O presente trabalho propõe o estudo da cromodinâmica quântica (QCD) através de simulações numéricas da teoria na rede. Inicialmente será apresentada a formulação de integral de trajetória da mecânica quântica para em seguida generalizar os resultados para a teoria quântica de campos. A teoria na rede exigirá a discretização do espaço-tempo e mostraremos como colocar os graus de liberdade bosônicos e fermiônicos na rede. Usaremos a formulação de Wilson para a ação de glúons e férmions na rede. Simulações numéricas em QCD na rede envolvendo férmions são consideravelmente complicadas e têm um custo computacional altamente limitante. Mais precisamente, não é possível simular a estrutura do vácuo fermiônico sem algum tipo de aproximação. Neste trabalho usaremos a aproximação quenched da QCD, que consiste em negligenciar os efeitos de loops de quarks do vácuo. Também empregaremos apenas dois sabores de quarks degenerados, correspondendo às duas espécies de quarks leves presentes na teoria. Ao longo deste trabalho serão exploradas as dificuldades encontradas em colocar quarks na rede e também será determinado o espectro de hádrons como uma aplicação de interesse. Contudo, também estudaremos um problema simples envolvendo a teoria de gauge pura na rede, isto é, calcularemos o valor esperado para o operador plaqueta. Este estudo servirá como um pré aquecimento antes de lidar com o problema mais desafiador do espectro de hádrons e também permitirá aprender algumas técnicas de simulação que serão utilizadas na determinação do espectro de hádrons, a saber, o método de Monte Carlo e os algoritmos banho térmico e sobre-relaxação, que servem para construir configurações de gauge (glúons). A interpretação dos resultados obtidos deverá ser realizada a partir da análise estatística dos dados. Estimaremos o tempo de termalização do operador plaqueta a partir da visualização da equilibração do resultado e usaremos o método bloco de dados para estimar o tempo de correlação das configurações de gauge. Essas estimativas serão importantes para decidirmos os parâmetros de simulação adequados para o espectro de hádrons, pois neste caso não teremos acesso à quantidade suficiente de dados para determinar o valor desses parâmetros. Para a determinação de erros estatísticos será usado o método de jackknife. O cálculo do espectro de hádrons envolve a inversão de uma matriz esparsa não positiva definida, mais precisamente, o operador de Dirac. Esta será a parte que mais consumirá tempo nas simulações e usaremos o algoritmo gradiente biconjugado estabilizado (Bi-CGStab) para a inversão. A determinação da massa de hádrons somente será possível após a fixação da massa experimental de algum hádron (usaremos o píon), e após a extrapolação quiral dos resultados, que será realizada a partir do método dos mínimos quadrados não linear. Ao final deste trabalho obteremos uma estimava para a massa do próton e do méson rho. / This work proposes the study of quantum cromodynamics (QCD) through numerical simulations of the lattice theory. Initially we will present the path integral formulation of quantum mechanics and then generalize these results to quantum field theory. The lattice theory requires the discretization of space-time and we will show how to put the fermionic and gluonic degrees of freedom on the lattice. We will use Wilson’s formulation for the action of fermions and gluons. Numerical simulations in lattice QCD with fermions are considerably complicated and their cost is highly limiting. More precisely, it is not possible to simulate the fermionic structure of the vacuum without any kind of approximation. We will use the quenched approximation in this work, which consists of neglecting the effects of vacuum quark loops. We also will employ only two flavors of degenerate quarks corresponding to the two species of light quarks present in the theory. Throughout the work we will discuss the difficulties related to putting quarks on the lattice and we will evaluate the hadron spectrum as an application of interest. However, we also must study a simple problem involving the lattice pure-gauge theory, i.e., we will compute the mean value of the plaquette operator. This will be a warm-up study before dealing with the more challenging problem of the hadron spectrum and will allow us to learn some simulation techniques that will be used in the hadron spectrum determination, namely the Monte Carlo method and the heat bath and overrelaxation algorithms, which are useful to build gauge configurations (i.e. gluon configurations). The interpretation of the results obtained should be performed using statistical analysis of the data. We will estimate the thermalization time of the plaquette operator from the visualization of the equilibration result and will estimate the correlation time of gauge configurations using the data blocking method. These estimates are important to decide the suitable simulation parameters for the hadron spectrum, because in that case we will not have access to a quantity of data large enough to determine the value of these parameters. For the statistical error determination we will use the jackknife method. The calculation of the hadron spectrum involves the inversion of a non positive definite sparse matrix, more precisely, the Dirac operator. This will be the most time-consuming step of the simulation and we will use the Bi-Conjugate Gradient Stabilized (Bi-CGRStab) algorithm to do the inversion. The determination of hadron masses will only be possible after fixing an experimental mass of some hadron (we will use the pion), and after the quiral extrapolation of the results, which will be performed using the non-linear least square method. At the end of this work we will obtain an estimate of the mass of the proton and of the rho meson.

