Neste trabalho estudamos uma nova classe de superpotenciais em mecânica quântica supersimétrica, os quais denominamos de modulares, por serem funções do módulo da coordenada x. O superpotencial de partida proposto é da forma x |x|. Esta ideia permite tornar solúvel exatamente, a energia zero, um incontável número de potenciais gerados por estas funções no âmbito da mecânica quântica supersimétrica. Exploramos algumas aplicações para estes superpotenciais, com ênfase para uma representação da molécula de amônia supersimétrica e, em particular, mostramos que um sistema muito estudado na literatura, gerado pelo superpotencial x 1/x, pode ser resolvido mais facilmente recorrendo-se à representação modular. Procuramos estudar as soluções exatas ou aproximadas - do espectro de energias dos Hamiltonianos parceiros supersimétricos utilizando metodologias adequadas ao respectivo caso, incluindo-se o conhecido potencial x4, sendo que o método variacional de coeficientes de funções foi o que melhor se adaptou ao estudo. Este método, pouco utilizado até o momento na literatura, permitiu não apenas resolver com excelente aproximação os primeiros níveis do sistema em estudo, como também comprovou a supersimetria do sistema modular. Mostramos também que em sistemas quânticos supersimétricos, a equação de Schroedinger pode ser colocada na forma da equação de Sturm-Liouville e apresentar soluções de polinômios ortogonais, sendo que a função-peso de tais polinômios é gerada pelo superpotencial. Uma breve abordagem da simetria PT envolvendo diretamente o potencial por nós proposto também foi investigada, e mostramos que o sistema é equivalente a um Hamiltoniano não Hermitiano com potencial V(z) = (z4). / In this work we study a new class of superpotentials in supersymmetric quantum mechanics, which we call modular because of their dependence on the modulus of the x coordinate. The starting superpotential is of the form x |x|. This idea helps make exactly solvable, at zero energy, several potentials generated by these functions in the context of supersymmetric quantum mechanics. We explore some applications for these superpotenciais, with emphasis on a representation of the supersymmetric ammonia molecule and, in particular, we show that a system generated by the superpotential x-1/x, widely studied in the literature, can be solved more easily making use to the modular representation. We also seek for spectral solutions exact or approximated - of the partners Hamiltonians based on the exact ground state wave function of zero energy including the conventional x4 potential. The use of the variational method of functions coefficients. These methods, rarely used to date in the literature, allowed not only solve with excellent approximation the first levels of the system under study, but also proved the supersymmetry of the modular system. The results were compared with others found in the literature. We also show that for supersymmetric quantum systems, the Schroedinger equation can be put in a form of the Sturm-Liouville equation, and so, orthogonal polynomials solutions can be find through a weight-function generated by the superpotential. A brief overview of the PTsymmetry of the system directly involving a modular model proposed was also investigated, and we show that this system is equivalent to the non-Hermitian Hamiltonian one with potential V (z) = z4.

Neste trabalho estudamos uma nova classe de superpotenciais em mecânica quântica supersimétrica, os quais denominamos de modulares, por serem funções do módulo da coordenada x. O superpotencial de partida proposto é da forma x |x|. Esta ideia permite tornar solúvel exatamente, a energia zero, um incontável número de potenciais gerados por estas funções no âmbito da mecânica quântica supersimétrica. Exploramos algumas aplicações para estes superpotenciais, com ênfase para uma representação da molécula de amônia supersimétrica e, em particular, mostramos que um sistema muito estudado na literatura, gerado pelo superpotencial x 1/x, pode ser resolvido mais facilmente recorrendo-se à representação modular. Procuramos estudar as soluções exatas ou aproximadas - do espectro de energias dos Hamiltonianos parceiros supersimétricos utilizando metodologias adequadas ao respectivo caso, incluindo-se o conhecido potencial x4, sendo que o método variacional de coeficientes de funções foi o que melhor se adaptou ao estudo. Este método, pouco utilizado até o momento na literatura, permitiu não apenas resolver com excelente aproximação os primeiros níveis do sistema em estudo, como também comprovou a supersimetria do sistema modular. Mostramos também que em sistemas quânticos supersimétricos, a equação de Schroedinger pode ser colocada na forma da equação de Sturm-Liouville e apresentar soluções de polinômios ortogonais, sendo que a função-peso de tais polinômios é gerada pelo superpotencial. Uma breve abordagem da simetria PT envolvendo diretamente o potencial por nós proposto também foi investigada, e mostramos que o sistema é equivalente a um Hamiltoniano não Hermitiano com potencial V(z) = (z4). / In this work we study a new class of superpotentials in supersymmetric quantum mechanics, which we call modular because of their dependence on the modulus of the x coordinate. The starting superpotential is of the form x |x|. This idea helps make exactly solvable, at zero energy, several potentials generated by these functions in the context of supersymmetric quantum mechanics. We explore some applications for these superpotenciais, with emphasis on a representation of the supersymmetric ammonia molecule and, in particular, we show that a system generated by the superpotential x-1/x, widely studied in the literature, can be solved more easily making use to the modular representation. We also seek for spectral solutions exact or approximated - of the partners Hamiltonians based on the exact ground state wave function of zero energy including the conventional x4 potential. The use of the variational method of functions coefficients. These methods, rarely used to date in the literature, allowed not only solve with excellent approximation the first levels of the system under study, but also proved the supersymmetry of the modular system. The results were compared with others found in the literature. We also show that for supersymmetric quantum systems, the Schroedinger equation can be put in a form of the Sturm-Liouville equation, and so, orthogonal polynomials solutions can be find through a weight-function generated by the superpotential. A brief overview of the PTsymmetry of the system directly involving a modular model proposed was also investigated, and we show that this system is equivalent to the non-Hermitian Hamiltonian one with potential V (z) = z4.

