Minimisation Fit

The straight line fits described above have a couple of disadvantages:

• There are candidate cell configurations which can hardly be fitted properly. This might decrease the Occupancy and thus make the identification more difficult.
• The treatement of cells that are close to a bremsstrahlung shower is not clear. A possibility would be to omit them for the fit, but then cases with only few cells hit become very difficult to detect. If those cells were always taken into account the linefit would sometimes produce incorrect results.

A more complex fit using MINUIT is used to make a precise fit of the muon trajectory.

If the entire calorimeter readout information was available offline one could calculate the probability for a muon to deposit a certain amount of energy in a cell. This could be used to define a function MINUIT has to minimize.

The data available to MUFFIN only contain zero-suppressed calorimeter information (see section 6.3.1). Also here bremsstrahlung showers pose a problem because the cell energy can be very high and consequently the probability for a muon to have caused such a signal very low.

The following function is minimised by modifying the trajectory parameters (offset and direction).

(34)

is the number of cells in the UCAL. is the probability for a cell to have a signal for a given trajectory. is the probability for a cell not to have a signal for a given trajectory. These probabilities are calculated using the estimated energy loss of a muon in the UCAL material and the energy cut value of each cell (see also section 6.8.4 for a detailed description).

By splitting up the sums into sums of cells hit and not hit by the fitted trajectory one can write:

(35)

The first sum is the number of cells that have not been hit, for those cells the probability for no signal and the sum is simply the number of cells which have no signal and have not been hit:

The fourth sum contains the cells which have not been hit, for those cells the probability for a signal so the term disappears.

So we have

(36)

This can be simplified by writing:

And so the function to be minimized is:

The individual terms are:

	The total number of cells hit by the fitted trajectory.
	The number of cells of the candidate which have a signal but are not hit by the fitted trajectory.
	The sum of the probabilities for recording no signal for the cells which have no signal but are hit by the fitted trajectory.
	The sum of the probabilities for recording a signal for the cells which have a signal and are hit by the fitted trajectory.

The minimization function in (6.4) expresses the following ideas:

• It does not really matter if a muon hits a cell and does not cause a signal as long as it only traverses a small volume of that cell: In that case the total number of cells hit increases by but at the same time the probability for not causing a signal in that cell is also close to .
• If a muon hits a cell with a signal it would be good to traverse so much volume that it is possible for the muon to cause a signal in the cell.

MINUIT modifies the line parameters direction and offset such that becomes minimal.

If a candidate cell (per definition with signal) is not hit by the trajectory increases by . There is however a reason why a cell might belong to a muon candidate but still not have been traversed by the muon: The reason for this is muon bremsstrahlung showers. Here a cell can have energy without being traversed by the muon. MINUIT will try to put the candidate trajectory through these cells too. Sometimes though this is geometrically impossible. MUFFIN tries to identify cells that have possibly only energy because there was a bremsstrahlung shower. It uses the algorithm described in section 6.4.1 and 6.4.2 to tag all cells that might have energy only because there was a bremsstrahlung shower. If such a cell is not hit by the trajectory fit to the cells is not increased by .

To get the start value for the track parameters MUFFIN chooses one of the following trajectory estimates depending on which estimate gives the best HitRatio:

• A muon chamber track.
• An inner track.
• The result of the linear regression line fit.