Particle-Flow and Program Concept

The energy flux of stable particles at the interaction point (IP) can be presented as a ``hedgehog'' of vectors in the direction space $\vec{\Omega}$:

$$E_{IP}(\vec{\Omega}) = \sum_{i}^{N_{IP}} \delta(\vec{\Omega}... ... m^{2}_{i}} = \sum_{i}^{N_{IP}} \delta(\vec{\Omega}_{i}) E_{i}$$ (1)

where: $N_{IP}$ is number of particles appearing at the IP with $\vec{p}_{i}$, $m_{i}$, $E_{i}$ - the momentum, mass and energy of each particle.

The charged tracks change their initial direction due to the bending in the strong magnetic field that leads to jet energy flux distortion at the calorimeter inner surface.

The energy flux at the calorimeter surface $E_{face}(\vec{\Omega},\vec{r})$ can be written as:

$$E_{face}(\vec{\Omega},\vec{r}) = \hat{\cal{T}} E_{IP}(\vec{\... ...sum_{k}^{N_{face}} \delta(\vec{\Omega}_{k}) E_{k}(\vec{r}_{k})$$ (2)

where: $\hat{\cal{T}}$ is the transport operator through the relatively small amount of matter (VTX + TPC); $N_{face}$ is the number of particles that reached the calorimeter surface; $\vec{r}_{k}$ is the particle impact point at the calorimeter surface and $\vec{\Omega}_{k}$ is the particle direction at the calorimeter surface. The operator $\hat{\cal{T}}$ includes: small multiple scattering collisions and small energy losses, particle decay, and the effect of motion in the magnetic field for charged particles.

The energy deposition distribution in the calorimeter volume $E_{dep.}(\vec{r})$ can be written as:

$$E_{dep.}(\vec{r}) = \hat{\cal{S}} E_{face}(\vec{\Omega},\vec{r})$$ (3)

where: $\hat{\cal{S}}$ is the shower transformation operator. It produces the spatial energy distribution in calorimeter volume from the initial particle energy flux at the calorimeter surface.

The operator $\hat{\cal{S}}$ can be formally represented as a sum of operators of separated particle types (ID) with different behaviour in the calorimeter.

$$\hat{\cal{S}} = \hat{\cal{S}}_{e} + \hat{\cal{S}}_{\gamma} +... ...hat{\cal{S}}_{\mbox{\tiny ID}} \delta(\mbox{\scriptsize ID}) }$$ (4)

The amplitude ($A_{ic}$) distribution in the calorimeter electronic cells can be written by the simplest operator $\hat{\cal{J}}$:

$$A_{ic}(\vec{r}_{ic}) = \hat{\cal{J}} E_{dep.}(\vec{r}) = \!\int\limits_{V_{ic}}\!\! C_{s} \; E_{dep.}(\vec{r}) \; dV$$ (5)

where: $C_{s}$ is the conversion coefficient that depends on the calorimeter sampling structure and particle ID;
$V_{ic}$ - electronic cell volume with $\vec{r}_{ic}$ as a center.

After substitutions we have the set of equations for $A_{ic}(\vec{r}_{ic})$ that are measurable amplitudes at the center $\vec{r}_{ic}$ of electronic cell volume $V_{ic}$:

$$A_{ic}(\vec{r}_{ic}) = \sum_{\mbox{\tiny ID}} \sum_{k}^{N_{... ...ny ID}}} \sqrt{\vec{p}^{2}_{k}(\vec{r}_{k})+m^{2}_{k}} \; dV$$ (6)

The number of such equations is equal to the number of calorimeter cells for event with non-zero amplitude in it.

The backward problem for this system of equations is the particle-flow reconstruction problem.
It was solved in some way for many different HEP detectors.

Analogs of such a kind of problem (successfully solved!) are the adjoint problem for the design of nuclear power reactors and the image reconstruction in the modern gamma-chamber in nuclear medicine.

Taking into account the definition of operator $\hat{\cal{S}}$ one can recognize well known calorimeter energy-flow formula in this equations, that is a sum of energy over all calorimeter cells: $\mathcal{E} = {\mathcal{E}}_{e} + {\mathcal{E}}_{\mu} + {\mathcal{E}}_{charg.had.} + {\mathcal{E}}_{\gamma} + {\mathcal{E}}_{neut.had.}$.

In reality any calorimeter cell can carry the signals from many initial particles!

A few more remarks on this problem:

These formulae are the strict mathematical definition of the bootstrap method for the fast simulation programs.

The equations can be rewritten as a sum of charged and neutral particles separately with the parameterization of shower operators $\hat{\cal{S}}$ to make an analytical investigation of its properties.

