Search for non-relativistic magnetic monopoles with IceCube
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Abstract
The IceCube Neutrino Observatory is a large Cherenkov detector instrumenting \(1\,\mathrm {km}^3\) of Antarctic ice. The detector can be used to search for signatures of particle physics beyond the Standard Model. Here, we describe the search for non-relativistic, magnetic monopoles as remnants of the Grand Unified Theory (GUT) era shortly after the Big Bang. Depending on the underlying gauge group these monopoles may catalyze the decay of nucleons via the Rubakov–Callan effect with a cross section suggested to be in the range of \(10^{-27}\) to \(10^{-21}\,\mathrm {cm^2}\). In IceCube, the Cherenkov light from nucleon decays along the monopole trajectory would produce a characteristic hit pattern. This paper presents the results of an analysis of first data taken from May 2011 until May 2012 with a dedicated slow-particle trigger for DeepCore, a subdetector of IceCube. A second analysis provides better sensitivity for the brightest non-relativistic monopoles using data taken from May 2009 until May 2010. In both analyses no monopole signal was observed. For catalysis cross sections of \(10^{-22}\,(10^{-24})\,\mathrm {cm^2}\) the flux of non-relativistic GUT monopoles is constrained up to a level of \(\Phi _{90} \le 10^{-18}\,(10^{-17})\,\mathrm {cm^{-2}\,s^{-1}\,sr^{-1}}\) at a 90 % confidence level, which is three orders of magnitude below the Parker bound. The limits assume a dominant decay of the proton into a positron and a neutral pion. These results improve the current best experimental limits by one to two orders of magnitude, for a wide range of assumed speeds and catalysis cross sections.
Keywords
Background Event Proton Decay Grand Unify Theory Magnetic Monopole Probability Density Distribution1 Introduction
In Grand Unified Theories (GUTs) [2] magnetic monopoles appear as stable, finite-energy solutions of the field equations [3, 4]. The predicted masses range from \(10^5\) to \(10^{17}\,\mathrm {GeV}\) [5, 6, 7, 8, 9] and their magnetic charges are integer multiples of the Dirac charge \( g_{\mathrm {D}} \). The lower part of the mass range up to \(\sim \!\!\!10^{13}\,\mathrm {GeV}\) refers to intermediate mass monopoles (IMMs) which arise from intermediate stages of symmetry breaking below the GUT scale. In contrast the superheavy monopoles with masses at the GUT scale may have been created during the phase transition associated with the spontaneous breakdown of the unified gauge symmetry in the early universe at \(\sim \!\!\!10^{-36}\,\mathrm {s}\) after the Big Bang [10]. The monopole mass and charge depend on the underlying gauge group, the symmetry breaking hierarchy, and the type and temperature of the phase transition in a particular GUT.
Many experiments have searched for relic magnetic monopoles, but there is no experimental proof for their existence. The current best limits for magnetic monopoles constrain their flux to a level of \(\sim 10^{-16}-10^{-18}\,\mathrm {cm}^{-2}\,\mathrm {s}^{-1}\,\mathrm {sr}^{-1}\) depending on the monopole speed and interaction mechanism [15, 16, 17, 18]. Consequently, searches for magnetic monopoles require very large detectors.
The IceCube Neutrino Observatory currently is the world’s largest neutrino detector. The primary goal is the detection of Cherenkov light from electrically charged secondary particles produced when high-energy astrophysical neutrinos interact in the surrounding matter [19]. However, IceCube can also be used to search for magnetic monopoles. Depending on their speed monopoles have different signatures in IceCube. Relativistic monopoles with a speed above the Cherenkov threshold, e.g. \(\beta \approx 0.76\) in ice, can be detected by the Cherenkov light they directly produce [20]. Non-relativistic monopoles that catalyze the decay of nucleons in the detector medium can, in contrast, be detected by the Cherenkov light from electrically charged secondary particles produced in subsequent nucleon decays along the monopole trajectory (Sect. 2.2). Therefore, different analysis strategies are needed in order to cover both detection channels. This paper presents the results of a search for non-relativistic magnetic monopoles which would catalyze the proton decays via the Rubakov–Callan effect in IceCube.
2 Monopole detection with IceCube
2.1 The IceCube detector
The IceCube Neutrino Observatory consists of the in-ice detector, IceCube, and the surface air shower detector, IceTop. It is located at the geographic South Pole. For the in-ice detector, \(1\,\mathrm {km}^3\) of the Antarctic ice, which is used as detection medium, has been instrumented. The detector consists of 86 strings equipped with 60 digital optical modules (DOMs) each. The DOM, the sensor of the IceCube detector, consists of a glass pressure housing enclosing a \(25.4\,\mathrm {cm}\) diameter Hamamatsu photomultiplier tube (PMT) with the electronics needed for signal digitization, and a set of LEDs for calibration purposes [21, 22]. Signals that pass a threshold of about \(0.25\) photo-electrons are digitized and recorded. This process is called a DOM launch or for simplicity a hit in the following. Two hits are labeled as hard local coincidences (also called HLC pair), if their time difference is less than \(1\,\mathrm {\upmu s}\) and the corresponding DOMs are nearest or next-to-nearest neighbors on the same string. The recorded data is sent to the surface and a trigger algorithm evaluates the time and position of the hits and decides whether they form an event. For example, for relativistic particles a simple multiplicity trigger requiring eight HLC hits within a time window of \(5\,\mathrm {\upmu s}\), called SMT8, is used. The DOMs are deployed at depths between \(1450\) and \(2450\,\mathrm {m}\) [23]. At depths below \(2100\,\mathrm {m}\), eight inner strings are placed with smaller separations from each other and thus form a region of denser instrumentation. Together with seven central standard strings they form DeepCore, a low-energy subdetector [24]. The construction of IceCube was completed December 16, 2010 but data taken during intermediate construction stages were already used for physics analyses during earlier years. One of the two presented analyses uses data taken from May 2009 to May 2010, when IceCube was operating in its 59-string configuration (IC-59). The other analysis uses the fully installed detector.