Wir untersuchen numerisch und teilweise analytisch die kompakte Quantenelektrodynamik auf dem Gitter mit Wilson-Fermionen. Dabei konzentrieren wir uns auf zwei wesentliche Teilprobleme der Theorie: der Einfluss von Eichfeld-Moden mit verschwindendem Impuls in der Coulomb-Phase und die Effizienz von verschiedenen Monte-Carlo-Algorithmen unter Berücksichtigung dynamischer Fermionen. Wir zeigen, dass der Einfluss der Null-Impuls-Moden auf die eichabhängigen Gitter-Observablen wie Photon- und Fermion-Korrelatoren nahe der kritischen chiralen Grenzlinie innerhalb der Coulomb Phase zu einem Verhalten führt, das vom naiv erwarteten gitterstörungstheoretischen Verhalten abweicht. Diese Moden sind auch für die Abschirmung des kritischen Verhaltens der eichinvarianten Fermion-Observablen nahe der chiralen Grenzlinie verantwortlich. Eine Entfernung dieser Null-Impuls-Moden aus den Eichfeld-Konfigurationen führt innerhalb der Coulomb-Phase zum störungstheoretisch erwarteten Verhalten der eichabhängigen Observablen. Die kritischen Eigenschaften der eichinvarianten Fermion-Observablen in der Coulomb-Phase werden nach dem Beseitigen der Null-Impuls-Moden sichtbar. Der kritische Hopping-Parameter, den man aus den invarianten Fermion-Observablen erhält, stimmt gut mit demjenigen überein, der aus den eichabhängigen Observablen extrahiert werden kann. Wir führen den zweistufigen Multiboson-Algorithmus für numerische Untersuchungen im U(1)-Gittermodell mit einer geraden Anzahl von dynamischen Fermion-Flavour-Freiheitsgraden ein. Wir diskutieren die geeignete Wahl der technischen Parameter sowohl für den zweistufigen Multiboson-Algorithmus als auch für den hybriden Monte-Carlo-Algorithmus. Wir geben theoretische Abschätzungen für die Effizienz dieser Simulationsmethoden. Wir zeigen numerisch und theoretisch, daß der zweistufige Multiboson-Algorithmus eine gute Alternative darstellt und zumindestens mit der hybriden Monte-Carlo-Methode konkurrieren kann. Wir argumentieren, daß eine weitere Verbesserung der Effizienz des zweistufigen Multiboson-Algorithmus durch eine Vergrößerung der Zahl lokaler Update-Schleifen und auch durch die Reduktion der Ordnungen der ersten und zweiten Polynome zu Lasten des sogenannten 'Reweighting' erzielt werden kann. / We investigate numerically and in part analytically the compact lattice quantum electrodynamics with Wilson fermions. We studied the following particular tasks of the theory: the problem of the zero-momentum gauge field modes in the Coulomb phase and the performance of different Monte Carlo algorithms in the presence of dynamical fermions. We show that the influence of the zero-momentum modes on the gauge dependent lattice observables like photon and fermion correlators within the Coulomb phase leads to a behaviour of these observables different from standard perturbation theory. These modes are responsible also for the screening of the critical behaviour of the gauge invariant fermion values near the chiral limit line. Within the Coulomb phase the elimination of these zero-momentum modes from gauge configurations leads to the perturbatively expected behaviour of gauge dependent observables. The critical properties of gauge invariant fermion observables upon removing the zero-momentum modes are restored. The critical hopping-parameter obtained from the invariant fermion observables coincides with that extracted from gauge dependent values. We implement the two-step multiboson algorithm for numerical investigations in the U(1) lattice model with even dynamical Wilson fermion flavours. We discuss the scheme of an appropriate choice of technical parameters for both two-step multiboson and hybrid Monte Carlo algorithms. We give the theoretical estimates of the performance of such simulation methods. We show both numerically and theoretically that the two-step multiboson algorithm is a good alternative and at least competitive with the hybrid Monte Carlo method. We argue that an improvement of efficiency of the two-step multiboson algorithm can be achieved by increasing the number of local update sweeps and also by decreasing the orders of first and second polynomials corrected for by the reweighting step.

