Showering Likelihood

The showering likelihood algorithm is intended to separate rings into two differ- ent types: a showering type and a non-showering type. The former type describes the diffused ring patterns created by electrons, positrons and gamma-rays due to the effects of electromagnetic showering and multiple scattering. See Fig. 7.4 for an ex- ample of how a showering ring appears in the event display. The latter type describes the sharper-edged ring patterns generated by more massive charged particles, such as muons and charged mesons, including kaons. See Fig. 7.5 for an example of how a non-showering ring appears in the event display.

The Cherenkov opening angle provides another feature of distinction between showering and non-showering rings. The Cherenkov angle is expected to always be maximal (42◦_{) for the lighter, showering type particles, but it may be smaller for more}

massive particles if they are not traveling at highly relativistic speeds or after they have lost energy through ionization. The fuzziness or crispness of the light pattern

Figure 7.4: Example of a showering ring.

and the Cherenkov opening angle are both exploited simultaneously by the showering likelihood algorithm.

The showering likelihood calculation depends on the ability to formulate expected charge distributions for the two ring type hypotheses. The details of those expected charge distributions will be explained first, followed by the description of the likeli- hood itself.

Expected Charge Distributions

The expected charge distributions describe the amount of charge expected to be seen in in each PMT given that the ring was produced by an electron (qexp_{(e)) to}

represent the showering type, or a muon (qexp_{(µ)) to represent the non-showering}

type. They are defined as:

q_iexp(e) = αeQexp(pe, θi)

 R ri 1.5 1 exp(ri L) f (Θi) + qiscat, (7.10) q_iexp(µ) = αµ sin2θxi ri sin θxi + ri· dθ dx|x=xi
+ q knock i ! 1 exp(ri L) f (Θi) + qiscat, (7.11) where:

αe, αµ : normalization factors

ri : distance from the vertex to the i-th PMT

θi : opening angle between the i-th PMT direction

and the ring direction

L : light attenuation length in water

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

of the photon incidence angle Θi

R : radius of the virtual sphere (16.9 m) Qexp_(p

e, θi) : expected p.e. distribution from an electron as a

function of the the electron momentum and the opening angle

x : position of the muon along its track

xi : position of the muon along its track where

Cherenkov photons are emitted toward the i-th PMT

qscatt

i (qiknock) : expected p.e.s for the i-th PMT from scattered

photons (knock-on electrons)

θ (θxi) : Cherenkov opening angle of the muon at track

position x (xi)

The expected p.e. distribution distribution for an electron, Qexp_(p

e, θi), was ob-

tained through Monte Carlo studies. Note that scattered photons are accounted for as well as direct photons by qscat

i .

The expected p.e. distribution for a muon is calculated analytically. The sin2θ de- pendence arises from the Cherenkov angle dependence of the intensity of the Cherenkov photons. The term r(sin θ + r(dθ/dx)) takes into account the shrinking size of the

Cherenkov angle as the particle loses momentum while it travels through the water.

Likelihood Calculation

The showering (e-like) and non-showering (µ-like) likelihood functions for the n-th ring are defined as:

Ln(e, µ) = Y θi<(1.5×θC) prob qiobs, q exp i,n(e, µ) + X n′_6=n q_i,nexp′ ! , (7.12)

where the product is over the PMTs inside the Cherenkov cone of the n-th ring (1.5 × θC). qiobs is the number of observed p.e.s in the i-th PMT, q

exp

i,n(e or µ) is the

expected number of p.e.s in the i-th PMT coming from the n-th ring when assuming the n-th ring was produced by either an electron or a muon. q_i,nexp′ is the same quantity,

only from the n′_{-th ring. The function prob(q}obs

i , qiobs) gives the probability of detecting

qobs

i p.e.s in the i-th PMT given the expected amount, q exp

i . The q exp

i,n(e) and q exp

i,n(µ)

expectation values are optimized by altering the direction and opening angle of the n-th ring to yield the maximum likelihood value.

The likelihood is translated into a χ2 _{parameter to allow it to be combined with}

another estimator that uses the Cherenkov opening angle. The χ2 _{value is shown}

below:

χ2

n(e, µ) = −2 log Ln(e, µ) + const. (7.13)

The probability for a ring to be of a particular type based on the light pattern is then given by the following:

P_npattern(e, µ) = exp ₋(χ

2

n(e, µ) − min[χ2n(e), χ2n(µ)]) 2 2σ2 χ2 n ! , (7.14)

where σ_χ22 n =

√

2N is the approximate resolution of the χ2 distribution, N being the number of PMTs used in the calculation.

The probability for a ring to be of a particular type based on the Cherenkov opening angle is given by:

P_nangle(e, µ) = exp ₋ θ

obs n − θnexp(e, µ)
2 2(δθn)2 !