269L-2021

Research:

Research Topic: Top Quark Properties at LHC

Working with Aaron Beisaw under Dr. Kaustubh Agashe

2/5/21

Today we have completed our first meeting with Dr. Agashe. Our research is mainly based on his previous research on two-body decay kinematics. Details can be found here: https://arxiv.org/abs/1209.0772.

In our scenario, the parent particle B is produced by particle collision. B will decay into two particles a and A, and a is a massless particle. We are able to calculate the energy of a in the rest frame of B via energy and momentum conservation. This energy relation can be expressed in terms of the mass of A and B.

Then by Lorentz transformation with a metric signature, we can find the energy of a in the lab frame.

It gives:

Hence we are able to express the lab frame energy of a in terms of its rest frame energy. θ here is the angle between particle a and boost direction.

2/12/21

This week we continue working on building the model. In our model, both γ and θ vary from event to event, thus we can get a probability distribution. A very essential assumption has been made on our model. We assume parent B is unpolarized, and therefore cosθ distribution is constant.

This implies we will obtain a rectangular distribution for a fixed angle. The shape of this rectangle, of course, is depended on how the mother particle is been boosted. I show the max and min of this rectangular distribution. These two endpoints will tell us the symmetric property of distributions. We showed that this distribution isn't symmetric in the linear scale but it is symmetric in the log scale. The proof is given here.

2/19/21

To obtain the overall distribution, we have to consider all possible boosts act on the mother particle. In this case, we stack up the rectangles with various boots. Because the area of the rectangle is normalized(event number is fixed), therefore large boosts will cause a wide but short rectangle because the particle is likely to be boosted into higher energy.

We define x = ELab/Erest. Such stacking up can be expressed as f(x), where f(x) is the total distribution function:

Taking the first derivative to find the extremum of f(x):

After some analysis, we find the shape of the energy distribution for massless particle a. It is a Poisson distribution.  Details are shown below:

2/26/21

This week we finish building our model. We generalized our energy relations with ma isn't equal to 0. Besides, we also examine an interesting case where the lower bound xmin exceeds the peak, i.e. xmin > 1. For bottom from top quark decay (e.g., light cold particle), γb =15 , γB critical <500. Thus the collider energy is ~ 200Tev (2*MT*500).

It turns out in order to make it happen, we must first own a very high-energy collider. Thus such a theoretical scenario wouldn't occur in our near future. Notes are attached for anyone interested. 

Also, we are now moving into the simulation part to check our model. We will use Madgraph to run the simulation. However, I currently facing difficulties when using Madgraph because it doesn't seem to produce distributions automatically. Also, I am not able to install Pythia-pgs and Delphes in Madgraph from the cluster as well.

3/5/21

Thanks to Dr. Jabeen's help, today I successfully installed Pythia-pgs and Delphes in Madgraph from the cluster. The solution is trying to download them in a much newer version of CMSSW. The procedure and commands of downing can be found on this website: https://twiki.cern.ch/twiki/bin/view/CMSPublic/MadgraphTutorial.

Then I generate p p > t t~. While launching this process, I turned on the PYTHIA 6 and DELPHES. 

I run for 10,000 events. It took a while to generate the whole process.

3/9/21

It turns out we don't have to use PYTHIA and Delphes for our current step. Simulations via Madgraph are sufficient now. Now I have to come up with some Mathematica codes which may help us to successfully generate the histogram. It seems that MadAnalysis might be also helpful, but somehow I am not able to download directly inside the cluster.

3/24/21

In short, no progress has been made in the last few weeks. I tried to come up with a Mathematica code that students in the previous years used to generate the distribution, but I failed. Then I am trying to write a C code that reads and plots the simulation data. Hope this time will work. 

3/31/21

Last weekend I successfully installed the MadAnalysis in my cluster. I download it to my desktop and then scp it to my cluster. I found the following tutorials are very helpful to learn MadAnalysis:

• https://www.physics.sjtu.edu.cn/madgraphschool/sites/www.physics.sjtu.edu.cn.madgraphschool/files/MadAnalysis_MadGraphSchool.pdf

• https://indico.cern.ch/event/656211/contributions/2756855/attachments/1547520/2429331/fuks_MA5.pdf

This time, using MadAnalysis, I am able to produce the histogram successfully. I first simulated p p > t t~ > w+ b w- b~ in Madgraph with a event number = 10,000 times. Then importing the unweight.lhe Madgraph produced to MadAnalysis. MadAnalysis would help to generate the histogram automatically. 

The general code I use for MadAnalysis is: 

import /path/to/the/lhe/file/lhefile.lhe ## import file ##

    plot E(b) 250 0 300 ## plot energy of b from 0 to 300 with bin number = 250 ##

    plot E(b~) 250 0 300

    submit ## run ##

Histogram 1 is the energy distribution for bottom quark b, and histogram 2 is the energy distribution for bottom antiquark, b~.

