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Shivam Sawarn

[WIS22] Predicting the electron Invariant mass from collision data

Shivam Sawarn, Delhi University

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Predicting the electron Invariant mass from the CERN dielectron collision data
by Shivam Sawarn
Hindu College, Delhi University

Abstract: The project aims at exploring various features of the dielectron collision using rigorous data analysis tools. The dataset consisting 100,000 dielectron events in the invariant mass range 2-110 GeV is also used to train the neural network. The neural network is used on test dataset to predict the invariant mass of the electron.

Introduction

Since the 1970s, particle physicists have described the fundamental structure of matter using an elegant series of equations called the Standard Model. The model describes how everything that we observe in the universe is made from a few basic blocks called fundamental particles, governed by four forces. Physicists use the world’s most powerful particle accelerators and detectors to test the predictions and limits of the Standard Model. Over the years it has explained many experimental results and precisely predicted a range of phenomena, such that today it is considered a well-tested physics theory. Scientists believe that we may see a unification of forces in the high energy spectrum which is why the collision at high energies are of interest.
There are four fundamental forces at work in the universe: the strong force, the weak force, the electromagnetic force, and the gravitational force. They work over different ranges and have different strengths. Despite its name, the weak force is much stronger than gravity but it is indeed the weakest of the other three. Three of the fundamental forces result from the exchange of force-carrier particles, which belong to a broader group called “bosons”. Each fundamental force has its own corresponding boson – the strong force is carried by the “gluon”, the electromagnetic force is carried by the “photon”, and the “W and Z bosons” are responsible for the weak force.

Z- Boson

Discovered in 1983 by physicists at the Super Proton Synchrotron at CERN, The Z boson is a neutral elementary particle which along with its electrically charged cousin, the W - carries the weak force .
The weak force is essentially as strong as the electromagnetic force, but it appears weak because its influence is limited by the large mass of the Z and W bosons . Their mass limits the range of the weak force to about 10 - 18 metres, and it vanishes altogether beyond the radius of a single proton .
Physicists working with the Gargamelle bubble chamber experiment at CERN presented the first convincing evidence to support this idea in 1973. Neutrinos are particles that interact only via the weak interaction, and when the physicists shot neutrinos through the bubble chamber they were able to detect evidence of the weak neutral current, and hence indirect evidence for the Z boson .
Although more time and analysis is needed to determine if this is the particle predicted by the Standard Model, the discovery of the elusive Z bosons set the stage for this important development . The 2013 Nobel Prize in Physics has been awarded to two of the theorists who formulated the Higgs mechanism, which gives mass to fundamental particles like electrons.

Theory

Particle theorists seldom include the

’s and

ℏ

’s in their formulas. One is just supposed to fit them at the end, to make the dimensions come out right i.e. one may assume

c=ℏ=1

Location of a particle is described by a 4-component coordinate vector :

=(t,x,y,z)

Particle kinematics is described by 4-component vector:

=(E,

)

Energy and 3-component momentum of a particle moving with velocity ‘v’ are given by:

E=γmandp=γmv,whereγ=

From Mass-Energy Equivalence,

where,

is the invariant mass of the electron-positron pair. There is a special axis defined through the geometry of particle physics experiments, namely the Beam axis - the axis parallel to the incoming beams. This axis is uniquely defined. Usually, this beam axis is chosen to be the z-axis. In most cases, the position, where the beams are brought to collision, is pretty well-known which is used to fix an “origin” of the coordinate system. Pseudorapidity is a function of the production angle with respect to a beam axis. It is defined as:

η=-lntan



where polar angle θ describes the angle of a particle with respect to the z-axis, η =0 means the produced particle is perpendicular to the beam axis and high pseudorapidity means it is closer to the beam. It is a good approximation of the true relativistic rapidity when a particle is relativistic. It entangles energies and longitudinal momentum. It is useful to have other quantity that characterises momenta in collider experiments. For symmetry reasons, the azimuthal angle φ is usually employed, thus we use Transverse momentum, defined by

⊥

It is clear that physical scattering amplitudes should better be independent of the frame they’re calculated or measured in. For 22 type collider experiment, calculation in Standard Model yields the following formula for the invariant mass ‘M’:

(cosh(

)-cos(

)

where

is the transverse momentum, η is the pseudorapidity, and ϕ is the azimuthal angle.

