Energy Reconstruction Methods in the IceCube Neutrino Telescope
IceCube Collaboration, M. G. Aartsen, R. Abbasi, M. Ackermann, J. Adams, J. A. Aguilar, M. Ahlers, D. Altmann, C. Arguelles, J. Auffenberg, X. Bai, M. Baker, S. W. Barwick, V. Baum, R. Bay, J. J. Beatty, J. Becker Tjus, K.-H. Becker, S. BenZvi, P. Berghaus, D. Berley, E. Bernardini, A. Bernhard, D. Z. Besson, G. Binder, D. Bindig, M. Bissok, E. Blaufuss, J. Blumenthal, D. J. Boersma, C. Bohm, D. Bose, S. Böser, O. Botner, L. Brayeur, H.-P. Bretz, A. M. Brown, R. Bruijn, J. Casey, M. Casier, D. Chirkin, A. Christov, B. Christy, K. Clark, L. Classen, F. Clevermann, S. Coenders, S. Cohen, D. F. Cowen, A. H. Cruz Silva, M. Danninger, J. Daughhetee, J. C. Davis, M. Day, C. De Clercq, S. De Ridder, P. Desiati, K. D. de Vries, M. de With, T. DeYoung, J. C. Díaz-Vélez, M. Dunkman, R. Eagan, B. Eberhardt, B. Eichmann, J. Eisch, S. Euler, P. A. Evenson, O. Fadiran, A. R. Fazely, A. Fedynitch, J. Feintzeig, T. Feusels, K. Filimonov, C. Finley, T. Fischer-Wasels, S. Flis, A. Franckowiak, K. Frantzen, T. Fuchs, T. K. Gaisser, J. Gallagher, L. Gerhardt, L. Gladstone, T. Glüsenkamp, A. Goldschmidt, G. Golup, J. G. Gonzalez, J. A. Goodman, D. Góra, D. T. Grandmont, D. Grant, P. Gretskov, J. C. Groh, A. Groß, C. Ha, A. Haj Ismail, P. Hallen, A. Hallgren, F. Halzen, et al. (188 additional authors not shown)
EEnergy Reconstruction Methods in the IceCube Neutrino Telescope
M. G. Aartsen b , R. Abbasi ac , M. Ackermann as , J. Adams o , J. A. Aguilar w , M. Ahlers ac , D. Altmann v , C. Arguelles ac ,J. Auffenberg ac , X. Bai ag,1 , M. Baker ac , S. W. Barwick y , V. Baum ad , R. Bay g , J. J. Beatty q,r , J. Becker Tjus j ,K.-H. Becker ar , S. BenZvi ac , P. Berghaus as , D. Berley p , E. Bernardini as , A. Bernhard af , D. Z. Besson aa , G. Binder h,g ,D. Bindig ar , M. Bissok a , E. Blaufuss p , J. Blumenthal a , D. J. Boersma aq , C. Bohm aj , D. Bose al , S. B¨oser k , O. Botner aq ,L. Brayeur m , H.-P. Bretz as , A. M. Brown o , R. Bruijn z , J. Casey e , M. Casier m , D. Chirkin ac , A. Christov w , B. Christy p ,K. Clark am , L. Classen v , F. Clevermann t , S. Coenders a , S. Cohen z , D. F. Cowen ap,ao , A. H. Cruz Silva as ,M. Danninger aj , J. Daughhetee e , J. C. Davis q , M. Day ac , C. De Clercq m , S. De Ridder x , P. Desiati ac , K. D. de Vries m ,M. de With i , T. DeYoung ap , J. C. D´ıaz-V´elez ac , M. Dunkman ap , R. Eagan ap , B. Eberhardt ad , B. Eichmann j ,J. Eisch ac , S. Euler a , P. A. Evenson ag , O. Fadiran ac , A. R. Fazely f , A. Fedynitch j , J. Feintzeig ac, ∗ , T. Feusels x ,K. Filimonov g , C. Finley aj , T. Fischer-Wasels ar , S. Flis aj , A. Franckowiak k , K. Frantzen t , T. Fuchs t , T. K. Gaisser ag ,J. Gallagher ab , L. Gerhardt h,g , L. Gladstone ac , T. Gl¨usenkamp as , A. Goldschmidt h , G. Golup m , J. G. Gonzalez ag ,J. A. Goodman p , D. G´ora v , D. T. Grandmont u , D. Grant u , P. Gretskov a , J. C. Groh ap , A. Groß af , C. Ha h,g ,A. Haj Ismail x , P. Hallen a , A. Hallgren aq , F. Halzen ac , K. Hanson l , D. Hebecker k , D. Heereman l , D. Heinen a ,K. Helbing ar , R. Hellauer p , S. Hickford o , G. C. Hill b , K. D. Hoffman p , R. Hoffmann ar , A. Homeier k , K. Hoshina ac ,F. Huang ap , W. Huelsnitz p , P. O. Hulth aj , K. Hultqvist aj , S. Hussain ag , A. Ishihara n , S. Jackson ac , E. Jacobi as ,J. Jacobsen ac , K. Jagielski a , G. S. Japaridze d , K. Jero ac , O. Jlelati x , B. Kaminsky as , A. Kappes v , T. Karg as ,A. Karle ac , M. Kauer ac , J. L. Kelley ac , J. Kiryluk ak , J. Kl¨as ar , S. R. Klein h,g , J.-H. K¨ohne t , G. Kohnen ae ,H. Kolanoski i , L. K¨opke ad , C. Kopper ac , S. Kopper ar , D. J. Koskinen s , M. Kowalski k , M. Krasberg ac , A. Kriesten a ,K. Krings a , G. Kroll ad , J. Kunnen m , N. Kurahashi ac , T. Kuwabara ag , M. Labare x , H. Landsman ac , M. J. Larson an ,M. Lesiak-Bzdak ak , M. Leuermann a , J. Leute af , J. L¨unemann ad , O. Mac´ıas o , J. Madsen ai , G. Maggi m , R. Maruyama ac ,K. Mase n , H. S. Matis h , F. McNally ac , K. Meagher p , M. Merck ac , T. Meures l , S. Miarecki h,g , E. Middell as , N. Milke t ,J. Miller m , L. Mohrmann as , T. Montaruli w,2 , R. Morse ac , R. Nahnhauer as , U. Naumann ar , H. Niederhausen ak ,S. C. Nowicki u , D. R. Nygren h , A. Obertacke ar , S. Odrowski u , A. Olivas p , A. Omairat ar , A. O’Murchadha l , L. Paul a ,J. A. Pepper an , C. P´erez de los Heros aq , C. Pfendner q , D. Pieloth t , E. Pinat l , J. Posselt ar , P. B. Price g ,G. T. Przybylski h , M. Quinnan ap , L. R¨adel a , M. Rameez w , K. Rawlins c , P. Redl p , R. Reimann a , E. Resconi af ,W. Rhode t , M. Ribordy z , M. Richman p , B. Riedel ac , S. Robertson b , J. P. Rodrigues ac , C. Rott al , T. Ruhe t ,B. Ruzybayev ag , D. Ryckbosch x , S. M. Saba j , H.-G. Sander ad , M. Santander ac , S. Sarkar s,ah , K. Schatto ad ,F. Scheriau t , T. Schmidt p , M. Schmitz t , S. Schoenen a , S. Sch¨oneberg j , A. Sch¨onwald as , A. Schukraft a , L. Schulte k ,O. Schulz af , D. Seckel ag , Y. Sestayo af , S. Seunarine ai , R. Shanidze as , C. Sheremata u , M. W. E. Smith ap , D. Soldin ar ,G. M. Spiczak ai , C. Spiering as , M. Stamatikos q,3 , T. Stanev ag , N. A. Stanisha ap , A. Stasik k , T. Stezelberger h ,R. G. Stokstad h , A. St¨oßl as , E. A. Strahler m , R. Str¨om aq , N. L. Strotjohann k , G. W. Sullivan p , H. Taavola aq ,I. Taboada e , A. Tamburro ag , A. Tepe ar , S. Ter-Antonyan f , G. Teˇsi´c ap , S. Tilav ag , P. A. Toale an , M. N. Tobin ac ,S. Toscano ac , M. Tselengidou v , E. Unger j , M. Usner k , S. Vallecorsa w , N. van Eijndhoven m , A. Van Overloop x ,J. van Santen ac, ∗ , M. Vehring a , M. Voge k , M. Vraeghe x , C. Walck aj , T. Waldenmaier i , M. Wallraff a , Ch. Weaver ac ,M. Wellons ac , C. Wendt ac , S. Westerhoff ac , B. Whelan b , N. Whitehorn ac, ∗ , K. Wiebe ad , C. H. Wiebusch a ,D. R. Williams an , H. Wissing p , M. Wolf aj , T. R. Wood u , K. Woschnagg g , D. L. Xu an , X. W. Xu f , J. P. Yanez as ,G. Yodh y , S. Yoshida n , P. Zarzhitsky an , J. Ziemann t , S. Zierke a , M. Zoll aj a III. Physikalisches Institut, RWTH Aachen University, D-52056 Aachen, Germany b School of Chemistry & Physics, University of Adelaide, Adelaide SA, 5005 Australia c Dept. of Physics and Astronomy, University of Alaska Anchorage, 3211 Providence Dr., Anchorage, AK 99508, USA d CTSPS, Clark-Atlanta University, Atlanta, GA 30314, USA e School of Physics and Center for Relativistic Astrophysics, Georgia Institute of Technology, Atlanta, GA 30332, USA f Dept. of Physics, Southern University, Baton Rouge, LA 70813, USA g Dept. of Physics, University of California, Berkeley, CA 94720, USA h Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA i Institut f¨ur Physik, Humboldt-Universit¨at zu Berlin, D-12489 Berlin, Germany j Fakult¨at f¨ur Physik & Astronomie, Ruhr-Universit¨at Bochum, D-44780 Bochum, Germany k Physikalisches Institut, Universit¨at Bonn, Nussallee 12, D-53115 Bonn, Germany l Universit´e Libre de Bruxelles, Science Faculty CP230, B-1050 Brussels, Belgium m Vrije Universiteit Brussel, Dienst ELEM, B-1050 Brussels, Belgium n Dept. of Physics, Chiba University, Chiba 263-8522, Japan ∗ Corresponding author Physics Department, South Dakota School of Mines and Technology, Rapid City, SD 57701, USA also Sezione INFN, Dipartimento di Fisica, I-70126, Bari, Italy NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
Preprint submitted to Journal of Instrumentation August 13, 2016 a r X i v : . [ phy s i c s . i n s - d e t ] F e b Dept. of Physics and Astronomy, University of Canterbury, Private Bag 4800, Christchurch, New Zealand p Dept. of Physics, University of Maryland, College Park, MD 20742, USA q Dept. of Physics and Center for Cosmology and Astro-Particle Physics, Ohio State University, Columbus, OH 43210, USA r Dept. of Astronomy, Ohio State University, Columbus, OH 43210, USA s Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark t Dept. of Physics, TU Dortmund University, D-44221 Dortmund, Germany u Dept. of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2E1 v Erlangen Centre for Astroparticle Physics, Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg, D-91058 Erlangen, Germany w D´epartement de physique nucl´eaire et corpusculaire, Universit´e de Gen`eve, CH-1211 Gen`eve, Switzerland x Dept. of Physics and Astronomy, University of Gent, B-9000 Gent, Belgium y Dept. of Physics and Astronomy, University of California, Irvine, CA 92697, USA z Laboratory for High Energy Physics, ´Ecole Polytechnique F´ed´erale, CH-1015 Lausanne, Switzerland aa Dept. of Physics and Astronomy, University of Kansas, Lawrence, KS 66045, USA ab Dept. of Astronomy, University of Wisconsin, Madison, WI 53706, USA ac Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center, University of Wisconsin, Madison, WI 53706, USA ad Institute of Physics, University of Mainz, Staudinger Weg 7, D-55099 Mainz, Germany ae Universit´e de Mons, 7000 Mons, Belgium af T.U. Munich, D-85748 Garching, Germany ag Bartol Research Institute and Dept. of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA ah Dept. of Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, UK ai Dept. of Physics, University of Wisconsin, River Falls, WI 54022, USA aj Oskar Klein Centre and Dept. of Physics, Stockholm University, SE-10691 Stockholm, Sweden ak Dept. of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794-3800, USA al Dept. of Physics, Sungkyunkwan University, Suwon 440-746, Korea am Dept. of Physics, University of Toronto, Toronto, Ontario, Canada, M5S 1A7 an Dept. of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA ao Dept. of Astronomy and Astrophysics, Pennsylvania State University, University Park, PA 16802, USA ap Dept. of Physics, Pennsylvania State University, University Park, PA 16802, USA aq Dept. of Physics and Astronomy, Uppsala University, Box 516, S-75120 Uppsala, Sweden ar Dept. of Physics, University of Wuppertal, D-42119 Wuppertal, Germany as DESY, D-15735 Zeuthen, Germany
