doi:10.1016/j.astropartphys.2004.06.003

Search for neutrino-induced cascades with AMANDA

M. Ackermann

, J. Ahrens

, H. Albrecht

, X. Bai

, R. Bay

, M. Bartelt

S.W. Barwick

, T. Becka

, K.H. Becker

, J.K. Becker

, E. Bernardini

D. Bertrand

, D.J. Boersma

,S.Bo

ser

, O. Botner

, A. Bouchta

O. Bouhali

,J. Braun

,C. Burgess

, T.Burgess

,T. Castermans

,D.Chirkin

B. Collin

, J. Conrad

, J. Cooley

, D.F. Cowen

, A. Davour

, C. De Clercq

T. DeYoung

, P. Desiati

, P. Ekstro

, T. Feser

, T.K. Gaisser

R. Ganugapati

, H. Geenen

, L. Gerhardt

, A. Goldschmidt

, A. Groß

A. Hallgren

, F. Halzen

, K. Hanson

, R. Hardtke

, T. Harenberg

T. Hauschildt

, K. Helbing

, M. Hellwig

, P. Herquet

, G.C. Hill

J. Hodges

, D. Hubert

, B. Hughey

, P.O. Hulth

, K. Hultqvist

S. Hundertmark

, J. Jacobsen

, K.H. Kampert

, A. Karle

, J. Kelley

M. Kestel

,L.Ko

pke

, M. Kowalski

, M. Krasberg

, K. Kuehn

, H. Leich

M. Leuthold

, I. Liubarsky

, J. Madsen

, K. Mandli

, P. Marciniewski

H.S. Matis

, C.P. McParland

, T. Messarius

, Y. Minaeva

, P. Mioc

inovic

R. Morse

,K. Mu

nich

, R. Nahnhauer

, J.W. Nam

, T. Neunho

ffer

P. Niessen

, D.R. Nygren

,H.O

gelman

, Ph. Olbrechts

C. Pe

rez de los Heros

, A.C. Pohl

, R. Porrata

, P.B. Price

, G.T. Przybylski

K. Rawlins

, E. Resconi

, W. Rhode

, M. Ribordy

, S. Richter

J. Rodrı

guez Martino

, H.G. Sander

, K. Schinarakis

, S. Schlenstedt

T. Schmidt

, D. Schneider

, R. Schwarz

, A. Silvestri

, M. Solarz

G.M. Spiczak

, C. Spiering

, M. Stamatikos

, D. Steele

, P. Steffen

R.G. Stokstad

, K.H. Sulanke

, I. Taboada

, L. Thollander

, S. Tilav

W. Wagner

, C. Walck

, M. Walter

, Y.R. Wang

, C.H. Wiebusch

R. Wischnewski

, H. Wissing

, K. Woschnagg

, G. Yodh

DESY-Zeuthen, D-15735 Zeuthen, Germany

Institute of Physics, University of Mainz, Staudinger Weg 7, D-55099 Mainz, Germany

Bartol Research Institute, University of Delaware, Newark, DE 19716, USA

Department of Physics, University of California, Berkeley, CA 94720, USA

Department of Physics, Bergische Universita

t Wuppertal, D-42097, Wuppertal, Germany

Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA

0927-6505/$ - see front matter

doi:10.1016/j.astropartphys.2004.06.003

Astroparticle Physics 22 (2004) 127–138

www.elsevier.com/locate/astropart

Universite

Libre de Bruxelles, Science Faculty CP230, Boulevard du Triomphe, B-1050 Brussels, Belgium

Division of High Energy Physics, Uppsala University, S-75121 Uppsala, Sweden

Department of Physics, University of Wisconsin, Madison, WI 53706, USA

Department of Physics, Stockholm University, SE-10691 Stockholm, Sweden

University of Mons-Hainaut, 7000 Mons, Belgium

Department of Physics, Pennsylvania State University, University Park, PA 16802, USA

Vrije Universiteit Brussel, Dienst ELEM, B-1050 Brussels, Belgium

Department of Physics, University of Maryland, College Park, MD 20742, USA

Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

Blackett Laboratory, Imperial College, London SW7 2BW, UK

Physics Department, University of Wisconsin, River Falls, WI 54022, USA

Department of Technology, Kalmar University, S-39182 Kalmar, Sweden

Departamento de Fı

sica, Universidad Simo

n Bolı

var, Caracas, 1080, Venezuela

Received 5 May2004; received in revised form 26 June 2004; accepted 30 June 2004

Available online 2 August 2004

Abstract

We report on a search for electro-magnetic and/or hadronic showers (cascades) induced byhigh-energyneutrinos in

the data collected with the AMANDA II detector during the year 2000. The observed event rates are consistent with the

expectations for atmospheric neutrinos and muons. We place upper limits on a diffuse flux of extraterrestrial electron,

tau and muon neutrinos. A flux of neutrinos with a spectrum

which consists of an equal mix of all flavors, is

limited to

(

) = 8.6

GeVcm

at a 90% confidence level for a neutrino energyrange 50 TeV to 5 PeV.

