**Introduction**

This page has details about how the parameterisation of the track-bump matching is progressing using real data charged pion and electron samples. The outline of this page is as follows:

- Plots of the raw delta(phi) and delta(theta), which are the angular differences in phi and theta between
the track intersection points on the Emc and the positions of bumps, for pions and electrons.
- Fits to the delta(phi) and delta(theta) charged pion distributions, split up into different cos(theta)
and momentum ranges.
- Parameterisation of the variables used in the above fits (pion and electron data).
- Plots of significance levels (based on the pion and electron hypotheses).
- Plots of matching efficiency vs cos(theta) and momentum.

The following pion data were obtained by running Bear (in 8.4.1) through the BetaPidCalib digi collections, while the electron data were obtained from Kanga skims (in 8.6.2). The pions and electrons were selected appropriately using the BetaPidCalib selectors and lists.

**Raw Pion delta(phi) and delta(theta) distributions**

Below is a complete set of plots that show the original detal(phi) and delta(theta) distributions between tracks and bumps for pion data. The data was divided up into separate momenutm ranges (shown in the table below), and for each momentum range, the angular coverage of the EMC was split up into 5 cos(theta) ranges, each range covering approximately 30 degrees. (Tracks with theta angles below 20 degrees were not considered). The distributions were made by considering the closest bump for a given track. These plots only contain tracks whose intersection points with the EMC were within 100 cm of at least one bump. For pions, if there was more than one bump closer than 20 cm, then only the highest energy bump was considered. This was done to try to accomodate for any split-offs that pions may have (e.g. a split-off may contain a larger fraction of the energy deposited by a pion, but it may not be the bump closest to the track intersection point).

Note that the pion distributions below also include fits that are described later. The cos(theta) ranges chosen were:

- -0.8 <= cos(theta) <-0.46
- -0.46 <= cos(theta) < 0.0
- 0.0 <= cos(theta) < 0.54
- 0.54 <= cos(theta) < 0.89
- 0.89 <= cos(theta) < 0.9397

**Raw Electron delta(phi) and delta(theta) distributions**

Below is a complete set of plots that show the original detal(phi) and delta(theta) distributions between tracks and bumps for electron data. The data was divided up into separate momenutm ranges (shown in the table below), and for each momentum range, the angular coverage of the EMC was split up into 5 cos(theta) ranges, each range covering approximately 30 degrees. (Tracks with theta angles below 20 degrees were not considered). The distributions were made by considering the closest bump for a given track. These plots only contain tracks that intersected with the EMC, and the chosen bump (that is closest to the track) was required to have a minimum energy of 50 MeV and have at least 4 digis in the cluster.

Note that the electron distributions below also include fits that are described later. The cos(theta) ranges chosen were:

- -0.8 <= cos(theta) <-0.46
- -0.46 <= cos(theta) < 0.0
- 0.0 <= cos(theta) < 0.54
- 0.54 <= cos(theta) < 0.89
- 0.89 <= cos(theta) < 0.9397

The general features of these distributions are that the mean values of the shifts in phi are not zero and increase as the momentum decreases, while the mean values of delta(theta) are almost zero. This is what was also seen in earlier Monte-Carlo studies.

** Parameterisations of the Fit Variables for Pion Data**

The idea here is that we want to be able to predict what the shifts (and spreads) in phi and theta should be for a given track (momentum, angle and charge), so that we can calculate likelihoods and significance levels. The following double gaussian function was used for the pion fits:

**f(x) = N * [exp( -0.5* ((x - mean)/narrowsigma)^2 ) + relNorm*exp( -0.5* ((x - mean)/widesigma)^2)]**

Below is the full set of parameterisations of the mean, spread values and relative normalisations (relNorms) of the delta(phi) and delta(theta) pion distributions for the same five cos(theta) ranges mentioned above, as a function of the reciprocal of the particle momentum (1/p). The 1/p values are negative for the negatively charged particles. (Note that the absolute normalisation values, N, don't need to be parameterised as they are not needed for likelihood and significance level calculations). Each momentum range was represented as a point at the mid-point of each range. The mean and sigma values were fitted to hyperbolic tangent functions of the form:

**f(x) = A tanh ( d ( x - a ) ) + c**

where A is the normalisation, d is some multiplicative factor, a is the value of 1/p at the
mid-point of the function and c is an offset.

The relative normalisations were just fitted by
linear interpolations between the various points.

- Mean value of delta(phi)
- Narrow sigma of delta(phi)
- Relative normalisation of delta(phi)
- Wide sigma of delta(phi)
- Mean value of delta(theta)
- Narrow sigma of delta(theta)
- Relative normalisation of delta(theta)
- Wide sigma of delta(theta)

** These parameterisations are now used in the EmcGeomTrkMatchMethod default matching method
in EmcReco V00-04-26.**

** Parameterisations of the Fit Variables for Electron Data**

The idea here is that we want to be able to predict what the shifts (and spreads) in phi and theta should be for a given track (momentum, angle and charge), so that we can calculate likelihoods and significance levels. The following double gaussian function was used for the electron fits:

**f(x) = N * [exp( -0.5* ((x - mean)/narrowsigma)^2 ) + relNorm*exp( -0.5* ((x - mean)/widesigma)^2)]**

Below is the full set of parameterisations of the mean, spread values and relative normalisations (relNorms) of the delta(phi) and delta(theta) electron distributions for the same five cos(theta) ranges mentioned above, as a function of the reciprocal of the particle momentum (1/p). The 1/p values are negative for the negatively charged particles. (Note that the absolute normalisation values, N, don't need to be parameterised as they are not needed for likelihood and significance level calculations). Each momentum range was represented as a point at the mid-point of each range. The mean and sigma values were fitted to hyperbolic tangent functions of the form:

**f(x) = A tanh ( d ( x - a ) ) + c**

where A is the normalisation, d is some multiplicative factor, a is the value of 1/p at the
mid-point of the function and c is an offset.