Abstract
Accurate measurement of neutrino energies is essential to many of the scientific goals of large-volume neutrino telescopes.The fundamental observable in such detectors is the Cherenkov light produced by the transit through a medium ofcharged particles created in neutrino interactions. The amount of light emitted is proportional to the deposited energy,which is approximately equal to the neutrino energy for ν e and ν µ charged-current interactions and can be used to seta lower bound on neutrino energies and to measure neutrino spectra statistically in other channels. Here we describemethods and performance of reconstructing charged-particle energies and topologies from the observed Cherenkov lightyield, including techniques to measure the energies of uncontained muon tracks, achieving average uncertainties inelectromagnetic-equivalent deposited energy of ∼
15% above 10 TeV.
Keywords:
Neutrino telescopes, Energy reconstruction, Water Cherenkov
1. Introduction
The IceCube neutrino observatory [1, 2] is a cubic kilo-meter photomultiplier (PMT) array embedded in glacialice at the geographic South Pole. The complete array ismade of 5160 downward-facing Hamamatsu R7081 photo-multipliers deployed on 86 vertical strings at depths be-tween 1450 and 2450 meters in the icecap. IceCube de-tects neutrinos by observing Cherenkov light induced bycharged particles created in neutrino interactions as theytransit the ice sheet within the detector; the energy andmomentum of these charged particles reflect the energyand momentum of the original neutrino.At the TeV energies typical of such neutrino telescopes,the primary neutrino interaction channel is deep-inelasticscattering with nuclei in the detector material. In both neutral and charged-current interactions, a shower of hadronsis created at the neutrino interaction vertex. In charged-current interactions, this shower is accompanied by an out-going charged lepton. This lepton, in particular for elec-trons, may also lose energy rapidly and itself trigger an-other overlaid shower. Cherenkov light is radiated by thisprimary lepton and any accompanying showers with a to-tal amplitude proportional to the integrated path length ofcharged particles above the Cherenkov threshold. This, inturn, is proportional to the total energy of these particles[3]. The light production from electromagnetic (EM) show-ers is both maximal and has low variance with respect todeposited energy [3]. As such, it forms a natural unit of re-constructed shower energy. This electromagnetic-equivalent2 ν e energy [GeV]10 E n e r g y i n c h a r g e d p a r t i c l e s [ G e V ] Charged current 10 ν e energy [GeV]Neutral current10 -3 -2 -1 Probability density [arb. units]
Figure 1: Energy deposited in Cherenkov-radiating parti-cles by deep-inelastic ν e -nucleon scatterings in ice [5]. Incharged-current interactions within the detector this de-posited energy is very nearly equal to the neutrino en-ergy. The larger spread to lower deposited energies in theright panel is due to neutral-current scattering; the rateof neutral-current interactions is approximately 3 timessmaller than that of charged-current interactions in theenergy region of interest [6].energy, in conjunction with the energies of detected out-going leptons, can then be used to infer the energy of theoriginal neutrino. In identifiable charged-current interac-tions (e.g. ν µ CC events), neutrino energy resolution forevents where the interaction vertex is observed is in princi-ple limited only by detector resolution. For neutral-currentevents, neutrino energy spectra can be inferred statisti-cally. A similar method can be used to estimate energyspectra for events where not all charged particles are con-tained within the detector [4], such as muons produced incharged-current ν µ interactions an unknown distance out-side the detector.Electromagnetic showers are produced in ν e charged-current interactions from the outgoing electron, in τ de-cays, and along muon tracks from muon bremsstrahlungand pair production interactions. At high energies ( (cid:38) ν e interac-tions (Table 1, Fig. 1), but the distribution is quite broadfor neutral-current interactions, in which the outgoing neu-trino carries a large and highly variable fraction of theenergy out of the detector.IceCube events have two basic topologies: tracks andcascades (Table 1, Fig. 2). Tracks are made predomi-nantly by muons, either from cosmic-ray air showers or ν µ charged-current interactions. Cascades are those eventswithout visible muon tracks and are formed by particleshowers near the neutrino vertex. These are produced in ν e charged-current and all-flavor neutral-current interac-tions. The particle showers in cascade events have typicallengths of 10 m (Fig. 3), which are not in general resolvablewith the 17 m vertical inter-PMT spacing and 125 meterhorizontal inter-string spacing of the IceCube array. Asa result, it is not possible to separate ν e charged-currentinteractions from neutral-current interactions.For both track and cascade events, we discuss the gen-eral approach to energy reconstruction and provide exam-ples of the most commonly used algorithms. All energyreconstruction methods described here are based on thelinearity of light yield with energy loss and use the com-mon likelihood model described below. Performance dataprovided are meant to characterize the behavior of theenergy reconstruction only and, except when noted other-wise, are given assuming that the topology of the events,in particular direction and position, are known exactly.This controls for uncertainties induced by positional anddirectional reconstructions and shows the intrinsic uncer-tainties of the reconstruction being discussed. Althoughthe resolutions shown here are typical of energy resolu-tions in IceCube physics analyses, some variation shouldbe expected due to uncertainties from topology reconstruc-tions, which will tend to worsen the resolution, as well asfrom selection of well-reconstructed events in the analyses,which tends to improve it. To show typical performancein physics analyses, we include the final-level resolutionsof the algorithm being discussed for a recent example Ice-Cube analysis at the end of each section.In addition to performance of energy reconstruction insimulation, we also discuss relevant calibration issues indata. Accurate measurement of energies requires correctinference of incident Cherenkov photon fluxes from the dig-itized photomultiplier signals. We demonstrate this hereusing verification of the PMT anode current reconstructionand single-photoelectron calibration (Sec. 4), the PMTquantum efficiency, optical transmissivity of the DOM,3nteraction Signature E vis /E ν ; E ν = 1 TeV E ν = 10 TeV E ν = 100 TeV ν e + N → e + had. Cascade 94% 95% 97% ν µ + N → µ + had. Track (+ Cascade) 94% 95% 97% ν τ + N → τ + had. → had. Cascade/Double Bang < < < ν τ + N → τ + had. → µ + had. Cascade + Track < < < ν l + N → ν l + had. Cascade 33% 30% 23%Table 1: Neutrino interactions with nucleons in IceCube. E vis denotes the median fraction of the neutrino energydeposited in any present primary lepton and in the EM-equivalent energy of a hadronic cascade at the vertex. Incharged-current interactions (top section of table), nearly all the energy of the interacting neutrino is deposited in suchlight-producing particles. In neutral-current interactions (bottom), a large fraction of the neutrino energy leaves withthe outgoing neutrino (Fig. 1) [5]. Note that some of E vis may escape the detector: muon tracks at these energieshave lengths of multiple kilometers, and τ leptons will decay before ranging out, depositing only a fraction of E vis inthe detector. Events in IceCube are observed as a combination of cascades (near-pointlike particle showers) and longtracks, as are left predominantly by muons. “Double Bang” refers to two cascades joined by a short track, a signature ofcharged-current ν τ interactions at high energies ( (cid:38) τ are resolvable in IceCube. Due to the long lengths of muon tracks above 1 TeV, most observed neutrino-induced muonshave production vertices outside the detector and the initial hadronic cascade is not observed. (a) A muon that started in the detector and deposited 74 TeVbefore escaping, carrying away its remaining energy. (b) A cascade that deposited 1070 TeV in the detector. Its en-ergy can be determined directly since the cascade is completelycontained in the instrumented volume. Figure 2: Examples of neutrino event topologies in IceCube from [12]. Each panel is a schematic view of the detector,with each photomultiplier represented by a sphere whose volume is proportional to the collected charge. The smallerupper panels show projections of the detector along its z , x , and y axes, respectively.4 C h e r e n k o v p h o t o n s p e r . m
100 TeV10 TeV1 TeV 100 GeV10 GeV1 GeV Shower energy [GeV]100000150000200000250000300000 P h o t o n s p e r G e V Figure 3: Longitudinal distributions of electromagneticcascades in ice simulated with GEANT4 [13]. The to-tal light output of the cascade is directly proportional toits energy, while the length of the shower increases onlylogarithmically with energy. The angular profile of lightemission is nearly identical across the entire energy range[3]. The range in positions of the shower maximum (be-tween 1 and 5 meters from the vertex here) is small enoughthat the showers can be considered pointlike on the scaleof IceCube instrumentation. Hadronic showers have nearlyidentical properties when viewed with IceCube [10]. and overall energy scale using low-energy muons (Sec. 5.1),and linearity of response over a wide range of photon fluxesusing a dedicated calibration laser (Sec. 5.2). These estab-lish the validity of the models used in the reconstructionsand complement previous calibration measurements of theIceCube instrumentation [1, 14].
2. Likelihood Model
The near-constant light emission profile of both elec-tromagnetic and hadronic showers and the linear scaling oflight output with energy allow the use of such showers asfundamental units of energy reconstruction by scaling theexpected light output of a simulated event (a “template”)to match observed data. We then estimate a shower’s en-ergy deposition E by comparing the observed number ofphotons in a PMT k to the expectation Λ for a templateevent with some reference energy (usually 1 GeV) [4]. Thetemplate functions are typically evaluated from tabulated[15] Monte Carlo simulation [16] of light propagation inthe ice sheet [17, 18], although limited-accuracy analyticapproximations (Sec. 3.1) or direct real-time Monte Carlosimulation can also be used. These template functionstake into account the expected detector response as wellas position-dependent light propagation properties due towind-deposited particulate layers deep in the glacier [18]and will be described in more detail in the following sec-tion.The number of detected photons is expected to followa Poisson distribution with mean λ = Λ E . Then the like-lihood L for an energy E resulting in k detected photonsfrom an event producing Λ photons per unit energy canbe evaluated as follows: L = λ k k ! · e − λ λ → E Λ= ( E Λ) k k ! · e − E Λ ln L = k ln ( E Λ) − E Λ − ln ( k !) . (1)Maximizing this with respect to energy, and adding thecontributions from all DOMs (digital optical modules):0 = ∂ (cid:80) ln L ∂E = (cid:80) DOMs j ( k j Λ j /E Λ j − Λ j )= (cid:80) k j /E − (cid:80) Λ j ∴ E = (cid:80) k j / (cid:80) Λ j . (2)The generalization allowing additional contributions (e.g.PMT noise) is to replace the substitution λ = E Λ in Eq. 1by λ = E Λ + ρ , where ρ is the expected number of noisephotons. The likelihood (Eq. 1) then becomes:ln L = k ln ( E Λ + ρ ) − ( E Λ + ρ ) − ln ( k !) . (3)Maximizing with respect to E , as in Eq. 2:0 = (cid:80) ( k j Λ j / ( E Λ j + ρ j ) − Λ j ) (cid:80) Λ j = (cid:80) k j Λ j / ( E Λ j + ρ j ) . (4)5nlike Eq. 2, this does not have a closed form solutionfor E since Λ no longer cancels in the first term and E can therefore not be factored out. Solutions can, however,be easily obtained using gradient-descent numerical mini-mization algorithms.Timing can also be included in this formulation by di-viding the photon time arrival distributions (see Sec. 4)for bright events into multiple time bins and interpreting k and Λ as the light per time bin instead of per PMT. Theformulation of the Eq. 3 is identical under this change.Use of detailed timing provides small increases in energyreconstruction performance in the single-source case. It isprimarily only important in more complex situations suchas using unfolding to estimate energies of multiple simul-taneously emitting light sources (Sec. 8.2). Timing is alsoused when only parts of the observed charge distributionare usable due to, for example, saturation in the photomul-tipliers and digitizers since it allows the saturated portionsof the readout to be masked out during the fit.
3. Methods to Compute Light Yields
The ability to reconstruct E relies on correct compu-tation of the light-yield scaling function Λ. This functiondepends on the positions of the observing photomultiplier( (cid:126)x p ), the position of the event vertex ( (cid:126)x ν ), the orientationof the event ( θ, φ ), and, when using timing information, thetime the particle was at (cid:126)x ν and the time of observation.The typical observation distance a few scattering lengthsaway from the source, the complex wavelength dependenceof light propagation, and the inhomogeneous optical prop-erties of the ice [17, 18] make a precise analytic form forΛ impossible. For applications requiring speed more thanaccuracy, an approximate form for the light yield can bederived. Final reconstructions depend on tabulated re-sults of Monte Carlo simulation of in-ice light propagation[16] smoothed with a multi-dimensional spline surface [15].Direct use of Monte Carlo, without pre-tabulation, is alsopossible but computationally prohibitive in almost all ap-plications. An approximation for point-like spherically-symmetricemission (an approximation to an electromagnetic shower)or uniform emission at the Cherenkov angle along an infi-nite track (an approximation to a minimum-ionizing muon)is possible when computational speed is essential. In thelimit of little scattering, photons propagate in straightlines away from the source and photon density decreasesexponentially with distance due to absorption. Near thesource this results in a 1 /r dependence for a point sourceand 1 /r for a cylindrical source like a muon ( r being thedistance of closest approach). At larger distances, prop-agation of photons enters a diffusive regime in which themotion of the photons can be approximated with a randomwalk, changing the photon density to exp ( − r/λ p ) /r for a distance to source [ m ] f l u x f r o m one pho t on [ m - ] homogeneous ice -9 -8 -7 -6 -5 -4 -3 -2 -1 -4 -3 -2 Figure 4: Approximations to the observed light level asa function of distance for point light sources. Lines showEq. 5. Results from a Monte Carlo simulation of photontracking are shown with black points. The insert at thetop right shows small distances. These plots are made as-suming homogeneous optical properties of the ice, whereasthe actual glacial ice has depth-varying properties [18].point source and exp ( − r/λ p ) / √ r for a cylindrical source.The characteristic “propagation” length ( λ p ) is defined viathe absorption and effective scattering lengths, λ a and λ e : λ p = (cid:112) λ a λ e / µ ( r ) = n A · π e − r/λ p λ c r tanh( r/λ c ) , where λ c = λ e ζ , ζ = e − λ e /λ a (point source), µ ( r ) = l A · π sin θ c e − r/λ p √ λ µ r tanh √ r/λ µ , where (cid:112) λ µ = λ c sin θ c (cid:113) πλ p (track). (5)In these expressions, n is the number of photons emit-ted by a point source and l the number of photons permeter from the uniform track source. The quantity A is theeffective photon collection area of the receiving sensor and θ c is the Cherenkov angle. We have verified these formu-lae with Monte Carlo photon tracking [19]. For the typicalvalues of λ a =98 m, λ e =24 m, the description can be fur-ther improved by using a fitted value of λ ∗ p = λ p / .
