A Program for SU(N) Color Structure Decomposition into Multiplet Bases Using Wigner 3j and 6j Coefficients

The increased capacity of elementary particle accelerators raises the demand for the simulation data of the experiments. One of the bottlenecks in the simulations is the QCD color structure calculation, which is usually treated using non-orthogonal and overcomplete sets of bases. The computational cost could be decreased significantly if orthogonal bases, such as the multiplet bases, were used instead. However, no computation tool performing calculations using these bases is available yet. In this thesis, we present a Mathematica program as proof-of-principle demonstrating the color structure decomposition into the multiplet bases. For a given amplitude, the corresponding multiplet basis states can be created and the scalar product between the amplitude and each of the basis states can be evaluated whenever the required Wigner 6j coefficients are available. The program offers tools for visualization of the tensor expressions in the birdtrack notation as well as a syntax similar to how the tensor expressions would be defined on paper. The available functions and replacement rules allow performing operations on SU(Nc) tensor expressions including index contraction, tensor conjugation, and scalar product of tensors.It might be stunning to realize that researching the smallest constituents of the world we see requires building the largest constructions people have created. On the border of Switzerland and France one can find the greatest of the examples, the Large Hadron Collider, a more than eight kilometers in diameter large ring filled with a vacuum where scientists let bunches of around 100 million protons collide almost 40 million times per second. Despite all the great discoveries (you might have heard of the Higgs boson found in 2012, for example), particle physicists often claim that further explorations of the fundamentals of the Universe require building even larger colliders and increasing the number of collisions even more. However, something is often left out in this demand for faster, bigger, and stronger. In order to find new unknown physics, we must obtain just as much data from the computer simulations as we get from the experiments. Like with two fingerprints, we analyze all the different curves and shapes in the graphs obtained from these two data sets. Any discrepancy found in them would give a clue on where to search for discoveries such as new types of elementary particles, or even change our view on how our Universe works. My contribution to particle physics is connected to speeding up the methods of mathematical simulations of the collision data. A calculation of each clash of particles, like a collision of two asteroids, involves keeping track of a tremendous mess of collision products --- where they fly, how fast and how they interact with each other. What is even more complicated to calculate are predictions about what kind of particles get to be created in each of these collisions. This is determined by the laws of \textit{Quantum Field Theory}, a theory stating that all particles can be viewed as energetic bumps in some invisible fields spanning the whole Universe. The theory predicts how particles get created and destroyed in the interaction points like waves on a drum membrane when it is hit. %Some of the particles created in these tiny interaction points attract each other so strongly that they combine long before hitting the detectors. These particles are called quarks and gluons and this immensely strong force is called, well, the strong force. An even more peculiar feature of this force is that unlike the electromagnetic charge, the strong force charge comes in three types. Gluons and quarks interact differently depending on what charge they possess. Even though simplified methods to simulate particle collisions approximately exist, for exact calculations, all of the possible color charge combinations have to be considered. This makes one of the greatest bottlenecks of the whole simulation process and is a challenge that this thesis offers a possible solution to. Some of the particles created in these tiny interaction points attract each other so strongly that they combine long before hitting the detectors. These particles are called quarks and gluons and the force that creates these immensely strong bounds is called, well, the strong force. An even more peculiar feature of this force is that unlike the electromagnetic charge, the strong force charge comes in three types. Gluons and quarks interact differently depending on what charge they possess. Even though simplified methods to simulate particle collisions approximately exist, for exact calculations, all of the possible color charge combinations have to be considered. This makes one of the greatest bottlenecks of the whole simulation process and is a challenge that this thesis offers a possible solution to. In my thesis, I computationally implemented a new technique that uses abstract mathematical objects called multiplet bases that could potentially speed up the strong force calculations. We have already validated our method for the simplest collisions. However, the highest hope is to update the multiplet bases method to be able to calculate collisions where eight or more gluons and quarks appear and where the speed differences would become more significant. When this is achieved, it should immensely increase the capacity of simulating particle collisions. In this way, physicists would be able to search for even more complicated processes and in this complexity maybe some great discoveries hide