2.2 The Rubakov–Callan effect
Non-relativistic magnetic monopoles would themselves be too slow to emit Cherenkov light when propagating through the IceCube detector. However, relativistic charged secondary particles, produced in monopole interactions with the surrounding matter, can produce Cherenkov light and thus can be detected by the IceCube detector.
Illustration of a proton decay into a positron and a neutral pion catalyzed by a GUT monopole
Illustration of the signature of a non-relativistic magnetic monopole (green) catalyzing nucleon decays (red) along its track in IceCube. The resulting cascades with mean distances \(\lambda _{\mathrm {cat}}\) are symbolized by orange rays
Searches for slow monopoles based on the Rubakov–Callan effect have been pioneered with the underground detectors IMB and Kamiokande-II [41, 42] and the underwater detectors in Lake Baikal [43, 44, 45]. A similar search has also been performed with AMANDA, the predecessor of IceCube [46].
During the commissioning of the full detector (IC-86) in May 2011, a dedicated trigger for slow particle signatures (Slow-Particle Trigger, Sect. 3.1) in DeepCore was implemented. The denser instrumentation of DeepCore allows IceCube to detect monopoles of low light emission, i.e. with rather large values of mean distances \(\lambda _{\mathrm {cat}}\) between induced catalysis points. In 2009, the deployment of the first DeepCore strings was still ahead. Due to the larger spacing and the lack of an appropriate trigger, IC-59 was blind for large \(\lambda _{\mathrm {cat}}\). For smaller \(\lambda _{\mathrm {cat}}\) the mentioned drawbacks were balanced by the larger geometrical area compared to DeepCore.
2.3 Simulation of magnetic monopoles
The signal expectation was determined from Monte Carlo simulations of magnetic monopoles in IceCube, while the background expectation was determined from experimental data itself, with only supplementary simulations.
IceCube simulation includes particle injection and propagation, taking into account appropriate particle interactions, as well as the full detector response to the generated Cherenkov photons.
The arrival directions of magnetic monopoles are assumed to be isotropic. The starting points of simulated monopole tracks are generated randomly on a disc of fixed size. The distance of the plane is fixed with respect to the DeepCore detector but its orientation is random. It is assumed that the magnetic monopoles are not substantially decelerated along their track and their velocity is constant [47].
The distances between the catalyzed nucleon decays are simulated as a Poisson process with a mean free path \(\lambda _{\mathrm {cat}}\) along the monopole track. Each nucleon decay is simulated as an electromagnetic cascade with an energy of \(940\,\mathrm {MeV}\), corresponding to the benchmark detection channel (Eq. 3). The simulation and propagation of the Cherenkov light from these cascades is done with the software package Photonics [48] using the ice model described in [49] for the IC-59 analysis and an improved version described in [50] for the IC-86/DeepCore analysis.
Background noise in the DOMs has to be superimposed on the signal. This noise consists of uncorrelated random noise, mostly from radioactive decays in the DOMs and correlated noise because of after pulses and signals from atmospheric muons. For the IC-59 analysis, the random noise is simulated as a Poisson process and the atmospheric muons are simulated using the software package CORSIKA [51] based on a 5-component model for cosmic rays with the hadronic interaction model SIBYLL [52] and the Hörandel flux model [53]. For the simulation of noise in the IC-86/DeepCore analysis the detector response of simulated monopole signals is superimposed with random and correlated noise hits from experimental data. These noise hits were recorded with a fixed rate trigger (FRT) that was implemented to measure and analyze background noise in the detector. More details on the FRT data are given in Sect. 3.2.
Event display of a simulated monopole with \(\beta = 10^{-3}\) and \(\lambda _{\mathrm {cat}} = 1\,\mathrm {cm}\) with superimposed background noise. The black line represents the monopole track. The DOMs are shown as tiny black dots. The color code illustrates the time scale from red for early times to blue for later times. The radii of the colored spheres scale with the number of recorded photo-electrons
3 Search for magnetic monopoles with the slow particle trigger
The experimental data set was recorded between May 2011 and May 2012 with a dedicated slow particle trigger applied to DeepCore. In this period the live time of the detector was 351 days, with a total number of approximately 50 million triggered events.
3.1 The Slow-Particle Trigger
Multiple IceCube triggers are implemented in the software of the data acquisition system [22]. Most of them are sensitive to signatures of relativistic particles, e.g. muons, so they have little sensitivity to non-relativistic magnetic monopoles. Only for the case of very bright magnetic monopoles the large amount of light can frequently prompt triggers for relativistic particles. This case is described in Sect. 4.1.
Illustration of the SLOP trigger. The times and positions are arbitrary. The x- and y-axis correspond to spatial coordinates and the color bar corresponds to a time scale. a List of all HLC pairs. For the trigger algorithm only the position and time of the first hit of each HLC pair is used. b The two HLC pairs (orange) with a time difference \(\Delta t < t_{\mathrm {proximity}}\) are removed. c All combinations of three HLC pairs, called triplet, with a time difference \(\Delta t_{\mathrm {ij}} \in [t_{\mathrm {min}},t_{\mathrm {max}}]\) between two pairs are built. d The cuts on the quality criteria \(\Delta d\) and \(v_{\mathrm {rel}}\) remove two more triplets. If the remaining triplets overlap in time and fulfill \(n{\text {-}}\mathrm {triplet} \ge n_{\mathrm {min}}{\text {-}}\mathrm {triplet}\), a trigger is generated and the full detector data within the time span from the first to the last HLC pair of the triplets is recorded [54]
The SLOP-Trigger searches for time isolated local coincidences in nearby DOMs caused by subsequent nucleon decays along the monopole trajetory. These coincidences have to be consistent with a straight particle track of constant speed.