07, anda corresponding increase in normalization by 26% (Fig.4). Since the optical properties of the ice vary with depth,the values of 1 /λ a , 1 /λ e are taken as averages of the localvalues between the emitter and receiver.In practice, considerable errors are introduced by theanalytic expressions above due to various effects that theyignore: for example, the directionality of the cascade, which6s not really an isotropic source of photons. These approx-imations result in poor performance when using Eq. 1,which assumes a high-quality representation of the ex-pected light output and the dominance of statistical uncer-tainties. When using these analytic approximations (e.g.for reasons of computational speed), a variant on Eq. 1 isrequired with wider tails to cover the approximation un-certainty. We incorporate this by convolving Eq. 1 with aprobability distribution G on the mean light-yield λ , whichwas chosen empirically to be: G µ ( x ) = const. x · (cid:0) e − wy + ( y/σ ) (cid:1) − , with y = ln xµ . (6)The parameter w is a “skewness” parameter, which allowsfor larger over-fluctuations (e.g. in case of a large brems-strahlung loss along a muon track—see Sec. 8). A higher-quality but slower parametrization of Λ isobtained using a multi-dimensional spline surface [15] fitto the results of Monte Carlo simulations of many differ-ent source configurations. This provides a high-qualityparametrization of light propagation and is, in general,used for all final analysis results.For an approximately point-like source like an electro-magnetic (EM) shower, Λ depends on nine parameters.For the case of IceCube, however, symmetries of the de-tector, in particular the approximate azimuthal and lat-eral translational symmetry of light propagation, allow theparametrization of Λ in terms of six: the depth and zenithangle of the source, the displacement vector connectingit to the receiver, and the difference between the time oflight detection and production. Fig. 5 shows the parame-terization evaluated at a single depth and zenith angle forvarious receiver displacements.The resulting tables are approximately 1 GB in sizeand take on order 1 µ s to evaluate for a particular source-receiver configuration (much longer than the approxima-tion described above), but are sufficiently accurate thatEq. 1 can be used directly with full knowledge of lightpropagation in ice. Using splines also allows analytic eval-uation of likelihood gradients, useful when fitting for ge-ometric parameters (Sec. 6), as well as the possibility ofconvolving the timing distributions with additional effectssuch as the PMT transit time spread. The same approach,albeit with a different parametrization, can also be usedto tabulate light yield from sources with other geometries,like minimum ionizing muons.
4. Waveform Unfolding
IceCube uses waveform-recording digitizers to collectdata from photomultiplier tubes (PMTs) [1]. For thesedata to be used in Eq. 1, these waveforms must be trans-formed to a reconstructed number of photons per unit time. The PMT output is AC-coupled by a toroid trans-former through a set of pulse-shaping amplifiers to fourdigitizers: three high-rate short-duration custom modules(Analog Transient Waveform Digitizers or ATWDs) record-ing the first 420 nanoseconds of the waveform at threegains with a typical sampling period of 3.3 ns and a con-tinuous pipelined digitizer (the fast ADC or fADC) witha 25 ns sampling period. The shape applied by the ampli-fiers is much wider than the intrinsic width of the PMTpulse. The resulting recorded waveforms (Fig. 6) are thena linear combination of the characteristic shaping func-tions of the amplifiers with timing and amplitude relatedto the charge collected at the PMT anode. The unit usedfor this collected charge, the photoelectron (PE), is definedas the most likely deposited charge from a single photon, ∼ . Note that, due to the shape of the PMT charge re-sponse function [14] and our discriminator settings, themean deposited charge of triggering photons is 14% lowerthan the most likely value.A non-negative linear simultaneous unfolding of all dig-itizers (in this case, using the Lawson-Hanson NNLS algo-rithm [20]) can then be applied using the shaping functionsas a basis to recover the collected PMT charge as a func-tion of time (Fig. 6). Charge resolution obtained by thismethod, which is dominated by the width of the charge re-sponse of the Hamamatsu R7081 photomultiplier [14, 21],is typically around 30% at the single photon level (Fig.7). As photon statistics accumulate (Sec. 5.2), this un-certainty is reduced by averaging over many electron cas-cades, and PMT charge counting is dominated by purelyPoissonian effects, as in Eq. 3. For our purposes, it is suf-ficient to approximate the charge resolution by allowingthe photon count ( k ) in Eq. 1 to take non-integer values,replacing the k ! normalization term with Γ( k + 1). At lowamplitudes, where only a few electron cascades contribute,the contribution of the single photoelectron (SPE) widthis maximal (30%), but the Poisson uncertainties on photoncollection remain larger. As a result, our pure-Poissonianapproximation can be used for the distribution of collectedcharges at all amplitudes (Fig. 8).Relative timing from the electronics and unfolding istypically accurate to around 1 ns for consecutive non-overlapping pulses on a single DOM while the ATWD isactive. This is comparable to the transit time spread of thePMT and degrades to 8 ns when only the fADC data areavailable. Pulse timing in complicated waveforms can besubstantially more uncertain (up to 10 ns for the ATWD),but still correctly reproduces the timing distribution ofphoton cascades at the PMT anode even at very high to-tal amplitudes (Sec. 5.2). This allows the use of a Poissonlikelihood (Eq. 1) for particle reconstruction in events atall energies.7
20 40 60 80 100 120 140 160 180Observation angle [deg]10 -4 -3 -2 -1 E x p e c t e d c h a r g e [ P E / G e V ] Observation distance
20 m50 m 100 m150 m (a) Total observed light level as a function of radial distancefrom the source and observation angle with respect to thesource direction. While scattering in the ice washes out thepeak at the Cherenkov angle, the direction of the source re-mains visible as an asymmetry even at large distances. T i m e d e l a y p r o b a b ili t y [ / n s ]
50 m 100 m 150 mObservation angle
40 deg130 deg (b) Normalized time distribution of detected photons at differ-ent distances for two observation angles. Photons detected atthe Cherenkov angle have generally experienced the least scat-tering, and so are detected earlier and more closely bunched intime than those detected at other angles.
Figure 5: Distribution of detectable photons obtained from a Monte Carlo simulation of a horizontal, 1 GeV electro-magnetic cascade in the upper part of the IceCube detector. Both the number and time distribution of photons dependon the direction of the cascade, here oriented in the direction of observation angle 0. The distributions shown are madefrom spline tables (Sec. 3.2); directionality is neglected when using the first-guess approximation of Sec. 3.1. -6-5-4-3-2-1 0 1 11600 11700 11800 11900 12000 12100 0 0.2 0.4 0.6 0.8 1 1.2 A DC V o l t age ( m V ) P ho t oe l e c t r on s Time (ns)1.02 PhotoelectronsReconstructed PhotoelectronsATWD 0Best-fit ATWD 0fADCBest-fit fADC (a) Unfolding of a simple waveform containing one detectedphoton, showing good agreement between the best-fit recon-struction and the data in all active digitizers. Both the totalreconstructed amplitude (top label) and number of pulses (redline) agree with the single photon interpretation of these data. -8-7-6-5-4-3-2-1 0 1 11400 11600 11800 12000 12200 12400 0 0.2 0.4 0.6 0.8 1 1.2 1.4 A DC V o l t age ( m V ) P ho t oe l e c t r on s Time (ns)4.03 Photoelectrons (b) Unfolding a more complicated waveform, illustrating thesmooth transition between digitizers and handling of pileup.The far-right pulse was recorded after the ATWD buffer wasfilled and is reconstructed from fADC data only. The heightsof the first two pulses show the PMT amplification variance.
Figure 6: Examples of waveform unfolding in data from the IceCube detector for both simple and complex waveforms.The lines marked
Best-fit are predictions of the various digitizer read-outs given the reconstructed PMT hits. For aperfect reconstruction, and with no noise in the data, these lines would exactly match within the digitizer step (typically0.15 mV in the highest-gain ATWD and 0.1 mV in the fADC). The vertical lines with crosses at the top represent thetimes and amplitudes of the unfolded pulses relative to the right-hand axis.8
0 0.5 1 1.5 2 2.5 D en s i t y ( A U ) Photoelectrons SLCHLC
Figure 7: The charge distribution of photomultiplier read-outs (mostly single photoelectrons) measured in the Ice-Cube detector. The width of this distribution arises fromstochasticity in the electron cascade process in the PMT[14]. The 0.2 PE trigger level is clearly visible on the left.The two readout modes shown (HLC and SLC, for “hard”and “soft” local coincidence [1]) are based on an in-moduletrigger decision; HLC hits are more likely to be physicsevents than noise, and so have longer readouts with moredetailed waveforms. The charge resolution of both read-out modes is equivalent for isolated single-photoelectronpulses; the high-charge tail in the HLC charge distributionis caused by the higher photon densities typically neededto satisfy the HLC trigger condition.
