Cluster ID – The concept
The general idea of Cluster ID is to begin with a cluster builder of some kind. The most natural place to begin is by forming clusters of hits that are contiguous. In a projective tower geometry one can imagine the following algorithm. Start with a hit cell as a seed for a cluster. Look at its 26 neighboring cells (to the sides, below and on all the diagonals). Include in the cluster any such neighbor that is hit. Iterate this process with each of the hits as it is added to the cluster. This is called a contiguous-hits cluster. Now choose a new hit cell that does not belong to any previous cluster and use it as a seed until all hit cells belong to exactly one cluster. It is not essential for the Cluster ID approach to use this particular form of cluster builder as we will see more clearly later but we use it to elucidate the concepts involved.
Once the first step is completed and a set of clusters has been defined by a cluster builder, the goal is to understand the origin of each cluster. For example, was it created by a single photon interaction, a pion interaction, a fragment from another interaction, or does it contain hits from two particles, for example two overlapping photons or a photon shower which has been intersected by the minimum ionizing track of a pion which interacts much deeper in the calorimeter? There are many possibilities for the origin of each cluster. The key assumption of the Cluster ID approach is that each type of cluster has a set of unique measurable characteristics that can be used to identify its origin with some level of certainty.
To illustrate this idea consider the difference between how a photon showers and how a pion showers. The photon’s shower is controlled by the radiation length of the calorimeter material and is typically about 10% of the interaction length of the material. The interaction length controls the showering of a hadronic particle. Thus, a photon in contrast to a pion will begin its shower early in the detector and again because of the comparatively short radiation length will interact repeatedly in a short distance producing a small compact cluster of hits. The pion will interact on average much deeper in the calorimeter and its charged secondaries will also travel interaction length distances before they interact. This results in a shower that is much less dense than that of a photon. The same is also true of a neutral hadron such as a K0-long. Thus, if we have a technique to measure shower properties that capture these ideas we can separate photons from hadrons. To separate charged and neutral hadrons we have the difference that the pion hadronic shower is preceded by a minimum ionizing track that always begins with hits in the first layer of the calorimeter, thus giving a method to also distinguish the pion from the K0L.
One can envision similar ideas to distinguish other types of clusters. A single photon shower has a cigar shape and overlapping photons that do not lie on exactly the same axis will have the shape of two cigars overlapping. Various techniques can be developed to identify this case. Similar ideas apply to pions piercing photons. In the case of overlapping pion showers there will be two minimum ionizing tracks beginning at the first layer as part of the cluster. If a pion overlaps with a neutral hadron one can consider other kinds of approaches. One can “scan” across the whole cluster and look for two concentrations of energy (or hits) with a low energy (or hit) density region in between. One can also associate the pion with a track with well measured momentum and see if the energy of the cluster which contains both the pion and the neutral hadron is too large to have been due solely to the pion.
Of course, for each of these techniques one can imagine special cases where they won’t work. For example the two minimum ionizing tracks of a two pion overlap could also be a single pion with a backscattered particle or even a neutral hadron with two backscattered particles. All these special cases and many more need to be considered. But the efficacy of the approach is only to be found by quantitative work. Surely there is no technique that works correctly every time. The question is, quantitatively how well does each technique work?
There are at least two approaches to using these identifying properties. One way is with a set of cuts for each cluster type hypothesis and the other way is letting a neural net assign probabilities for each hypothesis. In the cuts approach, for example, consider using the property of density of energy in a volume that contains the cluster. A plot of that quantity for photons of various energies will be some kind of bell-shaped distribution with a mean and width. The same distribution for neutral hadrons will have a much lower mean and a width such that a cut at some point between the two mean values will serve to distinguish the two types at some level of efficiency. If a second property is measured which also has discriminating power such as first hit layer, then for the photons the mean and width of that number will only overlap the low end of the distribution for the neutral hadron. Now the two cuts can be adjusted to optimize efficiency of identification. Additional properties can be added to the mix until the problem of making the cuts has become quite multi-dimensional and a neural net is a better candidate than using cuts. Although this illustration has been the separation of photons and neutral hadrons in the real application one is interested in separating many types at the same time.
To further elucidate the concept of Cluster ID we emphasize it is critical that we develop a set of properties that are measured for all the clusters in the calorimeter. Each type of cluster is identified by its differences with all the other types. In fact, one develops a set of properties iteratively by beginning with a set of properties and testing its correctness at identification, then looking at cases where one type is misidentified as another type and trying to add to or modify the set of properties. Contrast this with the energy flow approach where a large set of hits (those found by track extension) are simply ignored. In other words, in ClusterID there is no separate pion or photon identification algorithm. They are all identified in relation to each other simultaneously.
As with any calorimeter reconstruction technique, the goal is to use the technique investigate how well various physics measurements of interest can be made. Then having chosen a specific algorithm and a specific physics measurement one can vary the physical calorimeter parameters such as segment size, material, etc to optimize the design for the particular algorithm used to make that particular physics measurement. It is reasonable to expect that a different algorithm applied to a different physics measurement will also lead to the same design optimization but we have not progressed that far in our studies to do more than speculate about it. But that is the kind of question that points the direction for further study.