A Search for Solar Neutralino Dark Matter with the AMANDA-II Neutrino Telescope

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Thomas Burgess

A Search for Solar Neutralino Dark Matter

with the amanda-II Neutrino Telescope

Department of Physics

Stockholm University

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c

Thomas Burgess 2008

ISBN 978-91-7155-597-7

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Abstract

A relic density of Weakly Interacting Massive Particles (WIMPs) re-maining from the Big Bang constitutes a promising solution to the Dark Matter problem. It is possible for such WIMPs to be trapped by and accumulate in gravitational potentials of massive dense objects such as the Sun. A perfect WIMP candidate appears in certain supersymmetric extensions to the Standard Model of particle physics, where the light-est supersymmetric particle is a neutralino which can be stable, massive and weakly interacting. The neutralinos may annihilate pair-wise and in these interactions neutrinos with energies ranging up to the neutralino mass can be indirectly produced. Hence, a possible population of dark matter neutralinos trapped in the Sun can give rise to an observable neutrino ux.

The Antarctic Muon And Neutrino Detector Array, AMANDA, is a neutrino telescope that detects Cherenkov light emitted by charged particles cre-ated in neutrino interactions in the South Pole glacial ice sheet using an array of light detectors frozen into the deep ice. In this work data taken with the AMANDA-II detector during 2003 are analyzed to measure or put upper bounds on the ux of such neutrinos from the Sun. In the analysis detailed signal and background simulations are compared to measurements. Background rejection lters optimized for various neu-tralino models have been constructed. No excess above the background expected from neutrinos and muons created in cosmic ray interactions in the atmosphere was found. Instead 90% condence upper limits have been set on the neutralino annihilation rate in the Sun and the muon ux induced by neutralino signal neutrinos.

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1 Introduction

There is a theory which states that if ever anybody discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. There is another theory which states that this has already happened.

Douglas Adams (1952 - 2001)

Since the 1930's numerous astronomical observations have provided evidence that the amount of observed luminous matter is not enough to explain the dynamics of stars and galaxies. The observations can be explained by introducing additional dark matter, about six times as much as the luminous matter. A few percent of the dark matter can be explained as undetected normal matter, but the main part needed remains to be explained. A promising solution to the problem is a relic density of Weakly Interacting Massive Particles (WIMPs) remaining from the Big Bang. A perfect WIMP candidate appears in certain supersymmetric extensions of the Standard Model of particle physics, where the lightest supersymmetric particle is a neutralino which can be stable, massive and weakly interacting.

Because the neutralinos are Majorana particles they may annihilate pair-wise and produce standard model particles. The WIMPs in the Milky Way dark matter halo may scatter and lose energy and become trapped by and accumulate in gravitational potentials of massive dense objects, such as the Sun, the Earth or the galactic center. Neutrinos are indirectly produced in the annihilation interactions inside the neutralino accumulation, which provides a neutrino point source. The analysis presented in this thesis searches for such a neutralino induced neutrino ux from the Sun using the AMANDA neutrino telescope.

The Antarctic Muon And Neutrino Detector Array, AMANDA, is a neutrino tele-scope that detects Cherenkov light emitted by charged particles created in neutrino interactions in the South Pole glacial ice sheet using an array of light detectors frozen into the deep ice. The bulk of the instrumented volume makes a cylinder roughly 500 m in height and 200 m in diameter. The direction of the incoming neutrinos can be reconstructed using the observed hit patterns. In this work 150 days of data taken with the AMANDA-II detector during 2003 is analyzed to measure or put upper bounds on the ux of such neutrinos from the Sun.

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1.2 Author's contribution

Apart from the analysis presented in this thesis I have worked on several other AMANDA/IceCube related projects including:

• The rst study on the prospects of a solar neutralino search with AMANDA made in collaboration with Christin Wiedemann.

• Development and implementation of an algorithm to analyze the peak hit amplitude distribution used in the time calibration and on-line detector mon-itoring system in collaboration with Christin Wiedemann.

• Part of the team that started development of the standard AMANDA data-analysis framework sieglinde. Amongst other things I wrote the coding standards document, end user documentation, an automated man-page like documentation system, an XML steering le parser, several event observable calculation modules and implemented the cats reconstruction algorithm (in collaboration with Peter Steen).

• Testing and some development of AMASIM in collaboration with Stephan Hun-dertmark.

• Active part in early discussions and development of IceCube simulation soft-ware, including porting sieglinde coding standards to IceCube.

• Developed and implemented PSInterface a generic interface between de-tector and photon propagation simulations, which has been used to compare and debug photon propagation simulations and now is a standard tool used in IceCube simulation and and reconstruction software.

• Implemented a new memory management system for the photon propagation

simulation Photonics in collaboration with Johan Lundberg.

• Developed WimpEventF2k which takes simulated WimpSim events and places them around AMANDA so they can be used in AMASIM.

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1.2 Author's contribution 3 • Assembly (in Stockholm) and installation (at the South Pole) of upgraded SWAMP (SWedish AMPliers) system which is part of the AMANDA surface electronics. Work took place during the austral summer seasons 20012002, 20022003 and 20042005.

• Various outreach activities including SEASA [1] and a booklet with South Pole travel letters to swedish school children [2]

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2 Dark Matter

Each pound of dark matter weighs over 10,000 pounds Professor Farnsworth, Futurama

The rst evidence of Dark Matter was observed in the 1930's when studying the dynamics of distant galaxies [3]. The mass inferred from the observed velocity distribution greatly exceeded the mass estimated from the total light output. The conclusion was that a large fraction of the mass in the galaxies was not luminous. Numerous current measurements strongly supports this conclusion. One of the most well-known evidences for dark matter comes from numerous observations of the rotation velocities of stars in spiral galaxies. Observations show that the orbital velocities do not decrease with the square-root of the distance from the galactic center as expected from Kepler's third law. Instead the rotation curves atten out and become almost constant even far outside the edge of the luminous matter in the galaxy. This behavior can be expected if the galaxy is surrounded by a halo of dark matter.

Although observational evidence is abundant the nature of dark matter remains unknown. Some theories, such as MOdied Newtonian Dynamics, MOND [4], aim to avoid the dark matter by modifying the theory of gravity, but these theories appear both unnatural and unlikely, and fail to explain the missing matter on all scales. From the theory of Big Bang nucleo-synthesis strict limits are set on the baryon density of the Universe, thus normal matter such as brown dwarfs cannot make up more than a tiny fraction of the dark matter [5].

From current knowledge of cosmological parameters from the combined results from WMAP and SDSS LRG data [6] we know that the total energy density of the Universe in units of critical energy is Ωtot = 1.003+0.010+0.009, where the total

en-ergy density has contributions from dark enen-ergy ΩΛ= 0.761+0.017−0.018 and matter Ωm=

0.239+0.018−0.018, and the matter energy density has contributions from baryonic matter

Ωb = 0.0416+0.0019−0.0018 and dark matter ΩDM = 0.197+0.016−0.015. Together this means an

essentially at universe with dominated by dark energy and dark matter, both of unknown origins, and less than 5% normal baryonic matter. An alternative con-tribution to the dark matter could be a relic density of stable particles, created in thermal equilibrium in the hot early Universe and frozen out as the Universe cooled down. Depending on cross-sections and particle masses the particles may leave equilibrium at relativistic velocities as hot dark matter, or at lower velocities

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weakly interacting cold dark matter [8]. The most promising dark matter constituent appears to be cold dark matter in the form of Weakly Interacting Massive Particles, WIMPs.