We present bounds for specific extraterrestrial neutrino flux predictions. Several of these models are ruled out.

PACS:

95.55.Vj; 95.85.Ry; 96.40.Tv

Keywords:

Neutrino telescopes; Neutrino astronomy; AMANDA

1. Introduction

The existence of high-energyextraterrestrial

neutrinos is suggested bythe observation of

high-energy cosmic rays and gamma rays. Obser-

vation of neutrinos could shed light on the produc-

tion and acceleration mechanisms of cosmic-rays,

which for energies above the ‘‘knee’’ (10

eV) re-

main not understood. Cosmic rays are thought

to be accelerated at the shock fronts of galactic ob-

jects like supernova remnants, micro-quasars, and

in extragalactic sources such as the cores and jets

of active galactic nuclei (AGN)

[1]. High-energy

protons accelerated in these objects maycollide

with the gas and radiation surrounding the acceler-

ation region, or with matter or radiation between

the source and the Earth. Charged pions, pro-

duced in the interaction, decayinto highlyener-

getic muon neutrinos and muons which further

decayinto electron neutrinos. Fermi acceleration

of charged particles in magnetic shocks naturally

leads to power-law spectra,

, where

is typi-

callyclose to 2. Hence, the spectrum of astrophys-

ical neutrinos is harder than the spectrum of

atmospheric neutrinos (

3.7

) potentiallyallow-

ing to distinguish the origin of the flux (see for

example [2]).

For a generic astrophysical neutrino source, one

expects a ratio of neutrino fluxes

1:2:0. Due to neutrino vacuum oscillations this ra-

tio changes to

1:1:1 bythe time the

Corresponding author. Address: Physics, Lawrence Berke-

ley National Laboratory, 1 Cyclotron Road, Berkeley 94720,

USA. Tel.: +1 510 495 2488; fax: +1 510 486 6738.

E-mail address:

mpkowalski@lbl.gov (M. Kowalski).

128

M. Ackermann et al. / Astroparticle Physics 22 (2004) 127–138

neutrinos reach the Earth

[3]. Recentlya search

with the AMANDA detector was reported

[4],

resulting in the most restrictive upper limit on

the diffuse flux of muon neutrinos (in the energy

range 6–1000 TeV). Clearly, a high-sensitivity to

neutrinos of all neutrino flavors is desirable. The

present paper reports on a search for a diffuse flux

of neutrinos of all flavors performed using neu-

trino-induced cascades in AMANDA.

2. The AMANDA detector

AMANDA-II

[5]

is a Cherenkov detector

consisting of 677 photomultiplier tubes (PMTs)

arranged on 19 strings. It is frozen into the Antarc-

tic polar ice cap at a depth ranging mainlyfrom

1500 to 2000 m. AMANDA detects high-energy

neutrinos byobservation of the Cherenkov light

from charged particles produced in neutrino inter-

actions. The detector was triggered when the num-

ber of PMTs with signal (hits) reaches 24 within a

time-window of 2.5

The standard signatures are neutrino-induced

muons from charged current (CC)

interactions.

The long range of high-energymuons, which leads

to large detectable signal event rates and good

angular resolution results in restrictive bounds on

neutrino point-sources [6].

Other signatures are hadronic and/or electro-

magnetic cascades generated byCC interaction

and

. Additional cascade events from all

neutrino flavors are obtained from neutral current

interactions. Good energyresolution, combined

with low-background from atmospheric neutrinos

makes the studyof cascades a feasible method to

search for extraterrestrial high-energyneutrinos.

3. Update on cascade search with AMANDA-B10

Before the completion of AMANDA-II, the

detector was operated in a smaller configuration.

The results for the search of neutrino induced cas-

cades in 130.1 effective days of the 10-string

AMANDA-B10 detector during 1997 have been

reported before [7]. The same analysis has been ap-

plied to 221.1 effective days of experimental data

collected during 1999. The AMANDA detector

in 1999 had three more strings than in 1997, yet

data from these strings were not used in this anal-

ysis, so that the detector configuration used in the

1999 neutrino induced cascade search is verysim-

ilar to that of 1997.