Collider experiment

We take up a derived dataset having presence of two electrons with invariant mass between 2-100 GeV.

This is a figure of an event recorded from Compact Muon Solenoid(CMS) detector of Large Hadron Collider (LHC) used to extract the data of particles and record in the '.csv' file used below. Each track shown by a distinct line contains the information of various particles for example muon, electron, neutrino and photon tracks in the figure here later used to analyze different phenomena. The figure to the right of event is the Feynman diagram of the process where Z particle is the intermediate vector boson.

Importing the dataset:

In[]:=

cern=CloudImport["https://www.wolframcloud.com/obj/Stoicishiv/dielectron.csv"]

Out[]=

{{Run,Event,E1,px1,py1,pz1,pt1,eta1,phi1,Q1,E2,px2,py2,pz2,pt2,eta2,phi2,Q2,M},{147115,366639895,58.7141,-7.31132,10.531,-57.2974,12.8202,-2.20267,2.17766,1,11.2836,-1.03234,-1.88066,-11.0778,2.14537,-2.34403,-2.07281,-1,8.94841},

⋯99997⋯

,{146511,523243830,54.4622,11.3526,11.8809,51.924,16.4328,1.8678,0.808132,-1,8.58671,0.378009,3.07828,8.00705,3.10141,1.67717,1.44861,1,4.6967},{146511,524172389,7.64,0.886162,5.4789,-5.25033,5.5501,-0.842662,1.41044,1,52.1088,16.8075,-4.6051,49.1084,17.427,1.75925,-0.267427,-1,36.5043}}

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Preprocessing

Removing ‘Event’ and ‘Run’ column:

In[]:=

cernd=Drop[cern,0,2]

Out[]=

{{E1,px1,py1,pz1,pt1,eta1,phi1,Q1,E2,px2,py2,pz2,pt2,eta2,phi2,Q2,M},{58.7141,-7.31132,10.531,-57.2974,12.8202,-2.20267,2.17766,1,11.2836,-1.03234,-1.88066,-11.0778,2.14537,-2.34403,-2.07281,-1,8.94841},

⋯99997⋯

,{54.4622,11.3526,11.8809,51.924,16.4328,1.8678,0.808132,-1,8.58671,0.378009,3.07828,8.00705,3.10141,1.67717,1.44861,1,4.6967},{7.64,0.886162,5.4789,-5.25033,5.5501,-0.842662,1.41044,1,52.1088,16.8075,-4.6051,49.1084,17.427,1.75925,-0.267427,-1,36.5043}}

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First two columns were just meant for the identity of the data included, hence removed.

Removing extra keywords:

In[]:=

cernd[All,KeyMap[Replace["px1 "->"px1"]]];

Identifying non-numerical data:

In[]:=

CountDistinct[Cases[cernd,{___,"nan"|"N/A",___}]]

Out[]=

Replacing non-numerical with the mean of the column it lies in:

In[]:=

filtercern=cernd/."nan"->Mean[cernd[[2;;200,17]]]

Out[]=

⋯99997⋯

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Extracting a numerical dataset:

In[]:=

filtercernd=filtercern[[2;;100001,All]];

Statistical summary of the dataset:

In[]:=

ResourceFunction["StatisticsSummary"][filtercernd]

