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10
Shivam Sawarn
[WIS22] Predicting the electron Invariant mass from collision data
Shivam Sawarn, Delhi University
Posted
4 months ago
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2 Replies
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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.
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,
3
.
1
0
1
4
1
,
1
.
6
7
7
1
7
,
1
.
4
4
8
6
1
,
1
,
4
.
6
9
6
7
}
,
{
1
4
6
5
1
1
,
5
2
4
1
7
2
3
8
9
,
7
.
6
4
,
0
.
8
8
6
1
6
2
,
5
.
4
7
8
9
,
-
5
.
2
5
0
3
3
,
5
.
5
5
0
1
,
-
0
.
8
4
2
6
6
2
,
1
.
4
1
0
4
4
,
1
,
5
2
.
1
0
8
8
,
1
6
.
8
0
7
5
,
-
4
.
6
0
5
1
,
4
9
.
1
0
8
4
,
1
7
.
4
2
7
,
1
.
7
5
9
2
5
,
-
0
.
2
6
7
4
2
7
,
-
1
,
3
6
.
5
0
4
3
}
}
l
a
r
g
e
o
u
t
p
u
t
s
h
o
w
l
e
s
s
s
h
o
w
m
o
r
e
s
h
o
w
a
l
l
s
e
t
s
i
z
e
l
i
m
i
t
.
.
.
Preprocessing
Removing ‘Event’ and ‘Run’ column:
I
n
[
]
:
=
c
e
r
n
d
=
D
r
o
p
[
c
e
r
n
,
0
,
2
]
O
u
t
[
]
=
{
{
E
1
,
p
x
1
,
p
y
1
,
p
z
1
,
p
t
1
,
e
t
a
1
,
p
h
i
1
,
Q
1
,
E
2
,
p
x
2
,
p
y
2
,
p
z
2
,
p
t
2
,
e
t
a
2
,
p
h
i
2
,
Q
2
,
M
}
,
{
5
8
.
7
1
4
1
,
-
7
.
3
1
1
3
2
,
1
0
.
5
3
1
,
-
5
7
.
2
9
7
4
,
1
2
.
8
2
0
2
,
-
2
.
2
0
2
6
7
,
2
.
1
7
7
6
6
,
1
,
1
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.
2
8
3
6
,
-
1
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0
3
2
3
4
,
-
1
.
8
8
0
6
6
,
-
1
1
.
0
7
7
8
,
2
.
1
4
5
3
7
,
-
2
.
3
4
4
0
3
,
-
2
.
0
7
2
8
1
,
-
1
,
8
.
9
4
8
4
1
}
,
⋯
9
9
9
9
7
⋯
,
{
5
4
.
4
6
2
2
,
1
1
.
3
5
2
6
,
1
1
.
8
8
0
9
,
5
1
.
9
2
4
,
1
6
.
4
3
2
8
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1
.
8
6
7
8
,
0
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8
0
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1
3
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8
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5
8
6
7
1
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9
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0
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8
2
8
,
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0
0
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3
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6
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7
1
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1
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4
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6
1
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4
.
6
9
6
7
}
,
{
7
.
6
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,
0
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8
8
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1
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2
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5
.
4
7
8
9
,
-
5
.
2
5
0
3
3
,
5
.
5
5
0
1
,
-
0
.
8
4
2
6
6
2
,
1
.
4
1
0
4
4
,
1
,
5
2
.
1
0
8
8
,
1
6
.
8
0
7
5
,
-
4
.
6
0
5
1
,
4
9
.
1
0
8
4
,
1
7
.
4
2
7
,
1
.
7
5
9
2
5
,
-
0
.
2
6
7
4
2
7
,
-
1
,
3
6
.
5
0
4
3
}
}
l
a
r
g
e
o
u
t
p
u
t
s
h
o
w
l
e
s
s
s
h
o
w
m
o
r
e
s
h
o
w
a
l
l
s
e
t
s
i
z
e
l
i
m
i
t
.
.
.
First two columns were just meant for the identity of the data included, hence removed.