Signal simulation for the analysis of 1999 data

was improved to the standards reported in this let-

ter. No events were found in the 1999 experimental

data after all selection criteria had been applied.

We will present results supposing a background

of zero events.

Using the procedure explained in this letter we

obtain an upper limit on the number of signal events

90%

= 2.75 at a 90% confidence level, from which

we calculate the limit on the flux of all neutrino fla-

vors. Assuming a flux

consisting of an

equal mix of all flavors, one obtains an upper limit

90%

=8.9

GeVcm

. In the calcula-

tion of this limit we included a systematic uncer-

taintyon the signal detection efficiencyof ±32%.

About 90% of the simulated signal events for this

limit have energies between 5 and 300 TeV, while

5% have lower and 5% have higher energy. Differ-

ences between this result and the one obtained with

1997 experimental data[7]are due to the larger live-

time in 1999 and improved simulation.

4. Data selection and analysis for AMANDA-II

The data set of the first year of AMANDA-II

operation comprises 1.2

triggered events col-

lected over 238 days between February and

November, 2000, with 197 days live-time after cor-

recting for detector dead-time.

The background of atmospheric muons was

simulated with the air-shower simulation program

CORSIKA (v5.7)[8]using the average winter air

densityat the South Pole and the QGSJET had-

ronic interaction model [9]. The cosmic raycom-

position was taken from [10]. All muons were

propagated through the ice using the muon

propagation program MMC (v1.0.5)[11]. The sim-

ulation of the detector response includes the prop-

agation of Cherenkov photons through the ice as

well as the response of the PMTs and the surface

electronics.

M. Ackermann et al. / Astroparticle Physics 22 (2004) 127–138

129

Besides generating unbiased background

events, the simulation chain was optimized to the

higher energythreshold of this analysis. By

demanding that atmospheric muons passing

through the detector radiate a secondarywith an

energyof more than 3 TeV, the simulation speed

is increased significantly. A sample equivalent to

920 days of atmospheric muon data was generated

with the optimized simulation chain.

The simulation of

and

events was done

using the signal generation program ANIS (v1.0)

[12]. The simulation includes CC and neutral cur-

rent (NC) interactions as well as

production

in the

channel near 6.3 PeV (Glashow reso-

nance). All relevant neutrino propagation effects

inside the Earth, such as neutrino absorption or

regeneration are included in the simulation.

The data were reconstructed with methods de-

scribed in Ref. [7]. Using the time information of

all hits, a likelihood fit results in a vertex resolu-

tion for cascade-like events of about 5 m in the

transverse coordinates (

) and slightlybetter in

the depth coordinate (

). The reconstructed vertex

position combined with a model for the energy

dependent hit-pattern of cascades allows the

reconstruction of the energyof the cascade using

a likelihood method. The obtained energyresolu-

tion in log

lies between 0.1 and 0.2. The per-

formance of the reconstruction methods have

been verified using in situ light sources.

Eight cuts were used to reduce the background

from atmospheric muons bya factor

. The

different cuts are explained below. The cumulative

fraction of events that passed the filter steps are

summarized in Table 1.

Since the energyspectrum of the background is

falling steeplyone obtains large systematic uncer-

tainties from threshold effects in this analysis.

For example, an uncertaintyof ±30% in the pho-

ton detection efficiencytranslates up to a factor

±1

uncertaintyin rates. Such effects can explain

the discrepancies of Table 1 in passing efficiencies

between atmospheric muon background simula-

tion and experimental data. However, as will be

shown later, the threshold effects are smaller for

harder signal-like spectra.

At the lowest filter levels (cuts 1 and 2), varia-

bles based on a rough

first-guess

vertex position

reconstruction are used to reduce the number of

background events byabout a factor of 30. It is

useful to define the time residual of a hit as the

time delayof the hit time relative to the time ex-

pected from unscattered photons. The number of

hits with a negative time residual,

early

, divided

bythe number of all hits,

hits

, in an event should

be small. This first cut criterion is effective since

earlyhits are not consistent with the expectation

from cascades, while theyare expected from long

muon tracks. Cut 2 enforces that the number of

the so called direct hits,

dir

(photons having a

time residual between 0 and 200 ns), is large.