The relative normalisations were just fitted by
linear interpolations between the various points.

- Mean value of delta(phi)
- Narrow sigma of delta(phi)
- Relative normalisation of delta(phi)
- Wide sigma of delta(phi)
- Mean value of delta(theta)
- Narrow sigma of delta(theta)
- Relative normalisation of delta(theta)
- Wide sigma of delta(theta)

** These parameterisations are now available in the EmcGeomTrkMatchMethod default matching method
in EmcReco V00-04-39.**

**Significance Level Distributions**

**Pions**

Below are plots that show the phi, theta and total significance level distributions for charged pions, using the above pion track match parameterisations, for three angular regions (BB = backward barrel, FB = forward barrel, EC = end-cap). The total SL values are equal to (phiSL)*(thetaSL)*(1 - log(phiSL*thetaSL)).

0.2-0.3 GeV pi- SL | 0.3-0.4 GeV pi- SL | 0.4-0.5 GeV pi- SL | 0.5-0.6 GeV pi- SL |

0.6-0.8 GeV pi- SL | 0.8-1.2 GeV pi- SL | 1.2-2.0 GeV pi- SL | >= 2.0 GeV pi- SL |

0.2-0.3 GeV pi+ SL | 0.3-0.4 GeV pi+ SL | 0.4-0.5 GeV pi+ SL | 0.5-0.6 GeV pi+ SL |

0.6-0.8 GeV pi+ SL | 0.8-1.2 GeV pi+ SL | 1.2-2.0 GeV pi+ SL | >= 2.0 GeV pi+ SL |

**Electrons - using pion parameterisations**

Below are plots that show the phi, theta and total significance level distributions for electrons, using the above
**pion** track match parameterisations, for three angular regions (BB = backward barrel, FB = forward barrel, EC = end-cap).
The total SL values are equal to (phiSL)*(thetaSL)*(1 - log(phiSL*thetaSL)).

**Electrons - using electron parameterisations**

Below are plots that show the phi, theta and total significance level distributions for electrons, using the above
**electron** track match parameterisations, for three angular regions (BB = backward barrel, FB = forward barrel, EC = end-cap).
The total SL values are equal to (phiSL)*(thetaSL)*(1 - log(phiSL*thetaSL)).

**Matching efficiencies**

Plots of how the matching efficiency varies with cos(theta) and momentum are given below
for pions (using the pion parameters). The efficiency was taken to be the fraction of pion tracks
(each at least 100 cm from a bump)
that had total matching significance levels at or above the 10^{-6} level.

- Matching efficiency for pi- vs momentum
- Matching efficiency for pi+ vs momentum
- Matching efficiency for pi- vs cos(theta)
- Matching efficiency for pi+ vs cos(theta)

The plots below show the matching efficiency for electrons using the pion track match parameterisations.
The efficiency was taken to be the fraction of electron tracks that intersected the EMC
that had total matching significance levels based on the pion hypothesis at or above the 10^{-6} level.

- Matching efficiency for e- vs momentum (pi hypo)
- Matching efficiency for e+ vs momentum (pi hypo)
- Matching efficiency for e- vs cos(theta) (pi hypo)
- Matching efficiency for e+ vs cos(theta) (pi hypo)

The plots below show the matching efficiency for electrons using the electron track match parameterisations.
The efficiency was taken to be the fraction of electron tracks that intersected the EMC
that had total matching significance levels based on the electron hypothesis at or above the 10^{-6} level.
The matching efficiency for the low momentum electrons based on the electron parameters is rather low owing to
the low statisics preventing us from calibrating the low energy delta(phi) and delta(theta) distributions properly.

- Matching efficiency for e- vs momentum (elec hypo)
- Matching efficiency for e+ vs momentum (elec hypo)
- Matching efficiency for e- vs cos(theta) (elec hypo)
- Matching efficiency for e+ vs cos(theta) (elec hypo)

The plots below show how the **pion** matching efficiency varies with different total significance level cut values,
based on the **pion hypothesis**, for the three angular regions (BB, FB and EC).

The plots below show how the **electron** matching efficiency varies with different total significance level cut values,
based on the **pion hypothesis**, for the three angular regions (BB, FB and EC).

The plots below show how the **electron** matching efficiency varies with different total significance level cut values,
based on the **electron hypothesis**, for the three
angular regions (BB, FB and EC).

E-Mail: J.Back@qmw.ac.uk

Created : 4

Last significant update: 6