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where θobs

n and δθn are the reconstructed opening angle of the n-th ring and the fitting

error, respectively, and θexp

n (e or µ) is the expected opening angle of the n-th ring

based on the reconstructed electron-like or muon-like momentum.

The final probability calculation is simply the product of the light pattern prob- ability calculation and the Cherenkov angle probability calculation:

P (e, µ) = Ppattern_{(e, µ) × P}angle(e, µ). (7.16) The final showering likelihood variable used to determine whether a ring is of a showering or non-showering type is defined as:

Lshow ≡p− log P (µ) − p− log P (e). (7.17)

Positive values of Lshowindicate a preference for the non-showering type, and negative

values indicate a preference for the showering type.

7.1.4

Ring Momentum

To calculate the momentum of the particle which produced the Cherenkov ring, the integrated charge within a 70◦ _{half-opening angle is used. This integrated charge}

RT OTn = GMC Gdata     α X θi,n<70◦ −50 ns<ti<250 ns 

q_i,nobsexpri L cos Θ_i f (Θi)  − X θi,n<70◦ Si     , (7.18) where: α : normalization factor

Gdata, GMC : relative PMT gain parameter for data and MC

θi,n : opening angle between n-th ring direction and

i-th PMT direction

ti : residual (= time of flight subtracted) hit time

of i-th PMT

L : light attenuation length of water

ri : distance from vertex position to i-th PMT

f (Θi) : correction function for PMT acceptance as a

function of photon incidence angle Θi

Si : expected amount of p.e.s from scattered pho-

tons for i-th PMT

The summation is restricted to a time window spanning −50 ns to +250 ns, where 0 ns represents the peak of the residual hit time distribution. The purpose of this win- dow is to exclude light that may have originated from muon-decay electrons created by muon decay.

After determining the RT OT value of a ring, the corresponding momenta for various particle hypotheses is calculated by linearly extrapolating between points on the look-up table shown in Fig. 7.6. This typically means finding the showering-like

momentum (equivalent value for electron, positron, or gamma), and the muon-like momentum. For the purposes of this study, a third particle type was added to the list: the charged kaon. To generate the points in the look-up table, Monte Carlo simulations of single ring events for each of the particle types were generated at various true momentum values, and the mean RTOT value for the corresponding true momentum was calculated and entered into the table. The kaon points were generated specially for this study.

Figure 7.6: RT OT - momentum look-up table. The blue, upper set of crosses are for electrons, the red, middle set of crosses are for muons, and the black, lower set of crosses are for charged kaons.

The energy scale stability of the detector was tested by observing the mean re- constructed energy of stopping cosmic ray muons and the decay electrons which they produced (see Section 3.7.3 for details). It varied within ±0.88% during the SK-I runtime.

The absolute energy scale was adjusted separately by observing the number of photoelectrons generated by through-going cosmic ray muon events. A variety of

calibration data were used to check the scale, such as stopping cosmic ray muons, the decay electrons they produce, and the reconstructed invariant mass from π0 _particles

produced in atmospheric neutrino interactions. By comparing these data samples to Monte Carlo simulations, the absolute calibration error was estimated to be less than ±0.74% for the SK-I data taking period.

Ring Separation

In order to properly reconstruct the momenta of particles in a multi-ring event, the charge collected by the PMTs must be correctly divided amongst the different rings. This process is called ring separation. To perform the ring separation, a likelihood function is used which describes the likelihood of the charge observed in a given PMT to belong to a particular ring:

L = X θi′ ,n<70◦ log prob qi′, X n′ αn′· q_iexp_′·n′ !! , (7.19)

where qi′ is the observed amount of charge in the i′-th PMT, q_iexp′_,n′ is the expected

amount of charge in the i′_{-th PMT coming from the n}′_{-th ring, prob(q}obs

i , q

exp

i ) is the

probability function for detecting qobs

i charge in the i-th PMT given the total expected

amount qiexp, and αn′ is the scaling factor for each ring. The scaling factor is used as

an optimization parameter, increasing or decreasing the amount of expected charge from each ring such that the total amount expected best matches the observed value.

The observed p.e.s in the i-th PMT belonging to the n-th ring are obtained as:

q_i,nobs = q_iobs αn· q

exp i,n

P

n′αn′ · q_i,nexp′

, (7.20)