Cut 3 is a requirement on the reduced likelihood

parameter resulting from the standard vertex fit,

vertex

< 7.1 (see also [7]). Note, that the likelihood

Table 1

Cumulative fraction of triggered events passing the cuts of this analysis

# Cut variable Exp. MC

Atm.

early

hit

< 0.05 0.058 0.033 0.94 0.63

dir

> 8 0.030 0.016 0.89 0.57

vertex

< 7.1 0.0027 0.0012 0.39 0.35

energy

vs.

reco

0.0018 0.00077 0.35 0.26

60 <

reco

< 200 0.0010 5.9

0.28 0.18

reco

vs.

reco

8.6

5.1

0.26 0.15

> 0.94 9.7

4.8

0.040 0.091

reco

> 50 TeV 8

2.8

0.029

Values are given for experimental data, atmospheric muon background Monte Carlo (MC) simulation, atmospheric

simulation and

signal simulation with an energyspectrum

. The flavor

was chosen to illustrate the filter efficiencies, since interactions of

always lead to cascade-like events.

130

M. Ackermann et al. / Astroparticle Physics 22 (2004) 127–138

parameter is, in analogyto a reduced

, defined

such that smaller values indicate a better fit result,

hence a more signal-like event. In a similar man-

ner, the resulting likelihood value from the energy

fit,

energy

, is used as a selection criterion (cut 4).

However, since the average value of

energy

has

an energydependence, the cut value is a function

of the reconstructed energy,

reco

. Cut 5 on the

reconstructed

coordinate,

reco

, was introduced

to remove events which are reconstructed outside

AMANDA and in regions where the simulation

of the ice properties for photon propagation is

insufficient. While the upper boundarycoincides

roughlywith the detector boundary, the lower va-

lue is about 100 m above the geometrical border of

the detector. Restricting

reco

improves signifi-

cantlythe description of the remaining experimen-

tal data (for example the reconstructed energy

spectrum)

[13]. Onlyevents reconstructed with a

radial distance to the detector

-axis,

reco

<100

m, are accepted (cut 6), unless their reconstructed

energies lie above 10 TeV. For each decade in en-

ergyabove 10 TeV one allows the maximal radial

distance to grow by75 m. This reflects the fact that

the cascade radius,

increases as a function of en-

ergy, while the expected amount of background

decreases.

Three discriminating variables are used to form

the final qualityparameter

1. The value of the reduced likelihood parameter

resulting from the vertex fit,

vertex

. Note that

this variable has been used previouslyin cut 3.

2. The difference in the radial distance of the ver-

tex position reconstructed with two different

hit samples,

. While the first reconstruction

is the regular vertex reconstruction using all

hits, the second reconstruction uses onlythose

hits outside a 60 m sphere around the vertex

position resulting from the first reconstruction.

Since the close-byhits typicallycontribute most

to the likelihood function, their omission allows

to test the stabilityof the reconstruction result.

If the underlying event is a neutrino-induced

cascade, the second reconstruction results in a

vertex position close to that of the first recon-

struction. In case of a misidentified muon event,

removing hits located close to the vertex typi-

callyresults in a significantlydifferent recon-

structed position.

3. The cosine of the angle of incidence cos

reconstructed with a muon-track fit. The

muon-track fit assumes for the underlying like-

lihood parameterization that the hit pattern

originates from a long range muon track. The

fit allows to reconstruct correctlya large frac-

tion of the atmospheric muons.

The final qualityparameter is defined as a like-

lihood ratio:

Þþ

;

where

runs over the three variables.

(h = s for

signal and h = b for background) are probability

densityfunctions defined as

Þ¼

Þþ

ÞÞ

(

) corresponds to the proba-

bilitydensityfunctions of the individual variables

for background due to atmospheric muons

and signal consisting of a flux of

with a spectral

slope

Þ/

. Theyare obtained from

simulations.

The distributions of the individual variables as

well as of the likelihood ratio

are shown in

Fig. 1 for experimental data, atmospheric muon

background and signal simulations. The experi-

mental distributions of

and

vertex

approxi-

matelyagree with those from the simulation

while the distribution of cos

shows some larger

deviations. The deviation reflects a simplified

description of the photon propagation through

the dust layers in the ice [13]. The experimental

distribution is not perfectlydescribed bythe

atmospheric muon simulation, which is mainlyre-

lated to the mismatch in the cos

distribution.

The related uncertainties in the cut efficiencies

are included in the final results.

At this stage of the event selection one is left

with events due to atmospheric muons, which hap-

pen to radiate (mostlythrough bremsstrahlung) a

We define the cascade event radius as the direction

averaged distance from the vertex at which the average number

of registered photon-electrons is equal to 1.