A good WIMP candidate must by only weakly interacting in order not to be lu-minous, stable enough to remain from the Big Bang, and massive enough not to have been detected in accelerator searches while lighter than 50 TeV/c2 _{not to over-close}

the Universe. Several suitable WIMP candidates arise from various extensions to the standard model of particle physics, typically the WIMP candidate is the lightest of a family of new particles predicted by the model. The analysis in this thesis is a search for a signal arising from neutralinos, χ, which are the lightest particle in some supersymmetric extensions to the standard model of particle physics. Other suggested WIMP candidates include the lightest Kaluza-Klein particles in theories Universal Extra Dimensions theories [9], and the lightest inert Higgs particle in the Inert Doublet Model theories [10].

2.1 Supersymmetry and neutralinos

SUperSYmmetry (SUSY) is an extension to the Standard Model (SM) of particle physics that introduces a symmetry such that for each fermionic degree of freedom there is a bosonic degree of freedom. Supersymmetry is theoretically appealing be-cause it oers a solution to the hierarchy problem, i.e. the ne tuning needed in calculation of the Higgs mass, by the introduction of terms canceling correction terms in the mass that otherwise may become divergent. Furthermore it helps the unication of gauge couplings at high energy. In this work the Minimal Super-symmetric extension to the Standard Model (MSSM) is tested. In MSSM each SM particle has one SUSY partner, with identical quantum numbers except for the spin. In addition there are two Higgs doublets instead of one as in the SM, which give rise to ve Higgs bosons. In table 2.1 SM particles and corresponding SUSY partners are shown.

Accelerator searches have so far not discovered SUSY particles, and one may conclude that SUSY is broken such that SUSY particles are more massive than the

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2.1 Supersymmetry and neutralinos 7 Table 2.1: Normal particles and their superpartners in the MSSM.

Normal particles/elds _{Interaction eigenstates Mass eigenstates}Supersymmetric particles/elds

Symbol Name Symbol Name Symbol Name

q = d, c, b, u, s, t quark q˜L, ˜qR squark q˜1, ˜q2 squark

` = e, µ, τ lepton `˜L, ˜`R slepton `˜1, ˜`2 slepton

ν = νe, νµ, ντ neutrino ν˜ sneutrino ν˜ sneutrino

g gluon g˜ gluino g˜ gluino

W± H− H+ W-boson charged Higgs ˜ W± ˜ H₁− ˜ H₂+ wino higgsino ) ˜ χ±_1,2 chargino B W3 γ, Z0 H0 1 H0 2 H₃0    B-eld W3-eld neutral Higgs ˜ B ˜ W ˜ H₁0 ˜ H0 2 bino wino higgsino      ˜ χ0_1,2,3,4 neutralino

energies probed so far. Often one introduces a discrete symmetry, R-parity, which is +1 for normal particles and -1 for superpartners, and is given by

R = (−1)3B+L+2S (2.1)

where S is spin, B baryon number and L lepton number. If R-parity is conserved, superparticles cannot decay into normal standard model particles, and hence heavy superparticles decay into lighter superparticles and the Lightest Supersymmetric Particle, LSP must be stable. In SUSY models where the LSP is a neutralino gives a perfect WIMP candidate, with the neutralino as stable, massive and weakly interacting particle. Neutralinos are linear combinations of the supersymmetric particles the bino, wino, and neutral higgsinos,

˜

χ0_i = Ni1B + N˜ i2W˜3 + Ni3H˜10+ Ni4H˜20, (2.2)

where Nij are the diagonalized neutralino mass matrix elements. For simplicity the

lightest neutralino ˜χ0

1 is written χ and called the neutralino χ.

The neutralino is a Majorana particle and is hence its own anti-particle, there-fore it may pair-annihilated and produce annihilate and produce standard model particles, with the most important decay modes being

χχ −→ q ¯q, `

+_`−_{, W}+_W−_{, Z}0_Z0_{, Z}0_H0 1,

Z0H₂0, H₁0H₃0, H₂0H₃0, H±W± . (2.3)

In the non-relativistic limit neutralinos cannot annihilate directly into neutrinos, however a neutrino signal can be created in the decay of the annihilation products.

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over time neutralinos are accumulated in the center of the objects. The accumu-lation increases the neutralinoneutralino collision probability, and with time an equilibrium between annihilation and capture is reached. Thus uxes of neutralino annihilation products are expected from compact objects. For the analysis presented in this thesis searches for such a ux of neutrinos from the core of the Sun.

2.2 Detecting neutralinos

Current limits from accelerator SUSY searches place a lower limit on the neutralino mass, mχ > 46GeV/c2 [11]. The cosmologically interesting region 0.025 < Ωχh2 < 1

places an upper limit on the neutralino mass of around 10 TeV [12].

Accelerator neutralino searches aims to directly produce neutralinos and do not actually measure the neutralinos in the dark matter halo. Neutralino dark matter searches can be done using direct and indirect measurements. Several experiments, including CDMS [13] and XENON10 [14], have been conducting direct searches in which one tries to measure the recoil as neutralinos that travel through the detec-tor scatter of target nuclei. The indirect neutralino dark matter searches aims at measuring the neutralinos indirectly through some of their annihilation products such as antiparticles, gamma rays and neutrinos. Many experiments perform such searches, amongst others neutralino induced antimatter is searched for with the PAMELA satellite [15], gamma-rays with air Cherenkov telescopes like MAGIC [16] and the upcoming GLAST satellite [17]), and neutrinos the subject of this the-sis with neutrino telescopes such as AMANDA [18], Super-Kamiokande [19] and ANTARES [20].

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3 Neutrino astronomy with AMANDA

Great God! This is an awful place. . .

Capt. Robert Scott (18681912) journal entry made after losing the race against Amundsen to discover the South Pole.

3.1 Detecting high energy neutrinos

Neutrinos cannot be detected directly but can be studied indirectly through their interactions. In charged current neutrinonucleon interactions

ν`(¯ν`) + N −→ `−(`+) + X , (3.1)

a neutrino reacts with a quark in a nucleon N and produces a charged lepton ` and an hadronic shower X, see gure 3.1(a). In neutral current neutrinonucleon interactions

ν`(¯ν`) + N −→ ν`(¯ν`) + X , (3.2)

a neutrino scatters o a nucleus N and a hadronic shower X is produced, see g-ure 3.1(b). By observing the particles produced in the reactions, the properties of the interacting neutrino can be inferred.

The charged current neutrinonucleon interaction results in signatures, depend-ing on the lepton avor. All events have hadronic cascades at the interaction point in common. In electron neutrino events the electron produces a short (tens of me-ters) electromagnetic cascade in coincidence with the hadronic cascade. In muon neutrino events muons are produced which can travel several kilometers. The tau leptons produced in tau neutrino interactions have a very short lifetime, 0.29 ps, and hence will decay shortly after production. In the decay a secondary cascade can be produced. For tau neutrino events in the PeV range the separation between the neutrino interaction and tau decay is about 50 m on average, which may lead to a spectacular signature with two cascades (double-bang event). In neutral cur-rent events there is no charged lepton, but still a hadronic cascade starting at the interaction point. In gure 3.2 the dierences in interactions for all neutrino avor are depicted. The muon direction is better correlated with the incoming neutrino direction than the direction of the electromagnetic or hadronic cascade.