Removing extra keywords:
I
n
[
]
:
=
c
e
r
n
d
[
A
l
l
,
K
e
y
M
a
p
[
R
e
p
l
a
c
e
[
"
p
x
1
"
-
>
"
p
x
1
"
]
]
]
;
Identifying non-numerical data:
I
n
[
]
:
=
C
o
u
n
t
D
i
s
t
i
n
c
t
[
C
a
s
e
s
[
c
e
r
n
d
,
{
_
_
_
,
"
n
a
n
"
|
"
N
/
A
"
,
_
_
_
}
]
]
O
u
t
[
]
=
8
5
Replacing non-numerical with the mean of the column it lies in:
I
n
[
]
:
=
f
i
l
t
e
r
c
e
r
n
=
c
e
r
n
d
/
.
"
n
a
n
"
-
>
M
e
a
n
[
c
e
r
n
d
[
[
2
;
;
2
0
0
,
1
7
]
]
]
O
u
t
[
]
=
{
{
E
1
,
p
x
1
,
p
y
1
,
p
z
1
,
p
t
1
,
e
t
a
1
,
p
h
i
1
,
Q
1
,
E
2
,
p
x
2
,
p
y
2
,
p
z
2
,
p
t
2
,
e
t
a
2
,
p
h
i
2
,
Q
2
,
M
}
,
{
5
8
.
7
1
4
1
,
-
7
.
3
1
1
3
2
,
1
0
.
5
3
1
,
-
5
7
.
2
9
7
4
,
1
2
.
8
2
0
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-
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.
2
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6
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1
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6
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2
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3
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0
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8
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2
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1
,
-
1
,
8
.
9
4
8
4
1
}
,
⋯
9
9
9
9
7
⋯
,
{
5
4
.
4
6
2
2
,
1
1
.
3
5
2
6
,
1
1
.
8
8
0
9
,
5
1
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4
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,
{
7
.
6
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,
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8
8
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1
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4
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2
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5
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8
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2
6
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2
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1
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4
4
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1
,
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2
.
1
0
8
8
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1
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.
8
0
7
5
,
-
4
.
6
0
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4
9
.
1
0
8
4
,
1
7
.
4
2
7
,
1
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7
5
9
2
5
,
-
0
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2
6
7
4
2
7
,
-
1
,
3
6
.
5
0
4
3
}
}
l
a
r
g
e
o
u
t
p
u
t
s
h
o
w
l
e
s
s
s
h
o
w
m
o
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s
h
o
w
a
l
l
s
e
t
s
i
z
e
l
i
m
i
t
.
.
.
Extracting a numerical dataset:
I
n
[
]
:
=
f
i
l
t
e
r
c
e
r
n
d
=
f
i
l
t
e
r
c
e
r
n
[
[
2
;
;
1
0
0
0
0
1
,
A
l
l
]
]
;
Statistical summary of the dataset:
I
n
[
]
:
=
R
e
s
o
u
r
c
e
F
u
n
c
t
i
o
n
[
"
S
t
a
t
i
s
t
i
c
s
S
u
m
m
a
r
y
"
]
[
f
i
l
t
e
r
c
e
r
n
d
]
O
u
t
[
]
=
C
o
l
u
m
n
1
C
o
l
u
m
n
2
C
o
l
u
m
n
3
C
o
l
u
m
n
4
C
o
l
u
m
n
5
C
o
l
u
m
n
6
C
o
l
u
m
n
7
C
o
l
u
m
n
8
C
o
l
u
m
n
9
C
o
l
u
m
n
1
0
C
o
u
n
t
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
N
o
n
N
u
m
e
r
i
c
0
0
0
0
0
0
0
0
0
0
M
e
a
n
3
6
.
4
3
6
5
0
.
1
3
5
8
9
7
0
.
1
8
2
2
9
1
-
1
.
5
0
8
0
4
1
4
.
4
1
2
2
-
0
.
0
6
4
0
9
5
5
0
.
0
2
1
6
1
4
3
-
0
.
0
0
5
4
8
4
4
.
0
0
2
9
-
0
.
0
0
3
9
8
3
5
7
S
t
a
n
d
a
r
d
D
e
v
i
a
t
i
o
n
4
1
.
2
1
6
2
1
3
.
4
0
5
1
3
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4
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0
3
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1
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0
3
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1
2
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3
8
8
7
1
.