M. Ackermann et al. / Astroparticle Physics 22 (2004) 127–138

131

large fraction of their energyinto a single electro-

magnetic cascade. The reconstructed energy

corresponds to that of the most energetic second-

ary-particle cascade produced in the near vicinity

of the detector. To optimize the sensitivityof the

analysis to an astrophysical flux of neutrinos, a

further cut on the reconstructed energy,

reco

was introduced.

The sensitivityis defined as the average upper

limit on the neutrino flux obtained from a large

number of identical experiments in the absence

of signal

[14,15]. The sensitivitywas calculated

for a flux of

with spectrum

. A flux of

was used for optimization, since

-induced events

always have cascade-like signatures. The sensitiv-

ityis shown in

Fig. 2

as a function of the

reco

and

cut.

> 0.94 and

reco

> 50 TeV were cho-

sen in this two dimensional optimization proce-

dure such that the average upper limit is lowest.

With these cuts the expected sensitivityfor an

spectrum of electron neutrinos is 4.6

(

/GeV)

GeV

0.02

0.04

0.06

0.08

0. 1

0.12

0.14

5.8

6.2 6.4 6.6 6.8 7

vertex

entries (norm. to 1)

experiment

atm. MC

0.02

0.04

0.06

0.08

0.1

0.12

1 0.80.60.4

0.2

0.2 0.4 0.6 0.8 1

cos

entries (norm. to 1)

experiment

atm. MC

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 5 10 15 20 25 30

[

]

entries (norm. to 1)

experiment

atm. MC

entries (norm. to 1)

experiment

atm.

0.0

0.0 0.1 0.2 0.3

0.4

0.6 0.7 0.8 0.9 1

0.5

∆ρ

Fig. 1. Normalized distribution of the three input variables

vertex

, cos

and

as well as the resulting likelihood variable

Shown are experimental data as well as atmospheric muon and signal MC simulations after cut 6.

Likelihood Parameter L

log

reco

/GeV)

4.7E07

5E07

6E07

7E07

4. 2

4. 4

4. 6

4. 8

5. 2

5. 4

0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98

Fig. 2. Optimization of final cuts. The sensitivityfor the diffuse

flux of

is shown as a function of cuts on

reco

and

. The

coefficient next to the contour lines correspond to the average

upper limit in units of (

/GeV)

GeV

132

M. Ackermann et al. / Astroparticle Physics 22 (2004) 127–138

The energyspectra of experimental data as well

as signal and background simulations after all but

the final energycut are shown in

Fig. 3. Note that

the energyspectrum begins at 5 TeV, since this

is the lowest energyfor which the optimized back-

ground simulation is applicable. The number of

events due to simulated atmospheric muons was

normalized to that observed in the experiment.

One experimental event passes all cuts, while

events are expected from atmospheric

muons and a small contribution from atmospheric

neutrinos.

The spectrum as obtained from simulation of

atmospheric muons passing cut 7 was normalized

to the number of experimental events resulting in

an expectation of 0

events due to atmos-

pheric muons. The three main sources to the error

are given bylimited statistics of simulated atmos-

pheric muon events (the error of

was deter-

mined using the Feldman–Cousins method

[15]),

uncertainties in the cut efficiency(±20% obtained

from variation of the cuts) and limited knowledge

of the ice properties (±12% obtained from varia-

tion of the ice properties in the simulation). The

total error was obtained byadding the individual

errors in quadrature.

The predicted event number from atmospheric

neutrinos simulated according to the flux of Lipari

[16] is 0

, where the uncertainties are mainly

due to uncertainties in ice properties (error of

±0.03 obtained from variation of the ice properties

in simulation), and in detection efficiencies of

Cherenkov photons (

obtained from variation

of the photon detection sensitivityin the simula-

tion). The theoretical uncertainties in the flux of

atmospheric neutrinos was estimated to be about

25%

[17]

and is small when compared with the

other uncertainties. Again, the total error was

obtained byadding the individual errors in

quadrature.

The uncertaintyin the detection efficiencyof

neutrino events from an astrophysical flux with a

spectral index

2 are estimated to be not larger

than 25%. Because of the flatter energyspectrum,

the uncertainties related to the energythreshold

(such as the photon detection efficiency) result in

smaller uncertainties in rate when compared to

the uncertainties found for atmospheric neutrino

events. The main sources of error are again uncer-

tainties in the simulation of the ice properties

(±15%) and the detection efficiencies of the Cher-

enkov photons (±20%).

The experimental event which passed all selec-

tion criteria is shown in Fig. 4.

