In this thesis the proposed exact results for low energy effective N = 2 supersymmetric SU(N) Yang-Mills gauge theory coupled to Nf fundamental matter multiplets in four dimensions are considered. The proposed exact results are based upon the work of Seiberg and Witten for low energy effective four dimensional M = 2 supersymmetric SU[2) Yang-Mills gauge theory coupled to Nf fundamental matter multiplets. The testing and matching of the proposed exact results via supersymmetric instanton calculus are the motivation for two studies. Firstly, we study the ADHM construction of instantons for gauge groups U(N) and SU(2) and for topological charge two and three. The ADHM constraints which implicitly specify instanton gauge field configurations are solved for the explicit exact general form of instantons with topological charge two and gauge group U[N). This is the first explicit and general multi-instanton configuration for the unitary gauge groups. The U[N) ADHM two-instanton configuration may be used in further tests and matching of the proposed exact results in low energy effective M =2 supersymmetric SU(iV) Yang-Mills gauge theories by comparison with direct instanton calculations. Secondly, a one-instanton level test is performed for the reparameterization scheme proposed by Argyres and Pelland matching the conjectured exact low energy results and instanton predictions for the instanton contributions to the prepotential of low energy effective M = 2 supersymmetric SU [N) Yang-Mills gauge theory with Nf = 2N mass-less fundamental matter multiplets. The constants within the reparameterization scheme which ensure agreement between the exact results and the instanton predictions for general N > 1 are derived for the entire quantum moduli space. This constitutes a non-trivial test of the proposed reparameterization scheme, which eliminates the discrepancies arising when the two sets of results are compared.

We evaluate the prompt atmospheric neutrino flux at high energies using three different frameworks for calculating the heavy quark production cross section in QCD: NLO perturbative QCD, k(T) factorization including low-x resummation, and the dipole model including parton saturation. We use QCD parameters, the value for the charm quark mass and the range for the factorization and renormalization scales that provide the best description of the total charm cross section measured at fixed target experiments, at RHIC and at LHC. Using these parameters we calculate differential cross sections for charm and bottom production and compare with the latest data on forward charm meson production from LHCb at 7TeV and at 13TeV, finding good agreement with the data. In addition, we investigate the role of nuclear shadowing by including nuclear parton distribution functions (PDF) for the target air nucleus using two different nuclear PDF schemes. Depending on the scheme used, we find the reduction of the flux due to nuclear effects varies from 10% to 50% at the highest energies. Finally, we compare our results with the IceCube limit on the prompt neutrino flux, which is already providing valuable information about some of the QCD models.

This thesis presents an improved method to calculate renormalization constants in a regularization invariant momentum scheme using twisted boundary conditions. This enables us to simulate with momenta of arbitrary magnitude and a fixed direction. With this new technique, together with non-exceptional kinematics and volume sources, we are able to take a statistically and theoretically precise continuum limit. Thereafter, all the running of the operators with momentum scale is due to their anomalous dimension. We use this to develop a practical scheme for step scaling with off shell vertex functions. We develop the method on 16³ × 32 lattices to show the practicality of using small volume simulations to step scale to high momenta. We also use larger 24³×64 and 32³×64 lattices to compute renormalization constants very accurately. Combining these with previous analyses we are able to extract a precise value for the light and strange quark masses and the neutral kaon mixing parameter BK. We also analyse eigenvectors of the domain wall Dirac matrix. We develop a practical and cost effective way to compute eigenvectors using the implicitly restarted Lanczos method with Chebyshev acceleration. We show that calculating eigenvectors to accelerate propagator inversions is cost effective when as few as one or two propagators are required. We investigate the technique of low mode averaging (LMA) with eigenvectors of the domain wall matrix for the first time. We find that for low energy correlators, pions for example, LMA is very effective at reducing the statistical noise. We also calculated the η and η′ meson masses, which required evaluating disconnected correlation functions and combining stochastic sources with LMA.

This thesis discusses recent advances in precision calculations of quantum chromodynamics and their application to the Large Hadron Collider (LHC) physics program and beyond. The first half of the thesis is dedicated to the study of vector boson fusion Higgs (VBF) production; fully differential at the next-to-next-to-leading order level (NNLO), and inclusively at next-to-next-to-next-to-leading order (N³LO). Both calculations are performed in the structure function approximation, where the VBF process is treated as a double deep inelastic scattering. For the differential calculation a new subtraction method, "projection-to-Born", is introduced and applied. We study VBF production in a number of scenarios relevant for the LHC and for Future Circular Colliders (FCC). We find NNLO corrections after typical cuts of 5-6% while differential distributions show corrections of up to 10-12% for some standard observables. For the inclusive calculation we find N³LO corrections at the order of 1-2‰. The second half of the thesis presents recent results on the matching of fixed order calculations with parton showers. We first present the POsitive Weight Hardest Emission Generator (POWHEG) method for matching next-to-leading order (NLO) calculations with parton showers. We then proceed to apply it to the case of vector boson fusion ZZjj production and discuss the results for scenarios relevant for the LHC and a possible FCC. In order to present the matching of a NNLO calculation with a parton shower, we next discuss the Multi-Scale Improved NLO (MiNLO) procedure. By applying a reweighting procedure to MiNLO improved Drell-Yan production, we obtain a generator which is NNLO accurate when integrated over all radiation while providing a fully exclusive description of the final state phase space. We compare the calculation to dedicated next-to-next-to-leading logarithm resummations and find very good agreement. The generator is also found to be in good agreement with 7 and 8 TeV LHC data.