Determination of the Diffusion coefficient from the theoretical calculations of the Brownian motion

Literatūrā atrodamas vairākas difūzijas koeficienta no Brauna kustības rēķināšanas metodes. Šajā darbā tiek apskatītas un ar skaitliskām simulācijām pārbaudītas pārklājošos, nepārklājošos un nekorelēto pārvietojumu metodes kā arī vairākas lineārās regresijas metodes, gan ar trajektorijas mērījumu kļūdu, gan bez. Tika atrasts, ka pārklājošos intervālu metode no apskatītajām sniedz vismazāko rezultāta kļūdu. Papildus tam tiek atrasta sakarība, kas saista izmērītās trajektorijas punktu skaitu ar iegūtā difūzijas koeficienta kļūdu un mērījuma kļūdu.In the literature it is possible to find several methods for computing the diffusion coefficient from a Brownian motion experiment. In this thesis the methods of overlapping, non-overlapping intervals and uncorrelated intervals as well as several methods of linear regression are being examined both with and without the measurement error. From all of the reviewed methods it was found that the method of overlapping intervals gives the smallest standard deviation from the true diffusion coefficient. Additionally a relationship between the number of measured trajectory points, the error of the obtained diffusion coefficients and the error of measurement is obtained

Review of the measurements of the strong coupling constant in CMS at 13 TeV

The strong coupling constant is the least known of the coupling constants in the standardmodel. Nevertheless it appears in the calculations of cross sections of all the processes at the LHC. We present a review of the strong coupling constant measurements conducted at the CMS experiment, focusing on those performed at a center-of-mass of 13 TeV

Central exclusive production of top quarks

This summer student project at CERN was dedicated to exclusive production of top quarks. For proton colliders exclusive production means that colliding protons after the collision do not get destroyed but through photons (25% of times) or gluons (75% of times) loose some of their momentum and create in our case top and anti top quark pair (see Feynman diagram, figure 1). Due to the momentum loss, these protons are more bent in the magnetic field of the accelerator and escape the beam. There is no detector in the CMS to detect protons but as these protons are declined from the beam only slightly, it is possible to detect them in special detectors called Roman pots (RP) owned by the TOTEM experiment [1]. RP are aligned on both sides of the interaction point. From the position in the RP where the proton was detected and knowing how protons interact with the magnetic field bending the proton beam, it is possi-ble to calculate the fractional momentum loss ξ of a proton - how much momentum it lost after the collision. The proton interaction with the magnetic field can be described by the dispersion value Dx so that xRP = Dx(ξ)ξ If the energy loss of both protons is known, it is also possible to reconstruct the invariant mass of the tt¯system by a formula: mtt¯ = √sξ1ξ2, (1) where s is the centre of mass energy. The exclusive production is a new research field at CERN because it uses data from detectors owned by two different experiments - CMS and TOTEM. Only in a recent years a joint experiment called CMS-TOTEM Precision Proton Spectrometer (CT-PPS) was created and so far only one paper on the exclu-sive production has been published. It provided an evidence for the semi exclusive µµ− pair production -an event where only one of the protons was detected [2]. One of the problems for obtaining a good results for the exclusive top quark production is a large number of the background protons in the RP which could be solved by using the low pile up (low PU) data. The goal of this project was to make the first examination of data from the low PU run of 2018. At the time of doing this project the dispersion value Dx for the low pile up run of 2018 was not known but was to be calculated soon. This work would ensure the usage of these data for the search of central exclusive top quark production as soon as all of the information about the dispersion of the protons for this run would be received. The main tasks was to calculate the resolution of tt¯system for the 2018 data, improve the selection criteria, look for a shift of the PT for the central system at low PT scale and look if there are any problems with the data

Measurement of the double-differential inclusive jet cross section in proton-proton collisions at s\sqrt{s} = 5.02 TeV