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neutrino) converts an up (down) quark to a down (up) quark by exchange of a

W−(W+_{) boson. A `}+_(`−₎

lepton is created and the nu-cleus produces a hadronic shower.

ters of the nucleon through

a Z0 _{boson exchange and}

a hadronic shower is pro-duced.

Figure 3.1: Charged and neutral current neutrinonucleon interactions.

The neutrinonucleon cross-sections increase with energy. The charged current interaction cross-section is higher than the neutral current cross-section, and further-more it is higher for neutrinos than for anti-neutrinos because of helicity constraints. For the analysis in this thesis which depend on good pointing resolution only CC muon neutrino events are used.

In Cosmic Ray, CR, interactions in the atmosphere large amounts of muons and neutrinos are created. When the CR primary particles interact with nuclei in the atmosphere short lived π and K mesons are created which in turn give rise to atmospheric muons and neutrinos through the decays

p + X → π±(K±) + Y

- µ±+ ν_µ(¯ν_µ)

The cosmic rays arrive at the Earth almost isotropically and over a large range of energies. Since the muons only can penetrate a limited range the atmospheric muon ux in the detector only comes from above the detector, whereas the atmospheric neutrino ux is near isotropic. At the depths of the detector the atmospheric muons outnumber muons from atmospheric neutrino interaction by a factor of 106_{. These}

atmospheric muons and neutrinos are both backgrounds to a possible solar neu-tralino signal. The dierent event types and angles are summarized in gure 3.3. The background of atmospheric muons can be removed by using the Earth as a muon lter and only considering upwards moving muons. Since the Sun is below

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3.1 Detecting high energy neutrinos 11

Figure 3.2: Neutrino interaction signatures. In all examples a hadronic cascade is produced at the interaction point.In gure (A) an electron neutrino interacts and an electromagnetic shower is produced. The electron direction is close to the neutrino direction. In gure (B) a muon neutrino interacts and a long ranging muon is produced in a direction close to the neutrino direction. In gure (C) a tau neutrino interacts and a short range tau lepton is created. When the tau decays a secondary shower may be produced, and the double shower signature can be used to identify tau events. In gure (D) a neutrino scatters of a nucleus in neutral current reaction, leaving no other visible signature than the hadronic cascade.

the horizon half of the year, the improvement in signal from background separation comes at the price of losing part of the experiment live-time.

Cherenkov radiation

A charged particle traversing a dielectric medium produces photons in the electro-magnetic interaction with the medium. If the particle travels at a speed v faster than the speed of light in the medium c/n(λ) constructive interference occurs at a certain angle θc and a coherent electromagnetic shock wave is formed, as shown

in gure 3.4. This phenomenon is called Cherenkov radiation. From the criterion v > c/n(λ) the minimum energy of a Cherenkov emitting particle is given by

Emin = mc

2

p1 − β2 =

mc2

p1 − n(λ)−2 , (3.3)

where m is the particle rest mass, λ the photon wavelength, and n the refractive index of the medium. The AMANDA experiment is sensitive to photons with wave-lengths in the range from 300 nm to 600 nm. In this range the South Pole ice has the refractive index n = 1.33, and the minimum energy for Cherenkov emitting muons is Emin

µ = 160 MeV, while for electrons it is Eemin = 21 MeV. The Cherenkov angle

θc is given by

cos θc =

c n(λ)vµ

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Figure 3.3: Sources of muons in a neutrino telescope. Signal neutrinos from neu-tralino annihilations in the Sun νχarrive (at the South Pole) with zenith angles ±23◦

around the horizon. When high energy Cosmic Ray (CR) particles collide with nu-clei in the atmosphere muons µatm and neutrinos νatm are produced. Some of the

atmospheric neutrinos will interact and produce muons in the detector. Because of the limited muon range atmospheric muons only arrive from above the detector, whereas the atmospheric neutrinos arrive from all angles.

where n(λ) is the refractive index of the medium, vµ the speed of the particle, and c

the speed of light in vacuum. For ultra-relativistic muons in AMANDA the Cherenkov angle is θc ≈ 41.2◦.

The number of Cherenkov photons N emitted per unit length and wavelength interval dλ can be calculated using the Frank-Tamm formula [21]

d2_N dxdλ = 2παZ2 λ2 1 − 1 β2_n2 , (3.5)

where Z is the particle charge, α the ne structure constant, λ the wavelength and Z the particle charge. Integrating over the sensitive wavelength region, the number of photons emitted per centimeter is found as

Z 600 nm 300 nm d2_N dxdλdλ = 2παZ 2 1 − 1 β2_n2 −1 λ 600 nm 300 nm = 332 cm−1, (3.6)

Ice properties

When the Cherenkov radiation propagates through ice it is aected by scattering and absorption on air bubbles and dust particles. Scattered light arrive later to a detector than unscattered light, and the time delay depends on the distance between the source and the detector and on the eective scattering length λeff

λeff =

λgeo

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3.2 The AMANDA-II neutrino telescope 13

Figure 3.4: Cherenkov radiation. A charged particle traveling with the velocity vµ

in a medium where the speed of light is c/n. When vµ> c/n the disruptions of the

electromagnetic elds medium interfere constructively in the direction θc, giving rise

to a conical wavefront.

where λgeo is the average distance between scattering centers and hcos θi the average

cosine scattering angle. For dust particles in the South Pole ice the average cosine scattering angle is 0.94, and for air bubbles it is 0.75 [22, 23]. The absorption in the ice is described by the absorptivity λ−1

a which is the fraction of light absorbed per

unit length.

As shown in gure 3.5 scattering and absorption are highly depth and wavelength dependent. At shallow depths, above 1400 m, the abundance of air bubbles causes very short scattering lengths, whereas deeper and older ice has nu such air bubbles. In the deep ice the average eective scattering length is ∼ 20 m. The ice in the holes where the detector strings have been deployed have high abundance of air bubbles and an eective scattering length of approximately 60 cm. For wavelengths between 200 nm and 500 nm the average absorptivity in the deep ice is λ−1

a = 10

−2 _m−1_.

3.2 The AMANDA-II neutrino telescope

The Antarctic Muon And Neutrino Detector Array, AMANDA, is a neutrino telescope situated at the geographic South Pole. Construction of AMANDA has been done in several steps. In the austral summer season 19931994 the prototype experiment AMANDA-A was installed at depths of 8001000 m below the ice surface. The pro-totype was a proof of principle for an ice Cherenkov neutrino detector, but it was found that the scattering length in this ice was too short because of air bubbles in the ice at these depths [24]. The nal AMANDA-II was deployed at depths below 1500 m and was extended gradually from 1995 to 2000 when the detector was

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com-Figure 3.5: Maps of optical scattering and absorption for deep South Pole ice. The depth dependence between 1100 and 2300 m and the wavelength dependence between 300 and 600 nm for the eective scattering coecient (left) and for ab-sorptivity (right) are shown as shaded surfaces, with the air bubbles contribution to scattering and the pure ice contribution to absorption superimposed as (partially obscured) steeply sloping surfaces. The dashed lines at 2300 m show the wavelength dependencies: a power law due to dust for scattering and a sum of two components (a power law due to dust and an exponential due to ice) for absorption. The dashed line for scattering at 1100 m shows how scattering on air bubbles is independent of wavelength. The slope in the solid line for absorptivity at 600 nm is caused by the temperature dependence of intrinsic ice absorption, Figure and caption taken from [22].

pleted. A much larger detector, IceCube, that contains AMANDA is currently under construction at the South Pole, with expected completion in 2011. During the years 2000-2005 AMANDA-II detected 4282 high quality atmospheric neutrino events from the northern hemisphere [25].