In this thesis, we study the soft gluon corrections to hard wide-angle scattering processes due to a virtual gluon exchange (one-loop order) and the emission of up to two gluons. Our primary aim is to determine the ordering condition that should be used to dress a hard scattering process with corrections due to gluon emissions and a Coulomb (Glauber) gluon exchange. We find that, due to an elegant cancellation of many Feynman diagrams, a specific ordering variable should be used to order the transverse momentum of the exchanged Coulomb gluon with respect to the real emissions. Furthermore, in the case of the scattering process accompanied with a single emission, we find that the radiative part of the loop correction satisfies the same ordering condition as the Coulomb gluon contribution. Based upon the assumption that the ordering condition continues at higher orders, we conjecture an expression for the soft corrections to a general hard scattering process due to one-loop and any number of gluon emissions.

Quelques aspects de la physique des mésons B ont été étudiés par la<br />simulation numérique de la Chromodynamique Quantique sur réseau, qui est une approche non perturbative - basée sur les principes premiers de la Théorie Quantique des Champs - de calculer les fonctions de Green de la théorie. <br />Les couplages $\hat{g}$ et $\tilde{g}$ paramétrant le Lagrangien chiral effectif qui décrit les interactions entre mésons lourd-légers et pions mous ont été calculés au-delà de l'approximation quenched (à N_f=2). L'opérateur<br />$\bar{q}\gamma_\m \gc q$ a été renormalisé non perturbativement en utilisant les identités de Ward chirales. On trouve <br />$\hat{g}=0.4\div 0.6$ et $\tilde{g}=-0.1\div-0.3$. <br />La masse du quark étrange a été estimée par<br />une simulation unquenched (à N_f=2): elle a été renormalisée<br />dans le schéma non perturbatif RI-MOM. On obtient en faisant le raccordement avec le schéma MSbar m_s^{MSbar}(2 GeV)=101\pm 8^{+25}_{-0} MeV. <br />Une méthode a été proposée pour évaluer sur le réseau les facteurs de<br />forme associés aux transitions semileptoniques à recul nul <br />B \to D^{**} décrites dans le cadre de la Théorie Effective des Quarks Lourds. La constante de renormalisation de l'opérateur $\bar{h}\gamma_i \gc D_j h$ a été calculée à l'ordre d'une boucle de la théorie des perturbations.<br />On trouve \tau_1/2(1)=0.3\div 0.5 et \tau_3/2(1)=0.5\div 0.7. <br />Enfin le paramètre de sac associé à l'amplitude de mélange B_s-anti B_s a été évalué, en choisissant une action pour le quark étrange qui vérifie la symétrie chirale à maille a du réseau finie. De cette manière les erreurs systématiques, provenant des mélanges de chiralité, sont nettement réduites lors de la renormalisation de l'opérateur à 4 fermions $\bar{h}\gamma_{\m L}q\bar{h}\gamma_{\m L}q$. On obtient dans l'approximation quenched $B_{B_s}=0.92(3)$.