Out[]=

	Column 1	Column 2	Column 3	Column 4	Column 5	Column 6	Column 7	Column 8	Column 9	Column 10
Count	100000	100000	100000	100000	100000	100000	100000	100000	100000	100000
NonNumeric	0	0	0	0	0	0	0	0	0	0
Mean	36.4365	0.135897	0.182291	-1.50804	14.4122	-0.0640955	0.0216143	-0.00548	44.0029	-0.00398357
StandardDeviation	41.2162	13.405	13.4703	51.6037	12.3887	1.46214	1.79956	0.99999	46.7511	13.1274
Min	0.377928	-250.587	-126.079	-840.987	0.219629	-4.16538	-3.14158	-1	0.4725	-233.73
25%	8.45786	-5.23381	-5.27735	-15.8599	3.77073	-1.28392	-1.52706	-1	11.0555	-4.79534
Median	21.717	0.141338	0.099092	-0.312987	12.9678	-0.0611785	0.034324	-1	25.2646	-0.035638
75%	50.0023	5.71442	5.648	13.2121	20.0189	1.1444	1.56234	1	66.9251	4.81942
Max	850.602	134.539	147.467	760.096	265.578	2.62297	3.14142	1	948.375	227.33
Mode	—	—	—	—	—	-2.15485	1.45766	-1	—	—
columns 1–10 of 17

This summary affirms the dataset we have preprocessed from ‘count’ and ‘non-numeric’ parameters of each column. Several more features are given in the table which gives an idea of the dataset.

Dataset Features

◼

Energy

Plotting different features of energy for both the particles:

In[]:=

GraphicsRow[{Labeled[GraphicsRow[{ListPlot[filtercernd[[All,{1,17}]],Frame->True,FrameLabel->{"E1","M"},PlotLabel->"Distribution of Energy in Mass Range"],Histogram[filtercernd[[All,1]],GridLinesAutomatic,ChartStyleEdgeForm[None],PlotTheme->"Scientific",PlotLabel"Distribution of Energy(GeV)"]},ImageSize800],"Particle 1"],PairedSmoothHistogram[filtercernd[[All,1]],filtercernd[[All,9]],PlotRange{{0,200},All},FillingAxis,FillingStyleGreen,AxesLabel{"E1","E2"},ImageSizeLarge,PlotLabel"Energy Probability density distribution of both particles"]},ImageSizeFull]

Out[]=

Energy of the particle 1 lies mostly below 50 GeV, but is distributed uniformly in the mass range. This is due to decay of the electrons in several light-particles.

Distribution of energies of both the particles are quite symmetrical.

◼

Momentum

Plotting different features of the momentum(x, y and z components) of particles and their combined scatter plot:

In[]:=

GraphicsRow[{Labeled[GraphicsGrid[{{ListPlot[filtercernd[[All,{2,17}]],Frame->True,FrameLabel->{"px1","M"}],Histogram[filtercernd[[All,2]],GridLinesAutomatic,ChartStyleEdgeForm[None],PlotTheme->"Scientific"]},{ListPlot[filtercernd[[All,{3,17}]],Frame->True,FrameLabel->{"py1","M"}],Histogram[filtercernd[[All,3]],GridLinesAutomatic,ChartStyleEdgeForm[None],PlotTheme->"Scientific"]},{ListPlot[filtercernd[[All,{4,17}]],Frame->True,FrameLabel->{"pz1","M"}],Histogram[filtercernd[[All,4]],GridLinesAutomatic,ChartStyleEdgeForm[None],PlotTheme->"Scientific"]}},ImageSizeLarge],"Distribution of components of momentum in Mass Range for particle 1 and their distribution respectively(GeV)"],ListPointPlot3D[filtercernd[[All,2;;4]],AxesTrue,ColorFunction"Rainbow",PlotLabel"3D Scatter plot of momentum"]},ImageSizeFull]

Out[]=

Distribution of components of momentum in mass range is quite symmetrical in

and

direction. Their distribution in both directions are quite symmetrical.

on the other hand has different distribution in mass range from other two, specially in mass range 60-80 GeV. The distribution of

is almost symmetrical, but some upper-hand for the negative(-z) direction particles.

◼

Transverse momentum

Plotting transverse momentum of particles:

In[]:=

GraphicsRow[{Labeled[GraphicsRow[{ListPlot[filtercernd[[All,{5,17}]],Frame->True,FrameLabel->{"pt1","M"},PlotLabel->"Distribution of

in Mass Range"],Histogram[filtercernd[[All,5]],GridLinesAutomatic,ChartStyleEdgeForm[None],PlotTheme->"Scientific",PlotLabel"Distribution of transverse momentum(GeV)"]},ImageSize800],"Particle 1"],PairedSmoothHistogram[filtercernd[[All,5]],filtercernd[[All,13]],PlotRange{{0,100},All},FillingAxis,FillingStyleRed,PlotLabel"Transverse Momentum Probability density distribution for both particles"]},ImageSizeFull]

Out[]=

Transverse momentum is the resultant of momentum(x and y) components. The distribution gives the idea that momentum transfer between the beam and the particle is highest below 10 GeV.