5. Energy Scale Calibration
The energy reconstruction capabilities presented hererequire linear behavior of the photomultiplier tubes andreadout electronics over several orders of magnitude in col-lected charge. Precise understanding of light propagationin the ice with horizontal instrumentation spacing similarto the average absorption length of light ( ∼
125 m) is alsorequired, as are verification of purely Poissonian fluctua-tions in collected charge as in Eq. 1 and calibration of theabsolute energy scale of the detector. Due to the natureof the IceCube detector as a naturally occurring volumeof ice, the verification of these properties of the detectormust be conducted in situ.Calibration of the IceCube detector relies on built-in reference electronics, calibration LEDs integrated intoeach DOM, in-ice calibration lasers, observations of dustconcentrations and optical properties in the side walls ofice boreholes taken during IceCube construction [22], aswell as observed physics data. The modeling of light prop-agation in ice [17, 18] is conducted using the LED and side-wall dust observations. Final energy scale calibration usesminimum-ionizing muons as a standard brightness can-dle in order to probe the detector volume with Cherenkovlight. Calibration laser data is then used to establish thelinearity of the detector’s light response and thereby theenergy ladder used to reconstruct high-energy particles.
Minimum-ionizing muons, created in atmospheric cosmic-ray showers, are well-suited for energy scale calibration be-cause they have constant known light emission, are abun-dant, and leave well-defined tracks in the detector. Ob-taining a large sample of ∼
100 GeV single muons, theirpositions and directions are reconstructed to high preci-sion. The observed PMT charges are then compared toexpected values, similar to the energy reconstruction pro-cedure in Eq. 1.To isolate a sample of events we first make cuts ontrack quality parameters, including the number of on-timePMT hits and the fit quality of the track reconstruction.We then select low-energy single muons by searching fortracks that deposit little light in the outer strings andappear to stop in the detector fiducial volume. Finally,we require the tracks to be inclined 45 ◦ -70 ◦ with respectto the straight downgoing direction to ensure the muon’sCherenkov cone is incident on the active side of our PMTs,which face down towards the bottom of the glacier. Thesecuts provide a sample of 70,000 events in 30 days of datataken with IceCube in its 79-string configuration. MonteCarlo studies of the detector response to cosmic ray airshowers [23] show this sample consists of >
95% singlemuons with a median energy of 82 GeV at the detec-tor center. Distributions of event observables show goodagreement between simulation and experimental data.To calibrate the energy scale, we focus on a subset ofIceCube DOMs in the deep part of the detector where9 n, q ( PE ) P r o b a b ili t y d e n s i t y λ =2 . P ( n | λ ) P ( q | λ )˜ P ( q | ¯ q λ ) n, q ( PE ) P r o b a b ili t y d e n s i t y λ =10 .
70 80 90 100 110 120 130 n, q ( PE ) P r o b a b ili t y d e n s i t y λ =100 . Figure 8: An approximation to the distribution of charge responses given a mean number of detected photons ( λ ) at thePMT photocathode. The stepped line shows the Poisson probability of ejecting n electrons from the photocathode, given λ . The solid line shows the numerical convolution of this Poisson distribution with the charge response function (Fig. 7),yielding the given number of photoelectrons (PE) at the anode. The dashed line shows the analytic approximation weuse to this convolution, using the extension of the Poisson probability distribution to real numbers q . The shift in thepeak centers between the stepped and dashed/solid lines (between n and q ) is related to our choice of the definition of λ with respect to the PMT quantum efficiency and the shape of the charge response functions of the PMTs [14].the glacial ice is exceptionally clear (absorption lengths of ∼
200 m [17, 18]). For each DOM in this region, we firstreconstruct the muon track while excluding all informa-tion from the DOM in question. Simulation studies showthis procedure successfully reconstructs the muon direc-tion and the track-DOM distance within ∼ ◦ and ∼
10 mof the true direction and position. After binning the ob-served charge based on the track-DOM distance, we find a (cid:46)
5% average excess of charge in data compared to nom-inal values (Fig. 9) with up to 9% deviations at certaindistances.
The IceCube detector includes two 337-nm pulsed ni-trogen lasers [24] that are operated through adjustable op-tical attenuators to produce pulses of light that are iden-tical except for the number of emitted photons and cor-respond approximately to the light output of electromag-netic showers in the 1-100 PeV range. This strictly linearbehavior can be used to verify the linearity of the DOMelectronics and the photon-counting procedure (Sec. 4), ex-tending the energy scale established at low photon countswith minimum-ionizing muons to the regime where theDOM collects thousands of photons.If the DOM response is linear then the time distribu-tions of PMT charge in response to laser pulses with dif-ferent attenuation settings must be scaled copies of eachother (Fig. 10). Further, the total charge collected over agiven time window must be proportional to the intensity ofthe pulse. Since the pulses trigger DOMs both close to thelaser and those hundreds of meters away, the linearity ofthe DOM response both in charge and timing structure can be established from fractions of to many thousands of PEper recorded waveform (Fig. 12). Any non-linearity in ei-ther the hardware or photon reconstruction would appearin two possible ways: as a distortion of the total charge vs.attenuator setting (see Fig. 12), and as distortions in theshapes of the waveforms at different amplitudes as a resultof different instantaneous photon amplitudes at differentpoints in the waveform.In addition to the linearity of the waveforms, and theability to interpret waveforms containing many photoelec-trons statistically as in Eq. 1, the repeatability of the cal-ibration laser pulses allows tests of the statistical uncer-tainty in light collection. At levels above the 0.2 pho-toelectron discriminator [1], the distribution of measuredcharges in the DOMs is well-described by a Poisson dis-tribution (Fig. 11), verifying the likelihood model used inthe reconstructions (Eq. 1).
6. Cascades
Reconstruction of the energies of the isolated cascadesproduced in ν e and neutral-current events is the simplestscenario. As the light deposition pattern for such events isindependent of energy on the scale of IceCube instrumen-tation (Fig. 3), the likelihood model and scaling formulae(Eq. 1) are obeyed exactly for cascade events. Thus, theenergy of a cascade with a known orientation and vertexcan be recovered using a template cascade by numericalmaximization of Eq. 3. The likelihood L ( E ) (Eq. 3) isstrictly concave, with a single maximum, and can be opti-mized easily with any standard algorithm; here we use the10 .00.51.01.52.02.53.0 A v e r a g e C h a r g e ( P E ) DataSimulation S c a l e d A b s o l u t e C h a r g e Std. DOM Efficiency+10% DOM Efficiency-10% DOM EfficiencyData
Figure 9: Absolute charge measurements with minimum-ionizing muons. Top: average observed charge vs. dis-tance from the DOM to the reconstructed muon track,shown for both data and simulation. Due to the selectionof minimum-ionizing, single muons, the observations aredominated by single photoelectrons. Bottom: the averagecharge vs. track-DOM distance normalized to the chargeexpected from standard IceCube simulation. Muon dataand simulation are shown. As a proxy for an altered en-ergy scale, simulations with altered DOM efficiencies areshown for comparison. The observed charge is slightlyhigher than nominal values but below the charge expectedfor a DOM efficiency increased by 10% (upper band). Er-ror ranges reflect statistical uncertainties on the shownsample only without including statistical uncertainties inthe simulation dataset to which they were normalized. -3 -2 -1 P h o t o e l e c t r o n s / n s P r o b . d e n s i t y [ a r b . un i t s ] Mean total charge [PE]278.499.669.7 21.1 7.0 3.4
Figure 10: Top: charge collected at a single DOM as afunction of time in response to calibration laser flasheswith 6 different transmittance settings at a distance of246 m from the laser. Bottom: the same distributionsas the top, but normalized to each transmittance setting.The charge distributions are scaled copies of each other,whether derived from isolated pulses (lowest dashed line)or high charge bunches (highest solid line).11 E v e n t s p e r P E P ( q | λ =8 .
