This is my personal repository of my work on COMET tracking using various machine learning approaches. This work was carried out with Alex Rogozhnikov at Yandex Data Factory. The simulation data was produced using the ICEDUST software framework. I orchestrated the simulations themselves across various batch computing farms, namely the ones at CC-IN2P3, Imperial College London, IHEP, and Tianhe-2.
The Coherent Muon to Electron (COMET) a next-generation particle physics experiment is designed to investigate charged lepton flavour violation (CLFV) by searching for muon to electron conversion on an aluminium nucleus. This process is not allowed in the Standard Model of particle physics, but has very good sensitivity to Beyond the Standard Model physics.
COMET will take place in two phases. The first phase is designed to probe muon to electron conversion 100 times better than the current limit. This target limit will look for a single event in 1015 events. To give some scale to this search, we could reach a similar sensitivity if we looked at one event per minute since the beginning of the universe (13.8 billion years ago).
Unfortunately, we do not have 13.8 billion years for our search. To combat this, the COMET experiment is designed to probe millions of events per second for our elusive signal of new physics. This leads to a high intenisty environment, i.e. one with many many particles flying around in the detector.
COMET is designed to transform a high intensity proton beam (read: many protons per second) into the ideal environment to watch for our signal process. To do so, it employs a clever collection of magnets, targets, and filters to create a high intensity muon beam. These components form the "beamline" of the experiment.
The red arrow in the figure above represents the direction of the incoming proton beam. These protons hit a fixed target and interact with the target material to produce pions. This target is surrounded by a large magnet, whose magnetic field is designed to direct the pions towards the curved part of the diagram above.
These pions are unstable particles, and naturally decay into muons while they are flying through the curved section. The curved section is also a collection of magnets, which are designed to filter away and undesirable particles. This delivers a muon beam to the detector region, at the bottom left end of the diagram above. The bottom right shows two of the detector systems in use for Phase-I.
Inside the detector region, the muon beam is collides with a number of thin, fixed aluminum disks. Some of the muons lose enough momentum to stop inside the target. Once they do, they form muonic atoms, where a muon replaces an electron inside an aluminium atom in the target. This is the condition needed from which the signal process can occur. This condition happens millions of times per second, and for the vast majority of the time, we expect non-signal things to happen. Below, we can see the muons in green colliding the with silver stopping target disks. The disks are surrounded by the detector.
The signal process would yield an electron with much higher energy than the background processes. To spot the signal, we need to spot this electron. Both the detector and the stopping target are surrounded by a large magnet, which causes particle trajectories to curve and fly in a helix shape. The radius of the curve of the helix is larger for a particle with a high momentum (or energy).
With enough momentum, a particle enters the detector, which is the cylindrical volume. This cylindrical volume contains an array of coaxial wires. As the particles pass these wires, they deposit a small amount of electric charge on the wire, which is then readout and saved to a file. These are referred to as hits on the wire.
This section describes the algorithm constructed in this jupyter notebook. Please see the notebook for more plots and for the code itself.
The track finding algorithm must group signal hits in a given event so that the track fitting algorithm can fit a trajectory to the hits. As a first step, the proposed algorithm filters out all background hits. The image below shows a signal track, blue, leaving the aluminium target in the middle, entering the cylindrical detector, and leaving hits on the wires it passes. All of the hits in red are from background particles.
A hit is characterized by three main "local features" in this geometry:
- The amount of energy deposited,
- The time which the hit occurred,
- The radial distance of the hit from the target.
By design, the amount of energy deposited is already a great feature for classification. Many of the background particles leaving the red hits are protons, which deposit more energy than electrons. Traditionally, physicists would cut on this feature as the basis of a classification algorithm. The picture below compares the normalized distributions of energy depositions for signal and background particles. Note the logarithmic x-axis. The performance of the algorithm will often be compared to the performance of only using this feature, and not considering any others.
Classifying using this set of local features provides signifiant gains over
using only the energy deposition. To harvest even more classification power
from these features, the properties of neighbouring hits are also considered.
For each hit, this add four more features:
- The energy deposition on the wires to the left and right, if any
- The timing of the hit on the wires to the left and right, if any.
The classification power of these features come from the fact that signal-like hits are often flanked by other signal-like hits. These four features, combined with the original three local features, define a total set of 7 features, which I will refer to as the "neighbour feature" set.
Lets illustrate the rest of the algorithm by way of an example. First, lets start with our unfiltered, unlabelled event:
Needless to say, its not at all obvious to the naked eye if this event contains a signal track, and which of these points form a signal track shape. Lets add some labels to make it more obvious, where as before, blue is signal, red is background:
The signal points are surrounded by background points. The first stage of classification is now used to improve the situation. To this end, a GBDT is trained over the 7 neighbour features. Each hit is classified, where a score of 1 corresponds to a signal-like hit. The fill of each hit is then scaled to this score, such that outlines with no fill mean a background-like response, whereas full circles indicate a signal like response:
We can see in this event, this vastly improves our ability to spot the signal hits. Most of the background points are filtered out, while most of the signal like hits remain. With that said, there are still collections of background hits that are well separated from the signal track pattern.
To fix these isolated background points, a shape-feature is created for each hit using a circular hough transform. Essentially, the signal track radius is fed into the algorithm. The space of potential track centres is discretized. Each hit uses the signal track radius to determine which of the potential track centres its track could have originated from. In this way, each hit "votes" on its favourite track centres. This vote is weighted by the hits response from the first stage of the algorithm, such that signal-like hits get a higher vote. Graphically, we can picture this voting process as below. In this image, the orange points are the potential track centres. Their fill is weighted by how many votes they get. Each green circle corresponds to one hit, where the overlap of this green circle with a track centre awards that track centre with votes:
We can see that in this event, there are two distinct circles in the signal like track (due to the stereometry of the detector, which is not explained here). The algorithm is able to detect both of these as likely track centres. The trick now is to invert the transformation to allow the most likely track centres to pick out the hit points that they correspond to. The "best" track centre can be determined by exponentially reweighing the voting score for each track centre, then to invert the mapping. Graphically, this looks like:
With each track centre now voting on its favourite hit points, each hit point can be assigned a score that describes "how likely the hit point is to be a signal-track radius away from the most likely track centres." This defines our 8th feature. Combined with the 7 neighbour features, our new feature set is called the "track feature" set. The algorithm now trains a new GBDT over these 8 features. As before, we can visualize the output of this classifier on the sample event by weighting each hit by the response to the final algorithm. Here we drop the hit outlines and only leave the fills:
We can see that the background hits are nearly all suppressed, with the exception of a few around the signal track, whereas the signal hits are nearly all retained. Success!
Finally, here are a few plots that summarize the algorithm. Lets start with the ROC curves. I've phrased these curves in terms of signal hit retention (i.e. true positive rate) vs. background hit rejection (i.e. 1 - false positive rate), and zoomed the axis range in on the regions we care about.
This plot compares the effectiveness of:
- The baseline of cutting on the energy deposition alone
- A GBDT on the local features
- A GBDT on the local + neighbour features
- A GBDT on the local + neighbour + track features
As we can see, introducing more features consistently improves our classification abilities.