respect to other limits. again. Nuclearites with masses in excess of 1014 GeV and 3 typical velocities of order of 10− c reach underground detectors. Correspondingly, the sensitivity to the flux of 9.2 Strange Quark Matter nuclearites is roughly of the same order as that to slow monopole fluxes catalysing nucleon decays. Results for Using energy and symmetry arguments, it has been spec- the MACRO experiments [352] and ongoing studies for ulated that strange quark matter (SQM), a hypotheti- the ANTARES detector [353] addressing fluxes of order cal form of matter with roughly equal numbers of up 10 16 10 15 cm 2sr 1s 1 underline the high potential (u), down (d), and strange (s) quarks, could be the true − − − − − − for a corresponding reinterpretation of IceCube analyses. ground state of QCD [346,347,348]. For a plasma of quarks in thermodynamical equilibrium it might be energetically preferable to condense into a phase containing strange 9.3 Long-Lived Charged Massive Particles quarks instead of ordinary matter with neutrons (udd) and protons (uud). Many extensions of the Standard Model predict the exis- An approximate thermodynamical calculation with tence of long-lived charged massive particles (CHAMPs). massless quarks and neglecting strong interactions shows These particles occur naturally in scenarios where the that the average kinetic energy per quark in ordinary bulk decay of charged particles is limited by (approximate) matter could be reduced in bulk SQM by a factor of about discrete symmetries and involves final states that have 0.89 [347]. Therefore, it is feasible that the extra “penalty” only very weak couplings. Analogously to muons, these paid by the presence of more massive strange quarks CHAMPs have a reduced electro-magnetic energy-loss in is over-compensated by the reduction in energy density. matter due to the suppression of bremsstrahlung by the However, the meta-stable state of protons, neutrons, and rest mass. Still, they may be detected by their Cherenkov 40 emission and, due to the long range, even with an en- produced stau NLSPs are parallel tracks in the Cherenkov larged effective detection area. detector [357]. In the following we will consider SUSY breaking sce- The detection efficiency of stau pairs, i.e., coincident narios where the right-handed stau is the next-to-lightest parallel tracks, depends on the energies and directions SUSY particle (NLSP). However, most of the arguments of the staus, as in the case of single events, but also on apply equally well to other scenarios of CHAMPs with a their separation. A large fraction of staus reaching the decay length larger than other experimental scales (see, detector will be accompanied by their stau partner from e.g., Ref. [354]). If -parity is conserved the stau NLSP the same interaction. However, not all of the stau pairs R can only decay into final states containing the LSP. De- might be seen as separable tracks in a Cherenkov tele- pending on the mass and coupling of this particle the scope if they emerge from interactions too close to or lifetime of the stau NLSP can be very long and, in some θ also too far from the detector. The opening angle τeτe be- cases, it can be considered as practically stable on experi- tween staus can be estimated by the initial opening angle mental time-scales. between the SUSY particles in deep inelastic scattering. In the case of a neutralino LSP, the stau NLSP can be The separation of stau tracks in the detector is then given very long-lived if its mass is nearly degenerated with that as x 2 ∆` tan(θ /2), where ∆` is the distance of the ' τeτe of the neutralino [355]. However, there are also super- interaction to the detector center. weakly interacting candidates for the LSP in extensions Double tracks with low track separation are difficult of the MSSM, which provide the long life-time of the to distinguish from single muons copiously produced NLSP more naturally. Possible scenarios include SUSY either directly in air showers or from atmospheric neu- extensions of gravity with a gravitino Ge LSP, SUSY ver- trinos. A required minimum track separation in IceCube sions of the Peccei-Quinn axion and the corresponding of 150 m was found to be necessary to suppress these axino LSP [356], and the MSSM with right-handed chiral muons, due to the geometry of the detector [364]. A pos- neutrinos and a right-handed sneutrino LSP. sible background to the stau pair signal produced in neu- Staus produced in SUSY interactions of EHE neutrinos trino interactions consists of parallel muon pair events could traverse Cherenkov telescopes at the level of a few from random coincidences produced by upgoing atmo- per year, assuming that extragalactic neutrino fluxes are spheric neutrinos. However, this is bounded from above close to the existing bounds [357,358,359,360,361,362] by the number of muons arriving within a coincidence or prompt atmospheric neutrino fluxes close to upper time-window requiring them to be almost parallel and theoretical estimates [361]. The leading-order SUSY con- is several orders of magnitude below the stau pair event 0 12 tribution consists of chargino χ or neutralino χ exchange rate with N + (10 )N (Ref. [365]). Muon pairs µ µ . O − µ between neutrinos and quarks, analogous to the parton- from charged current muon-neutrino interactions involv- level SM contributions shown in Fig. 11. The reactions ing final state hadrons that promptly decay into a second produce sleptons and squarks, which promptly decay muon are expected to be more likely. This has been es- into lighter stau NLSPs. timated in Ref. [360] for the production and decay of Cosmic ray interactions in the atmosphere could also charmed hadrons. produce a long-lived stau signal in neutrino telescopes [363] The rate of stau pairs from neutrino production is (see also Ref. [361]). At energies above 104 GeV the de- largely uncertain and depends on the SUSY mass spec- cay length of weakly decaying nucleons, charged pions trum. If the SUSY mass spectrum close to observational and kaons is much larger than their interaction length in bounds (see, e.g., Ref. [41]), the rate might reach a few the air. Therefore, as they propagate in the atmosphere events per decade in cubic-kilometer Cherenkov tele- the probability that a long-lived hadron (h) collides with scopes [357,359,360,361]. SUSY SM a nucleon to produce SUSY particles is just σh /σh , Experimentally, the situation over the years has be- SUSY where σh is the cross section to produce the SUSY par- come ever more challenging. The SUSY