◼

Pseudorapidity

Plotting pseudorapidity of the particles:

In[]:=

GraphicsRow[{Labeled[GraphicsRow[{ListPlot[filtercernd[[All,{6,17}]],Frame->True,FrameLabel->{"eta1","M"},PlotLabel->"Distribution of pseudorapidity in Mass Range"],Histogram[filtercernd[[All,6]],GridLinesAutomatic,ChartStyleEdgeForm[None],PlotTheme->"Scientific",PlotLabel"Distribution of pseudorapidity"]},ImageSize800],"Particle 1"],PairedSmoothHistogram[filtercernd[[All,6]],filtercernd[[All,14]],PlotRangeAll,FillingAxis,ColorFunction"Rainbow",PlotLabel"Pseudorapidity Probability density distribution for both particles"]},ImageSizeFull]

Out[]=

The graph of both the particles shows that particle 2 is less uniformly distributed around beam axis i. e. some are way too close and far than the particle 1. Particle 1 is uniformly distributed around the beam axis i. e. having almost same value of η.

◼

Charge

Visualizing the charge distribution of both particles:

In[]:=

PairedSmoothHistogram[filtercernd[[All,8]],filtercernd[[All,16]],PlotRangeAll,FillingAxis,PlotStyleBlue,PlotLabel"Charge distribution for both particles"]

Out[]=

This graph is amazingly symmetrical being exactly same distribution of charges giving an idea that electrons and anti-electrons are equal constituent of both particles 1 and 2.

◼

Mass

Plotting mass distribution with bin width set to 100:

In[]:=

Labeled[Histogram[filtercernd[[All,17]],100,GridLinesAutomatic,ChartStyleEdgeForm[None],PlotLabel"Distribution of M - The invariant mass of two electrons (GeV)"],{Rotate["Frequency",90Degree],""},{Left,Bottom}]

Out[]=

Frequency

It has a concentration on the minimum value of 2 and two further peaks can be observed.

◼

Numerical Features

Correlation matrix is the best way to observe the correlation between different coefficients of the data.

Plotting the correlation matrix of the dataset:

In[]:=

cor=Correlation[filtercernd[[All,1;;17]]];plotlist=NumberForm[#,1]&/@(cor//Flatten);collabels={"E1","px1","py1","pz1","pt1","eta1","phi1","Q1","E2","px2","py2","pz2","pt2","eta2","phi2","Q2","M"};rowlabels=collabels;rowticks=Thread[{Range[17],rowlabels}];colticks=Thread[{Range[17],collabels}];MatrixPlot[cor,Epilog->{Black,MapIndexed[Text[#1,#2-1/2]&,Transpose@Reverse@Partition[plotlist,17],{2}]},FrameTicks->{rowticks,colticks}]

Out[]=

The parameters closely related are:

◼

E1(2) and pt1(2)

◼

phi1(2) and py1(2)

◼

eta1(2) and pz1(2)

◼

pt1(2) and M

Visualizing scatter plot of first three closely related set of parameters of both particles:

In[]:=

GraphicsGrid[{{Labeled[GraphicsRow[{ListPlot[filtercernd[[All,{1,5}]],Frame->True,FrameLabel->{"E1","pt1"},PlotLabel->"Corr coefficient - 0.7"],ListPlot[filtercernd[[All,{9,13}]],Frame->True,FrameLabel->{"E2","pt2"},PlotLabel->"Corr coefficient - 0.6"]},ImageSizeLarge],"Correlation between E and pt"]},{Labeled[GraphicsRow[{ListPlot[filtercernd[[All,{7,3}]],Frame->True,FrameLabel->