The sensitivityof the detector to neutrinos can

be characterized byits effective volume,

eff

,or

area,

eff

, remaining after all cuts are applied.

eff

represents the volume, in which neutrino interac-

tions are observed with full efficiencywhile

eff

represents the area with which a neutrino flux

can be observed with full efficiency. While the con-

cept of

eff

is more intuitive because it relates to

the geometrical size of the detector, the concept

eff

is more convenient for calculations of neu-

trino rates (see Eq.2in Section 5).

Fig. 5shows

eff

as obtained from simulation

for all three neutrino flavors as a function of the

neutrino energy. The effective volume has been

averaged over all neutrino arrival directions. As

can be seen,

eff

rises for energies above the thresh-

old energyof 50 TeV. Above PeV-energies

eff

de-

creases for

and

, an effect related to both

4 4.5 5 5.5 6 6.5

log

reco

/GeV)

events / 197 d / 0.2

experiment

atm.

signal MC (

)

Fig. 3. Distributions of reconstructed energies after all but the

final energycut. Shown are experimental data, atmospheric

muon simulation and a hypothetical flux of astrophysical

neutrinos. The final energycut is indicated bythe line with the

arrow.

M. Ackermann et al. / Astroparticle Physics 22 (2004) 127–138

133

reduced filter efficiencies and neutrino absorption

effects. In the case of

, the volume saturates be-

cause of regeneration effects:

and be-

cause of the event

event topology: there is an

increase in detection probabilityfor CC

interac-

tions (with energies above

GeV) because the

cascade from the hadronic vertex and the cascade

arising from the subsequent tau decayare sepa-

rated far enough in space to be detected as inde-

pendent cascades. The light from the cascade

which is further awayfrom the detector is thereby

attenuated enough not to influence the reconstruc-

tion and selection procedures which were optim-

ized for single cascades.

Fig. 6

shows

eff

as obtained from simulation

for all three neutrino flavors as a function of the

neutrino energy. Note that

eff

is small because

of the small neutrino interaction probability,

which is included in the calculation of

eff

(but

not in

eff

~ 200 m

Fig. 4. The experimental event which has passed all selection criteria is displayed from the side (left) and from above (right). Points

represent PMTs, and shaded circles represent hit PMTs (earlyhits have darker shading, late hits have lighter shading). Larger circles

represent larger registered amplitudes. The light pattern has the sphericityand time profile expected from a neutrino induced cascade.

The arrow indicates the length scale.

0.2

0.4

0.6

0.8

1.2

eff

[

0.01 x km

]

w/o earth

with earth

log

(E /GeV)

Fig. 5. Effective volume for

and

interactions as a

function of the neutrino energy. The effective volume is shown

without including Earth propagation effects (full line) and with

Earth propagation effects (dashed line).

134

M. Ackermann et al. / Astroparticle Physics 22 (2004) 127–138

The detector sensitivityvaries onlyweaklyas a

function of the neutrino incidence angles. How-

ever, because of neutrino propagation effects effec-

tive area and volume are suppressed for neutrinos

coming from positive declinations.

The effect of the resonant increase of the cross-

section for

at the Glashow resonance is not in-

cluded in

eff

shown in Fig. 6. For energies between

6.7

and 10

6.9

GeV the average effective area

including all effects from propagation through

Earth and ice is

eff

. Because of the large

interaction cross-section, the effective area con-

verges rapidlyto zero for positive declinations.

For negative declinations, it is in good approxima-

tion independent of the neutrino incidence angle.

5. Results

Since no excess events have been observed

above the expected backgrounds, upper limits on

the flux of astrophysical neutrinos are calculated.

The uncertainties in both background expectation

and signal efficiency, as discussed above, are in-

cluded in the calculation of the upper limits. We

assume a mean background of 0.96 with a Gaus-

sian distributed relative error of 73%, and an error

on the signal detection efficiencyof 25%. For a

90% confidence level an upper limit on the number

of signal events,

90%

= 3.61, is obtained using the

Cousins–Highland [18] prescription implemented

byConrad et al. [19], with the unified Feldman–

Cousins ordering [15]. Without anyuncertainties

the upper limit on the number of signal events

would be 3.4.

The effective area can be used to calculate the

expected event numbers for anyassumed flux of

neutrinos of flavor

(

model

;

eff

;

with

being the live-time. If

model

is larger than

90%

, the model is ruled out at 90% CL. Table 2

summarizes the predicted event numbers for differ-

ent models of hypothetical neutrino sources.