Thesis (M.Sc.) - Carleton University, 2008. / Includes bibliographical references. Also available in electronic format on the Internet.

La première partie de mon travail porte sur le développement d'un algorithme utilisant les données enregistrées par CMS afin de détecter la présence de déformations particulières susceptibles d'affecter le trajectographe de CMS. La méthode s’appuie sur le biais que ces déformations induisent dans la mesure de l'impulsion des électrons. Sa fiabilité a été prouvée par simulation Monte-Carlo. Elle a été utilisée pour la validation des données 2012. La seconde partie de mon travail consiste en une analyse statistique des données de collisions prises par le détecteur CMS en 2012, afin de mettre en évidence la présence de particules neutres et à long temps de vie se désintégrant en un électron et un muon. L'observation d'un tel processus constituerait une preuve de l'existence de nouvelle physique. En l'absence d'observation statistiquement significative, une méthode Bayesienne est utilisée pour interpréter ce résultat en terme de contraintes sur un modèle supersymétrique (MSSM). / The first part of my work concerns the development of an algorithm using CMS data to automatically detect the presence of particular deformations than can occur in the CMS tracker. The method takes advantage of the bias that such deformations induce in the measurement of electron impulsion. The reliability of the method has been proved using Monte-Carlo simulation. The algorithm was then used to certify that none of the considered deformations affected the data recorded in 2012.The second part of my work consists of a statistical analysis of the data recorded by CMS, in order to look for evidence of the presence of exotic long-lived particles. The latter could manifest themselves through their decay to an electron and a muon. Such an observation would provide a strong clue of the existence of new physics. In the absence of any observation statistically significant, a Bayesian method is used to interpret this result in term of constraints on a supersymmetric model (MSSM).