The AMANDA-II detector consists of Optical Modules (OMs) which detect light using highly sensitive Photo Multiplier Tubes (PMTs). The array consists of in total 677 OMs distributed on 19 strings that have been deployed in holes drilled with hot water to depths down to 2500 m. As shown in gure 3.6 the majority of the OMs are contained in a vertical cylinder 200 m in diameter between the depths of 1500 m and 2000 m below the ice surface. The depths of the modules in each string are shown in gure 3.7, and the horizontal arrangement of the strings is shown in gure 3.8. String 17 got stuck during deployment and never made it down to its

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3.2 The AMANDA-II neutrino telescope 15 target depth.

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Figure 3.6: Schematic of AMANDA-II neutrino telescope. The AMANDA-II experiment consists of 19 strings with a total of 677 optical modules deployed between 1.5 km and 2.5 km depths in the ice sheet at the South Pole.

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3.2 The AMANDA-II neutrino telescope 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2400 2200 2000 1800 1600 1400 1200 1000 Depth (m) String number

Figure 3.7: Depth below surface of optical modules for each string in AMANDA-II.

−100 −50 0 50 100 −100 −50 0 50 100 (m) (m) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

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to the trigger and a Time to Digital Converter (TDC) through a discriminator. The TDC saves the times when the pulses exceed and fall below the discriminator thresholds for up to eight pulses within a 32 µs buer. The delayed channel is connected to a peak Analog to Digital Converter (pADC) which saves the highest measured amplitude during which the pADC gate is open.

The main AMANDA trigger is the majority trigger which requires a certain number of hit OMs within a sliding time-window. In 2003 the majority trigger threshold was set to 24 hit OMs within 2.5 µs. The string trigger is an additional trigger for low multiplicity events which requires N hit optical modules to be hit out of a set of M consecutive modules on a string with in a time-window, as shown in gure 3.10. In 2003 the string trigger was set to 6 out of 9 hits in strings 1-4 and 7 out of 11 hits for strings 5-19 both within 2.5 µs. Once a trigger condition has been met the ADC gate opens for 9.8 µs, and after 10 µs a common stop is sent so both the TDC buer and pADC are read out and stored with a GPS time-stamp.

Figure 3.9: Schematic of AMANDA surface electronics.

The trigger settings are a trade-o between low energy threshold and detector dead time. In addition to the previously described data acquisition a parallel system

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3.2 The AMANDA-II neutrino telescope 19 has been added to AMANDA, the TWR DAQ. This system digitizes the prompt pulse from the SWAMPs and ORBs using a Transient Waveform Recorder, TWR, and the digitized pulses are evaluated in a software trigger. The TWR systems is much more exible and dead-time less which allows for a reduced low energy threshold. However, for 2003 data the system was not ready for use in a solar neutralino search.

Calibration

Once a PMT has detected a photon it takes some time for the pulse to reach the surface electronics and rise above the discriminator threshold. In addition to travel time in the cables pulse shapes are also widened as they propagate through the electronics. Thus the time the TDC measures, tTDC is delayed as compared to the

actual hit time,thit, by T0 + α/

√

A where T0 is the cable delay time, A the hit

amplitude and α the time slewing correction factor. The time calibration constants T0 and α change between years because surface hardware and settings are changed

during maintenance, tuning and upgrades.

The time calibration is done by ring a Nd:YAG laser at the surface through optical cables that are attached to diuser balls deployed with the OMs. The transit time and amplitude from the laser is pulses are measured and the estimated total uncertainty in calibrated hits is 5 ns. An alternative approach is to use the ver-tically downwards traveling atmospheric muons that are abundant in experimental data. This method produces results consistent with but less accurate than the laser method, but has the advantage that it can be done without special runs that require active personnel on the South Pole.

The geometrical location of the OMs in the ice was recorded during the de-ployment of the strings, and has been veried using laser and atmospheric muon measurements.

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Figure 3.10: String trigger principle. Hits in optical modules are marked with dashed circles. The trigger condition is fullled when X out of Y consecutive optical modules in a string are hit within a time window Tstrt. In the gure the string trigger

condition 3 out of 5 is fullled for the left hand string, but not for the right hand string. In the 2003 setup the string trigger 6 out of 9 was used for strings 1-4, and 7 out of 11 for the remaining strings. In both triggers the time window 2.5 µs was used.

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4 Simulation

. . . but if you have so advanced simulations, why do you bother making the experiment?

One of my students

Detailed Monte Carlo simulations of physics events and detector response are required to understand and analyze experimental data. Furthermore, comparisons of experimental data and simulations of known backgrounds are important in under-standing the detector response. Simulations are also used to model possible signal scenarios that this analysis searches for. The AMANDA simulation chain can be divided into three parts: event generation, particle propagation, and detector response, as shown in gure 4.1. First, particles are generated in the vicinity of the detector. Next, the generated particles are propagated through the ice and secondary parti-cles are generated. Finally, the light output from the partiparti-cles is generated and the resulting detector response to the light is simulated.

4.1 Event generators

The physics event generators uses some source model and produces physics events near the detector. In this work the generators WimpSim [26] with WimpEventF2k were used for the neutralino signal simulation, and ANIS [27] and dCORSIKA [28] for the atmospheric neutrino and muon backgrounds.

Neutralino signal

The neutralino signal is generated with the WimpSim software package which gen-erates the neutrino signal in the center of the Sun, transports the neutrinos to the detector and produces neutrinonucleon interactions in the ice. The WimpSim events are distributed in the ice and translated to the AMANDA system with the software WimpEventF2k(written by the Author of this thesis).

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Figure 4.1: Simulation software layout. Physics event generator output from ANIS, dCORSIKA or WimpSim is fed into MMC where secondary particles is added. The MMC output is then fed into AMASIM which uses Photonics for the light simulation and then simulates the detector response. AMASIM output is compared to real experiment data.

WimpSim

WimpSim [26] is a recently (2007) developed WIMP neutrino signal generator software package. It consists of two parts, WimpAnn and WimpEvent1_{. The rst part, WimpAnn,}

generates the WIMP neutrinos in the core of the Sun and then propagates them to a distance of 1 A.U. from the Sun. The second part, WimpEvent, propagates the neutrinos from WimpAnn through the Earth to a given detector location where a neutrinonucleon interaction is forced. WimpSim handles WIMP annihilations in the Earth as well as in the Sun, but for this analysis only events from the Sun were considered.

WimpAnn uses DarkSUSY [29] and PYTHIA [30] to simulate neutrino emission from WIMP annihilations in the Sun. Neutrinohadron interactions during the prop-agation from the center of the Sun to a distance of 1 A.U. are simulated using nusigma [31] 2_{. The simulation includes ν}

τ regeneration. The regeneration happens

when a charged current ντ interaction produces a τ, which in turn decays and thus

producing a new ντ. This process reduces the probability that the ντ are absorbed

which enhances the total neutrino ux to the detector. This process is simulated using PYTHIA.

1_{For this analysis the versions WimpAnn-v2.06 and WimpEvent-v2.06 were used.}

2_{For this analysis the software versions DarkSUSY-v4.1.6-lite, PYTHIA-v6.400 and}

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4.1 Event generators 23 WimpEvent also uses nusigma to propagate the neutrinos through the Earth to the detector. Both WimpAnn and WimpEvent handles full three avor neutrino oscillations during the propagation of neutrinos [26]. For the neutral and charged current neutrino (and antineutrino) proton or neutron interactions nusigma uses the CTEQ6-DIS parton distributions [32]. Neutrino interaction targets are assumed to be nonisoscalar (i.e. they may have dierent numbers of protons and neutrons).