The SLOP-Trigger is illustrated in Fig. 4. Specific values for the different trigger parameters are listed in Table 1. It is based on local coincidences of hits (HLCs, Sect. 2.1). For the trigger, the position and time, defined by the first hit of the HLC pair, of all HLC pairs are stored in a list (Fig. 4a). Since muons pass the detector within \(\sim 5\,\mathrm {\upmu s}\) they produce several HLC pairs within a short time. By removing all HLC pairs with time differences \(\Delta t < t_{\mathrm {proximity}}\) from the list, muon hits are efficiently rejected (Fig. 4b).
The remaining HLC pairs are searched for every combination of three HLC pairs, the triplets (Fig. 4c). The time difference between any two HLC pairs within a triplet has to be in the range \([t_{\mathrm {min}},t_{\mathrm {max}}]\). Furthermore only triplets that match a track-like signature are kept. Therefore two quality criteria are required: the contributing HLC pairs have to be ordered along a line and the time differences have to be consistent with a constant speed (Fig. 5).
Trigger conditions of the SLOP-trigger [54]
| Parameter | Value |
|---|---|
| \(t_{\mathrm {proximity}}\) | \( 2.5\,\mathrm {\upmu s}\) |
| \(t_{\mathrm {min}}\) | \(0\,\mathrm {\upmu s}\) |
| \(t_{\mathrm {max}}\) | \(500\,\mathrm {\upmu s}\) |
| \(\Delta d\) | \(\le \) 100 m |
| \(v_{\mathrm {rel}}\) | \(\le \) 0.5 |
| \(n_{\mathrm {min}}{\text {-}}\mathrm {triplet}\) | \( 3\) |
| \(L_{\mathrm {max}}\) | \(5\;\mathrm {ms}\) |
Illustration of a triplet. All three HLC pairs are defined by the position \((\varvec{x_1},\varvec{x_2},\varvec{x_3})\) and the time \((t_1,t_2,t_3)\) of the first hit of an HLC pair. The trigger observables are the distances \((\Delta x_{21},\Delta x_{32},\Delta x_{31})\) and time differences \((\Delta t_{21},\Delta t_{32},\Delta t_{31})\)
3.2 Background study for the SLOP data
Event duration distribution of an experimental 2 days data set (green). The trigger rate is \(2.1\,\mathrm {Hz}\). The maximum is at about \(750\,\mathrm {\upmu s}\). For comparison the event duration distributions of the generated background events (black) is superimposed. The number of entries is normalized to one
\(n{\text {-}}\mathrm {triplet}\) distribution of the experimental test data sample (blue) in comparison to a distribution of simulated monopoles with \(\beta = 10^{-3}\) and \(\lambda _{\mathrm {cat}} = 1\,\mathrm {cm}\) (green). In addition, an exponential function is fitted to the tail of the experimental distribution for \(n{\text {-}}\mathrm {triplet} \ge 15\) (red)
Figure 6 shows the distribution of event durations of SLOP triggered events. Typical durations are of the order of milliseconds, whereas the other IceCube triggers have typical durations of a few microseconds.
Illustration of the generation of background events by reshuffling experimental data measured by a fixed rate trigger (FRT). FRT events have a length of \(10\,\mathrm {ms}\). They are split into \(10\,\mathrm {\upmu s}\) snippets. The snippets are shuffled randomly to build new \(10\,\mathrm {ms}\) frames. Then the SLOP trigger algorithm is applied
To understand the underlying random processes for the background events we developed a method to generate a high statistics sample of background events by reshuffling experimental events recorded with the FRT. The FRT fires at fixed time intervals (e.g., every 30 s), and DOM data from the entire detector are recorded over a time interval of 10 ms. The resulting events contain all types of random and correlated backgrounds, and highly unlikely any signal.
The FRT events of \(10\,\mathrm {ms}\) length were split into snippets of \(10\,\mathrm {\upmu s}\), which were then randomly re-ordered to form new \(10\,\mathrm {ms}\) events. The newly assembled events are then passed to the SLOP trigger algorithm (Fig. 8). This way, a total of 400 s of FRT data were re-shuffled to generate a background sample of about 25 days of live-time equivalent. The generated sample closely resembles the experimental SLOP-triggered events. Figure 6 compares the event duration of the generated background events to the SLOP-triggered events in 2 days of experimental data. The method reproduces the measured event duration distribution reasonably well over several orders of magnitude. For shorter event duration the distribution of the generated data sample tends to be below the distribution of the experimental test data sample. This is expected because this method cannot correctly model noise hits that are correlated over time scales larger than the length of the \(10\,\mathrm {\upmu s}\) snippets. Below \(10\,\mathrm {\upmu s}\) the triplets are characterized by the same DOM combinations due to the low statistics of the FRT events. The overall good agreement indicates that correlated noise is a subdominant effect and is only relevant for short time scales. We will presume later that different triplets due to correlated noise are themselves based on largely independent sets of HLC pulses. Therefore, for large values of \(n{\text {-}}\mathrm {triplet}\) the contribution from correlated noise triplets is added as a random process similar to the triplets from uncorrelated noise.
Comparison of the \(n{\text {-}}\mathrm {triplet}\) distributions of the experimental test data set (green) and the generated background events (black). Triplets caused by HLC pairs fulfilling \(\Delta t_{21}\) or \(\Delta t_{32} \le 50\,\mathrm {\upmu s}\) are not taken into account (cleaned)
3.3 Background model for the SLOP data
\(n{\text {-}}\mathrm {triplet}\) distributions of experimental SLOP data. The blue distribution corresponds to a \(\sim \) 15 % higher noise rate than the red one. Both distributions are normalized to one. The solid lines show the results of the background model fit and the fit parameters \(\mu \) and \(p\) are shown in the boxes
With this ansatz it is possible to describe the distribution of \(n{\text {-}}\mathrm {triplet}\) with only three parameters \(P_0, \mu \) and \(p\). Figure 10 shows the fit of this model to two normalized, experimental \(n{\text {-}}\mathrm {triplet}\) distributions which are based on SLOP data corresponding to different noise rates. Since the distributions are normalized only \(\mu \) and \(p\) have to be fit. The background model well describes the \(n\mathrm {triplet}\) distributions over several orders of magnitude. Moreover the increase in the noise rate is reflected in an increase of the value of \(\mu \), which depends on the noise rate. In summary it can be confirmed that the background events from the SLOP trigger are dominated by random noise.