International audienceThe inclusive jet cross section is measured as a function of jet transverse momentum $p_\mathrm{T}$ and rapidity $y$. The measurement is performed using proton-proton collision data at $\sqrt{s}$ = 5.02 TeV, recorded by the CMS experiment at the LHC, corresponding to an integrated luminosity of 27.4 pb$^{-1}$. The jets are reconstructed with the anti-$k_\mathrm{T}$ algorithm using a distance parameter of $R$ = 0.4, within the rapidity interval $\lvert y\rvert$$\lt$ 2, and across the kinematic range 0.06 $\lt$$p_\mathrm{T}$$\lt$ 1 TeV. The jet cross section is unfolded from detector to particle level using the determined jet response and resolution. The results are compared to predictions of perturbative quantum chromodynamics, calculated at both next-to-leading order and next-to-next-to-leading order. The predictions are corrected for nonperturbative effects, and presented for a variety of parton distribution functions and choices of the renormalization/factorization scales and the strong coupling $\alpha_\mathrm{S}$

Measurement of the double-differential inclusive jet cross section in proton-proton collisions at s= \sqrt{s} = 5.02 TeV

The inclusive jet cross section is measured as a function of jet transverse momentum $p_{\mathrm{T}}$ and rapidity $y$. The measurement is performed using proton-proton collision data at $\sqrt{s} =$ 5.02 TeV, recorded by the CMS experiment at the LHC, corresponding to an integrated luminosity of 27.4$\,\text{pb}^{-1}$. The jets are reconstructed with the anti-$k_{\mathrm{T}}$ algorithm using a distance parameter of $R=$ 0.4, within the rapidity interval $|y| <$ 2, and across the kinematic range 0.06 $< p_{\mathrm{T}} <$ 1 TeV. The jet cross section is unfolded from detector to particle level using the determined jet response and resolution. The results are compared to predictions of perturbative quantum chromodynamics, calculated at both next-to-leading order and next-to-next-to-leading order. The predictions are corrected for nonperturbative effects, and presented for a variety of parton distribution functions and choices of the renormalization/factorization scales and the strong coupling $\alpha_\mathrm{S}$.The inclusive jet cross section is measured as a function of jet transverse momentum $p_\mathrm{T}$ and rapidity $y$. The measurement is performed using proton-proton collision data at $\sqrt{s}$ = 5.02 TeV, recorded by the CMS experiment at the LHC, corresponding to an integrated luminosity of 27.4 pb$^{-1}$. The jets are reconstructed with the anti-$k_\mathrm{T}$ algorithm using a distance parameter of $R$ = 0.4, within the rapidity interval $\lvert y\rvert$$\lt$ 2, and across the kinematic range 0.06 $\lt$$p_\mathrm{T}$$\lt$ 1 TeV. The jet cross section is unfolded from detector to particle level using the determined jet response and resolution. The results are compared to predictions of perturbative quantum chromodynamics, calculated at both next-to-leading order and next-to-next-to-leading order. The predictions are corrected for nonperturbative effects, and presented for a variety of parton distribution functions and choices of the renormalization/factorization scales and the strong coupling $\alpha_\mathrm{S}$

Measurement of the double-differential inclusive jet cross section in proton-proton collisions at s\sqrt{s} = 5.02 TeV

International audienceThe inclusive jet cross section is measured as a function of jet transverse momentum $p_\mathrm{T}$ and rapidity $y$. The measurement is performed using proton-proton collision data at $\sqrt{s}$ = 5.02 TeV, recorded by the CMS experiment at the LHC, corresponding to an integrated luminosity of 27.4 pb$^{-1}$. The jets are reconstructed with the anti-$k_\mathrm{T}$ algorithm using a distance parameter of $R$ = 0.4, within the rapidity interval $\lvert y\rvert$$\lt$ 2, and across the kinematic range 0.06 $\lt$$p_\mathrm{T}$$\lt$ 1 TeV. The jet cross section is unfolded from detector to particle level using the determined jet response and resolution. The results are compared to predictions of perturbative quantum chromodynamics, calculated at both next-to-leading order and next-to-next-to-leading order. The predictions are corrected for nonperturbative effects, and presented for a variety of parton distribution functions and choices of the renormalization/factorization scales and the strong coupling $\alpha_\mathrm{S}$