Nas últimas duas décadas a teoria dos sistemas eletrônicos correlacionados teve enorme progresso, que sustentou o paralelo desenvolvimento da pesquisa experimental dos sistemas de férmions pesados. Dada a complexidade do problema proposto pelas correlações fortes, diversas técnicas complementares de cálculo foram desenvolvidas no período. O presente plano se propõe a explorar uma extensão de uma das mais antigas, a técnica do grupo de renormalização numérico (GRN), tratando perturbativamente o modelo de Kondo para uma impureza magnética em um hospedeiro metálico. É bem conhecido que a expansão perturbativa de propriedades físicas, como a susceptibilidade, em termos do acoplamento de troca diverge logaritmicamente próxima da temperatura de Kondo. A abordagem do GRN para isto considera a transformação discreta, T[HN] = HN+1, onde {HN} é uma seqüência de Hamiltonianos. Neste trabalho, para regularizar a expansão da susceptibilidade, usamos um procedimento alternativo considerando a transformação contínua análoga, T&#948z[HN(z)] = HN(z+&#948z), onde z é um parâmetro arbitrário que generaliza a discretização logarítmica do GRN. Ao contrário do procedimento de Wilson, nós esperamos que este novo procedimento possa ser mais facilmente aplicável a Hamiltonianos mais complexos, complementando a diagonalização numérica. / In the last two decades the theory of electronic correlated systems has had an enormous progress, which has sustained the parallel development of the experimental research in heavy fermion systems. Given the complexity imposed by the strong correlations, several techniques appeared. The present work explores an extension of one of the oldest, the Numerical Renormalization Group (NRG), treating perturbatively the Kondo model for a magnetic impurity in a metallic host. It is well known that perturbative expansion of physical properties, like susceptibility, in terms of the exchange coupling diverges logarithmically near the Kondo temperature. The NRG approach for this consider the discrete transformation, T[HN] = HN+1, where {HN}, is a sequence of Hamiltonians. In this work we use an alternative procedure to regularize the expansion, using an analogous continuum transformation T&#948z[HN(z)] = HN(z+&#948z), where z is an arbitrary parameter which generalizes the NRG logarithmic discretization. Unlike Wilson\'s procedure, we hope this new one can be easily applicable to more complex Hamiltonians, complementing the numerical diagonalization.

NIL

In this project, we examine how integration over matrices is performed. We investigate and develop a method for calculating matrix integrals over the set of real square matrices. Matrix integrals are used for calculations in several different areas of physics and mathematics; for example quantum field theory, string theory, quantum chromodynamics, and random matrix theory. Our method consists of ways to apply perturbative Taylor expansions to the matrix integrals, reducing each term of the resulting Taylor series to a combinatorial problem using Wick's theorem, and representing the terms of the Wick sum graphically with the help of Feynman diagrams and fat graphs. We use the method in a few examples that aim to clearly demonstrate how to calculate the matrix integrals. / I detta projekt undersöker vi hur integration över matriser genomförs. Vi undersöker och utvecklar en metod för beräkning av matrisintegraler över mängden av alla reell-värda kvadratiska matriser. Matrisintegraler används för beräkningar i ett flertal olika områden inom fysik och matematik, till exempel kvantfältteori, strängteori, kvantkromodynamik och slumpmatristeori. Vår metod består av sätt att applicera perturbativa Taylorutvecklingar på matrisintegralerna, reducera varje term i den resulterande Taylorserien till ett kombinatoriellt problem med hjälp av Wicks sats, och att representera termerna i Wicksumman grafiskt med hjälp av Feynmandiagram. Vi använder metoden i några exempel som syftar till att klart demonstrera hur beräkningen av matrisintegraler går till.

Nas últimas duas décadas a teoria dos sistemas eletrônicos correlacionados teve enorme progresso, que sustentou o paralelo desenvolvimento da pesquisa experimental dos sistemas de férmions pesados. Dada a complexidade do problema proposto pelas correlações fortes, diversas técnicas complementares de cálculo foram desenvolvidas no período. O presente plano se propõe a explorar uma extensão de uma das mais antigas, a técnica do grupo de renormalização numérico (GRN), tratando perturbativamente o modelo de Kondo para uma impureza magnética em um hospedeiro metálico. É bem conhecido que a expansão perturbativa de propriedades físicas, como a susceptibilidade, em termos do acoplamento de troca diverge logaritmicamente próxima da temperatura de Kondo. A abordagem do GRN para isto considera a transformação discreta, T[HN] = HN+1, onde {HN} é uma seqüência de Hamiltonianos. Neste trabalho, para regularizar a expansão da susceptibilidade, usamos um procedimento alternativo considerando a transformação contínua análoga, T&#948z[HN(z)] = HN(z+&#948z), onde z é um parâmetro arbitrário que generaliza a discretização logarítmica do GRN. Ao contrário do procedimento de Wilson, nós esperamos que este novo procedimento possa ser mais facilmente aplicável a Hamiltonianos mais complexos, complementando a diagonalização numérica. / In the last two decades the theory of electronic correlated systems has had an enormous progress, which has sustained the parallel development of the experimental research in heavy fermion systems. Given the complexity imposed by the strong correlations, several techniques appeared. The present work explores an extension of one of the oldest, the Numerical Renormalization Group (NRG), treating perturbatively the Kondo model for a magnetic impurity in a metallic host. It is well known that perturbative expansion of physical properties, like susceptibility, in terms of the exchange coupling diverges logarithmically near the Kondo temperature. The NRG approach for this consider the discrete transformation, T[HN] = HN+1, where {HN}, is a sequence of Hamiltonians. In this work we use an alternative procedure to regularize the expansion, using an analogous continuum transformation T&#948z[HN(z)] = HN(z+&#948z), where z is an arbitrary parameter which generalizes the NRG logarithmic discretization. Unlike Wilson\'s procedure, we hope this new one can be easily applicable to more complex Hamiltonians, complementing the numerical diagonalization.