3.4 Reconstruction of a monopole track
Distribution of the reconstructed speeds for two different simulated monopole speeds. At the top the distributions for monopoles with \(\lambda _{\mathrm {cat}}=1\,\mathrm {m}\) and at the bottom for monopoles with \(\lambda _{\mathrm {cat}}=1\,\mathrm {cm}\) are shown. For comparison, reconstructed experimental data corresponding to a live time of 8 h are plotted. The three dotted black lines show the true speeds and the speed of light. All distributions are normalized to one
This reconstruction algorithm is simple, robust, and fast, while still yielding a sufficient accuracy. It also allows us to approximate the monopole direction by the direction of \(\varvec{v}\). The mean difference between the true and reconstructed direction varies between \(\sim 11^{\circ }\) and \(\sim 20^{\circ }\) depending on the monopole speed and the mean free path \(\lambda _{\mathrm {cat}}\).
3.5 Event selection and background reduction
For this first IceCube analysis of SLOP data a robust approach based on \(n{\text {-}}\mathrm {triplet}\) as the single final selection criterion and the determination of the expected background from experimental data was chosen.
Probability density distributions of \(n{\text {-}}\mathrm {triplet}\) for events with larger reconstructed speed (top) and for events with smaller reconstructed speed (bottom). In black the distributions of 1 year experimental data are shown. The signal distributions are shown with decreasing \(\lambda _{\mathrm {cat}}\) in blue, red, and green. The final cuts on \(n{\text {-}}\mathrm {triplet}\) are shown by the dashed black line
These selection cuts were defined before unblinding the full experimental data. Here, an iterative two step procedure was chosen. First, 10 % experimental data was unblinded with the selection determined by the aforementioned experimental 2 days data sample. After no signal or unexpected background was observed the same procedure was applied to the full experimental data.
\(n{\text {-}}\mathrm {triplet}\)-speed distribution of 1 year of experimental data. The final cuts on \(n{\text {-}}\mathrm {triplet}\) are shown by the dashed red lines. The boundary between the two speed regions is shown by the dashed black line
3.6 Results
\(n{\text {-}}\mathrm {triplet}\) distributions for events with larger reconstructed speed (top) and for events with a smaller reconstructed speed (bottom). The black data points show the distributions of the full experimental data. The expected signal is shown with decreasing \(\lambda _{\mathrm {cat}}\) in blue, red and green. The fitted functions \(P(n \,\arrowvert \, \mu ,p)\) are shown in purple and the final selections on \(n{\text {-}}\mathrm {triplet}\) as dashed black lines
Probability density distribution of the expected number of background events. The median is shown by the dashed red line. The quantiles \(Q_{0.16}\) and \(Q_{0.84}\) are shown by the dash-dotted red line
4 Search for very bright magnetic monopoles with the IC-59 array
The search for magnetic monopoles presented in this section uses data taken during the season 2009–2010 when IceCube was running in its 59-string configuration. This analysis used the data taken with the standard IceCube triggers. The standard trigger that is used for highly energetic relativistic particles is a simple multiplicity trigger (SMT), which requires at least eight HLC hits within a sliding time window of 5 \(\upmu s\) (SMT-8). Other triggers are optimized for relativistic particles with lower energies. Data are recorded over at least the time interval over which the trigger condition of any of the triggers is fulfilled. For HLC hits, the full PMT waveforms are digitized and recorded [22]. Not all triggered events were transmitted to the Northern hemisphere by satellite. Events of various categories (e.g. track-like, cascade-like, very bright events, etc.) have been selected by various online filters at the South Pole [57, 58]. Although the filters are optimized for relativistic particles, they may accept bright monopole events if a sufficient number of DOMs are hit. This analysis uses the cascade and high-energy filters, which have the best acceptance for non-relativistic monopoles. The total live time of this data set is 311.25 days, with an average rate of selected events of 85.5 Hz. The efficiency of this filter selection with respect to the multiplicity trigger is above 75 % for monopoles of \(\beta = 10^{-3} \) and \(\lambda _{\mathrm {cat}} = 1\,\mathrm {mm}\).
4.1 Selection of very bright magnetic monopoles
Slow monopoles with a catalysis cross section \(\sigma _{\mathrm {cat}}\) much larger than \(10^{-23}\,{\mathrm {cm^{2}}}\) appear as very bright tracks. Simulations of the detector response to such tracks show that the multiplicity condition is fulfilled over most of the monopole crossing time, or that successive triggers occur close enough in time for the recording intervals to overlap. So, a large fraction of a monopole’s catalysis signature would be captured in a single event, if \(\sigma _{\mathrm {cat}}\) is sufficiently high. For \(\sigma _{\mathrm {cat}} < 10^{-23}\) cm\(^{2}\) monopoles still yield multiple triggers, but the triggers occur less frequently, so that the signature is often split up into several subevents. The smaller the cross section, the more the monopole event splits up and the larger are the gaps between the subevents. Eventually, the signal becomes indistinguishable from the background. Therefore, this analysis focuses on catalysis cross sections above \(10^{-23}\) cm\(^{2}\). For monopoles with such high \(\sigma _{\mathrm {cat}}\), the IC-59 analysis achieves a better sensitivity than the analysis using the SLOP trigger. This is simply because the IC-59 array had a much larger detection volume than the DeepCore array available to the previously described analysis. Future monopole searches will use data taken after 2012, when the SLOP trigger was operating on the full IC-86 array. These analyses will take advantage of both the large detection volume of the full IC-86 array and the high efficiency of the SLOP trigger.