The output from WimpSim are events with the incoming neutrino and outgoing lepton and hadronic shower. The directions of the outgoing particles will be deviate with respect to the neutrino directions depending on the neutrino energy, outgoing particle type, target mass, and collision kinematics. Scattering angles, ψ, decrease with neutrino energy and are typically smaller for muons than for hadrons. Over the year the Sun goes from +23◦ _{above the horizon to −23}◦ _{above the horizon and}

back, for this analysis the portion of the year with the Sun below the horizon was used. The distribution of scattering angles and Sun position is shown in gure 4.2.

) o ( ψ 0 20 40 60 80 rate (a.u.) -3 10 -2 10 -1 10 500 GeV soft χ : M µ ψ 3000 GeV hard χ : M µ ψ 3000 GeV hard χ : M X ψ ) o ( θ 60 80 100 120 rate (a.u.) 0 0.02 0.04 0.06 0.08 0.1 0.12

Figure 4.2: WimpEvent angles. The left panel shows the distribution of scattering angles, ψ, for the particles produced in neutrinonucleon interaction. The right panel shows the Sun zenith angle distribution at the South Pole.

Since the SUSY parameters are unknown, a large range of possible masses and annihilation channels must be considered. Accelerator searches and cosmological arguments put limits on the mass to the range 46 < Mχ < 104 GeV/c2 [11, 12]. To

cover the range separate simulations for the following neutralino masses are made: Mχ = 50, 100, 250, 500, 1000, 3000, 5000 GeV/c2.

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χχ → τ+_τ−_{. The neutrino energy spectra in the detector are nearly at up to the}

neutralino mass for hard channels, and rapidly falling with energy with a mean of 1/10 of the neutralino mass for the soft channels. The muon energy spectrum is slightly softer than the neutrino spectrum, on average the muon energy is 1/3 of the neutralino mass for hard channels and 1/12 of the neutralino mass for soft channels. Figure 4.3 shows the neutrinonucleon interaction particle energy spectra.

Figure 4.3: The left panel shows the muon energy spectra for 250 and 500 GeV/c2

for hard and soft channels.

The neutrino oscillations were simulated with parameters from the central values presented in [33] for neutrino oscillations with no CP violation and normal neutrino

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4.1 Event generators 25 mass hierarchy:

θ13= 33.2◦, θ12= 0.0◦, θ23= 45.0◦,

∆m2₂₁ = 8.1 · 10−5 eV2, ∆m2₂₃= 2.2 · 10−3 eV2 . (4.1)

All WimpEvent events interact with a nucleon near the detector. However, since the interaction probability depends on energy, target, and neutrino type each event has an associated weight factor w. At the energy scale used in this analysis the cross section for neutrinonucleon interactions for anti neutrinos is about 2/3 of that for neutrinos. The dependence of the weights on the neutrino type and energy is shown in gure 4.4. The weights for lepton and hadron tracks are constructed so that the volumetric ux Γ per annihilation can be written

Γ = 1 N N X i wi interactions cm3_annihilation , (4.2)

where N is the number of annihilations simulated. The weights for neutrinos are constructed to give an area ux Φ in [neutrinos/cm2_{/annihilation] instead.}

(GeV) ν E 0 200 400 600 800 1000 1200 1400 ) -3 w (interactions m 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -30 10 × µ ν µ ν

Figure 4.4: Weights, w, as a function of neutrino energy, Eν for νµ WimpSim events

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uniformly generated random number. With ρ and the oset angles ψ` and ψX all

angles the detector system angles, azimuth and zenith, is determined. Because of the oset angle the zenith angle distribution is wider for outgoing particles than for neutrinos, and furthermore the widening increases with decreasing neutrino energy. The geometry of the WimpSim events and zenith angle distributions for neutrinos and muons for high and low mass neutralino signal is shown in gure 4.5.

(a) Neutrino (ν) interacts and a lepton (`) and hadronic shower (X) are created in a plane tilted (with angle ρ) with scattering an-gles ψ relative to the neutrino.

θ cos -0.6 -0.4 -0.2 0 0.2 0.4 0.6 a.u. 0 0.2 0.4 0.6 0.8 1 3000 GeV hard χ : M ν θ 3000 GeV hard χ : M µ θ 50 GeV soft χ : M µ θ

(b) Comparison of muon and neutrino

WimpEventF2k zenith angles for high and low mass neutralino signal from the Sun.

Figure 4.5: Geometry of WimpEventF2k events

The interaction vertex position is randomized within a generation volume enclos-ing the sensitive detection volume. The generation volume can either be a vertical cylinder or a muonbox. The cylindrical volume is set to the sensitive detection vol-ume and the stretched so events on the edge of the volvol-ume fail to trigger the detector. Optionally the stretching can depend on the incoming neutrino zenith angle. In the case of Earth WIMP simulations with near vertical muon tracks a smaller volume

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4.1 Event generators 27 only stretched downwards is used. The muonbox is a box that is aligned with the neutrino direction with sides so that it just encompasses the sensitive volume, and stretched towards the neutrino direction. The volumes are shown in gure 4.6.

(a) Cylindrical generation volume. The sensitive volume is a vertical cylinder with radius R and height H, oset by

z0 along the z-axis. The generation

vol-ume for Solar WIMP events is the sensi-tive volume stretched by a distance L in

radius and L upwards and downwards,

for Earth WIMP events it is stretched by

Ldownwards.

(b) Muonbox volume. The volume is a box aligned with the neutrino direction, with sides

Lmin, D and S just encompassing the sensitive

volume and with the length stretched a distance

Lin the neutrino direction.

Figure 4.6: Generation volumes in WimpEventF2k.

In both cases the stretch length can be set to a xed value or to a parametrization of the maximum muon path length. The parametrization is found by assuming only continuous energy losses, and solving the energy loss equation

dE/dX = a + bE =⇒ Lµ(E) = S ρblog 1 + Eb a , (4.3)

where E is the muon energy, S = 1.2 a scaling factor to make the range conservative, ρ = 0.917kg/dm3 _{is the average ice density, and a = 0.260 GeV/mWe and b = 0.360·}

10−3 /mWe constants received by a t to detailed muon simulations as described in [34]. The muon range function is compared to the lengths of simulated muons in gure 4.7. By using a zenith angle and/or energy dependent generation volume,

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(GeV) µ E 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000

Figure 4.7: Energy versus track length for muons. The solid line is Lµ as given by

equation (4.3), the dots are muons simulated with MMC. Most muons stop before the maximal muon range because of stochastic energy losses.