Thereby, the spectral forms of

and

are as-

sumed to be the same (the validityof this approx-

imation is discussed in

[13]). Furthermore, full

mixing of neutrino flavors is assumed, hence

= 1:1:1 as well as a ratio

Electron neutrinos contribute about 50% to the

total event rate, tau neutrinos about 30% and

muon neutrinos about 20%. For the sum of all

neutrino flavors the various predicted fluxes are

shown in Fig. 7.

Table 2

Event rates and model rejection factors (MRF) for models of astrophysical neutrino sources

Model

model

2.08 0.811 1.28 4.18 0.86

SDSS [20]

4.20 1.91 2.77 8.88 0.40

SS Quasar [21]

8.21 3.57 5.30 17.08 0.21

SP u [22]

33.0 13.0 20.5 66.6 0.054

SP l [22]

6.41 2.34 3.98 12.7 0.28

Ppp + p

[23]

5.27 1.57 2.86 9.70 0.37

[23]

0.84 0.40 0.56 1.80 1.99

MPR [24]

0.38 0.18 0.25 0.81 4.41

The assumed upper limit on the number of signal events with all uncertainties incorporated is

90%

= 3.61.

0.5

1.5

2.5

eff

[

]

w/o earth

with earth

log

(E /GeV)

Fig. 6. Effective area for

and

interactions as a function

of the neutrino energy. The effective area is shown without

including Earth propagation effects (full line) and with Earth

propagation effects (dashed line).

M. Ackermann et al. / Astroparticle Physics 22 (2004) 127–138

135

The models byStecker et al.

[20]

labeled

‘‘SDSS’’ and its update

[21]

‘‘SS Q’’, as well as

the models bySzabo and Protheroe

[22]

‘‘SP u’’

and ‘‘SP l’’ represent models for neutrino produc-

tion in the central region of Active Galactic Nu-

clei. As can be seen from

Table 2, these models

are ruled out with

90%

model

0.05–0.4. Further

shown are models for neutrino production in

AGN jets: a calculation byProtheroe

[23], which

includes neutrino production through p

and pp

collisions (models ‘‘P pp + p

’’ and ‘‘P p

’’) as well

as an evaluation of the maximum flux due to a

superposition of possible extragalactic sources by

Mannheim et al. [24] (model ‘‘MPR’’). The latter

two models are currentlynot excluded.

For a neutrino flux of all flavors with spectrum

one obtains the limit:

GeVcm

For such a spectrum, about 90% of the events

detected have neutrino energies between 50 TeV

and 5 PeV, with the remainder equallydivided

between the ranges above and below. The limit is

shown in

Fig. 7

as a solid line ranging from 50

TeV to 5 PeV.

To illustrate the energydependent sensitivityof

the present analysis we restrict the energy range for

integration of Eq. (2) to one decade. Byassuming

a benchmark flux

Þ¼

?ð

log

ÞjÞ

where

= 1/(GeVcm

ssr)

represents the unit flux and

the Heaviside step-

function (restricting the energyrange to one dec-

ade), one obtains the number of events for a given

central energy

event

(

). The limiting flux at

the energy

is then given by

90%

(

90%

event

(

). The superposition of the limiting

fluxes as a function of the central energyis shown

inFig. 8. For a flux

the analysis has its

largest sensitivityaround 300 TeV.

The mentioned strong increase in effective area

at the energyof the Glashow resonance allows set-

ting of a limit on the differential flux of

at 6.3

PeV. Re-optimizing the final energycut for events

log

(E /GeV)

(

)

[

GeV s

]

4 4.5 5 5.5 6 6.5 7 7.5 8

Σν

Fig. 8. Illustration of the energydependency. The curved solid

line represents a superposition of the limiting fluxes of a series

of power law models

restricted to one decade in

energy. The limit for a flux

without energyrestriction

is shown for comparison. The range of the dashed line

represents the range of energies in which 90% of the signal

events are detected while the embedded solid line represents the

energyrange in which 50% of the signal events are detected,

with the remainder equallydivided between the ranges above

and below.

SDSS

SP l

P p

SS Q

MPR

log

(E /GeV)

(

)

[

GeV s

]

4 4.5 5 5.5 6 6.5 7 7.5 8

Fig. 7. Flux predictions for models of astrophysical neutrinos

sources. Models represented bydashed lines are excluded bythe

results of this work. Models fluxes represented bydotted lines

are consistent with the experimental data. The labels are

explained in the text. The solid line corresponds to the upper

limit on a flux

136

M. Ackermann et al. / Astroparticle Physics 22 (2004) 127–138

interacting through the Glashow resonance results

in an optimal cut,

reco

> 0.3 PeV. No experimen-

tal event has been observed in that energyrange,

which results in an upper limit on the number of

signal events of

90%

= 2.65 assuming ±25% uncer-

tainties in the signal expectation and negligible

background expectation. The limit on the flux at

6.3 PeV is

3PeV

Þ¼

GeV

Throughout the present paper, flux ratios at

production of 1:2:0 are assumed to result, via neu-

trino mixing, in ratios of 1:1:1 at the Earth

[3].