If one reduces the cell volume up to zero the overlapping still would take place for the shower volume.

At this stage of the investigation of the equations they are not solved directly, but as it was said early, they were solved indirectly in many HEP experiments.

TESLA calorimeter reconstruction program SNARK is one of such a solution.

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Figure 12: The well measured particle momentum substitutes the pretty random energy spread in the calorimeter volume. This leads to decreasing of the energy fluctuations in general. The vector subtraction from the overlapped showers is more effective in comparison with the scalar subtraction due to the implication of momentum vector.

The idea of solution is based on the shape of the shower operator $\hat{\cal{S}}_{\mbox{\tiny ID}}$ straightforward. If one have the ``exactly'' (with very good accuracy) measured particle energy and its direction - so, one can take this knowledge and looks for the calorimeter hits along the predicted particle direction and separates this part of event from the others, that is clear but for the fine calorimeter cell structure only.

Figure 13: The Particle-Flow algorithm in the reconstruction is: to substitute carefully the particle energy measured by calorimeter for the energy injected by well measured charged tracks along the predicted shower development direction, for each individual charged particle in every event.

The algorithm starts from the finding the charged track core. The track core is the collection of the calorimeter hits which are at the distances less then one cell size from the extrapolation of the TPC measured track (helix curve) into the calorimeter volume.

Such a procedure works well for the muon track with energy more then 2 GeV (at 3 Tesla), for the lower energy a helix can be replaced by more sophisticated curve with taking into account the energy losses. It also works well for the primary ionization part of the hadron track (that is about 20 % of whole deposited energy in the hadron shower). It also works for the electromagnetic shower but due to the fact that the electromagnetic shower is rather short ($<$ 20 cm) and transversally compact in W-Si structure of TESLA ECAL.

The track core collects almost all hits for muon track, the main part of hadron track primary ionization and rather big part of the electromagnetic shower. The longitudinal distribution and average density of hits can be tested for different particle hypothesis even at the level of track core collection.

The next step of the procedure is based on the previous knowledge and estimations; it is the hit collection around the track core at the distances of two cell sizes - called first shell (see picture).

The particle identification is repeated for the new cluster that includes as a track core as a first shell and ... so on.

The iterative procedure is finished when the collected energy become to be ``equal'' to the input particle momentum. For the muon hypothesis procedure is stopped just after the first step. The word ``equal'' means equal inside the calorimeter energy resolution, more exactly it is particle momentum plus/minus tuning parameters that depend on input energy, calorimeter sampling and particle hypothesis. We will not describe here in details all cases that program trying to resolve during the iterative procedure - its number is about 35 for now. There are different particle hypothesis and their combinations with taking into account the showers overlapping (one can see the text of program).

At the end of loop around charged tracks all hits belong to them are collected in clusters, labeled and they will not use more at the next step of the reconstruction.

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The program starts the procedure to collect and separate the neutral part of event after the collecting all clusters for charged particles.

This algorithm applied the hit histograming technique in $\Theta - \Phi$ space to find at first the super-clusters then clusters inside the super-clusters.

Then it collects additional hits around the core of found clusters and it try to create the for every neutral particle as hadron as electromagnetic one.

The super-clusters in $\Theta - \Phi$ coordinate space are constructed by combining all calorimeter hits which are within a certain distance $\Delta_1$ in $\Theta - \Phi$ space. If two super clusters overlap, the energy inside the overlapped region is assigned to both super-clusters weighted by the total energy of the respective super-cluster. The properties of the super-cluster are calculated: energy, center position and angular radius.

Clusters inside the super-cluster are reconstructed by combining hits assigned to the super-cluster which are within a certain distance $\Delta_2<\Delta_1$ in $\Theta - \Phi$ space. For the overlapping clusters the same method as above is applied. Then the cluster properties are calculated: energy, number of hits, hit density, center position, principal cluster axes, parameters of electromagnetic shower hypothesis.

Each cluster is assigned to a particle type based on its predicted and calculated shape parameters for gamma or hadron hypothesis. The remaining hits are joined to the closest cluster. The clusters whose main axes are overlapping within a certain window are merged into one cluster. Finally all cluster properties are recalculated.

This procedure is done for both ECAL and HCAL separately. ECAL and HCAL clusters are joined if they match a certain criteria.

Separation of overlapping clusters.

Assuming that the cluster profile in transverse direction can be described by an exponential law, the cluster shape in a logarithmic $\theta-\Phi$ space is simple a cone. If two clusters overlap, the energy of cells in the overlap region is assigned to both clusters, weighted by the linear distance to the respective cluster axis. In the current version of the software up to six clusters are allowed to overlap, at most three in one point.