200 220 240 260 280 300 320 340Collected charge [PE]0102030405060 E v e n t s p e r P E P ( q | λ =270 . Figure 11: Total charge collected in response to 1882 cal-ibration laser flashes at maximum transmittance in twoDOMs at distances of 297 m (top panel) and 194 m (bot-tom panel). In both cases, the fluctuations in collectedcharge follow the predicted distribution from Eq. 1.Non-monotonic Maximum Likelihood algorithm (NMML)[25] as in Sec. 8.2.The deposited energy from such neutrino interactions isnearly identical to the neutrino energy for charged-current ν e interactions (Fig. 1). For neutral-current interactions(all flavors), much larger variations are possible due to theunseen outgoing neutrino and the reconstructed depositedenergy is a lower limit on the neutrino energy. This is trueboth because of the potentially large amount of missingenergy in the outgoing neutrino in neutral current inter-actions and due to the ∼
15% lower light yields typical ofhadronic showers [10]. IceCube is not capable of resolvingthe differences between these event types and all quotedcascade energies, which are given as the reconstructed de-posited energy using an EM shower template, are thereforelower limits on the energies of the neutrinos that producedthem.The deposited energy resolution for contained ν e eventsis dominated by statistical fluctuations in the collectedcharge at low energies, improving from 30% at 100 GeVto 8% at 100 TeV, at which point the extension of theshower begins to distort the reconstruction (see Fig. 13).This high-energy limit on resolution is similar to uncertain-ties in the modeling of scattering and absorption in the icesheet [17, 18], which contribute a 10% systematic uncer-tainty to the energy. Since we do not perform analysis-level event selection in this article, events shown in Fig. 13are distributed across the entire detector. Below 100 GeV,events outside the densely-instrumented DeepCore subar-ray [26] typically only deposit a few photons, resulting inthe figure reflecting generic IceCube performance ratherthan that typical for a DeepCore-specific low energy anal-ysis that would exclude these few-photon events. -2 -1 Filter transmittance10 -1 T o t a l c o ll e c t e d c h a r g e [ P E ] (a) Charge collected in ten typical DOMs as a function of filtertransmittance. The observed PMT charge shown on the ver-tical axis is proportional to the number of collected photons,which is in turn proportional to the filter transmittance shownon the horizontal axis. Differences between DOMs are due todifferent distances from the laser calibration source. N o . D O M s [ a r b . s c a l e ] τ =0 . σ =4 . τ =0 . σ =3 . τ =0 . σ =2 .
10 5 0 5 10 N o . D O M s [ a r b . s c a l e ] τ =0 . σ =1 .
10 5 0 5 10Deviation from linearity [%] τ =0 . σ =1 .
10 5 0 5 10 τ =1 . σ =0 . (b) Fractional deviations of observed light amplitudes from pre-dictions, assuming a linear scaling, in 276 DOMs that trig-gered in response to calibration laser pulses at six differentfilter transmittance settings ( τ in each panel). Each entry ineach histogram is the proportional deviation of the collectedcharge from a best-fit line like those shown in (a). The linearfit effectively pivots around the highest transmittance setting(lower right panel). The largest deviations are at the lowesttransmittance setting (upper left panel), and are on the scaleof 5%. Figure 12: Linearity measurements using the calibrationlaser source. The PMT, digitization electronics, and pho-ton reconstruction procedure (discussed in Section 4) re-spond linearly to photon fluxes spanning 4 orders of mag-nitude.12 R e c o n s t r u c t e d e n e r g y [ G e V ] Mean deposited energy [GeV]010203040 σ [ % ] ν e -inducedPurely electromagneticElectromagnetic (no extension) -3 -2 -1 Probability density [arb. units] (a) Reconstructed energy deposition of cascade events as afunction of true energy deposition. Reconstructed energy [GeV]0.000.050.100.150.200.25 P r o b . d e n s . (b) Slices of the reconstructed-energy distribution shown in (a)at fixed true total energy depositions of 10 , 10 , 10 , 10 , and10 GeV.
Figure 13: Performance of cascade energy reconstructionon simulated ν e events with known vertices and directionsthroughout the IceCube array. The horizontal axis in (a)shows the mean electromagnetic-equivalent deposited en-ergy (correcting any hadronic component of the showersto have the energy of an EM shower with the same lightyield). The inherent variance in the energy fraction in thehadronic energy in the neutrino interaction contributes tothe reconstruction uncertainty shown in the lower panel of(a) (solid line) relative to purely electromagnetic showers(dashed line). Fluctuations in the light yield and chargedparticle content of hadronic showers also increase the un-certainty in the energy measurement at all energies (differ-ence between dashed and solid line). Above 100 TeV, theresolution of the single-cascade template method is limitedby the unmodeled longitudinal extension of the showers(dashed vs. dotted lines). The light-yield templates (Λ) in Eq. 1 are independentof energy but are functions of the event topology, in par-ticular the neutrino interaction vertex position and showerorientation (Fig. 5). Simultaneous maximization of thelikelihood in E , direction, and vertex position, throughthe influence of the last two on Λ, allows reconstructionof these parameters as well. In this full-reconstructioncase, energy resolution is very similar (see Fig. 14) and asystematics-dominated angular resolution on the order of15 ◦ is achieved for energies of (cid:38)
100 TeV.Computational performance of cascade reconstructionis greatly enhanced by using a standard numerical mini-mizer for the topological parameters and then employinga second internal minimization algorithm to solve for thebest-fit value of E at each iteration. This exploits therelatively long time required to evaluate Λ relative to themultiplication Λ E as well as the fact that the sub-problemof energy reconstruction is nearly linear. Since solving for E , given Λ, requires less CPU time than evaluating Λ,which is constant with energy, this procedure reduces by 1the effective dimensionality of the problem. Methods andperformance for angular and positional reconstruction ofEM and hadronic showers will be addressed in more detailin a future publication.
7. Tracks: Muons and Taus
Measurement of the energies of particles producing through-going tracks is more complicated than in the case of cas-cades. At low energies ( (cid:46)
100 GeV), the range of muonsin ice is short enough that all muon energy is deposited inthe detector and a calorimetric approach can be taken. Atthe higher energies on which this paper is focused, muonstypically have a range longer than the length of the de-tector. This presents two immediate complications. First,the muon’s point of origin (the neutrino vertex for muonsproduced in ν µ interactions) is unknown for events start-ing outside the detector and so a measurement of muonenergy at the detector can provide only a lower bound onthe muon’s energy at production, the quantity of interestfor reconstructing muon neutrino energies. Second, themuon energy must be estimated only from the propertiesof the light emitted by the muon during the portion of itstrack in the detector, in particular the differential energyloss rate ( dE/dx ).Above the minimum-ionizing regime ( (cid:38) CTruth.energy/I3Units.GeV10 R e c o n s t r u c t e d e n e r g y [ G e V ] Mean deposited energy [GeV]010203040 σ [ % ] -3 -2 -1 Probability density [arb. units]10 Mean deposited energy [GeV]0102030405060 A n g u l a r e rr o r [ d e g r ee s ] Quantile
Figure 14: Reconstruction performance for ν e events in de-posited energy (top) and direction (bottom) at final levelfor a recent IceCube analysis searching for astrophysicalneutrino events at typical energies of 100 TeV [12]. En-ergy resolution includes the effects of the event selectionin the analysis as well as the effects of uncertainty inducedby the vertex and angular reconstruction (bottom). Theresults are generally similar to those shown over the equiv-alent energies in Fig. 13, although with a larger populationof outliers due to limited vertex resolution. Event selectioneffects, along with reduced photon statistics, worsen angu-lar resolution at the left near the lower energy thresholdof the analysis. deviate substantially from the ideal case of constant av-erage energy loss. Muon neutrinos that undergo charged-current interactions inside the instrumented volume haveno energy losses before the interaction vertex, followed bya hadronic cascade at the interaction vertex and stochasticlosses along the outgoing muon track. Charged-current ν τ interactions above ∼ τ itself, in turn followed by a cascade or muonfrom the τ decay.
8. Muon Energy Loss Reconstruction
The simplest approach to reconstruction of muon en-ergy losses is analogous to reconstruction of cascade ener-gies: finding the best-fit scaling of a template muon to theobserved light deposition. The significant event-to-eventvariation in muon topologies from large stochastic lossesalong the track causes bias and poor resolution (Fig. 15a,[28]) when this approach is taken. Avoiding this problemrequires a segmented reconstruction that can measure thevariations of energy loss along the track.
One approach to track segmentation is to assign eachphotomultiplier to one of several segments and then use thetemplate-muon method (Eq. 1) in each segment [28, 29].This results in several averages taken over sections of thetrack, allowing the identification of outliers that distortthe global average. Typically, these segments are ∼
100 min length, similar to the inter-string spacing in the IceCubearray. When reduced to differentially small segments, eachcontains only one PMT; as they reach the size of the detec-tor the result converges to that from the single-templatemethod. This approach is described in more detail in [28].An overview is given here.When using spatial separation of the light deposition,the track is divided into bins bordered by planes perpen-dicular to the track, effectively treating each bin as a sep-arate detector (Fig. 16). The light from each PMT inthe bin is treated in isolation from the remainder of thedetector, and the energy of the track within the bin is de-termined using a single template muon following the sameapproach as for an unsegmented muon reconstruction.Many factors influence the choice of bin size, with theobjective to create bins as independent from each other aspossible. IceCube uses a sparse grid of photomultiplierswith sizable distances between PMTs, and each typicallyobserves light from only a limited part of the muon track.The length of the optimal segmentation depends on theoptical properties of the medium and expected correlationscale of the observed light. This is related to the expectedenergy of the muons: high-energy muons create more lightthat travels further through the ice, increasing the scaleof correlations between segments. The typical absorption14 R e c o n s t r u c t e d e n e r g y l o ss [ G e V ] Single template10 Unfolding10 Muon energy loss [GeV]0.00.51.01.5 ∆ l og E ( % C . L . ) Single templateUnfolding -3 -2 -1 Probability density [arb. units] (a) Total energy loss reconstructed by the single-templatemethod analogous to the cascade reconstruction method ofSec. 6 (top panel) and by the unfolding method of Sec. 8.2(middle panel). The bottom panel shows, for each energy-lossbin, the 1 σ range of energy losses reconstructed for events inthat bin. Reconstructed energy loss [GeV]0.000.050.100.150.200.25 P r o b . d e n s . Single templateUnfolding (b) Slices of the distribution of reconstructed total energy lossshown in (a) at fixed true total energy loss of 10 , 10 , and 10 GeV.
