Bigcal Clusters

The analyzer output for BigCal is a 56 by 32 matrix with energies for each block. For energy reconstruction, we first define a cluster as a 5 by 5 matrix of the blocks surrounding the most energetic one.

To find clusters, the block with maximum energy is identified (parent block). Next, the energies of blocks surrounding the parent block are recorded, as illustrated schematically in Figure \ref{fig:cluster}, forming a 5 by 5 matrix which is defined as a cluster. As the detector consists of two parts (Protvino and RCS) with different block sizes, there can be mixed clusters where the matrix is not geometrically symmetric, with an example shown on Figure \ref{fig:cluster}. When a cluster is identified, the energies of the 25 blocks are set to zero in the original 56 by 32 matrix, and the procedure is repeated to find additional clusters.

The criteria for cluster identification are:

• Minimum energy of the block must be greater than 10 MeV

• Minimum energy of parent block equal to 50 MeV

• Minimum cluster energy equal to 150 MeV

• Minimum number of blocks in the cluster with energies greater than 10 MeV must be greater than 2

It was found during the analysis that the traditional methods of coordinate reconstruction, e.g. Eq. \ref{trEq}, where \(X_i\) and \(E_i\) correspond to the centroid coordinate and energy of the block \(i\), which is used to calculate the energy weighted cluster centroid in the \(X\) coordinate, were failing, especially when the number of cells involved was small. \[\begin{aligned} \langle X\rangle =\sum_i{\frac{X_i/E_i^2}{1/E_i^2}}%\nonumber \\ % \langle X\rangle =\sum_i{\frac{X_i/E_i^2}{1/E_i^2}} \label{trEq}\end{aligned}\] The PMTs gain calibration was being affected by poor coordinate reconstruction. During the gain calibration for BigCal we noticed that \(\pi^0\) mass, reconstructed from two photon events drifted with photon energy. This effect could come either from an unexpected drift in gain parameters, from wrong coordinate reconstruction done with to the conventional clustering algorithms, or from the complicated dependence on gain parameters. To avoid the problem with incorrect coordinate reconstruction and to take into account the cuts on minimum energy of the BigCal block we decided to use a neural network (NN) method.

As input to the NN we used the energy of all 25 (5 by 5) blocks in a cluster, and the parent block’s row \(iX\) and column \(iY\) (in the 32 by 56 blocks matrix ). Using CERN’s GEANT3 simulation tool [\ref{ref:here}], we generated six million electron events and six million photon events to train the network. The photons were generated to obtain the right calibration of the BigCal detector. Photon clusters are different from electron clusters due to the effect of the target’s magnetic field on electrons, and the fact that photon showers start about one radiation length deeper in the lead glass than electron showers.

The neural network is based on a standard ROOT package [\ref{ref:here}] (TMultiLayerPerceptron). The NN has one hidden layer with 10 neurons and three outputs, \(dX\), \(dY\) and \(dE\), as shown in Figure \ref{fig:NN} where \(dX\) and \(dY\) are coordinate corrections to the parent block’s center and \(dE\) is an energy correction to the total energy of the cluster.

The neuron training functions was chosen to be a Gaussian and the learning method to be KBFGS. Figure \ref{fig:NNCOR} shows the coordinate and energy resolutions before and after corrections. Coordinate reconstruction resolution using the NN is about 3 times better than that of conventional methods.

We also observed that to remove the dependence of the reconstructed pion mass on cluster energy we need to use \(E_{clust}=\sum E_i\dot g_i+dE\) instead of \(E_{clust}=\sum E_i\dot g_i\) where \(g_i\) is the gain parameter for the block and \(dE\) is the output of the NN .

The NN provided the corrected coordinates at BigCal. Due to the magnetic field, the angles for charged particles at the target are different from the angles obtained from a straight line connecting the target coordinate to BigCal’s coordinate. To correct for the effect of the field, electron events were generated using GEANT for different field orientations and the correction to the straight line angles was obtained using the following fit function\[\begin{aligned} %\lefteqn{\nonumber (\theta_t,\phi_t) = (\theta_s,\phi_s)\cdot180/3.1415926{}}\\ \lefteqn{\nonumber (\theta_t,\phi_t) = (\theta_s,\phi_s)\cdot180/\pi{}}\\ {}& + (p_1+p_2\cdot\theta_s+p_3\cdot\phi_s+p_4\cdot\theta_s^2+p_5\cdot\phi_s^2+p_6\cdot\theta_s\cdot\phi_s)\nonumber \\ {}& (p_7+p_8/E+p_9/E^2)(p_{10}+p_{11}\cdot X_r+p_{12}\cdot X_r^2) \label{eq:fieldfit}\end{aligned}\] where \(\theta_s,\phi_s\) are angles reconstructed using the straight line approximation, \(\theta_t\),\(\phi_t\) are angles at the target, \(E\) is the energy of the cluster, and \(X_r\) and \(Y_r\) are raster coordinates. Figures \ref{fig:th_para} and \ref{fig:th_perp} show polar angle reconstruction using the straight line approximation and the fit procedure, respectively. Figures \ref{fig:phi_para} and \ref{fig:phi_perp} show the corresponding results for azimuthal angle reconstruction. The angular resolutions obtained using the NN fit are 0.5 degree for the polar angle and 1 degree for the azimuthal one.

To calibrate BigCal’s PMT gains we used neutral pion events. The events were selected from the data by choosing events with two neutral clusters (no Cerenkov signal). In addition we applied cuts on minimum energy of the cluster, \(E_{cluster}>0.6\) GeV, and on the number of non-zero energy cells in a cluster, \(N_{cell}>4\), for improved position reconstruction. Using the NN the energies of the clusters were corrected for arbitrary gain parameters, and the invariant mass of the events was calculated. The invariant mass of the event was assigned to the most energetic block in the cluster. The assumption is that the most energetic block (as in average it carries more than 50 percent of the cluster energy) is responsible for any deviation from the known pion mass. The position of the mass peak in a histogram of the mass distribution of the events was fitted and the centroid value divided by the neutral pion mass. This ratio was taken as the new gain parameter. The procedure was iterated until the parameters converged. The neural network was crucial in obtaining the correct angle and energy corrections. Figure \ref{fig:big} illustrates the pion mass resolution obtained by this procedure. This resolution is directly proportional to the cluster energy resolution of BigCal.

During the early part of the experiment BigCal opened the trigger. As a result, the ’s timing depended on which of BigCal’s rows had triggered the event. The top panel of Figure \ref{fig:cerbig} shows the uncorrected   time distribution versus BigCal’s triggering rows, for one of the   mirrors. The region of the peaks corresponds to those rows included in the geometric projection of this mirror onto BigCal. To decrease the background after the cut cerenkov TDC the time distribution for each row was fitted with a Gaussian and the peak shifted to zero. This procedure decreased the width of the one dimensional time distribution and therefore decreases the background. The bottom panel of Figure \ref{fig:cerbig} shows the timing distribution after the alignment.