Introduction

The muon fitter attempts to fit high-energy muons that travel through the PSUP. The fitter only works on single muon events with muon energies in excess of about 10 GeV.

Muons events, unlike others, not point-like interactions but rather are tracks. This fitter assumes that the muons are high-energy, and move in a straight line through the detector, moving at c. There are 5 independent parameters that fully specify a single muon track. The muon fitter fits to:

• Time the muon left the PSUP
• Position of the exit point on the PSUP (Polar coordinates of theta,phi)
• Direction vector. (Polar coordinates of eta, psi)

The exit point is defined as the point of intersection between the muon track and a sphere of radius 840.7cm from the center (the average radius of the PMT positions). Errors are also given on these measurements, but they are not particularly reliable.

subsectionAlgorithm

Because muon events are qualitatively different from other kinds, the operation of the fitter must be gone into in some gory detail. Whereas other event types set off a hundred or so PMTs, high-energy muons will typically set off 4000 of them. After this has been culled down to a list of about 1000 tubes, the fitter then makes an educated guess of the exit point, and then uses PMT timings to try to find the fit.

Most of the tubes that fire have inaccurate times because they are triggered by light that has gone through reflections and scattering. It was found that the tubes that collected the largest charges tend to have accurate times. Thus, we wish to cut away tubes with small measured charge.

Muon events are not equally bright, however, even with monoenergetic muons. This is because muons can perform high-momentum-transfer knock-on collisions with electrons, creating two relatavistic particles making Cherenkov light along the same track, and thus increasing the quantity of Cherenkov light. The fitter adapts it's charge cut limit to the event by using:

$q_{cut} = {{\rm Total\ charge} \times {\rm Charge\ bound\ ratio} \over {\rm Total\ Number\ of\ Tubes}}$

where the 'charge bound ratio' is an arbitrary constant with a value of 1.4.

After making this cut, there still remain some tubes with incorrect times. A further cut can be made by chopping the last few ( 10) tubes arbitrarily. This is optional, but can slightly improve the probability of a good fit.

The main fit stage is quite sensitive to the initial fit parameters, so starting with a good guess is nessesary to get a good final fit. The fitter starts it's guess by calculating the center-of-mass of the charge distribution. That is, it takes each pmt location and wieghts by the charge:

$\vec X_{c.m.} = {\sum\limits_i q_i \vec x_i \over \sum\limits_i q_i}$

This vector is then normalized to the diameter of the PSUP (840.7 cm) and is taken as the intial guess for the exit vertex. This guess tends to be within 100cm of the true exit position. The error on this guess is directly correlated to the (as yet unknown) track impact parameter. The exit time is taken as being the last (uncut) tube to fire. This can be taken to be the n-th last tube to fire if one has not cut tubes arbitrarily as above.

The track direction is not guessed a priori, so instead a search is performed. The fitter searches through several trial values of the zenith and azimuthal track directions as specified by the input titles bank. Each trial fit goes through a small number of iterations to try to find the best local minimum. The track direction is searched on a (theta,phi) grid with a default grid spacing of 14 degrees. The exit position is also searched on simultaneously, creating a 4-d grid. This proceedure is obviously quite time consuming, so only 25 different exit points are tried by default. This results in a total 4d grid of 5x5x25x12 points.

For each of these trials, a closed-form solution is used to minimize the exit time parameter. The figure of merit used is the $\chi^2$/tube, integrated over all tubes with $\vert t_{fit}-t\vert < t_{window}$.

Fitting is done by using the Numerical Recipies routine MRQMIN. It searches on five parameters: the exit time of the muon $t_f$, the polar coordinates of the exit position on the PSUP $\vec x_f (\theta , \phi)$, and the polar coordinates of the track direction $\hat v (\eta, \psi)$. For each PMT in the list, the time is predicted for light that came from the Cherenkov cone of the track:

$t_{i,fit} = t_f + {1\over c}[\vec r_p \sqrt{n^2-1} - h]$

where

$h = \hat v \dot (\vec x_f - \vec x_i)$

$\vec r_p = \vert\vec x_f - \vec x_i + h \hat v \vert$

$\sqrt{n^2-1} = \tan \theta_c$

Figure 17.1: Fitter Geometry

The $\chi^2$ is summed over all the tubes. When the fit converges, this value is compared to the target $\chi^2$. If the fit isn't good enough (i.e. $\chi^2$/tube hasn't reached it's defined target, 0.5 by default), the fitter starts throwing out tubes with the greatest contribution to the $\chi^2$, stopping when either the target is reached or when the limit for cutting tubes is reached.

The fitter typically fits to within 50cm radius of the true exit vertex, with an track direction error of less than 5 degrees. However, the fitter will occasionally mis-fit an event where the track has a very large impact parameter (i.e. it only skims the outer 1m of the PMT sphere).

These mis-fits can be identified by looking at the ratio of NHITS to fitted track length, i.e. $2 \sqrt((840cm)^2 - p^2)$ If this ratio falls below about 4 hits/cm, the track is a misfit and can be cut.

Track mis-fits tend to occour when either the impact parameter is very large (>800 cm) or when the muon energy is low enough to make scattering and stopping a factor.