Figure 15: Total energy loss within IceCube from sim-ulated single muons passing through the detector recon-structed by the single-template method, analogous to thecascade reconstruction method of Sec. 6, and by the un-folding method of Sec. 8.2. The continuous-loss approxi-mation inherent in the single-muon template method be-comes steadily worse as the energy deposition increases,whereas the resolution of the unfolding method improvesas photon statistics accumulate, reaching a full width of20% above a deposition of 1 PeV. Figure 16: Diagram of spatial binning of energy loss alonga track, used in spatial-separation-based energy depositionreconstruction methods. Within each bin, separated bydashed lines, (cid:104) dE/dx (cid:105) is determined using a single tem-plate muon as though the photomultipliers in each binconstituted an independent detector.length for light in the IceCube array, averaged over thedifferent optical properties at different depths in the icesheet [17, 18], is ∼
125 m. This is approximately the de-tector’s inter-string spacing and is used as a typical valuefor the scale of track segmentation in spatial-separation-based reconstruction methods.
Variations of muon energy loss can happen at scalesmuch smaller than the usual segmentation scale used inspatial separation. Due to the lack of physical segmen-tation in the detector, light from single bright stochasticlosses can travel distances longer than the size of the seg-ments used and be detected simultaneously with photonsfrom nearer parts of the muon track. Every PMT readoutis then a combination of light from everywhere along thetrack emitted at many places within the detector.The fact that this combination is linear, and that theindividual stochastic losses take the form of electromag-netic showers (Sec. 6), makes unfolding these contributionstractable. We have already considered energy reconstruc-tion in the presence of multiple overlaid light sources inSec. 1 when discussing the inclusion of PMT noise andcan generalize Eq. 3 to allow the additional sources to beshowers of variable energy by replacing the substitution λ → E Λ + ρ, (7)with another where the expected photon count ( λ ) is thesum of photons from multiple sources ( λ i ) and the noiserate ( ρ ): λ → (cid:88) sources i E i Λ i + ρ. (8)15ere each E i is the energy deposition by a particular sub-source i and Λ i is the expected light yield in a particularphotomultiplier and time bin from light source i . Ourlikelihood (Eq. 1) can then be rewritten in terms of vectoroperations:ln L = k ln (cid:0) E i Λ i + ρ (cid:1) − E i Λ i − ρ − ln ( k !)= k ln (cid:16) (cid:126)E · (cid:126) Λ + ρ (cid:17) − (cid:126)E · (cid:126) Λ − ρ − ln ( k !) , (9)and summing over time bins j : (cid:80) j ln L = (cid:80) j k j ln (cid:16) (cid:126)E · (cid:126) Λ j + ρ j (cid:17) − (cid:80) j (cid:16) (cid:126)E · (cid:126) Λ j − ρ j (cid:17) − (cid:80) j (ln k j !) . (10)Like Eq. 3, this has no analytic maximum for (cid:126)E , but canbe solved to first order (the approximately Gaussian errorregime applicable at high energies): k j = (cid:126)E · (cid:126) Λ j + ρ j . (11)Introducing the matrix Λ for the predicted light yieldat every point in the detector from every source positionat some reference energy, this can be rewritten in terms ofa matrix multiplication: (cid:126)k − (cid:126)ρ = Λ · (cid:126)E. (12)Equation 12 can be inverted by standard linear alge-braic techniques to find the best-fit (cid:126)E . We want, however,to incorporate additional physical constraints. In partic-ular, negative energies are unphysical and should not bepresent in the solution even though a matrix inversion mayoften yield solutions with negative terms. The solutionis to use a non-negative least squares algorithm [20] toachieve a high-quality fit to the data (Fig. 17) by use ofa linear deposition hypothesis with possible light sourcesevery few meters along the track. This first-order linearsolution can be further refined to exactly maximize Eq. 9by the use of algorithms used in positron emission tomog-raphy reconstructions. Here we use the Non-MonotonicMaximum Likelihood (NMML) algorithm [25], achievingresolution on total deposited energy along muon tracks of ∼ −
15% (Fig. 18), comparable to that achieved fordeposited energies with cascade events (Fig. 13).Additional physical constraints can be included by theuse of regularization terms in the likelihood (Eq. 9). Al-though most uses for regularization (preventing ringing,in particular) are eliminated by the non-negativity con-straint, such terms can still be useful as additional weakpenalties. We use Tikonoff regularization to accomplishtwo goals. Adding a term proportional to the norm of thefirst derivatives of the dE/dx vector (a first-order penalty)can be used to restrict fluctuations in muon energy loss.More commonly we apply an extremely weak ridge penaltyon || dE/dx || to break inherent degeneracies in the solutionto Eq. 9 between nearby low-energy events and distant
100 1000 10000 100000 1e+06 0 500 1000 1500 2000 2500 3000 3500 d E / d X ( G e V / m ) Time (ns)Differential Energy Reconstruction of 5 PeV Muon in IC-86Total Reconstructed Energy Loss: 108.8 TeVTotal True Energy Loss: 107.9 TeVMonte Carlo TruthReconstructed
Figure 17: Reconstruction of the energy deposition of asimulated 5 PeV single muon using unfolding with a 15meter cascade spacing. For this event, the total recon-structed energy loss within the detector differs from thetrue value by less than 1%.high-energy events when only small numbers of photonson the detector boundary are observed. In such cases, theweak penalty causes the fit to prefer the nearby, low-energysolution.
9. Interpretation of Segmented Energy Losses
The mean energy loss rate of a muon ( (cid:104) dE/dx (cid:105) ) isroughly proportional to its energy above ∼ (cid:104) dE/dx (cid:105) must be used. A simple average(top panel of Fig. 19) provides poor muon energy resolu-tion with large non-Gaussian tails due to statistical biasfrom large stochastic losses in the detector. The segmentedenergy loss rates computed in the previous section, how-ever, can be used to develop more robust estimators. Themost common approach is to use the truncated mean in-stead of a straight average to reduce the effects of outliers;other techniques that use information about the likelihoodof large losses may further improve resolution. All the observables discussed here (total energy loss,mean energy loss rate, truncated mean energy loss rate)are related to the energy of the underlying muon but arenot themselves energies. In order to be able to discuss theresolving power of these observables, we require a way tomeasure the resolution of proxy observables with disparateranges and units. We can construct such a measure foreach proxy observable by simulating events and building16 R e c o n s t r u c t e d e n e r g y [ G e V ] Mean deposited energy [GeV]010203040 σ [ % ] ν µ -induced10 -3 -2 -1 Probability density [arb. units] (a) Reconstructed total energy deposition as a function of truetotal energy deposition. Reconstructed energy loss [GeV]0.00.10.20.30.40.5 P r o b . d e n s . (b) Slices of the reconstructed total energy distribution shownin (a) at fixed true total energy depositions of 10 , 10 , and10 GeV.
Figure 18: Reconstruction, using unfolding with 2.5 metercascade spacing, of the total energy deposition of simulated ν µ interactions with known directions and vertices that areinside the instrumented volume. Observation of such in-detector starting muon events allows positive identificationas a charged-current ν µ interaction where all energy fromthe neutrino is deposited in the detector. High precisionreconstruction of charged particle energies in such eventsthen allows neutrino energy reconstruction limited only byinstrumental resolution. -1 Monte Carlo truth10 -1 R e c o n s t r u c t e d d E / d x [ G e V / m ] Single template10 Muon energy at detector entry [GeV]10 -1 Unfolding10 -3 -2 -1 Probability density [arb. units] (a) Mean energy loss rate determined from true Monte Carlo in-formation (top panel), the single-muon template method (mid-dle) panel, and multi-source unfolding (bottom panel). -1 Reconstructed dE/dx [GeV/m]0.000.050.100.150.200.250.30 P r o b . d e n s . Monte Carlo truthSingle template Unfolding (b) Slices of the reconstructed energy loss rate distributionsshown in (a) at fixed muon energies of 10 , 10 , and 10 GeV.