In dieser Arbeit werden Gross-Neveu Modelle mit einer endlichen Anzahl von Fermiontypen auf einem zweidimensionalen Euklidischen Raumzeitgitter betrachtet. Modelle dieses Typs sind asymptotisch frei und invariant unter einer chiralen Symmetrie. Aufgrund dieser Gemeinsamkeiten mit QCD sind sie sehr gut geeignet als Testumgebungen für Fermionwirkungen die in großangelegten Gitter-QCD-Rechnungen benutzt werden. Das Schrödinger Funktional für die Gross-Neveu Modelle wird definiert für Wilson und Ginsparg-Wilson Fermionen. In 1-Schleifenstörungstheorie wird seine Renormierbarkeit gezeigt. Die Vier-Fermionwechselwirkungen der Gross-Neveu Modelle habe dimensionslose Kopplungskonstanten in zwei Dimensionen. Die Symmetrieeigenschaften der Vier-Fermionwechselwirkungen und deren Beziehungen untereinander werden diskutiert. Im Fall von Wilson Fermionen ist die chirale Symmetrie explizit gebrochen und zusätzliche Terme müssen in die Wirkung aufgenommen werden. Die chirale Symmetrie wird durch das Einstellen der nackten Masse und einer der Kopplungen bis auf Cut-off-Effekte wiederhergestellt. Die kritische Masse und die symmetriewiederherstellende Kopplung werden bis zur zweiten Ordnung in Gitterstörungstheorie berechnet. Dieses Resultat wird in der 1-Schleifenberechnung der renormierten Kopplungen und der zugehörigen Betafunktionen benutzt. Die renormierten Kopplungen werden definiert mit Hilfe von geeignete Rand-Rand-Korrelatoren. Die Rechnung reproduziert die bekannten führenden Koeffizienten der Betafunktionen. Eine der Kopplungen hat eine verschwindende Betafunktion. Die Rechnung wird mit dem vor kurzem vorgeschlagenen Schrödinger Funktional mit exakter chiraler Symmetrie, also Ginsparg Wilson Fermionen, wiederholt. Es werden die gleichen Divergenzen gefunden, wie im Fall von Wilson Fermionen. Unter Benutzung des regularisierungsabhängigen, endlichen Teils der renormierten Kopplungen werden die Verhältnisse der Lambda-Parameter bestimmt. / Gross-Neveu type models with a finite number of fermion flavours are studied on a two-dimensional Euclidean space-time lattice. The models are asymptotically free and are invariant under a chiral symmetry. These similarities to QCD make them perfect benchmark systems for fermion actions used in large scale lattice QCD computations. The Schrödinger functional for the Gross-Neveu models is defined for both, Wilson and Ginsparg-Wilson fermions, and shown to be renormalisable in 1-loop lattice perturbation theory. In two dimensions four fermion interactions of the Gross-Neveu models have dimensionless coupling constants. The symmetry properties of the four fermion interaction terms and the relations among them are discussed. For Wilson fermions chiral symmetry is explicitly broken and additional terms must be included in the action. Chiral symmetry is restored up to cut-off effects by tuning the bare mass and one of the couplings. The critical mass and the symmetry restoring coupling are computed to second order in lattice perturbation theory. This result is used in the 1-loop computation of the renormalised couplings and the associated beta-functions. The renormalised couplings are defined in terms of suitable boundary-to-boundary correlation functions. In the computation the known first order coefficients of the beta-functions are reproduced. One of the couplings is found to have a vanishing beta-function. The calculation is repeated for the recently proposed Schrödinger functional with exact chiral symmetry, i.e. Ginsparg-Wilson fermions. The renormalisation pattern is found to be the same as in the Wilson case. Using the regularisation dependent finite part of the renormalised couplings, the ratio of the Lambda-parameters is computed.