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Report
Analysis of the Test Data of CDC Prototype-4 for the COMET Experiment
Amangaliyev Temirlan Mirlanuli
Department of Physics, Faculty of Science, KazNU named after al-Farabi
It is known that the Standard Model (SM) cannot be considered as an ultimate theory of
elementary particle physics. It should be an approximation of a deeper theory of the structure of
the microworld. Such theories are collectively referred to as "new physics" or "beyond the
Standard Model". Charged lepton transition has big potential to get new physics beyond the SM.
So, the COMET (Coherent Muon to Electron Transition) experiment is pointed to search for
violation of charged lepton flavor conversion in a presence of nuclear. All of all it is really
important to choose a high qualitative detector with small dead time, good space resolution and
etc…
Introduction
Cylindrical drift chamber (CDC) is an ionization chamber used in COMET experiment. It
composes of two conducting wires: cathode and anode, and medium gases. Once an energetic
particle pass through the chamber, it will collide into a scintillator and give the first signal. It will
also interact with the gases, ionizing it. With the help of electric field applied via the anodes,
secondary and tertiary ionizations will occur, giving an avalanche. The ionized electrons are then
attracted to the closest anode wire, thus triggering the second signal. The time difference
between the two signals is called the drift time which indicates the time ionized electrons used to
travel from the track to the wire.
So, in next picture showed geometry of prototype, which has 7 sense layers + 2 guard layers
(1 is dummy) (6 layers can be used for analysis with 1 layer as test layer),stereo-angle : ± 4 deg
tilted layer by layer, sense wire : φ25μm (Au-W), field wire : φ126μm (Al)
Geometry of detector of Spring -8 beam test
So, in my data analysis I was happy to use run with finger like scintillator with applying 2300 V
of high voltage.
Setup of experiment
Gradient tool
During the solving problems of mathematics or physics sometimes we have to find global
minimum of non-linear function, which is not trivial problem, all of all I decided to use gradient
method. Here I am going to give short idea of gradient tool without mathematical proof.
The basic idea of the method is to carry out optimization in the direction of steepest descent, and
this trend is set antigradient:
x (k+1) = x k − λk ∇f(x k )
Where λk is actually is step of iteration, so here there are three possible way to choose the step
- constant, in which case the method may diverge;
- fractional increments, imply that length during the descent is divided into a certain number;
- steepest descent;
The convergence of the gradient descent with constant step.
Let , λk = λ = cosnt the function f is differentiable and bounded below. Let condition of
Lipschitz for gradient is ‖f ′ (x) − f ′ (y)‖ ≤ L‖x − y‖. Let 0 < λ < 2/L ,then lim f ′ (x k ) = 0,
(k+1)
k→∞
k
f(x
) < f(x ) for any choice of the initial approximation. Under the conditions of the
theorem ensures the convergence of the gradient method comes to the bound or directly to
f(x ∗ ): lim f ′ (x k ) = x ∗ , f ′ (x ∗ ) = 0.
k→∞
However, in practice the methods of steepest descent are usually saddle points and finds a local
minimum of the objective function. Selection of the optimal step is usually take value like 0.01.
However in my case I decided to use step cutting instead of const step .
This picture illustrate geometrical picture of method. At each step, we shift the vector
antigradient, "a reduction in lambda times"
Tracking
Analysis Procedure in short was done by next order
Raw data
Track
Finding
Track Fitting
• Check the quality of data ,noise level, Adc sum, Tdc
distribution etc.
• Select “Good ”TDC Hits , such as discarding hits with
small ADC sum or too long driftTime
• Garfield simulation as input of XT relation
• Error minimization with gradient tool
First of all before analysis I did Garfield simulation to define X-T relation and to define Drift
velocity vs applied voltage dependence. So, I got
XT realation
Drift velocity vs applied
1st method (Atis tracking tool )
This method is pretty long( in meaning of usage) , however easy to write its script in C++/C.
Tracking tool is to calculate the fitted line on two wings separately. Then it is possible to create
two plane from
these two lines. The intersection is therefore the trajectory of the electron
One big difficulty is that the plane equations are somewhat complicate due to the x-y curvature
of each layer. So it was approximated that they are flat. And the plane equations in this
approximation are (We swap the coordinate to match with the right-hand rule):
0 = bL x ′ + aL (y ′ − λz ′ ) + 1
0 = bR x ′ + aR (y ′ + λz ′ ) + 1
The extra parameter λ is actually with being chamber's length in –z direction. Since the trajectory
is in the x direction, we can express the trajectory as a parametric vector equation x with being
the parameter:
It was defined that the definition of an error in 2 dimensional fitting is :
2nd tracking tool (Tima )
In this method error definition is differ from previous one. As it is showed in next picture
And 3 dimensional error is next
χ
2 (a,
b, c, d) = ∑ (ri −
i
|axi + byi + cz + d|
√a2 + b 2 + c 2
2
)
Here we have to note that all xi and yi are function of layer and z component. In a case of finger
like scintillator z component vary from -30 to +30.
For formula was written special directed script and algorithm is showed in next picture
Result :
Here we can see 2 dimensional(x-y plane) view of track
Where red line is track, circles are drifting circles
Spatial resolution was defended as σ is the fitted sigma of residual distribution (Measured
error). Actually by residue = hitR - fitR (hitR is measured drift distance fitR is closest
approach to the reconstructed track)
σ =160um mean=0,013mm
Additionally I did some Garfield simulation, all difficulty was in writing of gas definition.
Energy loss vs Pressure particles in mixed helium and ethane gas with ratio 50/50
Most probable energy loss of three particles in mixed helium and ethane gas with ratio 50/50
Potential of wire
Possible drift lines particles in mixed helium and ethane gas with ratio 50/50