The WimpSim event weights wi (see equation (4.2)) can be used to calculate the

physical number of events Nph from N generated events in a volume V by

Nph = V N

X

i

wi. (4.4)

In WimpEventF2k eventwise generation volumes Vi are introduced, thus the

corre-sponding number of observed physical events Nobs is

Nobs = N X i δiViwi where δi = 1 event observed

0 event not observed . (4.5)

The number of events in a super set volume Vgen ⊇ Vi that encloses all individual Vi

gives Ngen = Vgen N X i wi, (4.6)

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4.1 Event generators 29 and thus the eective volume Veff is

Veff = Nobs Ngen Vgen = PN i δiViwi PN j wj . (4.7)

Using Veff and the average muon length at generation level hLµi an eective area

Aeff can be written

Aeff =

Veff

hLµi

. (4.8)

Atmospheric muon background dCORSIKA

When cosmic rays enter the Earth atmosphere they may interact and produce par-ticle showers. Among the parpar-ticles produced in the atmospheric interactions only muons and neutrinos penetrate the atmosphere and deep ice so that they can be detected in AMANDA. These atmospheric muons are simulated using dCORSIKA3_which

is an AMANDA specialized adaption of the air shower generator code CORSIKA [35]. The simulation includes hadronic interactions in the atmosphere, decays of un-stable particles, tracking of particles taking ionization energy losses, scattering and the impact of the Earth's magnetic eld into account. The main modications in dCORSIKA are adaptions for a large cylindrical detector instead of a surface detector, only saving primary particles and muons that can reach the detector, and alterations in order to take into account the atmospheric prole and magnetic eld at the south pole.

In the simulations cosmic ray particles are injected in the atmosphere from an E−2.7energy spectrum with primary particle composition taken from [36]. Hadronic interactions are simulated using the QGSJET model [37]. Typically particles are generated with energies in the range 6 · 102 _{GeV to 10}11 _{GeV in the zenith angle}

range 0◦ _{to 90}◦_{. The produced muons are distributed on a surface just below the ice}

surface. Optionally events are re-sampled so that they are redistributed a chosen number of times. Only muons that enter a cylindrical volume surrounding the detector are kept for further processing 4_.

Atmospheric neutrino background ANIS

The neutrinos produced in the atmospheric interactions are simulated using the software ANIS All Neutrino Interaction Simulation [27]5_{. This program is able to}

generate events of any neutrino avor. Neutrinos are generated in the atmosphere

3_{For the work presented in this thesis dCORSIKA-v6.2041 [28] was used}

4_{Typically, as in the case of the analysis presented in this thesis, this cylinder is of height 800 m}

and radius 400 m

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4.2 Propagators

The generated particles were propagated with MMC [34], and the light output was propagated using Photonics [40] through AMASIM [41].

Lepton propagation MMC

When a muon propagates through matter it undergoes energy losses due to ioniza-tion, bremsstrahlung, photonuclear interactions, and pair production. In gure 4.8 the energy loss as a function of muon energy is depicted. The light from secondary particles produced in the energy losses make up most of the light emission from a muon track.

These processes are simulated in the muon propagation simulation software pack-age MMC Muon Monte Carlo [34, 42] 7 _{The simulation includes muon decay, cross}

section reductions due to the LPM eect [43], and scattering. MMC can propagate muons with energies in the range from 10−1 _{GeV to 10}11 GeV. MC also handles

propagation through several dierent media. Dierent media denitions are used for air, ice and bedrock, as shown in gure 4.9.

MMC divides the propagation into three separate regions: 1) before entering, 2) inside, and 3) after leaving the sensitive detector region (see gure 4.9). The sensitive detection region is taken to be a vertical cylinder surrounding the physical AMANDA detector with height and radius determined so that it is unlikely to detect light from secondary particles produced outside this cylinder 8

In the rst stage the muon energy is determined just before entering the sensitive detector region. Here stochastic energy losses larger than 5% of the muon energy are simulated, but lower energy losses are treated with a continuous energy loss

approx-6_{For this analysis the volume has a radius 800 m and a height extending from 2000 m before}

the detector center to 1000 m after the detector center

7_{For the analysis presented in this thesis the version MMC-v1.4.1 was used.}

8_{For the analysis presented in this thesis the radius 400 m and height 800 m was used for the}

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4.2 Propagators 31

Figure 4.8: Muon energy losses as a function of muon energy, adapted from [34]. At low energies ionization (ioniz) dominate the energy losses, whereas at higher energies discrete losses in the form of bremsstrahlung (brems), e+_e− _{pair (pair) production}

and photonuclear interactions (photo) dominate.

imation. In the next stage the muon is propagated through the sensitive detector region, and light from the muon and secondary particles can be detected. Secondary particles from stochastic energy losses above 0.5 GeV are treated as individual par-ticles and are kept in le output for later processing steps, whereas emission below 0.5 GeV is treated as continuous energy losses. In the nal stage the stopping point of the muon is determined as an estimation from the average muon range.

Photon propagation

Cherenkov light produced by particles traversing the ice is subject to scattering and absorption due to bubbles and dust in the ice. Typical scattering and absorption lengths in the ice are tens and hundreds of meters, respectively. The Cherenkov light is not emitted isotropically and thus the light detection eciency of the OMs depends on track and PMT orientation. Because of this, detailed photon propaga-tion simulapropaga-tions has to be conducted in order to get photon ux and arrival time

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Figure 4.9: Denitions of MMC propagation regions and propagation media. The box represents the detector, and the arrow represents a muon. The media are dened as spherical shells of dierent material and density. In region 1 the muon energy when entering the detector is determined. In region 2 more detailed simulations are done where secondary particles down to 0.5 GeV are created. In region 3 the stopping point of the muon is found.

distributions.

For the purpose of photon propagation the codes PTD [44] and Photonics [40] have been created. Photonics which is used in the analysis in this thesis is more recently developed and provides a much more detailed simulation than PTD. Fur-thermore PSInterface was created (by the author of this thesis) to form a common abstraction layer between the photon simulation implementation and the end user. The photon simulations determine the following distributions as a function of track and OM geometries:

Mean amplitude The expected number of photoelectrons detected in a PMT (depending on the simulation it may have to be scaled by PMT quantum eciency and area).

Hit time delay The additional time, compared to the expected time for a direct hit, to detect a photon in an OM due to the photons scattering in the ice. Hit probability The probability of a hit given a time delay.

The track geometry contains information about the track particle type (muon, shower, etc.), a point and time somewhere on the track, the track zenith and az-imuth angles, and in some cases the track length and energy. The OM geometry contains information about positions and orientations (up or down).

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4.2 Propagators 33 Photon propagator Photonics

The software package Photonics is used to simulate the propagation of photons from particle tracks to photo electrons registered in optical modules.

The simulation is done in two steps. In the rst step tables with the light response in the detectors are calculated, and in the second step then the tables are used in the detector simulation. This way there is no need for time consuming on-line photon propagation in the detector simulation. The table approach has the drawback that in order to accommodate the heterogeneity with necessary accuracy the Photonics tables become very large, in the order of 17 Gb.

The tables contain mean amplitude and hit time delay distributions as a function of the particle direction (θ, φ), the point of closest approach between the track and optical module ρ, the length from the track vertex to the point of closest approach L, and the optical module coordinate.

The generation of the tables is done by generating photons from sources according to the source wavelength and angular distributions. For the analysis presented in this thesis the sources are either the single Cherenkov emission along a muon track, or shower events of many short Cherenkov tracks. By creating tables with many single Cherenkov emissions added together extended muon tracks can be simulated. Next the paths of the photons are tracked through the ice. The tracking takes in to account scattering and absorption input from the ice model [22] to accommodate for the heterogeneous ice. The ux of photons are then recorded in the tables. The detector simulation AMASIM or PSInterface the interfaces to Photonics to read these tables.