La matière sombre est un mystère dans le domaine de l’astrophysique depuis déjà plusieurs années. De nombreuses observations montrent que jusqu’à 85 % de la masse gravitationnelle totale de l’univers serait composée de cette matière de nature inconnue. Une théorie expliquant cette masse manquante considérerait les WIMPs (Weakly Interacting Massive Particles), particules stables, non chargées, prédites par des extensions du modèle standard, comme candidats. Le projet PICASSO (Projet d’Identification des CAndidats Supersymétriques à la matière Sombre) est une expérience qui tente de détecter directement le WIMP. Le projet utilise des détecteurs à gouttelettes de fréon (C4F10) surchauffées. La collision entre un WIMP et le noyau de fluor crée un recul nucléaire qui cause à son tour une transition de phase de la gouttelette liquide à une bulle gazeuse. Le bruit de ce phénomène est alors capté par des senseurs piézoélectriques montés sur les parois des détecteurs. Le WIMP n’est cependant pas la seule particule pouvant causer une telle transition de phase. D’autres particules environnantes peuvent former des bulles, telles les particules alpha où même des rayons gamma . Le système d’acquisition de données (DAQ) est aussi en proie à du bruit électronique qui peut être enregistré, ainsi que sensible à du bruit acoustique extérieur au détecteur. Finalement, des fractures dans le polymère qui tient les gouttelettes en place peut également causer des transitions de phase spontanées. Il faut donc minimiser l’impact de tous ces différents bruit de fond. La pureté du matériel utilisé dans la fabrication des détecteurs devient alors très importante. On fait aussi appel à des méthodes qui impliquent l’utilisation de variables de discrimination développées dans le but d’améliorer les limites d’exclusion de détection du WIMP. / Dark matter has been a mystery for astrophysicists for years now. Numerous observations have shown that up to 85 % of the gravitation mass of the universe is made of this unknown type of matter. One of the theories explaining this missing mass problem considers WIMPs (Weakly Interacting Massive Particles), neutral stable particles predicted by extensions of the standard model, as possible candidates. The PICASSO experiment (Project In Canada to Search for Supersymetric Objects) tries to detect this particle directly. The technique uses superheated droplet detectors, with freon (C4F10) as the active medium. When a WIMP hits the fluorine nucleus, it creates a nuclear recoil, which in turn triggers a phase transition from a liquid droplet to a gaseous bubble. The acoustic noise of this event is then recorded by piezoelectric transducers mounted on the walls of the detector. There are however other particles than the WIMPs that can trigger this phase transition. Alpha particles, or even gamma rays can create bubbles. The Data Acquisition System (DAQ) is also subject to electronic noise that can be picked up, and to acoustic noise coming from an exterior source. Fractures in the polymer holding the droplets in place can also trigger spontaneous phase transitions. There is therefore a need to minimize the impact of these background noises. The level of purity of the ingredients used in detector fabrication then becomes very important. Digital processing methods are also used to develop discrimination variables that improve the limits of detection of the WIMP.

La matière sombre est un mystère dans le domaine de l’astrophysique depuis déjà plusieurs années. De nombreuses observations montrent que jusqu’à 85 % de la masse gravitationnelle totale de l’univers serait composée de cette matière de nature inconnue. Une théorie expliquant cette masse manquante considérerait les WIMPs (Weakly Interacting Massive Particles), particules stables, non chargées, prédites par des extensions du modèle standard, comme candidats. Le projet PICASSO (Projet d’Identification des CAndidats Supersymétriques à la matière Sombre) est une expérience qui tente de détecter directement le WIMP. Le projet utilise des détecteurs à gouttelettes de fréon (C4F10) surchauffées. La collision entre un WIMP et le noyau de fluor crée un recul nucléaire qui cause à son tour une transition de phase de la gouttelette liquide à une bulle gazeuse. Le bruit de ce phénomène est alors capté par des senseurs piézoélectriques montés sur les parois des détecteurs. Le WIMP n’est cependant pas la seule particule pouvant causer une telle transition de phase. D’autres particules environnantes peuvent former des bulles, telles les particules alpha où même des rayons gamma . Le système d’acquisition de données (DAQ) est aussi en proie à du bruit électronique qui peut être enregistré, ainsi que sensible à du bruit acoustique extérieur au détecteur. Finalement, des fractures dans le polymère qui tient les gouttelettes en place peut également causer des transitions de phase spontanées. Il faut donc minimiser l’impact de tous ces différents bruit de fond. La pureté du matériel utilisé dans la fabrication des détecteurs devient alors très importante. On fait aussi appel à des méthodes qui impliquent l’utilisation de variables de discrimination développées dans le but d’améliorer les limites d’exclusion de détection du WIMP. / Dark matter has been a mystery for astrophysicists for years now. Numerous observations have shown that up to 85 % of the gravitation mass of the universe is made of this unknown type of matter. One of the theories explaining this missing mass problem considers WIMPs (Weakly Interacting Massive Particles), neutral stable particles predicted by extensions of the standard model, as possible candidates. The PICASSO experiment (Project In Canada to Search for Supersymetric Objects) tries to detect this particle directly. The technique uses superheated droplet detectors, with freon (C4F10) as the active medium. When a WIMP hits the fluorine nucleus, it creates a nuclear recoil, which in turn triggers a phase transition from a liquid droplet to a gaseous bubble. The acoustic noise of this event is then recorded by piezoelectric transducers mounted on the walls of the detector. There are however other particles than the WIMPs that can trigger this phase transition. Alpha particles, or even gamma rays can create bubbles. The Data Acquisition System (DAQ) is also subject to electronic noise that can be picked up, and to acoustic noise coming from an exterior source. Fractures in the polymer holding the droplets in place can also trigger spontaneous phase transitions. There is therefore a need to minimize the impact of these background noises. The level of purity of the ingredients used in detector fabrication then becomes very important. Digital processing methods are also used to develop discrimination variables that improve the limits of detection of the WIMP.