4.2 IC-59 background reduction
The high-energy and cascade filters provide a data sample with about \(10^9\) events. The vast majority of these events are down-going atmospheric muons. This background is reduced using a set of straight cuts in a first step. These cuts are based on the time and location of the detected Cherenkov photons. Contrary to the IC-86/DeepCore analysis, whose cut parameters where defined using the time and location of DOM launches or HLC pairs, this analysis uses a feature extraction algorithm on the PMT waveforms, which reconstructs the constituent PMT pulses caused by individual photo electrons. In a second step a Multivariate Analysis is adopted to reduce the background further.
- 1.
The event duration \(\Delta {{t}}\) defined as the time difference between the last and first pulse registered by a DOM in an event.
- 2.
The reconstructed speed v from the line fit.
- 3.
The number of clusters (\(N_{\mathrm {clusters}}\)), which is defined by the reconstructed pulses on all DOMs sorted into groups of pulses which occur close in space and time. Each such group is called a cluster and the total number of these clusters in an event is used as a cut variable. Bright signal tracks tend to have a higher number of clusters than atmospheric muon background events.
- 4.
The total number of photo-electrons collected in the whole detector divided by the event duration, \(Q_{\mathrm {tot}} / \Delta {\mathrm {t}}\).
- 5.
Median of the distance between clusters along the reconstructed track.
- 6.
The center of gravity (COG) of the event, defined as the average spatial coordinates of all hits.
- 7.
Mean distance of the hit DOMs to the center of gravity (COG) of the event divided by the event duration.
- 8.
Number of clusters divided by the event duration.
- 9.
Number of simple multiplicity triggers divided by the number of strings with hit DOMs
Signal efficiencies, data reduction factors and data rates before and after each cut for both \(\sigma _{\mathrm {cat}}\)( \(\lambda _{\mathrm {cat}}\)). For \(\sigma _{\mathrm {cat}}=1.7\cdot 10^{-22}\,\mathrm {cm^{2}}\) the corresponding applied cuts are “Cut 1 to Cut 6” which are described in subsection 4.2. For \(\sigma _{\mathrm {cat}}=1.7\cdot 10^{-23}\,\mathrm {cm^{2}}\) the applied cuts are: Cut 1, Cut 2 and Cut 6
| Before the cut | Cut 1 | Cut 2 | Cut 3 | Cut 4 | Cut 5 | Cut 6 | |
|---|---|---|---|---|---|---|---|
| \(\sigma _{\mathrm {cat}}=1.7\cdot 10^{-22}\,\mathrm {cm^{2}}, \lambda _{\mathrm {cat}}=1\,\mathrm {mm}\) | |||||||
| \(\beta =10^{-2}\) | 57 % | 43.5 % | 42.3 % | 41.9 % | 41.9 % | 41.8 % | 33.6 % |
| \(\beta =10^{-3}\) | 75.4 % | 45.3 % | 41.1 % | 41 % | 41 % | 39.8 % | 34.3 % |
| Experiment: reduction factor | – | \(8 \cdot 10^{-4}\) | \(7.8\cdot 10^{-5}\) | \(6.6 \cdot 10^{-5}\) | \(5.7 \cdot 10^{-5}\) | \(5.3\cdot 10^{-5}\) | \(4.8 \cdot 10^{-5}\) |
| Experiment: rate (s\(^{-1}\)) | 85.5 | \(6.8 \cdot 10^{-2}\) | \(6.7 \cdot 10^{-3}\) | \(5.6 \cdot 10^{-3}\) | \(4.9 \cdot 10^{-3}\) | \(4.5 \cdot 10^{-3}\) | \(4.1 \cdot 10^{-3}\) |
| \(\sigma _{\mathrm {cat}}=1.7\cdot 10^{-23}\,\mathrm {cm^{2}}, \lambda _{\mathrm {cat}}=1\,\mathrm {cm}\) | |||||||
| \(\beta =10^{-2}\) | 41 % | 17.2 % | 17 % | 13.9 % | |||
| \(\beta =10^{-3}\) | 43 % | 3.5 % | 3.23 % | 3.1 % | |||
| Experiment: reduction factor | – | \(1.4 \cdot 10^{-3}\) | \(3.9 \cdot 10^{-4}\) | \(3.45 \cdot 10^{-4}\) | |||
| Experiment: rates (s\(^{-1}\)) | 85.5 | \(1.2 \cdot 10^{-1}\) | \(3.3 \cdot 10^{-2}\) | \(2.95 \cdot 10^{-2}\) | |||
4.3 Signal expectations
Data are divided into two sets according to the monopole track brightness (i.e. the catalysis cross section). The cross section values for which we optimized the analysis and derive flux limits are \(\sigma _{\mathrm {cat}} = 1.7 \cdot 10^{-22}\, \mathrm {cm^{2}}\) and \(\sigma _{\mathrm {cat}} = 1.7 \cdot 10^{-23}\, \mathrm {cm^{2}}\), which correspond to \(\lambda _{\mathrm {cat}}=1\,\mathrm {mm}\) and \(\lambda _{\mathrm {cat}}=1\,\mathrm {cm}\), respectively.