Photon simulation interface PSInterface

The photon simulation interface PSInterface was written as an abstraction layer between the photon simulation implementation and the end user to ensure that all the dierent implementations can be accessed in a uniform way. Using PSInterface the same code can use any implementation except during the initialization stage. The interface is intended for use both as a library within other software and as a stand alone package interactively usable within ROOT [45]. PSInterface has been used to debug, compare and evaluate dierent simulation implementations, for ex-ample as shown in gure 4.10, for the main IceCube simulation software icesim, and for particle track reconstruction procedures in icerec. PSInterface is a pure abstract C++ interface base class, with a number of implementing classes for various photon simulation implementations, as shown in gure A.3. Each class that extends PSInterface must implement the methods shown in gure A.4 so that coordinates, mean amplitudes, time delays and hit probabilities can be accessed uniformly.

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Figure 4.10: Comparison of PTD (left) and Photonics (right) using PSInterface. The gures shows the light ux as seen in a PMT integrated over all time delays for a horizontal shower. Both gures shows an up down asymmetric due the PMT downward orientation. The discontinuities in the PTD gure are due to poor inter-polation. The forward peak in the PTD gure is caused by a missing Jacobian factor in the shower light distribution. This comparison was done in collaboration with Johan Lundberg.

4.3 Detector response AMASIM

The nal step in the chain of simulation software is the detector response simulation which is done using AMASIM9_{. A owchart of the main detector response simulation}

in AMASIM is depicted in gure 4.11.

For a given particle track AMASIM uses Photonics to get the expected number of detectable photo electrons, ¯NP.E., in each OM in the array. This value is scaled with

the eective PMT area Aeff and relative OM sensitivity ε, and the number of hits

is found by sampling a Poissonian distribution with expectation of ¯NP.E. · Aeff · ε.

For each hit Photonics is used again to sample the hit time delay distributions and generate a hit time. Additional random hits due to dark noise and after pulses in the OMs are also added to the event. The noise rate, as well as the after pulse time delay and probability are tuned for each OM.

Hit amplitudes are randomly picked from an experimentally measured single photo electron response distribution and scaled to t dierent OM types. Hits close in time are joined together as one hit, with a new amplitude as the sum of the individual hit amplitudes. For each hit a prompt and a delayed pulse are generated

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4.3 Detector response AMASIM 35 from experimentally measured pulse shapes scaled with the hit amplitude. The pulse shape used depends on the type of OM simulated. The pulses of each hit are added together in a waveform. In the process the waveform is subjected to the response function and saturation in the amplier. Like in the experiment the delayed waveform is used to determine the peak amplitude, and the prompt waveform is used to determine hit times.

These hits are used as input to the trigger simulation. Two types of triggers are simulated, the string trigger (with pre-scaling) and the multiplicity trigger. During 2003 the multiplicity trigger requirement was at least 24 hits, and the string trigger was condition was set to at least 6 hits in 9 consecutive OM's for strings 1-4 and 7 out of 9 hits for strings 5-19. The string trigger was pre-scaled so only half of the triggers were kept. Both trigger conditions apply to the hits withing a 2.5 µs wide sliding time window.

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5 Reconstruction methods

Science is what we understand well enough to explain to a computer. Art is everything else we do.

Donald E. Knuth, Foreword to the book A = B, 1996

In order to do neutrino astronomy with the observed hit patterns from particle interactions in the ice, physics properties such as particle energies and directions must be reconstructed. In AMANDA a good directional accuracy is only possible us-ing events with high energy muons. The pointus-ing accuracy of the reconstruction is limited due to the angular dierence between the neutrino and the muon direction, which is approximately 0.7◦_{· E}−0.7

ν where Eν is the neutrino energy Eν in TeV [46].

Electron and tau events produce short tracks that do not contain enough directional information for an accurate track reconstruction, but if contained they give a rea-sonable energy reconstruction. Muon events, however, are rarely contained in the detector, and thus a proper reconstruction of the energy is not possible. For this analysis the main purpose is to nd a neutrino signal coming from the Sun, and hence a good pointing accuracy is very important, which is the reason the analysis only considers muon events.

As shown in gure 5.1(a), a muon track is dened by a space point r0 = (x0, y0, z0)

where the particle is at a time t0, and the zenith and azimuthal angles (θ, φ) for its

direction. As the muon travels through the detector it emits Cherenkov radiation at an angle θc. A Cherenkov photon emitted at a point rc may be detected at a time

tobs in an optical module located at rOM, see gure 5.1(b).

Ignoring scattering and time jitter in the detector, the photons are expected to arrive at a time given by the geometrical time distance tgeo, which is given by the sum

of the time it takes the muon to travel to the point rc from r0 and the time it takes

the photon to travel in a straight line to rOM. The residual time due to scattering

and time jitter is given by tres = tobs− tgeo. The time jitter causes a Gaussian spread

around tres = 0 and the scattering causes a tail of large tres values. Additional hits

due to secondary particles contribute to the tail of the tres distribution. Randomly

distributed noise hits are seen as general increase in the rate. In gure 5.2 the eects of these factors on tres are summarized.

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(a) Muon track denition. A muon track is dened by the particle position at a certain time r0(x0, y0, z0, t0) and the

di-rection in spherical coordinates (θ, φ)

(b) Trackoptical module

geome-try. The particle travels along

the track from r0 and

pro-duces a Cherenkov photon in the

Cherenkov angle θc at a point rc.

The photon is later detected in an

optical module located at rOM.

Figure 5.1: Denitions for describing the geometry of a muon track producing a hit in an optical module.

5.1 First guess reconstruction methods

At early stages of the analysis data is dominated by downward going atmospheric muons, hence a fast reconstruction that accurately determines whether events travel upwards or downwards is required. For this purpose the fast reconstruction methods called Direct Walk [48] and Direct Wimp [49] are used. More elaborate reconstruc-tions are made on the remaining events using the JAMS [50] method as a rst guess for several maximum likelihood reconstructions. The reconstruction methods used for the analysis presented in this work are described in more detail below.

Direct Walk

Direct Walk, DW, is a rst guess algorithm that uses pattern recognition to nd the hits which are caused by most likely unscattered photons. The algorithm selects pairs of hits that are separated by more than 50 m and fulll the requirement that the dierence in the two hit times is |δt| < dOM/c + 30 ns, where dOMis the distance

between the OMs. For each hit pair a track element is dened, consisting of a vertex determined as the center of the two hits and a direction along the line connecting the hits (see gure 5.3).

Next the Number of Associated Hits, NAH, for the track element is determined

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5.1 First guess reconstruction methods 39

Figure 5.2: Hit time residuals, tres, adapted from [47]. The distribution is widened

by the time jitter (top left) with a Gaussian distribution of width σj. A high rate

of uncorrelated noise hits (top right) raises the distribution for all tres. Scattering of

light on the way to the optical module increases the travel time and higher values of tres are obtained. The average amount of scattering increases with the distance

between the track and module (bottom right). Emission from secondary particles (bottom left) adds to the tail at high tres.

d < 25(tres+30)1/4m with respect to the track element. Poor quality reconstructions

are rejected by requiring NAH ≥ 10 and σL ≡

q

N_AH−1P

i(Li − hLi)2 ≥ 20m, where

Li is the distance between the track vertex and the point of closest approach to the

optical module and hLi is the average of all Li. Often this method produces many

valid track candidates (TC).

The highest quality track candidates are selected by a requirement on the quality parameter QTC ≥ 0.7 · Qmax, where QTC = min(NAH, 0.3 · σL+ 7) and Qmax is the

maximal QTC. Finally for each remaining track candidate the number of neighboring

track candidates within a 15◦ _{degree cone is counted. The nal track hypothesis}

is taken as the average of all track candidates in the search cone with the most candidates.