Next we brieflydiscuss possible caveats. In case

of neutrino production through pp collisions one

expects as manyneutrinos as anti-neutrinos––a

fact not altered byneutrino oscillations. However,

in case of neutrino production through the reac-

tion p

, the ratio of neutrinos to anti-

neutrinos is different as onlyenergetic electron

neutrinos are produced. The

produced in the

decays of neutrons carry only very small fractions

of the parent energies, and hence for most neutrino

spectra contribute negligible to the high-energy

flux of neutrinos. A larger flux of

will result

from neutrino oscillations. Assuming the neutrino

oscillation parameters currentlyfavored byexper-

imental data (see

[25]

for a recent analysis):

=33

=45

and

one expects the

following effects on the flux of neutrinos and

anti-neutrinos from p

production:

Hence after oscillations,

would contribute

1/15

of the total flux of neutrinos (in contrast to 1/6 in

the case of pp interaction).

For CC and NC interactions at the rather high-

energies relevant for this analysis, the detection

probabilityfor anti-neutrinos becomes similar to

that of neutrinos and hence the distinction of the

production modes is not so important. However,

since only

interact through the Glashow reso-

nance, both neutrino production mechanism and

neutrino oscillations have to be taken into account

when translating the above limit to the flux of

the source.

6. Conclusion

We have presented experimental limits on dif-

fuse extragalactic neutrino fluxes. We find no evi-

dence for neutrino-induced cascades above the

backgrounds expected from atmospheric neutrinos

and muons. In the energyrange from 50 TeV to 5

PeV, the presented limits on the diffuse flux are

currentlythe most restrictive. We have compared

our results to several model predictions for extra-

galactic neutrino fluxes and several of these models

can be excluded.

Results from the first phase of AMANDA, the

10-string sub-detector AMANDA-B10, have been

reported in[7]and an update to the analysis was

presented above. Compared to AMANDA-B10,

the analysis presented here has a nearly ten times

larger sensitivity, mainly achieved through using

the larger volume of AMANDA-II and byextend-

ing the search to neutrinos from all neutrino

directions.

The limits presented here are also more than a

factor of two below the AMANDA-B10 limit ob-

tained bysearching for neutrino-induced muons

[4]and roughlyas sensitive as the extension of that

search using AMANDA-II 2000 data [26].

(Assuming a neutrino flavor ratio of 1:1:1, the

numerical limits on the flux of neutrinos of a

specific flavor (e.q.

) reported in the literature

are 1/3 of the limits on the total flux of neutrinos.)

The limits obtained from a search for cascade-like

events bythe Baikal collaboration [27] are about

50% less restrictive than the limits presented here.

With the present analysis one obtains a large

sensitivityto astrophysical neutrinos of all flavors

and in particular to electron and tau neutrinos.

Hence, given the large sensitivityto muon neutri-

nos of other search channels, AMANDA can be

considered an efficient all-flavor neutrino detector.

Acknowledgments

We acknowledge the support of the following

agencies: National Science Foundation––Office of

Polar Programs, National Science Foundation––

Physics Division, University of Wisconsin Alumni

Research Foundation, Department of Energy, and

M. Ackermann et al. / Astroparticle Physics 22 (2004) 127–138

137

National EnergyResearch Scientific Computing

Center (supported bythe Office of EnergyRe-

search of the Department of Energy), UC-Irvine

AENEAS Supercomputer Facility, USA; Swedish

Research Council, Swedish Polar Research Secre-

tariat, and Knut and Alice Wallenberg Founda-

tion, Sweden; German Ministryfor Education

and Research, Deutsche Forschungsgemeinschaft

(DFG), Germany; Fund for Scientific Research

(FNRS-FWO), Flanders Institute to encourage

scientific and technological research in industry

(IWT), and Belgian Federal Office for Scientific,

Technical and Cultural affairs (OSTC), Belgium;

IT acknowledges support from Fundacio

n Vene-

zolana de Promocio

n al Investigador (FVPI), Ven-

ezuela; DFC acknowledges the support of the NSF

CAREER program.

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