4
6
2
1
4
1
.
7
9
9
5
6
0
.
9
9
9
9
9
4
6
.
7
5
1
1
1
3
.
1
2
7
4
M
i
n
0
.
3
7
7
9
2
8
-
2
5
0
.
5
8
7
-
1
2
6
.
0
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9
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8
4
0
.
9
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0
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2
1
9
6
2
9
-
4
.
1
6
5
3
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-
3
.
1
4
1
5
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-
1
0
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4
7
2
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-
2
3
3
.
7
3
2
5
%
8
.
4
5
7
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6
-
5
.
2
3
3
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1
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5
.
2
7
7
3
5
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1
5
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5
9
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3
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0
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3
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1
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2
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3
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2
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1
.
5
2
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0
6
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1
1
1
.
0
5
5
5
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4
.
7
9
5
3
4
M
e
d
i
a
n
2
1
.
7
1
7
0
.
1
4
1
3
3
8
0
.
0
9
9
0
9
2
-
0
.
3
1
2
9
8
7
1
2
.
9
6
7
8
-
0
.
0
6
1
1
7
8
5
0
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0
3
4
3
2
4
-
1
2
5
.
2
6
4
6
-
0
.
0
3
5
6
3
8
7
5
%
5
0
.
0
0
2
3
5
.
7
1
4
4
2
5
.
6
4
8
1
3
.
2
1
2
1
2
0
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0
1
8
9
1
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1
4
4
4
1
.
5
6
2
3
4
1
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6
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9
2
5
1
4
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8
1
9
4
2
M
a
x
8
5
0
.
6
0
2
1
3
4
.
5
3
9
1
4
7
.
4
6
7
7
6
0
.
0
9
6
2
6
5
.
5
7
8
2
.
6
2
2
9
7
3
.
1
4
1
4
2
1
9
4
8
.
3
7
5
2
2
7
.
3
3
M
o
d
e
—
—
—
—
—
-
2
.
1
5
4
8
5
1
.
4
5
7
6
6
-
1
—
—
c
o
l
u
m
n
s
1
–
1
0
o
f
1
7
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:
I
n
[
]
:
=
G
r
a
p
h
i
c
s
R
o
w
[
{
L
a
b
e
l
e
d
[
G
r
a
p
h
i
c
s
R
o
w
[
{
L
i
s
t
P
l
o
t
[
f
i
l
t
e
r
c
e
r
n
d
[
[
A
l
l
,
{
1
,
1
7
}
]
]
,
F
r
a
m
e
-
>
T
r
u
e
,
F
r
a
m
e
L
a
b
e
l
-
>
{
"
E
1
"
,
"
M
"
}
,
P
l
o
t
L
a
b
e
l
-
>
"
D
i
s
