Muon Energy reconstruction in IceCube and neutrino flux measurement

Dmitry Chirkin, University of Wisconsin at Madison, U.S.A.,MANTS meeting, fall 2009

Muon Energy reconstruction in IceCube

• parameterization of light pattern created by a muon

• fitting of event data to this light pattern

• calibration of the fitted parameter to get the muon energy

IceCube DOM

3 ATWD channels with gains ¼/2/16Up to 12 s combined waveform length

Up to 200-300 p.e./10 ns charge resolution

Number of photons vs. muon energyIn ice muon energy loss is dE/dx=a+bE with

a=0.26 GeV / mwe (1 mwe = 1/0.917 m of ice)b=0.36 10-3 / mwe

A bare muon generates Cherenkov photons

This is about 32440 Cherenkov photons per meter of muon track at visible wavelengths.

From Geant-based simulations each cascade left by a muon generates as much light as a bare muon with the length of track of

4.37 m E / GeV for electromagnetic cascades3.50 m E / GeV for hadronic cascades

For a typical muon the average is ~ 4.22 meters E/GeV

A typical cascade emits 4.2232440 = 1.37 105 photons/GeV for a muon track the “photon density parameter” isArea . Nc [m] = 32440 [m-1] (1.22+1.36 . 10-3 E/[GeV]) . 492.10 cm2 . F

= 2107.84 [m] (1.22+1.36 . 10-3 E/[GeV])

F=PMT efficiency, glass/gel transmission, etc.

Parameterization of the photon field left by a muon

The flux function: total expected number of photons μ arriving at each OM.

The parameterization of the flux function (as used by an icetray module MuE) is based on the following premises:

μ = Nl · μ0(d), where Nl is the average number of photons emitted per unit length of a muon track and d is the distance to the track. precise if the muon track is infinite and emits the same number of Cherenkov photons per unit length anywhere along its track.

In the immediate vicinity of the track:

Far:

We can stitch these together with

Other flux function parameterizations

• based on PDF evaluated by explicit photon propagation simulations with photonics, which take into account the exact ice structure

semi-infinite muon parameterization light saber (uniform cascades along track)

• the above treatment employs layered ice treatment as well, but through the average scattering and absorption approximations.

• fitting to decreasing amount of light along track• fitting to segments of the muon track• single OM energy estimates along track

Fitting data to the parameterized photon field

likelihood function for the track hypothesis used in event reconstruction is:

The total number of photons observed by an OM is:

the corresponding expectation is:

since

In the presence of systematic uncertainties in the flux function, expression above can be integrated over the possibilities allowed by the uncertainties, or one employs the 2 sum minimization instead, with errors accounting for both statistical and systematic uncertainties.

Energy calibration with simulation and resolution

Muon true (simulated) energy at the closest approach point to the center of gravity of hits in the event (weighted with charge)

Energy proxy: reconstructed number of cherenkov photons per unit length times effective area of the PMT

~ 0.3

dE/dx vs. number of Cherenkov photons

• reconstructing dE/dx: a convenient approximation number of Cherenkov photons is almost proportional to dE/dx

• final “calibrated” energy parameter is what is most convenient to one’s analysis: Rate of energy loss, or dE/dx: best, e.g., for muon bundles Muon energy at closest approach point to center-of-gravity of hits

Muon energy reconstruction

Conclusions:

• several light parameterization schemes exist

• various fitting algorithms are used

• Energy resolution of ~ 0.3 in log10(E [GeV]) is normally achieved

Neutrino energy spectrum unfolding

• event selection

• parameter distributions

• smearing/unfolding matrix

• summary of unfolding techniques

• verifying the unfolding algorithm

• measuring the neutrino spectrum

Event selection

8548

eve

nts

4492

eve

nts

854

8 e

ven

ts

449

2 e

ven

ts

2290

eve

nts

229

0 e

ven

ts

275.5 days of IceCube (22 strings) taken in 2007

My own framework for applying cuts: SBM (subset browsing method)

30 parameters identified to separate signal and backgroundStep 1: constructs surface separating signal from backgroundStep 2: additional requirements for similarity with simulated signal

atmospheric satmospheric s

(sim

ula

ted

s

and

s)

90 – 180o

90 – 120o

120 – 150o

150 – 180o

~90% 95% 99% purity

Muon energy resolution

Precision of the energy measurement: reconstructed vs. simulated true:

~ 0.3 in log10(E)

True (from simulation) muon energy distribution

reconstructed muon energy distribution

simulationdata

Parameter distributions

Reconstructed zenith angle distribution

datasimulation

datasimulation

Center of gravity (COG), or “average” event depth

Point-spread function (PSF):Median angular resolution is ~ 2o.

2400 2200 2000 1800 1600center of gravity depth [m]

horiz

onta

l

vert

ical

up

Neutrino energy from reconstructed muon energy

Transformation/unfolding matrix

What we have:

muon energy at detector with 0.3 in log10(E) resolution

and its zenith angle with ~1.5o resolution

What we want:

muon neutrino energy distribution

The transformation matrix is known from the simulation and relates muon and neutrino numbers:

m=An

Unfolding methods

Performance of the following unfolding methods was studied:

Simple inversion and no-regularization 2 and likelihood minimization

SVD (singular value decomposition):• regularizing with the 2nd derivative of the unfolded statistical weight• regularizing with the 2nd derivative of the unfolded log(flux)

This is the selected method as it has the best behavior for:constant spectral index regularization term goes to 0best identification of deviations from the given

spectrum• also added the likelihood term describing fluctuations in the unfolding matrix

Bayesian iterative unfolding:• with and without smoothing of the unfolding matrix

Statistical uncertainties

The following method is selected:

Expand the regularization term in the vicinity of the minimum: constant term sum of first derivatives, creating a bias for counts in each bin sum of second derivatives, which tightens the minimum

Introduce modified likelihood function by keeping the Poisson sum, and only the bias term from the regularization term (so that the minimum found during the unfolding does not change). However, do not include the sum of second derivatives of the regularization term.

Vary the unfolded counts in each bin (independently) till modified likelihood function increases by ½.

Testing for bias, diffuse E-2 flux

Testing for spectral index, charm contribution

rqpm

Errors from belt construction, ½-likelihood estimate

From 1000 simulations For a single representative simulation

Including fluctuations of the smearing matrix

Unfolded data For a single representative simulation

9938 15 4.6 2.1 0.3

cf. AMANDA-II 2000-3: ~ 1.2

1.9 0.5 0.1

preliminary

Unfolded data at 2 different quality levels

preliminary preliminary

Unfolded data with only events in the top or bottom

preliminary preliminary

Conclusions and Outlook

• Despite some residual problems in detector simulation, agreement with Barr. et al. (Bartol) muon neutrino flux is demonstrated

• Improving the simulation is actively pursued, and the result with reduced systematic (and smearing matrix statistical) uncertainties is forthcoming

preliminary