models typically SM ticles X (gluinos or squarks) and σh the Standard Model studied to be in reach of colliders, today have lower lim- cross section of the hadron with nuclei in the atmosphere its of their mass scales of the order of TeV from the non- (which is also the total cross section to a good approxima- observation at the LHC [41]. This leads to significantly SM tion). However, since σh is above 100 mb, it is apparent reduced expected opening angles (respectively, track sep- that this probability will be very small and that it would arations) of the double tracks and low production cross be much larger for a neutrino propagating in matter. sections in these models. However the possibility exists The energy of charged particle tracks observed in neu- that a model beyond the current Standard Model is re- trino detectors is determined by measuring their specific alised in nature with properties that escape observation energy loss, dE/dx. For muons with energies above an at current collider detectors. Hence, the flux of double energy of about 500 GeV, the energy loss rises linearly tracks has been probed generically with IceCube [364]. with energy. However, since the energy loss depends on It turns out, a flux of hundreds of double tracks with a the particle Lorentz boost, a high-energy stau is practi- separation of above 150 m may pass undetected (Fig. 38). cally indistinguishable from a muon with reduced en- This is largely due to the challenge of requiring two very ergy, E /E m /m . A smoking-gun signal for pair- faint tracks to be reconstructed with high accuracy while µ τe ∼ µ τe 41 10 Summary IceCube opened a new window to study the non-thermal universe in 2013 through the discovery of a high-energy neutrino flux of astrophysical origin, and a first iden- tification of a high-energy neutrino point source may be possible through joint multi-messenger 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 pro- vides a probe of Standard Model interactions at energies Fig. 38 Upper limit on the flux of the double tracks at IceCube assuming a uniform arrival direction distribution for up-going only marginally covered or inaccessible by particle accel- tracks [364]. erator experiments. The continuous flux of atmospheric neutrinos allows to study standard neutrino flavour oscil- lations at a precision that is compatible with those of ded- the detector is optimised for higher light yields and single icated oscillation experiments. Moreover, the very long tracks or cascades. Due to generally higher noise rates, a oscillation baselines (thousands of kilometres for atmo- similar analysis in sea water neutrino telescopes may be spheric neutrinos or giga-parsec in astrophysical neutri- even more challenging. Being the first such exploration, nos) and the very high neutrino energies (up to PeV) can the potential for improvements is manifold including probe feeble deviations from the standard three-flavour the implementation of a dedicated double track trigger, oscillation scenario, that are otherwise undetectable. Ice- IceCube detector upgrades that allow identification of Cube has indeed provided strict limits on the allowed smaller track separations and the analysis of all years of parameter space for an additional light “sterile” neutrino available data. state with no Standard Model interactions, reducing con- siderably the range of allowed values of the new oscilla- 2 2 tion variables sin 2θ24 and ∆m41. Similar analyses can set limits on the degree of Lorentz invariance violation, an effect that can be factorised in terms of operators propor- 9.4 Fractional Electric Charges tional to powers of the neutrino energy. The energy reach of IceCube allows to probe higher dimension operators The Standard Model intrinsically does not constrain the than previous experiments. elementary charge but observationally it appears as a Neutrinos are also valuable indirect messengers of physical constant, i.e., all observed colour-singlet par- dark matter annihilation and decay in the Earth, Sun, ticles have integer multiples of elementary charge. As Milky Way, local galaxies, or galaxy clusters. In general, outlined in section8 magnetic monopoles would provide neutrino emission does not suffer from large astrophysi- a mechanism to explain electrical charge quantisation. In cal fore- and backgrounds like electromagnetic emission GUT theories the resulting quantisation is driven by the and does not suffer from deflections in magnetic fields minimum possible magnetic charge rather than directly. like cosmic rays. Neither indirect dark matter detection Free fractionally charged states hence are often predicted with neutrinos shares the same systematic uncertainties in multiples of 1/2 (e.g., in the Pati-Salam model [366]) or from astrophysical or particle physics inputs with other 1/3, but other and smaller fractions are possible. Beyond search techniques. In this way, indirect limits on dark this, fractional charges may exist in composite objects matter properties from neutrino observations are comple- with large (1012 GeV) confinement scales, probably also mentary to indirect searches with other messengers or contributing to dark matter [367]. direct searches with accelerator or scattering experiments. Experimentally, fractionally charged particles in cos- Furthermore, neutrinos can be visible from very distant mic rays can be observed though the anomalously low dark matter sources like galaxy clusters and also probe energy loss and lower light emission. Discarding one the interior of compact sources like our Sun. Besides prob- event from the signal region, the lowest upper limits on ing dark mater capture or self annihilation cross sections, the flux of such particles have been posed by a study of IceCube has current world leading limits on the dark the MACRO experiment [368] assuming a simple charge- matter lifetime, extending the range of the dark matter squared scaling of the energy loss. Due to its size, it is masses probed up to two orders of magnitude with re- expected that IceCube can reach considerably improved spect to results from Cherenkov telescopes. sensitivity to the signature of fractionally charged parti- Neutrino telescopes are also sensitive to a variety of cles in future analyses. exotic signatures produced by rare particles, like Big Bang 42 relics, passing through the detector and emitting direct or 16. J. Ahrens et al. (AMANDA Collaboration), Nucl. Instrum. indirect Cherenkov light, as well as luminescence. 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