The search of small deviations from Lorentz symmetry as a possible window into new physics at very high energies has been a very active field of theoretical and experimental research in the past decades. The Standard Model Extension (SME) is an effective field theory that includes in the Standard Model (SM) and the Einstein-Hilbert actions all operators that violate Lorentz symmetry, while still preserving other relevant symmetries of these theories, allowing a systematical search for low energy physical effects of these Lorentz violations (LV). Negative results in experimental searches are then converted into very stringent limits for these LV, which constrain possible quantum gravity theories that may induce such LV. The LV operators in the SME can be divided into two sets: the minimal ones are leading by dimensional analysis, since they have mass dimension less or equal to four; non minimal operators have higher dimension, they are subleading as a general expectation, but sometimes they induce low energy effects that might be measured. In this research proposal, we intent to study the generation of non minimal LV operators as perturbative corrections, originating from some more fundamental, sometimes minimal, LV operators. Such mechanism for generation of minimal LV has a rich history in the field, ofter plagued with nontrivial technical issues such as ambiguities in the calculation of Feynmann diagrams, but the generation of non minimal LV, particularly in the gravity sector, is still a quite unexplored topic. We propose to help fill this gap by: extending previous work related to generation of LV couplings in axion physics, and extending this line of research for non minimal LV couplings in gravity.
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