4/3/2021

This week I ran the energy distribution for E(b) and E(b~) again but set event number = 100,000 this time. I have also made E_b only up to3 00 GeV so that position of the peak is more clear to see. 

To change events number, open run_card.dat in Madgraph when run the simulation using the run command. Then hit Insert to edit the file. You can then change 10,000 to arbitrary numbers of events you want to simulate. After change the event number, I hit Esc to stop editing, typing :w to save the file, and then typing :q to close the file.

There are the updates:

It is very obvious that both these two distributions peaked around 70 GeV. This is what the theory has predicted. We can yield a numerical value for this invariance rest energy of the b quark by knowing the mass of the top quark, the mass of W boson, and the mass of b quark.  These values can be obtained inside the run card for each simulation.

4/10/21

To simulate collision creates by different collider energy levels, go to run_card.dat in Madgraph.  Notice the collider energy level = sum of the beam energy. Normally, Madgarph will simulate collision events in 13TeV.

To put outcomes from various events into one plot, we can use the following command to superpose different plots in MadAnalysis before submit:

set main.stacking_method = normalize2one

I set the energy of the collider to be 630 GeV and 1980GeV. This process can be done in the run card after I launch the simulations.  The number of events is 100,000. I generate the probability distribution for both E(b) and PT(b). Here are two plots I generated:

The peaks of E(b) for both 630 GeV and 1980 GeV are centered around 68 GeV. While that is not the case for PT(b). Examing our model via only two different collider energy isn't very persuasive, and the next step for me is to generate distributions produced by various collider energy ranging from 7 TEV to 100TEV. I am still not being able to figure out how to plot distribution in the log space. I will meet with Dr.Jabeen on Monday to seek possible solutions.

4/12/2021

Today Aaron and I met with Dr.Jabeen and we are able to successfully generate the distribution in log space. Dr.Jabeen suggested us to do the plot, not inside the MadAnalysis, via Root. Use find . f-name "*.C" command, we can find C files MadAnalysis produced. 

Then Emacs selection_0.C, change SetLogx(0) to SetLogx(1).

Save this C file. We can now use Root to help us generate the plot, just simply type the root -l command. Make sure to type cmsenv beforehand. The plot of energy distribution generated in log space is:

In log(x) space, the peak of the two distributions is centered and symmetric around 68GeV. This plot confirms our model. Thanks Dr. Jabeen for this helpful workaround!

4/16/2021

Today I generate the histogram for various collider energies. For some unknown reason, I wasn't able to use firefox to open the html. MadAnalysis created. I use root command to open it instead. Here is the plot I generated:

It is quite obvious that for the energy distribution ranging from 630GeV to 100TeV, they are peaked around 70 GeV. This invariance in energy has confirmed our model. If we plot them in log[x] space, we find the distributions are symmetric about 70 GeV, as we have already predicted.

Dr. Agahse today suggests that for low collider energies (630GeV and 1980GeV), the dominant process is longer p p but p p~. This is because quark-antiquark scattering has a much higher cross-section than quark due to the extra possibility of annihilation at low energy (more explanations in https://physics.stackexchange.com/questions/8954/what-are-the-main-differences-between-p-p-and-p-bar-p-colliders). 

To simulate pp, go to run_card.dat in Madgraph again. This time just changes 1 to -1.

4/27/2021

This week I superposed the plot I generated. For collider energy < 2TeV, I chose p p~ collision. p p collision for energy > 2TeV. The generated plot is:

We still observe that the peak is invariance at 68 Gev and b quark energy distributions are symmetric on the log scale. Despite the top quark would be boosted due to different collider energies, hence produce b quark with various speeds. The peak remains at 68 GeV, remarkably. 

In addition, as we have shown, this invariance in peak only applies to energy. This invariance law no longer holds for transverse momentum distribution. Therefore, our simulations have verified our two-body decay model.

4/30/2021

Today we have our last meeting with Dr. Agashe this semester. This meeting is mainly about presentation and showcase, but he also mentions the advantages and disadvantages of our model. 

Advantages:

• The information of the parent particle in the rest frame can be extracted from the energy distribution of the light mass child particle a in the lab frame;

• It only relies on one particle in the two-body decay, and we can still obtain the information of the parent particle even the other child particle is invisible;

• This model enabled us to calculate the mass of the top quark, as was done at the LHC during the analysis of the 8 TeV dataset.

Disadvantages:

• Our analysis doesn’t include radiation from the bottom quark because this extra radiation will convert a two-body to a three-body decay;

• Model is valid only when the beyond the standard model is negligible.

5/10/2021