Top Distribution of the event duration \(\Delta {\mathrm {t}}\), for experimental data and simulated bright monopoles with \(\lambda _{\mathrm {cat}}\) = 1 mm (i.e. \(\sigma _{\mathrm {cat}} =1.7\cdot 10^{-22} \mathrm {cm^{2}}\)), before applying a cut. The green histogram represents monopoles with \(\beta =10^{-3}\), the blue histogram with \(\beta =10^{-2}\). The gray histogram represents the data. The red dashed line marks the value of the chosen cut which is set at \(\Delta {\mathrm {t}} > 30\,\upmu \mathrm{s}\). Bottom The same for the reconstructed speed v with a cut at v \(< 9\cdot 10^{-3}\) m/ns. Histograms are normalized to 1
Relativistic single-muon tracks have a reconstructed speed v around \(0.3\,\mathrm {m/ns}\). Events having passed the preceding cut on the time duration \(\Delta {\mathrm {t}}\) are enriched with coincident muons from uncorrelated air showers, resulting in a lower v (see Fig. 16, bottom). For monopoles, the speed v is close to the simulated values. Cutting at v \(< 9\cdot 10^{-3}\) m/ns (corresponding to \(\beta < 3\cdot 10^{-2}\)) reduces the background by another order of magnitude.
Further cuts on variables 3–6 reduce the background by another factor two. In total after this first set of cuts the data rate is reduced by a factor \(5\cdot 10^{-5}\) while the signal efficiencies only drop to 33.6 and 34.3 % for \(\beta =10^{-2}\) and \(\beta =10^{-3}\), respectively. Data reduction factors, rates and signal efficiencies before and after each applied cut are presented in Table 2.
Same as Fig. 16 but with \(\lambda _{\mathrm {cat}}=1\,\mathrm {cm}\), i.e. 10 times lower. The red dashed line marks the value of the chosen cut which is set at \(\Delta {\mathrm {t}} > 28\,\upmu \mathrm{s}\) for the event duration and v \(< 1.5\cdot 10^{-2}\) m/ns for the speed. Histograms are normalized to 1
4.4 IC-59 final cut optimization
To optimize the sensitivity for bright monopoles a Multivariate Analysis is used. It classifies each event by a Boosted Decision Trees (BDT) score in the range \([-1,+1]\) [59, 60]. A BDT score of \(-1\) characterizes a background-like event whereas a BDT score of \(+1\) characterizes a signal-like event.
For the analysis, a sample of 10 % of all experimental data (burn sample) was divided into two equally sized sets. BDTs have been trained on each combination (\(\beta \),\(\lambda _{\mathrm {cat}}\)) of the signal Monte Carlo and on 50 % of the corresponding burn sample, using combinations of the variables described above. The sensitivity was estimated from the other 50 % of the burn sample by fitting an exponential function to the tail of the BDT score distribution. Over a large range of the BDT scores, the fit describes the data rather well. Still, its extension into the signal region has no strict physical justification.
The final cut on the BDT scores for each combination of (\(\beta \),\(\lambda _{\mathrm {cat}}\)) is obtained by using the Model Rejection Factor (MRF) method [56]. For the chosen high catalysis cross sections the limits for three (\(\beta , \lambda _{\mathrm {cat}}\)) combinations are significantly better or comparable to those of the IC-86/DeepCore analysis. The fourth combination (\(\beta =10^{-3}, \lambda _{\mathrm {cat}}=1\) cm) is not competitive because the optimal cut results in 42 expected background events for the full data sample.
4.5 Results
Distribution of the BDT scores, after unblinding, for data and signal with \(\lambda _{\mathrm {cat}}=1\) mm, and speed \(\beta =10^{-3}\). The dot dashed line shows the optimized cut on the BDT score obtained from the Model Rejection Factor method. One event survived the BDT cut and is compatible with the background
Number of expected and observed events per year for every (\(\beta , \lambda _{\mathrm {cat}}\)) parameter combination. \(N_{\mathrm {expected}}\) is derived from the integral of the fitted BDT scores with an exponential. The integral ranges are from the BDT cut value to unity. The errors on the number of expected background event are 1\(\sigma \) errors derived from a toy Monte Carlo experiment.
| \(\mathrm{BDT}_{\mathrm {cut}}\) | \(N_{\mathrm {expected}}\) | \(N_{\mathrm {observed}}\) | |
|---|---|---|---|
| \(\sigma _{\mathrm {cat}}=1.7\cdot 10^{-22}\mathrm {cm^{2}}, \lambda _{\mathrm {cat}}=1\,\mathrm {mm}\) | |||
| \(\beta =10^{-2}\) | 0.46 | \(0.6^{+0.2}_{-0.1}\) | 0 |
| \(\beta =10^{-3}\) | 0.48 | \(4.8^{+0.7}_{-0.6}\) | 1 |
| \(\sigma _{\mathrm {cat}}=1.7\cdot 10^{-23}\mathrm {cm^{2}}, \lambda _{\mathrm {cat}}=1\,\mathrm {cm}\) | |||
| \(\beta =10^{-2}\) | 0.5 | \(3.0^{+0.6}_{-0.5}\) | 0 |
| \(\beta =10^{-3}\) | Not sensitive | Not sensitive | Not sensitive |
5 Systematic uncertainties
The calculation of upper flux limits takes into account the statistical and systematic uncertainties in the background and signal predictions. Because the number of expected background events is estimated from experimental data, only the statistical uncertainties of the fit parameters of the background model are relevant.
For signal the imperfect detector description is taken into account. For example in case of the IC-86/DeepCore search, the random noise leads to an increase of \(n{\text {-}}\mathrm {triplet}\) for signal events. Furthermore the optical light detection efficiency is important. This efficiency takes into account the cumulative effect of the light yield of nucleon decays, where a single electromagnetic cascade is simulated instead of several daughter particles, the light propagation through the ice and its detection by the DOMs. These effects result in an uncertainty of the detection efficiency for magnetic monopoles which is used to derive the upper limits.