t
r
i
b
u
t
i
o
n
o
f
E
n
e
r
g
y
i
n
M
a
s
s
R
a
n
g
e
"
]
,
H
i
s
t
o
g
r
a
m
[
f
i
l
t
e
r
c
e
r
n
d
[
[
A
l
l
,
1
]
]
,
G
r
i
d
L
i
n
e
s
A
u
t
o
m
a
t
i
c
,
C
h
a
r
t
S
t
y
l
e
E
d
g
e
F
o
r
m
[
N
o
n
e
]
,
P
l
o
t
T
h
e
m
e
-
>
"
S
c
i
e
n
t
i
f
i
c
"
,
P
l
o
t
L
a
b
e
l
"
D
i
s
t
r
i
b
u
t
i
o
n
o
f
E
n
e
r
g
y
(
G
e
V
)
"
]
}
,
I
m
a
g
e
S
i
z
e
8
0
0
]
,
"
P
a
r
t
i
c
l
e
1
"
]
,
P
a
i
r
e
d
S
m
o
o
t
h
H
i
s
t
o
g
r
a
m
[
f
i
l
t
e
r
c
e
r
n
d
[
[
A
l
l
,
1
]
]
,
f
i
l
t
e
r
c
e
r
n
d
[
[
A
l
l
,
9
]
]
,
P
l
o
t
R
a
n
g
e
{
{
0
,
2
0
0
}
,
A
l
l
}
,
F
i
l
l
i
n
g
A
x
i
s
,
F
i
l
l
i
n
g
S
t
y
l
e
G
r
e
e
n
,
A
x
e
s
L
a
b
e
l
{
"
E
1
"
,
"
E
2
"
}
,
I
m
a
g
e
S
i
z
e
L
a
r
g
e
,
P
l
o
t
L
a
b
e
l
"
E
n
e
r
g
y
P
r
o
b
a
b
i
l
i
t
y
d
e
n
s
i
t
y
d
i
s
t
r
i
b
u
t
i
o
n
o
f
b
o
t
h
p
a
r
t
i
c
l
e
s
"
]
}
,
I
m
a
g
e
S
i
z
e
F
u
l
l
]
O
u
t
[
]
=
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:
I
n
[
]
:
=
G
r
a
p
h
i
c
s
R
o
w
[
{
L
a
b
e
l
e
d
[
G
r
a
p
h
i
c
s
G
r
i
d
[
{
{
L
i
s
t
P
l
o
t
[
f
i
l
t
e
r
c
e
r
n
d
[
[
A
l
l
,
{
2
,
1
7
}
]
]
,
F
r
a
m
e
-
>
T
r
u
e
,
F
r
a
m
e
L
a
b
e
l
-
>
{
"
p
x
1
"
,
"
M
"
}
]
,
H
i
s
t
o
g
r
a
m
[
f
i
l
t
e
r
c
e
r
n
d
[
[
A
l
l
,
2
]
]
,
G
r
i
d
L
i
n
e
s
A
u
t
o
m
a
t
i
c
,
C
h
a
r
t
S
t
y
l
e
E
d
g
e
F
o
r
m
[
N
o
n
e
]
,
P
l
o
t
T
h
e
m
e
-
>
"
S
c
i
e
n
t
i
f
i
c
"
]
}
,
{
L
i
s
t
P
l
o
t
[
f
i
l
t
e
r
c
e
r
n
d
[
[
A
l
l
,
{
3
,
1
7
}
]
]
,
F
r
a
m
e
-
>
T
r
u
e
,
F
r
a
m
e
L
a
b
e
l
-
>
{
"
p
y
1
"
,
"
M
"
}
]
,
H
i
s
t
o
g
r
a
m
[
f
i
l
t
e
r
c
e
r
n
d
[
[
A
l
l
,
3
]
]
,
G
r
i
d
L
i
n
e
s
A
u
t
o
m
a
t
i
c
,
C
h
a
r
t
S
t
y
l
e
E
d
g
e
F
o
r
m
[
N
o
n
e
]
,
P
l
o
t
T
h
e
m
e
-
>
"
S
c
i
e
n
t
i
f
i
c
"
]
}
,
{
L
i
s
t
P
l
o
t
[
f
i
l
t
e
r
c
e
r
n
d
[
[
A
l
l
,
{
4
,
1
7
}
]
]
,
F
r
a
m
e
-
>
T
r
u
e
,
F
r
a
m
e
L
a
b
e
l
-
>
{
"
p
z
1
"
,
"
M
"
}
]
,
H
i
s
t
o
g
r
a
m
[
f
i
l
t
e
r
c
e
r
n
d
[
[
A
l
l
,
4
]
]
,
G
r
i
d
L
i
n
e
s
A
u
t
o
m
a
t
i
c
,
C
h
a
r
t
S
t
y
l
e
E
d
g
e
F
o
r
m
[
N
o
n
e
]
,
P
l
o
t
T
h
e
m
e
-
>
"
S
c
i
e
n
t
i
f
i
c
"
]
}
}
,
I
m
a
g
e
S
i
z
e
L
a
r
g
e
]
,
"
D
i
s
t
r
i
b
u
t
i
o
n
o
f
c
o
m
p
o
n
e
n
t
s
o
f
m
o
m
e
n
t
u
m
i
n
M
a
s
s
R
a
n
g
e
f
o
r
p
a
r
t
i
c
l
e
1
a
n
d
t
h
e
i
r
d
i
s
t
r
i
b
u
t
i
o
n
r
e
s
p
e
c
t
i
v
e
l
y
(
G
e
V
)
"
]
,
L
i
s
t
P
o
i
n
t
P
l
o
t
3
D
[
f
i
l
t
e
r
c
e
r
n
d
[
[
A
l
l
,
2
;
;
4
]
]
,
A
x
e
s
T
r
u
e
,
C
o
l
o
r
F
u
n
c
t
i
o
n
"
R
a
i
n
b
o
w
"
,
P
l
o
t
L
a
b
e
l
"
3
D
S
c
a
t
t
e
r
p
l
o
t
o
f
m
o
m
e
n
t
u
m
"
]
}
,
I
m
a
g
e
S
i
z
e
F
u
l
l
]
O
u
t
[
]
=
Distribution of components of momentum in mass range is quite symmetrical in
x
and
y
direction. Their distribution in both directions are quite symmetrical.