For some tracks, especially low energy tracks, DW may fail to produce a valid reconstruction. For this purpose a variant, called Direct Wimp or DWimp, was developed with less strict requirements on the track candidates. In DWimp the hit pair separation requirement is lowered from 50 m to 35 m for hits on dierent strings,

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Figure 5.3: Direct Walk track element. Two hit optical modules located at r1 and

r2 are connected by a line. A track element is formed in the center between r1 and

r2 with the direction along the line between the OMs.

and from 50 m to 10 m for hits in the same string. Furthermore, the requirement on NAH is lowered from 10 hits to 8 hits. DWimp accepts less energetic tracks at

the price of poorer angular resolution.

JAMS

JAMS - Just Another Muon Search is a more elaborate pattern recognition based rst guess method. A schematic of the principle of JAMS is shown in gure 5.4.

JAMS is based on the idea that when rotating the hit positions (x, y, z) in a coordinate system aligned with the true track direction ˆrtrue, so that x0, y0 is in

a plane perpendicular to ˆrtrue while z0 is along ˆrtrue, the hits will form a

Gaus-sian cluster in x0

, y0. Furthermore hits will cluster in time along the muon direc-tion along the z0 _{axes. If the rotation is done in a dierent direction however the}

clusters will be smeared. Clusters are dened by counting the number of neigh-boring hits for each hit. A neighbor is dened as a hit fullling the requirement r =p(∆x0₎2_{+ (∆y}0₎2_{+ c}2_(∆t0₎2 _{< r}

max, where ∆x0, ∆y0, ∆t0 are the dierences in

coordinates between two hit, and rmax is a congurable value. JAMS requires at

least 7 hits to fulll the requirement to keep a cluster for a track hypothesis. Using the angles from the hypothetical track direction and the average of the hits on the cluster, a rst guess track hypothesis (x, y, z, θ, φ) is found. Next the rst guess is rened with a simplied likelihood reconstruction. Finally a quality parameter for each cluster is determined by feeding event topological variables into a neural

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5.1 First guess reconstruction methods 41

Figure 5.4: Principle of JAMS pattern recognition reconstruction. Tracks are found by rotating the hit patterns and searching for directions where the hit patterns form clusters. The found track hypotheses are improved using a simplied likelihood reconstruction, and then ordered by their reconstruction qualities. The best 3 tracks are saved for further processing.

network trained to separate high and low quality reconstructions. The topological variables include the number of early hits, the number of late hits, and the fraction of hits outside a 50 m cylinder around the track. The clusters are sorted by the quality parameter and the best 3 track candidates are saved for further analysis steps.

JAMS performance is slower but more accurate when compared to Direct Walk, but faster and less accurate when compared to the full likelihood reconstruction. JAMS has proved to be very eective in correctly rejecting tracks that the likelihood reconstructions falsely reconstructs as upwards going events. Since it examines many directions it is less prone to be confused by coincident muon events than the likelihood reconstruction.

An attempt to use the ideas in JAMS with a more advanced and steerable hier-archical clustering algorithm and angular grid has been made in CATS Cascade Analysis of Tracks and Showers, written by the author of this thesis. In CATS the

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L =

Nhits Y

i=1

p (xi|α) , (5.1)

where Nhits is the total number of observed hits in an event.

By varying the hypothesis so that L is maximized the most probable track hy-pothesis is found. In practice it is more convenient to minimize the negative loga-rithm of the likelihood instead of maximizing the likelihood,

− log L = −

Nhits X

i=1

log p (xi|α) . (5.2)

The Pandel distribution is a probability density function that includes the eects of scattering and absorption has been parametrized with a Gamma distribution [51]

p (tres, ρ) = 1 N (ρ) τ−dλtd/λ−1res Γ (d/λ) e −₍tres(1_τ+_nλac )+ρ/λa) , (5.3)

where ρ is the closest distance between track and optical module, λa the absorption

length in ice, n the refractive index of the ice, λ and τ free parameters, and the distance dependent normalization constant is N(ρ) = exp(ρ/λa) (1 + τ c/λa)

−ρ/λ

.

The constants τ = 557 ns, λ = 33.3 m, and λa = 98 m are found from photon

propagation simulations using an averaged ice model.

The Pandel parametrization does not include the electronic jitter and may also diverge at small tres when ρ < λs. The Patched Pandel distribution is an extension

that aims to solve these problems by convolving the Pandel distribution with a Gaus-sian distribution, centered at tres = 0 and with the width σj from the time jitter.

The analytical convolution turned out to be very slow and instead a smooth tran-sition function between the Gaussian distribution and the Pandel distribution was introduced by using a third order polynomial P(tres) to join the two distributions.

Now the new patched Pandel distribution is written as ˆ p (tres, ρ) =    G(tres), tres≤ 0 P(tres) 0 < tres <p2πσj p(tres, ρ) tres≥p2πσj . (5.4)

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5.2 Likelihood based reconstruction 43 Noise is added by the addition of a constant probability to the patched Pandel distribution.

The best track hypothesis α, given a set of hits with time residuals tres and

closest distances ~ρ with respect to α, is found by minimizing − log L(tres, ~ρ |α) = −

Nhits X

i=1

log ˆp (tres,i, ρi|α) . (5.5)

The minimization is done using the Simplex minimization algorithm [52].

Iterative reconstruction

The likelihood reconstruction suers from frequent ndings of local minima instead of the global minimum. The local minima may be caused by symmetries in the de-tector, multiply scattered hits arriving at unexpected times, and uncorrelated hits from random noise. Also in some cases the minimizer gets stuck at extreme zenith angles. A good rst guess seed track reduces these problems, but still misrecon-structions are not uncommon. The iterative reconstruction aims to nd the global minima by performing several consecutive reconstructions with (θ, φ) randomized in a cone around the track with the lowest negative log likelihood . With enough itera-tions the procedure frequently nds a global minimum, but at the cost of increased computing time.

Bayesian weighted reconstruction

The maximum likelihood reconstruction consider all track hypothesis equally prob-able, even though most tracks recorded in data are downwards going atmospheric muons. Using a Bayesian approach to the reconstruction the belief that most events are downwards going can be included in the reconstruction [53]. Assuming the probability density function of observing a track α is given by g(α) and using Bayes theorem to write the conditional probability density function h(α|x) of observing α α given an observation x yields

h(α|x) = R p(x|α)g(α)ˆ ˆ

p(x|α)g(α)dx. (5.6)

The probability density function g(α), as shown in gure 5.5, is obtained from the zenith angle distribution in simulations of atmospheric muons. In principle g(α) could depend on other track parameters, but in practice the zenith angle is the most important observable to reject atmospheric muon background.

The denominator in equation (5.6) is a constant, and hence can be ignored in the minimization of the negative log likelihood. With this the negative Bayesian

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) θ cos( -1 -0.5 0 0.5 1 -1 10 1 10

Figure 5.5: Zenith angle weight function g (α) versus cos θ used in the Bayesian reconstruction.

weighted logarithmic likelihood is written as − log LB(α|tres, ρ) = − Nhits X i=1 log ˆp (tres,i, ρi|α) ! − g(α) (5.7)

The probability density function g(α) can be viewed as a weight to the normal likelihood, and has the eect that tracks with poor likelihood will be considered down-going. This way atmospheric muon background events misreconstructed as upward going events may be rejected more eciently than in the normal likelihood reconstruction.

A Search for Solar Neutralino Dark Matter with the AMANDA-II Neutrino Telescope

Thomas Burgess