We consider a symmetric scalar theory with quartic coupling in 4-dimensions. We show that the 4 loop 2PI calculation can be done using a renormalization group method. The calculation involves one bare coupling constant which is introduced at the level of the Lagrangian and is therefore conceptually simpler than a standard 2PI calculation, which requires multiple counterterms. We explain how our method can be used to do the corresponding calculation at the 4PI level, which can not be done using any known method by introducing counterterms. * carrington@brandonu.ca †

Techniques based on n-particle irreducible effective actions can be used to study systems where perturbation theory does not apply. The main advantage, relative to other non-perturbative continuum methods, is that the hierarchy of integral equations that must be solved truncates at the level of the action, and no additional approximations are needed. The main problem with the method is renormalization, which until now could only be done at the lowest (n=2) level. In this paper we show how to obtain renormalized results from an n-particle irreducible effective action at any order. We consider a symmetric scalar theory with quartic coupling in four dimensions and show that the 4 loop 4-particle-irreducible calculation can be renormalized using a renormalization group method. The calculation involves one bare mass and one bare coupling constant which are introduced at the level of the Lagrangian, and cannot be done using any known method by introducing counterterms. * carrington@brandonu.ca †

Arsenic is one of the most pervasive environmental toxins. It enters our water and food supply through many different routes, including the burning of fossil fuels, the application of arsenic-based herbicides, and natural sources. Using a density functional theory (DFT) cluster approach, the mechanism of arsenic (III) S-adenosylmethionine methyltransferases and various selenium-containing analogues was investigated. Notably, the methylation of arsenic by arsenic (III) S-adenosylmethionine is proposed to be a way to remove arsenic from contaminated water or soil. From the DFT cluster results, it was found that the selective substitution of the active-site Cys44, Cys72, and Cys174 residues with selenocysteines had a marginal effect on the barrier for CH 3 transfer. Specifically, the average Gibbs activation energy was calculated to be only 4.2 kJ mol –1 lower than the Gibbs activation energy of 107.4 kJ mol –1 for the WT enzyme. However, importantly, it was found that with selective mutation, the methylation process becomes considerably more exergonic, where the methylation reaction can be made to be 26.4 kJ mol –1 more exergonic than the reaction catalyzed by the WT enzyme. Therefore, we propose that the selective substitution of the active-site Cys44, Cys72 and Cys174 residues with selenocysteines could make the process of methylation and volatilization more advantageous for bioremediation.