p
z
on the other hand has different distribution in mass range from other two, specially in mass range 60-80 GeV. The distribution of
p
z
is almost symmetrical, but some upper-hand for the negative(-z) direction particles.
◼
Transverse momentum
Plotting transverse momentum of particles:
I
n
[
]
:
=
G
r
a
p
h
i
c
s
R
o
w
[
{
L
a
b
e
l
e
d
[
G
r
a
p
h
i
c
s
R
o
w
[
{
L
i
s
t
P
l
o
t
[
f
i
l
t
e
r
c
e
r
n
d
[
[
A
l
l
,
{
5
,
1
7
}
]
]
,
F
r
a
m
e
-
>
T
r
u
e
,
F
r
a
m
e
L
a
b
e
l
-
>
{
"
p
t
1
"
,
"
M
"
}
,
P
l
o
t
L
a
b
e
l
-
>
"
D
i
s
t
r
i
b
u
t
i
o
n
o
f
p
t
i
n
M
a
s
s
R
a
n
g
e
"
]
,
H
i
s
t
o
g
r
a
m
[
f
i
l
t
e
r
c
e
r
n
d
[
[
A
l
l
,
5
]
]
,
G
r
i
d
L
i
n
e
s
A
u
t
o
m
a
t
i
c
,
C
h
a
r
t
S
t
y
l
e
E
d
g
e
F
o
r
m
[
N
o
n
e
]
,
P
l
o
t
T
h
e
m
e
-
>
"
S
c
i
e
n
t
i
f
i
c
"
,
P
l
o
t
L
a
b
e
l
"
D
i
s
t
r
i
b
u
t
i
o
n
o
f
t
r
a
n
s
v
e
r
s
e
m
o
m
e
n
t
u
m
(
G
e
V
)
"
]
}
,
I
m
a
g
e
S
i
z
e
8
0
0
]
,
"
P
a
r
t
i
c
l
e
1
"
]
,
P
a
i
r
e
d
S
m
o
o
t
h
H
i
s
t
o
g
r
a
m
[
f
i
l
t
e
r
c
e
r
n
d
[
[
A
l
l
,
5
]
]
,
f
i
l
t
e
r
c
e
r
n
d
[
[
A
l
l
,
1
3
]
]
,
P
l
o
t
R
a
n
g
e
{
{
0
,
1
0
0
}
,
A
l
l
}
,
F
i
l
l
i
n
g
A
x
i
s
,
F
i
l
l
i
n
g
S
t
y
l
e
R
e
d
,
P
l
o
t
L
a
b
e
l
"
T
r
a
n
s
v
e
r
s
e
M
o
m
e
n
t
u
m
P
r
o
b
a
b
i
l
i
t
y
d
e
n
s
i
t
y
d
i
s
t
r
i
b
u
t
i
o
n
f
o
r
b
o
t
h
p
a
r
t
i
c
l
e
s
"
]
}
,
I
m
a
g
e
S
i
z
e
F
u
l
l
]
O
u
t
[
]
=
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:
I
n
[
]
:
=
G
r
a
p
h
i
c
s
R
o
w
[
{
L
a
b
e
l
e
d
[
G
r
a
p
h
i
c
s
R
o
w
[
{
L
i
s
t
P
l
o
t
[
f
i
l
t
e
r
c
e
r
n
d
[
[
A
l
l
,
{
6
,
1
7
}
]
]
,
F
r
a
m
e
-
>
T
r
u
e
,
F
r
a
m
e
L
a
b
e
l
-
>
{
"
e
t
a
1
"
,
"
M
"
}
,
P
l
o
t
L
a
b
e
l
-
>
"
D
i
s
t
r
i
b
u
t
i
o
n
o
f
p
s
e