Two-particle Bose-Einstein correlations and their Lévy parameters in PbPb collisions at sNN\sqrt{s_\mathrm{NN}} = 5.02 TeV

Two-particle Bose-Einstein momentum correlation functions are studied for charged-hadron pairs in lead-lead collisions at a center-of-mass energy per nucleon pair of $\sqrt{s_\mathrm{NN}}$ = 5.02 TeV. The data sample, containing 4.27$\times10^{9}$ minimum bias events corresponding to an integrated luminosity of 0.607 nb$^{-1}$, was collected by the CMS experiment in 2018. The experimental results are discussed in terms of a L\'evy-type source distribution. The parameters of this distribution are extracted as functions of particle pair average transverse mass and collision centrality. These parameters include the L\'evy index or shape parameter ($\alpha$), the L\'evy scale parameter ($R$), and the correlation strength parameter ($\lambda$). The source shape, characterized by $\alpha$, is found to be neither Cauchy nor Gaussian, implying the need for a full L\'evy analysis. Similarly to what was previously found for systems characterized by Gaussian source radii, a hydrodynamical scaling is observed for the L\'evy $R$ parameter. The $\lambda$ parameter is studied in terms of the core-halo model.Comment: Submitted to Physical Review C. All figures and tables can be found at http://cms-results.web.cern.ch/cms-results/public-results/publications/HIN-21-011 (CMS Public Pages

Measurement of the double-differential inclusive jet cross section in proton-proton collisions at s\sqrt{s} = 5.02 TeV

International audienceThe inclusive jet cross section is measured as a function of jet transverse momentum $p_\mathrm{T}$ and rapidity $y$. The measurement is performed using proton-proton collision data at $\sqrt{s}$ = 5.02 TeV, recorded by the CMS experiment at the LHC, corresponding to an integrated luminosity of 27.4 pb$^{-1}$. The jets are reconstructed with the anti-$k_\mathrm{T}$ algorithm using a distance parameter of $R$ = 0.4, within the rapidity interval $\lvert y\rvert$$\lt$ 2, and across the kinematic range 0.06 $\lt$$p_\mathrm{T}$$\lt$ 1 TeV. The jet cross section is unfolded from detector to particle level using the determined jet response and resolution. The results are compared to predictions of perturbative quantum chromodynamics, calculated at both next-to-leading order and next-to-next-to-leading order. The predictions are corrected for nonperturbative effects, and presented for a variety of parton distribution functions and choices of the renormalization/factorization scales and the strong coupling $\alpha_\mathrm{S}$

Measurement of the double-differential inclusive jet cross section in proton-proton collisions at s\sqrt{s} = 5.02 TeV

International audienceThe inclusive jet cross section is measured as a function of jet transverse momentum $p_\mathrm{T}$ and rapidity $y$. The measurement is performed using proton-proton collision data at $\sqrt{s}$ = 5.02 TeV, recorded by the CMS experiment at the LHC, corresponding to an integrated luminosity of 27.4 pb$^{-1}$. The jets are reconstructed with the anti-$k_\mathrm{T}$ algorithm using a distance parameter of $R$ = 0.4, within the rapidity interval $\lvert y\rvert$$\lt$ 2, and across the kinematic range 0.06 $\lt$$p_\mathrm{T}$$\lt$ 1 TeV. The jet cross section is unfolded from detector to particle level using the determined jet response and resolution. The results are compared to predictions of perturbative quantum chromodynamics, calculated at both next-to-leading order and next-to-next-to-leading order. The predictions are corrected for nonperturbative effects, and presented for a variety of parton distribution functions and choices of the renormalization/factorization scales and the strong coupling $\alpha_\mathrm{S}$