Figure 19: Mean energy loss rate of through-going muonsas a function of the muon energy at the point where itenters the detector volume. While the unfolding methodreproduces well the fluctuations in the true energy lossrate above a few TeV (the similarity in (cid:104) dE/dx (cid:105) between“Monte Carlo truth” and “Unfolding”) , these fluctuationslimit the usefulness of the mean loss rate as a proxy forthe energy of through-going muons.17 log ( E true / GeV) O b s e r v a b l e O -3 -2 -1 Probability density [arb. units] 0.05 0.10 P ( O | ˆ E true ) log ( E reco / GeV) P ( E r ec o | ˆ E t r u e ) σ =0 . Figure 20: A construction for determining the energy res-olution of a proxy observable. Muons of a given energyˆ E true (vertical dashed line at log ( E true / GeV) = 5) willproduce a distribution P ( O | ˆ E true ) of the proxy observable O (right panel). Each O , in turn, has an associated dis-tribution of true energies P ( E true | O ). The distributionof best-fit energies E reco for muons with a given ˆ E true ,shown as a solid line in the bottom panel, is given by (cid:82) O P ( E true | O ) P ( O | ˆ E true ) dO . The width of this distribu-tion is a measure of the energy resolution of the proxyobservable (in this case, log ( dE/dx ) from the “singletemplate” method) for muons of the given energy.up a joint distribution of muon energy and the proxy ob-servable like the one shown in Fig. 20. The relationshipbetween the muon energy and the observable is not one-to-one. The proxy observable can take on a wide rangeof values for muons of the same energy, and each valueof the observable can in turn arise from a range of muonenergies. This ambiguity can be captured by construct-ing a confidence interval as shown in Fig. 20. In essencethis confidence interval measures the vertical width of theobservable distribution in proportion to the overall slopeof the joint distribution. In the minimum-ionizing regime,where the energy loss rate becomes nearly independentof energy, the distribution must be quite narrow in orderto separate muons of different energies. At high energies,however, it can become wider while maintaining an equiv-alent resolving power. By discarding energy losses from segments of the muontrack with the highest loss rates, it is possible to obtaina more robust measurement of the typical (cid:104) dE/dx (cid:105) of the Muon energy at detector entry [GeV]0.250.300.350.400.450.50 E n e r g y r e s o l u t i o n : σ l og E µ True ∆ E/ ∆ x Single templateSingle template (widened PDF)
Truncated mean (segmented detector)Truncated mean (unfolded losses)
Figure 21: Comparison of the energy resolution of vari-ous methods on simulated through-going muons using themeasure illustrated in Fig. 20. The true mean energy lossrate is shown for purposes of comparison only and pro-vides a poor energy proxy no matter how precisely it canbe reconstructed.track (the truncated mean approach [28, 30]). Rejectingthe bins with the largest energy losses removes outliersfrom the average and the variance in the calculation of (cid:104) dE/dx (cid:105) and the muon energy is therefore reduced. Thisapproach can be applied to the results of segmented dE/dx reconstructions using both spatial separation (Sec. 8.1,[28]) and unfolding (Sec. 8.2).This reconstruction technique provides substantiallybetter estimates of the muon energy than deposition-onlyor single-muon-template estimates using either segmentedmeasurement of dE/dx (Fig. 21). Similar performancecan also be achieved by widening the upper end of thephoton counting probability densities (Sec. 3.1, here with w = 10) in the single-template method to achieve an ef-fective truncated mean (“single template (widened PDF)”in Fig. 21). Although these resolutions are computed inan idealized case, in which the muon position and direc-tion are known exactly, typical resolutions at the analysislevel, given standard muon geometry reconstructions [31],exhibit similar behavior (Fig. 22). Analysis of energy loss topologies is currently a subjectof ongoing work and will be discussed in detail in futurepublications. Two possible applications of this informationare discussed below.When treated in detail, the rate and amplitude of high-energy stochastic losses may provide more detail aboutmuon energy than a (cid:104) dE/dx (cid:105) measurement [29]. The mostinteresting case is muons emitted from in-detector ν µ inter-actions where all charged particles are observed and un-18 R e c . e n e r g y [ G e V ] Muon energy at detector entry [GeV]0.200.250.300.350.400.45 σ l og E µ Single template (widened)
IceCube-79 Muon AnalysisBenchmark -3 -2 -1 Probability density [arb. units]
Figure 22: Energy resolution for a sample IceCube muonanalysis at final level. The slight reduction in resolutionat high energies compared to the “benchmark” level (fromFig. 21) arises from convolution of the intrinsic energy res-olution (“benchmark”) with the finite resolution of the an-gular and positional reconstruction used in this analysis.ambiguous flavor identification is possible. Accurate de-termination of the energy of the outgoing muon, takinginto account the event topology, may allow neutrino en-ergy resolution for these events to approach the depositedenergy resolution (Fig. 18).Detailed energy loss topologies are also useful beyondmuon energy measurements for identification of interac-tions and determination of neutrino flavor. Electron neu-trino charged-current interactions and all neutral-currentinteractions have nearly point-like energy deposition (Sec. 6),while muons have extended tracks, potentially beginningwith a bright hadronic cascade at the neutrino vertex.Charged-current ν τ interactions can have a variety of sig-natures, with tracks from taus (length ∼
50 m (cid:0) E (cid:1) )and potential decay muons as well as hadronic cascadesassociated with the neutrino vertex and tau decay. High-resolution segmented dE/dx reconstruction, as developedhere, may allow positive identification of these type ofinteractions even when the τ track has a length shorterthan the detector instrumentation spacing, which resultsin an otherwise cascade-like light pattern in the detector(Fig. 23).
10. Conclusion
Using the techniques described here, IceCube achievesaverage deposited energy resolution in all channels of ∼
15% (depending on the event selection used in individualanalyses, better resolutions are possible). This is limitedprimarily by systematic uncertainties above a few TeV.Some of these may be reduced in the future, in particularthose due to modeling of light propagation and shower (a) A simulated charged-current ν τ interaction creating ahadronic cascade and a τ which decays after 50 m to a secondcascade. The black spheres indicate the energy depositions ofthe two cascades. On a macroscopic level, this event looks verycascade-like. d E / d X ( T e V / m ) Distance (m)Differential Energy Reconstruction of Contained Tau in IC-86Total True Energy Loss: 598 TeVTotal Reconstructed Energy Loss: 626 TeVMonte Carlo TruthReconstructed (b) Differential energy reconstruction showing two separatedpeaks, a distinct signature of ν τ interactions. Detailed dE/dx reconstruction may allow identification of these interactionswith decay lengths smaller than the scale of the detector in-strumentation. Figure 23: Example of a simulated charged-current ν τ in-teraction with subsequent decay producing a second cas-cade (also known as “Double Bang”; see Table 1).19evelopment. Others, such as the inherent variance ofhadronic shower light yield, are fundamental limitationsto IceCube’s performance. Deposited energy resolutionis highest for near-pointlike largely electromagnetic par-ticle showers, such as those produced in ν e interactions,for which resolution is limited by shower fluctuations overmost of IceCube’s energy range.For extended objects, such as muon tracks, precisionreconstruction and localization of energy loss topology arepossible. Work is on-going to use this information fully forhigh-quality muon energy reconstruction, in particular inthe case of muons produced in ν µ charged-current interac-tions, where neutrino flavor can be measured directly andall charged particles are visible. As this work on using thedetailed energy loss patterns of muons to measure their en-ergies continues, a variety of alternative observables can beused, optimized for different use cases. Searches for muonneutrinos interacting within the detector, where the visi-ble energy is dominated by the contained hadronic cascadeat the vertex, typically use integrated deposited energy asan observable, as in [12] and Fig. 18. For searches focusedon higher energy uncontained events, the cascade at thevertex is unobservable and muon energy is a more robustindicator of neutrino energy than the deposited energy inthe detector. Such analyses [32] typically use one of thestochastic-loss-filtering muon energy estimators shown inFig. 21. A wealth of additional information is provided bythese topological reconstructions and the likelihood modeldescribed here: in addition to reconstruction of muon en-ergies, topologies can also be used to study muon energyloss processes at very high energies and for particle iden-tification. Acknowledgments
We acknowledge support from the following agencies:U.S. National Science Foundation-Office of Polar Programs,U.S. National Science Foundation-Physics Division, Uni-versity of Wisconsin Alumni Research Foundation, theGrid Laboratory Of Wisconsin (GLOW) grid infrastruc-ture at the University of Wisconsin - Madison, the OpenScience Grid (OSG) grid infrastructure; U.S. Departmentof Energy, and National Energy Research Scientific Com-puting Center, the Louisiana Optical Network Initiative(LONI) grid computing resources; Natural Sciences andEngineering Research Council of Canada, WestGrid andCompute/Calcul Canada; Swedish Research Council, SwedishPolar Research Secretariat, Swedish National Infrastruc-ture for Computing (SNIC), and Knut and Alice Wal-lenberg Foundation, Sweden; German Ministry for Ed-ucation and Research (BMBF), Deutsche Forschungsge-meinschaft (DFG), Helmholtz Alliance for AstroparticlePhysics (HAP), Research Department of Plasmas with Com-plex Interactions (Bochum), Germany; Fund for Scien-tific Research (FNRS-FWO), FWO Odysseus programme,Flanders Institute to encourage scientific and technologicalresearch in industry (IWT), Belgian Federal Science Policy Office (Belspo); University of Oxford, United Kingdom;Marsden Fund, New Zealand; Australian Research Coun-cil; Japan Society for Promotion of Science (JSPS); theSwiss National Science Foundation (SNSF), Switzerland;National Research Foundation of Korea (NRF); DanishNational Research Foundation, Denmark (DNRF). N.W.was supported by the NSF GRFP.
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