Babar/CM2 A-to-Z at Manchester

James Werner

Discriminate function for High Energy Particles Identification.

Discriminate function is a mathematical function of available information to put together the same type of things (classification process). Discriminate functions have been used for precocious diagnosis, increasing survival time and patient quality of life. It is a binary classification, exhaustively studied in the literature. This function contains the disease dynamics used to classify the patients with as little false negative diagnosis as possible. If its value is greater than zero then it means that the patient is ill, otherwise healthy. The results of discriminate function are 2 disconnected clusters: one with ill patients, and one with healthy patients.

Discriminate functions could be used in High Energy Physics to reduce the necessary effort to identify events of a reaction and minimise background.

Discriminate functions are based on intrinsic invariant characteristics of the system under study. Physical systems have several intrinsic characteristics: mass, charge, fine structure interaction, quantum numbers, etc. Each one of them describes an attribute of the system. These intrinsic characteristics are consequences of invariance. For example, conservation of linear momentum is a consequence of invariance under space displacement. Conservation of energy is consequence of invariance under time displacement. Conservation of angular momentum from invariance under spacial rotation. Conservation of mass from invariance of the Lorentz transformation.

You have to keep in mind that mass shows the amount of matter the particle has and is invariant for each type of particle. Pions have mass 140 MeV, e+- mass 0.5 MeV, photons 0 MeV, etc. Discriminate function in this case is a mathematical function (or transformation) that relates 2 or more physical variables to create clusters (regions of the plot where the particles with same characteristics are together).

Energy x Momentum Bi-parametric Histogram (EMBH).

The fundamentals of HEP established the relation between mass, momentum and energy:

E*E = p*p c*c + m*m c*c

Usually, c=1 and mass, momentum and energy are in MeV. Hence,

E*E = p*p + m*m (eq.1)

The discriminate in this case can be obtained from a plot of E against p, that will put particles with same mass together, in different regions of the plane. It will show if a particle belongs to one cluster or not. This will allow the physicist to know a priori what particle produced the track, and helps in the event selection. It is not possible to obtain m from eq.1 because we do not know the detector transfer function. The transfer function is the mathematical model of the detector. Usually it is obtained through the detector response to a known excitation (in our case, to a set of known decays). The study of the discriminate function will allow us a better understanding of the BaBar detector.

It is mandatory to measure momentum and energy from independent detectors. The BaBar experiment provides energy information from the Electromagnetic calorimeter (photons total energy transferred to the detector) and momentum measurements from the drift chamber ( p=0.3 B rho where B is the magnetic field and rho is the radius of curvature).

Plotting Energy as a function of Momentum aggregates different particles in groups with the same mass. If a particle momentum and energy matches one cluster, there is a great probability to identify the particle knowing the particles that are member of that cluster.

Figs 1 to 7 shows several EMBH. Fig 1 to 3 shows EMBH obtained from GeneratorQAAp report imported to Excel for all possible decays (even those not measured by the BaBar detector), for Kaons and Pions. Fig 4 and 5 come from Monte Carlo simulation, and Fig. 6 and 7 come from experimental data. The diagonal cluster contains electrons (particles almost without mass: 0.5Mev). The horizontal cluster are Muons because they transfer always 200MeV to the calorimeter for any momentum. The remaining clusters are charged pions and Kaons.

Fig.1 Energy x Momentum for all particles in the report. There are particles with lifetime almost nothing, and several others that the BaBar detector cannot measure in this plot. This graphic shows the general behaviour of mass for several particles (hyperboles).

Fig.2 Energy x Momentum for Kaons. Obtained from GeneratorQAAp report converted to Excel (there is no error due to file format, bugs in the program, or other mistakes!) and only Kaon events were plotted.

Fig.3 Energy x Momentum for Pions. Obtained from GeneratorQAAp report converted to Excel (there is no error due file format, bugs in the program, or other mistakes!) and only pion events were plotted.

Fig.4 Energy x Momentum from SP-1005-Tau11-R14.tcl (10,000 evts).

Fig.5 Detail in the region 0-1 Gev in the SP-1005-Tau11-R14.tcl dataset in Fig.4.

Fig.6 Energy x Momentum for Tau 1N dataset. It contains 18,700,000 events. It is possible to see the huge amount of background everywhere.

Fig.7 Energy x Momentum, but with colours and not considering the background (there is a threshold for background colour).

The use of discriminate function overcomes the problem of the different positions between the clusters obtained from the theory and from experimental data and MC forecast. This difference is due to the BaBar Detector transfer function.

Another important point is the energy distribution of events in the same cluster is not random. It depends on the previous decays, and the total available energy. This is a second step identification process, and will be analysed later to discriminate between different decays and same charged tracks.

How to identify events using EMBH.

The use of EMBH for events selection should follow the steps:
  1. Draw an EMBH diagram from the experimental data. Depending the statistics of the target particle, a few thousand events are enough. Do not use equation 1, because the detector's transfer function is not considered in it. Background from different decays will be taken in account.
  2. Model the cluster regions through mathematical functions (this is the descriminate function). There are two possible approaches:
    a. Obtain the mathematical function for the maximum value of the cluster centre and evaluate if the track belongs to the cluster by the distance between the point and the perpendicular of the maximum function.
    b. Model the boundaries of the cluster, and evaluate each if event is inside the cluster or outside.
  3. Evaluate the reaction of interest, to obtain how many tracks and what particles are expected.
  4. For each event, verify if the number of tracks match. If not, abandon the event.
  5. For each track from an event, verify what cluster it belongs to, and identify the particle that made the track. If the particle comes from the reaction of interest, verify the remaining tracks. If not, abandon the event.
  6. After selecting a set of events that came from your reaction, plot the energy distribution of each different particle from the selected events, and identify the energy peaks for each different reaction channel to confirm the events selected. If there is only one reaction producing these particles, there will be only one peak.
  7. If necessary, other criteria could be applied to the selected events.
The conventional process to identify events through the missing energy obtained from momentum and energy conservation is error prone. The consequence is a high number of background events and consequential signal/noise degradation. EMBH has potential better conditions because it uses several discriminate for each event selection, reducing background or wrong selection.

What are the clusters????

You can see 2 clusters and 2 possible particles: K and Pi.

What is the correct match?

Now we have to select events and identify these events to create a training dataset to model the discriminate function using Genetic Programming. For each particle there will be a different discriminate function that will tell us the probability the track is a Pi or a K.

I will generate 1,000 events using Moose (with contains the detector transfer function), and compare the events from the report with the recorded values in the dataset. This will allow me to identify with 100% certainty every track and every photon, generate the training dataset, and be sure about the analysis program.

Soon in this page!!!

Further developments.

This approach has potential not only in event selection, but also in Level 3 Trigger discrimination and event selection to form skimmed data sets at Tier 0/SLAC.
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Copyright 2004 Manchester University
Feedback to: jamwer@hep.man.ac.uk