The impact of these uncertainties on the flux limits is estimated by simulating monopoles with simulation parameters changed within their estimated uncertainties. The uncertainties of the superimposed background noise, the light yield of nucleon decays and the light propagation through ice are estimated by their differences in the detection efficiencies of signal simulations taking into account different approaches (Sect. 2.3). For the IC-86/DeepCore analysis the superimposed noise can be described by random and correlated noise hits from experimental data or noise simulated as a Poisson process and atmospheric muons simulated using the software package CORSIKA. Since for the IC-59 analysis no unbiased experimental data exists the background noise can be simulated by a noise generator that also takes into account correlated noise hits. For reasons of simplification the proton decay is simulated as a single electromagnetic cascade with an isotropic direction which is valid as long as the mean free path is much smaller than the IceCube spacing. Due to kinematics in the proton decay (Eq. 3) two back-to-back electromagnetic cascades with an isotropic direction have to be simulated. The uncertainties due to this simplification are estimated by the differences between both approaches. For the light propagation through ice the two ice models described in [49, 50] are used. The uncertainty of the optical efficiency of DOMs can be estimated as \(\pm 10~\%\). Signal simulations based on optical DOM efficiencies varied by \(\pm 10~\%\) are compared with simulations based on the default settings. The differences in the detection efficiencies are used as an estimate for the uncertainty.
The impact of different systematic uncertainties on the detection efficiencies of magnetic monopoles depending on the mean free path \(\lambda _{\mathrm {cat}}\) and the monopole speed \(\beta \). The first column shows the impact of different assumptions for the superimposed background noise. Also the uncertainties of the simplified nucleon decay simulation (second column), the optical DOM efficiency (third column) and the optical ice properties (fourth column) are shown
| \(\lambda _{\mathrm {cat}}\;(\mathrm {m})\) | Noise simulation | Nucleon decay simulation | Optical DOM efficiency | Optical ice properties | ||||
|---|---|---|---|---|---|---|---|---|
| \(\beta =10^{-2}\) | \(\beta =10^{-3}\) | \(\beta =10^{-2}\) | \(\beta =10^{-3}\) | \(\beta =10^{-2}\) | \(\beta =10^{-3}\) | \(\beta =10^{-2}\) | \(\beta =10^{-3}\) | |
| IC-86 | ||||||||
| 3.0 | +29 %\(/-\)2 % | +23 % | +36 % | \(-\)16 % | +36 %\(/-\)23 % | +46 %\(/-\)29 % | +20 %\(/-\)11 % | +72 % |
| 1.0 | +52 % | +6 %\(/-\)11 % | +27 % | +1 %\(/-\)11 % | +14 %\(/-\)7 % | +16 %\(/-\)20 % | \(-\)17 % | +12 % |
| 0.3 | +31 % | +5 %\(/-\)8 % | +15 % | +1 %\(/-\)8 % | \(\pm \)10 % | \(\pm \)11 % | \(-\)16 % | +8 %\(/-\)2 % |
| 0.1 | +19 %\(/-\)1 % | +3 %\(/-\)4 % | \(\pm \)7 % | +1 %\(/-\)6 % | +5 %\(/-\)11 % | +8 %\(/-\)6 % | \(-\)15 % | +1 %\(/-\)6 % |
| 0.03 | +17 %\(/-\)2 % | +1 %\(/-\)5 % | +9 %\(/-\)4 % | +1 %\(/-\)4 % | +10 %\(/-\)5 % | +6 %\(/-\)3 % | \(-\)9 % | \(-\)7 % |
| 0.01 | +15 %\(/-\)4 % | \(-\)4 % | +11 %\(/-\)2 % | \(\pm \)2 % | +12 %\(/-\)1 % | +5 %\(/-\)0.3 % | \(\pm \)5 % | \(-\)10 % |
| 0.001 | +15 %\(/-\)4 % | \(-\)4 % | +11 %\(/-\)2 % | \(\pm \)2 % | +12 %\(/-\)1 % | +5 %\(/-\)0.3 % | \(\pm \) 5 % | \(-\)10 % |
| IC-59 | ||||||||
| 0.01 | +2 % | – | \(-\)5 % | – | +9 %\(/-\)6 % | – | +7 % | – |
| 0.001 | \(-3~\%\) | \(+5~\%\) | \(-\)3 % | \(-\)1 % | \(-\)5 % | +9 %\(/-\)2 % | +2 % | +4 % |
The statistical uncertainties of the calculated effective areas for different mean free path \(\lambda _{\mathrm {cat}}\) and speed \(\beta \) for the IC-86 and IC-59 analyses
| \(\lambda _{\mathrm {cat}} \mathrm{(m)}\) | Statistical uncertainties | ||
|---|---|---|---|
| \(\beta = 10^{-2}\) (%) | \(\beta = 10^{-3}\) (%) | ||
| IC-86 | 3.0 | \(9\) | \(10\) |
| 1.0 | 8 | 5 | |
| 0.3 | 8 | 2 | |
| 0.1 | 8 | 2 | |
| 0.03 | 9 | 2 | |
| 0.01 | 10 | 2 | |
| 0.001 | 12 | 2 | |
| IC-59 | 0.01 | 0.2 | – |
| 0.001 | 0.1 | 0.1 | |
For the calculation of the final flux limits we perform high statistics computer experiments. In each we randomize the effect of each systematics effect \(R_{\mathrm {i}}\) according to its specific uncertainty. For each parameter combination \(\beta \) and \(\sigma _{\mathrm {cat}}\) this results in the effective probability density distribution for the relative change of the detection efficiency \(R\) taking into account all uncertainties.
Probability density distribution for systematic signal uncertainties for \(\beta =10^{-2}\) monopoles for the IC-86/DeepCore analysis
6 Flux limits
The flux limits on non-relativistic magnetic monopoles are calculated assuming an isotropic flux and the proton decay channel \(p\rightarrow \mathrm{e}^+ \pi ^0 \) (Eq. 3) with the catalysis cross section \(\sigma _{\mathrm {cat}}\), which depends on the speed \(\beta \) (Eq. 4). Using the quantity \(\hat{l}\) (Eq. 6) the flux limits can also be expressed without assuming a specific decay channel.