u
d
o
r
a
p
i
d
i
t
y
i
n
M
a
s
s
R
a
n
g
e
"
]
,
H
i
s
t
o
g
r
a
m
[
f
i
l
t
e
r
c
e
r
n
d
[
[
A
l
l
,
6
]
]
,
G
r
i
d
L
i
n
e
s
A
u
t
o
m
a
t
i
c
,
C
h
a
r
t
S
t
y
l
e
E
d
g
e
F
o
r
m
[
N
o
n
e
]
,
P
l
o
t
T
h
e
m
e
-
>
"
S
c
i
e
n
t
i
f
i
c
"
,
P
l
o
t
L
a
b
e
l
"
D
i
s
t
r
i
b
u
t
i
o
n
o
f
p
s
e
u
d
o
r
a
p
i
d
i
t
y
"
]
}
,
I
m
a
g
e
S
i
z
e
8
0
0
]
,
"
P
a
r
t
i
c
l
e
1
"
]
,
P
a
i
r
e
d
S
m
o
o
t
h
H
i
s
t
o
g
r
a
m
[
f
i
l
t
e
r
c
e
r
n
d
[
[
A
l
l
,
6
]
]
,
f
i
l
t
e
r
c
e
r
n
d
[
[
A
l
l
,
1
4
]
]
,
P
l
o
t
R
a
n
g
e
A
l
l
,
F
i
l
l
i
n
g
A
x
i
s
,
C
o
l
o
r
F
u
n
c
t
i
o
n
"
R
a
i
n
b
o
w
"
,
P
l
o
t
L
a
b
e
l
"
P
s
e
u
d
o
r
a
p
i
d
i
t
y
P
r
o
b
a
b
i
l
i
t
y
d
e
n
s
i
t
y
d
i
s
t
r
i
b
u
t
i
o
n
f
o
r
b
o
t
h
p
a
r
t
i
c
l
e
s
"
]
}
,
I
m
a
g
e
S
i
z
e
F
u
l
l
]
O
u
t
[
]
=
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:
I
n
[
]
:
=
P
a
i
r
e
d
S
m
o
o
t
h
H
i
s
t
o
g
r
a
m
[
f
i
l
t
e
r
c
e
r
n
d
[
[
A
l
l
,
8
]
]
,
f
i
l
t
e
r
c
e
r
n
d
[
[
A
l
l
,
1
6
]
]
,
P
l
o
t
R
a
n
g
e
A
l
l
,
F
i
l
l
i
n
g
A
x
i
s
,
P
l
o
t
S
t
y
l
e
B
l
u
e
,
P
l
o
t
L
a
b
e
l
"
C
h
a
r
g
e
d
i
s
t
r
i
b
u
t
i
o
n
f
o
r
b
o
t
h
p
a
r
t
i
c
l
e
s
"
]
O
u
t
[
]
=
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:
I
n
[
]
:
=
L
a
b
e
l
e
d
[
H
i
s
t
o
g
r
a
m
[
f
i
l
t
e
r
c
e
r
n
d
[
[
A
l
l
,
1
7
]
]
,
1
0
0
,
G
r
i
d
L
i
n
e
s
A
u
t
o
m
a
t
i
c
,
C
h
a
r
t
S
t
y
l
e
E
d
g
e
F
o
r
m
[
N
o
n
e
]
,
P
l
o
t
L
a
b
e
l
"
D
i
s
t
r
i
b
u
t
i
o
n
o
f
M
-
T
h
e
i
n
v
a
r
i
a
n
t
m
a
s
s
o
f
t
w
o
e
l
e
c
t
r
o
n
s
(
G
e
V
)
"
]
,
{
R
o
t
a
t
e
[
"
F
r
e
q
u
e
n
c
y
"
,
9
0
D
e
g
r
e
e
]
,
"
"
}
,
{
L
e
f
t
,
B
o
t
t
o
m
}
]
O
u
t
[
]
=
F
r
e
q
u
e
n
c
y
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:
I
n
[
]
:
=
c
o
r
=
C
o
r
r
e
l
a
t
i
o
n
[
f
i
l
t
e
r
c
e
r
n
d
[
[
A
l
l
,
1
;
;
1
7
]
]
]
;
p
l
o
t
l
i
s
t
=
N
u
m
b
e
r
F
o
r
m
[
#
,
1
]
&
/
@
(
c
o
r
/
/
F
l
a
t
t
e
n
)
;
c
o
l
l
a
b
e
l
s
=
{
"
E
1
"
,
"
p
x
1
"
,
"
p
y
1
"
,
"
p
z
1
"
,
"
p
t
1
"
,
"
e
t
a
1
"
,
"
p
h
i
1
"
,
"
Q
1
"
,
"
E
2
"
,
"
p
x
2
"
,
"
p
y
2
"
,
"
p
z
2
"
,
"
p
t
2
"
,
"
e
t
a
2
"
,
"
p
h
i
2
"
,
"
Q
2
"
,
"
M
"
}
;
r
o
w
l
a
b
e
l
s
=
c
o
l
l
a
b
e
l
s
;
r
o
w
t
i
c
k
s
=
T
h
r
e
a
d
[
{
R
a
n
g
e
[
1
7
]
,
r
o
w
l
a
b
e
l
s
}
]
;
c
o
l
t
i
c
k
s
=
T
h
r
e
a
d
[
{
R
a
n
g
e
[
1
7
]
,
c
o
l
l
a
b
e
l
s
}
]
;
M
a
t
r
i
x
P
l
o
t
[
c
o
r
,
E
p
i
l
o
g
-
>
{
B
l
a
c
k
,
M
a
p
I
n
d
e
x
e
d
[
T
e
x
t
[
#
1
,
#
2
-
1
/
2
]
&
,
T
r
a
n
s
p
o
s
e
@
R
e
v
e
r
s
e
@
P
a
r
t
i
t
i
o
n
[
p
l
o
t
l
i
s
t
,
1
7
]
,
{
2
}
]
}
,
F
r
a
m
e
T
i
c
k
s
-
>
{
r
o
w
t
i
c
k
s
,
c
o
l
t
i
c
k
s
}
]
O
u
t
[
]
=
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:
I
n
[
]
:
=
G
r
a
p
h
i
c
s
G
r
i
d
[
{
{
L
a
b
e
l
e
d
[
G
r
a
p
h
i
c
s
R
o
w
[
{
L
i
s
t
P
l
o
t
[
f
i
l
t
e
r
c
e
r
n
d
[
[
A
l
l
,
{
1
,
5
}
]
]
,
F
r
a
m
e
-
>
T
r
u
e
,
F
r
a
m
e
L
a
b
e
l
-
>
{
"
E
1
"
,
"
p
t
1
"
}
,
P
l
o
t
L
a
b
e
l
-
>
"
C
o
r
r
c
o
e
f
f
i
c
i
e
n
t
-
0
.
7
"
]
,
L
i
s
t
P
l
o
t
[
f
i
l
t
e
r
c
e
r
n
d
[
[
A
l
l
,
{
9
,
1
3
}
]
]
,
F
r
a
m
e
-
>
T
r
u
e
,
F
r
a
m
e
L
a
b
e
l
-
>
{
"
E
2
"
,
"
p
t
2
"
}
,
P
l
o
t
L
a
b
e
l
-
>
"
C
o
r
r
c
o
e
f
f
i
c
i
e
n
t
-
0
.
6
"
]
}
,
I
m
a
g
e
S
i
z
e
L
a
r
g
e
]
,
"
C
o
r
r
e
l
a
t
i
o
n
b
e
t
w
e
e
n
E
a
n
d
p
t
"
]
}
,
{
L
a
b
e
l
e
d
[
G
r
a
p
h
i
c
s
R
o
w
[
{
L
i
s
t
P
l
o
t
[
f
i
l
t
e
r
c
e
r
n
d
[
[
A
l
l
,
{
7
,
3
}
]
]
,
F
r
a
m
e
-
>
T
r
u
e
,
F
r
a
m
e
L
a
b
e
l
-
>