Upper limits on the flux of non-relativistic magnetic monopoles depending on the speed \(\beta \) and catalysis cross section \(\sigma _{\mathrm {cat}}\) of the IC-59 analysis and IC-86/DeepCore analysis. The dashed lines are limits published by the MACRO experiment [16]. Here, MACRO 1 is an analysis developed for monopoles catalyzing the proton decay. MACRO 2 is the standard MACRO-analysis, which is sensitive to monopoles ionizing the surrounding matter. Additionally, the IceCube limits are shown as a function of \(\hat{l}\), which is proportional to the averaged Cherenkov photon yield per nucleon decay (not valid for MACRO limits)
Assuming monopoles are the dominant part of Dark Matter, i.e. the relic mass density of monopoles is similar to the Dark Matter mass density, our most stringent flux limits constrain the monopole mass to be at least of the order of the Planck mass \(m_{\mathrm {pl}}=1.22 \cdot 10^{19}\,\mathrm {GeV}\) [62]. This implies that monopoles with masses significantly smaller than the Planck mass do not contribute dominantly to the Dark Matter mass density.
Indirect searches for monopole induced proton decays set very strong bounds on monopoles with non-relativistic speeds, e.g. the limits from Super-Kamiokande [63] assuming gravitational trapping of monopoles in the sun. Also a variety of bounds based on observations of neutron stars, white dwarfs, and gas giants have been obtained. These bounds range from \(\sim \!\!\!\!10^{-18}-10^{-29}\,\mathrm {cm}^{-2}\,\mathrm {s}^{-1}\,\mathrm {sr}^{-1}\) and depend on the catalysis cross sections as well as on details of the assumed astrophysical scenarios [64, 65, 66]. Although the direct IceCube searches are not as stringent as indirect searches the former are not affected by astrophysical uncertainties. Thus the direct IceCube limits can be considered as a robust upper bound on the monopole flux, if the Rubakov–Callan effect is realized in nature.
7 Summary and outlook
Data taken from May 2011 until May 2012 with a dedicated slow-particle trigger and for the brightest monopoles data taken from May 2009 until May 2010 with standard IceCube triggers were analyzed. The analysis, which is based on data of the slow-particle trigger, was developed by using simulated monopole events and experimental data to estimate the background properties. For this first analysis of such a signal in IceCube a robust approach based on a single final selection criterion and the comparison between the number of expected background events and observed experimental events is chosen. Using the experimental data, the number of expected background events can be estimated to \(n_{\mathrm {b}} = 3.2^{+1.8}_{-1.1}\).
The IC-59 analysis based on standard IceCube triggers is sensitive only for bright monopoles with \(\sigma _{\mathrm {cat}} > 1.7 \cdot 10^{-23}\,\mathrm {cm^2}\). The analysis used Boosted Decision Trees (BDT) to discriminate between monopole signal and background. The expected number of background events is derived from a fit of the BDT scores tails with an exponential function for each (\(\beta \),\(\lambda _{\mathrm {cat}}\)). The number of observed events after unblinding is \(1\) for an expected background of \(4.8_{-0.6}^{+0.7}\). This event contains multiple coincident muons, which renders it compatible with a background event. The obtained flux limits for \(\beta =10^{-2}\) and \(\lambda _{\mathrm {cat}} =0.01\) m, \(0.001\) m from the IC-59 analysis are better than the ones from the IC-86/DeepCore analysis because of the bigger effective area. For \(\beta =10^{-3}\) the limits are comparable since the standard IceCube triggers are less sensitive to the monopole signal in comparison to the dedicated slow-particle trigger.
In both analyses no monopole signal has been observed. Thus, the limits on the flux of non-relativistic magnetic monopoles—catalyzing the proton decay—are improved by about more than one order of magnitude in comparison to MACRO [16] for most of the investigated parameter space and reach down to about three orders of magnitude below the Parker limit.
Since May 2012 the dedicated slow-particle trigger has been updated to the full IceCube detector. From this upgrade, we expect an improvement in sensitivity by roughly an order of magnitude [67]. This gain is supplemented by improvements of the data selection which have been developed after completion of this analysis. Examples are the implementation of a Kalman-filter-based HLC hit selection, which improves the angular and speed reconstruction, and the implementation of an event selection based on a Boosted Decision Tree [68].
Notes
Acknowledgments
We acknowledge the support from the following agencies: U.S. National Science Foundation-Office of Polar Programs, U.S. National Science Foundation-Physics Division, University of Wisconsin Alumni Research Foundation, the Grid Laboratory Of Wisconsin (GLOW) grid infrastructure at the University of Wisconsin - Madison, the Open Science Grid (OSG) grid infrastructure; U.S. Department of Energy, and National Energy Research Scientific Computing Center, the Louisiana Optical Network Initiative (LONI) grid computing resources; Natural Sciences and Engineering Research Council of Canada, WestGrid and Compute/Calcul Canada; Swedish Research Council, Swedish Polar Research Secretariat, Swedish National Infrastructure for Computing (SNIC), and Knut and Alice Wallenberg Foundation, Sweden; German Ministry for Education and Research (BMBF), Deutsche Forschungsgemeinschaft (DFG), L.B. was funded by the DFG Sonderforschungsbereich 676, Helmholtz Alliance for Astroparticle Physics (HAP), Research Department of Plasmas with Complex Interactions (Bochum), Germany; Fund for Scientific Research (FNRS-FWO), FWO Odysseus programme, Flanders Institute to encourage scientific and technological research in industry (IWT), Belgian Federal Science Policy Office (Belspo); University of Oxford, United Kingdom; Marsden Fund, New Zealand; Australian Research Council; Japan Society for Promotion of Science (JSPS); the Swiss National Science Foundation (SNSF), Switzerland; National Research Foundation of Korea (NRF); Danish National Research Foundation, Denmark (DNRF).
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